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2.29: Numerical Fluid Mechanics

Project  Report  Numerical  Investigation  of  Turbulence  Models  

Mostafa Momen                        

May  2015  Massachusetts  Institute  of  Technology  

 

           

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Numerical Investigation of Turbulence Models

Term Project: Mostafa Momen  

1. Introduction:

Most real-world flows occurring in nature and engineering applications are turbulent. The boundary layer in the earth’s atmosphere, interstellar gas clouds, boundary layers on aircraft wings, natural gas flow in pipes, and water flow in rivers are all examples of turbulent flows. Turbulence happens when the Reynolds number of a flow is high, typically once Re > O(1000). Reynolds number is the ratio of the inertial to viscous forces, hence when the inertial forces are dominating; the flow regime will become turbulent. Furthermore, when the small and large scales of the motion or dissipating and energy containing scales in a flow are well separated from each other the turbulence occurs. To simulate these highly nonlinear turbulent flows numerically, there are three primary methods. Among three available approaches Direct Numerical Simulation (DNS) is the most accurate one but also most computationally expensive (sometime impossible) way in high Reynolds flows. Reynolds-Averaged Navier-Stokes (RANS) models, which only solve for the mean flow and model all turbulent scales are the least accurate and the cheapest methods at the same time which are good for some engineering applications. The DNS grids are very fine and in the size of the Kolmogorov length scale so this method requires enormous computational resources for highly turbulent flows. Thus in most engineering applications, RANS models are used. Large-eddy simulation (LES) technique, which consists of resolving governing equations for the energy containing large scales and modeling the smaller scales, lies between these two approaches. In the following study, we will examine a finite-volume (FV) DNS code with some RANS turbulence models versus finite-difference LES results and analyze their differences. To do so, we will employ the methods and notions learned in class to develop the numerical anlaysis.

2. RANS Closure Models: In the RANS approaches to turbulence, all of the unsteadiness is averaged out implying that all unsteadiness is regarded as part of the turbulence. The RANS and energy equations cannot be solved without information about the various correlation terms that make up the stress tensor. Thus, to close these equations one must model all the turbulence part and Reynolds stresses. It is well known that these terms, which represent turbulent diffusion, are much larger than those corresponding to laminar diffusion except in the immediate vicinity of a wall, and in turbulent wall boundary layers, wakes, jets and more complex flows, these turbulent diffusion terms are of similar magnitude to the convective terms. Early approaches to turbulence modeling include the mixing length assumptions of Prandtl (1945) and eddy-viscosity assumptions of Boussinesq (1877) for wall boundary

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layers and jets. Kolmogorov (1942) and Rotta (1962) proposed models based on partial-differential equations but, in the absence of digital computers, could not solve them (Cebeci 2005). In general, the RANS based turbulence models are divided into zero, one and two equation models according to the number of the equations that they solve to obtain the turbulent viscosity. The zero-equation models also called algebraic models are calculated directly from the flow variables. An alternative approach for preliminary turbulence modeling is to construct (differential) transport equations for some of the turbulence quantities and to model higher-order terms, which turn out to be triple correlations (Fletcher 1991). There are several two-equation models, which are based on solving differential equations. Three of the most popular and widely used models are the k-ε model of Jones and Launder (1972), the k-ω model of Wilcox (1998) and the SST model of Menter (1994) which blends the k-ε model in the outer region and k-ω model in the near wall region. We may write the main RANS based turbulence models according to their structure as in the following:

a) Zero-equation models (Algebraic models) 1- Cebeci-Smith model 2- Baldwin-Lomax model 3- Johnson-King moel 4- A roughness dependent model 5- Smagorinsky model

b) One equation models

1- Prandtl’s one-equation model 2- Baldwin-Barth model 3- Spalart-Allmaras model 4- Rahman-Siikonen-Agarwal model

c) Two equation models

1- k-ε models 2- k-ω models 3- Realisability issues

Here we focus on two approaches and show the results of the simulations for these two RANS models:

2.1 Smagorinsky model This is a simple model that computes the turbulent viscosity based on the local derivatives of the velocity field and the local grid size as the following:

(1)

This formula was first proposed by Smagorinsky (1964) for the subgrid scale eddy viscosity and is sensitive to the grid sizes.

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2.2 k-ε model The k-ε model is one of the most popular turbulence models and is the most common model used in CFD to simulate mean flow characteristics in turbulent flows, even though it does not perform well in some cases such as when pressure gradients are very large. The Reynolds stresses might be written as:

′ui ′uj = νT

∂ui

∂x j

+ ∂u j

∂xi

⎛

⎝⎜

⎞

⎠⎟ −

23δ ijk (2)

where k is the turbulent kinetic energy (TKE) defined as :

k = 1

2′ui ′ui (3)

The exact k-ε equations include many unknown terms. For a much more practical approach, the standard k-ε model that is proposed by Launder and Spalding (1974) is used to minimize the unknowns of the problem and present a set of equations which can be applied to a large number of turbulent applications. Thus, this model contains two model equations for the TKE and the dissipation as:

Model equation for k :

∂k∂t

+∂ kui( )∂xi

= ∂∂x j

ν +νT

σ k

⎛⎝⎜

⎞⎠⎟∂k∂x j

⎡

⎣⎢⎢

⎤

⎦⎥⎥+ Pk − ε

(4)

The second model equation may be expressed as below:

Model equation for ε :

∂ε∂t

+∂ εui( )∂xi

= ∂∂x j

ν +νT

σε

⎛

⎝⎜⎞

⎠⎟∂ε∂x j

⎡

⎣⎢⎢

⎤

⎦⎥⎥+Cε1

εk

Pk −Cε 2

εkε (5)

where Pk =2νT.EijEij is the rate of production of TKE and the final eddy viscosity would be:

νT = C

µ

k 2

ε (6)

These two equations (4) and (5) are nonlinearly coupled equations that we need to solve together.

     

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3. Numerical Simulations: Three numerical simulations have been performed to investigate the turbulence models. First and third cases are implemented using 2.29 FV code in different boundaries and second case is done with the help of a 3D LES code on a homogenous boundary.

a) Case 1: 2D FV on a homogenous terrain By employing the 2.29 FV code a wall-bounded flow is simulated with the following numerical details and the numerical domain could be viewed in Fig. 1:

- Grid size: 64×64 - Boundary Conditions: No-slip condition on the ground, Time-dependent BC’s

elsewhere with Periodic BC in horizontal directions - Size of domain: Y=100m, X=2π×100m - Time of simulation: 5s with Dt =0.01s - ν =1.695E-3 (almost 0.01 of the air and real-world atmospheric boundary layer) - Wind profiles: perturbed logarithmic profiles

Fig.1. Numerical domain of case 1 showing velocity profiles

A time-dependent boundary condition is used and a random noise with about 10% of the mean value is added to the wind profiles to make the flow regime turbulent. The formation of turbulent eddies are clear in Fig. 1 near the ground. If one increases the perturbations, eddies and thus the turbulence would grow accordingly. All the involving gradients in equations (4) and (5) are calculated in the main loop of fluid solver. And then two k-ε model equations are treated fully explicitly at each time-step and solved simultaneously to obtain the following results. Numerical Stability: Note that with these conditions sometime the explicit scheme becomes unstable and the solutions grow in time (especially when the generated noises are high). This instability spreads to the whole domain through dispersion terms and blows up all the solutions. Hence if the scheme yields unstable solutions at just one single point, it will blow up the whole domain in the next steps due to spatial dependences. One way to tackle this problem is to use implicit schemes considering that the equations are coupled and nonlinear which might make it more challenging.

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Fig.2. TKE results obtained from the FV DNS code (left) and k-ε equation (right) in case 1

Figure 2 depicts two TKE plots resulted directly from the FV code (left panel) and also the TKE computed from the k-ε equations (right panel). As it is clear from above, they are not exactly the same. They should not be necessarily identical since the k-ε equations are not exact and this approach employs a simplistic assumption about dissipation equation. However, they are both in the same order of magnitude and are somehow correlated. The k-ε model is more sensitive to the time-varying boundary conditions and as one could observe there are some local maxima around the left BC. To understand why these two are different, the production terms of the DNS and k-ε equations (Pk) which are the largest terms in the TKE budget equations are computed and plotted in Fig. 3.

Fig.3. TKE production term for FV DNS code (left) and k-ε model, PK (right)

As one could observe, the TKE productions are different in two cases, indicating why the TKE in Fig.2 must not be the same. That is because the real production term is different with the production term resulted from equation (5). Now, turbulent viscosity resulted from the k-ε model in equation (6) is plotted against the Smagorinsky model in Fig. 4 to demonstrate their differences. This figure shows that these tow models are yielding different eddy viscosity values due to their distinct structure, however they both peak near the boundary where the turbulence is high. This is true, but as one could see the Smagorinsky model is dependent on the local velocity derivatives which are highest near the wall and is sensitive to the grid sizes, while in k-ε, eddy viscosity depends on the TKE and dissipation values of equations (4) and (5). Thus, k-ε result somehow looks similar to the TKE depicted in the right panel of Fig.2.

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Fig.4. Turbulent viscosity from Smagorinsky (left) and k-ε models (right)

b) Case 2: 3D LES on a homogenous terrain

Using a large-eddy simulation code, a wall-bounded flow similar to case 1 is simulated with the following numerical details:

- Grid size: 64×64×64 - Boundary Conditions: No-slip condition on the ground, Periodic in horizontal

directions, Stress free top BC - Size of domain: Y=100m, X=2π×100m, Z=2π×100m - Time of simulation: 2 hour warm-up + 20 hour to reach to quasi-steady state

condition with Dt =0.1s - Wind profiles: 3D Ekman profiles - Type of simulation: Coriolis forcing with geostrophic wind velocity of about 8m/s - Finite difference methods: fully-explicit 2nd order Adams-Bashforth for time-

advancement, and pseudo spectral approach in horizontal direction with 2nd order centered-difference in vertical direction

- Subgrid scale model used: Lagrangian scale-dependent dynamic model - Courant-Friedrichs-Levy condition for stability ~ 0.3

To compare the previous results with a LES code, case 2 is implemented as above. Figure 5 displays the LES results after 22-hour simulation in quasi-equilibrium state. The left panel of this figure shows the TKE results which are analogous to Fig.2 in terms of the order of the magnitude of the results and also the fact that both peak near the ground that confirm previous results.

Fig.5. LES results: TKE (left) and Reynolds stresses (right)

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Furthermore, this figure also indicates the plot of the dominating Reynolds stresses, which again peak near the wall. Note that all results in this figure are averaged in the z-direction (other horizontal direction) and that is why they look smoother than the previous results. Figure 6 demonstrates the output eddy-viscosity of LES code, which is determined from the following equation:

νT =R213 + R

223

dudy

⎛⎝⎜

⎞⎠⎟

2

+ dwdy

⎛⎝⎜

⎞⎠⎟

2

(7)

(8

Fig.6. Turbulent viscosity of LES code

As one could observe from above, the turbulent viscosity peaks almost in 50m above the ground, which is different from Fig. 4 where it is highest about 5m above the boundary. This difference could be due to the inherent differences between the LES, DNS and RANS models used here as well as the distinct numerical setup of the problem in case 1 and 2. The sum of Reynolds stresses in LES is shown in Fig.5 that peaks near the ground as expected. However, the velocity gradients or strain terms in equation (7) are also high near the wall and that is why in Fig.6 eddy viscosity does not peak near the ground. Nevertheless, the value of the eddy-viscosity in both cases is in the same order of magnitude and that qualitatively confirms the previous results.

c) Case 3: FV on inhomogeneous terrains One of the recent research problems that could be investigated further using these codes is studying the mean and turbulence characteristics in inhomogeneous surfaces, which is the case in most of the real-world situations. Investigating the wind effects and pollution dispersion on urban geometries and canyons is one of the current research efforts. Moreover, understanding the physical mechanisms behind the katabatic and anabatic winds on the steep slopes or mountains which are usually driven by heat and occur every day in sloping areas is vey important. Figure 7 demonstrates two examples of such flows.

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Fig.7. Flow field around a building (left) and mountaintop (right) “courtesy of U of Tokyo

and UAF”

Hence to that end, we have used the FV code in case 1 and added a mask layer with an elliptical equation near the ground in the middle to simulate the wind effects on a hill. Note that this process is implemented conveniently in FV code; However its implementation is a little challenging in finite difference codes and may require additional schemes such as immersed boundary methods. We have defined the following numerical domain in Fig. 8. All other numerical details are the same as case 1 except the additional mask layer in the middle.

Fig.8. Numerical domain of case 3 showing velocity profiles

First results are the TKE plots depicted in Fig. 9. Similar to case 1, both of the plots are analogues with little differences due to the structure of the models. We also observe that both graphs peak near the boundaries and also over the hill. The second result compares the Smagorisnky model with k-ε eddy-viscoity in such slope flows. Figure 10 displays that they are not the same again, but both are high near the wall. The Smagorinsky model even shows higher peak on top of the inhomogeneity due to high velocity gradients.

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Fig.9. TKE results obtained from the FV DNS code (left) and k-ε equation (right) in case 3

Fig.10. Turbulent viscosity from Smagorinsky (left) and k-ε models (right) in case 3

4. Conclusion:

The 2.29 FV code was used to produce turbulent flows. To that end logarithmic wind profiles were imposed and a random noise was added to their values at each time-step. Two turbulence models, including Smagorinsky and k-ε, were implemented and tested in two cases using this FV code. The results were satisfactory showing peaks for the TKE and production terms near the ground. k-ε model is relatively easy to implement in a fully explicit way, however it becomes challenging in implicit schemes since it includes two nonlinearly coupled equations. On the other hand, k-ε model uses simplistic dissipation equation and works poorly for flows with strong separations and rotating flows.  Case 2 employed a 3D large-eddy simulation code to demonstrate the differences among the 2D FV and 3D LES results. The TKE graphs looked similar and the eddy-viscosity magnitudes were in the same order though there were some differences in their profiles. The last case was aimed to show further practical applications, which are the focus of some of the recent research efforts. The mean and turbulence dynamics on inhomogeneous terrains could be investigated using these numerical codes. In fact in highly turbulent flows, which most of times is the case particularly in geophysical flows, the implementation of DNS is impossible considering the current computational resources. Thus, the provided idea here in using the turbulence models to convert DNS to RANS code will be very helpful for further research developments.

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5. Future work:

In order to change the DNS code into RANS, one needs to convert the scalar viscosity defined in the code into a space-dependent turbulent viscosity resulted from the k-ε model in an efficient way. This could be implemented using appropriate optimized algorithms and then the 2.29 FV code could be used in higher Reynolds number and lower viscosity flows. We might also improve the generated turbulence mechanism by using temporally and spatially correlated noises. This will enhance the quality of the generated perturbations and they should be verified with some other results too. After implementing these two steps, the created RANS code could be used in many research disciplines that include high Reynolds flows such as wind over inhomogeneous terrains such as the one investigated in case 3 of this project. References: Boussinesq, J.: Theorie de l'ecoulement tourbillant, Mem. pres. Acad. Sc. XXIII, 46, 1877.

Cebeci, T., J. P. Shao, F. Kafyeke, and E. Laurendeau, 2005: Computational Fluid Dynamics for Engineers. 328ff. pp.

Ferziger, J. H., and I. M. Peric: Computational Methods for Fluid Dynamics. 3rd ed, 2002.

Fletcher, C. A. J.: Computational Techniques for Fluid Dynamics. 2nd ed. 439 pp, 1991.

Jones, W. P. and Launders, B.E.: The Predicition of Laminarization with a Two-Equation Model of Turbulence. Int. J. Heat and Mass Transfer 15, 301-314, 1972.

Kolmogorov, A.N.: Equations of Turbulent Motion of an Incompressible Fluid, Izv. Akad. Nauk. SSR Ser. Phys. 6, 56, 1942. (English translation, Imperial College, Mech. Eng. Dept. Report ON/6, 1968).

Menter, F.R.: Two-Equation Eddy Viscosity Turbulence Models for Engineering Applications. AIAA J. 32,

1299 1310, 1994. Prandtl, L.: Uber ein neues Formelsystem fur die ausgebildete Turbulenz, Nachrichten von der Akad. der

Wissenschaft in Gottingen, 1945.

Rotta, J.C.: Turbulent Boundary Layers in Incompressible Flow, Prog. Aero. Sci. 2, 1, 1962. Wilcox, D. C: Turbulence Modeling for CFD. DCW Industries, Inc., 5354 Palm Drive, La Canada, Calif.,

1998.