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Proceedings of COBEM 2005 18th International Congress of Mechanical Engineering

Copyright © 2005 by ABCM November 6-11, 2005, Ouro Preto, MG


NUMERICAL CHARACTERIZATION OF A TURBULENT

BACKWARD-FACING STEP FLOW


Cleber Spode

Federal University of Uberlândia, School of Mechanical Engineering, 38400-902 Uberlândia – MG, Brazil

cspode@mecanica.ufu.br


Rubens Campregher

Federal University of Uberlândia, School of Mechanical Engineering, 38400-902 Uberlândia – MG, Brazil

campregher@mecanica.ufu.br


Aristeu da Silveira Neto

Federal University of Uberlândia, School of Mechanical Engineering, 38400-902 Uberlândia – MG, Brazil

aristeus@mecanica.ufu.br


Abstract. The adiabatic three-dimensional turbulent flow over a backward-facing step has been studied here by the Finite Volume Method using parallel processing techniques applied to the incompressible Navier-Stokes equations. The classical flow over the backward-facing step is a benchmark for new fluid dynamics codes due to the fact that, despite its simple geometry, it presents a complex generation of three-dimensional structures, influencing the transition phenomenon and properties such as characteristics frequencies of vortex emission and re-attachment length. Based on the step high (h) and the free stream velocity (U0) the flow was simulated at Reynolds 5100 for an expansion ratio of 1.20. According to the literature, backward-facing step flow having such characteristics present a critical Reynolds number around 748. A Large Eddy Simulation technique with the Classical Smagorinsky model, adding the van Driest damping function to the near wall eddy-viscosity, was applied. Topological results of the flow are presented, identifying Kelvin-Helmholtz instabilities during the transition period. A second order scheme in space and time, with centered differences for the diffusive and advective terms and the three-time-level method for the time-dependent term was adopted. Results for the re-attachment length around 7h, for velocity and for the Reynolds stress tensor profiles presents good agreement against Direct Numerical Simulation and against experimental data.


Keywords: Turbulence Modeling, Parallel Processing, Backward-Facing Step Flow.


1. Introduction


Turbulent flows with attached and detached regions are common in Engineering applications, both in internal and external flows such as diffusers, combustors and channels with sudden expansions, and in external flows like around airfoils and buildings. In these situations, the adverse pressure gradient or the geometry forces the boundary layer to detach from the wall. The flow can latter reattaches forming a recirculation bubble. The backward-facing step, with its simple geometry can simulate all these phenomena. Furthermore, experimental and numerical data are vastly documented for values of reattachment length, Strouhal number, Reynolds stress tensor, mean velocity, and pressure coefficient.

Kuehn (1980), Durst & Tropea (1981) and Ötüngen (1991) studied the effects of expansion ratio () on the reattachment length () and they found that increases when grows. Armanly et al. (1983) studied the Reynolds number effect on the reattachment length and observed that even grows until reaching Re=1200 (based in the step high h and free stream velocity U0). In the range the reattachment length remains constant. Some inquires about the Kelvin-Helmholtz instabilities and its transition have been performed. Silveira Neto et al. (1993) have made a detailed topological description of the flow structures behind a backward-facing step, identifying coherent structures with characteristic frequencies. Friedrich and Arnal (1990) also studied the backward-facing step flow through Large-Eddy Simulation. A fine statistic analysis using Direct Numerical Simulation was done by Le, Moin and Kim (1997) that obtained excellent agreement against experimental data of Jovic and Driver (1994). The Reynolds number was 5100 and the authors have presented profiles of mean velocity, Reynolds stress tensor, reattachment length, pressure coefficient and characteristic frequencies. These data are used to validate the simulations in the present work. Investigations of the flow velocity profiles and turbulence intensities in the recovery region were conducted by Bradshaw and Wong (1972) and Kim, Kline and Johnston (1978). The experiments showed that the mean velocity were not fully recovery at more then 50 step heights behind the separation zone.

The Direct Numerical Simulation (DNS) can capture every scale presented in the flow but demands very fine meshes and high order schemes to do that. The Large-Eddy Simulation (LES) diminishes the mesh refinement by modeling the small scales of the flow, thus describing a more bulk flow behavior. Furthermore, LES presents excellent description of free shear flows, but it has difficulties on reproducing the near wall flow. Damping functions, as van Driest, can be adopted to solve the excessive near wall eddy viscosity produced by the algebraic LES models, improving its performance in the description of flows with both wall and free shear flows regions. On the other hand, the Reynolds Average Navier-Stokes method (RANS) has been used with success to simulate boundary layer flows, but it can not tackle detached and very transient flow structures.

The aim of the present work is to evaluate the Large-Eddy Simulation of the backward-facing step flow at Reynolds number 5100 using parallel processing and validate the numerical code developed in this benchmark case. Results for velocity and Reynolds stress tensor profiles, reattachment length, pressure coefficient and flow structures are presented.


2. Mechanical Modeling


2.1. Computational Domain


The computational flow domain used is shown in the Fig. 1. The domain consists of a streamwise length , including a inlet section , vertical height and spanwise with , where is the step high. The coordinate system is placed at the lower step corner, from which the reattachment length is measured.





Figure 1. Schematic view of the flow domain.


The expansion ratio is 6 and the Reynolds number is defined by:





The employed mesh is non uniform in the y direction and constant in x and z directions. A total of 384 computational cells are used in the x direction and 32 cells in the z direction. In the vertical direction, it is used 96 cells with non-uniform distribution, with 36 cells placed within the step (). The grid spacing in wall units are , , and , based on the inlet boundary layer shear velocity.


2.2. Numerical Method


The flow is modeled by the incompressible filtered Navier-Stokes equations, applying the Large-Eddy Simulation with the Smagorinsky model.








The left side of Eq.  represents the transient and advective terms, respectively. In the right side are the pressure gradient and the viscous term and the Eq.  represents the continuity.

The is the molecular viscosity plus the eddy viscosity computed by the Smagorinsky model, as shown in Eq. .





Where represents the strain rate tensor and the filter is defined as .

The Classical Smagorinsky model produces excess of eddy viscosity close to surfaces. The van Driest damping function has been adopted to correct this fail, as shown in Eq. .





where is the distance from the wall in viscous wall units , is the shear velocity , is the shear stress at the wall, is a constant usually taken to be approximately 25 (Ferziger and Peric, 1999) and is the Smagorinsky constant.

The equations are solved using a Parallel Finite Volume code, with second order centered differences for the advective and diffusive terms and the three-time-level temporal scheme. The arrangement is collocated, with the Rhie & Chow interpolation to prevent problems with checkerboard pressure fields and the SIMPLEC algorithm is used to the pressure-velocity coupling.


2.3. Boundary Conditions and Time Advancement


No-slip boundary conditions are applied to the bottom walls and in the spanwise direction, the flow is assumed to be statistically homogeneous and periodic condition is used. A no-stress wall is applied at the top boundary, the velocity at this wall is





The inlet flow consist of a mean velocity profile U(y) obtained from Jovic and Driver (1994) on which is superimposed a white noise of amplitude 1.25% U0 . The boundary layer thickness is about . To the exit it is applied a convective boundary condition (Eq. ).





Pauley, Moin & Reynolds (1988) showed that, for unsteady problems, the convective boundary condition is suitable for moving vortical structures out of the computational domain. This problem has been solved applying a damping function for the velocity in the end of the domain, according to Souza et al. (2002).

The time advancement using the three-time-level scheme is implicit and the time step adopted is which is five times bigger then the used in the Le, Moin and Kim (1997) DNS. The total simulation time is , and approximately is discarded to allow the passage of initial transients and to reach the statistical established regime of the flow. The time used to compute the statistics is , with 12000 samples (one sample every 5 time steps).


2.4. Parallel Processing


The parallelization technique adopted is based on the domain division and distribution of the load amount the computers of a cluster (Baker & Smith, 1996). Each machine is responsible for its computational domain, and the information is shared between the domains as shown in Fig. 2. The MPICH parallel library and IFC compiler are used and the program runs in the LTCM-UFU Cluster Beowulf of 10 Pentium 4(2.8GHz/1.5GbRAM) linked by a 1 Gbps net.





Figure 2. Parallel domain division.


3. Results


Transient results, identifying the initial Kelvin-Helmholtz instabilities and the formation of coherent structures and multiplicity of scales are shown. Results of velocity and Reynolds stress tensor profiles, pressure coefficient and reattachment length are presented in stations of interest in the central plane (z=W/2), comparing results of no-model and LES modeling against the experimental data of Jovic & Driver (1994).


3.1 Transient Aspects of the Flow


Kelvin-Helmholtz instabilities are induced in the inflectional velocity profile generated by the free stream that flows over the step presenting a mixing layer behavior. The structures starts quasi-two-dimensional, and quickly transition, see Fig. 3. Detailed topological features analysis in the backward-facing step flow was done by Silveira Neto et al. (1993) and by Delcayre and Lesieur (1997), identifying similar flow structures.





Figure 3. Iso-surface of vorticity (), colored by the pressure – LES case.


Figure 4 shows the tridimensional character of the flow in the statistical established regime, with counter-rotating cells before the mean reattachment length.




Figure 4. Counter-rotating cells. Plane Z x Y , , .

3.2. Mean Velocity and Reynolds Stress Tensor Profiles


Velocity and Reynolds Stress Tensor profiles are presented in four stations of interest, in the recirculation zone () , close to the mean reattachment point (), after the reattachment location () and far from the reattachment location (), where the boundary layer is attached and in developing process. The numerical results are plotted with the experimental data of Jovic & Driver (1994). Figure 5 shows the u mean velocity (U/U0).




Figure 5. Mean velocity profiles.


The mean velocity presents good agreement with experimental data, especially in the recirculation zone, where the profile is practically identical to the experiment. A minor discrepancy is observed in the free stream region, with numerical data overprediction. This behavior was also observed in the DNS results of Le, Moin and Kim (1997). The profile is underestimated, indicating that the numerical reattachment length is greater than 6.0h. The subsequence profiles are in good agreement despite small differences in the near wall region for the x/h=10 profile. The turbulence modeling plays a important hole in the simulation, providing better results for the u mean, reducing the difference between numerical and experimental data. It is important to remember that the measurements of Jovic & Driver (1994) has a mean velocity  2% accuracy and the Reynolds stresses were measured with  15% accuracy.

The RMS profiles of the longitudinal and vertical velocity fluctuations , and the Reynolds stress component are compared with experimental data in Figure 6. There are good, with small overprediction in the vertical component that also were observed in the DNS results of Le, Moin and Kim (1997). Surprising, the no-model results are better in the vertical component, but for the other components the turbulence modeling demonstrated its importance and caught more details, especially in the component, were the peak in the x/h=4 was captured, with under prediction for the no-model result. In a general form, the no-model results underpredict and the LES results overpredict the fluctuations, but the shape of the profiles are the same.











Figure 6. Reynolds stress tensor components.


3.3. Pressure coefficient and reattachment length


The pressure coefficient is defined as





where is the reference pressure extracted from the free stream flow. Good agreement against the experimental data is observed (see Fig. 7). There is a small difference near the reattachment due the fact that the reattachment is overestimated. In the recovery region the pressure coefficient converge to the experiment, fact that Le, Moin and Kim did not observe in the DNS simulation, where was obtained a slightly superior value for the pressure coefficient in the recovery region.





Figure 7. Step-wall pressure coefficient comparison.


The reattachment length is a important result in the backward-facing step flow that resume in one value only the complex interactions of flow structures. The p.d.f method are used to determine the reattachment location, , in which the mean reattachment point is indicated by the location of 50% forward flow fraction. The p.d.f results for the LES computation is shown in Fig. 8.





Figure 8. p.d.f. flow fraction - LES computation.


The reattachment point is and the comparison with the no model and DNS computation and experimental results are shown in Tab. 1.


Table 1. Reattachment length.







Exp. Jovic & Driver (1994)

6.0  0.15

Le, Kim and Moin (1997)

6.28

No model

7.5

LES – Cs=0.1

6.95


The reattachment length confirms the importance of the turbulence modeling, converging to values near the experimental and DNS data. The turbulent kinetic energy has your maximum value at the reattachment location, and in this context, a more accurate description was obtained with LES modeling resulting in a smaller reattachment length. Better results can be reached with the refinement of the mesh in the streamwise direction. Le, Moin and Kim (1997) had reduced in 10% the reattachment length with a improve of 60% in the number of cells in the spanwise direction in their computations.


4. Conclusions


The numerical simulation of a turbulent backward-facing step flow was performed to evaluate and validate the parallel numerical code developed. Coherent structures had identified Kelvin-Helmholtz instabilities, reproducing a mixing layer behavior. Statistical results show good agreement with data of Jovic & Driver (1994). The hole of the turbulence modeling was evidenced in the budgets of mean velocity and Reynolds stress tensor components, with little discrepancies in the reattachment zone. The reattachment length is satisfactory which agrees to within 13% of experimental data. All these results confirm the good description of the flow by the numerical code developed and validate it for future cases.


5. Acknowledgements


The authors Cleber Spode e Aristeu da Silveira Neto thank the financial support of CnPQ and Rubens Campregher of CAPES financial support.


6. References


Armaly, B. F., Durst, F., Pereira, J. C. F., Schönung, B., 1983, “Experimental and theorical investigation of backward-facing step” Journal of Fluid Mechanics 127, 473-496.

Baker, L. Smith, B.J., 1996, “Parallel Programing”, Computing McGraw-Hill.

Bradshaw, P., Wong, F. Y. F., 1972, “The reattachment and relaxation of a turbulent shear layer”, Journal of Fluid Mechanics 52, 113-135.

Delcayre, F., Lesieur, M., 1997, “Topological Features in the reattachment region of a backward-facing step”, First AFOSR International Conference on DNS and LES, Ruston, LA USA.

Durst, F., Tropea, C., 1981, “Turbulent, backward-facing step flows in two-dimensional ducts and channels”, Proceedings fo Third International Symposium on Turbulent Shear Flows, University of California, Davis, 18.1-18.5.

Ferziger, J.H. and Perić, M., 1999, “Computational Methods for Fluid Dynamics”, Springer.

Friedrich, R.,Arnal,M., 1990, “Analysing turbulent backward-facing stepflow with the lowpass filtered Navier-Stokes equations”, J. of Wind Eng. Industrial Aerodynamics 35, 101-128.

Kim, J., Kline, S. J., Johnston, J. P., 1978, “Investigation of separation and reattachement of a turbulent shear layer: Flow over a backward-facing step”, Rep. MD-37. Thermosciences Division, Dept. of Mech. Eng, Stanford University.

Le, H., Moin, P., Kim, J., 1997, “Direct numerical simulation of turbulent flow over a backward-facing step”, Journal of Fluid Mechanics 330, 349-374.

Jovic, S., Driver, D., 1994, “Backward-Facing Step Measurements at Low Reynolds Number, Reh=5000”, NASA Technical Memorandum 108807.

Kuehn, D. M., 1980, “Some effects of adverse pressure gradient on the incompressible reattaching flow over a rearward-facing step”, AIAA Journal 18, 343-344.

Ötügen, M. V. 1991., “Expansion ratio effects on the separated shear layer and reattachment downstream of a backward-facing step”, Experiments in Fluids 10, 273-280.

Pauley, L. L., Moin, P., Reynolds, W. C., 1988, “A numerical study of unsteady laminar boundary layer separation”, Rep. TF-34. . Thermosciences Division, Dept. of Mech. Eng, Stanford University.

Silveira Neto, A., Grand, D., Metais, O. Lesieur, M., 1993, “A Numerical investigation fo the coherent vortices in turbulence behind a backward-facing step”, Journal of Fluid Mechanics 256,1-25.

Souza, L., Mendonça, M., Medeiros, M., Kloker, M., 2002 “Three Dimensional Code Validation for Transition Phenomena”, III Escola de Primavera de Transição e Turbulência – Florianópolis.


7. Responsibility notice


The authors are the only responsible for the printed material included in this paper.

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