Page 362 - Computational Fluid Dynamics for Engineers

Compressible

«fctf Navier-Stokes Equations

12.1 Introduction

The compressible Navier-Stokes equations represent the highest level of math-
ematical modeling for describing aerodynamic flows commonly used in engi-
neering applications. These include: the aerodynamic flow around aircraft, the
internal flow occurring in turbomachinery; the flow developing in ducts in the
presence of heat transfer; or a combination of these such as the external aero-
dynamic flow impinging on a wing leading edge surface heated by internal hot
air emanating from the engine compressor stages to avoid ice accumulation.
The methods developed by the research community for the solution of these
equations for specific applications have been building on the advances made on
solving the simplified forms of the equations, since the compressible Navier-
Stokes equations do not introduce fundamentally different flow characteristics.
Analysis of these equations shows that their types are a mix of all types studied
in the previous chapters. Since the viscous terms transform the inviscid first-
order partial differential equations into second-order, the momentum and energy
equations are parabolic in time and space, but elliptic in space when steady
state conditions are reached. The continuity equation is, however, hyperbolic
in space and time. Therefore, since the methods for addressing each of these
issues have been discussed in the previous chapters, emphasis is placed on the
specific difficulties associated with the engineering applications of the Reynolds-
Averaged compressible Navier-Stokes equations in this chapter (Section 12.2)
and their methods of solution. In particular, the MacCormack scheme adapted
for viscous flows is discussed in Section 12.3, the Beam-Warming method in
Section 12.4 and the finite-volume method in Section 12.5. Application on the
model problem of Section 11.5, that is the sudden expansion laminar duct flow,
is discussed in Section 12.6 along with the detailed description of the computer
programs given in Appendix B.