Reynolds` Averaging and Simple Turbulence Models Lecture Thurs

Turbulent Models
DNS – Direct Numerical Simulation
Solve the equations exactly
Possible with today’s supercomputers
Upside – very accurate if done correctly
 You get way more information than you normally need
 Length scales must be resolved down to the smallest
turbulent eddy throughout the domain
 Therefore, requires millions of cells and becomes
Large Eddy Simulation (LES)
◦ Assume that the large eddies in the flow are
dependent on the geometry and specific flow
◦ The smaller eddies are all similar and can be
modeled independently of geometry
◦ Less compute-intensive than DNS
◦ Gives more information than an averaged technique
◦ Still yields more information than normally required
for engineering applications
Turbulence Models based on the Reynolds
Averaged N-S Equations (RANS)
◦ Developed first
◦ The most general approximation
◦ Still in the widest use for engineering problems
(okay, arguably…)
◦ We will derive
simple models
the RANS and introduce a few
Reynolds decomposition
Mathematical rules for flow variables f and g, and independent
variable s
Incompressible Newtonian Fluid:
Incompressible: density is constant
Newtonian: stress/strain rate is linear and described by:
Into these equations, substitute for each
variable, the average and fluctuating
composition, by the Renolds decomposition,
And so forth….
Time average the equations
Rearrange using the relationships presented
Replace the strain tensor term with the mean rate of the strain tensor:
And rearrange some more…..
to isotropic
stress from
mean pressure
Change in mean
momentum of fluid
element owing to
Unsteadiness in the
mean flow and the
convection by the
mean flow
The viscous
The Reynolds
Mean body
The Reynolds stress is the apparent stress owing to the
fluctuating velocity field
The Reynolds Stress term is non-linear and is
the most difficult to solve – so we model it!
First, and most simple model, proposed by
Joseph Boussinesq, was the Eddy Viscosity
model. Simply increase the viscous stress by
some proportional amount to account for the
Reynolds’ stresses. Works very well for
axisymmetric jets, 2-D jets, and mixing
layers, but not much else.
Ludwig Prandtl introduced the concept of the
mixing length and of a boundary layer.
'Original Image courtesy of Symscape‘
Still based on the concept of eddy viscosity
However, the eddy viscosity varies with the
distance from the wall
Very accurate for attached flows with small pressure gradients.
k-Є is one of a class of two-equation models
The first two-equation models were k-l, based
on k, the kinetic energy of turbulence, and l,
the length scale
More commonly in use now, however, are k-Є
models, Є being turbulent diffusion
Application of the model requires additional
transport equations for solution
Turbulent viscosity:
k production term Pk
Pb models the effect of buoyancy
•Prt is the turbulent Prandtl number for energy (default 0.85)
•β is the coefficient of expansion
Model constants: C1Є = 1.44, C2Є = 1.92, Cμ = 0.09, σk = 1.0, σЄ = 1.3
Launder, B.F., and Spalding, D.B.,
Mathematical Models of Turbulence,
Academic Press, London and New York, 1972.

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