S.-H. Peng and W. Haase (Eds.): Adv. in Hybrid RANS-LES Modelling, NNFM 97, pp. 271–278, 2008. springerlink.com © Springer-Verlag Berlin Heidelberg 2008

Numerical Simulation of the Dynamic Stall of a NACA 0012 Airfoil Using DES and Advanced OES/URANS

Modelling

G. Martinat1, Y. Hoarau2, M. Braza1, J. Vos3, and G. Harran1

1 Institut de Mécanique des Fluides de Toulouse, France 2 Institut de Mécanique des Fluides et Solides de Strasbourg, France

3 Computational Fluid and Structure Engineering, Switzerland

Abstract

This paper provides a study of the dynamic stall of a pitching NACA 0012 airfoil at 106 Reynolds number by means of numerical simulation. A 2D study is carried out comparing three OES and URANS turbulence models (URANS, URANS k-ω SST and k-ε OES). Then a 3D computation is performed using DES Spalart modelling. URANS k-ω SST is providing the best results for 2D computations.

1 Introduction

The predicition of dynamic stall phenomenon at high Reynolds number is a crucial need in aeronautics and more specifically in rotorcraft dynamics. In this context, the forced unsteadiness (organised motion) interacts non lienarly with the fine scale, random turbulence and produces a strong irreversibility effect that usually leads to hysteresis loops in the aerodynamics coefficients versus the angle of incidence curves. Under these conditions of strong non-equilibrium turbulence, standard modelling approaches are often insufficient to predict the dynamic stall at high Reynolds number. Under these above effects, the stall angle is found higher than the normal static stall one. The applications of these flows occur in turbomachinery and in helicopter rotorblades as well as in wind turbune airfoils. It is important to have a good prediction of the dynamic stall to ensure efficiency for the design. A comprehensive review of the dynamic stall is described in (Mc Croskey, 1981 and Mc Croskey, 1982).

2 Turbulence Modelling

2.1 Macrosimulation Approaches for Unsteady Flows

The periodic nature of the flow past an oscillating airfoil allows us the definition of phase averaged quantities. The flow is classically decomposed into a mean component, a periodic fluctuation and a random fluctuation (Reynolds, 1971):

272 G. Martinat et al.

Ui = Ui + ˜ U i + ˜ u i . The phase averaged quantities is then: Ui = Ui + ˜ U i (Cantwell and Coles, 1984). Figure 1 shows the energy spectrum obtained on the experiment on a circular cylinder at high Reynolds number with LDV and PIV (Perrin and al, 2006) compared to the decompositionUi = Ui + ˜ u i . Due to a non linear

interaction of chaotic with organised structure, the slope of the fluctuation spectrum in the inertial part is different than the one of turbulence in equilibrium. As a consequence, production is not equal to dissipation like in URANS equilibrium turbulence modelling, but instead we need to reconsider the turbulence time and length scales.

Fig. 1. Velocity spectrum in a cylinder wake at Reynolds=140000. Experimental data from PIV (Perrin and al, 2006) and LDV (Djeridi and al, 2003).

In this context of advanced URANS methods, EMT2/IMFT has developed the Organised Eddy simulation (O.E.S) approach (Braza and al, 2006). This consists in distinguishing the structures to be resolved from the one to be modelled on the basis of their physical nature, organised or chaotic and not on their size (this is the case in LES approach).

The advantages of this approach are the robustness at high Reynolds number wall bounded flows and the fact that the method is not intrinsically three-dimensional. From the second order moement closure DRSM (Launder and al, 1975) a modified two equation model has been derived, where the turbulence length scales have been modified in the sense of evaluation of the Cμ eddy diffusion coefficient and of the damping turbulence law towards the wall (Braza and al, 2006 and Jin and Braza, 1994). In addition, a tensorial OES eddy-viscosity model has been derived to capture the non-equilibrium turbulence (Bourguet and al, 2007) where the Cμ eddy diffusion coefficient varies according to a directional criterion of stress-strain misalignment. A schematic representation of OES compared to LES modelling is given in figure 2, while the description of the OES approach can be found in Braza et al, (2006).

Numerical Simulation of the Dynamic Stall of a NACA 0012 Airfoil 273

Fig. 2. Schematic representation of OES modelling versus URANS and LES approaches

In this context, isotropic OES modelling is derived from k-ε Chien model (Chien, 1982) which is described by the following equation system:

( )

( )4

2

2

2

1 2 2

2

2

2

t t

UyC

t t

t

Dk k U k

Dt y y y y

D U keC C f

Dt y y k y k y

kC f

ν

ε

μ

νν ν ν ε

ε ε ε ε νν ν ν

ν με

−

⎛ ⎞⎡ ⎤∂ ∂ ∂= + + − −⎜ ⎟⎢ ⎥∂ ∂ ∂⎣ ⎦ ⎝ ⎠⎡ ⎤

⎛ ⎞⎡ ⎤∂ ∂ ∂ ⎢ ⎥= + + − +⎜ ⎟⎢ ⎥ ⎢ ⎥∂ ∂ ∂⎣ ⎦ ⎝ ⎠ ⎢ ⎥⎣ ⎦⎛ ⎞

= ⎜ ⎟⎝ ⎠

where 3

1Uy

Cf e ν

μ

−= − and

3

1Uy

Cf e ν

μ

−= − .

Modification for O.E.S are given by the following equations:

21 exp( 0.0002 0.000065

0.02

f y y

C

μ

μ

+ += − − −

=

This model was compared to experimental study provided by IRPHE laboratory on NACA 0012 at 20° of incidence for a 105 Reynolds number. Good results were

274 G. Martinat et al.

obtained on the test cases (figure 3). In the present study, the isotropic version of the OES modelling has been used and the computations were performed with NSMB solver (Vos and al, 1998).

Fig. 3. Instantaneous iso-vorticity lines a t*=3 using OES modelling and mean velocity profile in the recirculation region for a NACA0012 at 20° of incidence and Reynolds number of 105

2.2 DES Modelling

As said in Travin and al, 2000, “A Detached-Eddy Simulation is a three-dimensionna numerical simulation using a single turbulence model, which functions as a sub-grid scale model in regions where the grid density is fine enough for a Large-Eddy Simulation, and a s a Reynolds-Averagemodel in regions where it is not”.

The DES length scale is chosen according to the following equation:

( )min ,DES RANS DESl l C= × Δ

where CDES is the DES constant calibrated by means of homogeneous, isotropic turbulence spectrum, and Δ is the largest dimension of the elementary control volume cell, Δ=max(Δx,Δy,Δz).

For the one equation Spalart-Allmaras model (Spalart and Allmaras, 1992), it gives:

( )min ,DES DESl d Cω= × Δ

where dω is the distance from the wall. The consequence of the length scale modification is an increase of the dissipation term in the eddy-viscosity transport equation:

2

11 22

bt

DES

C vD C f f

lν

ω ωκ

⎛ ⎞⎛ ⎞= − ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

%.

Numerical Simulation of the Dynamic Stall of a NACA 0012 Airfoil 275

3 Physical Analysis of the Dynamic Stall Around a Pitching Airfoil at 106 Reynolds Number

The term dynamic stall usually refers to unsteady separation and stall phenomena on airfoils that are forced to execute time dependant motion. If the angle of attack oscillates around a mean value which is of the order of the static stall angle, large hysteresis cycle develop in the aerodynamic forces and moment. Indeed, during the upstroke motion, the effect of adversure pressure gradient is limited, driving to a dynamic stall angle which largely exceeds the static one with aerodynamics coefficients that also greatly exceeds their staic counterparts. During the downstroke part of the motion, the effect of adverse pressure gradient is reinforced, leading to a reattachment incidence angle which is lower than in the static case.

For this study, the airfoil performs a sinusoidal pitching motion around the quarter chord point. The mean pitch angle is α0=15° and the amplitude of pitch is Δα=10°. The reduced frequency which is based on half chord length is k=0.1.

The meshgrid used is O topology structured mesh, it has 285 cells in the I direction and 169 cells in the J direction. For the three dimensional study, the two dimensionnal meshgrid is extruded and it has 40 cells in the K direction. The meshgrid was provided by TU Berlin and validated on a 106 Reynolds number NACA 0012 at 20° of incidence case. The solver uses third order upwind-roe space scheme and dual time stepping with second order implicit backwards time-scheme. The time step varies along the computation by using constant CFL numbers. Experimental data are available from Mc Alister and al, 1982.

3.1 Two-Dimensional Computations

K-ω SST, k-ε Chien and k-ε OES are two dimensionally performed and compared. Figure 4 shows the isovorticity fields for 21 different angles of incidence. From 5° to 16° upstroke, the flow remains attached to the profile. From 20° to 25° we can observe the birth of the leading edge vortex and its growth until it is shed at 25° angle of incidence, immediately followed by a tailing edge vortex which birth occurs at 22° angle of incidence upstroke. From 25° to 12° downstroke, the flow is fully stalled and begins reattachment for 8° of incidence downstroke.

As seen on figure 5 the three models seems to have a similar behavior. OES modelling do predict the best angle of dynamic stall but overestimate the lift and drag coefficients. K-ε Chien gibes accurate results with a good prediction of lift, a small overprediction of drag and a delay on the predicted stall angle. The k-ω SST is giving the most accurate results with a good estimation of the stall angle and good prediction of the lift and drag coefficient during the upstroke motion. The three models are overestimating the lift coefficient during the downstroke motion.

276 G. Martinat et al.

Fig. 4. Iso-vorticity fields using k-ω SST modelling for 5, 6, 7, 8, 12, 16, 20, 21, 22, 23, 24, 25 degrees angle of attack upstroke and 24, 23, 22, 21, 20, 16, 12, 8 and 6 degrees downstroke

Numerical Simulation of the Dynamic Stall of a NACA 0012 Airfoil 277

Fig. 5. Hysteresis loops on lift and drag coefficient for the three models tested on the two dimensionnal computations

3.2 Three Dimensional Computations

Three dimensional were performed using DES-Spalart approach. Those computations were done using 8 NEC SX8 processors in a first time and then 16 AMD opteron processors.

Computations are not long enough to converge hysteresis cycles and vorticity fields are the same as in the two-dimensionnnal computations. Figure 6 compares the vorticity fields obtained with two dimensional computations using URANS k-ω SST modelling and with three dimensional computations using DES-spalart modelling.

Fig. 6. Comparison of the vorticity field at 23.5° of incidence upstroke for three-dimensional DES-Spalart modelling (left) and two dimensional k-ε Chien modelling (right)

4 Conclusion and Prospects

In this study, the behavior of three different models is shown on the flow around a pitching airfoil and compared to experimental results. k-ω provided the best results

278 G. Martinat et al.

among all compared to k-ε Chien and to OES modelling. Three dimensional results are not completely converget yet to shoh hysteresis cycles but seem to be promising regarding results into the upstroke phase of the motion which are comparable to those obtained with two dimensional study.

In the future, three dimensional computations with DES-Spalart will be finished and compared to some obtained with DDES and URANS.

References

Bourguet, R., et al.: Anisotropic eddy viscosity concept fot strongly detached unsteady flows. AIAA Journal 45 (2007)

Braza, M., Perrin, R., Hoarau, Y.: Turbulence properties in the cylinder wake at high Reynolds number. Journal of fluids and Structures 22 (2006)

Cantwell, B., Coles, D.: An experimental study of entrainment an transport in the turbulent wake of a circular cylinder. Journal of fluid mechanics 136 (1984)

Djeridi, H., et al.: Near-wake turbulence propertiesaround a circular cylinder at high Reynolds number. Flow turbulence and combustion 71 (2003)

Jin, G., Braza, M.: A two equation turbulence modelfor unsteady separated floxs around airfoils. AIAA Journal 32 (1994)

Launder, B.E., Reece, G.J., Rodi, W.: Progress in the development of a Reynolds-stress turbulence closure. Journal of fluid mechanics 68 (1975)

McAlister, K.W., et al.: An experimental study of dynamic stall on advanced aifoil sections. NASA/ TM-84245 (1982)

Mc Croskey, W.J.: Unsteady airfoils. Annual review of fluid mechanics 14 (1982) Mc Croskey, W.J.: The phenomenon od dynamic stall. NASA report: NASA/TM-81264 (1981) Menter, F.R.: Zonal two equation k-ω turbulence models for aerodynamics flows. AIAA Paper,

93-2906 (1993) Perrin, R., et al.: Phase-averaged measurements of the turbulence properties in the near wake of

a cyrcular cylinder at high Reynolds number by 2C-PIV and 3C-PIV. Experiments in fluids 42 (2007)

Reynolds, W.C., Hussain, A.K.M.F.: The mechanics of an organized wave in turbulent shear flow. Part 3. Theoretical models and comparison with experiments. Journal of Fluid Mechanics 54 (1971)

Spalart, P.R., Allmaras, S.R.: A one equation turbulence modelm for aerodynamics flows. AIAA Paper, 92-0439 (1993)

Travin, A., et al.: Detached eddy simulation past a circular cylinder. Flow, turbulence and combusiton 63 (1999)

Vos, J., et al.: Recent advances in aerodynamics inside the nsmb (Navier Stokes Multi Blocks) consortium. AIAA Paper, 1998-0802