18.5.3 Convective Fluxes

Roe Flux-Difference Splitting Scheme

The inviscid flux vector ${\mbox{\boldmath$F$}}$ appearing in Equation  18.5-6 is evaluated by a standard upwind, flux-difference splitting [ 296]. This approach acknowledges that the flux vector ${\mbox{\boldmath$F$}}$ contains characteristic information propagating through the domain with speed and direction according to the eigenvalues of the system. By splitting ${\mbox{\boldmath$F$}}$ into parts, where each part contains information traveling in a particular direction (i.e., characteristic information), and upwind differencing the split fluxes in a manner consistent with their corresponding eigenvalues, we obtain the following expression for the discrete flux at each face:

Here $\delta {\mbox{\boldmath$Q$}}$ is the spatial difference ${\mbox{\boldmath$Q$}}_R - {\mbox{\boldmath$Q$}}_L$. The fluxes ${\mbox{\boldmath$F$}}_R = {\mbox{\boldmath$F$}}\left({\mbox{\boldmath$Q$}}_R\right)$ and ${\mbox{\boldmath$F$}}_L = {\mbox{\boldmath$F$}}\left({\mbox{\boldmath$Q$}}_L\right)$ are computed using the (reconstructed) solution vectors ${\mbox{\boldmath$Q$}}_R$ and ${\mbox{\boldmath$Q$}}_L$ on the "right'' and "left'' side of the face. The matrix $\vert{\rm\hat{A}}\vert$ is defined by

where $\Lambda$ is the diagonal matrix of eigenvalues and M is the modal matrix that diagonalizes ${\Gamma}^{-1}$A, where A is the inviscid flux Jacobian ${\partial {\mbox{\boldmath$F$}}}/{\partial {\mbox{\boldmath$Q$}}}$.

For the non-preconditioned system (and an ideal gas) Equation  18.5-10 reduces to Roe's flux-difference splitting [ 296] when Roe-averaged values are used to evaluate ${\Gamma} \,\vert{\rm\hat{A}}\vert$. At present, arithmetic averaging of states ${\mbox{\boldmath$Q$}}_R$ and ${\mbox{\boldmath$Q$}}_L$ is used.

In its current form, Equation  18.5-10 can be viewed as a second-order central difference plus an added matrix dissipation. The added matrix dissipation term is not only responsible for producing an upwinding of the convected variables, and of pressure and flux velocity in supersonic flow, but it also provides the pressure-velocity coupling required for stability and efficient convergence of low-speed and incompressible flows.

AUSM+ Scheme

An alternative way to compute the flux vector ${\mbox{\boldmath$F$}}$ appearing in Equation  18.5-6 is by using a flux-vector splitting scheme [ 53]. The scheme, called Advection Upstream Splitting Method (AUSM), was first introduced by Liou and Steffen in 1993 [ 202]. The AUSM scheme first computes a cell interface Mach number based on the characteristic speeds from the neighboring cells. The interface Mach number is then used to determine the upwind extrapolation for the convection part of the inviscid fluxes. A separate Mach number splitting is used for the pressure terms. Generalized Mach number based convection and pressure splitting functions were proposed by Liou [ 201] and the new scheme was termed AUSM+. The AUSM+ scheme has several desirable properties:

1.   Provides exact resolution of contact and shock discontinuities

2.   Preserves positivity of scalar quantities

3.   Free of oscillations at stationary and moving shocks

The AUSM+ scheme avoids using an explicit artificial dissipation, by proposing a numerical flux of the form:

Here $m_{f}$ is the mass flux through the interface, which is computed using the fourth order polynomial functions of the left and right side (of the interface) Mach numbers.

ANSYS FLUENT utilizes an all-speed version of the AUSM+ scheme based on the low-Mach preconditioning.

Low Diffusion Roe Flux Difference Splitting Scheme

In order to reduce dissipation in LES calculations, ANSYS FLUENT uses a modified version of the Roe Flux Difference Splitting scheme, called the Low Diffusion Roe Flux Difference Splitting scheme. The scheme includes low Mach number preconditioning, in which the artificial dissipation term has been reduced [ 43] through the use of a hybrid scheme that combines a central scheme and a second-order upwind scheme (Roe's Flux Difference scheme).

The low diffusion discretization must be used only for subsonic flows. For high Mach number flows, you should switch to the second-order upwind scheme.

The low diffusion discretization is only available with the implicit-time formulation (dual-time-stepping). When running LES with the explicit time formulation, you will need to use the second-order upwind scheme.

To learn how to apply the different convective fluxes, see this section in the separate User's Guide.