Justin Weinmeister Honors Project MECH 342

1

Honors Project

Mech 342

Justin Weinmeister

11/29/2015

Lift Produced from an Airfoil under Ground Effect using Kutta‐Joukowski Theorem

Original Proposal

This project will investigate the change in lift on an airfoil when it is near the ground. The project

will solve this by using the Kutta‐Joukowski theorem to solve the lift on an airfoil. For a standard airfoil,

this would entail creating a stream function containing a uniform potential, doublet, and vortex.

Conformal mapping is then used to create a Joukowskian airfoil. The lift per unit length of the airfoil can

then be calculated by solving this stream function. To simulate ground effect, a mirror airfoil will be

placed below the original airfoil to create a boundary. This boundary is created from the counter‐

rotating vortex of the mirror airfoil. This will create a new stream function which will be solved using

MATLAB. The deliverables should include a plot showing the streamlines of the flow across the airfoil

and a plot of the lift produced from the airfoil based on its height above the simulated ground.

Change in Experiment

The original proposal could not be solved as stated due to discovery of limitations of potential

flow theory. In order to map a circular cylinder into a Joukowski airfoil, the circle must be located near

the origin of the original coordinate system. With this study locking the origin of the coordinate system

in place and changing the location of the circular cylinder, the shape of the Joukowski airfoil would

change based on its location. The results of this study, though, still apply to an airfoil in ground effect as

the governing equations are the same for the cylinder and airfoil. The trend of the coefficients of lift will

be the same though they may not match those of the equivalent Joukowski airfoil.

Revised Problem Statement

MATLAB will be used to calculate the change in the coefficient of lift on a rotating circular

cylinder under ground effect in a uniform flow. Potential flow theory of an inviscid fluid will be used to

calculate the change in lift on the cylinder. The problem will be evaluated at multiple heights to

determine if the coefficient of lift increases or decreases as the cylinder is brought closer to the ground.

Introduction

The problem illustrates the effect of the ground on the lift of an airfoil as the airfoil’s distance to

the ground is decreased. This is a result of the Kutta‐Joukowski theorem of potential flow. Potential flow

theory can describe a rotating circular cylinder in a uniform flow by the superposition of the potential

function for a uniform flow, a doublet, and a vortex. The potential function for this superposition is

shown in equation 1.

2

1 cos 1

This rotating circular cylinder produces lift in the flow due to the circulation of the fluid around

the cylinder creating a favorable pressure distribution. The rotating circular cylinder accurately

represents an airfoil because the governing fluid effects are similar.

The Kutta‐Joukowski theorem states that this rotating circular cylinder can be transformed to

represent an airfoil by conformal mapping of the cylinder. The Kutta condition is then applied so that

the rear point of the airfoil is a stagnation point as otherwise a stagnation point occurs on the top

surface of the airfoil. Using the Kutta‐Joukowski theorem, a number of Joukowski airfoils can be

produced that can be solved to find the lift per unit length of an airfoil. The potential flow theory of this

Joukowski airfoil is the same as that of the rotating circular cylinder with only the geometry mapped to a

different plane in order to better represent the shape of real airfoils. Because the potential flow theory

for both cases is the same, the change in the coefficient of lift for both cases will behave similarly. By

solving this problem, it can be determined if the floating feeling that pilots experience as they try to land

a plane is the result of an increase in the coefficient of lift or due to other factors.

Results

The potential flow problem was solved by differentiating the potential function to find the

velocity potential function. This velocity potential function was then evaluated over a mesh of points

that contained the circular cylinder described by the potential function. This gave the magnitude of the

velocity of the fluid at every mesh point. The function was evaluated by using chosen parameters for the

angle of attack, freestream velocity, freestream pressure, cylinder radius, cylinder location, and

circulation. These parameters were not based on any physical model as the value of the coefficient is

not important for this calculation, only its trend. The coefficient of pressure was then evaluated at every

point in the mesh using equation 2.

1 2

The coefficient of pressure was then numerically integrated along the boundary of the upper

cylinder. First the bottom was integrated, then the top. This was done by calculating the mesh point at a

finite step away from the previous step along the boundary of the cylinder. The velocity at this point was

then calculated using the velocity potential function in order to get the coefficient of pressure at this

point. The coefficient of lift for this finite step is the product of the finite length multiplied by the

coefficient of pressure at this point. This process was conducted along the entire boundary of the

cylinder. The resultant coefficient of lift for the cylinder is the sum of these individual coefficients.

Figure 1 shows the trend of the coefficient of lift of an infinitely long rotating circular cylinder

under ground effect in a uniform flow. The results are shown as true separation of the center of studied

3

circular cylinder from the ground. Each point represents a change of 0.2 in seperation.

Figure 1

The results show that there are two points that do not fit the general trend of the rest of the

data. The point for a separation of zero indicates when the studied circular cylinder is overlapped with

the counter‐rotating circular cylinder at the origin. This point can be disregarded because this

configuration is physically impossible to accomplish. The next point, at a separation of 0.2, can also be

disregarded as it describes the situation when the cylinders are non‐overlapping but touch at the origin.

Flow should not be able to pass between the two circular cylinders in this situation and the data point

can be disregarded. The remainder of the data shows that the coefficient of lift asymptotically

approaches a value of 0.5 quickly with increasing separation.

Figures 2, 3, 4, and 5 show the circular cylinder with a separation of 0, 1, 3, and 9, respectively.

Each plot shows the streamlines as dashed black lines with the stagnation lines as solid red. The

coefficient of pressure for the field is also displayed as a color map.

4

Figure 2: Separation of 0


5


6

It can be seen that in the zero separation case the function approximates that of a non‐rotating

circular cylinder in a uniform flow which generates an area of high pressure on the leading and trailing

edges of the cylinder with areas of low pressure on the top and bottom. The streamlines are evenly

distributed.

In the case of a separation of 1, the streamlines are now deflected up and over the circular

cylinder with an area of high pressure below the cylinder extending all the way to the ground and an

area of low pressure above the cylinder. Stagnation points on the cylinder occur below the cylinder and

on either side of the bottom edge. The streamlines have a slight deformation all the way to the ground.

In the case of a separation of 3, the streamlines are still deflected in the same shape as for a

separation of 1. The areas of high and low pressure are located in the same locations, though the region

of high pressure no longer extends all the way to the ground. The streamlines are no longer deflected all

the way to the ground, though there is only a very small region where they are not deflected. The


7

stagnation points are located in the same position as for the case of a separation of 1, indicating the

applied circulation is constant.

In the case of a separation of 9, the streamlines are again deflected in the same shape and the

pressure regions are similar to those of a separation of 3. It can also be seen that the streamlines are

nearly horizontal for much of the distance between the cylinder and the ground. This indicates that the

ground is having very little effect on the lift of the cylinder. Again, the stagnation points are located in

the same location, indicating that the applied circulation is constant.

Analysis

This calculation of the potential flow of an inviscid fluid around a rotating circular cylinder under

ground effect shows that the lift produced will increase as the cylinder is brought closer to the ground.

This change in the coefficient of lift is the result of a favorable pressure distribution as all other

parameters are held constant. This calculation approximates ground effect on an airfoil and it can be

deduced that the lift on an airfoil will increase as the airfoil is brought closer to the ground.

This result agrees with literature1 that the coefficient of lift of an airfoil in ground effect at a low

angle of attack increases as the airfoil is brought closer to the ground. The study of the circular cylinder

is not advanced enough to compare different angles of attack as the symmetric geometry of the cylinder

does not simulate the non‐symmetric geometry of the airfoil. Additionally, most studies of ground effect

do not using potential flow theory as lifting line theory is more advanced and better approximates the

lift on an airfoil in under ground effect. The lifting line theory shows that an airfoil in ground effect has a

much more favorable lift‐to‐drag ratio as the wind‐tip vortex and ground interaction greatly reduces the

drag on the airfoil. This effect is span‐wise. The lifting line theory also shows that there is a small

increase in the coefficient of lift for an airfoil due to chord‐wise effects. This agrees with the results of

the calculations.

References

1. See appendix C

Appendix

A. MATLAB code for iteration of program used to generate trend plot.

B. MATLAB code for generation of streamline plots

C. Paper discussing ground effect on an airfoil using potential flow

11/29/2015 iterate_pressure

file:///U:/Mech%20342/Honors%20Project/html/iterate_pressure.html 1/4

Contents

Parameters

Symbolic parameters and flows

Calculating Complex potential and velocity

Numerical function

Mesh

Numerical Results

Calculate lift

Plot

clc;clear;

eval = 45;Cl = 0:eval;

for iterate = 0:eval

Parameters

alpha = 0.0*pi/180; % Angle of attackU = 1; % Freestream velocityP = 1; % Freestream pressurer0 = 0.2; % Radius of doubletz0 = 0.0 + 1i*(iterate/5); % Origin of featureGamma = ‐1; % Circulation


syms s_Usyms s_alphasyms s_r0syms s_z0syms s_Gammasyms s_zsyms s_Fsyms s_W

%Uniform flowsyms s_F_uniform %Complex potential for uniform flows_F_uniform = s_U*exp(‐1i*s_alpha)*s_z;

%Point vortexsyms s_F_vortex %Complex potential for vortexs_F_vortextop = ‐1i*s_Gamma/(2*pi)*log(s_z ‐ s_z0);s_F_vortexbottom = ‐1i*s_Gamma/(2*pi)*log(s_z + s_z0);

%Doubletsyms s_F_doublet %Complex potential for doublet (circular cylinder in uniform flow)s_F_doublettop = s_U*s_r0^2/(s_z ‐ s_z0);s_F_doubletbottom = s_U*s_r0^2/(s_z + s_z0);


s_F = s_F_uniform + s_F_doublettop + s_F_doubletbottom + s_F_vortextop ‐ s_F_vortexbottom; % Complex potentials_W = diff(s_F, s_z); % Complex velocity

Numerical function

matlabFunction(s_F, 'file', 'genfunc_complexpotential', 'vars', [s_U s_alpha s_z s_z0 s_r0 s_Gamma]);matlabFunction(s_W, 'file', 'gnefunc_complexvelocity', 'vars', [s_U s_alpha s_z s_z0 s_r0 s_Gamma]);

Mesh

xmin = ‐2;xmax = 2;x_spacing = 201;ymin = ‐10;ymax = 10;y_spacing = 1001;x = linspace(xmin,xmax,x_spacing); %x‐coordinates

Appendix A



y = linspace(ymin,ymax,y_spacing); %y‐coordinates[X,Y] = meshgrid(x,y); %Calculation area

z = X + 1i.*Y; %Complex definition of calculation area

Numerical Results

F = genfunc_complexpotential(U, alpha, z, z0, r0, Gamma); %Complex potentialW = genfunc_complexvelocity(U, alpha, z, z0, r0, Gamma); %Complex velocity

PSI = imag(F); %Streamlinesu = real(W + 0.0.*z); %x‐component of velocityv = ‐imag(W + 0.0.*z); %y‐component of velocityq2 = u.^2 + v.^2; %Magnitude of velocity squaredCp = 1 ‐ q2./U.^2; %Coefficient of pressure

Calculate lift

%Integrate along bottom first from trailing edge to leading edgexc = (r0 + 0.001); %x‐coordinate of trailing edgeyc = imag(z0); %y‐coordinate of trailing edgeClp = 0;%Calculate matrix index at start x‐y coordinate for complex velocity %mesh ax = round(((x_spacing ‐ 1)/2) + xc/((xmax ‐ xmin)/(x_spacing ‐ 1))); ay = round(((y_spacing ‐ 1)/2) + yc/((ymax ‐ ymin)/(y_spacing ‐ 1)));Cp1 = Cp(ay,ax);

for idx = 0:0.01:pi %Define starting points x0 = xc; y0 = yc; %Define starting Coefficient of pressure Cp0 = Cp1; %Calculate new coordinates based on angle xc = 0 + (r0 + 0.001)*cos(idx); yc = imag(z0) ‐ (r0 + 0.001)*sin(idx); z = xc + 1i*yc; W = genfunc_complexvelocity(U, alpha, z, z0, r0, Gamma); u = real(W); v = ‐imag(W); q2 = u^2 + v^2; Cp1 = 1. ‐ q2/U^2; Cp = 0.5*(Cp0 + Cp1); x1 = xc; dx = x1 ‐ x0; y1 = yc; dy = y0 ‐ y1; dl = sqrt(dy^2 + dx^2); Clp = Clp + dl*Cp;

end%Integrate along top from leading edge to trailing edgexc = (‐r0 ‐ 0.001); %x‐coordinate of leading edgeyc = imag(z0); %y‐coordinate of leading edge

for idx = pi:0.01:2*pi %Define starting points x0 = xc; y0 = yc; %Define starting Coefficient of pressure Cp0 = Cp1; %Calculate new coordinates based on angle xc = 0 + (r0 + 0.001)*cos(idx); yc = imag(z0) + (r0 + 0.001)*sin(idx); z = xc + 1i*yc; W = genfunc_complexvelocity(U, alpha, z, z0, r0, Gamma); u = real(W); v = ‐imag(W); q2 = u^2 + v^2; Cp1 = 1. ‐ q2/U^2; Cp = 0.5*(Cp0 + Cp1); x1 = xc; dx = x1 ‐ x0; y1 = yc; dy = y0 ‐ y1; dl = sqrt(dy^2 + dx^2); Clp = Clp + dl*Cp;end

Cl(iterate + 1) = Clp;



end

xaxis = linspace(0,9,(eval + 1));Cl

Cl =

Columns 1 through 7

‐4.9519 0.5475 0.6672 0.6333 0.6082 0.5908 0.5781

Columns 8 through 14

0.5686 0.5612 0.5553 0.5505 0.5466 0.5432 0.5403


0.5378 0.5357 0.5337 0.5321 0.5305 0.5292 0.5280


0.5268 0.5258 0.5249 0.5240 0.5232 0.5225 0.5218


0.5212 0.5206 0.5201 0.5196 0.5191 0.5186 0.5182


0.5178 0.5174 0.5170 0.5167 0.5164 0.5161 0.5158


0.5155 0.5152 0.5150 0.5147

Plot

scrsz = get(groot, 'ScreenSize');figure('Position',[1 scrsz(4)/2 1000 900])plot(xaxis,Cl,'‐ok','MarkerFaceColor','k')xlabel('Seperation')ylabel('Coefficient of Lift')title('Coefficient of Lift for Rotating Cylinder in Ground Effect')



Published with MATLAB® R2015a

http://www.mathworks.com/products/matlab/

11/29/2015 groundeffectairfoil_pressureintegration

file:///U:/Mech%20342/Honors%20Project/html/groundeffectairfoil_pressureintegration.html 1/4

Contents

Parameters



Numerical function

Mesh

Numerical Results

Plot

Calculate lift

clc;clear;

Parameters

alpha = 0.0*pi/180; % Angle of attackU = 1; % Freestream velocityP = 1; % Freestream pressurer0 = 0.2; % Radius of doubletz0 = 0.0 + 1i*1; % Origin of featureGamma = ‐1; % Circulation


syms s_Usyms s_alphasyms s_r0syms s_z0syms s_Gammasyms s_zsyms s_Fsyms s_W

%Uniform flowsyms s_F_uniform %Complex potential for uniform flows_F_uniform = s_U*exp(‐1i*s_alpha)*s_z;

%Point vortexsyms s_F_vortex %Complex potential for vortexs_F_vortextop = ‐1i*s_Gamma/(2*pi)*log(s_z ‐ s_z0);s_F_vortexbottom = ‐1i*s_Gamma/(2*pi)*log(s_z + s_z0);

%Doubletsyms s_F_doublet %Complex potential for doublet (circular cylinder in uniform flow)s_F_doublettop = s_U*s_r0^2/(s_z ‐ s_z0);s_F_doubletbottom = s_U*s_r0^2/(s_z + s_z0);


s_F = s_F_uniform + s_F_doublettop + s_F_doubletbottom + s_F_vortextop ‐ s_F_vortexbottom; % Complex potentials_W = diff(s_F, s_z); % Complex velocity

Numerical function

matlabFunction(s_F, 'file', 'genfunc_complexpotential', 'vars', [s_U s_alpha s_z s_z0 s_r0 s_Gamma]);matlabFunction(s_W, 'file', 'gnefunc_complexvelocity', 'vars', [s_U s_alpha s_z s_z0 s_r0 s_Gamma]);

Mesh

xmin = ‐2;xmax = 2;x_spacing = 201;ymin = ‐2;ymax = 2;y_spacing = 101;x = linspace(xmin,xmax,x_spacing); %x‐coordinatesy = linspace(ymin,ymax,y_spacing); %y‐coordinates[X,Y] = meshgrid(x,y); %Calculation area

z = X + 1i.*Y; %Complex definition of calculation area

Appendix B



Numerical Results

F = genfunc_complexpotential(U, alpha, z, z0, r0, Gamma); %Complex potentialW = genfunc_complexvelocity(U, alpha, z, z0, r0, Gamma); %Complex velocity

PSI = imag(F); %Streamlinesu = real(W + 0.0.*z); %x‐component of velocityv = ‐imag(W + 0.0.*z); %y‐component of velocityq2 = u.^2 + v.^2; %Magnitude of velocity squaredCp = 1 ‐ q2./U.^2; %Coefficient of pressure

Plot

scrsz = get(groot, 'ScreenSize');figure('Position',[1 scrsz(4)/2 1000 900])cont = linspace(‐6,1,30);contourf(X,Y,Cp,cont,'LineColor','none')colormap defaulthbar = colorbar;title(hbar, 'Coefficient of Pressure');hold on;contour(X,Y,PSI,80,'‐‐k','LineWidth',1);contour(X,Y,PSI,[0.0 0.0],'‐r','LineWidth',2);hold offaxis equal;xlabel('x');ylabel('y');



Calculate lift

%Integrate along bottom first from trailing edge to leading edgexc = (r0 + 0.001); %x‐coordinate of trailing edgeyc = imag(z0); %y‐coordinate of trailing edgeCl = 0;%Calculate matrix index at start x‐y coordinate for complex velocity %mesh ax = round(((x_spacing ‐ 1)/2) + xc/((xmax ‐ xmin)/(x_spacing ‐ 1))); ay = round(((y_spacing ‐ 1)/2) + yc/((ymax ‐ ymin)/(y_spacing ‐ 1)));Cp1 = Cp(ay,ax);

for idx = 0:0.01:pi %Define starting points x0 = xc; y0 = yc; %Define starting Coefficient of pressure Cp0 = Cp1; %Calculate new coordinates based on angle xc = 0 + (r0 + 0.001)*cos(idx); yc = imag(z0) ‐ (r0 + 0.001)*sin(idx); z = xc + 1i*yc; W = genfunc_complexvelocity(U, alpha, z, z0, r0, Gamma); u = real(W); v = ‐imag(W); q2 = u^2 + v^2;



Cp1 = 1. ‐ q2/U^2; Cp = 0.5*(Cp0 + Cp1); x1 = xc; dx = x1 ‐ x0; y1 = yc; dy = y0 ‐ y1; dl = sqrt(dy^2 + dx^2); Cl = Cl + dl*Cp;

end%Integrate along top from leading edge to trailing edgexc = (‐r0 ‐ 0.001); %x‐coordinate of leading edgeyc = imag(z0); %y‐coordinate of leading edge

for idx = pi:0.01:2*pi %Define starting points x0 = xc; y0 = yc; %Define starting Coefficient of pressure Cp0 = Cp1; %Calculate new coordinates based on angle xc = 0 + (r0 + 0.001)*cos(idx); yc = imag(z0) + (r0 + 0.001)*sin(idx); z = xc + 1i*yc; W = genfunc_complexvelocity(U, alpha, z, z0, r0, Gamma); u = real(W); v = ‐imag(W); q2 = u^2 + v^2; Cp1 = 1. ‐ q2/U^2; Cp = 0.5*(Cp0 + Cp1); x1 = xc; dx = x1 ‐ x0; y1 = yc; dy = y0 ‐ y1; dl = sqrt(dy^2 + dx^2); Cl = Cl + dl*Cp;end

Cl

Cl =

0.5908

Published with MATLAB® R2015a

http://www.mathworks.com/products/matlab/

RMIT School of Aerospace, Mech. & Manufacturing Eng. AERO2365 Thesis / Project 2007

AERO2365 Thesis Paper Analysis of Wing in Ground Effect using Potential Flow Theory

Sridhar Ravi1 & A/Prof. Hadi Winarto1 1School of Aerospace, Mech. & Manufacturing Eng., RMIT University, VIC, 3001, AUSTRALIA

Abstract An aircraft, flying close to ground or water surface experiences remarkable increase in Lift and a substantial decrease in Drag. WIG (Wing in Ground Effect) crafts put this phenomenon to use, having very small aspect ratio wings and flying close to ground; they are constantly under the influence ground effect, making them very effective. This thesis analyses the ground effect phenomenon through a theoretical perspective by applying Potential Flow Theory, in order to obtain a qualitative and quantitative idea of its implications the aerodynamic attributes of a wing. Thin Airfoil theory and Numeric Lifting Line theory were applied to study Chord-wise and Span-wise ground effect respectively. Results indicate; influence of chord wise ground effect, is maximum at low angles of attack and the phenomenon doesn’t always contribute increase the overall lift coefficient. On the contrary, influence of span-wise ground effect increases with increase in angle of attack and the induced drag almost halves when height above ground is about 10% of chord length. It was also concluded that the Lift to Drag ratio nearly doubles when flying very close to ground, which is a substantial increment.

Introduction The thesis aims at using basic ideas of theoretical aerodynamics to explain ground effect phenomenon. By making some intelligent approximations and extensions, a relatively accurate idea of the degree of influence ground effect has on performance of a representative wing can be gauged. Work conducted and results from this thesis project would not only interest people interested in WIG craft construction but also aerodynamicists, the thesis gives very realistic predictions of variations of parameters like Cl, Cd of a wing in and out of ground effect. Ground effect in aircraft can be divided into span-wise and chord-wise ground effect. Chord-wise ground effect is primarily responsible for increase in lift[1]. Though this fact is known, there isn’t a clear idea as to the reasons and the magnitude of lift increase that is experienced by an aircraft when in acute ground effect. A successful explanation and investigation of chord-wise ground effect would also greatly assist in better prediction of an aircraft’s take off and landing performance. Span-wise ground effect is responsible for decrease in induced drag.

Chord-Wise Ground Effect Thin Airfoil Theory is the best possible means to address Chord-wise ground effect. Thin airfoil theory being the most basic and fundamental tool for chord-wise aerodynamic analysis of infinitely long wings serves as an ideal tool for this analysis. Concepts of Thin Airfoil Theory are well established, hence, this section will directly start by presenting the general solution of a flat plate modeled with “n” lumped vortices and “n” control points, out of ground effect. It should be noted that for the case of this thesis, the analysis is conducted on a flat plate. The general equations for “n” panels. The equations are as follows

1 2 n

1 1 1 2 1

1 2 n

2 1 2 2 2

1 2 n

1 2

F o rC 1; s in ( )2 2 2

F o rC 2; s in ( )2 2 2

..

..

..

F o rC n ; s in ( )2 2 2

n

n

n n n n

Ѓ Ѓ Ѓ Vr r r

Ѓ Ѓ Ѓ Vr r r

Ѓ Ѓ Ѓ Vr r r

απ π π

απ π π

απ π π

+ + ………… = − ×× × × × × ×

+ + ………… = − ×× × × × × ×

+ + ………… = − ×× × × × × ×

Where; “rij” refers to the distance between the “j” vortex and the “i” control point. On solving the “n” simultaneous equations for the various “gamma” we can get the solution for the entire inclined flat plate. A computer program that provides a solution for the Cl and circulation distribution of an inclined flat plate has been scripted and thoroughly analysed through its results. Consequently, suitable derivations were made such that the general thin airfoil theory for a flat plate “flying” at infinite height, would be expanded to involve ground effect. That is, once the derivation is completed, the lift coefficient should not only depend on the angle of attack but also on altitude of flight. The exact concept can be extended to involve ground effect as follows; Starting off with the basic case of a single lumped vortex at quarter chord point a corresponding control point at three-quarter chord point at an angle alpha to the on coming flow, we place a mirror of the same, at a distance 2h from the trailing edge. The setup is depicted in the figure below;

Figure 1. Single Lumped vortex in Ground Effect The control point has not been placed on the image chord as it is unimportant. By placing the above setup in Cartesian coordinates, we see that if the coordinates of the vortex in the chord is given as

1Γ =(x,y) then the coordinates of its mirror vortex can be given as

1MΓ =(x,-y). The strength of each vortex can be given as the

following 1 1MΓ =−ΓApplying Neumann boundary condition and summing up velocities normal to the chord and equating to zero will yield;

Where; 11'W is the velocity induced by the mirror vortex at control

point “C”. β is the angle between

11'W and the horizontal axis, or ground. Now, when flying at a height “h” above the ground, the strength of vortex is estimated to be;

11' 11sin( ) sin( ) 0W W Vβ α α× − − + × =

1 1'1

1 1'

sin( ) 2sin( )

r rVr r

α πβ α×

Γ = − × × × ×× − −

Appendix C

Analysis of Wing in Ground Effect

22( ) 1 yyb

⎛ ⎞Γ = Γ −⎜ ⎟⎝ ⎠

( ) ( )eff n i ny yα α α= −

1( ) ( )2n ny V c clΓ = ⋅ ⋅ ⋅

( )input old new oldDΓ = Γ + Γ −Γ

The general equation for solving the for a chords divided into “n” panels at a height “h” to the ground inclined at an angle “alpha” is given as follows;

Where; “Self” refers to “rij” is the distance between the “j” vortex and the “i” control point. “Image” refers to “rij’” is the vertical component with respect to the chord, of the normal of the distance between the image of the “j” vortex and the “i” control point. On solving the “n x n” matrix above, the strengths of all the vortices are obtained. These can be plotted depending on their location on the chord. This plot depicts the overall chord wise circulation distribution when the wing is flying at a height “h” above the ground, measured from its trailing edge. It is possible to solve for up to three lumped vortex with its corresponding control points and an image at a distance “h” above, by hand. After which, it becomes very tedious and the susceptibility to make an error greatly increases. . It is hence a necessity to develop a computer program that would perform the same task with much greater efficiency. These new equations that address the chord-wise ground effect have been incorporated into the existing program and a new program is created. This new program carries out calculations of the wing, represented, as a flat plate in ground effect and out of ground effect. This would give the user a comparison between the two scenarios. The program demands the input of angle of attack, altitude of flight and number of panels from user. Once the lift coefficient of the wing has been calculated, a plot of the circulation distribution over the wing chord is generated. The plot shows locations on the wing chord where the influence of ground effect is most experienced and otherwise. Using the program extensive data has been gathered in order to effectively show the variation of different aerodynamic parameters with altitude of flight. The most important analysis that was conducted was the variation of lift coefficient with angle of attack at various altitudes of flight and variation of lift coefficient with altitudes at different angles of attack; this is unlike out of ground effect case, where Lift Coefficient is solely dependant on angle of attack. Span-Wise Ground Effect Span-wise ground effect is responsible for the significant reduction in induced drag experienced by the aircraft[2]. Similar to the manner in which chord-wise ground effect was addressed, span-wise ground effect too would be analysed in a two fold process. The first objective is to theoretically simulate a simple wing using lifting line theory and study the different aerodynamic attributes of the same. As the simulation of a wing using concepts of Lifting Line Theory, needs to be done numerically; MATLAB, is chosen as the most suitable due to its relatively simple command line as wells as its stability and accuracy Numeric Lifting line theory starts off with assuming a representative wing. A plain rectangular wing is assumed. The wing should be inclined at an angle “alpha” to the on coming flow. A suitable wing section has been assumed and a suitable lift distribution is also taken. For simplicity, an elliptical lift distribution can be assumed and at low angles of attack. As a matter

of convention, the wing is to be placed aligned with the horizontal axis and the tips reaching b/2 and –b/2 of the horizontal axis respectively. The mathematical equation to plot the elliptical lift distribution over

the wing is given by;

Where; ( )yΓ = Circulation at each specific point (y) along the spa of the wing Γ =The maximum circulation i.e. the circulation at y=0 in the center of the wing b= wing span, as earlier assumed. The wing is then divided into a number of span wise sections. The greater the number of divisions, the better it would be for approximations and plotting the lift distribution over the wing. On diving the wing into “k” sections; they would yield “k+1=n” points along the span[3]. The equation to calculate the induced angle of attack at every position all over the wing is given by; On performing the above integration an induced angle of attack is obtained for each “n” point on the wing. In the induced angle of attack equation above “ys” refers to each point on the wing. Hence, the initial circulation distribution needs to be integrated piece-wise to calculate the influence of each section “k” on every point. This effective angle of attack is the angle at which the air is actually hitting that specific point on the wing as seen by that point. On comparing the effective angle of attack to the airfoil section’s Cl Vs AOA curve; the new Cl for each point can be got. From the new found Cl for each point, its respective new circulation can also be calculated, using the relation from the Kutta- Joukowski theorem; c= chord at the specific point (in the case of no taper, the chord is constant all through the span of the wing) Cln= is the new lift coefficient got in the previous step V= velocity of the airflow The circulation at each point ( ( )nyΓ ) is compared to the initially assumed gamma based on the elliptical lift distribution, for each point. “If the two values don’t match to less than 5 decimal figures then a new input for the circulation distribution is generated. The new input is generated using the relation; “D” is a suitably chosen step size. Typically the value of D is about 0.05 This cycle needs to go on until; the stipulated accuracy is reached at each point on the wing.[4] The Lift coefficient as well as the induced drag can be easily calculated. The second half of this segment of the project deals with simulation of ground effects using Lifting line theory through an extension of the existing knowledge from the previous step. As the simulation of ground effect is achieved by extending the Lifting Line Theory, there aren’t any specific equations that can be used, hence, the equations need to be derived based on the basic ideas. Simulating span-wise ground effect follows a near ditto procedure as adopted in the chord-wise ground effect case. If a wing is flying at an altitude “h” above the ground, by placing the mirror image of the wing at a distance 2h below, the ground effect boundary condition can be enforced. The total induced angle of attack will be a sum of two contributions. One being the induced angle of attack due to the

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1 2 n

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ForC1; self image self image self image sin( )

ForC2; self image self image self image sin( )......

ForCn; self image self image self image si

Ѓ Ѓ Ѓ V

Ѓ Ѓ Ѓ V

Ѓ Ѓ Ѓ V

α

α

+ + + +………… + =− ×

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circulation on the wing influencing every point on the wing, second is the influence of the circulation present on the image. The coordinates of every point on the image would be known as we know the coordinate of the wing flying at a height “h” above the ground with a span of “b”. the circulation plot of the image would also be just the image, hence, dЃ/dy for a specific point on the image will be equal to –( dЃ/dy) for its respective mirror point on the wing. Figure 2. Front view of Span-wise ground effect Therefore, the induced angle of attack due to the circulation present on the image at a certain point “i” on the wing can be estimated as follows. rij’ is the normal component with respect to the wing, of the perpendicular to the distance between point “i” on the wing and every point “j” on the image. dЃ/dy for every point “j’” on the image will be equal to –dЃ/dy of “j” located on the wing. It is vital that the direction of the induced velocity and hence the induced angle of attack is calculated with care, as at certain points of the wing there is a negative induced velocity and positive else where due to the strength, orientation and position of the circulation on the image. Once the ground effect is simulated and the program is running smoothly; then a comparative study between the aerodynamic characteristics of the simple wing derived in the first half of the project and the wing under adverse ground effect can be done. This can give a very useful insight on the influence of ground effect on the different parameters of the representative wing. The change in the Lift produced and the rate of change of drag to the distance from the ground the wing is flying can all be studied and quantified. As a final step both Span-wise and Chord-wise ground effect programs were amalgamated in the following manner;

Chord-Wise Results For Chord-Wise ground effect analysis, a flat plate with chord length unity and modelled using 20 panels is assumed. IGE and OGE stand for In Ground Effect and Out of Ground Effect respectively. Below are the results obtained from the analysis conducted for chord-wise ground effect;

Figure 3. CL(IGE/OGE) variation with (h/c)

Figure 4. CL(IGE/OGE) variation with AOA

One of the striking aspects noticeable in the graphs is that, the relative increase in the lift coefficient between IGE and OGE is profound at low angles of attack. The reason IGE/OGE is greatest when angle of attack is low is that; at low angles of attack, the distance between image vortex and control point is minimum. Hence, the image vortex has maximum influence and in turn greatly affects the overall lift coefficient. We also see that the (IGE/OGE) tends to unity as (h/c) increases. Indicating that, the results break down to OGE condition at higher altitudes, as expected.

Figure 5. Induced velocity of mirror vortex in same direction as induced velocity due to vortex on wing In the figure above the normal component of the induced velocity by image vortex, with respect to the flat plate is in the same direction as to the induced velocity to vortex on flat plate. This would lead to an increased induced velocity at the control point, i.e. contribution of vortex on flat plate as well as image. Hence, it leads to a reduction in lift coefficient. Therefore, Ground effect doesn’t always lead to an increase in overall Cl. Span-Wise Results A rectangular wing with aspect ratio 6 is taken. A lift curve slope of 5.7723 is assumed, to be coherent with NACA0015 airfoil section. It should be noted that all analysis was done in the linear region of the lift curve, i.e. maximum angle of 10˚ was used. A

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14

b

ib ij

ddy dy

V rα

π−

Γ

=× × ∫


lot of data and plots were generated to yield a number of very interesting results;

Figure 6 Cdi(OGE/IGE) variation with (h/c)

Figure 7. Cdi(OGE/IGE) variation with AOA In figure 7 we see that the induced drag coefficient seems to drop considerable as (h/c) decreases. At about 0.4 of chord we see that Cdi IGE is only half of CdiOGE. Though this is slightly different from Wieselsberger’s prediction, the relationship between Cdi(IGE/OGE) depends on the aspect ratio of the wing too. Hence, Wieselsberger’s prediction might be valid for a different aspect ratio wing. We also note that angle of attack has no influence on Cdi(OGE/IGE), at a specific height. Lift to drag ratio at various angles of attack at different distances from the ground was calculated using the program yielded the following plot. In the graphs, both (L/D)IGE and (L/D)OGE are plotted together.

Figure 8. (L/D) variation with AOA There is a remarkable increase in the lift to drag ratio as (h/c) decreases. We can see that (L/D) almost doubles at (h/c)=0.1 at alpha ten degrees. Though it is quite difficult to maintain a (h/c) of 0.1, it shows the sort of improvement one can get due to span wise ground effect Span-wise Chord-wise Combined Application On combing the two programs, a new set of analysis was conducted using the following properties; a flat plate rectangular wing with aspect ratio 6 is taken. Thirty lumped vortex panels were used to model the flat plate. Number of iterations was set to 40. From the results obtained, the variation of CL(IGE/OGE) was very similar to the plots in Figure 3&4. Therefore, unlike in span-wise ground effect analysis when CL(IGE/OGE) did not vary with AOA, in this case it did. Then most interesting plots obtained are the Cdi(OGE/IGE) plots, as can be seen below;

Figure 9. Cdi(OGE/IGE) as a function of (h/c) @ different AOA

Figure 10. Cdi(OGE/IGE) as a function of AOA @ different (h/c)

Figure 11. (L/D) as a function of AOA at different (h/c) Under the combined influence of span-wise and chord-wise ground effect the (L/D) ratio greatly increases, however, the Cdi(OGE/IGE) is not a function of AOA as well as (h/c) therefore, it is not possible to make a comparison of the results above with Weiselsberger’s prediction. Conclusion Overall lift coefficient of the flat plate exponentially increases with decrease in distance from ground. How,wever, it was a striking revelation that the influence of span wise ground effect remained constant on the Cl and Cdi at a given (h/c) over a range of AOA. Induced Drag coefficient drops with decrease in proximity to ground due to span-wise ground effect.On the combined application of span-wise and chord-wise ground effect, Cl(IGE/OGE) and Cdi(OGE/IGE) become a function of AOA as well as (h/c). (L/D) ratio increases exponential with decrease in (h/c) and increase in AAO. It is not possible to draw a comparison with Weiselsberger’s prediction as it ignores the relationship between Cdi(OGE/IGE) and Angle of Attack. References 1. Wing In Ground effect aerodynamics. 1996-2003

[cited 17/8/07]; Available from: http://www.se-technology.com/wig/index.php.

2. Ariyur, K.B., Prediction of Dynamic Ground Effect Forces on Fixed Wing Aircrafts. 2005: U.S.A. p. 1.

3. John D.Anderson, j., Fundamentals to Aerodynamics. Third ed. 2001: McGraw Hill.

4. Karanjgaokar, N., Numeric Lifting Line Theory. 2003, Indian Institute of Technology: Mumbai. p. 4.

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Justin Weinmeister Honors Project MECH 342