Comparison of Different Turbulent Models for Backward -facing Step Flow

Contents

Introduction

Eddy viscosity model

Range of application of viscosity model

RANS Single Equation model: Spalart- Allmaras

RANS Two-Equation Model: (k – $\epsilon$

) model

RANS Two-Equation Model: ( $k–\omega$

) model

Limitations of Eddy -viscosity Models

Reynolds Stress Model (RSM): Reynolds Based Models

Range of application of RSM

Shortcoming of RSM

Case Analysis

Background

Near Wall treatment methods

Result and discussion

Conclusion

References

.

Introduction

Turbulence is the complex type of fluid motion, which makes it definition even more difficult. There are many definitions as for example, the basic definition by Von Karman is “Turbulence is an irregular motion which in general  are seen  in fluids, gaseous or liquid, when they flow past solid surfaces or even when neighboring streams of the same fluid flow past over one another”.(Wilcox) .The Equations for turbulence Fluctuations are obtained by Reynolds de-composition where the flow variables are express in the form of averaged mean value and the fluctuation about the mean .

                                                    $\varphi ={\varphi }^{‘}+\tilde{\varphi }$

                                         (1)

Where $\varnothing$

is the instantaneous scalar quantity, $\overline{\mathit{\varnothing }}$

is the mean value and $\acute{\varnothing }$

is the fluctuation. Substituting the Eq (1) into the Continuity and Navier Strokes Equation one derives what is called Reynolds Average Navier Strokes Equation.

$\frac{\partial {\bar{\mathrm{U}}}_{\mathrm{i}}}{\partial \mathrm{x}}=0$

$\frac{\partial {\bar{U}}_i}{\partial t}+\frac{\partial }{\partial x_j}\left({\bar{U}}_i{\bar{U}}_j\right)=–\frac{1}{\rho }\frac{{\partial }_P^{–}}{\partial x_i}+\nu \frac{{\partial }^2{\bar{U}}_i}{\partial x_j\partial x_j}–\frac{\partial }{\partial x_j}\overline{\left(u_i^{‘}{u}_j^{‘}\right)}\mathit{ }\left(3\right)$

Where ${\bar{U}}_i$

is the mean velocity, $U_i^{‘}$

is the fluctuating velocity, $\rho$

the density of the fluid  $\nu$

is the kinematic viscosity, while the term $\overline{U_i^{‘}{U}_j^{‘}}$

is the Reynolds-stress tensor.

${\tau }_{\mathit{ij}}=\overline{U_i^{‘}{U}_j^{‘}}$

   is a symmetric tensor which has six independent components. The number of unknown quantities (3-velocity components and six stresses and pressure is larger than the number of the available equations (continuity and NS) where  no of variables is not equal to no of equations which need to be closed leading  to closure problems .The  method  used to resolve this problem of “closure “ can be done either  by use of Boussinesq eddy -viscosity or the by calculating the of the induvial Reynolds stress using differential transport equation.

Eddy viscosity model

 In eddy viscosity model the Reynolds stress are modelled as follows

${\tau }_{\mathit{ij}}=\overline{U_i^{‘}{U}_j^{‘}}=\frac{2}{3}k{\delta }_{\mathit{ij}}–{\nu }_t\left(\frac{\partial {\bar{u}}_i}{\partial x_j}+\frac{\partial \overline{\mathit{Uj}}}{\partial x_j}\right)$

                                                       (4)                                                                                                 

$k=\frac{1}{2}\overline{U_i^{‘}{U}_j^{‘}}\mathit{ }$

$\omega =\raisebox{1ex}{$\epsilon $}\!\left/ \!\raisebox{-1ex}{$k$}\right.\mathit{ }$

Where k is the turbulent kinetic energy and ${\nu }_t=\frac{{\mu }_t}{\rho }$

is the turbulent or the eddy viscosity, $\omega$

is the specific dissipation rate

$\frac{\partial {\bar{u}}_i}{\partial t}+\frac{\partial }{\partial x_j}\left({\bar{U}}_i{\bar{U}}_j\right)=–\frac{1}{\rho }\frac{\partial \bar{P}}{\partial x_i}+\frac{\partial }{\partial x_j}\left[\left(v+{\nu }_t\right)\frac{\partial {\bar{u}}_i}{\partial x_j}\right]\mathit{ }$

(5)

 Each turbulent model calculates the ${\nu }_t$

differently, the one equations model considers as a characteristic velocity the square root of the turbulent kinetic energy and assign algebraically the length scale.

${\nu }_t=C_v\sqrt{k}\mathit{l }\left(6\right)$

Whereas the two-equation model such as $k–\epsilon$

and $k–\omega$

compute both the characteristic velocity and length and predicts the value of ${\nu }_t$

which is given as

K $–\epsilon$

model

$v_t=\left\{C_{\mu }{f}_{\mu }\frac{K^2}{\epsilon }\right\}$

                                                                                     (7)

$k–\omega$

model

$v_t=\left\{\alpha \frac{k}{\omega }\right\}$

                                                                                           (8)                                                   

Where $C_{\mu },\alpha$

are constant, $f_{\mu }$

being a damping function, $\epsilon$

is the turbulent kinetic energy dissipation rate and $\omega$

the dissipation per unit turbulence kinetic energy

Range of application of viscosity model

RANS Single Equation model: Spalart- Allmaras

This one equation model is not memory intensive,  work with poor mesh , stable with good convergence and its application can be found on internal and external flows and boundary layers flow under pressure gradient , but the limitations of the models are they works poorly with 3d flows , flows involving  strong separation , decaying turbulence and shear flows . (engineering.com, n.d.).

RANS Two-Equation Model: (k – $\epsilon$

) model

Standard k-epsilon, RNG K-epsilon, Realizable k -epsilon

Standard (k- $\epsilon$

) performs poorly for complex flows which includes strong stream line curvature, Adverse pressure gradient and separation. This is suitable for initial screening for alternative design and for initial iterations, whereas the RNG ( $k–\epsilon$

) can solve complex flows and realizable ( $k–\epsilon$

) offers a better computational efficiency over RNG ( $k–\epsilon$

) (www.fluentusers.com, n.d.)

RANS Two-Equation Model: ( $k–\omega$

) model

Standard ( $k–\omega$

), SST ( $k–\omega$

)

This model is appropriate for turbomachinery simulations and where strong vortices are present. This model works for swirling flows and near the wall region but overpredicts separations. converge is difficult as compared to $(k–\epsilon )$

model and are sensitive to initial conditions.  $\mathit{SST}\left(k–\omega \right)$

model is widely used in aerospace applications. This model can be applied to viscous affected region without any modifications. SST(k-w) is superior on its capability to predict separation and reattachment, better as compared to Standard  $k–\epsilon$

and $k–\omega$

. (www.engineering.com, n.d.)

Limitations of Eddy -viscosity Models

Although the eddy viscosity model like  $k–\omega$

and $k–\epsilon$

model is widely used in the engineering applications they have a significant shortcoming when a complex, real-life turbulent flows are encountered. (www.cfd-online.com/wiki/Turbulence _Modeling , n.d.) The Applications of the Boussinesq Approximations leads to failure to predict anisotropy of the normal stresses, to model secondary and swirling flows and to account for streamline curvature effects (Turbulence Modelling for CFD).

Reynolds Stress Model (RSM): Reynolds Based Models

The Reynolds stress model also known as Reynolds Transport Model, are higher level closure problem generally known as second order closure problems. This modelling originates from the work by chou (1935) and Rotta (1951). In this approach the individual Reynolds Stress, $\rho \overline{u_i^{‘}{u}_j^{‘}}$

, are calculated. The Reynolds stress model closes the Reynolds Average Navier Stokes equations where the transport equation are solved for Reynolds stresses, along with the equation for the dissipation rate. The exact transport equations for the transport of the Reynolds stress $\rho \overline{u_i^{‘}{u}_j^{‘}}$

,  may be written as follows.

       (7)

Or 

Local time derivate + $C_{\mathit{ij}}=D{T}_{\mathit{ij}}+D_{L,\mathit{ij}}+P_{\mathit{ij}}+{\varphi }_{\mathit{ij}}–{ϵ}_{\mathit{ij}}+F_{\mathit{ij}}$

Where $C_{\mathit{ij}}$

is the Convection term, $D{T}_{\mathit{ij}}$

is turbulent diffusion , $D_{L,\mathit{ij}}$

is the Molecular diffusion , $P_{\mathit{ij}}$

is the stress  production , ${\varphi }_{\mathit{ij}}$

pressure strain ${ϵ}_{\mathit{ij}}$

is dissipation and $F_{\mathit{ij}}$

is the production by the system rotation. Out of these terms $C_{\mathit{ij}}$

, $D_{L,\mathit{ij}}$

, $P_{\mathit{ij}}$

, $F_{\mathit{ij}}$

does not require modelling whereas $D{T}_{\mathit{ij}}$

, ${\varphi }_{\mathit{ij}}$

, and ${ϵ}_{\mathit{ij}}$

requires modelling.

Range of application of RSM

The Reynolds stress Model are suitable for complex 3d flows with strong streamline curvature, Strong Swirl Rotation for e.g. rotating flow passage, curved duct, cyclones). Flow with sudden changes in the mean strain rate, secondary flow and Buoyant flow. Reynolds stress model are superior performance compared to eddy viscosity model in these cases, but it comes at the cost of reduced numerical robustness, increased in the computational time.

Shortcoming of RSM

The accuracy of the RSM depends on how accurately the, pressure strain and dissipation is modelled which are complicated, and are often  responsible for compromising the in predictions of RSM. The RSM rely on ( $\mathit{\epsilon \; or\; \omega }$

) and inherit deficiencies resulting from the assumption in these equations. The accurate prediction of flow separation is problematic when $\epsilon$

– equation model is used. In order to avoid these issues, a Reynold stress model has been implemented a model that uses the $\omega$

– equation, and it is showed later in this case analysis the Reynolds Stress Omega based model show a better result in predictions of reattachment length as compared to other RSM model.  Although, Reynolds stress model are more suited to complex flows theoretically, however in practical they are often not superior to two-equation model, which will also be proven later in this case analysis.

Case Analysis

Background

 Here a case analysis is performed using a Geometry of a backward facing- step where the flow is computationally simulated   with different turbulence models available.  The Plane backward step flow is very complex, even though the geometry is simple it includes recirculation, flow separation, reattachment, Adverse pressure gradient, boundary layer redevelopment  (S.P.YUAN, 1998) . The separated flow , reattachment, promotes to  pressure fluctuation ,structure vibration  and also shows  unsteadiness in  structure with a large scale vortex in the separated shear layer and low frequency motion around the reattachment  with fluctuation of an instantaneous reattachment point (Troutt, 1984).Many investigators have compared their model with (Driver, 1985) which is widely used benchmark to evaluate turbulence model ability to predict reattachment location of  flow

In order to compute the flow over the backward-facing step, (0°) a commercially available CFD, FLUENT, is used. The turbulence model used in this study are standard $k–\epsilon$

model, RNG  $k–\epsilon$

model, RKE $k–\epsilon$

model, RSM, standard $k–\omega$

model and SST $k–\omega$

.

Near Wall treatment methods

 The case has been analyzed using both the Enhanced wall treatment and Non -Equilibrium wall function.  The enhanced wall treatment combines a two-layer model with so called enhanced wall function. In two-layer model, the whole domain is subdivided into a viscosity -affected region and fully turbulent region. The near wall mesh is kept at $y^{+}\approx 1$

which is very fine enough to resolve the viscous sublayer. At $y^{+}\approx 1$

,  the enhanced wall treatment is identical to traditional two-layer zonal model. (Fluent ).In Fluent model that uses $\omega$

, the near wall treatment is not available because the near wall treatment that is used is a $y^{+}$

which is an insensitive method that automatically behaves either as a viscous sublayer resolving treatment or as a wall function depending on how fine or coarse the near wall mesh is . (Fluent ). Similarly, the Non-equilibrium wall function is used because of its capability to partly account for effects of pressure gradients. The non- Equilibrium wall functions are recommended for use in complex flow involving separation reattachment and impingement where mean flow and turbulence is subjected to severe pressure gradients and changes rapidly. (Fluent)

Figure 2mesh for backward facing step with 0-degree wall-angle

The mesh used for present computation is quadrilateral mesh with a cell of 21750 and with a minimum orthogonality quality of 1. The average velocity is 41.7 m/s and Reynold’s number are of 36,00. The wall boundaries condition was applied with no slip condition and the flow was assumed to be incompressible

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Result and discussion

The reattachment length for different turbulence models are shown in table 1 and is being compared with values by Driver and Seegmiller. (Driver, 1985) also validated with (KIM, GHAJAR, & L.FOUTCH, 2005) .in the above journal the author has also compared results using standard wall function which has been avoided in this report and the comparison has been done using Non-equilibrium wall function and enhanced wall function .

The turbulence model RKE, and RNG with non-equilibrium wall function showed a good result when compared to the experimental data. However, the RSM and SKE underpredicted the reattachment length. For model employing Enhanced wall function all turbulence models of  $k–\epsilon$

  overpredicted the reattachment length whereas RSM underpredicted.  Similarly, RSM when modelled using Stress Omega which required no wall treatment showed a good result whereas the SKW and SKE (no wall treatment) overpredicted the reattachment length.

                             Table 1: Comparison of reattachment length.

                                          Figure 3 Reattachment point for different model using enhanced wall treatment.

                                          Figure 4Reattachmnent for different turbulence model using Non- equilibrium wall function

Conclusion

In Practical RSM model are not often superior to two –equation model. Reynolds stress model that implement model (Reynolds Stress Omega) shows a better result in predictions of reattachment length as compared to other RSM model. The two-equation mode is widely used for engineering applications for its robustness some success has been achieved with two -equations models however failure is still common for many applications that involve strong curvature, buoyancy, strong swirl rapid compression and expansion.  Among the difference turbulence model available there always has been a tradeoff between the computational efficiency and solution accuracy and not every model is suitable for every type of flow, so a best judgment should be used.

References

• (n.d.). Retrieved from engineering.com.
• (n.d.). Retrieved from www.fluentusers.com.
• (n.d.). Retrieved from www.engineering.com.
• (n.d.). Retrieved from www.cfd-online.com/wiki/Turbulence _Modeling .
• (n.d.). In D. C. Wilcox, Turbulence Modelling for CFD (p. 229). DCW industries.
• (n.d.). Retrieved from Fluent .
• Driver, D. a. (1985). AIAA journal. Features of reattaching Turbulent Shear Layer in Divergent Channel Flow , 163-171.
• KIM, J.-Y., GHAJAR, A. J., & L.FOUTCH, C. T. (2005). Comparsion of near- wall treatment methods for high reynolds number backward-facing step flow . International journal of computational fluid dynamics, 493-500.
• S.P.YUAN, R. &. (1998). NEAR-WALL TWO-EQUATION AND REYNOLDS STRESS MODELING OF BACKSTEP FLOW , 283-298.
• Troutt, T. S. (1984). Organised Strucutres in a Reattaching Seperated flow field , 413-427.
• Wilcox, D. C. (n.d.). Turbulence Modeling For CFD. DCW Industries.