Interdisciplinary Applied Mathematics

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For steady internal flows, specifying the back-pressure at the exit of the domains constitutes a difficulty, since the characteristic decomposition method is based on p, (pu), (pv), and E. In the implementation of backpressure we have used the calculated values of velocity and temperature at the exit of the channel. Then, the back-pressure is imposed indirectly by calculating the density corresponding to the calculated temperature and the desired back-pressure. This implementation results in uniform backpressure with good accuracy and eliminates numerical boundary layers at the channel exit.


Implementation of Slip Boundary Conditions


The numerical implementation of equations (2.19) and (2.20b) is somewhat complicated due to the mixed-type (Robin) boundary conditions. An explicit (in time) implementation of equation (2.19) at time level (n + 1)At (neglecting for simplicity the temperature term) is as follows:










where ai denotes the weights necessary to obtain the time-accuracy O(AtJ) with At the time step. However, explicit treatment of boundary conditions is an extrapolation process, and thus it is numerically unstable, e.g., for relatively high values of Knudsen number.


We have determined through numerical experimentation that the overall Navier-Stokes solution with explicitly implemented velocity slip boundary conditions becomes unstable when the calculated slip amount (Us — Uw) at a certain time step is sufficiently large to cause a sudden change of the sign of wall vorticity in the next time step. This empirical finding can be readily justified by considering the following argument. For a linear Couette flow (see Chapter 3)    with    driving velocity    U0    and    local    gas    velocity    U1    at a

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