Having written the semi-discrete form of the governing equations for each control volume as in equation (2.33), the residual R(Ui)needs to be evaluated to perform time marching. In finite volume method this cell centroid residual calculation requires evaluation of inviscid and viscous fluxes at each face of the control volume. Hence the solution methodology employed for the calculation of face fluxes has vital role in deciding the accuracy of the final result. The convective and viscous flux calculation methodologies adopted in the present solver are discussed below.
Spatial discretization 30
2.4.1 Calculation of convective fluxes
The unsteady compressible inviscid flow governing equations are hyperbolic in nature. The property of nonlinear hyperbolic system of equations allows discontinuities in the solution even if the initial condition is smooth. Therefore concerned numerical discretization schemes, based on the information propagation direction, are known as upwind schemes. Here in the present study seven different upwind schemes, which belong to either of the following two categories are considered.
• Flux-vector splitting
• Flux difference splitting
The flux-vector splitting methods can be viewed as the first level of upwind schemes, since they only account for the direction of wave propagation. The flux-vector splitting schemes de- compose the vector of the convective fluxes into two parts according to the sign of characteristic variables. The second category, flux-difference splitting schemes is based on the solution of the local the Riemann (shock tube) problem. The upwind schemes that are incorporated in the present solver along with their references and flux formulations are given in Appendix A. Each of these schemes have advantages and limitations, which are explored through various numerical studies in next chapter.
2.4.2 Reconstruction and implementation of second order spatial accu- racy
Upwind schemes require flow variables to be computed on the either sides of the faces of a control volume. These states can be assigned by extrapolating cell centroid values directly to the control volume faces, if only first order accuracy in spatial discretization is expected. But for enhancing the accuracy of results one should implement higher order extrapolation techniques in the solver. In the present solver second order accuracy in spatial discretization is achieved by implementing piecewise linear reconstruction method suggested by Barth and Jespersen [70]. In this reconstruction technique, it is assumed that solution is piecewise linearly distributed in the
Spatial discretization 31 control volume. For a cell-centered scheme, Barth and Jespersen calculated the left and right states for a control surface using the following relations,
UL =Ui+ϕ(∇Ui·r~L) UR=Ui+ϕ(∇Uj·r~R)
(2.34)
where∇Uiis the gradient of any flow variableU at the cell centerI , which is given by,
∇Ui = ∂U
∂x,∂U
∂y T
(2.35) The termϕis the limiter function. Here,r~Landr~Rare the vectors pointing from the cell-centroid to the face centroid as shown in the figure 2.2.
FIGURE2.2: Linear reconstruction for the cell centered scheme
Another important fact in this contest is that without a flux limiter, solution of second or higher order schemes suffers from oscillations in the neighborhood of discontinuities. This situation is generally referred to as monotonicity lose. So care must be taken in the implementation of reconstruction to ensure the monotonicity while, reconstructing left and right variable values.
This is achieved by the use of limiters. At strong discontinuities, limiter will reduce the slope to zero to prevent the generation of a new extremum, thus at the region of discontinuity, solution becomes first order accurate in order to ensure monotonicity. Therefore popular Venkatakrishnan limiter [71, 72] implementation is used in the present solver. Venkatakrishnan’s limiter offers
Spatial discretization 32 monotonic solution as well as better convergence to steady state. According to this methodology the limiter function is calculated as follows.
ϕ=
1
∆2
"
(∆1,max2+ε2)∆2+ 2∆22∆1,max
∆21,max+ 2∆22+ ∆1,max∆2+ε2
#
if ∆2 >0
1
∆2
"
(∆1,min2 +ε2)∆2+ 2∆22∆1,max
∆21,max+ 2∆22+ ∆1,max∆2+ε2
#
if ∆2 <0
1 if ∆2 = 0
(2.36)
where,
∆1,max=Umax−Ui
∆1,min=Umin−Ui
(2.37)
whereUmaxandUmin are the maximum and minimum values of all surrounding cellsj, includ- ing the celliitself, and are given by;
Umax=max(Ui, maxjUj) Umin =min(Ui, minjUj)
∆2 =∇Ui·r~L
For avoiding the division by zero,∆2 is generally specified as,
∆2 =|∆2|+w
where, w is approximate machine accuracy, ~rdenotes the vector from the cell centroid to the corresponding face centroid. The parameterε2 is evaluated as,
ε2 = (K∆h)3
Calculation of viscous fluxes 33 whereKis a constant of order one (generally taken as 2) and∆his square root of area in 2D and cube root of volume in 3D. The solution accuracy and convergence obtained with Venkatakrish- nan’s limiter largely depends on theKvalue. Setting the value ofK = 0, results in full limiting, which in turn leads to poor convergence. Whereas the use of largeK values leads to no-limiting condition, thereby offers spurious solutions. So fine turning of theK is required to maintain op- timum trade off between convergence and solution accuracy. Although the Venkatakrishanan’s limiter offers substantial improvement in solution accuracy and convergence, the computational cost for the calculation of this limiter function is relatively high. The additional computations of the parameters involved in this limiter function necessitate additional time and storage. These pitfalls of the Venkatakrishnan’s are tolerated to aim the solution accuracy and convergence.