Solution of the Navier-Stokes Equations

Transcription

1 Numercal Flud Mechancs Fall 2011 Lecture 25 REVIEW Lecture 24: Soluton of the Naver-Stokes Equatons Dscretzaton of the convectve and vscous terms Dscretzaton of the pressure term Conservaton prncples Momentum and Mass Energy t 2 2 v.( t ). v 0 2 v v p v g 2 2 u j p p g. r. u ( p e gxe e ) 3 3 x v v dv ( v. n) da p v. n da (. v). n da : v p. v g. v dv 2 2 CV CS CS CS CV S p e nds. j Choce of Varable Arrangement on the Grd Collocated and Staggered Calculaton of the Pressure v t 2 2. p p...( v v). v. g..( v v) p uu j x x x x j Numercal Flud Mechancs PFJL Lecture 25, 1

2 REVIEW Lecture 24, Cont d: Soluton of the Naver-Stokes Equatons Pressure Correcton Methods: ) Solve momentum for a known pressure leadng to new velocty, then ) Solve Posson to obtan a corrected pressure and ) Correct velocty, go to ) for next tme-step. A Smple Explct Scheme (Posson for P at t n, then mom. for velocty at t n+1 ) A Smple Implct Scheme Nonlnear solvers, Lnearzed solvers and ADI solvers Implct Pressure Correcton Schemes for steady problems: terate usng Outer teratons: Inner teratons: Numercal Flud Mechancs Fall 2011 Lecture ( ) n n n n n uu j j p u u t + x j x j x n1 1 1 ( ) n n p uu j j x x x x j x j m1 m m m u δp u δp δ u δ δp δ u * * m m* m 1 * but requre m m m and 0 m m m m m m* u u u A u b A u b A u b δx δx δx δx δx δx m m* u m m δp A u b m* u δx Projecton Methods: Non-Incremental and Incremental Schemes Numercal Flud Mechancs PFJL Lecture 25, 2

3 TODAY (Lecture 25): Naver-Stokes Equatons and Intro to Fnte Elements Soluton of the Naver-Stokes Equatons Pressure Correcton Methods Implct Pressure Correcton Schemes for steady problems: terate usng Outer teratons and Inner teratons Projecton Methods: Non-Incremental and Incremental Schemes Fractonal Step Methods: Example usng Crank-Ncholson Streamfuncton-Vortcty Methods: scheme and boundary condtons Artfcal Compressblty Methods: scheme defntons and example Boundary Condtons: Wall/Symmetry and Open boundary condtons Fnte Element Methods Introducton Method of Weghted Resduals: Galerkn, Subdoman and Collocaton General Approach to Fnte Elements: Steps n settng-up and solvng the dscrete FE system Galerkn Examples n 1D and 2D Numercal Flud Mechancs PFJL Lecture 25, 3

4 References and Readng Assgnments Chapter 7 on Incompressble Naver-Stokes equatons of J. H. Ferzger and M. Perc, Computatonal Methods for Flud Dynamcs. Sprnger, NY, 3 rd edton, 2002 Chapter 11 on Incompressble Naver-Stokes Equatons of T. Cebec, J. P. Shao, F. Kafyeke and E. Laurendeau, Computatonal Flud Dynamcs for Engneers. Sprnger, Chapter 17 on Incompressble Vscous Flows of Fletcher, Computatonal Technques for Flud Dynamcs. Sprnger, Chapters 31 on Fnte Elements of Chapra and Canale, Numercal Methods for Engneers, Numercal Flud Mechancs PFJL Lecture 25, 4

5 Methods for solvng (steady) NS problems: Implct Pressure-Correcton Methods Smple mplct approach based on lnearzaton most useful for unsteady problems It s not accurate for large (tme) steps (because the lnearzaton would then lead to a large error) Should not be used for steady problems Steady problems are often solved wth an mplct method (wth pseudo-tme), but wth large tme steps (no need to reproduce the pseudo-tme hstory) The am s to rapdly converge to the steady soluton Many steady-state solvers are based on varatons of the mplct schemes They use a pressure (or pressure-correcton) equaton to enforce contnuty at each pseudo-tme steps, also called outer teraton Numercal Flud Mechancs PFJL Lecture 25, 5

6 Methods for solvng (steady) NS problems: Implct Pressure-Correcton Methods, Cont d For a fully mplct scheme, the steady state momentum equatons are: Wth the dscretzed matrx notaton, the result s a nonlnear algebrac system n1 n1 n1 n 1 n ( uu j) j p x j x j x u u The b term n the RHS contans all terms that are explct (n u n ) or lnear n u n+1 or that are coeffcents functon of other varables at t n+1, e.g. temperature Pressure s wrtten n symbolc matrx dfference form to ndcate that any spatal dervatves can be used The algebrac system s nonlnear. Agan, nonlnear teratve solvers can be used. For steady flows, the tolerance of the convergence of these nonlnear-solver teratons does not need to be as strct as for a true tme-marchng scheme Note the two types of successve teratons: n1 u A u b n1 n1 u Outer teratons: (over one pseudo-tme step) use nonlnear solvers whch update the n 1 elements of the matrx u n1 A as well as u (uses no or approxmate pressure term) Inner Iteratons: lnear algebra to solve the lnearzed system wth fxed coeffcents n1 Numercal Flud Mechancs δp δx PFJL Lecture 25, 6

7 Methods for solvng (steady) NS problems: Implct Pressure-Correcton Methods, Cont d Outer teraton m (pseudo-tme): nonlnear solvers whch update the elements m* m* of the matrx as well as : m1 m* u The resultng veloctes do not satsfy contnuty (hence the *) snce the RHS s obtaned from p m-1 at the end of the prevous outer teraton. m* m Hence, needs to be corrected. The fnal needs to satsfy u A u Inner teraton: After solvng a Posson equaton for the pressure, the fnal velocty s calculated usng the nner teraton (fxed coeffcent A) Fnally, ncrease m to m+1 and terate (outer, then nner) u Ths scheme s a varaton of the prevous tme-marchng schemes: Man dfference s that terms n RHS can be explct or mplct n outer teraton m* u A u b m* m1 u u Numercal Flud Mechancs m* δp δx * * m m m m m m u δp δ u δ δp δ u δ u A u b and 0 A u b A u b δx δx δx δx δx δx m m m m m m m m m u u u m* u A u b m m u m* δp δx m PFJL Lecture 25, 7

8 These schemes that frst construct a velocty feld that does not satsfy contnuty, but then correct t usng a pressure gradent are called projecton methods : Methods for solvng (steady) NS problems: Projecton Methods The dvergence producng part of the velocty s projected out One of the most common methods of ths type are agan pressure-correcton schemes m m* m m1 Substtute u u u' and ' n the prevous equatons p p p Varatons of these pressure-correcton methods nclude: SIMPLE (Sem-Implct Method for Pressure-Lnked Equatons) method: Neglects contrbutons of u n the pressure equaton SIMPLEC: approxmate u n the pressure equaton as a functon of p SIMPLER and PISO methods: terate to obtan u There are many other varatons of these methods: all are based on outer and nner teratons untl convergence at m (n+1) s acheved. Numercal Flud Mechancs PFJL Lecture 25, 8

9 Projecton Methods: Example Scheme 1 Guermond et al, CM-AME-2006 Non-Incremental (Chorn, 1968): No pressure term used n predctor momentum equaton Correct pressure based on contnuty Update velocty usng corrected pressure n momentum equaton * n n j j * u u t u n1 n1 1 n1 n1 * n1 * ( uu) + ; 0 xj x j D n1 n1 p u u t n1 n1 x p 1 * n1 p u ; 0 n1 u x x t x n D 0 x n1 p u u t x n1 Numercal Flud Mechancs Note: advecton term can be treated: - mplctly for u* at n+1 (need to terate then), or, - explctly (evaluated wth u at n), as n 2d FV code and many others PFJL Lecture 25, 9

10 Projecton Methods: Example Scheme 2 Guermond et al, CM-AME-2006 Incremental (Goda, 1979): Old pressure term used n predctor momentum equaton n 1 n Correct pressure based on contnuty: p p p' Update velocty usng pressure ncrement n momentum equaton n1 n1 n 1 n1 * n n j j * u u t u n1 * n1 * ( uu) p + ; 0 x j x j x D n1 n1 n n1 p p u u t n1 n n1 n x p p 1 * n1 p p u 1 ; 0 n u x x t x n D 0 x n1 u u t p x p n Notes: - ths scheme assumes u =0 n the pressure equaton. It s as the SIMPLE method, but wthout the teratons - As for the non-ncremental, the advecton term can be explct or mplct Numercal Flud Mechancs PFJL Lecture 25, 10

11 Other Methods: Fractonal Step Methods In the prevous methods, pressure s used to: Enforce contnuty: t s more a mathematcal varable than a physcal one Fll the RHS of the momentum eqns. explctly (predctor step for velocty) The fractonal step methods (Km and Mon, 1985) generalze ADI But works on term-by-term (nstead of dmenson-by-dmenson). Hence, does not necessarly use pressure n the predctor step Let s wrte the NS equatons a n symbolc form: u u ( C D P) t n 1 n where C, D and P represent the convectve, dffusve and pressure terms The equaton s readly splt nto a three-steps method: u u C t * n u u D t ** * u u P t n1 ** In the 3 rd step, the pressure gradent must satsfy the contnuty equaton Numercal Flud Mechancs PFJL Lecture 25, 11

12 Fractonal Step Methods, Cont d Many varatons of Fractonal step methods exsts Pressure can be a pseudo-pressure (depends on the specfc steps, what s n u **, P ) Terms can be splt further (one coordnate at a tme, etc) For the tme-marchng, Runge-Kutta explct, drect 2 nd order mplct or Crank-Ncholson scheme are used Lnearzaton and ADI s also used Used by Cho and Mon (1994) wth central dfference n space for drect smulatons of turbulence (Drect Naver Stokes, DNS) Here, we wll descrbe a scheme smlar to that of Cho and Mon, but usng Crank-Ncolson Numercal Flud Mechancs PFJL Lecture 25, 12

13 Fractonal Step Methods: Example based on Crank-Ncholson In the frst step, velocty s advanced usng: Pressure from the prevous tme-step Convectve, vscous and body forces are represented as an average of old and new values (Crank-Ncolson) Nonlnear equatons terate, e.g. Newton s scheme used by Cho et al (1994) Second-step: Half the pressure gradent term s removed from u *, to lead u ** u u t Fnal step: use half of the gradent of the stll unknown new pressure New velocty must satsfy the contnuty equaton (s dvergence free): Takng the dvergence of fnal step: ** * 1 n n * * n H( u ) H( u) p u u t 2 x p 2 x Once p s solved for, the fnal step above gves the new veloctes n n1 1 ** 1 p 2 x u u t p x x t x n1 ** 2 ( u ) n Numercal Flud Mechancs PFJL Lecture 25, 13

14 Fractonal Step Methods: Example based on Crank-Ncholson Puttng all steps together: n * n n1 1 n H( u ) H( u) 1 p p 2 2 x x n u u t To represent Crank-Ncolson correctly, H(u * ) should be H(u n+1 ) However, we can show that the error s 2 nd order n tme and thus consstent wth C-N s truncaton error: subtract the frst step from the fnal step, to obtan, n1 n 2 n 1 * t p p t p u u 2 x x 2 x t Wth ths, one also obtans: n1 n ' n1 * t ( p p ) t ( p ) u u 2 x 2 x whch s smlar to the fnal step, but has the form of a pressure-correcton on u * Fractonal steps methods have become rather popular Many varatons, but all are based on the same prncples (llustrated by C-N here) Man dfference wth SIMPLE-type tme-marchng schemes: SIMPLE schemes solve the pressure and momentum equatons several tmes per tme-step n outer teratons Numercal Flud Mechancs PFJL Lecture 25, 14

15 Incompressble Flud Vortcty Equaton (revew Lecture 6) Vortcty Naver-Stokes Equaton curl of Naver-Stokes Equaton Numercal Flud Mechancs PFJL Lecture 25, 15

16 Streamfuncton-Vortcty Methods For ncompressble, 2D flows wth constant flud propertes, NS can be smplfed by ntroducng the streamfuncton ψ and vortcty ω as dependent varables Streamlnes: constant ψ Vortcty vector s orthogonal to plane of the 2D flow 2D Contnuty s automatcally satsfed: In 2D, substtutng knematc condton: v u u, v and y x x y u v 0 x y v u u and v n y x x y x y ( ω v) leads to the The vertcal component of the vortcty equaton leads: 2 2 u v 2 2 t x y x y Numercal Flud Mechancs PFJL Lecture 25, 16

17 Streamfuncton-Vortcty Methods, Cont d Man advantages: Pressure does not appear n ether of these questons! NS has been replaced by a set of 2 coupled PDEs Instead of 2 veloctes and 1 pressure, we have only two dependent varables Explct soluton scheme Gven ntal velocty feld, compute vortcty by dfferentaton Use ths vortcty ω n n the RHS of the dynamc equaton for vortcty, to obtan ω n+1 Wth ω n+1 the streamfuncton ψ n+1 can be obtaned from the Posson equaton Wth ψ n+1, we can dfferentate to obtan the velocty Contnue to tme n+2, and so on One ssue wth ths scheme: boundary condtons Numercal Flud Mechancs PFJL Lecture 25, 17

18 Streamfuncton-Vortcty Methods, Cont d Boundary condtons Boundary condtons for ψ Sold boundares are streamlnes and requre: ψ = constant However, values of ψ at these boundares can be computed only f velocty feld s known Boundary condtons for ω Nether vortcty nor ts dervatves at the boundares are known n advance u wall y wall Vortcty at the wall s proportonal to the shear stress, but the shear stress s often what one s tryng to compute x y.e. one-sded dfferences at the wall: 2 n For example, at the wall: / wall wall snce Boundary values for ω can be obtaned from, but ths usually converges slowly and can requre refnement Dscontnutes also occur at corners Numercal Flud Mechancs PFJL Lecture 25, 18

19 Streamfuncton-Vortcty Methods, Cont d Dscontnutes also occur at corners for vortcty v u The dervatves and x y are not contnuous at A and B Ths e.g. means specal treatment for v u x y refne the grd at corners Vortcy-streamfuncton approach useful n 2D, but s now less popular because extenson to 3D dffcult In 3D, vortcty has 3 components, hence problem becomes as/more expensve as NS Streamfunctom s stll used n quas-2d problems, for example, n the ocean or n the atmosphere A C B D Image by MIT OpenCourseWare. Numercal Flud Mechancs PFJL Lecture 25, 19

20 Artfcal Compressblty Methods Compressble flow s of great mportance (e.g. aerodynamcs and turbne engne desgn) Many methods have been developed (e.g. MacCormack, Beam-Warmng, etc) Can they be used for ncompressble flows? Man dfference between ncompressble and compressble NS s the mathematcal character of the equatons Incompressble eqns: no tme dervatve n the contnuty eqn: They have a mxed parabolc-ellptc character n tme-space Compressble eqns: there s a tme-dervatve n the contnuty equaton: They have a hyperbolc character: Allow pressure/sound waves How to use methods for compressble flows n ncompressble flows? Numercal Flud Mechancs.( v) 0 t. v 0 PFJL Lecture 25, 20

21 Artfcal Compressblty Methods, Cont d Most straghtforward: Append a tme dervatve to the contnuty equaton Snce densty s constant, addng a tme-rate-of-change for ρ not possble Use pressure nstead (lnked to ρ va an eqn. of state n the general case): 1 p t u x where β s an artfcal compressblty parameter (dmenson of velocty 2 ) Its value s key to the performance of such methods: The larger/smaller β s, the more/less ncompressble the scheme s Large β makes the equaton stff (not well condtoned for tme-ntegraton) p t Methods most useful for solvng steady flow problem (at convergence: 0 ) To solve ths new problem, many methods can be used, especally All the tme-marchng schemes (R-K, mult-steps, etc) that we have seen Fnte dfferences or fnte volumes n space Alternatng drecton method s attractve: one spatal drecton at a tme 0 Numercal Flud Mechancs PFJL Lecture 25, 21

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