User`s Manual for UGKS1D and UGKS2D Codes
Contents
1. calculate heat flux q with h b W ht bt with q H B W ja print with q ELS B Wr calculate h and b t Figure 1 3 update cell averaged value The equation for updating h and b can be obtained from Eq LII MES Iie gp ae aie nif nn hg 52 mtl 1 2 fi1y2 dt gt aa ee a E a At bp gaa pnt prrtl gtr _ pn At bik bik Meta r a Paa ffa dt gt J o 1 7 Boundary condition 9 Only isothermal wall boundary condition with complete accommodation is discussed Assuming left wall x 1 2 The boundary condition described here is quiet simple the incoming distribution function is directly obtained through interpolation This kind of boundary treatment is sufficient for Knudsen number case Kn 0 075 1 0 10 0 For flow in continuum regime one need to use the same method as the inner region to calculate the incoming distribution function and flux First obtain hi bi by one sided interpolation from the interior region For example n Ax Lk 01kg Second calculate the density at the wall with the condition that no particle penetrating the wall rti peti f UGwd dt f f uf d dt 0 D apurhi K Y apupe Aw lur Uw where gw Pw Aw Uw are the variables at the wall which gives Pw d The corresponding reduced Maxwellian distribution at the wall H BY is also obtained Third2. calculate flux of conservative variables related to go 4 calculate flux of conservative variables related to g and fo l calculate flux of distribution functions Figure 1 2 interface flux calculation Reconstruct initial distribution L i 1 2 if up lt 0 see Eq L25 there is no need to store the left and right values separately We define the variable Take h as example Since we take value from h only if up gt 0 and take value from fig 2 k only h hik i41 2 Li Oe KE uk 29 hi 1 2 k h hi 1 k Ti _ T541 2 C 1 ks uk lt 0 and similarly g one Uk 2 0 ae Cae uk lt 0 In the program they are written as h h h O541 2 k O44 A ug O741 1 A ue and hiti 2 b hie i412 Bio s A ug hitik ein i4 1 2 0 41 4 1 H uz Calculate Wo Wo is calculated from Eq 1 16 with hr hit1 2 k5 bk bi41 2 k Then the primary variables is obtained from the relation the expression for below only holds for equilibrium state poo K 1 po0 po 4 poEo 3 UG Vo po po Uo The heat flux is calculated by Eq LIS with hk hi41 2 4 bk bi41 2 4 U Uo Calculate a a The macroscopic slope is approximated by ow ae Wo Wi ow one Wisi Wo Ox Ox X Ti41 2 Vi Ti41 Vj41 2 and the three components of a a are calculated from 2 a ea K 1 Ox 2o jJ Ox Ox 2 iy Ly pl a 1 3
3. gbamu Be He T 2Xo and then evaluate H A B corresponding to gt by Eq 1 A 1 Pr d 2 Hee PDA oy lt ta Uo EA 0 4 1 Pr B Ou 20 70 wens ek See 0 The flux of conservative variables related to g is X aruk Ht F Mt Daku Hi Y aku HE akur Be The flux of conservative variables related to fo is D QkUkh i 1 2 k 2 OnE 1 2 k Fp Mt Y akUuhiyi 2k Mts D akuko R 2k 4 akuhiqij2 k d akUkbiti 2 k 62 ORURO a 79 4 5 axe ag The flux of conservative variables is n l t Puse ff faspubdedt Fos Fy E p s Calculate the flux of distribution functions The flux of reduced distribution function h is calculated by yeti n l t J fpe J Firxudgat tr tr Mtiuk Hp Hf 1 Mtouz om abu Hy 503 Un Ba H uz 1 Mtouz afm aku Hy 503 Ue Ba 1 A ux 1 Mtzuz am Asup Hk 343 UkHk Ba 2h ae Mtaughi4i 2 k Mt5UkTi41 2 k The flux of reduced distribution function b is calculated by n l r f f a2 dt f J fi 1 2 kukdEdt i n Mtiuk Bk Br 1 Mtu Gz a u Bp 508 Uk Brt lt t gt m H uz 1 Mtzu2 of a u Bp 503 Uk Bat lt gt m 1 A ug 1 Mtzup aB Agu Bk z43 ukBr lt Et gt i Mtgurbj41 2 Mtsugoy tyak 1 6 Update cell averaged value The procedure is shown in Figure store W and calculate H B T calculate W t by Eq LI2 and H B r
4. _ z T lt U 1 x20 a a and A have the same form 6 l 2 2 a a azu asz u amp where a1 2 a3 are local constants Inserting Eq L23 and Eq L24linto Eq 22 one obtains f 0 t uz 1 e 7 go g r 1 e 7 te 7 a H ug a 1 H ux ungo r t r 1 e7 7 Ago 1 25 e UT F ajzr Untoie H lur Fajar uatoitie 1 H ux Gi 1 2 0 fi 1 2 k gt where 9 1 2 z is the first three terms related to equilibrium distribution fai 2 k is the last term related to the initial non equilibrium distribution Here go or Wo in Eq can be obtained by applying the compatibility condition at x 0 t 0 ee Oe which gives Wo f sovae f fo 0 0 un gd 1 26 Then a a A are obtained from the slope of conservative variables aw 2 aw d agod 1 2 fewe FP fiwe 1 27 oW B 1 2 T f Agovd 1 28 5 The time derivative of W can be calculated via the compatibility condition rt Awa iad 0 dt i x 0 t 0 which gives oF a Hu a 1 H u ugoydz ee 1 5 2 The numerical procedure The flow chart of the numerical procedure is shown in Figure From reconstruction obtain hj41 2 n bi 1 2 k from Eq 1 26 obtain Wo from Eq obtain a a t from Eq L29 obtain OW dt from Eq 1 28 obtain A calculate collision time and some time integration terms
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6. o o o o p Experiment Normalized density and temperature 0 itisiliriilririlirriliriiliriiiliriiliiii 6 8 em zj lapi Figure 2 1 Argon shock structure at Ma 8 0 2 3 UGKS2D Code 2 3 1 Differences with 1D For 2D problem many expressions need to be slightly changed For example K 2 1 0 2 AUU V T where v is particle velocity in y direction V is macroscopic velocity in y direction The relation between K and y becomes K 4 IEK The reduced Maxwellian distribution becomes B is not changed sc A u U 2 4 v V 2 13 The collision invariants are Y 1 u v 1 2 u v And the expressions for macroscopic variables are correspondingly changed For example the nondimensionalized pressure is calculated via K 2 2 p fe U w V jhdu bdu When calculating the flux the slopes related to Maxwellian become 1 a a azu azv asz lu u g and the components are calculated via 2 E K 2 A p u ne 5 2 202 avg OE 4 KF 2 r 0 2o j Ox ue On J ay 59 tas Voas 5 UR v EEP a 2 3 2 Lid driven cavity problem The test case included in UGKS2D code is Lid driven cavity problem 3 The argon gas is enclosed by four walls to form a rectangular shape The upper wall is moving in tangential direction with velocity Uw other walls are stationary All walls are kept at a constant temperature Tw and full a
7. 19 1 4 Time step and reconstruction The time step is determined by the CFL condition Ax At CFL 1 20 U c 120 where CFL is the CFL number c is the speed of sound The macroscopic velocity U c can also be replaced by max u In the program the van Leer limiter is used for the reconstruction For example the slope of h at the i th cell and k th particle velocity is s1 s2 mt 1 21 PEI nee of sign s1 sign s2 where s hik hi 1 k i ii S2 hiti k hi k i 1 Xi The slope of b is calculated in the same way 1 5 The algorithm Take the interface 2 41 2 0 at t 0 as example 4 1 5 1 The calculation of interface fluxes Here the original distribution function is used for illustration From Eq LIO the integral solution at the cell interface is 1 f i f O t ur l PF alt up Ee Ode e 7 fo urt 0 ur 1 22 0 The initial distribution function around the interface fo is L fit1 2 k T Oik x lt 0 1 23 TE sop Oi 1 k T gt 0 fo z 0 uk f where la 2 k fa J2 k are the reconstructed initial distribution functions at the left and right side of the interface The Maxwellian distribution around the interface in f is approximated by Taylor expansion g x t u goll 1 H 2 a a H ala a At 1 24 where go is the Maxwellian distribution at x 0 t 0 and Hfz is the Heaviside function 0 0 Hja
8. calculated through the same method described in section In the UGKS1D code the reference state is L lmfp 1 Tos T Po P1 where Imfp 1 is the upstream mean free path There are several important options in the code e method_output the method to output the solution If set to ORIGINAL the original nondimensionalized values will be written to the result file If set to NORMALIZE the density will be normalized by p p p1 p2 p and the temperature is also normalized by the same way e mu_ref the reference viscosity coefficient Hoo which can be specified directly or calculated through the provided function get_mu This function calculate the viscosity coefficient through Sat at 2 J 7 oo K CO Hoo TaS 2w 7 2w 12 where a w are coefficients related to molecule model The default settings in the code is for Argon shock structure calculation at Ma 8 0 see 2 e Argon gas with Prandlt number Pr 2 0 3 0 and variable hard sphere model w 0 72 e Mach number 8 0 e Newton Cotes integration with 100 velocity points ranges from 15 to 15 e Computation domain is 50 times of the upstream mean free path e Cell size is half of the upstream mean free path e Output solution at t 250 Figure P I shows the comparison of the simulated density with the experimental measurements 9 and also temperature profile Ma 8 w 0 72 Argon shock structure 0 8 0 6 0 4 0 2 T UGKS p UGKS
9. designed for human modification Examples of transparent image formats include PNG XCF and JPG Opaque formats include proprietary formats that can be read and edited only by proprietary word processors SGML or XML for which the DTD and or processing tools are not generally available and the machine generated HTML PostScript or PDF produced by some word processors for output purposes only The Title Page means for a printed book the title page itself plus such following pages as are needed to hold legibly the material this License requires to appear in the title page For works in formats which do not have any title page as such Title Page means the text near the most prominent appearance of the work s title preceding the beginning of the body of the text The publisher means any person or entity that distributes copies of the Document to the public A section Entitled XYZ means a named subunit of the Document whose title either is precisely XYZ or contains XYZ in parentheses following text that translates XYZ in another language Here XYZ stands for a specific section name mentioned below such as Acknowledgements Dedications Endorsements or History To Preserve the Title of such a section when you modify the Document means that it remains a section Entitled XYZ according to this definition The Document may include Warranty Disclaimers next to the notice which states that t
10. of lt u gt lt o is from oo to 0 1 lt u gt lt 0 zerfe VAU LeU 2 Var n l ett gt lt 0 U lt yt gt lt 0 t lt u gt lt o lt ul gt zo U lt ul gt lt 0 23 Moments of lt u E w gt There are three components for 1D problem lt u gt lt E gt lt u Ey gt lt uti gt lt gt 4 lt ut gt lt gt lt u gt lt t gt Moments of lt au y gt There are three components for 1D problem 1 lt au gt a lt u W gt a lt u tty gt 548 lt unt gt lt ure gt 24 Bibliography N 1 Kun Xu and Juan Chen Huang A unified gas kinetic scheme for continuum and rarefied flows Journal of Computational Physics 229 20 7747 7764 2010 Kun Xu and Juan Chen Huang An improved unified gas kinetic scheme and the study of shock structures IMA Journal of Applied Mathematics 76 5 698 711 2011 Juan Chen Huang Kun Xu and Pubing Yu A Unified Gas kinetic Scheme for Continuum and Rarefied Flows II Multi dimensional Cases Communications in Computational Physics 12 3 662 690 2012 Kun Xu Meiliang Mao and Lei Tang A multidimensional gas kinetic BGK scheme for hypersonic viscous flow Journal of Computational Physics 203 2 405 421 2005 Kevin H Prendergast and Kun Xu Numerical Hydrodynamics from Gas Kinetic Theory Journal of Computational Physics 109 1 53 66 1993 Kun Xu A Gas Kinetic
11. 0 Po Ox Ox _ 1 0p lf K 1 a 9 na 5 UR 2o as Calculate OW 0t and A From Eq L29 the time derivative of W is calculated from Ow Fp Po lt av ub gt gt o lt auy gt lt 0 where lt gt is the moments of Maxwellian distribution function The detail definition and calculation can be found in 6 and also Appendix A Moments of Maxwellian distribution function A is calculated in the same way as a a using Eq Calculate collision time and some time integration terms From Eq L 5Jand Eq LI9 the collision time is 2 Po T Hoo 7 Some time integrals used in the evaluation of flux are listed below gett Mt e Tdt r 1 eo in nti Mt f te t T dt rAte T rMta sae Mt I 1 e dt At Mta ar Mt 1 r 1 e 7 te 7 dt r Mt Mts t n 4 1 1 Mts r t r 1 e dt she Mt tr Calculate the flux of conservative variables related to go n l Theoretically f H J Ji 2updEdt can be calculated analytically But the integration related to gt is too complex and will be calculated with numerical integration Only the terms related to go will be integrated analytically here F Mtipo lt uw gt Mt2po lt atu gt o lt a uty gt lt 0 Mtzpo lt Aut gt Calculate the flux of conservative variables related to g and fo First evaluate Hg By corresponding to go by Eq Sa K Hk po
12. BGK Scheme for the NavierStokes Equations and Its Connection with Artificial Dissipation and Godunov Method Journal of Computational Physics 171 1 289 335 2001 J Y Yang and Juan chen Huang Rarefied Flow Computations Using Nonlinear Model Boltzmann Equations Journal of Computational Physics 120 2 323 339 1995 C K Chu Kinetic Theoretic Description of the Formation of a Shock Wave Physics of Fluids 8 1 12 1965 H Alsmeyer Density profiles in argon and nitrogen shock waves measured by the absorption of an electron beam Journal of Fluid Mechanics 74 3 497 513 1976 Benzi John Xiao Jun Gu and David R Emerson Effects of incomplete surface accommodation on non equilibrium heat transfer in cavity flow A parallel DSMC study Computers amp Fluids 45 1 197 201 2011 B Shizgal A Gaussian quadrature procedure for use in the solution of the Boltzmann equation and related problems Journal of Computational Physics 41 2 309 328 1981 25
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14. User s Manual for UGKS1D and UGKS2D Codes Ruijie Wang and Kun Xu August 10 2012 Copyright 2012 Ruijie Wang Kun Xu Permission is granted to copy distribute and or modify this document under the terms of the GNU Free Documentation License Version 1 3 or any later version published by the Free Software Foundation with no Invariant Sections no Front Cover Texts and no Back Cover Texts A copy of the license is included in the section entitled GNU Free Documentation License Contents 1 5 1 The calculation of interface fluxes 2 UGKS Code Differences with 1D 2 3 2 _Lid driven cavity proble 2 3 3 Other informatio 10 FUTURE REVISIONS OF THIS LICENSH 7 AGGREGATION WITH INDEPENDENT WORK ADDENDUM How to use this License for your documents Appendix A Moments of Maxwellian distribution functio Bibliography 11 11 11 12 12 12 13 13 14 15 17 17 18 18 19 20 20 21 21 21 21 22 22 23 25 il Chapter 1 Unified Gas Kinetic Scheme This chapter describes the Unified Gas Kinetic Scheme presented in I 2 This is a 1D formulation The 2D formulation with directional splitting is presented in 3 which can be extended to develop a truly multidimensional formulation see 4 1 1 Model equation The model equation is the BGK Shakhov model In one dimensional case Tu 2 1 1 Ot Ox rT at where f is the dis
15. ccommodation is assumed The gas is initially at rest with the same temperature as the wall Figure P 2 shows the schematic of the problem Uw Tw _ ___ gt Tw Argon Tw Tw Figure 2 2 Schematic of Lid driven cavity problem In the code the default test case is for Tw 273K Uw 50m s Kn lmfp L 0 075 where lm fp is mean free path and L is the domain length Choose the initial condition as reference state the settings are e Second order interpolation e VHS model for collision time and HS model for reference state 14 e Prandtl number Pr 2 0 3 0 Knudsen number Kna 0 075 at reference state e Tw 1 Uw 0 15 e L 1 with 45x45 grids e Gaussian quadrature with 28x28 velocity points Figure 2 3 shows the result for the above setup and compared with the DSMC solution I0 Kn 0 075 Temperature Kn 0 075 Heat flux 275 00 274 75 274 50 274 25 274 00 0 8 273 75 273 50 273 25 273 00 272 75 272 50 0 6 Y 0 4 0 2 a TE o ee 0 2 0 4 0 6 0 8 0 2 0 4 0 6 0 8 x x a Temperature contour b Heat flux z Kn 0 075 Vertical center line Kn 0 075 Horizontal center line UGKS H 0 15 2020s oo00 DSMC 0 8 i i M Q17 L r os 0 05 Y p V U 0 0 4 L 0 05 M 0 1 0 2 i UGKS ais 0000 DSMC fi il i pot f i fi fi fi fi fi 2 0 02 04 06 08 029 95 oa 06 OB U Uw X c U velocity along vertical center line d V velocity along horizontal center line Figur
16. e 2 3 Cavity flow at Kn 0 075 In the temperature contour the black lines are from DSMC the white lines and background contour are from UGKS 2 3 3 Other information The Gaussian quadrature used in the code is from Table Ia of Ii which is better than Gaussian Hermite quadrature in high Knudsen number case But for the cavity problem with Kn gt 1 Newton Cotes formula of 61x61 velocity grids with velocity range u v 4 4 can avoid oscillating in the solution which happens using Gaussian quadrature and second order interpolation For example the setting for Newton Cotes integration subroutine init vartiable declarations 15 real kind RKD umin vmin declare smallest discrete velocity umin 4 0 vmin 4 0 largest discrete velocity Global variables umax 4 0 vmax 4 0 Inumber of velocity points Global variables unum 61 vnum 61 call init_velocity_newton unum umin umax vnum vmin vmax set the velocity space lother commands end subroutine init On Intel Core Quad Processor Q9450 12M Cache 2 66 GHz 1333 MHz FSB with openmp enabled the computation time of the above setup is about 16 minutes with openmp With 65x65 physical space the computation time is about 35 minutes 16 GNU Free Documentation License Version 1 3 3 November 2008 Copyright 2000 2001 2002 2007 2008 Free Software Foundation Inc lt http fsf org gt Everyone is permitted to copy and distribute
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23. lculated via p W a yf 1 7 p 5 iu 0 e 1 8 a fU Dlu U f 1 9 where pF is total energy An integral solution of the BGK Shakhov model can be constructed by the method of characteristics 5 f a t u ah fra tu ETETA ee OT foe u t t t u 1 10 where 2 x u t t is the particle trajectory and ff is the initial gas distribution function at t 1 2 Solution algorithm For the numerical computation in addition to the discretization of physical space and time the velocity space is also discretized That is the distribution function is for some discrete particle velocities instead of continuous velocity space from oo to oo as in 6 Then the moments of the non equilibrium distribution function are calculated through numerical integration the moments of equilibrium distribution are still calculated using analytical integration The discretization of the velocity space is determined by the choice of numerical integration method In the finite volume approach if trapezoidal rule is used for the approximation of collision term Eq becomes mp Lf oe de Se fir atal f TE fiyaa ae 2 pee P 1 11 where fip and F are cell averaged distribution function of the i th cell and k th discrete particle velocity uz at time t and t respectively Ax is the cell length and At is the time step fj 1 2 and ond po fi 1 2 are the fluxes of the distribution fu
24. ll be put into the bin directory and the compiled manual pdf will in the doc directory 11 2 1 2 Running Just type the program name to run it under the bin directory no input file or data is required The code will generate two files e hst record the convergence history every 10 iterations e dat record the final result of the program Both files are in tecplot format and will be put into current working directory 2 1 3 Other information The comments in the program are written in doxygen format and can be used to generate documentation of the code But the doxygen configuration file is not included and the documentation generation is not tested This program uses for version control and the source repository is published on GitHub To obtain the code via git use the following command git clone git github com lainme UGKS git 2 2 UGKS1D Code Once the flow condition before the normal shock is known upstream the flow variables after the shock downstream can be calculated by normal shock relation M2 y 1 2 Mo ily l 2yMz y 1 pR __ Q 1 M p yA MZ 2 T 1 2 M M 1 T MGA where M p1 T are upstream Mach number density and temperature Mo p2 Tz are downstream Mach number density and temperature For shock structure calculation the upstream and downstream conditions are specified at the two bound aries via ghost cell And the flux across the interfaces are
25. ly the distribution function at the boundary interface is expressed by same holds for bx hy HY Hux hi 1 Huy Finally the flux across the wall is calculated by Yo apurhs Fiy2 At 5 aku hg X ang ugha unde and 2 1 fio pdt Atughe tr grti F jo pdt Atuxbr tn 10 Chapter 2 UGKS Codes 2 1 Usage 2 1 1 Compiling A makefile is provided to compile the program under Linux If you are using any IDE e g Visual Studio use the compiling function provided by the IDE The makefile can be used to compile the two codes UGKS1D and UGKS2D and also this manual There are several requirements for the makefile to work e Fortran Compiler either ifort or gfortran supporting Fortran 2003 e Bash shell e Latex only for compilation of the manual It requires hyperref parskip amsmath amssymb fullpage appendix graphicx subfigure xcolor listings or minted packages and also bibtex dvips and ps2pdf By default the make command will compile both UGKS1D and UGKS2D with openmp and ifort This behavior can be changed by specifying the target and passing parameters 1 Compile both UGKS1D and UGKS2D with openmp and ifort make 2 Only compile UGKS1D make 1D 3 Only compile UGKS2D make 2D 4 Compile both UGKS1D and UGKS2D but WITHOUT openmp and WITH gfortran make OMP no FC gfortran 5 Compile the manual make manual 6 Clean the compilation make clean The executables wi
26. mes two similar equations for the update of h and b respectively The overview flow chart of the solution algorithm in one iteration is shown in Figure calculate time step Y reconstruction calculate slop of h b in each physical cell h calculate interface flux of h b W calculate W t1 calculate Att pt and 77 calculate h 1 prt Figure 1 1 solution algorithm in one iteration 1 3 Nondimensionalization In the program the following nondimensionalization is used i t Up T n P f T _ p ta Ux Coo x La P pos Tes P m PooC2 q a h A b A E H PooC3 Pest Co Poo on Pa relics The free stream variables are related through L Coo V2RT oo too a gt Aco 1 02 In the following all variables are nondimensionalized but we will drop the for simplicity After nondi mensionalization and using the reduced distribution function the expressions for macroscopic variables become p fran X arhe pU tau So arhgur 1 16 pE fran baw T akhku S arbe 2 2 x K 1 z 25 fe U hdu fian X an ux U hy So andr 1 17 a 5 fe U u U 2hdu fe Ubn D apluk U uk U hp 4 N ar ur U bx where a x is the weight of the numerical integration at the k th particle velocity The summation is over all the discrete particle velocity 1 18 The equation of state is 1 1 aT NE 1 19 p 5pT T 1
27. nction across the cell interface Tik are modified n 1 equilibrium distribution 7 and T are particle collision time Multiplying the collision invariants to Eq and make integration over the velocity space the evolution of conservative flow variables becomes n n 1 W Wet Ag Fi 1 2 F 41 2 1 12 o where F f praza In order to update the distribution function in Eq LII there are three unknowns to be obtained the interface gas distribution function f the modified equilibrium distribution f and collision time 7 at the next time level The flux f is calculated using the integral solution Eq at the cell interface Since f t and r 1 have one to one correspondence to the macroscopic variables they are obtained by using the updated conservative variables in Eq L 12 In order to remove the dependence of distribution functions on the internal degree of freedom the reduced distribution function 7 8 is used in real computation which is defined as n f fdt b f eja 1 13 and the reduced modified equilibrium distributions are ht H HAt b B B where the corresponding reduced Maxwellian distribution g becomes oo y 2 es K ode ew Ba Ogig FH 1 14 and the corresponding terms related to gt becomes H T gt d et U q 2 u U K 5 H oe 1 15 BY T E gtd k 5p u U q 2 u U K 3 B Then the update of f using Eq L11 beco
28. ns Front Cover Texts and Back Cover Texts replace the with Texts line with this with the Invariant Sections being LIST THEIR TITLES with the Front Cover Texts being LIST and with the Back Cover Texts being LIST If you have Invariant Sections without Cover Texts or some other combination of the three merge those two alternatives to suit the situation If your document contains nontrivial examples of program code we recommend releasing these examples in parallel under your choice of free software license such as the GNU General Public License to permit their use in free software 22 Appendix A Moments of Maxwellian distribution function In the program the moments of Maxwellian distribution function is frequently used and they are usually obtained from subroutines The moments of Maxwellian distribution function is defined as p lt gt fia and have the property that lt uE gt lt u gt lt E gt where m n are integers Moments of amp _ K 4 3K K K 1 lt gt 5 lt E gt Fr Moments of u The integration limits of lt u gt is from oo to ov lt u gt l lt ul gt U n 1 lt u gt U lt u gt 42h cums The integration limits of lt u gt o is from 0 to oo 1 lt u gt gt 0 zerfe VAU 1 e7 U 1 0 lt ul gt s9 U lt u gt s9 gt gt t37 n 2 n 1 n 1 n lt u gt so U lt u zato St gt so The integration limits
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33. tice grants a world wide royalty free license unlimited in duration to use that work under the conditions stated herein The Document below refers to any such manual or work Any member of the public is a licensee and is addressed as you You accept the license if you copy modify or distribute the work in a way requiring permission under copyright law A Modified Version of the Document means any work containing the Document or a portion of it either copied verbatim or with modifications and or translated into another language A Secondary Section is a named appendix or a front matter section of the Document that deals exclusively with the relationship of the publishers or authors of the Document to the Document s overall subject or to related matters and contains nothing that could fall directly within that overall subject Thus if the Document is in part a textbook of mathematics a Secondary Section may not explain any mathematics The relationship could be a matter of historical connection with the subject or with related matters or of legal commercial philosophical ethical or political position regarding them The Invariant Sections are certain Secondary Sections whose titles are designated as being those of Invariant Sections in the notice that says that the Document is released under this License If a section 17 does not fit the above definition of Secondary then it is not allowed to be desi
34. tribution function u is particle velocity T p is particle collision time p is the dynamic viscosity p is the pressure and ft is the modified equilibrium distribution function The modified equilibrium distribution is given by fr g 1 a Pre a 5 6pRT 9 9 1 2 where g is the Maxwellian distribution Pr is the Prandtl number c is the random velocity q is heat flux R is gas constant and T is the temperature The Maxwellian distribution for 1D problem is K 1 g p gt ey 1 3 T where p is density A m 2kT m is molecule mass k is Boltzmann constant U is the macroscopic velocity K is the number of internal degree of freedom and amp For example a monatomic gas at 1D problem has K 2 to account for the motion in y z direction and v w where v w are particle velocity in y z direction The relation between K and the ratio of specific heat is K Kta 1 4 K 1 The dynamic viscosity can be calculated from Sutherland s law or hard sphere HS variable hard sphere model VHS T Ww H Href z 1 5 1 where ref is the reference viscosity and Tef is the reference temperature w is the index related to HS or VHS model The collision term meets the requirement of conservative constraint or compatibility condition f nua o 1 6 where w 1 u 1 2 u 7 is the collision invariants and d dude The macroscopic variables can be ca
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