INTERNATIONALMechanical Engineering and Technology (IJMET), ISSN 0976 – International Journal of JOURNAL OF MECHANICAL ENG...
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) V...
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) V...
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) V...
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) V...
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) V...
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) V...
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) V...
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) V...
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) V...
Upcoming SlideShare
Loading in …5
×

Estimation of stability derivatives of an oscillating hypersonic delta wings with curved

426 views

Published on

0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
426
On SlideShare
0
From Embeds
0
Number of Embeds
2
Actions
Shares
0
Downloads
4
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

Estimation of stability derivatives of an oscillating hypersonic delta wings with curved

  1. 1. INTERNATIONALMechanical Engineering and Technology (IJMET), ISSN 0976 – International Journal of JOURNAL OF MECHANICAL ENGINEERING 6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEME AND TECHNOLOGY (IJMET)ISSN 0976 – 6340 (Print)ISSN 0976 – 6359 (Online) IJMETVolume 3, Issue 3, September - December (2012), pp. 483-492© IAEME: www.iaeme.com/ijmet.asp ©IAEMEJournal Impact Factor (2012): 3.8071 (Calculated by GISI)www.jifactor.com ESTIMATION OF STABILITY DERIVATIVES OF AN OSCILLATING HYPERSONIC DELTA WINGS WITH CURVED LEADING EDGES Asha Crasta 1, S. A. Khan2 1. Research Scholar, Department of Mathematics, Jain University, Bangalore, India Email: excelasha1@rediffmail.com 2. Principal, Department of Mechanical Engineering, P.A. College of Engineering, Mangalore, India Email: sakhan06@gmail.com ABSTRACT In the present study hypersonic similitude has been used to obtain stability derivatives in pitch and roll of a delta wing with curved leading edges for the attached shock case. A strip theory is used in which strips at different span-wise locations are independent. This combines with the similitude to give a piston theory. The present theory is valid only for attached shock case. Effects of wave reflection and viscosity have not been taken into account. Some of the results have been compared with those of Hui et al, Ghosh and Liu & Hui. Results have been obtained for hypersonic flow of perfect gas over a wide range of mach numbers, incidences and sweep angles. Keywords: Attached shock wave, Curved leading edges, delta wings, Hypersonic, Pitch ,Roll 1. INTRODUCTION The analysis of hypersonic flow over flat deltas (with straight leading edge and curved leading edge) over a considerable incidence range is of current interest with the advent of space shuttle and high performance military aircrafts. The knowledge of aerodynamic load and stability for such types is a need for simple but reasonably accurate methods for parametric calculations facilitating the design process. The dynamic stability computation for these shapes at high incidence (which is likely to occur during the course of reentry or maneuver) is of current interest. When descending shock waves which are usually strong and can be either detached or attached. The idea of hypersonic similitude is due to Tsien [1], who investigated the 2-D and axi-symmetric irrotational equations of motion. Sychev’s [2] large incidence hypersonic similitude is applicable for wings of extremely small span. Cole and Brainard [3] have given a solution for a delta wing of very small span on large incidence. Messiter [4] has found a solution, in the realm of thin shock layer theory, for steady delta wing with detached shock case at small incidence based on hypersonic small disturbance theory. Pike [5] and Hui [6] 483
  2. 2. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEMEhave given theories for steady delta wings in supersonic/hypersonic flow with attachedshocks. For 2-D flow exact solutions were given by Carrier [7] and Hui [6] for the case of anoscillating wedge and by Hui[8] for an oscillating flat plate, which is valid uniformly for allsupersonic Mach numbers and wedge angles or angles of attack with attached shock wave.Hui [9] calculated pressure on the compression side of a flat delta. The numerical solutionsfor the attached shock case have been given by Babaev [10], Voskresenkii [11]. The role ofdynamic stability at high incidence during re-entry or maneuver has been pointed out byOrlik-Ruckemann [12]. The shock attached relatively high aspect ratio delta is often preferredfor its high lift to drag ratio. Hui and Hemdan [13] have studied the unsteady shock detached case in the context ofthin shock layer theory. Liu and Hui [14] have extended Hui’s [9] theory to a shock attachedpitching delta. Light hill [15] has developed a “Piston Theory” for oscillating airfoils at highMach numbers. A parameter δ is introduced, which is a measure of maximum inclinationangle of Mach wave in the flow field. It is assumed that M∞ δ is less than or equal to unity(i.e. M∞ δ ≤ 1) and is of the order of maximum deflection of a streamline. Light hill [15]likened the 2-D unsteady problem to that of a gas flow in a tube driven by a piston andtermed it “Piston Analogy”. Ghosh [16] has developed a large incidence 2-D hypersonic similitude and pistontheory. It includes Light hill’s [15] and Mile’s [17] piston theories. Ghosh and Mistry [18]have applied this theory of order of ¢2 where ¢ is the angle between the attached shock andthe plane approximating the windward surface. For a plane surface, ¢ is the angle between theshock and the body. The only additional restriction compared to small disturbance theory isthat the Mach number downstream of the bow shock is not less than 2.5. Ghosh[19] has obtained a similitude and two similarity parameters for shock attachedoscillating delta wings at large incidence. Crasta & Khan [20] have extended the Ghoshsimilitude to supersonic flows past a planar wedge. Further, Khan & Crasta [21] haveobtained stability derivatives in pitch and roll of a delta wing with curved leading edges forsupersonic flows .In the present analysis this similitude has been extended to shock attacheddelta wings with curved leading edges at large incidence for Hypersonic flows. The pressureon the lee surface is assumed zero.1.1 NomenclatureA γ(γ+1)/4AF, AH amplitude of full and half sine waveAR aspect ratioB [4/(γ+1)] 2C chord lengthCmα, Cmq stiffness &damping derivative in pitchClp rolling moment derivative due to rate of rollL rolling momentM ∞, U ∞ Free stream Mach number and velocityM2 Mach number behind the shockMp Piston Mach numberMs Shock Mach numberS1 Similarity parameters in hypersonic flowa∞ free stream sound velocity 484
  3. 3. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEMEb Semi spanh Non dimensional pivot position, x0/ck π/cm Pitching momentumn Power of X in the equation for leading edgeP Pressure on the wing surfaceP∞ Free stream pressureq Rate of pitchr distance along a ray from apext time in secondU, V velocity components in X, Y directionX, Y, Z body fixed reference systemX0 pivot position for pitching oscillationα Incidence angleβ Shock wave angleαo Mean incident for an oscillating wingγ Specific heat ratioδ Inclination of characteristic linesε Sweep angleθ Half wedge angleφ Angle between shock and wing in the stripφw angle between shock and wedgeρ∞ free stream density2. Analysis To get a curved leading edge we superpose a full sine wave and or half sine wave on astraight leading edge. X-axis is taken along the chord of the wing and the Z-axis isperpendicular to the chord in the plane of the wing.Equation of x-axis is z = o  2πx   πx Equation of full and half sine wave are Z = −a F sin   and Z = −a H sin    c   c  1Equation of straight L.E Z = x cot ∈ 2Where aF & aH are the amplitudes of the full & half sine waves and c is chord length of thewing. Hence the equation of the curved leading edge is  2πx   πx  Z = x cot ∈ − a F sin   − a H sin    c   c  3Area of the wing: C π 4A Area ABD = ∫ Zdx , Let k = . Hence, the wing Area = C 2 (cot ∈ − H ) 0 c π2.1 Strip theory A thin strip of the wing, parallel to the centerline, can be considered independent ofthe z dimension when the velocity component along the z direction is small. This has beendiscussed by Ghosh’s [19]. The strip theory combined with Ghosh’s large incidencesimilitude leads to the “piston analogy” and pressure P on the surface can be directly related 485
  4. 4. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEMEto equivalent piston mach no. Mp. In this case both Mp and flow deflections are permitted tobe large. Hence light hill piston theory [15] or miles strong shock piston theory cannot beused but Ghosh’s piston theory will be applicable. 1 P 2 2 = 1 + AM P + AM P ( B + M P ) 2 , Where P∞ is free stream pressure… 4 P∞2.2 Pitching moment derivatives Let the mean incidence be α 0 for the wing oscillating in pitch with small frequencyand amplitude about an axis X0. The piston velocity and hence pressure on the windwardsurface remains constant on a span wise strip of length 2z at x, the pressure on the lee surfaceis assumed zero. Therefore, the nose up moment is c m = −2 ∫ p.z.( x − x0 ) dx 0 5 2.3 Stiffness derivative The stiffness derivative is non-dimensionalized by dividing with the productof dynamic pressure, wing area and chord length. 2 ∂m − Cm α = ( ) 6 4AH ∂q α=α ρ∞U∞ 2C3 (cot ∈ − ) 0 π q =0Where ρ ∞ and U∞ are density and velocity of the free stream, and q is the rate of pitch(about x = x0) defined positive nose up. c − Cm = α 2 sin α 0 cos α 0 f ( S 1) 3 (cot ∈ − 4 A H ) 0 ∫ ( x cot ∈ −aF sin 2kx − aH sin kx ) (x-x0) dx 7 c πBy solving the above equation, we get sin α 0 cosα 0 f ( S 1) 2 1 − Cmα = [( − h) cot ε + {AF + AH .2.(2h − 1)} 8 4 AH 3 π (cot ∈ − ) π 1 (γ + 1) 2 ) ( B + S 12 ) 2 ]Where f ( S 1) = [ 2 S 1 + ( B + 2S 1 9` 2S 1by using above expression for stiffness derivative calculations have been carried out andsome of the results have been shown. 486
  5. 5. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEME2.4 Damping derivative The damping derivative is non-dimensionalzed by dividing with the dynamic Cpressure, wing area, chord length and characteristic time factor ( ). U∞ 2  ∂m  4AH  −Cmq = )  ∂q  4 ρ ∞ U ∞ C (cot ∈ − α =α 0 π q=0 10 ∂mSince m is given by integration to find ( ) differentiation within the integration is ∂qnecessary. 1  ∂p  P ( x − x0 ) 2) 2)2 ]  ∂q  α =α = A ∞ [ 2S 1 + ( B + 2S 1 (B + S1 11   q =0 0 a∞Substituting the value of the integral in the above equation sin α 0 f ( S 1) 4 1 1 4 − Cm q = [(h 2 − h + ) cot ε − ( 2h − 1) AF + 2( 2h 2 − 2h − 2 + 1) AH ] (cot ∈ − π4 AH ) 3 2 π π  12 2.5 Rolling Moment Derivative due to Rate of Roll Let the roll be p and rolling moment is L, defined according to the right hand systemof reference  c Z = f ( x)  ∴ L = 2 ∫  ∫ pzdz dx   13 0 0  The local piston Mach number normal to the wing surface is given by z (M P ) = (M ∞ ) sinα0 p 14 a∞ The roll-damping derivative is non-dimensionalised by dividing with the product ofdynamic pressure, wing area, and span and characteristic time factor∴ − Cl p = 1 4AH C3 15 ∫U∞C³b(cot ∈ - ) ∞ π sin α 0 f ( S1 ) cot 3 ∈ AF AH 2 − Cl p = [ + cot 2 ∈ − (π − 4) + 4 AH 12 2π π 3 (Cot 2 ε − cot ε ) π 1 2 4 16 AFAH 16 16 cot ∈ ( AF 2 + AH ) − AH 3 − − AF 2 AH ] 4 9π 9 π 2 15π. 487
  6. 6. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEMEWhere S1 = M ∞ sinα 0 (γ + 1) ( B + 2S 12 )f ( S 1) = [ 2S1 + 1 ] 17 2S 1 2 2 (B + S1 )2.6 Results and Discussions The wing geometry for a full and half sine wave (convex and concave) is shown inFig.1 and 2. Stiffness and damping derivatives in pitch have been calculated for a full sinewave and half sine wave (Fig. 3, 4, 5, 6) using present theory and have been compared withGhosh’s theory (Fig. 7 & 8). Results of present theory are better when compared to Ghosh asit uses straight leading edges where as in the present case wing has curved leading edges. Thepresent theory invokes Ghosh strip theory arguments. Hui et al also use strip theoryarguments where by flow at span wise station is considered equivalent to an oscillating flatplate flow; this flow is calculated by perturbing the known steady plate flow (oblique shocksolution) which serves as the basic flow for the theory. Hui et al have obtained closed formexpressions for stiffness and damping in pitch but have not calculated the rolling momentderivative. The present theory calculates rolling moment derivative also, since it makes use ofGhosh quasi steady theory which is simpler than both liu and Hui et al and brings out theexplicit dependence of derivatives on the similarity parameter S1. Concave shaped wing Convex shaped wingFig1: Wing geometry of half sine wave AH=0.1 with Sweep angle = 50 Fig2: Wing geometry of full sine wave AH=0.1 and AH=-0.1 with Sweep angle = 50 488
  7. 7. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEMEFig3&4: Variation of Stiffness derivative with pivot positionFig 5&6: Variation of Stiffness derivative with pivot position.Fig. 7&8: Variation of Stiffness and damping with pivot position 489
  8. 8. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEMEFig. 9 shows the variation of Stiffness and Damping derivative with angle of incidence.Stiffness derivative increases linearly up to angle of 30-35 degrees, it is seen that the dampingderivative increases with increase of angle of incidence and holds good up to a value of about35 degrees. Fig. 10 compares the damping of a wing with the results of Ghosh’s theory fordifferent incidences. Both theories show same trend. Close to shock- detachment the presenttheory is not valid. Fig. 11 shows that rolling damping decreases with Mach number initiallyand confirms the Mach number independence principle for large Mach numbers. Fig. 12 & 13 show variation of rolling derivative with sweep angle for a fulland half sine wave. As the sweep angle increases the rolling derivative decreases up to 75degrees for a full sine wave and up to 65 degrees for a half sine wave. The mathematical errorin Ghosh [19], due to this the factor Clp is decreased to one-forth. This has been identified andrectified (Fig 14.). Fig9: Variation of Stiffness and damping derivative with angle of incidenceFig10: Variation of damping derivative with angle of incidence 490
  9. 9. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep Dec (2012) © IAEME Sep-Fig 11: Variation of rolling derivative with Mach NumberFig 12&13: Variation of Rolling derivative with Sweep angleFig 14: Variation of rolling derivative with Aspect ratio 491
  10. 10. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 3, Sep- Dec (2012) © IAEME2.7 CONCLUSION In the present theory the similitude and piston theory of Ghosh have been extended toa flat wing with curved leading edges. There is a strong impetus for further research by takinginto account the effects of shock motion, viscosity, bluntness of the wing and the real gaseffects. The concepts of the present theory can be extended to axi-symmetric case, for whichanalytical or experimental data are quite limited.REFERENCES[1] Tsien .H. S, Similarity laws of hypersonic flow, Journal of Mathematical Physics, Vol. 25, 1946,pp. 247-251.[2] Sychev V. V., Three Dimensional Hypersonic Gas Glow Past Slender Bodies at High Angles ofAttack, Journal of Applied Mathematics and Mechanics, Vol. 24, August 1960, pp. 296-306.[3] Cole, J. D. and Brainerd, J.J. Slender wings at high angles of attack in hypersonic flow, ARSReprint 1980-1961.[4] Messiter, A. F. Lift of slender delta wings according to Newtonian theory, AIAA Journal, 1963, 1,pp. 794-802.[5] Pike. J, The pressure on flat and anhydral delta wings with attached shock waves, TheAeronautical Quarterly, November 1972, XXIII, Part 4, pp. 253-262.[6] Hui, W.H., Stability of Oscillating Wedges and Caret Wings in Hypersonic and Supersonic Flows, AIAA Journal, Vol. 7, Aug. 1969, pp.1524-1530.[7] Carrier, G.F. 1. 1949, The oscillating Wedge in Supersonic stream , Jr.Areo.Sci. Vol. 16, No.3,pp. 150-152, March.[8] Hui, W. H., Supersonic/hypersonic flow past on oscillating flat plate at high angles of attack ,ZAMP, Vol. 29, 1978, pp. 414-427.[9] Hui, W. H. Supersonic and hypersonic flow with attached shock waves over delta wings, Proc ofRoyal Society, London, 1971, A. 325, pp. 251-268.[10] Babev, D. A. 1963, Numerical solution of the problems of Supersonic flow Past the lowersurface of a delta wing , AIAA Jr., Vol. 1, pp. 2224-2231.[11] Vorkresenskii, G.P. 1968, Numerical solution of the problems of supersonic gas flow past anarbitrary surface of a delta wing in compression region , IZV Akad Nauk SSSR, mekh, zkidk, gaza,no.4, pp.134-142.[12] Orlik-Ruckemann, K. J., Dynamic stability testing of aircraft needs versus capabilities, Progressin the Aerospace Sciences, Academic press, N.Y., 1975, 16, pp. 431-447.[13] Hui, W. H. and Hemdan, H. T. Unsteady hypersonic flow over delta wings with detached shockwaves, AIAA Journal, April 1976, 14, pp. 505-511.[14] Lui, D. D. and Hui W. H., Oscillating delta wings with attached shock waves, AIAA Journal ,June 1977,15, 6, pp. 804-812.[15] Light Hill, M. J., Oscillating Aerofoil at High Mach Numbers, Journal of Aeronautical Sciences,Vol. 20, June 1953, pp. 402-406.[16] Ghosh K, A new similitude for aerofoil in hypersonic flow , Proc of the 6th Canadian congress ofapplied mechanics , Vancouver, 29th may-3rd June, 1977, pp. 685-686.[17] Miles, J. W., Unsteady flow at hypersonic speeds, Hypersonic flow, Butterworths ScientificPublications, London, 1960, pp. 185-197.[18] Ghosh, K. and Mistry B. K. Large incidence hypersonic similitude and oscillating non-planarwedges, AIAA Journal, August 1980,18, 8, pp. 1004-1006.[19] Ghosh K., Hypersonic large deflection similitude for oscillating delta wings, The Aeronauticaljournal,Oct.1984, pp. 357-361.[20] Asha Crasta and Khan S. A., High Incidence Supersonic similitude for Planar wedge,International Journal of Engineering research and Applications, Vol. 2, Issue5, Sep-Oct 2012, pp.468-471.[21] Khan S. A. and Asha Crasta, Oscillating Supersonic delta wings with curved leading edges,Advanced Studies in Contemporary mathematics, Vol. 20(2010) , No.3, pp.359-372. 492

×