Multi-Disciplinary Design of a Grid Fin for a Generic High Speed Missile

June 2020

Report By: Shakunt Tambe CID: 01196160

Supervisor: Dr. Paul Bruce Imperial College London

Second Marker: Dr. David Hayes MBDA Missile Systems, UK

Submitted to Imperial College London in partial fulﬁlment of the requirements for the Degree of Master of Engineering in Aeronautical Engineering Department of Aeronautics Imperial College London South Kensington SW7 2AZ Blank Page Abstract

The use of grid fins for high speed missiles has recently seen a resurgence in interest. This project investigates the aerodynamic performance and the inevitable trade-offs that must be made for producibility of a low drag grid fin that is more competitive with traditional planar fins. By adopting a three-stage computational approach, CFD studies were carried out to characterise the effect on aerodynamic performance from altering local geometrical parameters such as the leading and trailing edge sharpness, thickness to chord ratio, chord to span ratio and 3D modifications such as the effect of brazed joints and tapered trailing edges. For the geometry of a Vympel R-77 grid fin, the importance of the leading edge sharpness and overall thickness to chord ratio was significant in achieving an optimum lift to drag ratio. Flow choking at the supersonic design point (Mach 3) was not found to be of concern and a periodicity in performance is analysed by developing a generalised model of the shock structure and underlying flow physics. Superior performance was typically observed at smaller angles of incidence resulting from greater pressure differences between the upper and lower surfaces of the grid fin cell. The effects of 3D modifications, particularly brazing at the joints, resulted in up to a 33% performance degradation, prompting suggestions for alternative approaches. Lastly, preliminary heating estimates from theory, empirical methods and CFD were compared which can additionally inform future material choices and producibility decisions.

i Acknowledgements

Dr Paul Bruce has not only provided an outstanding level of project supervision and guidance but also an impeccable standard of teaching during my time at Imperial; his lectures have been some of the most enjoyable, clearly delivered and of course engaging topics that I have studied. I am grateful to Dr Bruce for ﬁrstly taking on this project, and then sharing his incredible knowledge, passion and invaluable advice throughout, without which completing this project would never have been possible.

I would also like to express a huge thank you to Dr David Hayes, Jeﬀ Thornton, Jeremy Murgatroyd and the rest of the team at MBDA who have allowed me the unique opportunity to complete this project and provided extremely helpful feedback and guidance.

Thanks also to Prof. Denis Doorly for providing an amazing level of support through personal tutorial sessions over the past four years at Imperial which have certainly helped me to overcome the challenges of this degree.

ii Contents

List of Figures ...... v

List of Tables ...... vi

Nomenclature ...... vii

1 Introduction ...... 1 1.1 Motivation...... 1

2 Background and Literature Review ...... 2 2.1 Missile Configurations ...... 2 2.2 Applications of Grid Fins...... 3 2.2.1 Vympel R-77 ...... 3 2.2.2 SpaceX Falcon 9 ...... 4 2.2.3 Orion Launch Abort Vehicle...... 4 2.2.4 Producibility of Existing Designs...... 5 2.3 Drag Reduction through Geometry Modification...... 6 2.4 Leading and Trailing Edge Geometry...... 9 2.5 Full Body CFD Studies...... 9 2.6 Unit Grid Fin Method...... 10 2.7 Theoretical Methods ...... 11 2.8 Flowfield around Grid Fins...... 11

3 Computational Approach ...... 12 3.1 Key Objectives ...... 12 3.2 Flight Conditions ...... 12 3.3 Conﬁguration and Vympel R-77 Geometry...... 12 3.4 Three Stage Approach...... 13 3.4.1 Stage 1 - Single Flat Plate...... 13 3.4.2 Stage 2 - Two Flat Plates...... 14 3.4.3 Stage 3 - Unit Grid Fin (3D) ...... 14 3.4.4 Beneﬁts of the Three Stage Approach ...... 15

4 Single Flat Plate ...... 16 4.1 Simulation Setup ...... 16 4.2 Mesh Convergence ...... 17 4.3 Aerodynamic Forces ...... 17 4.4 Validating Results...... 18 4.4.1 Identifying the Sources of Drag ...... 19 4.5 Higher Angles of Incidence...... 20 4.6 Eﬀect of Thickness to Chord Ratio...... 21 4.7 Eﬀect of Removing Leading or Trailing Edge Taper...... 21 4.8 Summary of Findings from a Single Flat Plate...... 22

5 Two Flat Plates ...... 23 5.1 Simulation Setup ...... 23 5.2 Small Angles of Incidence ...... 23 5.2.1 Detailed Analysis of Varying Chord to Span Ratio...... 24 5.3 Higher Angles of Incidence...... 27 iii 5.4 Mach Number Variation...... 28 5.5 Summary and Limitations of the Two Flat Plates Analysis...... 29

6 Unit Grid Fin ...... 30 6.1 Simulation Setup ...... 30 6.2 Mesh Convergence ...... 31 6.3 Aerodynamic Forces ...... 31 6.3.1 Validation...... 33 6.3.2 Transonic Performance...... 34 6.3.3 Vortical Structures ...... 35 6.4 Eﬀects of 3D Geometry Modiﬁcations...... 36 6.4.1 Aerodynamic Performance...... 36

7 Heating Estimates ...... 38 7.1 Theoretical and Empirical Methods...... 38 7.2 Results...... 38

8 Conclusion ...... 39 8.1 Summary ...... 39 8.2 Producibility and Further Work...... 40

References ...... 41

Appendices ...... 46

A Vympel R-77 Missile ...... 46

B Grid Fin Drawing (Vympel R-77) ...... 47

C Drag Reduction through Geometry Modiﬁcation ...... 48

D Flight Conditions and Prism Layer Parameters ...... 49

E Shock Expansion Theory for a Single Flat Plate ...... 50

F STAR-CCM+ Settings for CFD Simulations ...... 52 F.1 Single Flat Plate...... 52 F.2 Two Flat Plates ...... 53 F.3 Unit Grid Fin ...... 54

G Wall Coordinates ...... 55 G.1 Single Flat Plate...... 55 G.2 Two Flat Plates ...... 56 G.3 Unit Grid Fin ...... 57

H UGF 3D Geometry Modiﬁcations ...... 58

I Calculation of Heating Estimates ...... 60

iv List of Figures

1 A Russian R-77 missile with tail mounted grid fins [1]...... 1 2 Canard, wing, tail and split-canard control missile configurations, each shown with grid fins.2 3 Falcon 9 with grid fins mounted at the top of the first stage [22]...... 4 4 Four, body folding titanium grid fins on the first stage of the SpaceX Falcon 9 launch vehicle [29]...... 5 5 Swept grid fins at the base of a scale model of the NASA Orion LAV [28]...... 5 6 Local sweep applied to individual grid fin cell members [31]...... 6 7 A folded sheet metal and mandrel curing approach to producing grid fins as given in [32].. 6 8 Flowfield structure around a single grid fin cell at different speeds [37]...... 7 9 A schematic of the flow approaching a blunt, unswept grid fin cell and a 30° sharp swept cell [41]...... 8 10 Section and edge variations in [48]...... 9 11 Flow structure and SBLI between a grid fin and the missile body at M = 3, α = 0° [40]. . 10 12 Surface pressure distribution showing the grid fin wake interacting with the aft missile body at M = 2.5, α = 5° [4]...... 10 13 Representation of a unit grid fin within a lattice framework [53]...... 10 14 A simplified drawing of the R-77 grid fin geometry showing a top and right view with key geometrical parameters labelled. A full drawing is provided in AppendixB...... 13 15 Stage 1 analysis...... 13 16 Stage 2 analysis...... 13 17 Stage 3 analysis...... 13 18 An illustration of the different LE and TE taper geometries used within the stage 1 analysis. All corners have a radius of 0.01t...... 14 19 An illustration of the UGF model showing the ‘tunnel’ domain and the pairs of periodic wall boundaries. Variable have the values presented in Table3...... 15 20 The mesh around a sharp single plate configuration...... 16 L 21 Plots of CL, CD and D convergence against varying number of cells within the domain. . . 17 22 Lift and drag coefficient against α for all four edge taper geometries. Linear theory (and skin friction corrected linear theory) also plotted for comparison...... 18 23 Shock expansion theory around the single flat plate (sharp LE and TE), shown at an α > 0. 18 24 Edge geometries used for verifying drag due to thickness. Dimensions in mm, not to scale. 19 25 CD against α for all taper geometries and the sharpened edge cases. Computed from CFD simulations and skin friction corrected linear theory...... 20 26 CL and CD against α for all four edge taper geometries, up to α = 20°. Linear theory (and skin friction corrected linear theory) also plotted for comparison at small angles...... 20 L t 27 D against varying c ...... 21 L 28 Plots of CL, CD and D against α with the effect of removing LE and/or TE taper...... 21 29 The simulation setup used for CFD analysis of the two flat plates...... 23 L c 30 Plots of CL, CD and D against varying s for small angles of incidence (α = 0, 3, 5°). . . . 24 31 An illustration of the flow structure between (a forward section of) two sharp edged flat plates c at an intermediate s (where flow choking has not occured) and α = 0°...... 25 32 Absolute pressure fields from CFD and corresponding CP distributions on the internal surfaces c of the two plates, shown for s = 1, 2, 4.5, 8, 30. All shown for M = 3, α = 5° ...... 26 L c 33 Plots of CL, CD and D against varying s for higher angles of incidence (α = 0, 3, 5, 10, 20°). 28 c 34 Absolute pressure for s = 2 at M = 3, α = 20°...... 28 L 35 Plots of CL, CD and D against varying Mach number (0.6 ≤ M ≤ 3) for a single flat plate and two flat plates in 2D...... 29

v 36 Volume mesh inside the tunnel domain, with a volume of refinement around the UGF. . . . 30 37 UGF surface mesh; oriented with flow left to right...... 30 L 38 Plots of CL, CD and D convergence against varying number of volume mesh cells within the ‘tunnel’ domain around the UGF. Simulated at M = 3, α = 10°...... 31 L 39 Plots of CL, CD and D against α for 0° ≤ α ≤ 20° from the 3D UGF simulations...... 32 40 Mach number contour plots for different free-stream Mach numbers at α = 5°. Shown in a lengthwise plane within the ‘tunnel’, 5mm offset from the centre of UGF with free-stream flow is left to right...... 33 41 Absolute pressure contours for M = 1.5, 2, 3 shown on a widthwise plane (s × s) centred within the UGF and tunnel domain. All at α = 5°...... 33 42 Validation of results against those presented by Miller et al. [48]...... 34 L 43 D against Mach number for the high subsonic, transonic and low supersonic regime. . . . 34 44 Mach number contours around the horizontal plate of the UGF at M∞ = 0.6, 0.7, 0.8, 1. . 35 45 Skin friction coefficient along the vertical UGF plate. Take at 5mm above the centre for M = 3, α = 5°...... 35 46 τw vectors at M = 3, α = 5°...... 36 47 Vortices from the Q-criterion at M = 3, α = 5°...... 36 48 Geometrical modifications applied to the TE of the UGF. Top view of TE shown...... 36 L 49 Percentage changes in CL, CD and D from an unmodified UGF at M = 3, α = 3°...... 37 L 50 Percentage changes in CL, CD and D from unmodified UGFs at M = 1.8, 2 and 3; all at α = 3°...... 37 51 Temperature distribution from CFD shown for a forward section of the horizontal UGF plate. 38 52 q˙w along the plate from using theoretical, CFD and empirical (graphical) methods...... 38 53 Alternative node locations for a UGF study [53]...... 40 A.1 A schematic diagram showing the layout of the Russian Vympel R-77 missile with tail mounted grid fins [69]. Parts are labelled in Table A.1...... 46 A.2 Close-up images of the R-77 grid fins [69]...... 46 B.1 An engineering drawing of the R-77 grid fin...... 47 C.1 A synopsis of the geometrical variations covered in several literature studies [47]...... 48 E.1 Identified oblique shocks and expansion fans used for application of the shock expansion theory to the single flat plate configuration at α = 3°...... 50 G.1 Histogram plots of wall y+ for different edge geometries at M = 3, α = 3°, over the whole single flat plate surface...... 55 + c G.2 Histogram plots of wall y at M = 3, α = 3° for the sharp edged two plates with s = 1, 3, 8, 20...... 56 G.3 Histogram plots of wall y+ at α = 5° and M = 0.9, 1.5, 2, 3 for the unit grid fin 3D simulation. 57 H.1 An exhaustive illustration of all geometrical modifications applied to the TE of the UGF. All views are isometric taken from the TE...... 58 ∗ I.1 τw, Cf and St found from theory, CFD and through the empirical method given in [68]. . 62 List of Tables

1 A summary of the identified aerodynamic flow features through grid fins at different regimes; based on [37]...... 11 2 Conditions calculated using the ISA at an altitude of 10km above mean sea level...... 12 3 Values of the geometrical parameters shown in Fig 14...... 13 5 4 Comparison of CL for the sharp edged plate at M = 3, Re = 9.71 × 10 from CFD, linear theory and shock expansion theory...... 19 A.1 Identified components presented in Figure A.1...... 46 vi Nomenclature s Span between grid fin cells, m

2 M Mach number Sref Reference area, m

2 α Angle of incidence, ° Sw Wetted area, m

∆s First prism layer height St Stanton number

δ Boundary layer thickness, m T Temperature, K

2 q˙w Heat transfer rate, kW/m t Thickness of grid ﬁn web element, m c T ∗ Reference temperature, K s Chord to span ratio L uτ Friction velocity, m/s D Lift to drag ratio t V c Thickness to chord ratio ∞ Free stream velocity, m/s

γ Speciﬁc heat ratio, γ = 1.4 for air xt Leading or trailing edge length, m

ν Kinematic viscosity, m2/s y+ Wall distance coordinate

3 ρ∞ Free stream density, kg/m ALM Additive Layer Manufacture

τw Wall shear stress, Pa CFD Computational Fluid Dynamics a Speed of sound, m/s CNC Computer Numerical Control c Chord length, m GF Grid ﬁn

CD0 Zero lift drag coeﬃcient ISA International Standard Atmosphere

CDi Induced drag coeﬃcient LAV Launch Abort Vehicle

CD (Total) Drag coeﬃcient LE Leading edge

Cf Skin friction coeﬃcient RANS Reynolds-Averaged Navier Stokes

CL (Total) Lift coeﬃcient SBLI Shock Boundary Layer Interaction

CP Pressure coeﬃcient TE Trailing edge n Number of total prism layers UGF Unit grid ﬁn

P Pressure, Pa USAF United States Air Force

Re Reynolds number VOR Volume of Reﬁnement

vii 1 Introduction

Grid fins (also commonly referred to as lattice fins) are an alternative lift generation and control device for missiles and spacecraft as compared with traditional planar fins. Grid fins typically consist of an outer frame with an internal grid structure of thin, small chord sections as shown in Fig.1, aligned perpendicular to the oncoming flow.

Grid fins were commonly seen on several intermediate range missiles and rockets produced within the Soviet Union during the 1970s such as the SS-12, SS-20 and Soyuz N1 launch vehicle intended for the Soviet moon program. Driven by modern day requirements for missiles to have compact overall packaging for use in internal weapons bays of next generation aircraft, there has been a renewed interest in grid fin design and development. More recently, notable uses of grid fins can be found in the American GBU-43/B (MOAB) missile and the Rus- Figure 1: A Russian R-77 missile with tail mounted grid fins [1]. sian Vympel R-77, which is a supersonic ramjet powered air-to-air missile. SpaceX has also had in interest in the use of grid fins for their hypersonic reusable launch vehicles such as the Falcon 9 booster, configured to have four tail mounted grid fins to control roll, pitch and yaw movement [2] and having the ability to be stowed when not in use. 1.1 Motivation The unique aerodynamic and structural characteristics of grid fin architectures bring several advantages over conventional planar fins. A near zero hinge moment is observed allowing downsizing of actuator components giving weight reductions and more compact packaging; a smaller centre of pressure variation over a wide Mach number range; a high strength to weight ratio and increased stall angle of attack [3–5].

Grid fin applications for control and stability during atmospheric re-entry such as those seen on the SpaceX Falcon 9, have primarily focused on material selection and structural design to ensure the grid fin would be capable of withstanding the thermal environment during re-entry and would be able to effectively translate control inputs to stabilise the vehicle. However, a major drawback of grid fins as compared with planar fins of similar lift characteristics, is the higher drag values; often up to four times that of a planar fin, especially at transonic and low supersonic speeds [4–6]. Minimising this drag has not been a key performance driver for applications such as SpaceX whereas, when designing a grid fin for a high speed missile, this would be crucial to maximise range and provide stability at supersonic speeds.

Given that minimising drag is a key performance driver for such missile applications, an ideal grid fin design would have infinitely sharp leading and trailing edges as well as the thinnest possible section [7]. However, in practice, manufacture of such a design would be impossible and so, this project investigates several design attributes and tradeoffs such as the variation of geometry, which would be valuable to the designer in order to balance the aerodynamic performance with producibility. Aerodynamic analysis conducted in this project will allow the best compromise to be made depending on the manufacturing methods, materials and processes available to achieve the desired performance and hence, assess the relative importance of specific geometry and aerodynamic variations in producing the optimal grid fin. 1 2 Background and Literature Review

Given the typical context of grid fin use in missiles and defence applications, especially within the Soviet Union and United States Air Force (USAF), the availability of model-specific technical and performance data within the public domain is limited. However, more recent commercial applications such as in the SpaceX Falcon 9 allow for investigation into grid fin manufacture methods, geometry variations and possible performance drivers for the design. In addition to this, generic grid fin performance has been explored through various experimental and CFD studies allowing for comparison and validation against results produced in this project. This also allows more fundamental aerodynamic theory to be applied and thus, provide more readily available methods for evaluating the performance and producibility tradeoffs in grid fin designs.

2.1 Missile Configurations Similar to aircraft, several variations of missile configurations and layouts exist. These variations are primarily driven by the performance, packaging and thermal requirements of the missile being designed. The positioning of the fins along the missile body and their function is commonly varied and is an important consideration regardless of the type of fin used.

By varying fin position, size and function, a limitless number of designs are possible even within a specified Mach number or flight condition as presented in [8]. However, the most commonly seen layouts are wing control, canard control or tail control configurations [9–11]. Within each, the fin actuation is usually combined to give three degrees of freedom for the missile; roll, pitch and yaw (as well as in-flight longitudinal and lateral stability). The three common configurations, along with a novel split-canard configuration, are illustrated in Fig2.

Active grid fin(s)

(b) Wing control (a) Canard control

Figure 2: Canard, wing, tail and split-canard control missile conﬁgurations, each shown with grid ﬁns.

As discussed in-depth in [9, 12, 13], each of the configurations in Fig2 has advantages and drawbacks. Canard control which consists of active fins located on the aft section typically offers better manoeuvrability at small α but experiences large flow separation as α is increased to higher angles. Canard mounted grid fins would still require large tail fins (likely planar) to overcome the destabilising effect. Overall, the canard configuration is best suited to smaller missiles which do not require roll stabilisation [12] and will not experience large deflections of the grid fins. It is common to design canard controlled missiles with no roll control since roll stabilisation on planar tail fins adds complexity and cost [14]. An alternative is to place grid fins at the tail along with a planar finned canard as is investigated in [14], showing an improvement in roll effectiveness at supersonic speeds (M = 1.5, 3).

2 Wing control involves a large actuated fin mounted close to the center of gravity with fixed stabilising fins at the tail and aft sections. Despite being an early design, wing control still benefits from one of the fastest response characteristics as compared with canard or tail control [12], aiding seeker performance to minimise tracking error and be highly manoeuvrable [13]. A major disadvantage is packaging of the missile due to the large dimensions of the fin though use of a grid fin, as shown in Fig 2b, would help to mitigate against this by stowing against the body. The interaction of vortices shed from a grid fin with the tail mounted fins also needs to be considered carefully and often requires iterative optimisation of the fin location [12].

Tail control is the most commonly used configuration of modern missiles and benefits from good manoeuvrability even at high α due to the large moment arm between the centre of gravity and the tail mounted fins. Due to tail deflection being in the opposite direction to the missile angle of attack, slower response is seen as compared with other configurations since the initial lift force is in a direction opposite to the desired one [12]. A common challenge with tail control layouts is the limited packaging space for control mechanisms and hence, employing grid fins rather than planar fins would mitigate against this since grid fins typically produce smaller hinge moments as compared with planar fins [3,5]. As shown in Fig1, the Vympel R-77 / AA-12 along with most other grid finned missiles of the past have utilised tail mounted grid fin control configurations. As a result of this, many existing CFD studies on grid fin performance, as subsequently discussed, assume a tail grid fin configuration.

In addition to these three commonly discussed configurations, many novel configurations also exist as given in [8]. One of these is a split-canard control layout as shown in Fig 2d. Such a configuration, albeit with planar fins, has been used on missiles such as the Kegler AS-12, Vympel R-73, Vympel R-60 and R550 Magic. A split canard layout overcomes some of the challenges of a single canard by generating vortices to accelerate the flow to the second canard and delay the onset of separation, enabling a higher α limit [13]. The downwash produced by the first canard postpones the leading-edge vortex breakdown to higher angles of attack [15]. Such a design may be of particular interest for a supersonic missile with grid fins in place of a second canard, perpendicular to the oncoming flow; grid fins perform best in high supersonic regimes rather than low supersonic or transonic [6] and hence this additional acceleration from the first canard would have potential for improving performance.

2.2 Applications of Grid Fins In order to appreciate the design considerations that have been made for grid fins and understand producibility motivations, it is useful to focus on specific applications of grid fins for missiles and spacecraft which have already been designed, manufactured, tested and are fully operational rather than purely conceptual studies. The grid fins used on the Russian R-77 Vympel missile are useful to analyse as they have been designed for operation in similar thermal and atmospheric conditions to those presented as requirements in [7]. Also, the application of grid fins to the SpaceX Falcon 9 booster provide a useful insight into how producibility decisions are altered for a commercial space application. Nevertheless, given the specialised defence and space related applications, material properties, specifications and design drawings are not readily available in the public domain. Therefore, model-specific details presented in this project have been cited where appropriate and are mostly a combination of technical specifications presented in various published sources.

2.2.1 Vympel R-77 The Vympel R-77 missile (also known as the AA-12 Adder) is designed as a medium range air-to-air missile, suitable for launch from under wing platforms such as that on the Russian MiG-29 aircraft [16]. The typical flight regime for the R-77 is supersonic, between Mach 2 and Mach 4, with up to a 40° angle of attack and a range of up to 70km [17]. As shown in Fig1, the cross (‘ ×’) configuration of the grid fins allows for roll, pitch and yaw motions with each fin being individually retractable by a motorised hinge [17]. The ability to retract 3 against the body of the missile is seen on several of the Soviet missiles presented earlier and allows for ease of transportation and storage; the fins are typically in the extended position when the missile is suspended from the aircraft wing as they form the primary control surfaces and hence are required at all phases of flight.

The grid fin design employed on the R-77 is typical of a generic design such that it consists of a straight edged frame, with a regularly spaced internal lattice structure. The overall chord length of the grid fin is constant throughout and each cell within the lattice has equal spacing. The cells are divided such that each grid division is diagonal across the rectangular frame, with a tapering seen towards the base where the span reduces and frame thickens for structural rigidity. In this diagonal orientation, the grid fin is composed of 2 cells (width) by 4 cells (height). The base of the grid fin is attached to the body of the missile using a single hinge, with a ramped fairing at the base covering the exposed hinge components [18]. A detailed labelled diagram of the R-77 layout as well as images of the grid fin are given in AppendixA. The geometry of the R-77 grid fins and its particular application to this project [7] is discussed in greater detail in Section 3.3.

Due to the generic nature of the R-77 grid fins, with no cell or chord-wise variations in geometry, it is often used as a basis for CFD studies and literature concerning grid fin aerodynamic analysis; The R-77 / AA-12 missile is specifically used for investigating aerodynamic performance in [18–20] and is discussed in [21] within a review of comparing approaches to CFD studies on grid finned missiles. A scale model of the R-77 missile is used in [19] in order to compare theoretical predictions (based on shock expansion theory) with prior experimental data from wind tunnel tests for the normal and lateral forces along the grid fin, pitching moments and vortex interactions of the missile body with the grid fin cells. Variations of this configuration that include altering the cell size and grid fin shape are also presented for comparison in [19].

2.2.2 SpaceX Falcon 9 The integration of grid fins on the first stage of the SpaceX Falcon 9 reusable launch vehicle has the same purpose as on the R-77 missile, to provide a manoeuvrable control surface device [23]. However, the thermal and atmospheric environment that they have been designed for varies largely from the application to missile systems; SpaceX grid fins are designed for hypersonic flight speeds and for deployment during the re-entry phase [24] meaning that the fins must withstand much higher temperatures and multiple uses without need for replace- ment. The configuration employed on the Falcon 9 is shown in Fig 3 and Fig4 with four foldable heat-resistant grid fins mounted at the top of the first stage section [25, 26]. The folding design drives the requirement for the fins to have a sweep in the chord-wise direction allowing better integration around the Falcon 9 body and minimising drag (by minimising exposed frontal area) when folded. During re-entry, cold gas thrusters are used to correct the vehicle’s attitude Figure 3: Falcon 9 with grid fins before the fins are deployed; this is then followed by deployment of mounted at the top of the first stage the four composite landing legs as it approaches the touchdown point [22]. [24].

2.2.3 Orion Launch Abort Vehicle As presented in [27, 28] and shown in Fig5, grid fins were also designed for use on the NASA Orion Launch Abort Vehicle (LAV). An interesting aspect of this design is that the grid fins are optimised for small length : diameter ratios which is different to typical missile bodies. A wide Mach number range (0.5 < M < 2.5) is explored in [28] with a particular focus on transonic performance and the effect of sweep to improve pitch 4 stability of the LAV as compared to without grid fins. Both [27] and [28] present experimental results from wind tunnel tests (of a 6% scale model), while CFD results are also compared in [27] for a range of twelve different Mach numbers and for 0° < α < 15°. Although produciblity and integration with the LAV is not explored in depth, the results presented in [28] show the differences between a (‘+’) and (‘×’) configuration as well as geometrical sweep of the fins (over the whole Mach range) which would provide starting points when considering producibility and manufacture. Based upon the CFD data generated from [27] and the experimental data in [28], an aerodynamic database is now available to the designer of a generic missile, if meeting the length : diameter ratios of the LAV and at flight conditions compatible with those in [7].

Figure 4: Four, body folding titanium grid Figure 5: Swept grid fins at the base of a fins on the first stage of the SpaceX Falcon scale model of the NASA Orion LAV [28]. 9 launch vehicle [29].

2.2.4 Producibility of Existing Designs An important motivation for the use of grid fins on SpaceX Falcon 9 is the ability for them to be reused on multiple flights and hence be non-ablative. During atmospheric re-entry, the exterior surface of the Falcon 9 experiences temperatures up to 1850°C[30] and hence stagnation points on the grid fins would likely be much hotter. Therefore, a single piece cast and cut titanium alloy is used for each fin [26, 29]. The high strength : weight ratio, corrosion resistance and high melting point allow titanium to be the ideal choice for grid fins on space vehicles despite the relatively high cost. Typically, casting allows for benefits such as a consistent quality cross section, high dimensional accuracy and a superior surface finish even when thinner sections are used, ideal for optimising grid fin aerodynamic performance which is sensitive to surface geometry and dimensions. More recently, SpaceX has announced the move to welded steel grid fins on future projects such as the Starship Super Heavy. This decision is motivated by the fact that the Starship’s grid fins would be much larger (8-10 times the surface area) and hence the cost and feasibility of casting titanium at that scale is a challenge. The stainless steel alloy has been designed to have similar structural properties despite the reduced heat capacity [29]. The construction of a welded grid fin does however reduce the possibility of variations between cells as new tooling would be necessary (which is often at a high cost) and would ultimately produce a surface finish, particularly at joins, inferior to that of a casting process.

While limited information is available regarding the manufacturing process used on the Vympel R-77 grid fins, a common challenge (also a key objective of this project [7]) is the trade off between produciblity and aerodynamic performance. The desire for thin surface elements with perfectly sharp leading edges and the associated high manufacturing costs for a grid fin based on the R-77 is highlighted in [31]. Further, the possibility of using thin metal sheets within each cell is discarded as this would likely not withstand the thermal and mechanical loads present at supersonic flight speeds and would be difficult to produce accurate leading and trailing edges on.

5 Therefore, [31] explores various methods of reducing wave drag (which would be inherent of more typical production methods) such as through local sweep of fin cells, altering intersection geometries and varying sharpness of the leading edges based on available manufacturing techniques; results from experimental tests at 4 ≤ M ≤ 6 are presented for three different geo- Figure 6: Local sweep applied to individual grid metries. Most notably, up to a 20% improvement in lift fin cell members [31]. to drag ratio and 38% in zero lift drag is reported for local sweep applied to the grid fin [31] (shown in Fig6) which informs the impact of producibility on performance, depending on the chosen process.

Modern day capabilities with composite materials and additive layer manufacturing (ALM) means that new methods of producing grid fins are becoming possible and hence, would be able to better meet the geometrical demands to achieve optimum aerodynamic performance. Although patented, ideas presented in [32] and shown in Fig7 include a metallic origami-like structure for the internal framework produced from thin metal sheets cut by laser or CNC water jet methods. Also presented is a method to produce a composite grid fin from a mandrel based curing process giving elongated extrusions from which the final fin is sliced. An alternative, economical moulding process to reduce cost of manufacture as compared with water jet cutting or machining from solid metal billet is discussed in [33] which still allows for complex geometries and the use of composites to achieve the optimum performing design. ALM, such as by the use of metal 3D printing is also becoming a popular option to produce the intricate geometries of grid fins [34]. ALM also has the key benefits of reduced need for quality control, superior surface finishes, being highly accurate and being able to alter the infill density to reduce overall weight of the fin, further minimising hinge moments. ALM for grid fins may also have benefits in improving the heat transfer Figure 7: A folded sheet metal and mandrel curing within the component as compared to brazed joints approach to producing grid fins as given in [32]. which is explored through simulation in [35].

2.3 Drag Reduction through Geometry Modiﬁcation Based on linear aerodynamic theory, drag forces acting upon the complete missile can be decomposed into zero lift drag and induced drag [36].

CD = CD0 + CDi (1) Induced drag is often the largest component and is affected by design characteristics, such as the variation of leading and trailing edge geometries or the sectional dimensions of the finite lifting surface. The zero lift drag can be further separated as presented in [11]; the empirical relationship given in Eq2 is used for zero lift drag computation to account for interference effects of missile components and the variation in skin roughness over the body.

CD0 = 1.25 CD0 + CD0 + CD0 + CD0 (2) nose body ﬁns tail

6 The contribution of zero lift drag from the (grid) ﬁns can then be further separated into wave drag and skin friction drag.

CD0 = CDskin friction + CDwave (3) fins fins fins Many empirical relationships exist for predicting skin friction drag at supersonic speeds based on length scaled Re. Assuming turbulent flow, an (incompressible) skin friction coefficient relationship given in [11] is −2.58 CDskin friction = 0.455 (log10 Re) , to which a compressiblity correction and reference area scaling is then applied. Most importantly, [11] gives the relationships in Eq4 for the wave drag arising from a leading edge for a double wedge supersonic aerofoil (assuming a perfectly sharp edge), which acts as the grid fin section.

B t 2 S C = w Dwave (4) ﬁns β c Sref √ B = c/xt β = M 2 − 1 where 1−xt/c ,

Several studies have been performed on the effects of varying grid fin cell geometry in order to minimise the total drag and make it more comparable to that of planar fins. These have not only been limited to variations in leading and trailing edge sharpness but have also focused on the local and global sweep of lattice cells and variations of cell size, which would ultimately drive producibility decisions. A combination of experimental and CFD results for subsonic [3, 38–40], transonic [3, 40–46] and supersonic regimes [3–5, 40–43, 46] have been conducted. Typically, grid fins have the poorest performance at transonic flight speeds due to a large standoff bow shock occurring from choking of the individual grid fin cells; the flow field at subsonic, transonic (choked), low supersonic and Figure 8: Flowfield structure around a single grid fin high supersonic speeds for a single cell is shown cell at different speeds [37]. in Fig8.

Similar to the application of SpaceX in Fig4, adding sweep to the grid fin aids packaging and minimises drag. By assuming a sharp leading edge swept fin (30° sweep) and a blunt edged unswept fin, [41] presents the experimental results from wind tunnel testing at transonic and supersonic speeds (0.75 ≤ M ≤ 1.7). As shown in Fig9, a standoff bow shock is observed for both, with the swept configuration reducing the strength and causing two, offset shocks rather than one; the overall CD saw a reduction of up to 30% by adding sweep. Similar studies by Zeng [43] and Wang et al. [42] use a CFD approach to investigate a sharp edge sweep for 0.905 ≤ M ≤ 2 with similar results showing a CD reduction, especially at transonic choking cases. Out of the eight geometries presented in [42], the largest CD reduction of 54.8% against a blunt unswept fin is seen at M = 1.7. However, [46] shows the results of the above but with the addition of accounting for the flow interaction with the full missile body. The full missile body is simulated along with the sweep effects of the grid fin at M = 0.9, 1.2, 2 and a smaller overall drag reduction is observed (up to 13%), indicating that flow interaction with the body is significant. Interactions of the boundary layer on the body with the incident shock waves from the grid fins (SBLI) likely cause a notable increase in drag which is not explored in [42].

7 Figure 9: A schematic of the ﬂow approaching a blunt, unswept grid ﬁn cell and a 30° sharp swept cell [41].

Existing studies [23, 38, 39, 44] on variation of cell geometry are mostly in the subsonic or transonic regime. However, [3] explores cell geometry variation up to M = 3.5 which is more suitable for the requirements of the project [7]. A more generic, octagonal shaped fin configuration is employed in [3, 39] whereas a shape more similar to the R-77 grid fin is used in [23, 44] albeit with scaling and simplifications applied where experimental tunnel testing has occured. As presented in [39], using smaller cells within the grid fin frame (i.e more cells and more surfaces), leads to an increase in normal and axial forces, however, the increase in drag coefficient is better explored in [3]. At M = 3, Washington et al. [3] shows that a coarse-celled grid fin has a CD twice that of a planar fin whereas a fine cell density causes a CD four times that of a planar fin. This is particularly significant as despite resulting in increased normal and axial forces (hence being more efficient as control devices), a trade-off with increased drag is observed as the number of cells and hence number of surfaces inside the grid fin frame is increased. A ‘transonic bucket’ is also reported in [3] which correlates to the drag increase at transonic regimes and a corresponding drop in normal force produced by the grid fin. This transonic regime (with choking in [3] occurring at M ≈ 0.75) is therefore of great concern when designing grid fins at high speeds since they will often transition through these transonic Mach numbers. The critical Mach number (Mcr) for the onset of transonic choking with varying cell geometries, web thicknesses and area ratios is reported in [44] and is in the range of 0.65 < M < 0.87; a more coarse fin or increasing thickness of the section results in increase of the observed Mcr. Nevertheless, the effect of changing α on Mcr L is not explored in [44]. Despite varying the cell geometry, the reported CL, CD and D shows similar trends in all cases in [23, 38] at the subsonic speeds tested. It is observed that CL shows a linear behaviour with α, up to a maximum α after which it begins to fall; CD shows a rapid increase at higher α (above ≈ 10°) and hence L there is a peak observed in the D plots indicating an optimum α for the given grid fin geometry and flight conditions. The CFD analysis in [23] also explores the variation in cell geometry through approximating each cell as a NACA 0012 symmetric aerofoil as well as then as a rectangular flat plate. It is important to note however, the analogy to a flat plate is more appropriate to a supersonic high speed simulation than is the (subsonic optimised) NACA aerofoil due to it is thickness and large nose radius. Pressure coefficient plots in [23] show good agreement in trends by using a blunt flat plate aerofoil as an approximation to the real grid fin section and this is most likely due to the lack of an oblique shock from the leading edge (at the subsonic speeds tested) meaning that a leading or trailing edge sharpness is not necessary for adequate performance.

Altering geometric parameters (frame shape, frame thickness, web thickness, cell size) provides a reliable and consistent drag reduction technique and enables grid ﬁns to exceed planar ﬁn performance [47]; a summary of many of the studies discussed above is provided in [47]. An image showing geometrical variations used within these is given in AppendixC for a range of missile and spacecraft based applications.

8 2.4 Leading and Trailing Edge Geometry Miller et al. [48] explore several different combinations of grid fin sections, leading edges and trailing edges, shown in Fig. 10 as techniques of drag reduction. These are investigated through wind tunnel tests for 0.5 ≤ M ≤ 2.5 and −8° ≤ α ≤ 20° with a length averaged Re between 5.35×10−6/ft and 8.05×10−6/ft depending on the tunnel speed. Chord lengths for the grid fin sections are kept constant whilst the leading and trailing edge angles are varied between 17° and 90° (blunt edge) as well as the length of the edge along the chord. The chord sections are investigated independently as well with a grid fin frame attached; Figure 8 in [48] shows

that the highest CD0 is observed for a sharp edge with a frame, followed by a blunt edge without a frame. The lowest drag is seen for a half diamond wedge aerofoil. This indicates that the effect of the frame is also an Figure 10: Section and edge variations in [48]. key consideration for grid fin drag and hence may also benefit from a producibility analysis to minimise the thickness and minimise drag.

2.5 Full Body CFD Studies When considering the aerodynamic performance of grid fins, especially at supersonic speeds, SBLIs are present along each grid fin cell as well as along the missile body. Predictions for the drag reduction performance of geometrical modifications are often overestimated when this interaction is not considered. Burkhalter et al. [49] presents a non-linear theoretical method for evaluating grid fin performance and many of the methods used, including an extension of the vortex lattice solution and prediction of forces accounting for the contribution from the missile body as well as the grid fin. The interaction of the missile body is also considered in [4, 18] such as through the distribution of surface pressure contours at the junction of the grid fin attachment and the skin on the missile body. Additionally, body mounted ramp fairings are investigated in [18] for up to Mach 2, as a successful method of reducing body interaction and blockage effects. Furthermore, the SBLIs inside the cells were more evident as the thickness of the web was increased. It is worth noting that the L3 configuration of the grid fin in [18] is geometrically similar to that of the Vympel R-77. Both a cruciform type (‘+’) and cross type (‘×’) configuration of grid fins is investigated in [4] through full body CFD simulations at M = 2.5. The SBLI between the missile body and a mounted grid fin at M = 3 is shown in Fig. 11

Some of the first viscous full body CFD simulations on grid finned missiles were conducted by DeSpirito et al. [50, 51]. Both of these studies, at M = 2.5, showed good agreement with wind tunnel experiment and employed symmetry planes on a cruciform (‘+’) configuration grid finned tail control missile. Both also clearly show the complex shock structure occurring through the grid fin cells via the pressure coefficient contours. The ogival shaped nosecone also produces oblique shocks and the interaction of these with the fins can be seen in [51].

As seen in Fig. 11, the wake region of the grid fin can produce complex flow features including strong vortices and a recirculating region which eventually interacts with the body of the missile. CFD simulations of this wake interaction at M = 1.5, 2.5, 3.0 are included in [14] and [52]. As discussed in [14], the strength of the vortical structures in the wake grows rapidly, especially when the grid fins are deflected, causing a large increase in drag. Therefore, it may be argued that aside from optimising the grid fin geometry and balancing 9 produciblity to reduce drag, the grid fins should be placed as far aft as possible to minimise the effect of these vortices on the overall drag force, especially when a ramped fairing is not present [18]. Fig. 12 shows the interaction of the grid fin wake with the aft missile body.

web bow shock web wake vortices

flow direction base recirculating region recompression FIN near wake recompression shock shock boundary layer

BODY separation recirculation area reattachment

Figure 11: Flow structure and SBLI between a grid Figure 12: Surface pressure distribution showing ﬁn and the missile body at M = 3, α = 0° [40]. the grid ﬁn wake interacting with the aft missile body at M = 2.5, α = 5° [4].

2.6 Unit Grid Fin Method Whilst a full body simulation is valuable in understanding the interaction of flow features, especially in the boundary layer and the wake, it is often too computationally expensive especially at a preliminary design stage [53]. Additionally, the corresponding mesh required to be able to fully capture the flow across the whole missile requires several iterations to optimise. As proposed by Dikbas et al. [53], the unit grid fin (UGF) approach offers a suitable alternative to analysing and predicting the aerodynamic forces around grid fins by simplifying the problem to only a portion of the fin. The validation of this approach is presented in [53] against a cruciform (‘+’) configuration before the idea is extended to a planarform configuration at Mach numbers across the transonic and supersonic regime ( M = 0.7, 1.2, 1.5). As found by Dikbas et al., the method is most accurate for supersonic regimes with reduced accuracy at transonic speeds due to the lack of ability to fully capture bow shock effects. The unit Figure 13: Representation of a unit grid fin grid fin approach is shown in Fig. 13 where a central region within a lattice framework [53]. of the grid fin lattice is isolated for the computation. Hence, the effect of the frame, connecting rods and body is not included so a correction can then be employed to calculate the effective angle of attack as given in Eq5. 2 r0 αeff = α 1 + (5) r1r2 where r0, r1, r2 are the radii to missile body surface, top of connecting rod and top of frame respectively. A comparison of Mach contours of a UGF against a full grid fin at M = 2.5 are shown in [53] and show good agreement of flow features allowing this to be a faster, more readily accessible alternative to full body CFD simulations. 10 2.7 Theoretical Methods Although the flow features of grid fins are extremely complex and sensitive to free-stream Mach number and conditions, theoretical prediction methods are available as a method of calculating global forces and moment coefficients by simplifying the grid fin. A modified Evvard’s theory for supersonic flight regimes is presented in [37]. This relies upon modifying a linear aerodynamic panel theory to account for grid fin frames and end plates. The grid fin element is separated into regions by Mach lines and then discretised into smaller rectangular elements such that the pressure distribution and hence normal and axial forces can be computed. While this theory is not able to account for complex geometries and complex flow features such as SBLIs, Kretzchmar et al. [37] found an acceptable comparison in the values calculated for horizontally placed fins with no deflection or angle of attack. Therefore, the theory has not been validated for non-zero α and it cannot fully capture flow physics inside each grid fin cell. Actuator disk theory has also been applied to grid fins by Reynier et al. [54]. Whilst this also is able to compute global forces and moments, even at small angles of attack, Reynier et al. concludes that the theory is limited as soon as vortical flow develops along the body and hence local Mach number variations occur. These theoretical methods are therefore unsuitable for reliable prediction of lift and drag forces as well as capturing the flow features around grid fins on a supersonic missile. CFD and experimental testing on scale models remain the most reliable methods of obtaining aerodynamic forces.

2.8 Flowfield around Grid Fins Based upon the literature reviewed in 2.3 and 2.5, the expected flow field around a typical grid fin can be summarised as given in Table1. It must be noted that the Mach numbers identified in Table1 are only an indication based on previous studies and hence would be highly dependant on the chosen grid fin geometry, thermal environment and atmospheric conditions of flight.

Table 1: A summary of the identified aerodynamic flow features through grid fins at different regimes; based on [37].

Mach number Flow features M<0.7 • No supersonic regions in local ﬂow. Subsonic • Lift and drag show little sensitivity to geometry or cell pattern variation. 0.72 • Shocks do not impinge adjacent cells. Supersonic • SBLIs present on missile body and in wake. • Performance is sensitive to angle of attack.

11 3 Computational Approach

3.1 Key Objectives Based upon the requirements and work packages given in [7], the literature reviewed in Chapter2 and the timeline for the project, the key objectives for aerodynamic and producibility analyses are summarised as follows.

• To understand and calculate aerodynamic forces and flow features around missile mounted grid fins at the specified flight conditions primarily through the use of CFD. • To synthesise the effects of grid fin geometry parameters (edge sharpness, span, chord, taper / sweep) on aerodynamic performance across a range of flight conditions. • To inform producibility decisions based upon the aerodynamic analysis that will ultimately make grid fins more competitive to traditional planar fins. 3.2 Flight Conditions The baseline conditions considered for a generic high speed missile were at an altitude of 10km above mean sea level with a cruising speed of M = 3 [7]. From this, the complete atmospheric and thermal environment was computed using the assumption of an International Standard Atmosphere (ISA) model which was a reasonable approximation to exact conditions given that the flight would still be within the troposphere. The conditions at M = 3 are given in Table2 and were calculated using an implementation of the ISA in MATLAB. The MATLAB code is provided for reference in AppendixD and has been used for all subsequent calculations of flight conditions.

Table 2: Conditions calculated using the ISA at an altitude of 10km above mean sea level.

Parameter Value Ambient temperature, T (K) 223.15 Speed of sound, a (m/s) 299.46 Pressure, P (Pa) 26436.30 Density, ρ (kg/m3) 0.4127

3.3 Configuration and Vympel R-77 Geometry As explored in Chapter2, several configurations for missile fin layouts exist depending on the application and performance requirements. Although novel concepts such as the use of grid fins on a split-canard layout have never previously been explored in literature and offer unique benefits to exploit the advantage of grid fins, a more conventional tail control model was ultimately chosen. The tail control configuration shown in Fig. 2c comprises of grid fins mounted aft of the missile body with additional planar fins mounted forward for stability. A tail control layout is most commonly observed in existing literature and hence would allow for a means of validating CFD results and making them reliable. Since the focus of the project is based upon a generic missile and that the key objectives relate to the grid fin performance rather than control or overall missile performance, a tail control layout, with a large number of experimental and CFD studies available for validation, was used. The baseline geometry for the grid fin was based on that of the Vympel R-77 grid fin [7] which also features grid fins mounted in the tail control configuration. Given the limited availability of model-specific engineering drawings to capture dimensions and detailed features, a number of sources were used to generate a compiled data-set of dimensions which would be most similar to that of the R-77. Initially, photographs of the R-77 missile were studied to gain an appreciation for the relative scale of the grid fin elements and cell layout. Following this it was noted that [19–21, 55] all presented dimensions for the R-77 12 grid fins (albeit some were scaled models). Nevertheless, slight differences in dimensions were seen and this is likely attributed to the authors also making best estimates for dimensions rather than having a verified source. Therefore, the dimensions from these were averaged to generate a final set of suitable geometrical dimensions. As shown in Fig. 14 the geometry can be summarised to have a rectangular frame, with a 4 × 2 cell layout and a tapered 2 cell base. The cells are square with a span s and material thickness between cells t; the overall chord length of the fin, c is also identified. Thes values of theset parameters are given in Table3. c

Table 3: Values of the geometrical parameters shown in Fig 14.

Geometrical Parameter (mm) Chord length, c 38.50 Span, s 45.00 Thickness, t 0.675 Figure 14: A simpliﬁed drawing of the R-77 grid ﬁn geometry showing a top and right view with key geometrical parameters labelled. A full drawing is provided in AppendixB.

A full engineering drawing showing a complete set of dimensions for the R-77 grid fin is provided in Appendix B. 3.4 Three Stage Approach Although a single, full body or full fin CFD model is a potential method for obtaining the relevant aerodynamic forces and effects of geometry changes, it would lead to a high computational cost, increased dependence on the mesh quality and risk being too heavy a CFD exercise for the time available. Also, this would loose the importance of understanding the flow physics and produciblity trade-offs [7]. Therefore, an alternative three stage approach, as shown in Figs. 15, 16 and 17 was taken.

3D effects c t M M ∞ LE/TE ∞ s c M ∞

Figure 15: Stage 1 analysis. Figure 16: Stage 2 analysis. Figure 17: Stage 3 analysis.

3.4.1 Stage 1 - Single Flat Plate The first stage of the aerodynamic analysis involved a two dimensional model with a single flat plate placed inside of a circular domain of radius 10c. The single flat plate is representative of a single grid fin web element. This allowed a detailed analysis of the following effects: 13 1. The effect of varying the leading edge (LE) and trailing edge (TE) taper such that the effective sharpness of the edges was altered. The geometrical variations used for the LE and TE are given in Fig. 18.

t 2. The effect of varying thickness to chord ratio c to understand the consequences of using more slender grid fin elements. Using a single flat plate approximation to a grid fin element also allowed for validation of results using theoretical methods such as thin-aerofoil linear Ackeret theory and shock expansion theory to prove the reliability of the CFD results. As shown in Fig. 18, four variations of the edge geometry were used; a sharp edge, one-third edge, two-third edge and blunt edge. All of these had a taper length of 2t and a radius of 0.01t was added to corners to allow for better mesh convergence and simulate a realistic radius that would inherently be present if such parts were produced using methods such as ALM. The taper length and corner radius were chosen as reasonable assumptions given the generic nature of the design and would likely require refinement at later design stages once exact producibility capabilities are known.

Sharp edge One third edge Two third edge Blunt edge t t 2t/3 t t t/3 2t 2t 2t

Figure 18: An illustration of the diﬀerent LE and TE taper geometries used within the stage 1 analysis. All corners have a radius of 0.01t.

3.4.2 Stage 2 - Two Flat Plates The next stage of the CFD analysis involved another two dimensional model in a similar circular domain of radius 10c but with two flat plates as shown in Fig. 16. This was an approximation to the cross section of a complete cell within the grid fin with spacing s and chord c. This two flat plate approximation allowed analysis of the following effects:

c 1. The eﬀect of varying the chord to span ratio s and hence varying the cell size within the grid ﬁn. This is representative of varying the local cell geometry.

2. An analogy to the classical example of the design of a ramjet intake. This allowed understanding of the shock interference effects (as a result of M) between the two plates and hence the onset of choked flow within the cell corresponding to poor aerodynamic performance. Conversely, as shown in Fig.8, it also allowed the understanding of where least shock interaction between the plates would occur and where best grid fin performance is likely to be observed for the chosen geometry of the R-77 grid fins.

3.4.3 Stage 3 - Unit Grid Fin (3D) The UGF approach [53] shown in Fig. 19 was used for the three-dimensional (3D) CFD model to gain an understanding and visualisation of the 3D effects that would affect aerodynamic performance of the grid fin and hence inform produciblity. In particular: 1. The effect of shock waves in 3D such as at the intersection of the horizontal and vertical plates and at the LE and TE. 14 2. The effect of geometrical modifications in 3D such as local sweep of the grid fin cells, taper or simulating brazing of the joints if such a grid fin was produced using traditional methods.

Although the UGF approach can be applied to different portions of the whole grid fin such as the corner with the outer frame, or at the base which joins the hinge, a central section as highlighted in Fig. 13 was chosen. This was because it would best represent generic grid fin performance and would predict flow physics on the majority of the fin surface. Also, as a preliminary or generic design project, consideration of specific frame shape and integration with the hinge was outside the defined project scope and hence including such effects was not deemed important. Based on [53] and shown in Fig. 19, the UGF in a (‘+’) configuration is placed centrally within a cuboid shaped ‘tunnel’ domain of length 20c. The ‘tunnel’ walls are chosen to be coincident with the UGF and hence are square with side length s. The UGF of plate thickness t is bounded by two pairs of periodic wall boundary conditions, simulating the effect of a larger grid fin made of an infinite number of such (‘+’) sections. The ‘tunnel’ has one inflow and one outflow face as shown.

Outflow s t UGF Periodic c Pair 1 s Tunnel c 20 Periodic Inflow Pair 2

Figure 19: An illustration of the UGF model showing the ‘tunnel’ domain and the pairs of periodic wall boundaries. Variable have the values presented in Table3.

3.4.4 Benefits of the Three Stage Approach The three stage approach brought several benefits over conducting a full 3D model of an entire grid fin (and t c missile body). Most importantly, the identified features (LE/TE taper, optimum c , optimum s , choked flow onset, 3D shock effects and 3D TE shape) could be investigated in isolation of each other; the effect of each was clearly visible without having unnecessary additional variables. Moreover, conducting the 2D studies of stage 1 and stage 2 allowed easy validation of results against well known aerodynamic theory. This provided confidence in the results especially given that experimental testing was not conducted as part of this project which would be crucial in validating a 3D CFD model of the entire fin. Simcenter STAR-CCM+ (version 13.04.011 for 2D models and version 2019.2 for 3D models) was used as the primary tool for gathering CFD results. By conducting the three stage approach it also meant that a greater number of simulations could be run as each required fewer computational resources; mesh complexity for cases such as the 2D simulations was significantly lower than the UGF 3D case. This also meant that debugging any errors was quicker due to having a simpler model than would be with a full grid fin and missile body simulation. As suggested in [7, 55], the main flight regime of the missile at M = 3 and an altitude of 10km was considered as the baseline. However, additional cases between M = 0.6 and M = 3 at the same altitude, were also explored. Angles of incidence between α = 0° and α = 20° were used as reasonable assumptions for what a missile grid fin may experience in the majority of flight phases.

15 4 Single Flat Plate 4.1 Simulation Setup As per the dimensions given in Table3, a single flat plate was placed within a 2D circular domain for the stage 1 analysis and a subtract operation applied to remove the volume of fluid occupying this flat plate from the domain. As seen in Fig 20, a polygonal mesh was used in the far field and a prism layer mesh used close to the wall (plate surface) to be able to capture boundary layer effects in the viscous simulation. CFD settings were largely based upon suggestions given in [55–57] for the flow conditions and thermal environment identified in Table2. Most notably, a Reynolds-Averaged Navier Stokes (RANS), ideal gas and coupled flow model was used with a coupled implicit solver and K − ω SST turbulence model. Full details of CFD settings are given for reference in Appendix F.1. Many of these were as recommended in [56, 58]; a turbulent flow over the plate was assumed at M = 3 [57] with full chord length based Re = 9.7183 × 105 .

(b) Leading edge of sharp conﬁguration.

(a) Circular domain.

Figure 20: The mesh around a sharp single plate configuration. It is commonly a rule of thumb to use a domain which is 10× larger than the characteristic length of the body [57]. Regardless, the radius of the circular domain was investigated and it was found that when using a radius smaller than 10c, certain simulation cases (such as a large α or subsonic M) became unstable, likely due to the domain being too small to capture all of the flow features. Also, varying the domain radius whilst also checking for mesh convergence became a circular argument. To this end, the circular domain was chosen to be 10c for the 2D simulation cases and the mesh settings refined accordingly. As given in AppendixD, Re was calculated based on the full chord length of the plate (38.5mm). In order to define and assess the suitability of the prism layer mesh, the wall function approach was used with the K − ω SST turbulence model to have a target wall y+ ≤ 1 around the flat plate [57]. By using the Prandtl one-seventh power law, the skin friction coefficient (Cf ), wall shear stress (τw) and corresponding first prism layer thickness (∆s) was found. − 1 Cf = 0.026Re 7 (6) s r 2 τw 0.5Cf ρV∞ uτ = = (7) ρ ρ y+ν ∆s = where y+ = 1 for initial setting (8) uτ Based on the method in [57], the prism layer total height was set to be that of the boundary layer thickness δ for which many empirical formulae based on Re in turbulent flow exist; the one used is given below. 0.385c δ = 1 = Total height of prism layers (9) Re 5 16 The number of layers (n) within the prism mesh was calculated using a simple geometric series for a stretch factor of 1.5 between adjacent layers. ln δ + 1 n = 2∆s − 1 (10) ln(1.5) Once the prism layer was defined, the wall y+ value over the entire plate was monitored in each simulation and mesh values adjusted accordingly to maintain a wall y+ ≤ 1 and thus be able to fully capture the boundary layer effects. Histogram plots of these are given for reference in Appendix G.1.

4.2 Mesh Convergence To assess the number of volume mesh cells required in the domain, a mesh convergence study shown in Fig. 21 was carried out on a sharp edged conﬁguration. A sharp edge was chosen since it had the fewest number of cells across the LE and TE out of the four taper geometries so would provide a benchmark for convergence.

0.077 0.0152 5.3

5.25 0.0765 0.015 5.2

0.076 0.0148 5.15

5.1 0.0755 0.0146 5.05

0.075 0.0144 5 0 5 10 15 0 5 10 15 0 5 10 15 104 104 104 (a) Lift coeﬃcient. (b) Drag coeﬃcient. (c) Lift to drag ratio.

L Figure 21: Plots of CL, CD and D convergence against varying number of cells within the domain. The optimum number of cells within the domain was selected when increasing the number of cells (which results in a more computationally expensive solution) did not alter the value of the measured variable [57]. This was a mesh independent solution since the value had converged so further mesh refinement only added computational cost without improving results. As shown in Fig 21, the selected configuration (×) is with L 42703 cells and had the best trade off between cost and accuracy for CL, CD and D . Once the optimum mesh was configured, it was important to have a consistent and reliable method for determining the convergence of solutions. The solutions subsequently presented were obtained by ensuring residuals had fallen to their minimum with little or no oscillation between consecutive iterations; values of the monitored variables (e.g. CL, CD) were constant (up to 5 decimal places) and that no convergence errors were displayed.

4.3 Aerodynamic Forces

Using the LE and TE taper geometries shown in Fig. 18, CL and CD over the single plate was recorded and is plotted in Fig. 22 for 0° ≤ α ≤ 5°. All four edge geometry configurations were applied to both the leading and trailing edges of the single flat plate and as seen in Fig. 22a, show little variation in CL. At lower angles of incidence, the performance from each configuration is almost identical with a some deviation seen as α is increased towards 5°. At α = 5°, the sharp edge has an 11% increase in CL as compared with the blunt edge case, with 1/3 and 2/3 edges being accordingly in between. Nevertheless, the trend in CL follows a linear fashion for all four configurations as seen by comparison with the CL estimation computed using linear Ackeret theory which is also plotted and discussed further in Section 4.4. As mentioned in Section 1.1 and in [7], a more distinct difference in performance between the edge geometries is observed with CD in Fig. 22b. Here, the sharp edge configuration produces the lowest drag, across the whole α range tested, with the blunt 17 edge having the highest drag indicated by the largest CD values out of the four configurations. Given that the plates act like symmetric aerofoils, the zero-lift CD (at α = 0°) of a blunt edged plate is 208% higher than that of a sharp edged case; this difference reduces to the blunt configuration having a 99% higher CD at α = 5° compared with the sharp case indicating that especially at lower angles of incidence, the drag coefficient is extremely sensitive to the edge geometry. The 1/3 and 2/3 edge follow a similar convention to the CL performance in Fig. 22a such that as the edge bluntness increases, CD increases. The plots in Fig. 22 further emphasise the importance of LE/TE geometry in designing an optimal grid fin with the lowest drag. 0.14 0.05 0.12

0.1 0.04

0.08 0.03

0.06 0.02 0.04 0.01 0.02

0 0 0 1 2 3 4 5 0 1 2 3 4 5

(a) CL against α. (b) CD against α.

Figure 22: Lift and drag coeﬃcient against α for all four edge taper geometries. Linear theory (and skin friction corrected linear theory) also plotted for comparison.

4.4 Validating Results When performing simulations using CFD, it is crucial to validate the results (in addition Expansion to a mesh convergence study) to prove their Expansion reliability and have confidence in drawing Oblique shock conclusions. Given that experimental testing Oblique shock t c of the single plate configuration was not per- Oblique shock formed in this project, theoretical methods Oblique shock were the main tool used for validation. Of these, shock expansion theory and linear flat Expansion plate (Ackeret) theory were utilised since a Expansion 2D model was employed. Linear theory is a suitable method for validation given the slen- Figure 23: Shock expansion theory around the single flat derness of the geometry tested, resembling plate (sharp LE and TE), shown at an α > 0. that of an ideal flat plate (with infinitesimal thickness) and the low angles of incidence (α ≤ 5°) for which linear theory is most suited to [59]. Shock expansion theory is an exact, yet more cumbersome method which assumes inviscid flow and calculates pressure distributions resulting from oblique shocks and Prandtl-Meyer expansion fans. Given the small radius of 0.01t applied to corners, shock expansion theory still provides a good approximation of locations of oblique shocks and expansions fans. Both theoretical methods assume isentropic flow outside the shock which is a reasonable assumption given that no sources of heating or additional dissipation such as propulsion systems were considered. An illustration showing the application of shock expansion theory to the single flat plate case is shown in Fig. 23. The lift and drag coefficient were calculated using linear theory as follows.

18 4α 4α2 CL = √ (11) CD = √ (12) M 2 − 1 M 2 − 1

In Fig 22, CL matches the trend from linear theory well, especially the sharp edged case which shows least deviation from the linear theory values. Although CD shows a matching trend to linear theory, it under-predicts the values compared to those obtained from CFD. This is discussed further in Section 4.4.1. A comparison of CL from all three methods for the sharp edge geometry is given in Table4. A canonical calculation of CL and CD using the shock expansion method is provided for reference in AppendixE. 5 Table 4: Comparison of CL for the sharp edged plate at M = 3, Re = 9.71 × 10 from CFD, linear theory and shock expansion theory.

α CFD Linear Theory Shock Expansion Theory 0 0.0000 0.0000 0.0000 1 0.0254 0.0247 0.0254 2 0.0508 0.0493 0.0511 3 0.0764 0.0740 0.0768 4 0.1019 0.0987 0.1025 5 0.1273 0.1234 0.1285

4.4.1 Identifying the Sources of Drag Following on from Equation3, at supersonic speeds, drag can be decomposed into three main components; drag due to skin friction, lift-induced drag (also known as vortex drag) and drag due to thickness (also known as zero-lift wave drag) [36, 60].

CD = CD + CD + CD (13) friction lift-induced thickness

In order to validate the results for CD, the sources of drag contributing in Fig. 22b had to be verified. Linear theory in Equation 12 gives the lift induced drag. Since linear theory assumes inviscid flow, a skin friction drag − 1 correction was applied based on an empirical relationship given in [61] such that CD = 0.0284Re 8 . friction This corrected value of linear theory, marked as ‘Linear theory*’ is plotted in Fig. 22b. As seen, there is still a difference in CD between corrected linear theory and the sharp edged plate, which arises from the drag due to thickness. To verify this, two further geometries shown in Fig. 24 (sharper edge and sharpest edge respectively) were simulated and their results are plotted in Fig. 25. c = 38.5 c = 38.5 10 t = 0.675 19 t = 0.675

(a) ‘Sharper’ edge. (b) ‘Sharpest’ Edge.

Figure 24: Edge geometries used for verifying drag due to thickness. Dimensions in mm, not to scale. From Fig. 25a, increasing the sharpness of the LE and TE (over the sharp case in Fig. 18) and hence reducing the thickness, results in CD closer to that of corrected linear theory; this is further emphasised in Fig. 25b. The ‘sharpest’ edge performs even closer to linear theory than the ‘sharper’ edge, indicating that if the plate was inﬁnitesimally thin, it would have CD exactly matching (skin friction corrected) linear theory. To this 19 end, the 2D CFD model for the single plate was deemed to be validated and reliable since results could be replicated with theoretical methods and any discrepancies clearly accounted for.

0.06 0.025

0.05 0.02 0.04

0.03 0.015

0.02 0.01 0.01

0 0.005 0 1 2 3 4 5 0 1 2 3 4 5

(a) CD against α for all tested geometries. (b) CD against α showing the speciﬁc eﬀect of sharpness.

Figure 25: CD against α for all taper geometries and the sharpened edge cases. Computed from CFD simulations and skin friction corrected linear theory.

4.5 Higher Angles of Incidence Often, missile grid fins may experience higher deflection angles as a result of large control inputs or correcting from steeper attitudes. After validating the models for small α using theoretical methods, higher angles up to α = 20° were simulated as recommended in [55]. The results of CL and CD are presented in Fig. 26. The performance of the four tested geometries continues a linear trend in CL. The slight CL benefit seen with a sharp edge in Fig. 22a is further emphasised such that at α = 20°, a sharp edged configuration has a 6.3% higher CL than with a blunt edge. On the contrary, CD in Fig. 26b shows a trend that the performance of each configuration becomes less differentiated as α is increased. This suggests that optimising the edge geometry L is more important for overall D at smaller α values, such as if the grid fins were used purely for stability with little or no deflection.

0.6 0.25

0.5 0.2

0.4 0.15 0.3 0.1 0.2

0.05 0.1

0 0 0 5 10 15 20 0 5 10 15 20

(a) CL against α. (b) CD against α.

Figure 26: CL and CD against α for all four edge taper geometries, up to α = 20°. Linear theory (and skin friction corrected linear theory) also plotted for comparison at small angles.

20 4.6 Effect of Thickness to Chord Ratio As discussed in Section 3.4.1, a useful outcome of the single t 8 plate study is the effect of varying c and so the results from L t CFD simulations for D are presented in Fig. 27. To vary c between 0.0043 and 0.070, c was fixed to 38.5mm and only 6 (t) was altered to avoid having to re-mesh the domain (domain radius = f(c)) and conduct additional convergence studies. The sharp edged case at M = 3, α = 3° was chosen and is plotted 4 in Fig. 27; the (×) point shows the chosen geometry of the R-77 t as identified in Table3. It is clear that as c is decreased and the 2 L 0 0.02 0.04 0.06 0.08 plate made more slender, D shows significant improvement. For example, halving the thickness from 0.675mm to 0.3375mm L causes a 26% increase in D ; the performance is more sensitive L t t Figure 27: D against varying c . to changes at smaller c indicated by the distribution showing a decaying trend.

4.7 Eﬀect of Removing Leading or Trailing Edge Taper The results presented in Fig. 22 assume the same edge geometry on both the LE and TE. The eﬀect of the removing edge taper only on one of these was explored and the results are presented in Fig. 28.

0.14 0.045

0.12 0.04

0.1 0.035

0.08 0.03

0.06 0.025

0.04 0.02

0.02 0.015

0 0.01 0 1 2 3 4 5 0 1 2 3 4 5

(a) CL against α. (b) CD against α. 6 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0 1 2 3 4 5

L (c) D against α. L Figure 28: Plots of CL, CD and D against α with the eﬀect of removing LE and/or TE taper.

21 By comparing the performance of one edge taper with that of both edges tapered, the leading edge has a more significant contribution to achieving an optimum performance. Considering CD, a sharp LE blunt TE has only a small increase in drag from a sharp LE sharp TE configuration. However, when the LE is blunt and TE is sharp, the drag produced is comparable to that of a blunt LE blunt TE case, which is a much greater. This trend is also observed in CL although less prominent; removing TE taper has little degradation of CL L however a much poorer performance is seen with LE taper removed. Comparing the D performance in Fig. L 28c, a sharp LE blunt TE has a 109% higher D at α = 5° than a blunt LE sharp TE geometry. The difference diminishes as α → 0°.

4.8 Summary of Findings from a Single Flat Plate Through CFD simulations of the single flat plate configuration, the two main objectives highlighted in Section t 3.4.1, of understanding the effect of edge taper geometry and understanding the effect of c on the overall performance, have been achieved. At small angles of incidence, the choice of LE and TE taper geometry L has little effect on CL however a significant difference is seen with CD indicating that overall D is sensitive to the type of edge geometry. Indeed, a sharp edged flat plate produces the lowest drag and by comparing the results from CFD with linear theory (skin friction corrected) and validating the components of this drag force, a minimum drag single flat plate would have the largest possible taper distance (xt)[7]. A Busemann grid fin, composed of a Busemann biplane section is explored in Section 5 of [62]; by taking advantage of increased sharpness from a Busemann section and the benefit of shock wave cancellation, this configuration is proposed to have a significant drag reduction (at M = 3) as compared with a standard sectioned grid fin [62]. This agrees with the ‘sharpest’ plate, most representing the Busemann section, having a notably lower CD as L t compared to a blunt plate. Optimum D is achieved by then also minimising c . For produciblity, this means t that for a minimum drag single flat plate, a slender geometry (small c ) with a sharp edge would be ideal. Practically, a perfectly sharp edge and infinitely slender geometry is impossible and hence a compromise in performance is required when producibility is considered; a 2/3 edge still shows a significant improvement in performance from a blunt edge geometry and would likely be more achievable to produce.

The importance of the LE has been demonstrated for a single flat plate such that the LE contributes more to overall drag as compared with the TE. Consequently, it may be beneficial to only require the LE to have a precise sharpness with a blunt TE which would only bring a marginal drag penalty. At higher angles of incidence, the difference in performance between the types of edge taper geometries begins to diminish. This suggests that if larger α is of more importance, the type of edge geometry has a smaller impact on performance. Although the impact of the TE is minimal on CL and CD, it is likely of interest for aeroacoustics. Since a TE (especially at high M) energetically sheds vortices, the TE taper will affect the frequency of these and hence the acoustic profile of the grid fin. The oscillation frequency, characterised by the Strouhal number, will depend on the TE taper geometry. Such fluid-structure interactions are important considerations to avoid resonance with other components which could be catastrophic. Nevertheless, these considerations are beyond the scope of this project and would typically be explored at a later design stage.

t Although the performance of an isolated single flat plate with varying edge taper geometries and c ratios can now be understood, it does not yet capture all of the (2D) effects present in grid fins. As seen in Fig.8, a vital consideration is the interaction of shock waves within the individual cells. As discussed in Section2, transonic performance and the onset of choked flow often leads to huge increases in drag which typically supersede drag contributions from edge taper geometries. Therefore, the next stage of the analysis develops from local single plate geometry to the global geometry of two flat plates to make further suggestions on producing an optimum performance grid fin with minimum drag.

22 5 Two Flat Plates 5.1 Simulation Setup The 2D simulation for the single flat plate was extended to incorporate the two flat plates as shown in Fig. 29 and based on the dimensions given in Table3. Although nearly identical to the single plate, a full breakdown of CFD program settings are given for reference in Appendix F.2. The circular domain of radius 10c as shown in Fig. 29a was maintained, with a polygonal mesh in the far field and a prism layer mesh on each of the flat plates. Given that the span, s (the spacing of the two plates) is much smaller than the radius of the domain, a mesh convergence study was not considered necessary especially since similar M, α and Re were simulated.

s 10c

(a) An illustration of the simulation setup (Not to scale). (b) Mesh around the two plates in the circular domain.

Figure 29: The simulation setup used for CFD analysis of the two flat plates. Following the motivation behind the three stage approach, the LE and TE taper geometry was fixed to a sharp edge as given in Fig. 18. This was so that the effects of edge geometry were isolated from the effects that were c unique to the two plate simulation; s ratio and the shock interaction between the two plates. Nevertheless, M and α were varied in similar ranges to the single plate case to observe how shock interactions and consequent pressure distributions along the shock train were dependent on the incoming flow conditions.

5.2 Small Angles of Incidence

L The effect of varying chord to span ratios on CL, CD and D for small α = 0, 3, 5° is shown in Fig. 30. Since the flat plates act like symmetrical aerofoils, no lift is generated at α = 0°. However, even at α = 0° for c s <≈ 10 a non-zero CD is observed. In this range of chord to span ratios, there is no interaction of oblique shocks between the two flat plates and this is a zero lift drag coefficient. Comparing this value with that for a sharp edged single flat plate given in Fig. 22b, they are of the same magnitude. This is a source of validating the two plate model and gives confidence in the reliability of subsequent results (it should be noted that the CD for the two flat plates is normalised with a reference area twice that of the single plate configuration). For c c α = 0°, as s is increased, a reduction in CD is observed with a minimum at s = 25. This can be attributed to the shock wave cancellation between the two plates generating lower drag much like the Busemann biplane c concept given in [62]. Beyond this, a sharp rise in drag is seen as s is increased to 30. This correlates with a standoff bow shock and choked flow within the cell. Explanation of the flow physics behind these trends is given in more detail in Section 5.2.1.

As the angle of incidence is increased, the overall changes in CL and CD are more pronounced and trends c are shifted to the left on plots such that corresponding features occur at lower s values. In the α = 5° case, c the initial drop in CD between 2 ≤ s ≤ 4 causes a 24.1% reduction in drag whereas in the α = 3° case this c is a 18.6% reduction. However, more significant is the fact that at lower chord to span ratios s < 4 , the L effects of CL dominate due to larger changes in magnitude than CD and hence, D is mainly influenced by L c lift rather than drag. D then varies periodically for 5 < s < 20 depending on the structure of the shock c wave reflections and their impingements between the two plates. As s increases further beyond this, to the 23 L point where the flow chokes, the effects of drag then dominate and cause D to degrade. Here, the effects of CL are small and after the periodicity decays, little change in CL occurs as seen in Fig. 30a. Therefore, L the dominating influence of lift or drag in overall performance D of the two plate configuration is highly c sensitive to the chosen s . It is useful to note that for the geometry of the Vympel R-77 grid fin as presented in c c Table3, s = 0.86 and hence flow behaviour can be assumed similar to that at s = 1 due to the lack of shock interaction between the plates. 0.14 0.026

0.12 0.024

0.1 0.022 0.02 0.08 0.018 0.06 0.016 0.04 0.014 0.02 0.012 0 0.01 -0.02 0.008 0 5 10 15 20 25 30 0 5 10 15 20 25 30

c c (a) CL against s . (b) CD against s . 6 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 -0.5 0 5 10 15 20 25 30

L c (c) D against s . L c Figure 30: Plots of CL, CD and D against varying s for small angles of incidence (α = 0, 3, 5°).

5.2.1 Detailed Analysis of Varying Chord to Span Ratio

c c It was predicted that at low s and especially at s = 0.86 for the R-77 geometry, no shock interaction would occur between the top and bottom plate. This can be explained by a crude approximation of the plates being infinitesimally thin and sharp such that the flow deflection angles are small and a Mach wave is assumed −1 1 to have formed from each LE. Based on this, for M∞ = 3 and calculating the Mach angle µ = sin M , c s = 2.83 is required before the Mach wave from one plate begins to impinge on the other. From obtaining the c CFD results in Fig. 32 (at M = 3, α = 5°) it is seen that at s = 1 (Fig. 32a), the oblique shocks indeed do not interact over the entire chord length of the plates. This is supplemented by the plot of CP along the internal c surfaces of the two plates in Fig. 32b showing a uniform distribution. At s = 2 (Fig. 32c), the oblique shocks begin to interact and the corresponding CP profile shows an asymmetrical distribution between the plates. This agrees with the prediction since in the real model, a small deflection angle is no longer true and hence a large pressure wave and oblique shock is present from the LE rather than a Mach wave, resulting in a smaller c s for shock interaction to occur than predicted by the Mach angle calculation. To explain the flow physics c and the structure of the ‘shock train’ at intermediate s , a general model illustrated in Fig. 31 (for a truncated forward section) is useful.

24 SBLIs top plate with sharp LE and TE

+ + + + + + ------+ + high pressure low pressure oblique shock compressions ...... small bow shock expansions due to corner radius low pressure high pressure ------+ + + + + + - -

bottom plate with sharp LE and TE

Figure 31: An illustration of the flow structure between (a forward section of) two sharp edged flat plates at c an intermediate s (where flow choking has not occured) and α = 0°. At M = 3, α = 0° and supplemented by the CFD results in Fig. 32, several characteristic flow features can be identified. For a corner radius of 0.01t applied to the LE and TE, a small local bow shock is present which is then followed by an oblique shock as the flow is deflected along the edge taper. At the end of the taper length, a Prandtl-Meyer expansion fan is observed. For α = 0° this is symmetrical for the second plate and results in the oblique shocks and expansions crossing at the centreline. When the expansions and shocks impinge on the opposite plate, a SBLI must occur as the boundary layer along the plate is still growing. c Although the SBLI effects were not investigated in detail, from looking at CFD results such as for s = 4.5 in Fig. 32e, a small local separation bubble is seen with the pressure contours. Following the reflection of the expansions, compression fans are observed and once again impinge on the opposite plate. This repeated expansion-compression trend is seen along the whole length of the plate, with the original oblique shock reducing in strength and being smeared by the fans as it is reflected throughout. This ‘shock train’ formed within the two plate configuration is analogous to that observed in the exhaust of an under expanded jet with a repeating cell unit [63]. Although simplified and truncated for only a forward portion of the flat plates in Fig. 31, this behaviour continues for the full chord length. A useful validation of this can be made with the results presented for M = 2.5 in [64]; a similar CP contour plot with a series of expansion and compression fans is captured and the effect of varying the leading edge taper angle is also investigated. The shock interaction within the web of a grid fin (although in 3D) is also discussed in [19]. Here, Theerthamalai et al. explore how variation in α causes eventual shock detachment from the inner grid fin surfaces and hence attribute this to an increase in CD. It should be noted however that this experimental behaviour was carried out on a complete grid fin with a different geometry to that of the Vympel R-77.

-0.2

0.2

0.4

0.6 0 0.2 0.4 0.6 0.8 1

c C c = 1 (a) Absolute pressure for s = 1. (b) P distribution for s .

-0.2

0.2

0.4

0.6 0 0.2 0.4 0.6 0.8 1

c (c) Absolute pressure for s = 2. c (d) CP distribution for s = 2. 25 -0.2

0.2

0.4

0.6 0 0.2 0.4 0.6 0.8 1

c (e) Absolute pressure for s = 4.5. c (f) CP distribution for s = 4.5. -0.2

0.2

0.4

0.6 0 0.2 0.4 0.6 0.8 1

c (g) Absolute pressure for s = 8. c (h) CP distribution for s = 8. -0.2 0.2 0.6 1 1.4 1.8 0 0.2 0.4 0.6 0.8 1

c = 30 (i) Absolute pressure for s . c (j) CP distribution for s = 30.

Figure 32: Absolute pressure ﬁelds from CFD and corresponding CP distributions on the internal surfaces of c the two plates, shown for s = 1, 2, 4.5, 8, 30. All shown for M = 3, α = 5°

From looking at the flow structure in Fig. 31 and corresponding CP plots and contours in Fig. 32, the trends L in CL, CD and D observed in Fig. 30 can be justified. In general, the expansion and compression fans cause alternating regions of high and low pressure within the cell. By returning to derivation of lift coefficient 1 R x=TE from integrating the difference in pressure distributions, CL = c LE ∆CP dx, an equal distribution of high and low CP results in CL = 0 as is seen with the α = 0° case. As the angle of incidence is increased, this distribution of CP is no longer symmetric between the two plates and thus causes the measured force coefficients to vary depending on the shock structure; this explains the non-zero CL with α > 0°. When a L c drop in D is seen for 2 ≤ s ≤ 4, as this is dominated by the changes in CL, this is likely due to a change in the number of shock reflections and hence high and low pressure regions on the inner surfaces of the two plates. If there is a greater distribution of low pressure such as on the lower plate inner surface, this overcomes the effects of the (smaller) high pressure regions on the top plate causing a smaller overall Σ∆CP and hence c smaller CL. As s increases, so do the number of shock reflection regions. This is seen by comparing the c c cases of s = 4.5 and 8 in Fig. 32. At s = 4.5, only one distinct high pressure region exists on the top plate inner surface whereas two exist on the bottom plate inner surface (this is clearly identified in the CP plot of c Fig. 32f). At s = 8, the top plate inner surface has three high pressure regions while the bottom plate inner surface has four. Therefore, such variation in the number of high and low pressure regions between the top c L and bottom plates with increasing s causes the periodicity seen with the D behaviour.

As identified in Chapter2 and particularly Table1, off-design transonic behaviour is of importance to the designer as it often results in a bow shock in front of the grid fin causing a rapid increase in overall drag and a mass flow spillage outside the internal cells. Alternatively, even in supersonic regimes, a similar bow shock 26 c phenomenon is observed by increasing s to the point where the flow between the plates becomes choked, causing an unstart behaviour and hence the ejection of a (bow) shock. This behaviour is characterised by reducing the inlet cross sectional area in [37] which is analogous to reducing the span s and hence increasing c c s . By looking at the increase in drag from Fig. 30b after s = 25 it is clear that the shock is no longer L ‘swallowed’ between the plates which causes the drag rise and D degradation. From the CFD results presented c in Fig. 32i, the flow becomes fully choked with a standoff bow shock at s = 30. The corresponding CP plot in Fig. 32j shows that an identical, high pressure distribution is present on the inner surfaces of both plates and this indicates that the ‘shock train’ structure is no longer present. In addition to the increase in drag force, choked flow will also cause a loss in fin effectiveness such that forces on the hinge and normal forces against the missile body will increase [37]. Although not considered in this project, future design iterations and experimental studies on a complete fin accounting for such forces may benefit from a choked flow force correction factor presented in [37] and given in Eq. 14 which can quantify this reduction in effectiveness. r γ+1 2 γ−1 Aexit γP0ρ0 γ+1 Correction Factor = (14) Acapturedρ∞V∞ c For choking resulting from increasing s , an analogy to the intake of a ramjet can be considered where off design performance and an ejected shock causes an unstart behaviour and is controlled by increasing the throat area to ‘swallow’ the shock, ideally holding it aft of the throat [65]. To calculate a theoretical prediction c for the s at the onset of choked flow, as is also given in [45], a combination of isentropic and normal shock flow relations can be used [59, 65]. In the following, the conditions (•)1 are directly behind the ejected normal shock, (•)2 is at the cross section (with dimension s) through the leading edges and (•)T is the cross section at the end of the LE taper with dimension (s − t). As before, (•)∞ are free stream conditions identified in Table2. s 2 (γ − 1)M∞ + 2 M1 = 2 (15) 2γM∞ − (γ − 1)

−(γ+1) γ+1 2(γ−1) γ−1 2 2(γ−1) A1 γ + 1 1 + M = 2 1 (16) AT 2 M1 s At the onset of unstart, A1 = A2 and hence M1 = M2. From this, we get that s−t = 1.39 and therefore c s = 16. This can also be computed using the Kantrowitz limit which is also derived from isentropic relations: −1 2 −0.5 2 γ−1 s (γ − 1)M∞ + 2 2γM∞ − (γ − 1) = 2 2 (17) s − t (γ + 1)M∞ (γ + 1)M∞ c This also gives a s = 16 for the onset of unstart within the two plates. However, by looking at the CFD results c presented thus far, it is seen that for small α, s ≈ 25 is required for the onset of choked flow with fully choked c flow only occurring at s ≈ 30. Although performing more simulations within this range would improve c the accuracy of predicting the exact s for choking, a difference between CFD and the theoretical methods is clearly present. The difference is attributed to the assumption in theory made of a unstarted case, with a single normal shock which is held directly aft of end of the LE taper as opposed to the complex ‘shock train’ structure. Also, boundary layer effects and SBLIs seen in Figs. 31 and 32 are not accounted for. Therefore, c the theoretical method is found to under-predict the value of s at which choking begins. A similar comparison of CFD against isentropic formulae and the Kantrowitz limit is presented in [66] for a biplane configuration which has similarity with the two flat plates at α = 0°. Regardless, it clear that for M = 3, flow choking and c unstart is not a concern at any realistic s and certainly not for the R-77 geometry.

5.3 Higher Angles of Incidence

As per the recommendations in [55], higher incidence angles were also explored. The results for CL, CD and L D are presented in Fig. 33 below. 27 0.6 0.22 6 0.2 5.5 0.5 0.18 5 4.5 0.16 0.4 4 0.14 3.5 0.12 3 0.3 0.1 2.5 0.08 2 0.2 1.5 0.06 1 0.1 0.04 0.5 0.02 0 0 0 -0.5 0 5 10 15 20 25 30 0 5 10 15 20 25 30 0 5 10 15 20 25 30

c c L c (a) CL against s . (b) CD against s . (c) D against s . L c Figure 33: Plots of CL, CD and D against varying s for higher angles of incidence (α = 0, 3, 5, 10, 20°). L At higher α, an overall degradation in performance is observed. The D plot in Fig. 33c shows that optimum c L performance across the majority of realistic s values occurs for α = 5°. When α = 10°, D values are lower c for the entire s range. Further reduction from this occurs for α = 20°. This indicates that for the two plate configuration, an optimum angle of incidence would lie between 5° and 10° at the tested flow conditions. L c L Additionally, the periodicity observed in D at intermediate s begins to attenuate; a near flat D response is c seen at α = 20° across the entire s range. This shows that along with an optimum performance for α = 5°, c there is also a peak in this periodicity and thus an increased sensitivity to s . To this end, it can be deduced L c that at an α which delivers optimum performance for the two plate configuration, D is highly sensitive to s ; L at off-design points, this sensitivity then reduces with a flattening of the D response.

While for the α range presented in Fig. 30 the effects L of lift dominate the D behaviour, as α is increased, the L effects of drag begin to dominate D . This is seen by the results given in Fig. 33b which show a significant increase in CD for α = 10° and 20° as compared with smaller incidences. Also, the drag produced at small c s supersedes the drag even in choked flow. This can be explained by observing the pressure distribution in the wake and behind the oblique shocks as shown in Fig. 34. Comparing this with Fig. 32c, due to the large α, the wake produced from the top plate interacts with the majority of the inner surface of the c Figure 34: Absolute pressure for s = 2 at M = bottom plate meaning that the oblique shock effects 3, α = 20°. are no longer dominant. When there is no internal c shock interaction (i.e. at low s ) yet this high pressure wake is present, it produces the high drag force. Such wake behaviour does not exist at lower α and so high c c drag at low s is only present for large incidences. Once the plates are closer, with s increased, the repeated shock structure returns and the wake effects become less prominent.

5.4 Mach Number Variation Although the results presented so far are for M = 3, variation of Mach number was also explored as per [55] and in relevance to the typical flight regimes of a grid finned missile. As mentioned previously, transonic behaviour typically produces the poorest performance and quantifying this for the two plate configuration allows comparison against design point performance. From Fig. 35, a two plate configuration has a peak in CD around M = 0.9, as expected. Nevertheless, the changes in CL at subsonic M are greater in magnitude 28 L than those in CD and therefore, the optimum D performance for both a single and two plates is observed in the subsonic regime. As M becomes supersonic, above M = 1.4, the performance of the two flat plates is identical to that of a single plate, which is typically well understood. The interaction of shock structures beyond M = 1.4 causes the two plates to perform like a single flat plate. However, it must be noted that a periodic condition (to represent a full grid fin) is not applied to the two plate model discussed in this chapter and hence aerodynamic forces include contributions from outer surfaces of the plates. Therefore, this study should be treated independently to a grid fin configuration and only aids to supplement understanding of the shock structure interaction in the internal region between the plates. 0.8 0.06 18 0.055 0.7 16 0.05 0.6 0.045 14 0.5 0.04 12 0.4 0.035 10 0.3 0.03 0.025 8 0.2 0.02 6 0.1 0.015 0 0.01 4 0.6 1 1.4 1.8 2.2 2.6 3 0.6 1 1.4 1.8 2.2 2.6 3 0.6 1 1.4 1.8 2.2 2.6 3

L (a) CL against M. (b) CD against M. (c) D against M.

L Figure 35: Plots of CL, CD and D against varying Mach number (0.6 ≤ M ≤ 3) for a single ﬂat plate and two ﬂat plates in 2D.

5.5 Summary and Limitations of the Two Flat Plates Analysis

c The objectives for the stage 2 analysis set out in Section 3.4.2 on how s affects performance (CL and CD) c of the two flat plates and consequently, how high s may result in flow choking with high drag can now be summarised. After extending the validated single plate CFD model to two plates, it was found that optimum L performance (for the R-77 geometry) occurs at typically low α. Here, D shows a periodicity for intermediate c c c s ; low s is dominated by the effects of lift whereas high s is dominated by drag arising from choked flow and a standoff bow shock. By observing the general flow structure and corresponding pressure distribution, the repeating shock behaviour within the inner surfaces creates differing regions of high and low pressure, causing L L the D periodicity. Extending this to higher α, performance is seen to degrade and poor D is characterised by c the wake generated behind the plates. At low α, flow is fully choked for s ≈ 30 and can also be estimated using theoretical methods. Mach number variation shows performance similar to a single flat plate for M > 1.4, and poor transonic performance around M = 0.9. An optimum is seen in the subsonic regime.

The two flat plate CFD model has limitations when predicting flow behaviour for grid fins. Since a periodic c condition is not applied, forces from free outer surfaces are also measured. Hence, trends seen for s sensitivity to particular α values may not be representative of a full grid fin and thus, the two plate model is more powerful as an independent study of the shock structure within a single grid fin cell and how it is affected by c s . To fully understand flow behaviour for a (unit) grid fin, a 3D simulation is necessary which can explore shock patterns in 3D as well as the effect on performance from global geometrical variations (informing c t produciblity) rather than the local variations, such as s , c and edge taper geometry carried out up to this point in the 2D simulations and analyses.

29 6 Unit Grid Fin

6.1 Simulation Setup The configuration and dimensions shown in Fig. 19 were implemented using the methods given [53] with a sharp edged UGF. In order to accurately capture flow behaviour near the UGF (and in its immediate wake), yet avoid too fine a mesh which would add computational cost, a volume of refinement (VOR) was placed within the ‘tunnel’ as shown in Fig. 36. For an origin placed centrally in the UGF, the VOR extended 50mm forward and 150mm behind this point and was configured to have a finer volume and surface mesh than the far field. The surface mesh inside the VOR and around the UGF is shown in Fig. 37.

Figure 36: Volume mesh inside the tunnel domain, with a volume of refinement around the UGF. Along with a surface mesher, a trimmed cell mesher in the fluid volume and a prism layer mesher on the UGF for boundary layer effects were used. Based on the recommendations in [56, 57] a trimmer mesh was chosen over the tetrahed- ral mesh used in [53] primarily for its speed and stability in turbulent simulations. Although a polyhedral mesh would also suf- fice for capability and speed, a Figure 37: UGF surface mesh; oriented with flow left to right. trimmer mesh better suited the geometry and provided more stable results. Contrary to the 2D simulations, a Spalart-Allmaras (SA) turbulence model was used for the 3D simulation. This was based on the recommendation in [53] and its advantage of being a one-equation transport model which is suited to external aerodynamic flows for the Re concerned. The SA model was also less computationally expensive than the K − ω SST and provided faster solution convergence [58]. Solution convergence was observed using the same methods as for the 2D simulations as well as monitoring the wall y+ to ensure boundary layer flow physics had been fully captured; histogram plots of these are given for reference in Appendix G.3.

As shown in Fig. 19, periodic (touching) pairs of wall boundaries were used within the ‘tunnel’ to simulate an infinitely large grid fin with repeating units of the UGF [55]. When setting up the periodic conditions in STAR-CCM+, several difficulties were faced and required manipulation of the mesh parameters to ensure 30 that the opposite pairs of walls had a successful conformal match (i.e. the same number and size of cells so that they could act as periodic boundaries). To resolve this, some of the rectifications involved the use of an automated mesh rather than manually setting mesh parameters to improve the conformal match; setting inflow and outflow as free stream walls rather than a pressure inlet / outlet; removing the use of a surface wrapper which caused mesh deformation near the walls (hence no conformal match); the use of purely translational periodic walls (i.e. no rotational periodicity). A measure of successful (conformal) periodic walls besides contour plots was that the number of volume mesh cell boundaries on opposing walls was identical [56]. In addition to these settings, a three-dimensional viscous, turbulent, RANS model with coupled flow solver was selected. The coupled flow model was set with a 2nd order implicit discretisation for numerical stability with 20 a courant number λ = M [58]. Full details of CFD parameters are given for reference in Appendix F.3.

6.2 Mesh Convergence Similar to the 2D models, it was important to conduct a mesh convergence study to ensure that the results L were independent of the mesh and hence, reliable. The results for CL, CD and D for varying mesh size are shown in Fig. 38. 0.1315 0.035 3.86

0.131 3.84 0.1305 0.0345 3.82 0.13 3.8 0.1295 0.034 3.78 0.129

0.1285 0.0335 3.76 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 106 106 106 (a) Lift coeﬃcient. (b) Drag coeﬃcient. (c) Lift to drag ratio.

L Figure 38: Plots of CL, CD and D convergence against varying number of volume mesh cells within the ‘tunnel’ domain around the UGF. Simulated at M = 3, α = 10°. Using the same approach described in Section 4.2, the chosen number of cells (×) was a trade-off between computational cost and solution accuracy. As seen in Fig. 38a, the lift and drag coefficients begin to converge 6 L 6 after 2 × 10 cells. From looking at D in Fig. 38c, results after 2 × 10 cells have variations that are smaller in magnitude and the amplitude of these begins to decrease as the number of cells approaches 3 × 106. Ultimately, a configuration with 2, 749, 079 cells was chosen as any further increases in cell number led to notable increases in computational resources without significant improvement in result accuracy. Also, given that the length of the domain (20c) was much larger than the dimension of the UGF (c), increases in the number of cells mostly affected the far field region. Therefore, the variations plotted in Fig. 38 used a systematic approach of first increasing the cell number within the VOR (until convergence) and then keeping this fixed (at 18% of the base size) to then reduce the global base size (which increased number of cells everywhere).

6.3 Aerodynamic Forces Once the mesh parameters for the UGF CFD model were fixed, a range of simulations were run to understand the aerodynamic performance at different flight conditions. By primarily varying M and α for the range L suggested in [55], CL, CD and D are plotted in Fig. 39.

31 0.5 0.2

0.4 0.15

0.3 0.1 0.2

0.05 0.1

0 0 0 5 10 15 20 0 5 10 15 20

(a) CL against α. (b) CD against α. 5

-1 0 5 10 15 20

L (c) D against α. L Figure 39: Plots of CL, CD and D against α for 0° ≤ α ≤ 20° from the 3D UGF simulations.

For the case of M = 3, a linear trend in CL is observed as the angle of incidence is increased. Both the CL and CD trends at M = 3 match those from the 2D simulations well and hence, indicate that the 2D simulation may be a valuable tool especially in preliminary design given its significantly lower computational cost. As M is varied across subsonic, transonic and low supersonic values, the CD trends are replicated, with M = 0.9 and 1.5 having the highest (and near identical) drag. However, at M = 1.5 the lift generated across L the entire α range is highest and contributes to having a superior D performance. It can be deduced from Fig. 39a that the lift curve slope, atleast in supersonic cases, is inversely proportional to the Mach number, ∂CL 1 L ∂α ∝ M . From the D performance in Fig. 39c, a peak in performance is seen for all Mach numbers between L α = 5° and α = 10°. This agrees with the D performance from the two plate configuration in Fig. 33c. L The best overall D performance is seen with M = 1.5 while the transonic M = 0.9 performs the worst, as expected. To analyse this further, the Mach number contours plotted in Fig. 40 are useful. Following the discussion for the two flat plate simulations, at M = 3 for the representative R-77 geometry, no interaction of the shock structure is seen between adjacent grid fin cells. Therefore, the contours in Fig. 40d show an uninterrupted repeating cell in the wake comprised of expansions and compressions with lowest drag seen for M = 3 in Fig. 39b. In this 3D model, the interaction between the oblique shock from the vertical UGF plate LE with that from the horizontal plate LE is also of importance. At M = 3 the oblique shock angle is too small and no such interaction occurs. However, for smaller Mach numbers the oblique shocks (and hence subsequent expansion/compression fans on the plate surface) begin to interact within each quadrant of the UGF and cause an increase in drag. In Fig. 40, when such interaction of LE oblique shocks occurs, it causes a smearing of the contours evident from looking at the M = 3 contours against those of smaller M cases. This smearing effect can also be seen on the absolute pressure contours in Fig. 41. At M = 3, the contours 32 are more closely spaced (indicating the lack of such smearing) and in all four quadrants of the visualisation plane there is a majority high pressure distribution. At smaller M, the contours smear and spread out causing a greater pressure difference between upper and lower quadrants. Looking at M = 1.5, a particularly distinct pressure difference (with mostly low pressure in upper quadrants and high pressure in lower quadrants) is L observed and this contributes to the superior CL and therefore optimum D for M = 1.5.

(a) M∞ = 0.9, α = 5°. (b) M∞ = 1.5, α = 5°.

Figure 40: Mach number contour plots for different free-stream Mach numbers at α = 5°. Shown in a lengthwise plane within the ‘tunnel’, 5mm offset from the centre of UGF with free-stream flow is left to right.

(a) M∞ = 1.5 (b) M∞ = 2 (c) M∞ = 3

Figure 41: Absolute pressure contours for M = 1.5, 2, 3 shown on a widthwise plane (s × s) centred within the UGF and tunnel domain. All at α = 5°. 6.3.1 Validation Both qualitative and quantitative validation methods were used. A majority of authors in literature investigate grid fins with different geometries to the R-77 or measure forces with a full missile body. Additionally, hinge forces or pitching moments rather than lift and drag, which were the main foci of this project, are usually documented. Therefore, qualitative (visual) comparison of Mach and CP contour plots such as those in [18, 40, 50, 52] still allowed a way of ensuring the simulation results were sensible. The best source of visual validation was against Dikbas et al. [53]. Although a full missile body interference correction as well as a different UGF geometry is considered, Figure 13 of [53] showed good comparison with the contour plots presented in this chapter; the profile of shocks, Mach number distributions and periodic boundaries could be validated. Additionally, studies such as [44–46] allowed qualitative validation of the transonic behaviour. Quantitative comparison in Fig. 42 with the experimental results of Miller et al. [48] could be done since similar Re and M = 2.5 were used. By comparing the CD against M plots for the ‘F2’ frame type (sharp edged configuration) in Fig. 42b with Fig. 42a, the trends match well, with both having a peak at M = 0.9. 2 Miller et al. also plots CD against CL at M = 2.5; comparing this with the CFD results at M = 2 in Fig. 33 42c again matches the trend well; the error in values can be attributed primarily to differing geometry between the R-77 and that used in [48].

0.035

0.03

0.025

0.02

0.015

0.01 0.5 1 1.5 2 2.5 3

(b) CD against M at α = 0° from [48] (adap- C M α = 0 (a) D against at ° from CFD ted). 0.16

0.12

0.08

0.04

0 0 0.04 0.08 0.12 0.16

2 2 (d) CD against CL from [48] (c) CD against CL from CFD

Figure 42: Validation of results against those presented by Miller et al. [48]. 6.3.2 Transonic Performance As identified in Chapter2, transonic performance is of interest as grid fins on a missile will usually 5 transition through this regime while accelerating to 4.5 the design Mach number [45]. In Fig. 40a, a su- 4 personic pocket is seen on the upper surface of the horizontal plate of the UGF. Investigating this fur- 3.5 ther, the Mach number contours for subsonic and 3 transonic regimes in Fig 44 show that the supersonic 2.5 pocket grows as the Mach number tends to transonic values. The supersonic pocket is analogous to that 2 seen on the upper surface of a supercritical aerofoil; 1.5 a sonic line encompasses a series of expansions and 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 compressions with a near-normal shock wave at the rear [63]. The sonic line and normal shock at the L Figure 43: against Mach number for the high sub- rear of the pocket are clearly captured in the contour D sonic, transonic and low supersonic regime. plots, providing another means of validating the accuracy of the CFD model. This can be associated with the critical Mach number Mcr such that for the CFD results, an Mcr ≈ 0.6 is observed when local flow first becomes supersonic. This correlates closely with the Mcr ≈ 0.75 noted in [44, 45]. After the L critical mach number is reached, drag begins to increase, causing a drop in D ; this is again analogous to drag L divergence in supercritical aerofoils [63]. Supplementing Fig. 39c, Fig. 43 shows that poorest D performance occurs at M = 0.9 with a rapid improvement as flow becomes supersonic. This agrees well with literature; 34 grid fins typically perform best in subsonic and supersonic regimes. As M → 0.9, both a bow shock and supersonic pocket grow, along with the normal shock at the rear becoming inclined more towards the TE. Even at M = 1.5, evidence of this near-normal shock is seen however, it is a weak shock and thus still permits L optimum D performance.

(a) M∞ = 0.6, α = 5°. (b) M∞ = 0.7, α = 5°.

Figure 44: Mach number contours around the horizontal plate of the UGF at M∞ = 0.6, 0.7, 0.8, 1.

6.3.3 Vortical Structures For future design stages, the aerodynamic interaction of the UGF with the outer grid fin frame 0.015 and the missile body is pertinent. The vortices shed from the UGF, shown in Fig. 47 (from the Q- 0.01 criterion), would directly interact with such com- 0.005 ponents and thus their identification is useful in designing an overall minimum drag grid fin. A vor- 0 tex peeling behaviour is seen at the central joint 0 0.2 0.4 0.6 0.8 1 of the horizontal and vertical UGF plates and this vortex is amplified in the wake as it ‘unrolls’ and rotates outwards from the centre. Additionally, Figure 45: Skin friction coefficient along the vertical symmetrical vortices are observed at ≈ 0.4c which UGF plate. Take at 5mm above the centre for M = rotate away from the UGF surface. The behaviour 3, α = 5°. of these is similar to vortices on a delta winged aircraft; a vortex sheet remains attached until the TE. The oblique shocks seen in Fig. 40 also correlate with the shape of the vortices and hence indicate that the vortices are sensitive to free-stream flight conditions. Wall shear stress (τw) vectors in Fig. 46 show that towards the TE as the boundary layer develops, overall τw and hence Cf reduces. A plot of Cf taken at 5mm above the centre on the vertical plate is shown in Fig. 45. x A drop in Cf is seen across the shock and expansion fan locations and this is evident at c ≈ 0.3 in Fig. 45 and in the vector plot. The highest τw is observed along the LE and particularly on the upper surface given that α > 0°. Therefore, for producibility and future design iterations, these regions of high skin friction contribute to producing the most drag. When material and process selection is considered, these drag sources should be focused upon such as to minimise surface roughness and ensure geometry features are produced accurately. 35 Figure 46: τw vectors at M = 3, α = 5°. Figure 47: Vortices from the Q-criterion at M = 3, α = 5°.

6.4 Effects of 3D Geometry Modifications As mentioned in Section 3.4.3, 3D geometry modifications and their aerodynamic effects were a key objective of the UGF study and can inform producibility in a similar manner to the investigation of LE against TE sharpness in the 2D analysis. Therefore, as shown in Fig. 48, various geometrical modifications to the TE were investigated in order to understand their effects on grid fin performance and hence quantify their importance in producibility considerations. This is particularly useful for packaging concerns since global TE modifications may enable specific shaping of the grid fin such as for internal weapon bays; understanding the aerodynamic trade-off of these is essential. Outwards taper Inwards taper Brazed joint ‘Taper X out’ ‘Taper X’ ‘Braze X’ X X X

Scalloped fine Scalloped coarse ‘ ‘ ALL AT Cosine mult.’ Cosine single’ TRAILING EDGE

Figure 48: Geometrical modiﬁcations applied to the TE of the UGF. Top view of TE shown.

The variations shown in Fig. 48 are representative of the types but are not exhaustive of dimensional changes; for example, the taper was investigated for different values ofX as is subsequently presented. Five basic types of modifications were simulated; a braze along the central joint of the UGF of varying sizes; inwards and outwards tapers, and a coarse/fine scalloped edge (formed of cosine functions). Full drawings of each are provided for reference in AppendixH. It should be noted that the italicised labels in Fig. 48 are used for plots of results (i.e. Braze 3 indicates a brazed joint of 3mm).

6.4.1 Aerodynamic Performance As given in Fig. 49, the results from each of the TE types were compared with an unmodified UGF at L M = 3, α = 3° and hence percentage differences in CL, CD and D plotted. Although the scalloped (cosine) edges give up to a 1.5% CL increase, this is outweighed by their increase in drag, as is the case for all other TE L variations. Therefore, all of the modifications resulted in poorer overall D performance as compared with an L unmodified UGF. D was most sensitive to the effects of the brazed joints and these caused the biggest drop in performance. Typically, a brazing joint of up to 3mm is common on manufacturing of such components and L in this regard, causes a dramatic 33% decrease in D mainly contributed by a 49% increase in drag. Although reducing the size of the brazed joint improves performance, brazing remains the biggest source of drag out 36 of the tested variations at M = 3. This is likely because brazing causes an increase in overall frontal area of the UGF as opposed to other modifications which only alter pressure distributions between upper and L lower surfaces. Surprisingly, the effect of inward against outward TE taper is not opposing; both result in D degradation and this is primarily due to a smaller difference in pressure between the upper and lower surface of the horizontal plate of the UGF as compared with an unmodified geometry.

Taper 10 out Taper 10 out Taper 10 out Taper 5 out Taper 5 out Taper 5 out Taper 2 out Taper 2 out Taper 2 out Cosine mult. Cosine mult. Cosine mult. Cosine single Cosine single Cosine single Taper 10 Taper 10 Taper 10 Taper 5 Taper 5 Taper 5 Taper 2 Taper 2 Taper 2 Braze 3 Braze 3 Braze 3 Braze 2 Braze 2 Braze 2 Braze 1 Braze 1 Braze 1 -1 -0.5 +0 +0.5 +1 +1.5 +0 +10 +20 +30 +40 +50 -35 -30 -25 -20 -15 -10 -5 +0

C L (a) L (b) CD (c) D L Figure 49: Percentage changes in CL, CD and D from an unmodified UGF at M = 3, α = 3°. As previously mentioned, M = 3 results in an oblique shock which does not interact with the adjacent grid fin cell. Therefore, the effects of four taper modifications were investigated for slower Mach numbers and are presented in Fig. 50. Indeed, smaller Mach numbers cause poorer CL performance particularly for a 10mm L inward taper. However, when comparing D it is clear that these effects are balanced by the drag contributions L and thus result in the net D effect being largely independent of M. This reinforces the conclusion that the TE modifications, specifically the taper, do not have significant effect on the shock structure along the UGF and rather only affect the surface CP distributions between the upper and lower surfaces of the UGF.

Taper 10 out Taper 10 out Taper 10 out

Taper 10 Taper 10 Taper 10

Taper 2 out Taper 2 out Taper 2 out

Taper 2 Taper 2 Taper 2

-15 -10 -5 +0 -10 -5 +0 +5 +10 -9 -8 -7 -6 -5 -4 -3 -2 -1 +0

C L (a) CL (b) D (c) D L Figure 50: Percentage changes in CL, CD and D from unmodified UGFs at M = 1.8, 2 and 3; all at α = 3°. It can be concluded that the 3D geometrical modifications to the UGF TE mostly result in large degradation of L D and hence, should be avoided unless their packaging benefits outweigh the aerodynamic impact. A brazed joint has the most notable drag contribution and thus, further producibility studies would likely be required to minimise its frontal area or to look at alternative joining methods. Unlike the 2D single plate analysis, the TE geometry (albeit not the edge taper itself) does have a significant contribution to aerodynamic performance and in particular, the pressure difference between upper and lower quadrants of the UGF which is the main cause for the performance sensitivity.

37 7 Heating Estimates In addition to aerodynamic performance, the material choices are governed by their ability to withstand aerodynamic heating. At a preliminary or generic design stage, it is useful to gather first estimates of this heating as result of the thermal environment and flight conditions expected for the R-77 grid fin so that appropriate materials and producibility processes can be shortlisted. 7.1 Theoretical and Empirical Methods

Using the theoretical methods from [67], estimates of Stanton number (St) and hence heating rate q˙w were found. Although complete calculations are given in Appendix.I, the general method is summarised as follows. By assuming an adiabatic wall, the wall temperature was calculated and from this, Eckert’s reference temperature T ∗ which is an average temperature value within the boundary layer. From T ∗, the average wall shear stress and hence skin friction were computed. Lastly, the Chilton-Colburn analogy was used to ﬁnd St from which q˙w was found. These calculations were performed for assumptions of both a laminar and turbulent boundary layer.

For comparison, a ‘rapid approximation’ method given in [68] was also performed. This graphical method relies on finding St from pre-plotted curves of Re and enthalpy for both turbulent and laminar cases. This provides a simple two step approximation to q˙w based on only knowing the Re and the enthalpy at the wall and edge of the boundary layer. Thus, it is most suited for such preliminary estimates and avoids cumbersome calculation or the need for CFD modelling. 7.2 Results Using theory, T ∗ = 463K and 478K for the laminar and turbulent cases respectively. From looking at Fig. 51, the average boundary layer temperature predicted from the CFD model matches this well. For these preliminary estimates, both CFD and theory assume an adiabatic wall (UGF). Also, it has been assumed that the temperature at the edge of the boundary layer is the same as that in the free-stream. From comparing the heating rate in Fig. 52, a good match is seen between all three methods; a turbulent boundary layer is a more realistic assumption shown by its closer match to q˙w calculated from CFD than a laminar case. For the turbulent theoretical prediction, the heating rate tends to ≈ 200 kWm−2 along the plate though a sharp rise is seen towards the LE as expected. Most significantly, the empirical method (shown interpolated) from [68] also provides a reasonable approximation of heating rate and tends to an error of ±35% with CFD. Given the speed and simplicity of computation with this method, it provides a sensible starting point for the designer without having to create, test and run a CFD model. In Fig. 52, the stagnation point heating is not shown and is often of greatest influence to LE (stagnation point) material choice. By using the Sutton-Graves formula for stagnation point convective heating based on a nose radius of 1%t, a value of 3.12 × 104 kWm−2 is found. This being drastically higher than the heating along the rest of the UGF suggests the need for the LE to be made of a different material to the rest of the grid fin.

103

102

0 0.2 0.4 0.6 0.8 1

Figure 51: Temperature distribution from CFD Figure 52: q˙w along the plate from using theoretical, shown for a forward section of the horizontal UGF CFD and empirical (graphical) methods. plate. 38 8 Conclusion 8.1 Summary This project investigated grid fins for missiles as an alternative lift generation device to traditional planar fins. Although grid fins have been used on missiles of the past and in space applications, recent resurgence in interest has stemmed from their aerodynamic benefits and their superior packaging capabilities which are especially relevant to applications such as internal weapon bays of next generation fighter aircraft. Through conducting CFD studies, this project was able to predict aerodynamic performance and characterise the flow through grid fins. In order to produce the optimal, low drag grid fin, the aerodynamic trade-off from the effects of varying geometrical parameters such as the leading and trailing edge sharpness, thickness to chord ratio, cell spacing and trailing edge taper across a variety of flight conditions was quantified. This enables future produciblity decisions to best compromise optimum aerodynamic performance with the material and manufacturing process capabilities that are available.

By segmenting the study into the three stage computational approach, the aerodynamic effects could be isolated from each other and individual recommendations can now be made. From the single plate analysis, it was found that the sharpness of the leading and trailing edges is directly related to the drag produced by the grid fin. Although the edge sharpness had little effect on CL, CD benefited from having the sharpest edge possible, especially at lower angles of incidence. Indeed, an infinitely sharp edged grid fin produces the lowest drag however, a reasonably similar performance is seen with a 2/3 edge which is likely more realistic for manufacture. The importance of the edge sharpness diminishes as the angle of incidence is increased. By using a systematic approach for isolating such changes in geometry, it was also found that the aerodynamic performance is highly sensitive to the leading edge sharpness whereas trailing edge sharpness has a much smaller contribution to lift and drag of a grid fin. Thus, from the single plate analysis, the optimum grid fin would have a sharp L leading edge as well as the smallest possible thickness to chord ratio, to deliver the best overall D performance.

By then extending this to two flat plates to simulate a cell within the grid fin, it can now be concluded that L L smaller incidences typically deliver optimum D and at these, D is highly sensitive to the spacing of the cell c s . For the R-77 geometry, the spacing of 0.86 is at an optimum and no interaction of the oblique shocks c within each grid fin cell occur at M = 3, meaning drag is minimised. Increasing s causes a repeating shock structure within the cell and leads to periodicity in the performance. At the intended Mach numbers, choking c and unstart are not of any concern and these only occur for highly unrealistic s = 30 or if Mach number is significantly reduced to transonic values.

To optimise computational resources and simplify the analysis, the unit grid fin approach was used and conveyed the effects in 3D as well as quantified the impacts of 3D geometrical modifications. For the grid L fins of the Vympel R-77, agreeing with the two flat plates, optimum D was observed at 5° < α < 10° and rather with M = 1.5 than M = 3. This is characterised by the pressure differences in each quadrant of the UGF. Worst performance is seen at M = 0.9 in the transonic regime where a supersonic pocket was seen to grow on the UGF, analogous to a supercritical aerofoil. Highest skin friction is present at the leading edge and a reduction of wall shear stress along the chord is accompanied by a rolling sheet of vortices which would likely interact with the grid fin frame. All of the TE geometrical modifications resulted in L significant degradation of D , especially the effects of brazing along the central joint. Such modifications should be avoided as far as possible unless their producibility benefits clearly justify their aerodynamic penalty.

Given that most literature focuses on performance of theoretical and ideal grid fins or focuses on the interaction of these with a missile body, this project has been able to approach grid fin aerodynamic performance from an alternative point of view. This project has provided a novel understanding of the isolated effects of geometrical modifications on aerodynamic performance and thus most importantly, a view on how these may inform a 39 decision for real world producibility. This trade-off between producibility and aerodynamic performance has also been explored with preliminary heating estimates which can influence future material and manufacturing process selection. 8.2 Producibility and Further Work Now that the aerodynamic performance is quantified, appropriate producibility decisions and trade-offs can be made and will be largely dictacted by the capabilities available. To best balance this trade-off, a set of realistic recommendations can be summarised as follows: • Aim to produce a two third edge geometry on the leading and trailing edges given that infinite sharpness is not feasible. • Prioritise the surface finish and accuracy of the leading edge over the trailing edge. • Design the grid fin web to have the minimum possible thickness to chord ratio. • A chord to span ratio (cell spacing) given by the Vympel R-77 is at no risk of choking at supersonic speeds. • Aim to avoid brazing at the central joints of the cells as this carries large drag penalties and instead explore alternative methods of assembly. • Avoid the implementation of trailing edge 3D modifications for packaging as these carry a significant performance cost. • Prioritise the material choice for the leading edge either through a different material than the rest of the grid fin or by use of a coating / shielding which is able to withstand the aerodynamic heating without contributing further to skin friction drag from its surface roughness.

Since this project was a preliminary study for a generic application of grid fins, several areas of further work can be identified. Firstly, this project only focused on aerodynamic performance within the central region of the grid fin and did not con- sider the effects of a frame and the drag generated from vortices that may arise at this intersection between the frame and grid fin web. As identified in Fig. 53, this project focused on the behaviour in the node 1 region whereas nodes 2 and 3 are also important considerations given that producibility of these joints with the frame would be critical. Therefore, future studies should aim Figure 53: Alternative node to expand the UGF method given in [53] to these other node loca- locations for a UGF study [53]. tions.

The main method of analysis used in this project was CFD. Thus, use of the three stage approach allowed validation against theory on simplified sections especially since a full fin-body simulation was not conducted as is found in literature. Therefore, future studies should aim to firstly validate and then build upon the CFD results by conducting experimental tests on scaled grid fin models in a supersonic wind tunnel. This will give a better understanding of flow physics such as the SBLIs and vortical structures especially at these frame-web intersections through photogrammetry techniques. Experimental testing will also allow the impact of any material selection or manufacturing process (used for producing the scale model) to be quantified; CFD is able to simulate ideal geometries which are not always feasible in real world application. Wind tunnel tests will also enable the preliminary heating estimates to be validated and extended by taking actual temperature measurements along the grid fin and at these web-frame intersections.

Lastly, further CFD studies could be carried out on a complete grid fin to understand the effects of global geometrical changes such as global sweep, global chord variation and differing cell sizes within the web. From this, the hinge forces and actuator sizing can then be determined as the design is iterated further. 40 References

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45 Appendices

A Vympel R-77 Missile

Figure A.1: A schematic diagram showing the layout of the Russian Vympel R-77 missile with tail mounted grid ﬁns [69]. Parts are labelled in Table A.1

Table A.1: Identified components presented in Figure A.1. Label Component 1 Active radar guidance system 2 Body tube 3 Warhead 4 Ramjet 5 Fairing 6 Fuse 7 Mount 8 Navigation and control system 9 Autopilot system 10 Stabilisation (planar) fins 11 Thermal battery 12 Actuator linkage / gearbox 13 Grid fins

(a) (b)

Figure A.2: Close-up images of the R-77 grid ﬁns [69].

46 B Grid Fin Drawing (Vympel R-77) 4 3 2 1

F 287.053 127.954 F 255.036 38.500 63.920

E 45.000 0.675 E

A D D A A-A 1 : 1 0.675

C C

45.000

B B DATE: MARCH 2020 SCALE:1:5

TITLE: GRID FIN GEOMETRY (BASED ON VYMPEL R-77) 38.500 DRAWN BY A4 A SHAKUNT TAMBE A

DIMENSIONS ARE IN MILLIMETERS SHEET 1 OF 1

SOLIDWORKS4 EducationalFigure Product. For B.1: Instructional An3 engineering Use Only. drawing of2 the R-77 grid ﬁn. 1

47 C Drag Reduction through Geometry Modiﬁcation

Figure C.1: A synopsis of the geometrical variations covered in several literature studies [47].

48 D Flight Conditions and Prism Layer Parameters

Listing 1: MATLAB code written for the calculation of ﬂight conditions based on the International Stand- ard Atmosphere as well as an initial estimate of prism layer properties based on empirical skin friction relationships. 1 %Flight conditions and mesh properties calculator. 2 %Written byS.Tambe, January 2020. 3 4 clear; clc; 5 6 h=10000; %altitude of 10km AMSL 7 M=input('Enter Mach number:'); %Mach number 8 9 [T, a, P, rho]=atmosisa(h) %compute the Temperature, speed of sound, pressure and density using ISA 10 mu=(0.1456E−5)*(sqrt(T)/(1+110/T)); %dynamic viscosity using Sutherlands law 11 V=M*a %Velocity inm/s 12 c=38.5e−3; % flat plate chord in metres 13 Re=(rho*V*c)/mu %Reynolds number 14 nu=mu/rho %kinematic viscositym^2/s 15 16 Cf=0.026*(Re^(−1/7)) %Prandtl's one seventh power law to calculate skin friction coeff. 17 tw=0.5*Cf*rho*V^2; %wall shear stress 18 u_tau=sqrt(tw/rho); 19 yplus=1; %target wally+ 20 deltaS=yplus*nu/u_tau %distance of first prism layer centre point from wall. 21 delta=0.385*c*Re^(−1/5) %boundary layer thickness 22 23 %calculating initial number of prism layers(n). 24 %usinga geometric series 25 sf=1.5; %stretch factor − a ratio between thickness of prism layern andn −1. 26 n=(log((delta/(2*deltaS))+1)/log(sf))−1 %number of prism layers(an initial guess)

49 E Shock Expansion Theory for a Single Flat Plate

The shock expansion theory is a theoretical method which can be applied to the single flat plate (sharp edged) configuration by assuming the geometry and flow field (for the sharp case) shown in Fig. E.1. The method is based on that given in [36, 59] and is shown below as a representative example for the M = 3, α = 3° case. It should be noted that this can be applied canonically to other angles of incidence and Mach numbers as appropriate.

O-S P-M Exp 1 α = 3º 2 P-M Exp δ 3 O-S M∞=3 2 δ δ w 3 δ 4 1 = 0º δ 5 δ 4 δ δ 8 6 8 O-S 5 δ 7 6 7

P-M Exp O-S (not to scale) P-M Exp

Figure E.1: Identified oblique shocks and expansion fans used for application of the shock expansion theory to the single flat plate configuration at α = 3°.

−1 1 For a sharp LE and TE plate with geometry as given in Fig. 18, the wedge angle δw = tan 4 . For the regions numbered 1 to 8 as shown, with the incoming ﬂow turn angle δ1 = 0° and at α = 3°, the remaining ﬂow turning angles can be found.

• δ2 = δ7 = +11.04° • δ3 = δ6 = −3° • δ4 = δ5 = −17.04°

Region 1 → 2 across oblique shock: (Using the oblique shock relations) P2 M1 = 3, M2 = 2.4537, ν = 38.033°, = 0.0629 P02 P2 = 2.1997 (where ν is the Prandtl-Meyer angle). P1

Region 2 → 3 across expansion fan: (Using isentropic ﬂow relations) P3 ν3 = ν2 + (δ2 − δ3) = 52.073°, M3 = 3.123, = 0.0227 P03 P3 = P3 P03 P02 P2 = 0.7923 P1 P03 P02 P2 P1

Region 3 → 4 across expansion fan: (Using isentropic ﬂow relations) P4 ν4 = ν3 + (−δ4 + δ3) = 66.113°, M4 = 4.025, = 0.0064 P04 50 P4 = P4 P04 P03 P3 = 0.2227 P1 P04 P03 P3 P1

Region 1 → 5 across oblique shock: (Using the oblique shock relations) P5 M1 = 3, M5 = 2.1500, ν = 30.426°, = 0.1011 P05 P5 = 3.1861 P1

Region 5 → 6 across expansion fan: (Using isentropic ﬂow relations) P6 ν6 = ν5 − (δ5 − δ6) = 44.466°, M6 = 2.7393, = 0.0404 P06 P6 = P6 P06 P05 P5 = 1.2743 P1 P06 P05 P5 P1

Region 6 → 7 across expansion fan: (Using isentropic ﬂow relations) P7 ν7 = ν6 + (δ7 − δ6) = 58.506°, M7 = 3.4985, = 0.01314 P07 P7 = P7 P07 P06 P6 = 0.4141 P1 P07 P06 P6 P1

Computing total lift force: L = ∆P × length × vertical component L = (P7 − P2)(2t cos(δw)(cos(δ2)) + (P 6 − P 3)(c − 4t)(cos(α)) + (P5 − P4)(2t cos(δw))(cos(−δ4)) L For values given in Table3, L = 491.9229N, CL = 2 = 0.0768 0.5ρV∞S

Computing drag force (inviscid): D = ∆P × length × horizontal component D = (P7 − P2)(2t cos(δw)(sin(δ2)) + (P 6 − P 3)(c − 4t)(sin(α)) + (P5 − P4)(2t cos(δw))(sin(−δ4)) D For values given in Table3, D = 42.0414N, CD = 2 = 0.0066 0.5ρV∞S

To compare the drag force from shock expansion theory with that from CFD, an empirical skin friction correction was applied since shock expansion theory assumes inviscid ﬂow. The method given above was repeated for α = 0° → 5° in order to generate the values shown in Table4.

51 F STAR-CCM+ Settings for CFD Simulations

F.1 Single Flat Plate The following is a detailed overview of the simulation settings used for the single plate CFD analysis within a 2D circular domain. The settings shown below are specifically for a sharp LE and TE plate simulation at M = 3, α = 3°. Note: the settings were refined accordingly for different flow conditions.

Meshers: • Polygonal, prism (wall thickness mode with geometric progression layer stretching) • Base size = 5 × 10−5m • CAD projection = active • Target surface size = 100% of base • Minimum surface size = 10% of base • Surface growth rate = 1.2 • Number of prism layers = 18 • Prism layer near wall thickness = 9 × 10−7m • Prism layer total thickness = 9.406 × 10−4m • Custom surface control active with target surface size 0.02m • Total volume mesh cells = 42703

Physics models: • All y+ treatment • Coupled energy • Coupled ﬂow - Implicit 2nd order • Ideal gas • K − ω SST turbulence • RANS • Turbulent • Steady • Two dimensional • Initial conditions: velocity = [898.3899 ∗ cos(3 ∗ 3.14/180), 898.3899 ∗ sin(3 ∗ 3.14/180), 0.0], Static temperature = 223.15K; Gauge pressure = 0Pa • Reference pressure = 26436.0 Pa

Region boundaries: • Domain = free stream. Flow direction [cos(3 ∗ 3.14/180), sin(3 ∗ 3.14/180), 0.0]; Mach number = 3; Static temperature = 223.15K; Turbulence intensity = 0.01 • Flat plate = Wall (adiabatic)

Solvers: • Partitioning • Wall distance • Coupled implicit: Courant number = 6.67 (20/M); ‘Grid Sequencing’ and ‘Expert Driver’ active. • K-Omega turbulence • K-Omega turbulence viscosity

52 F.2 Two Flat Plates The following is a detailed overview of the simulation settings used for the two plate CFD analysis within a 2D circular domain of radius 10c. The settings shown below are specifically for a simulation at M = 3, α = 3° c with s = 2. Note: the settings were refined accordingly for different flow conditions and configurations.

Meshers: • Polygonal, prism (wall thickness mode with geometric progression layer stretching) • Base size = 5 × 10−5m • CAD projection = active • Target surface size = 100% of base • Minimum surface size = 10% of base • Surface growth rate = 1.2 • Number of prism layers = 18 • Prism layer near wall thickness = 9 × 10−7m • Prism layer total thickness = 9.406 × 10−4m • A custom surface control per ﬂat plate each with target surface size 0.02m • Total volume mesh cells = 83618

Derived parts (for generating CP plots on inner surfaces): • ‘BottomUpper’ - Upper surface of bottom ﬂat plate; parallel to the chord of the plates with a vertical 1 coordinate −9.625mm (i.e − 4 c) • ‘TopLower’ - Lower surface of top ﬂat plate; parallel to the chord of the plates with a vertical coordinate 1 +9.625mm (i.e − 4 c) Solvers: • Partitioning • Wall distance • Coupled implicit: Courant number = 6.67 (20/M); ‘Grid Sequencing’ and ‘Expert Driver’ active. • K-Omega turbulence • K-Omega turbulence viscosity

53 F.3 Unit Grid Fin The following is a detailed overview of the settings used for the 3D CFD simulation of the unit grid fin (UGF) placed within the cuboid shaped ‘tunnel’ domain of length 20c and side length s. The settings shown below are specifically for a simulation at M = 3, α = 3°. Note: the settings were refined accordingly for different flow conditions and configurations.

Meshers: • Surface, Trimmed cell, prism (wall thickness mode with geometric progression layer stretching) • Base size = 0.0045m • CAD projection = active • Target surface size = 100% of base • Minimum surface size = 10% of base • Surface growth rate = 1.3 • Number of prism layers = 19 • Prism layer near wall thickness = 9 × 10−7m • Prism layer total thickness = 9 × 10−4m • Maximum cell size = 100% of base • Post mesh optimisation = disabled • A custom surface control on the UGF: Target surface size = 10% of base; Minimum surface size = 5.5% of base • A custom volume control for VOR: Custom isotropic trimmed cell size = 18% of base; Custom surface remesher size = 18% of base • Total volume mesh cells = 2749079 • Cell faces on periodic pair 1 = 34015 • Cell faces on periodic pair 2 = 33637 Physics models: • All y+ treatment • Coupled energy • Coupled flow - Implicit 2nd order • Ideal gas • Spalart-Allmaras turbulence with 2nd order convection • RANS • Turbulent • Steady • Two dimensional • Initial conditions: velocity = [898.3899 ∗ cos(3 ∗ 3.14/180), 898.3899 ∗ sin(3 ∗ 3.14/180), 0.0], Static temperature = 223.15K; Gauge pressure = 0Pa • Reference pressure = 26436.0 Pa Region boundaries: • Inflow, Outflow = free stream. Flow direction [cos(3∗3.14/180), sin(3∗3.14/180), 0.0]; Mach number = 3; Static temperature = 223.15K; Turbulence intensity = 0.01 • UGF = Wall (adiabatic) • Domain walls = Periodic paired interfaces (as shown in Fig. 19). Translational periodic with translation [0,0,-0.045]m between pairs. Derived parts: • (Various planes used for visualisation) Solvers: • Partitioning • Wall distance • Coupled implicit: Courant number = 6.67 (20/M); ‘Grid Sequencing’ and ‘Expert Driver’ active. • Spalart-Allmaras turbulence • Spalart-Allmaras viscosity 54 G Wall Coordinates

G.1 Single Flat Plate While running CFD simulations on the single flat plate case, the mesh suitability was assessed a number of ways including the value of the wall coordinate y+. Given the turbulent simulation and the turbulence model used, a target y+ ≤ 1 was desired across the whole surface of the flat plate to ensure the boundary layer effects were fully captured. Below, in Fig. G.1, four representative examples of such y+ monitors are shown with histograms showing the frequency of the values around the flat plate surface. The plots shown are for the M = 3, α = 3° case for each of the edge taper geometries however similar plots were monitored for all test cases.

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(a) Sharp edge M = 3, α = 3°. (b) One third edge M = 3, α = 3°. 1100 1200 1000 900 1000 800 800 700 600 600 500 400 400 300 200 200 100 0 0 0 1 2 3 4 0 1 2 3 4

Figure G.1: Histogram plots of wall y+ for diﬀerent edge geometries at M = 3, α = 3°, over the whole single ﬂat plate surface.

55 G.2 Two Flat Plates Since the single plate CFD model was extended to a two ﬂat plate case, the same method of monitoring the value of the wall coordinate y+ was used. The target y+ ≤ 1 was now desired across the whole surface of each ﬂat plate and hence in Fig. G.2, four representative examples of such y+ histograms are shown with frequencies of values for each plate. The plots shown are with M = 3, α = 3° sharp edged plates for a c small, representative selection of chord to span ratios s = 1, 3, 8, 20. However, this method was used in all c + simulations of the two plate model, across all M, α and s variations with the same target y .

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c c (c) M = 3, α = 3°, s = 8. (d) M = 3, α = 3°, s = 30.

+ c Figure G.2: Histogram plots of wall y at M = 3, α = 3° for the sharp edged two plates with s = 1, 3, 8, 20.

56 G.3 Unit Grid Fin For the 3D simulation of the unit grid ﬁn, the value of the wall coordinate y+ was monitored throughout the simulations to ensure that the boundary layer eﬀects were captured appropriately. For the Spalart-Allmaras turbulence the desired value was still y+ ≤ 1 though it was more forgiving than the K − ω SST model and hence values up to 3 were acceptable [56, 57]. The plots in Fig. G.3 show representative examples of y+ histograms over the entire UGF surface. The plots are shown for M = 0.9, 1.5, 2, 3 however this method was used in all simulations across all M and α variations with the same target y+. CFD settings given in Appendix F.3 were altered to achieve this accordingly.

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(a) UGF with M = 0.9, α = 5°. (b) UGF with M = 1.5, α = 5°. 104 104 2.5 2.5

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Figure G.3: Histogram plots of wall y+ at α = 5° and M = 0.9, 1.5, 2, 3 for the unit grid ﬁn 3D simulation.

57 H UGF 3D Geometry Modiﬁcations

Given below in Fig. H.1 is an illustration showing the complete set of geometrical modiﬁcations made to the three-dimensional unit grid ﬁn. ALL ARE ISOMETRIC VIEWS FROM TRAILING EDGE

(Unmodified) Braze 1mm Braze 2mm

Braze 3mm Cosine single Cosine multiple

Taper 2mm Taper 5mm Taper 10mm

Taper 2mm out Taper 5mm out Taper 10mm out

Figure H.1: An exhaustive illustration of all geometrical modiﬁcations applied to the TE of the UGF. All views are isometric taken from the TE.

The brazed joint was simulated in the CAD models using a fillet of the given dimension (i.e. a braze 1mm 58 was a 1mm fillet). Additionally, the ends of these fillets were rounded to smooth their profile with the LE and TE as far as possible without creating drafted sections. The relevance of dimensions for the taper geometries are identified using theX in Fig. 48.

For the cosine TE proﬁles, the exact equations used for modelling these are given below for reference.

Cosine single: 2π y = 2.5 cos x + 36 (18) 45 Cosine multiple: y = cos (x) + 37.5 (19) Where x is the axis along the chord (x = 0 is at the LE).

59 I Calculation of Heating Estimates

Listing 2: MATLAB code written to compute preliminary heating estimates using theoretical methods. The code also loads data from CFD and the empirical (graphical) method which can then be plotted for comparison. 1 %Code for Prelim. heating estimates from theoretical methods, empirical 2 %(graphical) methods and as found in CFD results. 3 %Written byS.Tambe. May 2020. 4 clear; clc 5 6 %free stream and flight conditions 7 Minf=3; 8 h=10000; %m, altitude of 10km AMSL 9 [Tinf, ainf, Pinf, rhoinf]=atmosisa(h); %compute the Temperature, speed of sound, pressure and density using ISA 10 Ue=Minf*ainf; %free stream velocity 11 mu=(1.458E−6)*((Tinf^(1.5))/(Tinf+110.4)); %dynamic viscosity using Sutherlands law 12 gamma=1.4; %ratio of specific heats 13 T0=(1+0.5*(gamma−1)*Minf^2)*Tinf; %stagnation temperature 14 R=287; %J/kgK 15 Pr=0.71; %Prandtl number for air 16 Cp=1005; %J/kgK specific heat capacity at constant pressure 17 18 %Plate coordinates 19 plate_start=0.5E−3; %nominal 0.5mm at start to avoid error with0 divide 20 plate_end=38.5E−3; %38.5mm chord length 21 x_coords=linspace(plate_start,plate_end); %create array of dist. coords 22 23 %% From Lecture Course/ Textbook Theory 24 %Laminar case 25 disp('Laminar boundary layer calculations...... ') 26 r=Pr^0.5 %recovery factor 27 Te=Tinf %temperature at edge ofb.l 28 Taw_lam=Te+r*(T0−Te) %adiabatic wall temperature 29 Tstar_lam=Te*(0.45+(0.55*(Taw_lam/Te))+(0.16*r*((gamma−1)/2)*Minf^2)) %averageT * temp inb.l 30 rhostar_lam=Pinf/(R*Tstar_lam) %density based onT * 31 mustar_lam=(1.458E−6)*((Tstar_lam^(1.5))/(Tstar_lam+110.4)) %viscosity based onT * 32 tau_w_lam=(0.332*rhostar_lam^(0.5)*Ue^(1.5)*mustar_lam^(0.5)); %wall shear stress 33 tau_w_lam=tau_w_lam.*(x_coords.^(−0.5)); %multiplied by the coord along plate 34 Cfstar_lam=tau_w_lam./(0.5*rhostar_lam*Ue*Ue); 35 Ststar_lam=(0.332*rhostar_lam^(−0.5)*Ue^(−0.5)*mustar_lam^(0.5)*Pr^(−2/3)); % Stanton no 36 Ststar_lam=Ststar_lam.*(x_coords.^(−0.5)); %multiplied by the coord along plate 37 qwdot_lam=Ststar_lam.*(rhostar_lam*Ue*Cp*(Taw_lam−Te)); 38 39 %Turbulent case 40 disp('Turbulent boundary layer calculations...... ') 41 r=Pr^(1/3) %recovery factor 60 42 Te=Tinf %temperature at edge ofb.l 43 Taw_turb=Te+r*(T0−Te) %adiabatic wall temperature 44 Tstar_turb=Te*(0.45+(0.55*(Taw_turb/Te))+(0.16*r*((gamma−1)/2)*Minf^2)) %averageT * temp inb.l 45 rhostar_turb=Pinf/(R*Tstar_turb) %density based onT * 46 mustar_turb=(1.458E−6)*((Tstar_turb^(1.5))/(Tstar_turb+110.4)) %viscosity based on T* 47 tau_w_turb=(0.0296*rhostar_turb^(0.8)*Ue^(1.8)*mustar_turb^(0.2)); %wall shear stress 48 tau_w_turb=tau_w_turb.*(x_coords.^(−0.2)); %multiplied by the coord along plate 49 Cfstar_turb=tau_w_turb./(0.5*rhostar_turb*Ue*Ue); 50 Ststar_turb=(0.0296*rhostar_turb^(−0.2)*Ue^(−0.2)*mustar_turb^(0.2)*Pr^(−2/3)); % Stanton no 51 Ststar_turb=Ststar_turb.*(x_coords.^(−0.2)); %multiplied by the coord along plate 52 qwdot_turb=Ststar_turb.*(rhostar_turb*Ue*Cp*(Taw_turb−Te)); 53 54 %% CFD data 55 load M3CFDskinfrictiondata.mat % load skin friction CFD data for M3 alpha0 56 57 x_coordsCFD=M3CFDskinfriction(:,1); %extract coordinates 58 x_coordsCFD=x_coordsCFD+0.0191; %shift coordinates to match theory 59 Cf_cfd=M3CFDskinfriction(:,2); %extract Cf data 60 tau_w_cfd=Cf_cfd*0.5*rhoinf*Ue*Ue; %wall shear stress 61 St_cfd=Cf_cfd.*(0.5*((Pr)^(−2/3))); %Chilton Colburn 62 qwdot_cfd=St_cfd*rhoinf*Ue*Cp*(Taw_turb−Te); %heat rate from CFD results 63 64 %% Empirical from NACA paper(graphical method) 65 load empirical_NACApaperdata_St.mat %load the data found using graphical method 66 67 %calculate heat rate from empirical St data 68 for i=1:length(St_empturbcomp) 69 qwdot_empturb(i)=St_empturbcomp(i)*rhostar_turb*Ue*Cp*(Taw_turb−Te); 70 end 71 72 %% Sutton−graves stagnation point temp formula 73 74 k=1.7415E−4; %a constant 75 Reff=0.01*0.675E−3; %effective'nose' radius in metres. 1% oft 76 qconv_stag=k*sqrt(rhoinf/Reff)*(Ue^3); 77 qconv_stag=qconv_stag/1000 %KW/m^2 78 79 %% now plot data...

For reference, the plots generated from the above code are provided below. Since the heating rate q˙w is ∗ derived from St which is consequently derived from Cf and hence τw, the plots show the same trends as seen in Fig. 52. Thus, these were excluded from the main body of the report.

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∗ Figure I.1: τw, Cf and St found from theory, CFD and through the empirical method given in [68].

62 “Do something because you really want to do it. If you’re doing it just for the goal and don’t enjoy the path, then I think you’re cheating yourself.” -Kalpana Chawla