Page 363 - Computational Fluid Dynamics for Engineers

354 12. Compressible Navier-Stokes Equations

12.2 Compressible Navier-Stokes Equations

The compressible Navier-Stokes equations, which are in fact five scalar conser-
vation equations, Eqs. (2.2.17)-(2.2.22) contain five unknowns: the density, the
three components of velocity and the energy. Their solution can be obtained by
Direct Numerical Simulation (DNS) which requires that all the physical scales
imbedded within the flow under study are captured. Since the smallest scale is
the Kolmogorov scale, and that it scales with the Reynolds number of the flow,
the grid requirements to capture these scales become so large that the time to
obtain converged flow solutions becomes impractical with the current available
computer power. One way to reduce the computing requirements is to model the
smallest scales and to compute only the larger ones, giving rise to Large Eddy
Simulation (LES). Yet the required computing power remains large and most
computational techniques use the Reynolds-Averaged form of the compressible
Navier-Stokes equations (RANS) with turbulence models (Chapter 3) to rep-
resent the Reynolds stress and turbulent heat flux terms with their solutions
depending on the accuracy of the models.

12.2.1 Practical Difficulties

Apart from the accuracy of turbulence models, the solution of the RANS equa-
tions presents several difficulties. One is that the turbulence model introduces
additional equations whose solutions can be rather involved if turbulence models
based on transport equations are used rather than the zero-equation models.
Another problem arises for high-Reynolds number flows. Since compressible
flows are often associated with high-speed flows, the boundary-layer region is
confined to a very small distance away from the solid surface. In order to accu-
rately compute the strong gradients present inside the boundary layer, several
grid points need to be placed inside it. Typically, around 20-30 grid points are
needed, and the first one must be within y + < 1 to capture the laminar sublayer
region (y + being the normal wall distance in non-dimensional boundary-layer
coordinates defined in Section 3.5). If a formula for local skin-friction coefficient
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Cf for a flat plate [1] is used for a Reynolds number of 10 , this requirement
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translates into the first point being located at a distance of 10~ c away from
the wall, c being the associated length scale of the surface (Fig. 12.1). Not only
is this distance smaller than typical CAD system tolerances, which means that
CAD surfaces must be "repaired" to mathematically close them, but it leads
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to high-aspect ratio cells (up to 10 ) within the boundary-layer region. This
introduces additional stiffness in the numerical algorithm, since we are now in
presence of waves travelling at dissimilar speeds in the streamwise and normal
flow directions. In addition, since most codes have formal second-order accurate
discretization on smooth grids, grid stretching and skewness can produce high
truncation errors leading to excessive dissipation. This puts severe restrictions