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Pipe Flow Reference Manual - UTC

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1. 2 2 2 Vm Vout Vout H H H H t C 24 J out hy 2g C 2g 7 Vn 1s approximated by assuming that ym Yout res JO PEE ay i Vn An D mnYm 1 5 15 0 0 42 14 0 0 94ms 7 5 Q 2 0 v 1 out aa OR S n 2 a Ae ge ON oe U 7 26 v 0 045 m 1 27 2g 6 Econ T Gair 7 28 A l 2 K 1 fou Gin pal QourAm Oout 90 2 0 78 45 m e E i 751 ae 0 567 MOUSE PIPE FLOW Reference Manual 105 Flow Resistance 7 5 4 Substituting values to the Equation 7 25 and calculating H yields H 15 89m 7 29 The deviation between the MOUSE simulation and manual calculation result is due to the fact that MOUSE calculates v by using the following area in the pipe D T Aout 4 Preismann slot area The table shows which setups have been used for the calculation and also which head loss types are included C 9 G1 Gc2 are all head loss coeffi cients due to contraction and correspond to the types a b and c Table 7 1 Test matrix for implementation fo Head Loss Type Contraction Test Setup Direction Cy Drop a Cool Oer Geo Heate Hm 1 I X 15 81 15 82 2 I X 15 87 15 87 3 D X 15 87 15 87 4 ID X X 15 89 15 90 5 I X X 15 95 15 94 6 ID X X 15 87 15 87 7 HD X X 15 94 15 94 8 HD X X 16 00 15 99 9 IV X X X 16 02 16 02 10 IV X X X 16 03 16 02 Implementation
2. aoaaa aaa aaa 84 BOUNDARY CONDITIONS o 85 FLOW RESISTANCE 0 0 0 0 2000000000002 eee 89 7 1 Friction Losses in Free Surface Flow Links 89 7 1 1 Numerical Description 89 7 1 2 The Friction Resistance Described by the Manning Formula 90 7 1 3 Depth variable Manning coefficient 91 7 1 4 Colebrook White Formula for Circular Pipes 93 7 1 5 Hazen Williams Equation 94 7 2 Head Losses in Manholes and Structures Introduction 94 7 3 Standard MOUSE Solution F A Engelund 95 7 3 1 Head Loss atthe Node Inlet 95 7 3 2 Head losses atthe outlet from a node 96 7 3 3 Implementation of the Total Energy Loss Computation 99 7 4 An Alternative Solution Based on Weighted Inlet Energy Levels 100 7 5 Selecting an Appropriate Local Head loss Computation 100 7 5 1 Constitutive Parameters of Head Loss Computation Options 101 7 5 2 Default Computational Options 2 102 7 5 3 Example Node Outlet Head Losses Variation as Function of Head Loss Coefficient Mode 103 7 5 4 Implementation of Head Loss Description in Kinematic Wave Simu lations 106 SOME SPECIAL TECHNIQUES 109 8 1 Surface Flooding s sac4 eee4dee 444 bs Se BEd ew SG 109 82 Sealed Nodes seasea e686 0e Ge ed wee Se De ee 110 8 3 Spilling Nodes
3. q the discharge pr m of the weir m2 s H upstream water level above the crest m Cy the level discharge coefficient for the sharp crested weir obtained as Cy 2 3C see section 2 4 1 Overflow weirs ll p 26 34 MOUSE Functions as The coefficient Cy is given in Table 2 2 for different values of the weir height divided by the water level above the crest w H Table 2 2 Variation of Cy for different values of w7 H Cy Value of w H 4 7 0 673 0 757 0 761 wpis 0 707 1 n In the interval from w H 0 05 to 0 1 the coefficient Cy is interpolated linearly between 0 761 and 0 757 In the interval from w H 0 1 to 0 2 the coefficient is interpolated linearly between 0 757 and 0 673 Ignoring the energy loss from the upstream section to the weir section the energy equation reads 2 2 pagis a 2 20 2g H w 2g y y where y the distance from the sill level to the surface at the weir crest m g the Coriolis factor Wy the vertical contraction coefficient E the energy level m The depth at the weir crest is considered to be critical i e y y 2 3E This assumption is very rough because the streamlines are curved As a consequence the depth over the crest will be less than the critical depth In the context of the present implementation curvature of the streamlines is ignored since the expression is only used to evaluat
4. MOUSE PIPE FLOW Reference Manual 47 a Modelling the Physical System If pumps are present in the model set up it might be necessary to introduce relatively small time steps 5 10 sec 2 4 4 Flow Regulation In computational terms the flow regulation differ fundamentally from the weir orifice and pump function by the fact that the control is simulated within the pipe connecting two nodes and NOT by replacing the pipe with a functional relation This means that the conduit connecting the two specified nodes is treated by the algorithm as a normal link The flow is controlled by setting the general equation coefficients at the control loca tion first upstream Q point in the pipe The control function is specified as a function of water level in a control node A The control is applied only within the specified range of water levels and if the water level is outside the specified range an unregulated flow applies Therefore it is important that the specified range covers all expected water levels at point 4 Otherwise a sharp transition between the Q defined by the control function and natural unregulated discharge would occur at the range bounds causing numerical instabilities The following expression determines the flow mint O H Qnat for H lt H4 lt H min max Oe else 2 46 Onat where Qreg applied regulated discharge m3s Oy discharge defined by the regulation function m3s Onat na
5. b 2a tan amp 27 out 2 360 oe 2 where 0 is the angle between the centrelines of the inlet outlet and Dou is the diameter of the outlet pipe In the case of a change in elevation the effective flow area is diminished with a factor drop_factor which is equal to 1 when the inlet flows directly into the outlet and 0 when there is no interception between the incoming jet calculated conservatively without the entrainment angle and the out let In between these two conditions the drop factor is interpolated line arly The effective flow area is then interpolated as A pow Sjer 1 drop_factor 1 drop_factor 1 2 Am 2 5 For a straight inlet outlet with no change in elevation the formula gives that the effective flow area equals the jet area The manhole volume contributes to the overall system volume and is included in the computations If the water level exceeds the ground elevation H then surface flooding occurs consequently followed by appropriate treatment by the model see section 8 1 Surface Flooding p 109 Structures basins This type of nodes is associated with arbitrarily shaped structures of sig nificant volume non circular manholes tanks reservoirs basins and natural ponds Structure geometry is defined by a table of data sets min two related to monotonously increasing elevations containing the following H elevation m A cross section area used in calcula
6. 2 624 02 cece ee eee ee Ae Re eS 110 8 4 PressureMains 0000002 E ea e a a AR y 111 8 9 Dry Conduits s s sere ee oie ts Aa E a a 112 Sse 9 NOMENCLATURE lt os 5 cieczehe cheek eRedelsa ties haces 113 10 REFERENCES 2 seces 44 Ad Be hee RAS Gd Ra eS 119 8 MOUSE MOUSE PIPE FLOW Reference Manual 10 MOUSE 1 A GENERAL DESCRIPTION The MOUSE Pipe Flow Model is a computational tool for simulations of unsteady flows in pipe networks with alternating free surface and pressu rised flow conditions The computation is based on an implicit finite dif ference numerical solution of basic 1 D free surface flow equations Saint Venant The implemented algorithm provides efficient and accurate solu tions in multiply connected branched and looped pipe networks The computational scheme is applicable to vertically homogeneous flow conditions which occur in pipes ranging from small profile collectors for detailed urban drainage to low lying often pressurised sewer mains affected by the varying water level at the outlet Hydrodynamics of pris matic open channels can also be simulated Both sub critical and supercritical flows are treated by means of the same numerical scheme that adapts according to the local flow conditions Nat urally flow features such as backwater effects and surcharges are pre cisely simulated Pressurised flow computations are facilitated through implement
7. Agj 1 the surface area between grid points j and j 2Ax distance between points j and j Substituting for the finite difference approximations in Equation 4 3 and rearranging gives a formulation of the following form n 1 n 1 aQ BA y 8 4 7 where a B and y are functions of b and 6 and moreover depend on Q and h at time level n and Q on time level n 4 3 2 Momentum equation The momentum equation is centred at O points as illustrated in Figure 4 4 70 MOUSE Numerical Scheme eas The derivatives of Equation 3 7 are expressed as finite difference approximations in the following way n l n ao Q Q ce Lot 4 8 1 I nts 2 nts ya iad a2 e i ae j 1 i Se 3 4 9 Ox 2Ax 7 n 1 n n 1 n ae thii hn th oh 2 2 Ox 24x ne A2 xj Timestep Axi Ax j t Gridpoint jl j j 1 Figure 4 4 Centring of the momentum equation in the Abbott scheme For the quadratic term in Equation 4 9 a special formulation is used to ensure the correct sign for this term when the flow direction is changing during a time step D f OF O U go 4 11 MOUSE PIPE FLOW Reference Manual 71 Sez Numerical Solution of the Flow Equations in MOUSE Link Networks where n 1 2 n 1 2 p QQ 4 12 Q Q 7 97 As standard fis set to 1 0 With all the derivatives substituted by finite difference approximations and appropriately rearranged the momentum equation can be written in th
8. It was mentioned above that numerical errors in connection with the numerical solution of the kinematic wave equations produce a diffusive dampened wave motion If the pressure term is included in the equation of momentum then a damping term will automatically be included in the equations the correct solution is a dampened wave motion The momentum equation for diffusive wave approximation reads ga g40 gAly 3 22 By retaining the pressure term 0h dx in the computation it is possible to implement the downstream boundary conditions and thus consider back water effects The diffusive wave approximation is therefore from a theo retical and practical point of view a better approach than the kinematic wave approximation The computational basis for the diffusive wave approximation is in princi ple identical to the one applied for the dynamic wave approximation for Froude number Fr gt 1 supercritical flow Further more for stability rea 62 MOUSE Dynamic Wave Approximation sons a moving average in time is applied to the slope of the water surface Oh ox in order to dampen the short periodic fluctuations This means that only relatively steady backwater phenomena compared to the time step are resolved 3 6 Dynamic Wave Approximation 3 6 1 General The general flow equations form the best theoretical foundation for a flow model because the full equation of momentum makes it possible to describe all forces af
9. nodes and the boundary conditions are required at both end of the con duit at each time step throughout the computation In some situations boundary conditions are specified as unique relations of two flow variables e g stage discharge relation i e as hydraulic bounda ries in certain points These are defined as functions i e as a part of the system description In other cases proper boundary conditions are constructed by the model as a consequence of current flow situation and of various user specified dis turbances in form of e g adding or extracting water controlling the flow adding energy pumping or as effect of external water level These distur bances may be constant stationary or time variable By default MOUSE supplies all necessary boundary conditions founded on the topology and geometry of the system Therefore the simulations can be run even if no boundary conditions of the other type are specified by the user With respect to the volume balance in the system two groups of boundary conditions can be distinguished 1 External boundary conditions describing the interaction of the modelled system with its surroundings 2 Internal boundary conditions describing relations between certain parts of the model The external boundary conditions comprise the following At manholes and structures 1 Constant inflow or extraction Q const 2 Time variable inflow or extraction O Q t 3 Computed inflow
10. thus allowing correct simulation of fast transients and backwater profiles The dynamic flow description should be used where the change in inertia of the water body over time and space is of importance This is the case when the bed slope is small and bed resistance forces are relatively small 2 Diffusive wave approach which only models the bed friction gravity force and the hydrostatic gradient terms in the momentum equation This allows the user to take downstream boundary conditions into account and thus simulate backwater effects The diffusive wave description ignores the inertia terms and is therefore suitable for back water analyses in cases where the link bed and wall resistance forces dominate and for slowly propagating waves where the change in iner tia is negligible 3 Kinematic wave approach where the flow is calculated on the assumption of a balance between the friction and gravity forces This means that the kinematic wave approach cannot simulate backwater effects Thus this description is appropriate for steep pipes without backwater effects Which Flow Description Depending on the type of problem the most appropriate description can be selected All three approaches simulate branched as well as looped net works The dynamic wave description is recommended to be used in all cases except where it can be shown that either the diffusive or kinematic descriptions are adequate The diffusive and kinematic wave appr
11. w 4 2 38 Y 2g y 2g The momentum equation from section 2 to 3 can be written as 2 yy 2 y Leet ea ED 2 gy 2 gy 2 sree where the shear stress on the bottom from section 2 and 3 is neglected The contracted overflow area can then be expressed by applying the verti cal contraction coefficient given as y Y Wo 44 MOUSE Functions as By rearranging the two equations and eliminating one of the two unknowns q and y2 the combined equation reads Ty Bee OT I 1 2 W Y Wy 503 YalG 0 2 40 g G a Introducing the constants C 1 w9 1 y3 and C 1 y 7 1 wo 2 the equation can be reduced to a second degree polynomial in the form 1 1 TCi Cyt Cy w2 y1 ZCay 0 2 41 Introducing A 1 4 C B Ch C C w w2 y 1 4 Cy3 2 42 it can be shown that the only realistic solution for the second degree poly nomial is the negative one So y can be expressed as _ B NJ4B Adc 2 43 Jy TA and the discharge can then be derived from equation 2 38 The solution is sensible to the selection of the vertical contraction coeffi cient The contraction coefficient must be determined so that smooth tran sition between free and submerged underflow is maintained For a certain range of contraction coefficient values only imaginary solu tions to the Equation 2 43 exist In such cases i e as long as the com bined energy and momentum equation fail to deli
12. Area MOUSE Classic Engelund Classic Water depth Km 0 25 Full MOUSE Classic Engelund Modified Classic Velocity head Km 0 25 Full Flow Through Manhole Classic Water depth Km 0 25 Calculated No Cross Section Changes Weighted Inlet Energy No head loss calculation Weighted Inlet Ener na Water depth nia Km 7 5 3 Example Node Outlet Head Losses Variation as Function of Head Loss Coefficient Mode In this example a simple sewer system consisting of two pipes two man holes and one outlet is constructed Tests for different head loss types a b and c have been performed with various modifications in flow direc tion or drop height or both Table 7 1 shows a complete test matrix Four variants of the model setup have been constructed 1 Straight sewer pipelines with no drops and no changes in directions I A change in direction is introduced in variant I IW A drop is introduced in variant I IV A drop and a change in direction are introduced in variant I A definition sketch of the setups I IV is shown in Figure 7 6 The manual calculation example corresponds to test No 4 in the test matrix MOUSE PIPE FLOW Reference Manual 103 Flow Resistance Figure 7 6 Example definition sketch In the performed tests the value of the HEADLOSS COEFFICIENT has been set to 0 5 for all three types a b and c The head loss coe
13. Colebrook White friction resistance can only be used if an implicit friction formulation is activated Use of the Colebrook White formula must be restricted to circular pipes only Also the Colebrook White formula is fully valid for full flowing pipes 7 1 5 Hazen Williams Equation The Hazen Williams equation is an empirical formula which relates the flow of water in a pipe with the physical properties of the pipe and the pressure drop caused by friction It is used in the design of water pipe sys tems such as fire sprinkler systems water supply networks and irrigation systems The Hazen Williams equation has the advantage that the coeffi cient C is not a function of the Reynolds number but it has the disadvan tage that it is only valid for water and it not able to account for temperature or viscosity Q k CAR oe 7 11 where k is a conversion factor for the unit system k 1 318 for US customary units k 0 849 for SI units C is the roughness coefficient R is the hydraulic radius S is slope of the energy line head loss per length of pipe 7 2 Head Losses in Manholes and Structures Introduction The general flow equations are valid only for continuous conduits where in principle the only resistance to the flow originates from the bottom and side wall friction Hydraulic conditions in nodes i e at manholes and structures take the role of boundary conditions for computation of the flows in the con
14. FLOW IN LINKS 3 1 Saint Venant Equations General Computations of the unsteady flow in the links MOUSE Pipe Flow Model applied with the dynamic wave description performs by solving the verti cally integrated equations of conservation of continuity and momentum the Saint Venant equations based on the following assumptions the water is incompressible and homogeneous i e negligible variation in density the bottom slope is small thus the cosine of the angle it makes with the horizontal may be taken as 1 the wavelengths are large compared to the water depth This ensures that the flow everywhere can be regarded as having a direction parallel to the bottom i e vertical acceleration can be neglected and a hydro static pressure variation along the vertical can be assumed the flow is sub critical Super critical flow is also modelled in MOUSE but using more restrictive conditions The general form of the equations takes the form as follows Conservation of Mass continuity equation 00 24 _ 9 3 1 Ox t Conservation of Momentum momentum equation 2 a0 OQ Ar oy a7 Ox gaa gAl gAly 3 2 where Q discharge m3s A flow area m2 y flow depth m g acceleration of gravity ms 2 x distance in the flow direction m t time s MOUSE PIPE FLOW Reference Manual 53 a Description of Unsteady Flow in Links a velocity distribution coefficient I
15. Generals oraa a a i i ae e a a P iA edi a a Tae a aS A a 59 3 4 2 Implementation ooa aaa 00 000 60 3 5 Diffusive Wave 0 00000 62 3 6 Dynamic Wave Approximation ooo 63 3 6 1 General aoaaa aaa a 63 3 6 2 Supercritical flow simulations with dynamic wave approximation 63 3 7 Flow Description in Links Summary 64 3 7 1 Inventory aaa 0 0002 eee 64 3 7 2 Which Flow Description 64 4 NUMERICAL SOLUTION OF THE FLOW EQUATIONS IN MOUSE LINK NET WORKS 67 4 1 General aaa aaa a 67 4 2 Computational Grid oaa 67 4 3 Numerical Scheme aoaaa aaa a 68 4 3 1 Continuity Equation aa aaa a 69 4 3 2 Momentum equation 00 70 6 MOUSE 44 The Double Sweep Algorithm 0000005 72 4 4 1 Branch matrix 2 0 2 020 000 0000 000 eee eee 72 45 Stability Criteria cd ca Loon 24 Seeded ide oe A ee Bee 2 75 4 6 Optimising the Simulation Time Step At 76 4 6 1 Automated Self adaptive Time Step Variation 77 4 6 2 Criteria Controlling the Self adaptive Time Step Variation 77 4 7 Mass Continuity Balance 2000000200 81 4 7 1 Improved Continuity Balance for Links 81 4 7 2 User Defined Minimum Water Depth 82 INITIAL CONDITIONS 020 0 0202020 0 0 000000000000000 83 5 1 Default Initial Conditions aoaaa aa a 83 5 2 Initial Conditions provided by Hotstart
16. Hydraulic Engineering Vol 116 No 11 November 1990 4 _ Cunge J A and Wegner M 1964 Integration numerique des equa tions d ecoulement de Barre de Saint Venant par un schema implic ite de differences finies Application au cas d une galerie tantot en charge tantot a surface libre La Houille Blance No 1 5 __ F A Engelund og FI Bo Pedersen Hydraulik Den Private Ingeniorfond Danmarks Tekniske H jskole ISBN 87 87245 64 7 In Danish MOUSE PIPE FLOW Reference Manual 119 a References 120 MOUSE
17. O shaped pipe the dimension to be specified is the width D m and for the Egg shaped pipe the dimension to be specified is the cross section height The non standard link cross sections can be specified and maintained through the Cross section Editor Cross sections are distinguished as opened and closed i e open channels on the one side and pipes and tunnels on the other 16 MOUSE Links aS The data required for description of a non standard cross section can be entered in a raw form either in a X Z or in Height Width format please refer to the user guide which gives six options in total Pairs of X Z co ordinates in a counter clockwise direction Z X Z Open X Z Closed x Figure 2 2 X Z types of cross sections Pairs of H W co ordinates in an upwards direction H H W Open H W Closed W Figure 2 3 H W types of Cross sections MOUSE PIPE FLOW Reference Manual 17 Modelling the Physical System The raw geometrical data are then automatically processed in order to create tables with parameters suitable for flow computations Such a table contains 50 data sets covering the range from the lowest to the highest point specified in equal increments The parameters in the table are W surface width m L height relative depth m A cross section area m R AP hydraulic radius m In case of a closed link MOUSE automatically provides an appropr
18. Supplying Boundary Conditions Examples to be continued MOUSE PIPE FLOW Reference Manual 87 Boundary Conditions MOUSE Friction Losses in Free Surface Flow Links a 7 FLOW RESISTANCE 7 1 Friction Losses in Free Surface Flow Links 7 1 1 Numerical Description Head losses caused by the resistance in free surface flow links are intro duced as a friction slope term into the momentum equation see section 3 2 Implementation of the Saint Venant Equations in MOUSE p 54 The friction slope yis equal to the slope of the energy grade line and is defined as I oo 7 1 where t tangential stress caused by the wall friction Nm 2 p density of water kgm 3 R hydraulic radius m A P where P is the wetted parameter The friction slope can be derived as a function of an appropriate combina tion of the flow parameters Q A and R and the water and conduit wall properties v k Generally the friction slope can be expressed as l f O 7 2 where f is a generalised friction factor By these means the friction slope is explicitly determined as a function of instantaneous values of local flow parameters A more stable formulation is achieved through an implicit description of the friction term It is derived from a variational principle at a grid point j as I I Odly 1 0 OF 7 3 MOUSE PIPE FLOW Reference Manual 89 Flow Resistance 7 1 2 T
19. This is normally not the case with closed conduits where the value of conveyance drops in the region near the top of the sec tion For such cases when raw data are input MOUSE adjusts the hydrau 18 MOUSE Nodes as lic radius so that the limiting conveyance for the cross section corresponds to the actual conveyance value for the full profile When closed cross sec tion data are input in the processed form attention should be paid in the upper region of the profile so that decreasing conveyance is avoided 2 3 Nodes 2 3 1 General Description Points associated with link ends and junctions are called nodes Each link is actually defined with exactly two nodes Depending on the position in a network layout a node is associated with one or more links In the later case a node is called a junction An arbitrary number of links can be attached to a junction thus allowing construction of arbitrary network lay outs 2 3 2 Types and Definition of Nodes Every node in a network is defined by its identification max 25 charac ters and its x and y co ordinates m Exception is storage nodes which do not require co ordinates Further according to the type of node an appropriate set of parameters is required Circular Manholes Circular manhole is a vertical cylinder defined by the following parame ters Aout bottom elevation m H op surface elevation m Dm diameter m K outlet shape types 1 9 D
20. a generalisation it is assumed that the effective flow area in the man hole equals the cross section area of the jet at the outlet This is valid in the case of no change in direction from inlet to outlet It is calculated as D 2 2 m 6 8 pD E E pe Di 1 2 a tan 2T 2 2 in la Agi A et where D is the diameter of the inlet pipe So far the alternative formula is only applicable in MOUSE for manholes with one inlet and one outlet However the implementation includes the possibility for a change in elevation and a change in flow direction from inlet to outlet 4 ee pee Figure 2 5 Manhole with one inlet one outlet and a change in flow direction In the case of a change in flow direction the effect of the jet at the outlet will gradually diminish with increasing angle The effective flow area is therefore linearly interpolated between the full cross section area of the manhole A and the area of the jet Ajen as the angle increases The distance a from the point where the jet intercepts the manhole to the centreline of the inlet see Figure 2 5 is conservatively calculated as half the diameter of the inlet D thus neglecting the entrainment angle of 6 8 Din 2 3 aii 2 3 MOUSE PIPE FLOW Reference Manual 21 Modelling the Physical System The distance b from the point where the side of the outlet enters the man hole to the centreline of the inlet is approximated with
21. and con servation of momentum The equations are rewritten and solved for q and h points Conservation of mass h point CQ 24o 2 51 50 MOUSE Functions 2 4 8 Valves Conservation of momentum q points SO A gs gAly gly 2 52 Q the flow A the flow cross section y the depth of water I the friction slope lI the bottom slope of the canal Taking into account the continuous discharge over the weir Equation 2 52 is modified to LaLa 2 53 In branches with long weirs the pair of equations 2 52 and 2 53 is solved The implementation of long weirs will include the possibility of specify ing a variation of the weirs crest along the weir for example a sloping weir crest For this type of weirs MIKE URBAN will divide the weir into a series of smaller sections equal to the number of computational points and apply an average crest level in each of these sections Implementation in MIKE URBAN It is possible to define a valve between any internal nodes but not at an outlet The valve will be topologically fully defined with two node identi fiers defining the upstream node FROM and downstream node TO The definition of the upstream and downstream nodes does not restrict the direction of the flow because the valve function allows the flow in both directions depending on the current hydraulic conditions Practically this means that if the pressure level in the downstream node
22. and therefore inherently yields approximate results The following description covers weirs where it is acceptable to assume a constant water depth at the weir crest MOUSE also supports a weir where this assumption is not acceptable This type of weir is called a long weir please refer to the section con cerning this type of element Definition of an Overflow Weir General Overflow weirs structures can be specified in nodes defined either as manholes or as structures but not at an outlet A weir is topologically fully defined with two node identifiers defining the upstream node FROM and downstream node TO The definition of the upstream and downstream nodes does not restrict direction of the flow because the weir function allows the flow in both directions depending on the current hydraulic conditions Practically this means that if the water level in the downstream node is higher than the water level in the upstream node then the water flows backwards i e the computed flow rates are given a negative sign If an overflow structure discharges out of the contemplated pipe system then the downstream node identifier is left unspecified empty 26 MOUSE Functions The relation between the water level in the structure or manhole and the released discharge can be defined as a specific O H relation or the built in overflow formula can be used In the later case the discharge is calculated
23. critical depth m Yn normal depth m In the later case the outlet is considered to be a free outlet meaning that the outlet water level does not influence the flow in the adjacent link Otherwise the model applies the specified water level with the corre sponding backwater effect and a possibility for reverse flow 2 4 Functions Functions are used for the calculation of the flow between two nodes or in specified links according to the functional relation and the hydraulic con ditions at relevant points in the system MOUSE PIPE FLOW Reference Manual 25 a Modelling the Physical System There can be more functions defined simultaneously between two nodes of the network One or more functions can be defined in a link between the two nodes 2 4 1 Overflow weirs The overflow structures are normally found in sewer systems with pur pose to lessen the hydraulic load in the pipe system during extreme flow conditions by allowing a part of the flow to be spilled to a recipient Also overflow structures can be used for internal distribution of the flow within the pipe system According to hydraulic conditions two different types of overflow are possible e free overflow e submerged overflow The free overflow is the more frequent of the two types and the present conceptualisation is therefore concentrated on this phenomenon The com putation of the submerged overflow is based on the same concept as the free overflow
24. equation represent the inertia forces local and convective acceleration while the third term represents pressure forces The two terms on the right hand side of the equation represent gravity and friction forces respectively The velocity distribution coefficient accounts for an uneven velocity dis tribution across a section and corresponding difference in the actual momentum compared to those obtained with an average velocity It is defined as a fras 3 5 QO Assuming that the bottom slope J is small y 0 then Z can be expressed as a function of the water depth and water surface gradient i e Oy oh Mi aa oe 2 0 MOUSE PIPE FLOW Reference Manual 55 Description of Unsteady Flow in Links 3 3 It is thus possible to use the height A above a certain reference level as the dependent variable instead of the water depth y The equation of momentum can hence be written as 2 aa ag AZA ah _ T T ga gAl 3 7 Pressure and gravity forces can be expressed in one term only as oh gA P 3 8 The friction slope Ipis equal to the slope of the energy grade line and is introduced into the equation using the Manning s formulation for more details see section 4 Numerical Solution of the Flow Equations in MOUSE Link Networks p 67 Modelling The Pressurised Flow The full flow capacity of a closed conduit pipe can be defined as a dis charge at which the flow depth is equal to the condu
25. etc A list of all possible limiting factors is given below If any of these criteria are exceeded i e if the generated variation is too large then a revised solution is calcu lated The revised solution is obtained as a linear interpolation between the last two simulation results the previous time step solution and the solution with preliminary time step so that all specified criteria are fulfilled The different criteria which control the variation in the time step are out lined below The user has the option to modify the individual criteria through variation in the parameters All of these parameters must be defined in the DHIAPP INI file 4 6 2 Criteria Controlling the Self adaptive Time Step Variation Resolution of the Boundary Conditions The time step is limited by the excessive errors generated due to the differ ence in the boundary time series resolution In case of relatively fine reso lution of boundary time series application of long time steps may e g MOUSE PIPE FLOW Reference Manual 77 Sez Numerical Solution of the Flow Equations in MOUSE Link Networks cause volume errors The maximum allowed error in the boundary condi tions is given by e lt QacceptLimitRel Bvar 4 22 where s is the largest error between the given and simulated boundary conditions see Figure 4 9 Bvar is the value of the given boundary condi tions and QacceptLimitRel is a user specified value given in the DHI APP INI fil
26. follows the Manning function up to a value of y D 0 8 see Figure 3 3 According to the kinematic wave theory Q Qj will not increase further after the pipe runs full as the pressure grade line is assumed to remain par allel to the pipeline In reality however pressurised flow often gives rise to an increased pressure gradient and thus an increased flow rate The kinematic wave theory is therefore not suitable for computations of pres surised flow without special adaptations y D Mouse Degree of Filling 1 0 0 9 0 8 0 7 0 6 0 5 0 4 0 3 02 01 Q aQ full 01 0 2 03 0 4 05 06 07 08 0910 Figure 3 3 The Degree of Filling function applied in MOUSE In order to make an approach to pressurised flow the following assump tion has been made An increase in pressure gradient gives rise to an increased flow rate according to 1 3 21 MOUSE PIPE FLOW Reference Manual 61 Description of Unsteady Flow in Links 3 5 lis the remaining part of the pipe length This correction corresponds to an empirical deviation from the kinematic wave theory so that the pressure grade line is no longer parallel to the pipe slope 10 of the excess pressure is now used to increase the pressure gra dient see Figure 3 4 Pressure Gradeline Manning Gradeline 0 9 y D B58 AAA l Figure 3 4 The assumption that 10 of the excess pressure is used to increase the pressure gradient Diffusive Wave
27. for all Courant numbers In practice however this is restricted because the numerical implementation and the accuracy criteria impose some additional limitations The most conservative condition for a correct and stable solution of the implemented finite difference scheme is the velocity condition v AtS Ax 4 21 The automatically generated computational grid fulfils this condition 4 6 Optimising the Simulation Time Step At The computational efficiency of any discrete time numerical simulation algorithm is highly dependent on the time step applied in the simulations In turn the feasible time step in a concrete situation depends on apart from the inherent performance properties of the computational scheme the dynamics of the flows in the simulated network It is therefore desira ble to optimise the algorithm so that in conditions of variable flow dyna mics as usually occur during the simulated interval the total computational effort is minimised while preserving stable and accurate computations MOUSE optimises the simulation time step by e The automated self adaptive time step variation This is controlled by the actual hydraulic and operational conditions within the entire model area throughout the numerical simulation This concept can be applied in connection with the Dynamic and Diffu sive flow descriptions while it cannot be used with the Kinematic flow description In this context it is important to note that
28. is defined as a sealed node then the maximum water level at a node is set to the ground surface In this case the pressure will rise without any water on the ground surface The following relations are valid Hn Pr for Pa lt Hop 8 1 and H H op for Pp gt Hop where Hm is the water level in the node m Pm is the pressure level in the node m Hop is the ground level for the node m 8 3 Spilling Nodes Any manhole or basin can be defined as spilling If the water level in a node defined as a spilling node reaches the ground level the water will start spilling irreversibly out of the system The flow will be computed using the free overflow formula according to the following for Hn lt Hop t AP 8 2 Ospi gt 0 for Hn gt H op AP O i Relative Weir Coefficient 0 63 B 2g Hy Hop AP op where Qspii the spill discharge m s B a conceptual spill width m H the water level in the manhole m Hop the ground level in the manhole m AP the Buffer Pressure Level for the spill m g the acceleration of gravity ms Relative WeirCoefficient the linear scaling coefficient for the spill 110 MOUSE Pressure Mains The level i e head at which the spill starts can be controlled by option ally specifying the Buffer Pressure Level as a relative elevation above or below the ground surface default value 0 For circ
29. manhole and all downstream water levels of the inflowing conduits are the same often leads to overestimates of the energy loss at the inlet In many cases the wetted cross section area in the inlet pipe is smaller than in the manhole leading to almost entire loss of the kinetic energy of the incoming flows which is not the case This problem is reduced by applying the effective flow area in the manhole but this is available in MOUSE only for circular pipes and for the flow through man holes i e with one pipe in and one pipe outflow An alternative solution is available which fully ignores the energy loss at the inlet For a flow through manhole this practically means that the energy level in the manhole is set to be equal as at the downstream end of the inlet pipe For manholes with multiple inlets the energy level is calcu lated as the weighted average of the inlet flows i e large flows contribute most to the energy level Thus in this formulation the total loss at the manhole is concentrated computationally at the outlet and can be fully controlled by the user Without doubt this approach proves valuable for some specific situations particularly for the flow through manholes with normal flow conditions However due attention must be paid for cases with high inlet energy levels e g a small pipe with high velocity flow entering a large basin In such a case the energy level of otherwise still water in the basin would b
30. of Contraction HLC Type b interprets the specified value as the outlet contraction coefficient Conn see Equation 7 17 This means that the model ignores the geometrical relations between the node and the outlet links outlet shape and applies the specified value directly as the Con The contraction losses in the outlet links are then computed by multiplying the velocity head in the respective link by the cong The total head loss for an outlet link is computed as a sum of the contraction direction and elevation loss Selection of Total HLC Type c interprets the specified value as the total outlet head loss This means that the model completely ignores the geometry of the node links and applies the specified value Total HLC directly as the Coun the same for all outlet links at the node The total head losses in the outlet links are then computed by multiplying the velocity head in the respective link by the specified C Effective Node Area This parameter is only relevant for MOUSE classic computational method and for flow through manholes in circular pipes In all other cases the default total wetted node area is applied The following choices are availa ble Total wetted area calculated as product of diameter and water depth for manholes and red from the basin geometry table 4 for basins Typically results in overestimate of local loss in a node Calculated Effective Area The effective area in a manhole i
31. of Head Loss Description in Kinematic Wave Simulations When applying the kinematic wave approximation the head loss descrip tion in nodes is based on the same equations as described above However in order to reduce the computational time the energy losses are computed once for a number of different flow conditions and tabulated for use during the simulation In cases where there is more than one inlet link in a manhole the losses are calculated on the basis of the assumption that the flow in each link relative to the flow in the other inlet links is propor tional to the corresponding full flow capacity This assumption affects the 106 MOUSE Selecting an Appropriate Local Head loss Computation a energy losses due to changes in elevation and direction only when these losses are different for the different inlet links MOUSE PIPE FLOW Reference Manual 107 a Flow Resistance 108 MOUSE Surface Flooding maa 8 SOME SPECIAL TECHNIQUES 8 1 Surface Flooding If the water level in a manhole or a basin reaches the ground level an arti ficial inundation basin is inserted above the node The surface area of this basin is gradually over one meter increased from the area in the manhole or the basin to a 1000 times larger area thus simulating the sur face inundation The maximum level of inundation is 10 meter above the specified ground level When the outflow from the node surmounts the inflow the
32. orifice downstream node TO Basic Geometrical Assumptions Bottom is considered horizontal both in the sections upstream and down stream from the orifice The upstream overflow crest height w is calculated as the distance between the orifice invert level and the bottom level of the upstream node Similarly the overflow crest height from downstream w is given as the distance between the orifice sill level and the bottom level of the down stream node Other parameters are described in the following text or illustrated on drawings MOUSE PIPE FLOW Reference Manual 31 a Modelling the Physical System Approximation of Arbitrary Geometrical Shapes An orifice opening is defined as a closed polygon through the MOUSE cross section editor Any form of convex and concave shapes is allowed as long as there are no intersected arcs see Figure 2 9 2 2 Figure 2 9 Examples of an illegal left and correct definition of an orifice poly gon For the computational purpose a polygon is cut into a number of narrow rectangles slices which approximate the shape of an orifice see Figure 2 10 WL downstream BL upstream BL downstream Figure 2 10 _ Illustration of a general shape orifice 32 MOUSE Functions as For the given upstream and downstream water levels flow through the orifice is computed as an integral of the flows through individual slices with the total flow corrected for latera
33. the variable X is one of the three cross section variables and the meaning of Max X depends on the value of Crosscheck If Crosscheck is given as then Max X is the maximum value of the actual parameter over the cross section while a value of Crosscheck which is equal to 2 means that Max X is given as the actual value of the respective cross section parameters However the check is carried out only if the relative depth in the cross section is larger than the variable CrossLowDepthLimit The check of these limitations is carried out at the end of a time step simula tion If limitations are violated then the solution is scaled down with respect to dt The default value of Max VarCrossConstant is 0 03 if Crosscheck is 1 and CrossLowDepthLimit is 0 04 Variation in Courant Number In the dynamic flow conditions the Courant number see section 4 5 Sta bility Criteria p 75 is continuously changing from time step to time step In order to avoid stability and accuracy problems the Courant number is limited by C lt MaxCourant where C ra 4 28 V is flow velocity and dx the distance between two computational grid points Check ofthis limitation is carried out after the simulation of a time step If the limitation is violated the solution is scaled down with respect to dt Recommended value of MaxCourant specified in DHIAPP INI file is 20 60 Weir oscillations If the storage volume in one of the nodes connecting a wei
34. BY DHI MOUSE Pipe Flow Reference Manual MIKE BY DHI 2011 Please Note Copyright This document refers to proprietary computer software which is protected by copyright All rights are reserved Copying or other reproduction of this manual or the related programs is prohibited without prior written consent of DHI For details please refer to your DHI Software Licence Agreement Limited Liability The liability of DHI is limited as specified in Section HI of your DHI Software Licence Agreement IN NO EVENT SHALL DHI OR ITS REPRESENTA TIVES AGENTS AND SUPPLIERS BE LIABLE FOR ANY DAMAGES WHATSO EVER INCLUDING WITHOUT LIMITATION SPECIAL INDIRECT INCIDENTAL OR CONSEQUENTIAL DAMAGES OR DAMAGES FOR LOSS OF BUSINESS PROFITS OR SAVINGS BUSINESS INTERRUPTION LOSS OF BUSINESS INFORMATION OR OTHER PECUNIARY LOSS ARISING OUT OF THE USE OF OR THE INA BILITY TO USE THIS DHI SOFTWARE PRODUCT EVEN IF DHI HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES THIS LIMITATION SHALL APPLY TO CLAIMS OF PERSONAL INJURY TO THE EXTENT PERMITTED BY LAW SOME COUN TRIES OR STATES DO NOT ALLOW THE EXCLUSION OR LIMITA TION OF LIABILITY FOR CONSEQUENTIAL SPECIAL INDIRECT INCIDENTAL DAMAGES AND ACCORDINGLY SOME PORTIONS OF THESE LIMITATIONS MAY NOT APPLY TO YOU BY YOUR OPENING OF THIS SEALED PACKAGE OR INSTALLING OR USING THE SOFTWARE YOU HAVE ACCEPTED THAT THE ABOVE LIMITATIONS OR THE MAXIMUM LEGALLY APPLICA BLE SUBSET O
35. F THESE LIMITATIONS APPLY TO YOUR PUR CHASE OF THIS SOFTWARE Printing History December 2004 October 2007 July 2010 MOUSE CONTENTS noe MOUSE PIPE FLOW Reference Manual 9 1 A GENERAL DESCRIPTION 0 2 00 00 000004 11 2 MODELLING THE PHYSICAL SYSTEM 13 2 1 The Model Elements Inventory 000 13 22 WINKS se ose ete Be ee ee es Be ees nae es 13 2 2 1 General Description 2204 13 2 2 2 Specification of a Link 14 23 NOES e e 6 taae a e a r Ge a eee ee a bee Paws 19 2 3 1 General Description ooo a 19 2 3 2 Types and Definition of Nodes 19 2 4 Functions 0 petada bes iya doe dadi peo idara 25 2 4 1 Overflow weirs aooaa aaa a 26 2 4 2 Orifice Function aaa aaa a 31 2 4 3 Pump Function aaa aaa a 46 2 4 4 Flow Regulation aoaaa aaa eee es 48 2 4 5 Non return Valve 0 0 00000 eee 48 2 4 6 Combined Regulation non return valve regulation 49 2 4 7 Long WINS ask ae 6 he Ge dda ee owe ee G4 GO bed 49 24 8 Valves 0 0 000 00 ee 51 3 DESCRIPTION OF UNSTEADY FLOW IN LINKS 53 3 1 Saint Venant Equations General 53 3 2 Implementation of the Saint Venant Equations in MOUSE 54 3 3 Modelling The Pressurised Flow aoaaa aaa 56 3 4 Kinematic Wave Approximation ooo 59 3 4 1
36. Loss Computation Theoretically the total energy loss at the outlet from the node expressed as a function of the velocity head in the outlet pipe can be as high as the available energy level in the node The limiting case occurs e g with com pletely clogged outlet K gt with no flow in the outlet pipe However in computational reality in order to preserve a robustness of the computation various additional limitations could be introduced With respect to that MOUSE offers two possibilities The first limitation relates the maximum head loss to the depth in the out let pipe 5 232 Vi Va AE min A O Cu 3 35 7 18 It also introduces the limitation on the total head loss coefficient as Cou EG lt 1 0 7 19 These limitations have caused that the computed head losses and the cor responding flow conditions around nodes in some cases were inexact Due to the advances in the computational implementation the limitation from Equation 7 19 could been removed allowing the total head loss for the outlet pipe j being computed as 2 Vi AE min ziro p Clevel Gconir y gt i 36 7 20 The limitation of the total head loss coefficient to 1 0 is however still present MOUSE PIPE FLOW Reference Manual 99 a Flow Resistance 7 4 An Alternative Solution Based on Weighted Inlet Energy Levels The assumption applied in the MOUSE standard solution that the water level in the
37. a tion between relative depth and the slot width as implemented in MOUSE is given in Table 3 1 Table 3 1 Relation between relative depth and slot width y D Bgo D D 1m 0 98 0 36 1 00 0 19 1 10 0 0166 1 20 0 0151 1 50 0 0105 gt 1 50 0 0100 The default slot width can be modified for individual links through the ADP file 3 4 Kinematic Wave Approximation 3 4 1 General The flow conditions in steep partly full pipelines are mainly established by the balance between gravity forces and friction forces Consequently the inertia and pressure terms in the momentum equation are less domi nant Accelerations are comparably small and the flow is almost uniform so that the kinematic wave approximation is a reasonable approach MOUSE PIPE FLOW Reference Manual 59 Description of Unsteady Flow in Links The momentum equation reduces to gAl gAly 3 18 i e the friction slope is equal to the bottom slope uniform flow condi tions In MOUSE the Manning s formula for uniform flow is used and the momentum equation reads O MAR I 3 19 The kinematic wave is independent of the downstream conditions mean ing that disturbances only propagate downstream The kinematic wave description can therefore only be applied in cases when the flow is inde pendent of the downstream conditions which is the case in supercritical flow Froude s number Fr gt 1 The analysis of the characteristics of the kinem
38. a constant time step is simply a restricted case of these concepts 76 MOUSE Optimising the Simulation Time Step At a 4 6 1 Automated Self adaptive Time Step Variation The automated self adaptation of the simulation time step 1s performed during the running simulation Such on the fly calculation of the time step is performed through a three step procedure e Before the actual time step is taken a preliminary value of the time step is calculated on the basis of the following The instantaneously time step is increased by a user specified fraction the time step acceleration Acceptance of this time step is validated through checking the resolution of boundary conditions and pump operations see below Finally the suggested time step is validated with respect to user specified minimum and maximum values The minimum and maximum values and acceleration factors are specified as a part of the simulation configuration If the maximum and mini mum values of dt are equal the program will use a constant time step e The preliminary hydrodynamic solution is calculated with the prelimi nary time step value e Based on an assessment of the preliminary solution a judgement is made whether the used time step is acceptable or not The user has the opportunity to specify numerous different limitation factors such as a maximum allowed variation in the water level in grid points a maxi mum allowed variation in the courant number
39. arbitrary number of pressure main net works and there is no limitation on the number of elements in each sub network Pressure main networks must always converge down to one receiving manhole which is called the tail node The tail node is the point of transi tion between domains where the hydraulic solution is based on the St Venant equation and the special pressure main model The computation of the special pressure main sub models uses the maxi mum of the water level in the St Venant governed domain and the water level at the tail nodes as downstream boundary conditions As default it is assumed that the tail node water level is equal to the maximum of the up vert level of all pressure pipes attached to the tail node but the user can change this default value The upstream pressure main network must be linked with the St Venant controlled domain through pumps The pressure mains feature can handle an unlimited number of pumps attached to one pressure main net work but the solution feature can only handle networks where the upstream link to the St Venant domain is modeled by pumps 8 5 Dry Conduits If parts of the sewer system dry out during the simulation then the model artificially maintains a minimum water depth in those conduits corre sponding per default to 5 o of the characteristic dimension of the con duit diameter for circular pipes or max 5 mm This is necessary with regards to the numerical sta
40. atic wave approximation reveals that a solution obtained for partly filled pipes is physically unreal istic as the characteristic wave speed 0Q 0A increases with increasing depth in a circular pipe filled for less than 60 and decreases with increasing depth when the pipe is filled for more than 60 This points that an uncritical use of the kinematic wave approach can lead to incorrect results caused by an unrealistic deformation of the propagating wave The kinematic wave is by nature undamped The flow rate and the water depth will therefore remain unchanged for an observer moving down stream with the velocity 0Q A Generally it is not realistic to neglect pressure and inertia terms in the momentum equation in most real flow situations Therefore the kinematic wave approximation has to be used with care 3 4 2 Implementation The computations of the kinematic wave approximation in MOUSE are facilitated with the so called degree of filling function The filling function can be determined from the Manning s formula assuming uniform flow conditions i e I Ip MAR F Z z a Mar 3 20 full My yA gn fun 60 MOUSE Kinematic Wave Approximation a where suffix full indicates values corresponding to a filled pipe and y D indicates the degree of filling This theoretically determined filling function has an over capacity at y D gt 0 9 The filling function applied in MOUSE does not include this over capacity but
41. ation of a narrow slot as a vertical extension of a closed pipe cross section Free surface and pressurised flows are thus described within the same basic algorithm which ensures a smooth and stable transition between the two flow types The complete non linear flow equations can be solved for user specified or automatically supplied boundary conditions In addition to this fully dynamic description simplified flow descriptions are available Within the Pipe Flow Model advanced computational formulations ena ble description of a variety of pipe network elements system operation features and flow phenomena e g e flexible cross section database including standard shapes e circular manholes e detention basins e overflow weirs e pump operation e passive and active flow regulation e constant or time variable outlet water level constant or time variable inflows into the sewer network MOUSE PIPE FLOW Reference Manual 11 ea A General Description e head losses at manholes and basins e depth variable friction coefficients The features implemented in conceptualisation of the physical system and the flow process enable realistic and reliable simulations of the perform ance of both existing sewer systems and those under design 12 MOUSE The Model Elements Inventory maa 2 MODELLING THE PHYSICAL SYSTEM 2 1 The Model Elements Inventory Elements available for definition of a numerical
42. ative depth the water depth divided by the height e g by diameter for circular pipes HO is the relative depth before the attempted time step and H7 is the relative depth at the end of the time step AH is the difference in the relative depth through the time step The Water LevDiffMaxRel value can be user controlled from DHIAPP INI file If limitation is violated at any H point in the model then the obtained solu tion is scaled down with respect to dt The default value of WaterLevDiffMaxRel is 0 3 which corresponds to a maximum relative change of 30 Variation of Cross Section Parameters The variation of cross section parameters A R23 and B where A is the cross section area R is the hydraulic radius and B the width of water sur face can be included as additional criterion for limiting the simulation time step Whether the check on the cross section parameters is to be acti vated or not is specified through the variable Crosscheck in the DHI APP INI file the value 0 means that this is de activated while the values 1 or 2 mean that the check is activated in one of the two available variants If the check on the cross section parameters is activated then it is carried out in all H grid points The variation in the three cross section parameters is limited by AX lt MaxVarCrossConstant Max X 4 27 MOUSE PIPE FLOW Reference Manual 79 Sez Numerical Solution of the Flow Equations in MOUSE Link Networks where
43. bility in the solution of the flow equations This correction practically means artificial generation of water i e some water volume is added to the system As a consequence of that the conti nuity status report shown at the end of the simulation does not give a fair impression of the accuracy of the simulation 112 MOUSE Dry Conduits maa 9 NOMENCLATURE a b c zquasi constants in a modified continuity equation around a node A B Zquasi constants in a generalised continuity equation around a node a the speed of sound in water with actual pipe walls rigidity ms a vertical distance from the point where the jet intercepts the manhole to the centreline of the inlet a the speed of sound in water for absolutely rigid pipe walls ms a the speed of sound along pipe walls ms A cross section area m7 Afow effective flow area in a manhole m Aje crosssection area of the jet at the point of interception with the manhole m A crosssection area of the wet part of the manhole m Afu cross section area at full pipe flow m A the cross section area without excess pressure m A structure wetted cross section area m Ag structure water surface area m Ao j surface area between grid points j and j Ao j Surface area between grid points j and j b vertical distance from the point where the side of the outlet enters the manhole to the centreline of the inlet b sur
44. ction m distance between two computational points m node co ordinates m depth m depth in a contracted section m 116 MOUSE Dry Conduits maa 1 2 3 depth in upstream central and downstream section m Ye critical depth m y normal natural depth m y D the relative water depth Z generalised flow variable substituting and Q a Coriolis velocity distribution coefficient a B y coefficients in finite difference equations Cy total calculated node head loss coefficient for outlet conduit j Cai calculated node head loss coefficient due to change of direction Celevationcalculated node head loss coefficient due to change of elevation Ccontr j Calculated outlet contraction head loss coefficient for outlet conduit j 0 weighting coefficent of the numerical scheme horizontal angle between inlet conduit i and outlet conduit j p water density kgm Po density of water for a free surface flow kgm 3 tangential stress caused by the wall friction Nm v kinematic viscosity m2s y vertical contraction coeff MOUSE PIPE FLOW Reference Manual 117 Nomenclature 118 MOUSE Dry Conduits maa 10 REFERENCES 1 MOUSE User Manual and Tutorial DHI 1999 2 Abbott M B Computational Hydraulics Elements of the Theory of Free Surface Flows Pitman 1979 3 Pedersen F B Mark O Head Losses in Storm Sewer Manholes Submerged Jet Theory Journal of
45. d be noted that the compatibility of discharge values at the transition from the free overflow equation to the free underflow equation must be secured Theoretically this transition should take place at the moment where the upstream water level touches the top of the gate This point is difficult to define as the water level is drawn down towards the contracted section Another complication is the fact that the underflow equation is accurate only for upstream depths considerably exceeding the depth of the gate opening For this reason the transition is simply assumed to take place at an upstream water level equal to the top of the gate while the difference between overflow and underflow equations is fully corrected in the under flow computation at that level This requires a correction in the free under flow equation through the use of a correction coefficient For increasing upstream water levels this correction coefficient is gradu ally reduced as follows Crew Ce 020 2 36 new where c aeter hee C 2 37 l woN 2gE with E and C taken at the top of the gate level For increasing upstream levels the discharge coefficient approaches the constant value C usually taken as 0 608 The free flow equations require a further correction based on the pressure distribution at the outflow side There are two extreme cases the jet can either emanate surrounded by free atmosphere like an orifice or it can have fu
46. der to reduce the amount of water generated in conduits due to the changes of surface width as function of water depth i e to improve the continuity balance the Taylor expansion of the general continuity equa tion 3 1 has been applied Since the surface width is assumed to be con stant during two time steps the continuity equation can be rearranged as 120 yoh nT 4 30 where A is the water level m and w is the surface width m The term ise in the equation above can be expanded in a Taylor series W OX z P n 1 22100 A aE aan MOUSE PIPE FLOW Reference Manual 81 Sez Numerical Solution of the Flow Equations in MOUSE Link Networks where 0 represents the time centering of the numerical scheme and n and n refer to the simulation time steps This modification is applicable only for conduits with relatively smooth changes of the surface width As the width for arbitrary pipes and pipes from the cross section database may vary in a very unpredictable way the Taylor expanded equation is only applied to standard pipes 4 7 2 User Defined Minimum Water Depth Further means of controlling the volume continuity balance for links with no or little water are provided as user controlled minimum water depth for links running dry or with very little flow The default minimum water depth can be modified in the DHIAPP INI file In this file two parameters can be changed e BRANCH_MIN_H_REL 20 This is the relative minimum wa
47. duits In turn hydraulic conditions in a node depend on the flows in the inlet and outlet conduits These hydraulic conditions expressed in terms of the energy conservation principle are calculated as water levels and velocity heads The calcula tion is based on the mass continuity and formulation of more or less advanced energy relation between the node and the neighbouring links with inclusion of some energy losses caused by local flow disturbances at different locations in the node 94 MOUSE Standard MOUSE Solution F A Engelund The implemented solution ensures that mutual dependence of the flows in links and hydraulic conditions in nodes are correctly resolved even for complex branched and looped conduit networks Energy losses in junctions are of the same order of magnitude as those caused by the pipe wall friction Knowledge about the magnitude of these energy losses based on experimental data is very limited but some theo retical results are available e g ref 3 Importance of a detailed evalua tion of these losses is related to the relative length of the links I D and grows with relative shortening of the conduits 7 3 Standard MOUSE Solution F A Engelund A simplified computational model for energy losses in junctions imple mented in MOUSE is based on F A Engelund s energy loss formulae see ref 5 Furthermore a critical depth formulation with approximation of critical flow condit
48. e po s eT boundary t t dt Figure 4 9 Resolution of the boundary conditions The boundary resolution criteria is tested on all time series defined as boundary or results from a runoff simulation the CRF file However the test is only applied to boundary conditions which are larger than QlowLimitM3s a minimum flow threshhold value The default value of QacceptLimitRel is 0 1 and QlowLimitM3s 0 01 Variation in the Operation of the Pump Flow The variation in the pump flow through one time step is limited by AQ lt MaxPumpFlowVar Q 4 23 where AQ is the variation in the pumped flow Q is the current value of the pumped flow and MaxPumpFlowVar is the user specified maximum rela tive variation The default value of MaxPumpFlowVar is 0 1 which corresponds to a 10 maximum variation in the pumped flow during one time step 78 MOUSE Optimising the Simulation Time Step At a It should be noted that this test also implies that the simulation is always decelerated down to the minimum time step whenever a pump is switched ON or OFF Variations in the water level in grid points The variation of the water level in all H grid points is limited by the fol lowing functions AH lt WaterLevDiffMaxRel H 4 24 for H lt WaterLevDiffMaxRel H1 gt HO AH lt H for H lt WaterLevDiffMaxRel H1 lt HO 4 25 AH lt WaterLevDiffMaxRel for H gt WaterLevDiffMaxRel 4 26 where H is the rel
49. e TO If the pump discharges out system then the downstream node identifier TO is left unspecified empty The pump operation is specified by defining the range of operation start level Hstaro m and stop level Hstop mM and one of the two available relations in a form of tabulated pairs of values 1 AH m and Qpump m s or 2 H m and Qpump m s The Q um H table consists of min two data sets there is no upper limit Intermediate values are linearly interpolated Variables H H star and Hstop denote water level in a pump sump basin pump wet well node Relation 1 correlates water level in the pump sump basin and the pump discharge O A if H lt H stop or if H lt H O amp start 2 44 else 0 Relation 2 defines the pump performance as a function of the water level difference between the two nodes O AH if H lt H stop or if H lt H O ump start 2 45 else 0 A number of pumps with different operation strategies can be simultane ously defined between the two nodes As the pump performance can be quite significant even during the start up it has been necessary to dampen the pump dynamics in order to sustain the numerical stability The dampening is obtained by centring the pump rate backwards in time so that the pump performance does not instantane ously reach the full capacity but instead the pump discharge is gradually increased over some time steps
50. e calculated as equal to the energy level of the approaching flow i e much higher than realistic with erroneous results as a consequence 7 5 Selecting an Appropriate Local Head loss Computation In some cases results from using different approach for node head loss calculation can be considerably different and due attention must be paid to the selection of the most appropriate approach The head loss calculation for individual nodes can be controlled by select ing one of the available options there are nine options available in MOUSE and five options in MIKE URBAN However in MIKE URBAN user can create any number of options as needed Each option is characterised by the fundamental computational principle and by a number of parameters which control the behaviour of the algo 100 MOUSE Selecting an Appropriate Local Head loss Computation a rithm or the size of loss coefficient In MIKE URBAN the existing options can be modified and new options created in Outlet Head Loss dialog Furthermore the actual head loss calculation for individual nodes can be both in MOUSE and in MIKE URBAN controlled by a local speci fication of various relevant parameters 7 5 1 Constitutive Parameters of Head Loss Computation Options The following parameters constitute a definition of head loss calculation option Computation Method Three different methods are avialable e MOUSE Classic Engelund described in section 7 3 e Weig
51. e following form ahi FRO Era S 4 13 where a f A m a 4 14 8 AA Ax At a q v 0 hi p O77 OP hiy p ps1 4 4 The Double Sweep Algorithm 4 4 1 Branch matrix As shown earlier the continuity equation and momentum equation can be formulated in a similar form compare Equation 4 7 and Equation 4 13 Using instead of h and Q the general variable Z which thus becomes A in grid points with odd numbers and Q in grid points with even numbers the general formulation will be BZ yz 8 4 15 n aZ j Gt J a Writing the appropriate equation for every grid point a system of equa tions is obtained for each conduit branch in the network constituting the branch coefficient matrix as illustrated in Figure 4 5 Applying a local elimination the branch coefficient matrix can in princi ple be transformed as shown in Figure 4 6 below It is thus possible to 72 MOUSE The Double Sweep Algorithm maa express any water level or discharge variable within the branch as a func tion of the water levels in the upstream and downstream nodes e g man holes H and H i e h h A Hp 4 16 and similarly Q O A H3 4 17 Manhole 1 Manhole 2 Manhole 3 Gq Bo Energy E Q 4 H Mom E Q amp 2 bo g2 Cont E Q 83 h Gq B4 4 Os NS 5 Energy E Q Figure 4 5 Branch matrix with coefficients derived from the node energy level momentum and c
52. e functions located in the link see sections 2 4 2 Orifice Function p 31 2 4 3 Pump Function p 46 and 2 4 4 Flow Regulation p 48 are present in the model In the flow regulation restriction only positive flow is affected by the regulation Similarly the non return valve function allows only positive flow MOUSE PIPE FLOW Reference Manual 15 a Modelling the Physical System By convention positive flow values represent the flow in the direction from upstream to downstream node Link Cross Sections As a built in feature MOUSE supports four different pipe cross section types Any other non standard pipe tunnel or open channel can be described through the Cross section database facility by specifying the geometric shape of the cross section or a table of geometrical parameters MOUSE includes the following standard pipes 1 Circular pipe 2 Rectangular pipe B H 3 O shaped pipe H B 1 125 1 4 Egg shaped pipe H B 1 5 1 Egg shaped pipe O shape pipe 1 125D H 4 gt B 2 3D B D Figure 2 1 MOUSE egg shaped cross sections Note the difference in selec tion of the characteristic dimension D Any of the four standard pipe cross sections is fully defined by specify ing the pipe type and characteristic dimension s While for the circular and rectangular shape this is straightforward attention should be paid for the definition of the egg shaped cross sections For the
53. e the effect the velocity term have on the coefficient Cy The effect of curved streamlines is indeed incorporated in the coefficients C and Cz MOUSE PIPE FLOW Reference Manual 35 Modelling the Physical System By inserting y 2 3E on the right hand side of Equation 2 20 the follow ing relation is obtained 2 E 5E od gt g V se E 2eE C EJ2gE 2 21 2ew Z i where Cpg the energy discharge coefficient for the sharp crested weir Since the discharge q can be expressed either via the water level above the crest upstream of the weir or the energy level at the upstream section the following relation between the level discharge coefficient and the energy discharge coefficient can be derived 3 2 2 z ca 1 1 2 22 2gH w H As it can be seen from the relation above the coefficient Cy takes several effects into account One effect is the change of the velocity term in the energy equation v 2g For large values of w is the upstream energy level E approximately equal to the depth over the crest H and Cz is equal Cy For smaller values of w the upstream velocity term becomes more important and Cz and Cy will deviate from each other The other effects are the curved streamlines the change in the Coriolis coefficient a the vertical contraction coefficient w the surface tension and the friction The latter effects influence both Cz and Cy By moving from a g H relation to a q relation t
54. ea in the case of the excess pressure pg y D approximately equals to Ax A 1 822 3 12 a where A the area without excess pressure and a is given as E e a 2 3 13 Po D with E the Young s modulus of elasticity Nm 2 e the pipe wall thickness m The a has the dimension ms and is in the order of 1400 ms for most concrete pipes Combining these equations yields A A p Q 1 1 _ 840 3 at p a S aala a a t on where eta 3 15 Jaisa represents the speed of sound in water considering the compressibility of water and the deformation of the pipe wall Equation 3 10 can now be written as 00 2 ap 840 dy _ 4 3 16 58 MOUSE Kinematic Wave Approximation The analogy with the continuity equation can thus be maintained in case that the fictitious slot width b is specified as A Deion BS 3 17 a a is in the order of 1000 ms for most pipes In order to obtain a smooth transition between the free surface flow com putations and pressurised flow computations it is required to apply a soft transition between the actual pipe geometry and the fictitious slot Such a smooth transition has been designed based on a series of tests with various slot configurations The slot configuration thus obtained ensures stable computations without affecting the accuracy significantly The applied slot width is larger than the theoretical value The default rel
55. ed by copy and paste operations from the MOUSE650 OUT file This ASCII file is generated by every computation with the MOUSE Pipe Flow Model Before using the ADP file the Manning number parameters for the selected lines must be modified i e values for bottom and top of pipe Manning numbers and possibly the variation exponent must be adjusted for the pipes or canals where varying Manning numbers are to be used 7 1 4 Colebrook White Formula for Circular Pipes In 1939 Colebrook and White derived an approximate formula which uni fies the description of the turbulent flow in both rough and smooth circular pipes This formula is extensively used for the computation of flow resis tance in predominantly full flowing pipe networks According to Colebrook and White the friction factor fis computed itera tively using one of the several formulations known from the literature The formula implemented in MOUSE reads os 2 k F cw cew n Erei 7 9 where k the equivalent wall roughness m R the hydraulic radius Re the Reynolds number CW CW2 CW3 cw4 empirical constants The default values of the constants cw cwg are cw 6 4 cw 2 45 cw3 3 3 cw4 1 0 The default values can be modified through DHIAPP INI file The actual friction slope is calculated by using the following relation 2 on E ee 7 10 2g4 R 2g4 R MOUSE PIPE FLOW Reference Manual 93 a Flow Resistance The
56. efinition of the outlet shape is connected with calculation of head losses in nodes see section7 2 Head Losses in Manholes and Structures Intro duction p 94 MOUSE PIPE FLOW Reference Manual 19 Modelling the Physical System MANHOLE DATA coordinate Y coordinate Diameter m Ground Level mabs_ Invert Level mabs Critical Level mabs Outlet Shape Figure 2 4 MOUSE manhole Flow conditions in a manhole are an important element of the overall flow description The following parameters are calculated H water level in a manhole m Vm Velocity calculated per default as Q Y 2 1 e Em Hott Dm i e uniform velocity distribution is assumed The flow area calculated as above gives a very conservatively low esti mate of the velocity head and hence a conservative energy loss in the man hole causing higher water levels in the manholes than observed in reality An alternative formula for a more realistic calculation of the flow area in manholes is also available however only for flow through manholes with one inlet pipe and one outlet pipe The alternative formulation is based on the assumption that the inflow behaves like a submerged jet which entrains water from the ambient fluid and increases the discharge through the manhole The angle of entrainment is approximately 6 8 The cross section area of the jet thus depends on the distance from the inlet MOUSE Nodes eas As
57. face width m b storage width m MOUSE PIPE FLOW Reference Manual 113 Nomenclature b slot B width of Preissmann slot m overflows width m C AR cross section conveyance m8 3 C Cy Cr Cy D out Courant number Coefficient of discharge Coefficient of discharge energy based Coefficient of discharge level based pipe diameter m diameter of the inlet pipe m diameter of the manhole m diameter of the outlet pipe m drop _ factor factor diminishing the effective flow area in a manhole due to e E exp E f drop in elevation the pipe wall thickness m energy level just upstream overflow m Manning s number variation exponent default 1 00 the Young s modulus of elasticity Nm coeff for flow direction change default f 1 g 9 81 constant acceleration of gravity ms Fr h H Froude s number water level m cross sections elevation relative to bottom m pumps water level in a pump sump m overflows water level just upstream the overflow m 114 MOUSE Dry Conduits AH overflows entrance energy loss m pumps level difference between two nodes Hf regulation water level at the control point A m Anon node bottom elevation m H water level in a node m H min Ang regulation water levels at the control point A defining the range in which the regulation is to be applied m Hu water surface ele
58. fecting the flow conditions However larger compu tational load in comparison with the kinematic and diffusive wave appro ximations involves correspondingly larger CPU time for the same analysis Additionally difficulties are present when simulating the super critical flow conditions 3 6 2 Supercritical flow simulations with dynamic wave approximation The full Saint Venant equations 3 1 and 3 2 are applicable in the dynamic wave approximation only for sub critical flow conditions i e for Froude number Fr lt 1 In supercritical flow conditions the equations are reduced to the diffusive wave approximation In the sub critical regime the contribution of the inertia terms 0Q 0t and 0 0 4 is gradually Ox taken out by a reduction factor according to Figure 3 5 Reduction Facter 4 1 0 pe Oo 0 5 1 0 1 5 Froude Number Figure 3 5 Gradual reduction of momentum terms during transition to supercritical flow Similarly the differential equation is gradually centred upstream as the influence of the upstream conditions increases according to the same function MOUSE PIPE FLOW Reference Manual 63 Sez Description of Unsteady Flow in Links 3 7 3 7 1 3 7 2 Flow Description in Links Summary Inventory The MOUSE Pipe Flow Model provides a choice between 3 different levels of flow description approximations 1 Dynamic wave approach which uses the full momentum equation including acceleration forces
59. fficients for drop in the setup HI and IV is 0 4 inlet pipe is 0 6 m above the bot tom in manhole B The head loss for direction in the setup II and IV is 0 25 angle between pipes are 45 The example also includes calculation of the friction loss in the down stream pipe Manual Head Loss Calculation Assumptions The water level in the inlet pipe is assumed equal to the water level in the manhole This implies that the expansion loss at the inlet is automatically assumed All calculated energy losses are assumed to occur at the outlet pipe i e E E pipe AH 7 21 manhole or expressed by using the notation in Figure 7 6 2 2 2 v v v Z y Z 4 Z t Ct 22 2g Ym m 2g Yout out a 2g 7 GC a coefficient expressing the total outlet energy loss see section 7 3 2 Head losses at the outlet from a node p 96 Data Discharge Q 2 0 ms Diameter in outlet pipe Dou 10 m 104 MOUSE Selecting an Appropriate Local Head loss Computation a Diameter in manhole Dy 15 m Velocity in outlet pipe Q is capacity assumed Vou 2 55 ms Length of outlet pipe L 50 0 m Manning number M 70 m s Water level in outlet Hou 15 0 m Bottom level in manhole Zn 14 0 m Head shape loss coefficient K 0 5 Friction loss in outlet pipe from manhole to outlet 2 2 hy ot 5 08 473 7 23 MAR 70 0 7854 0 25 The water level in the manhole H can be found from
60. he link 7 3 2 Head losses at the outlet from a node All the individual losses in a node except the inlet loss calculated by the model are added up at the outlet separately for each outlet link The outlet loss for the link j is assumed to be proportional to the velocity head in the outlet link j 2 i 14 k where amp are individual head loss coefficients for link j calculated on the basis of geometrical set up of the node and flow distribution among the links attached to the node The model distinguishes among the following losses e Change in flow direction e Change in elevation e Loss due to contraction at outlet Loss due to change in flow direction This loss is a function of the angles between the inlet and outlet links and distribution of the discharge in the inlet and outlet links as shown in Figure 7 3 and Figure 7 4 96 MOUSE Standard MOUSE Solution F A Engelund ss Pipe 1 Og Yy M4 13 ym gt 03 V3 A3 Pipe 3 23 oe Q2 2 A2 Pipe 2 Figure 7 3 Manhole consisting of 2 inlet links and 1 outlet link Q3 V3 A3 Figure 7 4 Manhole consisting of 1 inlet link and 2 outlet links Based on the generalised notation the calculation of the head loss coeffi cient is performed individually for each outlet link as follows O 6 air ee L 7 15 9 90 where 7 stands for inlet links and j stands for outlet links Loss Due to Change in Elevation Vertical changes in flow directi
61. he variation in the discharge coefficient should be expected to be smaller The energy level is given as 2 2 Bane ee 2 23 E 2g H w where B Coefficient of the relation between energy and water level 36 MOUSE Functions a By combining Equations 2 23 with 2 22 B can be expressed as wsio 2 24 1 H and the coefficient Cz can be expressed as C Cp 5 2 25 By The table below shows the relation between Cy Cz B and q for H 1 for different values of w H showing indeed that the coefficient Cg shows less variation than the values for Cy Table 2 3 Relations between Cy Ce B q E for different values of w1 H i w H ree E B QforH 1 E 1 71B3 Starting with values for H and w given the energy level can be derived by iteration The iteration implemented in the program is based on a Newton Raphson technique The discharge over the weir can then be determined by inserting the energy level into equation 2 21 Submerged Overflow The submerged overflow is identified when the downstream water level influences the discharge over the weir and water surface is free i e the upper of the gate is not in contact with the water surface as can be seen in Figure 2 11 The submerged overflow case will be applied when the wgH 1 0 and AH H lt 1 3 where w is the height of the orifice and H is the water depth above the sill level MOUSE PIPE FLOW Reference Ma
62. his results in n 0 n Lj lj Odiy eo HOI Oly HQ O 2O 0 PC a The coefficient O determines the time weighting of the scheme For stabi lity reasons the coefficient should be above 0 5 The recommended also default value is 1 0 i e a fully forward time weighting of the scheme MOUSE provides an optional choice between the explicit and implicit flow resistance description through the DHIAPP INI file see relevant doc umentation The explicit description is selected per default The Friction Resistance Described by the Manning Formula The classic explicit application of the Manning s formula reads as _ _ lQ l 7 MAR 7 5 with the friction factor 1 f a MARY 7 6 where M is the Manning number A the area and R the hydraulic radius Usage of the O O instead of Q2 facilitates computations of the reverse flow The Manning s number M or n 1 M is the parameter used as a measure of the conduit s wall roughness Default values are given in section 2 2 2 Specification of a Link p 14 The implicit formulation of the Manning s formula is obtained by the dif ferentiation of f with respect to A which results in f LOM op fOd AFOR dh Madh Adh 3ROh a 90 MOUSE Friction Losses in Free Surface Flow Links a and substituting the derivative into the Equation 7 4 7 1 3 Depth variable Manning coefficient Per default MOUSE assumes a constant Manning s number
63. hted Inlet Energy Method described in section 7 4 e No Head Loss Calculation The first two are described in detail in respective chapters The third option ignores all local losses Regardless of the shape of the outlets geometrical set up of the junction and distribution of flows among inlet and outlet conduits water levels in the junction and the outlet conduit are set equal as if there is no change of geometry and the flow conditions between the junction and outlet conduit This literary means that this option should be applied only where there is no change in cross section If inappropriately applied inconsistent results may be generated On the contrary this option can be recommended for use if an artificial node is introduced somewhere on a straight section of a conduit where actually no losses occur Maximum Loss Limit This parameter is of relevance for both MOUSE Classic and Weigthed Inlet Energy computation methods It actually sets the limitation on the maximum computed headloss to the water depth or the velocity head in the outlet pipe according to equations 7 18 to 7 20 Loss Coefficient The available loss coefficient types distinguish three different interpreta tions of the specified head loss coefficient Selection of Km Type a interprets the specified value as the outlet shape coefficient K see Equation 7 17 MOUSE PIPE FLOW Reference Manual 101 Flow Resistance 7 5 2 Selection
64. hydrograph Q Q0 4 Weir discharging out of the system Q O A 5 Pump discharging out of the system Q Q A Application of negative inflows extraction should be done with due care because extraction of more volume than the system can supply would end up with error in computations MOUSE PIPE FLOW Reference Manual 85 Boundary Conditions At outlets 1 Constant outlet water level H const 2 Time variable outlet water level H H t 3 O H relation at the outlet O Q H Internal boundary conditions can be defined as follows At manholes and structures 1 Weir discharging to another manhole or structure Q O H where H stands for energy level above the weir crest in case of a free overflow and for difference of energy levels upstream and downstream of the weir in case of a submerged overflow If an alternative formulation for the weir is specified with a user specified O H relation such conditions should be provided that the overflow is always free i e that holds the unique relation between the water level and the flow 2 Pump discharging to another manhole or structure Q Q H or Q Q AH where H stands for water level in the manhole or structure and AH level difference between the two manholes or structures associated with the pump Some of the listed boundary conditions are illustrated in Figure 6 1 86 MOUSE Initial Conditions provided by Hotstart a OUTLET Figure 6 1
65. iate slot for pressurised flow computations see section 3 3 Modelling The Pressurised Flow p 56 Intermediate values are linearly interpolated The first set of values is associated with depth equal to zero y 0 and the last set with the maximum specified value relative to the bottom For open channels MOUSE will compute the flow as long as the water level is below the lower end of the cross section If this level is exceeded the computation will be stopped unless extrapolation of cross section is specified in the DHIAPP INI file For closed conduits MOUSE allows an unlimited raise of pressure i e Preissmann slot is extended indefinitely in the height Processed data for a cross section is specified as a table with depth Y width B area A and hydraulic radius R Conveyance is computed automatically by MOUSE as C AR2 3 The processed cross section data table for an open cross section should cover the whole range of the expected oscillation of the water surface If the water surface exceeds the maximum specified elevation in the table the computation is stopped For closed cross sections the processed data table has to cover the entire range from the bottom to the top of the cross section MOUSE adds the Preissmann slot see ref 4 automatically To ensure the computational stability the cross section conveyance should be maintained monotonously increasing or at least constant with increase of water level
66. ions is used in MOUSE for simulation of a free inlet to a manhole 7 3 1 Head Loss at the Node Inlet It is assumed that the water levels in the inlet conduit and in the manhole or structure are the same This assumption implies that the energy loss of the flow entering and expanding in the node amounts to the difference of the velocity heads in the inlet conduit i and the node m respectively 2 AE 7 12 Essentially one dimensional analysis in MOUSE relies on this simplifica tion also in nodes with multiple inlet and outlet conduits i e where mix ing of flows of different energy levels occurs Therefore in some extreme cases where head losses in nodes play a crucial role for the correct solu tion it is advisable to perform a more detailed analysis in order to assess the approximation errors inherent to this approach In a case of a free inlet of a sub critical flow i e when the water level in the junction is lower than the critical depth level in the inlet link the water level in the link is assumed to be equal to the critical depth For different MOUSE PIPE FLOW Reference Manual 95 Flow Resistance cross sections appropriate approximations are applied e g for a circular pipe as follows Q JP where D diameter of the circular pipe m Similarly in a case of a low water level in the junction with supercritical flow steep inlet links the downstream water level is set equal to normal depth in t
67. is higher than the pressure level in the upstream node then the water flows backwards i e the computed flow rates are given a negative sign All valves can be defined either to be non return valves meaning that only flow in the posi MOUSE PIPE FLOW Reference Manual 51 Modelling the Physical System tive direction is allowed or non restricted valves allowing flow in both directions The valve will apply the flow equation 2 54 O A m 2 54 where g the gravity constant k the flow factor which depends on the opening of the valve A full open flow area of the valve AH the energy drop over the valve An assumption for this equation is that the valve is located in a pipe which is running under pressure However should the system run under non pressurized flow conditions the flow area A in equation 2 54 of the valve is reduced by a linear reduction for non pressurized flow conditions A d a3 2 55 where d the depth of the flow depth G the pressurized flow depth This means that the velocity head upstream and downstream of the valve is equal and equation 1 can be rewritten to 4 22 ae Ar 2 56 where h the drop in pressure over the valve MIKE URBAN supports full RTC control features of the new valve which means that it will be possible to define control algorithm for the opening of valve 52 MOUSE Saint Venant Equations General as 3 DESCRIPTION OF UNSTEADY
68. it height Any further increase of discharge fundamentally changes the conditions of flow i e basic assumptions for the derivation of the Saint Venant equations are not valid Namely the flow changes from the free surface flow to the pressu rised flow However it is possible to generalise the equations for free surface flow so that the pressurised flow in closed conduits is covered This is done by introducing a fictitious slot in the top of the conduit see Figure 3 2 The idea of introducing a fictitious slot was first presented by Preissmann and Cunge 1961 and has since been used by Cunge and Wegner 1964 see ref 4 The derivation can be obtained from the continuity equation which can be written as 2 drar WA axar 3 9 assuming the density of water p constant over the cross section 56 MOUSE Modelling The Pressurised Flow a qg Ala Figure 3 2 Pipe with a fictitious slot By partial differentiation is found So 3 10 ol PhS Dd IO L oD lop Lx DIa fod 2b For a circular pipe it can be shown that the density of the water can be approximated as p po 1 3 11 a where Po the density of water for a free surface flow kgm ag the speed of sound in water ms y the water depth m D the pipe diameter m MOUSE PIPE FLOW Reference Manual 57 Description of Unsteady Flow in Links Furthermore it can be shown that the cross sectional ar
69. l contraction Orifice Flow Regimes Basically there are four different types of flow regimes through an orifice i e for individual slice for the approaching flow in sub critical regime These are classified as e Free overflow e Submerged overflow e Free underflow and e Submerged underflow A definition sketch of the four types of flow regimes is shown on Figure 2 11 Further the theory distinguishes different forms of overflow jets depend ing on the geometrical and hydraulic relations In the current implementa tion equations for the ventilated jet for the free overflow and the momentum equation for the filled jet with a simplified correction for the downstream pressure for the submerged case have been adopted These types are the most common The solution for the approaching flow in super critical regime has not been implemented MOUSE PIPE FLOW Reference Manual 33 Modelling the Physical System Overflow free 3 Underflow free Overflow submerged 1 Underflow submerged Figure 2 11 Flow regimes through an orifice Free Overflow This flow regime is identified when the downstream water level has no influence on the discharge over the weir The water surface is free and the solution is therefore a pure free overflow weir solution The weir is considered to be ventilated and sharp crested The discharge over a unit width of a weir for a given water level is given by q Cy HJ2gH 2 19
70. ll contact with the bottom on the downstream side the vertical sluice gate 42 MOUSE Functions as In the first case the pressure over the height of the jet is approximately atmospheric In the other case the pressure follows a hydrostatic distribu tion The real situation usually is somewhere in between these two extreme cases and the flow through the gate is corrected for the influence from the pressure on the downstream side The underflow equation has been derived on the basis of experiments where the downstream bottom level is the same as the sill level of gates w2 0 This implies a hydrostatic pressure distribution in the contracted flow section With positive values for w drop structure however these pressures drop to lower values with nearly atmospheric pressure over the height of the jet In this case the discharge will be higher due to the lower counter pressure Comparison of the orifice flow equation and the under flow equation reveals that this difference may be up to 9 The same rea soning applies to some extent for the case of overflow where the discharge equation for the case of a free overfall w 0 is also based on hydrostatic pressure distribution assumption To cover most cases in a reasonable way therefore the free flow dis charges are increased by 5 for the case where the downstream water level is found below the crest level of the gate For the range of down stream water levels betwee
71. model in MOUSE are 1 Links e pipes standard and arbitrary cross sections e open channels arbitrary cross sections 2 Nodes e manholes e basins structures e storage nodes e outlets 3 Functions for description of certain physical components of sewer sys tems including e overflow weirs e orifices e pumps e non return valves e flow regulators 4 Controllable structures for the simulation of reactive or time depend ent operation real time control including e rectangular underflow gate with movable blade e rectangular overflow weir with changeable crest elevation Principles underlying the concept of controllable structures are described in the MIKE URBAN Collection System User Guide 2 2 Links 2 2 1 General Description Links in MOUSE Pipe Flow Model are defined as one dimensional water conduits connecting two nodes in the model The link definition allows MOUSE PIPE FLOW Reference Manual 13 Modelling the Physical System 2 2 2 that the dependent flow variables e g water levels and discharges can be uniquely described as functions of time and space A link is featured by constant cross section geometry constant bottom slope and constant friction properties along the entire length A straight layout is assumed MOUSE supports two classes of links e closed conduit links pipes e open channel links Closed conduits under certain hydraulic conditions may become pre
72. n the crest level and the upstream level the cor rection applied is reduced quadratically as the downstream water level is increasing The quadratic reduction follows from the quadratic relation between the integrated hydrostatic pressure force and the water depth Although the matrix of free flow discharges is set up for the complete range of downstream water levels up to the level which equals the upstream level it should be realised that some of these corrected dis charges are overwritten by new values for the submerged flow case Submerged Underflow The submerged underflow is identified when the upstream water level is above the gate level and the downstream water level influences the dis charge through the gate The threshold for swapping from free underflow to submerged underflow is for the simplification purpose defined at AH H 1 3 This ensures that the same criterion is applied both in the overflow and underflow cases and a consistency of the solution is main tained when w H approaches unity A definition sketch of the submerged underflow is shown in Figure 2 13 MOUSE PIPE FLOW Reference Manual 43 Modelling the Physical System hz Ya Ne Nps i y Figure 2 13 Definition of submerged underflow A combined energy and momentum formulation is applied the same prin ciple as for the submerged overflow If the energy loss from section 1 to 2 is ignored the energy equation reads 2 2 y w L y
73. nce the analysis For such cases the default initial depth can be reduced by setting the parameters BRANCH_MIN_H_REL controls the initial depth relative to the conduit size and BRANCH MIN H ABS controls the absolute depth of the ini tial water depth to appropriate values in the DHIAPP INI file If there are outlets in the system with initial water level specified higher than the outlet bottom a horizontal water surface is assumed extending inside the system until the point in the pipe system where the water level coincides with the bottom level see Figure 5 1 WATER LEVEL BRANCHES 1 1 1994 00 00 FIGPF16 PRF wom om om emo fem om om om om wa Figure 5 1 Initial conditions with backwater outlet MOUSE PIPE FLOW Reference Manual 83 ma Initial Conditions 5 2 Initial Conditions provided by Hotstart Realistic initial conditions can be specified by taking the water levels and discharges from previously calculated result file Flow conditions at any time level contained in the interval covered by the result file can be chosen as initial condition The result file used as a HOTSTART file has to be complete i e water levels and flows at all computational points have to be saved 84 MOUSE Initial Conditions provided by Hotstart a 6 BOUNDARY CONDITIONS Unique solution of the flow equations requires appropriate set of boundary conditions Flow equations are solved for each conduit between two
74. nual 37 a Modelling the Physical System The submerged overflow case is illustrated in Figure 2 12 also giving the meaning of the geometrical parameters used in the sequel ni h2 H ha y y y w Y Nb hba Y Figure 2 12 Definition fo submerged overflow Since the energy loss from section 1 to 2 is much smaller than the energy loss from 2 to 3 the energy loss is neglected i e E1 Ey The energy equation now reads 2 2 y w 7 y W t 4 2 26 Y 2g y 2g The momentum equation from section 2 to 3 can be written as 2 y e o Z2 4 42 zgo 2 zy 2 Van 38 MOUSE Functions where the shear stress on the bottom between section 2 and 3 is neglected The contracted overflow area can be expressed by applying the vertical contraction coefficient given as y Y V2 W gt There are two unknowns in these two equations By rearranging the equa tions and substituting the q actually g 2g from one of the equations into another the remaining unknown in the obtained equation is y gt The equation can be transformed into a 4t degree polynomial of a general form Caya Cay Coy Cyyg Cy 0 2 28 The polynomial is solved iteratively applying the Newton Raphson prin ciple The initial value of yz applied in the iterations is y y w w7 1 2 AH The iterative process terminates when y converges within the specified threshold or if the number of iterations exceeds the
75. of intersection Free Underflow The underflow is free if the issuing jet of the supercritical flow is open to the atmosphere and is not overlaid or submerged by tail water Following an approach similar to the one developed in the section related to free overflow the discharge through the opening e g gate can be expressed as q Crw 2gE 2 30 40 MOUSE Functions where q the specific discharge E the energy level upstream of the opening w the gate opening Cg the discharge coefficient with respect to the energy level The energy level at the upstream side can be expressed as E H 1 __ 8H 2 31 where H the upstream water level measured from the crest of the weir q the discharge w is the weir height at the upstream side Usually discharge is given as a function of the upstream water depth above the crest rather than by energy level q Cyw N2gH 2 32 with Cc C Cy c ia E 2 33 I paci a H w ie H ppi H where C is a constant representing the contraction coefficient of the jet Substitution of equation 2 33 into equation 2 32 leads to the expres sion 2 we H 144 H B 1 2 34 MOUSE PIPE FLOW Reference Manual 41 Modelling the Physical System Further the relationship between Cz and Cy may be derived as Cu Bs C 2 35 From above equations the underflow discharge can be computed How ever it shoul
76. on occur and cause energy losses if there is a difference in elevation between inlet and outlet link These losses are described considering the magnitude of the difference in elevation see Figure 7 5 MOUSE PIPE FLOW Reference Manual 97 Flow Resistance Figure 7 5 Manhole with a difference in elevation between inlet and outlet pipe The individual head loss coefficient is calculated according to the follow ing expression where the weighting relative to the flow rates in the inlet links relative to the outlet link is also included Q Z Z Z D Z D Clevel _ gt O D g D 7 16 i 1 If the calculated head loss coefficient is smaller than 0 a zero value is assumed Loss Due to Contraction The flow leaving the manhole and entering the outlet conduit is more or less contracted and due to subsequent expansion there occurs an energy loss The outlet head loss coefficient depends on the shape of the manhole outlet manhole and the link cross sections and distribution of flow among multiple inlet and outlet links MOUSE calculates the outlet head loss coefficient according to the fol lowing Ceanenti Km ajA 4 7 17 where MOUSE Standard MOUSE Solution F A Engelund 7 3 3 Km specified outlet shape coefficient for the node For relatively large basins K approaches Cooni An flow cross sectional area in the node m Implementation of the Total Energy
77. on the basis of a given structure geometry crest elevation structure width orientation relative to the flow crest type It is important that the width of the overflow is realistic compared to the physical dimensions of the man hole or structure E g an overflow width of 10 m in a manhole having a diameter of 2 m will inevitably cause numerical problems when the over flow is in function Q H Relation The user defined Q H relation consists of at least 2 pairs of tabulated val ues for water level above the weir crest H m and corresponding dis charge O m s Intermediate values are linearly interpolated The O H table has to fulfil certain conditions e the first H value has to be the overflow weir crest elevation e the H values have to be given in a monotonously increasing order e the largest H value given in the table shall not be less than the largest H value to be computed The model does not extrapolate beyond the tabulated values Built in Overflow Formula MOUSE provides two different methods for the computation of the free overflow e Flow computation based on the energy loss coefficient and weir orien tation This is applied if the field for the discharge coefficient on the weir dialog is left empty e Flow computation based on a standard rectangular overflow weir for mula with user specified discharge coefficient This is applied if a dis charge coefficient is specified Energy Loss Coefficient In ca
78. ontinuity equations 0yo Ofo 1 Bo 1yo 1 o 2 Bo 2yo 2 o 3 Bo 3yo 3 o 4 Bo 4yo 4Bo 5 Bo 5yo 5Bo 6Bo Eyo Bo Figure 4 6 Branch matrix after local elimination MOUSE PIPE FLOW Reference Manual 73 Sez Numerical Solution of the Flow Equations in MOUSE Link Networks The continuity equation around a node can in principle be expressed as n l n 1 n 1 n 1 n 1 ah DRsanch cQ As anch Opranch Fa FZ 4 1 8 node branch where a z are quasi constants If Equation 4 15 is substituted herein a global relation can be obtained AH BH Z 4 19 where A B to Z are quasi constants Equation 4 19 shows that the water level in a node can be described as a function of the water levels in the neighbouring nodal points It is there fore possible to set up a nodal point matrix at each time step using the coefficients from Equation 4 19 and the solution to the matrix yields by backward substitution the water levels in all nodal points at the next time step Figure 4 7 shows an example with 8 nodal points and 9 branches 8 7 8 RS 1 x 2 Sa x 3 x Sag x 4 me y 5 x x 6 x x 7 x x x 8 x x x Figure 4 7 Principle of a nodal matrix for a system with 8 nodes and 9 branches 74 MOUSE Stability Criteria maa The crosses in the matrix symbolise coefficients meaning that for instance the water level in node 4 can be expressed as a function of the water levels in node
79. over the link section height However in real situations conduit wall roughness often changes with water depth because different parts of the link cross section are exposed to quite different flow conditions during its lifetime This introduces difficulties in fitting the computed stage discharge curve based on a single M value specified for a link with the actual measured stage discharge relation This is usually related to old systems where significant sediment deposits and pipe wall erosion are present The MOUSE Pipe Flow Model accepts a specification of a non linear variation of Manning number with relative elevation water depth in the conduit Three parameters define the Manning s number variation bot tom value full flow value and a non linear exponent Intermediate values are calculated by a general expression exp Maci E Moott ats Mop Moon 2 7 8 where M act calculated Manning s number Moot Miop Manning s numbers specified for the conduit bottom and top respectively exp Manning s number variation exponent default y D z the relative water depth in a conduit The formula is used for relative depths h D in the interval 0 0 1 0 For relative depth gt 1 0 the Manning number is set to the Manning value The variation between Manningpot and Manning is controlled by the Variation Exponent The variation of the Manning number in relative terms is illustrated in Figure 7 1 An example of the va
80. ow using the same critical depth formulation in the case of a submerged overflow In this situation the head that is driving the flow is expressed as the differ ence between the upstream and downstream water surface elevations Weir Crest s Weir defined in this node Figure 2 8 Principle of submerged overflow The submerged weir flow is then with user specified level discharge coefficient approximated as 3 C B Je Am O eir Cy B J2g AWAD 2 16 30 MOUSE Functions as or with energy loss coefficient for orthogonal overflow weir 3 E 2 2 H Oeir B Je AH AH 2 17 c and for a side overflow weir 0 3 2 H AH Over B dg 2a AE 2 18 3 K 2 4 2 Orifice Function Orifice is an opening of any shape allowing water passage between other wise separated parts of the network Usually an orifice represents a flow restriction Like an overflow weir orifice is defined in MOUSE as a function between two nodes MOUSE supports the computation of flows through orifices of any shape in all possible flow regimes Further a rectangular orifice with moveable top is used for the simulation of controllable rectangular sluice gates Orifice functions can be specified in nodes defined either as manholes or as structures but not at an outlet An orifice is topologically fully defined with two node identifiers defining the orifice upstream node FROM and the
81. oximating logarithmic scaling is applied for the water levels while the scaling of the gate position is linear During the pre processing the following operations are executed e Grids for the full range of upstream and downstream water levels are generated The grid spacing depends on the local geometrical parame ters e Discharge from the dimensionless 4D table for the given upstream and downstream water level and if relevant gate position are read and interpolated e The unit discharges are scaled by multiplying the discharge by the upstream depth above the crest i e slice bottom to the power of 1 5 H15 e The discharge is corrected reduced for the effect of lateral contrac tion e The discharge for entire orifice is summed up The actual flow through an orifice in a given hydraulic situation is obtained during the simulation by interpolating the flow derivatives with respect to h h and wo in the 3 D table and inserting these directly into the MOUSE pipe flow algorithm By these means accuracy and stability of the computation is preserved even with very rapid water level changes and fast movement of the gate 2 4 3 Pump Function Pump functions can be specified in nodes defined either as manholes or as structures but not at an outlet A pump is topologically fully defined with 46 MOUSE Functions as two node identifiers defining the pump sump basin node FROM and the downstream recipient nod
82. oximations are simplifications of the full dynamic descriptions They are implemented to offer improved computational efficiency but should only be used when the omitted terms have insignificant influence When there is any doubt it is better to use the full dynamic description or trials should be undertaken to establish the dif ference between the alternative methods and advice sought from experi enced persons It is very important to have a solid understanding of the influence of the different terms 64 MOUSE Flow Description in Links Summary ea None of the three wave descriptions includes detailed hydraulic descrip tions of hydraulic jumps However the chosen formulations ensure a cor rect description upstream and downstream of the jump MOUSE PIPE FLOW Reference Manual 65 Description of Unsteady Flow in Links 66 MOUSE General as 4 NUMERICAL SOLUTION OF THE FLOW EQUATIONS IN MOUSE LINK NETWORKS 4 1 General The implemented algorithm solves the flow equations by an implicit finite difference method Setting the numerical scheme into the frame of the Double Sweep algorithm ensures preservation of the mass continuity and compatibility of energy levels in the network nodes The solution method is the same for each model level kinematic diffu sive and dynamic 4 2 Computational Grid The transformation of Equations 3 1 and 3 2 to a set of implicit finite difference equations is perfo
83. p bottom slope i friction slope The derivation of these equations is described in a number of textbooks and scientific papers The general flow equations are non linear hyperbolic partial differential equations The equations determine the flow condition variation in water depth and flow rate in a pipe or channel when they are solved with respect to proper initial and boundary conditions Analytical solutions are only possible in special cases with a rather limited number of applications therefore the general equations have to be solved numerically 3 2 Implementation of the Saint Venant Equations in MOUSE The Saint Venant equations can be rewritten as follows OQ OA _ Ax Er 0 3 3 and 2 aa 0Q AY 4 _ ae ee 3 4 with the same nomenclature as for Equations 3 1 and 3 2 The sketch of the system being described by the equations is presented in Figure 3 1 54 MOUSE Implementation of the Saint Venant Equations in MOUSE Figure 3 1 Sketch of the pipe section The equations above are valid for free surface flow only They can how ever be generalised to include flow in full pipes pressurised flow as dis cussed in section 3 3 Modelling The Pressurised Flow p 56 The continuity equation expresses that the volume of water 0Q which is added in pipe section of length Ox is balanced by an increase in cross sec tional area OA storage The first two terms on the left side of the momentum
84. propriate Local Head loss Computation p 100 This actually corresponds to the standard overflow formula for a rectangu lar notch OQweir Ca B 2g Hy 2 12 where C is a discharge coefficient expressed for an orthogonal weir as 3 e EE 2 13 2 K cae and for a side overflow weir Gee 2 14 E aak Where K is the head loss coefficient applied for the upstream manhole E g this method if used with K 0 5 sharp edged outlet is equivalent MOUSE PIPE FLOW Reference Manual 29 Modelling the Physical System to a standard weir formula with C4 0 7589 and C4 0 4582 for orthogo nal and for side weir respectively Please note that the crest type sharp or broad crested has no influence on the calculations The side overflow yields a smaller discharge for the same overflow level because in this case the kinetic energy of the approaching flow is excluded from the computations User Specified Discharge Coefficient If the method with default energy loss coefficient is not applicable for a particular weir the standard overflow formula Equation 2 12 is applied with a user specified level discharge coefficient Cy 2 3Cj which gives 3 Over Cy B J28 HY 2 15 This implies that the head loss coefficient specified for the weir node and the weir orientation are ignored in the weir computation Submerged Overflow The model calculates the flow rate for the submerged overfl
85. r is small weir oscillations might occur for free flow conditions This phenomenon results in a continuous change in flow direction over the weir until the instability is dampened In order to avoid this situation a criterion related to the change in water levels between up and downstream nodes around the weir is implemented The criterion relates to dt by dt AH AH possible dt AH 0 02 4 29 80 MOUSE Mass Continuity Balance maa where AH is the difference in water level between the two nodes con nected to the weir and n corresponds to the time step level The absolute allowed change of 0 02 m is hard coded in the program and cannot be con trolled by the user 4 7 Mass Continuity Balance Theoretically what concerns the mass continuity balance the applied computational scheme is inherently conservative for prismatic conduits with vertical walls In practical applications the continuity balance may be jeopardised in a number of situations such as e Relatively sharp changes of surface width due to rapid changes of water depth or a sharp change of cross section shape with depth This may be e g the case at relatively small depths in circular pipes and in arbitrary cross sections e Sharp changes in surface area of basins e Surcharge of manholes e Etc The scale of the problem is usually related to the length of the simulation time step 4 7 1 Improved Continuity Balance for Links In or
86. rial M N 1 M Code 5 Tron 70 0 0143 6 Ceramics 70 0 0143 7 Stone 80 0 0125 8 Other 50 0 0200 The default values can be edited by the user The modified default values are associated with the current project only i e will affect any simulation carried out with the MOUSE project file Also the default Manning number for any individual link can be overwritten by a user specified link specific value Longitudinal Profile A link is longitudinally defined by bottom elevations of the upstream and downstream end By default link bottom elevations are assumed to be equal to the adjacent node s bottom elevations The default setting can be over ruled by specification of the actual link end elevations but not below the node bottom Normally length of a link is calculated on the basis of the nodes co ordi nates The length computation will take into account if the link between the nodes is not straight Optionally for links connected to circular man holes it is possible to calculate the length from the manhole perimeter In cases where actual link length significantly deviates from the calculated value a user specified length can be supplied instead Longitudinal slope of a link is assumed constant It is calculated using link end elevations and the link length Specification of a node as upstream or downstream has in principle only a declarative meaning and does not affect the computations An exception is if th
87. riation is shown in Figure 7 2 with Manning M values MOUSE PIPE FLOW Reference Manual 91 Flow Resistance Variation exponert 1 0 j y Manni Nh ot Manning top Figure 7 1 Relative variation of the Manning number with relative depth h D Manning 1 100 90 000 90 000 90 000 90 000 90 000 90 000 1 000 90 000 90 000 90 000 90 000 90 000 90 000 0 900 89 686 89 374 88 461 87 000 84 300 81 870 0 800 89 338 88 691 86 833 84 000 79 200 75 360 0 700 88 949 87 934 85 100 81 000 74 700 70 290 0 600 88 506 87 086 83 238 78 000 70 800 66 480 0 500 87 991 86 117 81 213 75 000 67 500 63 750 0 400 87 373 84 977 78 974 72 000 64 800 61 920 0 300 86 597 83 580 76 432 69 000 62 700 60 810 0 200 85 540 81 743 73 416 66 000 61 200 60 240 0 100 83 830 78 929 69 487 63 000 60 300 60 030 0 000 60 000 60 000 60 000 60 000 60 000 60 000 Exponent 0 1 0 20 5 1 0 2 0 3 0 Figure 7 2 Variation of the Manning M for Manning 60 and Manning 90 with different values of the variation exponent The Manning number variation is specified through the ASCII file ADP The specified Manning numbers in the ADP file must follow the selected option for the Manning number convention Syntax of the format of the ADP files must be as shown in the DHIAPP INI and ADP Ref erence Manual 92 MOUSE Friction Losses in Free Surface Flow Links a The lines of the A4DP file related to the Manning number variation may be easily compil
88. rmed on a computational grid consisting of alternating Q and h points staggered grid i e points where the dis charge Q and water level h respectively are computed at each time step see Figure 4 1 The computational grid is generated automatically by the model or with user specified number of grid points The computational grid for a conduit contains an odd number N of Q and A points with A points at both ends The minimum number of computational points N in a conduit is 3 i e two h points and one Q point in between The points are all equally spaced with a distance x equal to Ax 4 1 l N 1 where is the conduit length On the basis of the input data and the specified time step the model auto matically generates a complete computational grid based on the velocity condition see section 4 5 Stability Criteria p 75 The velocity used in the calculation is a full flow velocity obtained from the Manning formula tion assuming completely filled conduit If the velocity condition can not be satisfied for the specified simulation time step which often happens with short and steep pipes then the model issues a warning with proposal for a shorter time step required for the condition to be satisfied MOUSE PIPE FLOW Reference Manual 67 ss Numerical Solution of the Flow Equations in MOUSE Link Networks The grid generated by the model can be altered individually for each con duit i e can be made more dense or
89. s 1 5 and 6 When the nodal point matrix has been solved the solution in the branches is found by backward local elimina tion The bandwidth of the nodal point matrix as indicated by the stippled lines depends on the order in which the nodal points are defined The bandwidth of the matrix in Figure 4 7 is equal to 5 The computational time required for solution of the nodal point matrix depends on the band width size and sharply increases with increasing bandwidth In order to minimise the computational time an automatic minimisation of the bandwidth is performed by internal perturbation of the nodal points The bandwidth displayed in Figure 4 7 for the network with 8 nodal points and the 9 branches could be reduced to 4 as shown in the matrix in Figure 4 8 1 4 5 6 7 2 3 8 RS 1 De W SRA x 2 x x x x Od X 3 x x x zo SNA x 4 x x x poa x T eee x K x x x 6 a x x x x 7 Salty x x x x 8 RSL aby x x Figure 4 8 Minimised matrix band width 4 5 Stability Criteria A criterion for a stable solution of the finite difference scheme is given by the Courant condition _ At v Jey C BDT 4 20 where v mean flow velocity ms Af time step s MOUSE PIPE FLOW Reference Manual 75 Sez Numerical Solution of the Flow Equations in MOUSE Link Networks Ax distance between computational points in the conduit m y water depth m Theoretically the implemented numerical scheme is unconditionally sta ble
90. s calculated on the basis of empirical formula see section 2 3 2 Types and Definition of Nodes p 9 This results in a significantly smaller area than full wet ted area and consequently with a more realistic flow calculation Reduced Calculated Effective Area The effective area in a manhole is fur ther reduced to 50 of the calculated effective area Default Computational Options The following tables provide an overview of available head loss calcula tion options in MOUSE and MIKE URBAN By comparing the two tables it is possible to identify the equivalent options During import of a MOUSE project into MIKE URBAN the nine options get imported under original names with prefix MOUSE_ 102 MOUSE Selecting an Appropriate Local Head loss Computation a MOUSE DEFAULT OPTIONS Method Loss Limit COeMCIeNt coer Effective Type Node Area Round Edged Water depth Km Sharp Edged Water depth Km 05 Orifice Classic Water depth Km 0 5 Full No head loss No CRS Changes 1 calculation nia na 0 nia Energy Loss Classic Velocity head Km 05 Full No CRS Changes 2 5o noa00SS n a na 0 nia calculation Effective Flow Area 1 Classic Water depth Km 0 26 Calculated r Reduc ed Effective F low Area 2 Classic Water depth Km 0 25 calculated Weighted Inlet Mean Energy Approach Energy Water depth Km 0 25 Full MIKE URBAN DEFAULT rer Coefficient Effective OPTIONS Method Lasstimk Type Node
91. se of a free overflow the water depth above the weir crest will be equal to the critical water depth Certain energy loss occurs with a magni tude depending on the structural configuration The overflow situation is schematised in Figure 2 7 MOUSE PIPE FLOW Reference Manual 27 Modelling the Physical System E Energy Loss Figure 2 7 Free Overflow In the critical flow section the Froude s number Fr equals to 1 and the critical flow condition can be written as Fr 1 2 7 where v mean flow velocity ms Ye critical depth m g 9 81 ms Conservation of energy between the upstream and critical cross section yields with 2 E y AE 2 8 with yt AE K 2g 2 9 where E energy level at the section just upstream the structure m AE entrance energy loss m K energy loss coefficient 28 MOUSE Functions Based on the energy conservation and critical flow principles discharge over a plane overflow having a structure width B m is calculated for a weir orthogonal to the flow axis 90 as 3 2 Quin Be de ea 2 10 and for a side overflow weir 0 3 2 2 Quer B dg se H 2 11 where H water depth above the weir crest level m K energy loss coefficient associated with the outlet head loss specified for the weir node see sections 7 2 Head Losses in Manholes and Structures Introduction p 94 to 7 5 Selecting an Ap
92. sparse according to the needs of the current application see documentation on ADP file Manhole 2 Manhole 3 Manhole 1 Figure 4 1 A section of the network with Computational Grid 4 3 Numerical Scheme The implemented numerical scheme is a 6 point Abbott scheme see ref 2 The scheme for the method is shown in Figure 4 2 t timestep n 1 n i 2 At n Centred 6 point Abbott scheme Figure 4 2 The flow equations are approximated by finite differences 68 MOUSE Numerical Scheme eas 4 3 1 Continuity Equation In the continuity equation the storage width b is introduced as A _ Oh a Dpt 4 2 giving CO p OF PE ba 0 4 3 As only Q has a derivative with respect to x the equation can be centred at an point see Figure 4 3 A2 xj Timestep A Ax itl 7 n 1 Gridpoint yl j 1 Figure 4 3 Centring of the continuity equation in the Abbott scheme a general ised scheme Note that in MOUSE Ax and Ax are always equal MOUSE PIPE FLOW Reference Manual 69 Sez Numerical Solution of the Flow Equations in MOUSE Link Networks The individual derivative terms in Equation 4 3 are expressed by finite difference approximations at the time level n 4 as follows Oit Qi O OO ee EE ax 2Ax 64 r Gaa n ah h h t At a b 1s approximated by A A a TO o j 1 b a ema re 4 6 where Ao the surface area between grid points j and j
93. specified number If the convergence is achieved the discharge can then be derived from equa tion 2 26 The value of y is rejected if the maximum number of iterations is exceeded or in the following cases e Ify gives a negative argument to the square root for the discharge e Ify2 gt yrwytw2 or y2 lt wp The equations applied above have some shortcomings At first the effect of the curved streamlines is not taken into account properly in contrast to the free overflow case which is derived from empirical expressions The curved streamlines will in this case give a different pressure distribution over the crest deviating from the hydrostatic pressure and the pressure will be smaller The curved streamlines will become less and less important the smaller the values of AH H are Secondly the contraction coefficient has a significant effect on the discharge e g this approach is very sensi tive to the choice of the vertical contraction coefficient The submerged overflow solution must be compatible with the free over flow at the transition between the two flow regimes In other words intro ducing the submerged solution at AH H 1 3 requires that the submerged discharge for this water level difference is equal to the free flow discharge This is not achievable in all cases and sometimes another pragmatic solu tion must be adopted for the transitional regime MOUSE PIPE FLOW Reference Manual 39 Modelling the Physical Sys
94. ssu rised In such a case the confinement of the flow fundamentally changes the environment in which the flow process takes place but the MOUSE Pipe Flow Model continues to perform the computations using the same flow description as for open channel flow This is possible because MOUSE furnishes actually closed conduits pipes with a fictitious slot Preissmann slot on the top of the cross section thus replacing a pipe with an open channel featuring a cross section shaped to approximate the hydraulic behaviour of a pressurised pipe Specification of a Link Specification of a Link requires specification of the associated nodes see paragraph 2 3 Nodes p 19 the link material longitudinal parameters and the cross section definition shape and size Link Material The parameter which characterises the link material is the link friction expressed as Manning s number M or n 1 M The link can be defined as constituted of one of 8 predefined material types Table 2 1 lists the available link materials with MOUSE default values for Manning s number Table 2 1 Manning s Numbers MOUSE Default Values Mouse Default Value MOUSE Material M N 1 M Code 1 Smooth Concrete 85 0 0118 Normal Concrete 75 0 0133 2 3 Rough Concrete 68 0 0147 4 Plastic 80 0 0125 14 MOUSE Links Ses Table 2 1 Manning s Numbers MOUSE Default Values Mouse Default Value MOUSE Mate
95. system Storage nodes are fully defined with the identification string alone The only other parameter associated with a storage node is the content of water the capacity is not limited currently stored in the storage node Water enters a storage node from any manhole or structure either through a weir gate orifice or a pump A storage node may be emptied by an emptying function 24 MOUSE Functions Outlets Outlets are nodes specified at locations where the modelled system inter acts with receiving waters External water volume is assumed so large that the outlet water level is not affected by the outflow from the sewer system As such outlets are appropriate for simulation of the sewer flow recipients river lake and sea An outlet can behave as an inlet which depends on the flow conditions in the link attached to the outlet and the water level specified at the outlet This means that the flow in both directions can occur Outlets are defined with the following parameters Ay o4 0utlet bottom elevation m H water surface elevation at outlet m Water surface elevation H can be specified as constant or as time dependent see section 6 Boundary Conditions p 85 Depending on the specified outlet water level the model applies the fol lowing elevation of the water surface H in the link adjacent to the outlet Hout for Aour2 Agony t min Ve Yn h else 2 6 Ayo min y Yn where Ye
96. tem Following the approximate rule as for the flow over a broad crested weir a flow reduction is introduced as soon as the difference between upstream and downstream water level is less than one third of the upstream water level The remaining submerged discharge is proportional to the square root of the difference in upstream and downstream water levels above the weir crest The free flow is taken from the sharp crested case as described above The flow in the submerged flow can be approximated as 7 eH forf lt i 2 29 3 where AH the water level difference between the upstream and downstream section qr the free flow at the level where AH 1 3 The implemented algorithm includes a combination of the parabolic and the momentum solution The parabolic solution is applied if the combined energy and momentum equation does not give applicable solutions for the given AHH i e if y is rejected The discharge is solved for decreasing values of AH H and for each value of AH H lt 1 3 is the combined energy and momentum equation evaluated As soon as the combined energy and momentum equation begin to give applicable solutions a swap from the approximate parabolic solution to the combined energy and momentum solution is performed The contraction coefficient will in this case be based on the criteria that the discharge applying the combined momentum and energy solution is the same as from the parabolic solution at the point
97. ter depth in promille of the characteristic dimension in a link e BRANCH_MIN_H_ABS 20 This is the absolute minimum water depth mm in a link The minimum water depth in a link will be set to BRANCH_MIN_H_REL calculated as promille of the link size e g pipe diameter or height of the open channel but never larger than BRANCH_MIN_H_ABS mm In the presented example the minimum water depth is set to 20 promille of the link size but with a maximum of 20 mm This means that for links smaller than 1 meter the minimum water depth is set to 20 promille of the link size For links larger than 1 meter the minimum water depth is kept at 20 mm independently of the link dimen sions 82 MOUSE Default Initial Conditions eas 5 INITIAL CONDITIONS The hydrodynamic computation is started from the flow conditions in the systems specified for time t 0 MOUSE provides two different options for establishment of proper initial conditions 5 1 Default Initial Conditions MOUSE automatically specifies the initial conditions establishing a default initial water depth equal to 0 5 of the characteristic dimension of the conduit diameter for circular pipes but not more than 0 005 m and flow rates are calculated based on the Manning formulation for uniform flow In case of dry weather flow applications the volume of artificially gene rated water may be significant compared to the dry weather flows This may compromise the volume bala
98. tion of the flow velocity in the structure assuming uniform velocity distribution m2 22 MOUSE Nodes as A water surface area used for calculation of volume m K outlet shape types 1 9 The first set of values corresponds to the structure bottom The last set cor responds to the surface level Intermediate values are linearly interpolated The H column can start at any value e g 0 0 for interpretation of H as depth in the basin The MOUSE Engine will associate the first H value to the bottom level of the node Definition of the outlet shape is connected with calculation of head losses in nodes see section 7 2 Head Losses in Manholes and Structures Intro duction p 94 A structure volume contributes to the overall system volume and is included in the computations If the water level raises above the highest elevation value in the table describing the structure geometry the program extends the basin geome try following the principle as described in section 8 1 Surface Flooding p 109 An example of a definition of a basin is given in Figure 2 6 MOUSE PIPE FLOW Reference Manual 23 Sse Modelling the Physical System 14 200 0 000 100 000 E 16 200 16 000 220 000 ai 19 200 49 000 220 000 fa Figure 2 6 Definition of a basin an example Storage Nodes Purpose of storage nodes is a controlled simulation of the surface flood ing i e controlled return of the water into the sewer
99. tural unregulated discharge obtained as an explicit estimate based on the known water levels in the previ ous time step on each side of the regulation point m3 s Ay water level at the control point 4 m Ain Tmax water levels at the control point A defining the range in which the regulation is to be applied m 2 4 5 Non return Valve The function for simulation of non return valves is included into the model structure identically as the flow regulation function 48 MOUSE Functions The flow is applied according to the following Q for Ay 2 FA gwen Cag else 2 47 0 where Q calculated discharge m s 1 Qreg applied discharge m3s Hig Hdown water levels at the computational points upstream and downstream respectively m 2 4 6 Combined Regulation non return valve regulation 2 4 7 A combination of the two previous functions results with min O Ha Qnar for H min Hy H mas and Hup S Hiown Qeg else 2 48 0 where Qreg applied discharge m3s OA discharge defined by the regulation function m3s OQnat E natural discharge obtained as an explicate estimate based on the known water levels in the previous time step on each side of the regulationpoint m3s Ay water level at the control point 4 m Ain Tina water levels at the control point A defining the range in which the regulation is to be applied m Hp Hdown water levels at the computational points
100. ular manholes the spill width B equals to 1 5 times the manhole diameter for the water level H H optAP With increasing water level the spill width B increases following the same functional relation as used for the basin area above surcharging nodes i e increases exponentially to approximately max 1000 times the manhole diameter see paragraph 8 1 For nodes defined as basins the spill width B is set equal to the square root of the basin surface area The spilling capacity of a spilling manhole can be controlled by specifying the Relative Weir Coefficient default value 1 8 4 Pressure Mains The pressure mains also referred to as rising mains in earlier versions of MOUSE feature is intended for modeling the permanently pressurized individual pipes or networks in connection to pumps Computationally MOUSE assumes that a rising main network always runs under pressure and therefore the reaction time within the rising main network is insignifi cant Solution in pressure mains is based on the two equations eo 8 3 Ox and 00 QV where Q discharge m3s A flow area m2 y flow depth m g acceleration of gravity ms x distance in the flow direction m t time s MOUSE PIPE FLOW Reference Manual 111 a Some Special Techniques l bottom slope Ip friction slope All nodes within the pressure main networks are assumed to be sealed MOUSE supports modeling of an
101. upstream and downstream respectively Long Weirs A long weir is an element which is able to simulate variations in the water depth above the weir crestalong the weir itself The long weir must be defined as a link between two channels of the nat ural channel type The weir is topographically defined by the two links defining the upstream link Source channel and the downstream link Destination channel The Location field is the upstream node of the MOUSE PIPE FLOW Reference Manual 49 Modelling the Physical System source channel and the To field is the upstream node of the destination channel Discharge over the weir is calculated at each h point in the upstream downstream channel Thus an internal weir linking two branches require that the number of computational points in the two branches is the same The weirs must always link two open channels Description of the Flow the Long Weir For free flow conditions the weir discharge per length of weir will be cal culated as 2 3 2 qs 3Cav2eH 2 49 where H the water level above the weir crest Cy the weir coefficient g the gravity constant For submerged weir the flow conditions are defined as 2 q AL TN i 2 50 3hy where AH the different between upstream and downstream water levels h the upstream depth over crest level hg height of the weir The Saint Venant Equations are solved for conservation of mass
102. vation at outlet m Agtarp Hsp Pumps start and stop level for a pump m Hipp node surface elevation m Fup gown OF H H water levels at the computational points upstream and down o M Mact M bott stream respectively m bottom slope m friction slope m wall roughness m overflows energy loss coefficient specified outlet shape coefficient for a node conduit length m pipe length which gives rise to pressurised flow m Manning number m 3s calculated Manning s number m 3s Manning s numbers specified for the conduit bottom m 3s MOUSE PIPE FLOW Reference Manual 115 Nomenclature Mpu Manning number at full pipe flow m 3s 1 n N q Q invers of manning number M number of grid points in a pipe specific discharge m s discharge m s Oyu full pipe flow for uniform flow conditions m s O A 4 regulation discharge defined by the regulation function m3s Ona regulation natural discharge m3s Qeg regulation applied discharge m s Oweir overflows discharge m3s R A Phydraulic radius m Ryu hydraulic radius at full pipe flow m t At time s computational time step s mean flow velocity ms flow velocity in a node ms gate opening distance from the overflow crest to the upstream bottom m distance from the overflow crest to the downstream bottom m distance in the flow dire
103. ver reasonable results the parabolic solution is applied similarly as for the transition between free and submerged overflow Practical Computational Aspects Computation of the flows through an orifice is based on a pre processed 4 D table containing the flows through a vertical slice of unit width com puted as a function of four dimensionless parameters wo H w H w2 H and AH H and using the equations described in previous paragraphs The unit flows are computed at discrete points determined by the following set of the dimensionless parameter values MOUSE PIPE FLOW Reference Manual 45 a Modelling the Physical System w H 0 00 0 05 0 10 0 30 0 50 0 80 1 00 w H 0 00 0 05 0 10 0 30 1 00 5 00 100 00 wH 0 00 0 05 0 10 0 30 1 00 5 00 100 00 AH H 0 00 0 01 0 04 0 09 0 16 0 25 0 36 0 49 0 64 0 70 0 80 0 85 0 90 0 95 1 00 This table is stored in a binary file MOUSE650 ORI and is supplied as a part of MOUSE installation At the simulation start MOUSE generates a structure specific 3 D table for each orifice where actual flows to be applied in the computation are stored This table of the size 28 x 28 x 10 contains discharges for all the combinations of 28 upstream and downstream water levels covering the full range of possible water levels When the algorithm is used for a gate the third dimension is used for 10 different gate openings A non equidistant scaling appr
104. water stored in the inundation basin re enters the system When the water level in the node increases and is above ground level the following is assumed During a time step the surface area in the basin is calculated using the water level from the start of the actual time step A situation like this is shown in Figure 8 1 If the water level passes through the transition region between the actual manhole or structure and the artificial basin this assumption leads to generation of water In Figure 8 1 the shaded area illustrates the generated volume of water Waer leva and area in timestep r 1 A 1000 Am Ground leve Generated volume fromtimestep n ton 1 Generated volume fromtimestep nto n 1 Water level and area intimestep n Figure 8 1 Simulation of the surface flooding When the increase of the water level during a time step is relatively small then the generated water volume is negligible If the water level is chang ing rapidly the generated volume of water is important and due to that an appropriate correction is built in the program to ensure no generation of water An alternative to the assumption of constant surface area during a time step is to introduce iterations in the simulation Iterations would signifi cantly increase the simulation time MOUSE PIPE FLOW Reference Manual 109 a Some Special Techniques 8 2 Sealed Nodes Any manhole or basin can be defined as sealed If a node

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