Turbulence Models and their Applications

TURBULENCE MODELS AND THEIR APPLICATIONS Presented by: T.S.D.Karthik Department of Mechanical Engineering IIT Madras Guide: Prof. Franz Durst

10th Indo German Winter Academy 2011

2

Outline 3 

Turbulence models introduction

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Boussinesq hypothesis

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Eddy viscosity concept

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Zero equation model

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One equation model

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Two equation models

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Algebraic stress model

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Reyolds stress model

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Comparison

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Applications

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Developments

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Conclusion Turbulence Models and Their Applications

Turbulence models 4

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A turbulence model is a procedure to close the system of mean flow equations. For most engineering applications it is unnecessary to resolve the details of the turbulent fluctuations. Turbulence models allow the calculation of the mean flow without first calculating the full time-dependent flow field. We only need to know how turbulence affected the mean flow. In particular we need expressions for the Reynolds stresses. For a turbulence model to be useful it:



must have wide applicability, be accurate, simple,

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and economical to run.

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Turbulence Models and Their Applications

Common turbulence models 5

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Classical models. Based on Reynolds Averaged Navier-Stokes (RANS) equations (time averaged):    

Zero equation model: mixing length model. One equation model Two equation models: k- style models (standard, RNG, realizable), k- model, and ASM. Seven equation model: Reynolds stress model.

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The number of equations denotes the number of additional PDEs that are being solved.

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Large eddy simulation. Based on space-filtered equations. Time dependent calculations are performed. Large eddies are explicitly calculated. For small eddies, their effect on the flow pattern is taken into account with a “sub-grid model” of which many styles are available.

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DNS Turbulence Models and Their Applications

Classification 6


Prediction Methods 7


Boussinesq hypothesis 8

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Many turbulence models are based upon the Boussinesq hypothesis.   

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It was experimentally observed that turbulence decays unless there is shear in isothermal incompressible flows. Turbulence was found to increase as the mean rate of deformation increases. Boussinesq proposed in 1877 that the Reynolds stresses could be linked to the mean rate of deformation.

Using the suffix notation where i, j, and k denote the x-, y-, and z-directions respectively, viscous stresses are given by:  ui u j    ij       x  x j i  



Similarly, link Reynolds stresses to the mean rate of deformation Turbulence Models and Their Applications

Eddy Viscosity Concept 9

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One of the most widely used concept Reynold’s stress tensor –  ij

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T

 U i U j    ui ' u j '   t    x xi j 

   

A new quantity appears: the turbulent viscosity or eddy viscosity (νt ). The second term is added to make it applicable to normal turbulent stress. The turbulent viscosity depends on the flow, i.e. the state of turbulence. The turbulent viscosity is not homogeneous, i.e. it varies in space. Turbulence Models and Their Applications

Eddy Viscosity Concept 10

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It is, however, assumed to be isotropic. It is the same in all directions. This assumption is valid for many flows, but not for all (e.g. flows with strong separation or swirl).

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The turbulent viscosity may be expressed as

 t  uclc (or ) lc2 / tc 

This concept assumes that Reynolds stress tensor can be characterized by a single length and time scales.


Major Drawbacks 11

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Interaction among eddies is not elastic as in the case for molecular interactions in kinetic theory of gases.

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For many turbulent flows, the length and time scale of characteristic eddies is not small compared with the flow domain (boundary dominated flows).

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The eddy viscosity is a scalar quantity which may not be true for simple turbulent shear flows. It also fails to distinguish between plane shear, plane strain and rotating plane shear flows.

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Successful – 2D shear flows. Erroneous results for simple shear flows such as wall jets and channel flows with varying wall roughness. Turbulence Models and Their Applications

Zero Equation Model - Mixing Length Model 12

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On dimensional grounds one can express the kinematic turbulent viscosity as the product of a velocity scale and a length scale:

 t (m 2 / s)   (m / s) (m) 

If we then assume that the velocity scale is proportional to the length scale and the gradients in the velocity (shear rate, which has dimension 1/s): U    y we can derive Prandtl’s (1925) mixing length model: U  t   2m y

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Algebraic expressions exist for the mixing length for simple 2-D flows, such as pipe and channel flow. Turbulence Models and Their Applications

Equations for mixing length 13

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Wall boundary layers lm  Ky

(y /   / K)

lm  

(y /   / K)

δ = boundary layer thickness y = distance from the wall λ = 0.09 K = Von-Karman Constant 

Developed pipe flows

lm y y  0.14  0.08(1  ) 2  0.06(1  ) 4 R R R

R = radius of the pipe or the half width of the duct Turbulence Models and Their Applications

Mixing Length Model Discussion 14

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Advantages:  Easy to implement.  Fast calculation times.  Good predictions for simple flows where experimental correlations for the mixing length exist.  Used in higher models Disadvantages:  Completely incapable of describing flows where the turbulent length scale varies: anything with separation or circulation.  Only calculates mean flow properties and turbulent shear stress.  Cannot switch from one type of region to another  History effects of turbulence are not considered. Use:  Sometimes used for simple external aero flows.  Pretty much completely ignored in commercial CFD programs today. Turbulence Models and Their Applications

One Equation Model 15

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Different transport equation for ‘k’ is solved.

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L is defined algebraically, in a similar manner as mixing length. u c  k1/ 2 ν t  lm k

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PDE for turbulent KE : Diffusion, Production and Dissipation terms

Di U i k    Pε xi xi where

Modeled Equation

u 'u ' p'u ' 1 j j i D  (   2ν  u 'u 'u ' ) i 3 ρ x 2 j j i i k    t  k   U j U i  U j k2 U   Ui     CD   t j xi xi   k  xi   xi x j  xi lc P  u 'u ' i j x i u' u' j j ε  ν x x i i Turbulence Models and Their Applications

One Equation Model Discussion 16

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Economical and accurate for:    

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Weak for:    

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Attached wall-bounded flows. Flows with mild separation and recirculation. Developed for use in unstructured codes in the aerospace industry. Popular in aeronautics for computing the flow around aero plane wings, etc.

Massively separated flows. Free shear flows. Decaying turbulence. Complex internal flows.

Characteristic length still experimental. Turbulence Models and Their Applications

Two Equation Models – The k-ε model 17

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Equations for k and ε, together with the eddy-viscosity stressstrain relationship constitute the k-ε turbulence model.

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ε is the dissipation rate of k.

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If k and ε are known, we can model the turbulent viscosity as: t     k

1/ 2

k 3/ 2





k2




The k-ε model 18

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K equation: Model (simplified) equation for k after using Boussinesq assumption by which the fluctuation terms can be linked to the mean flow is as follows:  U j U i  U j   t k k k  2k  Ui  t        t xi x j  xi xi  k xi xi xi  xi with  t  0.09

k2




Turbulent Dissipation 19

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We can define the rate of dissipation per unit mass ε as:

  2 eij '.eij ' 

Equation for ε D  Dv  Df  Dp  P1  P2  P3  P4  Y Dt where D εv  Diffusive viscous transport D fε  Diffusive transport by fluctuation D εp  Diffusive transport by pressure fluctuation Pε1  Production by deformation of mean flow field Pε2  Production by deformation of mean flow field Pε3  Production by gradient of mean vorticity Pε4  Production by vortex stretching Y  Viscous destruction


Turbulent Dissipation 20

D Dt

 uk '         2   p '    u'     x  x k x   x x  k k k i i  



  x k

v

D

 u ' u ' u ' u '  U  i k kl  i  2  l  x x x x  x l l i k  k 

1

P

2

P



f

D

p

D

2 2 2   u '  U u ' u ' u '   u 'i  i i k i k  2 u '  2  2   k x x x x x x  x  x  k l l k l k l l  

3

P


4

P

Y

Model Equation for ε 21

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A model equation for ε is derived by multiplying the k equation by (ε/k) and introducing model constants.  t  C

k2



U j       x j x j

 t   k   U i U j  U j   P  t   x xi  xi j 



    x j

  2  Cz 2    C z1 P K K 

Closure coefficients found emperically

 k  1 Cz 2  1.92 Cz1  1.44 C  0.09  z  1.33

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K- ε model leads to all normal stresses being equal, which is usually inaccurate. Turbulence Models and Their Applications

Applications 22

Flow on a backward facing step using k-epsilon model


Single and multiple jet flows using k-epsilon models 23


k-ε model discussion 24

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Advantages:   

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Relatively simple to implement. Leads to stable calculations that converge relatively easily. Reasonable predictions for many flows.

Disadvantages: 

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Poor predictions for:  swirling and rotating flows,  flows with strong separation,  axis symmetric jets,  certain unconfined flows, and  fully developed flows in non-circular ducts. Valid only for fully turbulent flows. Requires wall function implementation. Modifications for flows with highly curved stream lines. Production of turbulence in highly strained flows is over predicted. Turbulence Models and Their Applications

More two equation models 25

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The k-ε model was developed in the early 1970s. Its strengths as well as its shortcomings are well documented.

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Many attempts have been made to develop two-equation models that improve on the standard k-ε model.

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We will discuss some here:  k-ε RNG model. 

k-ε realizable model.

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k-ω model.

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Algebraic stress model.

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Non-linear models. Turbulence Models and Their Applications

RNG k- ε 26 

k- equations are derived from the application of a rigorous statistical technique (Renormalization Group Method) to the instantaneous Navier-Stokes equations.

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Similar in form to the standard k- equations but includes: 

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Additional term in  equation for interaction between turbulence dissipation and mean shear.

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The effect of swirl on turbulence.

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Analytical formula for turbulent Prandtl number.

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Differential formula for effective viscosity.

Improved predictions for: 

High streamline curvature and strain rate.

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Transitional flows, separated flows.

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Wall heat and mass transfer.

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Also, time dependent flows with large scale motions (vortex shedding)

But still does not predict the spreading of a round jet correctly. Turbulence Models and Their Applications

Realizable k-ε model 28

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Shares the same turbulent kinetic energy equation as the standard k- model. Improved equation for ε. Variable Cμ instead of constant. Improved performance for flows involving:  

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Planar and round jets (predicts round jet spreading correctly). Boundary layers under strong adverse pressure gradients or separation. Rotation, recirculation. Strong streamline curvature.


Realizable k- ε Cμ equations 30

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Eddy viscosity computed from.  t  C

k2



,

C 

1 A0  As

U *k

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U *  Sij Sij  ij ij

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1 A0  4.04, As  6 cos  ,   cos 1 6W 3 Sij S ji S ki ~ W ~ , S  Sij Sij S Turbulence Models and Their Applications



Realizable k- ε positivity of normal stresses 31

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Boussinesq viscosity relation: 2  ui u j  2 k  - k  ij ;  t  C  ui u j   t    x    j xi  3

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Normal component: 2 2 k U u 2  k  2C 3  x

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Normal stress will be negative if: k U 1   3.7  x 3C Turbulence Models and Their Applications

k-ω model 32

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This is another two equation model. In this model ω is an inverse time scale that is associated with the turbulence.

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This model solves two additional PDEs:  

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A modified version of the k equation used in the k-ε model. A transport equation for ω.

The turbulent viscosity is then calculated as follows: t 

k



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Its numerical behavior is similar to that of the k-ε models.

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It suffers from some of the same drawbacks, such as the assumption that νt is isotropic. Turbulence Models and Their Applications

The SST Model 33 

The SST (Shear Stress Transport) model is an eddy-viscosity model which includes two main novelties: 1. It is combination of a k-ω !model (in the inner boundary layer) and k-ε model (in the outer region of and outside of the boundary layer); 2. A limitation of the shear stress in adverse pressure gradient regions is introduced. Actual

Model

Non-linear models 34

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The standard k-ε model is extended by including second and sometimes third order terms in the equation for the Reynolds stresses. One example is the Speziale model:

3 2 k2 k  ij    ui ' u j '   k ij  C 2 Eij  4C D C2 2 * f ( E , E / t , u, U / x) 3  







Here f(…) is a complex function of the deformation tensor, velocity field and gradients, and the rate of change of the deformation tensor. The standard k-ε model reduces to a special case of this model for flows with low rates of deformation. These models are relatively new and not yet used very widely. Turbulence Models and Their Applications

Reynolds stress model 35

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RSM closes the Reynolds-Averaged Navier-Stokes equations by solving additional transport equations for the six independent Reynolds stresses.   

Transport equations derived by Reynolds averaging the product of the momentum equations with a fluctuating property. Closure also requires one equation for turbulent dissipation. Isotropic eddy viscosity assumption is avoided.

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Resulting equations contain terms that need to be modeled.

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RSM is good for accurately predicting complex flows. 

Accounts for streamline curvature, swirl, rotation and high strain rates.  Cyclone flows, swirling combustor flows.  Rotating flow passages, secondary flows.  Flows involving separation. Turbulence Models and Their Applications

Reynolds stress transport equation 36

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The exact equation for the transport of the Reynolds stress Rij: DR ij  P  D      ij ij ij ij ij Dt

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This equation can be read as:       

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rate of change of Rij  ui ' u j ' plus transport of Rij by convection, equals rate of production Pij, plus transport by diffusion Dij, minus rate of dissipation εij, plus transport due to turbulent pressure-strain interactions πij, plus transport due to rotation Ωij.

This equation describes six partial differential equations, one for the transport of each of the six independent Reynolds stresses. Turbulence Models and Their Applications

Reynolds stress transport equation 37

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The various terms are modeled as follows: 

Production Pij is retained in its exact form.

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Diffusive transport Dij is modeled using a gradient diffusion assumption.

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The dissipation εij, is related to ε as calculated from the standard ε equation, although more advanced ε models are available also.



Pressure strain interactions πij, are very important. These include pressure fluctuations due to eddies interacting with each other, and due to interactions between eddies and regions of the flow with a different mean velocity. The overall effect is to make the normal stresses more isotropic and to decrease shear stresses. It does not change the total turbulent kinetic energy. This is a difficult to model term, and various models are available. Common is the Launder model. Improved, non-equilibrium models are available also.



Transport due to rotation Ωij is retained in its exact form. Turbulence Models and Their Applications

RSM equations 38

U j  U i   Production exact : Pij    Rim  R jm x m x m   J

ijk Diffusive transport exact : D  ij x k J  u ' u ' u '  p( u ' u ' ) ijk i j k jk i ik j

 t     t Rij     div grad ( Rij )  Diffusive transport model : Dij  xm   k xm   k   t is the turbulent kinematic viscosity calculated in the standard way Turbulence Models and Their Applications

RSM equations continued 39

ui ' u j ' Dissipation exact :  ij  2 xk xk

Dissipation model :  ij  23  ij

 ui ' u j '   Pressure strain exact :  ij   p '     x j xi 



Pressure strain model :  ij   C1 ( Rij  23 k ij )  C2 ( Pij  23 P ij ) k P is the pressure

Rotational term (exact) :  ij   2 k ( R jm eikm  Rim e jkm ) eijk is  1, 0, or 1 depending on the indices

 k is the rotation vector Turbulence Models and Their Applications

Algebraic stress model 40



The same k and ε equations are solved as with the standard k-ε model.

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However, the Boussinesq assumption is not used.

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The full Reynolds stress equations are first derived, and then some simplifying assumptions are made that allow the derivation of algebraic equations for the Reynolds stresses.

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Thus fewer PDEs have to be solved than with the full RSM and it is much easier to implement.

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The algebraic equations themselves are not very stable, however, and computer time is significantly more than with the standard k-ε model.

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This model was used in the 1980s and early 1990s. Research continues but this model is rarely used in industry anymore now that most commercial CFD codes have full RSM implementations available Turbulence Models and Their Applications

Setting boundary conditions 41



Characterize turbulence at inlets and outlets (potential backflow).  



k-ε models require k and ε. Reynolds stress model requires Rij and ε.

Other options: 

Turbulence intensity and length scale.   



Turbulence intensity and hydraulic diameter. 



Length scale is related to size of large eddies that contain most of energy. For boundary layer flows, 0.4 times boundary layer thickness: l  0.4d99. For flows downstream of grids /perforated plates: l  opening size.

Ideally suited for duct and pipe flows.

Turbulence intensity and turbulent viscosity ratio. 

For external flows: 1

 / t

 10


Some Applications 42 

Turbulent Annular Flow


43

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The figures show plots of the normalized tangential velocity, for each of the turbulence models, plotted with the experimental data. RNG k - epsilon model produces the best results with the standard k epsilon model giving the worst but this variation is small compared to their deviations from the experimental data. The k-L mixing length model does lead to an answer which predicts the movement of the maximal tangential velocity from the inner wall to the centre of the annulus better than the other models. Where the implementation of the model fails is in its prediction of the flow near the walls. Turbulence Models and Their Applications

Flocculation tank – Analogy with flow over a back step 44

•In the middle of the channel, the flow separate due to the small step size of height h. The flow reattaches at about 7 times the step height further downstream - similar to the 180 degree bend in the flocculation tank where we have flow separation and reattachment downstream •Analyzed using K-ε, K-ω SST, K-ε realizable, K-ε RNG, RSM turbulence models and compared with experimental data. •Plotting the derivative du/dy, the change in direction of velocity in x direction with respect to y at the wall, the reattachment point is easily identified. At the wall, separated flow will give a negative du/dy, while reattaches flow has a positive du/dy value. Turbulence Models and Their Applications

Flocculation tank – Analogy with flow over a back step 45

Turbulenc e Model Reattach ment Ratio

•The K-ε model under-predicts the reattachment length. K-ω SST and K-ε gives the most accurate 0.195/0. 0.242/0.0 0.235/0.0 0.2/0.038 realizable representation of the back step flow with 038 = 38 = 6.37 38 = 6.18 = 5.26 reattachment length. However, from literature 5.13 reviews, K-ε realizable is more proven for a variety of types of flows. •Below in Figure, the stream contours (of the averaged velocity) of the Re=48,000 for the k-ε realizable model case closely approximate the experimental results. K-e

K-W SST

K-e realizable


RSM

Flow over airfoil 46

• Contour plots of the predicted turbulent viscosity around an airfoil obtained with four different steady-state turbulence models: an algebraic model, a oneequation model, and a duo of twoequation models. • While the Spalart-Allmaras and Chien kepsilon models are in rough agreement with each other, the SST and BaldwinLomax models predict a very different turbulent viscosity distribution. • If you looking solely at performance in the shear layer, you might want to

choose either the Spalart or Chien models. Turbulence Models and Their Applications

Comparison of RANS turbulence models 47

Model Zero Equation Model STD k-ε

RNG k-ε Realizable k-ε Reynolds Stress Model

Strengths

Weaknesses

Economical (1-eq.); good track Not very widely tested yet; lack of subrecord for mildly complex B.L. models (e.g. combustion, buoyancy). type of flows. Robust, economical, reasonably accurate; long accumulated performance data.

Good for moderately complex behavior like jet impingement, separating flows, swirling flows, and secondary flows. Offers largely the same benefits as RNG but also resolves the round-jet anomaly.

Mediocre results for complex flows with severe pressure gradients, strong streamline curvature, swirl and rotation. Predicts that round jets spread 15% faster than planar jets whereas in actuality they spread 15% slower. Subjected to limitations due to isotropic eddy viscosity assumption. Same problem with round jets as standard k-. Subjected to limitations due to isotropic eddy viscosity assumption.

Physically most complete model Requires more cpu effort (2-3x); tightly (history, transport, and coupled momentum and turbulence anisotropy of turbulent stresses equations. are all accounted for). Turbulence Models and Their Applications

Recommendation 48 

Start calculations by performing 100 iterations or so with standard k-ε model and first order upwind differencing. For very simple flows (no swirl or separation) converge with k-ε model.



If the flow involves jets, separation, or moderate swirl, converge solution with the realizable k-ε model.



If the flow is dominated by swirl (e.g. a cyclone or un-baffled stirred vessel) converge solution deeply using RSM and a second order differencing scheme. If the solution will not converge, use first order differencing instead.



Ignore the existence of mixing length models and the algebraic stress model.



Only use the other models if you know from other sources that somehow these are especially suitable for your particular problem (e.g. SpalartAllmaras for certain external flows, k-ε RNG for certain transitional flows, or k-ω). Turbulence Models and Their Applications

Other Numerical Methods: DNS (Direct Numerical Simulation) 49



Very accurate



High computing time



Works on small Reynolds number flows



Used to verify the turbulence model



Some arbitrary initial velocity field is set up and the Navier-Stokes equations are used directly to describe the evolution of this field over time. Turbulence Models and Their Applications

LES (Large Eddy Simulation) 50



More accurate than RANS



More computing time than RANS



Middle route between DNS and RANS



Can work on larger Reynolds number and more complex flows



Simulations for large scales and RANS for small scales



Both DNS and LES require to solve the instantaneous Navier-Stokes equations in time and three-dimensional space.



Hybrid Approach  RANS + LES Turbulence Models and Their Applications

Developments 51

Accuracy increases Complexity increases Computing time increases Usability decreases

Eddy Viscosity Models

Reynolds Stress Models


Probability Density Functions

Conclusion 52



Different turbulence models – strengths and weaknesses



Usage of models specific to certain flows



Applications – Where to use what?



Compromise between accuracy and computing power



Other developments


References 53



Turbulent Flows – Fundamentals, Experiments and Modeling, G.Biswas and V.Eswaran, Narosa Publishing House, 2002



Turbulence Modeling for CFD – Wilcox, D.C, 1993



Fluid Mechanics - An Introduction to the Theory of Fluid Flows, Franz Durst, Springer, 2008



Turbulence model validation - https://confluence.cornell.edu/ http://www.tfd.chalmers.se/doct/comp turb model Fluent – Modeling Turbulence

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54
