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

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

(7.23) 
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” ^{10} becomes larger than the thickness of the viscous sublayer.
At higher surface roughness, turbulent eddies are generated near the wall at a scale , rather than . The viscous eﬀects become negligible, causing the nondimensionalised distance to become in the log law Eq. (7.13 ). To reﬂect this, Eq. (7.13 ) is modiﬁed to a form

(7.24) 

(7.25) 
It is open to interpretation how to determine from roughness measurements of a surface. The parameter is sometimes split into , where is a measured sand grain roughness height and is a coeﬃcient that depends on the shape, consistency and packing of the roughness elements. Using that approach, values of often yield a good match between Eq. (7.24 ) and measured data.