Week 2 - Flow over a Cylinder.

Aim

• Explain the phenomenon of Karman vortex street.
• Simulate the flow over a cylinder ( with steady and unsteady case)
• Calculate the strouhal number for Re = 100
• Ccalculate the coefficient of drag and lift over a cylinder by changing the values of Re by 10,100,1000,10000 & 100000(Use steady solver)
• Discuss the effect of Reynolds number on the coefficient of drag. [ Results should be validated with any standard literature and error should be within 5 %]

Result requirements

1. In both approaches, show that the flow has converged. That is, what quantities or plots need to be looked at to determine that the flow has converged?
2. Plots of the coefficient of drag and lift.
3. Plots to show vortex shedding behind the cylinder.
4. Mention the referred material at the end of the challenge submission page.

Introduction

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 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.

As the flow Reynolds number (Re=UD/ν)">(Re=UD/ν) based on the free stream velocity (U)">(U), cylinder diameter (D)">(D) and kinematic viscosity ν of the fluid, changes from a creeping laminar flow value of the order of 0.1 to a turbulent flow of the Re of a million or even higher, variety of physical complexities start taking place. Steady laminar flow exists at Reynolds number between 5 and 40 with a pair of symmetric counter-rotating vortices formed behind the cylinder. Beyond a critical value of Re, depending on the other flow disturbances, a transverse oscillation sets in with loss of flow symmetry and vortices are shed from the cylinder surface. Between Re=190">Re=190 and 260, wakes of two dimensional cylinders are observed to become susceptible to a primary instability mechanism which leads to the amplification of three dimensional disturbances and eventually to the development of strong streamwise oriented vortical structures. These three-dimensional disturbances alter the structure and evolution of the wake vortices and as a result, even at low Reynolds number, two-dimensional simulations often fail to accurately predict even gross flow quantities like mean drag coefficient and root mean square (RMS) of the lift fluctuation.

Karman vortex street

Kármán vortex street (or a von Kármán vortex street) is a repeating pattern of swirling vortices, caused by a process known as vortex shedding, which is responsible for the unsteady sepertion of flow of a flow around blunt bodies.

Vortex shedding is an oscillating flow that takes place when a fluid such as air or water flows past a bluff (as opposed to 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 krman 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. This vibration is the cause for overhead power line wires humming in the wind, and for the fluttering of automobile whip radio antennas at some speeds. Tall chimneys constructed of thin-walled steel tubes can be sufficiently flexible that, in air flow with a speed in the critical range, vortex shedding can drive the chimney into violent oscillations that can damage or destroy the chimney.

Strouhal number

In dimensional analysis, the Strouhal number is a dimensionless number describing oscillating flow mechanisms. The parameter is named after Vincenc strouhal, a Czech physicist who experimented in 1878 with wires experiencing vortex shedding and singing in the wind.The Strouhal number is an integral part of the fundamentals of fluid mechanics.

The Strouhal number is often given as

where f is the frequency of vortex shedding, L is the characteristic length (for example, hydraulic diametre or the airfoil thickness) and U is the flow velocity. In certain cases, like heaving (plunging) flight, this characteristic length is the amplitude of oscillation. This selection of characteristic length can be used to present a distinction between Strouhal number and reduced frequency:

Geometry

Mesh

• Triangular mesh is used
• Mesh is further refined at cylinder edges with element size 0.2m
• Mesh size is set to 0.5
• 6 Inflation layers are added of thickness 0.272
• Created named selection for the boundaries as inlet x, outlet x and two symmetry

Setup

• Laminar viscous model is used
• Material is set to air of viscosity 0.02 and density of 1
• For the boundary condition the velocity at inlet is set as required (for the fisrt case it is set to 1m/s)
• Reference values have been changed as area as 2 square metres and length as 2 metres,(since it is a 2D simulation, when the cylinder is viewed from the inlet i would be a line of area 2 m^2 and same goes with length)and also the velocity is changed accordingly
• The solution method is set to SIMPLE scheme for steady state analysis and Coupled for transient state analysis
• To check the vortex flow I have created a monitor point, where vortex avarage option is selected from definitions at this point
• Drag coefficient and lift is selected from definitions to observe the changes
• A velocity contour is created from the results section
• Also to capture the velocity changes such as vortex shedding Solution animation is used and gave the option to capture the velocity contour at every 10 iterations

Outputs

Transient flow(Re = 100)

Vertex avarage of velocity

Lift

Coefficient of drag

• The vortex shedding starts when the vertex avarage velocity changes continues like a wave, we can also observe vortex shedding in Lift plots in same way
• Strouhal number is found using FFT(Stands for Fast fourier transformation) in Fluent. The obtained FFT plot is given below, So this plot is converted into excel form which will have two coloumns of values the coloumns represent Coefficient of lift and corrosponding Strouhal number , From this the maximuim magnittude of Cl (lift coefficient) is found and the corrosponding value will the strouhal number for the current problem

It is converted to an excel sheet which is given below

(title "Spectral Analysis of cl-rfile")
(labels "Strouhal Number" "Magnitude")

((xy/key/label "cl")
0.0000000e+00	4.4779037e-03
7.9840319e-03	1.6629067e-04
1.5968064e-02	1.9957649e-04
2.3952097e-02	2.2523115e-04
3.1936128e-02	2.5934514e-04
3.9920159e-02	3.0531461e-04
4.7904193e-02	3.7784420e-04
5.5888224e-02	4.7444663e-04
6.3872255e-02	5.9122685e-04
7.1856290e-02	7.2909635e-04
7.9840317e-02	8.9111016e-04
8.7824352e-02	1.0819740e-03
9.5808387e-02	1.3194800e-03
1.0379241e-01	1.6185646e-03
1.1177645e-01	2.0223351e-03
1.1976048e-01	2.5940770e-03
1.2774451e-01	3.6500723e-03
1.3572854e-01	4.6543796e-03
1.4371258e-01	2.0192415e-02
1.5169661e-01	1.7782271e-02
1.5968063e-01	9.5566260e-03
1.6766468e-01	4.7620693e-03
1.7564870e-01	3.0365917e-03
1.8363273e-01	2.2975595e-03
1.9161677e-01	1.8541630e-03
1.9960080e-01	1.5615014e-03
2.0758483e-01	1.3531264e-03
2.1556886e-01	1.1984465e-03
2.2355290e-01	1.0788265e-03
2.3153692e-01	9.8384544e-04
2.3952095e-01	9.0541027e-04
2.4750499e-01	8.4180303e-04
2.5548902e-01	7.8799773e-04
2.6347306e-01	7.4174267e-04
2.7145708e-01	6.9933920e-04
2.7944112e-01	6.6947658e-04
2.8742516e-01	6.3223334e-04
2.9540917e-01	5.8985653e-04
3.0339321e-01	5.6847150e-04
3.1137726e-01	5.5815338e-04
3.1936127e-01	5.3790107e-04
3.2734531e-01	5.1758200e-04
3.3532935e-01	4.9761351e-04
3.4331337e-01	4.8039021e-04
3.5129741e-01	4.6363793e-04
3.5928145e-01	4.4759925e-04
3.6726546e-01	4.3267140e-04
3.7524951e-01	4.1876009e-04
3.8323355e-01	4.0528851e-04
3.9121756e-01	3.9336804e-04
3.9920160e-01	3.8189287e-04
4.0718561e-01	3.7118845e-04
4.1516966e-01	3.6193078e-04
4.2315370e-01	3.5255161e-04
4.3113771e-01	3.4625106e-04
4.3912175e-01	3.3807789e-04
4.4710580e-01	3.2088760e-04
4.5508981e-01	3.1283981e-04
4.6307385e-01	3.0891201e-04
4.7105789e-01	3.0553658e-04
4.7904190e-01	3.0065372e-04
4.8702595e-01	2.9502562e-04
4.9500999e-01	2.9012377e-04
5.0299400e-01	2.8510729e-04
5.1097804e-01	2.7908376e-04
5.1896209e-01	2.7467226e-04
5.2694613e-01	2.6985985e-04
5.3493011e-01	2.6528211e-04
5.4291415e-01	2.6064349e-04
5.5089819e-01	2.5664485e-04
5.5888224e-01	2.5248647e-04
5.6686628e-01	2.4835745e-04
5.7485032e-01	2.4163976e-04
5.8283430e-01	2.3807296e-04
5.9081835e-01	2.3833863e-04
5.9880239e-01	2.3239947e-04
6.0678643e-01	2.3076410e-04
6.1477047e-01	2.2914914e-04
6.2275451e-01	2.2743021e-04
6.3073850e-01	2.2524798e-04
6.3872254e-01	2.2237956e-04
6.4670658e-01	2.1976786e-04
6.5469062e-01	2.1745801e-04
6.6267467e-01	2.1507457e-04
6.7065871e-01	2.1244203e-04
6.7864269e-01	2.1045853e-04
6.8662673e-01	2.0807475e-04
6.9461077e-01	2.0652104e-04
7.0259482e-01	2.0333740e-04
7.1057886e-01	2.0223003e-04
7.1856290e-01	1.9917122e-04
7.2654688e-01	1.9924215e-04
7.3453093e-01	1.9244000e-04
7.4251497e-01	1.9115831e-04
7.5049901e-01	1.9312110e-04
7.5848305e-01	1.9210094e-04
7.6646709e-01	1.8779801e-04
7.7445108e-01	1.8615114e-04
7.8243512e-01	1.8435878e-04
7.9041916e-01	1.8179975e-04
7.9840320e-01	1.7899687e-04
8.0638725e-01	1.7703579e-04
8.1437123e-01	1.7484790e-04
8.2235527e-01	1.7190416e-04
8.3033931e-01	1.7025668e-04
8.3832335e-01	1.6854411e-04
8.4630740e-01	1.6697324e-04
8.5429144e-01	1.6444179e-04
8.6227542e-01	1.6293665e-04
8.7025946e-01	1.6235559e-04
8.7824351e-01	1.5977748e-04
8.8622755e-01	1.5628790e-04
8.9421159e-01	1.5281960e-04
9.0219563e-01	1.5240045e-04
9.1017962e-01	1.5133283e-04
9.1816366e-01	1.5119810e-04
9.2614770e-01	1.5085525e-04
9.3413174e-01	1.5051373e-04
9.4211578e-01	1.4899267e-04
9.5009983e-01	1.4741771e-04
9.5808381e-01	1.4597259e-04
9.6606785e-01	1.4534363e-04
9.7405189e-01	1.4355066e-04
9.8203593e-01	1.4290745e-04
9.9001998e-01	1.4105586e-04
9.9800402e-01	1.4010663e-04
1.0059880e+00	1.3959360e-04
1.0139720e+00	1.3769868e-04
1.0219561e+00	1.3856156e-04
1.0299401e+00	1.3685362e-04
1.0379242e+00	1.3495589e-04
1.0459082e+00	1.3475887e-04
1.0538923e+00	1.3364899e-04
1.0618763e+00	1.3278438e-04
1.0698602e+00	1.3157151e-04
1.0778443e+00	1.3174220e-04
1.0858283e+00	1.3148939e-04
1.0938123e+00	1.3095149e-04
1.1017964e+00	1.3011973e-04
1.1097804e+00	1.2914777e-04
1.1177645e+00	1.2872870e-04
1.1257485e+00	1.2773465e-04
1.1337326e+00	1.2764214e-04
1.1417166e+00	1.2597386e-04
1.1497006e+00	1.2626727e-04
1.1576847e+00	1.2500357e-04
1.1656686e+00	1.2359639e-04
1.1736526e+00	1.2292850e-04
1.1816367e+00	1.2190016e-04
1.1896207e+00	1.1981817e-04
1.1976048e+00	1.1896653e-04
1.2055888e+00	1.1983241e-04
1.2135729e+00	1.2124057e-04
1.2215569e+00	1.2056262e-04
1.2295409e+00	1.1956705e-04
1.2375250e+00	1.1917000e-04
1.2455090e+00	1.1815147e-04
1.2534931e+00	1.1773626e-04
1.2614770e+00	1.1714750e-04
1.2694610e+00	1.1676679e-04
1.2774451e+00	1.1653310e-04
1.2854291e+00	1.1581980e-04
1.2934132e+00	1.1485301e-04
1.3013972e+00	1.1485572e-04
1.3093812e+00	1.1442659e-04
1.3173653e+00	1.1316230e-04
1.3253493e+00	1.1319041e-04
1.3333334e+00	1.1357363e-04
1.3413174e+00	1.1229639e-04
1.3493013e+00	1.1041718e-04
1.3572854e+00	1.1042860e-04
1.3652694e+00	1.1010830e-04
1.3732535e+00	1.1010949e-04
1.3812375e+00	1.1025351e-04
1.3892215e+00	1.1057185e-04
1.3972056e+00	1.0999897e-04
1.4051896e+00	1.0983582e-04
1.4131737e+00	1.0949438e-04
1.4211577e+00	1.0923170e-04
1.4291418e+00	1.0914942e-04
1.4371258e+00	1.0828472e-04
1.4451097e+00	1.0896126e-04
1.4530938e+00	1.0844298e-04
1.4610778e+00	1.0841517e-04
1.4690619e+00	1.0867360e-04
1.4770459e+00	1.0856955e-04
1.4850299e+00	1.0805746e-04
1.4930140e+00	1.0758661e-04
1.5009980e+00	1.0779114e-04
1.5089821e+00	1.0790252e-04
1.5169661e+00	1.0854984e-04
1.5249501e+00	1.0866184e-04
1.5329342e+00	1.0803452e-04
1.5409181e+00	1.0770700e-04
1.5489022e+00	1.0701928e-04
1.5568862e+00	1.0741116e-04
1.5648702e+00	1.0642370e-04
1.5728543e+00	1.0597512e-04
1.5808383e+00	1.0601390e-04
1.5888224e+00	1.0533075e-04
1.5968064e+00	1.0547535e-04
1.6047904e+00	1.0460995e-04
1.6127745e+00	1.0406422e-04
1.6207585e+00	1.0322381e-04
1.6287425e+00	1.0247451e-04
1.6367265e+00	1.0267957e-04
1.6447105e+00	1.0244823e-04
1.6526946e+00	1.0205869e-04
1.6606786e+00	1.0127118e-04
1.6686627e+00	1.0070228e-04
1.6766467e+00	9.9589502e-05
1.6846308e+00	9.8863311e-05
1.6926148e+00	9.8272205e-05
1.7005988e+00	9.7748132e-05
1.7085829e+00	9.6674456e-05
1.7165669e+00	9.6339594e-05
1.7245508e+00	9.5299278e-05
1.7325349e+00	9.5138166e-05
1.7405189e+00	9.4124902e-05
1.7485030e+00	9.3386501e-05
1.7564870e+00	9.2975322e-05
1.7644711e+00	9.2456430e-05
1.7724551e+00	9.2413269e-05
1.7804391e+00	9.1757203e-05
1.7884232e+00	9.0785201e-05
1.7964072e+00	9.0759786e-05
1.8043913e+00	9.1845344e-05
1.8123753e+00	9.1259048e-05
1.8203592e+00	9.1056812e-05
1.8283433e+00	9.1518014e-05
1.8363273e+00	9.1231203e-05
1.8443114e+00	9.1540642e-05
1.8522954e+00	9.1390502e-05
1.8602794e+00	9.1193084e-05
1.8682635e+00	9.1621281e-05
1.8762475e+00	9.1535912e-05
1.8842316e+00	9.1518385e-05
1.8922156e+00	9.2090377e-05
1.9001997e+00	9.1655071e-05
1.9081836e+00	9.0995054e-05
1.9161676e+00	9.1923284e-05
1.9241517e+00	9.0872963e-05
1.9321357e+00	8.9313362e-05
1.9401197e+00	8.9717891e-05
1.9481038e+00	9.0582427e-05
1.9560878e+00	9.0044879e-05
1.9640719e+00	9.0622554e-05
1.9720559e+00	9.1081609e-05
1.9800400e+00	9.1146452e-05
1.9880240e+00	9.1300193e-05
1.9960080e+00	9.1287722e-05
)

So the maximum value for Cl 2.0192415e-02 and the corrosponding value is 1.4371258e-01 which is our Strouhal number for this problem.

Steady state

For Re = 10

For Re = 100

For Re = 1000

For Re = 10000

For Re = 100000

Scaled Residuals and Vertex avarage Plots

For Re = 10

For Re = 100

For Re = 1000

For Re = 10000

For Re = 100000

• Looking at the above plots we can say if the solution got converged or not
• When a solution is converged the slope of the scaled residuals plots won't change
• Also we can check it with the vertex avarage of velocity behind the cylinder (here the monitor point is set up 8 metres from the centre of the cylinder), here the vertex avarage velocity drops and then it goes through cycle of values as we can observe it in the above plots, that represents the solution has converged

Plot for Lift

For Re = 10

For Re = 100

For Re = 1000

For Re = 10000

For Re = 100000

Coefficient of Drag

For Re = 10

For Re = 100

For Re = 1000

For Re = 10000

For Re = 100000

Validation of Cd value

Reynolds Number	Velocity	Cd(Real)	Cd(From plot)	Percentage error
10	0.1	3.355312	3.3419566	0.398
100	1	1.347501	1.3006097	3.479
1000	10	1.193902	1.1228868	5.948
10000	100	1.028611	1.0575571	-2.814
100000	1000	0.953672	0.93224271	2.247

• Looking at the above table we can also determine the effect of reynolds number on Coefficient of drag
• As Reynolds number is incresed the Cd value is getting lower and lower

Conclusions

• Steady and unsteady state simulations were done and it was noted that the unsteady approach for the problem where more accurate and smooth this is becouse unsteady state analysis uses and time dependent study while steady state is independent of time but when it comes to computational time and convergance rate the steady state calculation takes less time
• Vortex shedding can be observed from vertex avarage of valocity magnittude plot and lift plot, that is whenever the graph behaves like a wave that is where the vortec shedding starts
• Cd value decreases when the reynolds number is increased

References

Numerical simulation of laminar flow past a circular cylinder - ScienceDirect