Aerodynamics of Sphere

** The article below has been compiled from various sources online and through other various video from platform such as youtube to provide a proper understanding of aerodynamics and fluids. **

Understanding fluid

Fluid is understood as the homogeneous matter of the body where molecules are loosely bound together, which does not have any specific shape or form and can take the form of the container. The molecules in fluid are in a constant motion and are loosely bound together, This means that molecules collide with each other and with the surface of the solid object and hence there is transfer of kinetic energy with each other and with the solid surface.
This  mutual transfer of kinetic energy means that the aerodynamics can be viewed from two different perspectives. One with the  perspective of the fluid itself and other with the perspective of the solid. Each of these perspective poses a challenge, in the perspective of the fluid the major challenge is to explain the motions of all particles making up the fluid itself, whereas in the point of view of solid the challenge is to explain the force generated by the fluid .Often it is the net airflow that is important as it is looked from a more practical point of view.
Aerodynamics can be divided into two major categories based on the airflow, one being the internal airflow like aerodynamics experienced by a tube or pipe, where the air flows inside of the body, and other being the external airflow. Here we will  be more focused on aerodynamics outside of the body. To do the analysis of the body we generally keep the body stationary and pass air with the velocity (v) this is called the steam velocity and is uniform without any change or fluctuation in density or pressure. It is a vector quantity ie itt has both magnitude and velocity.
When an air flow begins from a great distance from a body and encounters the body the forward facing part of the body pushes the air laterally outwards relative to the free steam velocity. As The flow is forced laterally its velocity increases and after the flow has passed the maximum cross sectional area of the body the flow converges laterally inwards back to the original flow direction.
The continuous path made by the air molecules as it approaches , encounters and then passes the body is called streamline. It is a representation of the natural path followed by the molecules as it moves from the direction of the high pressure in the direction of low pressure . The pressure varies in different parts of the air mass hence the molecules at slightly different positions have different paths and the sum of all these paths represents the basic flow field  around the body. A body may contain a stagnation point which is a point in the body where the streamlines separate.  It has a characteristic that when air molecules hitting the sagination point come to complete rest. The airflow experienced near the stagnation point will experience reduction in velocity relative to free steam velocity. When the air mass has its velocity reduced from initial value the kinetic energy is turned into pressure. Since this pressure is derived from the motion of the air, it is represented by the letter Q.
Q = 12v2
 where  is the density of air and v is the free stream velocity. Hence It can be seen that Q will be maximum at the streamline impacting the stagnation point and will slowly decrease as we move away from the stagnation point until it becomes zero where the streamline is not in contact with the body.
The dynamic pressure acting on a small increment on a surface area of the body exerts a force , if the force is acting in the same direction as the free stream velocity it is referred to as the drag force, and The component of this force that acts right angle to the velocity direction is called lift. The following picture shows that force and the direction it acts .

A point to note is that the aerodynamics forces are all surface forces; we can see this if we add all the incremental forces generated by the dynamics pressure at everypoint. A given body subjected to aerodynamics  The flow field of one object having different size can be used to generalize the flow field of the similar type of objects at different sizes.

Aerodynamics of a sphere

Sphere is a symmetric shape and  is the same in all sides hence the angle of attack never varies. Even then the flow around the sphere can be more complex for  other reasons. One reason is that at the molecular level there are other physical forces that affect the motion of air molecules.
It comes down to gas composition, density and size of an object .  These things were recognized and hence by the scientist Osborne Reynolds a quantitative measure of this balance was created and  now is referred to as reynolds number.
Reynolds number is the ratio of inertial force to viscous force
Re = L =L
Where,

• ⍴ is the density of the fluid in kg.m-3
• 𝜇 is the dynamic viscosity of the fluid in N.s.m-2
• 𝜈 is the kinematic viscosity of the fluid in m2.s-1
• u is the velocity of a fluid with respect to the object in m.s-1
• L is the characteristic linear dimension in m
• if the Re > 2300  the flow is called laminar , and 2300<Re<4000 is said to be transient and if Re > 4000 it is said to be turbulent flow

Hence the aerodynamics differ from laminar , to turbulent flow. Considering the reynold number to be low then the streamlines would be closely attached to the spherical shape , it would be adjacent to the surface of the circular arc.  So in laminar flow the streamlining will separately radically from around the stagnation point on the nose and rejoin each other at a point 180 degree away from the sphere. 

If there is no viscosity in the fluid it can be shown that the  pressure distribution on both the front and back side will be equal and will have no drag.  hence drag force can be numerically represented as,

FD   =     drag
p   =     density of fluid
v    =     speed of the object relative to the fluid
CD    =     drag coefficient
A    =     cross sectional area

At molecular level the viscosity keeps the streamlined attached to the surface and as the velocity increases the air molecules tend to flow in the direction of free steam velocity and as the molecules cross the front half they begin to  detach from the surface and no longer converge.  This is called the detached flow.  And when the flow reaches the  critical balance between the momentum and viscosity it will change to turbulent flow which is referred to as the critical Reynolds number . i.e Re > 4000 as written above. As the Reynolds number increases the detachment point goes further back till the half point and since stable flow is not possible the detached flow will be unsteady. 

 

 

In the above figure taken from A comparative study of Football Aerodynamics by firxo alam , harun chowdhury, Hazim Morai and Franz Konstantin fuss, we can see two different types of football showing two different behaviors even though their shapes are similar.

For any solid object with the cross-sectional area (A) and dynamic pressure (Q) and multiplying them results in the maximum drag force experienced by the body but no body really experiences the maximum dynamic pressure over its entire face. It will only experience some fraction of that pressure and is referred to as drag coefficient.

F =Q* A *Cd



From the chart that has been drawn out from that study we can see that the drag coefficient has decreased as the Reynolds number increases. We can now say that as the Reynolds number increases the drag experienced by the body decreases . Especially in the smooth surface, drag is significantly decreased as the Reynolds number increases.

As free Steam Velocity approaches velocity of sound other complicated factor arises called the compressibility of air and the ratio of these two velocities is called The Mach number and is given by

m=v/c

As the free steam velocity approaches the M=1, it approaches the average random velocity of air molecules and as the local airflow exceeds the average velocity the air molecules cannot move fast enough and they pile up creating what we know as shock waves. And When Airflow around the body reaches m=1 or exceeds that value by even a slight amount it will be referred to as a transonic condition and in this condition the momentum and velocity are modified by the presence of shock waves. Approximately at M = 5 the air shockwaves start to converge and come back to stable form and the flow comes back to simple form. The velocity above the approximation is said to be hypersonic and its behavior to become simpler is called hypersonic invariance.

Magnus effect

The magnus effect is a phenomenon related to spinning objects moving through a fluid. The path of the object is deflected in such a manner that is not possible when the object is not spinning.

As the ball rotates the air going upward increases in velocity and thus creates a low pressure zone, whereas when the air goes downward the velocity is low compared to upward and thus pressure is increased. This creates a force pushing upward called the magnus force which intern creates lift.

Conclusion

Often looking at the aerodynamics of an object we tend to look at the net air flow which is not that difficult to explain but there is a lot more going on when trying how such air flow came to be. The aerodynamics of a sphere itself differ from condition to condition, it is different in laminar flow then compared to turbulent flow . This is because when looking at aerodynamics of a body we are not only looking at how the air flows around the body because it gets impossibly difficult to explain each and every molecules of air interacting with the body, so we use abstract variables like pressure , viscosity, free stream velocity, Drag coefficient, Reynolds number , mach number and many more. Sphere is practically the most simplest object to analyze, since it is equal on all axes but even then the airflow around the sphere can be highly complex. Due to the high level of geometric symmetry , a simple translating sphere cannot produce lift but a spinning sphere can displace enough air to produce lift although that lift is not ideal it is produced due to magnus effect.

 

References

Aerodynamics of Spheres,Larry Lemke,An examination of the flight characteristics of sphere,
Firoz Alam, Harun Chowdhury, Hazim Moria, Franz Konstantin Fuss,A comparative study of football aerodynamics,Volume 2, Issue 2,2010,