AIAA-2010-4290

American Institute of Aeronautics and Astronautics

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Prediction of Laminar-Turbulent Transitional Flow over Single and Two-Element Airfoils

Shirzad Hosseinverdi1 and Masoud Boroomand2

Nomenclature

AmirKabir University of Technology (Tehran Polytechnic), Tehran 15875-4413

The correlation-based transition model has been developed to predict precisely the laminar-turbulent transitional flow over single and two-element airfoils. At relatively low Reynolds number, laminar-turbulent transition has an important role on skin friction and the onset of separation and eventually on aerodynamic characteristics. For this reason, the accurate prediction of transition onset is fundamental to the modeling of such flows. This model is based on coupling the turbulence model with empirical and semi empirical correlations. The two-equation shear stress transport (SST) k- turbulence model of Menter was employed to determine accurately turbulent flow. To predict the transition onset, two different methods were used. The first method is the Cebeci & Smith correlation which is based on Michel's method and Smith e9-correlation. The second method is based on the linear stability theory, originally proposed by Smith and Van Ingen that is referred to the en method. The laminar boundary layer properties required for calculation of the transition onset and extent were calculated by solving a two-equation integral formulation with the pressure distribution of the Navier-Stokes solution as the input. Whereas the extent of transition calculated by using the intermittency function that is based on the work of Dhawan and Narasimha. The developed transition model was used to predict the incompressible transitional flow over the flat plate under zero pressure gradient, natural laminar flow airfoil NLF(1)-0416, S809 wind turbine airfoil and two-element NLR 7301 airfoil with trailing edge flap. For the cases of flow over flat plate, good agreement of skin friction between the transitional computed and the experimental data were observed. In the case of airfoil flows, the significant improvements were obtained in drag prediction by using the transitional computation in comparison with the fully turbulent simulation in the given experimental data.

C = airfoil chord Cf = skin friction coefficient Cd = dissipation coefficient CD = drag coefficient CL = lift coefficient CP = surface pressure coefficient d = normal wall distance H = shape factor H* = kinetic energy shape factor K = turbulent kinetic energy ReC = Reynolds number based on chord length, UC/ ReX = Reynolds number based on local distance, UeX/ Re = Reynolds number based on momentum thickness, Ue/ S = surface distance Ti = free stream turbulence intensity U = free stream velocity Xtr = transition onset 1 Student, Department of Aerospace Engineering, [email protected], Student Member AIAA. 2 Associate Professor, Department of Aerospace Engineering, [email protected], Senior Member AIAA.

40th Fluid Dynamics Conference and Exhibit28 June - 1 July 2010, Chicago, Illinois

AIAA 2010-4290

Copyright 2010 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


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y+ = non-dimensional wall-normal distance = absolute value of vorticity = boundary layer thickness = momentum thickness = molecular viscosity t = turbulent viscosity = stream wise distance = density k, = turbulent model constants = specific dissipation rate = intermittency function

Subscripts L = laminar T = turbulent Tr = transition = free stream 0 = critical

I. Introduction HE prediction of transitional flows which occur typically in aerospace applications has been objected of research in fluid mechanics and aerodynamics for about century. However, until today, a complete

understanding of this phenomenon and what physically is happening in transitional region have not been fully understood. Transitional boundary layer flows are important in many engineering applications, such as airfoil, wind turbine and turbomachinery. In the design of aerodynamic vehicles, it is critical that the design group have an accurate assessment of aerodynamic characteristics that are being considered. Errors in the aerodynamic coefficients will result in errors in performance estimations and economic projection of flying vehicles. One of these characteristics is drag. It is crucial that the drag value predicted from CFD simulations would be as close as possible to the actual flight values. As an example of drag effects on flying objects, a reduction of one drag count (CD=10-4) on a subsonic civil transport airplane means about 200 lb (one person) more in payload, Van Dam26. Drag over a wing can be divided into three major components: viscous, induced and wave drag. Induced drag is due to the induced downwash velocity by wing tip vortices; wave drag is result of the shock waves generated in transonic flow and the viscous drag generated by skin friction and thus it is directly related to viscous flow behavior. Induced and wave drag are generated by normal forces and they can be well predicted using Euler equations simulations. On the other hand, viscous drag can only be predicted by solving the Navier-Stokes equations. This study focuses on two-dimensional, incompressible transitional flow over flat plate and airfoils so the center of attention is on the viscous drag component. In the drag evaluation, one of the key factors is laminar-turbulent transition. There are number of different transition mechanisms which depend on the factors such as: turbulence level of the external flow, the pressure gradient along the laminar boundary layer, the geometrical details, the surface roughness and the free stream Mach number. In the aerodynamic flows, it is generally assumed that the most common transition mechanism is natural transition which is the result of flow instability (Tollmien-Schlichting waves or cross-flow instability), where the resulting exponential growth of two-dimensional waves eventually results in a non-linear break down to turbulence, Schlichting. Another transition mechanism that has also received heightened attention is called bypass transition. In turbomachinery application, the main transition mechanism is bypass transition imposed on the boundary layer by high levels of turbulence in free stream, Morkovin27. Free stream with high level of turbulence generated by upstream blade rows is an example of this type of flows. Another example of by pass transition that can occur in aircrafts flights is on the tail surfaces located in the wake of fuselage, or on flaps of a multi-element airfoil. Moreover, the well known important transition mechanism is separation-induced transition, Mayle13, where a laminar boundary layer separates under the influence of pressure gradient and transition develops within the separated shear layer (which may or may not reattach). Transition analysis is divided in two tasks; the first one is the prediction of transition onset while, the second one is evaluation of the transition extent. In the first category, various methods have been used to predict transition that can be classified in the following manner:

T


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A Direct Numerical Simulation (DNS) is performed by solving the full unsteady Navier-Stokes equations. Since there is no Reynolds averaging there is no requirement for turbulence closure by a turbulence model. In order to capture the small scales of turbulent a DNS computation requires extremely fine grid. Due to the extraordinary large computational requirements, DNS is clearly not yet at a stage where it can serves as a practical tool for engineering design applications. Because of the significant computational costs associated with DNS, a number of researchers have applied the concept of Large Eddy Simulation (LES) to transitional flows. One of the main problems with LES is that the predicted transition location is very sensitive to the choice of Smagorinsky constant which is used to calibrate the subgrid eddy viscosity. Germano et al.28 proposed a dynamic subgrid-scale model which computes the Smagorinsky constant locally. Nevertheless, the dynamic LES model is not a complete solution to the issues associated with applying LES to predict transitional flows. Another approach is the application of low-Reynolds number turbulence models. However, the ability of low-Reynolds turbulence models to predict transition seems to be coincidental. This is based on reproducing the viscous sublayer behavior, not on predicting transition from laminar to turbulent flow. It is now generally accepted that the use of turbulence models without any coupling to an intermittency equation appears to be very delicate and often unreliable method of predicting transition. In addition, low-Reynolds turbulence models can be applied to bypass transition and are therefore not suitable for aerodynamic flows. One of the most common ways of predicting boundary layer transition onset location is employing empirical correlations. The empirical correlations usually relate to the free stream turbulence intensity (Ti) and to the transition Reynolds number based on the momentum thickness Reynolds number (Ret). A typical example is the Mayle13 correlation which is based on fairly recent high-quality experimental data for transition onset. Another popular correlation is that of Abu-Ghannam and Shaw14. It is based on the older data but it can account for the effect of pressure gradient on the transition onset location. There are also other models such as the methods of Michel30, Granville31, Smith and Gamberoni8, Van Ingen9, and Van Driest and Blumer29. Of those that are widely recognized for their accuracy and ease of implementation in engineering applications are Michel's and Granville's methods. Michel based his method on relating the transition momentum thickness Reynolds number Ret to the local distance Reynolds number Rex through a single curve that matches fairly well with the experimental data for two-dimensional airfoils in incompressible flows. Later Cebeci & Smith1 in according with the Michel's method, proposed a similar curve that is compatible with the e9 method. Finally, one of more widely used method for predicting transition is so-called en method. This model is based on the local linear stability theory and the parallel flow assumption in order to calculate the growth of disturbance amplitudes from the boundary layer natural point to the transition location. Drela and Giles3 develop the semi-empirical model to predict the transition onset; this model is based on the Orr-Sommerfeld equation coupled with empirical correlations originally developed for two-dimensional steady incompressible flow. The other aspect of transition analysis process is that of the transition extent. That is the zone over which the boundary layer undergoes a change from a fully laminar flow to a fully turbulent flow. Such a region starts at the transition onset location and ends at the point where the flow is 100% turbulent. The region can be represented quantitatively by using intermittency functions. Dhawan and Narasimha4 have correlated the experimental data and proposed a generalized distribution function across the transition region. Chen and Thyson7 based their work on Emmons spot theory for predicting the transition and they have suggested an intermittency function to achieve a smooth transition from laminar into turbulent flow in the boundary layer. Later Cebeci2 added some modifications for the function, so that it will include cases of two-dimensional low-Reynolds-number flows where laminar separation bubbles increase in size and in occurrences. Edward et al.5 have developed a one- equation turbulence transition model that is based on both the Spalart-Allmaras one-equation turbulence model and a transition model that was developed by Warren and Hassan6. Intermittency function that used by Edward et al. consists of two components. One is based on the work of Dhawan and Narasimha and labeled as the surface-distance-dependent component. The other is a multidimensional component that is used to calculate the transition distribution normal to the surface. Then the two components are combined together to describe the transition zone distribution in the flow. It is to be concluded that by using the empirical and semi-empirical correlations developed to model both the onset and extent of transition, major time and effort can be saved in comparison with the percent accuracy to be gained by using more complex procedures. This work is focused on the modeling of the transition of laminar-turbulent boundary layer. To predict the transition onset location, the empirical correlation of Cebeci & Smith and semi-empirical correlation of en method are used. To capture the transition region the intermittency function of Dhawan and Narasimha with the surface-distance-dependent component of Edward et al. is used. The accuracy of developed transition model is assessed for a transitional flow over flat plate under zero pressure gradient, natural laminar flow airfoil NLF(1)-0416, S809 wind turbine airfoil and two-element NLR 7301 airfoil with trailing edge flap.


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II. Transition Modeling In a typical flow simulation using RANS solver coupled with a turbulence model, the effective turbulence viscosity is considered as

TLeff += (1)

Where, L and T are the laminar and turbulent viscosity coefficients, respectively. One of the methods of introducing transition in a fully turbulent boundary layer is by multiplying the turbulent viscosity by an intermittency function. Then the modified effective viscosity is equal to

TLeff += (2)

Thus for =0, the boundary layer is fully laminar and for =1, the boundary layer is fully turbulent. For any value in between zero and one, the flow is in the transition region. Turbulent viscosity T is obtained from a turbulence model and is calculated by using the onset prediction method and transition extent model.

A. Turbulence Model Among the several variations of widely used two-equation turbulence models, the shear-stress transport (SST) k- turbulence model of Menter19 was adopted to properly resolve the flow properties in turbulent regime. This model is a two-equation eddy-viscosity model which merges the k- model of Wilcox with a high Reynolds number k- model (transformed into the k- formulation). The transport equations for the turbulent kinetic energy and the specific dissipation rate of turbulent in Cartesian coordinate are as follows:

( ) ( ) kPxkku

xtk

kj

tkjj

*=

+

+

(3)

( ) ( ) ( )jj

ktj

tjj xx

kfPx

uxt

+=

+

+

112 2,12 (4)

Turbulent viscosity t and production of turbulent kinetic energy PK are computed thorough the following relations

( )211

,max faka

T =

(5)

j

iijijkkijjitk x

ukuuuuP

+=

32

32

., (6)

Auxiliary functions (f1, f2) are given by

=

4

22

2*14

,500,maxmintanhyCDk

yykf

k

(7)

=

2

2*2500,2maxtanh

yykf (8)

= 202 10,12max

jjk xx

kCD

(9)

In the above formulation, y is the normal-distance to the wall. The constants of model are: a1=0.31, *=0.09.


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Finally, the coefficients of , , k and are obtained from the relation 10: ( ) 2111 1 ff += (10)

Let 1 represents any constant in k- model (inner), 2 any constant in the transformed k- model (outer) and the corresponding constant of k- (SST) model. Constants of inner and outer models are given in table 1.

Table 1. Coefficient of outer and inner models Model k Inner (1) 5/9 3/40 0.85 0.5 Outer (2) 0.44 0.0828 1 0.856

B. Transition Onset Prediction Methods To predict the transition onset, two different methods were used. First model is the correlation based on Michel's

method and Smith's e9-correlation which derived by Cebeci & Smith. They suggest that the following correlation gives more accurate results than the empirical one-step model of Michel;

46.0,

,, ReRe

224001174.1Re trxtrx

tr

+= (11)

The above expression is derived for attached and incompressible flow. According to this method, the boundary layer development on the body is calculated for a laminar flow starting at the leading-edge of flow so that both Re and Rex can be determined. The location where the (Re, Rex) values interact this curve corresponds the onset of transition location.

The second method is based on linear stability theory and is referred to the en model that originally proposed by Smith and Gamberoni and Van Ingen. The en assumes that transition occurs when the most unstable Tollmien-Schlichting wave in the boundary layer has grown by some factor of en, where n is defined as follows:

( );~max xnn = ( ) =x

x idxxn

0

)(;~ (12)

Where is the frequency, x0() is the onset location of instability, -i is the spatial growth rate of the TS wave and n(x ;..) describe the amplitude growth of the disturbance along the surface. Given a velocity profile, the local disturbance growth rate can be determined by solving the Orr-Sommerfeld eigenvalue equations. The amplification factor is calculated by integrating the growth rate, usually the spatial growth rate, starting from the point of neutral stability.

Drela and Giles, using the Falkner-Skan profile family, have solved the Orr-Sommerfeld equation for the various shape factors and unstable frequencies. The logarithmic of the maximum amplification rate n is calculated by integrating the local amplification rate downstream from the point of instability, simply assumes that the slope of the amplification rate dn/dRe is only a function of local calculated shape actor H and is given by the corresponding self-similarity solution:

ReRe

Re

Re 0,d

ddnn = (13)

Where, Re is Reynolds number based on the boundary layer momentum thickness. The critical Reynolds number (Re,0) and the slope dn/dRe are expressed by the following formulas:

44.01

295.39.121

20tanh489.01

415.1Relog 0,10 ++

=

HHH (14)

[ ]{ } 5.02 25.0)65.45.1tanh(5.27.34.201.0Re

++= HHd

dn

(15)


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For non-similar flow, based on the properties of the Falkner-Skan profile family, the amplification factor with

respect to the spatial coordinate is determined as follow:

1121

ReRe

Re

2

e

eee

e

uddu

uddn

dd

ddn

ddn

+== (16)

An explicit expression for the integrated amplification factor then becomes:

d

ddnn =

0

)( (17)

Where 0 is the point at which Re=Re,0. The n factor is empirically determined from several experimental data and can vary from one flow situation to another. The free stream effect can be incorporated into the en method by the following correlation suggested by Mack10:

),01.0ln(4.243.8 Tin = 98.207.0 Ti (18)

C. Transition Extent Prediction The extent of the transition region is obtained by using an intermittency function that will divide the flow into

three zones: the laminar zone with a corresponding intermittency function value of zero, the fully turbulent zone with value of 1, and the transition zone that will have a value that varies between zero at the beginning of transition and progressively increase in the transitional region until it reaches unity in the fully turbulent region.

The intermittency function used in the current work is composed of two parts, a surface-distance-dependant component S based on the work of Dhawan and Narasimha and a multidimensional component b(x,y) of Edwards et al. that serves to restrict the applicable range of the transition model to boundary layers. The two components are blended together through the following equation:

[ ]1),(1),( += sb yxyx (19) The Dhawan and Narasimha expression S is defined along the surface of the geometry from the stagnation point:

)412.0exp(1 2=S (20)

,/)max( trss = 75.0

,Re0.9Re= trs (21)

The normal-distance-dependent function b is defined on the basis of the work of Edwards et al. as follows:

)tanh(),( 2= yxb (22)

[ ]

+

=tt

ttt

3

21 ),max(,0max (23)

==

+==

25

35.1221 10,,)(

,500 UtCtdC

td

t t (24)

b approaches to the one near the solid surface, and decays sharply to zero as the edge of boundary layer is approached. By using the intermittency function, the RANS equations are coupled with the transition model in following way:

=

tt

0

t

t

xxxx

>

(25)


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D. Calculation of Laminar Boundary Layer Properties According to the equations (11) to (21), the laminar boundary layer parameters, *, with the boundary layer

edge velocity ue, are required to calculate the transition onset and extent. The direct procedure to extract the boundary layer parameters is using the definition of displace and momentum thickness in the following way:

,)1(0

*

duu

e =

d

uu

uu

ee

)1(0 = (26)

Determination of boundary layer properties using the above equations and Navier-Stokes solution as the input are in some difficulties and are not accurate because firstly the boundary layer thickness is not well defined. Secondly, the turbulence starting from the transition point affects the integral parameters upstream. These results are differing from their fully laminar value in the boundary layer. So an alternative procedure is required for calculating these parameters. In the present work, the two-equation integral formulation based on the dissipation closure is chosen. The first equation is the Von Karman integral relation given by:

2)2( fe

e

Cddu

uH

dd

=++

(27)

Where is momentum thickness, H is shape factor, Cf is skin friction coefficient, ue is the velocity at the boundary layer edge and is he stream wise coordinate. The second equation is a combination of equation 27 and the kinetic energy thickness equation, and is given by:

22)1( **

*f

de

e

CHC

ddu

uHH

ddH

=+

(28)

Where H* is the kinetic energy shape parameter and Cd is the dissipation coefficient. For laminar flow, the two ordinary first order differential equations can be solved with the following closure relationships for H*, Cf and Cd respectively:

+

+

=

HH

HH

H2

2

*

)4(040.0515.1

)4(076.0515.1

4

4

>

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AIAA-2010-4290