Lecture 5.8 Flux Vector Splttng 1
Flux Vector Splttng The vector E n (5.7.) can be rewrtten as E = AU (5.8.1) (wth A as gven n (5.7.4) or (5.7.6) ) whenever, the equaton of state s of the separable form of p = f(e) (5.8.) In such a case, the flux vector E behaves as a homogenous functon of degree one, that s, E(λU) = λe(u) (5.8.3) Snce any functon whch satsfes (5.7.9) has the property that E(U) = E (U)U (5.8.4) E s the Jacoban, E allows the decomposton (5.8.1)
Flux Vector Splttng Now usng (5.7.5) and (5.7.6), we have 1 1 E = AU = PDP - U = P( C + + C - ) P - U (5.8.5) C + and C - are obtaned by splttng C C + contans the postve egenvalues of C C - contans the negatve egenvalues of C. 3
Flux Vector Splttng Rewrtng E as ( + -1 - - 1) ( + -) E= PC P + PC P U = E + E U (5.8.6) Rewrtng Euler equatons (5.7.) by usng Eq. (5.8.6), we have + - + - U ( E + E ) U E E + = 0 + + = 0 t x t x x (5.8.7) Here, E = PC P, E = PC P + + - 1 - - - 1 (5.8.8) 4
Steger-Warmng Flux Vector Splttng In Steger-Warmng splttng scheme, the flux terms are approxmated usng the nformaton from ther upwnd terms. That s, n the fnte dfference approxmaton, the dervatves of E + and E - are approxmated usng backward and forward dfferences, respectvely. Smlarly, n the fnte volume approxmaton, E + +1/ s equated to E + and smlarly, E - +1/ s equated to E - +1. The same also can be extended to the other terms for the problems n hgher dmensons. Lmtatons Snce the components of the splt fluxes are not contnuously dfferentable at sonc and stagnaton ponts, for the Steger- Warmng scheme, there can be some oscllatons n the obtaned solutons when the veloctes reach sonc or stagnaton ponts. 5
Van Leer Splttng Van Leer Modfed the Splttng such that 1. E (U) must be contnuous on U. E + (U) =E(U) f M 1andE - (U) =E(U) f M -1 3. E + and E - must have the same symmetrc propertes wth respect to M 4. The Jacobans E /U must be contnuous on U 5. E /U must have one zero egenvalue for M <1 6. Lke E, E must be a polynomal n M and of the possble lowest degree 6
Van Leer Splttng If E s modfed to æ æ 1ö æ 3 1 1 ö ö E= rcm rc M rc M M + + ç ç g ç g-1 è è ø è øø T (5.8.9) Then the splt vectors are gven by (for M < 1) T æ ö + 1 c + = ç r ( 1+ ) 1 ( + ( g-1) ) + g ( E ) + ( g ) E1 E c M E M ç4 g - 1 çè ø T æ ö - 1 c - =- ç r ( 1- ) 1 (( g-1) -) - g ( E ) - ( g ) E1 E c M E M ç 4 g - 1 çè ø (5.8.10) 7
Lmtatons of Van Leer Splttng Scheme 1. gt s dsspatve. The contact surfaces are not dentcal 3. It an lead to a large dsspaton partcularly n vscous regons 4. Total enthalpy s not preserved n steady Euler equatons. 8
AUSM Scheme Advecton Upstream Splttng Method (AUSM) schemes The am of AUSM s to combne the desrable attrbutes of both fnte dfference and fnte volume splttngs and smultaneously elmnatng ther weaknesses. Essentals Behnd AUSM Scheme Frst and foremost s the concept of upwndng. Second, the use of Remann problem n constructng the numercal flux n the fnte-volume settng. Thrd, the necessty of ncludng all physcal processes, as charactersed by the lnear (convecton) and nonlnear (acoustc) felds. Fourth, the realsaton of separatng the flux nto convecton and 9 pressure fluxes.
ThecoreoftheAUSMfamlyofschemesstherealsatonthat the convecton and acoustc waves ought to be treated as two physcally dstnct processes F = F (c) + F (p) where and æ1 ö () c F = my, m = ru, ç u Y=ç ç çèh ø F ( p) æ0 ö = p ç çè0 ø Here, the convectve flux F (c) contans the convectve mass flow rate m and the correspondng passve scalar quanttes n Y. The pressure flux F (p) contans nothng but the pressure term. 10
() c (, ) = (, ) Y (, ) f u u m u u u u Y 1 L R 1 L R 1 L R m 1 ( u, u ) 1 1 ( u ) ìïy ï L,f m1³ 0,otherwse L R =í ï ïî Y( ur ) ì rl,f u1³ 0 = u rlr=í ï ïï îrr,otherwse Clearly the defnton of the nterface convectve velocty u 1/ s a crtcal step, for whch we employ two functons, respectvely expressed n terms of egenvalues u ± a when u a, as envsoned n Van Leer s flux vector splttng. Note that usng the egenvalues as a bass for expressng the numercal fluxes s qute common n the upwnd formulaton, easly dentfable n all flux schemes mentoned above. Here, we wrte n terms of splt Mach-number functons. 11
+ - u1= a é 1 M( )( ML) M( )( MR) ù ê +, M m m ú L R = ë û u a L R 1 where the subscrpt m refers to the degree of polynomal used n M ± specfc defntons p = P M P + P M p + - ( m)( ) ( n)( ) 1 L L R R For two-dmensonal form the convectve terms F (c) can now be consdered as passve scalar quanttes convected by normal velocty V defned at the cell nterface. On the other hand the pressure flux terms are governed by acoustc wave speeds. The above equatons can be further smplfed as, F m p c (5.8.1) 1
where, 1 0 u p l x m V p v p ly H LR 0 The mass flux at the nterface denoted by subscrpt hasthe form of (5.8.13) m V a M L/ R L/ R where V s the convectve velocty (velocty normal to the edge) and ρ s the densty convected by V. 13
The cell nterface straddles two neghborng cells labeled by subscrpts L and R, respectvely, namely to the left and rght of the nterface. Snce the convectve flux s assocated wth the lnear feld of the system of convectve flux, the nterface densty s dctated by the drecton of n accordance wth dea of upwndng. That s, ρ L/R = ρ L If V > 0 (5.8.14) ρ L/R = ρ R Otherwse (5.8.15) Usng Mach number as workng varable, m am L f M 0 m am Otherwse R (5.8.16) (5.8.17) 14
Here, a 0.5*( al ar ) The nterface Mach number M s calculated as follows: V L / R M L / R a L / R u L / R ( l x ) v L / R V ( l ) (5.8.18) (5.8.19) (convectve velocty normal to the edge wth l normal vector and V L, V R beng the normal velocty across the edge calculated correspondng to left and rght edges respectvely) M ( V V ) L R a M0 mn(1, max( M, M) 0,1 where, M s the Mach number at the ext. y 0, 1 (5.8.0) (5.8.1) (5.8.) f a ( M0) M0( M0) 15
where K p ( PR PL M (4) ( M L) (4) ( M R) max( 1 M,0) fa a ( L R) 0 1 K p 1 ) (5.8.3) (5.8.4) Calculaton of s determned n a smple upwnd fashon. L R If m > 0 (5.8.5) Otherwse (5.8.6) 16
Calculaton of p p ( 5) ( Ml ) pl (5) ( M R) pr Ku(5) (5) ( L R)( faa1/ )( ur ul ) (5.8.7) The splt mach numbers functons μ + (4) and μ - (4) whch are polynomal functons of degree are calculated as follows: ( M) ( M M ) 1 (1) 1 ()( M) 4( M 1) and (5.8.8) (5.8.9) ( M) If M 1 (4) (1) (5.8.30) ( M ) (1 16 ) Otherwse (4) () () (5.8.31) 17
The splt pressure functons ς + (5) and ς - (5) are calculated as follows 1 (5.8.3) (5)( M) (1) f M 1 M (5) ( M) () ( M) 16M () Otherwse usng the parameters 3 3 3 1 f K 16 14 16 8 ( 4 5 a ),, wth 0 u 1 (5.8.33) The values of K p, K u and σ = 1.0 are chosen n all calculatons as K p = 0.5, K u =0.75andσ =1.0. The dffcultes n mplementng the AUSM on unstructured grd are overcome by choosng local orthogonal co-ordnate systems on the centers of all the edges of the dscretzaton. 18
Summary of Lecture 5.8 Flux vector splttng schemes for solvng compressble flow equatons are dscussed n ths lecture. END OF LECTURE 5.8 19