Chen Guoyi

Shanghai Experimental Foreign Language School, Shanghai 201210

Abstract

Explained the reason why the surface dimples are capable of lowering drag acting on a golf ball in motion through a fluid medium. Using Autodesk^® 3DsMax^® to construct golf ball surface model. Acquiring the correlation between concave degree of surface dimple in golf ball and drag force using COMSOL^® Multiphysics laminar flow analysis. Obtained the surface structure that could make the golf ball fly the farthest using MATLAB^® analysis.

Keywords: Flow drag; COMSOL^® Multiphysics Simulation; Golf ball.

Introduction

This experimental report was initiated when I was in Grade 10, joining a summer school conducted by the Department of Physics, National University of Singapore. The topic of this experimental report came from a question raised in that summer school, “Why can the dimples on the surface of a golf ball make it fly farther?” At that time, I was not able to give the correct answer, quite the opposite, I think those dimples would only bring more resistance to it. Consequently, I had the plan to continue my exploration into this issue afterwards.

In Grade 11, I designed an experiment to explore this topic by building a pendulum, the end of which was a 3D printed structure with dimples on the surface of bob of the pendulum. An air spray gun was used to accelerate its rotation before the structure was released, and the offset of the structure in movement was recorded by a stroboscopic camera from the top view to measure the magnitude of the lift caused by the dimples on the surface. Unfortunately, this experiment ended in failure. One of the reasons was that the structure just fell apart because it could not withstand the kinetic energy brought by the air spray gun. More importantly, I could not guarantee the exclusiveness of the experimental variables due to the insufficient accuracy of 3D printing.

Now that it seemed difficult to make an experiment in real world, I turned my attention to computer modelling. In Grade 12, I grasped certain knowledge of mathematics and physics and was able to use mathematical modelling and Multiphysics simulation to successfully explore this classic problem of fluid with the support of lecturers from Zhejiang University. Finally, I also obtained the surface structure that could make the golf ball fly the farthest.

In conclusion, this paper will investigate the drag reduction theory of a surface dimple structure of a golf ball in Part I, and then apply such a theory to determine the optimum surface dimple concave degree that makes the golf ball fly the farthest in Part II by using computer modelling.

Part I: Theoretical interpretations of golf balls’ drag reduction principles

Flight of objects such as golf balls was assumed in high school’s syllabus to take place through an ideal fluid with zero viscosity, and any moving object was imagined as free from the effects of viscous drags from the ideal fluid. For instance, when we throw a piece of feather and a golf ball simultaneously with the same force toward the same direction through an ideal fluid environment, theoretically speaking, the horizontal displacement of the feather would be greater than that of the golf ball. This is contrary to reality, whereby the golf ball would travel farther than the feather, showing just why the impact viscous drag exerts on the flying object must not be neglected.

However, the drag applied by a fluid on a moving object is more complicated than expected. Apart from the viscous drag, there is also pressure drag induced by flow field change. As shown in Figure 1, the fluid through which a golf ball moves is stratified, forming a fluid boundary layer, where the molecule velocity relative to the golf ball surface gradually increases as it is farther away from the surface.

Fig. 1: Visualising Fluid Flow Velocity with Hypsometric Plots
(Velocity increases with wavelength depicted in figure)

Should the object in flight travel at a relatively high speed, as shown in Figure 2, the fluid boundary layer may momentarily be attached to the spherical surface before segregating from it, forming a vortex behind the golf ball. Such a phenomenon is referred to as boundary layer separation, which is always followed by vortices. It is noteworthy that boundary layer separation is in essence the product of fluid viscosity.

Fig. 2: Visualising fluid flow with streamline plots in 3D

As can be observed from Figure 3, pressure intensity is comparatively low at the centre of the vortex. When a vortex appears, the neighbouring fluid section ahead of the golf ball has a pressure intensity greater than that behind it, leading to pressure drag acting along the direction opposite to its motion. Generally, the faster the golf ball travels, the earlier the boundary layer separation will occur. In which case the vortex behind the ball will have a wider area, and thus a stronger pressure drag is created.

Fig. 3: Visualising Fluid Flow Pressure with Hypsometric Plots
(Pressure increases with wavelength depicted in figure)

Looking at the question raised earlier, as to why the maximum flight range of a golf ball increases with rougher surfaces, the reason lies on the boundary layer, which prevents the tiny dimples on the ball’s surface from stronger viscous drags. Conversely, when adding dimples on the surface, as illustrated in Figure 4, the fluid flow around the golf ball becomes attached to the surface, effectively relieving the vortices behind the golf ball. If the fluid flow through the golf remains laminar, the front pressure intensity will be basically identical to that at the rear, i.e., negligible pressure drags. Hence, drags acting on the ball are fundamentally composed of the fluid’s viscous drag.

Fig. 4: Visualising Fluid Flow with Streamline Plots in 3D

Clearly, the dimples on the golf ball’s surface play a critical role in preventing premature boundary layer from early separation. As can be seen from Figure 5, as the fluid flow meets the front dimples, local separation takes place at the leading edge of these dimples.

Fig. 5: Fluid flow process near concave dimple

The unstable flow caused by the separation in turn gives rise to a fluctuation along a direction perpendicular to the flow, as denoted by the red wavebands in Figure 5. By modelling the local molecular motion trajectory inside the dimple, as shown in Figure 6, such a vibrational motion could increase the kinetic energy of the molecules and in turn lower the local pressure, allowing the fluid boundary layer to reattach at the trailing edge of the dimple and flow down continuously along the spherical surface. This mechanism then analogously repeats itself along the surface downstream until the resulting negative pressure can no longer absorb the fluid boundary layer on that position and beyond, hence giving rise to a complete boundary layer separation. To sum up, the dimples constantly separate and reattach the boundary layer locally, delaying the formation of a complete boundary layer separation on the golf ball’s surface. This is the drag reduction principle governing a golf ball’s surface dimples.

Fig. 6: Motion of red trail particles in figure 5

Through the above theoretical analysis and modelling, the reasons why the surface dimples are capable of lowering drag acting on a golf ball in motion through a fluid medium have been elaborated. Since the factors affecting maximum flight range vary dramatically over different cases. We adopted experimental methods especially modelling in order to obtain the optimal dimple configuration and depth for the maximum design flight range.

Part II: Investigation on optimum drag reduction structure of golf ball

Since the movement of the golf ball is more complicated when considering air drag, modelling is divided into three sections for investigation: motion equation modelling, surface structure modelling and fluid environment modelling. The correlation between the maximum distance of the golf ball and the surface drag coefficient of the sphere will be explored in the motion equation modelling section, while the correlation between the surface drag coefficient of the sphere and the surface dimple concave degree will be investigated during surface structure modelling and fluid environment modelling sections. Eventually, the three models will be combined to evaluate the optimum surface dimple concave degree that makes the golf ball move the farthest.

Section 1: Motion equation modelling

During motion equation modelling, only the translational movement of the golf ball in air-fluid is considered, and the golf ball is only subjected to the flow drag and vertical downward gravity in movement, which has the following definitions:

Flow drag:
- the sum of viscous drag and pressure drag of an object in the fluid environment. When the golf ball moves in the fluid, the direction of flow drag received by the golf ball is opposite to its instantaneous movement direction. There is a formula as follows:

${f_D} = \frac{1}{2}\rho {C_D}{v^2}A = b{v^2}$

Where
- ρ: Fluid density ( $kg\;{m^{ - 3}}$ )
- C_D: Surface drag coefficient (dimensionless)
- v: Velocity of fluid relative to the front of ball
- A: Frontal area of ball
- b: $b = \frac{1}{2}\rho {C_D}A$

Gravity:
- As the field of gravity near the ground can be regarded as a uniform gravitational field, the acceleration of gravity is a constant $g\;\left( {m \cdot {s^{ - 2}}} \right)$ ;

In addition, when constructing the experimental environment: the ambient temperature T and the pressure P have been kept constant.

Fig. 7: Motion of a golf ball in with and without air resistance

The force analysis of Figure 7 shows that the directions of drag received by the golf ball in the vertical direction during the ascending and descending stages are different, which should be considered separately. The motion equation of each stage is as follows:

Mathematical modelling is performed as per the above motion equation, and the relationship between the maximum horizontal displacement of the ball and its surface drag coefficient is studied using MATLAB^®.

When programming with MATLAB^®, below are the following conventions:

A MATLAB^® project is used to calculate the values of variables listed above.

function x =funx(b)

m =0.045;
g =9.8;
v0 =60;
theta =45*pi/180;

t1 = sqrt(m/(b*g))*atan(sqrt(m/(b*g))*v0*sin(theta));
ymax =(m/b)*log((cos(atan(v0*sqrt(b/(m*g))*sin(theta))-sqrt((g*b)/m)*t1))/(cos(atan(sqrt(b/(m*g))*v0*sin(theta)))));
t2 =sqrt(m/g/b)*acosh(exp(b*ymax/m))+t1;
x =m/b*log(1+b*v0*(t1+t2)*cos(theta)/m);

Then using MATLAB^®to draw the graph of function x against b in the domain ${ b|0 < b \le 5}$ .

Fig. 8: Graph of the maximum horizontal displacement x against b¹
(In the domain ${ b|0 < b \le 5}$ .)

The motion equation model points out the positive correlation between the maximum movement distance x of the golf ball and its surface drag coefficient C_D, that is, the smaller the surface drag coefficient C_D of the golf ball, the greater the maximum movement distance x.

The surface drag coefficient C_D of the golf ball and the surface dimple concave degree will be predicted during surface structure modelling section and fluid environment modelling section to find the relationship between the concave degree and the maximum movement distance x.

¹ $b = \frac{1}{2}\rho {C_D}A$ , $\rho$ : Fluid density (kg·m^-3), C_D: Surface drag (dimensionless), A: Frontal area of ball (m²).

Section 2: Surface structure modelling

In the surface structure modelling, the surface model data of the golf ball will be determined and substituted into the fluid environment model to obtain the drag coefficient C_D by means of equivalent substitution.

Therefore, in combination with the above motion equation model, the surface dimple concave degree that makes the golf ball fly the maximum distance is summarised. Autodesk^® 3DsMax^® is used to conduct 3D modelling on the golf ball surface. To facilitate the modelling analysis, it is assumed that the diameters of all dimples of the golf ball are equal.

The parameter definitions are shown in the figure below (Figure 9):

Fig. 9: Three-dimensional depiction of golf ball dimple

The model parameters are as follows:

Draw the function image of $\epsilon$ about $\delta r$ , and take a step length of 10^-3 m in to obtain 11 sets of discrete dimple depths and their corresponding concave degree coefficients as shown in the following table:

To obtain the surface drag coefficient C_D of the golf ball, we will establish a fluid environment model, and substitute the established golf ball model into the fluid environment modelling section for calculation.

Section 3: Fluid environment modelling

Since the surface drag coefficient C_D of the golf ball cannot be obtained directly in modelling, an equivalent substitution method is considered: In the Part I, the author elaborated on the principle of drag reduction of golf ball surface dimples is to prevent the fluid boundary layer from separating from the ball surface prematurely to avoid the formation of turbulence at the back end. Besides, when the surface drag coefficient C_D is larger, the fluid boundary layer is thicker, and the boundary layer is more difficult to separate. As a result, a fluid environment model is constructed as shown in the figure below:

Fig. 10: Side view of golf ball with dimples flowing through air
(Velocity decreases with wavelength depicted in figure)

As shown in Figure 10, the golf ball rotates anticlockwise at a constant angular velocity $\omega$ around the centre axis in the x direction in this model and is subjected to laminar flow with the same initial velocity from left to right. Due to Bernoulli’s Principle considered at a certain point within the fluid:

$p + \frac{1}{2}p{v^2} + \rho gh = C$

where:

p: Pressure (Pa)
v: Velocity (ms^-1)
ρ: Density of fluid (kgm^-3)
g: Earth Gravity (9.810ms^-2)
h: Height (m)
C: A constant

When a fluid is considered incompressible, the higher the fluid velocity, the lower the pressure. The upper surface of the rotating sphere has a higher flow velocity relative to the fluid, while the lower surface has a low flow velocity relative to the fluid, creating a pressure difference between the upper and lower surfaces.

At this point, the sphere receives an upward lift force f, and the magnitude of f is positively correlated with the flow velocity of the fluid on the upper and lower surfaces of the sphere. When the spin angular velocity of the sphere is constant, the relationship between the flow velocity (linear velocity) of the sphere surface and the rotation radius is as follows:

$v=R\omega$

In summary, when the thickness r of the fluid boundary layer increases, the rotation radius R is larger. According to the formula $v=R\omega$ , the flow velocity on the sphere surface is greater. Based on Bernoulli’s Principle, the higher the fluid velocity on the surface of the sphere, the greater the pressure difference between the upper and lower surfaces of the sphere. At this time, the upward lift force f received by the sphere is greater, thereby getting the conclusion of $[f \propto r]$ . Additionally, the thicker the fluid boundary layer, the greater the surface drag coefficient C_D, thus obtaining f is positive correlated with C_D.

The parameters for establishing the fluid model are shown in the Appendix II. The following results are obtained by substituting the 11 sets of golf ball models in Section 2 into the fluid environment:

To sum up, when the concave degree 𝜖 of the golf ball is in the range of 4.4 × 10^-3 to 5.15 × 10^-2 (𝛿𝑟 is in the range of 9.43 × 10^-5 m to 1.0943 × 10^-3 m), the upward lift force f received by the sphere is greater. Due to the wide domain of the above interval, a smaller step length in Section 3 will be used to find the optimum surface dimple concave degree in the domain interval of 9.43 × 10^-5 m to 1.0943 × 10^-3 m.

The golf ball model is established by using a smaller step length (step length: 10^-4 m) within the domain interval of 𝛿𝑟 in the range of 9.43 × 10^-5 m to 1.0943 × 10^-3 m. The golf ball parameters are shown in the following table:

The detailed outcomes of the experiment are shown in Appendix I.

Conclusion

The lift force – time data obtained from the fluid model are summarised in the following table:

Three-dimensional depiction of the data:

Meanwhile, we could obtain the maximum lift force corresponded to different models, as shown in figure 13.

Fig. 13: Three-dimensional depiction of the data

So far, we could obtain the conclusion: Model F experiences the maximum lift force amongst these models thus the maximum C_D as lift force and C_D are positively correlated with each other. Substituting this result back to equation (2-25) in Section 2.1 could prove that the horizontal distance covered by Model F is also the farthest. Consequently, the surface dimple concave degree that will allow the golf ball to fly the farthest is greater than Model E’s and smaller than Model G’s (2.3261×10^-2 < 𝜖 <3.2673 ×10^-2).

Limitation and improvements

Due to the limited calculation power of the devices involved in the project, we did NOT consider:

turbulent flow.
temperature change of the fluid.
change in shape of golf ball.
golf ball with different dimension.
the spin of ball.

While we are holding the project, we determined it is a theoretical experiment without verification in the real world. In the future, if we can get more data, I shall calculate Pearson Product-Moment Correlation Coefficient to analyse the degree of correlation between upward lift force and surface drag coefficient.

Appendix I & II

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致谢

人生十八载年华间，我读过的书，走过的路，遇见的人，引领我将这篇论文送到您的面前。何其有幸，您终于能看到这篇后记——属于我的“文末搁笔，思绪繁杂”的时间。

对该论题的研究贯穿着我的高中阶段的始与终，它是我对时光的一句答复，与对一段来之不易的成长的总结。这篇论文能够顺利地呈现在您的面前，得益于我遇见的人们。我谨藉此机会，感谢他们：

感谢我的指导老师杨超茗先生。他是一位实力超群的Rafflesian，本科与研究生阶段于英国牛津大学(University of Oxford, Wadham College)物理系深造。在他专业的指导与协助下,我们如约完成了论文的翻译与校对工作。他在学术与英语上对我的影响令我终生受用。

感谢Worldshaper Academy的董杉博士与杨巍博士在论文构思阶段给予我的点播，他们的建议很大程度上丰满了论文的骨干。感谢浙江大学谈之奕教授、张程伟讲师与戴晓霞讲师在研究阶段提供的支持，特别是使用MATLAB进行数学建模时提出的方法论让我受益匪浅。感谢浙江大学许辰宇博士对论文写作方面提出的建议，他是我最信任的兄长，他的反馈使得本论文的行文格式更加标准、书面与专业。

感谢我的高中数理化老师张诚博士、张威老师、梁正老师与龚钰老师。在他们孜孜不倦地教诲下，我逐步夯实了自己的物理数学基础，从而有资格开启对该论题的探索。其中，张诚博士曾任职于COMSOL集团，本论文在COMSOL^® Multiphysics 流场建模部分的改良与优化均来自他的建议。

感谢新加坡国立大学(National University of Singapore, NUS)苏重豪教授以及2019 NUS Physics Challenge Camp的组织团队。来自苏教授的推荐信、教授们有趣的课堂以及NUS物理系有爱的氛围……这些宝贵的收获与记忆成为了激励我拼搏的活化能(Activation Energy)，使我更坚定地向着“初心”砥砺前行。

感谢我的伯乐江茹老师与熊辉博士，感谢他们毫无保留地信任我，在我求学遇到困难时尽一切可能地帮助我脱离困境。感谢曾与我同路相伴的D同学，感谢她陪伴我度过迷茫与孤独的时光，祝愿她同样前程似锦。

在此，我还要感谢一位老朋友，他将我领入科学的大门，陪伴我阅读第一本书，带领我迈入人生的第一段路。

他就是您。

遥记儿时我会在您的书上乱涂乱画，您却保留我的“杰作”还教我如何握笔；那时我常常把做实验的小灯泡踩爆，却还如同吃铁饭碗般地当您的“助教”；

想当年顽劣的我用彩笔在白墙勾勒痕迹，您却不慌不忙教我滴定去污剂的配比；不曾想过您那本泛黄斑驳的日记，装载着多少我早已遗失的回忆……

一次次的物质反应，一次次的刨根又问底；

多想回到那次日全食，在您的肩头聆听天体的奥秘；

多想下课时再被您抱起，为期盼放学的学生们打铃。

如今，我已超过了做您学生的年龄；

也懂得了些许化学反应的机理；

这个世界真是奇妙又神奇，却还躲不过小时候您告诉我的“大道理”；

您的一生清贫，挥教鞭舞桃李；

那些桃李已四散去，在最适合的土地上发扬自己。

何其有幸，您终于能看到这篇后记

这篇论文能够顺利地呈现在您的面前，得益于遇见了你。

献给我的爷爷，陈家荣老师 (1951-2021)。

For my dear dear grandpa, Mr. CHEN Jiarong (1951-2021).

陈国毅

2022年3月28日

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