Partial Differential Equations - Modelling and ... - ResearchGate
Partial Differential Equations - Modelling and ... - ResearchGate Partial Differential Equations - Modelling and ... - ResearchGate
The von Neumann Triple Point Paradox 125 2.5 2 1.5 1 0.5 0 −2 −1 0 1 2 Fig. 9. The geometry of the M =1.04/11.5 ◦ Euler example. The insert indicates the region where extreme local grid refinement is performed. to exactly agree with this shock located at x =1.04. The lower boundary condition mimics symmetry about the x-axis for x0. The grid geometry can be seen in Figure 9. This problem is well outside the range where regular reflection solutions are possible. Refer again to the figure to see that its numerical solution (under the insert) clearly resembles single Mach reflection. However, Mach reflection (where three plane shocks meet at a point) is also not possible for a shock this weak [Hen87]. This example demonstrates a classic von Neumann triple point paradox. This problem is solved in self-similar coordinates by essentially the same high order Roe method discussed in the previous section. However, we simplify the Roe approach by again evaluating the Roe matrix at the midpoint, which for the Euler equations is only an approximation to the Roe average. Also, to avoid spurious expansion shocks, artificial dissipation on the order of O(|U r − U l |) is appended to the diagonal part of the Roe dissipation matrix in a field by field manner. We locally refine a very small neighborhood around the apparent triple point as done earlier. The full finest grid has eleven million grid points with 800 × 2000 = 1.6 × 10 6 (∆x ≈ 5 × 10 −7 ) devoted to the local refinement. We plot the sonic number M which is defined as follows. The eigenvalue corresponding to a fast shock in unit direction n for the self-similar Euler flux Jacobian is λ =(u − ξ,v − η) · n + c where ξ = x/t and η = y/t. Define r 2 = ξ 2 + η 2 and set n = (ξ,η)/r, u n =(u, v) · n to find ( un − r λ = c c ) +1 = c(1 −M) where M = r − u n . c
126 R. Sanders and A.M. Tesdall 0.3045 0.3045 0.304 0.304 0.3035 0.3035 0.303 0.303 0.3025 1.0385 1.039 1.0395 1.04 1.0405 (a) 0.3025 1.0385 1.039 1.0395 1.04 1.0405 (b) Fig. 10. A closeup of the Euler triple point. The sonic number M on the left and density ρ on the right. The dotted line on the left delineates the supersonic patch within the subsonic zone behind the Mach stem. 1.015 1.01 1.005 1 0.995 0.99 0 0.2 0.4 0.6 0.8 1 (a) 1.02 1.015 1.01 1.005 1 0.995 0.99 0.985 0 0.2 0.4 0.6 0.8 1 (b) Fig. 11. Vertical cross sections of M taken bottom-up slightly to the left of the Mach stem. On the left M =1.04/11.5 ◦ . The reflected shock is the large jump. Note the crossings at M = 1. On the right, a second example problem with a slightly stronger incident shock M =1.075/15.0 ◦ . The evidence of a sequence of shock/expansion wave pairs is stronger for this second example. When M < 1, the flow is called subsonic. When M > 1, the flow is called supersonic. In this sense, when crossing through a self-similar stationary shock, the fact that M crosses from subsonic to supersonic is nothing more than the entropy condition λ l >s>λ r . Figure 10 gives a sonic number contour plot (a) and density contours (b) in the triple point neighborhood. Clearly the evidence for Guderley Mach reflection in this example is not nearly as compelling as found for our earlier examples. However, these shocks are extremely weak. In recent work for a γ =7/5 gas, we slightly strengthened the incident Mach number, M =1.075, and obtained far more conclusive results. See the sonic number cross sections depicted in Figure 11.
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126 R. S<strong>and</strong>ers <strong>and</strong> A.M. Tesdall<br />
0.3045<br />
0.3045<br />
0.304<br />
0.304<br />
0.3035<br />
0.3035<br />
0.303<br />
0.303<br />
0.3025<br />
1.0385 1.039 1.0395 1.04<br />
1.0405<br />
(a)<br />
0.3025<br />
1.0385 1.039 1.0395 1.04<br />
1.0405<br />
(b)<br />
Fig. 10. A closeup of the Euler triple point. The sonic number M on the left <strong>and</strong><br />
density ρ on the right. The dotted line on the left delineates the supersonic patch<br />
within the subsonic zone behind the Mach stem.<br />
1.015<br />
1.01<br />
1.005<br />
1<br />
0.995<br />
0.99<br />
0 0.2 0.4 0.6 0.8 1<br />
(a)<br />
1.02<br />
1.015<br />
1.01<br />
1.005<br />
1<br />
0.995<br />
0.99<br />
0.985<br />
0 0.2 0.4 0.6 0.8 1<br />
(b)<br />
Fig. 11. Vertical cross sections of M taken bottom-up slightly to the left of the<br />
Mach stem. On the left M =1.04/11.5 ◦ . The reflected shock is the large jump.<br />
Note the crossings at M = 1. On the right, a second example problem with a<br />
slightly stronger incident shock M =1.075/15.0 ◦ . The evidence of a sequence of<br />
shock/expansion wave pairs is stronger for this second example.<br />
When M < 1, the flow is called subsonic. When M > 1, the flow is called supersonic.<br />
In this sense, when crossing through a self-similar stationary shock,<br />
the fact that M crosses from subsonic to supersonic is nothing more than the<br />
entropy condition λ l >s>λ r .<br />
Figure 10 gives a sonic number contour plot (a) <strong>and</strong> density contours (b)<br />
in the triple point neighborhood. Clearly the evidence for Guderley Mach<br />
reflection in this example is not nearly as compelling as found for our earlier<br />
examples. However, these shocks are extremely weak. In recent work for a<br />
γ =7/5 gas, we slightly strengthened the incident Mach number, M =1.075,<br />
<strong>and</strong> obtained far more conclusive results. See the sonic number cross sections<br />
depicted in Figure 11.
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