Turbulence scaling in the wall region + damping functions
The wall region of a (2D, incompressible) turbulent boundary layer displays a universal structure, which is known as the law of the wall. According to this concept, the properties of the flow can be expressed uniquely in terms of the so–called wall scaling (also referred to as the use of ‘wall units’). In addition, it is commonly assumed that across the wall region the total shear stress is approximately constant: constantvisc turb w
One element of this law of the wall is that the velocity profile of the mean flow satisfies an expression
of the following type:
( ) u f y; where: *; with: * /* w
u yvu y vv
a. Derive from this wall scaling, by means of an elementary dimensional analysis, the scaling of
the following (kinematic) properties of the turbulence:
1. the (specific) turbulent shear stress: / ‘ ‘turb u v
2. the eddy viscosity: t
3. the mixing–length: mixl
In the scaling scale only the kinematic properties that are relevant for the wall region (*v and ) are used; in other words: determine for each of the above variables how they are to be non–dimensionalised, such that: dimensionless property = function of y+.
b. Show for each of these properties how the corresponding “function of y+” is related to the gradient of the velocity profile of the mean flow, that is given by: ‘( ) df duf y dy dy
c. Consider the behaviour of the expressions found in (b), for the limit y+>>1, which constitutes the overlap region, in which the logarithmic velocity profile holds. Use the results to show that in this overlap region the original (i.e. non–scaled) kinematic turbulence properties mentioned in (a), are independent of viscosity.
The effect of the viscosity on the turbulence in the wall region is often described by using so–called ‘damping functions’, which are defined as: ‘effective value’ = damping function * ‘fully turbulent value’ where the ‘fully turbulent value’ corresponds to the behaviour of a variable in the overlap region, where the effect of the viscosity can be neglected.
In correspondence to what has been derived in (b), the damping functions for the eddy–viscosity and for the mixing–length can be derived from the shape of the velocity profile.
d. An accurate description of the velocity profile in the entire wall layer, including the viscous region, is given by Spalding’s implicit expression, which reads: 2 3 ( ) ( )1 2 6 B u u u y u e e u
Show that this expression satisfies the two familiar limits of the law of the wall:
y+<<1: u y (viscous sublayer)
y+>>1: 1 ln( )u y B
(overlap region)
e. – Derive the expressions for both damping functions (i.e., for the eddy viscosity and for the mixing length), as they follow from Spalding’s law of the wall.
– Investigate the behaviour of both damping functions for small values of y+ (i.e. determine the first term of the series expansion for small y+).
– Plot both damping functions for the interval 0 y+ 100. In the calculations use for the constants in the law of the wall the standard values κ = 0.41 and B = 5.0.
Hint: maintain in the derivations u+ as the independent variable, and make use of the relation that: 1/ /du dy dy du
