The interaction between slender body vortices and a single fin located down the axis of the body is investigated numerically for angle of attack of 30 deg. and Reynolds number of 6000. The present research includes a parametric study on the effects of fin axial and azimuthal positions on the development of the vortex system. A numerical method based on the pseudo-compressibility is used for solving the three-dimensional incompressible Navier-Stokes equations using Lower-Upper Symmetric Gauss-Seidel implicit scheme.

The numerical results show that the vortices remain very coherent and attached to the body until they reach the fin section where they become less coherent and begin to

separate from the body. Also, the result shows that the fin location does not affect the upstream development of the vortices but it does affect the location at which the vortices separate from the body. The effect of azimuthal fin positions was also investigated. As

azimuthal angle of the fin increased, the size of the vortex on the port side decreased,

but the starboard side vortex grew in size and moved across the leeward ray to the port

side. The computed results are found to agree well with the experimental data

A. Omar, Ashraf

Kim, Chongam

Rho, Oh Hyun

Universiti Putra Malaysia Institutional Repository

Pertanika 1. Sci. & Techno!. Supplement 9(2):117-125 (20Gl)ISSN: 0128-7680© Universiti Putra Malaysia PressNumerical Simulation of the Interaction betweenSlender Body Vortices and a FinAshraf A. Omar, Chongam Kim & Oh Hyun RhoDepartment of Aerospace Engineering, Seoul National University, Seou~ KureaABSTRAGrThe interaction between slender body vortices and a single fm located down the axis ofthe body is investigated numerically for angle of attack of 30 deg. and Reynolds numberof 6000. The present research includes a parametric study on the effects of fin axial andazimuthal positions on the development of the vortex system. A numerical method basedon the pseudo-compressibility is used for solving the three-dimensional incompressibleNavier-Stokes equations using Lower-Upper Symmetric Gauss-Seidel implicit scheme.The numerical results show that the vortices remain very coherent and attached to thebody until they reach the fin section where they become less coherent and begin toseparate from the body. Also, the result shows that the fin location does not affect theupstream development of the vortices but it does affect the location at which the vorticesseparate from the body. The effect of azimuthal fin positions was also investigated. Asazimuthal angle of the fin increased, the size of the vortex on the port side decreased,but the starboard side vortex grew in size and moved across the leeward ray to the portside. The computed results are found to agree well with the experimental data.Keywords: Slender body and f"m interaction, asymmetric vortices, steady state, laminar,pseudo-compressibility, LU-SGS algorithmsINTRODUCTIONIn numerous applications involving missile, rocket, helicopter, and aircraft aerodynamics,the interaction of vortical flow with control surface plays a very important role indetermining the agility and overall performance of craft. Such vortex-eontrol surfaceinteraction, therefore, continues to be the subject of investigation by both experimentaland computational modeling 1-3. At lower angle of attacks, these vortices are symmetric,but as the angle of attack increases, the vortices become asymmetric and have differentstrengths and locations with respect to the missile body. This vortex system may im-pingeon, or interact with fins or other control surface located further down the missile axis.The interaction is very complex in nature; it can change the pressure field, and,consequently, the loading on these surfaces. This will alter the effectiveness of thesurface. In addition, vortex induced loads are difficult to predict and cause changes inyaw, roll, and pitch motions which can be coupled in a nonlinear manner. Aninvestigation to understand the nature and outcome of such interaction is, therefore, ofgreat value.There has been quite a bit of recent work investigating the interaction betweenvortical flows and fin surfaces. Acharya et al. (1994) conducted experiments to examinethe interaction between nose vortex and single fin on various positions along bodysurface. Their results documented the interaction using flow visualization and meanpressure measurements on the fin surface, where varying degrees of symmetry, preservationor loss of coherence of the incident vortical structures are evident. an-Qian Chan's etal. (1997) experiment focused on investigating the asymmetric vortices by means ofseries of configuration; the body alone, the combination of the body and variouspositions of the wings. Their results include a prediction of vortex induced effect on theside force coefficient. Washburn et al. (1997), who examined the effect of tail locationAshraf A. Omar, Chongam Kim & Oh Hyun Rhoon the vortical flowfield in the wake of a sharp-edged delta wing, observed that noupstream effects on the development or trajectories of the vortices were seen as theposition of the tail was changed.The purpose of the present investigation is to simulate numerically the incidentvortex interaction for simplified configuration of a slender body with a single fin, whichwas considered experimentally (Acharaya et al. 1994). The investigation included aparametric study of the effect of the axial location and azimuthal positions of the fin.The flow was considered to be steady and laminar due to experimental conditions(Acharaya et al.1994). An accurate numerical description of the flow required a solutionof the in-eompressible Navier-Stoke equation. The numerical algorithm is based on themethod of artificial compressibility and uses a second order central difference schemewith a third order numerical dissipation (Ok 1993; Yoon and Kwak; Start E. et al.). Theflow streamlines and the pressure coefficient distribution on both the fin surface ispresented. The formation and growth of the vortices, the trajectories of these vorticesand the global structure of these vortices at fin section during the interaction are alsopresented. The computational results are compared with available experimental data.PROCEDUREGoverning EquationsThe three-dimensional incompressible avier-Stokes equation are solved using themethod of pseudo compressibility (Ok 1993; Yoon and Kwak; Start E. et al.), which addsa pseudo-time derivative of pressure p to the continuity equation, to make the resultantgoverning equations hyperbolic, is(1)where "t pseudo-time. The above equation can be written in curvilinear coordinates asthe following form.1 aQ a a a:r~=- a;(E-EJ- aYJ(F-FJ- a~(G-G.)=-R (2)J is the Jacobian of the transformation; Q is the vector of primitive variable; E, F and G,convective flux vectors: Ey , Fy , Gy , the flux vectors for the viscous term; R is defined asthe residual vector of these equations. The variable ;, YJ, and ~ describe a curvilinear,body fitted grid.Numerical AlgorithmThe inviscid flux terms are differenced using a second order central differencingscheme with a third order numerical damping. The viscous flux terms are differencedusing a second order central difference (Ok 1993). An implicit time integration schemeapplied to equation (2) is118Q "+I _Q"....::.----,_=- = _R"+1A"tPertanikaJ. Sci. & Techno!. Supplement Vo!. 9 o. 2, 2001(3)Numerical Simulation between Slender Body Vortices and a Finwhere the superscript n denotes quantities at the n-th pseudo-time iteration. Pseudo-time step ~"t in equation (3) was calculated by local time step in order to accelerate theconvergence. The Lower-Upper Symmetric Gauss-Seedily (LU-SGS) implicit scheme byYoon and Kwak was used. More detail of the LU-SGS scheme can be found in Yoon &Kwak.BODY GEOMETRIES AND GRIDSThe body geometry is defined by a 2.6 diameter ogive nose with slender forebody lengthwith the variable location of the fin. The axial locations of the vertical fin are x/d=4.9,and 7.1, respectively. The fin has a tapered and clipped delta shape (Fig. 1). This bodywas tested experimentally by Acharya et al. (1994). The results computed on one zoneC-O type grid (Fig. 2), which are generated by rotating a two-dimensional C-type gridencompassing the contours of the ogive slender around its longitude axis. Outerboundary of computational domain is located at R/d=28. This radial extent of theFig. 1. Surface grid for a tangent-ogive body with a fin at (x/d)f=7.1Fig. 2. FuU view of the computational gridPertanikaJ. Sci. & Techno!. Supplement Vo!. 9 No.2, 2001 119Ashraf A. Omar, Chongam Kim & Oh Hyun Rhocomputational domain ensures that a solution is less susceptible to the far fieldboundary condition. Two different physical grid points are used· for computation; gridA, as shown in Fig. 2, is consisted of 75x81x71 points for body with axial fin positionequal to 7.1, and grid B of 65x81x71 points with axial fin position equal to 4.9. Grid wasclustered to the body and fin surface to resolve boundary layer and circumferential gridspacing is concentrated on the leeward surface where vortices are expected to exist. Fora fin placed at azimuthal angle greater than zero (non-vertical fin), the zero azimuthalangle (vertical fin) grid was rotated to new azimuthal angle.Initial and Boundary ConditionFreestream values are specified at far boundaries, while out flow boundary is extrapolatedusing zeroth order extrapolation. Pressure is kept its free stream value. On a solidsurface, usual no slip condition is applied. The pressure at the solid surface is obtainedby setting the pressure gradient normal to the wall to be zero. Unknown value of Q onthe boundaries are updated explicitly.RESULTS AND DISCUSSIONThe computation was performed for angle of attack of 30 deg. and Reynolds number of6000 based on the model diameter. These selections were based upon the experimentalwork by Acharya et al. (1994).The convergence history summarized in Fig. 3 discloses that the vortical flowfieldresults for both cases indeed reach a steady state solution. Shown in Fig. 3 are theconvergence histories of residual, normal force, and side force. It is seen that ther=::;;:l~c<.>r=::;;:l~120r=;:;:l~Fig. 3. Convergence histories: <Pf = 0°PertanikaJ. Sci. & Technol. Supplement Vol. 9 o. 2, 2001xld =3.5x'd =6.6Fig. 4. Cross sectional flow streamlines:(x/d)f=7.1Fig. 5. Cross sectional flow streamlines:(x/d)f=4·9residual is dropped below four orders of magnitude within about 4500 iteration. Fig. 4shows the development of the flow along axial location of the body when the fin islocated at x/d=7.1. At axial location close to the nose (x/d=1.7), the pair of the vorticeshas formed symmetrical, coherent and attached to the body. They remain largelycoherent, and attached to the body through x/d=4.9. At x/d=6.9, near the leading edgeof the fin, the degree of asymmetry increases where the left side primary vortex starts toleave away from the body due to presence of the fin, and the right side vortex stillincreasing in size and attached to the body. At x/d=8.8, in the middle of the fin, theprimary vortices are far away from the body. At this position, secondary vortices appearso clearly and it is big in size due the interaction with comer flow. At x/d=9.4, trailingedge of the fin (not shown here), the vortices are keeping a same shape with littleincrease in the size. The comparison between the computational streamlines andexperimental visualization photograph by Acharaya et at. (1994) shows good agreement.The flowfield develops similarly with a fin placed at axial location equal to 4.9 diameteras shown in Fig. 5. As the fin is positioned closer to the nose, the vortices are smallerin size when they come in contact with the fin. Therefore, while the axial location of thefin does not affect the upstream development of the vortices, it does affect the locationat which vortices separate from the body. No vortex breakdown was observed in eitherthe present study or the experiment (Acharaya et al. 1994). A comparison of the locationof primary vortices center between the experiment (Acharaya et al. 1994) and thecomputation is shown in Figs. 6.a and 6.b. Good agreement is obtained. These figuresshow that the trajectories of the port and the starboard side vortices are almost the samein each, confirming the symmetry of the trajectories of the vortices. These figures showclearly that the axial location of the fin dose not affect the development of theflowfield.The surface pressure coefficient contours for the axial fin position of x/d=7.1 and4.9 are shown in Figs. 7.a and 7.b, respectively. In these figures, dashed line indicates anegative Cp, while solid line a positive Cpo Fig. 7.a shows Cp contour in both sides ofthe fin located at x/d=7.1. A large low-pressure region, which is seen between the finroot and half span, is the consequence of the primary and secondary vortices. As seenPertanikaJ. Sci. & Techno\. Supplement Vol. 9 No.2, 2001 121Ashraf A. Omar, Chongam Kiln & Oh Hyun Rho·:c :a. (x/d)f=7.1Fig. 6. Vortex center trajecturies: l/Jj=Oa. (x/d)f=7.1b. (xfd)f=4.9Fig. 7. Pressure coefficient contaurs on fin: l/Jj=O°in Fig. 4, at x/d=8.8, the center of the two primary vortices pass on either side of thefin slightly inboard of the half span with secondary vortices on the fin-body corner, theycre-ating a region of low pressure between the fin root and half-span. At half-span,contours show a rapid rise in pressure in spanwise direction due to flow toward the finsurface. The fin surface pressure coefficient contour shown in Fig. 7.b for fin at x/d=4.9is similar but the low pressure region is small in spanwise extent and close to fin rootdue to the small size and location of the primary vortex.A combination of high angle of attack and asymmetrical azimuthal angle of the finproduces very complex vortical flow on fin section. Therefore, one of the concerns of122 PertanikaJ. Sci. & Technol. Supplement Vol. 9 0.2,2001umerical Simulation between Slender Body Vortices and a Finthe present study is to investigate the effect of the azimuthal position of the fin on theflow development and pressure distribution contour on both sides of the fin. Thesolution obtained for the same bodies and flow conditions is described for vertical fin.The fin was rotated to several azimuthal positions; <jlf=6°, llo and 16°, where <jlf=O° refersto the vertical fin, <jlf being positive clock-wise. For each azimuthal position of the fin, thenumerical results reached steady flowfields.In the first case, the fin located at x/d=7.1, Figs. 8.a, 8.b and 8.c show the flowdevelopment along the missile body when the fin azimuthal positions are 6°, II 0 and16°, respectively. There are no apparent effects of the azimuthal fin position on thestructures and the trajectories of the vortices upstream of the fin section. The effect ofthe fin azimuthal position is only altered once they reach the fin section. The experimentalflow visualization by Acharya et ai. 1994 also showed a similar trend. At both axial finlocations «x/d) f=4.9 and 7.1), the azimuthal position of the fin changes the relativedistance between each vortex and the fin surface. The trajectories of the vortices on thefin are also affected by the azimuthal position of the fin. At axial location ahead of fin'sleading edge, the vortices appear very similar to those shown in Figs. 4 and 5(verticall(/d=IU~ Q4'T"""""........-a-_""",• - ....VoL-t- ..... ""'"•-~,iOr-•-02 •• .I I IQ4r---------------,16124a. (xld)f=7.1oc. ¥= 16"Fig. 8. Effect oj the fin azimuthalpositions on flow development:(x/d)J= 7.1b. (xfd)f=4.9Fig. 9. Variation oj normal Jorce coefficient on fin with 4JJPertanikaJ. Sci. & Techno\. Supplement Vo\. 9 No.2, 2001 123Ashraf A. Omar, Chongam Kim & Oh Hyun Rhofin). On the fin section, as azimuthal angle of the fin increases, the size of the vorticeson the port side decreases and the starboard side primary vortices grow in size and moveacross the leeward ray to the port side of the body.The effect of the vortex system can be clearly shown by studying the normal forceon the fin. Fig. 9 shows the variation of the normal force on the fin with azimuthalposition of the fin at each fin location. Shown are the components on the port side andstarboard side surface (Cfp and Cfs, respectively). Overall, computed results agree fairlywell with experimental data (Acharya et at. 1994). The outward normal to the port sideof the fin is assumed to be positive direction according to experimental condition(Acharya et al. 1994). For each fin position «x/d) f=4.9 and (x/d) f=7.1), the force isnearly anti-symmetric about <jlf=O, where total force is close to zero. As azimuthal positionof the fin (<jlf) increases, the magnitude of the efp on the port side decreases due to adecrease in the vortex effect on the fin while the magnitude of Cfs on the starboard sideincreases. Also, the total normal force coefficient on the fin shows that the degree ofasymmetry increases as azimuthal position of the fin increases. The variation of Cf issimilar at both axial fin positions but slightly different in the magnitude of the normalforce. As fin is placed at x/d=7.1, the normal force is larger than the case of (x/d)f=4.9.This difference is due to the size of the vortices, which grow in size with increasing x/d.CONCLUSIONThe numerical investigation of the interaction between slender body vortices and singlefin for angle of attack of 30 deg. and Reynolds number of 6000 is carried out. The studyshows that the vortices remain very coherent and attached to the body until they reachthe fin section. At this point, they become less coherent and begin to separate from thebody. The vortex center trajectories upstream of the fin were not influenced by the finlocation. However, the location at which the vortices were separated from the body wasaffected. The surface pressure contour on the both fm surfaces shows the influence ofthe vortices. A large low-pressure region is seen between the fin root and half-span dueto the existence of primary and secondary vortices. At half-span, the pressure increaserapidly in the spanwise direction due to flow toward the fin surface. There is noapparent effect of the azimuthal fin positions on the global structure and the trajectoriesof the vortices upstream of the fin section. On the fin section, as azimuthal angle of thefin increased, the size of the vortex on the port side decreased, but the starboard sidevortex grew in size and moved across the leeward ray to the port side. No vortexbreakdown was observed in either the present study or comparable experiments. Thecomputed results were found to agree well with the experiment data.NOTATIONCfnCfpCfsCnCpCyE, F, GEv, Fv, GvJn124Normal force coefficient on finNormal fin coefficient on port side of the fin (positive)ormal force coefficient on the starboard side of the fin (negative)Normal forcePressure coefficientSide forces coefficientInviscid flux vectorsViscous flux vectorsJacobianTime levelPertanikaJ. Sci. & Technol. Supplement Vol. 9 0.2,2001PQRu jXix/d(x/d)f~,T(,SLh<j>fNumerical Simulation between Slender Body Vortices and a FinPressureVector of primitive variablesResidualVelocityCarestain coordinateDimensionless distance from the noseDimensionless distance from the nose to leading edge of the finGeneralized curvilinear coordinatePseudo-time stepAzimuthal angle of the finREFERENCESACHARAYA, M., J. KJEDAISCH and M. MOHAMMAD HAsSAN. 1994. "Asymmetry vortical flows over slenderbodies with appendage," lIT Fluid Dynamics Research Center, TR, Chicago, IL, June 1994.HONAM OK. Development of an Incompressible Navier-5tokes solver and its application to thecalculation of separated flows. Ph.D Dissertation, Department ofAeronautical and Astronautics,University of Washington, 1993.ME DENHALL, M. R, and PERKINS, S. C., Jr. "Vortex Induced Characteristics of Missiles in UnsteadyManeu-vers," AIAA Paper 89-D344, Jan 1989.AN-QIAN CHAN, ZHEN-Yu WANG and ZHENG HUANG. Asymmetric Vortices and AUeviation on theSlender Bodies with and without Wmgs. AIAA Paper 97-D748, 1997.ROCERS START E. , DOCHAN KWAK and CETIN Knus. 1991. Steady and unsteady solutions of theincompressible Navier-5tokes equations. AIAA Juu:mal 29(4): 603-610.YOON SEOKKWAN and DOGHEN KWAK "Three dimension incompressible avier-5tokes Solver usingLower-Upper-Gauss-5eidel Algorithm." AIAA Juurnal 29(6): 874-875.WASHBURN, A. E., L. . JENKINS and M. A. FER/dAN. Experimental Investigation of Vortex-FinInteraction. AIAA Paper 93-0050, Jan. 1993.PertanikaJ. Sci. & Techno!. Supplement Vo!. 9 0.2,2001 125

Numerical Simulation of the Interaction between

Slender Body Vortices and a Fin

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Numerical Simulation of the Interaction between Slender Body Vortices and a Fin

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