Term Project Report CFD

8/9/2019 Term Project Report CFD

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Computational Fluid Dynamics of

Compressible Flows

MECH5304

21 April 2010

Dr. Edgar A. MatidaDepartment of Mechanical &

Aerospace Engineering

Term project report

Student: Aymen Sakka

Carleton ID: 100828756


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Abstract

The flying distance of a golf ball is influenced not only by its material, but also by the aerodynamics of

the dimples on its surface. By using ANSYS CFX, the aerodynamics characteristics and to the drag forcesover the golf balls were studied. The drag coefficient variation with different Re numbers was studied.The validation is done by comparison to a flow over a smooth sphere. The results qualitatively agreed

with the literature.

Introduction

Although poorly documented, golf is believed to have originated in the early 1400s [1]. It was first playedas a very casual game for which no standard rules existed. A wooden ball was used in conjunction withwooden clubs prior to 1618[1], when the featherie" (a ball made of stitched leather and tightly packedwith feathers) was introduced. The featherie was favored for its more forgiving feel on the hands of

players when it was struck and was used until 1848 when the invention of the Gutta surpassed thefeatheryinboth durability and cost. The Guttawas made of gutta-percha packing material which was

not brittle and became soft and moldable at 100C.The Gutta's pliability made it necessary to roll the ball on a smoothing boardin order to maintain itsshape and keep it free of imperfections which were created during normal play of the game. The smooth

Gutta was used for only a few years before players began to realize that balls that had not been wellmaintained and had many nicks and scratches had a much more favourable flight. Thus began the practice

of hammering the Gutta with a sharp-edged hammer in a regular pattern to increase the consistency of theball's play.In 1898 the first Balata ball was created by wrapping rubber thread around a solid rubbercore which

was then covered by a solid layer of rubber that later became known as th e ball cover. The Balata wasthe first sign of a modern age of golf technology for it allowed molds to be used to create consistent cover

patterns. In 1908 makers discovered the superiority of a regular dimple pattern over the haphazard gridpattern favoured by players at the time. Dimples are small indentations on the exterior of the golf ball.

They are typically round in shape and vary in diameter from 2-5mm in diameter and are about .2mmdeep. Modern golf balls pack anywhere from 300-450 dimples of varying size arranged in a regular

pattern on the outside of every ball [3]. Dimples have been one of the most influential developments in

golf ball design because they alter the dynamics of the balls flight in such a way that gives golfers asignificant amount of control over the height and shape of their shots.

b 0.725 mm

c 3.5868 mm

k 0.7 mm

Figure 1: geometry of the golf ball Figure 2: geometry of the golf ball dimples


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Figures. 1 and 2 show the geometry and boundary of a typical golf ball. The golf ball diameter is 42.6 mm

while the dimples diameter is 3.58681 mm. The golf ball has 389 dimples (See Figure 1).The domain size

is 600 mm 400 mm 400 mm in the x, y, and z-directions (See appendix-1- ). The fluid is air at

25C and the relative pressure is 1 atm. The inlet velocity is changed to vary the Reynolds number.Air properties at 25C and P= 1 atm:

= 1.18 /3 = 1.84 105

= 1.595 105 2

The root mean square of errors (RMS) was set to 10E-04, it is defined as: RMSerr = erri2ni=1

Methods

Meshing

The mesh details for the smooth sphere and the golf ball are listed in appendix-2-. Both of the meshes use

Delaunay surface meshing and advancing front and inflation meshing strategy. Advancing front method is

used for the volume mesh as well.

Figure 3: Surface and volume mesh of a smoothsphere

Figure 4: Volume mesh for the golf ball


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Under solver control, the advection scheme and the turbulence numerics are set to High resolution. The

automatic scaling was chosen for time and conservative for space. As for convergence criteria, the RMS

residuals were set to 10-5for the sphere simulations and to 10-4for the golf ball simulations.

The inlet boundary is defined by normal velocity whereas a zero gauge pressure condition was set at the

outlet. The domain walls are considered as free slip walls. The smooth sphere and the golf ball are

considered to have a smooth wall with a no slip condition to take into account the viscosity of the fluid,

which is air at 25 C.

ANSYS CFX theory

Two Equation Turbulence Models

Two-equation turbulence models are very widely used, as they offer a good compromise betweennumerical effort and computational accuracy. Two-equation models are much more sophisticated than the

zero equation models. Both the velocity and length scale are solved using separate transport equations(hence the term twoequation').The k- and k- two-equation models use the gradient diffusion hypothesis to relate the Reynolds stressesto the mean velocity gradients and the turbulent viscosity. The turbulent viscosity is modeled as the

product of a turbulent velocity and turbulent length scale. In two-equation models, the turbulence velocityscale is computed from the turbulent kinetic energy, which is provided from the solution of its transport

equation. The turbulent length scale is estimated from two properties of the turbulence field, usually theturbulent kinetic energy and its dissipation rate. The dissipation rate of the turbulent kinetic energy is

provided from the solution of its transport equation.

The k-epsilon and SST Models in ANSYS CFX

The present numerical simulation of the airflow distribution around a golf ball requires the use of varioustheoretical mathematical models based on fluid dynamics principles. The k-based Shear-StressTransport (SST) model was designed to give highly accurate predictions of the onset and the amount offlow separation under adverse pressure gradients by the inclusion of transport effects into the formulation

of the eddy-viscosity. The SST model has a slight additional cost over other two equation models since awall scale equation is also solved.The present model in CFX consists of the continuity equation, the momentum equation, and the energyequation. k is the turbulence kinetic energy and is defined as the variance of the fluctuations in velocity. Ithas dimensions of (L2T-2); for example, m2/s2. is the turbulence eddy dissipation (the rate at which the

Figure 5: Surface mesh of the golf ball


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velocity fluctuations dissipate), and has dimensions of k per unit time (L 2T-3); for example, m2/s3.These

equations employed in the present numerical model are presented below.

Continuity equation:

+ . = 0Momentum equation

( ) + . = + . +

Where = + In early research, turbulent model was applied in high Reynolds number incompressible flows. But it was

later experimentally proven that the air flow near the wall is associated with low Reynolds numbers.Therefore, the development of turbulence model for low Reynolds numbers has been an intensive focus

for research activities. One remedy to this scenario is to introduce a wall function so that the low

Reynolds number air flow near the wall and the high Reynolds number flow far away from the wall canbe simulated at the same time. In this paper, the turbulent model used is the amended standard - model

because it has been proven to give good predictions for complex flows.

The k- model is given as

() + . = .

+ +

()

+ . = .

+ 1

+ (1 + 3) 2 2

Where

= 2 .

=

= 1

= 2

= 12(+), 1 = 2

Calculation of the drag coefficient [4]

The drag equation is a practical formula used to calculate the force of drag experienced by an object

moving through a fluid. The force on a moving object due to a fluid due to Lord Rayleigh is


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= 12

2 F is the force of drag.

is the density of the fluid.

V is the velocity of the object relative to the fluid.A is the reference area.

is the drag coefficient (a dimensionless constant).The reference area A is the area of the projection of the object on a plane perpendicular to the direction of

motion (i.e. cross-sectional area).

Drag is the net force on the body in the direction of the flow. In the above diagram, the drag is the sum of

the forces on the wall in the horizontal direction, i.e. the sum of the pressure force and the viscous forcecomponents in the x direction. It is apparent from this that viscous force is not a pure shear force since italso has a small component in the normal direction, arising in part from a normal component in the

laminar flow shear stress.

The pressure and viscous moments are related to the pressure and viscous forces calculated at the Wall.

The pressure moment is the vector product of the pressure force vector and the position vector r. Theviscous moment is the vector product of the viscous force vector and the position vector r. i.e. where andare the pressure and viscous moments respectively. These are summed over all the surface elements in the

Wall.

It is important to note that forces do not include reference pressure effects. The pressure force iscalculated as the integral of the relative pressure over the wall area and not as the integral of the sum ofthe reference and relative pressures.

Results and discussion

F low over a smooth sphere

Early aerodynamics researchers were quite puzzled by the theoretical result stating that there is no drag ona sphere because it contradicted experimental measurements indicating that a sphere does generate drag.

The conflict between theory and experiment was one of the great mysteries of the late 19th century thatbecame known as d'Alembert's Paradox, named for famous French mathematician and physicist Jean leRond d'Alembert (1717-1783) who first discovered the discrepancy[5].

The reason d'Alembert's ideal theory failed to explain the true aerodynamic behavior of a sphere is that heignored the influence of friction in his calculations. The actual flowfield around a sphere looks muchdifferent than his theory predicts because friction causes a phenomenon known as flow separation. We

The CFX-Solver calculates the pressure and viscouscomponents of forces on all boundaries specified asWalls. The drag force on any wall can be calculated

from these values as follows:

Lift is the net force on the body in the directionperpendicular to the direction of flow. In the above

diagram, the lift is the sum of the forces on the wall inthe vertical direction, i.e. the sum of the pressureforce and the viscous force components in the ydirection.

Figure 6: Drag forces on a body


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can better understand this effect by studying the following diagram of the actual flow around a smooth

sphere. Here we see that the flowfield around the sphere is no longer symmetrical. Whereas the flowaround the ideal sphere continued to follow the surface along the entire rear face, the actual flow nolonger does so. When the airflow follows along the surface, we say that the flow is attached. The point atwhich the flow breaks away from the surface is called the separation point, and the flow downstream of

this point is referred to as separated. The region of separated flow is dominated by unsteady, recirculatingvortices that create a wake[5].

Although the values of critical Reynolds numbers are not exactly the same, the computational prediction

is acceptable as far as the overall trend is concerned. The drag coefficient plot as a function of Reynolds

numbers displays a slightly higher value for Re= 105than the chart in appendix-3-.

0.45

0.460.47

0.48

0.49

0.5

0.51

0.52

0.53

0.54

0.55

0.56

0 50000 100000 150000 200000 250000 300000

Dragcoefficient

Reynolds number

Figure 8: Drag coefficient variation as a function of increasing Reynolds number

Cd=f(Re)

Since the laminar boundary layer

around the smooth sphere separates so

rapidly, it creates a very large wake

over the entire rear face. Re-

circulating vortices in the wake are

pointing out from the main axis of the

flow. This large wake maximizes the

region of low pressure and, therefore,results in the maximum difference in

pressure between the front and rear

faces (Figure 7). This difference

creates a large drag like that seen

below the transition Reynolds number.

Figure 7: Pressure streamlines around

the smooth s here for V=1 m s


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- The transition to a turbulent boundary layer, on the other hand, adds energy to the flow allowing it

to remain attached to the surface of the sphere. Since separation is delayed, the resulting wake is

much narrower. This thin wake reduces the low-pressure region on the rear face and reduces the

difference in pressure between the front and back of the sphere. This smaller difference in

pressure creates a smaller drag force comparable to that seen above the transition Reynolds

number.

- These results tell us that causing a turbulent boundary layer to form on the front surface

significantly reduces the sphere's drag. For a given sphere diameter, a designer has only two

options encourage this transition, either increase the speed of the flow over the sphere to increase

the Reynolds number beyond transition or make the surface rough in order to create turbulence.

The latter case is often referred to as "tripping" the boundary layer.

Figure 9: Velocity streamlines around

the smooth sphere for V=100 m/s

Figure 10: Eddy viscosity contour plot for

the flow over the s here for V=100 m s

Figure 10: Pressure vortex around the

smooth sphere for V=20 m/s


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F low over a golf ball

In the case of a golf ball, increasing the speed is not an option since a golfer can only swing the club so

fast, and this velocity is insufficient to exceed the transition Reynolds number. That leaves tripping the

boundary layer as the only realistic alternative to reducing the drag on a golf ball. The purpose of the

dimples is to do just that--to create a rough surface that promotes an early transition to a turbulent

boundary layer. This turbulence helps the flow remain attached to the surface of the ball and reduces thesize of the separated wake so as to reduce the drag it generates in flight. When the drag is reduced, the

ball flies farther.

Figure 11: Velocity 3D streamlines around the golf ball for V=100 m/s

Air flows smoothly over the contours

of the front side and eventually

separates from the ball toward the

back side. The flying golf ball also

leaves behind a turbulent wake region

where the air flow is fluctuating or

agitated (Figure 11), resulting in lower

pressure behind it. The size of the

wake affects the amount of drag on

the object. Dimples on a golf ballcreate a thin turbulent boundary layer

of air that clings to the ball's surface.

This allows the smoothly flowing air

to follow the ball's surface a little

farther around the back side of the

ball, thereby decreasing the size of the

wake. See figure 12.Figure 12: Pressure 3D streamlines around the golf

ball for V=100 m s


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Moving in to about 45 degrees from the leading surface of the golf ball, we can see a number of trains of

vortices are developed (Figure 13). These lines of whirlpools follow the scallops of the dimples in the

direction of the airstream. In the behaviour of the air shown just around a pair of dimples (Figure 14), we

see the airflow evolve, from front to back within a single dimple, as the air detaches and shears away,

varying with the streams direction. We see that the cumulative effect of the vortices cause air to come

down onto the ball to delaying energy-wasting separation. Hence, Golf balls with dimples turn out to be

more slippery than smooth spheres.

Figure 15: Velocity vectors in the vortex core

region around the golf ball for V=100m/s

Figure 13: Velocity contour plot in the vortexcore region around the golf ball for V=100 m/s

Figure 14: Detaching vertices from dimples inthe vortex core region for V=100 m/s

As figure 15 shows, the velocity

vectors field is pointing to the center

of the wake. This convergence pattern

of the vertices tends to reduce the size

of the wake. In fact, the vectors are

pointing against the flow main stream

which allows the wake to have a

counter effect the drag force on the

ball.


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Turbulence around the golf ball has a beneficial effect on reducing the wake of an object. If the surface

air in the boundary layer becomes turbulent, the higher kinetic energy in the turbulent region will help the

air stick to the surface longer before separating (Figure 16). The result is lower form drag. Figure 17

illustrates the high level of turbulence by a maximum eddy viscosity values just behind and very close to

the golf ball.

Drag coefficient

Figure 16: Turbulence kinetic energy contour

plot around the golf ball for V=100m/s

Figure 17: Eddy viscosity contour plot around

the golf ball for V=100m/s

It is the difference between the high and

low pressure values that account for drag

forces a body experiences. In the case of

separated flow around a sphere the drag

force and hence drag coefficient is

dominated by form drag which depends on

the separation point on the sphere. Hence

anything that effects the location of the

separation point has a large effect on the

drag coefficient. For example, the dimples

on a golf ball cause the laminar boundary

layer to become turbulent sooner and this

moves the separation point rearward

decreasing the from drag and the drag

coefficientas shown in figure.

Figure 18: Pressure contour plot around

the golf ball for V=100 m/s


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Validation of the simulations

For validation, this study used a 3-D sphere. The turbulence model being validated is the shear stress

transport model. Drag coefficient is the lowest at the critical Reynolds number of 4104. After that, drag

coefficient will raise slowly with Reynolds number. The drag coefficient of the sphere starts to drop off at

a Reynolds number of 8104, but stays fairly constant afterwards. This corresponds to the transition of

air flow from laminar to turbulent. Figure 19 below shows the comparison of drag coefficients at differentReynolds numbers for the golf ball against the smooth ball.

A golf ball usually flies at a Reynolds number more than 105, which is near the critical Reynolds number.

Figure 19 shows that for Reynolds bigger than 105, the drag coefficient for the golf ball starts to decrease

with higher intensity than the smooth sphere. These results qualitatively agree well with each other.Although the values of critical Reynolds number are not exactly the same, the computational prediction is

acceptable as far as the overall trend is concerned.

The dimples, paradoxically, do increase drag at low Reynolds numbers. But they also increase "Magnuslift", that peculiar lifting force experienced by rotating bodies travelling through a medium. Magnus liftis present because a driven golf ball has backspin[6].

Domain convergence analysis

An attempt has been made to simulate different flows around the golf ball with different Reynolds

numbers with a domain 1.5 bigger in each direction. These simulations (that came up with a fatal error at

CFX-post except three cases) have an RMS error of 0.017. Through this small error, we cannot judge the

error of the other flows. Still, it is a good indication of the accuracy of the chosen domain and confirming

this choice stated in the literature [7].

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

1000 10000 100000 1000000

Dragcoeffic

ient

Reynolds number

Figure 19: Drag coefficient variation with Reynolds number

Smooth sphere

Golf Ball


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Conclusions

- Drag on a golf ball comes mainly from air-pressure forces. This drag arises when the pressure in

front of the ball is significantly higher than that behind the ball. The only practical way of

reducing this differential is to design the ball so that the main stream of air flowing by it is as

close to the surface as possible. This situation is achieved by a golf ball's dimples, which augment

the turbulence very close to the surface, bringing the high-speed airstream closer and increasingthe pressure behind the ball. The effect is plotted in figure 19, which shows that for Reynolds

numbers achievable by hitting the ball with a club, the coefficient of drag becomes lower for the

dimpled ball.

- The critical Reynolds number (Recr) holds the explanation for the fact that golf balls have

dimples. Recris the Reynolds number at which the flow transitions from a laminar to a turbulent

state. For a smooth sphere, Recris much larger than the average Reynolds number experienced by

a gold ball. The dimpled ball has a lower Recr and the drag is fairly constant for Reynolds

numbers greater than Recr. Therefore, the dimples cause Recrto decrease, which implies that the

flow becomes turbulent at a lower velocity than on a smooth sphere. This causes the flow to

remain attached longer on a dimpled golf ball, which implies a reduction in drag. As the speed of

the dimpled golf ball is increased, the drag decreases. This is a good property in a sport like golf

where the main goal is to maintain the ball in this post-critical regime throughout its flight.

- Dimpled surface causes air to grip the ball for a longer period of time before passing, creating

turbulence and a thickened boundary layer. A smoother surface will allow the air to flow easier

over the ball creating what is called laminar flow. Unfortunately, laminar flow, while initially

having less drag, is also prone to separation, which produces an increased drag.

- On the golf ball, the pressure drag is much larger than the skin friction, so adding dimples isbeneficial. There was a lot of pressure drag to be reduced so the increase in skin friction is anacceptable trade off.

- Computational Fluid Dynamics can be a powerful tool to investigate effects of dimple geometry

on the flow field around a golf ball and enable more efficient design process of dimple geometryfor less drag and longer flight distances.


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References

[1] Materials in sports equipment, volume 1, Mike Jenkins, 2003 ,Woodhead Publishing Ltd and CRC

Press LLC.

[2] "Flying Characteristics and Flow Pattern of a Sphere with Dimples", K. Aoki, A. Ohike, K.Yamaguchi and Y. Nakayama, Journal Of Visualization, vol. 6, no. 1, pp. 67-76, 2003.

[3] http://www.aerospaceweb.org,18 April 2010.

[4] ANSYS CFX 12.0 help, April 2010.

[5] Applied and Computational Fluid Mechanics, Scott Post, 2011 copyright (c) by Jones and Bartlettpublishers, LLC.

[6] A statistical study on reduction of drag force for golf balls, Takeyoshi Kimura and Mitsuru

Sumiyama, Memoirs of Fukui University of Technology, Vol.34, Part 1, 2004.

[7] Effects of golf ball dimple configuration on aerodynamics, trajectory, and acoustics, Chang-Hsien Tai

, Chih-Yeh Chao, Jik-Chang Leong, Qing-Shan Hong, Department of Mechanical engineering, National

Ping-Tung University of Science and Technology.
http://www.aerospaceweb.org/http://www.aerospaceweb.org/


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Appendices

Appendix-1-: Geometry of the domain

Problem domain

Golf ball sketch


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Golf ball surface and volume mesh


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Appendix-2- Mesh Inputs

Smooth sphere

Volume mesh for the smooth sphere

Face spacing

Option Angular resolution

Angular resolution 18 degrees

Minimum edge length (mm) 1.5

Maximum edge length (mm) 30

Radius of influence (mm) 0

Expansion factor 1.2

Sizing

Used advanced sizing On: proximity

Relevance center Coarse

Smoothing Medium

Transition Slow

Span angle center coarseProximity accuracy 0.5

Min size 9e-003 m

Max face size 3e-002 m

Max tet size 3e-002 m

Growth rate 1.5

Minimum edge length 9.3934e-003 m

Inflation

Inflation option Smooth transition

Transition ratio 0.77

Maximum layers 5

Growth rate 1.2

Inflation algorithm Pre

Total thickness 7 mm

Mesh statistics

Total number of nodes 59340

Total number of tedrahedral 307214

Total number of prisms 980

Total number of elements 308194


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Golf ball meshing

Mesh spacing

Sphere spacing

Option Angular resolution

Angular resolution [Degrees] 18

Minimum edge length [mm] 0.5

Maximum edge length [mm] 3Radius of influence [mm] 0


Dimple spacing



Maximum edge length [mm] 3

Radius of influence [mm] 0


Edge spacing



Maximum edge length [mm] 1

Radius of influence [mm] 0Expansion factor 1.2

Inflation

Inflation option Smooth transition

Transition ratio 0.77

Maximum layers 5

Growth rate 1.2

Inflation algorithm Pre

Sizing

Used advanced sizing On: Curvature

Relevance center Coarse

Smoothing Medium

Transition Slow

Span angle center Fine

Curvature Default (18.0)

Min size Default(4.1102e-004 m)

Max face size Default(4.1102e-004 m)

Max tet size Default(8.2204e-004 m)

Growth rate 1.2

Minimum edge length 9.3934e-003 m

Mesh statistics

Total number of nodes 531932

Total number of tedrahedral 1988165

Total number of pyramids 22456

Total number of prisms 317746

Totalk number of elements 2328367


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Appendix-4-Simulations numerical results

V 100 90 80 70 60 50 40 30 20 10 7 5 2

Re 3E+05 2E+05 2E+05 2E+05 2E+05 1E+05 1E+05 80125 53417 26708 18696 13354 5342

ereF

3.892 3.157 2.47 1.917 1.413 0.985 0.634 0.359 0.157 0.04 0.02 0.01 0.002

Cd 0.463 0.463 0.459 0.465 0.467 0.468 0.471 0.475 0.465 0.472 0.476 0.483 0.502

lf

ll

F3.818 3.09 2.461 - 1.403 0.984 0.639 0.368 0.157 0.041 0.021 0.011 0.002

Cd 0.454 0.454 0.457 - 0.463 0.468 0.475 0.486 0.466 0.489 0.506 0.525 0.605

Simulation results of the drag coefficient over a smooth sphere and dimpled golf ball

V 100 90 80 70 60 50 40 30 20 10 7 5 2

Re 3E+05 2E+05 2E+05 2E+05 2E+05 1E+05 1E+05 80125 53417 26708 18696 13354 5342

lf

ll

F4.021 - - - - 1.029 - - - - - - -

Cd 0.462 - - - - 0.48 - - - - - - - 0

Simulation results of the drag coefficient over a dimpled golf ball with a domain expansion factor of 1.5

( - ): Fatal error in CFX-post

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