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or model turbulent diffusivity for momentum (turbulent viscosity divided
by fluid density). Once turbulent viscosity is given, one almost solves the
turbulent flow and heat transfer problem.
∂u
∂u
1 ∂
∂u
1 ∂
u
+ v
=
μ
− ρu " v " =
(τ viscous + τ turb. )
(10.11)
∂x
∂y
ρ ∂y
∂y
ρ ∂y
∂T
∂T
1 ∂
∂T
1 ∂ (
)
""
""
u
+ v
=
k
− ρC p v " T " =
q molecular + q turb.
∂x
∂y
ρC p ∂y
∂y
ρC p ∂y
(10.12)
∂u
∂u
τ total = τ m + τ t = μ
+ ρε m
(10.13)
∂y
∂y
where −ρu " v " = ρε m (∂u/∂y), eddy diffusivity for momentum ε m = υ t =
(μ t /ρ), turbulence viscosity μ t = ρε m
∂T
∂T
""
""
""
q
= q + q = k
+ ρC p ε H
(10.14)
total
m
t
∂y
∂y
where −ρC p v " T " = ρC p ε H (∂T/∂y), eddy diffusivity for heat ε H = α t
∂u
τ total = ρ (ν + ε m )
(10.15)
∂y
∂T
""
q
= ρC p (α + ε H )
(10.16)
total
∂y
Turbulence Prandtl number
υ t
ε m
Pr t =
=
∼ 1
(10.17)
α t
ε H
So the unknowns are
• Turbulent viscosity μ t
• Turbulent diffusivity υ t = (μ t /ρ) = ε m
• Turbulent diffusivity for heat ε H = α t
The following shows how to determine the turbulent diffusivity for
momentum.
201
Turbulent Flow Heat Transfer