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342 11. Incompressible Navier-Stokes Equations

11.6 Model Problem: Laminar and Turbulent Flat Plate Flow

This section discusses the application of the INS2D code described in the pre-
vious section to a laminar and turbulent flow over a flat plate. This is a useful
test case to validate a CFD code since analytical solutions for a laminar flow
and experimental results for a turbulent flow are available.
Of the three modifications to the INS2D code of the previous section, the
first one is the modification of the grid in the ^/-direction. As discussed in sub-
section 7.4.2, for laminar flows, it is often sufficient to use a uniform grid in the
y-direction. For turbulent flows, however, a uniform grid is not satisfactory be-
cause the boundary layer thickness and dimensionless wall shear are much larger
in turbulent flows than laminar flows. Since short steps in y must be taken to
maintain computational accuracy when the wall shear is large, the steps near
the wall in a turbulent flow must be shorter than the corresponding steps in a
laminar flow under similar conditions. A convenient grid for this purpose is the
grid given by Eq. (7.4.2) in the real coordinate system. The y-convective flux is
then modified as

dy A Vj

where
(Vj+i-Vj-i)
A
2
varies from cell to cell. Similarly, the y-viscous flux is modified as

_ (Fy)ij+i/2 - (Fy)ij-1/2
dF v
(11.6.2)
dy Ayj
Calculations for turbulent flow with the first-order upwind scheme used in
the INS2D showed the need for higher accuracy. As a result it was replaced
by the third-order upwind scheme discussed in Section 5.5. This second mod-
ification is straightforward, but care must be taken at the boundary of the
computational domain as the computational stencil is larger than in the first-
order upwind method. In our case, zero-order extrapolation of the dissipation
is used at the boundaries.
A third modification is the introduction of a turbulence model to compute
turbulent flows. This is achieved by using the Cebeci-Smith turbulence model
described in subsection 3.2.1. The resulting subroutine is named CS.TURB and
is called before the fluxes are evaluated.
Once these modifications are made to the solution algorithm, it is necessary
to select appropriate boundary conditions consistent with the computational
domain. They were chosen as follows: