Computational Flow field Analysis around a Single Wrap-around Fin [308042]

Computational Flow field Analysis around a [anonimizat]*,1, Gurteg Singh NAGI1, Hrishabh CHAUDHARY1,

Palak SAINI1, Rakesh KUMAR1

*Corresponding author

1[anonimizat] (Deemed to be University),

Chandigarh, India,

[anonimizat]

DOI: 10.13111/2066-8201.2020.12.2.X

Received: 29 January 2020/ Accepted: xx XXXXX 2020/ Published: June 2020

Copyright © 2020. Published by INCAS. This is an “open access” [anonimizat]-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)

Abstract: A flow field aerodynamic analysis using Computational Fluid Dynamics (CFD) model of a single wrap around fin is performed at 0° angle of attack and its flow visualizations are presented in this paper for a subsonic to supersonic Mach range. A comparative study is performed by utilizing two WAF geometries aligned in opposite directions and with different edges (Sharpened & Blunt). Aerodynamic characterization has been conducted using a realizable κ-ϵ turbulence model utilizing a high-quality mesh to understand the nuances of the flow around the curved fin structure. The quantities of interest are the computed aerodynamic coefficients which have been validated with their precursors. [anonimizat] a finer way.

Key Words: CFD, Aerodynamics, Wrap-around fin

NOMENCLATURE

1. INTRODUCTION

Background. Wrap-around fins (WAFs) have been in use for tube launched projectiles due to their body enveloping ability. WAFs stowing capabilities and versatility have attracted many investigators in the past. However, instabilities which are inherent to WAFs due to their asymmetric geometry (curved surface) especially in the transonic regime have always been a key area of interest for the researchers. The aerodynamics of flow around the fin as well as the fin passage (area between the two adjoining fins) have been examined time and again for their aerodynamic improvement. It has been illustrated that at 0° angle of attack the pressure on both sides of the fins is different.[1] [anonimizat]. The Roll reversal that is observed in the transition from subsonic to supersonic speeds at 0° angle of attack is a [anonimizat]-yaw coupling.[2]–[5] A key aspect in these findings is the observance of shock structures near these fins which are deemed to be the reason for the loss of static stability of such projectiles.[4] Pressure differences on the concave and the convex side of the fins seemed to incubate the rolling moment. The previous studies show the existence of vortex at the root of the fin i.e. [anonimizat].

Computational Studies. [anonimizat] [1], [6]–[13] showed general agreement with the experimental studies. [4], [14]–[19] Bagheri[1], et al. [anonimizat]. Li[6], et al. estimated the aerodynamic characteristics of a [anonimizat] a 𝜅 – 𝜔 transition shear stress transport (SST) turbulence model in the Mach number range of 1.6 to 5. Liu[7], et al. compared the wrap-around fins with the planar fin missile model computing the lift, drag, roll coefficients of the missile in the Mach number range 0.6M to 3.0M. Their computations were based on one equation Spalart-Allmaras turbulent model and the main difference in was found in the rolling moment coefficients. Krishna[8], et al computed the flow field solutions of missiles with WAFs focusing on the leading edge of the fins and their fin passage in the Mach number range of 1.2 to 2.5. Mao [9], et al. conveyed the effects of setting an angle of WAFs on rolling and pitching moment of the missile. Rolling Moments were computed by S. Mandić [10] between angle 0°and 0.8°, similar work on computation of rolling moments using CFD was done by Paek[11], et al. using the Euler equations. Edge[12] successfully calculated the aerodynamic coefficients of the missile with WAFs using CFD in the Mach number range of 1.3 to 3.0.

Single Fin Model. Tilman[20]–[24], et al. both experimentally and computationally investigated a single wrap-around fin model at freestream Mach numbers [~2.8,2.9]. The characterization of flow around a single fin was performed numerically at freestream Mach numbers [~2.8,2.9M].[23], [24] These numerical studies were based upon the Baldwin and Lomax model of turbulence.[25] The predicted results of the inviscid simulations, especially the rolling moment coefficients [23] when multiplied by four, seemed to agree with the earlier inviscid calculations[26] which motivated to extend this research using recent computational methods. Sharma [27]–[30], et al. reviewed various CFD studies on WAFs[27] and performed a visual comparitative validation study of a single fin WAF[28]. A study was carried out using a single generic planar fin configuration[29] rma[30], and finally a comparison was carried out between a single planar fin and WAF.[30] In the comparison study, the aerodynamics of a single planar fin having blunt leading edge were compared with a single WAF having sharpened edges. Also, validation of a single fin missile model with a full four fin missile model was established. This study mainly dealt with static pressure and Mach number contours around the sharpened edged WAF model, which may be of limited value to the readers, therefore setting up a need for transparent communication. The geometric influence on the aerodynamics of WAF has been presented in the current study by comparing an oppositely aligned blunt edged WAF with a sharpened edged WAF. Hence, in the current study, contours of pressure coefficients and free stream velocities (non-dimensional results) were chosen to be presented along with comprehensive solver selection criteria. The current study compliments and expands on the lack of an insight to the solver selection (validation) and the computational setup. The major objectives of this paper were to

(1) Compare and understand the aerodynamics and flow behaviour on adapting a single blunt WAF aligned inversely in the flow direction;

(2) Present a comprehensive solver validation of a single fin missile model;

(3) Present an exhaustive flow field visualisation of pressure coefficients & relative Mach number contours (Non-Dimensional) along with their Cumulative aerodynamic coefficients.

2. NUMERICAL METHODOLOGY

Geometry A complete missile body with an ogive nose was cut in such a way that it formed a semi-cylinder like body, with a fin mounted at the tail end of the model and same as that of Sharma[27], [30] et al. The dimensions of this model were such that it represented a complete Missile body of that of Vitale[4] et al. The true length of the model is 0.147m with maximum height of the semi-cylinder at 0.00795m, with AR= ~1 of the fin. The thickness of the fin was kept at 0.00254m. Additional set of computations were performed involving an identical oppositely aligned blunt edged single fin WAF geometry model with AR= ~1. The dimensions of the semi-cylindrical model remained the same as that of the above-mentioned geometry however for the better understanding of the new model geometry changes are presented in the Figure1(a-c).

Meshing Details A completely structured Mesh involving multiple o-grids (an ICEM® Feature) consisting of Hexa-dominant elements was employed in the computations.[30] The mesh sensitivity was confirmed by comparing the drag coefficient values for four different mesh sizes at Mach 2.8. These mesh sizes varied from 1.33 million cells to 1.92 million cells. The minimum computed value of drag coefficient was 0.379623 for the mesh with 1.33million cells and the maximum value obtained was 0.38762103 for the mesh with 1.47million cells. Therefore, a mesh with 1.45 million cells was adopted for all the computations consisting of sharpened edged WAF. In the case of a blunt edged WAF a mesh with 0.7million cells was considered, this mesh had a remarkable Minimum Orthogonal Quality = 6.34240e-01, therefore a grid sensitivity test did not seem to be necessary in this case.

Flow and Solver Details The computational domain remained similar for both the models, which consisted of an inlet quarter sphere, the far-field and the outlet surface, which have already been defined earlier and the details of their flow parameters and the solver are in the scope of reference [30].

The Solution methods were based on choosing the most attractive parameters mentioned in the review of previous computational studies on WAFs. [27]

The summary of the flow conditions as well as the solver adaptions have been mentioned in the Table 1 & 2.

The fundamental governing equations remain the continuity equation, the momentum equation and the energy equation [27], [30], [31] and the turbulence closure was carried out with two equation Realizable – turbulence model. [32] The term “realizable” means that the model satisfies certain mathematical constraints on the Reynolds stresses, consistent with the physics of turbulent flows.[33]

An additional solution method approach was endorsed using Roe-FDS scheme for subsonic Mach number simulations.

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Figure 1: (a) Blunt Edged Single WAF model; (b) Span & Chord of the Blunt edged fin (in mm) (c) Front view of the Blunt Fin WAF Model

Table 1: Summary of the flow conditions applied in the numerical simulations that were carried out

Table 2: Solver Details

Solver Validation Study: Iterative convergence of the Normalized Residuals. A strict convergence criterion was set up for each computation, the initial solution guess was predicted utilizing the first order upwind discretization (flow Variables) in which all the residuals were brought below the order of .

In the case of sharpened WAF case the initial solution took around 8000 to 1200 iterations to converge. Subsequently, a second order discretization (for all variables) was applied to the initial guess in which all the residuals were brought down to the order of , this took additional 400 to 600 iterations.

The convergence of the normalized residuals in the case of Blunt edged WAF took lesser number of iterations.

On utilizing the first order upwind discretization (flow variables) all the residuals were brought to the order of , this took around 250 to 1000 iterations and subsequently on applying the second order of discretization, all the residuals were brought down to the order of , this took around 200 to 600 additional iterations.

Residual convergence history of the sharpened and blunt edged Single WAF respectively, have been reported in the Figure 2(a-d) for both subsonic and supersonic Mach regime. It must be mentioned here that Courant-Friedrichs-Lewy (CFL) number was ramping up was done in almost all the computations resulting in a speedier convergence process.

After gaining confidence in the residual convergence, in some cases (mostly supersonic) the second order of discretization was directly applied for convergence and the residuals were brought down to the order of , in around 250 iterations Figure 2(d)

Figure 2: Residual Convergence of Sharpened Edged Single WAF at

(a) Subsonic Mach Number (~0.6M); (b) Supersonic Mach Number (~2.6);

(c) Blunt Edged Single WAF at Mach 0.8; (d) Blunt Edged Single WAF at Mach 2.6

Selection of Turbulence Model and Validation of Computed Results. For the purpose of selection of turbulence closure model, a comparison study was carried out between the computed results of one equation Spalart-Allmaras (S-A) and the two-equation realizable 𝜅 − 𝜖. It was done by computing, comparing and analysing the obtained drag & moment coefficients (Sharpened edged WAF) with both experimental and computational results available from the prior studies. (Figures 3-6) The computed drag coefficients using the (S-A) and the two-equation realizable 𝜅 – 𝜖 models nearly overlapped each at some instances however the results from the later were virtually indistinguishable when compared to the experimental results at Mach 1, 1.2, 1.8, & 3. The obtained drag coefficients on comparing with the prior computational results, the realizable 𝜅 − 𝜖 results in lesser error reduction as compared to the S-A model. In order to reaffirm the confidence in the selection of turbulence model, the computed moment coefficients were also compared with the prior experimental and computational results. The change in the sign of rolling moment was more accurately captured at Mach ~1.2 using the two-equation realizable 𝜅 – 𝜖 model and at a much later Mach (~1.5) while using the S-A model of turbulence.

Figure 3: Comparison of the evaluated Drag Coefficients values at different Mach numbers from the two different turbulence models (i.e. Κ-ε & S-A) with the results from prior experimental studies

Figure 4: Comparison of the evaluated Drag Coefficients values at different Mach numbers from the two different turbulence models (i.e. Κ-ε & S-A) with the results from prior computational studies

Figure 5: Comparison of the evaluated Moment Coefficients values at different Mach numbers from the two different Turbulence models (i.e. Κ-ε & S-A) with the results from the prior experimental studies

Figure 6: Comparison of the evaluated Moment Coefficients values at different Mach numbers from the two different Turbulence models (i.e. Κ-ε & S-A) with the results from the prior computational studies

3. RESULTS

The Force & the Moment Coefficients The comparison of the force (Drag) coefficients and the moment coefficients of both Sharpened edges and the blunt edged WAFs have been presented in this section. The drag Coefficients of the complete missile models were calculated by choosing the reference area , and reference length , where R= 7.95mm. The basis of selection of this area and length’s values were based on previous studies [6], [11], [13], [30]

Along with the drag coefficients of the full single fin models, the fin drag coefficient (Fin alone) have also been compared and their values have been presented in Figure 7. The missile model with blunt WAF showed increment in the drag coefficient values when compared to the sharpened edged single fin WAF. The overall drag coefficient values of the blunt edged WAF model were 9.0781% to 62.49324% greater as compared to the Sharpened edged WAF with least difference in the value at Mach 1.4 and maximum at Mach 3.0. It is interesting to note that the Fin alone contribution towards the overall Drag Coefficients of the model was 18.7522% to 53.1864% in the case of blunt fin WAF. The maximum contribution of the fin alone drag was at Mach 1.0 and the least at Mach 3.0 in the case of Blunt WAF. On the contrast the fin alone drag contribution in the case of sharpened edged WAF varied from 37.9788% to 44.7710%, with the least value at Mach 0.8 and maximum at Mach 3.0. The average contribution of the Fin alone drag coefficient was 29.9600% in the case of blunt edged WAF and 42.1846% in the case of the sharpened Fin alone WAF. The sharpened edged single fin missile model indicated 37.4115% reduction the overall drag values. The computed roll moment coefficient values have been presented in the Figure 8. Prior to comparing and analysing the moment coefficient values, it should be made clear that the “Direction Changed” mentioned with blunt edged WAF means that the moment coefficients were multiplied with “-1” before plotting. This was done to compliment the effect of opposite alignment of the blunt WAF in the result values. On examining the roll moment coefficients, it can be observed that the sharpened edged WAF model exhibits change in moment direction at ~1.2M whereas in the case of Blunt edged WAF the moment coefficients change their sign ~1.6. These results are of significance as Mach 1.2 is the nearly the same Mach number at which the roll moment coefficient changed sign in the previous experimental studies. [27], [30] Also, for the case of blunt edged WAF, the critical Mach number (Mach at which the roll moment coefficients changed their sign/direction) is stated much later ~1.6M for a blunt shaped WAF.[12] At supersonic Mach numbers especially above Mach 1.2, the blunt WAF indicate stronger moment coefficient forces as compared to the sharpened edged WAF. The moment forces at these Mach numbers (greater than Mach 1.2) produced by the blunt edged WAF were on an average 66.97% stronger than the forces produced by the sharpened edged WAF. The sharpened edged WAF showed second roll reversal above ~2.2M however there were no indications in the roll reversal of blunt edged WAF up to Mach 3.0.

Figure 7: Comparison of Drag Coefficients of Sharpened & Blunt Fin WAF model

Figure 8: Comparison of Moment Coefficients of Sharpened & Blunt Fin WAF model

Flow visualizations: The flow visualizations obtained from the current computations of both Sharpened edged and the oppositely aligned blunt edged WAF are presented in this section. It can be observed from Figure 7 & 8, that the prominent Mach numbers for the both types of WAF configurations influencing the flow physics are in the range between Mach 0.8 to 1.8. Therefore, non-dimensional pressure and Mach number contours have been presented in this section for better perception of the flow phenomenon in the above-mentioned governing Mach range. Both the sharpened edged and the blunt edged WAF models have been considered simultaneously for better comparison at similar Mach numbers. The flow visualizations consist of five sections, out of which the first four sections are dedicate to the pressure coefficient contours. These pressure contours are presented as (i) Surface pressures on the missile model, (ii) Flow pattern at the mid-plane around the model, (iii) Pressure coefficient contours around the fins, starting from the root of the fin, till the top of the fin, (iv) Relative Mach number contours on the model surfaces. (visualizations presented for both sharpened and the blunt edged fins), (v) Pressure coefficient contours along the fins in the longitudinal direction starting from the fin tip leading edge till the trailing edge of the fin.

The first set of images consist of pressure coefficient contours on the surfaces of the WAF models, at Mach 0.8-1.8. These surface contours images consist of top view (on the left part of the image) and the side views (both convex & the concave side of the fins). (Figures 9-14) At Mach 0.8 both the sharpened & the blunt edged WAF exhibit higher pressures on the convex side of the fin, however these forces commence on the sharpened edged WAF prior to the blunt edged WAF. There is a considerable contrast in the pressure coefficients of both the models at Mach 1.0, the sharpened edged WAF depict high pressure rise ahead of the convex side, and a sudden pressure drop in the case of blunt edged WAF, though the pressures on the convex side in the blunt edged fin remain higher as compared to the blunt edged. At Mach 1.2 the surface pressures on both sides of the fins shift towards identical patterns, depicting equivalence of pressures on both sides of the sharpened edged fins. A significant pressure drop can be observed on the concave side of the blunt edged WAF at Mach 1.2. Identical pressure contours on the concave and convex surface of the sharpened WAF, initiate to disparate at Mach 1.4 however, the pressure contours incept to compliment each other at Mach 1.4 in the case of blunt edged WAF. At Mach 1.6 the high-pressure contours ahead of the leading edge of the sharpened fins become observable, reaffirming the change in the roll direction. It is at Mach ~1.6 where indistinguishable pressure contours appear on the concave and the convex side of the blunt edged WAF. Therefore, reaffirming the delayed critical Mach number at which the roll reversal takes place. High pressure region emancipates ahead of the blunt leading edge of the fin endorsing the roll reversal at Mach ~1.8. On considering the pressure contours on the either sides of the sharpened edged WAF inception of its second phase of roll reversal at Mach 1.8 is perceivable.

The second set of images consist of computed pressure coefficient contours along the midplane about sharpened fin missile model at various Mach numbers (Mach 0.8-1.8). Due to the asymmetry of the WAF, a 2-D side view of the flow physics at the midplane is very aptly represented in the following figures. (Figures 15-20) The variation in the drag coefficients are better justified on examining the pressure coefficient contours at the midplane. Mach 0.8 already shows the initiating of the formation of bow shock ahead of the leading edge of the blunt WAF which the reason for may be influencing the formation of high-pressure shock region at the nose of the missile body, such phenomenon is not visible in the case of sharpened edged WAF model. Normal shock waves appear at both the types of WAF, thus also ruling out the dis-ambiguity in the compared missile geometries. (excluding the fin) remarkable difference in the pattern of pressure contours can be observed ahead of the leading edges owing to their fin geometries. Oblique shock waves begin to appear at Mach 1.2, in both the sharpened and the blunt edged WAF, however much prominent in the case of the sharpened edged WAF. At Mach 1.2 the oblique shock waves seem to be towards the direction of the concave side of the fin in the case of blunt edged WAF. At Mach 1.4 the oblique shock waves emanating at the nose and the leading edge of the Sharpened WAF are quite prominent. The dissimilarity at the nose of missile in the blunt WAF which was observed at Mach 1.2 are now completely absent at Mach 1.4, which compliment the initiation of roll reversal. Following Mach 1.6 the oblique shock waves kick off to align with the flow and become much prominent at Mach 1.8.

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Figure 15: (a) Sharpened Edged (b) Blunt edged; Mach 0.8

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Figure 16: (a) Sharpened Edged (b) Blunt edged; Mach 1.0

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Figure 17: (a) Sharpened Edged (b) Blunt edged; Mach 1.2

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Figure 18: (a) Sharpened Edged (b) Blunt edged; Mach 1.4

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Figure 19: (a) Sharpened Edged (b) Blunt edged; Mach 1.6

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Figure 20: (a) Sharpened Edged (b) Blunt edged; Mach 1.8

The third set of images consist of Computed Pressure Coefficient Contours around the Sharpened Fin in the Chordwise Upward Direction starting from the root of the fin (left) to the top tip of the fin (right). (Figures 21-26) Five different positions were selected along the chord planes of the fin starting from the root of the fin till top edge of the fin. Major flow phenomenon occurs at the plane which coincides with the root of the fin. Bow shock formation is quite prominent in the case of blunt edged WAF. Another interesting phenomenon that can be observed is that sharpening of the leading or the trailing edge lessen the influence of the trailing edge on the fluid flow, especially moving in the upward (top) direction. Multiple shock patterns emerge on the WAF with blunt edges, starting at Mach 1.2 till Mach 1.6. The bow shock phenomena remain an attribute of a blunt edged WAF throughout the supersonic regime.

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Figure 21: (a) Sharpened Edged (b) Blunt edged; Mach 0.8

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Figure 22: (a) Sharpened Edged (b) Blunt edged; Mach 1.0

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Figure 23: (a) Sharpened Edged (b) Blunt edged; Mach 1.2

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Figure 24: (a) Sharpened Edged (b) Blunt edged; Mach 1.4

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Figure 25: (a) Sharpened Edged (b) Blunt edged; Mach 1.6

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Figure 26: (a) Sharpened Edged (b) Blunt edged; Mach 1.8

The fourth set of images consist of computed relative Mach number contours on the missile model surfaces at various Mach 0.8-1.8. (Figures 27-32) The flow visualizations demanded non-dimensional velocity contours as supplementary relative Mach number contours for the effective understanding of flow physics especially at the surface of single WAF model. The relative Mach number becomes greater than one even at Mach 0.8, in the case of sharpened edged WAF at a position where the sharpened edge terminates. The pattern is identical on both the convex and concave sides of the fin. Similar increase in the relative Mach number is visible just at the edge of the blunt WAF, the only difference being that it is minuscule on the convex side of the blunt edged fin. In the case of sharpened edged WAF there is a drop in the flow velocity just ahead of the fin leading edge, and the velocity is uniform on the fin surfaces. The blunt edged fin continues to show similar relative Mach number trends as before. (increase in relative Mach number at the leading edge and at area following the leading edge. At Mach ~1.2 the relative Mach number contours show identical triangle shaped patterns (converging nozzle shaped) on both sides of the sharpened edged fins. The blunt edged WAF show similar nozzle shaped contours at Mach 1.2 on both sides fin, however on the concave side of a diverging nozzle shaped contours also appears on the fin surface. Thereafter Mach 1.4, the flow speed on the surface of the sharpened edged fin is uniform with drop in pressure just ahead of the leading edge of the fin. In case of blunt edged WAF, at Mach 1.4, on the concave side of the fin a prominent converging diverging flow velocity pattern can be observed. The sharpened edged WAF exhibit increase of pressures at the leading edge of the fin, with a uniform flow velocity aft of the leading edge of the fin. At this Mach number (M ~1.6), the leading edge of the blunt edged fin, exhibit higher velocities than the rest of the aft fin surfaces. Also, along the root of the fin the flow velocity increases. Such phenomenon is not present in the case of sharpened WAF throughout the Mach range. The oblique shockwaves have no interaction with the fin surfaces aft of the sharpened leading edge of the fin. The bow shock formed ahead of the blunt edged WAF still influences the flow velocity at the root of the fin.

The fifth set of images consist of computed pressure coefficient contours around the fin in the longitudinal direction starting from the leading edge of the fin (top image) to the trailing edge of the fin (bottom image) at various Mach numbers. (Mach 0.8-1.8) (Figures 33-35)

This view is supremely important in determining the flow physics in the longitudinal direction along the fin.

The influence of the leading edge on the flow dynamics seem to confine till 0.25c longitudinally in the case of sharpened edged WAF whereas the influence of blunt edged fin on the flow physics remain till the trailing edge of the WAF.

The pressure coefficient contour depicts like patterns in the Mach range ~1.2-1.4 in the case of sharpened edged WAF whereas in the case of blunt WAF the critical Mach number ~1.6.

Change of pressure gradients on either side of the fin can also be observed for the sharpened and the blunt edged WAF at the above-mentioned respective Mach numbers.

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Figure 33: (a) Sharpened Edged (b) Blunt edged; Mach 0.8 (c) Sharpened Edged (d) Blunt edged; Mach 1.0

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Figure 34: (a) Sharpened Edged (b) Blunt edged; Mach 1.2 (c) Sharpened Edged (d) Blunt edged; Mach 1.4

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Figure 35: (a) Sharpened Edged (b) Blunt edged; Mach 1.6 (c) Sharpened Edged (d) Blunt edged; Mach 1.8

4. CONCLUSIONS

This paper presented the aerodynamic flow visualisations along with the aerodynamic coefficients for a single WAF mounted on a semi-cylindrical missile body. Two geometries of WAF were considered for better understanding of the flow dynamics in the vicinity of the WAF. These geometries were unique to each other in terms of their leading and trailing edges as well as their alignment with the flow. Choosing a single fin geometry for simulation helped in better meshing around the curved fin and saving the computation time. The aerodynamics results were computed using realizable two equations κ-ϵ turbulence. The anomalies associated with the WAFs can be attributed to the peculiar-pressure region near the root of the fin. The unequal pressure distributions on both sides of the fins caused the self-induced rolling moment at 0° angle of attack. The computed aerodynamic coefficients of the single fin model (both Blunt and sharpened edged) were compared with each other, also in addition, using the reference area and reference length, same as that used for a four-fin full missile model, with the previous experimental and computational studies. The computed values showed an agreement with the previous literature. Though these results do not affirm the possibility of computing four-fin performance based on a single-fin computation, it however, can be used for preliminary design of new fin geometries for the studies incorporating reduction of the drag and minimising or eliminating roll reversal. Some of the key findings from the computations are listed below:

The asymmetry of the fin geometry incubates roll moment, and the shape of the leading and trailing edge influence this rolling moment. The leading edge of the fin majorly influences the flow phenomenon.

The Drag and Roll moment coefficients of the single WAF model successfully validated, thus may be used as a benchmark for future single WAF computations.

The pressure concentrations at the leading edge on the fin root (missile juncture) incubated the shock structures. The influence terminates at the leading edge in the case of sharpened edged WAF, however in the case of blunt edged WAF, an influence in the flow pattern is vissible throughout the length of the fin.

The sharpening of edges of the WAF did not completely eliminate the roll reversal phenomenon, however when compared to the Blunt edged WAF, the moment forces in the case of sharpened edged WAFs were smaller.

The sharpening of the leading and trailing edges of WAF preponed the roll reversal activity. In the case os sharpened edged WAF the roll reversal is reported at Mach ~1.2 whereas in the case of blunt edged WAF the roll reversal is reported at M~1.6.

There is a motivation to compute different fin geometries using a single fin missile model, at zero-degree angle of attack.

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[33] * * * ANSYS, Chapter 24. Using the Solver, 2001, pp. 1–114.