298
9. Turbulent Flows
Reynolds number modifications of the k-E model have been proposed; see
Pate1 et al. (1985) and Wilcox (1993) for a review of some of these modifications.
Logarithmic
region
Fig. 9.11. The turbulent boundary layer: velocity profile as a function of distance
normal to the wall (dashed lines are from corresponding equations, solid line represents experimental data)
At high Reynolds number, the viscous sublayer of a boundary layer is so
thin that it is difficult t o use enough grid points to resolve it. This problem
can be avoided by using wall functions, which rely on the existence of a
logarithmic region in the velocity profile; the velocity profile of a turbulent
boundary layer is shown in Fig. 9.11. In the logarithmic layer, the profile is:
where Ut is the mean velocity parallel to the wall u, is the shear velocity
given by u, =
Here, 7, is the shear stress a t the wall, rc is called
the von Karman constant ( K = 0.41), B is an empirical constant related to
the thickness of the viscous sublayer (B z 5.5 in a boundary layer over a
smooth flat plate; for rough walls, smaller values for B are obtained) and n+
is the dimensionless distance from the wall:
It is often assumed that the flow is in local equilibrium, meaning the
production and dissipation of turbulence are nearly equal. If this is the case,
one can show:
u, = cA'~& .
(9.48)
From this equation and Eq. (9.46) we can derive an expression connecting
the velocity a t the first grid point above the wall and the wall shear stress:
9. Turbulent Flows
Reynolds number modifications of the k-E model have been proposed; see
Pate1 et al. (1985) and Wilcox (1993) for a review of some of these modifications.
Logarithmic
region
Fig. 9.11. The turbulent boundary layer: velocity profile as a function of distance
normal to the wall (dashed lines are from corresponding equations, solid line represents experimental data)
At high Reynolds number, the viscous sublayer of a boundary layer is so
thin that it is difficult t o use enough grid points to resolve it. This problem
can be avoided by using wall functions, which rely on the existence of a
logarithmic region in the velocity profile; the velocity profile of a turbulent
boundary layer is shown in Fig. 9.11. In the logarithmic layer, the profile is:
where Ut is the mean velocity parallel to the wall u, is the shear velocity
given by u, =
Here, 7, is the shear stress a t the wall, rc is called
the von Karman constant ( K = 0.41), B is an empirical constant related to
the thickness of the viscous sublayer (B z 5.5 in a boundary layer over a
smooth flat plate; for rough walls, smaller values for B are obtained) and n+
is the dimensionless distance from the wall:
It is often assumed that the flow is in local equilibrium, meaning the
production and dissipation of turbulence are nearly equal. If this is the case,
one can show:
u, = cA'~& .
(9.48)
From this equation and Eq. (9.46) we can derive an expression connecting
the velocity a t the first grid point above the wall and the wall shear stress: