This document summarizes a student's final term project analyzing fluid flow around a cannon ball. The student will use computational fluid dynamics (CFD) software to simulate air flow around a spherical cannon ball moving in a projectile motion. Key results to be obtained from the CFD analysis include boundary layer thickness, displacement thickness, momentum thickness, shape factor, drag coefficient, and velocity and pressure distributions. The student outlines the cannon ball geometry, meshing approach, governing equations, and parameters that will be analyzed to understand the transition between laminar and turbulent flow around the moving sphere.

1. THEORY OF VISCOUS FLOWS
COURSE # MECH5001
FINAL TERM PROJECT
NAME: SAUMITRA GOKHALE DATE : April 9th
2014
STUDENT # 100907042
TITLE: ANALYSIS OF FLUID FLOW AROUND A CANNON
BALL IN A PROJECTILE MOTION
1. ABSTRACT:
The object selected for analysis of fluid flow in the Final Term Report is a Cannon ball. Using
CFX a quasi three-dimensional air flow around cannonball in simulated. Appropriate input and
boundary conditions are applied and pertinent results including Boundary layer thickness,
Displacement thickness, Momentum thickness, Shape factor, drag coefficient and velocity &
pressure distribution are obtained.
2. INTRODUCTION:
A. Background:
A Cannonball, which is selected topic for analysis of fluid flow, is used in war from ancient
times. It was first used in 14th
century in Florence, Italy. But since then it took time to
develop as an effective weapon that can be used in large scale. In early 1700s cannon was a
common weapon that was used in European armouries and successfully became integral part
of the field artillery. [1]
Later in the civil war, various types of cannons were used, having more velocity and longer
range. However for firing at higher trajectories and short ranges, two types of Smoothbore
cannons, guns and howitzers were used. [2]
B. Literature search report:
A cannon ball is considered as a solid sphere in this project. A solid sphere has a great
importance in environment and technology world. Analysis of a fluid flow in a solid sphere
provides a background for analysis of more complicated situations involving un-steady flows,
non-uniform flows and no spherical bodies. [3]
There can be two types of flows related to sphere; either it can be a flow passing through a
stationery sphere or a sphere moving in the stationery fluid. In this project, since cannon ball
is moving in a projectile motion, second type of flow that is sphere moving in a stationery
fluid is considered. [3]

2. Consider sphere moving in a fluid as shown in Figure
1. By providing no slip conditions on sphere there
implies there is a zero velocity at the surface of the
sphere. As a result there will be existence of a velocity
gradient which is responsible for a shear stress on the
surface of the sphere. By taking summation of shear
surface over entire surface will denote total drag force.
[3]
Regarding velocity and pressure distribution, at the stagnation point velocity should be zero
and pressure should be maximum around the surface of the sphere. Significance of shape
factor is, it gives type of boundary layer for laminar boundary layer shape factor should be
around 2.6.
Type of boundary layer depends on Reynolds Number. It might be laminar or turbulent for
low and high Reynolds no. respectively. As discussed earlier, there will be velocity gradient,
since velocity at the surface is zero. It increases as it moves away from the surface; however
it will never be equal to free stream velocity. In a laminar boundary layer velocity changes
uniformly as it moves away from the surface. A point at which velocity equals to 99% of free
stream velocity will denote Boundary layer thickness (δ). On the other hand in turbulent
boundary layer, velocity change is characterised by unsteady flow, inside the boundary layer.
Displacement thickness (δ*
) is the distance the surface would have moved in the y-direction
to reduce the flow by volume equivalent to real effect of the boundary layer. Momentum
thickness represents a thickness of the free stream flow that has a momentum equal to the
momentum deficit in the boundary layer. [4][5]
C. Statement of the problem:
For analysis of a fluid flow around a cannon ball in a projectile motion, is carried out. As
cannon ball is fired from the cannon initial velocity of 50 m/s is assumed (in the proposal
9.75 m/s was assumed, slight modification is done). As it moves through the air in a projectile
motion, velocity of the cannon ball decreases till it reaches at the top and it increases as ball
descends. By taking particular time interval (dt) and obtaining tangential velocity in that
interval , type of fluid flow around a sphere, whether laminar or turbulent or does it changes
from laminar to turbulent as it moves in a projectile motion, can be analysed.
For obtaining velocity in particular time interval (dt), analysis of the projectile path that
cannon ball will follow is carried out. Apart from assuming velocity, angle at which ball is
fired is also assumed as 60°. Also gravitational acceleration taken as 9.8 m/s2
and time step
(dt) is taken as 0.1. all the details are included in the excel sheet and analysis is performed as
tabulated in the Table 1.
Figure 1 Flow over Sphere [3]

3. Velocity in X-direction is calculated by the formula Vx = V*Sin 60, since 60° is the angle at
which cannon ball is fired and velocity in Y-direction is calculated using Vy= V*cos 60.
Once ball is fired from the cannon with initial velocity V, its speed will drop due to
gravitational force (g) till it reaches the top and then it will gain speed due to gravitational
force till it reach ground. Similar types of results are obtained and can be verified from Table
1 by comparing resultant velocity (U) values. X and Y values denoted amount which is
traveled by the cannon ball in x and y direction respectively. From Table 2 it is evident that
cannon ball will travel approximately 216.5 meters in x-direction till it touches the ground
and maximum 29.43 meters in y-direction.
Table 1: Projectile path analysis
By plotting X and Y values as shown in Figure
2, path travelled by the ball can be obtained.
Once resultant velocity is calculated in each
time step, it is used as an inlet for simulating
the flow in that time interval. So that by
simulation type of flow as can be obtained.
Various boundary conditions which are
necessary for simulation is discussed as below:
As stated earlier, simulation is to be performed
on a quasi three-dimensional air flow around a
cannon ball which is considered as smooth Sphere. The diameter of the Cannon ball is
assumed as 10 cm. An appropriate size enclosure is created (20*diameter in both positive and
negative X,Y and Z direction).
Inlet conditions are assigned as velocity U= 50 m/s for dt=0 and U= 45.89161 m/s for dt= 1
second and so on. An outlet has zero gauge average static pressure. Other boundary
condition includes free slip on all domain boundaries that is an enclosure and no slip on
Sphere surface. Also, biased accumulation of nodes towards the wall should be generated in
CFX meshing. Air at 25 Celsius, 1 atm reference pressure and no turbulence and heat transfer
model to be created in CFX. Using above data, analysis of grid convergence using coefficient
of drag as the determining parameter is performed along with analysis of flow and plotting
other major parameters like Boundary Layer Thickness, displacement thickness etc.
Time(dt) (sec) Vx (m/s) Vy(m/s) U(m/s) x (m) y (m)
0 43.30127 25 50 0 0
1 43.30127 15.2 45.89161 43.30127 19.61
2 43.30127 5.4 43.63668 86.60254 29.42
3 43.30127 -4.4 43.52425 129.9038 29.43
4 43.30127 -14.2 45.57017 173.2051 19.64
5 43.30127 -24 49.50758 216.5064 0.01
Figure 2 Projectile motion

4. 3. METHODS:
A. Geometry & Meshing:
Sphere of diameter 10 cm (100 mm) is created and
then sufficient large enclosure is generated.
Dimension of an enclosure is 2000
mm*2000mm*2000mm. Resulting geometry is as
shown in Figure 3. Using Booleans operation area of
the sphere is subtracted from enclosure area and the
geometry then carried forward for meshing.
For meshing, new section plane is used and whole
geometry is cut into half (approximately) then using
named selection mesh is generated on the geometry.
For the mesh relevance centre is selected as course
at first then by using grid convergence and
changing it to medium and then to fine more
accurate results were obtained, Advanced size
function is kept on: proximity and curvature,
smoothing set as High. Inflation option is set as
smooth transition and maximum layers are
assigned as 20. Resulting mesh is as shown in
Figure 4.
B. Equations:
The next step is to use the equations that are
necessary for obtaining major parameters like
Boundary layer thickness, Displacement thickness,
Momentum thickness, Shape factor and the drag
coefficient.
Reynolds No. can be calculated as,
𝑅𝑒 =
𝜌∗𝑑∗𝑈
𝜇
……………….Equation (1)
Where, ρ = Density (kg/m³) = 1.185 kg/m³
d = Diameter of cylinder (m) = 10 mm = 0.01 m
U = Constant normal velocity (m/s) = 42 m/s
µ = Dynamic viscosity ( kg/m*s) = 1.8* 10-05
( kg/m*s)
Coefficient of Drag can be calculated as,
𝐶𝑑 =
2∗𝐹𝑑
𝜌∗𝑈∗𝑈∗𝐴
……………….Equation (2)
Where,
Fd= Drag force (N) and A = Area (m2
)
Figure 3 Geometry
Figure 4 Mesh

5. Navier-Stokes equation [7]:
Navier Stokes equation in common dimensional form for two dimensional incompressible
fluid flows can be written as shown below.
𝜕𝑢
𝜕𝑥
+
𝜕𝑣
𝜕𝑦
= 0
𝜕𝑢
𝜕𝑡
+ 𝑢
𝜕𝑢
𝜕𝑥
+ 𝑣
𝜕𝑢
𝜕𝑦
= [
𝜕𝑈
𝜕𝑡
+ 𝑈
𝜕𝑈
𝜕𝑥
] + 𝑔𝑥 𝛽 ( 𝑇 − 𝑇0) + 𝑣
𝜕2
𝑢
𝜕𝑦2
Where, U= U (x,t) is a free stream velocity just outside the boundary layer.
Boundary layer thickness, displacement thickness and momentum thickness can be obtained
using Equation 3, Equation 4 and Equation 5 respectively.
𝑢(𝛿) = 0.99𝑈 ………………….Equation (3)
By taking values of u/U and comparing the y value corresponding to 0.99, boundary layer
thickness can be obtained.
For calculating simulated displacement thickness following equation is used.
𝛿∗
= ∫ (1 −
𝑢
𝑈
𝛿
0
) 𝑑𝑦 = 𝛿 − ∫ (
𝑢
𝑈
)
𝛿
0
𝑑𝑦…………Equation (6)
And
Momentum thickness is given by,
𝜃 = ∫
𝑢
𝑈
(1 −
𝑢
𝑈
𝛿
0
) 𝑑𝑦…………Equation (7)
Once displacement thickness and momentum thickness is obtained, these results can be used
to calculate the Shape factor (H) which is important in determining nature of the flow and
can be calculated using following Equation (8)
𝐻 =
𝛿∗
𝜃
………………….. Equation (8)
4. RESULTS AND DISCUSSIONS:
A. Calculation of coefficient of drag:
Drag coefficient is calculated using Equation 1 and Equation 2. First, Reynolds no. is
calculated from Equation 1. Using that Reynolds no. reference value (approximate) for
coefficient of drag for sphere can be obtained from adjacent Figure 5.

6. After obtaining reference drag coefficient value, simulated drag coefficient value is obtained
from Equation 2. For each mesh size that is for course, medium and fine different values of
drag coefficients are obtained and they are compared with reference coefficient values
obtained from previous step. All the results are summarised in following Table 2 and results
are plotted in Figure 6 as below.
dt U(m/s) Re
(10^4)
Fine Medium Course
Reference Value
(approx)
Error
(%)
cd cd cd cd
0 50 32.37 0.32 0.318 0.3161 0.38 16.38
1 45.89161 29.71 0.275 0.275 0.27237 0.37 26.3
2 43.63668 28.256 0.2515 0.251 0.2502 0.36 30.5
3 43.52425 28.1837 0.25 0.2499 0.2485 0.35 29
4 45.57017 28.2134 0.271 0.27 0.2703 0.36 25
5 49.50758 32.058 0.314 0.31339 0.3133 0.37 15.3
Table 2: Summary of results
It is evident from the Table 2 that, as mesh size is changed from course to fine more accurate
results are obtained, since error mentioned in the last column is associated with either Course
mesh size values or medium mesh size values, which implies Fine mesh size gives most
accurate values.
B. Calculation of Boundary layer thickness, Displacement thickness and momentum
thickness:
By calculating Boundary layer thickness, displacement thickness and momentum thickness at
time interval (dt = 1,2,3,4 & 5) and refereeing to corresponding resultant velocity values,
nature of the flow on cannon throughout its projectile motion can be predicated Hence, for
calculation of Boundary layer thickness, displacement thickness and momentum thickness
surface of the sphere is divided in 5 lines, with first line at an angle (φ) = 0° and second line
Figure 5: Cd vs Re [6]
Figure 6: Cd and Grid Convergence effect

7. at an angle (φ) = 45° and so on. Each of those lines data then exported to excel. Then,
exported velocity u is divided by U and all the corresponding values are obtained. For time
interval dt =0, from Table 1 value of U is equal to 50 m/s, for time interval dt = 1, value of U
is equal to 45.89161 m/s and for time interval dt = 2, value of U is equal to 43.63668 m/s.
From Table 1 it is evident that after time interval dt = 2, cannon ball reaches its peak position
and starts to descend and flow will be almost similar as in early stages of projectile motion.
Hence, analysis for Boundary layer thickness, displacement thickness and momentum
thickness is limited to time interval dt = 2 and velocity U= 43.63668 m/s.
As discussed earlier, after obtaining u/U values, (1-u/U) can be obtained. By using Equation
6, displacement thickness δ* can be obtained. By multiplying (1-u/U) values with u/U and by
using Equation 7, momentum thickness (θ) can be calculated. Once δ* and θ is calculated,
shape factor, H is obtained from Equation 8.
Results of Boundary layer thickness, displacement thickness and momentum thickness are
plotted for each time interval as below.
C. Shape Factor calculation:
Shape factor results are tabulated is Table 3
as shown below.
dt H Reference H Error %
0 1.010122
1.4
27.84839
1 1.010022 27.8555
2 1.010089 27.85
Table 3: Shape Factor (H)
For Laminar flow Shape factor should be
around 2.6 and should be 1.4- 1.5 for
turbulent flow.
0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0.0014
0 50 100 150 200
δ*
Angle Ψ
Displacement Thickness
dt = 2
dt = 1
dt = 0
0
0.02
0.04
0.06
0.08
0.1
0.12
0 50 100 150
δ
Angle Ψ
Boundary Layer Thickness
dt = 2
dt = 1
dt =0
0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0.0014
0 50 100 150 200
ϴ
Angle Ψ
Momentum Thickness
dt = 2
dt = 1
dt = 0

8. D. Pressure and velocity distribution:
From pressure and contour plot in results, there was minimum velocity on the stagnation
point, on the other hand pressure at the stagnation point observed maximum. On the upstream
of the sphere away from stagnation point velocity will increase and pressure will drop till it
reaches end of the upstream of sphere.
5. CONCLUSIONS:
Using CFX a quasi three-dimensional air flow around cannonball in simulated. Cannonball is
assumed as a smooth sphere, also initial velocity of the cannonball is assumed as 50 m/s. First
analysis of cannon balls project motion is carried out thereby acquiring necessary input details for
example inlet velocity. Cannon balls projectile motion is divided into certain time interval (dt)
and analysis of fluid flow in that particular time interval is carried out.
Desired results were obtained for coefficient of drag with error ranging from 15.3 % to 30.5 %.
Maximum error was recorded in simulation for time interval dt = 2. On the other hand simulation
for time interval dt = 5 is recorded most accurate.
After calculating coefficient of drag, Boundary layer thickness, displacement thickness and
momentum thickness were calculated. The project successfully meets with the primary aim of
predicting nature of the flow of the cannon ball in a projectile path. From Boundary layer
thickness, displacement thickness and momentum thickness calculations, shape factor (H) is
calculated. The value of the shape factor obtained in between 1.010022 and 1.010122 thereby
predicting the flow remains turbulent in whole projectile path, since shape factor should be
around 1.4 in Turbulent flow. This fact can also be verified from Reynolds no. since Reynolds no.
is higher in this case, it is expected that flow is turbulent for higher Reynolds no.
6. REFERENCES:
[1]: http://www.motherbedford.com/Cannon.htm
[2]: http://www.treasurenet.com/forums/today-s-finds/92585-another-civil-war-cannon-ball.html
[3]:http://ocw.mit.edu/courses/earth-atmospheric-and-planetary-sciences/12-090-special-topics-
an-introduction-to-fluid-motions-sediment-transport-and-current-generated-sedimentary-
structures-fall-2006/lecture-notes/ch2.pdf
[4]: Lecture notes by Prof Edgar Matida
[5]: http://www.grc.nasa.gov/WWW/k-12/airplane/boundlay.html
[6]: http://www.symscape.com/files/pictures/sphere-sports/sphere-cd-re.png
[7]: F.M. White, "Fluid Mechanics", McGraw-Hill