Comparison of LES studies in backward-facing step usingChebyshev multidomain and Legendre spectral element methods

H. Kanchi∗ and F. Mashayek†

University of Illinois at Chicago, Chicago, IL, 60607, USA

K. Sengupta,‡

Boeing Research and Technology - India, Bengaluru, India

G. B. Jacobs,§

San Diego State University, San Diego, CA, 92182, USA

and

P.F. Fischer,¶

Argonne National Laboratory, Argonne, IL, 60439, USA

Large-eddy simulation (LES) has been increasingly applied to complex shear flows en-countered in practical engineering devices. Here, we present the comparison of LES studiesperformed using the explicit Chebyshev multi-domain method and the semi-implicit Leg-endre spectral element method. Two different backward-facing step configurations aresimulated at Reynolds numbers 5000 and 28, 000. A grid resolution study at Re = 28, 000is carried out to compare the number of spectral elements required by the two spectralmethods to resolve the flow field. Effect of synthetic turbulence at inflow is investigatedusing two techniques of generating inflow turbulence at Re = 28, 000. In the LES studyat Re = 5000, it is observed that a semi-implicit method requires small time step size tocorrectly predict transition to turbulence.

Nomenclature

E Number of spectral elementsN Number of solution pointsH Step heightLx Streamwise extent of geometryLi Length of inlet sectionLe Length of post-step sectionLy Spanwise extent of geometryLz Wall normal extent of geometryLr Reattachment lengthUz Mean inflow velocityU0 Reference velocityMa Mach numberTwall Wall temperatureP1 Inlet pressurep pressure at a grid point△t time step size

∗Graduate student, Department of Mechanical and Industrial Engineering, Chicago, IL 60607.†Professor, Department of Mechanical and Industrial Engineering, Chicago, IL 60607, AIAA Associate Fellow.‡Aerospace Research Engineer, Boeing Research and Technology - India, Bengaluru, India§Associate Professor, Department of Aerospace Engineering and Engineering Mechanics, CA, 92182, AIAA Member.¶Senior Computational Scientist, Mathematics and Computer Science Division, Argonne, IL 60439

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20th AIAA Computational Fluid Dynamics Conference27 - 30 June 2011, Honolulu, Hawaii

AIAA 2011-3557

Copyright © 2011 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

I. Introduction

Spectral element methods, first introduced by Patera,1 offer several attractive features that make themexcellent candidates for large-eddy simulation (LES) of practical flows.2,3 They combine the accuracy of thespectral schemes with the flexibility of the finite elements method, thereby allowing high-order discretizationof complex geometries.4,5 The spatial resolution can be conveniently altered either by increasing the numberof elements (h-refinement) or by increasing the polynomial order within the elements (p-refinement). Insmooth solution spaces, the method provides asymptotically exponential rate of spatial convergence with p-refinement. Low dispersion errors in these methods lead to high temporal accuracy, making them suitable forwave dominated, unsteady problems. Moreover, low degree of data connectivity between elements facilitatesefficient parallel implementation.

The complex flow morphology in a backward-facing step, resulting from the separation of the upstreamboundary layer at the step to form a free shear layer which subsequently reattaches on the lower wall,makes the flow a challenging test case for any LES code. Our group developed a spectral/hp element large-eddy simulation technique for compressible flows using a staggered grid Chebyshev multi-domain spectralmethod (CMSM).6,7 The CMSM code has been used to obtain results, in a backward facing step geometry,for various LES cases.8,9 All the previous results using the CMSM code were a systematic investigationconsidering only the flow of the gas. The CMSM code was part of an ongoing research work to develop amethod for simulating reacting flows. For low-Mach number simulations of reacting flow CMSM code wastoo expensive, so our group decided to move to an incompressible Legendre spectral code with low-Machnumber formulation. The code is called Nek5000 and was developed by Fischer et al.10

Modeling of only the small scale flow-physics as opposed to the entire spectrum of turbulence (as done inRANS), reduces the complexity of modeling in LES. The most common approach for modeling the sub-gridterms to close the filter Navier-Stokes equations are Smagorinsky model and dynamic Smagorinsky model.Spectral element filtering strategies for LES was studied by Blackburn and Schmidt.11 They investigatedthree different filtering techniques within two-dimensional spectral elements for the simulation of incom-pressible turbulent channel flow with a dynamic sub-grid model. Fischer and Mullen12 introduced a filteringtechnique to stabilize spectral element direct simulation. Levin et al.13 applied a two-step filtering procedureto control the growth of non-linear instabilities in their eddy resolving spectral element ocean model. A tri-angular spectral element-Fourier method was used by Karamanos14 for LES with an explicit sub-grid model.Recently, an alternate approach for large-eddy simulation, using explicit filtering of the solution without anysub-grid modeling, has been proposed in Refs.15–17

Before moving on to extend Nek5000 to simulate reacting flow cases, we wish to validate LES results ofNek5000 in a backward-facing step geometry for a non-reacting flow. In this paper, two different Reynoldsnumbers are considered for the LES studies, Re = 5000 and Re = 28, 000. Nek5000 results are comparedwith experimental data and CMSM results for the same configurations. One of the critical issues in thesimulation of spatially developing flow with turbulent inflow is the prescription of the inlet turbulence. Bothexperimental and numerical studies indicate a strong sensitivity of the results to the inlet turbulence.18,19

Two techniques of generating inlet turbulence are investigated: a stochastic model and uniform randomfluctuations. A description of the stochastic model can be found in Refs.20,21

II. Numerical methodology and computational domain

Numerical schemes can crucially affect the fidelity of large-eddy simulation. Numerical errors in spaceand time can smear the solutions, overly dissipate turbulence, and so lead to an inaccurate computation ofthe turbulent flow. Among the errors related to the numerical scheme, the diffusion and dispersion errors,have the most serious consequences on the accuracy of the solution. High-order methods, which have lowdispersion and diffusive errors are therefore necessary for accurate and reliable simulations. Essentiallytwo high-order schemes, compact finite difference and spectral/hp element, suitable for simulating practicalturbulent flows are in practice. This paper compares simulation results for two spectral/hp element LESmethodologies, namely CMSM and Nek5000.

II.A. Chebyshev multidomain spectral method (CMSM)

The staggered-grid Chebyshev multi-domain spectral method solves the unsteady compressible Navier-Stokesequations and energy equation, with appropriate boundary conditions. The three-dimensional computational

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domain is divided into non-overlapping hexahedral sub-domains. The staggered grid method uses two setsof grids, one for the solution (Chebyshev-Gauss grid) and one for computation of the fluxes (Chebyshev-Gauss-Lobatto grid). The solution vector is approximated on the (N +1)× (N +1)× (N +1) tensor-productof Gauss points with N th approximation order of Lagrange interpolating polynomial. The flux vectors arecomputed by reconstructing the solution at the (N + 2) × (N + 2) × (N + 2) tensor-product of Lobattopoints through interpolations using Lagrange polynomials. Once the solution values are interpolated to theLobatto grid the advective fluxes are computed. The interface points will have different flux values dueto discontinuity of solution values at the sub-domain boundaries. The patching of the advective fluxes atthe interface is done by solving a Reimann problem at the sub-domain boundaries. The viscous fluxes arecomputed in two steps. The solution interpolant at the Lobatto grids must be continuous for a uniquefirst derivative at the subdomain interfaces. This is ensured by a Dirichlet patching, or averaging of thesolution on both sides of the interface. After the Lobatto interpolants for the solution values are patched,their derivatives are computed at the Gauss points. The gradients are then interpolated back to the Lobattopoints. The interface condition for viscous fluxes and any Neumann boundary condition are applied at thispoint. Finally, the total flux is obtained by adding the inviscid and viscous parts. Once the total fluxesare computed at the Gauss-Lobatto grids, these are differentiated and evaluated at the Gauss grid, to givepointwise derivatives. Finally, the semi-discrete equation for the solution unknowns at the Gauss grid isadvanced in time explicitly using a 4th-order low storage Runge-Kutta scheme. A detailed description ofthe method applied to LES studies is given in Refs.8,9

II.B. Legendre spectral element method (Nek5000)

The Legendre spectral/hp element method (Nek5000) used here, solves the unsteady incompressible Navier-Stokes equations with energy equation and convection-diffusion equation for passive scalars, subject toboundary conditions and initial conditions. The spatial discretization in Nek5000 is based on spectral elementmethod. The computational domain is represented as a set of (disjoint) macro-elements with the solution andgeometry being approximated by higher-order polynomial expansions within each macro-element. Three-dimensional domains are broken up into hexahedral elements. Within each element, a local Cartesian meshis constructed corresponding to a (N +1)×(N +1)×(N +1) tensor-product Gauss-Lobatto Legendre (GLL)collocation points. The dependent variables are expanded in terms of N th order tensor-product (polynomial)Lagrangian interpolants through GLL collocation points. The transient Nek5000 simulations employs bothexplicit and implicit time-integration techniques, and the solution is updated to the new time level usingvarious combinations of multistep and multistage schemes. The convective terms are treated explicitly anddiffusion terms are treated implicitly using a q-order backward differentiation multistep scheme. A morecomplete description of the method is given in Refs.5

II.C. LES methodology

The most common approach for modeling the sub-grid terms to close the filterd Navier-Stokes equations areSmagorinsky model and dynamic Smagorinsky model. LES studies by Sengupta and coworkers9 at Re =5, 000 in an open backward facing step geometry have shown that modeling the sub-grid terms with dynamicSmagorinsky model gives similar results to those obtained by using an explicit filter. Using explicit filteringof the solution without any sub-grid modeling, has been proposed in Refs.15–17,22 Explicitly filtered (“nomodel”) LES acts to control the numerical noise at the grid cut-off and creates an implicit sub-grid dissipationmechanism in the flow. The filter is designed to eliminate grid-to-grid oscillations without affecting theresolved scales. Interpolant-projection filtering procedure is used to filter the solutions variables. In thismethod, the filtered variable of degree N is obtained by projecting the variable back and forth to a lowerorder approximation of degree M defined on a subset of the original nodal values. It is imperative to usea filter that does not affect the large scales of the flow but at the same time prevents the numerical noise(inherent in under-resolved simulations with high-order methods) from contaminating the solution at highwavenumbers. Sengupta and coworkers9 tested filters of different strengths to determine the most effectivefilter. They observed that the filter M = N−2 resolves the large scales accurately. In our study, an explicitlyfiltered (“no model”) LES scheme is employed with a filter strength of M = N − 2.

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Li

H

Le

Ly

x

yz

Lz

5H

Figure 1. Schematic of backward-facing step geometry for LES study.

II.D. Computational domain for Re = 5000

The computational domain is shown in figure 1. The open backward-facing configuration was experimentallystudied by Jovic and Driver.23 Later, Le et al.24 and Akselvoll and Moin25 performed direct numerical andlarge-eddy simulations, respectively of the flow. All variables are normalized by the step height (H) and meaninlet velocity (U0). The computational domain has a streamwise extent, Lx = 30 with an inlet section ofLi = 10. The vertical and spanwise extent of the domain are Lz = 6 and Ly = 4, respectively. The inflow hasa fully developed flat plate boundary layer profile. For both of the methods, the mean inflow velocity, U(z),is obtained from Spalart’s26 boundary layer simulation at Reθ = 670, where θ is the momentum thickness.The boundary layer thickness is δ99 = 1.2H. The Reynolds number based on the mean velocity (U0) and stepheight (H) is 5000. The stochastic model generates the fluctuations in the three velocity components from thecorresponding rms profiles calculated in Spalart’s simulation.26 Periodic boundary conditions are applied inthe spanwise direction and free-stream boundary condition is applied at the top in the wall normal direction.The outflow boundary conditions are different for the two methods. For CMSM, the outflow boundarycondition is set by the mean velocity profile from Le et al.24 for the streamwise component and other twovelocity components are set to zero. For Nek5000, zero gradient boundary condition is prescribed at theoutflow.

II.E. Computational domain for Re = 28,000

Schematic of the confined (wall at the top) backward-facing step geometry is shown in figure 1. The compu-tational domain is the same as that investigated by Akselvoll and Moin25 in their “high Reynolds number”study. All variables are normalized by the step height (H) and mean inlet velocity (U0). The streamwiseextent (Lx) is 30 which includes an inlet section (Li) of length 10. The inlet channel height of 4H gives anexpansion ratio of 1.25. The computational domain has a spanwise extent of Ly = 3 and vertical extent ofLz = 5. The Reynolds number based on the step heights and mean velocity is 28, 000, which is in the rangeof practical engineering interest. Simulation results are validated against the experimental data of Adams etal. 27 The inflow turbulence is generated using the stochastic model. The rms profiles of the three velocitycomponents, which are input to the model, were taken from Akselvoll and Moin.25 For both of the methods,streamwise velocity u(z) is initialized with laminar channel flow profile,

u(z) = −6

[(z − 1

4

)2

−(

z − 14

)](1)

for z > 1 and set to 0 for z < 1. The other velocity components are taken as zero. No-slip conditions areenforced in the wall normal direction and periodic boundary conditions are applied in the spanwise direction.

In Nek5000, zero gradient boundary condition is prescribed at the outflow. For CMSM, the temperatureat the wall (Twall) is set to 1/Ma2, while the density is set to 1. The pressure is thereafter computed from theequation of state. At the outflow in CMSM, the streamwise velocity is set to the averaged profile obtainedfrom Akselvoll and Moin,25 while the wall normal and spanwise velocities are set to zero. The pressureand temperature are considered uniform across the height of the combustor. The magnitude of the outflow

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pressure is derived from the pressure at the inlet using the following expression,

p = P1 −3Li

4Re− 12(Lx − Li)

25Re, (2)

where P1 is given by, P1 = Twall/γ. The pressure in the inlet channel is initialized by,

p = P1 −3x

4Re(3)

while that in the post expansion section is initialized as,

p = P1 −3Li

4Re− 12(x − Li)

25Re, (4)

The initial temperature is set to Twall throughout the domain.

III. Results and discussions

III.A. Grid resolution study at Re = 28,000

Table 1. CMSM and Nek5000 LES cases for Re=28,000 flow.

Method Case Mach E N Lr

CMSM Ma2-GRID1 0.2 2232 7 6.85Ma2-GRID2 0.2 3264 7 7.1

Nek5000 N-GRID1 — 2232 7 5.69N-GRID2 — 4116 7 5.91

LES with a high-order method is sensitive to the grid resolution. Hence, it is essential to test the effectof grid resolution and achieve grid independent results. The cases considered for the two methods in thisstudy are given in table 1.

0 0.5 10

1

2

3

4

5

z/H

0 0.5 10

1

2

3

4

5

Exp., Adams et al.LES, Ma2-GRID1LES, Ma2-GRID2

0 0.5 10

1

2

3

4

5

0 0.5 1U/U

0

0

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z/H

0 0.5 1U/U

0

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4

5

0 0.5 1U/U

0

0

1

2

3

4

5

(a) (b) (c)

(d) (e) (f)

Figure 2. Comparison of the mean streamwise velocity at Re = 28,000 for CMSM with experimental data. Profilesshown at (a) x = 3.2, (b) x = 4.5, (c) x = 5.9, (d) x = 7.2, (e) x = 12.2 and (f) x = 17.5.

The CMSM cases were simulated by Sengupta.21 It should be noted that all the experimental data andNek5000 results are for incompressible flow. Therefore, comparison with the results can only be made forlow Mach number, where compressibility effects are expected to be small. Hence, the Mach number is taken

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as 0.2 with the understanding that the quasi-incompressible flow would resemble the incompressible flowexperiment. As the first measure, we compare the computed averaged reattachment lengths (Lr) with theexperimental value. The reattachment lengths of 6.85H and 7.1H for the quasi-incompressible cases, Ma2-GRID1 and Ma2-GRID2, are 2.2% and 6.0% longer, respectively than the experimental value of 6.7H. Thehigher value for the finer grid is possibly due to “over-filtering” of the solution. To compare the simulationresults of Nek5000 with CMSM, we started with a case (N-GRID1) having the same grid resolution as thecase Ma2-GRID1. The two cases, N-GRID1 and N-GRID2, have a reattachment length of 5.69H and 5.89H,respectively. The reattachment length is under-predicted as compared to the experimental value. A highergrid resolution study (not shown) did not improve the prediction of average reattachment length. WithNek5000, being a semi-implicit method, for stability considerations a time step size of △t = 0.01H/U0 wasused for the simulations. A study with smaller time step size (not shown) also did not increase the computedreattachment length. In CMSM, the time stepping is explicit and for stability of the method a time stepsize of △t = 7.39 × 10−4H/U0 was used.

The mean streamwise velocity is shown at six different locations in figure 2 for CMSM. The case Ma2-GRID1, provides slightly better prediction of the experiment values over Ma2-GRID2. Excellent agreementis observed across the domain height, in the recirculation region (x = 3.2, 4.5, 5.9). The velocities areunder-predicted within the boundary layer after the reattachment (x = 7.2) and in the recovery zone (x =12.2, 17.5). The under-predictions are as a result of the partial resolution within the boundary layer. AsMa2-GRID1 provided better results, it will be used for comparison with results obtained from Nek5000simulations.

0 0.5 10

1

2

3

4

5

z

0 0.5 10

1

2

3

4

5

0 0.5 10

1

2

3

4

5

Exp. Adams et al.N-GRID1N-GRID2Ma2-GRID1

0 0.5 1U

0

1

2

3

4

5

z

0 0.5 1U

0

1

2

3

4

5

0 0.5 1U

0

1

2

3

4

5

(a) (b) (c)

(d) (e) (f)

Figure 3. Comparison of mean streamwise velocity for Nek5000 with CMSM and experimental data at Re = 28,000.Profiles shown at (a) x = 3.2, (b) x = 4.5, (c) x = 5.9, (d) x = 7.2, (e) x = 12.2 and (f) x = 17.5.

The average streamwise velocity profiles for N-GRID1 and N-GRID2 cases are compared with experi-mental data and Ma2-GRID1 results in figure 3. Excellent agreement is seen in all the sections just aboveand below the shear layer (z = 1). Velocity values are under-predicted in the recirculation zone at x = 4.5and around the reattachment region in x = 5.9. At x = 7.2, the values are over-predicted closer to thebottom wall. In the recovery region, (x = 12.2, 17.5), the prediction is accurate. In the top half of all the sixsections, there is under-prediction of values because of under-resolution of the grid. More spectral elementsare required to better resolve the solution in this region.

For Nek5000, a grid resolution study (not shown) was conducted and a grid with E = 4116 and N = 7(case N-GRID2) provided good validation with the experimental data. The mean velocity profiles for theN-GRID2 case compared with experimental data, N-GRID1 and Ma2-GRID1 are shown in figure 3. Atx = 4.5, 5.9, the prediction of N-GRID2 is closer to experimental data. Increasing the number of spectralelements has improved the prediction in the recovery region. The top half in all the six sections shows bettervalidation with experimental data. Overall Nek5000 shows somewhat better agreement with experiment as

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compared to CMSM. It is interesting to note that Nek5000 give good validation with E = 4116 elementsas compared to E = 2232 for CMSM. Nek5000 almost requires twice the number of spectral elements ascompared to CMSM, to resolve the flow field at Re = 28, 000.

0 0.05 0.1 0.15 0.20

1

2

3

4

5

z

0 0.05 0.1 0.15 0.20

1

2

3

4

5

0 0.05 0.1 0.15 0.20

1

2

3

4

5

Expt. Adams et al.N-GRID2Ma2-GRID1

0 0.05 0.1 0.15 0.2u

rms

0

1

2

3

4

5

z

0 0.05 0.1 0.15 0.2u

rms

0

1

2

3

4

5

0 0.05 0.1 0.15 0.2u

rms

0

1

2

3

4

5

(a) (b) (c)

(d) (e) (f)

Figure 4. Comparison of averaged streamwise normal fluctuations of Nek5000, CMSM and experimental data for Re= 28,000. Profiles shown at (a) x = 3.2, (b) x = 4.5, (c) x = 5.9, (d) x = 7.2, (e) x = 12.2 and (f) x = 17.5.

Figure 4 shows the comparison between root mean squared streamwise velocity (urms) for CMSM (Ma2-GRID1), Nek5000 (N-GRID2) and experimental data. For CMSM, there is good agreement with the referencedata. The peak values at x = 4.5 and x = 5.9 appear to be slightly under-predicted. The experimentaldata at the same time has considerable scatter around the peak. In the recovery region (x = 12.2, 17.5)the simulation over-predicts the rms values. This is due to the coarse resolution at the wall where theboundary layer is marginally resolved. For Nek5000, urms values agree well with the experimental databelow the shear layer except for x = 3.2, where the values are over-predicted close to the bottom wall. Forx = 3.2, 4.5, 5.9, 12.2, there is a large difference in the urms values of CMSM and Nek5000 close to the topwall (z = 4.5). Nek5000 requires more grid resolution in this region for better results. At x = 12.2 andx = 17.5, Nek5000 predicts the experimental data accurately near the top wall. The peak values are betterpredicted by CMSM than Nek5000.

III.B. Influence of inflow boundary condition

Inlet turbulence is often generated by imposing random fluctuations on the mean velocity. The time seriesgenerated by random fluctuations include all the frequencies and have no spatial or temporal correlations,thereby having little resemblance to the actual physics of the flow. Sengupta and coworkers9 studied theinfluence of inflow boundary condition by simulating an open backward-facing step geometry at Re = 5000using the CMSM code. Two different methods for generating the inflow turbulence were tested: uniformrandom fluctuations imposed on the mean velocity and fluctuations obtained using the stochastic model.20,21

They observed that imposing random fluctuations at the inlet to mimic inflow turbulence results in significantinaccuracies in the predictions downstream of the step. In LES study at Re = 5000 with Nek5000, it wasobserved that the two methods of generating inflow turbulence gave similar results (not shown). Here, weinvestigate the influence of inflow boundary condition for the Nek5000 case (N-GRID2) at Re = 28, 000,using two methods of generating inflow turbulence.

Figure 5 compares the mean velocity profile for the two inflow boundary conditions with experimentaldata. We see that imposing random fluctuations at the inlet to mimic inflow turbulence results in averagedstreamwise velocity values similar to those obtained with the stochastic model. At x = 5.2, the randomboundary condition gives a better prediction of the velocity in the recirculation zone compared to stochasticboundary condtion. The computed average reattachment length with random inflow boundary condition is

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0 0.5 10

1

2

3

4

5

z0 0.5 1

0

1

2

3

4

5

0 0.5 10

1

2

3

4

5

Exp. Adams et al.N-GRID2 - stochN-GRID2 - randMa2-GRID1

0 0.5 1U

0

1

2

3

4

5z

0 0.5 1U

0

1

2

3

4

5

0 0.5 1U

0

1

2

3

4

5

(a) (b) (c)

(d) (e) (f)

Figure 5. Comparison of averaged streamwise velocity for random and stochastic inflow fluctuations compared withexperiments. Profiles shown at (a) x = 3.2, (b) x = 4.5, (c) x = 5.9, (d) x = 7.2, (e) x = 12.2 and (f) x = 17.5.

6.49. This compares well with the experimental value of 6.7. The streamwise normal velocity fluctuations(not shown) also give same results for the two inflow boundary conditions. The average time taken per timestep on 384 processors for random boundary condition is 0.561 seconds and stochastic boundary condition is0.69 seconds. The random boundary condition is simple to program and runs faster on a parallel machine.In Nek5000, for high Reynolds number flow using random inflow boundary condition is advantageous.

III.C. Low Reynolds number (Re = 5000) simulation

In previous sections, we compared the Nek5000 with CMSM for a high Reynolds number case. It wouldbe helpful to know how these spectral methods compare at low Reynolds number (Re = 5000). Herewe compare the Nek5000 results for an open backward-facing step geometry with CMSM results, directnumerical simulations (DNS) results, and experimental data.

0 0.5 10

0.5

1

1.5

2

2.5

3

y

0 0.5 10

0.5

1

1.5

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3

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0.5

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1.5

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0 0.5 10

0.5

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3Exp. JD∆t = 0.01∆t = 0.001CMSM

0 0.5 1U

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1.5

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3

DNS Lee et al

x = 6 x = 10 x = 19x = 4

x = 7.5 x = 15x = 2.5x = 0.5

Figure 6. Comparions of averaged streamwise velocity of Nek5000 at different time step sizes with CMSM and exper-imental data for Re = 5,000.

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For both of the methods, Nek5000 and CMSM, the domain was discretized with E = 656 spectralelements with polynomial order N = 7. The mean streamwise velocity comparison for this case is shownin figure 6. As in the Re = 28, 000 case, the Nek5000 simulation was first conducted with a time step sizeof △t = 0.01H/U0. The mean streamwise velocity at this time step value is shown in figure 6. There isgood agreement between the Nek5000 simulation results, experimental data and DNS results downstreamof x = 4.0. In the recovery region, values of Nek5000 are over-predicted compared to DNS results at x = 15and experimental results at x = 19. At x = 2.5, Nek5000 does not predict the transition to turbulence whileCMSM predicts the transition very well.The time step size for CMSM is about △t = 1.4 × 10−3H/U0.

Various grid resolution and parametric studies were conducted with Nek5000 to investigate transition toturbulence close to the step; none could predict the transition. Choi and Moin28 showed that very large timesteps (implicit scheme) cause the turbulence to decay to laminar state. Use of large time steps implies thatthe small scales can have large errors, which can corrupt the solution. In LES at low Reynolds numbers, thesmall scales in regions of locally low Reynolds numbers have larger effect on the large scales. Hence, resolvingthese small scales at low Reynolds numbers are important compared to high Reynolds numbers. Thereforein Nek5000, to capture the small time scales at low Reynolds numbers, smaller time steps are requiredcompared to high Reynolds number cases. From figure 6 it can be seen that △t = 0.001H/U0 predicts thetransition to turbulence accurately and provides more accurate results as compared to the larger time step.At low Reynolds numbers, Nek5000 is stable for a larger time step; however to capture the small scales asmaller time step is required. This is not the case for CMSM as the stability criterion imposes a smallertime step value for explicit time stepping. At low Reynolds numbers the total computational time taken forthe whole simulation is comparable for both of the methods.

III.D. Comparison of computational time for CMSM and Nek5000

A direct comparison of computational time of the two spectral methods, one which solves compressibleNavier-Stokes equations (CMSM) with one which solves incompressible Navier-Stokes equations (Nek5000),is not possible. Comparison of computational time for the two methods can only be made for low Machnumber, where compressibility effects are expected to be small. Hence, CMSM is simulated with a Machnumber of 0.2. It should, however, be pointed out that using low Mach number increases the time step size.

Table 2. Comparison of computational time of CMSM and Nek5000 for one flow through.

Re Method Processors Computational time5000 CMSM 128 1.07 × 103

Nek5000 128 6.0 × 103

28, 000 CMSM 512 2.7 × 103

Nek5000 512 1.1 × 103

It is important that same grid resolution be simulated on same number of processors for the comparisonof computational time of the two methods. For Re = 5000 case, 656 spectral elements with polynomialorder 7 were used for both of the methods. The cases were run on 128 processors and the comparison ofcomputational time for one flow through of CMSM and Nek5000 is given in table 2. It is observed that, theexplicit method (CMSM) is about 5.5 times faster than the semi-implicit method (Nek5000). The advantageof using a larger time step for semi-implicit scheme with Nek5000 is lost, because of the requirement of smalltime steps to capture the small scales of turbulence. The comparison of computational time of CMSM andNek5000 at Re = 28, 000 was done for a case with E = 2232 and N = 7. 512 processors were used tosimulate the case. From table 2 it is observed that, Nek5000 is about 2.45 times faster than CMSM. WithNek5000, larger time steps can be taken at high Reynolds numbers while CMSM is constrained by smalltime step size imposed by explicit time stepping.

IV. Conclusions

The semi-implicit Legendre spectral element method (Nek5000) predicts the experimental and compu-tational data accurately. Nek5000 requires more elements compared to CMSM to resolve all the velocityscales at Re = 28, 000. However, Nek5000 requires the same number of elements as CMSM at Re = 5000

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to resolve the velocity scales. At high Reynolds numbers, larger time step size can be used in Nek5000 ascompared to CMSM. Hence, Nek5000 requires less computational time than CMSM to completely simulatethe Re = 28, 000 case. At low Reynolds numbers, Nek5000 requires small time step size to predict the tran-sition to turbulence. While in CMSM the time step size is imposed by the stability criterion and is of thesame order as Nek5000. However, time taken per time step by CMSM is much lower than that of Nek5000;therefore, CMSM is faster than Nek5000 at Re = 5000. In Nek5000 at Re = 28, 0000, it is computationallyless expensive to use random fluctuations to generate synthetic turbulence as compared to the stochasticmodel that is needed by CMSM.

Acknowledgments

The support for this work was provided by the U.S. Office of Naval Research under grant N00014-08-1-0612. The simulations were conducted on the NCSA Abe supercomputer and SDSC Trestles supercomputer.

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