Turbulent Models

 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

 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

 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

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 earlier Replace the strain tensor term with the mean rate of the strain tensor: And rearrange some more…..

Change in mean momentum of fluid element owing to Unsteadiness in the mean flow and the convection by the mean flow Mean body force Contribution to isotropic stress from mean pressure field The viscous stresses The Reynolds stress 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 P k

 P b models the effect of buoyancy Pr t is the turbulent Prandtl number for energy (default 0.85) β is the coefficient of expansion Model constants: C 1 Є = 1.44, C 2 Є = 1.92, C μ = 0.09, σ k = 1.0, σ Є = 1.3

 Wiki:Copyrights Wiki:Copyrights  Launder, B.F., and Spalding, D.B., Mathematical Models of Turbulence, Academic Press, London and New York,  Symscape‘