Al-khwarizmi Engineering Journal Al-Khwarizmi Engineering Journal, Vol. 4, No. 2, PP 59- 68 (2008) Investigation of Load and Pressure Distribution on Wing with Wake Rollup for Low Speed Aircraft Laith W. Ismail Department of Materials Engineering/ University of Technology (Received 8 April 2008; accepted 22 Julay 2008) Abstract The presented work shows a preliminary analytic method for estimation of load and pressure distributions on low speed wings with flow separation and wake rollup phenomena’s. A higher order vortex panel method is coupled with the numerical lifting line theory by means of iterative procedure including models of separation and wake rollup. The computer programs are written in FORTRAN which are stable and efficient. The capability of the present method is investigated through a number of test cases with different types of wing sections (NACA 0012 and GA(W)-1) for different aspect ratios and angles of attack, the results include the lift and drag curves, lift and pressure distributions along the wing span taking into the consideration the effect of the angles of attack and the aspect ratios on the wake rollup. The pressure distribution on the wings shows that there is a region of constant pressure on the upper surface of the wings near the trailing edge in the middle of the wing, also there is a region of flow separation on the upper surface of the wings. A good agreement is found between the presented work results and other from previous researches. These results show that the presented method is able to capture much of flow over wings feature like separation and wake rollup. Introduction In order to predict the aerodynamic forces of wings up to and beyond stall, including flow separation and wake rollup must be taken into account. Prandtl L. and Tietjens O. G [1] Defined separation of flow first in 1904; fluid particles near the solid surface are retarded by the friction force and by the adverse pressure gradient in the free stream. If the momentum is insufficient, the particles will be brought to rest at the separation point. Downstream of the point, the adverse pressure forces will cause reverse flow. The original boundary layer either passes over the region of recirculation fluid and re-attaches to the surface or forms a wake. In the case of finite wings, the wake is a form of vortex sheet leaves the wing. As a consequence of the pressure difference between lower and upper sides of the wing, a certain flow is caused around the lateral edges. Thus starting a pair of strong trailing vortices. The vortex sheet immediately begins to roll itself up into these vortices; and if eventually passes all of its vortices into a pair [2]. Several methods have been developed for wing problems taking into account the interaction between the several regions of flow around the wing. The present work is a combination of a higher order vortex panel method coupled with the numerical lifting line theory by means of iterative procedure including a model of separation and a model of wake rollup. Many studies, both theoretical and experimental have been done, affected the development of the present method. Jacob K. [3] used two-dimensional procedure combined with a lifting surface theory for the computation of the flow around wings with rear separation. The separation point was predicted by a boundary layer calculation for the attached part of flow. Good predictions for the lift were obtained and it was shown that the wake flow behavior strongly influences the drag. Cebeci T., Lark R.W., Chang K.C., Halsey N.D. and Lee K. [4] developed an interactive viscous-inviscid procedure for the Laith W. Ismail Al-Khwarizmi Engineering Journal, Vol. 4, No. 2, PP 59-68 (2008) 60 computation of viscous flow over three- dimensional wing with separation. Pressure distributions over a number of spanwise stations, and the lift curve of the complete wing were computed. Nilay [5] applied a new hybrid CFD/Discrete Vortex (CFD/VD) method in order to preserve vortices over long distances. This method couples Lifting-Line/Free-Wake method and research in the tip vortex and the wake of a fixed wing, the indication that the tip vortex shed from the wing tip is preserved over long distance when the hybrid CFD/DV method is applied, as a starting point before attempting to simulate rigid rotor flow field. Theory 1. Potential Flow A higher order vortex panel method is developed from the source-vortex panel method [6] and reformulated in a manner suitable for the present model. For potential flow, the Navier- Stokes equations are reduced to the Euler’s equation of motion: PVV t V     1 ).( … (1) where 0.  V (equation of continuity). The total potential of the flow field may be written as: V    …(2) Where  is the potential due to the uniform onset flow, and V the perturbation velocity potential due to the boundary surfaces. From eq. (2), the potential at arbitrary point (P) is:      N j jpanel jP j PPP ds s yxV 1 2 )( )sincos(     … (3) The vorticity at any point along the panel is given by: )()( 1 jj j j jj L s s    … (4) The velocity components u and v in x and y directions respectively at the control point of the th i panel due to the vortex distribution on the th j panel are calculated from [7]:               N j V P PiPiPi N j V P PiPiPi ij ij vV y vyxv uV x uyxu 1 1 sin),( cos),(     …(5) The surface velocity may be directly obtained from: NiforV ii ,...,1  The pressure is calculated from the velocities according to Bernoulli equation as shown below: Nifor V V C i Pi ,...,11 2           …(6) where     q PP C P 2. Boundary Layer The laminar separation criterion by Curle and Skan [8] is used, when the pressure gradient parameter is: 09.0 2 2  ds dV v e   …(7) The location of natural transition is obtained using Michel’s method [9] when: 46.0 2 Re Re 22400 1174.1Re s s tr        …(8) 3. Wake Shape Iteration The iteration loop for wake shape is the inner loop and involves the potential flow analysis; within this loop the separation point is fixed. A local free vortex sheet panel correction angle is computed from the following equations: sheetvortexfreelowerfor nV sheetvortexfreeupperfor nV i ii i sep ii i                         . sin . sin 1 1 …(9) 4. Rolling Up of the Trailing Vortex Sheet The wake behind lifting wings tends to follow the local velocity so that the wake will carry no loads. The vortices shed into the wake flow will eventually rollup near the wing tips to form single vortex patterns and tip vortices [5]. The trailing vortex sheet strength is equal to: dz d z  )( …(10) The circulation or the strength of the th i vortex located at ),( ii yz is:        m b z m b z m b z m b z i ii dz dz d dzg 2 2 2 2  …(11) Laith W. Ismail Al-Khwarizmi Engineering Journal, Vol. 4, No. 2, PP 59-68 (2008) 61 Where i z is the initial abscissa of the point vortex and 0 i y at the same time. The lift and induced drag coefficients are obtained from:       2/ 2/ 2/ 2/ sin)( 2 )( 2 b b iD b b L dzz aV C dzz aV C  …(12) The induced velocity i W of the th i vortex, induced by the th j vortex at ),( ij yz is: R g W j i 2  …(13) where 22 )()( jiji yyzzR  The components of i W are: i ji ii i ji ii W R zz Wu W R yy Wu       cos sin …(14) Adding the contributions from all vortices other than the th i vortex itself and replacing u by dt dz and v by dt dy then:           m ij jiji jiji m ij jiji jiji yyzz zzg dt dy yyzz yyg dt dz 22 22 )()(2 )()(2   …(15) Eq. (15) form a system of two simultaneous 1 st order ordinary differential equation: ),( ),( 2 1 yzF dt dy yzF dt dz   …(16) The solution can be obtained using the 4 th order Rung-Kutta method [10] and the location of each discrete vortex in the vortex sheet can be calculated. The new location of all vortices will represent the shape of the trailing vortex sheet behind the wing. Results and Discussion The capability of the present method for the estimation of load and pressure distributions on rectangular wings with wake rollup will be investigated in a number of test cases. These cases, with different types of wing sections (NACA 0012 and GA(W)-1). Fig. (1) illustrate the effect of aspect ratio on the lift curve, (a) for wing section NACA 0012, (b) for wing section GA(W)-1. It is found that the increase of lift curve slop with increasing aspect ratio. The stall for NACA 0012 occurs at about ( .deg18 ) for (AR=5), while for GA(W)-1 beginning at ( .deg20 ) for (AR=5). Fig. (2) illustrate the effect of aspect ratio on the drag polar, it is shown that the drag coefficient increases rapidly as the lift coefficient increased and the drag coefficient is inversely proportional to aspect ratio. A good agreement was found between the presented work data and others from [11] shown in Fig. (3). Figs. (4) and (5) shows the local lift coefficient Cl with reference to the total lift coefficient CL along the span. When the aspect ratio increases the lift distribution approaches more and more like rectangular distribution, for small aspect ratio the lift distribution is elliptic. Figs. (6) and (7) shows three-dimensional pressure distribution on the upper surface of the span for different aspect ratios. For all cases, the buildup in pressure near the leading edge towards the middle of the wing is shown. The suction on the upper surface is increase with increasing aspect ratio. Also, there is a region of constant pressure on the upper surfaces of the wings near the trailing edge in the middle of the wings. This indicates that there is a flow separation at these regions and this phenomenon is expected at high angles of attack. Fig. (8) illustrate the rolled-up wake sheet for rectangular wing with NACA 0012 airfoil compared at the same conditions from [12]. It is found that the spiral motion is observed on the wake element over the tip region during the rollup process, while the wake sheet remains relatively flat at the middle of the wing. A good agreement was found between the presented work results and others from [12]. Fig. (9) illustrate a comparison between the present method results for rectangular wing with NACA 0012 and others for [13] for the effects of angle of attack and aspect ratio shown in (a) and (b) on the rolled-up wake geometry. It is shown that the wake deformation is more pronounced at higher angle of attack and aspect ratio. A good agreement was found at this comparison. Laith W. Ismail Al-Khwarizmi Engineering Journal, Vol. 4, No. 2, PP 59-68 (2008) 62 Conclusions 1. The lift coefficient is predicted well by present method. The lift curve slope, the smooth variation with angle of attack is all good predicted. 2. The drag coefficient increases rapidly as the lift coefficient increased and the drag coefficient is inversely proportional to aspect ratio. 3. At low aspect ratio the stall occurs at higher angles of attack. Separation begins at the station of maximum local lift in the middle of the wings. 4. Finally, the developed method can be used successfully as a preliminary analysis tool for estimation of load and pressure distributions on low speed wings with wake rollup. Nomenclature AR Aspect ratio - a Wing area m 2 CD Wing drag coefficient - CL Wing lift coefficient - Cl Airfoil lift coefficient - CP Pressure coefficient - gi Strength of i th vortex m 3 /s.m L Length of a panel m N Number of surface panels - n Unit normal vector - P Pressure N/m 2 q Dynamic pressure N/m 2 s Re Reynolds number at surface distance - tr2 Re Reynolds number at momentum thickness - s Surface distance (distance along the panel) m t Time sec. u Velocity component in x- direction m/s V Velocity m/s Ve Velocity at the edge of the boundary layer m/s V Total velocity vector m/s v Velocity component in y- direction m/s x Chordwise coordinate m y Normal coordinate m z Spanwise coordinate m  Angle of attack deg. g  Geometric angle of attack deg.  Circulation m 2 /s  Vorticity strength m3/s.m sep  Vorticity strength at separation point m 3 /s.m 2  Momentum thickness m  Inclination of panel with x- direction deg.  Pressure gradient parameter -  Density kg/m3  Velocity potential m2/s 0 2 4 6 8 10 12 14 16 18 20 22 Angle of Attack (deg.) -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 W in g L if t C o e ff ic ie n t AR=5 AR=4 AR=3 2 4 6 8 10 12 14 16 18 20 22 Angle of Attack (deg.) 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 W in g L if t C o e ff ii c ie n t AR=5 AR=4 AR=3 (a) (b) Fig. 1. Aspect Ratio Effect on the Lift Coefficient Wing Section NACA 0012 (b) Wing Section GA(W)-1 Laith W. Ismail Al-Khwarizmi Engineering Journal, Vol. 4, No. 2, PP 59-68 (2008) 63 0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 Wing Drag Coefficient 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 W in g L if t C o e ff ic ie n t AR=5 Ar=4 AR=3 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 Wing Drag Coefficient -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 W in g L if t C o e ff ic ie n t AR=5 AR=4 AR=3 (a) (b) Fig. 2. Aspect Ratio Effect on the Drag Polar (a) Wing Section NACA 0012 (b) Wing Section GA(W)-1 (a) (b) Fig. 3. (a) Aspect Ratio Effect on Lift coefficient for Rectangular Wing (b) Aspect Ratio Effect on Drag Polar for Rectangular Wing [11] Laith W. Ismail Al-Khwarizmi Engineering Journal, Vol. 4, No. 2, PP 59-68 (2008) 64 (a) (b) Fig. 4. Spanwise Lift Distribution on a Rectangular Wing Section NACA 0012 (a) .deg6 g  (b) .deg12 g  (a) (b) Fig. 5. Spanwise Lift Distribution on a Rectangular Wing Section GA(W)-1 (a) .deg6 g  (b) .deg12 g  -1.00 -0.80 -0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00 1.20 2z/b -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 AR=5 AR=4 AR=3 L l C C -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 2z/b -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 AR=5 AR=4 AR=3 L l C C -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 2z/b -0.90 -0.60 -0.30 0.00 0.30 0.60 0.90 1.20 1.50 AR=5 AR=4 AR=3 L l C C -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 2z/b -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 AR=5 AR=4 AR=3 L l C C Laith W. Ismail Al-Khwarizmi Engineering Journal, Vol. 4, No. 2, PP 59-68 (2008) 65 (a) (b) Fig. 6. Pressure Distribution on the Upper Surface of Wing with NACA 0012 (a) AR=3 (b) AR=4. (a) (b) Fig. 7. Pressure Distribution on the Upper Surface of Wing with GA(W)-1 (a) AR=3 (b) AR=4. Laith W. Ismail Al-Khwarizmi Engineering Journal, Vol. 4, No. 2, PP 59-68 (2008) 66 0.0 0.2 0.4 0.6 0.8 1.0 1.2 2z/b -0.2 -0.1 0.0 0.1 0.2 2 y /b [12] Presented work Fig. 8. Rolling Up Tip Vortices of a Rectangular Wing with NACA 0012 for AR=8 and .deg5 Fig. 9. Angle of Attack and Aspect Ratio Effect on the Wake Section for Rectangular Wing NACA 0012 (a) AR=8 and .deg10 (b) AR=6 and .deg6 0.0 0.2 0.4 0.6 0.8 1.0 1.2 2z/b -0.4 -0.2 0.0 0.2 0.4 2 y /b [13] presented work (a) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 2z/b -0.4 -0.2 0.0 0.2 0.4 2 y /b [13] presented work (b) Laith W. Ismail Al-Khwarizmi Engineering Journal, Vol. 4, No. 2, PP 59-68 (2008) 67 References [1] Prandtl L. and Tietjens O. G. “Fundamentals of Hydro- and Aerodynamics”, Dover 216. [2] Sighard F. Hoerner “Fluid – Dynamic Lift”, Practical Information on Aerodynamic and Hydrodynamic Lift, 1985. [3] Jacob K. “Computation of the Flow around Wings with Rear Separation”, Journal of Aircraft, Vol.21, No. 2, February 1984, PP.97- 98. [4] Cebeci T., Lark R.W., Chang K.C., Halsey N.D. and Lee K. “Airfoils with separation and the resulting Wake”, Journal of Fluid Mechanics, Vol. 163, February 1986, PP. 323-347. [5] Nilay Sezer-Uzol “High – Accuracy Wake and vortex Simulations Using a Hybrid Euler/ Discrete Voortex Method”, M.Sc. Thesis, Department of Aerospace Engineering, the Pennsylvania State University, May 2001. [6] Moran J., Cole K. and Wahl D. “Analysis of Two-Dimensional Incompressible flows by a Subsurface Panel method”, AIAA Journal, Vol. 18, No. 5, May 1980, PP.526-533. [7] Moran J. “An Introduction to Theoretical and Computational Aerodynamics”, John Wiley & Sons, 1984. [8] Chang P. K. “Control of Flow Separation”, McGraw-Hill Book Company, 1976. [9] Leishman J. G., Galbraith R. A. and Hanna J. “Modeling of Trailing Edge Separation on Arbitrary Tow-Dimensional Aerofoils in Incompressible Flow Using an Inviscid Flow Algorithm”, Glasgow University Aero Report No. 8202, May 1982. [10] Chow C. Y. “An Introduction to Computational Fluid Mechanics”, John Wiley & Sons. 1979. [11] John J. Bertin and Michael L. Smith “Aerodynamic for Engineers”, Prentice-Hall, 1979. [12] Yeh and A. Plotkin “Vortex Panel Calculation of Wake Rollup Behind a Large Aspect Ratio Wing”, AAIA Journal, Vol. 24, No. 9, September 1986, PP 1417 – 1423. [13] Suciu and L. Morino “Nonlinear Steady Incompressible Lifting – Surface Analysis with Wake Roll-Up”, AIAA Journal, Vol. 15, NO. 1, January 1977. Laith W. Ismail Al-Khwarizmi Engineering Journal, Vol. 4, No. 2, PP 59-68 (2008) 68 بحث توزيع الحمل والضغظ على الجنبح مع التفبف دوامبت األثر لطبئرة ذات سرعة واطئة ليث وضبح اسمبعيل انغبيؼخ انزكُىنىعُخ/ قسى هُذسخ انًىاد الخالصة َزضًٍ انجؾش انؾبنٍ طشَقخ رؾهُم أونُخ ًَكٍ اسزخذايهب نؾسبة رىصَغ االؽًبل وانضغىط ػهً االعُؾخ فٍ انسشع انىاطئخ يغ وعىد ؽُش رى سثظ طشَقخ اششطخ انذوايبد يغ َظشَخ انخظ انؾبيم انؼذدَخ ثىاسطخ اسهىة ركشاسٌ . ظبهشرٍ أَفصبل انغشَبٌ وانزفبف دوايبد األصش رى ثُبء ػذح ثشايظ فٍ انؾبسىة ثبسزخذاو نغخ فىسرشاٌ ورًزبص هزِ انجشايظ ثكفبئزهب . يزضًُبً ًَىرط اَفصبل انغشَبٌ وًَىرط انزفبف دوايبد االصش . واسزقشاسَزهب نُست (NACA 0012 and GA(W)-1)رى ثؾش ايكبَُخ انطشَقخ انؾبنُخ يٍ خالل ػذد يخزهف يٍ انؾبالد ألعُؾخ راد يقبطغ يخزهفخ رضًُذ انُزبئظ يُؾُُبد انشفغ وانكجؼ ورىصَغ انشفغ وانضغظ ػهً طىل ثبع انغُبػ, رى االخز ثُظش االػزجبس رأصُش , ثبػُخ وصواَب هغىو يخزهفخ رىصَغ انضغظ ػهً االعُؾخ ثٍُ اٌ هُبك يُطقخ انضغظ انضبثذ ػهً انسطؼ انؼهىٌ . صواَب انهغىو وانُست انجبػُخ ػهً انزفبف دوايبد االصش قىسَذ انُزبئظ . نألعُؾخ قشة انؾبفخ انخهفُخ فٍ يُزصف انغُبػ, كًب وعذ اٌ هُبك يُطقخ اَفصبل نهغشَبٌ ػهً انسطؼ انؼهىٌ نالعُؾخ . انًسزؾصهخ يغ أخشي رى انؾصىل ػهُهب يٍ ثؾىس يُشىسح وقذ ظهش رقبسة عُذ ثُُهب انُزبئظ انًسزؾصهخ ثُُذ اٌ انطشَقخ انؾبنُخ قذ رًكُذ يٍ االؽبطخ ثًخزهف ظىاهش انغشَبٌ ػهً االعُؾخ يضم اَفصبل انغشَبٌ وانزفبف . دوايبد االصش