Computational Study Into the Flow Field Developed Around a Cascade Around NACA0012 Airfoil

8/13/2019 Computational Study Into the Flow Field Developed Around a Cascade Around NACA0012 Airfoil

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a__23l dLS VI R Comput. Methods Appl. Mech. Engrg. 167 (1998) 17-32

Computer methodsin appliedmechanics andengineering

Computational study into the flow field developed around a cascadeof NACA 0012 airfoils

N. Ahmed, B.S. Yilbas*, M.O. BudairMechanical Engineering Department, King Fahd University of Petroleum Minerals, Dhahran 31261, Saudi Aruhia

Received 16 January 1998; revised 26 January 1998

AbstractNumerical simulation of flow past airfoils is important in the aerodynamic design of aircraft wings and turbomachinery components.

These lifting devices often attain optimum performance at the condition of onset of separation. Therefore, separation phenomena must beincluded if the analysis is aimed at practical applications. Consequently, in the present study, numerical simulation of steady flow in a linearcascade of NACA 0012 airfoils is accomplished with control volume approach. The flow field is determined by solving two-dimensionalincompressible Navier-Stokes equations while the effects of turbulence are accounted for by the k--E model. Boundary layer developed atthe suction and the pressure surfaces of the airfoil is investigated together with relevant pressure contours for different angles of attack andsolidity. Separation point at the airfoil surface is predicted at high angles of attack. Pressure, lift and drag coefficients are computed and theresults are compared with the predictions of isolated single NACA 0012 airfoil as well as the data available in the literature. However, theleading edge rotation is also introduced to determine the effect of leading edge rotation on stall inception of isolated airfoil. It is found thatincrease in solidity increases the angle of attack at which separation occurs and pressure, lift and drag coefficients are highly influenced bythe angle of attack and the solidity. The results of leading edge rotation indicates that the drag coefficient reduces considerably while the liftcoefficient increases. 0 1998 Elsevier Science S.A. All rights reserved.

1 IntroductionDuring the first two decades of this century the need for an aerodynamic approach to the design of turbines

and compressors was gradually realized. The design of wings and isolated airfoils was well understood andattempts had been made to apply the isolated airfoil approach to turbomachines. As a result of systematicimprovement of the aerodynamic art and the compilation of cascade data, it became possible to design anefficient turbomachine. However, more work should be carried out to minimize some operating problems ofturbo-machines, such as stalling and surging of compressors. With the development of high-speed computers,Computational Fluid Dynamics (CFD) is emerging as an equally important tool as compared to experiments. Atpresent, CFD is rapidly growing and it can be anticipated that it will eventually replace the physical modeltesting in many of the cases. Consequently, studies into cascade using CFD becomes fruitful, since it minimizesthe experimentation.

Rhie and Chow [l] used numerical method to solve two-dimensional Navier-Stokes equations forincompressible flow. The k E model was applied to turbulent flow with and without trailing edge separation. Itwas shown that CFD results agreed with the experimental measurements for NACA 0012 airfoils for attachedflows at Re = 2.8 X lo6 and Re = 3.8 X 106. Jonnavithula et al. [2] conducted a numerical study for stallpropagation in axial compressors. In the numerical study compressor blades were assumed as an isolated linearcascade of airfoils and stall propagation was simulated using vortex tracking method. Experimental resultsindicated that computational predictions were qualitatively correct.

* Corresponding author.

004%7825/98/ 19.00 0 1998 Elsevier Science S.A. All rights reserved.PII: SO0457825(98)00104-2


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Davoudzadeh et al. [3] made a two-dimensional study for the unsteady flow in a linear cascade of J-79 statorblades. Prandtl mixing length model was used to account for the effects of turbulence and the study was focusedon the stall inception portion of the overall rotatin g stall problems. Use of higher order turbulence models wassuggested for future studies to reduce the discrepancies between computational predictions and measurements.Davidson and Rizzi [4] carried out a study to predict stall over a two-dimensional airfoil. They used a standardexplicit Runga Kutta time marching code. They showed that for high Reynolds number (Re = 2 X IOh), BaldwinLomax model failed to predict the stall whereas Algebraic Stress Model (ASM) did predict the stall at an angleof attack of 16 which was found to be in good agreement with experimental observations.

On the experimental side, Day [5] conducted studies using hot wires on two laboratory test compressors toinvestigate the process leading to the formation of finite amplitude rotating stall ceils. Flow was analyzed bothspatially and temporarily to show that model perturbations were not always present prior to stall. Hence,measurements confirmed that unsuspected importance of short length scale disturbances in the process of stallinception. Mathioudakis and Breugelmans [6] carried out measurement of the three-dimensional flow heldwithin a single stage compressor operating in deep rotating stall. The overall features of big stall cellscharacterized by return flows with high tangential velocity upstream of the rotor had been confirmed. Radialvelocities in the cells were found to be much smaller than the axial and circumferential components.

Hoffman [7] made measurements on the lift and the drag characteristics and associated flow field over thesuction surface of a NACA 0015 airfoil at Re = 2.5 X IO both with and without free stream turbulence (FST).The oil flow technique was used to visualize the flow patterns on the suction surface of the airfoil at differentflow intensities. He found that increasing the FST from 0.25% to 9% resulted in an increase in peak liftcoefficient of 30% with no measurable change in the slope of the lift coefficient against angle of attack curve.For the case of drag no appreciable change was noticed. Mehta et al. [S] obtained experimental results forseparated flow over NASA GA(w)-1 airfoil having 2% trailing edge thickness. The fully separated flow wasexamined in terms of surface pressure distribution, skin friction, mean velocity profiles and the boundary layerintegral properties.

Yilbas et al. [9] examined the stall behavior of an isolated compressor rotor experimentally. They showed thataveraged flow measurements during full span stall supported the view that the unstalled part of the blading mustoperate at flows beyond the partial stall zone region.

On the other hand, the stall cell and reversed flow are limiting factors at peak performance of the airfoils. Pastefforts in this field have met with limited success due to availability of the computing facilities when introducingmoving boundaries. In early days. Tenant [IO] presented an interesting analysis for the two-dimensional movingwall diffuser with a step change in area. He concluded that the boundary layer separation delayed when the wallvelocity exceeded the upstream flow velocity. An extensive test program was conducted by Modi et al. [I I] tocontrol boundary layer for NACA 63-21X airfoil through leading edge rotation. They showed that, by leadingedge rotation, lift increased considerably while some degree of reduction in drag was observed. On the otherhand, separation control using moving surf was conducted by Hassan and Sankar [ 121 using a numerical schemefor laminar flow conditions. They showed that leading edge rotation of NACA 0012 airfoil increased the lift andreduced the drag considerably, but the effect of turbulence was left obscure.

The present study consists of two parts. In the first part leading edge rotation of isolated NACA 0012 airfoil isconsidered, in this case, the effect of leading edge rotation on stall inception is investigated. Consequently, flowcharacteristics including flow field, lift and drag coefficients are computed for different rotational speeds andvariable surface area of leading edge rotation. It should be noted that the present study is limited to numericalsimulation of the problem, therefore, practicality of leading edge rotation is not primary concern. In the secondpart, simulation of steady flow in a linear cascade of NACA 0012 airfoils is introduced and the effects ofsolidity (c/s) and angle of attack (cy) on flow field are computed. In both cases, k E model is employed to takeaccount for the turbulence while parametric study is conducted to investigate the effects of solidity and the angleof attack on lift, drag and pressure coefficients. To develop foundations for the parametric study, two-dimensional incompressible flow passing through the linear cascade is taken into account. In addition,predictions obtained from the present study are compared with the results obtained from previous studies [I, 131.

2. The governing equationsThe particular form of the general transport equation which governs the system under study is presented


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below. It is composed of a continuity and two momentum equations. It is assumed that no heat transfer takesplace in the system so that energy equation is not required. Further, the flow Mach number considered is lowenough to assume flow as incompressible. As a result of this, equation of state is not required. Therefore, thegoverning equations are:

Mass conservation

Momentum conservation(1)

where U, is the velocity component in the coordinate directions x,, p is the local pressure and p is the fluiddensity.2.1. Turbulence modeling

A two-dimensional k--E model is used for the solution of the conservation equations, for the kinetic energy ofturbulence and its rate of dissipation. The turbulence kinetic energy is given by:

The isotropic dissipation rate of the turbulent kinetic energy is given by

wherep, = turbulent viscosity, which is CWfPpk2

Second last term in Eq. (4), PE is the destruction rate and G is the rate of generation of turbulent kineticenergy and is given by

G=p,[(z+zJ2].At high Reynolds number where

(5)local isotropy prevails, the rate of dissipation, E is equal to the kinematic

viscosity times fluctuating vorticity. The isotropic dissipation rate E is defined as:au, du;=Tigg

An exact transport equation can be derived from the Navier-Stokes equation for fluctuating vorticity, and thusfor the dissipation, E. This contains complex correlation whose behavior is little known and for which fairlydrastic model assumptions must be introduced in order to make the equation tractable. The outcome of thismodeling is the c-equation (Eq. (4)). Together with the k-equation, e-equation forms the so-called k--Eturbulence model.

Generally, k--E model is valid in regions where the flow is entirely turbulent. Consequently, at high Reynoldsnumbers k--E model is used for the present computations and its constants are given in Table 1.Table IConstants for k-c modelCP C,l Cc2 q fP j; fi E0.09 I 44 1.92 I .o 1.3 I .o I .o 1 o 0.0


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20 N. Ahmed rt al. I Cwnput. Methods Appl. Mrch. Engrg. 167 1998) 17-32

2.2. The j e volume discretizationThe calculation domain is divided into a number of non-overlapping control volumes such that there is one

control volume surrounding each grid point. The differential equation is integrated over the control volume.Piecewise profiles expressing the variation of variable 4 are used to evaluate the required integrals. The result isthe discretization equation containing the values of 4 for a group of grid points. The discretization equationobtained in this manner expresses the conservation principle for the finite control volume just as the differentialequation expresses it for the infinitesimal control volume.2.3. The discretization procedure

As described earlier, partial differential equations are to be discretized into algebraic equations by usingappropriate approximation to obtain a numerical solution to the problem. The procedure followed is the FiniteVolume Method. We will describe it in general curvilinear coordinates for the genera1 transport equation for anorthogonal coordinate system. In vector notation, the general transport equation for steady-state situation isgiven by

v. (pU4) = v. (r*v4, + s (7)This equation is integrated over the finite control volume around note P, as shown in Fig. 1.Therefore

[V. W4)l dv d5 = sdvd5

Here, S, is the average value of S over the finite control volume and ,I+, u. are the components of velocityvector U in the orthogonal coordinate directions 5 and v, respectively. We represent the total flux across a faceof the finite control volume by J for convenience. Focusing our attention on east face:

(10)It is clear that total flux is composed of a convective flux and a diffusive flux. We represent them by C, and

De,, respectively.

NW i NEn..____ _________._____~___...........

(11)

Fig. I. A finite control volume.


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(12)

To get the linear algebraic equations, the source term is approximately linearizedX =&f@/? (13)

Hence, we haveJ, - J,v + J,, - J, = 6% + s, &) Arl A6 (14)

To make further progress, it is necessary to make profile assumptions about the variation of 4 within thefinite control volume. For the diffusion flux a linear profile can be assumed. This results in the centraldiscretization, i.e.

(15)Central discretization is usually not appropriate for the convective flux and may result in non-physicaloscillations in the solution. To make the discretization compatible with physical reality a hybrid scheme is used.Depending on the cell Peclet number it uses either an upwind or central discretization for the convective flux C,.Cell Peclet number is defined as

(16)Using Hybrid Scheme:

C,=pu, 2& + tip Ag , if -2


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3 Grid generation and calculation procedure3.1. Grid generation

For the accuracy of the numerical scheme it is important that grid clustering should be located at largegradient regions in the flow. The grid was generated using algebraic equations for the boundary nodes andLaplace equation for the interior nodes with an effort to minimize the non-smoothness of the grid but at thesame time having grid clustering at regions of larger gradients to obtain an economical and accurate solution.Fig. 2 shows the grid generated for the cascade of NACA 0012 airfoils.

The iterative method was found to be very sensitive to the smoothness and orthogonality of the meshgenerated. For non-smooth meshes heavy under-relaxation was required to prevent divergence of the solution.This large under-relaxation reduces the convergence rate with the consequence of increased computationaleffort.

Fig. 2. Grids used in the computation (a) Size 80 X 62, stagger angle = 30 and L./S = 0.55 for infinite cascade; (b) sire = 80 X 62, staggerangle = 30 and c/s = 0.83; (c) size = 80 62 for isolated airfoil.


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3.2. Boundary conditionsTo solve the governing equations, comprising the model, boundary conditions are needed at each part of the

domain boundary. The problem is solved in x-z plane and infinite linear cascade is considered.The magnitudes of u and w velocities are specified. The incoming flow has been considered to be turbulence

free. As a result the values of k and E have been taken to be zero. For pressure boundary condition either thevalue of pressure at a boundary is needed or the value of flow rate perpendicular to the boundary is to bespecified 1141. At the inlet the incoming flow rate to the domain is specified.

u, v and m are specified and k = E = 0; and it is assumed that at the exit boundary, convection of flowvariables is much larger than the diffusion so that no effect of the downstream values on the upstream flow field.As a result no boundary condition is needed at the exit boundary [ 141. To make our assumption strongly valid,we take the exit boundary at so large a distance (5 chord length) from the airfoil that there is no circulating flowat the exit boundary.

a4=O and -=0dYwhere C applies to all variables.

No slip boundary condition is imposed at the solid boundary. For turbulence quantities, Law of Wall is usedto determine their values in the first cell adjacent to the wall. These values serve as boundary conditions for therest of the domain.3.2.1. LUM?of the wall boundary conditions for standard k--E model

If the law of the wall is applied, then, we suppose that the first computational point close to the wall (P) is inthe turbulent sublayer. At this point the velocity UP is parallel to the boundary and has a logarithmic variation:

u*, called the friction velocity, and yl, representing a dimensionless normal distance from point P to the wall,are defined as I/Z

andPY,,YY,: = yy

where r,, is the shear stress at the wall, is the Von Karman constant, E is a roughness parameter and Y,, is theactual distance from the point P to the wall.

It is the value of the dimensionless distance y,: that sets the limits between the different sublayers. For theturbulent layer y, is approximately between 10 and 400.The nose rotation has been considered to explore its potential advantages. The isolated airfoil at threedifferent angles of attack ( 13, 15, 16) has been analyzed. Five percent of the chord on each of the suction andpressure surfaces of the airfoil has been specified with a wall velocity such that at the upper surface, thisvelocity is in the direction of the flow whereas at the lower surface it is in the direction opposite to that of theflow. The free stream velocity of the incoming flow is set at 50 m/s. Two values of tangential velocities as 50m/s and 150 m/s, have been considered for the leading edge rotation velocity.3.3. Calculation procedure

For the general variable 4 the solution to the discretized algebraic equations can be obtained using eitherdirect or iterative methods. Direct method needs the algebraic equations to be linear. If, however, the equationsare non-linear then an iterative method is necessary.

A very effective algebraic equation solver is the Tri Diagonal Matrix Algorithm. In this method, algebraic


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24 N. Ahmed et al. I Com~mt. Methods Appl. Med. Etzgrg. 167 (1998) 17-32

equations for a row of nodes are solved simultaneously using Thomas Algorithm. Hence, the boundary pointinformation is carried in a single iteration for that row. This process is carried out for each row successively.

Most of the time we used this method for the solution of our problem. However, equation for pressurecorrection was solved using whole field method, since a simultaneous satisfaction of the continuity in the wholedomain increased the convergence rate. Usually, several hundred sweeps were required to obtain a convergedsolution.

If the pressure field is given, the solution to the momentum equations can be obtained by employing themethod described above. Moreover, unless the correct pressure is employed, the resulting velocity field obtainedfrom the solution of the momentum equations unable to satisfy the continuity equation. However, no explicitequation for pressure is given. This is particularly true for an incompressible flow. In this regard, severalmethods are available. Consequently, the SIMPLE procedure is used, which is basically an iterative process.

Let a tentatively calculated velocity field based on a guessed pressure field p* is denoted by UT, nz. Let thecorrect pressure p is obtained from:

p =p* +p (26)The corresponding correction in velocities n;, uh can be introduced in a similar manner:

UC= UT + Ll; (27)

Making certain assumptions, the velocity correction formula for east face of the mesh element, for example, isgiven by:

Now, discretizing the continuity equation and using the velocity correction formulas, one can obtain anequation for pressure correction:

(A, - S,,)p, = A,>p,: A,P: + 4~: Am p:. SC, (30)Thus, we have obtained an equation for pressure correction or in turn for pressure. The important steps to

compute the flow properties are as follows:Guess the pressure field p*.Solve the momentum equations to obtain UT, u;.Solve the pressure correction equation.Calculate p by adding p to p .Solve equations for other variables 4 (e.g. turbulence kinetic energy) if they have a coupling withmomentum equations.Treat the corrected pressure as a new guessed pressure p . Return to step 2 and repeat the whole procedureuntil a converged solution is obtained.

4. esults and discussionsThe flow in a cascade of NACA 0012 airfoils is studied at Re = 3.24 X 10. An H-type grid was used with

80 X 62 points per passage. Over the airfoil surface 49 points were distributed. In order to increase the accuracyof the calculations and for economy of computations, a high grid refinement was introduced near the solidboundary. The grid independent test was conducted and its results are shown in Fig. 3. Consequently, selectionof 80 X 60 mesh points gives sufficient accuracy for the present case. The front and rear outer boundaries arelocated at 4 and 5 chord distances away from the body, respectively. Single passage periodicity assumption isused to simulate the infinite cascade. At the inlet, the incoming mass flow is specified together with velocitycomponents and turbulence quantities. At the exit, the velocity components and turbulence parameters areextrapolated from the inner solution by assuming that the first derivatives of the flow properties are zero. Theangle of attack is varied from 0 to 24 degrees gradually while solidity (c/s) ranges from 0.55 to 0.83.


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0.1 0. 2 0.3 0. 4 0. 5 0.6 0.7 0.8 0. 9 1

X/CFig. 3. Grid independent test results.

Fig. 4 shows the velocity field around the isolated single NACA 0012 airfoil for three different angles ofattack. It is obvious that boundary layer thickness is considerably small at the leading edge and increases alongthe chord. At the upper and lower surfaces of the airfoil, nearly symmetric boundary layer profiles are obtainedat angle of attack (a) of 3 as expected. As the angle of attack increases the symmetry disappears andconsiderable difference occurs. The development of free shear layer at the trailing edge is quite apparent. Thisfree shear layer diminishes in downstream because of the diffusion process. The spilling of the flow is evident athigh angles of attack, because of the development of high pressure region on the lower surface of the airfoil atthe leading edge. This results in a high velocity to be developed in this region. Therefore, the large differencebetween the pressures at the two surfaces of the airfoil generates a lift. As the angle of attack reaches to 16 andabove, the flow cannot remain attached to the airfoil surface in the down stream because of a high turning angleand thickening of the boundary layer occurs. In this case, it detaches from the surface, i.e. separation is resulted.

Fig. 5 shows the pressure coefficients for the two cases of leading edge rotation at different angles of attack. Itis evident that up to the extent of 25% chord length, small increase in the pressure difference between the upperand the lower surfaces occur. Moreover, in comparison to the case of without rotation, the pressure on both theupper and lower surfaces has increased somewhat, but the difference in pressure is almost identical for the twocases (i.e. with and without rotation). It is observed that the effect of leading edge rotation is more pronouncedfor a higher angle of attack (16) as compared to a lower angle of attack. It is quite apparent that a highrotational velocity causes high pressure difference to occur between the two surfaces. Hence, a rotating velocityof 150 m/s is found to be more promising to increase the lift.

However, another case has been considered in which rotational velocity has been set at 150 m/s, whereas the11% chord length has been specified with a wall velocity. Fig. 6 shows the velocity vector for this case at anangle of attack of 16 which can be comparable to no leading edge rotation. It is observed that there is anincreased flow spilling at the leading edge. The separation point has shifted downward towards the trailing edgeand a reduction in the wake size is quite evident. However, there is no significant effect of rotation on theseparation region and still large separation occurs close to the leading edge, therefore, the effect of leading edgesurface rotation seems to be very localized. If the separation could be delayed to a significant extent then thismethod could be very advantageous.

Fig. 7 shows the pressure coefficients for the two cases at different angles of attack. It can be seen that suctionpeak has increased in magnitude for 11% leading edge chord rotation. This has a positive effect on the liftgenerated. At the trailing edge, the pressure difference is essentially the same. On the pressure surface weobserve a pressure kink. This is due to the fact that airfoil surface has a velocity upstream of this point whereasit has zero velocity downstream of this point.


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6

(b)

Fig. 8 shows the lift and drag coefficients for the two cases. The case of leading edge rotation has beensimulated for only three different angles of attack since at lower angle of attack there is no separation, so there isno effect of delaying of separation by this practice. It is observed that there is an increase in the lift in the caseof the rotating nose as compared to the non-rotating nose airfoil. It is further observed that lift increases with theincrease in the percentage of the length of the chord that rotates. The slope of the curve corresponding to theleading edge rotation nose airfoil retains the same slope as at lower incidence% i.e. similar to the case obtainedfrom the experiment [ 11. On the other hand, decrease in drag occurs and the reduction in drag increases withincreasing angle of attack. The reduction in drag is more pronounced for 11 chord surface rotation at leadingedge. At 16 angle of attack the reduction in drag is quite signiticant. The slope of the drag curve reduces or therotating surface length increases. This is due to the separation point moving further down stream as the rotatingsurface length increases. In the case of full surface rotation drag coefficient reduces considerably while lifecoefficient increases.

Fig. 10 shows the flow field around the intinite cascade of NACA 0012 airfoils for C./S of 0.55 and 0.83,respectively. It is evident that increasing the solidity (c/s) effects the boundary layer developed around theairfoils. Separation starts earlier in the case of low solidity (c/s). The flow spilling at the leading edge is


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28 N. Ahmcd ef al. i Compuf. Methods Appl. Mech. Engrg. 167 (19981 17-.Z.?

0 01 02 03 0.4 0.5 06 07 0.8 09 1x/C

Ftg. 7. Pressure coefficient along the chord length for different rotatmg surface to chord ratio.

0 2 4 6 6 10 12 14 16 18a

Fig. 8. Lift and drag coefficients with angle of attack for different rotating surface to chord ratto.


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Fig. 9. Pressure coefficient along chord when full surface rotating

L ---LuC/H - A _-_/_A_-_ _a--- - AI Alpha=22 de.3,, _ --Zc-u-r------ A ___Y-- - - --c ACISd.55,l_~_------- _ __ u - - NYU- -M

Flow field for infinite cascade at (Y= 22 and stagger = 30, for solidity ratios of (a) c/s = 0.55 and (b) c/s = 0.83


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-*-- Cascade c/s = 0.830 Experiment Isolated ailroilll]

0 2 4 6 8 IO 12 14 16 18 20a

Fig. 12. Effect of cascade solidity ratio on (a) hft coefficient and (b) drag coefficient

decreases as the solidity increases. This is due to the pressure suppression as a result of closely spaced airfoils inthe cascade. Isolated single airfoil has the largest suction peak.

Fig. 12 shows the lift coefficient corresponding to different c/s and angle of attack. Lift coefficient increaseswith increasing of angle of attack. The maximum obtainable lift becomes small as the solidity increases. This isdue to the loss in suction peak at the upper surface as this is influenced by the pressure suppression of theneighboring airfoil. The angle of attack corresponding to separation point moves toward relatively higher valuesas the solidity increases. When comparing the present predictions with the previous results [ 1,131, they are ingood agreement. However, the drag coefficient increases as the angle of attack increases and it reduces slightlywith the increase in solidity ratio. Isolated airfoil has the largest drag for a given angle of attack. It is evidentthat as the separation is reached the drag increases sharply, whereas the lift drops for isolated airfoil. Moreover,cascade data show increasing trend for both drag and lift coefficients as the angle of attack increases towards20. This is because no separation is resulted due to pressure suppression effect of neighboring airfoils. Whencomparing cascade data with previous study [13], it is evident that both results are in agreement.

5 ConclusionsA simulation of the flow field around the isolated single with and without leading edge rotation and a linear

cascade of NACA 0012 airfoils is carried out using a control volume scheme. Attached and separated flows havebeen computed using the k E turbulence model. Lift, drag and pressure coefficients have been determined.


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32 N. Ahrnrd ct cd. I Comput. Methods Appl. Mech. Etqrg. I67 19988) 17-32

The conclusions derived for the flow field studied are as follows:High degree of flow spilling occurs at the leading edge of the airfoil for high angle of attack. This, then,results in separation at the trailing edge. The point of separation moves towards the leading edge when c/sincreases. The effect of pressure suppression is quite evident.With the increase in incidence the adverse pressure gradient attains large values.The boundary layer thickness increases towards the trailing edge as the angle of attack increases. The rateof this increase reduces at low solidity.In the case of leading edge rotation, separation delays at high angle of attacks and diminishes as the totalsurface rotates.

The conclusion obtained from the pressure, lift and drag coefficients may be listed as follows:Pressure coefficient on the suction surface at the trailing edge attains higher values with incidence and witha decrease in the solidity. The present predictions give closer results to experimental findings as comparedto previous studies.Lift coefficient reduces as large separation occurs. However, this has not been predicted to be drastic asbeing observed in experiments. Further. the incidence angle at which a drop in the lift occurs has a slightlylarger value as compared to available experimental data. This may be due to the steady state analysisemployed in the computation. In this case once the separation occurs, the vortex developed stays as it formsrather than detaches from the surface and forming the vortex shed. Therefore, the lift coefficient predictedimmediately after the separation may not agree well with experimental results. However, the present resultsobtained for isolated airfoil give closer values to experimental findings as compared to previous results.As the solidity increases, the incidence at which maximum lift is obtained, increases.It is found from the simulation results of leading edge rotation that slight increase in lift, but considerabledecrease in drag occur in the case of nose rotation. However, the lift coefficient increases drastically whiledrag coefficient reduces as the full surface of the airfoil is rotated, in this case. separation vanishes.

Acknowledgment

The authors acknowledge the support of King Fahd University of Petroleum and Minerals, Dhahran, SaudiArabia for this work.

References[ I] C.M. Rhie and W.L. Chow. Numerical study of the turbulent Ilow past an airfoil with trailing edge separation. AIAA J. 21( I I) (1983)

152%1532.[2] Thangam S. Jonnavithula and F. Ststo, Computational and experimental study of stall propagation m axial compressors, AIAA .I.

(1990) 1945-1952.[3] F. Davoudzadeh, N.S. Liu, S.J. Shamroth and S.J. Thoren, Navier-Stokes solution of the turbulent flow lields about an isolated airfoil,

AIAA J. 26 (I 990) 242-25 1.[4] L. Davidson and A. Rizri. Navier-Stokes stall predictions using an algebraic Reynolds stress model, J. Space Craft Rockets 29(6)(1992) 794-800.[S] I.J. Day, Stall inception in axial flow compressors, J. Turbomachinery I IS (1993) l-9.[6] K. Mathioudakis and F.A.E. Breugelmans, Three-dimensional flow in deep rotating cells of an axial flow compressor, J. Propulsion

Power 4 (1988) 263-269.[7] J.A. Hoffman, Effects of free stream turbulence on the performance characteristics of an airfoil, AIAA J. 29(9) ( 1991 ) 1353-13.54.[8] J.M. Mehta and S. Goradia, Experimental studies of the separated flow over a NASA GA(w)-1 airfoil, AIAA J. 22(4) ( 1984) SS2-554.[9] Ali Koc, B.S. Yilbas and E. Baltacmglu, Some observations on stall behavior of an isolated compressor rotor, Mech. lncorp Engineer

(1992) 7-14.[IO] J.S. Tennant, A subsonic diffuser with moving walls for boundary layer control. AIAA J. I I (1973) 240-242.[I I] V.J. Modi, J.L.C. Sun. T. Akutsu, P. Lake, K. McMilan, P.G. Swinton and D. Mullins, Moving-surface boundary-layer control for

aircraft operation at high incidence, AIAA J. Aircraft 1X( I I ) (198 I) 963-968.[ 121A.A. Hassan and L.M. Sankar, Separation control using moving surf a numerical simulation, AIAA J. Aircraft 29(l) (1992) I31 - 139.[I31 D.W. Zingg and G.W. Johnston, Interactive airfoil calculations with higher-order viscous flow equations, AIAA J. 29(7) (1991)

1033-1040.[141 C. Hirch, Numerical Computation of Internal and External Flows, Wiley Series in Numerical Methods in Engineering, Vol. 2 (JohnWiley and Sons Ltd., (Reprint), England, 1991).

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Computational Study Into the Flow Field Developed Around a Cascade Around NACA0012 Airfoil