Vorticity Confinement Applied to Induced Drag Prediction And

VORTICITY CONFINEMENT APPLIED TO INDUCED DRAG PREDICTION AND

THE SIMULATION OF TURBULENT WINGTIP VORTICES FROM FIXED AND

ROTATING WINGS

A Thesis

Presented to

The Graduate Faculty of The University of Akron

In Partial Fulfillment

of the Requirements for the Degree

Master of Science

Kristopher C Pierson

May, 2014

VORTICITY CONFINEMENT APPLIED TO INDUCED DRAG PREDICTION AND

THE SIMULATION OF TURBULENT WINGTIP VORTICES FROM FIXED AND

ROTATING WINGS

Kristopher Pierson

Thesis

Approved: Accepted:

______Advisor Dean of the College Dr. Alex Povitsky Dr. George K. Haritos

______Faculty Reader Dean of the Graduate School Dr. Scott Sawyer Dr. George R. Newkome

______Faculty Reader Date Dr. Minel Braun

______Department Chair Dr. Sergio Felicelli

ABSTRACT

To improve accuracy of CFD simulations by combating numerical viscosity, the vorticity confinement (VC) approach was applied to tip vortices shed by stationary wings for the prediction of induced drag by far-field integration in Trefftz plane as well as the simulation of tip vortex evolution. Vorticity confinement schemes are first evaluated and compared using 2D Euler simulation of a Taylor vortex. The effect of numerical dissipation is shown to cause the vortex to spread and decay without the presence of physical dissipative terms; VC counteracts this effect and maintains the vortex strength.

For 3D simulations, the coefficient of confinement was determined for various flight parameters. Dependence of vorticity confinement parameter on the flight Mach number and the angle of attack was evaluated. The aerodynamic drag results with VC are much closer to analytic lifting line theory compared to integration over surface of wing.

To apply VC to viscous and turbulent flows, it is shown that in many cases VC does not affect the physical rate of dissipation of line vortices for temporal and viscous scales relevant to wingtip vortex simulations. VC was applied to the simulation turbulent wingtip vortices originating from fixed and rotating wings. The six Reynolds stress turbulence model combined with VC most accurately matched experimental results of tip vortex evolution.

iii

ACKNOWLEDGEMENTS

I would like acknowledge the financial support and guidance from DAGSI/AFRL Ohio

Student-Faculty Research Fellowship in 2011-2013 sponsored by Drs. J. Camberos and

L. Downie, Army Research Office in 2012-2013 by ARO program director Dr. F

Ferguson, and ASEE/AFRL summer faculty grant (AFRL/WPAFB, Dayton, OH) in

2012. Also, DAGSI/AFRL was the sponsor of previous work at the University of Akron from 2009-2011 that was built upon in the present research. Finally, I would like to thank Dr. Povitsky for his guidance and support that made this research possible.

TABLE OF CONTENTS Page

LIST OF TABLES ...... vii

LIST OF FIGURES ...... viii

NOMENCLATURE ...... x

CHAPTER

I. INTRODUCTION ...... 1

1.1 Background ...... 1

1.2 Finite Volume Method ...... 4

1.3 Upwind Schemes and TVD ...... 5

1.4 Vorticity Confinement ...... 8

1.5 Wake-Integral Method ...... 12

1.6 Turbulence Modeling ...... 13

1.7 Research Goals...... 15

II. SIMULATION OF 2-D VORTICES USING VORTICITY CONFINEMENT AND TVD ...... 17

2.1 2D Code Description...... 17

2.2 Vorticity Confinement implementation and Formulations ...... 20

2.3 Results of Second Order Euler Simulations without TVD ...... 21

2.4 Comparison of TVD Schemes ...... 26

2.5 Decay of Line Vortex, Viscous Test Case ...... 29

III. INDUCED DRAG PREDICTION SIMULATIONS USING WAKE- INTEGRAL COMBINED WITH VORTICITY CONFINEMENT...... 33

3.1 Implementation of VC in ANSYS FLUENT ...... 33

3.2 Lifting Line Theory...... 34

3.3 Problem Description ...... 35

3.4 Results ...... 37

3.5 Differentiable Limiter ...... 42

IV. TURBULENT SIMULATIONS OF FIXED WINGTIP VORTICES USING VORTICITY CONFINEMENT ...... 46

4.1 Evaluation of Viscous Drag ...... 46

4.2 Results of Turbulent 3D Fixed Wing Simulations ...... 48

4.3 Comparison to Experiements ...... 51

V. TURBULENT SIMULATIONS OF WINGTIP VORTEX COMBINED WITH VORTICITY CONFINEMENT ...... 57

5.1 Grid and Setup ...... 57

5.2 Inviscid Results ...... 59

5.3 Turbulence Model Results ...... 61

5.4 Winglet Modeling and Results...... 63

VI. CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK ...... 66

BIBLIOGRAPHY ...... 69

APPENDICES ...... 72

APPENDIX A. FLUX SPLITING FOR STEGER WARMING SCHEME ...... 73

APPENDIX B. 2D MATLAB CODE ...... 75

APPENDIX C. ANSYS FLUENT UDF ...... 82

LIST OF TABLES

Table Page

3.1 Solver settings, model settings and solution methods for inviscid simulations .. 37

3.2 Drag coefficients by surface integration and wake-integral method compared to lifting line theory ...... 39

3.3 Tuned confinement parameters ...... 39

4.1 Flow characteristics for boundary layer in flat plate ...... 47

4.2 Results for near-field and far-field integration for viscous flow about flat plate, error is determined by comparison to Blasius solution for shear stress ...... 48

4.3 Shear force and induced drag results for turbulent simulations ...... 51

5.1 Physical problem specifications for rotating wing simulations ...... 58

5.2 Computational parameters for rotating wing simulations ...... 58

vii

LIST OF FIGURES

Figure Page

2.1 Positive flux extrapolation used in second order scheme ...... 19

2.2 First order simulation of Taylor vortex ...... 23

2.3 Upwind second-order simulation of Taylor vortex with and without VC ...... 24

2.4 Results of second order calculations using VCF2 with c=4.5 ...... 26

2.5 Comparison of TVD Schemes, c=4.0 ...... 28

2.6 Comparison of TVD Schemes, c=5.0 ...... 29

2.7 Decay of line vortex, second-order minmod TVD simulation comparison of epsilon values; Δx=Δy=0.1 ...... 32

2.8 Decay of line vortex, second-order minmod TVD simulations comparison of epsilon values; Δx=Δy=0.016 ...... 32

3.1 Computational domain for 3D fixed wing ...... 35

3.2 Residual convergence for AoA = 4 Degrees and Mach = 0.3 ...... 37

3.3 Induced drag coefficient for various Trefftz plane locations for inviscid simulations ...... 41

3.4 Contours of vorticity without VC enabled comparing differentiable and minmod limiter ...... 43

3.5 Contours of vorticity with VC enabled comparing differentiable and minmod limiter ...... 44

3.6 Contours of vorticity at two chord lengths from trailing edge using differentiable limiter without VC ...... 45

viii

4.1 Contours of effective viscosity produced by Spalart-Allmaras turbulence model for fixed wing simulation with VC implemented ...... 48

4.2 Induced drag at Ma=0.3 for turbulent simulations with a comparison of S-A and RANS results of induced drag at various Trefftz planes with (W/) and without (W/O) VC ...... 50

4.3 Percentage of induced drag contribution to total drag ...... 51

4.4 Interface of structured and unstructured zones in mesh II ...... 52

4.5 Experimental Results of vortex velocity ...... 53

4.6 Results of computations using mesh I ...... 54

4.7 Results of computations using mesh II ...... 55

5.1 Computational domain for rotating wing simulations ...... 58

5.2 Rotating wing simulation results, contours of vorticity with and without VC ... 60

5.3 Swirl velocity profiles from rotating wing simulations with and without VC ... 60

5.4 Swirl velocity of wingtip vortex from turbulent simulation and experiments .... 62

5.5 Geometry of winglet for rotating wing ...... 64

5.6 Contours of vorticity results for simulation of rotating wing with winglet ...... 65

NOMENCLATURE

Δt Temporal step size, s Δx Special step size, m

α Angle of attack

ε Confinement parameter, ms-1

µ Dynamic viscosity, kg m-1 s-1

-1 -1 휇푇 Turbulent viscosity, kg m s ν Kinematic viscosity, m s-1

ρ Density, Kg m-3

τ Shear stress, N m-2

Г Circulation, m2 s-1

Ф TVD operator 휔⃗⃗ Vector of vorticity

푛⃗ Vorticity confinement direction vector 푠 Vorticity confinement contraction term

푢⃗ Vector of Cartesian velocities

푤⃗⃗ Vorticity confinement two stencil term

퐹 Vector of flux terms, x-direction

퐺 Vector of flux terms, y-direction

푈⃗⃗ Vector of conserved variablies a Wave speed

푎0 Lift slope

x c Coefficient of confinement e Specific internal energy, J Kg-1

푒푠푝 Span efficiency factor ℎ Characteristic cell size, m

ℎ푒 Enthalpy r Radial distance, m t Time, s u Cartesian velocity in x-direction, m s-1 AR Aspect ratio

퐶퐷 Coefficient of drag

퐶퐿 Coefficient of lift M Mach number P Pressure, Pa

Re Reynolds Number

CHAPTER I

INTRODUCTION

1.1 Background

Accurate simulation of tip vortices can provide important information to aerodynamic designers, allowing them to improve designs to create more efficient air vehicles. The research presented in this thesis was motivated by the need to improve accuracy in simulating the convection of a wingtip vortex for both fixed and rotating wings using computational fluid dynamics (CFD). CFD provides an opportunity to lower development costs of new air vehicles while simultaneously improving their performance through computerized optimization schemes that help designers test thousands of configurations without the expense of wind tunnel testing. However, these simulations come with a new cost associated with them, known as computational cost.

Computational cost is the amount of computing time and memory usage associated with running a CFD simulation as well as the cost of qualified personnel who perform and analyze the simulations.

Two applications for the simulation of wing-tip vortices are presented here. One of these applications determines the amount of lift-induced (or simply induced) drag for fixed wing aircraft. The second application simulates a vortex created by rotating wings such as a helicopter rotor. Accurately performing these simulations can be a computationally expensive task.

Aerodynamic forces from a CFD simulation are most commonly determined by the integration of pressure and shear stress over the surface of the aerodynamic body.

This method, known as the near field technique, generally provides with accurate lift prediction; however, accurate drag prediction can be difficult to obtain barring a sufficiently refined grid at the near-surface region of the aircraft. This loss of accuracy in the prediction of induced drag occurs because it involves integration of pressure in areas of strong pressure gradients near wing tips, where tip vortices are formed. Recent computations [1] and the computations presented here have shown that the near-field method over-predicts (approximately double) the induced drag even if reasonably refined grid is adopted.

A coarse grid will not represent the curvature of the wing surface correctly and will lead to small errors in the direction of the acting pressure and, consequently, the resultant force which, when integrated, add together to create significant error in the result. With the near-field method this problem could, in principle, be resolved by increasing grid density and by using special features of grid generation such as dense boundary layers of grid point. This can be problematic for preliminary aerodynamic design as moderate-sized grids should be used to keep computational costs to at a reasonable level.

Alternatively, drag can be predicted from a technique known as the far-field or wake-integral technique. This technique applies a control volume surrounding the wing and employs conservation of mass and momentum over the control volume. This integration can be simplified under assumption of 1-D flow in far-field to include only integration over a cross-flow plane within the wake; this plane is known as the Trefftz

2 plane [2]. This overcomes many of the issues of drag prediction present in the near field.

Additionally, this technique decomposes the components of drag into induced and profile drag. Accurate prediction of drag allows researchers to improve the lift-to-drag ratio by using optimized wing configurations which allows for more efficient aircraft.

The upwind discretization for convective terms utilized in many commercial CFD programs causes numerical smearing of the solution at a much greater rate than physical dissipation. While this improves numerical stability, it is detrimental to the application of the wake-integral method. As they convect downstream, vortices are smeared and the wake-integrated drag is under-predicted for the fixed wing application. Vorticity confinement can be used as a means to improve the results of the wake-integral method for prediction of induced drag without the computational cost of a more refined grid. The

Vorticity confinement (VC) method captures concentrated vortices as they convect downstream through the addition of a body force term in the momentum equation [3,4,5].

This work is a major extension of a previous investigation into the wake-integral method presented in Ref. 5. Previously, the wake-integral method was combined with

VC to improve drag prediction for fixed-wing aircraft. That research focused on the implementation of VC combined with the wake-integral method and its application to a test case for a low aspect ratio wing at a Mach number of 0.3 and an angle of attack

(AoA) of 4 degrees for inviscid simulations. In order to extend the VC method to a wide range of Mach numbers and AoAs, a parameter known as the coefficient of confinement

(c) must be optimized for various cases. The current study aims to determine this range of c. To extend upon the previous investigation into inviscid aerodynamics, the VC method is applied to viscous simulation together with various turbulence models. In

3 order to apply turbulence models to a simulation utilizing VC, an investigation must first be conducted to show that VC does not affect viscous decay of a vortex for temporal and viscosity scales relevant to wingtip vortices. Additionally, Total Variation Diminishing

(TVD) schemes which are used to improve stability of second-order schemes are investigated to determine the effect TVD has on over-confinement, the range of c and the accuracy of applied VC.

For the optimization of a helicopter blade design, the modeling of blade-tip vortices is crucial [6,7]. The shedding of vortices has an effect on the performance of a helicopter as the vortices interact with various parts of the aircraft. As the vortex is shed from the blades, it interacts with other blades, the fuselage, landing gear and tail rotor causing vibration, noise and affecting the aerodynamics of the other rotors. The numerical experiments presented in this thesis show that there is an unacceptable level of numerical dissipation if an unstructured grid is employed. The numerical investigation presented aims to minimize and counteract numerical dissipation so that vortex strength may be accurately modeled over long distances that the vortex travels before impingement.

1.2 Finite Volume Method

The basis for CFD simulations is the discretization of the Navier-Stokes equations. The Navier-Stokes equations describe the motion of a fluid substance and are derived from Newton’s second law of motion (conservation of momentum) [8]; when combined with continuity (conservation of mass) and conservation of energy, the system of equations can describe a wide variety of fluid flow problems. Viscous forces, such as turbulent shear stress in the turbulent regime, are expressed through the addition of

4 various turbulence models such as Spallart- Allmaras and k-ε among other models depending on the nature of the problem [9].

Commercial CFD software typically discretizes these governing equations by the finite volume method (FVM). The FVM solves the governing equations by discretizing the flow-field into many control volumes where the flow properties are assumed to be homogenous within each volume [9]. Integration over the surface of the control volume yields equations that can be solved algebraically using iterative schemes; these individual cells are referred to as finite volumes. The most straightforward way of the accuracy of a

CFD simulation is to decrease the size of the finite volumes; however, this leads to a drastic increase in the amount of computational resources required [9].

When solving fluid flow problems using CFD, the difficulty of modeling a curved surface with discrete gridding causes significant error in areas of high pressure gradient.

The direction of pressure forces acting on a curved wall boundary may not be accurate as they are computed by surface integration over the flat faces of the grid. This issue affects the accuracy when predicting drag by the near-field technique.

1.3 Upwind Schemes and TVD

In the FVM, the flux terms of mass, momentum and energy at the cell face are defined by natural variables at the cell center. Consider the 1-D wave equation, Equation

1.1, as an example:

휕푢 휕푢 + 푎 = 0 (1.1) 휕푡 휕푥

Applying the FVM to this equation, the flux terms can be seen to represent the spatial derivative in Equation 1.2:

푛+1 푛 퐹 1 − 퐹 1 휕푢 푢 − 푢 휕푢 푖+ 푖− ≈ 푖 푖 = −푎 = 2 2 (1.2) 휕푡 ∆푡 휕푥 ∆푥

The flux of a first order upwind scheme is defined as a constant through neighboring

1 1 1 cells, that is 퐹푖+ = 퐹푖 for positive 푎. The flux terms (퐹푖+ & 퐹푖− ) can be defined by 2 2 2 using piecewise extrapolation to increase the order of the scheme. Please see Section 2.1 for the definition of the flux terms used for the 2D code in this thesis and Figure 2.1 which demonstrates flux extrapolation.

The first order scheme is numerically stable for a Courant–Friedrichs–Lewy

푎∙∆푡 ≤ 1 condition (CFL) of less than one ( ∆푥 for the wave equation) and is not susceptible to oscillation; however, it introduces a large amount of error into the solution in the form of numerical dissipation. The result of the numerical dissipation is smoothing of the solution and a more stable simulation compared to other methods such as central difference [9]. To extend to second order accuracy and reduce the amount of numerical dissipation, the scheme interpolates through two neighboring cells. Refer to section 2.1 for formulas describing the interpolation. Second order schemes may contain a flux limiter, denoted by 훷, which serves to increase stability by preventing oscillations which can develop using second order schemes. The scheme reverts to first-order when oscillations in the flow-field are detected.

Whether first or second order, the discretization of the Navier-Stokes equations using upwind schemes leads to the solution of a partial differential equation which is analytically different from the original equation. The concept is clearly demonstrated by

Anderson by applying to the 1-D wave equation (Equation 1.1) [9]. As seen in Equation

1.3, the discretization is first-order forward in time and first-order backward in space:

푢푡+∆푡 − 푢푡 푢푡 − 푢푡 푖 푖 + 푎 푖 푖−1 = 0 (1.3) ∆푡 ∆푥

After some manipulation (Ref. [9], pg. 103), it can be seen that the discretization leads to the solution of not the original equation, but of a modified equation with terms that act as artificial viscosity and dispersion.

휕푢 휕푢 푎∆푥 휕2푢 푎(∆푥)2 휕푢3 + 푎 = (1 − 휐) + (3휐 − 2휐2 − 1) (1.4) 휕푡 휕푥 2 휕푥2 6 휕푥3

+ 푂[(∆푡)3, (∆푡)2(∆푥), (∆푡)(∆푥)2, (∆푥)3]

Similar to what is seen in Equation 1.4 occurs from the discretization of the

Navier-Stokes equations. As such, artificial viscosity deteriorates the vortex structure that is shed by the wing in the same way that physical viscosity would; although it can become the dominate dissipative term in the simulation, it is a purely numerical phenomena. As a consequence, artificial viscosity can be detrimental when attempting to accurately model the effect of the vortex on other segments of the air vehicle as well as causing inconsistencies with the wake-integral drag prediction.

Since artificial viscosity is a function of truncation error, first-order upwind schemes exhibit a much high level of numerical dissipation compared to second-order upwind schemes. This makes the second-order scheme preferable to the first order scheme. However, first order schemes have the advantage of being monotonically preserving, with no new local maxima or minima created; this suppresses oscillations, which occur in second-order or higher schemes, and increases stability [9].

In order to increase the stability of second order schemes, total variation diminishing (TVD) flux limiters are used to reduce the calculation to first order in regions of oscillation. The choice of the flux limiter can have a significant effect on the level of

7 numerical dissipation of a particular scheme. Several flux limiters were implemented and evaluated in this thesis to quantify this property.

1.4 Vorticity Confinement

The vorticity confinement (VC) method was initially developed by John Steinhoff and his co-authors in [10,11]. The method adds a term to the momentum equation with acts as an anti-diffusive term. One of the initial applications of the VC method was the simulation of interacting vortical structures concentrated over only a few grid cells. It is noted that while the vortex behaves as it physically would in a Lagrangian simulation when isolated over long time scales, they behave as the solution of Euler equations when they interact [11]. The solution of Euler equations of conservation of mass, momentum and energy must be solved to simulate the generation of the vortex, so a purely

Lagrangian approach cannot be used. At the same time, the vortex scale is much smaller than that of the air vehicle and must be resolved with gridding which is relatively coarse and highly diffusive for the vortex which is created at the wing tip. By maintaining the vortex, convective effects of the vortex, such as on other regions of flow and the air vehicle structure, may be studied with low computational costs.

The initial formulation of VC, referred to herein as vorticity confinement formation one (VCF1), was developed to maintain a vortex by using a fixed constant known as the confinement parameter (ε). Seen in Equation 1.5, the confinement parameter controls the strength of the vorticity confinement source term added to the momentum equation:

휕푢⃗ 1 = −(푢⃗ ∙ 훻)푢⃗ − 훻푃 + 휇훻2푢⃗ − 휀푠 (1.5) 휕푡 휌

The 휀 term defines the strength of confinement while the vector 푠 defines the direction and increase strength based on the level of vorticity; for additional information on the VC scheme implemented, refer to Section 2.2. When applied to a 1-D simulation,

VC effectively balances the numerical diffusion terms in the modified equation (see

Equation 1.4) through the addition of a counter-diffusive source term which results in the solution of the original equation. However, for 2-D and 3-D simulation, the level of numerical diffusion depends on the orientation of the flow-field with the gridding; VC applies the necessary amount of counter-diffusion in a more precise way than approximating it in the various Cartesian directions independently.

For VCF1, the larger the size of ε, the stronger the steady-state form of the vortex.

Using dimensional analysis, it was determined that the confinement parameter has units of velocity [4]. To apply VC in a more general manner, a formulation for the confinement parameter which utilizes a true constant, c, as the only user input is adopted.

This reduces the dependence on flow and mesh properties as ε is determined dynamically by mesh size and the gradient of vorticity [4]. This adaptive method is referred to as vorticity confinement formulation two (VCF2). Equation 1.6 shows the definition of ε for VCF2:

휀 = 푐ℎ2|∇|휔⃗⃗ || (1.6)

While the VCF2 formulation significantly improves the dependence on an arbitrary constant, it has been shown in the current study that it does not completely remove the need to tune the constant.

An alternative VC scheme, known as VC2, was also developed by Steinhoff. One of the main reasons for the development of VC2 was that the original VC scheme merely

‘approximately’ conserved momentum whereas the VC2 formulation explicitly conserves momentum because the VC terms in the momentum equation have a spatial derivative operator[12,13]. Both VC1 and VC2 add an additional term, 휀푠 , to the momentum equation. The main advantage of VC2 is that due to the definition of 푠 , the confinement term can be expressed in derivative form if the diffusion terms in Equation 1.6 are written in terms of vorticity as shown in Equation 1.7:

휇훻2푢⃗ − 휀푠 = ∇ × (휇휔⃗⃗ − 휀푤⃗⃗ ) (1.7)

This achieves the goal of having a conservative (derivative) form of VC which guarantees momentum conservation. The equation can be expressed this way due to the definition of 푠 as 푠 = ∇ × 푤⃗⃗ ; additionally, the VC2 term 푤⃗⃗ is defined through the harmonic mean of vorticity of the computational stencil which allows for a smooth application of VC [3,12,14].

According to Steinhoff, momentum conservation makes VC2 more accurate for the preservation of vortices over long trajectories. However, “Only for very accurate long-term trajectory determination of vortices convecting in a slow background velocity field has the momentum-conserving VC2 formulation been found to be necessary” [3]. It is noted on page 60 in [9] that the conservative property has been actively debated during the history of CFD and that many useful finite-difference equations that don not have the conservative property prove to be more accurate in some sense than those that do.

Additionally, with the main advantage of VC2 being that it is a conservative scheme, it would be very difficult to develop an adaptive confinement parameter similar to Equation

1.5 that would maintain the conservative property [15]. This limits the practical

10 application of VC2 to structured gridding whereas VC1 can be applied to either structured or unstructured gridding.

VC1 was noted to converge faster than VC2 and has been implemented more often than VC2 without detriment to the results due to its non-conservative nature

[1,3,4,5,11,12,13]. The application of VC presented in this thesis involves the preservation of the vortex only up to 12 chord lengths with a Mach 0.3 to Mach 0.6 mean flow-field for fixed wing applications and the preservation of a tip vortex up to 180 degree wake age for rotating wing applications; these application do not involve ‘very slow’ background flow. As such, VC1 was chosen to be implemented for these applications.

Another formulation of VC, adaptive vorticity confinement, is presented in [16].

While previous work focused on limiting the dependence of the confinement terms on flow variables and mesh properties, adaptive VC eliminates the need for proportionality constant which is required for VC1 and VC2. Adaptive VC achieves this elimination by defining the confinement term as a function of the velocity field. However, in principle the method requires that the amount of numerical viscosity be known before the simulation is conducted, which is not a reasonable assumption for CFD problems using unstructured grids and complex geometry. The authors is [16] remedy this by estimating artificial viscosity by the difference between upwind and central discretization, which works well when implemented with a first-order upwind scheme, though it requires a threshold limiter to be stable. The threshold limiter is required because the confinement term’s definition results in the equivalent of replacing the stable upwind scheme with an

11 unstable central scheme [17]. Also, recent results have shown that the formulation results in dramatic over-confinement when used with second-order upwind schemes [17].

With regard to vorticity confinement, this thesis focuses on the study of the optimal value of confinement parameter in the VCF2 formulation. For fixed wing drag prediction applications, a wide range of Mach numbers and angles of attack (AoAs) in the subsonic flow regime are investigated. The optimal value of confinement parameter when applied to simulation of hovering rotors is investigated as well. The application of the VC method is extended to viscous flows and the effect of VC on viscous decay of vortices is studied. Also, the effect of total variation diminishing (TVD) schemes on the robustness of the method is examined to avoid over-confinement.

1.5 Wake-Integral Method

The wake-integral method is a far-field drag prediction technique which can be used to determine drag in both experimental and computational settings during the investigation of wing configurations. Far-field techniques are an alternative to the more traditional near-field surface integration method for determining drag from a CFD simulation as well as an alternative to experimental drag prediction using strain gages.

Surface integration determines aerodynamic forces by integrating pressure and shear stress over the surface of the aerodynamic body. Drag prediction is dependent on accurate description of wing curvature in regions with high pressure gradients, which is poorly modeled baring an extremely refined grid. These errors have a detrimental effect on drag prediction, producing error that is an order of magnitude more for drag than for lift for AoAs that an aircraft typically operates at [18].

Rather than integrating over the surface of the aircraft, the wake-integral method is built on the control volume approach. The technique applies a control volume surrounding the aircraft and employs conservation of mass and momentum over the control volume. This integration can be simplified under assumption that axial velocity is no longer changing (1-D flow assumption). When 1-D flow is satisfied, far-field drag prediction can be performed integration over a single plane within the wake which perpendicular to the mean flow; the plane is known as the Trefftz plane [2]. The experiments presented show that this plane should be at least 5 chord lengths if only a wing profile is being evaluated. If all components of the air vehicle are being evaluated, the Trefftz plane will be even further from the wings. This method overcomes issues such as representing curvature of the wing and large pressure gradients on its surface which negatively affect the accuracy of the near field technique.

In the application of the wake-integral technique researchers in [1,18,19] have shown that the wake-integral approach slightly under predicts drag in comparison to lifting line theory. Additionally, the results have shown that the drag prediction is not consistent and becomes smaller as the Trefftz plane is moved further downstream. This phenomenon is caused by the presence of high levels of numerical dissipation which are present in solvers used in commercial CFD software. In order to counteract numerical dissipation, VC is applied in conjunction with the wake integral method to provide consistent results for drag prediction.

1.6 Turbulence Modeling

When extending any simulation to include viscous terms in the regime of

Reynolds numbers which would constitute turbulent flow, the selection of a model of

13 turbulence is of upmost importance. While the Navier-Stokes equations contain all of the information necessary to solve viscous simulations, the required cell and time scales that must be solved make direct numerical simulation (DNS) of turbulence impractical [9].

Turbulence models serve the purpose of closing the Reynolds-averaged Navier-

Stokes (RANS) Equations. The closure problem arises because of the need to average velocities so that a steady state solution can be obtained. The RANS continuity

(Equation 1.8) and momentum equations (Equation 1.9), written in Cartesian tensor form are given as [20]:

휕휌 휕 + (휌푢푖) = 0 (1.8) 휕푡 휕푥푖

휕 휕 휕푝 휕 휕푢 휕푢 2 휕푢 휕 푖 푗 푙 ′ ′ (휌푢푖) + (휌푢푖푢푗) = + [휇 ( + − 훿푖푗 )] + (−휌푢̅̅̅푖̅푢̅̅푗) (1.9) 휕푡 휕푥푗 휕푥푖 휕푥푗 휕푥푗 휕푥푖 3 휕푥푙 휕푥푗

′ ′ The additional term that appears, −휌푢̅̅̅푖̅푢̅̅푗, represents the effect of turbulence and are known as Reynolds stresses; they are typically modeled by additional PDEs. One method of modeling this term makes use of the Boussinesq assumption, which relates the

Reynolds stresses to the mean velocity field by Equation 1.10:

2 휕푢 ′ ′ 푘 ̅ ̅−휌푢̅̅̅̅̅̅푖̅푢̅̅푗 = 2휇푇푆푖푗 − 훿푖푗 (휇푇 + 휌푘) (1.10) 3 휕푥푘

For turbulence models following the Boussinesq assumption, it is assumed that turbulent viscosity is an isotropic property [20].

The other subset of turbulence models does not follow the Boussinesq assumption; instead, scalar equations are solved for each Reynolds stress term which allows for anisotropic turbulent viscosity. However, this comes at a cost of solving six

14 additional partial differential equations. In addition, another scale-determining equation must be solved, typically for turbulent dissipation rate or specific dissipation rate [20]. A three dimensional Reynolds stress model will have a total of 7 addition PDEs to solve compared to the addition of one or two equations which is typical for turbulence models following the Boussinesq equation. All current models of turbulence have limitations and it is important to verify that the model is accurately reflecting the physical world. In the present work, VC is initially investigated for its effect on the simple case of viscous decay of line vortex using temporal and viscosity scales relevant to tip vortices. VC is then combined with turbulence models in 3D simulations to evaluate its interaction with the more complex turbulence models. For rotating wing simulations, recent experimental results are available and will be compared to the simulations with combined VC and turbulence models.

1.7 Research Goals

The goal of this research is to extend the work of induced drag prediction in [5] for

Mach 0.3 and AoA of 4 degrees to a wide range of Mach numbers and AoAs, extend the application of VC from Euler to Navier-Stokes simulations and to apply VC to Navier-

Stokes simulation using RANS turbulence model to simulate the evolution of a vortex from a rotating wing. For fixed wing drag prediction, the wake-integral method combined with VC may be able to replace the more common wing surface integration.

For the practical application of this method, the degree of dependence of VC on the confinement term must first be investigated to determine the most effective way to apply the method.

In order to extend the application of VC from Euler to Reynolds averaged Navier-

Stokes simulations, it must first be determined whether VC has an effect on physical dissipation of a vortex. Chapter II examines this with a 2-D simulation of a line vortex with the same time and viscous scales as a turbulent tip vortex. The line vortex results are compared with the analytical solution to determine whether VC has an effect on physical dissipation for these scales.

Chapter III contains the results of fixed wing simulations for various Mach numbers and AoAs. In Chapter III, the dependence of the constant coefficient of confinement on vortex strength is also analyzed. Chapter IV contains the application of

VC to fixed wing simulations including turbulence models. Chapter V is the result of the simulation of a tip vortex generated by rotating wing by Navier-Stokes with the addition of VC. For the rotating wing, experimental results are available for direct comparison to computational result.

CHAPTER II

SIMULATION OF 2-D VORTICES USING VORTICITY CONFINEMENT AND TVD

2.1 2D Code Description

A two-dimensional, compressible Euler/Navier-Stokes code was written in

MATLAB to determine the optimal values of VC parameters and to test the effect of non- differentiable versus differentiable TVD limiter together with VC. The code was constructed using structured finite-volume discretization with Steger-Warming flux splitting, upwind discretization of convective terms, central discretization of viscous terms and first-order explicit marching in time. The two dimensional inviscid Euler equation is defined by [9,21]:

휕푼 휕푭 휕푮 + + = 0, (2.1) 휕푡 휕푥 휕푦 where U represents the vector of conserved variables and F and G represent the vectors of fluxes, defined in Equation 2.2 [9,21]:

휌 휌푢 휌푣 휌푢 휌푢2 + 푝 휌푢푣

푼 = (휌푣) 푭 = ( 휌푢푣 ) 푮 = ( 휌푣2 ) (2.2) 휌푒 휌푢ℎ푒 휌푣ℎ푒

The fluxes are split into positive and negative components by the Steger-Warming flux splitting scheme. The basis of the splitting is determined by the eigenvalues of the

Jacobian matrices A and B, which can be used to define the flux variables as seen in

Equation 2.3:

푭 = [퐴]푈 (2.3) 푮 = [퐵]푈

The Euler equations are known to be a hyperbolic system of equations, thus a similarity transformation exists such that:

−1 [푇] [퐴][푇] = 휆퐴 (2.4) −1 [푇] [퐵][푇] = 휆퐵

The eigenvalue matrices are divided into two separate matrices, one containing only positive eigenvalues and one containing negative eigenvalues, the positive and negative fluxes are then determined from Equation 2.4 [9]:

+ − + −1 − −1 [퐴] = [퐴 ] + [퐴 ] = [푇][휆퐴 ][푇] + [푇][휆퐴 ][푇] (2.5) 푭 = 푭+ + 푭− = [퐴+]푼 + [퐴−]푼

The definition is similar for the G flux term. Please refer to Appendix A for the definition of the eigenvalues and the positive and negative flux terms.

The original equation, Equation 2.1, now represented in split-flux notation with the addition of the vorticity confinement term now becomes:

휕푼 휕푭+ 휕푭− 휕푮+ 휕푮− + + + + + εs = 0 (2.6) 휕푡 휕푥 휕푥 휕푦 휕푦

The fluxes in Equation 2.6 are upwinded based upon the sign of the eigenvalues, with fluxes with positive eigenvalues backward differenced and fluxes with negative eigenvalues forward differenced. The code is capable of using first or second order flux approximations and has the ability to include TVD limiters for the second order scheme.

The first-order scheme approximates the fluxes in a piecewise-constant fashion, shown in Equation 2.7:

+ − 퐹 1 = 퐹 + 퐹 (2.7) 푖+ 푖 푖+1 2

The second-order scheme approximates the fluxes through linear extrapolation as in

Equation 2.8:

+ + + − − − 퐹 1 = 퐹 + 0.5훷(퐹 − 퐹 ) + 퐹 + 0.5훷(퐹 − 퐹 ) (2.8) 푖+ 푖 푖 푖−1 푖+1 푖+1 푖+2 2

Figure 2.1 demonstrates the extrapolation for the positive fluxes. While the first-order

+ + 퐹 퐹 1 code would simply use the value of 푖 to calculate the positive flux at 푖+ , the second 2

+ + 퐹 1 = 퐹 + order scheme extrapolates to the edge of the cells with the use of 푖+ 푖 2

+ + 0.5훷(퐹푖 − 퐹푖−1).

Flux Extrapolation 10 + + 퐹 1 퐹 1 푖+ 2 푖− 2 8

2 𝑖 − 2 𝑖 − 1 𝑖 𝑖 + 1

0 1 2 Cell Number 3 4

Figure 2.1: Positive flux extrapolation used in second order scheme

The flux limiter, 훷, can be defined numerous ways to achieve TVD. The flux limiter is defined as a function of the fluxes, seen in equation 2.9, in order to determine if there are oscillations present.

+ + + 퐹푖+1 − 퐹푖 Φ 1 = 푓 ( ) (2.9) 푖+ + + 2 퐹푖 − 퐹푖−1

The limiters tested were the non-differentiable minmod limiter, Equation 2.10, and the differentiable van Albada limiter, Equation 2.11 [9]:

푓(푥) = max(0, min(1, 푥)) (2.10)

푥2 + 푥 푓(푥) = (2.11) 푥2 + 1

For simulations which include viscous terms, the viscous terms in the Navier-

Stokes equation are determined by the second-order central discretization of fluxes. The magnitude of viscosity, however, was determined by the Spallart-Allmaras model of turbulence by a 3-D simulation of tip vortices using the commercial CFD software

ANSYS FLUENT (see Figure 4.2 in Section 4.2). For the complete code including the vorticity confinement implementation, refer to Appendix B.

2.2 Vorticity Confinement implementation and Formulations

Both VCF1 and VCF2 were implemented and tested using the code. As shown in

Equation 2.12, the Vorticity confinement (VC) method captures concentrated vortices as they convect downstream through the addition of a body force term in the momentum equation [1,3,4,5]:

휕푢⃗ 1 = −(푢⃗ ∙ 훻)푢⃗ − 훻푃 + 휇훻2푢⃗ − 휀푠 (2.12) 휕푡 휌 where ε and s are defined as:

휀 = 푐ℎ2|훻|휔⃗⃗ || (2.13)

푠 = 푛⃗ × 휔⃗⃗ (2.14)

The confinement parameter, ε, is either set as a constant or can be dynamically determined by Equation 2.13. The constant ε formation is first formulation of VC

(VCF1) while ε determined by Equation 2.13 is the second formulation of VC (VCF2)

[4]. The dimensionless coefficient, c, is the constant coefficient of confinement parameter used for tuning the strength of confinement, h is the length scale of a particular grid cell (finite volume) and ⃗ω⃗ is the vector of vorticity. The direction of vector n⃗ for

Equation 2.15 is determined by:

훻|휔⃗⃗ | 푛⃗ = (2.15) |훻|휔⃗⃗ ||

Depending on the gridding used, the definition of h can have a significant effect on the determination of ε; however, for isotropic gridding forms such as average of edge-lengths surrounding a point, volume to surface ratio of a grid cell give similar results [4].

2.3 Results of Second Order Euler Simulations without TVD

A convected Taylor vortex was simulated in order to compare first-order and second order schemes and to determine the appropriate value of the confinement parameter for second order schemes. The tangential velocity of the Taylor vortex is given by Equation 2.16:

Γ 푈 = 푟 exp(0.5(1 − 푟2)) (2.16) 2휋

The grid stepping was Δx=Δy=0.2. The total time of the simulation was t=0.1s consisting of 2000 iterations. The vortex convected across the flow-field in the x direction at Mach 0.3.

The results of the first order calculations without VC and with VCF1 are shown in Figure 2.2. The scheme was highly diffusive; when the simulation completed, the vortex was nearly entirely smeared away by numerical dissipation.

Further first order simulations which show the results of the first order calculations combined with higher values of confinement parameter were carried out in

[22].When the order of numerical scheme is increased to second order, the results show that the total reduction of the maximum tangential velocity without application of VC is

30%. The lower level of numerical dissipation leads to a lower value of ε to properly maintain the vortex. While the first order simulations was significantly under-confined for ε=30, the second order simulations, shown in Figure 2.3, required only ε=3.6. The lower epsilon value also leads to a smaller source term added to the momentum equation.

This is desirable, as the smaller source term interferes less with the physical viscous terms of the simulation.

Velocity -5 -4 -3 -2 -1 0 1 2 3 4 5 -2

-4

-6

-8 Position 1st Order Scheme 1st Order Scheme With VC, Epsilon =30 Original Vortex (a)

2.5 25 2.5 0.5

2 2 0.45

1.5 20 1.5 0.4

1 1 0.35

0.5 15 0.5 0.3

0 0 0.25

-0.5 10 -0.5 0.2

-1 -1 0.15

-1.5 5 -1.5 0.1

-2 -2 0.05

0 0 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13

(b) (c) Figure 2.2: First order simulation of Taylor vortex: (a) velocity profiles; (b) vorticity contours and velocity vectors of initial vortex; (c) vorticity contours and velocity vectors of convected vortex

Swirl Velocity of Vortex, Initial vs Convected, Eps=0 Swirl Velocity of Vortex, Initial vs Convected, Eps=3.6 8 8

6 6

4 4

2 2

0 0

X Velocity (m/s) -2 X Velocity (m/s) -2

-4 -4

-6 -6

-8 -8 -10 -8 -6 -4 -2 0 2 4 6 8 10 -10 -8 -6 -4 -2 0 2 4 6 8 10 Y Coordinate Y Coordinate (a) (b)

Contours of Vorticity, Initial Vortex Contours of Vorticity, 2nd Order Calculations Eps = 0 3 25 4 25

3 2 20 20 2

1 X= 0 1 X= 10.8 Y= 0 15 Y= 0 15 Level= 25.7073 Level= 14.5952 0 0 Y Coordinate Y Coordinate 10 10 -1 -1

-2 5 5 -2 -3

-3 0 -4 0 -3 -2 -1 0 1 2 3 7 8 9 10 11 12 13 14 15 X Coordinate X Coordinate (c) (d)

Contours of Vorticity, 2nd Order Calculations Eps=3.6 4 25

20 2

1 X= 10.8 Y= 0 15 Level= 23.5687 0

Y Coordinate 10 -1

-2 5

-3

-4 0 7 8 9 10 11 12 13 14 15 X Coordinate (e) Figure 2.3: Upwind second-order simulation of Taylor vortex with and without VC: (a) tangential velocity without VC; (b) tangential velocity with VC (ε=3.6), figures (a) and (b) compare initial vortex and vortex after convection simulation; (c-e), contours of vorticity; (c) initial profile; (d) without VC; (e) with VC

The method was further extended to include the VCF2 formation. The constant coefficient of confinement parameter is set as c=4.5. Figure 2.4a shows contours of ε determined by Equation 2.13, Figure 2.4b shows the contours of vorticity for the vortex after a simulation time of t=0.1s and Figure 2.4c is the tangential velocity of the vortex compared to the initial vortex. The simulations were conducted with a Courant number of 0.2 resulting in approximately 1700 time steps. These results are similar to the second order simulation using VCF1. It is necessary to implement VCF2 for 3D simulations using unstructured gridding as the size of the cell correlates to the amount of numerical dissipation. VCF2 considers the grid size and adjusts the confinement parameter, ε, to correspond to the cell size and to vorticity gradient.

Contours of Vorticity, 2nd Order Calculations, VCF2 C=4.5 Contours of Eps, 2nd Order Calculations, VCF2 C=4.5 4 25 4 4

3 3 3.5 20 2 2 3

1 X= 10.8 1 Y= 0 15 2.5 Level= 22.2565

0 0 X= 11.6 2

Y= 0 Y Coordinate 10 Y Coordinate -1 -1 Level= 4.0569

1.5 -2 -2 5

1 -3 -3

-4 0.5 -4 0 7 8 9 10 11 12 13 14 15 7 8 9 10 11 12 13 14 15 X Coordinate X Coordinate (a) (b)

Swirl Velocity of Vortex, 2nd Order Calculations, VCF2 C=4.5 8

X Velocity (m/s) -2

-4

-6

-8 -10 -8 -6 -4 -2 0 2 4 6 8 10 Y Coordinate (c) Figure 2.4: Results of second order calculations using VCF2 with c=4.5: (a) contours of ε value; (b) contours of vorticity at t=0.1s; (c) tangential velocity, initial vortex and convection results

2.4 Comparison of TVD Schemes

While the pure second-order scheme is properly confined with a relatively low value of confinement parameter, ε, or constant coefficient of confinement parameter, c, it is easily over-confined if the proper value is not selected. This leads to a nonphysical solution where the vortex becomes stronger as time increases or to the failure of the

26 simulation altogether. In order to increase the fidelity of the scheme, VC was combined with various TVD schemes to test the effect on the simulation.

The simulation shown in Figure 2.5 utilized VCF2 with c=4.0. The initial vortex and gridding was the same as the vortex investigated in section 2.3; however, it was a stationary vortex with no cross-flow. The results show that the purely second order simulation experienced significant over-confinement with the maximum velocity of the vortex doubled compared to the initial profile. Additionally, the velocity gradient in the vortex core can be seen to increase significantly. The first order simulation continued to be significantly under-confined with much more numerical diffusion than any of the second-order schemes. It is also shown that the proper confinement parameter is affected by the mean flow. The previous simulation was properly confined with c=4.5 while the stationary vortex became over-confined with c=4.0.

2.5

1.5

0.5 First Order 0 Second Order -5 -3 -1 1 3 5 Minmod -0.5 Van Albada Velocity Velocity (Normalized) Original Vortex -1

-1.5

-2

-2.5 Position

Figure 2.5: Comparison of TVD Schemes, c=4.0

Figure 2.6 shows the results of c=5.0; here, the second order scheme failed completely due to over-confinement. The van Albada differentiable limiter (Equation

2.11) also began to show signs of over-confinement, though it was able to complete the simulation. The minmod limiter (Equation 2.10) is less affected by the selection of the value of c and the results are similar compared to c=4.0. This indicates that the minmod non-differentiable limiter is more stable compared to the differentiable van Albada limiter, which is desirable. This leads to a larger range of c where the vortex would be properly confined for the minmod limiter; however, this range will be higher than that of the differentiable limiter which will lead to a larger source term, which is not desirable.

1.5

1.0

0.5 First Order Minmod 0.0 Van Albada -5 -3 -1 1 3 5 Original Vortex

Velocity Velocity (Normalized) -0.5

-1.0

-1.5 Position

Figure 2.6: Comparison of TVD Schemes, c=5.0

2.5 Decay of Line Vortex, Viscous Test Case

In order to extend vortex simulations to include physical viscous terms, the effect of

VC on viscous decay was evaluated through simulation of the decay of a line vortex for the temporal and viscous scales relevant to tip vortices traveling from the wing tip to the

Trefftz plane. An analytical solution exists for the decay of a line vortex. The initial velocity profile of the vortex is given in Equation 2.17:

Γ 푈 = (2.17) 2휋푟

The second-order upwind scheme with minmod TVD was used to solve the Navier-

Stokes equations for various confinement parameters; these solutions may be directly compared to the analytical solution for decay of a line vortex [23]:

푟2 Γ − 푈 = (1 − 푒 4휐푡) (2.18) 2휋푟

In Equation 2.18, 휐 is kinematic viscosity, t is time and r is the radius.

The temporal and viscous scales were determined based upon a 3-D turbulent simulation of vortices created by a fixed wing. For that simulation, the speed of the fluid relative to the aircraft was Mach 0.3, which is approximately 100 m/s, and an angle of attack of 4 degrees. The time for the vortex to travel 10 chord lengths, typical for 3-D

chord t = 10 ∗ = simulations conducted in this thesis, is therefore approximately 0.1s ( U

10∗1m -2 m = 0.1s). The maximum level of total viscosity is not greater than 10 (refer to 100 ⁄s

Section 4.2). The product 휐푡 determines the total viscous decay of the line vortex and is

-3 -3 evaluated to be less than 10 (tυT< 10 ).

The effect of flow speed on this product is analyzed to determine whether it can be considered general for the various flow speeds analyzed for 3-D simulations. In

Equation 2.19, the velocity fluctuations, ̅푢̅푖̅푢̅̅푗, are roughly proportional to the mean flow,

U [8,9].

휕푢푖 휏푖푗 = 휌푢̅̅푖̅푢̅̅푗 = 휌휈푇 (2.19) 휕푥푗

From Equation 2.19, the turbulent viscosity scales proportionally with the speed, 휈푇~U.

The fluid particles time from wing tip to Trefftz plane, t, is inversely proportional to U

(t~1/U). Therefore, 휐푡 for simulation of tip vortices relevant to subsonic flight is not greater than 10-3.

The initial grid for the simulation used grid step size of Δx=Δy=0.1. The results shown in Figure 2.7 show that VC, implemented using VCF1, has a slight effect on the results of the decay of the line vortex for a confinement parameter, ε, ranging from 0 to

15. With VC implemented, the computed velocity tends to follow the analytical velocity as the core of the vortex is approached; however the effect is minimal and the velocity decreases near the core without reaching more that 0.2 that of the analytic velocity.

If the grid step size is refined six times, the results in Figure 2.8 show that VC improves the simulation and the simulated velocity corresponds more closely to the analytical velocity with maximum simulated velocity reaching 0.5 of the analytical maximum. The values of epsilon selected would results in over-confinement if applied to the inviscid second-order simulation of the Taylor vortex presented in Section 2.3.

Consequently, the selection of confinement parameter based upon inviscid simulations will have negligible effect on the simulation of physical viscous phenomena for the temporal and viscous scales relevant to the modeling of flow between wing tip and

Trefftz plane. The effect of these results is that confinement parameters can be determined by 3-D inviscid simulations (shown in Chapter 3) and applied to 3-D viscous simulations in Chapter 4.

1 0.9 0.8 0.7 0.6 Analytical 0.5 Minmod No VC 0.4 Minmod ε=5 0.3 Minmod ε=10 Velocity Velocity (Normalized) 0.2 Minmod ε=15 0.1 0 0 1 2 3 4 5 Position

Figure 2.7: Decay of line vortex, second-order minmod TVD simulation comparison of epsilon values; Δx=Δy=0.1. Velocity normalized by maximum analytical value. Analytical solution determined by Equation 2.18.

1 0.9 0.8 0.7 0.6 Analytical 0.5 Minmod No VC 0.4 Minmod ε=5 0.3 Velocity Velocity (Normalized) Minmod ε=15 0.2 0.1 0 0 0.5 1 1.5 2 2.5 3 Position

Figure 2.8: Decay of line vortex, second-order minmod TVD simulations comparison of epsilon values; Δx=Δy=0.016. Velocity normalized by maximum analytical value. Analytical solution determined by Equation 2.18.

CHAPTER III

INDUCED DRAG PREDICTION SIMULATIONS USING WAKE-

INTEGRAL COMBINED WITH VORTICITY CONFINEMENT

3.1 Implementation of VC in ANSYS FLUENT

ANSYS FLUENT is commercial CFD software used widely in industry for various applications in the analysis of fluid flow. It has the capability to simulate inviscid, laminar or turbulent flows as well as incompressible or compressible flows. It is an unstructured solver; for compressible flows, the energy equation must be solved in addition to equations for mass and momentum.

FLUENT allows for the addition of source terms in the various equations it solves through the use of a User Defined Function (UDF). VC is implemented in FLUENT through the UDF. The UDF is written in the C coding language using macros and functions provided by FLUENT to access solver data or to add terms to the flow equations.

The UDF created for implementation of VC first analyzes the flow and calculates additional flow variables not solved by FLUENT which are needed for the implementation of VC using the DEFINE_ADJUST macro. The additional variables solved for include vorticity in vector form, vorticity magnitude, gradient of vorticity magnitude and characteristic cell size h. The characteristic cell size is determined by the

33 difference in the maximum and minimum node values in a given cell in the three

Cartesian directions as shown in Equation 3.1:

2 2 2 ℎ = √(푥푚푎푥 − 푥푚푖푛) + (푦푚푎푥 − 푦푚푖푛) + (푧푚푎푥 − 푧푚푖푛) (3.1)

The εs term shown in Equation 2.12 can then be calculated and added to the momentum equations in FLUENT using the DEFINE_SOURCE macro.

3.2 Lifting Line Theory

In order to determine the accuracy of the proposed numerical approach, the results are compared to those obtained by Pradtl’s lifting line theory. According to lifting line theory, induced drag can be determined by [24]:

2 퐶퐿 퐶퐷,푖 = (3.2) 휋푒푠푝퐴푅

In equation 3.2, 퐶퐷,푖 is induced drag coefficient, 퐶퐿 is lift coefficient, 푒푠푝 is the span efficiency factor which accounts for non-elliptic circulation distribution and AR is the aspect ratio of the wing. For finite length wings, 퐶퐿 is defined by [24]:

푎0 퐶퐿 = ( 푎 ) 훼 (3.3) 0 ( ) 1 + (휋퐴푅) 1 + 휏

In Equation 3.3, 푎0 represents the lift slope for infinite wings, 휏 is a factor to account for finite wing geometry and 훼 is the angle of attack. For the NACA0012 wing investigated, 푎0 is 2휋, AR is 6.67, 휏 is 0.065 and 푒푠푝 is 0.94 [24].

For comparison to lifting line theory for simulation where compressibility effects cannot be neglected, the Prandtl-Glauert correction is utilized to determine the lift slope, coefficient of lift which is then used to determine the lift slope [25]:

푎0 푎0,푐표푚푝 = (3.4) √1 − 푀2

The inviscid calculations involve the same assumptions used in developing lifting line theory and are therefore useful in determining the accuracy of the drag prediction methods in the listed cases.

3.3 Problem Description

A NACA0012 airfoil profile was used to construct a 3D wing with an aspect ratio of 6.67. The computational domain, shown in Figure 3.1a, consisted of 1.38 million cells and extended 14 chord lengths away from the wing. The region of refinement was designated from the tips of the wing to the edge of the computational domain as seen in

Figure 3.1b; this is the area that the tip vortex tube is expected to be.

(a) (b) (c) Figure 3.1: Computational domain for 3D fixed wing: (a) grid geometry, the fixed wing is highlighted green; (b) the region of grid refinement for trailing vortices; (c) a slice of the grid showing the refined cells in the wake of the wing

Both pressure and density-based solvers were investigated to find the method with better convergence properties. The convergence was monitored by the residual monitoring provided with ANSYS FLUENT which present scaled residuals. After

35 discretization, the conservation equation for a general variable 휙 at a cell P can be written as [20]:

푎푝휙푃 = ∑푛푏 푎푛푏휙푛푏 + 푏 (3.5)

The variable and 푎푃 represents the center coefficient and 푎푛푏 represents the influence coefficients for the neighboring which is used for the unstructured solver. The variable b is the influence of the source terms and the boundary conditions [20]. The residual computed is the imbalance in Equation 3.5 summed over all cells in the domain. This is referred to as an unscaled residual. For further information regarding the unstructured solver used in ANSYS FLUENT, refer to [20].

The residuals presented by ANSYS FLUENT is known as a scaled residual and is given by:

∑ |∑ 푎 휙 + 푏 − 푎 휙 | 푅휙 = 푐푒푙푙푠푃 푛푏 푛푏 푛푏 푃 푃 (3.6) ∑푐푒푙푙푠푃 |푎푃휙푃|

For the pressure based and density based solvers, there was small difference in the computational results; however, for Mach 0.3 and AoA of 4 Degrees the density-based solver did not converge as quickly and was more oscillative in terms of the behavior of the residuals (see Figure 3.2). The pressure-based solver was used exclusively in subsequent cases. Second-order upwind schemes were used in the solver to match the truncation error with the h2 term in Equation 2.13. FLUENT model settings for inviscid computations are summarized in Table 3.

(a) (b) Figure 3.2: Residual convergence for AoA = 4 Degrees and Mach = 0.3: (a) residual plot for density-based solver; (b) residual plot for pressure-based solver

Table 3.1: Solver settings, model settings and solution methods for inviscid simulations

FLUENT Settings Solver and Models

Solver Pressure Based

Viscous Model Inviscid

Density Ideal Gas

Energy Equation On Solution Methods

Pressure-Velocity Coupling SIMPLE

Density, Momentum, Energy Second Order Upwind

Pressure Standard

3.4 Results

As seen in Figure 3.3, without application of VC the wake integral method provides different results for drag coefficient, CD, when integration is conducted at different Trefftz planes. When VC is used to counteract numerical diffusion and an

37 optimal confinement parameter is selected, the results can become practically independent of Trefftz plane location or at least show a marked improvement.

Lifting line theory was used as the basis of comparison to determine the accuracy of the near-field and far-field techniques. Table 3.2 shows that surface integration consistently over-predicted the drag force, sometimes causing the drag coefficient to be over two times the value for lifting line theory. The wake-integral method consistently under predicted the drag force but with much less error in comparison to surface integration. The drag coefficient was typically under predicted by 25%, with higher accuracy at lower Mach numbers and lower accuracy at higher Mach numbers.

Although the formulation of VC is designed to account for vortex strength and grid density, it was still necessary to tune the confinement parameter for individual cases.

The optimal confinement parameter was relatively consistent for different Mach numbers of the same AoA as seen in Table 3.3. As the AoA becomes larger, the confinement parameter becomes more consistent. This consistency allows for a faster and more accurate tuning for the proper confinement parameter. Even when the confinement parameter was not large enough to completely counteract numerical diffusion, the results were still notably better than those without VC implemented.

Table 3.2: Drag coefficients by surface integration and wake-integral method compared to lifting line theory

Drag Coefficient

M=0.3

Surf. Int. Wake-Int. AoA LLT Surface Int. Wake-Int. with VC Error % Error %

2 0.00136 0.0048 0.00118 253 -13

4 0.00544 0.0116 0.0043 113 -21

6 0.0122 0.021 0.0104 72 -15

10 0.034 0.052 0.027 53 -21

M=0.5

2 0.0018 0.0055 0.00133 206 -26

4 0.0072 0.013 0.0055 81 -24

6 0.016 0.0238 0.0121 49 -24

M=0.6

2 0.0021 0.0143 0.00153 581 -27

4 0.0084 0.0143 0.0061 70 -27

Table 3.3: Tuned confinement parameters

Confinement Parameters

AoA M=0.3 M=0.5 M=0.6

2 0.14 0.14 0.14

4 0.075 0.07 0.065

6 0.0525 0.05

10 0.04

0.00125 0.0060

0.0055 0.00120 0.0050

0.00115 0.0045

0.0040 Drag Coefficient Drag

Drag Coefficient Drag 0.00110 0.0035

0.00105 0.0030 0 4 8 12 0 4 8 12 Trefftz Plane (Chord Length) Trefftz Plane (Chord Length) W/O VC C = 0.075 C = 0.12 C = 0.14 W/O VC C=0.075 (a) (b)

0.0115 0.029 0.0110 0.028

0.0105 0.027 0.026 0.0100 0.025 Drag Coefficient Drag Drag Coefficient Drag 0.0095 0.024 0.0090 0.023 0 4 8 12 0 4 8 12 Trefftz Plane (Chord Lengths) Trefftz Plane (Chord Lengths) W/O VC C = 0.055 C = 0.065 W/O VC C=0.025 C=0.035 C=0.05 C=0.0525 C=0.04 C=0.045 (c) (d)

0.00144 0.0058 0.00140 0.0056 0.00136 0.0054 0.00132 0.0052 0.00128 Drag Coefficient Drag Drag Coefficient Drag 0.00124 0.0050

0.00120 0.0048 0 4 8 12 0 4 8 12 Trefftz Plane (Chord Lengths) Trefftz Plane (Chord Lengths) No VC C=0.12 No VC C=0.070 C=0.075 (e) (f)

0.0124 0.00160

0.0120 0.00155 0.00150 0.0116 0.00145 0.0112 0.00140 Drag Coefficient Drag Drag Coefficient Drag 0.0108 0.00135

0.0104 0.00130 0 4 8 12 0 4 8 12 Trefftz Plane (Chord Lengths) Trefftz Plane (Chord Lengths) C=0.055 No VC C=0.05 No VC C=0.12 (g) (h)

0.0064

0.0062

0.0060

0.0058

Drag Coefficient Drag 0.0056

0.0054 0 4 8 12 Trefftz Plane (Chord Lengths) No VC C=0.065 C=0.070 (i) Figure 3.3: Induced drag coefficient for various Trefftz plane locations for inviscid simulations: (a) AoA = 2° and M=0.3; (b) AoA = 4° and M=0.3; (c) AoA = 6° and M=0.3; (d) AoA = 10° and M=0.3; (e) AoA = 2° and M=0.5; (f) AoA = 4° and M=0.5; (g) AoA = 6° and M=0.5; (h) AoA = 2° and M=0.6; (i) AoA = 4° and M=0.6

The VC technique is most useful at higher AoAs as the amount of numerical diffusion that exists in these cases is substantially larger. For low AoAs, induced drag is not as significant as profile drag, though still not negligible. A reasonably conservative approach could use C = 0.0475 for all simulation for AoA lower than 6 Degrees and

0.0375 for all AoAs between 6 and 10 degrees. Though this would not completely eliminate numerical diffusion, it would make certain that there is no violation of physics

41 provided by over confinement and would still be a significant improvement over no confinement.

3.5 Differentiable Limiter

In an extension of the investigation conducted in Section 2.4, a simulation utilizing the differentiable limiter available in ANSYS FLUENT was conducted to determine the coefficient of confinement. The 4 degree AoA, Mach 0.3 case was investigated previously for the minmod TVD limiter. The coefficient of confinement for that case was determined to be c=0.075. From Section 2.4, it is expected that the differentiable limiter will produce less numerical dissipation and will therefore require a lower value for c.

Figure 3.4 shows the comparison of results for both the differentiable limiter and the minmod limiter without VC applied. It can be observed that the differentiable limiter has a lower level of numerical dissipation as the level of vorticity reduces at a lower rate than the minmod limiter.

(a) (d)

(b) (e)

Figure 3.4: Contours of vorticity without VC enabled comparing differentiable and minmod limiter; (a-c) Minmod Limiter; (d-e) Differentiable limiter; (a,d) 2 Chords (2C) from trailing edge; (b,e) 5C from trailing edge; (c,f) 8C from trailing edge; (g), scale for contour plots

Since the level of dissipation is less for the differentiable limiter, it is expected that the c value for proper confinement will be less than that for the minmod limiter. It was found that c=0.0525 works well for confinement of the differentiable limiter which is

30 percent lower than the value for the minmod limiter. Figure 3.5 shows a comparison of the differentiable and minmod limiter with VC enabled. VC works well to preserve the vortex present at 2 chord lengths from the wing for both cases if the correct coefficient of confinement is used.

(a) (d)

(b) (e)

Figure 3.5: Contours of vorticity with VC enabled comparing differentiable and minmod limiter: (a-c) minmod Limiter; (d-e) differentiable limiter; (a,d) 2 chords (2C) from trailing edge; (b,e) 5C from trailing edge; (c,f) 8C from trailing edge; (g) scale for contour plots

Although the differentiable limiter produces less numerical dissipation, there are additional considerations when selecting a model. Figure 3.6 shows the presence of numerical artifacts in the differentiable limiter simulation. These type of artifacts may be captured unintentionally by VC which will increase their strength and affect the stability of the simulation. The minmod limiter did not produce these types of artifacts.

Additionally, the coefficient of confinement requires more precise tuning to properly confine the tip vortex.

Figure 3.6: Contours of vorticity at two chord lengths from trailing edge using differentiable limiter without VC

CHAPTER IV

TURBULENT SIMULATIONS OF FIXED WINGTIP VORTICES USING

VORTICITY CONFINEMENT

4.1 Evaluation of Viscous Drag

As the first step in extending this technique to turbulent flow-field, the accuracy of the wake-integral technique in determining viscous simulation was evaluated. A flat plate with a laminar boundary layer was modeled for comparison of near-field and far- field integration techniques to the analytical Blasius solution. The shear stress on a 2D plate can be determined by the expression:

2 −1/2 휏0 = 휌푢00.332푅푒푥 (4.1) 휌푢 푥 푅푒 = 0 푥 휇

In Equation 4.1, 푢0 represents the free stream velocity outside of the boundary layer, 휌 represents the fluid density, 휇 represents the dynamic viscosity and 푥 represents the horizontal location on the flat plate. The shear force can be determined by integrating on

푥 over the length of the flat plate.

Table 4.1: Flow characteristics for boundary layer in flat plate

Flow Characteristics

Density (휌) 10 kg/m3

Viscosity (휇) 0.001 kg/(m*s)

Inlet Velocity (푢0) 1 m/s

ReL 10000

The computational results are compared with this analytical calculation to determine the accuracy of the drag prediction techniques in determining drag due to shear stress. The integration for the analytical results can be seen below:

1 1 1 2 2 2 휇 −1 2 휇 1 1 ∫ 휌푢00.332 ( ) 푥 2푑푥 = 2휌푢00.332 ( ) (1 2 − 0 2) = 0.0664 푁 푥=0 휌푢0 휌푢0

The computational setup consisted of a 1 meter long flat plate with an inlet on the left side. The top and right side of the domain were set as outflow boundary conditions and the plate wall was set as a no-slip wall boundary. The fluid properties and flow characteristics are given in Table 4.1. In order to determine the effect of grid density on the accuracy of the integration techniques, the simulation was run with several grid refinement levels. The results are presented in Table 4.2. Near-field integration was generally more accurate and was less sensitive to grid density. This is in contrast to pressure forces, which are more accurately predicted by the wake-integral technique.

Since the results for pressure and viscous forces are decomposed, near-field integration may be utilized to determine viscous forces on the wing and the wake-integral method may be used in the calculation of induced drag.

Table 4.2: Results for near-field and far-field integration for viscous flow about flat plate, error is determined by comparison to Blasius solution for shear stress

Grid Shear Stress Results (N)

Grid X Nodes Y Nodes Near % Error Far % Error

1 200 75 0.0715 7.68 0.0754 13.55

2 100 40 0.0724 9.04 0.0657 1.05

3 100 20 0.0724 9.04 0.0777 17.02

4 50 20 0.0728 9.64 0.0882 32.83

4.2 Results of Turbulent 3D Fixed Wing Simulations

As discussed in Chapter II, VC does not affect viscous dissipation in the range of vorticity confinement coefficients selected to counteract inviscid numerical diffusion.

This analysis was conducted using effective viscosity levels taken from FLUENT simulations seen in Figure 4.1. Therefore the same confinement parameters as for inviscid computations (listed in Table 3.3) were selected.

(a) (b) (c) (d)

Figure 4.1: Contours of effective viscosity produced by Spalart-Allmaras turbulence model for fixed wing simulation with VC implemented. Here, AoA is 4 degrees and Ma=0.3. Turbulent viscosity is between 0.001 and 0.0025. Location of cross-sections: (a) 2 chords downstream from trailing edge of wing (2C); (b), 5C; (c), 8C; (d) represents the values of the contours.

The two equation K-ε model, one-equation Spalart-Allmaras (S-A) and seven- equation Reynolds Averaged Navier-Stokes (RANS) turbulence models were investigated to determine the most applicable turbulence models for these simulations.

As seen in the AoA = 4 degrees simulation in Figure 4.2, the RANS and S-A turbulence models produced similar results with regard to induced drag. The K-epsilon model of turbulence had a non-physical effect on induced drag and was not used for subsequent simulations.

As AoA (α) increases, induced drag increase according to lifting line theory by α2.

This can be seen in the results shown in Figure 4.2. The results are not expected to be constant at the individual Trefftz plane locations even when numerical dissipation is completely counteracted by VC because of physical viscous dissipation and the results show this. The level of effective viscosity for the Spalart-Allmaras model of turbulence was approximately 70 percent higher than for the six Reynolds stress model.

51 210 50 49 190 48 47 170 46 45 150 44 43 Integral, Induced Drag Force (N) Force Drag Induced Integral,

- 130 Integral, Induced Drag Force (N) Force Drag Induced Integral,

- 42 0 4 8 12 0 4 8 12 Trefftz Plane (Chord Lengths) Wake Wake Trefftz Plane (Chord Lengths) SA W/O VC SA W/ VC SA W/O VC SA W/ VC RANS W/O VC RANS W/ VC (a) (b)

450 1150 440 1100 430 420 1050 410 1000 400 950 390 900 380 370 850 Integral, Induced Drag Force (N) Force Drag Induced Integral, - Integral, Induced Drag Force (N) Force Drag Induced Integral,

- 360 800 0 4 8 12 0 4 8 12 Wake

Wake Trefftz Plane (Chord Lengths) Trefftz Plane (Chord Lengths) SA W/O VC SA W/ VC SA W/O VC SA W/ VC RANS W/O VC RANS W/ VC RANS W/O VC RANS W/ VC (c) (d) Figure 4.2: Induced drag at Ma=0.3 for turbulent simulations with a comparison of S-A and RANS results of induced drag at various Trefftz planes with (W/) and without (W/O) VC: (a) AoA = 2°; (b) AoA = 4°; (c) AoA = 6° ; (d) AoA = 10°. K-ε model is used for AoA of 4 degrees (sub-figure b) only.

Table 4.3 shows the results of the shear stress evaluation on the surface of the wing for the Turbulence models investigated. Figure 4.3 is the percentage of drag caused by shear force to the overall drag experience by the finite wing. As the AoA increases to

10 degrees for the taken aspect ratio, the effect of induced drag increases to 80-90% of the total drag. Given the importance in design for accurately determining drag force,

50 these high AoA regions benefit the most from VC as it allows the wake-integral method to provide a more accurate and consistent result.

Table 4.3: Shear force and induced drag results for turbulent simulations

Shear Force & Induced Drag (N)

AoA RANS Spalart-Allmaras K-Epsilon Induced Drag

2 159 301 47

4 179 291 395 190

6 150 282 420

10 145 255 1050

100 90 80 70 60 50 40 30 20 Percentage Induced Drag Induced Percentage 10 0 2 4 6 8 10 Angle of Attack Spalart-Allmaras RANS

Figure 4.3: Percentage of induced drag contribution to total drag

4.3 Comparison to Experiements

Experimental tip-vortex experiments were conducted using a NACA0012 airfoil profile with a rectangular planform and blunt tip were performed in a wind tunnel test by

Davenport et al.[26]. The fixed wing had a chord length of 0.203 m and an aspect ratio of 8.66. The angle of attack for the experiment was 훼 = 5∘ with a freestream velocity of

푚 푈∞ = 38.1 ⁄푠. The geometry and flow conditions were replicated using ANSYS

DesignModeler; two meshes were created to show the effect of mesh parameters on the simulation of tip vortices. The 7 equation, 6 Reynolds stress (RANS) and Spalart-

Allmaras (S-A) turbulence models were investigate for accuracy in vortex evolution.

An unstructured mesh, referred to herein as mesh I, with the same meshing parameters (though slightly different geometry) as the mesh in the previous section was created and resulted in 1.5 million cells. The other mesh, referred to herein as mesh II, combined an unstructured zone with a structured zone in the region of vortex convection.

In addition to the structured region, the mesh was also well refined at the wing tips to improve the simulation of vortex generation. The grid interface between the structured and unstructured zones can be seen in Figure 4.4. The mesh setup resulted in the creation of a grid with 5.1 million cells.

Figure 4.4: Interface of structured and unstructured zones in mesh II

It was determined using inviscid simulations that the appropriate coefficient of confinement for Mesh I was c=0.04. Mesh I was found to be too coarse to compute the

52 vortex core accurately. There is a high gradient of velocity that is present in the vortex core, as seen in experimental data in Fig. 4.5, and it is necessary to have a well refined mesh to properly model the vortex peak velocity and core size. The velocity and position 푚 values were normalized by mean flow (푈∞ = 38.1 ⁄푠) and chord length (0.203m), respectively. It can be seen in Fig. 4.6 that while vorticity confinement removed the numerical dissipation for the simulation, it did not accurately represent peak velocity or velocity gradients; Fig. 4.6a show a decay of velocity between 5 and 10 chords (5C and

10C) downstream while Fig.4.6b shows little decay which is qualitatively similar to the experimental data (Fig. 4.6).

0.3 5C 0.2 10C

0.1

0 -0.3 -0.15 0 0.15 0.3 -0.1

Velocity (Normalized) -0.2

-0.3 Position (Normalized)

Figure 4.5: Experimental results of vortex velocity [26]

0.15 0.15 5C 5C 0.10 10C 0.10 10C

0.05 0.05

0.00 0.00 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -0.05 -0.05

Velocity Velocity (Normalized) -0.10 Velocity (Normalized) -0.10

-0.15 -0.15 Position (Normalized) Position (Normalized)

(a) (b) Figure 4.6: Results of computations using mesh I: (a) RANS turbulence model without VC; (b) RANS with VC Mesh II required a grid interface be used in ANSYS FLUENT which interpolated the values between the structured and unstructured zones. It can be seen in Fig. 4.7 that this resulted in an oscillation in the vortex velocity. Despite the oscillation in the vortex velocity, the grid shows an advantage in simulating the velocity gradient and peak velocity. Figure 4.7c,d were simulated using the S-A model of turbulence. The coefficient of confinement for mesh II was determined to be c=0.015; the structured grid experiences considerably less numerical dissipation compared to unstructured which leads to a reduction in the value of c.

One of the reasons that the Spalart-Allmaras model is overly diffusive is that the model is vorticity based. Laser sheet flow visualization experiments indicate that a tip vortex has three distinct regions, an inner laminar region, a transition region and an outer turbulent region [27]. Thus, turbulence models that produce a high level of turbulent viscosity in the vortex core result in a rapid decay of vortex strength compared to experiments [28].

0.25 0.25 5C 5C 0.20 0.20 10C 0.15 0.15 10C 0.10 0.10 0.05 0.05 0.00 0.00 -0.05 -0.5 -0.25 0 0.25 0.5 -0.05 -0.5 -0.25 0 0.25 0.5 -0.10 -0.10

Velocity Velocity (Normalized) -0.15 Velocity (Normalized) -0.15 -0.20 -0.20 -0.25 -0.25 Position (Normalized) Position (Normalized)

(a) (b)

0.25 0.25 5C 5C 0.20 0.20 0.15 10C 0.15 10C 0.10 0.10 0.05 0.05 0.00 0.00 -0.05 -0.5 -0.25 0 0.25 0.5 -0.05 -0.5 -0.25 0 0.25 0.5 -0.10 -0.10

Velocity Velocity (Normalized) -0.15 -0.15Velocity (Normalized) -0.20 -0.20 -0.25 -0.25 Position (Normalized) Position (Normalized)

Figure 4.7: Results of computations using mesh II: (a) RANS turbulence model without VC; (b) RANS turbulence model with VC; (c) SA turbulence model without VC; (d) SA turbulence model with VC

Figure 4.7a shows the results of The RANS model without VC. The results show a reduction in the velocity gradient and it can be seen that the vortex core to spreads from

5C to 10C downstream; this was not observed in the experimental results and is the effect of numerical dissipation. VC. Figure 4.7b show the results of the RANS model combined with VC. The method has the least amount of vortex decay in terms of velocity gradient

55 and peak velocity and compares better to experimental results than any other method tested.

CHAPTER V

TURBULENT SIMULATIONS OF WINGTIP VORTEX COMBINED

WITH VORTICITY CONFINEMENT

5.1 Grid and Setup

The blade is a NACA 2415 airfoil profile with a collective pitch of 4.5 degrees.

There is a center rod in the flow field which a single rotor is attached to. The blade length is 9.12 chords with a tip Mach number of M = 0.26, the chord length is 0.045 meters. The rotation of the blade was setup by frame motion with a rotational velocity of 214.4 radians/s.

The geometry for the simulation of a hovering rotor was set up to compare results to [29]. The grid is refined in the region that the vortex travels up to 90 degrees, which can be seen in Figure 2.1. The grid has a total cell count of 5,041,659 cells. The relevant physical parameters for the simulation are given in Table 2.1 and the simulations parameters are given in Table 2.2. Constant density was used as the Mach number does not exceed 0.3 which is considered the threshold for density effects to be significant [8].

Table 5.1: Physical problem specifications for rotating wing simulations

Aspect Ratio 9.12 Chord Length 0.045 m Tip Mach Number 0.26 Chord Reynolds Number 272,000 Collective Pitch 4.5 Degrees Wing Profile NACA2415

Table 5.2: Computational parameters for rotating wing simulations

FLUENT Settings Solver and Models Solver Pressure Based Viscous Model Inviscid Density Constant Energy Equation Off Solution Methods Pressure-Velocity Coupling SIMPLE Density, Momentum, Energy Second Order Upwind Pressure Standard Confinement Parameter 0.035

(a) (b) (c) Figure 5.1: Computational domain for rotating wing simulations: (a,b) grid geometry and mesh refinement zone for the convection of tip vortex; (c) mesh cross-section in region of refinement

The VCF2 method (see section 2.2) was implemented into the commercial software ANSYS FLUENT. The goal was to determine the proper confinement 58 parameter to combine with turbulence models. It was determined that the proper confinement parameter for this simulation to be c = 0.035.

5.2 Inviscid Results

The results show VC to maintain the vortex strength through wake age of 90 degree, which was the target distance for maintaining the vortex. The wake age is used to define the position of the vortex. For example, wake age of 45 degrees represents the vortex when the blade has traveled 45 degrees from the position the vortex was shed.

The radius of the rotor path is 9.12 chord lengths; for comparison with results for the stationary wing (Chapters III & IV), 45 degrees corresponds to 7.16 chord lengths away

휋 ∗ 9.12 chords = 7.16 chords from the trailing edge (4 ). Figure 5.2 shows a comparison of the results without (left column) and with VC (right column) at wake ages ranging from 3 degrees to 90 degrees. It can be clearly seen that without VC the vortex dissipation is already severe at 45 degrees.

With VC enabled, the vortex maintains its strength in terms of vorticity through

90 degrees; swirl velocity is also well maintained when VC is implemented, which can be seen in Figure 5.3; for the figures shown, swirl velocity is normalized by tip speed. As with the fixed wing case, VC is now applied to turbulent simulations using the confinement parameter found to properly confine the inviscid case.

(a) (d)

(b) (e)

Figure 5.2: Rotating wing simulation results, contours of vorticity with and without VC: (a-c) vorticity without VC; (d-f) vorticity with VC enabled with c=0.035; (a,d) 3 degrees; (b,e) 45 degrees; (c,f) 90 degrees.

0.3 0.3

0.25 0.25

0.2 0.2

0.15 0.15 0.1 0.1 0.05 0.05 Swirl Swirl Velocit (Normalized)

0 Swirl Velocity (Normalized) 0 0 0.2 0.4 0 0.2 0.4 Radius (Normalized by Chord) Radius (Normalized by Chord) 3 Degrees 45 Degrees 90 Degrees 3 Degrees 45 Degrees 90 Degrees (a) (b) Figure 5.3: Swirl velocity profiles from rotating wing simulations with and without VC: (a) swirl velocity at various wake ages without VC enabled; (b) swirl velocity at various wake ages with VC enabled

5.3 Turbulence Model Results

For the rotating wing, VC was combined with both the Spalart-Allmaras and six

Reynolds Stresses model and compared to experiments to determine the most appropriate turbulence model to pair with VC for the simulation of wing-tip vortices. The results of the velocity profiles are shown in Figure 5.4. The Spalart-Allmaras simulation did not respond well to the presence of VC and the results in Figure 5.4b show that there is little difference of the velocity profile with and without VC enabled.

The results for the 6 Reynolds Stress model, however, showed considerable improvement. It can be seen in Figure 5.4a that the peak velocity deceases by 16 percent when traveling from wake age 3 degrees to 45 degrees. For the six Reynolds stress model without VC enabled, the peak velocity reduces by 43 percent at this point.

Additionally, without VC enabled, the slope of the velocity in the core of the vortex is reduced considerably. With VC enabled for The six Reynolds Stress model computes a reduction in the peak velocity by only 20 percent, which is much closer to the experimental results which showed a reduction of 12 percent [29]. The slope of the velocity in the core of the vortex does not show much change for 3 degrees to 45 degrees which is consistent with experimental results.

0.4 0.4 0.35 0.35 0.3 0.3 0.25 0.25 0.2 0.2 0.15 0.15 0.1 0.1 0.05

0.05 Swirl Velocity (Normalized) Swirl Swirl Velocity (Normalized) 0 0 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 Position (Normailzed by Chord) Position (Normalized by Chord) 3 Degrees with VC 45 Degrees with VC 3 Degrees 45 Degrees 3 Degrees without VC 45 Degrees without VC (a) (b)

0.4 0.4 0.35 0.35 0.3 0.3 0.25 0.25 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 Swirl Swirl Velocity (Normalized) Swirl Swirl Velocity (Normalized) 0 0 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 Position Position 3 Degrees, VC On 3 Degrees, VC Off 3 Degrees(7 Eq.) 45 Degrees (7 Eq.) 45 Degrees, VC Off 45 Degrees, VC On 3 Degrees (SA) 45 Degrees (SA) (c) (d) Figure 5.4: Swirl velocity of wingtip vortex from turbulent simulation and experiments: (a) experimental results [29]; (b) SA turbulence model results with and without VC; (c) RANS turbulence model results with and without VC; (d) Turbulence models comparison with VC enabled

One of the reasons that the Spalart-Allmaras model is overly diffusive is that the model is vorticity based. Laser sheet flow visualization experiments indicate that a tip vortex has three distinct regions, an inner laminar region, a transition region and an outer turbulent region [27]. Thus, turbulence models that produce a high level of turbulent viscosity in the vortex core results in a more rapid decay of vortex strength compared to experiments [27,28]. Additionally, high levels of numerical viscosity will produce the

62 same effect and results in premature decay of the vortex. Since the Spalart-Allmaras model determines turbulent viscosity based upon vorticity, the core of the vortex experience excessive dissipation when simulated using this model. Additionally, VC has the effect of increasing vorticity; this causes the turbulence model to increase turbulent viscosity which effectively counteracts the effect of VC.

Post-processing of the results show that effective viscosity in the region of the tip vortex is approximately 20 times larger when using the Spalart-Allmaras model compared to the 6 Reynolds stress model. This is in contrast with the fixed wing results where the level of effective viscosity was only 70 percent higher for the Spallar-Allmaras model compared to the 6 Reynolds stress model. The time and viscous scales up to a wake age of 180 degrees produced by the 6 Reynolds stress model are comparable to those evaluated in Section 2.5 to determine the effect of VC on viscous decay. In Section

2.5, it was noted that at these levels VC has a negligible effect on physical viscous dissipation.

5.4 Winglet Modeling and Results

By altering the geometry of the wing tips, the tip vortices can be redirected and/or reduced in magnitude, which in return improves the performance of the wing [7].

Helicopter wing tips are mostly designed with the intent to both improve performance and reduce the noise and vibration generated. Not only does a rotor blade have to deal with vortices being produced off of its own tip, but also the vortices that were produced earlier by other blades, known as blade-vortex interaction (or BVI for short) [30]. Senior design student Thomas Sams [31] conducted an investigation into winglet geometry for his senior design project over the course of the 2012-13 academic year.

The objective of the senior design project was to study the effects of different winglet designs on helicopter rotor blades using computation fluid dynamics. The performance of a winglet design was classified based upon the winglets effect on lift, drag, and both location and magnitude of the tip vortex throughout its rotation. A final winglet design was chosen based on these criteria and is shown below in Figure 5.5 [31].

Figure 5.5: Geometry of winglet for rotating wing

A simulation of the winglet design in Figure 5.5 is conducted with the intent of simulating the tip-vortex evolution. The setup is identical to what is described in Section

5.1 but with the addition of the winglet. The evolution of the vortex up to 90 degree wake age using 1st order, 2nd order and 2nd order with VC schemes combined with the 7 equation, 6 Reynolds stress turbulence model is investigated for the winglet design. The proper value for confinement was found to be the same as the case without a winglet

(c=0.035).

(a) (d) (g)

(b) (e) (h)

(j) (c) (f) (i) Figure 5.6: Contours of vorticity results for simulation of rotating wing with winglet: (a- c) contours of vorticity for 1st order simulation; (d-f) contours of vorticity for second- order simulation; (g-i) contours of vorticity for second-order simulation with VC; The first row (a,d,g) is at 3 degrees wake age, the second row (b,e,h) is at 45 degrees wake age and the third row (c,f,i) is at 90 degree wake age; (j) is the scale for the contour plots.

Though the strength and the path of the vortex may be altered by the addition of the winglet, the vortex is expected to evolve qualitatively similar to the tip vortex without the winglet. Figure 5.6 shows contours of vorticity, a lower level of vorticity represents a smaller velocity gradient for the swirl velocity. At a wake age of 45, the gradient of the swirl velocity at the vortex core should not be altered much compared to at 3 degrees as seen in Figure 5.4a. The first and second order simulations without VC show a high level of dissipation compared to the simulation using vorticity confinement. In order to study the effects of blade-vortex interaction, the vortex should be well maintained until it impinges upon the next blade. Without VC, any attempt to study this would require a specially designed grid with a well refined structured grid in order to capture the vortex interaction as in [29].

CHAPTER VI

CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK

Far-field integration results in more accurate prediction of induced drag than near- field pressure integration in comparison to lifting line theory as a benchmark; however, due to numerical dissipation, the results are dependent upon the Trefftz plane chosen for integration. Vorticity Confinement (VC) is investigated as a means to eliminate the dependence of induced drag results upon Trefftz plane. VC is first shown to significantly reduce numerical dissipation for 2D simulations of a convected Taylor vortex. Two formulations of VC, VCF1 and VCF2, were investigated for their ability to maintain the vortex in an inviscid flow field. Since unstructured gridding is used for the determination of induced drag, VCF2 was applied to the 3D simulations because the VCF2 confinement term is adaptive based upon grid density.

2D results showed that second order schemes combined with TVD limiters have a greater level of stability than a pure second-order scheme when combined with vorticity confinement. The minmod limiter showed the greatest level of numerical stability and did not experience over-confinement when the van Albada differentiable limiter experienced over-confinement and the pure second-order scheme failed. In 3D simulations, the differentiable limiter produced numerical artifacts around the vortex region that were not present in the simulation using the minmod limiter.

The application of VC to far-field evaluation of induced drag was extended from previous work [5] to subsonic Mach number flows up to Mach 0.6 and AoAs up to 10 degrees. It was found that there is a different coefficient of confinement associated with each case. The confinement parameter was found to be dependent upon AoA and, to a lesser degree, Mach number. For an optimal confinement parameter, VC eliminates the dependence of induced drag on Trefftz plane.

To apply VC to simulations of tip vortex evolution, the effect of VC on viscous decay was first evaluated using 2D simulations of the decay of line vortex. For the time and viscosity scale relavent to that for fluid particle convection from wing tip to Trefftz plane, VC has only a minor effect on the viscous decay of line vortex and served to improve the simulations when the grid was well refined. Since VC does not significantly affect physical viscous dissipation when the proper confinement parameter is applied, it is therefore used combined with Reynolds-averaged Navier-Stokes (RANS) turbulence models to eliminate numerical dissipation associated with upwind schemes while retaining physical phenomena such as viscous and turbulent dissipation.

Tip vortex simulations for fixed wings showed the need for well refined structured gridding to capture peak velocities as well as velocity gradients. It was shown that the Spallart-Allmaras (SA) model of turbulence resulted in an unphysical decay of the vortex strength while the six Reynolds stress model combined with VC compared best to experimental data of all computational methods tested.

VC was applied to a simulation of a rotating wing. It was found that the first and second order upwind schemes are too diffusive to accurately simulate the evolution of a wing tip vortex for wake ages relevant to blade vortex interaction. When combined with

VC, the six Reynolds stresses turbulence model most accurately simulates the evolution of a wing-tip vortex whereas VC does not significantly improve results when SA model is applied. This occurs, at least in part, because the SA model produces turbulent viscosity dependent upon vorticity. The high vorticity in the center of the vortex causes the model to produce higher levels of turbulent viscosity than it should to model the phenomenon. As VC serves to increase vorticity, the SA model becomes even more diffusive in the presence of VC.

Although the dependence on a user defined coefficient is lessened when using

VCF2, the method still requires significant tuning when applying it to various AoAs. It was shown that the proper coefficient of confinement did not change with the addition of a winglet meaning that VCF2 is a viable option for the design of winglets to improve performance. It is recommended that further investigations into VC focus on more automated methods such as adaptive vorticity confinement described in [16]. This way,

VC can be applied to a simulation without the need for a user defined coefficient of confinement.

BIBLIOGRAPHY

[1] T. Snyder and A. Povitsky, “Vorticity Confinement Technique for Drag Prediction”, AIAA Paper 2010-4936, Presented at the 28th AIAA Applied Aerodynamics Conference. Expanded version accepted to AIAA J of Aircraft in November 2013.

[2] K. Karamcheti., Principles of Ideal-Fluid Aerodynamics, 2nd edition, Robert E. Krieger Publishing Company, 1980.

[3] J. Steinhoff, N. Lynn and L. Wang, “Computation of high Reynolds number flows using vorticity confinement: I. formulation,” preprint, University of Tennessee Space Institute, March 2005.

[4] R. Lohner, C. Yang and R. Roger, “Tracking vortices over large distances using vorticity confinement,” in 24th Symposium on Naval Hydrodynamics, (Fukuoka, Japan), 2002.

[5] T. Snyder, “A Coupled Wake-Integral/Vorticity Confinement Technique for the Prediction of Drag Force,” Master's Thesis, Dept. of Mechanical Engineering, The University of Akron, Akron, OH, 2012.

[6] A. Brocklehurst and G. Barakos, “A Review of Helicopter Rotor Blade Tip Shapes,” Progress in Aerospace Sciences, vol. 56, 2013, pp. 35-74.

[7] S. A. Ning and I. Kroo, “Multidisciplinary Considerations in the Design of Wings and Wing Tip Devices,” Journal of Aircraft , vol. 47, no. 2, 2010, pp. 534-543.

[8] R. W. Fox, P. J. Pritchard and A. T. McDonald. Fox and McDonald's Introduction to Fluid Mechanics, 8th ed., 2011.

[9] J. Tannehill, D. Anderson and R. Pletcher, Computational fluid mechanics and heat transfer. Taylor and Francis, 2nd ed., 1997.

[10] J. Steinhoff, W. Yonghu, T. Mersch, and H. Senge, “Computational vorticity capturing: Application to helicopter rotor flow,” in 30th Aerospace Sciences Meeting and Exhibit, (Reno, NV), AIAA, January 1992. AIAA Paper 92-0056.

[11] J. Steinhoff and D. Underhill, “Modification of the Euler equations for vorticity confinement: Application to the computation of interacting vortex rings,” Phys. Fluids, vol. 6, pp. 2738–2744, 1994.

[12] J. Steinhoff, M. Fan, L. Wang and W. Dietz, “Convection of concentrated vortices and passive scalars as solitary waves,” Journal of Scientific Computing, vol. 19, pp. 457–478, 2003.

[13] J. Steinhoff, N. Lynn, W. Yonghu, M. Fan, L. Wang and B. Dietz, “Computation of high Reynolds number flows using vorticity confinement: II. Results,” preprint, University of Tennessee Space Institute, March 2005.

[14] M. Costes, T. Renaud and B. Rodrigues, “Application of Vorticity Confinement to Rotor Wake Simulations,” Journal of Engineering Systems Modelling and Simulation, vol. 4, nos. 1/2, pp. 102-112, 2012.

[15] M. Costes, “Analysis of the Second Vorticity Confinement Scheme,” Aerospace Science and Technology, vol. 12, pp. 203-213, 2008.

[16] S. Hahn and G. Iaccarino, “Towards adaptive vorticity confinement,” in 47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition, (Orlando, FL), January 2009. AIAA paper 2009-1613.

[17] A. Povitsky and K. Pierson, “Vorticity Confinement Technique for Preservation of Tip Vortex of Rotating Blade,” in 31st AIAA Applied Aerodynamics conference, (San Diego, CA), 2013. AIAA paper 2013-2420.

[18] R. M. Cummings, M. B. Giles and G. N. Shrinivas, “Analysis of the elements of drag in three-dimensional viscous and inviscid flows,” in AIAA-96-2482-CP, AIAA, 1996.

[19] S. Monsch, “A Study of Induced Drag and Spanwise Lift Distribution for Three Dimensional Inviscid Flow Over a Wing,” Master's Thesis, Dept. of Mechanical Engineering, Clemson University, Clemson, SC, 2007.

[20] FLUENT Reference Manual, Software Release Ver. 14, ANSYS Inc., Canonsburg, PA, 2012.

[21] B. Parent, “Positivity-Preserving Flux-Limited Method for Compressible Fluid Flow”, Computers & Fluids, vol. 44, pp. 238–247, 2011.

[22] A. Povitsky and K. Pierson, “Vorticity Confinement Applied to Turbulent Wing Tip Vortices for Wake-Integral Drag Prediction,” in the 21st AIAA Computational Fluid Dynamics conference, (San Diego, CA), July 2013. AIAA paper 2013-2574.

[23] P. K. Kundu and I. M. Cohen. Fluid Mechanics. Academic Press, 4th ed., 2008.

[24] J. D. Anderson Jr., Fundamentals of Aerodynamics. McGraw-Hill, 4th ed., 2007.

[25] H. E. Murray, “Comparison with Experimental Methods of Predicting the Lift of Wings in Subsonic Compressible Flow,” Technical Note No. 1739, Langley Aeronautical Laboratory, (Langley Field, Va.), 1948.

[26] Devenport, W.J., Rife, M.C., Liapis, S.I., and Follin, G.J., “The Structure and Development of a Wing-tip Vortex,” Journal of Fluid Mechanics, Vol. 312, 1996.

[27] M. Ramasamy, “Contributions to the Measurement and Analysis of Helicopter Blade Tip Vortices,” Doctoral Dissertation, Dept. of Mechanical Engineering, University of Maryland, College Park, MD, 2004

[28] N. Komerath, O. Wong, B. Ganesh, “On the Formation and Decay of Rotor Blade Tip Vortices,” in the 34th AIAA Fluid Dynamics Conference and Exhibit, (Portland, OR), June 2004. AIAA paper 2004-2431.

[29] K. Duraisamt and J.D. Baeder, “High Resolution Wake Capturing Methodology for Hovering Rotors,” Journal of the American Helicopter Society, vol. 52, no. 2, pp. 110-122, 2006.

[30] C. M. Copland, “The Generation of Transverse and Longitudinal Vortices in Low Speed Wind,” Doctoral Dissertation, Dept. of Mechanical Engineering, University of Glasgow, Glasgow, Scottland, UK, 1997.

[31] T. Sams, Design and Evaluation of Winglets for Helicopter Blades, Senior Design Project, Mechanical Engineering Dept., Univ. of Akron, Akron, OH, May 2013. Advisor: A. Povitsky.

APPENDICES

APPENDIX A

FLUX SPLITING FOR STEGER WARMING SCHEME

휆 1 푢 휆1 푢 λ퐴 = = [ ] 휆2 푢 + 푐 [ 휆3] 푢 − 푐

푣 푣 λ = [ ] 퐵 푣 + 푐 푣 − 푐

λ퐴 and λ퐵 show the eigenvalues for flux splitting for the x and y directions, repectively.

+ − The positive and negative values are split into matrices 휆퐴 and 휆퐴 , for subsonic flow this leads to matrices:

푢 + 푢 휆 = [ ] 퐴 푢 + 푐 0

0 − 0 휆 = [ ] 퐴 0 푢 − 푐

The flux terms, 푭+ and 푭− are then given by:

훼+ + 푢훼 + 푐휆2 + + 푭 = 푣훼 푢2 + 푣2 휆 훼+ + 푢푐휆 + 푐2 2 [ 2 2 훾 − 1]

훼− − 푢훼 − 푐휆3 − 푣훼− 푭 = 푢2 + 푣2 휆 훼− − 푢푐휆 + 푐2 3 [ 2 3 훾 − 1]

+ 훼 = 2(훾 − 1)휆1 + 휆2

− 훼 = 휆3

훾 is the heat capacity ratio and 푐 is the speed of sound. The G flux terms similar to F with u replaced with v and v replaced with u.

APPENDIX B

2D MATLAB CODE

clear all hold off % this code model 2-D unsteady flow with Uniformly convected vortex for % Time period = 0.1 % Euler or Navier-Stoke equations may be solved % finite-volume second order using minmod limiter spatial discretization (Steger-Warming) CFL=0.2; totalstep=500 % coefficient of confinement ,c c=4.0 XL=-15; XR=15; YL=-15; YUP=15; epsilon=30 this1=1 % Mach number of initial mean flow Ma=0. Ma=0; % amount of circulation of vortex GR=-50.; % angle of inclination of mean uniform flow (no mean flow alpha = 0) alpha=0.; % speed of sound. m/s c_0=347; % heat capacity, J/Kg K c_p=1000.; % temperature_initial (K) T_0=300.; % gamma g=1.4; % atmospheric pressure, Pa p_atm=101325.; % gas constant R=287.; % Prandtl number Pr=0.7; % number of intervals 500 x 200 (step dx=dy=1/16 similar to Stanford) NX=90; NY=90; % grid step DX=(XR-XL)/NX; DY=(YUP-YL)/NY; % x & y-ccordinate discretized x=linspace(XL,XR,NX+1); y=linspace(YL,YUP,NY+1);

[x1,y1]=meshgrid(x,y);

%DX and DY varients DX1=1./DX; %DX reciprical DX2=DX1^2; %DX reciprical squared DY1=1./DY; %DY reciprical DY2=DY1^2; %DY reciprical squared

% integration in time CFL=0.8 CHANGE IT IF NEEDED t=0; % vector of conservative variables (initialization) W=zeros(NX+1,NY+1,4); % next-step vector of conservative variables WNEW=zeros(NX+1,NY+1,4); % vectors of fluxes & other various parameters (initialization); FP=zeros(NX+1,NY+1,4); FM=zeros(NX+1,NY+1,4); GP=zeros(NX+1,NY+1,4); GM=zeros(NX+1,NY+1,4);

FP2=zeros(NX+1,NY+1,4);

FM2=zeros(NX+1,NY+1,4); GP2=zeros(NX+1,NY+1,4);

GM2=zeros(NX+1,NY+1,4); phiFP=zeros(NX+1,NY+1,4); phiFM=zeros(NX+1,NY+1,4); phiGP=zeros(NX+1,NY+1,4); phiGM=zeros(NX+1,NY+1,4); hu=0 A=zeros(NX+1,NY+1); U=zeros(NX+1,NY+1); U9=zeros(NX+1,NY+1); V=zeros(NX+1,NY+1); P=zeros(NX+1,NY+1); T=zeros(NX+1,NY+1); VORT=zeros(NX+1,NY+1); U9=zeros(NX+1,NY+1); GR_VORT_X=zeros(NX+1,NY+1); GR_VORT_Y=zeros(NX+1,NY+1); VORT_MAG=zeros(NX+1,NY+1); eps_array=zeros(NX+1,NY+1); GRM=zeros(NX+1,NY+1); BL=0;

for j=1:NY+1 for i=1:NX+1 W(i,j,1)=1.2; T(i,j)=T_0; P(i,j)=R*W(i,j,1)*T_0; A(i,j)=(g*T(i,j)*R)^0.5; U(i,j)=Ma*(g*T(i,j)*R)^0.5*cos(alpha);

V(i,j)=Ma*(g*T(i,j)*R)^0.5*sin(alpha); % CALCULATE ROTATIONAL VELOCITY of Initial Line vortex % vortex is added to uniform flow arithmetically r=sqrt(x(i)*x(i)+y(j)*y(j)); % ZERO linear core see SNYDER THESIS

%UR=GR/(2*pi*r); UR=GR/(2*pi)*r*exp((1.-r*r)/2.); if (r>0) U9(i,j)=UR*y(j)/r; % add convection velocity and vortex velocity U(i,j)=U(i,j)+UR*y(j)/r; V(i,j)=V(i,j)-UR*x(i)/r; end W(i,j,2)=U(i,j)*W(i,j,1); W(i,j,3)=V(i,j)*W(i,j,1); W(i,j,4)=c_p*T_0+W(i,j,1)*0.5*(U(i,j)^2+V(i,j)^2); end end

% number of steps Tstep=200; Time = 0 %------t_vec=zeros(Tstep,1);

% FVM Calculation begins %while Istep < 0 while Time < 0.1 for i=1:NX+1 for j=1:NY+1 % positive fluxes - x-direction FL1=U(i,j); FL2=FL1; FL3=U(i,j)+ A(i,j); FL4=U(i,j)-A(i,j); if FL1>0 FP(i,j,1)=2*(g-1)*FL1 +FL3; FP(i,j,2)=2*(g-1)*FL1*U(i,j)+FL3*(U(i,j)+A(i,j)); FP(i,j,3)=2*(g-1)*FL1*V(i,j)+FL3*V(i,j); FP(i,j,4)=(g-1)*FL1*(U(i,j)^2+V(i,j)^2)+... see next line 0.5*FL3*((U(i,j)+A(i,j))^2+V(i,j)^2)+...see next line 0.5*A(i,j)^2*((3-g)/(g-1))*FL3; else FP(i,j,1)=FL3; FP(i,j,2)=FL3*(U(i,j)+A(i,j)); FP(i,j,3)=FL3*V(i,j); FP(i,j,4)=0.5*FL3*((U(i,j)+A(i,j))^2+V(i,j)^2)+...see next line 0.5*A(i,j)^2*((3-g)/(g-1))*FL3; end % positive fluxes - y-direction GL1=V(i,j); GL2=GL1; GL3=V(i,j)+ A(i,j); GL4=V(i,j)-A(i,j); if GL1>0 GP(i,j,1)=2*(g-1)*GL1+GL3; GP(i,j,3)=2*(g-1)*GL1*V(i,j)+GL3*(V(i,j)+A(i,j)); GP(i,j,2)=2*(g-1)*GL1*U(i,j)+GL3*U(i,j); GP(i,j,4)=(g-1)*GL1*(U(i,j)^2+V(i,j)^2)+... see next line 0.5*GL3*((V(i,j)+A(i,j))^2+U(i,j)^2)+...see next line 0.5*A(i,j)^2*((3-g)/(g-1))*GL3;

77 else GP(i,j,1)=GL3; GP(i,j,3)=GL3*(V(i,j)+A(i,j)); GP(i,j,2)=GL3*U(i,j); GP(i,j,4)=0.5*GL3*((V(i,j)+A(i,j))^2+U(i,j)^2)+...see next line 0.5*A(i,j)^2*((3-g)/(g-1))*GL3; end % negative fluxes - x-direction if FL1>0 FM(i,j,1)=FL4; FM(i,j,2)=FL4*(U(i,j)-A(i,j)); FM(i,j,3)=FL4*V(i,j); FM(i,j,4)=0.5*FL4*((U(i,j)-A(i,j))^2+V(i,j)^2)+0.5*FL4*A(i,j)^2*((3- g)/(g-1)); else FM(i,j,1)=2*(g-1)*FL1+FL4; FM(i,j,2)=2*(g-1)*FL1*U(i,j)+FL4*(U(i,j)-A(i,j)); FM(i,j,3)=2*(g-1)*FL1*V(i,j)+FL4*V(i,j); FM(i,j,4)=(g-1)*FL1*(U(i,j)^2+V(i,j)^2)+... see next line 0.5*FL4*((U(i,j)-A(i,j))^2+V(i,j)^2)+0.5*FL4*A(i,j)^2*((3-g)/(g- 1)); end % negative fluxes- y direction if GL1>0 GM(i,j,1)=GL4; GM(i,j,3)=GL4*(V(i,j)-A(i,j)); GM(i,j,2)=GL4*U(i,j); GM(i,j,4)=0.5*GL4*((V(i,j)-A(i,j))^2+U(i,j)^2)+0.5*GL4*A(i,j)^2*((3- g)/(g-1)); else GM(i,j,1)=2*(g-1)*GL1+GL4; GM(i,j,3)=2*(g-1)*GL1*V(i,j)+GL4*(V(i,j)-A(i,j)); GM(i,j,2)=2*(g-1)*GL1*U(i,j)+GL4*U(i,j); GM(i,j,4)=(g-1)*GL1*(U(i,j)^2+V(i,j)^2)+... see next line 0.5*GL4*((V(i,j)-A(i,j))^2+U(i,j)^2)+0.5*GL4*A(i,j)^2*((3-g)/(g- 1)); end for k=1:4 FP(i,j,k)=0.5*W(i,j,1)/g*FP(i,j,k); GP(i,j,k)=0.5*W(i,j,1)/g*GP(i,j,k); FM(i,j,k)=0.5*W(i,j,1)/g*FM(i,j,k); GM(i,j,k)=0.5*W(i,j,1)/g*GM(i,j,k); end

dt=0.01; dtmax=CFL/(DX1*(abs(U(i,j))+A(i,j))+DY1*(abs(V(i,j))+A(i,j))); if dtmax

dt=dtmax; end

end

for i=2:NX-1 for j=2:NY-1 for k=1:4 if FP(i,j,k) ~= FP(i-1,j,k) rFP(i,j,k)=(FP(i+1,j,k)-FP(i,j,k))/(FP(i,j,k)-FP(i-1,j,k)); else rFP(i,j,k)=1; end if FM(i+2,j,k) ~= FM(i+1,j,k) rFM(i,j,k)=(FM(i+1,j,k)-FM(i,j,k))/(FM(i+2,j,k)-FM(i+1,j,k)); else rFM(i,j,k)=1; end if GP(i,j,k) ~= GP(i,j-1,k) rGP(i,j,k)=(GP(i,j+1,k)-GP(i,j,k))/(GP(i,j,k)-GP(i,j-1,k)); else rGP(i,j,k)=1; end if GM(i,j+2,k) ~= GM(i,j+1,k) rGM(i,j,k)=(GM(i,j+1,k)-GM(i,j,k))/(GM(i,j+2,k)-GM(i,j+1,k)); else rGM(i,j,k)=1; end

end

end end

for i=5:NX-5 for j=5:NY-5 for k=1:4 phiFP(i,j,k)=max(0,min(1,rFP(i,j,k))); phiFM(i,j,k)=max(0,min(1,rFM(i,j,k))); phiGP(i,j,k)=max(0,min(1,rGP(i,j,k))); phiGM(i,j,k)=max(0,min(1,rGM(i,j,k))); end end end

% Second order flux calculation for i=2:NX-2 for j=2:NY-2 for k=1:4 FP2(i,j,k)=FP(i,j,k)+phiFP(i,j,k)*0.5*(FP(i,j,k)-FP(i-1,j,k));

FM2(i,j,k)=FM(i+1,j,k)-phiFM(i,j,k)*0.5*(FM(i+2,j,k)-FM(i+1,j,k));

GP2(i,j,k)=GP(i,j,k)+phiGP(i,j,k)*0.5*(GP(i,j,k)-GP(i,j-1,k));

GM2(i,j,k)=GM(i,j+1,k)-phiGM(i,j,k)*0.5*(GM(i,j+2,k)-GM(i,j+1,k));

end end end

for i=3:NX-2 for j=3:NY-2 for k=1:4 W(i,j,k)=W(i,j,k)-dt*(DX1*(FP2(i,j,k)-FP2(i-1,j,k)+FM2(i,j,k)-FM2(i- 1,j,k))+... DY1*(GP2(i,j,k)-GP2(i,j-1,k)+GM2(i,j,k)-GM2(i,j-1,k))); end % viscous term U-monentum W(i,j,2)=W(i,j,2)+dt*hu*(DX2*(U(i-1,j)-2*U(i,j)+U(i+1,j))+... DY2*(U(i,j-1)-2*U(i,j)+U(i,j+1))); % viscous terms V-momentum W(i,j,3)=W(i,j,3)+dt*hu*(DX2*(V(i-1,j)-2*V(i,j)+V(i+1,j))+... DY2*(V(i,j-1)-2*V(i,j)+V(i,j+1))); % viscous term energy transport W(i,j,4)=W(i,j,4)+dt*hu/Pr*(DX2*(T(i-1,j)-2*T(i,j)+T(i+1,j))+... DY2*(T(i,j-1)-2*T(i,j)+T(i,j+1))); %%%%%%%add VC%%%%%%%%

GRM(i,j)=(GR_VORT_X(i,j)^2+GR_VORT_Y(i,j)^2)^0.5;

% VCF2

if GRM(i,j)>0 GR_VORT_X(i,j)=GR_VORT_X(i,j)/GRM(i,j); end

if GRM(i,j)>0 GR_VORT_Y(i,j)=GR_VORT_Y(i,j)/GRM(i,j); end

%eps_array(i,j)=epsilon; eps_array(i,j)=c*(DX^2+DY^2)*GRM(i,j); % add extra vorticity confinement terms everywhere if abs(VORT(i,j))>=0. A9=dt*eps_array(i,j)*GR_VORT_Y(i,j)*VORT(i,j); W(i,j,2)=W(i,j,2)+A9; W(i,j,3)=W(i,j,3)-dt*eps_array(i,j)*GR_VORT_X(i,j)*VORT(i,j); end if abs(A9)>vc_max vc_max=abs(A9);

80 end

%%%%%%%add VC%%%%%%

U(i,j)=W(i,j,2)/W(i,j,1); V(i,j)=W(i,j,3)/W(i,j,1); T(i,j)=(W(i,j,4)-W(i,j,1)*0.5*(U(i,j)^2+V(i,j)^2))/c_p; if T(i,j)<10. T(i,j)=10.; end A(i,j)=(g*T(i,j)*R)^0.5; P(i,j)=R*W(i,j,1)*T(i,j);

end end vort_max = 0; for i=3:NX-1 for j=3:NY-1 VORT(i,j)=0.5*(DX1*(V(i+1,j)-V(i-1,j))-DY1*(U(i,j+1)-U(i,j-1))); VORT_MAG(i,j)=abs(VORT(i,j)); if VORT_MAG(i,j)>vort_max vort_max=VORT_MAG(i,j); end end for i=3:NX-1 for j=3:NY-1 GR_VORT_X(i,j)=0.5*DX1*(VORT_MAG(i+1,j)-VORT_MAG(i-1,j)); GR_VORT_Y(i,j)=0.5*DY1*(VORT_MAG(i,j+1)-VORT_MAG(i,j-1));

end end

end

Time = Time + dt Istep= Istep + 1

end

APPENDIX C

ANSYS FLUENT UDF

#include "udf.h" enum { mag_w, };

DEFINE_ADJUST(name,domain) { Thread *t; cell_t c; Node *node; int n; face_t f; real x; real y; real z; real xmax; real ymax; real zmax; real xmin; real ymin; real zmin; real xc[ND_ND]; thread_loop_c(t,domain) { begin_c_loop(c,t) { C_UDMI(c,t,0)=C_W_G(c,t)[1]-C_V_G(c,t)[2]; C_UDMI(c,t,1)=C_U_G(c,t)[2]-C_W_G(c,t)[0]; C_UDMI(c,t,2)=C_V_G(c,t)[0]-C_U_G(c,t)[1]; }

82 end_c_loop(c,t) } thread_loop_c(t,domain) { begin_c_loop(c,t) { C_UDSI(c,t,mag_w)=sqrt(C_UDMI(c,t,0)*C_UDMI(c,t,0)+C_UDMI(c,t,1)*C_UDMI( c,t,1)+C_UDMI(c,t,2)*C_UDMI(c,t,2)); } end_c_loop(c,t) } thread_loop_c(t,domain) { begin_c_loop(c,t) { C_UDMI(c,t,3)=C_UDSI_G(c,t,mag_w)[0]; C_UDMI(c,t,4)=C_UDSI_G(c,t,mag_w)[1]; C_UDMI(c,t,5)=C_UDSI_G(c,t,mag_w)[2]; } end_c_loop(c,t) } thread_loop_c(t,domain) { begin_c_loop(c,t) { C_UDMI(c,t,6)=sqrt(C_UDMI(c,t,3)*C_UDMI(c,t,3)+C_UDMI(c,t,4)*C_UDMI(c,t,4)+ C_UDMI(c,t,5)*C_UDMI(c,t,5)); } end_c_loop(c,t) } thread_loop_c(t,domain) { begin_c_loop(c,t) { c_node_loop(c,t,n) { node = C_NODE(c,t,n); x = NODE_X(node); y = NODE_Y(node); z = NODE_Z(node); if (x > -1E06 && n == 1) { xmax = x; } if (x > xmax && n != 1) {

83 xmax = x; } if (x < 1E06 && n == 1) { xmin = x; } if (x < xmin && n != 1) { xmin = x; } if (y > -1E06 && n == 1) { ymax = y; } if (y > ymax && n != 1) { ymax = y; } if (y < 1E06 && n == 1) { ymin = y; } if (y < ymin && n != 1) { ymin = y; } if (z > -1E06 && n == 1) { zmax = z; } if (z > zmax && n != 1) { zmax = z; } if (z < 1E06 && n == 1) { zmin = z; } if (z < zmin && n != 1) { zmin = z; } }

C_UDMI(c,t,7) = sqrt((xmax-xmin)*(xmax-xmin)+(ymax-ymin)*(ymax-ymin)+(zmax- zmin)*(zmax-zmin));

} end_c_loop(c,t) } thread_loop_c(t,domain) { begin_c_loop(c,t) { C_UDMI(c,t,8)= 0.0325*C_UDMI(c,t,7)*C_UDMI(c,t,7)*C_UDMI(c,t,6); C_CENTROID(xc,c,t); if (xc[0] > 2) { C_UDMI(c,t,9) =(C_UDMI(c,t,8)*C_R(c,t))*((C_UDMI(c,t,4)*C_UDMI(c,t,2)/C_UDMI(c,t,6))- (C_UDMI(c,t,5)*C_UDMI(c,t,1)/C_UDMI(c,t,6))); C_UDMI(c,t,10) =(C_UDMI(c,t,8)*C_R(c,t))*((C_UDMI(c,t,5)*C_UDMI(c,t,0)/C_UDMI(c,t,6))- (C_UDMI(c,t,3)*C_UDMI(c,t,2)/C_UDMI(c,t,6))); C_UDMI(c,t,11) =(C_UDMI(c,t,8)*C_R(c,t))*((C_UDMI(c,t,3)*C_UDMI(c,t,1)/C_UDMI(c,t,6))- (C_UDMI(c,t,4)*C_UDMI(c,t,0)/C_UDMI(c,t,6))); } C_UDMI(c,t,12)=C_UDMI(c,t,9)+C_UDMI(c,t,10)+C_UDMI(c,t,11); } end_c_loop(c,t) }

}

#define EPS 0.0225 DEFINE_SOURCE(xmom_source, c, t, dS, eqn) { real source; real xc[ND_ND]; C_CENTROID(xc,c,t); if (xc[0] > 2) { source =C_UDMI(c,t,9); dS[eqn]=0; } else source = dS[eqn] = 0.; return source; } DEFINE_SOURCE(ymom_source,c,t,dS,eqn) {

85 real source; real xc[ND_ND]; C_CENTROID(xc,c,t); if (xc[0] > 2) { source =C_UDMI(c,t,10); dS[eqn]=0; } else source = dS[eqn] = 0.; return source; } DEFINE_SOURCE(zmom_source,c,t,dS,eqn) { real source; real xc[ND_ND]; C_CENTROID(xc,c,t); if (xc[0] > 2) { source =C_UDMI(c,t,11); dS[eqn]=0; } else source = dS[eqn] = 0.; return source; }