Turbulent Models. DNS – Direct Numerical Simulation ◦ Solve the equations exactly ◦ Possible...
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Transcript of Turbulent Models. DNS – Direct Numerical Simulation ◦ Solve the equations exactly ◦ Possible...
DNS – Direct Numerical Simulation◦ Solve the equations exactly◦ Possible with today’s supercomputers◦ Upside – very accurate if done correctly◦ Downsides:
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 unmanageable
Methods to solve the N-S Equations
Large Eddy Simulation (LES)◦ Assume that the large eddies in the flow are
dependent on the geometry and specific flow parameters
◦ 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
Methods to solve the N-S Equations (con’t)
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 the RANS and introduce a few simple models
Methods to solve the N-S Equations (con’t)
Reynolds decomposition
Tools you will need in the Derivation of RANS
Mathematical rules for flow variables f and g, and independent variable s
Incompressible Newtonian Fluid:
Starting point… N-S Equations of motion
Incompressible: density is constantNewtonian: stress/strain rate is linear and described by:
Into these equations, substitute for each variable, the average and fluctuating composition, by the Renolds decomposition,
Derivation of the RANS (con’t)
And so forth….
Time average the equations
Rearrange using the relationships presented earlier
Derivation of the RANS (con’t)
Replace the strain tensor term with the mean rate of the strain tensor:
And rearrange some more…..
RANS for stationary, incompressible flow, Newtonian fluid
Change in meanmomentum of fluidelement owing toUnsteadiness in themean flow and the convection by the mean flow
Mean body force
Contributionto isotropic stress frommean pressurefield
The viscous stresses
The Reynoldsstress
The Reynolds stress is the apparent stress owing to thefluctuating 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.
Turbulence Models – Eddy Viscosity
Ludwig Prandtl introduced the concept of the mixing length and of a boundary layer.
Turbulence Models – Prandtl Mixing Length
'Original Image courtesy of Symscape‘ http://www.symscape.com/.
Still based on the concept of eddy viscosity However, the eddy viscosity varies with the
distance from the wall
Turbulence Models – Prandtl Mixing Length (con’t)
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
Turbulence Models – k-Є model
Pb models the effect of buoyancy
Transport equations for k and Є (con’t)
•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
http://www.cfd-online.com/Wiki/CFD-Wiki:Copyrights
Launder, B.F., and Spalding, D.B., Mathematical Models of Turbulence, Academic Press, London and New York, 1972.
Symscape‘ http://www.symscape.com
Sources