AdvancedTraining_3_Turbulence_Modelling.pdf
Governing Equations
-We now have enough equations to fully  describe our system
-Mass
-Momentum
-Energy

Turbulence Modelling
Theory and Practice of Modelling
Turbulent Flow in Flotherm
-Flomerics Ltd 2001
www.aji87.cn

Turbulence
.To model a turbulent flow, the temporal terms of the conservation equations would have to have a time step (dt) small enough to capture all turbulent fluctuations on even the smallest time scales.

. All physical dimensions of the control volume cells (dxi) terms would have to be as small as that known as the Kolmogarov scale, which decreases non-linearly with an increase in Reynolds number.

.Substitute back into the governing equations
.Gives extra terms called Reynolds stress (from the momentum equations) and Reynolds flux (from the energy equation)

.We could write equations for the Reynolds stresses and Reynolds Fluxes and solve them in conjunction with the other governing equations
.In FLOTHERM, we make use of simplifying assumptions

Turbulence Modelling
.The simplest view of the behaviour of a turbulent fluid is that it is just like that of a laminar one but with an increased viscosity
.This viscosity may vary from place to place.

-Physically, turbulence increases the mixing and heat transfer in the fluid
-We represent this as increased viscosity
-Increased thermal conductivity

Turbulence
-We can now write:
-Where:
- μ = Laminar viscosity
- μt = Turbulent viscosity

Turbulence Models
-This concept is usually attributed to Boussinesq (1877) and is called the "EFFECTIVE-VISCOSITY HYPOTHESIS".
-One important implication is that turbulent shear stresses are proportional to velocity gradients.
-The hypothesis has been known for many years not to be valid in all circumstances; but it is often so close to the truth that it is very widely used.

-Turbulent fluids have an effective viscosity
-Prandtl expressed as follows:
- μ = ρ l 2 |?u/?y|

-l is the mixing length
-Mixing length characterises the local structure of the turbulence
-May relate to the geometry of the system

Turbulence Models
-The main limitation imposed is that the eddy viscosity is the same in all directions at any point.
-This may not be true of turbulent
viscosity, which is effectively a property of the flow.

Models
-Turbulence models predict turbulence viscosity
-FLOTHERM has two forms of model:
-Zero equation model
-Two equation model

Zero Equation
-Turbulent viscosity depends on an algebraic function of flow properties and geometry
-FLOTHERM has:
-Automatic Algebraic Model (default)
-Revised Algebraic Model

Two Equation
-The two equation model uses two differential transport equations to predict the eddy viscosity on a cell by cell basis.
-FLOTHERM has:
-Standard KE Model
-Revised KE Model

Automatic Algebraic
-Based on LVEL model of Spalding et al
-Sets turbulent viscosity based on wall distance and local fluid velocity
-Turbulent viscosity varies from cell to cell in the bulk flow
-No cap on viscosity in bulk fluid

-We define:
-dimensionless velocity parallel to the wall, u+
-dimensionless distance from the wall,y+
-These are based on the wall shear stress and fluid properties

Automatic Algebraic
-Assume that there is a universal relationship between u+ and y+
-This is called the Law of the Wall
-Determine the effective viscosity

Automatic Algebraic
-The wall distance at each point in the solution domain is computed
-The Law of the Wall is applied to calculate the effective viscosity

Revised Algebraic
-The revised algebraic turbulence model uses a single value of turbulent viscosity in the bulk of the fluid (i.e. outside of wall boundary layers).

Standard KE
-This turbulence model calculates two variables:
-the kinetic energy of turbulence (k)
-the dissipation rate of k (denoted ε).
-Physically

-Method proposed by Harlow and Nakayama (1968)
-Solve transport equations for k and ε

Revised KE
-Revised KE incorporates a correction factor that reduces viscosity in near-wall cells
-BUT, it does not work with prisms in FLOTHERM
-Generally, best avoided unless you have good reason to use it

Transitional Flow
-None of the turbulence models considers transitional flow

Separating Flow
-Separating flow is better handled by the KE models
-Separating flow is not handled well by algebraic models
-Important for impinging jets and for stagnation regions, if heat transfer in these regions dominates

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