Week 2 - Flow over a Cylinder.
Flow over a Cylinder: Steady and Transient Analysis in Ansys Fluent.
Objective:
Simulate the flow over a cylinder for steady and transient cases and observe the phenomenon of Karman vortex street. Understanding the vortex shedding for different Reynolds numbers by changing the inlet velocities accordingly. Calculate the coefficient of drag and coefficient of lift for each case. Calculate Strouhal's number for the transient case.
Theory:
Flow over cylinder: In spite of extensive experimental and numerical studies almost over a century, flow around a circular cylinder still remains a challenging problem in fluid mechanics, where intensive investigations are continued even today to understand the complex unsteady dynamics of the cylinder wake flow. Cross-flow normal to the axis of a stationary circular cylinder and the associated problems of heat and mass transport are encountered in a wide variety of engineering applications. Both experimental measurements and numerical computations have confirmed the onset of instability of the wake flow behind a cylinder beyond a critical Reynolds number, leading finally to a kind of periodic flow identified by definite frequencies, well-known in the literature as the Von Karman vortex street. In the case of laminar flow past cylinders with regular polygonal cross-section, the flow usually separates at one or more sharp corners of the cross-section geometry itself, forming a system of vortices in the wake on either side of the mid symmetry plane. On the other hand, for a circular cylinder, where the point of flow separation is decided by the nature of the upstream boundary layer, the physics of the flow is much more complex than what its relatively simple shape might suggest.
Vortex shedding: Vortex shedding is an oscillating flow that takes place when a fluid such as air or water flows past a bluff (as opposed to a streamlined) body at certain velocities, depending on the size and shape of the body. In this flow, vortices are created at the back of the body and detach periodically from either side of the body forming a Von Karman vortex street. The fluid flow past the object creates alternating low-pressure vortices on the downstream side of the object. The object will tend to move toward the low-pressure zone. If the bluff structure is not mounted rigidly and the frequency of vortex shedding matches the resonance frequency of the structure, then the structure can begin to resonate, vibrating with harmonic oscillations driven by the energy of the flow.
Strouhal number: In dimensional analysis, the Strouhal number is a dimensionless number describing oscillating flow mechanisms. For large Strouhal numbers (order of 1). viscosity dominates fluid flow, resulting in a collective oscillating movement of the fluid "plug": For low Strouhal numbers (order of 10 and below), the high-speed, quasi-steady-state portion of the movement dominates the oscillation. Oscillation at intermediate Strouhal numbers is characterized by the buildup and rapidly subsequent shedding of vortices.
The Strouhal number is often given as `S_t=(f*D)/U`
Where.
f= frequency of vortex shedding.
D= Characteristic diameter of the cylinder.
U= Fluid flow velocity.
Co-efficient of drag: In fluid dynamics, the drag coefficient (C) is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment, such as air or water. It is used in the drag equation in which a lower drag coefficient indicates the object will have a less aerodynamic or hydrodynamic drag. The drag coefficient is always associated with a particular surface area. The drag coefficient of any object comprises the effects of the two basic contributors to fluid dynamic drag: skin friction and form drag. The drag coefficient of a lifting airfoil or hydrofoil also includes the effects of lift-induced drag. The drag coefficient of a complete structure such as an aircraft also includes the effects of interference drag.
The drag coefficient C is defined as `C_d=(2*F_d)/(rho u^2 A)`
`F_d`= the drag force, which is by definition the force component in the direction of the flow velocity.
`rho`= mass density of the fluid.
`u`= is the flow speed of the object relative to the fluid.
`A`= the reference area.
Co-efficient of lift: The lift coefficient (C₁) is a dimensionless coefficient that relates the lift generated by a lifting body to the fluid density around the body, the fluid velocity, and an associated reference area. A lifting body is a foil or a complete foil-bearing body such as a fixed-wing aircraft. C is a function of the angle of the body to the flow, its Reynolds number, and its Mach number. The section lift coefficient refers to the dynamic lift characteristics of a two dimensional foil section, with the reference area replaced by the foil chord.
where
L is the lift force,
S is the relevant surface area and
q is the fluid dynamic pressure, in turn, linked to the fluid density, and to the flow speed.
Procedure:
Geometry:
Computational Mesh:
1. Select the edges of geometry and named them as inlet, outlet, wall, and symmetry.
2. Now select the body and go to mesh and click on generate a mesh with mesh size 0.25 m. To ensure triangular mesh, go to mesh> insert > select triangular mesh type.
3.After generating mesh,it is seen that size of cirle looks distorted. Go to mesh > sizing > select circular edge > deffinition >type> no,of divisions >36.
4.To capture better results near cylinder wall, go to mesh > inflation > select the body and edge > inflation option: 1st layer thickness >1 st layer height:5e-3 m >maximum layer :6
Solving:
1. Select laminar flow inside the viscous model.
2.As we wanted to customize material property for simulation, go to materials fluent database > select the material (air) > copy it> rename the name of the material.
Keeping Density: 1 kg/m^3 and Viscosity: 0.1kg/m-s for all the cases which I will be simulating below.
Boundary Conditions:
Outlet: Gauge pressure = 0 Pa.
wall: No-slip condition.
3. Method of discretization: Simple
4. A monitor point is situated at 10 m in the x-direction from the axis of the cylinder to keep track of velocity.
5. Create reports/plots for drag, lift, velocity wrt cylinder.
6. Create plots for the velocity at the monitor point to analyze the vortex shedding.
7. Use hybrid initialization for steady-state and standard for transient state simulations.
Steady-state simulation:
1. Reynolds number = 100.
Velocity = 5 m/s.
No of iterations: 500.
Velocity Contour:
Residual plot:
Velocity-magnitude plot:
Coefficient of lift plot:
Coefficient of drag plot:
Transient simulation:
1.Reynolds number = 100.
Velocity = 5 m/s.
Standard initialization from the inlet is used.
Time-step: 0.1 s.
No. of time-steps: 1500
Velocity Contour:
Residual plot:
Velocity-magnitude plot:
Coefficient of lift plot:
Coefficient of drag plot:
Strouhal Number:
The Strouhal number is a dimensionless number describing oscillating flow mechanisms.
The Strouhal number is often given by the formula: St = f*d/u
Here, we have calculated the Strouhal no. using Ansys fluent.
1.Go to fluent solver > results > fft analyser > load lift coefficient fiel > select plot FFT
Here, the x-axis denotes Strouhal number and the y-axis denotes magnitude.
2.To get the magnitude of the Strouhal number,
go to results > FFT analyzer> click on write FFT to file.
Open the file and the 1st column denotes the frequency and the 2nd column denotes the Strouhal number.
Strouhal Number for current simulation and conditions: 0.170
Velocity contour animation:
Observations:
The transient simulation is more realistic since it takes account of the parameter-time in the governing equation. For the transient process like vortex shedding, it's better to use transient simulation to get realistic output. From around 28000 iterations onwards for the steady-state simulation, we can observe a periodic oscillatory for the residuals which implies that the solution is converged. Also check the quantity of interest which is drag coefficient, lift coefficient, and avg. vertex velocity and above graphs show that all the quantities are in steady-state hence we can say the solution is converged.
Validation:
B.N. Rajani, A. Kandasamy, Sekhar Majumdar,
Numerical simulation of laminar flow past a circular cylinder,
Coefficient of drag for the Re=100 case in the paper is=1.33 and the coefficient of drag for the simulated case is= 1.37. [Results should be validated with any standard literature and error should be within 5% and here it is 2.97%].
Part-2: Steady-state simulations for Re=10, 100, 10000, 100000, and 1000000. Viscosity is kept constant at 1 kg/m-s and velocity is varied to vary Re.
1.Reynolds number = 10.
Velocity = 0.5 m/s.
No of iterations: 500.
Velocity contour:
Residual Plot:
Velocity-magnitude plot:
Coefficient of lift plot:
Coefficient of drag plot:
2. For Reynolds number = 100, it has already been shown in the steady-state analysis above.
3. Reynolds number = 1000.
Velocity = 50 m/s.
No of iterations: 500.
Velocity contour:
Residual plot:
Velocity-magnitude plot:
Coefficient of lift plot:
Coefficient of drag plot:
4. Reynolds number = 10000.
Velocity = 500 m/s.
No of iterations: 500.
Velocity contour:
Residual plot:
Velocity-magnitude plot:
Coefficient of lift plot:
Coefficient of drag plot:
5. Reynolds number = 100000.
Velocity = 5000 m/s.
No of iterations: 500.
Velocity contour:
Residual plot:
Velocity-magnitude plot:
Coefficient of lift plot:
Coefficient of drag plot:
5. Reynolds number = 1000000.
Velocity = 50000 m/s.
No of iterations: 500.
Velocity contour:
Residual plot:
Velocity-magnitude plot:
Coefficient of lift plot:
Coefficient of drag plot:
Results:
Values of coefficient of drag and lift for respective Reynolds number.
Conclusions:
In the first case, the flow over a cylinder is simulated for both Steady and unsteady cases with Re = 100, and with the results generated the phenomena of Von Karman Vortex Street are also observed. Necessary plots are plotted in order to observe that the flow is converged and the vortex shedding is happening. Contours are generated in order to observe the flow over a cylinder. The Strouhal number is calculated for the transient case. The result of the coefficient of drag is validated with a research paper which is mentioned in the references below. In the second part, six different simulations i.e (Re = 10, 100, 1000, 10000, 100000, 1000000) are simulated to observe the behavior of the drag and lift coefficients by changing the Reynolds number. In order to observe the behavior of the drag coefficients for the simulations, the velocity of the flow is varied to change the Reynolds number and the rest of the parameters are kept constant. For Re=10. it is observed that there is no formation of vortex shedding and as we increase the Reynolds no. from 100 to 1000 the vortex shedding street is effectively seen. and for Re = 10000 to 1000000, the vortex shedding is seen to be diffusing and not that prominently seen as in the cases between Re=100 to 1000. As the Reynolds no. increases the coefficient of drag decreases and Lift coefficient increases but for Re= 10000 things get slightly changed and the drag sightly increases and for Re = 100000 lift is slightly decreased.
References: B.N. Rajani, A. Kandasamy, Sekhar Majumdar,
Numerical simulation of laminar flow past a circular cylinder,