Simulation of a Multiple-Rotor Aircraft in Hover Using the

Universidad de los Andes

Simulation of a Multiple-Rotor Aircraft in HoverUsing the Unsteady Vortex Lattice Method

A thesis submitted in partial fulfillmentof the requirements for the degree of

Master of Science

Author:Eng. Juan Diego Colmenares F.

Advisors:Omar D. Lopez, PhD.

Assistant Professor,Universidad de los Andes,

Colombia

Sergio Preidikman, PhD.Professor, Universidad

Nacional de Cordoba,Argentina

Dedicated to God, my family, and my girlfriend.

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Acknowledgments

First of all, I would like to acknowledge my advisors, Prof. Omar Lopez for his funda-mental role in supporting my work and for believing in me, and Prof. Sergio Preidikman,for sharing his knowledge and his enthusiasm in the study of aeronautics. I would alsolike to acknowledge the Pablo Neruda Program for funding and promoting this type ofcollaborative work between Latin American institutions. Thanks to the Department ofMechanical Engineering and the Faculty of Engineering at Universidad de los Andes forfunding airplane tickets. I want to thank the Preidikman family for their hospitality andthe attention received during my visit to Argentina. Many thanks to PhD candidateMarcos Verstraete and Dr. Bruno Rocca, they were both very generous in answeringquestions and giving me important information for the debugging of my code. A specialrecognition to the MOX team of the Faculty of Engineering at UniAndes, who kept theclusters up an running and solved all my technology related problems.

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Contents

1 Introduction 7

1.1 Rotor wake modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 CFD and Hybrid methods . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3 Comprehensive Rotocraft Modeling . . . . . . . . . . . . . . . . . . . . 12

2 Objectives 14

2.1 Main objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Specific objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Theoretical Framework 15

3.1 Potential Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 General solution of Laplace’s equation . . . . . . . . . . . . . . . . . . 16

3.3 Kelvin’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.4 The Kutta Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.5 Vortex sheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.6 Vortex lines and Helmholtz theorems . . . . . . . . . . . . . . . . . . . 20

3.7 Biot-Savart law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.8 Viscous vortex core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.9 Bernoulli’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

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4 Unsteady Vortex-Lattice Method, UVLM 25

4.1 Rigid body kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2 The numerical problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.3 Wake convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.4 Vortex-Lattice method on arbitrary bodies . . . . . . . . . . . . . . . . 30

4.5 Aerodynamic Forces and Moments . . . . . . . . . . . . . . . . . . . . 31

4.6 Solution Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5 GUAVLAM Code 36

5.1 Key Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.2 Improvement of the Rotor Stability in Hover . . . . . . . . . . . . . . . 37

5.3 Test Case 1: Potential Flow Around a Sphere . . . . . . . . . . . . . . 39

5.4 Test Case 2: Normal Force on a Flat Plate . . . . . . . . . . . . . . . . 40

5.5 Test Case 3: Isolated Helicopter Rotor in Hover . . . . . . . . . . . . . 41

6 Results 47

6.1 Generic two-rotor VTOL aircraft . . . . . . . . . . . . . . . . . . . . . 47

7 Conclusions 60

A Evaluation of Terms for the Calculation of Pressure Jump 67

B GUAVLAM Code Structure 70

B.1 GUAVLAM Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

B.2 GUAVLAM Derived Types . . . . . . . . . . . . . . . . . . . . . . . . . 76

C GUAVLAM Input file format 81

C.1 CaseControl file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

C.2 Geometry .geom file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

C.3 Geometry .wake file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

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List of Figures

1.1 Sketch of the tip-vortex and vortex sheet from a single-blade rotor. Ob-tained from [16] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2 Multiple-trailer wake model with a simulation of the tip vortex consoli-dation. Taken from Ref. [20]. . . . . . . . . . . . . . . . . . . . . . . . 13

3.1 Graphic representation of the Kutta condition. . . . . . . . . . . . . . . 18

3.2 Vortex tube. Taken from [21]. . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Velocity at point P due to vortex distribution in volume V. . . . . . . . 22

3.4 Schematic representation of the Biot-Savart law for a finite straight vor-tex filament. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.5 Comparison of different vortex core models. . . . . . . . . . . . . . . . 24

4.1 Vortex ring schematic representation. . . . . . . . . . . . . . . . . . . . 25

4.2 Frames of reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.3 Discretized representation of a rotor blade. . . . . . . . . . . . . . . . . 29

4.4 Diagram of the solution algorithm used by UVLM. . . . . . . . . . . . 35

5.1 Planform view of a blade and its wake. . . . . . . . . . . . . . . . . . . 38

5.2 Tangential component of the potential velocity field around sphere. . . 39

5

5.3 Normal force coefficient for a flat plate of AR=1. . . . . . . . . . . . . 40

5.4 Planform of the discretized Caradonna blade. . . . . . . . . . . . . . . 41

5.5 Sectional thrust coefficient for the Caradonna rotor. . . . . . . . . . . . 42

5.6 Thrust coefficient at different collective pitch angles. . . . . . . . . . . . 43

5.7 Wake structure of a rotor of AR = 6 after 9 revolutions. . . . . . . . . . 44

5.8 Evolution of thrust coefficient with (dashed line) and without (solid line)root vortex for collective pitch angles 5, 8 and 12. . . . . . . . . . . . 45

5.9 Evolution of the wake structure with (top) and without (bottom) rootvorticity for the case α = 12. . . . . . . . . . . . . . . . . . . . . . . . 46

6.1 Test configurations: (0,0,0); (0,0,10); (0,10,0); (10,0,0) . . . . . . . . . . 48

6.2 Vertical force coefficient from the test cases. . . . . . . . . . . . . . . . 50

6.3 Vertical thrust over the number of blades. Geometric configuration (0,0,0). 51

6.4 Moments about the Y axis. . . . . . . . . . . . . . . . . . . . . . . . . 52

6.5 Geometry with 3 blades per rotor and fully developed wake for the (0,0,0)configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6.6 Evolution of force coefficients with 2 blades per rotor. . . . . . . . . . . 54



6.9 Evolution of moment coefficients with 2 blades per rotor. . . . . . . . . 57



B.1 Wake corner and wake structure without corner. . . . . . . . . . . . . . 80

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Chapter 1

Introduction

Vertical and short take-off and landing (V/STOL) aircraft have been of interest tothe aeronautic and aerodynamics research community for many years. Their abilityto reach unconventional areas make them ideal for certain applications. However, fewconfigurations have been successful, as it is difficult to understand the aerodynamicinteractions during hover, transition (from vertical to forward flight and vice versa)and near-ground flight [31]. Rotorcraft have been most successful in this area, with thehelicopter being the predominant VTOL aircraft, but their complex aerodynamics areyet to be fully understood.

In recent years, there has been a great effort to create miniature air vehicles (MAV)and unmanned aerial vehicles (UAV) with VTOL capabilities, which can be used formilitary applications, such as reconnaissance and patrol, or civilian applications, suchas surveillance and search-and-rescue missions. Some configurations that have beenimplemented successfully include quadrocopters [18], ducted fans [30, 53, 28] and tiltrotors [39]. As more complex designs appear, it is important to have the adequate toolsto analyze the aerodynamics of such vehicles.

With the increase in computational power over the last decades, numerical meth-ods have become a fundamental part of the design process of aircraft. They allow theanalysis of different configurations, reducing the cost of experimental setups. Thesemethods vary in complexity and computational cost, depending on the developmentalstage of the design. During early stages, inexpensive methods are used to provide anidea of the aerodynamic performance of the aircraft.

For a fair estimate of forces and moments acting on lifting surfaces, methodsbased on potential flow, like the source panel and vortex-lattice method, prove to be

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computationally inexpensive and reliable, allowing the user to test different designs andconfigurations in a short amount of time. In these methods, the surface of the immersedbody is discretized into panels with unknown source/doublet strength, or in the case ofvortex-lattice, unknown circulations. Since the velocity field of the potential flow mustalways be tangent to the surface, a non-penetration boundary condition is applied atthe centroids of the panels in order to determine the source strengths or circulations.

The main difference between source/doublet panels and vortex-lattice method isthat the former is used to model thick bodies, while the latter is usually applied tomodel thin lifting surfaces, although studies show that pure vortex-lattice methodsmay also produce acceptable results for the flow around arbitrary closed bodies [2, 12].

Modeling wake structure is the most difficult part of modeling rotary-wing aero-dynamics. The wake is often represented by vortex filaments in both source panels andvortex-lattice methods [21]. Vortex methods have been heavily used for the analysis ofrotorcraft because of their ability to model the wake structure and predict loads andperformance in good agreement with experimental data [1].

The present thesis discusses the implementation of the General Unsteady Aero-dynamics Vortex Lattice Method (GUAVLAM) code, which is based on the unsteadyvortex lattice method (UVLM) [22, 32]. Results from the simulation of a generic two-rotor VTOL aircraft in hover flight are discussed. The geometry includes a fuselageand is simulated in various geometric configurations of the rotors in order to study itsinfluence on the aerodynamic behavior. The model was validated for the case of anisolated rotor in hover flight using experimental data from Caradonna & Tung [6]. Thesimulations serve to show the versatility and capabilities of the GUAVLAM code.

1.1 Rotor wake modeling

Initial experiments to characterize the wake structure of rotary wings were conductedby Gray [15, 16], who identified a high-strength tip-vortex and an inboard vortex-sheetwhere all vorticity is confined. These structures are depicted in Fig. 1.1. Trailingvortex filaments extend parallel to the local free-stream velocity, while shed vortexextend radially and are perpendicular to the trailing vortex.

For the simulation of the structure of the wake, three approaches may be taken: aprescribed wake geometry, iterative relaxation techniques and time-marching free-wakemethods.

8

Figure 1.1: Sketch of the tip-vortex and vortex sheet from a single-blade rotor. Obtainedfrom [16]

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A prescribed or rigid wake geometry has the advantage of low computationalcost. Blade loads calculated with rigid wake models that do not account for wakecontractions are generally in great disagreement with experimental results, which iswhy Landgrebe [23] developed a semi-empirical methods which uses experimental datato position the wake and thus accounts for wake contraction. The main disadvantageof this method is that it is not truly predictive because it needs previous data of wakegeometry.

Free-vortex methods include those based on iterative relaxation techniques andtime-marching schemes. The tip-vortex and inbound vortex-sheets are represented bydiscrete vortex segments, which are allowed to move with local particle velocity torender the wake force-free [32]. A velocity field is induced by the circulation of thevortex segments and it is calculated through repetitive integration of the Biot-Savartlaw [1].

Relaxation based free-vortex methods assume the periodicity of the rotor wake.The wake distorts freely from an initial solution under the self-induced velocity field,which is usually a prescribed wake solution [1]. The position vectors are a weightedaverage old and new vector positions, so that the wake position is “relaxed” until itremains unchanged after several iterations. A relaxation method was implemented byScully [41] after he failed to obtain convergent results for hover performance usingtime-marching schemes, using a semi-infinite vortex cylinder to model the far-wakecondition. Miller [26] used Weissinger-L lifting surfaces to model the rotor blades andan approach similar to Scully’s to model the wake. It was shown that the inclusionof of the root vortex had negligible effect on rotor hover performance analysis. Theuse of weighted averages improved the overall stability of the method. An eigenvalueanalysis performed by Quackenbush et al. [33] later showed the intrinsic instability ofrotor wakes in hover [4]. A more recent the work on relaxation methods was done byBagai [3] on a pseudo-implicit predictor-corrector scheme.

Time-marching free-vortex wake models capture unsteady wake aerodynamics. Inthese methods, induced velocities are calculated at each time step and the new positionof the wake is obtained through time integration. Some initial attempts to model hoverperformance failed to obtain convergent results, which is why the aforementioned relax-ation techniques were developed. In recent papers, high-order time integration [50] andpredictor-corrector schemes [5] have been implemented to predict performance duringhover and maneuvering flight. In order to reduce non-physical instabilities in the initialwake, Chung et al. [9] presented a slow starting method, in which the rotor speed isgradually increased from zero to the desired final angular velocity. This method reducesthe strong circulation of the starting vortex caused by an impulsively started rotor.

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Vortex-lattice methods based on time marching schemes have been used success-fully for simulating the load distribution and wake structure of rotary wings. In thesemethods, wake is modeled by the trailing vortices (parallel to the blade-tip vortex)and the shed vortices (perpendicular to the trailing vortices). Padakannaya [29] useda vortex-lattice method to study the rotor-vortex interaction problem, establishing thefeasibility of a simple vorte-lattice method to predict unsteady forces due to complexvortex interactions. Rottgermann [37] coupled vortex-lattice methods with a field panelmethod to compute the transonic non-linear effects at the blade tip. This method con-sists on discretizing the outer domain into a small Cartesian grid on which the transonicflow is solved using finite difference method, achieving excellent numerical results forthe load distribution near the blade tip at high Mach numbers. Roura et al. [38] im-plemented different models for the near wake and far wake regions, where the nearwake is modeled by a vortex lattice representing the full inbound vortex sheet and tip-vortex, and the far wake is modeled by a single concentrated tip-vortex, thus reducingcomputational cost.

A hybrid method combining vortex-lattice and sources was implemented for mod-eling flow around arbitrary closed bodies. In this method, the source strengths areprescribed and only the circulation of the vortex lattices are calculated. Pure vortex-lattice method may also produce acceptable results for some cases [2].

1.2 CFD and Hybrid methods

During the last decades, the increase in computational power has been exponential.This has allowed researchers to find solutions of the full Navier-Stokes equation forthe flow field surrounding rotor and complete rotocraft. Computational fluid dynamics(CFD) uses finite difference, finite elements or finite volumes to solve the equationsbased on an Eulerian formulation, as opposed to the Lagrangian formulation of vortexmethods.

Chen & McCroskey [8] were among the first to solve the Euler equation for arotor wake. The results showed good agreement with experimental data but the wakeexhibited considerable numerical diffusion, failing to retain the vorticity beyond onerotor revolution. In 1992, the TURNS code was developed [43] using thin-layer Navier-Stokes equations to capture the rotor wake. It showed acceptable results in comparisonwith experimental data, predicting flow separation seen in the tip region in experiments.

There has been great effort to improve the accuracy of these methods. Severalresearchers used different strategies to minimize the numerical dissipation of vorticity,

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such as unstructured meshes for local grid refinements [47], overset meshes [48], andvorticity confinement techniques [46]. A more extensive review on the evolution of CFDmethods for the analysis of rotorcraft can be found in Ref. [13].

Other methods use Lagrangian free-vortex wake model to represent the far wakeand solve the flow equations in a small grid region near rotor blades using CFD methods.This reduces numerical diffusion because the vortex core size is specified for the free-vortex wake model. These hybrid formulation which embed vortex methods into CFDcodes were adopted by Steinhoff & Ramachandran [45] and was later implemented inthe HELIX-I code. Other researchers have used similar methods to reduce numericaldiffusion. Sezer-Uzol & Long [42] developed an approach to preserve vortices over longerdistances in a coupled Euler/free-wake method, producing more accurate results for thetip vortex in the far wake than what is seen in most CFD codes.

Although this kind of methods have great potential for the future, they are stillvery computationally expensive for the unsteady analysis of rotorcraft and for com-prehensive rotorcraft analysis which is needed for flight simulators and exploratoryrotorcraft design.

1.3 Comprehensive Rotocraft Modeling

Several commercial codes exist for the comprehensive analysis of rotorcraft based onvortex methods. These codes allow the coupling of mechanical aspects of rotor blademodeling, such as trimming, flapping, aeroelastic flutter, etc., with aerodynamic models.

NASA has used an aeromechanics analysis code called CAMRAD II to performcalculations on tilt-rotors and compare with experimental data from the Tilt RotorAeroacoustic Model (TRAM DNW) [19]. An study on the influence of different wakemodels using CAMRAD II determined that the multiple-trailer wake model with a sim-ulation of the tip vortex formation process (consolidation), shown in Fig. 1.2, producesgood correlation with TRAM measurements for both performance and airloads [20].Yeo and Johnson [52] used this model to conduct a parametric analysis on the ro-tor/wing interference in a tilt-rotor and a quad tilt-rotor in forward flight, in whichthey found that a reduction in rotor tip speed increased the aircraft lift-to-drag ratio.

The Comprehensive Hierarchical Aeromechanics Rotorcraft Model (CHARM) codehas demonstrated the power of vortex methods in the study of helicopters, tilt-rotorsand other experimental designs, such as the VectoRotor hybrid VTOL vehicle [35],giving preliminary results for engine power and flight control requirements for suchaircraft. Recent development of the CHARM code allow its usage for real-time flightsimulations [34].

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Figure 1.2: Multiple-trailer wake model with a simulation of the tip vortex consolidation.Taken from Ref. [20].

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Chapter 2

Objectives

Given the context that has been presented over the previous chapters, the problemsolved in this thesis is:

Implementing a numerical method with relatively low computational costfor the simulation of a conceptual two-rotor VTOL aircraft in hover flight.

In order to solve that problem, the following objectives have been achieved.

2.1 Main objective

Simulation of a conceptual two-rotor aircraft in hover flight using the unsteady vortex-lattice method.

2.2 Specific objectives

1. Generate a mesh of the discretized geometry: blades and fuselage.

2. Implement the UVLM using Fortran as a programming language, taking advan-tage of external libraries for high performance.

3. Plot the bound and free vortex sheets as well as pressure contours on the surfaceof the aircraft using the program Tec360.

4. Simulate different geometric configurations for establishing tendencies of the effectof the geometry on aerodynamic performance.

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Chapter 3

Theoretical Framework

The purpose of this chapter is to introduce some concepts and equations to the readeras an overview of the theory behind potential flow methods.

3.1 Potential Flow

The problem of simulating flow around a body may be approached in many ways. Avery simplified model assumes that the flow is inviscid and incompressible. In orderto comply with the law of conservation of mass, the velocity field of an incompressibleflow must satisfy the continuity equation (Eq. 3.2).

∂V

∂t+ V · ∇V = f − ∇p

ρ(3.1)

∇ ·V = 0 (3.2)

The vorticity (ζ) is defined as the curl of the velocity field.

ζ = ∇×V (3.3)

By taking the curl of the Euler equation, a partial differential equation for therate of change of vorticity is obtained (see Eq. 3.4). After some algebraic manipulation,

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it can be demonstrated that for an inviscid flow a particle that is initially non-rotating,stays that way throughout its path.

Dζ

Dt=∂ζ

∂t+ V · ∇ζ = ζ · ∇V = 0 (3.4)

This has an important implication: if the flow is uniform and irrotational (ζ = 0everywhere), the solenoidal velocity field can be expressed as the gradient of a potentialfield (Eq. 3.5).

V = ∇Φ (3.5)

Therefor, the problem is reduced to finding the solution of the Laplace equation,shown in Eq. 3.6.

∇ ·V = ∇ · ∇Φ = ∇2Φ = 0 (3.6)

Approximate solutions of the potential and velocity field are obtained using nu-merical methods based on a discrete distribution of singularities, which will be discussednext.

3.2 General solution of Laplace’s equation

An incompressible, inviscid and irrotational flow field is described by Laplace’s equa-tion (Eq. 3.6). For a submerged body, the velocity component normal to the body’ssurface must be zero (Eq. 3.7). This is equivalent to imposing a homogeneous Neumanncondition on the potential field across the surface boundary.

(V −VLS) · n = ∇Φ · n = 0 (3.7)

where V is the absolute velocity of the fluid particle, VLS is the velocity of the surface,and n is a unit vector normal to the surface.

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The second condition is that the disturbance due to the body motion should decayfar from the body (Eq. 3.8), therefor, the velocity field away from the body should beequal to the free-stream velocity (V∞).

lim|rB|→∞

(∇Φ−V∞) = 0 (3.8)

where rB is the position vector of a fluid particle with respect to a body-fixed frame ofreference.

The methodology for the solution of Laplace’s equation is found in Ref. [21]. Thebasic solutions are singularities in the velocity field, such as sources, sinks and doublets.The general solution is a superposition of the basic solutions. The problem is thenreduced to finding the distribution of such elements in order to satisfy the boundaryconditions.

Other basic solutions to Laplace’s equation are possible, including point vortexand vortex quantities. A point vortex is a two-dimensional element that producestangential velocities only. The circulation (or vortex strength) is defined as the lineintegral of the velocity on a curve around the origin:

Γ =

∮C

V · dl (3.9)

In two dimensions, the velocity and potential field described by the point vortex areshown in Eq. 3.10 and Eq. 3.11.

V =Γ

2πr(3.10)

Φ = − Γ

2πθ (3.11)

These singularities are used for solving two-dimensional potential flow problems.For three-dimensional problems, source/doublet panels or vortex filaments may be usedfor the solution of potential flow.

3.3 Kelvin’s Theorem

Considering the circulation around a fluid curve, the rate of change of the circulationmay be expressed as:

DΓ

Dt=

D

Dt

∮c

V · dl =

∮c

DV

Dt· dl +

∮c

V · DDt

dl (3.12)

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The term∮

V · DDtdl in Eq. 3.12 disappears, and the equation can be written in

terms of the fluid acceleration since DV/Dt = a, which can be substituted using theEuler equation (Eq. 3.1).

DΓ

Dt=

∮c

a · dl =

∮c

(−∇p

ρ+ f

)· dl (3.13)

We assume that the body forces (f) that act on the fluid are conservative, sothat the work done by f around a closed path is zero. The term

∮cDVDt· dl is a perfect

differential around a closed path, which is also zero. Therefore, it can be stated thatthe circulation around the body and the wake is constant through time (Eq. 3.14). Thisis called Kelvin’s theorem, after the British scientist who published it.

DΓ

Dt= 0 (3.14)

3.4 The Kutta Condition

The solution of the Laplace equation is given by a distribution of singularities thatwill ensure the boundary conditions. However, another condition is required for theuniqueness of the solution in the presence of sharp edges. This condition is based onphysical principles of the flow around sharp edges [27]. The Kutta Condition states:

The flow leaves the sharp trailing edge of an airfoil smoothly and the velocitythere is finite.

Figure 3.1: Graphic representation of the Kutta condition.

The direction in which the velocity leaves the airfoil is given by the bisector linebetween both sides of the trailing edge. To achieve this, the pressure difference acrossthe trailing edge is forced to be zero:

∆pT.E = 0 (3.15)

For unsteady flows, the Kutta condition requires that all vorticity generated atthe trailing edge must be shed to the wake and allowed to move freely at local fluidparticle speed [32].

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3.5 Vortex sheets

Vortex sheets are ideal surfaces on which vorticity is infinite. Across vortex sheets,the tangential component of velocity is discontinuous. An integral formulation of thekinematic relationship between the velocity field and the vorticity can be demonstrated[51], given by the differential equation, Eq. 3.4. After several simplifications, thevelocity field at a point, given by position vector r, associated with a distribution ofvorticity bound by the volume V can be written as:

V(r, t) =1

4π

∫V

ζ(r0, t)× (r− r0)

|r− r0|2dV(r0) (3.16)

where r0 is the position vector of a material point on the vortex sheet.

If we define vortex sheets as a region where vorticity is confined to a thin, surface-like region of thickness ε, then the vortex sheet strength is given by:

limε→0

|ζ|→∞

εζ = γ (3.17)

The vortex sheet strength is tangent to the surface at every point and, because ofthe selenoidal poperty of vorticity, then:

∇V S · γ = 0 (3.18)

where ∇V S is a two-dimensional operator on the surface of the vortex sheet. Substi-tuting Eq. 3.17 in 3.16, it can be stated that the velocity field induced by the vortexsheet is:

V(r, t) =1

4π

∫SV S

γ(r0, t)× (r− r0)

|r− r0|2dS(r0) (3.19)

As we can see from Equation 3.19, solving this integral can be very complicatedas γ may vary along the surface. Also, these are vector quantities, so the directions ofthese vectors have to be determined.

Using Stoke’s theorem, one can relate the vortex strength of a vortex sheet to thecirculation on the boundaries [17]. This means, that a vortex sheet can be modeled byanother singularity, which is called vortex ring, that is the curve around the surface ofthe sheet.

V =

∫SHEET

γ × r

|r|3dS = Γ

∮RING

dl× r

|r|3(3.20)

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Figure 3.2: Vortex tube. Taken from [21].

where dl is the differential portion of the curve on the boundary of S.

This equality simplifies the problem of working with vortex sheets, since the ringsare composed of straight segments, called vortex filaments whose directions can bespecified, making it a scalar problem. To understand how there rings affect the potentialflow, the concept of vortex filaments is introduced.

3.6 Vortex lines and Helmholtz theorems

Vortex lines are field lines that are parallel to the vorticity vector: ζ× dl = 0, where dlis a segment along the vortex line. Vortex lines passing through a closed curve in spaceform a vortex tube (see Fig. 3.2), and a vortex tube of infinitesimal cross-sectional areais called a vortex filament.

The divergence of the curl of a vector field is always zero, therefor, the divergenceof the vorticity is zero: ∇ · ζ = ∇ · (∇ ×V) = 0. Consider a region Ω enclosed by asurface S. From application of the divergence theorem, Eq. 3.21 is obtained.

∫S

ζ · ndS =

∫Ω

∇ · ζdV = 0 (3.21)

Apply Eq. 3.21 on the surface of the vortex tube shown in Fig. 3.2. Since the vor-ticity is parallel on the wall Sw, its contribution vanishes and we are left with Eq. 3.22.

∫S

ζ · ndS =

∫S1

ζ · ndS +

∫S2

ζ · ndS = 0 (3.22)

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At each instant in time, the quantity of ζ is the same for any cross-sectionalsurface of the tube. Let C be any closed curve that surrounds the tube and lies on itswalls, the relationship between the circulation Γ and the vorticity ζ can be established.

Γ =

∮C

V · dl =

∫S

ζ · ndS = constant (3.23)

Eq. 3.23 has an important implication: a vortex filament cannot end in the fluidsince a cross sectional area of zero would lead to an infinite value for the vorticity.This is useful for modeling purposes, since one can define a vortex filament with fixedcirculation, zero cross-sectional area and infinite vorticity. Based on these results, theHelmholtz theorems were developed for inviscid flows, which can be summarized asfollows [21]:

1. The strength (circulation) of a vortex filament is constant along its length.

2. A vortex filament cannot start or end in a fluid: it must form a closed path orextend to infinity.

3. The fluid that forms a vortex tubes will continue to form a vortex tube and itsstrength remains constants throughout its motion. This is a consequence of theinviscid flow assumption.

3.7 Biot-Savart law

The velocity field induced by a three-dimensional vortex filament can be calculatedusing the Biot-Savart law. Its derivation starts with the assumption that the flow isincompressible [21], ∇ ·V = 0, thus, the velocity field V can be expressed as the curlof an arbitrary vector field A.

V = ∇×A (3.24)

If we choose a vector field A such that ∇ ·A = 0, the vorticity can be expressedin terms of A as shown in Eq. 3.25.

21

Figure 3.3: Velocity at point P due to vortex distribution in volume V.

ζ = ∇×V = ∇× (∇×A)

= ∇(∇ ·A)−∇2A

= −∇2A (3.25)

The vorticity equation is reduced to Poisson’s equation for the vector field A(see Eq. 3.25). The field induced due to vorticity in a volume V (see Fig. 3.3)can beevaluated using Green’s theorem.

A =1

4π

∫V

ζ

|ro − r1|dV (3.26)

The velocity field can then be substituted into Eq. 3.26 to yield Eq. 3.27.

V =1

4π

∫V

∇× ζ

|ro − r1|dV (3.27)

For a vortex filament of finite circulation, the integrand of Eq. 3.27 can be substi-tuted in terms of the vortex strength Γ. Knowing that Γ = ζdS and dV = dSdl, thenthe Biot-Savart law is expressed in terms of the circulation as follows:

V =Γ

4π

∫L

1

|r0 − r1|3dl× (r0 − r1) (3.28)

22

Figure 3.4: Schematic representation of the Biot-Savart law for a finite straight vortexfilament.

This is a general representation of the Biot-Savart law for vortex filaments. A moreconvenient representation is available for a straight vortex filament of finite length, sinceit can be solved numerically in an efficient way. A schematic representation of Eq. 3.29is shown in Fig. 3.4.

V =Γ

4π

r1 × r2|r1 × r2|2

r0 ·(

r1|r1|− r2|r2|

)(3.29)

3.8 Viscous vortex core

There are several possibilities to overcome the singularity of Eq.(3.29). A first approachis to set a vortex core radius µ, inside of which the induced velocity is set to zero,commonly known as cutoff vortex core. Another model assumes rigid rotation insidethe vortex core radius, called the Rankine model, which tends to over predict theinduced velocities [11].

A swirl velocity profile was proposed by Scully [41] for a more realistic behavior.An improvement of this model, formulated by Vatistas et al [49], proposes the velocityprofile described by Eq. (3.30), where the value n = 1 results in the swirl velocity profileproposed by Scully.

VBS =Γ

4π

r0 × r1(|r1 × r2|2n + µ2n

)1/nr0 · (r1 − r2) (3.30)

According to Leishman et al. [24], Vatistas vortex model with n = 2 best fits theirexperimental data for rotary wings, therefor, it was the chosen viscous core model forthis thesis. A comparison of different vortex models is shown in Figure 3.5.

23

Figure 3.5: Comparison of different vortex core models.

3.9 Bernoulli’s equation

Once the velocity field of the fluid around the body has been found, the pressuredistribution on the surface of the body can be calculated using Bernoulli’s unsteadyequation (Eq. 3.31). The terms on the right hand side of the equation are referencevalues at infinity. In this case: E∞ = 0 and ∂Φ∞

∂t= 0.

[E +

∂Φ

∂t+V 2

2+p

ρ

]=

[V 2∞2

+p∞ρ

]ref

= H(t) (3.31)

After rearranging terms, Eq. 3.31 is used to estimate the pressure jump at anygiven point:

p− p∞ρ

=V 2∞ − V 2

2− E − ∂Φ

∂t(3.32)

24

Chapter 4

Unsteady Vortex-Lattice Method,UVLM

UVLM is a computational, low-order method for solving the unsteady potential flowaround a lifting surface. The body is descretized into quadrilateral vortex sheets orpanels1. Each panel has a control point where the non-penetration boundary conditionmust be satisfied. The control points are located at geometric centroid of the panels, asshown on Figure 4.1. The edges of the vortex rings are straight finite vortex filaments,which induce a velocity field. The normal vectors n are unitary vectors in the directionof the cross product of the diagonals.

Figure 4.1: Vortex ring schematic representation.

1Triangular elements might be needed for complicated geometries, however, quadrilateral panelstend to give better numerical results when compared with experimental data [22]

25

Figure 4.2: Frames of reference.

4.1 Rigid body kinematics

In UVLM, two coordinate reference frames are used: an inertial or ground-fixed frameand a moving or body-fixed frame. The current implementation poses the problem interms of the inertial frame of reference N , with base vectors (X1,X2,X3). A movingframe A is defined by an origin whose position vector with respect to the inertial frameis given by RA and base vectors (a1, a2, a3).

In Fig. 4.2, a schematic representation of the different frames of reference is shown.A point P that lies on the surface can be expressed in terms of the inertial frame ofreference:

R = RA + r (4.1)

Through vectorial differentiation, the velocity of the point on the inertial frameof reference is:

V = VA + v + ω × r (4.2)

where VA is the velocity at the origin A, Ω is the angular velocity of the body-fixedframe, and v is the velocity of the point relative to A.

26

Euler Angles

In order to completely describe the instantaneous orientation of a rigid body, attitudeparameters have to be established. There is an infinite number of attitude coordinatesto choose from, each with particular strengths and weaknesses. The most commonlyused sets of attitude parameters are the Euler angles (see Ref. [40]). They describethe attitude of a reference frame A relative to the frame N through sequential rotationabout the displaced body-fixed axes (a1, a2, a3) in terms of parametric angles (θ1, θ2, θ3).

The successive rotations are labeled based on the order of the axes about whichthe rotation takes place, e.g., successive rotations about a3, a2 and a1 axis is labeled(3-2-1). The order of the rotation affects the final orientation: a (3-2-1) rotation doesnot yield the same result as (1-2-3) rotation.

For aircraft applications, a yaw-pitch-roll orientation sequence is often used, la-beled (3-2-1). However, because of the geometries that were studied in the presentwork, a (3-1-2) rotation sequence was more convenient.

The orientation matrix [C(θ3, θ1, θ2)] is computed as successive multiplications ofsingle-axis rotation matrices [Mi(θi)], defined as:

[M1(θ1)] =

1 0 00 cos θ1 sin θ1

0 − sin θ1 cos θ1

(4.3)

[M2(θ2)] =

cos θ2 0 − sin θ2

0 1 0sin θ2 0 cos θ2

(4.4)

[M3(θ3)] =

cos θ3 sin θ3 0− sin θ3 cos θ3 0

0 0 1

(4.5)

The order of the matrix multiplication for the (3-1-2) sequence is:

[C(θ3, θ1, θ2)] = [M2(θ2)] [M1(θ1)] [M3(θ3)] (4.6)

To convert vector components from the body-fixed frame A to the inertial frameof reference N :

[R]N = [C(θ3, θ1, θ2)] [R]B (4.7)

27

Euler angle derivatives

The rate of change of the Euler angles is expressed in terms of the angular velocityvector ω, which defines the instantaneous rotational velocity of the A frame relative toN :

ω = ω1a1 + ω2a2 + ω3a3 (4.8)

Euler kinematic differential equations are derived to avoid having to integrate theorientation matrix directly given an ω time history. These equations vary accordingto the rotation sequence. For the case of (3-1-2) rotation, the following kinematicdifferential equation is used:

θ3

θ1

θ2

=1

cθ2

−sθ3 0 cθ3

cθ2cθ3 0 cθ2sθ3

sθ2sθ3 cθ2 −sθ2cθ3

ω1

ω2

ω3

(4.9)

where c and s are abbreviations of cos and sin respectively.

4.2 The numerical problem

The UVLM consists on finding the Γi circulations for each vortex filament of the dis-cretized geometry such that the velocity field V is always tangential to the surface, i.e.,the non-penetration boundary condition is imposed at the control points. However,computation time can be greatly reduced if it is assumed that circulation is constantfor vortex ring, so that each vortex ring i will have a loop circulation Gi (see Fig. 4.3)of constant strength. This also ensures the spatial conservation of circulation.

Vortex segments that are shared by two vortex rings have a circulation Γij thatis the difference between the circulations of the two adjacent rings.

Γij = Gi −Gj (4.10)

Since each element produces a disturbance velocity on the velocity field, the cir-culation of each element induces a velocity at the control points of the other elements.

28

The sum of the free stream velocity and the induced velocities must satisfy the non-penetration condition at each point.

NV∑j=1

(Vi,j + V∞ −VLS,i) · ni = 0 (4.11)

where n is the vector normal to the surface and VLS,i is the velocity of the lifting surfaceat the control point i. The subscript i, j means velocity induced by panel j on panel iand NV is the total number of vortex sheets. This summation can be separated intovelocities induced by the bound vortex sheets (LHS of Eq. (4.12)), velocities inducedby the wake (VW in RHS of Eq. (4.12)) and the free stream velocity V∞. The problemis expressed as a linear system of algebraic equations with a dense matrix, being thevortex ring circulations Gj unknown.

NBV∑j=1

ai,jGj = (VLS,i −VW,i −V∞) · ni (4.12)

where ai,j is the normal component of the velocity induced on panel i by panel j withunitary circulation, VW,i is the velocity induced by the wake on panel i, and NBV isthe number of bound vortex sheets.

Figure 4.3: Discretized representation of a rotor blade.

29

4.3 Wake convection

The geometry of the wake is formed by shedding the vortex segments formed at thetrailing edge with local fluid particle velocity, which will satisfy the unsteady Kuttacondition. The shed vortex is joined to the trailing edge by a pair of trailing vortex,forming a new wake vortex ring along with the new vortex filament formed at thetrailing edge. In order to maintain the spatial conservation of circulation and Kelvin’stheorem, loop circulation from the bound vortex ring at the trailing edge is passed onto the newly formed wake vortex ring. The process is depicted in Fig. 4.3.

Although some papers use predictor-corrector [5, 1] and/or high order schemes [50]to calculate the positions of the wake at each time step, the present method uses theexplicit Euler method for the convection of the fluid particles of the wake, expressedin Eq. 4.13, where t subscript represents the previous time step and R is the positionvector of a fluid particle on the wake. This low order scheme produces a more unstablewake structure, but provides good numerical results for the loads on the lifting surfaceswith low computational cost [22]. Also, some authors use curved vortex segments onthe wake [50]. In the current implementation, all vortex segments are straight andfinite.

Rt+1 = Rt + Vt∆t (4.13)

4.4 Vortex-Lattice method on arbitrary bodies

Although hybrid methods involving source panels and vortex-lattice show better agree-ments in the calculation of aerodynamic loads on bluff bodies [2], pure vortex-latticemethods may still be used when the estimation of loads on the bluff body is not criticaland non-penetration boundary condition needs to be imposed on the surface of thebody.

When using a pure vortex-lattice method, special care must be taken when dealingwith closed geometries. This is because the solution is unique for the circulations ofthe vortex segments, but not for the circulations of the vortex rings [44], since thereis an infinite set of solutions of Gi that could satisfy the unique solution of the vortexsegments. Therefor, the linear system of equations is redundant and the matrix formedby the influence coefficients ai.j becomes singular.

30

One way to eliminate the singularity of the problem is to impose circulation onone of the vortex rings, thus eliminating one of the equations from the linear system.Another way of attacking this problem is to use an iterative method for solving thelinear system of equations, allowing circulations to converge to one of the possiblesolutions nearest to the initial guess. In the present work, the first approach was taken.

4.5 Aerodynamic Forces and Moments

As mentioned on an earlier section, pressure distribution on the surface of a body canbe obtained from the velocity field using the unsteady Bernoulli equation:

[E +

∂Φ

∂t+V 2

2+p

ρ

]=

[V 2∞2

+p∞ρ

]ref

= H(t)

Since the current aerodynamic model does not include potential forces, the relativepressure at a fluid particle is given by:

p− p∞ρ

=V 2∞ − V 2

2− ∂Φ

∂t

Dividing each side by 12V 2∞ yields Eq. 4.14. This is equation is non-dimensional

and defines quantity known as the pressure coefficient Cp (Eq. 4.15):

p− p∞12ρV 2∞

= 1−(V

V∞

)2

− 2

V 2∞

∂Φ

∂t(4.14)

Cp =p− p∞12ρV 2∞

(4.15)

It is convenient to introduce non-dimensional variables in terms of a characteristicvelocity VC :

31

V ∗ = V/VC

∂Φ∗

∂t=

1

V 2C

∂Φ

∂t(4.16)

The pressure coefficient can be expressed in terms of these dimensionless variables:

∆Cpi = 1− V ∗2 − 2∂Φ∗

∂t(4.17)

From here on, the asterisks (*) will be omitted and it shall be assumed that allvariables are non-dimensional.

For thin lifting surfaces, we are interested in finding the pressure jump (∆Cp)across the surface. In UVLM, the pressure jump is computed at the control point ofeach discrete bound vortex sheet.

∆Cp = (Cp)L − (Cp)U (4.18)

In Eq. 4.18, the subscript U denotes a point right above the vortex sheet, andthe subscript L denotes a point right bellow the vortex sheet. This equation will berewritten in terms of the aforementioned dimensionless variables.

∆Cp = V 2U − V 2

L + 2

(∂Φ

∂t

∣∣∣∣U

− ∂Φ

∂t

∣∣∣∣L

)= VU ·VU −VL ·VL + 2

(∂Φ

∂t

∣∣∣∣U

− ∂Φ

∂t

∣∣∣∣L

)(4.19)

Eq. 4.19 is equivalent to Eq. 4.20 which is more convenient for implementationpurposes.

∆Cpi = −2 [(Vm −VCP) ·∆V]i − 2∂Gi

∂t(4.20)

32

where Vm is the non-dimensional mean velocity at the control point i, ∆V is the tan-gential velocity jump at c.p. i and Gi is the loop circulation of the respective vortexring. The pressure distribution is numerically integrated along the surface to yield theacting normal force an moment coefficients (Eq. 4.21).

Cf =NP∑i=1

∆CpiA∗i ni

Cm =NP∑i=1

∆Cpir∗iA∗i ni (4.21)

where r∗ andA∗i are non-dimensional with respect to the square of a characteristic lengthRC : A∗i = Ai/R

2C , r∗ = r/RC . This definition is convenient for the current analysis,

as the rotor radius R may be set as the characteristic length for the calculation of thethrust coefficient (Eq. 4.22) and torque coefficient (Eq. 4.23).

CT =T

ρπω2R4(4.22)

CQ =Q

ρπω2R5(4.23)

4.6 Solution Algorithm

A general implementation of the UVLM requires the following steps:

1. Load discretized geometry: the geometry consists on the coordinates of the nodes,coordinates of the control points, normal vectors, connectivity (location) matrixand the neighbor matrix of the panels. In the current implementation this is readfrom an ASCII text file with a native format.

2. Set all circulations to zero as initial condition.

3. Start time loop: Since this is a method for unsteady aerodynamics, circulationsare solved and the loads are calculated at each time step. The following steps areexecuted inside the time loop:

33

(a) Convection of the wake: Calculate the induced velocities on the wake usingthe Biot-Savart law. This involves two embedded loops: a first loop iteratesover the position vectors of each node at the wake, in which induced velocityis being computed; a second loop iterates over each vortex segment, boundand free. The fluid particles are moved at local velocity, as the circulationsof the vortex rings at the trailing edge are passed on to the newly formedvortex rings on the wake. At t = 0, there are no induced velocities, so therewill be no convection of the wake.

(b) Update position of the surfaces: The nodes are moved according to thekinematics of the problem. Once the nodes are positioned, control pointpositions and velocities are updated. The nodes at the trailing edge areconnected to the wake and a new wake filament forms at the trailing edge.

(c) Generate the matrix ai,j for the linear system the linear system in Eq. (4.12):When working with rigid bodies, the influence of a body upon itself remainsconstant, therefor, there is no need to recalculate influence coefficients, thusreducing computational cost.

(d) Calculate the right hand side of Eq. (4.12). At t = 0, there will be novelocities induced by the wake.

(e) Solve the dense linear system.

(f) Calculate the mean velocity Vm at each control point: Being the mean veloc-ity the sum of free-stream velocity V∞ and the velocities induced by boundand free vortex sheets.

(g) Calculate pressure coefficients at each panel (Eq.(4.20)).

4. Output data: the current code uses Tecplot’s TecIO library to generate binaryfiles for Tec360.

For more details on the computational model, the reader should check Ref. [22, 32].

34

Figure 4.4: Diagram of the solution algorithm used by UVLM.

35

Chapter 5

GUAVLAM Code

The General Unsteady Aerodynamics Vortex Lattice Method (GUAVLAM) Code hasbeen developed as a flexible UVLM code for general aerodynamics based on the methoddeveloped by Mook et al. [22]. This chapter will provide an overview of its mostimportant features, as well as the algorithm followed by the UVLM.

5.1 Key Features

GUAVLAM was written in Fortran95 and uses external libraries to enhance perfor-mance, such as BLAS, LAPACK and OpenMP directives.

Its key features are:

• Allows modeling of potential flow around thin lifting surfaces and arbitrary closedgeometries: Lifting surfaces are associated with a wake structure, while bluffbodies are not. Also, closed geometries are marked with a flag so that the programcan take the right path in order to avoid generating a singular influence matrix.

• All magnitudes are normalized: Magnitudes from the input file must be normal-ized with respect to the characteristic length, time step and velocity. The loadswill be dimensionless with respect to these reference values.

• The dense linear system of equations Ax = B is solved using the direct solverDGESV: This solver is driver routine from the LAPACK library that performsL-U factorization of matrix A and solve for x.

36

• Angular and linear velocity are independent for each surface: This allows thesimulation of multiple aircraft and multiple bodies.

• Shared memory parallelism using OpenMP directives: Parallel loops are executedfor the calculation of the velocity induced by each vortex filament.

• Binary Tecplot files as output: GUAVLAM implements the TecIO library togenerate binary files that contain the geometry, wake structure and pressure co-efficients at each time step. This greatly reduces size of the output files whencompared with ASCII format.

The current version of the code presents the following restrictions:

• Viscous effects not accounted for: This is a shortcoming of the UVLM. Dynamicstall and/or boundary layer models have not been implemented in this version.

• Static loads on bluff bodies: Pressure coefficient on the surface of bluff bodies doesnot account for the term ∂Φ/∂t. This means loads are calculated “statically” ateach time step.

• Constant velocities: Linear velocities (free-stream and surface) are constant through-out the simulation. Angular velocities start near zero and increase gradually untilthey reach the desired velocity (slow starting method). After that, they remainconstant.

An overview of the code structure and the input file format is available in Ap-pendix A.

5.2 Improvement of the Rotor Stability in Hover

The instability in hover performance of rotors is a well documented issue [4]. Twostrategies were taken in order to reduce this instability and achieve better convergenceof thrust and power values.

37

Figure 5.1: Planform view of a blade and its wake.

Treatment of the root vortex

In Section 1.1 it was stated that for hover rotor performance analysis, the effect ofthe root vortex in the wake was negligible. However, this cannot be generalized, sinceexperiments show that highly twisted blades may generate a strong root vortex [36].GUAVLAM allows for the inclusion or exclusion of the root vortex. By default, theroot vortex is included.

For the purpose of convention, the position of the root and tip vortices dependson the numbering of the wake vortex rings: the root vortex is located at the side of the1st vortex sheet of the wake, while the side of the last vortex sheet is the tip-vortexside. This is depicted in Fig. 5.1, where ns is the number of spanwise elements.

Slow starting method

A slow starting method was used for the rotation of the blades. It was shown byChung et al. [10] that for free-wake calculations on rotor blades, the impulsive startmethod causes non-physical instability of the initial wake. Also, the strong root-vortex

38

circulation causes vortices near the root to move upward because the velocity influencefrom the root is larger than the downward velocity. The slow starting method wasproposed to overcome the non-physical instabilities and achieve better convergence ofthe results.

This method consists on starting rotation at an angular velocity near zero, which isgradually incremented until the final angular velocity is achieved. This was implementedin the GUAVLAM code by specifying the number of time steps it takes for the angularspeed to reach the final value ωf . The magnitude of the angular speed is increasedlinearly according to the following formula:

ωt =min(NT, t+ 1)

NTωf (5.1)

where t is the current time step and NT is the number of time steps of the slow start.The term t+ 1 means the angular velocity cannot be zero unless ωf is zero.

5.3 Test Case 1: Potential Flow Around a Sphere

Figure 5.2: Tangential component of the potential velocity field around sphere.

39

This case shows the correctness of the code in modeling potential flow aroundbluff bodies. The geometry consists on a submerged sphere as shown in Fig. 5.2. Thefree-stream velocity has a magnitude of V∞ = 1 in the X direction.

Numerical result shows good correspondence with the analytic solution for thepotential velocity field around a sphere (see Fig. 5.2). Tangential velocity is under-predicted, which is typical in velocity-based vortex-lattice methods [2].

5.4 Test Case 2: Normal Force on a Flat Plate

Numerical results obtained by the code were compared with results from different au-thors [7] for the case of a flat plate with aspect ratio (AR) of 1 (square plate). Fig. 5.3shows the “steady state” normal force coefficient for this case at different angles ofattack. Numerical results from the current implementation are in good agreement withnumerical results from the cited authors.

Figure 5.3: Normal force coefficient for a flat plate of AR=1.

40

Figure 5.4: Planform of the discretized Caradonna blade.

5.5 Test Case 3: Isolated Helicopter Rotor in Hover

Experimental data was taken from Caradonna & Tung [6] in order to validate thecurrent model for rotary wings. The experimental setup consists of a rotor with twountapered and untwisted rectangular blades of aspect ratio AR = 6. The blades wereconfigured at a collective pitch of 5, 8 and 12, with a pre-cone angle of 0.5.

The simulations used a viscous core size of 0.5% rotor radius. The non-dimensionaltime step was chosen such that at each step the rotor rotates 10. The results shownin this section were obtained with a blade discretization of 15 panels in the spanwisedirection using a cosine spacing, and 6 panels in the chordwise direction using a uniformspacing (Fig. 5.4).

The implemented method shows good results when compared with experimentaldata, especially for the test case with blades rotating at 1250 RPM, which is commonlyused for the validation of vortex methods [50] and corresponds to a tip Mach numberequal to 0.439. Fig. 5.5 shows the spanwise distribution of sectional thrust coefficient.It is clear that the numerical prediction of the implemented model is in good agreementwith the experimental data except near the tip of the blade, where thrust is generallyoverpredicted.

Results for the thrust coefficient of the rotor at different collective pith angles areshown in Fig. 5.6. Good correspondence is observed between numerical and experi-mental results, however, an error as high as 11% was found for the case of 5 collectivepitch angle. In order to find an averaged thrust coefficient, the simulation was set to 9revolutions and the values from the last two revolutions were averaged. Fig. 5.7 showsthe structure of the wake after 9 revolutions of the rotor. It is clear that the far wakeshows the unstable behavior from its early stages, but the near wake shows the expectedhelicoidal shape.

Influence of the root vortex

It was found that the inclusion of the root vortex in the wake had negligible effect onthe “steady state” result of thrust coefficient, converging to an average value similar to

41

Figure 5.5: Sectional thrust coefficient for the Caradonna rotor.

42

Figure 5.6: Thrust coefficient at different collective pitch angles.

43

Figure 5.7: Wake structure of a rotor of AR = 6 after 9 revolutions.

that achieved with the model without root vorticity. Furthermore, simulations withoutroot vortex showed a more stable behavior of the thrust coefficient, as seen in Fig.5.8for different collective pitch angles.

The exclusion of the root vortex also shows a smoother evolution of the wake com-pared to that of the model with vorticity at the root, reducing the initial upwash causedby the strong root-vortex circulation [10]. A comparison of the two wake structures isshown in Fig. 5.9.

44

Figure 5.8: Evolution of thrust coefficient with (dashed line) and without (solid line) rootvortex for collective pitch angles 5, 8 and 12.

45

Figure 5.9: Evolution of the wake structure with (top) and without (bottom) root vorticityfor the case α = 12.

46

Chapter 6

Results

Numerical results are shown for a generic two-rotor aircraft. A parametric analysis isperformed in order to observe the influence of the number of blades per rotor and theorientation of the rotors on the acting forces and moments.

6.1 Generic two-rotor VTOL aircraft

The tested aircraft geometry consists on two rotors with rectangular, untwisted bladesof AR = 10. Simulations were run with 2, 3 and 4 blades per rotor for 8 revolutions.The collective pitch angle is 12. The origin of the inertial reference frame is set at apoint above fuselage, where moments are measured. For the non-dimensional analysis,the rotor radius R and the wingtip velocity ωR were chosen as characteristic length andvelocity respectively, which in terms of the simulation means that the non-dimensionalrotor radius and angular velocity are R∗ = 1.0 and ω∗ = 1.0 respectively. Forceand moment coefficients are non-dimensionalized with respect to the wingtip velocity,Vw = ωR. The time step was chosen such that at each step the rotor rotates 10. Theloads on the fuselage are not taken into consideration in the computation of coefficients.

The positions of the rotors are moved using Euler angle rotation in a (3,1,2) se-quence, that is, the first angle θ1 rotates about the Z axis, the second angle θ2 rotatesabout the X axis and the third angle θ3 rotates about the Y axis (see Fig. 6.1). Theinitial geometric configuration is at angles (θ1, θ2, θ3) = (0, 0, 0). Other three configu-rations were tested, each rotating 10 about a different axis, as shown in Fig. 6.1.

47

All simulations were run on the high performance computing cluster, Master-HPC-MOX, at Universidad de los Andes. The cluster uses the CentOS based RocksCluster distribution as operating system. In terms of hardware, Master-HPC-MOXconsists of one head node and six compute nodes, each equipped with eight 2.6GHzAMD Opteron

TMProcessor 6282 SE and 64 CPUs. The code was compiled using

open95 from the Open64 Compiler Suite. Each simulation used 16 OpenMP threads.

Figure 6.1: Test configurations: (0,0,0); (0,0,10); (0,10,0); (10,0,0)

48

Forces and Moments

The evolution of the force coefficients for the tested configurations are shown in Fig-ures 6.6, 6.7 and 6.8. It is evident that an oscillation of the aerodynamic loads causedby the interaction of the blades with the fuselage is present in all simulations, althoughthe amplitude of the oscillation is reduced by the presence of more blades.

Averaged values of the force coefficients were obtained from the last two rotorrevolutions. These averages are shown in Tables 6.1, 6.2 and 6.3.

Table 6.1: Force and Moment coefficients for the 2 blade rotor case.

Config. CfX CfY CfZ CmX CmY CmZ

(0,0,0) 1.9101e-04 -3.9782e-06 8.1502e-02 -5.4116e-05 -2.2226e-04 6.2660e-06(0,0,10) 1.4384e-02 3.1441e-06 8.0518e-02 -7.9645e-05 -1.8957e-04 3.3277e-06(0,10,0) 1.3434e-04 -1.0650e-05 8.0500e-02 -8.7241e-05 2.0223e-03 2.2643e-05(10,0,0) 1.6554e-04 -1.0908e-05 8.1450e-02 1.6259e-04 2.1027e-02 -8.1947e-06



(0,0,0) 1.7660e-04 1.7843e-05 1.0517e-01 7.0483e-05 -6.0770e-04 -1.5803e-05(0,0,10) 1.8405e-02 -2.2269e-05 1.0338e-01 1.1258e-04 -5.8458e-04 -2.2492e-05(0,10,0) 8.1681e-05 7.6946e-05 1.0434e-01 -1.1279e-04 2.4592e-03 3.6958e-06(10,0,0) 1.5428e-04 -1.9323e-05 1.0517e-01 8.4354e-05 2.6693e-02 -1.1644e-05



(0,0,0) 2.2866e-04 -4.4222e-07 1.2625e-01 5.2575e-05 -7.0795e-04 -2.4977e-05(0,0,10) 2.2157e-02 -2.2780e-05 1.2417e-01 -1.3684e-04 -7.9363e-04 1.2987e-05(0,10,0) 1.5818e-04 -1.7273e-05 1.2427e-01 -9.6831e-05 2.9113e-03 1.1303e-05(10,0,0) 2.5239e-04 -2.7285e-05 1.2614e-01 -2.3262e-04 3.2217e-02 9.6206e-06

In most cases, force in the Y direction is near zero (∼ 10−5), which is expected.A slight force is generated in the X direction for most cases (∼ 10−4), except for the(0,0,10) configuration, where a significant force appears in the X direction (∼ 10−2)

49

due to the orientation of the rotors. Predominant force component is in the vertical(Z) direction (∼ 10−1), as was expected. The vertical force coefficients of the differenttest cases is shown in Fig. 6.2. It is evident that geometrical configuration has minorinfluence on vertical thrust, while the number of blades has major influence. It shouldbe noted that thrust/blade ratio is diminished with the addition of blades, as shownby Fig. 6.3.

Figure 6.2: Vertical force coefficient from the test cases.

The evolution of the moment coefficients for the tested configurations is shownin Figures 6.9, 6.10 and 6.11. The motion of the blades causes oscillatory moments,especially about the Y axis (pitching moment). There is an erratic behavior of the roll(CmX) and yaw moments (CmZ), which should be zero for the symmetric case. Sincesymmetry is not imposed in the current model, numerical and/or physical asymmetryoccurs. In order to find an average value, values from all time steps were averaged,except for the pitching moment (CmY ) which was averaged only for the last two rotorrevolutions (see Table 6.1, 6.2 and 6.3).

50

Figure 6.3: Vertical thrust over the number of blades. Geometric configuration (0,0,0).

The simulations achieved interesting results for the pitching moments (CmY ), asshown in Fig. 6.4. Numerical results show that the base case generates a small nega-tive moment coefficient, as well as the (0,0,10) configuration. However, the remainingconfigurations generate significant positive moments about the Y axis. Also, there isa major influence of the geometrical configuration on the pitching moment, while thenumber of blades has a minor influence. An image of the geometry with three bladesper rotor and the fully developed wake for the configuration (0,0,0) is shown in Fig. 6.5.

51

Figure 6.4: Moments about the Y axis.

52

Figure 6.5: Geometry with 3 blades per rotor and fully developed wake for the (0,0,0)configuration.

53

Figure 6.6: Evolution of force coefficients with 2 blades per rotor.

54


55


56

Figure 6.9: Evolution of moment coefficients with 2 blades per rotor.

57


58


59

Chapter 7

Conclusions

The unsteady vortex-lattice method was implemented successfully in the GUAVLAMcode, which is designed for general unsteady aerodynamics and is not limited to thecase of rotary wings. Numerical results were obtained using GUAVLAM for the case ofan isolated rotor in hover flight and a generic two-rotor VTOL aircraft.

The geometry of an isolated rotor in hover presented in this thesis complies withthe characteristics of experimental setup from Caradonna & Tung [6]. The aerodynamicloads from the numerical results have a satisfactory agreement with experimental data.Although the method overpredicts thrust near the tip of the blade, the thrust coeffi-cients for all three collective pitch angles tested show good correspondence with theexperimental data, which was taken at a low Mach, high Reynolds number configura-tion. Results also show that for modeling rotors in hover flight the exclusion of rootvorticity has no impact on the final thrust and generates a smoother initial wake, eventhough this cannot be generalized for the case of highly-twisted blades.

A generic two-rotor aircraft was modeled with 2, 3 and 4 blades per rotor at acollective pitch angle of 12. For each geometry, 4 different configurations were tested,changing the positions and orientation of the rotors. The behavior of the forces andmoments are consistent with expected results: force component in the Z axis is domi-nant in all configurations, while other components tend to zero, except for the case of(0,0,10) configuration where the component on the X axis is significant. Also, there isa considerable positive pitching moment (Cmy) for the (10,0,0) and (0,10,0) configura-tion, and a relatively small negative Cmy for the remaining configurations, including thebase case. The unsteady loads reveal an oscillatory behavior due to the blade/fuselage

60

interaction. It was found that the number of blades has a greater influence on verti-cal thrust, while the geometrical configuration has a greater influence on the pitchingmoment.

The test cases show the versatility of the GUAVLAM code for modeling rotorcraftin different configurations, with and without fuselage. For future work, the genericVTOL aircraft presented here should be simulated in other hover flight configurations,changing the values of the parametric angles and finding the stability derivatives of theaircraft. The geometry could also be tested in other flight regimes, such as forwardflight or vertical ascent/descent.

The GUAVLAM code could be used for the simulation of helicopter flight. Forthis, the code should be coupled with a rotor trimming model in order to simulatethe motions of the blades during different flight regimes. To obtain better resultsfor rotary-wing problems, a far-wake model consisting of a single concentrated vortexfilament should be implemented in the future, reducing the computational cost of calcu-lating induced velocities on a large wake structure and eliminating the highly distortedfar-wake region seen in current simulations. Viscous dissipation models can also beincluded. For the case of symmetric problems, implementing the symmetry conditionwould reduce computational cost.

The current version of GUAVLAM defines constant free-stream and surface ve-locities. This is acceptable for the present application, but for truly general unsteadyaerodynamics the velocities should be implemented as a function of time. Also, futureversions of the code should account for the unsteady term of the pressure coefficient forcomputing the pressure distribution over closed bodies. Another enhancement would beto include deformable surfaces for the simulation of morphing aircraft, flapping wingsand/or aeroelastic behavoir. Finally, a mesh and input file generator is desirable tofacilitate the simulation of more complex geometries.

61

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[2] K. Asfar, D. Mook, and A. Nayfeh. Application of the Vortex-Lattice Techniqueto Arbitrary Bodies. Journal of Aircraft, 16(7):421–424, 1979.

[3] A. Bagai. Contributions to the Mathematical Modeling of Rotor Flow-fields Usinga Pseudo-implicit Free-wake Analysis. University of Maryland at College Park,1995.

[4] M. Bhagwat and J. Leishman. On the Aerodynamic Stability of Helicopter RotorWakes. American Helicopter Society Forum, 2000.

[5] M. Bhagwat and J. Leishman. Time-Accurate Free-Vortex Wake Model for Dy-namic Rotor Response. American Helicopter Society, Specialist Meeting, 2000.

[6] F. Caradonna and C. Tung. Experimental and Analytical Studies of a ModelHelicopter Rotor in Hover. Technical report, 1981.

[7] L. Ceballos and C. Gebhardt. Untitled manuscript. unpublished, 2008.

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[25] L. Long and T. Fritz. Object-Oriented Unsteady Vortex Lattice Method for Flap-ping Flight. Journal of Aircraft, 41(6):1275–1290, Nov. 2004.

[26] R. Miller. A Simplified Approach to the Free Wake Analysis of a Hovering Rotor.In DGLR Seventh European Rotorcraft and Powered Lift Aircraft Forum 15 p(SEEN 82-18119 09-01), 1981.

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[37] A. Rottgermann, B. Uhl, and S. Wagner. Compressible Potential Flow around aHelicopter Rotor in Arbitrary Motion. Computational Mechanics, 17(5):318–325,1996.

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[39] A. Sanchez, J. Escareno, and O. Garcia. Autonomous Hovering of a Two-RotorUAV, pages 59–78. John Wiley I& Sons, Inc., 2013.

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[41] M. Scully, M. I. O. T. C. AEROELASTIC, and S. R. LAB. A Method of Com-puting Helicopter Vortex Wake Distortion. ASRL TR 138-1. Defense TechnicalInformation Center, 1967.

[42] N. Sezer-Uzol. High Accuracy Wake And Vortex Simulations Using A HybridEuler/Discrete Vortex Method. Master’s thesis, Department of Aerospace Engi-neering, The Pennsylvania State University, 2001.

[43] G. Srinivasan. Flow field of a Lifting Hovering Rotor: A Navier-Stokes Simulation.NASA technical memorandum. National Aeronautics and Space Administration,Ames Research Center, 1990.

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[45] J. Steinhoff and K. Ramachandran. Free-Wake Analysis Of Compressible RotorFlows. AIAA Journal, 28(3):426–431, 1990.

[46] J. Steinhoff, W. Yonghu, T. Mersch, and H. Senge. Computational VorticityCapturing- Application To Helicopter Rotor Flows. In AIAA, Aerospace SciencesMeeting and Exhibit, 30 th, Reno, NV, page 1992, 1992.

[47] R. C. Strawn and T. J. Barth. A Finite-Volume Euler Solver For ComputingRotary-Wing Aerodynamics On Unstructured Meshes. Journal of the AmericanHelicopter Society, 38(2):61–67, 1993.

[48] R. C. Strawn, E. P. N. Duque, and J. Ahmad. Rotorcraft Aeroacoustics Com-putations with Overset-Grid CFD Methods. Journal of the American HelicopterSociety, 44(2):132–140, 1999-04-01T00:00:00.

[49] G. Vatistas, V. Kozel, and W. Mih. A Simpler Model for Concentrated Vortices.Experiments in Fluids, 11(1):73–76, 1991.

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[51] J. Wu and J. Thompson. Numerical solutions of time-dependent incompressiblenavier-stokes equations using an integro-differential formulation. Computers andFluids, 1(2):197 – 215, 1973.

[52] H. Yeo and W. Johnson. Performance and Design Investigation of Heavy LiftTilt-Rotor with Aerodynamic Interference Effects. Journal of Aircraft.

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66

Appendix A

Evaluation of Terms for theCalculation of Pressure Jump

Evaluation of VU ·VU −VL ·VL

The velocities relative to the lifting surface computed at the control points have nonormal component because the non-penetration boundary condition must be satisfied.The tangential velocity jump across the vortex sheet is caused by the presence of localvorticity and is expressed as:

∆V = VU −VL (A.1)

If the presence of local vorticity is omitted, a mean velocity Vm may be computedat the midpoint of the thickness of the vortex sheet.

Vm =VU + VL

2(A.2)

VU = Vm +∆V

2(A.3)

VL = Vm −∆V

2(A.4)

The difference VU ·VU −VL ·VL may be rearranged as:

VU ·VU −VL ·VL = (VU + VL) · (VU −VL) (A.5)

Substituting Eq. A.3 and Eq. A.4 into Eq. A.5 yields:

VU ·VU −VL ·VL = 2Vm ·∆V (A.6)

67

Evaluation of ∆V

Details on the derivation of ∆V can be found in Ref. [32]. This term is related to thelocal vorticity at the vortex sheet, which for quadrilateral elements can be computedas:

Γi =1

2(Γ1 + Γ2 + Γ3 + Γ4) (A.7)

where the numbered circulations represent the vortex strengths of the vortex segmentsthat surround the vortex sheet of index i.

Finally, ∆V can be expressed as:

∆V = − n× Γ

A(A.8)

where n is the unitary normal vector at the vortex sheet and A is the area of thatelement.

Evaluation of(∂Φ∂t

∣∣U− ∂Φ

∂t

∣∣L

)The derivative of the potential flow through time at a fixed point of position vector Ris given by:

∂Φ

∂t≈ Φ(R, t+ ∆t)− Φ(R, t)

∆t(A.9)

However, for this type of application it is rather difficult to evaluate ∂Φ∂t

directly.Instead, it is convenient to chose an arbitrary small displacement ∆R so that:

∂Φ

∂t≈ Φ(R + ∆R, t+ ∆t)− Φ(R, t)

∆t−∇Φ(R, t) · ∆R

∆t+HOT (A.10)

where HOT are the higher order terms. By calculating the limit of ∆t → 0 then weobtain:

∂Φ

∂t=DΦ

Dt

∣∣∣∣−∇Φ ·VCP (A.11)

68

where VCP is the velocity of the point fixed to the lattice (not a fluid particle), whichhappens to be the control point of that vortex sheet. It can be shown that

∇Φ(RU, t)−∇Φ(RL, t) = ∆V (A.12)

Moreover, the difference of potentials on both sides of the vortex sheet can becomputed by integrating velocity on a closed curve that passes over the leading edgefrom a point on the lower side of the sheet to the same point on the upper side of thesheet.

ΦU − ΦL =

∮C

V · dR = Γ (A.13)

From Stoke’s theorem it can be shown that Γ has the same circulation as thevortex ring circulation Gi that encloses the control point. Therefor, we have that:

[∂Φ

∂t

∣∣∣∣U

− ∂Φ

∂t

∣∣∣∣L

]i

=DGi

Dt−∆V ·VCP (A.14)

For numerical purposes, the substantial derivative DGi

Dtis approximated by a first

order finite-difference [32]:

DGi

Dt≈ Gi(t)−Gi(t−∆t)

∆t(A.15)

69

Appendix B

GUAVLAM Code Structure

This chapter presents an overview on the structure of the GUAVLAM code. The differ-ent Fortran modules used by the program are described, along with a brief descriptionof the functions and subroutines defined within the modules. The user defined Fortrantypes used by the code are also explained.

B.1 GUAVLAM Modules

Module uvlm wrappers

Contains the subroutines used by the main program within the time loop. It us usedby the main program.

1. Dependencies:

• Module uvlm.

• Module tecplot wrappers.

2. Type Definitions

• None.

3. Subroutines and Functions:

70

• geomTotalNp: Returns the sum of the number of bound vortex sheets ofeach surface.

• generateGeomIndex: Returns an array of 2 × NG, where NG is the totalnumber of surfaces. Each column contains indexes that map the panels froma surface to the influence coefficients matrix [Aij], the solution vector Gj andRHSj vector. For example: if a surface has indexes 5 and 9, that meansvalues from G5 to G9 map to the closed loop circulations of this surface.

• geomAI: Generates the influence coefficients matrix [Aij] for the linear systemof algebraic equations.

• geomRHS: Generates the right-hand side vector for the linear system of alge-braic equations.

• geomConvect: Updates the position of the surface and the nodes of the wake.It also passes circulation from bound vortex sheets at the trailing edge tothe newly formed wake vortex sheets.

• geomG: Maps the circulations in the solution vector Gj to the correspondingbound vortex sheets on each surface.

• geomMVel: Calculates the mean velocity Vm at each control point.

• geomLoads: Calculates the pressure jump at each bound vortex sheet.

• geomTecIO: Writes results to the Tecplot output file.

Module case module

Contains subroutines that read input files and write output files with data from thelast time-step, which can be used as starting point for later simulation. It us used bythe main program.

1. Dependencies:

• Module surface module.

2. Type Definitions

• None.


• initCase: Reads data from CaseControl input file.

71

• initGeom: Loops over .geom input files and Calls function loadGeom.

• loadGeom: Loads data from a .geom file to the corresponding variable oftype SurfaceType.

• loadWake: Loads data from a .wake file to the corresponding variable of typeWakeType.

• writeCase: Generates new CaseControl, .geom and .wake files with datafrom the last time-step, which can be used as input to restart the currentsimulation.

• writeGeom: Writes data from a variable of type SurfaceType to a .geomfile.

• writeWake: Writes data from a variable of type WakeType to a .wake file.

Module uvlm module

Contains basic functions for the unsteady vortex lattice method.

1. Dependencies:

• Module surface module.

• Library omp lib.

2. Type Definitions

• None.


• inducedVelocity: Calculates the velocities induced by a surface and itswake at a given set of points in space. This is used to calculate the inducedvelocities at the control points and at the fluid particles of the wake structure.Calls functions surfaceInfluence and wakeInfluence subroutines.

• fakeBiot: Calculates vector part of the velocity induced by a vortex filamentby applying the Biot-Savart law, i.e., excluding the term Γ/2π.

• surfaceInfluence: Calculates the velocity induced by a variable of typeSurfaceType at a given point in space. Uses OpenMP Parallel DO to loopover the vortex segments.

72

• wakeInfluence: Calculates the velocity induced by a variable of type WakeTypeat a given point in space. Uses OpenMP Parallel DO to loop over the vortexsegments.

• FSRHS: Calculates the right-hand side of the linear system of algebraic equa-tions due to the free-stream velocity.

• wakeRHS: Calculates the right-hand side of the linear system of algebraicequations due to the velocity field induced by the wake.

• infMat: Selects the portion of the [Aij] matrix that correspond to the influ-ence of surface A on surface B and generates the influence coefficients. Callsfunction influenceMatrix.

• influenceMatrix: Generates the Aij coefficients for a set of bound vortexsheets at a set of control points.

Module surface module

Contains the type definition SurfaceType and subroutines that modify the componentsof the instances of this type.

1. Dependencies:

• Module wake module.

• Module math routines.

2. Type Definitions

• SurfaceType: Contains data that describes the geometry of lifting surface,its movement and the circulations and loads at the bound vortex sheets.


• loadCalc: Computes the pressure jump across the bound vortex sheets ofa surface. Computes steady loads for bluff bodies and unsteady loads forlifting surfaces. Calls subroutine velocityJump.

• velocityJump: Computes the tangential velocity jump across the boundvortex sheets of a lifting surface.

• panelAreas: Computes the areas of the bound vortex sheets of a surface.

73

• convect: Updates positions of the wake nodes and the circulations of thewake vortex sheets.

• gammas: Computes vortex segment circulations Γ from the closed loop cir-culations G.

• updateG: Updates the closed loop circulations G for at the bound vortexsheets.

• updateCoordinates: Calculates the new positions of the surface nodes ateach time-step, given the instantaneous linear and angular velocities of thebody-fixed frame of reference.

• cPoints: Computes the coordinates of the control points and the unitarynormal vectors at each bound vortex sheet, in terms of the inertial frame ofreference.

• triangCPoint: Computes the centroid of a triangular element.

• quadPoint: Computes the centroid of a quadrilateral element.

• diagonals: Computes the diagonals of a quadrilateral element.

Module wake module

Contains the type definition SurfaceType and subroutines that modify the componentsof the instances of this type.

1. Dependencies:

• None.

2. Type Definitions

• WakeType: Contains data that describes the geometry of wake structure, itsmovement and the circulations at the wake vortex sheets.


• wakeVortexI: Generates a 2×NV array, where NV is the number of wakevortex segments, mapping the head and tail points of each vortex segmentto the nodes of the wake structure.

• wakeVortexG: Computes the vortex segment circulations Γ from the closedloop circulations G of the wake.

74

• wakeVortexPos: Updates the positions of the wake nodes at the trailingedge of the associated lifting surface.

• wakePanels: Generates the location matrix for the quadrilateral wake vortexsheets.

Module tecplot wrappers

Contains wrapper functions that facilitate the use of functions from the TecIO libraryto generate binary output files (.tec) to be processed by Tecplot360.

1. Dependencies:

• Intrinsic module ISO C BINDING, only C NULL CHAR.

• Library tecio64.a.

2. Type Definitions

• None.


• initTec360File: Initiates a file that will contain grid and solution data.

• newSurfaceZone: Writes a Tecplot zone of type FEQUADRILATERAL,containing the nodes of the surface lattice, the normal vector componentsand the pressure jump coefficients at each bound vortex sheet.

• newWakeZone: Writes a Tecplot zone of type FEQUADRILATERAL con-taining grid of the wake structure.

Module math routines

Contains auxiliary routines for general applications.

1. Dependencies:

• None.

75

2. Type Definitions

• None.


• rad2Deg: Function that converts the input argument from radians to degrees.

• deg2Rad: Function that converts the input argument from degrees to radians.

• eulerRot: Computes the transformation matrix for the euler rotation for agiven set of euler angles and rotation sequence. Returns the input argumentarray of dimensions 3×N multiplied by the transformation matrix.

• eulerDerivatives: Calculates the time derivatives of the euler angles giventhe angular velocity of the body-fixed frame of reference expressed in thermsof body-fixed coordinate system. This function computes time derivatives ofeuler angles only for the 3-1-2 rotation sequence.

• cross: Returns the cross product of two vectors.

• addOrigin: Sums an input column vector to each column of an input matrix.

• exten2mat: Joins two matrices in the column direction.

• norm2: Computes the L2 norm of a vector. This is deprecated when compilingwith Intel Fortran Compiler or other modern compilers that support Fortran2003 features, since it is already an intrinsic function.

B.2 GUAVLAM Derived Types

User defined data structures, called derived types in Fortran language, were imple-mented for the description of the discretized geometry and the respective wake struc-ture. Each surface in the simulation is represented by an instance of the data structureand the complete geometry of the problem is represented by an array of structures.This takes ideas from object-oriented programming and the features available in mod-ern Fortran95, which allows a more encapsulated and reusable source code [14, 25]. Itis important to understand these structures since most of their components need to beentered by the input .geom and .wake files.

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SurfaceType

Contains the geometry of the surfaces (lifting and non-lifting) and other relevant infor-mation about the bound vortex sheets. Its components are:

• NN: Number of nodes.

• NP: Number of panels (bound vortex sheets).

• NVS: Number of bound vortex segments.

• BLUFF: Boolean flag indicating whether the surface is closed or not. This is usedfor the generation of the influence coefficients matrix.

• COORDSYS: A 3 × 3 containing the base vectors of the body-fixed frame of ref-erence.

• ORIGIN: Array containing (X, Y, Z) coordinates of the body-fixed frame of refer-ence relative to the inertial frame of reference.

• VEL: Array with (U, V,W ) velocity components of the body-fixed frame of refer-ence (VB) relative to the inertial frame of reference.

• WVEL: Components (ω1, ω2, ω3) of the angular velocity of the body-fixed frameof reference relative to itself, i.e., ω = ω1b1 + ω2b2 + ω3b3.

• EANGLES: Instantaneous Euler angles in (3,1,2) sequence describing the orienta-tion of body-fixed frame of reference relative to the inertial frame of reference.

• NODESXYZ: a 3 × N matrix containing the (X, Y, Z) coordinates of each noderelative to the inertial frame of reference, where N is the number of nodes.

• NODESLOC: a 3 × N matrix containing the (X, Y, Z) coordinates of each noderelative to the body-fixed frame of reference, where N is the number of nodes.

• LOCMAT: Location (connectivity) matrix for the elements of the surface mesh.

• NBMAT: Matrix containing the indexes of the neighbors of each surface panel.

• CPXYZ: a 3×NP matrix containing the (X, Y, Z) coordinates of the control pointsrelative to the inertial frame of reference, where NP is the number of panels orcontrol points.

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• CPUVW: a 3 × NP matrix containing the (U, V,W ) velocity components of thecontrol points relative to the inertial frame of reference, where NP is the numberof panels or control points.

• NORMAL: a 3 × NP matrix containing the (X, Y, Z) components of the normalvectors at each panel relative to the inertial frame of reference, where NP is thenumber of panels.

• AREA: An array containing panel areas.

• MVEL: A 3 ×NP matrix containing the (U, V,W ) velocity components of meanvelocity (Vm) at control points relative to the inertial frame of reference, whereNP is the number of panels or control points.

• GAMMAS: A 4×NP matrix containing the vortex segment circulations, Γ, thatsurround each panel. This is used for the computation of tangential velocity jump(∆V ).

• VORTEX: A 7×NS matrix containing information about bound vortex segments:vortex strength (Γ), coordinates of the vortex starting point (P1) and ending point(P2) relative to inertial frame of reference.

• OLDG: Vortex ring circulations from the previous time step.

• NEWG: Vortex ring circulations from the current time step.

• LOAD: Pressure jump coefficient at each panel.

• CORE: Core size of the vortex segments.

• GROUP: Group number of the current surface. Groups relate surfaces that sharea common origin or frame of reference. The program will output the total sum-mation of forces and moments of all surfaces, as well as the summation of forcesand moments of surfaces belonging to each group separately. For the currentapplication, this was used to separate the loads of each rotor.

• SLOW: Number of time steps for the slow staring method. This may be deprecatedin future versions, where velocities are functions of time.

• WAKE: a pointer to an instance of type WakeType. This is initialized as null andallocated when a lifting surface is associated with a wake structure.

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WakeType

This data structure fully describes the wake structure, which is associated to an instanceof type SurfaceType. Its components are:

• NEN: Number of nodes at the trailing edge.

• NEP: Number of vortex segments at the trailing edge.

• NVS: Total number of vortex segments on the wake.

• NODESXYZ: a 3 × N matrix containing the (X, Y, Z) coordinates of each noderelative to the inertial frame of reference, where N is the number of wake nodes.

• NODESUVW: a 3 × N matrix containing the (U, V,W ) components of the localfluid particle velocity at each wake node.

• LOOPG: loop or vortex ring circulations G on the wake.

• VORTEXG: vortex strength of the vortex segments on the wake.

• VORTEXP: A 6×NV S coordinates of the vortex starting point (P1) and endingpoint (P2) relative to inertial frame of reference.

• LOCMAT: Location (connectivity) matrix for the elements of the surface mesh.

• VORTEXPI: A 2 × NV S containing indexes for the mapping of wake node co-ordinates to starting point P1 and ending point P2 of the vortex segments, i.e.,the coordinates of P1 and P2 for vortex segment i are described by nodes inthe NODESXYZ matrix with indexes VORTEXPI(1,i) and VORTEXPI(2,i) respec-tively.

• CORNERS: Array containing the indexes of wake nodes on the corners of thesurface. Panels on the corner of lifting surfaces may shed vorticity from thetrailing edge and from the wing tip. This means two vortex segments are shedwith the same vorticity and a “corner” node is generated. The extra vortexsegment is not eliminated, the corner node is simply moved to the position ofone the adjacent nodes to produce one zero-length segment and one connector asdepicted in Fig. B.1. This is consistent with the method developed by Mook etal. [22]. The array must have at least one element, even in applications wherewing-tip vorticity is not being convected.

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Figure B.1: Wake corner and wake structure without corner.

• CORMAP: Array containing the indexes of the wake nodes to which the wakecorners will be mapped. For applications where wing-tip vorticity is not convected,it must contain one element and its value is equal to that of CORNERS (the nodemaps onto itself).

• TELSSI: Array mapping wake vortex sheet of index i with the bound vortex sheetof index TELSSI(i).

• TELSNI: Array mapping wake node of index i with the surface node of indexTELSNI(i).

• TEWAKEI: Array that maps a vortex segment at the trailing edge of index iwith the corresponding free vortex sheet of index TEWAKEI(i). Free vortex sheetcirculation GW = G(TEWAKEI(i)) is used in the calculation of the vortex strengthof a segment on the trailing edge, Γt.e. = GB−GW , whereGB is the loop circulationof the bound vortex sheet associated to that vortex segment.

• START: Time step at which the wake structure will start to form. It is initializedto zero by default.

• ROOT: Boolean flag to indicate the inclusion (T) or exclusion (F) of the rootvortex. This is initialized to True by default.

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Appendix C

GUAVLAM Input file format

Input is read by the program from three different types of files: a CaseControl file, .geomextension files and .wake extension files. To view example input files, follow this link.

C.1 CaseControl file

This file is read by the program from the current directory. It contains informationabout global variables, like the number of time steps and free stream velocity, as well alist of .geom files that contain the geometries for the simulation. The structure of thisfile is as follow:

1. integer Number of Geometries: Total number of surfaces used in the simulation,lifting and non-lifting.

2. integer Number of Zones: Number of zones created in the Tecplot 360 file. Thisincludes all surfaces and their respective wakes.

3. integer Number of Groups: Groups are formed by surfaces that share a commonbody-fixed origin, where forces and moments will be measured and written to theoutput files. If no groups are specified, forces and moments will only be measuredwith respect to the inertial frame of reference.

4. integer Number of time steps: Total duration of the simulation.

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https://drive.google.com/folderview?id=0B9wpS25WqLGfNVAzOGpsUEpCckU&usp=sharing

5. integer Initial time step: This is used when restarting a case. Otherwise, itsvalue should be zero.

6. real Time step size (∆t): Non-dimensional time step size.

7. real Viscous core radius: Non-dimensional viscous core radius.

8. real Free-stream velocity: Three element array containing the components of thefree-stream velocity in terms of the inertial frame of reference.

9. character Tecplot File Name: Maximum length of 100 characters.

10. character List of Geometry Files (.geom): List containing a .geom for eachsurface used in the simulation.

C.2 Geometry .geom file

This file contains data for a variable of type SurfaceType (see Section B.2). The firstthree lines of the file must contain the following data in this order:

1. integer Number of Panels: data for the NP component of the SurfaceType

variable.

2. integer Number of Nodes: data for the NN component of the SurfaceType

variable.

3. integer Number of Vortex Segments: data for the NVS component of the SurfaceTypevariable.

These variables are used for memory allocation purposes. The components of theSurfaceType data structure are entered in arbitrary order by writing the name of thecomponent in capital letters (see Section B.2) followed by the values of that variable.For the ‘WAKE’ component, the following line should contain a character variablewith the name of the corresponding .wake file. If the surface is non-lifting, the WAKEcomponent is omitted.

Some components of the SurfaceType data structure will be initialized to zero orhave a default value, which means they can be omitted in the input file. All componentsare required when restarting a case.

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• Required: COORDSYS, ORIGIN, VEL, WVEL, EANGLES, NODESLOC, LOCMAT,NBMAT .

Additional information must be entered to create the Tecplot Zones. This is doneby declaring the ‘ZONES’ statement followed by an integer for the number of zones (“1”for non-lifting surfaces, “2” for lifting surfaces) and the names of the Tecplot Zones.

C.3 Geometry .wake file

This file contains data for a variable of type WakeType (see Section B.2). The first fourlines of the file must contain the following data in this order:

1. integer Initial time step for the wake convection: data for the START componentof the WakeType variable.

2. integer Number of nodes at the trailing edge: data for the NEN component ofthe WakeType variable.

3. integer Number of bound sheets at the trailing edge: data for the NEP compo-nent of the WakeType variable.

4. integer Number of Vortex Segments at the trailing edge: data for the NVScomponent of the WakeType variable. If the value entered is “-1”, the number ofvortex segments will be calculated automatically, which is the preferred option.

5. integer Number of corners: Number of corner nodes, equal or greater than 1.

Most components of the WakeType type will be initialized to default values. Thefollowing components are required by the code through the input file:

• Required: CORNERS, CORMAP, TEWAKEI, TELSSI, TELSNI .

For restarting a case, all information about the wake structure should be enteredthrough the input file.

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