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MIET2394 – Computational Fluid Dynamics [Assignment C]

Effect of lift and drag on plane wings flying in formation

Jacob Rowe, 3379909

16 October, 2014

1 INTRODUCTION

During World War 2 P-51’s flew in a V-shaped formation to be able to increase the view range of the squad to spot a

surprise attack from an enemy. This involved one wing flying directly behind another and the flow off the first wing

affecting the following wing.

The correlation between the location of the following wing in relation to the forward wing and the lift and drag on the

following wing is being studied by altering the distance that the wing is behind the forward wind and also how far the

trailing wing is below the horizontal level of the forward wing.

2 METHODOLOGY

To achieve the desired results, the airflow over a forward and trailing wing is being studied with ANSYS 14.5 with

conditions that simulate the planes at cruise speed and altitude, whilst also having to determine a mesh size which would

be the most economical to use in relation to computing time and still producing results which are accurate enough for

the study being done on the model.

2.1 Geometry

The 2 dimensional representation of the wing was constructed based on a set of co-ordinates mapping the form of the

aerofoil which were provided by Airfoil tools [1] (Table 1). The 40 points gave the model a maximum height of

0.070094m, a minimum height of -0.044088m and a length of 1m. These dimensions’ match that of the tip aerofoil of

the P-51. The entire wing is not being modelled because the increased complexities of the model would increase the

time needed for computations and while the results would be more specific, using an extruded tip aerofoil will produce

results that are not exact however it will provide results that are indicative of true life with a lower computation time.

The aerofoil of the tip has been scaled down in size from the real life chord length to be smaller and more practical to

work with in term of reducing computation time and the results would accurately scale up to the full, life sized situation.

Figure 1 depicts the sketch of the aerofoil with the relevant nodes included. The aerofoil was then extruded to have a

length of 1.5m. The wing is within a defined fluid domain with the dimensions presented in Table 2.

Following the initial simulation, a second aerofoil was added to the setup. The location of the second wing is defined

by the leading edge of the second wing, relative to the leading edge of the first wing. The second aerofoil was also

extruded to a length of 1.5m. Subsequent tests were conducted with the location of the second wing varying its location

in the x and y direction. Figure 2 shows a wire frame model of the extruded wing to be used as the bases of the testing.

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Table 1: Data set for construction of aerofoil

x(m) y(m)

0.005585 0.008086

0.022214 0.017194

0.049516 0.026597

0.086881 0.035767

0.133474 0.044323

0.188255 0.051949

0.25 0.058605

0.317329 0.063939

0.38874 0.067552

0.462635 0.070094

0.537365 0.0689

0.61126 0.062002

0.68267 0.050474

0.75 0.037072

0.811745 0.025099

0.866526 0.016164

0.913119 0.009482

0.950484 0.004793

0.977786 0.002019

1 0

0 0

0.005585 -0.00724

0.022214 -0.01418

0.049516 -0.0199

0.086881 -0.0253

0.133474 -0.03053

0.188255 -0.03502

0.25 -0.03852

0.317329 -0.04144

0.38874 -0.04293

0.462635 -0.04409

0.537365 -0.04354

0.61126 -0.03898

0.68267 -0.03016

0.75 -0.02057

0.811745 -0.01233

0.866526 -0.00633

0.913119 -0.00297

0.950484 -0.00144

0.977786 -0.00061

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Figure 1: Coordinate representation of base model

Table 2: Dimensions of the fluid domain.

Figure 2: A wire frame of the model being tested

Width 9.5m

Height 4.5m

Depth 5m

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2.2 Meshing

Mesh independence testing was not conducted due to the large amount of computational processing time that would be

required therefore, an appropriate sized mesh was used selected based on knowledge of meshing requirements. The bulk

mesh type around the wing is largely rectangular (Figure 3), while the mesh type over the walls of the system are

triangular (Figure 4). The mesh type around the boundary layer of the wing is also a triangular mesh (Figure 5). The

mesh was set to be expanding in size at a growth rate of 1.1 from the wing. This ensures that the mesh is finer near the

wings where a greater level of accuracy in the area of interest and that the computation time is not drastically increased

by running fine calculations away from the car body where it is not needed. This method of meshing produced a mesh

with an element count of 3,053,736 from 740,862 nodes. The mean skewness of the mesh is 0.2098 with a standard

deviation of 0.1167 which are acceptable values to work with.

Figure 3: Rectangular mesh around the car body

Figure 4: Triangular mesh over system wall

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Figure 5: Triangular mesh around Boundary Layer

2.3 Model Setup

The flow within the 2 dimensional cross section is incompressible and most of it is laminar. Therefor the flow can be

described on the most part by the 2D Navier-Stokes dimensionless equations.

u

x+v

y= 0

𝑢u

x+ 𝑣

u

y= −

p

x+

1

𝑅𝑒(

2𝑢

x2+

2u

y2)

𝑢v

y+ 𝑣

v

y= −

p

y+

1

𝑅𝑒(

2𝑣

x2+

2v

y2)

p = dimensionless pressure of the fluid

The properties of the model were set prior to the test and they were custom values not assumed by the CFD software.

The fluid being used for the flow is air at 25°C which is the standard temperature for running simulations involving air.

The temperature for this simulation has not been altered because the heat transfer is not being studied and does not affect

the other properties of the simulation. The fluid domain has a velocity of 116.23 m s-1 at the inlet with a relative pressure

of zero at the outlet. Other material properties for the fluid are defined in Table 3. The wing has a no slip boundary layer

applied while the walls of the fluid domain are frictionless as to not interfere with the resulting flow over the wing as.

Table 3: Material properties of fluid domain

Density [kg m^3] 0.9093

Dynamic Viscosity [Ns/m^2] 1.694

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3 RESULTS AND DISCUSSIONS

3.1 The single wing model

The single car model was simulated using two different turbulence models, K-Epsilon and K-Omega. The two different

models produced similar results however there are noticeable differences between the two contour plots. As shown in

Figure 7, the models show that in areas with localized high and low y-plus values are aligned. This is important because

it proves that there is consistency and accuracy within the models without having being forced to perform physical

experimentation to confirm that the basic theory in this situation is correct.

However, there are specific differences in the results produced by the two models. The K-Epsilon turbulence

model has a larger range of y-plus values. This model has a higher maximum and also a lower minimum value when

compared to the K-Omega plot. In the areas of high y-plus values on the model, the K-Epsilon model shows a more

specific, higher contour plot. It shows the same high values as the K-Omega model does, however, the K-Epsilon model

shows some higher values within the same area. This trend holds true for the areas with a low y-plus value with the K-

Epsilon model producing more specific plots with better defined areas with low y-plus values.

a) b)

Figure 7: Y-Plus contour plot on wing using a) K-Epsilon turbulence model b) K-Omega turbulence model

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The pressure contour plots for the two different models produce results that are almost identical (Figure 8). Both models

are typical and to be expected for pressure systems around wings. The high pressure area located on the leading edge

occurs due to that fact that the air flow imparts a direct normal force due to the direction of the airflow. The extreme

low pressure acting on the top of the wing and in comparison to the higher pressure on the underside of the wing, agrees

with basic theory of lift and flight in relation to wings and the need for a lower pressure above and a higher pressure

below the wing. The similarities in the two models indicates that in relation to pressure, it is irrelevant which turbulence

model is used.

The turbulence kinetic energy contour plot (Figure 9) shows only minimal turbulence for the K-Epsilon

turbulence model at the trailing edge. This is the location that some turbulent flow would be expected to occur although

it would be expected to be to a larger degree. The K-Omega turbulence model produces slightly more turbulence in the

same area and it also extends further behind the trailing edge. This is more typical of what is to be expected from the

flow behind the trailing edge based on knowledge of basic aerodynamics. While there is more turbulent flow in the K-

Omega model, the amount is still quite low. This can be accredited to the symmetry of the aerofoil and also to the low

angle of attack of the wing. The low amounts of kinetic energy produced indicate that there is no separation of flow

from the wing and no re-circulation zone. The turbulent flow produced by the K-Omega is more indicative of what

would be expected.

The prediction that there would be no re-circulation zone based on the turbulent energy is supported by Figure

10 which shows the stream line plot over the wing. The streamline plot shows that the flow over the wing remains

laminar and is only slightly disturbed by the presence of the wing in its flow. This can be accounted for by the

aerodynamic design which includes its gradual increase in thickness and to the small front profile of the wing.

The vector velocity plot (Figure 11) further supports the conclusion that there is no separation of flow from the

wing or any re-circulation zone. The velocity vectors show that the flow remains attached to the wing the entire way

across and behind the wing it re-establishes a similar flow pattern to that of before the flow was disturbed by the wing.

Again there is no noticeable difference between the plots produced by the two models for either the streamline

or velocity vector. Therefore, for these two types of data it does not matter which turbulence model is used. Due to the

large similarities in the results produced by the two models and the differences in the turbulent kinetic energy contour

plots, the K-Omega will be used for the simulations involving two wings.

a) b)

Figure 8: Pressure contour plot around wing using a) K-Epsilon turbulence model b) K-Omega turbulence model

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a) b)

Figure 9: Turbulence Kinetic Energy Plot over wing using a) K-Epsilon turbulence model b) K-Omega turbulence

model

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a) b)

Figure 10: 3D Velocity Streamline plot over wing using a) K-Epsilon turbulence model b) K-Omega turbulence model

a) b)

Figure 11: Velocity Vector plot over wing using a) K-Epsilon turbulence model b) K-Omega turbulence model

Important data to obtain from these simulations are the coefficients of lift and drag. These coefficients can be calculated

using the following equations respectively:

𝐶𝐿 =𝐹𝐿

0.5 ∗ 𝜌 ∗ 𝑈2 ∗ 𝐴 (1)

𝐶𝐷 =𝐹𝐷

0.5 ∗ 𝜌 ∗ 𝑈2 ∗ 𝐴 (2)

CL = Coefficient of Lift

CD = Coefficients of Drag

FL = Force in the positive Y direction

FD = Force in the positive X direction

Ρ = Air Density

U = Velocity

A = Frontal Area

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Using the values predefined for these simulations:

Ρ = 0.9093 kg m-3

U = 116.23 m s-1

A = (0.070094 + 0.044088) m * 1.5 m

While the following values were calculated though the software:

FL = 2196.84 N

FD = 150.729 N

The coefficient of lift and drag were able to be calculated as shown in table 4.

Table 4: Calculated coefficients of lift and drag for K-Epsilon and K-Omega turbulence models

The results calculated align with what is to be expected. The coefficient of lift is far higher than the coefficient of drag.

The coefficient of lift is high due to the angle of attack and the fact that the shape of the aerofoil for the wing was

designed to have a positive coefficient of lift. Logic tells us that increasing the angle of attack would directly increase

the coefficient of lift because the Lift force would greatly increase which would far outweigh the negative effects of

increasing the frontal area, however this is not a part of this study. Relative to the coefficient of lift, the coefficient of

drag is low due to the sleek design of the aerofoil. With a basic understanding of aerodynamics, it is possible to accept

that these results would be correct although it is not possible to confirm that without practical testing.

3.2 The double wing model

The model was altered with the addition of a second wing directly behind the leading wing at a distance of two chord

lengths (2m) between the leading edges of both wings. The trailing wing has the same dimensions as the forward wing

and is also at an angle of attack of 3°. Figure 12 shows the y-plus contour plot of the wings from the top, while Figure

13 shows the contour plot from the underside. The top of the wing has a range of higher y-plus values than on the

underside. Although this difference is not extremely large, it still indicates that there is a higher velocity airflow over

the wing when compared to that on the underside. The trailing wing has a slightly lower y-plus values on its underside

which indicates that the velocity of the flow at this point would be lower resulting in a slightly lower pressure and thus

decreasing the coefficient of.

Cl Cd

K-Epsilon 2.081608 0.146105

K-Omega 2.087775 0.143246

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Figure 12: Y-Plus contour plot of the top of the wings

Figure 13: Under view of the Y-Plus contour plot of wing

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The pressure contour plot for the two wings (Figure 14) shows that there is a lower pressure area under the wing of the

trailing wing when compared to that of the leading wing. Due to the disturbed flow coming off of the leading wing, this

lower pressure will result in a lower lift force and thus a lower lift coefficient. The smaller low pressure area above the

wing will result in there being slightly more pressure acting on the top side of the wing lowering the overall lift force

and therefore lowering the coefficient of lift further. The flow from the first wing also increases the pressure acting

directly on the upper side of the wing further lowering the coefficient of lift by reducing the net lift force. This pressure

contour plot demonstrates clearly that every aspect of the pressure on the wing is affected by the differences in the flow

coming off the leading wing when compared to the leading wing where the airflow has not been disturbed before

interacting with the wing.

The turbulent kinetic energy plot for the two wings (Figure 15) demonstrates some vital information relating to the

stability of the airflow going over the second wing. It clearly demonstrates that the flow the second wing is receiving

off the leading wing has been disturbed and is slightly turbulent when it reaches the wing. This supports the conclusions

drawn about the y-plus and pressure being affected by the flow of the leading edge because the flow is clearly not as

laminar as the flow for the leading wing.

The lift and drag of the second wing are drastically different to that of the leading wing, and intern the two different

wings will have varying coefficients of lift and drag values. The lift and drag for the trailing wing are calculated in the

same manner as for the leading wing. The lift and drag forces are obtained by calculations performed by the software

internally and the values obtained for the trailing wing are:

Figure 14: Pressure contour plot around wing

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Figure 15: Turbulence Kinetic Energy plot around spoiler

Fl = 639.996 N

Fd = 141.049 N

This calculation resulted I the following results:

CD = 0.6082 CL = 0.13405 These results differ to that of the leading wing with:

Fl = 2196.84 N

Fd = 150.729 N

This calculation resulted I the following results:

CD = 2.0878 CL = 0.1432

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The Coefficients of lift and drag for the trailing wing were clearly affected by the flow from the leading wing. With this

being confirmed, it is now viable to continue testing with alternate locations for the trailing wing to determine the most

efficient flying arrangement for formation flying.

3.3 Car-spoiler model design variations

The variation of the location of the trailing wing provides us the opportunity to analyse the correlation between the

location of the second wing and its coefficients of lift and drag. By comparing the results of the simulations run with

differing distances behind (x direction) the leading wing and distances below (y direction) the leading wing it will be

possible to plot a trend line with an equation that is reasonably indicative of what can be expected of the different

locations.

The first set of simulations that were run placed the trailing wing directly behind the leading wing (y=0) with distances

behind the leading wing of 1.5m, 2m, and 2.5m (x=1.5, 2, 2.5). The lift and drag coefficient are calculated in the same

way as prior and produced the results in Table 5.

Table 5: Distance behind the leading wing with corresponding CL and CD

These results were then used to create graphs demonstrating the correlation between the location of the trailing wing

and the coefficients of lift (Graph 1) and drag (Graph 2).

Graph 1

x (m) Cl Cd

1.5 0.608223 0.134046

2 0.636423 0.126922

2.5 0.673616 0.125463

0.13

0.135

0.14

0.145

0.15

0.155

0.16

0 0.5 1 1.5 2 2.5

Co

effi

cien

t o

f Li

ft

X Distance

Cl for Trailing wing location of y=0

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Graph 2

From these graphs it is possible to interpolate functions that can relatively accurately calculate the coefficient of lift or

drag dependant on the given x distance. Based upon a third order polynomial extrapolation of the data, the equation for

the coefficient of drag is:

𝐶𝐷 = 0.0113 ∗ 𝑥2 − 0.0539 ∗ 𝑥 + 0.1894

While the equation predicting the coefficient of lift is:

𝐶𝐿 = 0.018 ∗ 𝑥2 − 0.0066 ∗ 𝑥 + 0.5776

The second set of simulations that were run placed the trailing wing slightly below the leading wing (y=0.25) with

distances behind the leading wing of 1m, 1.5m and 2m (x=1, 1.5, 2). The lift and drag coefficient are calculated in the

same way as prior and produced the results in Table 6.

Table 6: Distance behind the leading wing while 0.25m below with corresponding CL and CD

These results produced graphs demonstrating the correlation between the location of the trailing wing and the

coefficients of lift (Graph 3) and drag (Graph 4).

0.124

0.125

0.126

0.127

0.128

0.129

0.13

0.131

0.132

0.133

0.134

0.135

0 0.5 1 1.5 2 2.5 3

Co

effi

cien

t o

f D

rag

X-Distance

Cd for Trailing wing location of y=0

x (m) Cl Cd

1 0.807397 0.143614

1.5 0.996845 0.145685

2 1.061935 0.141912

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Graph 3

Graph 4

The equation for the coefficient of drag is:

𝐶𝐷 = −0.0117 ∗ 𝑥2 + 0.0334 ∗ 𝑥 + 0.1219

While the equation for the coefficient of lift is:

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.5 1 1.5 2 2.5

Co

effi

cien

t o

f Li

ft

X-Distance

Cl for Trailing wing location of y=0.5

0.13

0.135

0.14

0.145

0.15

0.155

0.16

0 0.5 1 1.5 2 2.5

Co

effi

cien

t o

f D

rag

X-Distance

Cd for Trailing wing location of y=0.5

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𝐶𝐿 = −0.2487 ∗ 𝑥2 + 1.0007 ∗ 𝑥 + 0.0554

The third and final set of simulations that were run placed the trailing wing further below the leading wing (y=0.5) with

distances behind the leading wing of 1m, 1.5m and 2m (x=1, 1.5, 2). The lift and drag coefficient are calculated in the

same way as prior and produced the results in Table 7.

Table 7: Distance behind the leading wing while 0.5m below with corresponding CL and CD

These results produced graphs which further demonstrate the correlation between the location of the trailing wing and

the coefficients of lift (Graph 5) and drag (Graph 6).

Graph 5

x (m) Cl Cd

1 0.212759 0.157555

1.5 0.761366 0.14132

2 0.813534 0.133816

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.5 1 1.5 2 2.5

Co

effi

cien

t o

f Li

ft

X-Distance

Cl for Trailing wing location of y=0.5

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Graph 6

The equation for the coefficient of drag is:

𝐶𝐷 = 0.0175 ∗ 𝑥2 − 0.0761 ∗ 𝑥 + 0.2162

While the equation for the coefficient of lift is:

𝐶𝐿 = −0.9929 ∗ 𝑥2 + 3.5794 ∗ 𝑥 − 2.3738

Based on the graphical representation of the data gathered from the simulations it is clear that the location of the

trailing wing in relation to the leading wing is a crucial factor in the lift and drag coefficients for the trailing wing. The

results indicate that the optimal y distance for the trailing wing is 0.25m below the leading wing which relates to a

quarter of the tip chord length below the leading wing. At this height all three coefficients of lift were higher than for

any test at a different y value. The coefficient of lift was highest when x=1.5 however the difference between that

value and the one obtained for x=1 is almost negligible. The coefficients of drag for these varying distances behind the

leading wing have very little difference between themselves and are reasonable values to be flying with.

4 CONCLUSIONS

There were only slight differences between the two different turbulence models except in the results show in the

turbulent kinetic energy contour plot which gave enough evidence to choose the K-Omega turbulence model to be used

for all subsequent simulations.

Based on the results achieved it is clear that for aircraft which are cruising in formation, the location of the plane’s wings

in relation to those of the plane in front of them is a crucial factor in relation to the lift and drag of the aircraft and thus

inadvertently the fuel economy and range of the aircraft. The graphs produced clearly show this relation and also identify

0.13

0.135

0.14

0.145

0.15

0.155

0.16

0 0.5 1 1.5 2 2.5

Co

effi

cien

t o

f D

rag

X-Distance

Cd for Trailing wing location of y=0.5

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that there is an ideal location for the aircraft to fly in to minimise the adverse effects of the turbulent from the leading

aircraft.

5 REFERENCES

[1] P-51D TIP (BL215) AIRFOIL, Airfoiltools, Viewed 30/9/14

<http://airfoiltools.com/airfoil/details?airfoil=p51dtip-il>