Steady Vs Unsteady flow over a cylinder
Aim - Simulation of flow over the cylinder and analyze the von Karman vortex street at various Reynolds numbers.
Introduction and Theory - In fluid dynamics, vortex shedding is the oscillating flow that takes place when a fluid such as water or air moves past a bluff body. It takes place at certain velocities and depends on the size and shape of the body. The vortices created in the downstream of the body are at low-pressure(adverse pressure gradient) and the body will tend to move in the direction of low pressure(if not fixed at its location).
Source:https://en.wikipedia.org/wiki/K%C3%A1rm%C3%A1n_vortex_street
Physics behind the process:
Source:https://aerospaceengineeringblog.com/boundary-layer-separation-and-pressure-drag/
From the above figure, we can observe that, at the first contact of the fluid with the cylinder the fluid comes to rest, this is the first stagnation point. As the fluid progresses, it follows the curvature(due to centripetal force) and gets accelerated due to the falling pressure(favorable pressure gradient). However, as we approach close to the poles of the cylinder a rising pressure gradient is formed, and the velocity of flow slows down and stops following the curvature due to extra space available downstream of the body. This creates an adverse pressure gradient and the boundary layer separates. Thus downstream of the body, wakes are formed which is also called vortices/eddies. They are generated periodically and can be seen in laminar flows.
Governing Equation:
The frequency at which vortex shedding takes place for an infinite cylinder is related to the Strouhal number by the following equation:
`St= (fL)/U`
Where,
f-Vortex shedding frequency
L-Characteristic length
U-Flow velocity
Other related equations are:
Coefficient of drag:
The drag coefficient is a dimensionless quantity, that is used to quantify the drag or resistance of the object in a fluid environment such as air or water. Lower drag coefficient indicates the object will have less aerodynamic or hydrodynamic drag and vice-versa.
`C_d=(2F_d)/(ρU^(2 ) A)`
Where,
F_d-Drag Force
ρ-density of fluid
U-flow velocity
A-reference Area
Coefficient of Lift:
The lift coefficient is a dimensionless quantity, that relates lift generated by a lifting body to the fluid density around the body, the fluid velocity, and the associated area. A higher lift coefficient indicates the object will produce a greater lift and vice-versa.
`C_L=(2F_L)/(ρU^(2 ) A)`
F_L-Lift force
ρ-density of fluid
U-flow velocity
A-reference Area
Applications-In meteorology, the flow of atmospheric air over islands or mountains causes von Karman vortex streets.
Source:https://en.wikipedia.org/wiki/File:Vortex-street-1.jpg
CASE SETUP:
Pre-Processing
1.Computational Domain:
Using SpaceClaim, the following 2D computational domain was created.
2.Mesh Generation:
Mesh details are:
Element size:0.25 m
Methods:
1. All Triangles method
2. Edge sizing at cylinder walls (based on number of division:1 edge for every 10 degrees,36 edges total)
3. Inflation layers (Based on first layer thickness, first layer thickness:1e-05 m and 55 layers)
The purpose behind creating a body-fitted mesh is to create a fine mesh at the boundary of cylinder walls to capture the physics accurately.
Mesh quality:
The following figures show the mesh quality.
1.Element quality:
2.Aspect Ratio:
3.Skewness:
4. Orthogonal Quality:
3. Fluent setup
First, the mesh check operation was conducted to check if the mesh generated had any errors.
Second, under the physics section in general properties, the following properties were selected.
Solver-Pressure based.
Velocity formulation-Absolute
Time-Based on the case study choose(Steady or Transient)
2D Space- Planar
Reference values:
Model: Viscous model- laminar(Since for external flows, until Re-10^5 all flows are laminar.)
Material:
User-defined material:Density:1 kg/m^3
Viscosity:0.02
Boundary Conditions:
Velocity Inlet - Dependent on case study-
Pressure Outlet - Gauge Pressure(0 Pa)
Symmetry - Top and bottom walls of the computational domain
Walls - Cylinder walls set to walls BC
Wall motion - Stationary wall
Shear condition - No slip
Solutions-
Vertex average plot of velocity magnitude monitor point at 8m from the center of the cylinder.
Coefficient of drag and Coefficient of Lift plots.
Contour plots of velocity and pressure.
Post Processing and Results:
PART-I
For Re-100 ,steady-state simulation
For Re-100,transient state simulation:
PART-II
For Re-10,steady-state simulation:
For Re- 1000 ,steady-state simulation:
For Re-10,000 ,steady-state simulation:
For Re-1,00,000 , steady state simulation
For Re-10,00,000 , steady-state simulation:
Results and Conclusion:
Hence from the above table, we can conclude that the Coefficient of drag decreases as the Reynold number increases, this is because the boundary layer separation occurs later as the Reynolds number increases, decreasing the pressure drag on the body. Even though the skin friction drag increases as the flow becomes more turbulent, the decrease in pressure drag is greater than the increase in skin friction drag.
Also, we can notice from the velocity plots that the vortex shedding phenomena loses its strength and structure as we increase the Reynolds number. However, the pressure contours display an increased pressure in the computational domain as the Reynolds number increases.
Note on convergence: For the steady-state cases, we can conclude from the residuals graph that the simulation/solution has converged(due to fewer variations as the iterations progress )
However, residuals are not the only source to consult to define convergence. We can even check other quantities such as velocity, drag, and lift plots. If the peak remains at the same height and keeps on repeating periodically we can say that the solution has converged.
In the transient simulation, we can observe that the residuals graph shows a sawtooth pattern, but since the peak of the successive spike is constant with each time-step, the solution has converged.
Strouhal Number
The Strouhal number is not defined for steady-state simulation since no time parameter is involved.
From the graph of transient simulation, we can observe that the Strouhal Number is 0.17.
We can also calculate it manually from the lift plot,
Flowtime-70 seconds
Number of peaks or falls-6
f=6/70=0.085 Hz
Therefore,Strouhal number is
St= (0.085*2)/1
St=0.17
Hence validating the result we obtained numerically.