14th Australasian Fluid Mechanics Conference

Adelaide University, Adelaide, Australia

10-14 December 2001

Large Eddy Simulation of Flow Past a Circular Cylinder

H. M. Blackburn1 and S. Schmidt2

1CSIRO Building Construction and EngineeringPO Box 56, Highett, Victoria, 3190 AUSTRALIA

2Hermann-Fottinger-Institut fur StromungsmechanikTechnische Universitat Berlin, GERMANY

Abstract

Results are presented of large eddy simulations forsmooth subcritical flow past a stationary circular cylin-der, at a Reynolds number of 3900 where the wake is fullyturbulent but the cylinder boundary layers remain lam-inar. Results obtained using a spectral element discreti-sation are compared with those obtained with a finitevolume multiblock method and published experimentalresults.

Introduction

In order to find more general use for simulation of tur-bulent flows in industrial applications, large eddy sim-ulation (LES) techniques must be married with spatialdiscretisation methods that can deal with arbitrary ge-ometries. Here we compare results from two alternativeapplicable discretisation techniques, spectral element andmultiblock finite volume, applied to simulation of the flowpast a circular cylinder with Re = U∞D/ν = 3900, whereU∞ is the freestream flow speed, D is the cylinder diam-eter and ν is the (molecular) kinematic viscosity of thefluid.

Large eddy simulation is based on the idea of partition-ing a turbulent flow into resolved-scale and modelledturbulent fields through convolution with a filter kernel:this operation produces the resolved-scale velocity fieldu from the underlying complete turbulent field u.Theresolved-scale field u is that which is explicitly repre-sented on the computational mesh, i.e. the filtering op-eration is assumed to be implicit in the formulation. Fil-tering the incompressible Navier–Stokes equations leadsto

∂tu+∇·uu = −∇P + ν∇2u, (1)

where P = p/ρ. As in conventional turbulence modelling,the nonlinear terms are not closed, because the filtereddyad uu is not available in terms of the resolved com-ponents u. To overcome this problem, the concept ofa subgrid-scale stress (SGSS) τ is introduced, such thatuu = uu+ τ . Then the momentum equation becomes

∂tu+∇·uu = −∇P + ν∇2u−∇·τ , (2)

and the turbulence modelling task is to predict τ fromthe resolved velocity field u.

The model adopted here is an eddy-viscosity type basedon the “mixing length” approach, i.e. the Smagorin-sky model. The eddy-viscosity assumption is that theanisotropic part of τ is related to the resolved strain ratetensor S through a scalar eddy viscosity,

τ − tr(τ )1/3 = −2νtS = −νt

[

∇u+ (∇u)T]

, (3)

with the isotropic (scalar) part of τ subsumed in P . Then

the Smagorinsky model obtains the turbulent eddy vis-cosity from the strain rate magnitude,

νt = (CS∆)2 |S| = (CS∆)2 tr(2S2)1/2; (4)

the mixing length is the product of CS , a model constant,and ∆, a measure of the local grid length scale; here∆ = (∆x∆y∆z)1/3.

There are few comparable experimental results for theseReynolds numbers, since while experimental correlationsfor the variation with Reynolds number of global flowmeasures such as Strouhal number and base pressure co-efficient have been published [5], there have been com-paratively few studies of the velocity field of the cylindernear wake. A cylinder-surface pressure distribution fora nearby Reynolds number (Re = 3000) is available [4].Specifically for Re = 3900, Ong and Wallace [6] carriedout a hotwire traverse study of the wake flow, but toavoid bias problems associated with flow reversal, theyrestricted attention to distances greater then 3D down-stream of the cylinder centreline. Another study [3], re-ported by Beaudan and Moin [1], utilised PIV methodsto collect near-wake data, although these appear to besomewhat unreliable, as they often display unacceptabledeviations from expected wake-centreline symmetries.

Computational Methods

Our numerical methods have been previously described[2, 7, 8]; here we provide a brief review. The spec-tral element approximation employs a standard Gauss–Legendre–Lobatto collocation on two-dimensional ele-ments, with the extension to three dimensions made byFourier expansions in the cylinder-axis direction. Thecode employs a mixed explicit-implicit time integrationscheme of second order, and is parallel across two-dimen-sional data planes—each process carries a subset of thetotal number of data planes. The block-structured finitevolume calculation uses a centred second-order approxi-mation for evaluation of all terms. Time-integration forthe finite-volume method employs a second-order fully-implicit technique based on the SIMPLE algorithm, andis parallelised across blocks; each process carries a blockof data. Direct comparison of the two codes on prob-lems with identical geometries and numbers of nodes haveshown very similar ratios of CPU time to flow evolu-tion time, but since the spectral element discretisationis higher-order, it would be expected to display greaterspatial accuracy in a comparsion of this kind.

Mesh outlines are shown in Figure 1. The overall dimen-sions of the meshes are somewhat different, but the keyvalues of mesh cross-flow and axial extents are identical;y = ±7D and z = πD respectively. The spectral ele-ment mesh used 480 elements, each with 64 collocationpoints, providing each plane of data with 30,720 inde-

(a)

(b)

Figure 1: Simulation meshes (two-dimensional projec-tions); (a), spectral element mesh, 480 elements; (b) fi-nite volume mesh, 84 blocks. For both meshes, cross-flowand axial extents are y = ±7D, z = πD.

Source St Cpb Cd

Norberg [4, 5] 0.210 -0.88 0.99Spectral element LES 0.218 -0.93 1.01Finite volume LES 0.242 -1.10 1.07

Table 1: Global flow parameters from simulations com-pared to experimental correlations.

pendent mesh nodes. There were 48 planes of data inthe cylinder-axis direction and the total number of meshpoints was 1,474,560. The finite volume mesh had 84blocks, 32 axial planes of data, and a total of 855,040control volumes. While the spectral element mesh hasa greater number of mesh points, its streamwise extentis 48D as opposed to 19D for the finite-volume mesh,so the near-wake resolution for the two meshes may beconsidered comparable.

The turbulence model for both simulations used the sameSmagorinsky constant CS = 0.1, chosen primarily on thebasis of previous LES calculations, which have used val-ues in the range 0.065–0.1. For the spectral-element sim-ulation, van-Driest type wall damping [2] was also incor-porated to reduce the computed mixing lengths to zeroas the cylinder wall is approached.

Results

The simulations were run until an approximatelystatistically-steady state had been achieved, after whichcollection of statistics was initiated. In the statisticspresented below, the resolved velocity field is decom-posed into ensemble-average and fluctuating components:u = U + u

′ (all components are normalised by thefreestream value U∞). The streamwise, crossflow andspanwise (x, y, z) components of u are respectively u, vand w, and the coordinate system origin is located at thecentre of the cylinder.

The most basic global parameter for the flow past a circu-lar cylinder is the Strouhal number St = fD/U∞, wheref is the centre frequency of cross-flow wake velocity andthe fluctuating lift on the cylinder. Figure 2 illustratesthe time series of spanwise-averge lift coefficient derivedfrom the spectral element simulations. Two other global

Figure 2: Time series of coefficient of spanwise-averagelift resulting produced by the spectral element simula-tion. The Strouhal number St = fD/U∞ = 0.218.

Figure 3: Time-mean pressure coefficient on the cylindersurface as a function of angular position. , spectralelement simulation; , finite volume simulation; •, ex-perimental results for Re = 3000 [4].

parameters of interest are the base pressure coefficientand the drag coefficient. Table 1 shows the values of thesethree coefficients from the two simulations, compared toexperimental correlations of Norberg [4, 5]. For all threeparameters the values from the spectral element simu-lation are closer to the experimental correlations thanthose from the finite volume simulation.

Next we examine the time-mean pressure distribution onthe cylinder surface, presented in terms of the pressurecoefficient Cp(θ) = 2[P (θ) − P∞]/U2

∞. A comparisonbetween coefficients derived from the two simulations andexperimental measurements conducted at Re = 3000 [4]is presented in Figure 3. All three sets of data showpressure minima at θ ≈ 70◦, and mean separation pointnear θ = 90◦, however the spectral element results arecloser to the experiental values than those derived fromthe finite volume simulations, in agreement with valuespresented in table 1.

In turning to the velocity-based statistics, we may com-pare to results from the Re = 3900 experiments ofLourenco and Shih [3] and Ong and Wallace [6]. Thefirst point of comparision is the distribution of time-meanstreamwise velocity component U on the wake centre-line, shown in Figure 4. Where the experimental mea-surements overlap near x/D = 4, they are in reasonableagreement with one another, and with results from bothsimulations. Further upstream, where only Lourenco andShih’s results are available, the simulation results differ

Figure 4: Time-mean streamwise velocity on wake centre-line. , spectral element simulation; , finite volumesimulation; •, experimental results [3]; ¥, experimentalresults [6].

from one another, and the experimental values. Notethat the experimental results show a positive streamwisevelocity right at the base of the cylinder, which cannotbe valid. The mean flow-reversal length is x/D ' 1.7Dfor the experiments, 1.8D for the finite volume simula-tion and 2.1D for the spectral element simulation (whichis similar to the Smagorinsky-model results of Beaudanand Moin [1]). Time-mean streamlines of the near-wakeflows are presented in Figure 5, where the difference instreamwise extent of the recirculation bubbles reflects thedifference in zero-crossing locations in Figure 4.

The streamwise distributions of the velocity componentvariances are presented in Figure 6. In this case, no di-rectly comparable experimental results are available. Re-sults from the two simulations are similar, although somedifferences are evident. The peak values for the spectralelement simulation results occur somewhat further down-stream than they do for the finite volume results, which isvery likely related to the difference in mean recirculation-bubble lengths. An interesting detail here is the fact thatthe crossflow and spanwise velocity fluctuations retain fi-nite values until a quite thin boundary layer is enteredjust near the base of the cylinder.

Time-mean velocity component profiles at two stream-wise locations are shown in Figure 7. In the more up-stream location (x/D = 1.54), comparison can be madeonly with a single set of experimental results [3]. The dif-ferences in the minima of the streamwise velocity compo-nent U can be related to the differences in the centrelinevalues observed in Figure 4. The experimental resultshere suggest a thicker wake than predicted by either ofthe simulations. The cross-flow velocities are somewhatdifferent in detail at this location—also, note the lack ofsymmetry in the experimental values. At x/D = 3.00we can compare also to Ong and Wallace’s hotwire mea-surements [6], but we note that (as is the case for moststatistics at this overlap location) the two sets of experi-mental values are not in particularly good agreement.

We present in Figure 8 velocity variance and covarianceprofiles at x/D = 1.54 and 3.00. In general we expectmore variability than for the previous comparisons, as

x/D

y/D

-1 0 1 2 3-1.5

-1

-0.5

0

0.5

1

1.5

(a)

x/Dy/

D-1 0 1 2 3

-1.5

-1

-0.5

0

0.5

1

(b)

Figure 5: Streamlines of the time-mean flow field; (a),spectral element simulation; (b) finite volume simulation.

Figure 6: Variances of velocity components on wake cen-treline. (a), 〈u′u′〉, streamwise velocity component; (b),〈v′v′〉, crossflow velocity component; (a), 〈w′w′〉, span-wise velocity component. , spectral element simula-tion; , finite volume simulation.

these are higher-order statistics. At the more upstreamof the two locations, many of the differences between thetwo sets of simulation and single set of experimental re-sults can again be related to differences in the length ofthe mean recirculation bubble. At x/D = 3.00, the twosets of simulation results are generally in better agree-ment with one another than the two sets of experimen-tal results, with the exception of the crossflow variance〈v′v′〉, in which case the two sets of experiments give sim-ilar values, as do the two sets of simulations, but there isa distinct difference between experiment and simulation.

Figure 7: Time-mean velocity profiles at two downstreamlocations, x/D = 1.54 and x/D = 3.00. , spectralelement simulation; , finite volume simulation; •, ex-perimental results [3]; ¥, experimental results [6].

Discussion and Conclusions

It is difficult to decide which of the simulations or exper-iments examined here provides the most reliable statis-tics for this flow. The apparent lack of agreement be-tween the two sets of experimental measurements usedas bases for comparision of wake velocity data is rathersurprising, but this flow is notorious for its sensitivity tovariations in Reynolds number, boundary conditions andturbulence in the oncoming flow. Reliable experimentalresults will greatly assist future validation studies.

On the basis of the generally better agreement of theresults from the spectral element simulation to the ex-perimental correlations for global parameters such asStrouhal number, it appears likely that the spectral ele-ment simulation is somewhat more reliable than the finitevolume discretisation, at least with the spatial resolu-tions and turbulence model employed here. Simulationresults presented in [1] suggest some sensitivity to thevalue of CS employed, and we plan to investigate thisaspect in future work. In the final analysis, the resultspresented suggest that both of the simulation method-ologies presented hold promise for large eddy simulationof turbulent flows in complex geometries.

Acknowledgement

We gratefully acknowledge support of APAC and its staff.

References

[1] Beaudan, P. and Moin, P., Numerical Experimentson the Flow Past a Circular Cylinder at Sub-CriticalReynolds Number, Tech. Rep. TF-62, Dept Mech.Engng, Stanford University, 1994.

[2] Blackburn, H. M., Channel Flow LES with SpectralElements, in 13th Australasian Fluid Mechanics Con-ference, Monash University, 1998, 989–992.

[3] Lourenco, L. M. and Shih, C., Characteristics of thePlane Turbulent near Wake of a Circular Cylinder; aParticle Image Velocimetry Study (data taken fromBeaudan & Moin 1994).

[4] Norberg, C., Effects of Reynolds Number and Low-Intensity Freestream Turbulence on the Flow Arounda Circular Cylinder, Tech. Rep. 87/2, Dept Appl.Thermodyn., Chalmers Univ. Tech., 1987.

[5] Norberg, C., An Experimental Investigation of theFlow Around a Circular Cylinder: Influence of AspectRatio, J. Fluid Mech., 258, 1994, 287–316.

[6] Ong, L. and Wallace, J., The Velocity Field of theTurbulent Very Near Wake of a Circular Cylinder,Exp. Fluids, 20, 1996, 441–453.

[7] Schmidt, S., Grobstruktursimulation turbulenterStromungen in komplexen Geometrien und bei hohenReynoldszahlen, Ph.D. thesis, Technische UniversitatBerlin, 2000.

[8] Schmidt, S., Xue, L. and Thiele, F., Large-Eddy Sim-ulation of Turbulent Pipe Flow on Semi-StructuredGrids, in 13th Australasian Fluid Mechanics Confer-ence, Monash University, 1998, 651–654.

Figure 8: Velocity (co-)variance profiles at x/D = 1.54and x/D = 3.00. , spectral element simulation; ,finite volume simulation; •, experimental results [3]; ¥,experimental results [6].