A Comparison of Euler Finite Volume and Supersonic Vortex Lattice Methods Used During the Conceptual Design Phase of Supersonic Delta Wings

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A Comparison of Euler Finite Volume and Supersonic Vortex Lattice Methods used during the Conceptual Design Phase of Supersonic Delta Wings

THESIS

Presented in Partial Fulfillment of the Requirements for the Degree Master of

Science in the Graduate School of The Ohio State University

Daniel Guillermo-Monedero, B.S.

Graduate Program in Aeronautical and Astronautical Engineering

The Ohio State University

2020

Master’s Examination Committee:

Dr. Clifford A. Whitfield, Advisor

Dr. Richard J. Freuler

Dr. Matthew H. McCrink

Copyrighted by

Daniel Guillermo-Monedero

2020

Abstract

This thesis uses two different methods to analyze wings in supersonic flows with a focus on preliminary design. The primary goals of this study are to compare the Euler equations finite volume method and supersonic vortex lattice method in predicting surface pressure on wings, and to develop a low-order supersonic vortex lattice method as a baseline tool that can be extended for further wing design and analysis. The supersonic vortex lattice method uses vortical sources to model the flow on the boundary, which replicates the aerodynamic shape of interest, in an inviscid and irrotational flow field and obtain solutions. The Euler equations of flow can be discretized using finite volume methods and can be integrated over the volume of interest and solutions can be obtained over the surface.

These mathematically similar methods have a lot of differences in their numerical formulations, which can be critical in the design and analysis of wings. Hence, it is important to understand the key differences of these in order to develop a reliable baseline low-order design tool.

To compare these two methods, a flat plate subsonic leading-edge delta wing with a leading-edge factor (βcot(ΛLE)) of 0.6 will be modeled at the same conditions. The lattice method will be coded using MATLAB and the Euler equations will be solved using

ANSYS® Fluent. The differential pressure at the camber, aerodynamic coefficients, time to solve, effort to discretize, and other mathematical considerations for both methods are

ii compared. As expected, pressure results shown good congruency between both methods.

The lift coefficient and moment coefficient show around a 10% difference, while the drag shows the most difference at 30%. Convergence for the supersonic vortex lattice method happens immediately, on-the-order of seconds, while ANSYS® Fluent takes significantly longer, on-the-order of hours. In general, the Euler method is much harder to set up and to discretize (mesh) the domain, but it can show flow structures at any point of the domain.

With these results it can be concluded that the supersonic vortex lattice method, when used appropriately, has significant advantages and potential to be used as an effective baseline tool for linear modeling of wings during the early stages of design. The fast convergence, setup, and the few amounts of elements are the most notable strengths of the lattice method.

Furthermore, non-linearities can be superimposed in the method and could be used to analyze, for example, active and passive flow control or any geometry of wings.

iii

Dedication

To my family, friends, advisor, instructors, and everyone else that made this thesis

possible.

Acknowledgments

I would like to express my deep gratitude to Dr. Clifford Whitfield for his constant guidance and education through my undergraduate and graduate careers. I’m very grateful for his dedication, enthusiasm, and knowledge about design, fluid dynamics, and teaching.

He is always an excellent mentor and an exceptional educator for me and many other students at the Ohio State University.

I would like to thank my committee members, Dr. Rick Freuler and Dr. Matthew

McCrink, for their time and help on the final stages of my thesis.

I also would like to express my gratitude to the Mechanical and Aerospace

Department at the Ohio State University for the opportunity to be a teacher assistant and cover my first year expenses for my master’s degree. Thanks to my parents, Conchita

Monedero and Ernesto Guillermo, for the economic support in my last semester of the degree. And thank you to the Office of International Affairs for the grant that allowed me to finish this degree.

A huge thanks to my colleague and friend Rodrigo Auza-Gutierrez for his frequent help on computational fluid dynamics.

A very special recognition goes to my family and girlfriend whose constant support and motivation helped me achieve my very best every single day.

Lastly, I would like to acknowledge all my school, undergraduate, and graduate instructors. All of you have made invaluable marks in my life, which make me be the best student and engineer I can be.

Vita

May 13, 1995 ...... Born, Caracas, Venezuela

May 2017 ...... B.S. Aeronautical and Astronautical

Engineering, The Ohio State University

August 2018 to May 2019 ...... Graduate Teaching Assistant, The Ohio

State University

Fields of Study

Major Field: Aeronautical and Astronautical Engineering

vii

Table of Contents

Abstract ...... ii

Dedication ...... iv

Acknowledgments...... v

Vita ...... vii

List of Tables ...... x

List of Figures ...... xi

Chapter 1: Introduction ...... 1

1.1 Introduction ...... 1

1.2 Background ...... 6

1.2.1 Overview of Steady, Inviscid Supersonic Flows and Solution Methods for

Wings ...... 6

1.2.2 Singularity Distribution Methods Modeling and the Supersonic Vortex Lattice

Method ...... 7

1.2.3 Computational Fluid Dynamics Modeling...... 10

1.3 Motivation and Main Objectives...... 11

Chapter 2: Singularity Distribution Method: The Supersonic Vortex Lattice Method ... 13

2.1 Governing Equations and Boundary Conditions ...... 13 viii

2.2 Numerical Approach ...... 17

Chapter 3: CFD Modeling ...... 23

3.1 Governing Equations of Fluid Flow ...... 23

3.2 Computational Modeling and Boundary Conditions ...... 23

3.3 Fluid Domain and Grid ...... 26

Chapter 4. Results ...... 31

4.1 Supersonic Vortex Lattice Method ...... 31

4.2 Euler equations CFD ...... 37

4.3 Comparison of Methods ...... 41

Chapter 5. Conclusions ...... 45

5.1 Conclusions ...... 45

5.2 Future Work Recommendations ...... 46

References ...... 48

List of Tables

Table 1: Mesh Independence Study Results ...... 28

Table 2: Aerodynamics Coefficients for Delta Wing with Leading Edge βcot(ΛLE)=0.6 at

Mach=1.5 and AoA=2.5 deg...... 33

Table 3: Aerodynamics Coefficients for Delta Wing of Subsonic LE at Mach=1.5 and

AoA=2.5 deg. with Euler formulations...... 37

Table 4: Percentage Difference of Aerodynamic Coefficients for Delta Wing obtained with Euler Formulations and SVLM Methods...... 41

List of Figures

Figure 1: Types of Flow on Highly Swept Wings [1] ...... 3

Figure 2: Aerodynamic Investigative Tools [1] ...... 4

Figure 3: Classic Linear Supersonic Aerodynamic Design Process [1] ...... 5

Figure 4: Arbitrary Wing Imposed in z=0 plane ...... 14

Figure 5: Panel Representation of Flat Plate Delta Wings with Subsonic Leading Edge 15

Figure 6: Numerical Representation of Influence Factor R ...... 18

Figure 7: Picture of Fluid Domain with Boundary Conditions...... 26

Figure 8: Z-X Plane View of Meshing Domain with Dimensions ...... 27

Figure 9: Y-X Plane View of Meshing Domain with Dimensions ...... 27

Figure 10: Mesh Independence Study Plot ...... 29

Figure 11: Mesh of Fluid Domain ...... 30

Figure 12: Detailed View of the Mesh Near the Wing ...... 30

Figure 13: Flat Plate Delta Wing with Leading Edge βcot(ΛLE)=0.6 ...... 31

Figure 14: Pressure Coefficient Distribution of Flat Plate Delta Wing ...... 32

Figure 15: Top View of Pressure Coefficient Distribution of Flat Plate Delta Wing...... 32

Figure 16: Sub plots of Lift Coefficient vs. AoA, Drag Polar, Moment Coefficient abt.

Apex vs. AoA, and Lift-to-Drag vs. Lift Coefficients of Flat-Plate Delta Wing with

Subsonic Leading-Edge ...... 34

Figure 17: Lift-Curve Slope vs. Free-Stream Mach Number for Delta Wing with

Subsonic Leading-Edge ...... 34

Figure 18: Delta Wings with Different Leading-Edge Factors ...... 35

Figure 19: Lift-Curve Slope Envelope for Flat-Plate Delta Wings ...... 36

Figure 20: Pressure Contour on Suction Face of Delta Wing ...... 38

Figure 21: Pressure Contour on Pressure Face of Delta Wing ...... 38

Figure 22: Magnitude of Velocity Contour at X=0.5c with Tangential Velocity Vectors 39

Figure 23: Magnitude of Velocity Contour at X=0.75c with Tangential Velocity Vectors

...... 39

Figure 24: Magnitude of Velocity Contour at X=c with Tangential Velocity Vectors .... 40

Figure 25: Magnitude of Velocity Contour at X=1.25c with Tangential Velocity Vectors

...... 40

Figure 26: Delta Wing with Three Different Chord Locations ...... 42

Figure 27: Pressure Coefficient Plot at 25% of Chord for Delta Wing for Euler CFD and

SVLM ...... 42

Figure 28: Pressure Coefficient Plot at 50% of Chord for Delta Wing for Euler CFD and

SVLM ...... 43

Figure 29: Pressure Coefficient Plot at 75% of Chord for Delta Wing for Euler CFD and

SVLM ...... 43

xii

Chapter 1: Introduction

1.1 Introduction

Modeling three-dimensional supersonic flow features is very complex, making the estimation of aerodynamic characteristics for the design and analysis of commercial and military aircraft very challenging.

Supersonic vortices, shockwaves, and expansion fans are only a few flow features that make the analysis difficult. Complexity can increase even more when considering viscous interactions, such as in the instances of boundary-shockwave interaction, flow separation, and turbulence. Supersonic flow features encompass a significant Mach range. Commonly used as a rule-of thumb, the supersonic range is typically on-the-order of

Mach 1.2 to 5.0. Supersonic flows are bounded by the transonic and hypersonic flow regimes. The transonic regime, usually characterized to have a range of Mach 0.8 to 1.2, shows characteristics of both supersonic and subsonic flows. The hypersonic regime has a range of Mach number typically greater than 5.0, in which thermo-chemical reactions of the air become of importance. The focus of this discussion will be on the supersonic flow regime.

Kulfan provides an overview of the flows encountered in a supersonic flight vehicle envelope and guidelines for the design of supersonic aircraft [1]. For the design of supersonic flight vehicle wings, the general goal is to find a configuration that maximizes lift while minimizing drag. However, in the supersonic regime the drag penalties can be very large. The total supersonic drag can be divided into three components: the drag due to lift (induced drag), the wave drag due to thickness (shape drag), and the skin friction drag

(viscous or parasitic drag). Addressing each drag component in design introduces varying points of emphasis for the configuration layout. For example, the first two components of drag, induced and shape drag, emphasizes the importance of the overall shape and size of the wing, while the second component, viscous drag, introduces the importance of wing surface material selection along with the shape. Regardless, the overall design strategies and analysis techniques used for the general desired aerodynamic performance, 1 depend heavily on the required flight condition and the stage in which the wing design is currently in, i.e. the conceptual, preliminary, or detailed design phase.

During the early conceptual phase of supersonic wing design, with provided required and desired flight conditions known, there are general guidelines for the wing planform shape and airfoil selection that should be followed. Generally, a thin airfoil section (thin maximum thickness) and sharp leading and trailing edges are desired for a supersonic wing. A thin airfoil ensures the supersonic flow will turn into the freestream at a lesser angle, reducing the strength of the shock compared to a thicker subsonic airfoil. Furthermore, a sharp leading-edge helps to ensure that the shock will remain attached to the surface as an oblique wave. In comparison, a bow shock in front of a wing has a higher pressure loss (stronger shock) and hence, more wave drag. Even though a thin airfoil is a necessity in most supersonic wings, a sharp leading-edge is not essential if the wing has sweep.

Wing leading-edge sweep makes the flow behave as if the airfoil section is thinner, and the sweep determines the strategy that should be used for the wing’s design layout, which is based on the leading-edge normal component of free-stream Mach number. If the component of the free-stream Mach number normal to the leading-edge is supersonic, then the wing has a supersonic leading-edge. Conversely, if the leading- edge freestream normal component is subsonic, then the wing has a subsonic leading edge. This can be easily

-1 distinguished for delta wings by calculating the Mach wave angle, µ (µ = sin (1/M∞)), from the apex. If the leading-edge lies behind the Mach angle (µ > 90-ΛLE), then the wing has a subsonic leading-edge; on the contrary, a leading-edge in front of the Mach wave signifies that the edge will be supersonic.

Both types of leading-edge characteristics have pros and cons. Depending of the leading-edge type, the design considerations for of the wing can differ significantly. For example, a subsonic leading-edge wing in slower supersonic flow may incorporate a rounded leading-edge, and even a common subsonic airfoil shape, without incurring large drag penalties in supersonic flow due to the absence of a strong bow shockwave forming. For this reason, wings designed for extended supersonic flight usually are highly swept.

Figure 1 shows various types of flow characteristics for highly swept wings that are often encountered throughout supersonic flight operations.

Figure 1: Types of Flow on Highly Swept Wings [1]

For the wing operating at its designed condition, outlined in red on the left-hand side of Figure 1, the flow should be attached to the whole surface. If the wing was designed specifically with a supersonic trailing-edge, a shock wave will form along the upper surface of the wing’s trailing-edge to adjust the flow to free-stream flight conditions. As the angle of attack is increased for the thin, highly swept delta wing, leading-edge vortices will appear. The supersonic designed wing in an off-design flight conditions may encounter flow structures like supersonic shock/boundary layer interactions, flow separation, leading-edge vortices, vortex shedding, and vortex bursting. The flow characteristics are strongly dependent on the geometry and the flight conditions that the aircraft’s wing is specifically designed for. In order to obtain an understanding of the complex flow characteristics common in supersonic wing analysis and design, there are multiple aerodynamic analysis tools available, with their preferred use depending on the phase of design. The analysis tools are commonly used in conjunction, and each play a very important role in the common goal of developing an understanding of the fluid dynamics. A pictorial of the various fluid dynamic analysis tools that are commonly used is shown below in Figure 2.

Figure 2: Aerodynamic Investigative Tools [1]

Computational fluid dynamics (CFD) tools use computers to mathematically model the flow field.

These can be as complex as solving the full Navier-Stokes equations for any geometry, but CFD is often used with forms of simplified fluid dynamics (SFD) to reduce complexity of the problem being solved and draw conclusions from simplified models. Experimental fluid dynamics (EFD), like wind tunnel testing for example, is usually used with CFD in order to validate models and carry out experiments using computations instead of costly wind tunnel or flight test analysis. One thing CFD and EFD have in common is visual fluid dynamics (VFD). CFD requires post-processing in order to visualize and analyze the data obtained. EFD with point gages usually need similar visual tools to the CFD, but there are experimental visualization techniques like Schlieren imaging, particle image velocimetry, pressure sensitive paint, etc. that are commonly used.

Extremely complex fluid dynamic that are harder to understand can be analyzed using approximate fluid dynamics (AFD). For example, transonic analysis models of wings commonly use empirical data to obtain drag approximations. All these tools converge at late design stages and are verified and supplemented with real fluid dynamics (RFD) during flight tests.

During the initial phases of supersonic aircraft wing design, it is common to start with linear theory in CFD to determine geometric configurations and components of the flight vehicle that satisfies the initial 4 design requirements and constraints. The initial design is analyzed and optimized, continually introducing more complex and costly analysis procedures, such as wind tunnel testing and higher-order computational analysis procedures. Figure 3 shows an example of a process used during supersonic aerodynamic design.

Figure 3: Classic Linear Supersonic Aerodynamic Design Process [1]

For the linear modeling of the flight vehicle’s wing during the early stages of design, there are different CFD tools of varying modeling complexity that can be utilized. During the initial design of the wing, it is common to assume an irrotational flow with an inviscid domain to simplify the aerodynamic analysis procedure and reduce the time in predicting results. Applying these assumptions in the analysis will provide conservative results of the aerodynamic forces that are encountered at low angles of attack. One common procedure for solving an inviscid, irrotational supersonic flow is using supersonic vortex lattice method (SVLM). By removing the assumption of an irrotational flow, supersonic inviscid flows can be modeled using the Euler equations for the flow without the need of more physical restrictions. There are methods to solve the Euler equations that can be found in many commercially available CFD solvers. 5

However, the additional complexity required to solve the Euler equations, reduces the overall efficiency in computationally setting up and obtaining supersonic aerodynamic results.

In order to facilitate efficient aerodynamic analysis during the early stages of supersonic flight vehicle design, it is important to know the differences with the analysis formulations, as well as comparing the efforts in obtaining results and the accuracy of those results. The focus of this effort will be using CFD, both SVLM and Euler methods, as the primary aerodynamicist investigative tool (Figure 2), to help facilitate an overall understanding of utilizing simplified solvers for the aerodynamic performance of wings in supersonic flow that are commonly used during the early stages of design.

1.2 Background

This section contains an overview of the relevant fluid mechanics encountered in inviscid, supersonic flows, previous relevant work pertinent to potential modeling of supersonics flow, and the use of

CFD, both SVLM and Euler methods, to solve delta wings in external flows.

1.2.1 Overview of Steady, Inviscid Supersonic Flows and Solution Methods for Wings

The inviscid, or zero viscosity (μ = 0), supersonic flow is still a complex subject, but it is well- studied and covered by very acknowledged authors like John D. Anderson and John J. Bertin [2,3]. These types of flow are associated with high variations in density, pressure, temperature, and the presence of discontinuities in the field (shockwaves). However, steady (time independent) and zero viscosity supersonic flow can accurately be modeled for wings through various methods, as long as the analysis is limited to small angles of attack and a Mach number range that is within the supersonic region (1.2 < Mach < 5.0). It is also important to note, that the Reynolds number is also of significant importance when modelling these flows.

The methods used for steady inviscid supersonic flow analyses are limited to Reynolds numbers that are not large enough in which hydrodynamic characteristics are considerably stronger than the ones usually observed at moderate supersonic flows. For example, flows in which physico-chemical effects are strong and need for inclusions of molecular or thermo-chemical considerations into their modelling. 6

For some simplified flows and simple geometries, there exist closed-form analytical solutions with methods like conical flow, potential flow analysis, and Prandtl-Meyer expansions. However, for complex geometries numerical methods are necessary, and provide solutions that are generally congruent to the experimental counterparts [3]. The most accurate direct numerical simulations (DNS) that solve the volume- discrete Navier-Stokes, or Euler equations, can be very computationally expensive and time consuming.

Many finite-volume and finite-difference methods exist that decrease the computational requirements but increase significantly in complexity. However, for larger domains the requirements can still be excessive and, in some instances, unnecessary. In design, it is common to use lower-order methods to conduct timely preliminary analysis when comparing multiple wing geometries. Potential methods can be very useful in providing timely analyses; specifically, the panel methods, which are similar to boundary elements methods

(BEM) given that a mesh is only required on the boundary of the geometry to be studied, and the method does not need a full fluid volume to be discretized. The panel methods make use of Green’s Theorem, which enables a solution to be obtained at the boundary. These methods have their advantages and disadvantages in designing a supersonic wing, and it is important to carefully consider the formulation of the various methods to know the limitations and levels of accuracy of the predicted aerodynamic results.

1.2.2 Singularity Distribution Methods Modeling and the Supersonic Vortex Lattice Method

For the study of aerodynamics in inviscid supersonic flows, the use of singularity distribution methods (SDM) can be a very powerful tool in analyzing wing performance characteristics. Singularity distribution methods combines elementary flow solutions like sinks, sources, doublets, vortices, etc. to model the flow that is similar to the aerodynamic shapes of interest to generate solutions. Singularity distribution methods are developed by considering the small-perturbation supersonic equations (Equation 1.1), which can be transformed to Laplace’s equation by using a coordinate transformation. Hence, if the potential distribution is known, which is given by the geometry, then closed-form solutions of the flow field can be obtained. This type of analysis assumes that the free-stream velocity is much larger compared to the perturbation velocity components. A mathematical treatment of the elementary flows and the second-order, 7 hyperbolic, partial differential equation, which describes linearized supersonic flow, to obtain results for any wing can be seen in the report by Lomax et al. [4]. The field equation that describes the linearized supersonic flow is given as:

2 (푀∞ − 1)휑푥푥 − 휑푦푦 − 휑푧푧 = 0 (1.1)

where φ is the perturbation velocity potential and M∞ the free-stream Mach number. Furthermore, Lomax et al. states that the velocity potential due to a unit source (Equation 1.2A), a unit elementary horseshoe vortex

(Equation 1.2B), and a unit doublet (Equation 1.2C) are given as:

Source: 휑 = − 푄 (1.2A) 푟푐

Vortex: 휑 = − 푧푄푣푐 (1.2B) 푟푐

푧푄훽2 Doublet: 휑 = 3 (1.2C) 푟푐

where Q is the strength of the singularity and

2 2 2 2 1⁄2 푟푐 = {(푥 − 푥1) + 훽 [(푦 − 푦1) + 푧 ]} ,

2 2 훽 = 푀∞ − 1, and

푥 − 푥1 푣푐 = 2 2 (푦 − 푦1) + 푧

equations 1.2 satisfies Equation 1.1, where point (x1,y1,z1) is the location of the singularity.

There are many ways of using these sources of mass, velocity, and vorticity (Equations 1.2) to model supersonic and subsonic flows [3,5]. In fact, for simple wing planforms exact analytical solutions can be obtained using SDM [6, 7]. However, for this research, the interest lies in using distributions of horse-shoe vortices to obtain the induced flow (and consequently the pressure) on the surface of wings. This method is

8 more commonly known in literature as the vortex-lattice method (VLM). The general methodology used in this study is presented originally in the paper by Middleton and Carlson [8]. However, a latter version of this method that includes a more accurate treatment of the oscillations encountered on the surface pressures is provided by Carlson and Miller [9]. Further studies are carried out with this method to obtain the leading- edge thrust characteristics of different wings [10] and modified to obtain nonlinear aerodynamics characteristics of supersonic wings [11]. It is important to note that Carlson and Mack [10] presented a correction on the aft-element sensing technique shown previously in the report of Carlson and Miller [9].

There are codes that employ VLM for the analysis of the potential subsonic and supersonic regimes of arbitrary configurations [12, 13]. Additionally, VLM can be modified to incorporate higher-order mathematical parameters for additional accuracy; for example, such a modification is incorporated within the

PAN AIR software [13]. A slightly different, but novel, numerical approach that also uses discrete vortical additions to the flow can be shown by Vigano and Maddalena [14]. They use the Helmholtz velocity decomposition to transform the linearized supersonic flow equation (Equation 1.1) to the wave equation and derive and use vorticity-sources to model supersonic flows, an approach first shown by Ferri [15, 16]. The velocity decomposition states that the velocity vector field (푉⃗ ) can be resolved into the sum of an irrotational vector field and a divergence-free vector field, given mathematically as:

푉⃗ = ∇휑 + ∇ × 퐴 (1.3)

Where φ is the scalar potential and 퐴 is the vector potential. As Ferri explains, if the velocity vector is represented using the scalar potential alone (irrotational flow), then Equation 1.1 is transformed to the one- dimensional wave equation in cylindrical coordinates and solutions can be obtained.

1.2.3 Computational Fluid Dynamics Modeling

The Euler equations are another valid set of equations to describe inviscid flows. Equations 1.4 to

1.6 below represent the conservation of mass, momentum, and energy, respectively, in an inviscid field, written in differential form as:

휕휌 + ∇ ∙ (휌풖) = 푆 (1.4) 휕푡 푚

휕 (휌풖) + ∇ ∙ (휌풖풖) = −∇푝 + 휌품 + 푭 (1.5) 휕푡

휕 (휌퐸) + ∇ ∙ (풖(휌퐸 + 푝)) = −∇ ∙ ∑ (h 퐉 ) + 푆 (1.6) 휕푡 푗 푗 푗 ℎ

where the independent variables are x, y, z, and t; and ρ is the density, u is the velocity vector, Sm is any mass source, p is the static pressure, g is the gravity vector, F are any external body forces in vector form, E is the fluid energy defined as E=h-p/ρ+v2/2, h is the enthalpy, v is the velocity scalar, J is the diffusion flux in vector form, Sh is any heat of chemical reaction or any other heat sources, the subscript j is for any other species, and any bolded quantity is a vector.

The Euler equations are the subject of much investigation in the applied mathematics and engineering communities [17]. Solutions can be obtained through various numerical approaches and optimized in a case-by-case basis. These equations can also be used to solve for the flow field around many geometric configurations, and they are usually used to give insight into the fluid mechanics for many applications. Euler equations and solution methods that use finite volume and finite difference methods are also described by authors like John D. Anderson and J. Blazek [18, 19]. They show structured discretization and different methods to solve the resulting system of equations efficiently using advanced numerical methods.

The Euler Equations can be successfully discretized, and solutions can be obtained using finite- volume methods for one, two, or three-dimensions [20, 21], and in as the case of the current study, can be used to model supersonic delta wings [22]. The complex flow features surrounding delta wings adds significant challenges in modeling. Even for structured two-dimensional formulations, dissipative terms in 10 the discretization are sometimes needed in order to eliminate oscillations near high gradients or discontinuities (shocks) [23]. Furthermore, in one study by Chakravarthy and Ota, solutions for conical delta wings with a structured coarse mesh grid significantly differed from a fine grid [24]. If handled appropriately,

Euler equations provide reasonably good estimations for supersonic delta wing aerodynamic performance when compared to experimental data as shown by Volkov et al [25].

A good understanding of numerical methods and the physics present in supersonic flows are necessary to understand the differences between supersonic VLM and a finite volume discretization and how they can correctly be applied to solving for the wing’s overall pressure distribution. The two methods are very different, but the results should show congruency between them. Additionally, the fact that the Euler equations are more general than the potential equations, it is of interest to understand how the assumptions made in linear theory might affect predicting the flow and resulting aerodynamic forces acting on the wing.

1.3 Motivation and Main Objectives.

The motivation for this paper is focused on developing an understanding of CFD tools that are common and advantageous during the conceptual and preliminary design phases for supersonic wings. The primary objectives are to (1) understand and compare two common and readily available methods for estimating surface pressures on supersonic delta wings during the wing’s initial design, and (2) develop a low-order supersonic VLM baseline tool in which future wing design and analysis features can be developed and implemented, for example passive and active flow control.

The current paper will study two common CFD methods that are used for the analysis of supersonic wings. Comparisons will be made in obtaining design-guided conclusions for establishing a conceptual and preliminary CFD/SFD aerodynamic design and analysis tool. To start in meeting this task, a literature review of both methods, SVLM and Euler, was conducted and the summary was discussed in Chapter 1. With the literature review as a foundation, a further mathematical considerations and numerical procedures for both methods to solve the same delta wing under the same conditions is developed and discussed in Chapter 2 for the SVLM and Chapter 3 for the Euler equations. Results for each method, along with comparisons and

11 supportive discussions is presented in Chapter 4. Lastly, conclusions gathered through-out the study will be discussed in Chapter 5, including recommendations for future work.

Chapter 2: Singularity Distribution Method: The Supersonic Vortex Lattice Method

This chapter includes the mathematical and numerical description of the Supersonic Vortex Lattice

Method (SVLM).

2.1 Governing Equations and Boundary Conditions

Carlson and Miller [9] proposed a relatively simple numerical approach of the vortex-lattice method to obtain pressure distributions for wings of arbitrary profiles and planforms using horseshoe-vortex distributions, which adhere to small-perturbation theory assumptions. This method is also provided in detail by Bertin [3].

Consider a thin wing of arbitrary planform and profile imposed in a cartesian plane shown in Figure

4. The origin point lays in the apex and the positive x-axis is directed downstream, the y-axis faces spanwise to the right wing, and the z-axis directed normal to the x-y plane.

Figure 4: Arbitrary Wing Imposed in z=0 plane

If the wing can be assumed to have negligible thickness with its mean camber line laying close to the z=0 plane, then the pressure differential on a rectangular modified Cartesian coordinate system is given by:

4 휕푧 (푥,푦) 1 ∗ ∆퐶 (푥, 푦) = − 푐 + ∬ 푅(푥 − 푥 , 푦 − 푦 ) ∆퐶 (푥 , 푦 ) 푑훽푦 푑푥 (2.1) 푝 훽 휕푥 휋 푆 1 1 푝 1 1 1 1

where zc(x,y) is the mean camber z-coordinates. The function R(x-x1, y-y1) can be thought of as an influence function that relates the local loading at the point (x1,y1) to its Mach-cone of influence on the field and it’s given by:

푥−푥1 푅(푥 − 푥1, 푦 − 푦1) = 2 2 2 2 2 0.5 (2.2) 훽 (푦−푦1) [(푥−푥1) −훽 (푦−푦1) ]

The integral on Equation 2.1 represents the influence of a continuous distribution of horseshoe vortices originating from wing elements with vanishingly small chords and spans. The region of integration,

S, (shown in Figure 5 below ) stretches on the wing planform inside the Mach forecone originating from the field point (x, y). It is important to note that the integral appears to be divergent due to the singularity at y1=y inside the region of integration. This integrand, however, can be treated according to the concept of the generalization of the Cauchy principal value, as shown by the superscript * in the integral. Furthermore, 14 when considering a more general integrand, which stems from lifting-surface theory and outside the limiting case as z approaches to zero, it does not have a singularity at y1=y for z ≠0 [4].

Figure 5: Panel Representation of Flat Plate Delta Wings with Subsonic Leading Edge

Lomax, et.al. [4] contains a thorough derivation of Equation 2.1 with Equation 2.3. Equation 2.3 is a solution to solving the linearized supersonic perturbation velocity potential (Equation 1.1) in normalized form, which is the wave equation. The three-dimensional wave equation can be used with Green’s Theorem to derive Equation 2.3. This solution was originally developed by Volterra and is shown in the report by

Heaslet et. al [26].

1 휕 푥−푥1 휑(푥, 푦, 푧) = − ∬ {∆푤(푥1, 푦1)푎푟푐푐표푠ℎ 2 2 2휋 휕푥 푠 훽√(푦−푦1) +푧

푧(푥−푥1) −∆휑(푥1, 푦1) 2 2 }푑푥1푑푦1 (2.3) [(푦−푦1) +푧 ]푟푐

where Δφ is the jump in potential in space, Δw is the jump of vertical velocity (z-component of velocity) across the z=0 plane; and β and rc are consistent with the definitions presented in Chapter 1 for the whole paper.

The derivation of Equation 2.1 with Equation 2.3 is extensive and requires obtaining the induced velocity (u) of a distribution of elementary horseshoe vortices given by:

푥−푥 푧⋅ 1 1 휕 ( )2 2 푢 = ∬ 푦−푦1 +푧 ∇푢(푥 , 푦 )푑푥 푑푦 (2.4) 2휋 휕푥 푆 2 2 2 2 1⁄2 1 1 1 1 {(푥−푥1) +(1−푀∞)[(푦−푦1) +푧 ]}

With the definition of velocity potential, the derivative of Equation 2.4 is determined with respect to z on both sides of the equation and, also the limit as z approaches zero is taken. Finally, the evaluation of the integrals and limits in the resulting equation is very cumbersome and requires taking a derivative through an integral. This procedure is done with Hadamard’s concept of finite part of an integral, requiring the limit as z approaches zero to be taken first. It also requires the introduction of Cauchy’s generalized principal in part due to the singularities within the integrand. More information on the Cauchy Principal value and

Hadamard’s concept finite part of infinite integral implementations on the method derivation can be seen

Lomax, et.al. [4].

The boundary condition that imposes the flow to remain tangent to the surface of the wing has already been implemented in Equation 2.1, and determined mathematically using the perturbation potential:

( 휑푧 ) = 푑푧푠(푥,푦) (2.5) 푈∞ 푧=0 푑푥

16 where zs(x,y) are the wing’s surface coordinates, φz is the z perturbation potential, and U∞ is the free-stream velocity.

If a sharp subsonic trailing-edge is present, the Kutta condition must be applied. The Kutta condition ensures that the pressure at the sharp trailing-edge of lower and upper surfaces are equal and creates a stagnation point at the trailing-edge. So, in mathematical terms, imposing that

퐶푝푢푡푒 = 퐶푝푙푡푒 (2.6)

where Cp is the pressure coefficient and subscripts ute and lte represent the upper and lower trailing-edge points respectively. Equation 2.6 guarantees that the local lift at the sharp subsonic trailing-edge is zero.

2.2 Numerical Approach

The continuous Cartesian-like domain will be discretized with the grid system shown in Figure 5.

This will allow for the integral in Equation 2.1 to be replaced with a numerical algebraic summation. The region of integration, previously bounded by the wing leading edge and the Mach forecone originating at

(x,y), is replaced by a discrete domain with rectangular panels bounded by the discretized wing leading edge and the Mach forecone originating at (L,N) (shown in Figure 5, as an example). The inclusion of full and partial grid elements, through the coefficients A, B, and C, allows for a better representation of the leading- and trailing-edges and tend to reduce irregularities that arise in local surface slopes (or in local pressure coefficient in this case) for elements near the leading-edge. It is important to note that in the (x, βy) coordinate system, the Mach influence and fore cone angle is always 90o.

The numbers L1 and N1 replace the element of integration d(x1) and d(βy1) respectively. L and N are the elements associated with, and immediately in front of, the field point (x, βy). Specifically, L=x and

N= βy where x is rounded down to the nearest integer and βy rounded to the nearest integer. Consequently, the discrete element takes the shape of a flat rectangle that extends from βy=N-0.5 to N+0.5 and from x=L-

1 to L and the control point lays on the trailing-edge and midspan.

The influence function (Equation 2.2) will be averaged along the discrete element. It’s important to note, that by numerical evaluation it can be observed that R is relatively insensitive to variations in x1 and will be approximated, hence the function will be averaged along βy1 only. The difference (x-x1) will be represented at the midpoint of the element with L-L1+0.5. So, the averaged R function is given by:

2 2 2 2 √(퐿−퐿1+0.5) −(푁−푁1−0.5) √(퐿−퐿1+0.5) −(푁−푁1+0.5) 푅̅(퐿 − 퐿1, 푁 − 푁1) = 푅푒푎푙 { − } (2.4) (퐿−퐿1+0.5)(푁−푁1−0.5) (퐿−퐿1+0.5)(푁−푁1+0.5)

The behavior of R can be best seen on Figure 6 below.

Figure 6: Numerical Representation of Influence Factor R

For a given L-L1 set of elements, the spanwise summation of R values is zero, which ensures that the flow

field has equal amounts of upwash and downwash along the wing. At L-L1=0, where there is only one element

18 in the spanwise summation, the value of R is zero. In other words, the element has no influence on itself.

Lastly, it is clear to see that the biggest influence is for panels directly behind (N-N1=0) and one panel to the right or left (N-N1=1 or -1).

The numerical representation of Equation 2.1 is:

4 휕푧 (퐿, 푁) ∆퐶 (퐿, 푁) = − 푐 푝 훽 휕푥

푁푚푎푥 퐿−|푁−푁1| 1 + ∑ ∑ 푅̅(퐿 − 퐿 , 푁 − 푁 )퐴(퐿 , 푁 )퐵(퐿 , 푁 )퐶(퐿 , 푁 )∆퐶 (퐿 , 푁 ) 휋 1 1 1 1 1 1 1 1 푝 1 1 푁1=푁푚푖푛 퐿1=퐿퐿퐸

(2.5)

where the limits of L1 in the summation are those of the wing leading-edge LLE=1+[xLE], with the square brackets designating that only the whole-number part of the quantity enclosed of the Mach forecone at the selected N1 value is used. The limits of N1 are those where the Mach forecone intersects the leading-edge.

The vertical lines are the absolute value of the enclosed quantity.

The factors A(L1,N1), B(L1,N1), and C(L1,N1) are weighting factors for the partial consideration of wing leading-edge, trailing-edge, and tip elements, respectively, which allows for a better definition of the wing and range from 0 to 1. The exact definitions of these is given as:

퐴(퐿1, 푁1) = 0 퐿1 − 푥퐿퐸 ≤ 0 퐴(퐿1, 푁1) = 퐿1 − 푥퐿퐸 on 0 < 퐿1 − 푥퐿퐸 < 1 퐴(퐿1, 푁1) = 1 퐿1 − 푥퐿퐸 ≥ 1

퐵(퐿1, 푁1) = 1 퐿1 − 푥푇퐸 ≤ 0 퐵(퐿1, 푁1) = 1 − (퐿1 − 푥푇퐸) on 0 < 퐿1 − 푥푇퐸 < 1 퐵(퐿1, 푁1) = 0 퐿1 − 푥푇퐸 ≥ 1

0.5 푁1 = 푁푚푎푥 퐶(퐿1, 푁1) = for 1 푁1 ≠ 푁푚푎푥 19

In order to determine the differential pressure of an arbitrary field point, ΔCp(L,N), the calculations need to be carried out in a particular sequence. An apex to rearward sequence (increasing x or L) will ensure that every ΔCp(L1,N1), residing inside the Mach forecone originating at ΔCp(L,N), is readily defined and computed. However, the computed ΔCp(L,N) will still show large oscillations. Carlson and Miller [9] introduce an aft-element integral smoothing procedure. The procedure is outlined below:

1. Calculate and temporarily retain ΔCp for a given L=constant spanwise row of elements. Designate

these values as ΔCp,a(L,N).

2. Calculate and temporarily retain ΔCp for the following spanwise row of elements (L=constant +1)

but only those that directly downstream using the ΔCp,a (L=constant). Designate these values

ΔCp,b(L,N).

3. Calculate a final, smoothed ΔCp values for the L=constant row from a fairing of integrated ΔCp,a

and ΔCp,b results. For leading edge elements, defined by the inequality L - xLE(N) ≤ 1, and modified

from the original paper to reflect the corrections given in Carlson and Mack [10], the fairing

operation is given as

∆퐶 (퐿, 푁) = 1 [1 + 퐴(퐿,푁) ] ∆퐶 (퐿, 푁) + 1 [ 1 ] ∆퐶 (퐿, 푁) (2.6) 푝 2 1+퐴(퐿,푁) 푝,푎 2 1+퐴(퐿,푁) 푝,푏

For all other elements (defined as L - xLE(N) > 1) the fairing operation is:

∆퐶 (퐿, 푁) = 3 ∆퐶 (퐿, 푁) + 1 ∆퐶 (퐿, 푁) (2.7) 푝 4 푝,푎 4 푝,푏

If the aforementioned precautions and steps are followed, then the pressure distribution of a wing with negligible thickness that abides by the assumptions of inviscid, irrotational flow can be calculated for wings of arbitrary planform and profile.

Once the surface pressure distribution is known, the aerodynamic coefficients for the geometry can be obtained. First, however, new weighting factors for the partial consideration of wing leading-edge, trailing-edge, and tip elements need to be computed. The leading-edge (subscript LE in equations) field- point-element weighting factor has a range of 0 to 1.5 and is defined as:

∗ 퐴 (퐿, 푁) = 0 퐿 − 푥퐿퐸 ≤ 0 ∗ 퐴 (퐿, 푁) = 퐿 − 푥퐿퐸 + 0.5 on 0 < 퐿 − 푥퐿퐸 < 1 ∗ 퐴 (퐿, 푁) = 1 퐿 − 푥퐿퐸 ≥ 1

Similarly, the trailing-edge (subscript TE in equations) field-point-element weighting factor has a range of

0 to 1.5 and is given by:

∗ 퐵 (퐿, 푁) = 1 퐿 − 푥푇퐸 ≥ 0 ∗ 퐵 (퐿, 푁) = 1 − (퐿 − 푥푇퐸) on 0 > 퐿 − 푥푇퐸 > −1 ∗ 퐵 (퐿, 푁) = 0 퐿 − 푥푇퐸 ≤ −1

with the center-line or wing tip -grid-element weighting factor given as:

퐶∗(퐿, 푁) = 0.5 푁 = 0 ∗ 퐶 (퐿, 푁) = 1 on 0 < 푁 < 푁푚푎푥 ∗ 퐶 (퐿, 푁) = 0.5 푁 = 푁푚푎푥

A wing area (S) can be calculated using these weighting factors with the following summation:

푆 = 2 ∑푁=푁푚푎푥 ∑퐿=1+[푥푇퐸] 퐴∗(퐿, 푁)퐵∗(퐿, 푁)퐶∗(퐿, 푁) (2.8) 훽 푁=0 퐿=1+[푥퐿퐸]

where the square brackets designate that only the whole-number part of the quantity enclosed is used.

With the weighting factors and the area of the wing determined, the coefficients for lift, drag, and moment about x=0 can be calculated with the following equations, respectively:

퐶 = 2 ∑푁=푁푚푎푥 ∑퐿=퐿푇퐸[3 ∆퐶 (퐿, 푁) + 1 ∆퐶 (퐿 + 1, 푁)]퐴∗(퐿, 푁)퐵∗(퐿, 푁)퐶∗(퐿, 푁) (2.9) 퐿 훽푆 푁=0 퐿=퐿퐿퐸 4 푝 4 푝

푁=푁푚푎푥 퐿=퐿푇퐸 −2 3 1 퐶 = ∑ ∑ [ ∆퐶 (퐿, 푁) + ∆퐶 (퐿 + 1, 푁)] 퐷 훽푆 4 푝 4 푝 푁=0 퐿=퐿퐿퐸

∗ [3 휕푧푐 (퐿, 푁) + 1 휕푧푐 (퐿 + 1, 푁)]퐴∗(퐿, 푁)퐵∗(퐿, 푁)퐶∗(퐿, 푁) (2.10) 4 휕푥 4 휕푥

퐶 = 2 ∑푁=푁푚푎푥 ∑퐿=퐿푇퐸 (퐿)[3 ∆퐶 (퐿, 푁) + 1 ∆퐶 (퐿 + 1, 푁)]퐴∗(퐿, 푁)퐵∗(퐿, 푁)퐶∗(퐿, 푁) (2.11) 푀 훽푆푐 푁=0 퐿=퐿퐿퐸 4 푝 4 푝

It must be noted that an extra N-wise row of values after the trailing-edge (Lmax+1) containing zeros needs to be added so the evaluation of the coefficients at the trailing-edge can be done.

Even with the linearity, irrotationality, and inviscid assumptions, the flow can be modeled relatively accurately and timely for flows from around 1.2 ≤ 푀∞ ≤ 5. Nevertheless, it is a linear method and it does not consider contributions of the leading-edge-suction force, nor any flow separation effects. The supersonic vortex lattice method accounts only for the inclination of the normal force to the relative wing. This procedure was developed as a baseline design tool using MATLAB, and incorporated variability in defining geometry and panel discretization. Results are shown and discussed in Chapter 4.

Chapter 3: CFD Modeling

This Chapter will provide a description of the governing equations, boundary conditions, and the mesh used in the computational fluid dynamics.

3.1 Governing Equations of Fluid Flow

In order to best compare the potential flow method with a finite volume method, the steady, inviscid, three-dimensional compressible Navier-Stokes equations of the fluid flow, commonly referred as the Euler

Equations, will be solved. The mass and momentum conservation laws, obtained from the ANSYS® Fluent theory guide [27], are given in the conservation form by equations 3.1 and 3.2, respectively.

휕휌 + ∇ ∙ 휌풖 = 0 (3.1) 휕푡

휕 (휌풖) + ∇ ∙ (휌풖풖) = −∇푝 (3.2) 휕푡

Given that the flow is compressible, the density of the flow will be calculated using the ideal gas law and the coupling between flow velocity and static temperature is already incorporated in the conservation of energy equation given by:

휕 (휌퐸) + ∇ ∙ (풖(휌퐸 + 푝)) = −∇ ∙ (h퐉) (3.3) 휕푡

3.2 Computational Modeling and Boundary Conditions

As explained in chapter one, the computational analysis of supersonic flows is often complicated. One of the most common hurdles to overcome in the study of flat plate delta wings inviscid modelling is the presence 23 of discontinuities in the flow in the form of shockwaves. Shockwaves impede convergence given that the properties of the flow ‘jump’ across the discontinuity, which increase the Euclidean norm (residuals) of the solution and in some cases the solution may even diverge. Additionally, given the hyperbolic nature (in time) of the Euler equations in high Mach numbers, the sizing of the domain becomes of particular importance.

Domains that are not correctly sized may result in inaccurate solutions for external flows.

The compressibility effects and high Mach number requires a density-based solver to be utilized.

Hence, the Euler equations will be iterated to steady-state, in contrary to using a time independent formulation to describe the flow. Furthermore, the limited computational power provided, and the desire to obtain a faster solution dictates, that the use of an explicit time marching algorithm is required. The marching algorithm, however, has a higher risk of divergence compared to the implicit formulation [28]. In order to aid convergence, a smaller time step will be used by modifying the Courant-Friedrichs-Lewy (CFL) number until a monotonical reduction of the residuals is obtained (around 1000 iterations). The time step is computed from the CFL number and given by:

2∗퐶퐹퐿∗푉 ∆푡 = 푚푎푥 (3.4) ∑푓(휆푓 ∗퐴푓)

max where V is the cell volume, Af is the face area, and λf is the maximum of the local eigenvalues originating from preconditioned system of equations (Euler equations) being solved. After the first reduction of residuals are obtained, the time-step will be increased by using Fluent’s default CFL number for a faster convergence and iterated until the solution is converged.

Convergence will be determined with two conditions. The first one is by monitoring the scaled residuals of the energy equation, the three components of flow velocity equation, and the continuity equation until they stop decreasing in magnitude. Second, and to monitor a properly converged solution, the lift coefficient at the walls of the wings will be observed, and the solution will be considered done when lift coefficient at the wing surface is invariant for a significant number of iterations. This will signify that steady

24 state has been reached, given that the force encountered in the surface of the wing (lift) is not changing with each time-step.

The convective fluxes will be recovered using the Advection Upstream Splitting Method (AUSM), giving that it provides exact resolution of shock discontinuities [27]. The flux-vector splitting scheme computes the flux vector by calculating a cell interface Mach number based on the characteristic speeds of neighboring cells and then used to determine the upwind extrapolation for the convection part of the inviscid fluxes. This will be paired with a first order upwind spatial discretization for the flow and a Green-Gauss node-based for the derivatives of the convection and diffusion terms, which, as explained by the Fluent’s theory guide, is more accurate for irregular unstructured meshes. This node-based gradient evaluation uses the arithmetic average of the nodal values to calculate the value on the face,

̅ 1 푁푓 ̅ 휙푓 = ∑푛 휙푛 (3.5) 푁푓

where 휙̅푓 is the face value, Nf is the number of nodes in the face, and 휙̅푛 are the nodal values.

It is important to note that an effort was made to use a second order upwind spatial discretization, however it is believed that the high gradients found across the shock wave make the second order discretization diverge. This could be mitigated by adding artificial dissipation, but because a quick solution that compares to the potential formulations is desired a first order will be used instead.

The boundary conditions can be best seen in Figure 7. The problem is set to operate at a height of

15,000 meters, hence the pressures and temperatures at the inlet, outlet, and far-field walls will be set to such standard atmospheric conditions. Density will be obtained using the ideal gas law. The inlet boundary and the top, side, and bottom walls will be set to pressure far-field boundary condition with a Mach velocity of

1.5 flowing in the positive x-direction. The outlet boundary will be set to a pressure outlet. The wing surfaces will be defined as stationary walls with no normal flow velocity and the domain symmetry wall defined as symmetry boundary condition.

Figure 7: Picture of Fluid Domain with Boundary Conditions

3.3 Fluid Domain and Grid

The discrete representation of the wing carries obstacles that relate to the three-dimensionality of the approach, which are not encountered when producing a wing with the panel method approach. In order to produce oblique shockwaves that remain attached, a very thin plate of 0.1 m in thickness with a sharp leading-edge is created. Even though there will be some separation of the shockwave, the sharp edges help the wave remains as attached as possible. The fluid domain is only around half the delta wing due to its symmetric nature. The rectangular region extends 10 chord lengths from the apex, upper surface, and lower surface of the wing, and 15 chord lengths after the trailing-edge and wing tip, best seen in Figure 8 and Figure

9. The large size of the domain ensures that the far-field conditions imposed by the pressure far-field boundary conditions is successfully met.

Figure 8: Z-X Plane View of Meshing Domain with Dimensions

Figure 9: Y-X Plane View of Meshing Domain with Dimensions 27

The patch independent tetrahedral mesh for the present external flow problem was generated with

ANSYS® MeshingTM. Two closed surfaces wrap the halved wing to allow for a body of influence sizing using the smaller cuboid. Additionally, to increase cell quality at the walls, a sizing restriction was imposed at the surface of the wing. The sharp edges present in the wing necessitate a defeaturing of 0.02 meters to successfully mesh the surface and edges of the plate.

To obtain a suitable grid size, which accounts for the limitations in computational resources and that the results over the wing are independent of grid size, a grid independence study was done by comparing z- component of the integrated pressure on the surface of the wing obtained with different mesh sizes. Different meshes were obtained using ANSYS® Workbench 2019 by modifying all sizing restrictions evenly, increasing them by a factor of 1.5 for each mesh. The parametric study results are shown in Table 1. A plot of the number of cells and the z-component of the surface pressure on the wing is show in Figure 10. Only meshes 1 through 5 where plotted. Mesh 6 provides a mesh that has a lot of elements, but they have very bad quality, hence giving an inaccurate result that lies outside the trend. Decreasing the sizing restrictions even more than Mesh 1 would provide a mesh that is prohibitively too large (too many elements) and, as shown by Mesh 7, increasing the sizing restrictions more than Mesh 6 caused a mesh failure.

Table 1: Mesh Independence Study Results

Wing Trailing Body of Mesh Pressure (Z- Face edge Face Mesh Influence Max Mesh Mesh component) on Sizing Sizing Element Element Element Elements Nodes Wing surface Element Element Size (m) Size (m) Size (m) (Pa) Size (m) Size (m)

Mesh 1 0.25 0.05 7.5 75 75 7,264,975 1,288,660 8,406,601.30 Mesh 2 0.375 0.075 11.25 112.5 112.5 3,098,784 555,777 8,400,829.60 Mesh 3 0.5 0.1 15 150 150 1,726,094 311,763 8,392,035.60 Mesh 4 0.625 0.125 18.75 187.5 187.5 1,484,603 271,782 8,397,908.50 Mesh 5 0.75 0.15 22.5 225 225 1,077,037 196,733 8,428,680.90 Mesh 6 0.875 0.175 26.25 262.5 262.5 2,456,774 425,893 8,373,078.40 Mesh 7 1 0.2 30 300 300 1,983,664 341,219 Mesher Failure

Grid Independece Study 8,435,000 8,430,000 8,425,000 8,420,000 8,415,000 8,410,000 8,405,000

[Pa] 8,400,000 8,395,000

8,390,000 component) component) surface on Wing - 0 2 4 6 8 Millions

Presure Presure (Z Number of Mesh Elements

Figure 10: Mesh Independence Study Plot

From Figure 10, it can be observed how the pressure becomes relatively invariant after around 3 million cells. Hence, the selected grid, highlighted in Table 1 and shown in Figures 11 and 12, was used for the analysis. The results are shown and discussed in Chapter 4.

Figure 11: Mesh of Fluid Domain

Figure 12: Detailed View of the Mesh Near the Wing

Chapter 4. Results

The following chapter will contain results of the modeled flat plate delta wing with the vortex panel method coded in MATLAB and the Euler Volume discrete equations solved with ANSYS® Fluent.

4.1 Supersonic Vortex Lattice Method

A delta wing, shown in Figure 13, with the leading edge defined as βcot(ΛLE)=0.6 and in an airspeed of Mach number of 1.5 is discretized with the method outlined in Chapter 2. A maximum number of 50 panels (Nmax=50=βymax) was used in the span-wise direction, translating to a total number of 2176 of panels used in the simulation. The panel method obtained a solution of the surface pressure very quickly (in the order of seconds), and the results for the half the wing is shown in Figure 14 and Figure 15.

Figure 13: Flat Plate Delta Wing with Leading Edge βcot(ΛLE)=0.6

Figure 14: Pressure Coefficient Distribution of Flat Plate Delta Wing

Figure 15: Top View of Pressure Coefficient Distribution of Flat Plate Delta Wing 32

Using the calculated surface data, now the lift coefficient, drag coefficient, and moment about X=0 can be computed using Equation 2.9, Equation 2.10, and Equation 2.11, respectively. Results are seen in the table below for an angle of attack (AoA) of 2.5 degrees and at an airspeed of Mach 1.5.

Table 2: Aerodynamics Coefficients for Delta Wing with Leading Edge βcot(ΛLE)=0.6 at Mach=1.5 and

AoA=2.5 deg.

Aerodynamic Coefficients for Delta Wing

Lift Coefficient (CL) 0.1064

Drag Coefficient (CD) 0.0046

Moment Coefficient about apex (CM) 0.1153

With a simple multiplication, these results can be extended for an angle of attack range within small angle approximation. Hence, aerodynamic plots of the subsonic leading-edge wing with leading-edge factor of 0.6 ( βcot(ΛLE)=0.6) is provided in Figure 16:

Figure 16: Sub plots of Lift Coefficient vs. AoA, Drag Polar, Moment Coefficient abt. Apex vs. AoA, and

Lift-to-Drag vs. Lift Coefficients of Flat-Plate Delta Wing with Subsonic Leading-Edge

Considering the lift characteristics (Figure 16, top left), the results for different airspeeds can be obtained for this wing very easily and are shown in Figure 17:

Figure 17: Lift-Curve Slope vs. Free-Stream Mach Number for Delta Wing with Subsonic Leading-Edge 34

The effect of the leading-edge factor on the wing’s leading-edge can be best seen in Figure 18. It is important to note than the maximum number of spanwise panels was also modified to make all the delta wings the same length. The most notable takeaway from Figure 18 is how the leading-edge sweep angle

(ΛLE) increases with decreasing leading-edge factor for the same airspeed.

Figure 18: Delta Wings with Different Leading-Edge Factors

The most notable strengths of SVLM are the small number of elements used in the calculations and the fast-computational time it takes to resolve a wing. Allowing for the same calculations to be done for a vast number of different wings or big range of Mach numbers relatively fast. As an example, a full envelope of the family of flat plate delta wigs with different leading-edge factors (βcot(ΛLE)=leading-edge factor) and airspeeds in the supersonic flight regime is shown in Figure 19. All the wings had the same maximum number of spanwise panels of 50 (Nmax=50). 35

Figure 19: Lift-Curve Slope Envelope for Flat-Plate Delta Wings

The results of lift obtained for each wing independently are linear, as seen in Figure 16, top left, indicating that the method cannot predict non-linearities like trailing-edge separation and detached leading- edge vortices. Hence, for wings at off-design conditions, when the leading-edge factor is far away from 1, the results are very conservative. The drag coefficient is also expected to be lower than the experimental

(EFD) or ‘real’ results because the SVLM cannot predict viscous drag nor any drag caused by non-linearities or leading-edge vortices. However, in general the pressure solutions show very good congruency with linear theory as shown by the original paper of Carlson and Miller [9].

4.2 Euler equations CFD

The same flat plate delta subsonic leading-edge (with leading-edge factor of 0.6) at an angle of attack of 2.5 degrees was used for this analysis. The volume around the wing was discretized and solved using ANSYS® Fluent with the methods shown in Chapter 3. After around 2,100 iterations (about 3 hours) the simulation converged, the residuals dropped below the desired magnitude and the lift coefficient was unchanged for about 200 iterations. Results of the aerodynamic forces obtained from Fluent are given in

Table 3.

Table 3: Aerodynamics Coefficients for Delta Wing of Subsonic LE at Mach=1.5 and AoA=2.5 deg. with

Euler formulations

Aerodynamic Coefficients for Delta Wing

Lift Coefficient (CL) 0.1188

Drag Coefficient (CD) 0.0065

Moment Coefficient about apex (CM) 0.1259

With ANSYS® CFD-Post, the post-processing program, contours of values can be easily obtained at the wing surface. Pressure contours are shown in Figure 20 for the top surface and in Figure 21 for bottom surface of the delta wing.

Figure 20: Pressure Contour on Suction Face of Delta Wing

Figure 21: Pressure Contour on Pressure Face of Delta Wing

One of the biggest advantages of finite volume methods in general is that we also have solutions for the flow in the vicinity of the geometry analyzed. Hence, we can observe flow structures around the wing if 38 desired. Velocity contours at different locations in the vicinity of the wing are shown Figures 22 to 25. The vectors (not to scale) on the contours show the tangential velocity direction on the contour ZY-plane.

Figure 22: Magnitude of Velocity Contour at X=0.5c with Tangential Velocity Vectors

Figure 23: Magnitude of Velocity Contour at X=0.75c with Tangential Velocity Vectors 39

Figure 24: Magnitude of Velocity Contour at X=c with Tangential Velocity Vectors

Figure 25: Magnitude of Velocity Contour at X=1.25c with Tangential Velocity Vectors

From Figures 22 to 25 velocity vectors, a vortical structure can clearly be seen on the leading-edge of the delta wing. Vortices are expected on subsonic leading-edge delta wings, especially on off-design conditions.

4.3 Comparison of Methods

The first thing to note is the difference in percentage from the aerodynamic values obtained with the methods for the delta wing at AoA of 2.5o and Mach of 1.5.

Table 4: Percentage Difference of Aerodynamic Coefficients for Delta Wing obtained with Euler

Formulations and SVLM Methods

Percentage Difference of Aerodynamic Coefficients for Delta Wing

Lift Coefficient (CL) 11.01%

Drag Coefficient (CD) 34.23%

Moment Coefficient about apex (CM) 8.79%

The most surprising result is how close the moment coefficient values are. Differences in drag were expected because potential methods cannot predicting drag very accurately can be predicted with Euler FVM.

The lift coefficient is relatively close. The difference could be the leading-edge vortex, shown in the Euler equation results, that gives additional lift through the leading-edge suction force.

The differential pressure coefficients at 3 different X-locations (shown in Figure 26) of the delta wing are shown in Figure 27 for X=0.25C, Figure 28 for X=0.5C, and Figure 29 for X=0.75C.

Figure 26: Delta Wing with Three Different Chord Locations

Figure 27: Pressure Coefficient Plot at 25% of Chord for Delta Wing for Euler CFD and SVLM 42

Figure 28: Pressure Coefficient Plot at 50% of Chord for Delta Wing for Euler CFD and SVLM

Figure 29: Pressure Coefficient Plot at 75% of Chord for Delta Wing for Euler CFD and SVLM 43

The differential pressure coefficient at the camber shows very good agreeance between the methods.

The biggest discrepancy is at the leading-edge, where both methods show oscillations of the pressure coefficient. For the Euler method, this could be because the pressure gradient is the greatest at the leading- edge, where suction forces and vortices are present. For SVLM oscillations are expected near the leading- edge where the pressure spikes. Concluding remarks about both methods are discussed in the following chapter.

Chapter 5. Conclusions

Chapter 5 will finish off this paper with conclusions obtained from the results seen in Chapter 4 and future work recommendations.

5.1 Conclusions

Supersonic flows have very complex fluid flow characteristics like shockwaves, expansion fans, supersonic vortices, supersonic turbulence, vortex shedding, vortex bursting, and interactions between all these various components. For the preliminary design of supersonic wings, where it is desired to efficiently obtain aerodynamic characteristics for different wings, modeling to capture all of these non-linearities can be very complicated, time consuming, and computationally extensive. Hence, the use of simplifications on these models with CFD can be very valuable to save time and resources. Euler and SVLM are two commonly used CFD/SFD aerodynamicist tools that can provide good measures for lift and wave drag coefficients for the early stages of design. Euler finite volume formulations are conservation equations in an inviscid domain and SVLM use linear, inviscid, and irrotational equations that describe supersonic flow.

The SVLM is an easy and fast method that gives results that are very accurate to the linear theory and comparable to the Euler formulations. The underlying math is complicated, but the applications is very simple. Discretization is almost trivial and obtaining solutions are extremely computationally efficient, given that only the boundary is discretized. The results obtained with this method allows for the extension on multiple airflow and wing conditions and can even be easily extended to wings with similar geometry with little effort, while still being more efficient than the Euler method. The method also allows for non-linear behaviors (separation, leading-edge suction) to be added, due to the linear nature of the method.

In contrast, Euler methods are much harder to set up. Meshing can be very tedious and time consuming, and depending on the geometry, mesh independence studies may need to be carried out. This 45 could result in complicated geometries requiring a very large number of volume elements, which directly increases the computational cost of each analysis. A deep understanding of the fluid mechanics encountered need to be known a priori in order to select the right methods to solve the equations and to mesh correctly.

Convergence can be slow and is never guaranteed, especially for supersonic shock-capturing analyses.

However, all this work comes with big rewards. If an analysis is carried out correctly, fluid results for the entire domain can be analyzed, flow structures can be seen, and more information is available than that provided by SVLM.

In a design set-up, where multiple wings at multiple angles of attack and different airspeeds need to be simulated and only surface pressures and forces are needed, it is clear that SVLM is significantly more advantageous. Furthermore, SVLM can be extended to account for non-linearities, and could be adapted for additional analysis features, like passive and active flow control impacts, which would be a significant challenge for Euler methods, given the necessity to mesh every single design change. An automation and extension for design could also be done for Euler finite volume methods, if the computational expense is not an issue.

5.2 Future Work Recommendations

The supersonic vortex lattice method works well for a low-order baseline wing design and analysis tool, and has the potential for significant adaptations for more complex design features. Due to the linear nature of the method, features can be developed and super-imposed with the existing code. In addition to being able to account for any geometry and camber, further fluid mechanic features like leading-edge suction, separation, and viscous effects can be added to the method with relatively low effort. This powerful tool also has the potential to model passive and active flow control and can be adapted for many other aircraft design components and applications.

The results presented in this work only consider inviscid formulations. A good extension of this study would be the comparisons of the results with viscous formulations and experimental data in order to observe the magnitude of error in the prediction of the aerodynamic coefficient and surface pressure.

Furthermore, adding non-linear behaviors to SVLM and again comparing with wind tunnel data or/and viscous finite volume formulations, would also be advantageous.

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