Flow over a cylinder

AIM: To Understand and Simulate the Transient Flow over a Cylinder in SolidWorks and observe the variation of velocity and pressure in a domain for three different Reynolds Numbers.

INTRODUCTION:
Flow over a cylinder is a classical problem for understanding the flow over an obstacle. The geometry of the cylinder is simple and allows us to understand all the complexities that occur and turbulence for different Reynolds Numbers. Reynolds Number helps us to understand the flow patterns of the different fluid flow situations. The different range of Reynolds Numbers represents different flow patterns. At low Reynolds Numbers, the flow pattern is observed to be Laminar i.e., Smooth, Constant fluid motion. At higher Reynolds Numbers the flow pattern is observed to be Turbulent i.e., Chaotic eddies, Vortices, and Flow instabilities. In this project, we will try to understand the variation of velocity and pressure in certain flow directions for different Reynolds Numbers by simulating it with proper boundary conditions, Mesh sizing, Convergence criteria using SolidWorks Flow Simulation.

THEORY: 
Before going further let us understand some important concepts:
Reynolds Number: It is defined as the ratio of inertial forces to viscous forces within the fluid which is subjected to relative internal movement due to different fluid velocities. The region where these forces change behavior is known Boundary layer. This relative movement generates fluid friction, which is a factor in developing turbulent flow. Counteracting this effect is the viscosity of the fluid, which tends to inhibit turbulence. The Reynolds number quantifies the relative importance of these two types of forces for given flow conditions and is a guide to when the turbulent flow will occur in a particular situation. This ability to predict the onset of turbulent flow is an important design tool for equipment such as piping systems and aircraft wing.

With respect to laminar and turbulent flow regimes:

• laminar flow occurs at low Reynolds numbers, where viscous forces are dominant, and is characterized by smooth, constant fluid motion;
• turbulent flow occurs at high Reynolds numbers and is dominated by inertial forces, which tend to produce chaotic eddies, vortices, and other flow instabilities.

Mathematically Reynolds Number is defined as:
`Re=(rhovD)/mu=(vD)/nu`

`rho`= Density of the fluid.
`v`= Velocity of the fluid.
`D`= Diameter of the tube or cylinder.
`mu`= Dynamic viscosity.
`nu`= Kinematic viscosity.

Understanding the flow over a cylinder consider the following illustration:

From the above Illustrations, there are five terms that need to be discussed:

1. Stagnation Point.
2. Point of Maximum velocity.
3. Boundary-Layer.
4. Flow Separation.
5. Wake.

Boundary-Layer:
a boundary layer is the layer of fluid in the immediate vicinity of a bounding surface where the effects of viscosity are significant.
The viscous nature of airflow reduces the local velocities on a surface and is responsible for skin friction. The layer of air over the wing's surface that is slowed down or stopped by viscosity, is the boundary layer. There are two different types of boundary layer flow: laminar and turbulent.
Laminar boundary layer flow
The laminar boundary is a very smooth flow, while the turbulent boundary layer contains swirls or "eddies." The laminar flow creates less skin friction drag than the turbulent flow but is less stable. Boundary layer flow over a wing surface begins as a smooth laminar flow. As the flow continues back from the leading edge, the laminar boundary layer increases in thickness.
Turbulent boundary layer flow
At some distance back from the leading edge, the smooth laminar flow breaks down and transitions to a turbulent flow. From a drag standpoint, it is advisable to have the transition from laminar to turbulent flow as far aft on the wing as possible or have a large amount of the wing surface within the laminar portion of the boundary layer. The low energy laminar flow, however, tends to break down more suddenly than the turbulent layer.

Flow separation: 
It is important to understand when and why the flow separates when fluid flows through the obstacle or cylinder in this case. In boundary-layer, due to viscosity, there is a change in pressure along the surface of the body. This causes the boundary layer to be sensitive to the pressure gradient (as the form of a pressure force acting upon fluid particles). If the pressure decreases in the direction of the flow, the pressure gradient is said to be favorable.   In this case, the pressure force can assist the fluid movement and there is no flow retardation.  However, if the pressure is increasing in the direction of the flow, an adverse pressure gradient condition as so it is called exist.  In addition to the presence of a strong viscous force, the fluid particles now have to move against the increasing pressure force.  Therefore, the fluid particles could be stopped or reversed, causing the neighboring particles to move away from the surface.  This phenomenon is called the boundary layer separation or flow separation.

Wake: 
Consider a fluid particle flows within the boundary layer around the circular cylinder.  From the pressure distribution measured in an earlier experiment, the pressure is a maximum at the stagnation point and gradually decreases along the front half of the cylinder.  The flow stays attached in this favorable pressure region as expected.  However, the pressure starts to increase in the rear half of the cylinder and the particle now experiences an adverse pressure gradient.  Consequently, the flow separates from the surface and creating a highly turbulent region behind the cylinder called the wake.  The pressure inside the wake region remains low as the flow separates and a net pressure force (pressure drag) is produced.

Before understanding the other two terms, It is now a good time to understand the Flow over a Circular Cylinder. Let us assume the flow is ideal, then the stream function and velocity potential for this flow are:
`psi = U_inftyrsintheta - (Ksintheta)/r`
`phi = U_inftyrcostheta + (Kcostheta)/r`
The components of the velocity around the cylinder are given w.r.t `r`and`theta`:
`v_r = 1/r(delpsi)/(deltheta) = costheta(U_oo - K/r^2)`
`v_theta= -(delpsi)/(delr) = -sintheta(U_oo + K/r^2)`
The radial velocity is zero when
`K/r^2=U_oo`
If we consider this particular streamline as the surface of the cylinder then the radius of the cylinder `a` is given by,
`a^2=K/U_oo`
The equations of the streamline, velocity potential, and the velocity components are replaced by,
`psi = U_oor(1-a^2/r^2)sintheta`
`phi = U_oor(1+a^2/r^2)costheta`
`v_r = U_oo(1-a^2/r^2)costheta`
`v_theta = -U_oo(1+a^2/r^2)sintheta`
The velocity components on the surface of the cylinder are obtained by putting `r=a` in the above expressions. Accordingly,
`v_(rs) = 0`
`v_(thetas) = -2U_oosintheta`
As `sintheta` has a zero at `0^@` and `180^@` and maximum of one at `theta = 90^@ and 270^@`

Stagnation Point: 
From the results, stagnations points are where the velocity is minimum i.e., zero.

Point of Maximum Velocity: 
Point of Maximum velocity is termed where velocity is maximum i.e., at `theta = 90^@`and`theta = 270^@`.

The above equations are considered assuming that the flow is ideal with velocity `U`. But, in this project, we would like to observe the behavior of velocity and pressure for transient flow i.e., the velocity of the flow changes w.r.t time. It will be observed that Stagnation Point and Point of Maximum velocity will no longer be a point but will be observed as a region.

Before proceeding with the flow simulation it is important to have an understanding of how different Reynolds Numbers affect the flow:


Above is the Illustrative tabular data on how to flow pattern changes for different Reynolds numbers.

SIMULATION SETUP:
Before starting the simulation we need to prepare a solid cylinder by making a sketch and extrude it to a certain length using the feature tab in Solidworks. A solid cylinder of 50mm radius and a 100mm length is prepared. 
Pre-Processing
Now, Using Solidworks flow simulation add-in and using wizard feature in flow simulation tab, fluid flow properties, fluid flow type, reference flow velocity are selected. Also, set a time-dependent study for this model in the wizard. Set the reference line velocity to be 10m/s in positive X-direction. As the fluid selected is Air which has the following properties:
`rho = 1.225(kg)/m^3`
`mu = 1.81**10^-5(kg)/(m*s)`
at `Temperature=K`and`Pressure = Pa`.
Thus resulting in Reynolds number of Re = 33,839 and computational domain length of 0.45m
Solving
Total simulation time is decided by calculating the start time and end time that the flow takes. Total simulation time is taken slightly more than the end time of the flow. 
Start time is calculated as (Total computational domain length)/Fluid velocity. The end time is calculated as 2 times the start time.
In this case,
The start time is 0.045sec and the end time is 0.09sec. For good simulation animations let us run the simulation for 2sec.
Mesh size is set at LEVEL 6.
Now run the simulation.

This is just a baseline simulation and the values are taken above are baseline values. Now consider three cases where values of Reynolds Numbers are 20%, 40%, 100% of the baseline Reynolds Number value. The remaining values are calculated with the same computational domain length as baseline computational domain length value and are tabulated below:

Run the simulation for each case for 2sec.

RESULTS and OBSERVATION:

Case 1





OBSERVATIONS
As the Reynolds Number is 40,607, it is in the range of 300In which the flow regime is Subcritical, the wake is completely turbulent and has Laminar boundary layer separation.
By observing the flow in velocity contour plot, blue color shows that the velocity is zero near the surface at the left and right of the cylinder which is called stagnation points. As observed in velocity contours near the stagnation points, the contour before the stagnation point is observed to be green, and further, it is observed to be the velocity of the flow.
When observed to the right side of the flow it is blue near the surface indicating a stagnation point where velocity is zero, it is also observed that the contour changes its color from blue to lite greenish-blue indicating there's an increase in velocity and there is change again to blue indicating drop in velocity. Thus indicating that the flow is turbulent in the wake.
In Pressure contour, it is observed that pressure is maximum at the point where velocity is minimum and minimum at the point where velocity is maximum.

Case 2




OBSERVATIONS
The Pressure contour is observed to be similar to Case 1. 
The Maximum velocity contour i.e., viz in the red color region is observed to lesser than the contour which is in Case 1. Remaining contours are approximately similar to the contours in case 1 as the Reynolds Number is still in the range 300
Case 3





OBSERVATIONS
The Pressure contour is observed to be similar to Case 1 and Case 2. 
The Maximum velocity contour i.e., viz in the red color region is observed to lesser than the contour which is in Case 1 and Case 2. Remaining contours are approximately similar to the contours in case 1 and case 2 as the Reynolds Number is still in the range 300

CONCLUSION:
All flows examined in this flow fall under the uncritical flow regime around the cylinder. Therefore, at the macroscopic level of research, the results observed arbitrarily of velocity and pressure are more or less the same.

REFERENCES:

https://www.ecourses.ou.edu/cgi-bin/ebook.cgi?doc=&topic=fl&chap_sec=07.4&page=theory

https://www.slideserve.com/justus/13-42-lecture-vortex-induced-vibrations

https://en.wikipedia.org/wiki/Flow_separation#:~:text=Flow%20separation%20or%20boundary%20layer,a%20widening%20passage%2C%20for%20example.

http://www.eng.fsu.edu/~shih/succeed/cylinder/cylinder.htm

https://en.wikipedia.org/wiki/Boundary_layer

https://en.wikipedia.org/wiki/Reynolds_number

https://en.wikipedia.org/wiki/Potential_flow_around_a_circular_cylinder#:~:text=In%20mathematics%2C%20potential%20flow%20around,is%20transverse%20to%20the%20flow.&text=Unlike%20a%20real%20fluid%2C%20this,known%20as%20d'Alembert's%20paradox.