Page 112 - Introduction to Computational Fluid Dynamics

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4.9 OVERALL PROCEDURE
4.8.3 Free-Shear Flows May 25, 2005 11:7 91
In free-shear flows, the mixing length is given by
l m = β (y E − y I ), (4.111)
where the E boundary is free and the I boundary is the symmetry axis. The value of
constant β depends on the type of flow. According to Spalding [78] β = 0.09 for
a plane jet, β = 0.075 for a round jet, and β = 0.16 for a plane wake. In general,
however, β must be regarded as an arbitrary constant whose value is determined
from experiment.
When the e– model is used, Equations 4.105 and 4.106 are directly applicable.
However, because of the absence of a wall, there will be no region where Re t → 0.
√
2
2 2
Also, the wall-correction terms ∂ e/∂y and 2νµ t (∂ u/∂y ) vanish. As such, the
model will reduce to
µ t
= e,
e = µ + , S e = G − ρ , (4.112)
Pr t,e

µ t
= ,
= µ + , S = [C 1 G − C 2 ρ ] , (4.113)
Pr t, e
with C 1 = 1.44, C 2 = 1.92, C µ = 0.09, Pr t,e = 1.0, and Pr t, = 1.3. This set is
called the High Reynolds number (HRE) model.

4.9 Overall Procedure

4.9.1 Calculation Sequence
The previous sections have provided all the essentials to construct the calculation
procedure. This is listed in the following.

Evaluations at x 0
1.Choose x 0 , where the initial profiles (y j ) are specified for j = 1, 2, ..., JN
for the chosen JN.
2.Calculate r j knowing α (x 0 ).
3.Set x u = x 0 and evaluate ω j ( j = 1, 2, ..., JN) from specified u j for a chosen
u
u
u
value of ψ . This sets ψ and hence ψ .
EI
E
I
Begin a New Step
from appropriate
4.Choose x so that x d = x u + x. Calculate ρ j , µ j , and C p j
u
known functions of scalar . Specify or calculate ˙ m I or ˙ m E as described in
j
Section 4.6.
5.Choose relevant and calculate coefficients and source terms in Equation 4.43
using upstream values. Note that if = u, the pressure gradient for internal
and external flows must be appropriately evaluated. Now solve Equation 4.43
using TDMA.