Analysis of Laminar Flow over a Cylinder
Introduction
Analysis of flow around a cylinder is necessary as the flow patterns for some range of Reynolds numbers, the body may undergo structural vibrations due to vortex shedding. Vortex shedding is an oscillating flow that takes place when a fluid flows past a body at a certain range of Reynolds number which depends upon the flow velocity, fluid properties, and size and shape of the body. This vortex induced vibrations should be considered in designing of structures otherwise catastrophe may occur due to vibrations.
In this report, steady and unsteady simulations of laminar flow over a cylinder are performed and the vortex shedding phenomena is captured. The Reynolds number used for laminar flow simulations is 100. Strouhal number is a non-dimensional number which dictates the vortex shedding phenomenon.
Re = `frac{rhouD}{mu}`
where 'rho' is the density of the fluid, u is the velocity of flow, D is the diameter of the cylinder and 'mu' is the viscosity of the fluid.
Strouhal Number, S = `frac{fD}{u}`
where f is the frequency of shedding, D is the diameter of the cylinder, and u is the velocity of flow.
Numerical Model
(i) Geometry and Mesh:
The parameters of the geometry are shown below:
Diameter of Cylinder, D = 2 m
Inlet and outlet Length = 10D
Distance between Inlet and Cylinder center = 10D
Distance between Cylinder center and outlet = 20D
The below figure shows the geometry of the computational domain:
The mesh used for computations is shown in below figure:
A fine mesh is used for computations and near the cylinder wall, inflation layers are added so that near-wall effects are captured. The parameters of the mesh are given below:
Element Type - Triangular
Element Size - 0.25 m
Cell Count - 40512
The parameters of the inflation of mesh near the cylinder wall are shown below:
First Layer Thickness - 5e-03 m
Number of Layers - 8
Growth Rate - 1.2
The inflated mesh near the cylinder wall is shown in the below figure:
The mesh quality is shown in below figure, as we can see the quality of elements is greater than 10% and very less number of elements have the quality between 10% and 80%. So overall the quality of the mesh is good and we can proceed for further steps.
(ii) Setup and Results:
The properties of fluid used for simulations are shown below:
Density - 1 kg/m^3
Viscosity - 0.02 kg/m-s
As the Reynolds number used for the simulations is 100, the flow is laminar.
(a)Steady Simulation:
The equations solved are the continuity and momentum equations. The computations are performed using ANSYS FLUENT with pressure-based, steady and double precision solver. The SIMPLE algorithm is selected for the velocity-pressure coupling. The second-order upwind scheme is used for the space discretization for higher accuracy. The boundary conditions for the model are shown below:
Inlet - velocity, u = 1m/s, Normal to the boundary
Outlet - Gage pressure = 0 Pa
Wall - No slip condition
The above figure shows the residuals plot which indicates the convergence of steady simulation. It is observed that the residuals are not changing after 400 iterations. The below figure shows the variation of velocity at a point 8 m away from the cylinder towards the outlet. As we can see the stable fluctuations velocity are seen after 400 iterations. So the number of iterations for convergence is 400.
The above video shows the vortex shedding at Reynolds number 100 in steady-state simulation.
The below figure shows the velocity contour where it can be observed that on the right side of the cylinder wall there is a low-velocity region filled with blue color. The reason for the low-velocity region is observed from the streamlines figure below where one can observe the vortices to the right of the cylinder wall. The streamline figure also shows the wavy nature of the flow due to vortex shedding.
The above figure shows the streamlines of laminar flow in steady simulation. The below figure shows the variation of gauge pressure in the computational domain.
Strouhal Number Calculation:-
As the steady-state simulation is independent of the time domain and gives only the output result, so it is not possible to find the frequency of the vortex shedding. Therefore Strouhal number cannot be calculated in steady-state simulation.
(b)Unsteady Simulation:
The equations solved are the continuity and momentum equations. The computations are performed using ANSYS FLUENT with pressure-based, transient, and double precision solver. The SIMPLE algorithm is selected for the velocity-pressure coupling. The second-order upwind scheme is used for the space discretization for higher accuracy. The boundary conditions for the model are shown below:
Inlet - velocity, u = 1m/s, Normal to the boundary
Outlet - Gage pressure = 0 Pa
Wall - No slip condition
The time step used for the transient simulation is 0.1 second.
The above figure shows the residuals plot. The below figure shows the variation of velocity with time at a point 8 m away from the cylinder towards the outlet. As we can see the stable fluctuations of velocity are seen after 150 seconds. So after 150 seconds, the flow has become steady and the unsteady simulation is said to be converged.
The above video shows the vortex shedding at Reynolds number 100 in a transient simulation.
The below figure shows the velocity contour where it can be observed that on the right side of the cylinder wall there is a low-velocity region filled with blue color. The reason for the low-velocity region is observed from the above streamlines figure where one can observe the vortices to the right of the cylinder wall. The streamline figure also shows the wavy nature of the flow due to vortex shedding.
The below figure shows the variation of gauge pressure in the computational domain:
Strouhal Number Calculation:
In the transient simulation, the frequency(f) of the velocity fluctuations can be obtained from the Frequency spectrum plot below:
The Frequency spectrum plot is obtained by taking the Fast Fourier Transform (FFT) of time domain data of velocity. It is observed from the above FFT plot that the maximum peak is at 0.16983 Hz.
Strouhal Number, S = `frac{fD}{u}`=0.16983*2=0.33966`
Conclusions
The steady and unsteady simulations of laminar flow over a cylinder are performed and the vortex shedding phenomena is captured. The Strouhal number obtained in transient simulation is 0.33