## Abstract

The purpose of this paper is to initiate a 3-D non-spinning semi-circular missile model with a single generic planar fin and perform a computational analysis on it to understand the flow pattern around the fin. A freestream computational fluid dynamics analysis was done in the subsonic and supersonic Mach range, in which, the fluid behaviour was investigated using a two-equation turbulence model. A structured mesh was adopted to visualize the flow pattern around the fin. The aerodynamic coefficients were calculated for the model, and the predicted values were compared with the previous experimental results as well as the numerical results. This paper attempts to present all the possible flow visualizations which might help in better understanding of flow around missiles having planar fins. This work also attempts to establish a turbulent computational model for a single planar fin missile model for subsonic to supersonic range.

## Introduction

Planar fins are conventional missile fins which have been in use as a controlling and stabilizing surface for so many years. The geometry of these fins plays an important role in the aerodynamics of the missile. Initially incepted rectangular, these fins are now available in advanced shapes such as triangular, trapezoidal, sweptback etc., with adaptions in their leading and trailing edges as well. Computational fluid dynamics (CFD) has been an excellent tool in predicting and understanding the flow behaviour around various geometries as well as projectiles since past so many years.

The flow around a planar fin can often be compared with a flow around a blunt flat plate on a surface [1]. Hung and Buning [2], Hung and Kordullat [3] initially carried out simulations on a 3-D flat plate, in turbulent flow. Sedney and Kitchens [4] conducted experiments in supersonic regime involving spherical protuberances, the flow separation can be co-related to flow over blunt surfaces. In this study the detailed pressure and Mach number contours were presented. These tests suggested that CFD could effectively help in understanding the aerodynamics and basic fluid flow pattern. The planar fins are basic fins and have been often used as a benchmark for comparison with unconventional fins [5,6,7,8,9,10]. Due to their geometric symmetry they often show negligible rolling moments.

The single fin study on a missile shaped body was initially carried out by Tilman et al. [11,12,13,14,15] in which various numerical and experimental studies were carried out on a curved fin attached to a semi-cylindrical body which was designed in such a way that it represented a full missile body. These studies formed the base of the current simulations. In the current study a single planar fin is attached to a semi-cylindrical missile body, whereas in the previous studies on missiles having planar fins consisted of complete missile model with three or greater than three fins. These focused on overall missile aerodynamics rather than on the elemental flow characteristics around the attached fin. A CFD analysis of this single generic planar fin was carried out in Mach range of 0.4–3.0 M using a structured mesh and turbulence flow conditions at standard atmospheric conditions (** p** is the pressure: 101,325 Pascal,

**is the temperature: 288.15 K &**

*T***is the fluid density: 1.2250 \({\text{kg/m}}^{3}\) etc.) A two equations \(\kappa{-}\epsilon\) (realizable) turbulence model was utilized, and Second order of discretization was implied in the CFD computations. Detailed illustrations of flow have been presented in this paper to understand the flow characteristics around the fin. The visualisations consist of shock waves formations and pressure & Mach number contours. The drag and the rolling moment coefficients on a single fin at 0° angle of attack were also computed and presented in the later sections. Also, the results of the aerodynamic coefficients were validated and compared with the results from previous experimental and numerical studies on planar fins.**

*ρ*## Numerical methodology

### The pre-processing and the mathematical approach

The numerical methodology is the same as that of Sharma et al. [16,17,18] in which a density based solver with absolute velocity formulation was utilized to solve the mass momentum, energy species. It is well known, that the Navier–Stokes equations are decomposed into the mean and the fluctuating components the scalar quantities (component) of pressure, energy etc. are referred to as species. The fundamental governing equations remain the continuity equation, the momentum equation and the energy equation, however the Density is variable in compressible flow. For detailed computational setup which includes the numerically simulated flow conditions, the paper [16, 18,19,20] can be referred.

### Geometry

A single fin test model was designed based on a single fin wraparound test model [11,12,13,14,15]. This model can also be used for experimental analysis in the wind tunnel with cross-sectional area limitations. The test model (Fig. 1) was designed to represent a single fin planar configuration. The model fin has the same proportions as free-flight models based on full long nose missile model of Vitale et al. [21]. The detailed dimensions of a single fin model can be estimated referring to references [14, 16]. The aspect ratio of the planar fin is kept same as that of a single wrap-around fin. The total length of the model was 0.173628 m, where the radius of curvature of the missile shaped model (Deemed to be a cylinder) was taken as; r = 0.0159 m Within this 5r length, the shape of this model was made such that second-order continuity could be ensured in the longitudinal direction. This model had an actual length of 0.08140 m the height and the fin chord length were taken as 0.02035 m. The thickness of the rectangular shaped blunt fin was kept at 0.00254 m [16].

### Grid definition and the computational field

The grid generation is an essential as well as one of the most time-consuming steps in CFD. Though the unstructured mesh has proven to be reliable, and advantageous [22] however, a structured mesh was adopted according to the expected flow pattern. The computational mesh was generated using ICEM® CFD. The domain consists of a quarter-sphere in the front part followed by a semi-cylindrical body as shown in the Fig. 2(a–e). The inlet starts 0.3 m ahead of the model’s semi-cylinder point and the domain’s outlet is placed approximately five times the size of the model, behind this point.

This shape of the domain was chosen as it minimized the number of elements, while avoiding the creation of skewed elements. This has a direct impact on reducing the computational time of the simulations. The spherical part of the domain served two purposes: firstly, it allowed the mesh to accommodate the curved surface of the missile nose while avoiding skewed elements which can lead to inaccurate calculations. Secondly it reduced the number of redundant elements in the first half of the domain which are regions of undisturbed free stream flow. Blocking technique (ICEM®) was adopted in which multiple O-grids were created inside the domain, and complete mesh was hexa-dominant. To ensure the quality of the mesh orthogonality was imposed on all the elements in the domain.

## Results and discussion

### Solver and mesh validation

Pressure far field conditions were used in the boundary conditions. The freestream velocities with respect to the Mach numbers have been shown in Fig. 3, these freestream velocities in m/s were calculated in respect to the standard air conditions. For better convergence, each simulation was hybrid-initialized with external favourable settings and viscous turbulence simulations were run for 2200 iterations at least. The residuals convergence was strictly monitored, and the solution acceleration was achieved by changing the courant number. The solution was deemed to be converged only when all the residuals were brought down to the order of \(10^{ - 5}\) when using first order discretization for the initial solution and finally below \(10^{ - 3}\) while applying second order of discretization (Fig. 4).

The grid sensitivity was performed on three different meshes, the mesh with the least element size had 1.4 million cells, and subsequently results from the meshes with 3.4 million and 4.8 million cells were obtained. There was less than 1% deviation, in their computed results hence, mesh with 1.4 million cells was taken as a suitable grid size to save computational time. The mass flow rate in the flux reports was also computed for net mass balances.

Turbulence model Selection was accomplished by carrying out a comparative analysis between the one equation Spalart–Allmaras (S–A) and the two-equation realizable \(\kappa{-}\epsilon\) (Fig. 5). It was done by comparing the Drag coefficients values at three different Mach numbers and then compared with the previous results of the experimental studies. It can be clearly seen that the Drag coefficient values captured by the \(\kappa{-}\epsilon\) model are closer to the experimental results at each Mach number therefore further simulations were carried out with \(\kappa{-}\epsilon\) turbulence model.

### Verification and validation of the computed results

The converged solutions of the simulations showed agreement with the previous experimental as well as computational studies. The computed values of the drag coefficients in the Mach range of 0.4–3.0 are presented in the Fig. 5 and the rolling moment coefficients in Fig. 6. For the calculation of aerodynamic coefficients the reference area \(A_{Ref} = \pi R^{2}\) and the reference length is \(L_{Ref} = R\), was taken ** R** = ~7.95 mm [23,24,25].

In Fig. 5 the experimental results of [5, 9] and the computed values of drag coefficients were well within the acceptable range. In the case of Ref. [9] the Mach number range was from Mach 0.338 to Mach 0.908. The current computed values are following the trend of the drag coefficients. However, the computed values are not too indifferent from the existing experimental results. Reference [9] confirms accuracy of its data within 20%. Also, the difference in the magnitude of the results can also be attributed to the fact that the fins on the missiles have sharpened edges.

The rolling moment coefficients were compared with the computed values of the previous CFD studies (Fig. 6) [5, 7]. Though the net rolling moment should be theoretically negligible in the case of planar fins, miniscule values were obtained which were close to the earlier predicted values (Fig. 6).

### Computed shock structures and surface pressures

The flow visualizations of the turbulent computations are presented in this section. The missile surface pressures (top view) show the pressure contours over the missile surface the region ahead of the fin (Fig. 7). Distinct oblique shock waves were formed after Mach ~ 0.8 and the shocks became more prominent with the increase in the Mach number. The blunt leading edge of the planar fin especially the root of the fin’s leading edge seeds the drag. One Mach being a special number, at which the disturbance in the flow over the missile surface can be observed especially on the ogive nose part of the model. The pressure accumulation can be seen at Mach ~ 0.8 M, and shock formation at the root of the fin is equal on both the sides of the leading edge. This equal distribution of pressures and shocks is reflected in the complete Mach range. The computed values of rolling moments of the full missile and the fin (alone) are shown in the Figs. 8 and 9. The symmetry of the planar fins promotes zero rolling moment, and the magnitude of these rolling moments can be seen again in the Figs. 8 and 9. The maximum deviation of the computed rolling moment coefficient from zero is at − 0.009 at 0.8 M and 0.011 at 1.6 M. It is interesting to note that the rolling moment changes its sign at Mach 1 M. At Mach number ~ 2.2 M, there is absence of pressure variations on the nose of the missile body and the oblique waves originating at the leading edge of the missile become acuter.

To examine the flow near the fin, a side view of the missile model was selected in the computational domain with sectional plane cut through the centre of fin (Fig. 10a–d). Similarly, the computed Mach number contours have been shown in and Fig. 11a–d. Normal shock wave can be observed at Mach ~ 1.0 M. The shocks at the fin leading edge cause high pressure formation at the leading edge of the fin. Thus, due to flow disturbance the values of computed drag coefficients show change in magnitude (Figs. 12, 13). This pressure accumulation which starts getting reduced at Mach number ~ 1.6 M and this is also reflected in the computed drag coefficients values, after which they start to decrease. At Mach 1.6 the flow is completely supersonic and oblique shock waves are can be seen distinctly originating from the nose of the missile (Fig. 11). On the formation of oblique shock wave, there is loss of total pressure aft of the wave, and as the Mach number keeps on increasing, the angle of the oblique shock wave keeps on decreasing, the wave tries to align itself with the flow around the body, thus there is reduction in the drag coefficients which are reflected in the Fig. 13. On examining the shock wave angles, it can be observed that the angle of the oblique shock wave at Mach ~ 1.6 (Fig. 11b) is almost at 45° and the shock angle continues to reduce with increase in Mach number and therefore, a reduction in drag values can be observed. The drag values increase drastically around Mach number 1.0 and they continue to increase Mach number ~ 1.6. The drag coefficients of the planar fins decrease with increase in Mach number especially after Mach ~ 1.5, which can be seen in the experimental as well the computed results of drag coefficients (Fig. 5). Oblique shock wave appear at the nose of the model around Mach ~ 1.4 and it keeps getting stronger and acuter with the increase in Mach number.

The flow phenomenon is taking place mostly at the root of the fin and the flow disturbance keeps on decreasing as we go chord-wise in the upward direction of the fins. This is prominently shown in Fig. 11a–d, of Mach number contours. The Mach number contours are smooth over the surface of the fins especially after Mach number ~ 1.8 M.

Flow behaviour such as and the flow attachment and separation around the planar fin surface is covered in this section. 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. Figure 12 is presenting the Fin drag coefficients which might be helpful in correlating the contours of pressure coefficient around the fins in the Fig. 13a–d. For better understanding each chord plane is advancing in the upward Y-direction of the domain. The flow separation from the fin surface can be compared to that of a flow separation from the blunt surface with height to width ratio of ~ 0.13 [4]. The current single planar fin case has a height to width ratio of ~ 0.11. The increase in pressure can be seen at the fin leading edge in the Fig. 13a.

The flow aft of the trailing edge of the fin shows negligible disturbance till Mach 1. A bow shock appearance can also be observed around Mach ~ 1.2, the flow separation keeps decreasing thereafter with the increase in Mach number. This decrease in the angle with the increase of Mach number is not that rapid as in the case of oblique shock waves at the nose of the model. This can be assumed by the fact that the value of the Fin alone drag coefficient do not decrease rapidly after Mach ~ 1.2. (approximately 11%).

## Conclusions

This paper presented the aerodynamic flow visualisations and the aerodynamic coefficients of a single planar fin mounted on a semi-cylindrical missile shaped body. These aerodynamics results were successfully computed using a realizable two equations \(\kappa{-}\epsilon\) turbulence. Choosing a single fin geometry for simulation helped in building a structured meshing around the planar fin, saving the computation time and better characterizing of the flow around the fin.

The drag coefficients show Mach number dependency, and the rolling moment is negligible, however a change of sign is peculiar in the case of a planar fin. The negligible rolling moment can be attributed to the fact that the fin geometry is symmetric and the pressure contours on either side of the fins are almost mirror images to each other.

The geometry of the fins plays an important role in the missile aerodynamics. The majority of phenomenon takes place at the root of the fin, basically the fin missile juncture. The trailing edge does not have much of an effect on the flow pattern around the fins.

The flow visualisations presented in this paper gave good insight of the flow behaviour around the fins. The single fin model gave good predictions in terms of aerodynamic coefficients and this study can further be extended for dynamic analysis as well. These simulations can form the basis of comparison with the single unconventional fins in the future.

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Sharma, N., Kumar, R. Investigation of flow-field around a single generic planar fin using CFD.
*SN Appl. Sci.* **2, **63 (2020). https://doi.org/10.1007/s42452-019-1713-8

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### Keywords

- CFD
- Aerodynamics
- Planar fin
- Fluid mechanics