Page 320 - Computational Fluid Dynamics for Engineers

310 10. Inviscid Compressible Flow

(a) (b)
Fig. 10.11. Wave propagation for the ID Euler equation, (a) supersonic flow u > c, (b)
subsonic flow u < c.

domain, it is necessary to specify three boundary conditions, taken from phys-
ical data such as experimental values. We refer to these as physical boundary
conditions. If the subsonic waves of Fig. 10.11b travel into the computational
domain, it is necessary to specify two physical boundary conditions. The re-
maining boundary condition for the subsonic inflow case must be determined
numerically by extrapolation of the interior flow domain. We call these nu-
merical boundary conditions. Now, if the supersonic or subsonic waves travel
outside the computational domain, it is necessary to specify respectively zero
and one physical boundary conditions. The remaining boundary condition for
the supersonic and subsonic outflow case must be determined numerically by
extrapolation of the interior flow domain. Table 10.1 summarizes the results.
For example, the supersonic flow conditions entering the computational do-
main might be the freest ream Mach number, the total temperature and to-
tal pressure. For supersonic outflow conditions, zero-order extrapolation of the
primitive variables can be specified. For subsonic inflow and outflow conditions,
it is standard procedure to solve the Riemann variables used in the method of
characteristics to determine the boundary conditions:

along u : s = constant
along u + c: u + 2c/(7 - 1) = R\ (10.7.1)

along u — c : u — 2c/(j — 1) = i?2
were s is the entropy, u the flow velocity normal to the surface, c the speed of
sound and 7 the ratio of specific heat. The incoming and outgoing characteristics
are

Table 10.1. Physical and numerical boundary conditions for the ID Euler equations.
M > 1 M < 1

Physical conditions Inflow 3 Inflow 2
Outflow 0 Outflow 1
Numerical conditions Inflow 0 Inflow 1
Outflow 3 Outflow 2