LES of Separated Flows at Moderate Reynolds Numbers

Appropriate for Turbine Blades and Unmanned Aero

Vehicles

J. A. Domaradzki, F. Cadieux, G. CastiglioniDepartment of Aerospace Engineering, University of Southern California

Los Angeles, CA 90089-1191

M. Grilli, S. HickelLehrstuhl fur Aerodynamik und Stromungsmechanik, Technische Universitat Munchen

85748 Garching b. Munchen

T. Sayadi, S. BoseCenter for Turbulence Research, Stanford University

Stanford, CA 94305-3035

September 28, 2012

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1 The description of the problem

Reynolds numbers for ows in rotating machinery, for unmanned aerial vehicles (UAV), micro air vehicles(MAV), wind turbines, and propellers are usually low or moderate (based on wing/blade chord theyare typically less than 2 106 and often only on the order of few 104 105). Recent experimentalinvestigations of low Reynolds number aerodynamics [2, 7] reveal that such ows are often dominated bythe eects of ow separation that can greatly inuence lift and drag, and thus ight stability of UAV's,eciency of wind turbines, and unsteadiness in turbine ows which is instrumental in predicting high cyclefatigue (HCF) for turbomachinery components. In order to produce ecient designs or control schemes toreduce separation eects numerical prediction tools for such ows are needed. However, reliable numericalresults for such ows are dicult to obtain unless DNS are used. This is because such ows provide achallenging environment for turbulence models, consisting of a mixture of regions where the ow is laminar,transitional, non-equilibrium turbulent boundary layer, and an equilibrium turbulent boundary layer. Ourwork addresses the feasibility of LES for laminar separation bubble ows with the numerical resolutionsignicantly reduced compared with the DNS resolution, say to around 1% of DNS resolution, commonlyachievable for fully turbulent ows.

2 Flows considered

We have considered two benchmark ows: a boundary layer ow over a at plate and a ow over aNACA-0012 airfoil at incidence.

For a at plate boundary layer a procedure used successfully by other investigators [8, 1, 6] to induceseparation was followed. The computational domain is a rectangular box with a rigid lower wall on whichthe boundary layer ow evolves. A laminar Blasius boundary layer velocity prole with the free streamvelocity U0 is imposed at the inow. At the top boundary, a vertical suction velocity is imposed in a narrowslot oriented perpendicular to the mean ow direction. The suction produces an adverse pressure gradientthat causes ow separation. The ow then transitions to turbulence and reattaches. The Reynolds numberat the location of the peak suction velocity is Rex = 10

5. The full compressible Navier-Stokes equationsare solved for a perfect gas using sixth-order compact nite dierences [5] and an implicit-explicit timeintegration scheme. The numerical scheme is constructed on a structured curvilinear grid, and the variablesare staggered in space. Both DNS and LES were performed.

For a ow around a wing/blade the specic geometrical setting is that of a 3D airfoil at incidence forwhich detailed DNS results were obtained in [3]. The goal is to reproduce the laminar separation bubbleon a NACA-0012 airfoil at Rec = 5 104 at 5 deg of incidence. To simulate this case the code developedat TUM has been employed. The code solves the compressible three-dimensional Navier-Stokes using aconservative immersed boundary method for representing sharp interfaces between a uid and a rigid bodyon Cartesian grids [4].

3 Preliminary results

For a at plate boundary layer the following simulations were performed: a DNS benchmark case (CTRDNS), a wall-resolved LES with the dynamic Smagorinsky model (CTR LES), and a highly under-resolvedDNS (CTR UDNS). Parameters for these simulations are summarized in Table 1. In all cases simulationswere run until the separation bubble stabilized and turbulent ow was well established downstream of reat-tachment. Results were then averaged over multiple vortex shedding periods. The capability to predictaccurately at low computational cost the average skin friction, pressure coecient, and the location of sepa-ration and reattachment is of particular interest to airfoil and blade designers. For the case considered heresuch a capability is demonstrated through results shown in in Figs. 1 and 2. Satisfactory qualitative andquantitative agreement was achieved between the highly under-resolved DNS (CTR UDNS), wall-resolvedLES with the dynamic Smagorinsky model, and the fully-resolved DNS cases. However, the relatively good

1

CTR DNS CTR LES CTR UDNS

Nx 1536 512 240Ny 128 32 32Nz 300 140 90Ntotal 106 59.0 2.3 0.7% of CTR DNS 100 3.9 1.2x+ 9.7 26.4 57.0y+ at X = 7Y 0.5 1.0 1.6z+ 7.6 27.5 29.6

Table 1: Resolution and parameters for all cases run.

1 2 3 4 5 6 7

2

1

0

1

2

3

4

5

6x 103

x

C f

Figure 1: Wall coecient of friction. Symbols: CTRDNS, line: CTR LES with dynamic Smagorinskymodel, dashed line: CTR UDNS.

1 2 3 4 5 6 70

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

x

C pw

Figure 2: Coecient of pressure at the wall. Sym-bols: CTR DNS, line: CTR LES with dynamicSmagorinsky model, dashed line: CTR UDNS.

performance observed in the CTR UDNS no-model runs suggests that the numerical dissipation may playa signicant role and this issue must be investigated further.

For the NACA-0012 airfoil at 5 deg of incidence only results from 2D simulations are currently available(3D simulations are in progress). Two dimensional Cartesian grid has been generated through an adaptivemesh renement procedure and two dierent resolutions were used, coarse with the total number of 64; 827cells and ne with 128; 142 cells. For comparison, the baseline 2-D case of Jones et al. [3] used 1:6 106grid points. The ow has been simulated for around 13 time units, c=U , where c is the airfoil cord and Uthe free stream velocity. The instantaneous vorticity eld of the simulated ow is shown in Fig. 3. Thepresent 2-D simulations capture qualitatively features observed in high resolution DNS of Jones et al. [3],including the presence of a separated ow. The quantitative assessment of the results is made by comparingin Figs. 4 and 5 the friction and pressure coecients from our simulations with the corresponding data ofJones et al. [3] for their high resolution 2-D DNS case. While the comparison for the pressure coecient isreasonable the comparison for the friction coecient is poor overall. One important feature that appearsto be predicted relatively well is the location of the initial separation point, signied by the transition inCf from the positive to negative values on the suction side of the airfoil. However, the simulations do notpredict properly the secondary separation region at the mid-cord. We anticipate that 3D simulations at ahigher numerical resolution will allow to draw more authoritative conclusions concerning applicability ofLES to such separated ows.

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Figure 3: The vorticity eld for a ow over an airfoil.

Figure 4: Coecient of friction Figure 5: Coecient of pressure

References

[1] M. Alam and N.D. Sandham. Direct numerical simulation of `short' laminar separation bubbles withturbulent reattachment. J. Fluid Mech., 410:1{28, 2000.

[2] R. Hain, C.J. Kaehler, and R. Radespiel. Dynamics of laminar separation bubbles at low-Reynolds-number aerofoils. J. Fluid Mech., 630:129{153, 2000.

[3] L.E. Jones, R.D. Sandberg, and N.D. Sandham. Direct numerical simulations of forced and unforcedseparation bubbles on an airfoil of incidence. J. Fluid Mech., 602:175{207, 2008.

[4] M. Meyer, Devesa, A., S. Hickel, X.Y. Hu, and N.A. Adams. A conservative immersed interface methodfor large-eddy simulation of incompressible ows. J. Comput. Phys., 229:6300{6317, 2010.

[5] S. Nagarajan, S. K. Lele, and J. H. Ferziger. Leading-edge eects in bypass transition. Journal of FluidMechanics, 572:471{504, 2007.

[6] P.R. Spalart and M.K. Strelets. Mechanisms of transition and heat transfer in a separation bubble. J.Fluid Mech., 403:329{349, 2000.

[7] G.R. Spedding and J. McArthur. Span eciencies of wings at low Reynolds numbers. J. Aircraft,47:120{128, 2010.

[8] P.G. Wilson and L.L. Pauley. Two- and three-dimensional large-eddy simulations of a transitionalseparation bubble. Phys. Fluids, 10:2932{2940, 1998.

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