Flapping flightsimulationmanualv1.1

http://www.flappingwings.co.uk 1

FLAPPING FLIGHT SIMULATION V1.1

USER MANUAL

last revision: 30/07/12 Ben Parslew


Introduction

The flappingflightsimulation package can be used to simulate the physics of flapping-wing

flight. This includes simulating the flight of living organisms, such as birds and insects, and

also man-made flapping-wing air vehicles. This document describes the installation of the

flappingflightsimulation package, and details how to use the three applications that are

included in the package.

The applications are distributed as freeware and not commercial products. Therefore, I

cannot guarantee that they are free from bugs, and I would appreciate any feedback to point

on problems with the software functionality, or the data it generates. The applications are

intended for a mixed audience, and so for this reason many of the technical details of the

physics of flapping flight are ommited from this manual for clarity. Further detail can be

found in the references section, particularly in references [1] and [2].

The following three applications are used included in the flappingflightsimulation V1.1

package:

FLAPSIM:

An inverse dynamics application that takes user inputs of wing motion, and calculates the

forces, torques and mechanical power for a flapping-wing system.

WAKESIM:

A point vortex simulator that predicts the geometry of the wake shed from a flapping-wing,

using the input wing dynamics from FLAPSIM.

FLAPOPTIMISE:

A numerical optimisation tool that predicts the most energy efficient types of wing motion

in different flight conditions.


FLAPTOPTIMISEFLAPSIM

WAKESIM

Wing dynamics

Flight requirements

Force, torque and power Wake geometry Optimal wing motion

Interaction between the three applications.

Installation

The applications were developed using Matlab R2010b and compiled using the MATLAB

compiler version 4.14 for Windows (32bit), and can be run in Windows XP, Windows Vista

and Windows 7. The user is required to install the MATLAB Component Runtime (MCR)

version 7.14, which is available from www.flappingwings.co.uk.

Installation Instructions:

1. Download and run MCRInstaller.exe from www.flappingwings.co.uk to install the Matlab

Compiler Runtime V7.14; follow the MCRInstaller onscreen instructions

2. Download flappingflightsimulation.zip and extract the contents to a suitable location (e.g.

C:\...\Desktop)

3. Run FLAPSIM.exe and WAKESIM.exe from the flappingflightsimulation directory.


flappingflightsimulation_V1_1 folder contents.

If you receive the error message "Could not find version 7.14 of the MCR..." you may need

to edit the system path as follows:

3.1 Right-click on My Computer and select Properties

3.2 On the Advanced tab (Windows XP) or under Advanced System Settings (Windows 7)

click Environment Variables

3.3 Under System Variables highlight the Path variable and click Edit

3.4 In the Variable Value textboxt add a semicolon to the end of the text, and then type the

path to which the MCR library was installed (the default is C:\Program

Files\MATLAB\MATLAB Component Runtime\v74\runtime\win32).

3.5 Click OK, and reboot. FLAPSIM and WAKESIM should now run.

Notes:

-The first time that FLAPSIM, WAKESIM and FLAPOPTIMIMSE are run it may take up to a

minute to load the interfaces.


FLAPSIM

To start FLAPSIM just run FLAPSIM.exe directly from the flappingflightsimulation folder.

After initialising, the following three FLAPSIM windows will appear:

Three FLAPSIM windows

The FLAPSIM user interface provides you with tools for simulating flapping-wing flight. You

can define the size, shape and movement of the wings, and FLAPSIM will calculate how the

aerodynamic forces on the wing vary throughout the wingbeat.

The command history window keeps track of the simulations that you have run during your

current session, and also includes lists data imports and exports.

The animation window illustrates the flapping-wing model*. After setting up a simulation,

FLAPSIM can generate animations of flapping flight within this window.

The FLAPSIM interface is laid out as seven panels that contain the input options:

* The rendering window draws an accurate representation of the wing geometry that is used in the FLAPSIM

simulations. An ellipsoidal 'body' is included to provide context, but is not included in the simulations.

user

interface

command

history

animation

window


FLAPSIM User Interface Panels

The following sections describe all of the different options in FLAPSIM, organised by the title

of each panel in the FLAPSIM user interface:

Wing planform {1}

When you run FLAPSIM the default wing planform is the Rock Pigeon (Columbia livia.). To

change the shape of the wing, simply type the name of any other wing geometry file into the

wing planform textbox. Wing geometry files are located in the planforms folder, and are .mat

format†.

You can define your own custom wing geometries and add them to the planforms folder to

use in FLAPSIM. To define your own wing geometry all that is required is a list of datapoints

to define the coordinates of the outline of the wing. The data should be arranged in the

same format as those in the geometry files in the planforms folder.

† .mat files are data files that can be opened directly using MATLAB, or using freeware such as SciLab.

{1}

{2}

{5}

{4}

{3}

{6}

{7}


By selecting bird or insect, you can choose whether FLAPSIM treats the wing as being made

of two separate panels connected by a wrist joint, or as a single panel. The blue wing

planform image illustrates the difference between the double- and single- panel wing.

Mass and geometry {2}

The wing length defines the distance between the shoulder joint and the wing tip, measured

along the x-axis. As the wing planform geometry has already been defined, adjusting the wing

length causes also changes the wing chord length, causing the planform the be geometrically

scaled.

The wing mass and radius of gyration is used by FLAPSIM to calculate the amount of power

used by the wing in flapping. Some examples of typical values can be found in [3].

When a double-panel wing is being used, you can choose the relative size of each of the

panels by adjusting the hand:wing ratio.

Aerodynamics {3}

FLAPSIM calculates the aerodynamic forces on the wing using a method known as the Blade

Element theory. This method models the wing as a series of small strips, or elements.

Aerodynamic forces are calculated on each element, and then averaged to obtain the net

aerodynamic force on the wing at any point in time. Full details of this method and it's

implementation in FLAPSIM can be found in [2], but for the purpose of this guide a simple

overview will be sufficient to explain how the different aerodynamics options affect the

simulation results.

When calculating lift and drag, FLAPSIM calculates the orientation of the wing with respect

to surrounding wind. The FLAPSIM options for wing cl45, cd0, wing cd90 define the amount of

lift and drag that the wing can produce for a given orientation and wind speed. Increasing cl45

increases the amount of lift that the wing can generate, while increasing cd0 and cd90 increases

the amount of drag.


FLAPSIM calculates aerodynamic force, torque and power at instantaneous points in time

throughout the wingbeat. The duration between these points in time is known as a timestep.

Increasing the number of timesteps increases the accuracy of the simulation, but also

increases the time required for FLAPSIM to run a simulation. Using a greater number of

elements also increases the accuracy of the blade-element method, and again increases the

simulation time.

As well as the Blade Element Theory, FLAPSIM also uses the Momentum Theory, to model

the downwash of air created by the wings. This downwash, or induced velocity, is fed into the

Blade Element Theory to increase the accuracy by recalculating the lift and drag. FLAPSIM

allows you to define how many times this recalculation occurs by setting the number of

induced iterations; the default value of 20 is found to be suitable for most simulations.

Kinematics {4}

Some of the key options for FLAPSIM are found in the kinematics panel, which is used to

define exactly how the wings move, and also select the flight speed. As with the

aerodynamics panel, exact definitions of the kinematics options are described in [2]. The

following diagrams illustrate examples of flapping-wing flight with various kinematics:

Wing elevation and depression


Wing elevation and depression, with a negative stroke-plane

Wing elevation and depression, with pronation-supination


Wing elevation and depression, with retraction-extension

The frequency defines how many wingbeats the bird performs per second. Wing elevation

amplitude defines the amount that the wings elevate and depress, which is the up-and-down

flapping motion observable in the forward flight of birds. Pronation amplitude defines that

amount that the wings twist about the shoulder joint, while hand pronation is twisting about

the wrist joint. The stroke-plane angle is the inclination of the plane in which the wings flap,

measured with respect to a vertical axis. Wing extension amplitude is the ratio of how

extended the wings are on the upstroke, compared to the downstroke; birds typically

extend their wings fully on the downstroke, and then use a lesser-extension (or retraction)

on the upstroke.

If you select the option for parameterised kinematics, the elevation, pronation, hand

pronation and extension all vary sinusoidally with time, and the stroke-plane angle is

constant,. This is representative of the wing kinematics used by animals in most flight

conditions.


This sinusoidal motion can be modified by varying the downstroke fraction, which defines

the ratio of time spent on the downstroke to time spent on the entire wingbeat.

Alternatively, user-defined kinematics can be used, and the elevation, pronation, hand

pronation, extension and stroke plane can be defined as varying in any continuous manner

with respect to time. This is done by typing the name of a wing kinematics data file into the

kinematics panel. Kinematics data files are stored in the kinematics folder. New data files can

be added to this folder for use in FLAPSIM, but these should contain constant or periodic

time histories of the kinematic variables, in an equivalent format to the example kinematics

data file. The cruise speed is defined using the FLAPSIM interface for both parameterised

and user-defined kinematics.

Plot wing dynamics {5}

When you click plot in the plot wing dynamics panel, FLAPSIM draws a diagram to illustrate

the wing kinematics and aerodynamic forces as a simple set of frames. The number of

frames can be set explicitly before plotting.

Plotted wing dynamics


The red arrows are force vectors, showing the direction of aerodynamic force and the

relative magnitude between frames. Single panel wings have one force vector per wing, that

represents the sum of the aerodynamic forces on all of the wing blade elements at an

instance in time. The double-panel wings have two force vectors, which show the sum of

the forces on the arm and hand separately. The length of the force vectors can be scaled up

and down prior to plotting using the vec. scale option.

FLAPSIM includes two options for plotting the wing dynamics: phase and displacement. With

phase plotting frames are plotted as series of equally spaced images, while displacement

plotting spaces the images by the correct distance that would have been flown between

taking the two frames; in this respect, displacement plots are similar to taking a series of

photographs of a flapping-wing system with a fixed camera.

The plotted frames can be exported to various formats using the file menu on the rendering

window.

Animate {6}

As well as plotting still frames, FLAPSIM can also be used to produce an animation of the

wingbeat and force vectors. The animation can be viewed from the side, top, front or from

an isometric viewing angle, by selecting the according option on the animation panel. To

control the speed of animation, the number of frames per cycle can be varied, whereby

plotting more frames per cycle leads to slower animations. The number of cycles in the

animation can also be varied.


Wing dynamic data {7}

You can use FLAPSIM to generate plots of the wing dynamic data over a wingbeat. Clicking

plot in the wing dynamic data panel generates a series of graphs present the kinematics of

the wing that were defined in the kinematics panel, and also the aerodynamic forces and

torques, and the mechanical power consumption. These graphs are particularly useful for

making detailed inspections of the affects of varying the simulation input options. As with

the plotted wing dynamics frames, the graphs can be exported using the file menu on the

rendering window. The raw dynamic data generated by FLAPSIM, which is used to plot the

graphs, can be exported directly to a .mat file by clicking export in the wing dynamic data

panel. The .mat file will then appear in the wingdynamics folder.


FLAPOPTIMISE

To start FLAPOPTIMISE just run FLAPOPTIMISE.exe directly from the

flappingflightsimulation folder. After initialising, the following three WAKESIM windows will

appear:

Three FLAPOPTIMISE windows

The FLAPOPTIMISE user interface is designed for optimising wing kinematics. As with

FLAPSIM, you can define the wing size and shape, along with other physical parameters.

FLAPOPTIMISE can then calculate the most energy efficient wing motion for a given flight

condition, such as cruise or climb. The calculated kinematic data can then be exported and

used as an input to FLAPSIM to visualise the wing dynamics and plot the time histories.

The optimisation table provides a list of solutions that FLAPOPTIMISE has found to a given

problem. The table is updated live during optimisation.

The FLAPOPTIMISE interface is laid out as eight panels that contain the input options:

optimisation

table

user

interface

command

history


FLAPOPTIMISE User Interface Panels

The following sections describe all of the different options in FLAPOPTIMISE organised by

the title of each panel in the FLAPOPTIMISE user interface:

Wing planform {1}

see FLAPSIM

Mass and geometry {2}

This panel is similar to the mass and geometry panel in FLAPSIM, but with the addition of the

bird mass. This defines the total mass of the bird, including the wings. The reason why this

option is needed is because FLAPTOPIMISE considers the dynamics of the bird as a whole,

and not only the wings (as in FLAPSIM). In cruising flight, for example, FLAPOPTIMISE

calculates wing motions that generate the right amount of aerodynamic force to support the

weight of the bird.

{1}

{2}

{5}

{4}

{3}

{7}

{8}

{6}


Aerodynamics {3}

This panel is similar to the aerodynamics panel in FLAPSIM, but also includes the body area

and body Cd0. These inputs are used to calculate the drag acting on the body of the bird (that

is, the drag acting on the bird without including drag on the wings). The body drag is

calculated for the same reason as the weight, in calculating the dynamics of the bird as a

whole, and not only the wings.

The body area is an aerodynamic reference area which in FLAPOPTIMISE defines the frontal

area of the bird body. The body Cd0 is the body drag coefficient, which defines the amount of

drag that the body produces at a given speed. Bodies with higher Cd0 generate more drag at

a given speed.

Flight mode {4}

The flight mode panel contains four different options that can be selected to define a

particular flight condition. Cruise is defined as flight in which the mean aerodynamic and

gravitational forces are in equilibrium. This means the a cruising bird is flying at constant

speed. FLAPTOPTIMISE can predict the wing kinematics for cruise that use the lowest

amount of mechanical power (min. power cruise) or that achieves the maximum range (max.

range cruise)‡.

Non-cruising flight modes can also be simulated: minimum power flight with varying speed

(min. power accel.) or minimum power flight with varying altitude (min. power climb). Negative

acceleration values can be specified to simulate deceleration, and negative climb angles to

simulate descent.

‡ Maximum range is equivalent to the minimum ratio of mechanical power to cruise speed.


Kinematics constraints {5}

Constraints are used to limit the range of kinematics that FLAPOPTIMISE can explore.

Theoretically the optimisation could be performed with no constraints on some kinematic

parameters, however this tends to make many problems unwieldy. However, some

constraints are required for numerical stability - for example the extension amplitude must

have an upper limit less than or equal to one, otherwise the maximum extension would be

longer than the wing length.

Another use of the constraints is that individual kinematic parameters can be set at fixed

values by setting equal and upper lower bounds. So for example, the velocity upper bound

and lower bounds could be fixed at 10ms-1, and then the optimisation could be used to

predict the minimum power wing motion at this fixed cruise speed. Constraining kinematics

parameters to fixed values also increases the speed at which the FLAPOPTIMISE algorithm

finds optimal solutions.

Initialisation {6}

FLAPTOPIMISE uses a number of initial 'guess' solutions to initiate the optimisation§. With

more initial guess values the optimiser is more likely to find a more energy efficient solution.

However, this also increases the computation time for optimisation.

The mesh density defines the number of initial guesses per kinematic variable. The guesses

for a given variable are evenly spaced between its lower and upper bounds. So for example

if the lower and upper bounds of velocity (vel.) are set at 4 and 13, a mesh density of 4 will

yield initial guesses for velocity as 4,7,10 and 13. As FLAPOPTIMISE uses all combinations of

initial guesses for the different variables, the total number of guesses escalates quickly by

increasing the mesh density, so this should be done with some caution**.

§ These are the initial values in the optimisation space where the gradient-based numerical optimisation routine

starts from.

** The total number of initial guesses is equal to the number of unfixed kinematic parameters, raised to the

power of the mesh density


Random initial guesses can also be included, and these will be between the defined upper

and lower constraints for each variable.

Clicking optimise starts the optimiser, and this will run until each of the initial guesses for the

kinematic parameters has been optimised. All of the solutions obtained from the

optimisation are displayed in the optimisation table. The columns in the table are ordered as

follows: cost, velocity (vel.), frequency (freq.), elevation amplitude (elev.), pronation amplitude

(pro.), stroke plane (s.p.), extension amplitude (ext.), hand pronation amplitude (pro.h.),

downstroke fraction (d.f.) and flag. Cost is the variable that the optimiser is trying to

minimise, which can either be average mechanical power in Watts (for min. power cruise,

min. power accel. and min. power climb) or power:speed ratio in Joules/metre (for max. range

cruise). The flag shows whether a particular solution is valid or not††; valid solutions for

FLAPOPTIMISE have a flag of +1 or +2.

Ranked optimisation results {7}

As well as the optimisation table, results can also be viewed as a list ranked by the cost. This

provides the user with a quick view of the most optimal solutions. Only the top ten valid

solutions are included in the ranked results.

Optimised kinematics data {8}

The data from FLAPOPTIMISE can be exported using two methods. The export result button

can be used to pick a certain solution from the ranked results, and exported in a format that

is suitable for importing into FLAPSIM. Using this option, simply click to select a cell in the

desired row of the ranked results, and then click export result. The kinematic parameters will

then be saved as a .mat file, with the name in the textbox adjacent to the export result

button; this file is saved in the folder optimise_results.

†† Valid solutions are those that satisfy the optimisation constraints. An example of an invalid is solution for

cruise is one in which vertical component of the mean aerodynamic force generated by the wings is not equal

to the weight of the bird.


The other method of exporting data is to export the entire optimisation table, including

both valid and invalid solutions. Clicking export table causes the optimisation table to be

saved as a .mat file in the folder optimisation_tables. The exported tables are useful for

examining other potential kinematics that can be used for a given flight mode, that don't use

the minimum amount of mechanical power.

WAKESIM

To start WAKESIM just run WAKESIM.exe directly from the flappingflightsimulation folder.

After initialising, the following three WAKESIM windows will appear:

Three WAKESIM windows

The WAKESIM user interface is designed for importing dynamic data that has been

generated by FLAPSIM, and using this to simulate the wake.

The rendering window is used after simulation the wake to visualise how the wake is drawn

and animated.

The WAKESIM interface is laid out as five panels that contain the input options:

rendering

window

user

interface

command

history


WAKESIM User Interface Panels

The following sections describe all of the different options in WAKESIM organised by the

title of each panel in the WAKESIM user interface:

Wing dynamics {1}

The wing dynamics panel is used to select the filename of a wing dynamics .mat file, that are

exported from FLAPSIM. The file should be located in the wingdynamics folder.

The width of the body, defined as the distance between the shoulder joints, can also be set

in the dynamics panel. While the body width was not required in FLAPSIM, it plays an

important role in WAKESIM as it affects how the wake shed from the left and right wings

interact with each other.

Point vortex simulation {2}

WAKESIM simulates the wake using a method known as the Point Vortex or Vortex Blob

method. This method simulates two vortex filaments, each shed from a wing tip. WAKESIM

calculates the mutual and self-induced interaction between these filaments. This approach is

typically used in the analysis of helicopter wakes, and full details of the implementation can

be found in [2], [3].

{1}

{2}

{3}

{4}

{5}


The vortex filament model has several input options that are found in the point vortex

simulation panel. The core radius option is used to adjust the degree of damping due to fluid

viscosity. Increasing the core radius increases the amount of damping; with a small core

radius the wake simulation can become unstable.

The number of points/cycle is the number of sections that the vortex filaments are divided

into when WAKESIM performs the calculations. The number of timesteps/cycle is equal to

the number of timesteps that WAKESIM uses to simulate the wake per wingbeat. Increasing

the points/cycle and timesteps/cycle generally leads to more accurate simulations, but requires

more computation time. Wakes can be simulated for any number of wingbeat cycles by

adjusting the cycles; note that using a large number of cycles greatly increases the

computation time.

To visualise the geometry of the wake, WAKESIM also simulates a set of tracer particles

that are shed from the trailing edge of the wing. These tracers do not influence, but are

influenced by, the vortex filaments. They propagate downstream and allow you to plot and

animate the wake geometry. When a larger number of tracers that are used, the wake can

be visualised in more detail, however the computation time increases. For a double panel

wing the number of tracers released from the arm (arm tracers) and hand (hand tracers) are

defined separately, while for a single panel wing the total number of tracers in defined (wing

tracers).

Clicking simulate in the point vortex simulation panel initiates the point vortex simulation, and

upon completion automatically runs the tracer simulation.

View {3}

The view in the rendering window can be set to the side, top, front or isometric. The show

ground option switches on or off the illustration of the ground and shadow below the

flapping-wing system in the animation window. Show ground should be selected prior to

rendering.


Rendering {4}

You can render the wake using three different techniques. When sheet is selected, clicking

render causes WAKESIM to draw a surface by connecting all of the tracers to their

neighbours using straight lines. WAKESIM displays the sheet as an opaque, blue reflective

surface:

Choosing streams and clicking render causes WAKESIM to draw tubes that connect all

tracers released from a given position on the wing trailing edge. By choosing tracers and

clicking render, WAKESIM draws all of the tracers as small spheres. Rendering the tracers is

the most computationally demanding of the three methods of rendering.

WAKESIM also provides a seek toolbar to allow you to visualise a stationary image of the

wake at a given instance in time. When the slider is moved from left to right, the wake is

rendered at times from the start to the end of the simulation. Seek can be used without

clicking render, to view a single rendered frame. This is particularly useful if a large number

of tracers were simulated and rendering the entire animation would take a long time.

The rendered wake can be inspected using the tools at the top of the rendering window,

and the image can be exported using the file menu.

Animation playback {5}

Clicking animate causes a new window to open and the rendered wake geometry to be

animated. The speed at which the animation plays can be changed by setting the frames/sec.

The no. cycles played back during the animation can also be changed.

You can save the animated wake by clicking export. This generates a .mat file that contains a

Matlab structure that holds the movie; each element of the structure is a single frame of the

animataion. As well as the .mat file, an .avi‡‡ file is created that can be opened using a

number of common media players.

‡‡ This requires the Indeo 5 codec to be installed, which is not done by default in Windows 7. Without this

codec WAKESIM will not produce an .avi file, and error messages will appear in the command history window.


LIMITATIONS

In flappingflightsimulation V1.1 FLAPSIM is only capable of modelling flapping flight at a

constant speed. In future revisions acceleration effects will also be included [2].

FLAPSIM does not the dynamics of the body into account.

FLAPSIM includes some, but not all of the commonly referenced unsteady aerodynamic

effects. Added mass and rotational lift effects are captured.

In flappingflightsimulation V1.1 WAKESIM is unable to provide field data such as the

velocity or vorticity in the wake region. This will be included in future revisions.

WAKESIM only models the vortex filaments shed from the wingtips, and does not

include a model of bound or starting/stopping vortices.

Known Issues

For very small elevation angles FLAPSIM tends to overpredict the induced velocity

If either FLAPSIM or FLAPOPTIMISE are run when the other application is still open the

command window will issue a warning, however both applications continue to function

correctly.

WAKESIM requires an unrealistically large core radius to ensure numerical stability

[4],[5].

When exporting a result from FLAPOPTIMISE to FLAPSIM, the mean forces and

mechanical power predicted by FLAPSIM may differ slightly from the values predicted by

FLAPOPTIMISE. This is due to FLAPTOPTIMISE using a modified algorithm for deriving

the induced velocity.

Graphics problems may occur if the animation window (FLAPSIM) or rendering window

(WAKESIM) are positioned on a second monitor.

WAKESIM can only export .avi files if the Indeo 5 codec is installed. For windows 7

users this can be done using the following steps:


1. Open a 'command prompt' using a right click and select, "Run as Administrator"

2. Type: cd C:\Windows\SysWOW64 and press enter (32bit users can skip this step)

3. Type: regsvr32 ir50_32.dll and press enter

Release History, Updates & Bug Fixes

30/07/2012: flappingflightsimulation package V1.1 and manual made available online

28/07/2012: (FLAPSIM) Dropdown menu included to import kinematic parameters from FLAPOPTIMISE

27/07/2012: FLAPOPTIMISE completed

24/07/2012: (FLAPSIM) Switching between single- and double-panel wing now adjust aerodynamic control

point positions correctly in aerodynamics algorithm, as well as in interface diagram.

19/07/2012: (FLAPSIM) Input values of wing mass now imported correctly into FLAPSIM dynamics algorithm.

12/07/2012: (FLAPSIM) Corrected scaling of wing position vectors for retracting wings; now correctly

accounts for hand circumduction.

01/07/2012: flappingflightsimulation package V1.0 and manual made available online

References

[1] B. Parslew and W. J. Crowther, ‘Simulating avian wingbeat kinematics’, Journal of

Biomechanics, vol. 43, pp. 3191–3198, 2010.

[2] B. Parslew, ‘Simulating Avian Wingbeats and Wakes’, PhD Thesis, The University of

Manchester, 2012.

[3] C. van der Berg and J. M. V. Rayner, ‘The moment of inertia of bird wings and the

inertial power requirement for flapping flight’, The Journal of Experimental Biology, vol.

198, pp. 1655–1664, 1995.

[4] A. Bagai and J. Leishman, ‘Rotor free-wake modeling using a pseudo-implicit technique

including comparisons with experiment’, Journal of the American Helicopter Society, vol. 40,

pp. 29–41, 1995.

[5] M. Stock, ‘A regularized inviscid vortex sheet method for three dimensional flows with

density interfaces’, PhD dissertation, University of Michigan, 2006.

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