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TECHNICAL PAPERS

# Anisotropy-Invariant Mapping of Turbulence in a Flow Past an Unswept Airfoil at High Angle of Attack

[+] Author and Article Information
N. Jovičić

Institute of Fluid Mechanics, University of Erlangen-Nürnberg, D-91058 Erlangen, Germany

M. Breuer1

Institute of Fluid Mechanics, University of Erlangen-Nürnberg, D-91058 Erlangen, Germany

J. Jovanović

Institute of Fluid Mechanics, University of Erlangen-Nürnberg, D-91058 Erlangen, Germanybreuer@lstm.uni-erlangen.de

On the web page www.lstm.uni-erlangen.de/$∼$njovicic some of these animations representing the flow in the anisotropy-invariant map can be found.

1

Corresponding author.

J. Fluids Eng 128(3), 559-567 (Oct 21, 2005) (9 pages) doi:10.1115/1.2175162 History: Received February 17, 2005; Revised October 21, 2005

## Abstract

Turbulence investigations of the flow past an unswept wing at a high angle of attack are reported. Detailed predictions were carried out using large-eddy simulations (LES) with very fine grids in the vicinity of the wall in order to resolve the near-wall structures. Since only a well-resolved LES ensures reliable results and hence allows a detailed analysis of turbulence, the Reynolds number investigated was restricted to $Rec=105$ based on the chord length $c$. Admittedly, under real flight conditions $Rec$ is considerably higher (about $(35–40)∙106$). However, in combination with the inclination angle of attack $α=18$ deg this $Rec$ value guarantees a practically relevant flow behavior, i.e., the flow exhibits a trailing-edge separation including some interesting flow phenomena such as a thin separation bubble, transition, separation of the turbulent boundary layer, and large-scale vortical structures in the wake. Due to the fine grid resolution applied, the aforementioned flow features are predicted in detail. Thus, reliable results are obtained which form the basis for advanced turbulence analysis. In order to provide a deeper insight into the nature of turbulence, the flow was analyzed using the invariant theory of turbulence by Lumley and Newman (J. Fluid Mech., 82, 161–178, 1977). Therefore, the anisotropy of various portions of the flow was extracted and displayed in the invariant map. This allowed us to examine the state of turbulence in distinct regions and provided an improved illustration of what happens in the turbulent flow. Thus, turbulence itself and the way in which it develops were extensively investigated, leading to an improved understanding of the physical mechanisms involved, not restricted to a standard test case such as channel flow but for a realistic, practically relevant flow problem at a moderate Reynolds number.

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## Figures

Figure 1

Invariant map according to Lumley and Newman (8)

Figure 2

(a) Two-dimensional sketch of the geometric configuration including block boundaries (thick lines) and (b) x−y plane of the grid (only every 5th grid line of the coarser grid is shown)

Figure 3

Streamlines of the time-averaged flow field (left) and a zoomed airfoil nose region (right); Rec=105, α=18deg, 16.23∙106CVs

Figure 4

Distribution of the pressure coefficient Cp=(p−p∞)∕(0.5ρ∞u∞2) of the time-averaged flow; Rec=105, α=18deg

Figure 5

Streamlines of the time-averaged flow field obtained on a refined grid with about 23∙106 control volumes (left) and from experimental investigations (29) (right), Rec=105, α=18deg

Figure 6

Profiles of the tangential velocity at three different locations along the airfoil and one in the wake, present data, refined grid simulation and experimental data (29), Rec=105, α=18deg (x′ is aligned with the chord starting at the nose. dn denotes the wall-normal distance from the airfoil. The velocities are normalized by the undisturbed velocity u∞ and the coordinates by the chord length c, respectively.).

Figure 7

Distribution of the resolved (nondimensional) turbulent kinetic energy k; Rec=105, α=18deg. k is normalized by the square of the undisturbed velocity u∞.

Figure 8

Distribution of the resolved Reynolds stresses (a) u′u′¯ (b) v′v′¯ (c) w′w′¯ and (d) u′v′¯, Rec=100,000 and α=18deg. The stresses are normalized by the square of the undisturbed velocity u∞.

Figure 9

Anisotropy-invariant map of the whole flow region of interest

Figure 10

Anisotropy-invariant mapping of turbulence in the near-wall region of the flow past an airfoil at high angle of attack

Figure 11

Anisotropy-invariant mapping of the near-wake region close to the trailing edge of the airfoil

Figure 12

Anisotropy-invariant mapping of the transitional flow region at the airfoil nose where the separation bubble is formed: Certain locations along the streamline (thick white line) are marked together with their corresponding positions in the invariant map (grid: 23 million CVs)

Figure 13

Same as in Fig. 1, however, obtained on a coarser grid with approx. 16 million CVs

Figure 14

Distribution of urms′, vrms′, and wrms′ (normalized by the undisturbed velocity u∞) as a function of streamwise position along the streamline shown in Fig. 1

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