7.6 Alternative wall functions

The standard wall function described in Sec. 7.5 uses a function for eqn that is discontinuous at eqn, switching to eqn for eqn. A continuous wall function is available which evaluates eqn as eqn from a single equation describing the universal character of the velocity profile at high eqn,9

 ! + + 1- + X3 (--u+)n- y = u + E exp( u ) n! ; n=0 \relax \special {t4ht=
where eqn. The equation combines Eq. (7.11 ) and Eq. (7.13 ), “disabling” Eq. (7.13 ) at low eqn by subtracting low order terms from a polynomial expansion of eqn.

PICT\relax \special {t4ht=

The wall function is applied by solving Eq. (7.22 ) for eqn from eqn calculated from Eq. (7.9) using the near-wall cell centre height eqn. The friction velocity is calculated by eqn, where eqn is the near-wall cell velocity. Finally, eqn on the wall patch is calculated from a numerical interpretation of Eq. (7.20 ),

t = u2 =jrnuj ; \relax \special {t4ht=
where eqn is the surface-normal velocity gradient. An iterative method is required to invert Eq. (7.22) and to accommodate other nonlinearities, e.g. eqn is itself a function of eqn.

Rough wall function

The standard wall function in Sec. 7.5 is applicable to smooth walls so does not account for surface roughness. Roughness is significant when the roughness “scale” eqn10 becomes larger than the thickness of the viscous sub-layer.

At higher surface roughness, turbulent eddies are generated near the wall at a scale eqn, rather than eqn. The viscous effects become negligible, causing the non-dimensionalised distance to become eqn in the log law Eq. (7.13 ). To reflect this, Eq. (7.13 ) is modified to a form

 + 1 + + u = --ln(y )+ B u ; \relax \special {t4ht=
where eqn a roughness function, dependent on eqn. An intuitive model for eqn is11
+ -1 + u = ln(1+ r ): \relax \special {t4ht=
This rough wall function Eq. (7.24 ) reduces to Eq. (7.13 ) using the conventional eqn definition of Eq. (7.9 ) at eqn. As eqn, it reduces to Eq. (7.13 ) using eqn.

It is open to interpretation how to determine eqn from roughness measurements of a surface. The parameter is sometimes split into eqn, where eqn is a measured sand grain roughness height and eqn is a coefficient that depends on the shape, consistency and packing of the roughness elements. Using that approach, values of eqn often yield a good match between Eq. (7.24 ) and measured data.

9Brian Spalding, A single formula for the law of the wall, 1961.
10eqn’ is often used to denote roughness, but we use ‘eqn’ to avoid confusion with turbulent kinetic energy.
11Cyril Colebrook, Turbulent flow in pipes, with particular reference to the transitional region between smooth and rough wall laws, 1939.

Notes on CFD: General Principles - 7.6 Alternative wall functions