# Presentation: 183774

- Flux Numerical Methods

## Flux Basics    

- The finite-volume formulation of the conservation equations resulted in the equation
- where     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
- Where
- It was assumed that the flux was uniform over the cell face.

## Fluxes on a Hexahedral Cell    

- A hexahedral cell contains 6 quadrilateral faces, thus
- where again,

- Face *f*
*( f is an index for the face )*

- Area normal vector for face *f*

## Numerical Flux on a Cell Face

- The numerical flux on a cell face is
- The normal area vector             is usually easily defined for a quadrilateral or triangular cell face.  The focus of the rest of this discussion is on numerical methods for computing
- at a cell face.
- We first will assume that     is a known velocity for the cell face.

## 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, *Q**L* and *Q**R*.  Our objective is to find the cell face flux.
- One can define
- A consistency condition for the numerical flux is that if *Q**L* = *Q**R** *, then

## Central Difference Method

- A central difference method for computing the flux is simply
- 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.
- Methods for computing *D**(2)* and *D**(4)* vary, but generally use second and fourth-order differences with switches to handle variations in *Q*.

## Use of Central Difference Method

- For the Navier-Stokes equations, the viscous shear stress and heat flux terms in the viscous component **D***V* 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 **D***I* of the Navier-Stokes equation have a hyperbolic character.  This wave nature can be put to use to compute the flux using *upwind methods*.

## Upwind Methods

- We expressed the non-convective portion of the flux of the Navier-Stokes equation as
- This results in the cell-face flux being expressed as
- or
- We will now focus on computing the inviscid flux       using upwind methods.  The focus will be on the use of Roe’s Upwind Flux-Difference Splitting Method.

## Roe Upwind Flux-Difference Method

- The Roe upwind flux-difference method computes the inviscid flux as:
- where        is the flux difference computed as,

- Roe’s method is the default flux method in WIND.

## Roe Upwind Flux-Difference Method

- The       are the eigenvalues that represent the speed of the waves.  The (+) indicate positive eigenvalues and the (-) indicates negative eigenvalues.
- The       are the right eigenvectors that represent the direction of propagation of the waves.
- The         are the Riemann invariants and represent the strength of the wave,

## Roe Upwind Flux-Difference Method

- The differentials are computed as
- Flow properties at the face are computed using Roe-averaging
- Similar for computing *v*, *w*, and *h**t*.

## Higher-Order Projection    

- The choice of values of *Q**L* and *Q**R* 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*.

## 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 **,
- 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.

## Examples of Limiters    

- The possible functions (and theory) for limiters is varied.  A couple examples include:
- Superbee:
- Chakravarthy:
- Where *r* is some ratio of the flow properties and indicates the amount of variation in the solution.  An example is
- The  is a compression parameter  1    2, where a value toward 1 makes the limiter more dissipative.

## Flux Vector Splitting    

- An alternative to flux-difference splitting is flux-vector splitting that considers that the inviscid flux can be linearly separated
- van Leer’s flux-vector splitting has the general form of

## Other RHS Methods

- Other methods for the “right-hand-side (RHS)” that will not be discussed:
- Methods available for 3rd 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)

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