. Methods for the Detection of Improperly Discretised Grid Regions

1.2 Error Indicator Based on the Artificial Dissipation of Central Convection Schemes

The concept of artificial dissipation was originally developed for Euler solvers used to compute inviscid flows. Since inviscid flows have no natural damping mechan­ism, dispersive error terms can cause oscillations in the solution. To damp those os­cillations and reduce the dispersive error an artificial dissipation is introduced. The first artificial dissipation model that incorporates a linear combination of second and fourth order difference dissipation terms was introduced in [9].

For viscous flows, the Navier-Stokes equations provide physical terms that con­tain natural dissipation effects. However, those dissipative terms are only significant in the viscous shear layer. They are insignificant in flow regions that show character­istics of inviscid flows. Thus, in practice, artificial dissipation terms still have to be introduced for Navier-Stokes computations in order to stabilize the flow in regions with inviscid behaviour.

The artificial dissipation is added to the internal fluxes across the cell faces by modifying the governing equations. Considering the flux QF’ across a finite volume face F be

1

Qf’C = 2 [Fr(t) +F/(0], (1)

the artificial dissipation is introduced by adding the term -^aD to the RHS of equation (1) which yields

1 I

QF/ = ^Wr(i)+F,(i)}–aD.

For Navier-Stokes solutions on highly stretched structured meshes, different scal­ings of the artificial dissipation term both in streamwise and normal direction within the region of viscous flow are needed. For unstructurd meshes, directional scaling is difficult to achieve since no mesh coordinate line exists. In order to obtain an ad­equate scaling of the dissipation for highly stretched portions of the grid, the strategy described in [10] is followed. The scaling factor a is given as

c 4Фрфг, j a = XF—— ,

Фг, ї + Фе, j

with

XCF = vF ■ F| + af ■ F

being the maximum eigenvalue of the flux jacobian for the face F. The terms

Фгу = л/грі and <PFj =

are necessary to avoid excessive local numerical dissipation in cases of meshes with high-aspect-ratio cells. The term

relates the size of XF across the Face F to the total eigenvalue XC integrated over the entire control volume surrounding P where

XC = X v ■ F + aF ■ F|,

k=2

and n is the number of surrounding faces. The corresponding terms Xc and rF, j can be defined analogously.

The artificial dissipative flux across the dual face F corresponding to the edge connecting Pi and Pj is given by

Di, j = eF(2) (Wi – Wj) – eF(4) (V2Wi – V2Wj), (2)

where W is the vector containing the conservative variables p, pv and pE. The amount of artificial dissipation added to the scheme is controlled by the coefficients

eF(2) = k(2) max(Vi, Vj) ■ Sc2 and

ekF(4) = k(2) max(0, k(4) – ej(2)) ■ Sc4,

where

I (Pj – Pi)

= j^N( 0________

I {Pj+Pi)

j£N({)

represents a shock switch, where sc2 and sc4 contain some anisotropy corrections and N(i) is the set of neighbours of i. The factors sc2 and sc4 are introduced in order to avoid a dependency of the dissipation on the number of neighbours. The constants ki1’1 and kare generally user defined parameters in the range of -j < № < ^ and p < k< p. Since the amount of added dissipation is scaled with a ~ Xf where for |F| ^ 0, Xp goes to zero, grid converged solution are independent of the dissipation.

In principle this added dissipation has no physical means and thus can be re­garded as an unphysical term that introduces an error to the solution. For this reason, the approach presented here regards the added artificial dissipation of central schemes as a measure for the discretization error. Out of the dissipative fluxes added to the scheme, the contribution dpE to the energy equation is used as a measure for the discretization error. This contribution has been chosen due to the fact that all key flow variables are represented by the energy equation. The amount of added specific artificial dissipation per cell volume that is being calculated during the evaluation of fluxes is stored and given out as a separate variable of the flow field solution and thus can be evaluated during the post processing. The amount of added artificial dissipation scales with the element face size and it is sensitive to the skewness and the gradients of the adjacent flow field. It is noted here that a skewed cell has high added artificial dissipation as this is based on the cell volume.

This error indication method does not give an absolute value for the discretisation error since the exact solution of the flow problem is unknown. In fact its intention is to guide the grid developer in deciding where to further improve the grid.