LES of Separated Flows at Moderate Reynolds Numbers Appropriate for Turbine

Blades and Unmanned Aerial Vehicles

J. A. Domaradzki

Department of Aerospace Engineering University of Southern California Los Angeles, CA 90089-1191

The main goal of the proposed research is to determine if LES can be used to obtain accurate results for laminar

separation bubble flows at a small fraction of DNS resolution and computational cost.

1 The physical problem

Reynolds numbers for ows in rotating machinery, for unmanned aerial vehicles (UAV), micro air vehicles (MAV), wind

turbines, and propellers are low to moderate. Based on wing/blade chord, they are typically less than 2 106

and often are

only on the order of 104

to 105

. By comparison, civilian airplanes are characterized by Reynolds numbers ranging from a

few millions to 80 106

for the Boeing 747 at cruising velocity. Recent experimental investigations of low Reynolds

number aerodynamics [4, 3, 9] reveal several features of such ows that complicate their prediction compared with high

Reynolds number ows. Low Reynolds number flows are often dominated by the effects of flow separation. Separation

greatly influences lift and drag, and thus the flight stability of UAV's as well as the efficiency of wind turbines. It also

causes unsteadiness in turbine flows which is a determining factor in high cycle fatigue (HCF) of turbomachinery

components.

The physical origin of flow separation is qualitatively well understood: the attached laminar boundary layer developing on

a wing is subjected to an adverse pressure gradient due to the airfoils curvature which causes it to separate. Immediately

behind the separation point there is an eectively stagnant ow region, 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. This shear layer experiences

Kelvin-Helmholtz instabilities that develop into turbulence after generating the rst characteristic spanwise vortices.

Further downstream, the separated turbulent ow reattaches and gradually evolves into the classical turbulent boundary

layer. The above picture emerges from numerous experimental investigations, e.g. [4, 3, 9], as well as from direct

numerical simulations (DNS) results [7, 12, 8, 1, 5, 6].

2 The numerical diculties

In order to produce ecient airfoil or blade designs and control schemes to reduce separation eects and better predict

HCF, we need numerical prediction tools for laminar separation bubble flows. At present, reliable numerical results for

such ows are dicult to obtain unless costly DNS are used. This is because such ows provide a challenging

environment for turbulence models. They consist of a mixture of regions where the ow is laminar, transitional, then

evolves from a non-equilibrium turbulent boundary layer to an equilibrium turbulent boundary layer. The importance of

the Reynolds stress varies widely across these regions and within some of the regions as well. Physically, low Reynolds

number separation driven by an adverse pressure gradient as opposed to geometry is a non-equilibrium process. It involves

subtle interactions between viscous, advective, and pressure eects that can so far only be captured by solving the full

Navier-Stokes equations, i.e. using DNS.

However, DNS require substantial computational resources, long wall-clock runs, and long analysis times, e.g. Jones et al.

[5] used over 170 million grid points for a relatively simple 3-D conguration. A number of congurations and angles of

attack need to be quickly investigated to allow for the optimization of airfoil and turbine blade designs. In this case, a DNS

approach is not feasible. Other simulation options must be considered. One option is to employ RANS models, modied to

account for the reduction of the eddy viscosity around the separation region. This is an approach commonly used and

optimized for high Reynolds number, turbulent ows, but one that is inadequate for the separated ows of interest.

Another option is to employ LES techniques. For instance, Yang and Voke [15] reported LES results obtained with the

dynamic Smagorinsky model in good agreement with experiments for boundary-layer separation and transition caused by

surface curvature at Re =3, 450. Yet even for this relatively low Reynolds number, the critical issues in getting

Comment [FC1]: I would expand on the importance of unsteady vortex shedding this creates.

experimental agreement was high numerical resolution (472x72x64 mesh points) and a high order numerical method,

requirements dicult to satisfy in simulations of practical ows often performed with low order nite dierence or nite

volume methods (e.g. commercial codes). Similarly, Eisenbach and Friedrich [2] performed LES of ow separation on an

airfoil at high angle of attack at Re = 105

using cartesian grids. This case also required very high resolutions between 50

and 100 million mesh points. Therefore, the question remains: can LES produce suciently accurate results for laminar separation bubble ow with drastically reduced resolution, say around 1% of DNS resolution, commonly achievable for

fully turbulent ows?

3 The proposed project

The proposed project is to study a laminar bubble separation problem at moderate Reynolds numbers using LES. It

consists of two related sub-projects and, depending on available codes and funding, the order of preference is: both

sub-projects 1 and 2, or only sub-project 1.

3.1 Sub-project 1: Separation bubble in a ow over a at plate

A procedure used previously by other investigators to induce the separation in a ow over a at plate [12, 1, 8] will be

followed. The computational domain is a rectangular box with the rigid lower wall on which a boundary layer ow exists.

On the upper wall, stress-free boundary conditions are imposed together with a suction velocity in a narrow slot oriented

perpendicular to the ow direction. This produces an adverse pressure gradient that causes ow separation. This

conguration is attractive because (1) there exist two databases obtained in high resolution DNS with the detailed results

still likely available from the authors [1, 8]; (2) the Stanford boundary layer code used in Wu and Moin [13, 14] could be

easily adapted to simulate this problem; (3) a graduate student at USC, Francois Cadieux, has began his Ph.D. research on

this problem and would provide man-power before, during, and after the Summer Program to complete the project.

This option is motivated largely by a simple geometry and the ability to use high order numerical methods (e.g., we use a

pseudo-spectral code at USC) as well as the existence of well documented DNS results, all of which will allow to isolate

the eects of a SGS model from purely numerical issues faced by subproject 2.

3.2 Sub-project 2: Separation bubble in a ow over a NACA-0012 airfoil

A NACA-0012 ow would be simulated which is a more realistic problem, but for which LES encounters additional

issues, beyond those of turbulence modeling. For this specic geometrical setting of a 3D airfoil at incidence, the detailed

DNS results were obtained by [5, 6]. The project would attempt to reproduce a laminar separation bubble on a

NACA-0012 airfoil at Rec =5 104

at 5 degrees of incidence with resolution reduced drastically from that used in DNS.

Ideally, a curvilinear coordinates code would be used for this problem (if available). Another Ph.D. student, Giacomo

Castiglioni, has started simulations on this problem using the immersed boundary (IB) method code INCA developed in

Adams group at Munich. While we have made progress using INCA there are fundamental issues with the accuracy of the

IB method at the rigid boundaries which impacts the separation predictions. A curvilinear-coordinates-based simulation

would be useful in disentangling the numerical issues of the IB method from the modeling issues of LES. We could also

consider using Stanford IB codes if issues of numerical accuracy at the rigid boundaries have been addressed and are not

considered serious. Basically any validated numerical code would be of interest if the numerical errors are expected to be

much less than the eects of a SGS model.

Comment [FC2]: This acronym has not been previously defined

Comment [FC3]: This is also undefined, is there a reference that could be included?

3.3 The technical approach

For the summer program the best in-house model recommended by CTR would be used, likely a dynamic model,

providing a benchmark LES database. LES would first be performed with the resolution estimated based on experience

with fully developed turbulent channel ow, i.e. around 1% of the DNS resolution, and then the resolution would be

increased until LES results for the separated ow match the corresponding DNS data satisfactorily: Spalart and Strelets [8]

for the at plate conguration and Jones et al. [6] for the NACA-0012 airfoil. Subsequently, after the Summer Program,

we plan to use for the same problem the Truncated Navier Stokes (TNS) method developed at USC, which has been

investigated recently and implemented in LES of turbulent channel ow by Tantikul [10, 11]. TNS is expected to perform

well for separated ows because it does not use Navier-Stokes eqs. modied by an explicit SGS model; instead it uses N-S

solutions that are periodically modied based on criteria sensing the buildup of small scale energy.

References

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