Notes on Computational Fluid Dynamics: General Principles

[chapter contents][previous topic][next topic][index]

7.6 Alternative wall functions

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

! + + 1- + X3 (--u+)n- y = u + E exp( u ) n! ; n=0 \relax \special {t4ht=

(7.22)

where $\text{[math]}$ . The equation combines Eq. (7.11 ) and Eq. (7.13 ), “disabling” Eq. (7.13 ) at low $\text{[math]}$ by subtracting low order terms from a polynomial expansion of $\text{[math]}$ .

$PICT\relax \special {t4ht=$

The wall function is applied by solving Eq. (7.22 ) for $\text{[math]}$ from $\text{[math]}$ calculated from Eq. (7.9) using the near-wall cell centre height $\text{[math]}$ . The friction velocity is calculated by $\text{[math]}$ , where $\text{[math]}$ is the near-wall cell velocity. Finally, $\text{[math]}$ on the wall patch is calculated from a numerical interpretation of Eq. (7.20 ),

t = u2 =jrnuj ; \relax \special {t4ht=

(7.23)

where $\text{[math]}$ is the surface-normal velocity gradient. An iterative method is required to invert Eq. (7.22) and to accommodate other nonlinearities, e.g. $\text{[math]}$ is itself a function of $\text{[math]}$ .

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” $\text{[math]}$ ¹⁰ becomes larger than the thickness of the viscous sub-layer.

At higher surface roughness, turbulent eddies are generated near the wall at a scale $\text{[math]}$ , rather than $\text{[math]}$ . The viscous eﬀects become negligible, causing the non-dimensionalised distance to become $\text{[math]}$ 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=

(7.24)

where $\text{[math]}$ a roughness function, dependent on $\text{[math]}$ . An intuitive model for $\text{[math]}$ is¹¹

+ -1 + u = ln(1+ r ): \relax \special {t4ht=

(7.25)

This rough wall function Eq. (7.24 ) reduces to Eq. (7.13 ) using the conventional $\text{[math]}$ deﬁnition of Eq. (7.9 ) at $\text{[math]}$ . As $\text{[math]}$ , it reduces to Eq. (7.13 ) using $\text{[math]}$ .

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

⁹Brian Spalding, A single formula for the law of the wall, 1961.

¹⁰‘ $\text{[math]}$ ’ is often used to denote roughness, but we use ‘ $\text{[math]}$ ’ 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