40 6 Vol. 40 No ADVANCES IN MECHANICS Nov. 25, 2010 CFD *, NS (CD) (MBS),, (PS). PS MBS,,,, PS. PS MBS R x( λr x), R x λ Reynolds

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1 40 6 Vol. 40 No ADVANCES IN MECHANICS Nov. 25, 200 CFD *, 0090 NS CD) MBS),, PS). PS MBS,,,, PS. PS MBS R x λr x), R x λ Reynolds CFL. PS MBS,, PS,, PS FD) FV), Reynolds. CFD. PS, Benchmark.,.,,,, CFD),,, 3 ) [ 9]. Navier StokesNS) CFD,.. mathematical basic schemes, MBS), upwind) [ 0] ; Lax Wendroff [] MacCormark [2] MBS, QUICK [2,6,3] Fromm [,4] 9 0 [5]. MBS CFD DNSdirect numerical simulation) [6,5,6] ; [2],. MBS,, MBS [7 5], / CIP cubic interpolated pseudo-particle) [52 90]. [9 00], MBS, PS).,,, PS. : , : ) gaozhi@imech.ac.cn

2 , Godunov [7] flux difference splitting, FDS) [8,9] flux vector splitting, FVS) [20 22] piecewise parabolic method, PPM) [23] FDS FVS advection upstream splitting method, AUSM) [24,25]. Jameson [26 28].,.,. Harten [29] high resolution), total variation diminishing, TVD) flux limiter) TVD,. TVD,, TVD, Harten [30] essentially non-oscillatory, ENO),,, ENO. Shu [3] total variation bounded, TVB) ; Liu [32,33] weighted essentially non-oscillatory, WENO),, ENO,,. MBS MBS ) flux limiter [34 37] ) flux corrected [38 4] ) self-adjusting hybrid [42,43] ) reconstruction-evolution) [44,45] TVD, ENO. Laney [], ,,,,,. [46], TVD non-oscillatory, non-freeparameter dissipation difference, NND), [47,48] NND. [49] ENO ENOUENO), [50,5]. compact), [52 67]. Lale [52] ; [53,54] ; [55 59] Pade ENO WENO, [60, 6] TVB, ; [62] ; [63] ; [64 67]. [68 72] Boltzmann gas kinetics),, N S DSMC) [73]. Chang [74], SE) CE), CE/SE [75]. Wang [76 78] FS) QUICK ENO. [79 8], cubic interpolated pseudo-particle, CIP) constrained interpolation profile, CIP),. Ni [82,83]. Morris [84] Batten [85], NS ). [86 9], ) Taylor, ),, PDS), PDS,,

3 6 : CFD 609 Benchmark., [92,93] [94,95], [52 67] PFD [86 9], NS MBS), ) [9 94] [96 98], NS CD) MBS PDS) PVS). Benchmark [92 20]., : CFD MBS) MBS : I PDS) PVS), PDS PVS. 2 CFD MBS) ) ) ;,, MBS) [ 6]. MBS). f i f i MBS) MBS f i R cri 2CS 4CS 6CS 8CS 2h f i f i ) 3 2 2h [8f i f i ) f i2 f i 2 )] h [45f i f i ) 9f i2 f i 2 ) f i3 f i 3 )] 7.33 h [0.8f i f i ) 0.2f i2 f i 2 ) f i3 f i 3 ) f i4 f i 4 )] 0CS 2 520h [2 00f i f i ) 600f i2 f i 2 ) 50f i3 f i 3 ) 25f i4 f i 4 ) 2f i5 f i 5 )].20 US 2US 3US 5US 7US h f i f i ) 3 2h 3f i 4f i f i 2 ) 5 6h 3f i 6f i f i 2 2f i ) h 20f i 60f i 5f i 2 2f i 3 30f i 3f i2 ) 7 2 h 0.25f i f i 0.3f i f i f i 4 0.6f i 0.f i f i3 ) 9US 2 520h 504f i 2 520f i 840f i 2 240f i 3 45f i 4 4f i 5 680f i 360f i2 60f i3 5f i4 ).50 2CS 4CS 6CS h 2 f i 2f i f i ) 3 28h 2 f i2 6f i 30f i 6f i f i 2 ) 5 36h 2 4f i3 27f i2 08f i 70f i 08f i 27f i 2 4f i 3 ) 7

4 , CFD MBS, MBS d m J2 fx i ) dx m = j= J α j f ij ), α j Taylor ; J J 2, J 2 = J MBS, 2J ; J J 2 MBS ). Reynolds R cri ), MBS MBS, ),, [6]. CFD MBS,, [2], LES) ). [2]) MBS,, ;, Jameson [8,26 28]. Reynolds R cri 2, R cri 2, 3) R cri 2 ),,,., [] MBS, advanced)., MBS. 3 NS MBS 3., MBS [92 98],,,,, ) MBS, MBS PDS) PVS), MBS. [92 98] : MBS, ) u i ) ), ;, ; Taylor,, ; MBS PDS) PVS).,. ),,, f j, f j f j., ),. ) MBS),,

5 6 : CFD 6, ;, PDS) positive) PVS). ). PDS PDS, TVD ; TVD, [,28,29]. 2. PDS PVS PDS PVS Burgers,,. PDS PVS,,,,. : )., Reimann,.,, MBS)., [07 09], [5] Roe [8] AUSM [24,25],. 3.2 NS MBS, Navier- StokesNS) x u t u u x v u y = p ρ x u γ 2 x 2 2 u y 2 ) 2) 2) MBS), 2) MBS ij u n ij ) un ij 2 x ij [ α x )u n i,j un ij ) α x )u n ij un i,j )] u n i,j un i,j ) v [ α y )u n i,j un ij ) ij α 2 y y )u n ij ij un i,j )] = u n i,j un i,j ) [ p n i,j 2ρ x pn i,j ) γ u n i,j ij t un x 2 ij ] 2u n ij un i,j ) yij 2 u n i,j 2un ij un i,j ) 3), α x = sgnu n ij, α y = sgnvij n. 3), ), 2 2 );,,, 3.,. MBS, MBS. 2) MBS3) u n ij vij n ) γ ρ x t vij n v n ij y );, x t G txg ty u n ij u n ij ) G ux [ α x )u n i,j un ij ) u n α x )u n ij un i,j )] ij G 2 x cx u n i,j un i,j ) = ij [ G cx u n i,j un ij ) G cxu n ij un i,j )] G px p n i,j pn i,j [ ) G px p n i,j pn ij ) 2ρ x ij γ x 2 ij G pxp n ij pn i,j )] u n i,j 2un ij un i,j ) 4),

6 , G tx, G ux, G cx, G cx, G cx, G px, G px G px G tx = M T xm t m, G ux = M a m x m, m= m= m= G cx = M b m x m, G cx = M b m x m, m= m= G cx = M b m x m, G px = M c m x m, m= G px = M c m x m, G px = M c m x m m= m= 5) 5) NPRF), [92 98], : ) ) : x γ/u ij, NPRF, Reynolds R x = u ij x/γ ; ) L ) x,, γ/u ij, NPRF R x ; L/UL U ) x/u ij, γ/u 2 ij, NPRF R x u ij t/ x, u ij t/ x CFL. 2) :,,, u ij ), γ ;, 3),. 3) : )., ;, u i > 0, i, i), i, i ). [96 00], ) ) ), PS) PS. PS4), NPRF G ux G cx, NPRF G cx G cx, NPRF G px NPRF G px G cx. 4) :, γ/u ij, MBS, NPRF NPRF [92 98]. 5) : NS MBS, [08, 09]) ) ; 4), NS PS)., 4) PDS), MBS PDS), u ij 2 x ij γ x 2 ij G ux [ α x )u n i,j un ij ) α x )u n ij un i,j )] G cx u n i,j un i,j ) [G cxu n i,j un ij ) G cxu n ij un i,j )] = u n i,j 2un ij un i,j ) 6) 5) G ux, G cx, G cx G cx 6), u n i±,j ij Taylor, u ij γ ) m u x m = u n ) m ij u γ x 7)

7 6 : CFD 63, PDS, x n n =, 2, ) a n, b n, b n b n, 5) NPRF, NPRF 6) PDS, NPRF [92 98] G ux = M m= m )! R x ij sgnu ij ) m 8) G cx = 2 R2 x ij 30 R4 x ij 260 R6 x ij 9) G cx = 6 R x ij 360 R3 x ij 3 7! R5 x ij ! R7 x ij 3! R9 x ij G cx = 6 R x ij 360 R3 x ij 3 7! R5 x ij ! R7 x ij 3! R9 x ij 0) R xij = u ij x ij /γ. v u y = γ 2 u MBS y2, G uy [ α y )u n i,j un ij ) v ij 2 y ij α y )u n ij un i,j )] G cy u n i,j un i,j ) [G cyu n i,j un ij ) G cyu n ij un i,j )] = γ y ij u n i,j un ij un i,j ) ) 8) 0) x, x ij u ij y, y ij v ij, R xij R yij = v ij y ij /γ G uy, G cy, G cy G cy PDS), MBS), PDS t G txu n ij u n ij ) = M t m= T xm t m ) u n ij u n ij ) 2) 2) u n ij n Taylor, u ij, u t u u x x 2 u x 2 ) m m u t m = u2 ij u 3) γ t 2), t m T xm G tx G tx = R xij λ x 2 2 R x ij λ x ) 2 4) 720 R x ij λ x ) R x ij λ x ) 6 R xij = u ij x ij /γ, λ x = u ij t/ x ij, λ x x CFL. G ty, PDS) [ ] R xij λ x t 2 2 R x ij λ x ) 2 [ R yij λ y 2 2 R y ij λ y ) 2 ] u n ij u n ij ) 5) R yij = v ij y ij /γ, λ y = v ij t/ y ij y CFL PDS), MBS),, PDS 4)) 2ρ x ij G px p n i,j p n i,j) 2ρ x ij [G pxp n i,j pn i,j ) G pxp n ij pn i,j )] 6) 6) p n i±,j p n ij Taylor, u ij, p x u u x x 2 u x2 m p x m = uij γ ) m p x 7)

8 ), x m C m, C m Cm, NPRF G px, G px G px, p ρ x PDS 2ρ x ij 6 R2 x ij R4 x ij ! R8 x ij 3 7! R6 x ij ) p n i,j pn i,j ) 8) [ 2ρ x ij 2 R x ij 2 R2 x ij 6! R4 x ij ) 6 7! R6 x ij p n i,j pn ij ) 2 R x ij 2 R2 x ij 6! R4 x ij ) 6 7! R6 x ij )] p n ij pn i,j 9) p PDS) ρ y, 8) 9) x ij, u n ij, pn i±,j y ij, vij n p n i,j±, p PDS. ρ y u )p MBS PDS), PDS, PDS) q,, ),,, ; PDS 3.2. ), G ux [ α x )u n i,j un ij ) u α x )u n ij un i,j )] ij G 2 x cx u n i,j un i,j ) ij G cxu n i,j un ij ) G cxu n ij un i,j ) γ x 2 un i,j 2un ij un i,j ) G us q G cs q 2 G cs G cs)q = 20) NPRF) G us = M m= G cs = M m= f m x m, G cs = M m= g m x m, G cs = M m= g m x m, g m x m 2) 20) 6), m ) m ) m 2 u x m = uij u γ x uij q 22) γ G us = G ux, G cs = G cx, G cs = G cx, G cs = G cx 23) 23), 20) [ α x )u n i,j un ij ) u ij α x )u n ij un i,j )] = 2 x ij u n i,j un i,j ) [u n i,j un ij ) un ij un i,j )] γ x 2 u n i,j G 2un ij un i,j ) ux γ x 2 u n i,j G 2un ij un i,j ) cx [ γ q x 2 G u n i,j un ij ) cx ] u n i,j un ij ) G cx 24) PDS24) PDS20),. 3.3 NS PDS) NS, 2 u i i =, 2, 3) x 2 i, ), 2 u i x 2 i,, PDS 20) 24), q NS. PDS NPRF NPRF, 20) 24) 23).

9 6 : CFD 65 NS, NS PDS, 24) PDS. NS x, PDS) [ α x )u i,j u ij ) u α x )u ij u i,j )] ij 2 x ij u i,j u i,j ) [u i,j u ij ) u ij u i,j )] x 2 ij [ α y )u i,j u ij ) v ij α y )u ij u i,j )] = 2 y ij u i,j u i,j ) [u i,j u ij ) u ij u i,j )] u i,j 2u ij u i,j ) G ux u i,j 2u ij u i,j ) γ G cx [ γ y 2 ij 2ρ x ij G u i,j u ij ) cx G cx ] u i,j u ij ) G uy u i,j 2u ij u i,j ) G cy u i,j 2u ij u i,j ) [ G u i,j u ij ) cy G cx ] u i,j u ij ) G px G ux P i,j P i,j ) G px G cx P i,j P i,j ) G px P i,j P ij ) G cx G px G cx P ij P i,j ) q 24a) NS PDS 24a), x ; x PDS G ux F yx G uy F xy ) 24b), F yx F xy yx) xy), F yx F xy = 0 24c),,, 24c).,, PDS,,,. 3.4 ) PDS) PDS MBS) Reynolds NPRF)), NPRF G cx, G cx, G cx, G px, G px G px R x M, 7) 7), PDS M 2) ; NPRF G ux R M x, PDS M ). PDS MBS R x λ x ), λ x = u ij t/ x ij x CFL, NPRF G tx R xij λ x ) M, PDS M ). 2) PDS PDS, PDS, G ux 8)) R x M ij, M ), PDS, ; PDS MBS, MBS ), )., G ux R M x ij M, PDS [92,0,04]. 3) PDS, G cx, G cx, G cx 9) 0)) R x M ij, PDS M 2) ; ) PDS,

10 PDS Reynolds R cri PDS, PDS), Reynolds R cri [96, 97] 2), Burgers, ; PDS 3). PDS MBS,, R cri,, R cri.25 )., PDS PDS MBS, PDS 2),, ),, ; MBS, MBS, ). DPS). f i = 0, DP) DPS) C i N)ϕ i [C i N) C i N)]ϕ i C i N)ϕ i = 0 25) N, 3 4, 7 8, 2 DPS, C i± 3) = C i± 4) = ) 2 4 R x 48 R2 x 25a) C i± 7) = C i± 8) = 2 x) 4 R 48 R2 x ) 2 30 R2 x 20 7! R6 x 25b) C i± ) = C i± 2) = ) 2 4 R x 48 R2 x 20 7! R6 x 24 R2 x 30 R2 x ) 2 37 ) 2 4 2! R0 x 25c) 6 DPS C i N) > 0, C i N) > 0 25d) R x ) C i N) C i N) C i N) 25e) 25), Reynolds R x, 6 DPS, Reynolds R cri., 25) ϕ i = A B Ci C i ) i 25f) A B. 25f), 25d), R x, 6 DPS. DPS) : DP), i i ) i i ) i ± ) i ),,, DPS).,,,,,. DPS)., DP), i ),,, ; DPS Reynolds DPS Reynolds R cri 4).

11 6 : CFD CDS) SP) DP) PDS) R cri [96,97] 2CDS SP PDS DP PDS R cri CDS) PDS [96,97] 2CDS 4PDS 8PDS ) PDS 4 8 PDS), dϕ i dt = C i ϕ i ϕ i ) C i ϕ i ϕ i ), C i± 0 TVD, 26) Reynolds., Harten [29] TVD. MBS,, PDS), PDS, PDS [96,97,2]. 5) MBS, MBS ), PDS MBS NPRF, Reynolds. MBS PDS [99,00] u i 6 x G u [ α)3u i 3u i ) α) u i2 7u i 6u i )] ν x 2 u i u i ) u i 6 x G u2 [ α) 3u i 3u i ) α)u i 2 7u i 6u i )] ν x 2 u i u i ) = 0 26a) u i 2 x G u [ α)u i2 42 α)u i 3α 7)u i ] ν x 2 u i u i ) u i 2 x G u2 [ α)u i 2 42 α)u i 3α 7)u i ] ν x 2 u i u i ) = 0 26b) NPRF G u, G u, G u2 G u2 Reynolds., PDS 26a) 26b), Reynolds R cri, 4 [99, 00]; MBS ), MBS PDS, 4, PDS Burgers, ; PDS. [99, 00], 3 UDS ) QUICK, 3 UDS QUICK GUDS). DP),., [99, 00] [6] PDS) Reynolds R cri ; QUICK

12 [99], ) QUICK QUICK 8 7 R a R a 2 R 2 a NR N ) 8 3 N =, 2,, 6) 26c) QUICK R b R b 2 R 2 b N R N ) d) R c R b 2 R 2 c N R ) N 8 26e) 7 R Reynolds, a N, b N c N. [99] 26c) QUICK QUICK Reynolds R cri, 2 < R cri < 3, 4. 26d) 26e) DP) QUICK R cri., QUICK QUICK R cri. 26c) R cri ; QUICK R cri,,, 26d) 26e);, R cri = 26d) 26e)., SP) DP) PDS), Reynolds R cri PDS.,, PDS PDS. 4 MBS SP, DP) PDS) Reynolds R cri [99,00] 3US 2CS SP PDS DP PDS R cri QUICK 2CS SP PDS DP PDS R cri 8/ : 3US 2CS,. 6) NS MBS, MBS, [07 09, 5] 7) PDS MBS Peclet P x, P x = u x, D, PDS D NS PDS. 4 NS MBS PVS) FD MBS) PDS, FV MBS),. FV, ):, 2 [2] ; MBS); FV. midpoint rule), ; MBS.,,, QUICK, FROMM, [0 2]. 4.2 NS Ω MBS FVS) NS ρϕ) dω t S S µgradϕ nds ρϕu nds = S p nds Ω qdω 27), ϕ.,

13 6 : CFD 69 27) ρϕ) ) Ω p J t p j= [ m jf ϕ jf µ ϕ ] S jf p jf S jf = q p Ω p 28) ξ j )jf p J, jf p jp, 2. 2 p, jp, jf, p jp ξ j n, Mugaferiga [2]. Ω p p, m jf jf m jf = ρ jf u jf S jf, S jf = S jf n 29) jf jf, S jf, n jf. MBS, MBS Ω p [ ρϕ) t J j= J j= ) n p ρϕ) t ) n p ] 2 m jf [ α j )ϕ p α j )ϕ jp ] m jf [δ j ϕ p δ j )ϕ jp ] { µ d 2 j d j S jf ϕ jp ϕ p ) } S jf [δ j p p δ j )p jp ] q p Ω p = 0 30) α j = sgnu jf ; d j = d j 28) FV MBS 30) Ω p, m jf, d j d j p jp ) ; d m j m =, 2, ),.,,, FV MBS 30) PVS [ ρϕ) ) n ) ] n ρϕ) Ω p t p t p J 2 m jf G uj [ α j )ϕ p α j )ϕ jp ] m jf G cj [δ j ϕ p δ j )ϕ jp ] j= m jf [G c j δ j )ϕ jp G c j δ j ϕ p ] G pj [δ j p p δ j )p jp ] J G pj [δ j p p δ j )p jp ] j= [G pj δ j)p jp G pj δ jp p ] µ d j s jf ϕ jp ϕ p ) q p Ω p = 0 d 2 j 3), NPRF) G uj, G cj, G cj, G cj, G pj, G pj G pj G uj = M m= G cj = M m= a m d m j b m d m j G cj = M b mδ j d j ) m m= G cj = M m= G pj = M m= b m δ j ) m d m j C m d m j G pj = M C mδ j d j ) m m= G pj = M m= C m δ j ) m d m j 32), PVS 3) j p jp ), 33), 33),,

14 ;, 33). µ d j S jf ϕ jp ϕ p ) d 2 j 2 m jf G uj [ α j )ϕ p α j )ϕ jp ] m jf G cj [δ j ϕ p δ j )ϕ jp ] m jf [G cj δ j)ϕ jp G cj δ jϕ p ] G pj [δ j p p δ j )p jp ]S jf G pj [δ j p p δ j )p jp ]S jf q p Ω p = 0 [G pj δ j)p jp G pj δ jp p ]S jf 33) 3, PVS 33),,, PVS) 33), 33), FV MBS PVS, 32) G uj, G cj, G cj, G cj 33), ϕ p ϕ jp S jf Taylor, m jf m ϕ ϕ jf ξ m j m ) m ϕ mjf ξj m = ϕ jf = µs jf ujf γ ) m ϕ jf 34) 33), d m j [94,95,98] G uj = M m= R dj = u jf d j γ m )! R djsgnm jf ) m G cj = 2 2δ j )R dj 3! 6δ j 6δ 2 j )R2 dj 4! 4δ j 36δ 2 j 24δ3 j )R3 dj 5! 30δ j 50δ 2 j 240δ3 j 20δ4 j )R4 dj 35) 36) δ j = 2 G cj = 2 R2 dj 20 R4 dj 37), 33), δ j d j ) m δ j ) m d m j, G cj M) = M m= G cj M) = M δ j ) m R m dj ) m δ j m= δ j [ m! 2m ] m )! δ j δj m Rdj m [ 2m m )! δ j m m )! 38) ] 39) G cj G cj, S jf, u jf > 0 p, jp ; u jf < 0 p, jp PVS) 3.2, MBS, p jp ), PVS p jf = G pj [δ j p p δ j )p jp ] 40) p jf = G pj δ j)p jp G pj δ jp p 4) G pj, G pj G pj 32), G pj, G pj G pj PVS 40) 4), p p p jp jf Taylor, p jf ) u jf u u jf, ξj jf ) m p ξj m jf = mjf µs jf ) m p jf = ujf γ ) m p jf 42) 40), d m j

15 6 : CFD 62 32) C m, G pj G pj = 2 δ j δ j )R 2 dj 6 δ j δ j ) 2δ j ) R 3 dj 24 δ j δ j )6δ j 6δ 2 j )R4 dj 20 δ j δ j ) 2δ j )22δ j 20δ 2 j ) R 5 dj δ j= 2 = 8 R2 dj 92 R4 dj 43), 4), δ j d j ) m δ j ) m d m j 32) C m C m, G pj G pj G pj = δ jr dj 2 δ jr dj ) δ jr dj ) δ jr dj ) δ jr dj ) 5 44) G pj = δ j)r dj 2 δ j) 2 Rdj δ j) 3 Rdj δ j) 4 Rdj δ j) 5 Rdj 5 45) PVS) PVS, j PVS [ ρϕ) ) n ) ] n ρϕ) t G tj t)ω p t t G tj = 46) M T jm t m 47) m= ) n ρϕ) 47) 46), n t Taylor,, ρϕ) ϕ jf ϕ jf t ϕ jf, ξ j m ρϕ) t m = u 2 jf γ ) m ρϕ) t 48) 46), t m T jm, G tj t) = 2 R dj λ j 2 λ jr dj ) 2 49) 720 λ jr dj ) λ jr dj ) 6 R dj = u jf d j /γ, λ j = u jf t/d j, λ j CFL. PVS [ J ρϕ) ) n ) ] n ρϕ) t Ω p G tj t) t t j= FVS) 50) MBS 3.2.4), MBS, FVS, 33), [ αj R jf G uj ϕ p α ] j ϕ jp 2 2 ϕ jp ϕ p ) R jf G cj [δ j ϕ p δ j )ϕ jp ] [ R jf G cj δ j)ϕ jp G cj δ ] jϕ p G usj S pj d j G csj S pj d j [ G csj 2 δ j G csj δ j) ] S pj q pj R jf = u jf d j, S pj = d j q pj Ω p, γ ud j S jf G usj = f m d m j, G csj = g m d m j G csj = g mδ j d j ) m, G csj = g m δ j ) m d m j 5) J q pj = q p j= 52) G uj, G cj, G cj, G cj, G usj, G csj, G csj G csj 5), ϕ jp ϕ p Taylor, m ) m ) m ϕ mjf mjf ξj m = ϕ jf S pj 53) µs jf µs jf 5), G uj, G cj, G cj G cj G usj, G cjs, G cjs

16 G cjs, G usj = G uj, G csj = G cj G csj = 2 δ j) δ j G cj, G csj = 2δ j δ j G cj 54) 52) 54) 5) 3 PVS, G uj ϕ jp ϕ p ) G cj ϕ jp ϕ p ) G cj ϕ jp ϕ l ) G ϕ l ϕ p ) cj R dj αj 2 ϕ p α ) j ϕ jp 2 R dj δ j ϕ p δ j )ϕ jp ) R dj δ j ϕ p δ j )ϕ jp ) S pj q pj 55), PVS 55) PVS 5), PDS 24) PDS 20) ;, PVS 5) PDS 20), NS PVS PDS ), 24b). PVS55) PDS24), NS PVS PDS,, 24c) NS PVS) NS, NS PFV 5) 55), 5) 55) S pj q pj. PVS, NPRF) NPRF, NS PVS. 4.3 ), MBS ) PVS, : ) PVS PVS MBS NPRF, Reynolds R dj. NPRF G cj, G cj, G cj, G pj, G pj G pj RM dj, 34) 42), PVS M 2) ; NPRF G uj R M dj, PVS M ). PVS MBS λ j R dj, λ j = u jp t /d j CFL, G t R dj λ j ) M, 48), PVS M ). 2) PVS PVS., NPRT G uj 35)) NPRF G ux 8)). PDS, ; PVS, ; PVS Reynolds R cri. δ j PVS, PVS, PVS. QUICK, R cri 8/3, ;, PVS. 3) PVS, ) PVS. NPRF G cj 37)), 2CVS) PVS Reynolds R cri, 2.0, ) PVS, G cj G cj 38) 39)), PVS), R cri., PVS

17 6 : CFD 623 {{ J µ j= d 2 j { µ d 2 j {{ J µ j= d 2 j d j S jf m jf [ δ j ) 3 ) ] } 2 δ j δ j R x ϕ jp 3 d j S jf m jf [δ j 2 δ j ) ] } } δ j )R x ϕ p = 0 56) 2 d j S jf m jf [ δ j ) 3 ) 2 δ j δ j R dj 2 5 ) 6 δ j δ j R dj ) ) ] } 24 δ j δ j R dj ) 3 ϕ jp { µ d 2 j 3 d j S jf m jf [δ j 2 δ j ) δ j )R dj δ j ) 7 δ j ) 2 Rdj δ j ) δ j ) 3 R 3 8 dj] }} ϕ p = 0 57) δ j = /2, PVS 56) 57) {[ J µ d j S jf 2 m jf j= d 2 j {[ J µ d j S jf 2 m jf j= d 2 j [ µ d j S jf 2 m jf d 2 j dj) ] [ 4 R µ ϕ jp d 2 d j S jf j 2 m jf ) ] } 4 R dj ϕ p = 0 58) 4 R dj 24 R2 dj ) ] 92 R3 dj ϕ jp 4 R dj 24 R2 dj ) ] } 92 R3 dj ϕ p = 0 59) PVS 56) 57) : δ j /3 < δ j < 2/3 3/7 < δ j < 4/7, PVS 56) 57) ϕ jp ϕ p R dj., δ j /2, PVS, Reynolds R cri [98], 5 ;, Jameson [26 28] ) R cri 2; PVS [96,97]. PVS Burgers,, [98]. PDS [96, 97]). Jameson [26 28], PVS,, PVS Reynolds R cri,., PVS, 3 6 PVS, 3 5 PVS [98,2,5],,, TVD, 3 PDS. 5 2CVS), SP DP) PVS Reynolds R cri δ j = /2) [98] 5 2CVS SP PVS DP PVS R cri ) MBS MBS PVS,. MBS ),, MBS MBS, MBS, QUICK ), PVS, PVS

18 ) MBS QUICK ), PVS MBS Reynolds. PVS QUICK, PVS,, PDS,,. 6), PVS, MBS Peclet P x, PDS. 3 5 PDS) PVS) PDS PVS, Benchmark [92 8]., NS MBS MBS,, pressure-based) SIMPLE [2 23]. 3a)) Benchmark [24 26]), PVS SIMPLE NS : [3 5] [95,5 8] ) PVS, Ghia [24] Benchmark 4 7). 4 7 Benchmark. PVS MBS, PVS MBS, [4]. PVS PVS ;, [5 8], Ghia. 4 u y ) 5 v x )

19 6 : CFD u y ) 7 v x ) [4, 5] PVS SIMPLE., 30 3b)) Benchmark,, [27 29]). PVS,,. [4, 5] Benchmark, 6. 6 Nusselt Nu) Nusselt, 6. RayleighRa) , ; 8 Benchmark, PVS Benchmark. 8 PVS RayleighRa) 0 3, 0 4, 0 5, 0 6, u v., Benchmark,. 8, Ra,,,. 6 Nu [4,5] Ra Nu [3] [6] [4] [5] [7] [0] FEM) [0] DSC) PFV 0 3 Max ) ) ) ) ) ) Min ) ) ) ) ) ) Ave Max ) ) ) ) ) ) Min ) ) ) ) ) ) Ave Max ) ) ) ) ) ) Min ) ) ) ) ) ) Ave Max ) ) ) ) ) ) Min ) ) ) ) ) ) Ave Max ) ) ) ) ) Min ) ).298.0) ) ) Ave

20 力 626 学 进 展 200 年 第 40 卷 表 6 壁面最大 最小和平均 N u 数及其坐标位置比较 Ra Nu 文献 [3] 文献 [6] 文献 [4] 文献 [5] 文献 [7] [4,5] 文献 [0] FEM) 续) 文献 [0] PFV DSC) 2 07 Max ) ) ) Min ).06.0) ) Ave Max ) ) ) Min..59.0).245.0) ) Ave Max ) ) ) ) Min ).766.0).428.0) ) Ave 图 8 浮力驱动方腔流动形态随 RayleighRa) 数的演化, 瑞利数从上至下依次为 Ra= 03, 04, 05, 06, 07, 08 文献 [6 8] 把四阶中心型有限体积 PVS 与 Level Set 物质界面函数方法 [30,3] 相结合, 计 算了几个具有自由面的两相流问题: 二维溃坝流 动; 二维矩形腔 高为 4 宽为 ) 内互不相溶两液 体 初始交界面为水平面) 失稳 Rayleigh Taylor 不 稳定现象) 后, 界面变形运动的演化历史; 液滴萃 取化工系统中液滴变形运动和传质过程, 以及界 面张力梯度不均匀驱动的 Marangoni 对流效应对 液滴变形运动和传质过程的影响, 系统模拟了雷 诺数 Re = ρu L/µ), 韦伯数 W e = ρu 2 L/δ) 弗劳

21 6 : CFD 627 F r = U 2 /gl), ρ /ρ 2 µ /µ 2,, PVS Level Set. [5, 6] PVS Level Set,, ;,. Benchmark, PVS LevelSet 2, ), ; Benchmark,,,. [5], NS MBS, SIMPLE MBS, PVS; NS MBS PVS Bumps ), M 0.5, ;,,,,,. [07 09] Roe [8], PDS u t j x h j/2 h j /2 ) = Re x 2 [ ] u j u j ) u j u j ) G u,j/2 G u,j /2 h j/2 = 2 fj f j a j/2 j/2 u ), 60) f j = 2 u2 j, j/2u = u j u j 6) a j/2 = j/2 u f j f j ), j/2 0 au j ), j/2 = 0 62) G u,j/2 = R euj/2 6 m= Reuj/2 n )! sgnu ) n, j/2 = ρ j/2u j/2 x µ j/2 63) [07 09] PDS 60) Burgers M=2 4), Roe PDS, Roe,, Sjogreen, [3], PDS. [5] Navier StokesRANS), 4 AUSM AUSMPW [24,25], , AUSMPW ; 2 u x 2, 2 v y 2, 2 w z 2, 2 k x 2 ) G uj, G vj, G wj, G ej, G kj G wj, u, v w 3, e, k w, 35). [5] AUSMPW PVS, DLR WB M = ) ), AUSMPW PVS AUSMPW,, DLR WB, AUSMPW, UGS6 ) AUSMPW PVS, DLR WB DLR WB), NASA WB. [07, 08, 5] Roe RANS AUSMPW, MBS, ;, MBS, Roe AUSMPW.,,

22 , ) MBS, MBS PVS. 0 DLRWB 6 9 DLRWB C p.230 ) NS CD) PDS) PVS) NS CD MBS) ; PDS PVS MBS,,, PDS PVS MBS,, PDS PVS., MBS ),, ). CFD, PDS PVS, CFD. PDS PVS, PDS PVS. NS CD MBS PDS PVS, 6 : PDS PVS, PDS, PVS;, 3 PDS 3 PVS TVD. 6 Burgers Sjogreen,.,,.

23 6 : CFD 629 DP) PDS) PVS),. Roe AUSMPW PDS PVS) WB), PDS PVS,, AUSM, Jameson, TVD ) MBS, PDS PVS ;, PDS PVS ),., ; MBS PDS PVS,,,., MBS) PDS PVS,,,,,,., Boltzmann KdV-Burgers MBS. Laney C B. Computational Gasdynamics. Cambridge: Cambridge University Press Ferziger J H, Peric M. Computational Methods for Fluid Dynamics, 3rd edn. Berlin: Springer, ,.. :, Pantankar S V. Numerical Heat Transfer and Fluid Flow. London: Taylor & Francis, Tannehill J C, Anderson D A, Pletcher R H. Computational Fluid Mechanics and Heat Transfer, 2nd edn. London: Taylor & Francis, ). :, ,.. :, :, ,.. :, Spalding D B. A novel finite-difference formulation for differential expressions involving both first and second derivatives. Int. J. Num. Methods Eng., 972, 4: Lax P D, Wendroff B. Difference schemes for hyperbolic equations with high order of accuracy. Comm. Pure Appl. Math., 964, 7: MacCormack R W. The effect of viscosity in hypersonic impact cratering. AIAA paper , Lenard B P. A stable and accurate convection modeling procedure based on quadratic upstream Interpolation. Comput. Math. Appl. Mech. Engrg., 979, 9: Fromm J E. A method for reducing dispersion in convective difference schemes. J. Comput. Phys., 968, 3: ,,,. Hoam-Open CFD., 2007, 203): ,. CFD., 2002, 242): Godunov S K. A difference method for the numerical calculation of discontinuous solutions of hydrodynamic equations. Math. Sobrnik, 959, 47: Roe P L.Approximate Riemann solvers, parameter vectors and differe schemes. J. Comput. Phy., 98, 43: Osher S. Shock modeling in aeronautics I. In: Morton K W, Baines M J eds. Numerical Methods for Fluid Dynamics. New York: Springer, Van Leer B. Towards the ultimate conservative difference scheme V: a second-order sequal to Godunov s method. J. comput. Phys., 979, 32: Steger J L, Warming R F. Flux vector splitting of the inviscid gasdynamics equations with application to finite difference methods. J. Comput. Phys., 98, 402): Van Leer B. Flux-vector splitting for the Euler equations. Lecture Notes in Physics, 98, 70: Colella P, Woodward P. The piecewise parabolic method PPM) for gas-dynamic simulations. J. Comput. Phys., 984, 54: Liou M S, Steffen J C. A new flux splitting scheme. J. Comput. Phys., 993, 07: Liou M S. Ten years in the making-ausm family. AIAA paper , Jameson A, Schmidt W, Turkel E. Numerical solutions of the Euler equations by finite volume methods using Runge-Kutta time-stepping schemes. AIAA paper 8-259, Swanson R C, Turkel E. Artificial dissipation and central difference schemes for the Euler and Navier-Stokes equations. AIAA paper 87-07, Jameson A. Analysis and design of numerical schemes for gas dynamics I: artificial diffusion, upwind biasing, limiters and their effects on accuracy and multigrid convergence. Int. J. Comp. Fluid Dyn., 994, 4): 7 28

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27 6 : CFD 633 most important six PDS and PVS for the convection diffusion CD) equation are : sixth-order accurate upwind finite-difference PDS, dual perturbation DP) fourth-and eighth-order accurate central PDS, dual perturbation third- and fifth-order accurate interpolation approximation) finite volume FV) central PVS and sixth order accurate upwind PVS. This six schemes are absolute stable or absolute positive and are non-oscillatory schemes for any values of grid Reynolds number. In one dimensional case, this six schemes are TVD scheme for any values of grid Reynolds number. However, the same order MBS must use multi-nodes and oscillate on coarse grids. PDS and PVS can not only be directly used to calculate flow, but also act as a basic or starting scheme for reconstructing high resolution scheme by self-adjust numerical dissipation. The above six PDS and PVS and others have already been used to calculate incompressible flows, compressible flows, mass transfer and Marangoni convection in the cases of a falling drop, two phase flows and others, and some excellent numerical results are achieved. For example, PVS solve lib-driven and buoyancy-driven cavity flows and result in several new Benchmark solutions. The numerical perturbation algorithm and corresponding schemes are also called Gao s algorithm and Gao s schemes. Several subjects worthy of further study are discussed. The present method is also suitable for reconstructing MBS of other mathematical physics equations such as the simplified Boltzmann equation, magnetohydrodynamic equations, KdV-Burgers equation etc.) with coupling dynamics effects. Keywords computational fluid dynamics, numerical perturbation algorithm, mathematical basic scheme, perturbational finite difference scheme, perturbational finite volume scheme,, 937,. 960, ;, ; ;. : Navier-Stokes NS) PNS ),. / ISF) ISF ), ISF.,. CFD... : PNS ; ;.