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7.9.3 Calculation Procedure at Pressure Far-Field Boundaries

The pressure far-field boundary condition is a non-reflecting boundary condition based on the introduction of Riemann invariants (i.e., characteristic variables) for a one-dimensional flow normal to the boundary. For flow that is subsonic there are two Riemann invariants, corresponding to incoming and outgoing waves:

 R_\infty = v_{n_\infty} - \frac{2c_\infty}{\gamma - 1} (7.9-1)

 R_i = v_{n_i} + \frac{2c_i}{\gamma - 1} (7.9-2)

where $v_n$ is the velocity magnitude normal to the boundary, $c$ is the local speed of sound and $\gamma$ is the ratio of specific heats (ideal gas). The subscript $\infty$ refers to conditions being applied at infinity (the boundary conditions), and the subscript $i$ refers to conditions in the interior of the domain (i.e., in the cell adjacent to the boundary face). These two invariants can be added and subtracted to give the following two equations:

 v_n = \frac{1}{2}(R_i + R_\infty) (7.9-3)

 c = \frac{\gamma - 1}{4}(R_i - R_\infty) (7.9-4)

where $v_n$ and $c$ become the values of normal velocity and sound speed applied on the boundary. At a face through which flow exits, the tangential velocity components and entropy are extrapolated from the interior; at an inflow face, these are specified as having free-stream values. Using the values for $v_n$, $c$, tangential velocity components, and entropy the values of density, velocity, temperature, and pressure at the boundary face can be calculated.

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