Zero-equation methods

Computational methods that are based on Eqns (7.144) and (7.145) plus semi­empirical formulae for eddy viscosity are often known as zero-equation methods. This terminology reflects the fact that no additional partial differential equations, derived from the Navier-Stokes equations, have been used. Here the Cebeci-Smith method[52] will be described. It was one of the most successful zero-equation methods developed in the 1970s.

Most of the zero-equation models are based on extensions of Prandtl’s mixing – length concept (see Sections 7.10.4 and 7.10.5), namely:

The constant к is often known as the von Karman constant.

Three key modifications were introduced in the mid 1950s:

(1) Damping near the wall-. Van Driest*

An exponential damping function was proposed that comes into play as у —► 0. This reflects the reduction in turbulence level as the wall is approached and extends the mixing-length model into the buffer layer and viscous sub-layer:

іш = ку[ 1 – exp(-у+Мо)] Ao = 26

(2) Outer wake-like flow-. Clauser*

It was recognized that the outer part of a boundary layer is like a free shear layer (specifically, like a wake flow), so there the Prandtl-Gortler eddy-viscosity model, see Eqns (7.110) and (7.111), is more appropriate:

є — ax Ue6*

const.

where Ue is the flow speed at the edge of the boundary layer and 6* is the boundary – layer displacement thickness.

(3) Intermittency-. Corrsin and Kistler, and Klebannoff5

It was recognized that the outer part of the boundary layer is only intermittently turbulent (see Section 7.10.7 and Fig. 7.40). To allow for this it was proposed that єт be multiplied by the following semi-empirical intermittency factor:

Cebeci-Smith method

The Cebeci-Smith method incorporates versions of these three key modifications. For the inner region of the turbulent boundary layer:

Term (i) is a semi-empirical modification of Van Driest’s damping model that takes into account the effects of the streamwise pressure gradient; к — 0.4 and Damping Length:

л 26i/

~ V^- 11.8(i/17e/F3)dl7e/dx)

For the outer region of the turbulent boundary layer: (єт)0 = aUe6*jtr yc< у <6

where a = 0.0168 when Reg > 5000. yc is determined by requiring

(єт)і = (єт)о at y = yc

See Cebeci and Bradshaw (1977) for further details of the Cebeci-Smith method.

It does a reasonably good job in calculating conventional turbulent boundary- layer flows. For applications involving separated flows, it is less successful and one – equation methods like that due to Baldwin and Lomax* are preferred.

For further details on the Baldwin-Lomax and other one-equation methods, including computer codes, Wilcox (1993) and other specialist texts should be con­sulted.