Page 364 - Computational Fluid Dynamics for Engineers

12.2 Compressible Navier-Stokes Equations 355

Fig. 12.1. High-Reynolds number grid inside the viscous region.

on the grid quality and typically leads to an increase in discretized mesh points
which makes meshes of the order of 1 ~ 10 million points common for standard
aircraft configurations.

12.2.2 Boundary Conditions

The compressible Navier-Stokes equations also require boundary conditions.
The system of equations, which contains five variables in three-dimensions (den-
sity, velocity field and temperature) needs five variables to be specified along
the inflow and outflow boundaries (Table 12.1). In practice and as in incom-
pressible flows, the inviscid compressible flow boundary conditions can be used
when the far-field boundaries are placed far enough from the immerse body.
Table 12.1 shows the number of physical and numerical boundary conditions
to be used for each case, following the analysis of [2]. Three-dimensional Euler

Table 12.1. Physical and numerical boundary conditions for the 3D compressible flow-
equations.

Navier-Stokes Euler
M < 1 M > 1 M > 1 M < 1
Physical conditions Inflow 5 5 Inflow 5 4
Outflow 4 4 Outflow 0 1

Numerical conditions Inflow 0 0 Inflow 0 1
Outflow 1 1 Outflow 5 4