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Assessment of positivity- preserving capability of various schemes when solving the 1D test cases.*

Table 2 Assessment of positivity- preserving capability of various schemes when solving the 1D test cases.*

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List of 1D test cases and initial conditions.* @ Tn all cases, the computational domain is located within —0.5m < x < 0.5m. TABLE 1.
List of 1D test cases and initial conditions.* @ Tn all cases, the computational domain is located within —0.5m < x < 0.5m. TABLE 1.
FIGURE 1. Comparison between the proposed schemes and the POSFL schemes [11] on the basis of density profiles for test case #1. The density profiles are obtained using a 120-node grid at a time of 0.8 ms using a time step size such that the maximum Courant number within the domain is 0.1.
FIGURE 1. Comparison between the proposed schemes and the POSFL schemes [11] on the basis of density profiles for test case #1. The density profiles are obtained using a 120-node grid at a time of 0.8 ms using a time step size such that the maximum Courant number within the domain is 0.1.
FIGURE 2. Comparison between the proposed schemes and the conventional flux-limited methods on the basis of density profiles for test case #1. The density profiles are obtained at t = 0.8 ms using a time step size such that the maximum Courant number within the domain is 0.1.
FIGURE 2. Comparison between the proposed schemes and the conventional flux-limited methods on the basis of density profiles for test case #1. The density profiles are obtained at t = 0.8 ms using a time step size such that the maximum Courant number within the domain is 0.1.
Assessment of relative error of various schemes when solving some 1D test cases.*?-“4 TABLE 3.
Assessment of relative error of various schemes when solving some 1D test cases.*?-“4 TABLE 3.
FIGURE 3. Comparison between the first-order Steger-Warming method and the proposed method on the basis of pressure contours (in Pascals) for test case #8 after 100 iterations; the grid spacing is fixed to (1/50) m along both dimensions; the CFL number is fixed to 0.25.
FIGURE 3. Comparison between the first-order Steger-Warming method and the proposed method on the basis of pressure contours (in Pascals) for test case #8 after 100 iterations; the grid spacing is fixed to (1/50) m along both dimensions; the CFL number is fixed to 0.25.
@ The time step is set locally (resulting in a different value for each node) according to Eq. (62). Assessment of positivity-preserving capability of various schemes when solving the 2D and 3D test cases. TABLE 5.
@ The time step is set locally (resulting in a different value for each node) according to Eq. (62). Assessment of positivity-preserving capability of various schemes when solving the 2D and 3D test cases. TABLE 5.
FIGURE 4. Comparison between the first-order Steger-Warming method and the proposed method on the basis of pressure contours (in Pascals) for test case #9 after 300 iterations; the grid spacing is fixed to (1/300) m along both dimensions; the CFL number is fixed to 0.25. Interestingly, the Courant number needs to be lowered as the number of dimensions is increased in order to guarantee positivity-preservation. It is recalled that the time step needs to be set to less than half of the CFL condition when solving one-dimensional problems (see proof in Section 4.2). However, when solving multidimensional problems, setting the time step in this manner would sometimes result in negative densities or internal energies. Through trial and error, itis found that the CFL number must be set to no more than 1/4 and 1/6 in 2D and 3D respectively to guarantee positivity- preservation (see results in Table 5). Such a necessary reduction of the time step for higher number of dimensions is not specific to the proposed stencils. In fact, a similar trend can be observed for the first-order Steger-Warming method: as the number of dimensions is increased, the CFL number needs to be lowered to ensure that the determinative properties remain positive.
FIGURE 4. Comparison between the first-order Steger-Warming method and the proposed method on the basis of pressure contours (in Pascals) for test case #9 after 300 iterations; the grid spacing is fixed to (1/300) m along both dimensions; the CFL number is fixed to 0.25. Interestingly, the Courant number needs to be lowered as the number of dimensions is increased in order to guarantee positivity-preservation. It is recalled that the time step needs to be set to less than half of the CFL condition when solving one-dimensional problems (see proof in Section 4.2). However, when solving multidimensional problems, setting the time step in this manner would sometimes result in negative densities or internal energies. Through trial and error, itis found that the CFL number must be set to no more than 1/4 and 1/6 in 2D and 3D respectively to guarantee positivity- preservation (see results in Table 5). Such a necessary reduction of the time step for higher number of dimensions is not specific to the proposed stencils. In fact, a similar trend can be observed for the first-order Steger-Warming method: as the number of dimensions is increased, the CFL number needs to be lowered to ensure that the determinative properties remain positive.
FIGURE 5. Comparison between the proposed method, the conventional TVD scheme, and the first-order Steger- Warming scheme on th basis of steady-state pressure contours (in Pascals) for test case #11. Because the schemes ou method results in a much im lined herein achieve positivity-preservation by limiting the second-order terms more substantially than the standard TVD stencils, it may be argued that positivity- preservation is achieved at the expense of resolution. But such 1 loss in resolution is verified not to be substantial. Indeed, as would be expected from a second-order stencil, the proposec proved reso Steger- Warming method (see Figs. 3 and schemes to those obtained with the conventional flux-limited methods (that is, when the conventional methods did not yield negative internal energy or d that the extra amount of dissipation nec ensity ution of shockwaves and especially of expansion fans when compared to the first-ordet 4). Further, whenever it was possible to compare the results obtained with the proposed , negligible differences were observed. This is well illustrated in Fig. 5, where steady-state results of a Mach 3 flow in a wavy-wal pressure contours obtained with both approaches. Although not shown here, results obtained for various other test cases confirm when the conventional TVD schemes in channel are depicted. Clearly, there is little to no discernible difference between the essary for positivity-preservation is typically negligible, and only becomes significant troduce negative internal energies or densities.
FIGURE 5. Comparison between the proposed method, the conventional TVD scheme, and the first-order Steger- Warming scheme on th basis of steady-state pressure contours (in Pascals) for test case #11. Because the schemes ou method results in a much im lined herein achieve positivity-preservation by limiting the second-order terms more substantially than the standard TVD stencils, it may be argued that positivity- preservation is achieved at the expense of resolution. But such 1 loss in resolution is verified not to be substantial. Indeed, as would be expected from a second-order stencil, the proposec proved reso Steger- Warming method (see Figs. 3 and schemes to those obtained with the conventional flux-limited methods (that is, when the conventional methods did not yield negative internal energy or d that the extra amount of dissipation nec ensity ution of shockwaves and especially of expansion fans when compared to the first-ordet 4). Further, whenever it was possible to compare the results obtained with the proposed , negligible differences were observed. This is well illustrated in Fig. 5, where steady-state results of a Mach 3 flow in a wavy-wal pressure contours obtained with both approaches. Although not shown here, results obtained for various other test cases confirm when the conventional TVD schemes in channel are depicted. Clearly, there is little to no discernible difference between the essary for positivity-preservation is typically negligible, and only becomes significant troduce negative internal energies or densities.
In the latter, d is the number of dimensions, U is the vector of conserved variables, X is a generalized coordinate such that the spacing between adjacent grid points is 1, A; is the convective flux Jacobian (i.e. A; = dF;/dU), A; is the eigenvalue matrix, and L-' is the right eigenvector matrix.
In the latter, d is the number of dimensions, U is the vector of conserved variables, X is a generalized coordinate such that the spacing between adjacent grid points is 1, A; is the convective flux Jacobian (i.e. A; = dF;/dU), A; is the eigenvalue matrix, and L-' is the right eigenvector matrix.
A.2. Multidimensional Euler Equations in Generalized C urvilinear C oordinates The two-dimensional and three- dimensional Euler equations can be written in generalized curvilinear coordinates as follows:
A.2. Multidimensional Euler Equations in Generalized C urvilinear C oordinates The two-dimensional and three- dimensional Euler equations can be written in generalized curvilinear coordinates as follows:
In the latter, the flow speed, the contravariant velocity, the inverse of the metric Jacobian, the spatial derivative of the generalized coordinate, and the normalized spatial derivative of the generalized coordinate correspond to:
In the latter, the flow speed, the contravariant velocity, the inverse of the metric Jacobian, the spatial derivative of the generalized coordinate, and the normalized spatial derivative of the generalized coordinate correspond to:
where 6;; is the Kronecker delta and where a cyclic permutation of the indices is implied (with the minimum and maximum being 1 and d respectively).
where 6;; is the Kronecker delta and where a cyclic permutation of the indices is implied (with the minimum and maximum being 1 and d respectively).
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Computational Fluid MechanicsAerodynamicsCompressible Fluid Dynamics
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