Steady Vs Unsteady flow over a cylinder

VORTEX SHEDDING FROM CYLINDER

Aim:-

To perform Steady and unsteady numerical simulation over a cylindrical surface for the idealization of vortex shedding effect in laminar flow. Monitoring the result for coefficient of drag and lift, and calculating the Strouhal number for specific vortex shedding frequency of a cylinder.

Introduction:-

The vortex shedding phenomena is encountered in many practical engineering application and physical science, and it is important characteristic of flow past a bluff body. flow separated from a slid body consist of alternating vortices of strength that produce their own velocity field, superimposed on the free stream velocity. these shedding vortices cause a oscillating force component perpendicular to the direction of flow, this force can induce a vibration in the body. Vortex shedding behind a circular and square body is a benchmark problem that has been well investigated and addressed in a variety of literature. This phenomenon was first observed in aircraft aerofoils because aerofoils typically exhibit wake vortex shedding frequencies, they are at a considerably risk of fatigue damage . the damage caused by vortex shedding is particularly serve if the vortex shedding frequency approaches a natural frequency of the aerofoil. A similar phenomenon is known to occur in propeller and its particularly concerning in marine propellers because an intense degree of vortex shedding leads to periodic fatigue stress in the blades of the propeller that shorten their service life.

Vincenc Strouhal discovered that the audible tone of a wire singing in the wind is proportional to the quotient of the wind speed by the wire thickness and that each cross-section has a certain proportional coefficient in the noncritical range of Reynolds umber. This is known as the Strouhal number and it is an important parameter to evaluate the vortex shedding frequency. Theodore von karman linked this phenomenon to the stable and staggered vortices that form behind a cylindrical object, which was later termed as Karman vortex effect. Analysed the alternating shedding of vortices from a cylinder and discovered that this process is caused by interaction between vortices on the upper and lower surfaces of the cylinder , once a vortex on a given surface reaches a sufficient strength, it draws the opposing shear layer across the near wake, causing the vortices to shed alternatingly from the upper and lower surface.

Precise numerical simulation is conducted in current case on a flow over a cylinder to demonstrate the formation of Karman vortex street. We have conducted CFD simulation using ANSYS FLUENT in two dimensional flow around a cylinder . We investigate the lift and drag curve for variety of operating condition to reveal their correlation with the vortex shedding frequency.

Geometry modelling and mesh preparation :-

In this study we used a two dimensional model constructed in the ANSYS design modeller. We use a cylinder of diameter of 2 m and then design a enclosure the create a surrounding fluid the enclosure size used is 25m×35m ,a cylinder is placed in middle with along vertical and 8 m horizontal from start. As shown in Figure 1. After that 2D surface is generated and then design is updated for meshing.

Figure 1, Geometry

Figure 2, Mesh Size 200 mm.

In meshing we choose a quadrilateral mesh with a 8 inflection layer. The growth rate of the prismatic layer mesh was set to 1.2 to insure a smooth transition to outer zone. The suitable mesh is chosen after doing grid dependency and it is shown in Figure 2 . For getting accurate result of flow separation after passing on top and bottom layer of cylinder. Boundaries are defined as Velocity inlet, pressure outlet and cylinder wall and other two surface as symmetric because we don’t need wall on that side they are free and open surfaces.

Pre processing:-

Based on the mesh convergence suitable mesh size is selected for all the cases performed . now as we reach at defining numerical section in Fluent . for present simulation we are interested in vortex shedding for Reynolds number (Re) = 100. For Re we have the following equation from that we can get velocity at different Re value

(1)

first of all we choose the Steady state condition with pressure based solution. Then the material is chosen as air with a density of 1 kg/m^3, and viscosity as 0.02 kg/m-s. For our present case we are investigating a vortex shedding in Laminar flow so we choose viscous model as laminar. Boundary condition are defined as for inlet choose velocity inlet and from above equation (1) we calculate velocity as 1 m/s^2. Then for outlet it be pressure outlet with zero gauge pressure because we want calculate a atmospheric pressure. Cylinder wall will be specified as wall with no slip condition. Other two wall will be symmetry as shown in Figure 3,To check for convergence we have to create a monitor point behind the cylinder here we plot a monitor point at x=12m and y=10m. then at this monitor point create a surface report with vortex average velocity at monitor point, then create force report if drag and lift coefficient at cylinder wall. For calculating drag and lift coefficient we have to provide reference values for area =2m, density =1 kg/m^3, length =2 m, velocity 1 m/s and viscosity =0.02 kg/m-s. in solution method chose as Simple scheme for steady case and for transient Coupled scheme then problem is initialized as standard initialization and run the simulation for 300 iterations for steady case and time step of for transient case.

Figure 3, Boundary Name

Dependency of solution on the grid size:-

The mesh size often has an impact on the accuracy of the solution. The size has to be small enough in order to properly resolve the fields that are solved for. To quantitatively compare the result with three different sizes ,mesh sizes of 500, 300, 200 mm. The number of nodes used are tabulated in table 1 .

S.NO	Mesh size	Number of nodes
1	500	6162
2	300	15722
3	200	35108

Table 1.

Figure 4, Velocity Line behind the cylinder

Mesh convergence is tested by using velocity line parameter along the length of enclosure behind the cylinder. the Figure 4, show the comparison of 500,300 and 200 mm size mesh as we see if we shift to 50 mm to 300 mm a significant deviation of curve is noted ,same is deviation is also noted from shifting mesh size to 300 mm to 200 mm. we have also check for 100 mm size but it take much time nearly 4 times than 200 mm size mesh and the deviation between them is much low . hence we choose a mesh size of 200 mm for all our simulations, below table with drag coefficient, lift coefficient and number of node used are tabulated. Figure 5, shows mesh dependency data.

Figure 5, Mesh Dependency

Simulation Result:-

Based on the result of the mesh , a base size of 200 mm were used as the computational parameter of the simulation. The inlet velocity be 1 m/s and Re is at 100 and flow is laminar. The lift and drag of the cylinder were then obtained from these simulation. For steady state a iteration was defined as the abscissa to plot the relative curve.

Figure shows the changes in the lift curve from the beginning of the simulation until a steady state reached as shown in Figure 6 .two different stages can be identified from these plot at the beginning of the simulation the flow field is in an unsteady state, and the amplitude of the lift curve oscillations increase, and after some time oscillation stop changing and the flow field become stable. The lift coefficient is -0.11996439.

Figure 6, Coefficient of lift

Figure 8, illustrate the change in the drag coefficient the curve start dropping and then become steady without further oscillating the Drag coefficient is 1.435012.

Figure 7, Monitor point velocity Figure 8, Coefficient of Dag

Figure 7, show the velocity at monitor point which set behind the cylinder the velocity first start dropping then it start oscillating at a frequency, showing a positive result of generation of vortex with alternating velocity. Also from the convergence plot in Figure 10,we see that all variable first start decrease then become stable ,as solution processed the effect of vortex predominate and residual increase generate a vortex street and again stables itself its shows that vortex are generating and solution goes on indicate a solution is converged. Velocity chart is plotted on a line behind the cylinder as shown in Figure 9 .

Figure 9, Velocity chart on line Figure 10, Convergence Plot

Figure 11, Velocity contour

In present steady state case we have seen vortex shedding as the flow pass the cylinder without generating any vortices. from the Figure 11,the velocity contour we can see a clear evidence in fluctuating of velocity as it pass the cylinder generation of Karman vortex effect.

For Transient case

For transient simulation using 200 mm mesh size, we use Coupled method because it show better improvement in transient case. To converge the simulation when the mesh contain elements, number of time steps be 1800 and time step size of 0.1 s. so total time is 180 seconds. In Figure 12,shows the lift coefficient with flow time in seconds in the beginning of the simulation , the flow field is in an unsteady state ,and the amplitude of the lift curve oscillations increase, after some time the amplitude of the lift curve oscillation stop changing and flow become more stable. Figure 13, for drag coefficient the large drastic change is observed and then when flow field become stable ,it exhibited periodic oscillation. For monitor point the velocity curve first start reduce and then start oscillating as flow field become stable as in Figure 14.

Figure 12, Coefficient of lift

Figure 13, Coefficient of Draft Figure 14, Velocity at monitor point

Figure 15, Convergence Plot

From the convergence plot in Figure 15,we can see that first residual start decrease then when it jump to new time step we can see a step jump in residual again start decrease the when new time step start it again jump. As the solution time increase its value is decreasing means its reaching to convergence.

Oscillating force can induce a vibration in the body, considerably risk to fatigue damage in aerofoils and in propeller. So different conditions are needed to be tested and relation between Re and other output parameter is related in Table 2, here we plot a graph with a Re values ranging from 10,100,1000,10000 and 100000. For this Re VS Drag coefficient is plotted in Figure 16. As the Re increase the flow of liquid n the surface changes from laminar to turbulent as the liquid gain momentum and velocity.in turbulent flow the fluid around the surface sticks around the surface longer thus, there will be smaller wake at the rear end of the object. Smaller the wake of the object will give a high pressure at the rear end of object. Thus there will be a smaller pressure difference between the ends of the object and smaller pressure difference means a smaller pressure drag on object. Thus a smaller drag coefficient.

S.NO	Re	Velocity	Cl	CD
1	10	0.1	-0.00096	3.511714
2	100	1	-0.01199	1.435012
3	1000	10	-0.63608	1.295331
4	10000	100	-0.7364	1.119656
5	100000	1000	0.547535	0.866319

Table 2

Result from our simulation is validate with one of the standard result source from internet on cylindrical object as I Figure 17, and we found that both result show similar variation as the Re value increase draft curve decrease and at lower Re value it have a higher value than 1

Figure 16, Drag curve Vs Re

Figure 17, Drag curve Vs Re curve from Internet

To calculate the frequency of vortex shedding we need a variation of Lift coefficient (Cl) VS Time file ,from that data after reaching steady state calculate the time required for completing a definite number of cycle. Then divide the number of cycle by time required to complete the cycle. Thereby we can get Strouhal frequency (f) and we can calculate Strouhal number (St) by equation

(2)

D= Diameter of cylinder in m.

U= Free stream velocity in m/s.

In present case we can see variation in Lift curve from 120 to 140 flow time and the number of cycle completed was 1.5 therefore our frequency is 0.075. now from equation 2 we get a Strouhal number equal to 0.15.

CONCLUSION:-

We simulated vortex shedding in a laminar flow over an cylinder with Reynolds number 100. In this numerical study we investigate the effect of drag and lift coefficient on the cylinder and demonstrate the effect of vortex shedding in laminar flow. We found that the vortex formation characteristic in both Steady and transient case and theoretically found out the nondimensionalise Strouhal number. Also we discussed the effect of drag coefficient with Reynolds number. The drag coefficient decrease as the Reynolds number increase. As the value of Re increase the flow become laminar to turbulent and hence reduce in drag coefficient .