Week 2 - Flow over a Cylinder.

Aim

Simulate the flow over a cylinder and explain the Karman vortex street phenomenon. Understanding the vortex for different Reynolds numbers corresponding to varying inlet velocities. Create a monitoring point behind the cylinder at a distance of four times the diameter that can be used to calculate and analyze vortex shedding.

Introduction 

The flow around a circular cylinder is a simple flow and has been studied for long time.However,it is often by gaining a deeper understanding of simple phenomena that we learn useful lessons for wider contexts.This project shows some of the results of numerical simulations that modelled laminar flow around a circular cylider.An understanding of this flow can yield insights into the flow fields surrounding airplanes,submarines, buildings and bridges.

The laminar region to the turbulent region is governed by a dimensionless number called the Reynolds number, which is the ratio of inertial forces to viscous forces. This number is a measure of various flow criteria in an area that can be calculated from the fluid flow density, flow velocity, characteristic linear dimension depends on the geometry and dynamic viscosity of the fluid. Since the flow in the wake region can have a periodic motion, it can have a certain frequency at which it resonates. If this frequency coincides with the structural frequency of the object, it can cause a phenomenon called resonance. This frequency of motion can be given by another dimensionless number called the grater number, which is used to determine the frequency over the flow. The relationship between the Reynolds number and the grater number can be used to determine the time step for the simulation, since for a particular Reynolds number the grater number is given in many literatures.

Vortex-induced vibrations are one of the key aspects that can lead to the study of vortex shedding in real-world applications such as bridges, aircraft control surfaces, heat exchangers, and many others. Vibrations caused by vortex phenomenon due to flow dynamics need to be studied for very good scale product design and development. The Strouhal number is also used with another number in an axial flow turbine to determine the blade frequency during the flow period.

Strouhal's number:

In dimensional analysis, the Strouhal number is a dimensionless number describing oscillating flow mechanisms.

At large Strouhal numbers (order 1), the fluid flow is dominated by viscosity, which results in a collective oscillatory motion of the fluid "plug". For low Strouhal numbers (of the order of 10−4 and below), the oscillation is dominated by the high-speed, quasi-steady part of the motion. Oscillation at intermediate Strouhal numbers is characterized by the accumulation and rapid subsequent release of vortices.

The Strouhal number is often given as

Where,

f= frequency of vortex shedding

D= characteristic diameter of the cylinder

U'= fluid flow velocity

Reynolds number

The Reynolds number is a key parameter in fluid dynamics.It describes the fluid,geometry and flow.This allows us to compare different fluid systems.It is the measure of the ratio of inertial to viscous forces in the flow of a fluid and defined as ,

where,
$\rho$ = Density of the Fluid
U = Inflow Velocity
D = Diameter of Cylinder
$\mu$ = Dynamic viscosity of Fluid

As we inncrease the flow rate(and Reynolds Number),the flow develops into the variety of structures.After a perticular value of Reynolds number,the flow becomes unstable resulting in a moving wake,famously known as Karman vortex street.This increases drag.

Why does the wake form like this ?.

Near the wall, fluid slows down.Pressure increases as the wall falls away.This pushes the fluid back,which causes boundary layer separation.The separation of the boundary layer can distinguish the wake.The further upstream separation occurs,the more drag is experienced by the cylinder.

Now consider a stream of fluid flow past a circular cylinder.As the fluid passes the top most point of the cylinder,it finds itself
unable to negotiate the rear half of the cylinder where there is an uphill pressure gradient.Hence the fluid separates from the leading surface and the flow is separated individually with highly organized and typically consists of two sequences of vertices,one from each side of the body,with circulations of opposite signs.Depending on the shape and motion of the body more complicated patterns of vortices may arise.

In the wake of the cylinder,the flow pattern consists of an alternating system of vortices or regions of rotation.This repeating pattern of swirling vortices at the wake of the cylinder is known as Karman vortex street.

As stated above vortex street can be observed only over a given range of Reynolds Number.At sufficiently large reynolds number the vortex street persists for many cylinder diameters,but,it can break down far downstream and reorganise itself into a secondary structure.

Procedure

We are supposed to consider 2 cases for simulation,

Case 1) At Re = 100,simulate the flow with steady and transient method and calculate the Strouhal number.

Case 2) Compairing the fluid at steady case with different Reynolds number 10,100,1000,10000 & 100000.

Case 1 here we have take the Re=100 simulate in steady state condition 

Geometry 

Meshing part

Meshing involves creating a small discrete region for calculating the governing equations. The mesh should be fine enough to compensate for any changes that occur during the simulation. The mesh size used for the stationary and unsteady solvers will be the same, which is 0.25 m for the total domain. It is important to capture the flow around the cylinder as a bout of flow separation. An inflatable layer needs to be used to capture the flow near the cylinder. The inflation creates a hexahedral mesh near the cylinder, which causes the solution to converge and provide an accurate solution.

Before creating the inflatable layer, the edge of the cylinder had to be captured using a local edge-size meshing method. Mesh size or number of layers can be used as user input for edge sizing. In this case, the mesh size 250mm. First inflation layer is 5mm and no. of inflation layer is 6.

 

 

 

      

       

 

Setup Part

Setup parameters :

• Viscous model        - Laminar
• Solver type            -  Pressure bases
• Solving Scheme      - SIMPLE
• Initializing method  - Hybrid(for Steady case)
• No. of Iterations      - 1000

                       

   

                                            

 

 

 

result

     

                     Velocity Contour For Re = 100(Steady)                                                       Velocity Contour For Re = 100(Transient)

    

                  Pressure Contour For Re = 100(Steady)                                                   Pressure Contour For Re = 100(Transient)

 

                                  Animation of Velocity Contour Re = 100(Steady) 

 

                                  Animation of Velocity Contour Re = 100(Transient)

    

                                Coefficient of Drag(Transient)                                                                         Coefficient of Drag(Steady)

   

                                Coefficient of Lift(Transient)                                                                         Coefficient of Lift(Steady)

   

                                Scaled Residual Plot(Transient)                                                                             Scaled Residual Plot(Steady)

    

                             Vertex -Avg plot (Transient)                                                                               Vertex -Avg plot (Steady)

 

 

                                     Strouhal number(transient)                                                                      Strouhal number(Steady)

COMPARING THE DRAG, LIFT AND STROUHAL NUMBER

SL.NO	TYPE OF STATE	COEFFECIENT OF DRAG	COEFFECIENT OF LIFT	STROUHLE NUMBER
1	STEADY STATE	1.3384182	0.12879239	0.016000001
2	TRANSIENT STATE	1.3371158	0.16463899	0.159772

 

Conclusion

• In the first case, the flow over the cylinder is simulated for both steady and unsteady cases with Re=100 and Von Karman Vortex Street phenomena are also observed with the generated results.
• The necessary graphs are plotted to observe that the flow is converged and vortex shedding occurs.
• Contours are generated to observe the flow through the cylinder.
• By plotting the power spectral density vs. frequency, the Strouhal number is calculated.

CASE 2 Compairing the fluid at steady case with different Reynolds number 10,100,1000,10000 & 100000.

Reynolds number 10

FOR Reynolds number 10 the velocity is 0.25

   

                               contour plot for velocity                                                                 contour plot for pressure                                           

     

                                     scaled residual plot                                                                   vertex-avg plot

 

                                  Coefficient of Drag                                                                                 Coefficient of Lift

                                           Animation of Velocity Contour Re = 10

 

Reynolds number 100

FOR Reynolds number 100 the velocity is 2.5

 

 

                             contour plot for velocity                                                         contour plot for pressure

 

                                      scaled residual plot                                                                     vertex-avg plot

  

                          Coefficient of Drag                                                                                     Coefficient of Lift

 

                                       Animation of Velocity Contour Re = 100

 

Reynolds number 1000

FOR Reynolds number 1000 the velocity is 25

 

                         contour plot for velocity                                                        contour plot for pressure

 

                                     scaled residual plot                                                                         vertex-avg plot

 

                                  Coefficient of Drag                                                                              Coefficient of Lift

 

                                       Animation of Velocity Contour Re = 1000

Reynolds number 10000

FOR Reynolds number 1000 the velocity is 250

 

                             contour plot for velocity                                                                contour plot for pressure

 

                                     scaled residual plot                                                                                 vertex-avg plot  

  

                                  Coefficient of Drag                                                                            Coefficient of Lift 

            

                                                                Animation of Velocity Contour Re = 10,000

 

Reynolds number 1,00,000

FOR Reynolds number 1000 the velocity is 2500

  

     

                                  contour plot for velocity                                                               contour plot for pressure

 

                                      scaled residual plot                                                                           vertex-avg plot

  

                                           Coefficient of Drag                                                                          Coefficient of Lift 

                                                       Animation of Velocity Contour Re = 10,000

CAMPARING THE DRAG AND LIFT  FOR Re 10,100,1000,10000,100000

SL.NO	REYNOLDS NUMBER	COEFFECIENT OF DRAG	COEFFECIENT OF LIFT
1	RE 10	3.3248068	0.0029474131
2	RE 100	1.3384182	0.12879239
3	RE 1000	0.69777409	-0.023485219
4	RE 10000	0.83163624	0.2847105
5	RE 100000	0.82699989	0.42198444

Error with respect to the theoretical values from the reference material                                                            

Reynolds number	theoretical Cd	Numerical Cd	error
10	3.5	3.3248068	4.88%
100	1.4	1.3384182	3.5%

Effect of Re number on Drag 

As it can be seen through the results,the coefficient og drag keeps decreasing whe the reynolds number is increasing.This is because, the flow becomes turbulent for reynolds number greater than 2000,which means laminar flow though present is negligible leading to a lower skin friction drag.Also,for low reynolds number(Re=10),the alternating vortices cannot be visualized due to the low velocity values.In such case only boundary layer separation occured.

@ Re = 10

At very slow flow there is no separation of flow and no wake downstream of the cylinder,as wee have neglected viscosity.Since the flow
is symmetric from upstream to downstream,there is no drag on the cylinder.But this type of flow does not occur in nature,where there
is always some small amount of viscosity is present in the fluid.

@ Re = 100

Stable vortices are formed.The flow is separated but steady.The vortices gererate a high drag on the cylinder.

@ Re = 1000

As the flow velocity increases,the down stream vortices becomes unstable,separate from the body and are alternatively shed from the downstream.The wake is wide and generate large amount of drag.

@ Re = 10000

The flow velocity is increased even more and the periodic flow breaks down into chaotic wake.The flow is laminar and orderly while on the back side of the cylinder,it is turned into a chaotic wake causing drag.

@ Re = 100000

The boundary layer transioned into turbulent flow and vortices of different scales are being shed in the wake region.But since the separation point is slightly downstream from the laminar separation point,which cause a slightly smaller wake and the drag is than the corresponding laminar drag.

Conclusion 

• five different simulations i.e. (Re = 10,100,1000,10000,100000) are simulated to observe the behavior of drag and lift coefficients by varying the Reynolds number.
• In order to observe the behavior of the drag coefficients for the simulations, the flow velocity is varied to vary the Reynolds number and the other parameters are kept constant.
• As we know, if Reynold's number is less than 2000, then the flow in the pipe is laminar. Therefore we use the viscous model = laminar.
• For Re=10, no vortex formation was observed and in the remaining cases it is clearly observed.
• According to the basic conditions and parameters, the results are validated from the literature.
• As we refine the mesh, we need to run more iterations.
• From the above table it can be concluded that the error percentage is less than 5%
• The Karman Vortex Street phenomenon for flow over a cylinder is studied and observed in this project.
• Increasing velocity eventually brings the turbulent drag up to and even higher than laminardrag value,but during transition from laminar to turbulent ,there is a range of reynolds number,for which turbulent drag is less than laminar drag.
• Vortex shedding can be seen i the case of both steady and transient simulation.But transient simulation produces better and more realistic results.
•  Strouhal Number calculation is only possible in the case of transient simulations as the frequency changes with respect to time.

Reference 

https://www.sciencedirect.com/science/article/pii/S0307904X08000243

https://www.researchgate.net/publication/312324020_Topological_fluid_mechanics_of_the_formation_of_the_Karman-vortex_street

Simulation_of_Cross_Flow_Induced_Vibration.pdf

Numerical simulation of laminar flow past a circular cylinder - ScienceDirect