Interdisciplinary Applied Mathematics

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Cs = 1 4 1 (R+R~ + s+— S~) — cup; s = sup,

where the subscript “up” denotes the upstream values (i.e., values from the element that the flow is leaving). Assuming ideal gas and locally isentropic flow, the state of the gas is fixed by solving the local pressure and density using speed of sound and the entropy

p    2    ,

= constant; cs = jp/p,


and thus the conservative variables and local temperature can be calculated.

Next, we discuss inflow and outflow boundary conditions. The Euler equations require specification of three inflow and one outflow boundary condition for subsonic flow. However, these boundary conditions are not known ahead of time. Therefore, we select predicted inlet and exit stages, and perform again a characteristic decomposition. The only difference is that we select the left state as the specified inlet state and the right state as the calculated values at the inlet. Then, we perform the characteristic decomposition. Similarly at exit, the predicted state is specified as the right state, and the calculated state is chosen as the left state.

The wall boundary conditions must be designed to reflect incident pressure waves with high-order accuracy, resulting in minimal numerical entropy creation near the boundaries. One implementation is to impose zero

FIGURE 14.3. The domain of influence at a corner of four elements, and corresponding Riemann invariants.

normal velocity at the surface by modifying the local pressure to account for the changes in energy while the density remains unchanged. For a ho-mentropic (i.e., constant entropy in the entire flow domain) inviscid flow, simulation with p,Flow resulted in change in the entropy near the wall by « 10~6.    This is an indication    of    the    very    low    values    of    numerical    diffu

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