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Flux Numerical Methods 1 Flux Basics The finite-volume formulation of the conservation equations resulted in the equation ˆ dQ ˆ ˆ PF dt ˆ where F was the flux of the flow across the control surface resulting from the approximation of the surface integral. For a finite-volume cell, the flux was expressed as nf F Ff ˆ ˆ f 1 Where ˆ v g Q D n dS Ff f ˆ f It was assumed that the flux was uniform over the cell face. 2 Fluxes on a Hexahedral Cell A hexahedral cell contains 6 quadrilateral faces, thus nf 6 Area normal ˆ F F ˆ f 1 f vector for face f where again, n dS f ˆ ˆ v g Q D n dS ˆ Ff ˆ Ff f f Face f ( f is an index for the face ) 3 Numerical Flux on a Cell Face The numerical flux on a cell face is ˆ v g Q D n dS Ff f ˆ f The normal area vector n dS f is usually easily defined for a quadrilateral or ˆ triangular cell face. The focus of the rest of this discussion is on numerical methods for computing v g Q D f at a cell face. We first will assume that g is a known velocity for the cell face. 4 Numerical Flux on a Cell Face We first consider that we have the states of the flow on the “left” and “right” of the cell face, QL and QR. Our objective is to find the cell face flux. Ff Ff QL , QR ˆ ˆ One can define QL ˆ Ff FL Ff QL FR Ff QR ˆ ˆ ˆ ˆ QR n dS f ˆ A consistency condition for the numerical flux is that if QL = QR , then ˆ ˆ FL FR 5 Central Difference Method A central difference method for computing the flux is simply Ff 2 ˆ L ˆ 1 F F ˆ R The central-difference method works okay for elliptic components of the flux because there is no preferred direction for the propagation of information. A simple central difference is often unstable, especially in the presence of strong gradients. One solution is to add some second-order and fourth-order dissipation (artificial viscosity) to the flux. Ff 2 ˆ L ˆ 1 F F D ( 2) D ( 4) ˆ R Methods for computing D(2) and D(4) vary, but generally use second and fourth-order differences with switches to handle variations in Q. 6 Use of Central Difference Method For the Navier-Stokes equations, the viscous shear stress and heat flux terms in the viscous component DV of the non-convective component D are elliptic and those flux components can be computed with the central difference method. Similarly, the fluxes of the turbulence and chemistry equations can be computed using the central difference method. The convective portion of the flux and the pressure term in the inviscid component DI of the Navier-Stokes equation have a hyperbolic character. This wave nature can be put to use to compute the flux using upwind methods. 7 Upwind Methods We expressed the non-convective portion of the flux of the Navier-Stokes equation as D NS D INS DV NS This results in the cell-face flux being expressed as Ff ˆ v g Q DI DV NS NS NS or ˆ ˆ ˆ Ff FfI FfV ˆ I We will now focus on computing the inviscid flux F f using upwind methods. The focus will be on the use of Roe’s Upwind Flux-Difference Splitting Method. 8 Roe Upwind Flux-Difference Method The Roe upwind flux-difference method computes the inviscid flux as: ˆ 1 ˆ 2 ˆ 1 ˆ FfI FLI FRI Ff 2 ˆ where F is the flux difference computed as, ˆ ˆ ˆ F F F 5 F (m ) rm wm ˆ m 1 5 F (m ) rm wm ˆ m 1 Roe’s method is the default flux method in WIND. 9 Roe Upwind Flux-Difference Method The m are the eigenvalues that represent the speed of the waves. The (+) indicate positive eigenvalues and the (-) indicates negative eigenvalues. The rm are the right eigenvectors that represent the direction of propagation of the waves. The wm are the Riemann invariants and represent the strength of the wave, p w1 2 c w2 n1 w n3 u w3 n2 u n1 v p w4 n1 u n2 v n3 w c p w5 n1 u n2 v n3 w c 10 Roe Upwind Flux-Difference Method The differentials are computed as R L p pR pL u u R u L v vR vL w wR wL Flow properties at the face are computed using Roe-averaging 2 R L 1 / 2u L 1 / 2u R u L 1/ 2 R L 1/ 2R Similar for computing v, w, and ht. 11 Higher-Order Projection The choice of values of QL and QR have several option: 1) Use the values of the finite-volume cells to the “left” and “right” of the face. This is a zero-order evaluation and will result in a spatially first-order flux. 2) Use an extrapolation of neighboring finite-volume cells to form a first- order evaluation of Q at the face. This will result in a spatially second- order flux. 1 1 QL Qi ( Qi Qi 1 ) QR Qi 1 ( Qi 2 Qi 1 ) 2 2 QL QR Qi-1 Qi ˆ F f Qi+1 Qi+2 12 Variation Limiting The simple extrapolation formulas assume a smooth variation of Q; however, discontinuities in Q are possible (i.e. shocks). Need some mechanism to sense such discontinuities and limit the variation of Q in these extrapolation formulas. Modify the extrapolations by introducing a limiter , 1 QL Qi ( Qi Qi 1 ) 2 1 QR Qi 1 ( Qi 2 Qi 1 ) 2 This gets into the topic of TVD (Total Variational Diminishing) flux limiting methods, which we will not get into here. The essential role of the limiter is to make 0 in the presence of large variations, which make the flux spatially first-order. 13 Examples of Limiters The possible functions (and theory) for limiters is varied. A couple examples include: Superbee: (r ) max 0, min 2r ,1, min r ,2 Chakravarthy: (r ) max 0, min r , Where r is some ratio of the flow properties and indicates the amount of variation in the solution. An example is Qi 1 r Qi The is a compression parameter 1 2, where a value toward 1 makes the limiter more dissipative. 14 Flux Vector Splitting An alternative to flux-difference splitting is flux-vector splitting that considers that the inviscid flux can be linearly separated ˆ ˆ ˆ FfI F F van Leer’s flux-vector splitting has the general form of 1 u n1 vn 2 c ˆ F f mass v n2 vn 2 c w n3 vn 2 c q 2 vn 2 1vn 2 c 2 2 2 1 2 c M n 12 vn v n ˆ Mn vn f mass 4 c 15 Other RHS Methods Other methods for the “right-hand-side (RHS)” that will not be discussed: • Methods available for 3 rd to 5th –order spatial accuracy. • Roe’s method is modified to allow non-uniform grids. • Roe’s method as used in the OVERFLOW code is available. • Coakley method is available • HLLE method is available (similar to Roe’s method) Next 16