Numerical Dissipation and SGS Modeling in LES of

Laminar Separation Bubble Flows

F. Cadieux, J. A. Domaradzki

Department of Aerospace Engineering, University of Southern California

Los Angeles, CA 90089-1191

August 14, 2014

0

1 LES of a laminar separation bubble flow

Laminar separation bubbles (LSB) occur on a wide range of blades and airfoils used in rotating machinery,wind turbines, and propellers as well as unmanned aerial vehicles (UAV) and micro air vehicles (MAV).The flow over blades and airfoils operating at moderate Reynolds numbers (104 − 106) first undergoesseparation due to the adverse pressure gradient generated by surface curvature. The shear layer thentransitions to turbulence and reattaches, closing off a recirculation region - the laminar separation bubble.

LSB flows, consisting of a mixture of regions where the flow is laminar, transitional, non-equilibriumturbulent boundary layer, and an equilibrium turbulent boundary layer, provide a challenging environmentfor turbulence models. Typical Reynolds-averaged Navier-Stokes (RANS) turbulence modeling methodswere indeed shown to be inadequate [4]. Recent work demonstrated that accurate large edddy simulation(LES) of such flows are possible using only O(1%) of the resolution required by a direct numerical simulation(DNS) [2]. However, the performance of different subgrid scale (SGS) models could not be properly assessed.This is because the estimated numerical viscosity due to filtering which was necessary for stability was onthe same order of magnitude as the eddy viscosity provided by the dynamic Smagorinsky model [2]. Inthis work, the effects of filtering are analyzed and the performance of SGS models are evaluated using aspectral code to avoid contamination of results by numerical dissipation.

2 Method

A procedure used successfully by other investigators to induce separation over a flat plate boundary layerwas followed [5, 1, 4]. The spectral DNS by Spalart and Strelets [4] is used as a reference and benchmark.The computational domain is a rectangular box with a rigid lower wall on which the boundary layer flowevolves. A laminar Blasius boundary layer velocity profile with the free stream velocity U0 is imposed at theinflow. At the top boundary, a vertical suction velocity is imposed in a narrow slot oriented perpendicularto the mean flow direction. The suction produces an adverse pressure gradient that causes flow separation.The flow then transitions to turbulence and reattaches. The Reynolds number at the location of the peaksuction velocity is Rex = 105. See Fig.1. The flow is integrated in time using a collocated pseudo-spectralfractional time step code [3]. The negligible numerical dissipation of such a spectral code is instrumentalin allowing to evaluate the performance of different SGS models and the effects of filtering, while avoidingthe pitfalls of previous investigations that used inherently dissipative numerical methods and codes.

3 Preliminary results

The following simulations of a laminar separation bubble over a flat plate were performed: a highly under-resolved DNS (UDNS) along with a fifth and 7th order exponential (Gaussian) filtered under-resolvedDNS (F5-UDNS, and F6-UDNS respectively). LES results with different SGS models are forthcoming.Parameters for these simulations are summarized in Table 1.

In all cases simulations were run until the separation bubble stabilized and turbulent flow was wellestablished downstream of reattachment. Results were then time-averaged over multiple flow throughs.The capability to predict accurately at low computational cost the average skin friction, pressure coefficient,and the location of separation and reattachment is of particular interest to airfoil and blade designers. Forthe case considered here such a capability is demonstrated through results shown in Figs. 2 and 3. UDNSand F7-UDNS results both display a larger separation bubble than the benchmark which can be observedin the delayed sharp rise in the coefficient of pressure, as well as the skin friction going from negative topositive indicating a later reattachment point. By contrast, the F5-UDNS performs considerably better -the bubble size and location agrees well with the benchmark DNS.

These preliminary results demonstrate that the use of filters alone can provide enough dissipation to actas a substitute for an explicit SGS model. Separately, several explicit SGS models have been implementedin the code and LES are being currently performed. The dissipation provided by the exponential filters

1

DNS UDNS F5/F7-UDNS

Nx 1022 320 320Ny 120 32 32Nz 120 72 72Ntotal × 106 14.7 0.7 0.7% of DNS 100 5.0 5.0

Table 1: Resolution and parameters for all cases. [4]

will be compared to that provided by the explicit SGS models and the modeling procedure with the bestoverall performance will be identified.

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] F. Cadieux, J. A. Domaradzki, T. Sayadi, and T. Bose. DNS and LES of laminar separation bubblesat moderate Reynolds numbers. ASME J. Fluids Eng., 136(6), 2014.

[3] J. A. Domaradzki and R. W. Metcalfe. Stabilization of laminar boundary layers by compliant mem-branes. Phys. Fluids, 30(3):695–705, 1987.

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

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

2

Figure 1: Physical domain, boundary and inlet conditions used to investigate laminar separation bubbleflow

1 2 3 4 5 6 7−5

−4

−3

−2

−1

0

1

2

3

4

5x 10

−3

x/H

Cf

Figure 2: Wall coefficient of friction.

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/H

Cp

Figure 3: Coefficient of pressure at the wall.

Symbols: Spalart DNS, dash-dotted line: under-resolved DNS (UDNS), dashed line: 7th order filter (F7-UDNS), line: 5th order filter (F5-UDNS).

3