计算流体力学(CFD)文档——8. Turbulence modelling

=
w≡
u
2
¡
Then the eddy viscosity (turbulent stress ÷ mean velocity gradient) is
t≡
/ ∂U ∂y
=(
u
2
¡
)
×
y
u¡
= ( u¡ y)
Hence, in the log-law region,
t = u¢ y
(6)
8.2.3 General Stress-Strain Relationship
Relate this to the expressions for representative shear stress ( − uv ) and normal stress ( − u 2 ) above.
8.2.4 Other Turbulent Fluxes
It is common to assume a gradient-diffusion relationship between turbulent fluxes and the
The kinematic eddy viscosity t = t/ has dimensions of [velocity] × [length], which suggests
that it be modelled as
t = u0l0
(9)
On physical grounds, u0 should be a velocity scale reflecting the magnitude of turbulent
viscous stress and velocity gradient. In simple shear,
−
uv =
∂U t ∂y
(2)
µt is called an eddy viscosity or turbulent viscosity. The kinematic eddy viscosity is
8. TURBULENCE MODELLING1
8.1 Objectives in turbulence modelling 8.2 Eddy-viscosity models 8.3 Advanced turbulence models 8.4 Wall boundary conditions Summary Examples
The objectives of turbulence modelling are to specify the Reynolds stresses − uv , − u 2 , ...
and, if there are additional transported quantities, the turbulent fluxes vφ etc.
gradient of the corresponding mean quantity:
−
vφ =
∂ t ∂y
(7)
The turbulent diffusivity t is proportional to the eddy viscosity:
t= t
(8)
t
t is called a turbulent Prandtl number. Its value is approximately 1.0.
Eddy-viscosity models are widely used and popular because:
• they are easy to implement in existing viscous solvers; • extra viscosity aids stability; • they have some theoretical foundation in simple shear flows.
t = t/
(3)
The total mean shear stress (1) is then
=
∂U eff ∂y
(4)
where the effective viscosity eff is the sum of molecular and turbulent viscosities:
eff = + t
However, one should exercise caution because:
• there is little theoretical foundation in complex flows; • modelling turbulent transport is reduced to a single scalar
Reynolds stress can be represented accurately.
t and hence at most one
8.2.2 The Eddy Viscosity in the Log-Law Region
In the log-law region of a turbulent boundary layer the mean velocity gradient is
t is a hypothetical property of the flow and must be modelled.
(3) t varies with position.
(4) At high Reynolds numbers, t » throughout much of the flow.
Pope, S.B., 2000, “Turbulent flows”, Cambridge University Press.
CFD
8–1
David Apsley
Notes.
(1) This is a model!
(2)
is a physical property of the fluid and can be measured;
+
∂V ∂x
)
=2
t
∂U ∂x
−
2 3
k
where k is the turbulent kinetic energy. (Exercise: by “pattern matching”, write down
expressions
for
the
other
Reynolds
stresses.)
The
−
2 3
SPRING 2011
8.1 Objectives in Turbulence Modelling
The Reynolds-averaged Navier-Stokes (RANS) equations are transport equations for the mean fluid variables in a turbulent flow.
(5)
1 Better (but very mathematical) descriptions of turbulence models can be found in: Wilcox, D.C., 1998, “Turbulence Modelling for CFD”, 2nd ed, DCW Industries.
The stress-strain relationship (2) applies only to simple shear flows. In general, representative
shear and normal stresses are given by
− uv − u2
=
t
(
∂U ∂y
For simple wall-bounded flows, l0 is proportional to distance from the boundary (e.g. l0 = y). For free shear flows (e.g. jet, wake, mixing layer) l0 is proportional to the width of the shear layer. However, both of these are geometry-dependent. For greater generality, we need to relate l0 to local turbulence properties.
For wall-bounded flows a candidate for u0 is the friction velocity u = w/ . However, a more appropriate velocity scale in general is k1/2, where k is the turbulent kinetic energy.
CFD
8–3
David Apsley
8.2.5 Specifying the Eddy Viscosity
With the eddy-viscosity hypothesis, closure of the mean-flow equations now rests solely on the specification of t, a property of the turbulent flow.
in order to close the mean-flow equations.
8.2 Eddy-Viscosity Models 8.2.1 The Eddy-Viscosity Hypothesis
y U(y)
The mean shear stress has both viscous and turbulent parts. In simple shear: