LES of Laminar Separation Bubble FlowsJ. A. Domaradzki

Department of Aerospace EngineeringUniversity of Southern CaliforniaLos Angeles, CA 90089-1191

The primary goal of the proposed research is to extend previous ndings regarding laminar separationbubble (LSB) ow over a at plate (CTR Summer Program 2012) to a more realistic geometrical congu-ration of a NACA-0012 airfoil. A compelling issue raised by the previous results is the role of the numericaldissipation in LES of such ows; its quantication in coarse mesh LES constitutes the second goal of thisproposal. Overall, the proposed research should determine necessary requirements for both the numericalmethod and the spatial resolution to obtain reliable and accurate LES results for LSB ows.

1 Physical problem

Reynolds numbers based on wing/blade chord for ows in rotating machinery, for unmanned aerial vehi-cles (UAV), micro air vehicles (MAV), wind turbines, and propellers are typically less than 2 106 andoften only on the order of 104 to 105. By comparison, civilian aircraft are characterized by the Reynoldsnumbers ranging from a few million to 80 106 for the Boeing 747 at cruising velocity. Recent experimen-tal investigations of low Reynolds number aerodynamics reveal that low to moderate Reynolds number

ows over airfoils and turbine blades are often dominated by the eects of ow separation and turbulentreattachment [17, 15, 25]. Such phenomena greatly inuence the aerodynamic forces the airfoil or bladeis subjected to. They change the lift and drag characteristics and thus ight stability of UAVs, aectwind turbine eciency, and cause unsteadiness in turbine ows, which is a determining factor in high cyclefatigue (HCF) of turbo-machinery components.

LSB ows are qualitatively well understood thanks to numerous experimental investigations, e.g., [13,16, 3, 14, 21, 4, 27, 5, 6, 17, 15, 25] as well as from direct numerical simulations (DNS) results [20, 24, 1,22, 18, 19]. The attached laminar boundary layer developing on a wing or blade is subjected to an adversepressure gradient due to the airfoil's curvature, which causes it to separate. There is an eectively stagnant

ow region immediately behind the separation point, the so-called dead air region, followed by a reverse

ow vortex. The interface between the separated ow moving away from the wing and the recirculating

ow in the vicinity of the wing results in a shear layer with an inectional mean velocity prole. Thisshear layer experiences Kelvin-Helmholtz instabilities that develop into turbulence after rst generatingcharacteristic spanwise vortices. Further downstream, the separated turbulent ow reattaches, therebyclosing o the LSB, and gradually evolves into the classical turbulent boundary layer. The separationbubble's shape and size changes in time due to vortex shedding, making the problem inherently unsteady.

2 Numerical diculties

Numerical prediction tools for LSB ows are needed in order to produce more ecient airfoil or bladedesigns, to create control schemes to reduce separation eects, and to better predict HCF. The Reynolds-averaged Navier-Stokes (RANS) approach was shown to be inadequate for such ows by Spalart [24].Reliable and accurate simulation results for LSB ows are currently dicult to obtain unless DNS orlarge eddy simulation (LES) with very high, DNS-like resolution is used. Jones et al. [18] used over 170million grid points in their DNS for a relatively simple 3-D airfoil conguration. Yang and Voke [26]reported LES results with the dynamic Smagorinsky model that were in good agreement with experimentsfor boundary-layer separation and transition caused by surface curvature at Re = 3; 450. Yet even forthis relatively low Reynolds number, the critical requirements to obtaining experimental agreement was ahigh numerical resolution (472 72 64 mesh points) and a high order numerical method. Such stringentrequirements are rarely satised by commercial codes. Similarly, Eisenbach and Friedrich [12] performed

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LES of ow separation on an airfoil at high angle of attack at Re = 105 using between 50 and 100 millionmesh points. Performing DNS or LES at such high resolutions require substantial computational resources,long wall-clock runs and long analysis times. If a number of congurations and angles of attack are tobe quickly investigated in the context of airfoil or blade shape optimization, using such high resolutionsbecomes prohibitive.

During the CTR Summer Program 2012, Cadieux et al. studied a LSB ow over a at plate using DNS,under-resolved DNS, and LES with dierent SGS models [8]. Their results demonstrated the feasibility ofvery coarse LES for LSB ows (around 1-3% of DNS resolution) for this simple geometrical conguration.Later work also identied numerical dissipation as playing a signicant role in LES at such low resolutions,even when nominally high order schemes were used [7]. Another rare example of a low resolution LES of aLSB ow is given by Almutairi et al. [2]. However, their LES method is a simple application of an explicitlter. Separately, we have also performed extensive LES simulations of a ow over a NACA-0012 airfoilwith an Immersed Boundary (IB) code INCA developed in Adams' group at the Technical Universityof Munich [9]. We were able to match the DNS benchmark of Jones [18] using a near-DNS resolution,but at coarse LES resolutions the accuracy of the Immersed Boundary method at rigid boundaries wasnot sucient to obtain agreement for the separation and reattachment points. These results collectivelypoint to two issues: (1) the extension to realistic geometrical congurations is non-trivial and (2) coarseresolution LES can be heavily inuenced by the numerical method and associated numerical dissipation.Addressing these two issues is of paramount importance to promote the use of very coarse LES as a designtool for industrial applications.

3 The proposed project

We propose to study LSB ows at moderate Reynolds numbers with a realistic geometrical congurationusing LES, and to quantify the numerical dissipation in such simulations.

The at plate simulations performed during the previous CTR summer program [8] would be extendedto a more more realistic conguration: a ow over a NACA-0012 airfoil at 5 degree angle of attack. Agraduate student at USC, Giacomo Castiglioni, has already performed extensive LES for this conguration[9]. Detailed DNS results for this specic case were obtained by Jones et al. using a compressible code ingeneralized curvilinear coordinates [18, 19]. Tabulated DNS data for the pressure and the friction coecienthave been obtained from the authors and are available for comparison. We propose to use the same CTRcode [23] that was used for the at plate LES in Summer 2012. Giacomo Castiglioni is already familiarwith this code through conversations with Dr. Taraneh Sayadi at the Munich Summer Program in 2013and contacts with Prof. Lele's current students. Simulations with the generalized curvilinear coordinatesCTR code would be useful in disentangling the numerical eects of IB methods from the eects of LESmodels. Furthermore, since in the previous summer program the CTR code demonstrated the capability tosimulate accurately a LSB at around 1-3% of DNS resolution we hope that the same resolution reductionwill be possible for the NACA-0012 airfoil conguration (which was not possible for the IB method).

One surprising conclusion from our work is that the numerical dissipation plays a signicant role in LESof LSB ows [7, 9]. In simulations performed with the IB method, the best agreement with the benchmarkdata was obtained using a dissipative WENO scheme without SGS model rather than a non-dissipativecentral dierence scheme with a SGS model. In the at plate simulations, under-resolved DNS provideda better agreement with the benchmark data than LES with several dierent SGS models tested. Forthe CTR code [23], a crude estimate of the numerical dissipation was made and it was determined thatit was principally due to the ltering needed to match solutions of the implicit time stepping used in thevicinity of the wall and the explicit time stepping used in the rest of the domain [7]. Clearly, knowingthe implicit numerical dissipation and how it compares/interacts with the explicit SGS dissipation shouldlead to improvements in LES capabilities. Hence, we plan to use a more rigorous method for estimatingthe numerical dissipation and apply it to the analysis of both separated ows (the at plate and theNACA-0012 airfoil).

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3.1 Technical approach

During the summer program, the NACA-0012 airfoil case would be run using the CTR code developed byProf. Lele's group [23]. We would use the same approach as developed for the at plate ow: to performrst LES and under-resolved DNS with a resolution around 1% of the benchmark DNS resolution of Joneset al., and keep increasing the resolution until LES results for the separated ow match satisfactorily thebenchmark DNS data [19].

The eects of the numerical dissipation would be assessed in two ways. We believe that performingruns with only explicit time-stepping, i.e., eliminating any explicit ltering procedures, would ensure aspectral-like behavior. Such simulations would be expensive but need to be performed only for limitedtime intervals, starting from restart les in runs performed with the production code. Having resultsobtained from dissipative and non-dissipative runs, both started from the same initial condition, allows tocompute the numerical dissipation by comparing the local energy decay rates for both cases (see [11, 10]).Another way, which does not depend on having an access to a non-dissipative code, relies on computingthe residual of the energy rate equation for a selected subdomain, i.e., a dierence between dE=dt andintegrated uxes through the subdomain boundaries. This method is currently being developed and hasbeen tentatively tested on the Taylor-Green vortex ow with promising results.

J.A. Domaradzki would participate as a PI guiding two graduate students from USC, G. Castiglioniand F. Cadieux. G. Castiglioni would be involved in simulating the NACA-0012 airfoil ow and F. Cadieuxin the analysis of the numerical dissipation, rst for the at plate problem for which data are availablefrom the previous summer program, and then for the airfoil problem once the data are generated.

4 Financial requests:

We request funding only to cover housing expenses at Stanford for the duration of the program. Basedon the previous program's housing rates we'd like to request a housing allowance of $3,000 for the PI and$1,000 for each students, i.e., the total of $5,000 for this project.

References

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[8] F. Cadieux, J. A. Domaradzki, T. Sayadi, T. Bose, and F. Duchaine. DNS and LES of separated owsat moderate Reynolds numbers. In Proceedings of the 2012 Summer Program, pages 77{86. Centerfor Turbulence Research, 2012.

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[26] Z. Yang and P.R. Voke. Large-eddy simulation of boundary-layer separation and transition at a changeof surface curvature. J. Fluid Mech., 439:305{333, 2001.

[27] Serhiy Yarusevych, Pierre E. Sullivan, and John G. Kawall. Coherent structures in an airfoil boundarylayer and wake at low Reynolds numbers. Physics of Fluids, 18(4):044101, 2006.

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