The k Equation
Turbulence kinetic energy k equation is used to determine the turbulence velocity scale: 
where Pk is the rate of production and ε is the dissipation rate.
Production actually refers to the rate at which kinetic energy is transferred from the mean flow to the turbulent fluctuations (remember the energy cascade). Pk is the turbulent stress times mean strain rate, so physically it is the rate of work sustained by the mean flow on turbulent eddies
Obviously Pk needs to be modeled due to the presence of Rij in the term
Direct Numerical Simulation (DNS)
In DNS, the 3D unsteady Navier-Stokes equations are solved numerically by resolving all scales (both in space and in time)
For simple geometries and at modest Reynolds numbers, DNS has been done successfully. For example, for a simple turbulent channel flow between two plates:
Reτ = 800, N = (Reτ)9/4 = 10,000,000 (cells), Δt = 10-5 sec.
DNS is equivalent to a “numerical wind tunnel” for conducting more fundamental turbulence research
For practical engineering purposes, DNS is not only too costly, but also the details of the simulation are usually not required.
Two general engineering approaches to modeling turbulence: Large-Eddy Simulation (LES) and Reynolds Averaging Navier-Stokes (RANS) models
Turbulent Heat Transfer
The Reynolds averaging of the energy equation produces a closure term and we call it the turbulent (or eddy) heat flux:
– Analogous to the closure of Reynolds stress, a turbulent thermal diffusivity is assumed:
– Turbulent diffusivity is obtained from eddy viscosity via a turbulent Prandtl number (modifiable by the users) based on the Reynolds analogy:
Similar treatment is applicable to other turbulent scalar transport equations
The Spalart-Allmaras Turbulence Model
A low-cost RANS model solving an equation for the modified eddy viscosity,
Eddy viscosity is obtained from
The variation of
Mainly intended for aerodynamic/turbomachinery applications with mild separation, such as supersonic/transonic flows over airfoils, boundary-layer flows, etc.
RANS Models – Standard k–ε (SKE) Model
Transport equations for k and ε
SKE is the most widely-used engineering turbulence model for industrial applications.
Robust and reasonably accurate; it has many submodels for compressibility, buoyancy, and combustion, etc.
Performs poorly for flows with strong separation, large streamline curvature, and high pressure gradient
RANS Models – k–ω Models
Belongs to the general 2-equation EVM family. Fluent 12 supports the standard k–ω model by Wilcox (1998) and Menter’s SST k–ω model (1994).
k–ω models have gained popularity mainly because:
– Can be integrated to the wall without using any damping functions
– Accurate and robust for a wide range of boundary layer flows with pressure gradient
Most widely adopted in the aerospace and turbo-machinery communities.
Several sub-models/options of k–ω: compressibility effects, transitional flows and shear-flow corrections.
Menter’s SST k–ω Model Background
Many people, including Menter (1994), have noted that:
– The k–ω model has many good attributes and performs much better than k–ε models for boundary layer flows
– Wilcox’ original k–ω model is overly sensitive to the free stream value of ω, while the k–ε model is not prone to such problem
– Most two-equation models, including k–ε models, over-predict turbulent stresses in the wake (velocity-defect) regions, which leads to poor performance in predicting boundary layers under adverse pressure gradient and separated flows
– The basic idea of SST k–ω is to combine SKW in the near-wall region with SKE in the outer region
Menter’s SST k–ω Model Main Components
The SST k–ω model consists of
– Zonal (blended) k–ω / k–ε equations (to address item 1 and 2 in the previous slide)
– Clipping of turbulent viscosity so that turbulent stress stay within what is dictated by the structural similarity constant (Bradshaw, 1967) - addresses the overprediction problem
The resulting blended equations are:
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