Page 97 - Computational Fluid Dynamics for Engineers

3.2 Zero-Equation Models 83

3.2 Zero-Equation Models

The zero-equation, often referred to as algebraic eddy viscosity and/or mixing
length models, are used to model the Reynolds shear stress term in the momen-
tum equations. For a two-dimensional incompressible laminar and turbulent
flow, the continuity and momentum equations (2.4.33) and (2.4.34), respec-
tively, are solved subject to given initial conditions and boundary conditions
with — u'v' represented by Eq. (3.1.2). For a limited range of flows, it is possi-
ble to specify the turbulent viscosity of Eq. (3.1.2) as the mixing length of Eq.
(3.1.1) in closed algebraic expressions. Examples include:
£/6 = 0.09 for a plane jet
= 0.075 for a round jet
= 0.16 for a plane wake
= 0Aly/6 for the near wall region of a boundary layer
= 0.09 for the outer region of a boundary layer
The restricted range of the correlations should be noted, together with the
simplicity of the approach. No differential equations are solved for turbulence
quantities, and turbulence transport is, consequently, not represented.

3.2.1 Cebeci-Smith Model

For external flows, a popular zero-equation model is the Cebeci-Smith (CS)
model. This model assumes that a turbulent boundary layer can be represented
in terms of eddy viscosities for inner and outer layers. The corresponding func-
tions are empirical and based on limited ranges of experimental data; the range
of data is, however, extensive so that the full algebraic formulation is the most
general available at present.
In the inner and outer regions of a boundary layer on a smooth surface, with
or without mass transfer, the eddy viscosity s m is written as [2,3]
Inner region: 0 < y < y c

du
\£m)i — ' 7tr (3.2.1)
dy
Here the mixing length / is given by

l = K,y\l- exp - - | ) 1 (3.2.2a)
(
where K — 0.40 and A is a damping-length constant, which may be represented
by
1
A = 2 6 ^ " \ JV = (l-11.8p+) /2 P + = ^ ^ r , (3.2.2b)
Outer region: y c < y < 6