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FIGURE 4. Comparison between the proposed method (using arithmetic averaging) and the Yee-Roe second-order scheme (using Roe aver- aging) for the Riemann problem test case at a time of 0.8 ms.

Figure 4 Comparison between the proposed method (using arithmetic averaging) and the Yee-Roe second-order scheme (using Roe aver- aging) for the Riemann problem test case at a time of 0.8 ms.

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where the notation (-)’” denotes the property (-) at the previous iteration count. Then, after substituting R and S from Eq. (32) and (33) the latter 4 equations become:
where the notation (-)’” denotes the property (-) at the previous iteration count. Then, after substituting R and S from Eq. (32) and (33) the latter 4 equations become:
The reason why the Yee-Roe flux has been rewritten in Steger-Warming flux-vector splitting form is to obtain positivit preserving conditions on the limiter matrix, which can be derived following the approach outlined in Ref. [12]. This wou yield the following ranges on the limiter matrices (see A ppendix A for a derivation): Thus far, the Yee-Roe flux has not been modified. Indeed, when the eigenvalues A+ outlined in Eqs. (10)-(11) and the limiter matrices Y* outlined in Eqs. (56)-(57) are substituted in Eq. (55), the flux at the interface corresponds exactly to the conventional Yee- Roe flux outlined in Eq. (46). Similarly, to obtain an expression for the ¥~ diagonal limiter matrix, multiply all terms within Eq. (53) by L;,, rewrite in tensor form, and isolate w~:
The reason why the Yee-Roe flux has been rewritten in Steger-Warming flux-vector splitting form is to obtain positivit preserving conditions on the limiter matrix, which can be derived following the approach outlined in Ref. [12]. This wou yield the following ranges on the limiter matrices (see A ppendix A for a derivation): Thus far, the Yee-Roe flux has not been modified. Indeed, when the eigenvalues A+ outlined in Eqs. (10)-(11) and the limiter matrices Y* outlined in Eqs. (56)-(57) are substituted in Eq. (55), the flux at the interface corresponds exactly to the conventional Yee- Roe flux outlined in Eq. (46). Similarly, to obtain an expression for the ¥~ diagonal limiter matrix, multiply all terms within Eq. (53) by L;,, rewrite in tensor form, and isolate w~:
FIGURE 1. Density obtained with the proposed method at a time of 0.6 s for the 3D Noh test case; the entropy correction factor 6 is set to 0.3.
FIGURE 1. Density obtained with the proposed method at a time of 0.6 s for the 3D Noh test case; the entropy correction factor 6 is set to 0.3.
FIGURE 2. Density obtained with the proposed method at a time of 1 s for the 2D Sedov test case; the entropy correction factor 6 is set to 0.3.
FIGURE 2. Density obtained with the proposed method at a time of 1 s for the 2D Sedov test case; the entropy correction factor 6 is set to 0.3.
FIGURE 3. Comparison between the proposed method (using arithmetic averaging), the Y ee-Roe second-order scheme (using Roe averaging), and the Steger-Warming flux limited scheme on the basis of skin friction coefficient at the wall for the boundary layer test case; one node in two is shown.
FIGURE 3. Comparison between the proposed method (using arithmetic averaging), the Y ee-Roe second-order scheme (using Roe averaging), and the Steger-Warming flux limited scheme on the basis of skin friction coefficient at the wall for the boundary layer test case; one node in two is shown.
FIGURE 5. Comparison between the proposed method (using arithmetic averaging), and the Yee-Roe scheme (using Roe averaging) for the Mach 2.5 channel flow test case on the basis of the steady-state density contours (in kg/m*) obtained using a 61 x 61 mesh.
FIGURE 5. Comparison between the proposed method (using arithmetic averaging), and the Yee-Roe scheme (using Roe averaging) for the Mach 2.5 channel flow test case on the basis of the steady-state density contours (in kg/m*) obtained using a 61 x 61 mesh.
Related topics:
Computational Fluid MechanicsAerodynamicsCompressible Fluid DynamicsLES and DNS
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