Page 109 - Introduction to Computational Fluid Dynamics

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2D BOUNDARY LAYERS
are many variants, all turbulence models of this type stem from a dimensionally
correct representation
µ t ∝ ρ l v , (4.90)

where v is the representative velocity fluctuation scale in the transverse direction
y and l is a representative length scale. Two turbulence models used extensively for
boundary layer calculations are described in the following.
4.8.1 Mixing Length Model

Since v is responsible for transverse momentum transfer, it may be written in
dimensionally correct form as

∂u

v = l m (4.91)
∂y
so that

∂u
µ t = ρ l 2 , (4.92)
∂y
m
where l m is called Prandtl’s mixing length. Now, because the velocity gradient can
be evaluated from the solution of the momentum equation, l m must be prescribed to
complete evaluation of µ t . Kays and Crawford [33], after extensive investigations
of a variety of wall-boundary-layer flows have prescribed the following formulas:

y y
⎧ +
κ y 1 − exp − , for < 0.2, (4.93)
⎪
⎪
A + δ
⎨
l m =
y
⎪
⎩ 0.085δ for ≥ 0.2, (4.94)
⎪
δ
where y is the normal distance from the wall, δ is the velocity-boundary-layer
thickness and κ = 0.41. Further,

yu τ τ w ∂u
+
y = , u τ = ,τ w = µ | w . (4.95)
ν ρ ∂y
Finally, the value of A is sensitised to effects of suction or blowing and local
+
pressure gradient in a generalised manner as

−1
+
+
A = 25 a v + bp / 1 + c v + + 1 , (4.96)
+
w w
where
3
+
+
p = µ dp
τ ρ −0.5 , v = v w , (4.97)
w
w
dx u τ
and a = 7.1, b = 4.25, and, c = 10.0. If p > 0 then b = 2.9 and c = 0.
+