 Measurement of the pressure distribution on a circular cylinder

Akash Trivedi
CID: 00637814
Personal Tutor: Dr. P. Robinson
Submission Date: 29/11/2011
The purpose of this experiment was to show that potential flow theory is not valid for flow around a circular cylinder and determine the effects of changing Reynolds number on drag. Viscosity causes the laminar and turbulent boundary layers to separate at 80 and 105 degrees respectively. Tripping the boundary layer effectively increases the Reynolds number by approximately 100% allowing the “drag bucket” region to be analysed. In this region, there is a 49.05% reduction in the corrected drag coefficient. The large blockage ratio of 0.22 also led to a reduced drag coefficient.

Introduction
The flow around a circular cylinder has been the subject of many experiments throughout the history of fluid dynamics. In this experiment, the solution to d’Alambert paradox via Prandtl’s viscous effects idea will be shown by calculating the lift and drag coefficients of the body, taking blockage effects into account. Additionally, the effect of Reynolds number on the boundary layer transition and the drag will be discussed.

Theory
Potential flow is dictated by three assumptions; the flow is inviscid, irrotational and incompressible. From vector calculus, it is shown that the curl of a gradient is equal to zero, therefore. This also means that the vorticity, or the curl of the velocity field is zero, i.e. . Hence, potential flow is given by the fact that the velocity field is equal to the gradient of the velocity potential, .

The d’Alambert paradox arises due to the inviscid assumption resulting in a cylinder with zero drag. Real flow is viscid as Prandtl correctly noticed and as such has a boundary layer for which if the pressure gradient is too adverse, will cause the flow to separate resulting in drag. The drag, and therefore the CD, is affected by the Reynolds number as it determines the boundary layer transition.
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4648200131445431482513144518478501314451543050131445
Figure 1: Coefficient of drag against Reynolds number for a circular cylinder. 1
As evident from Figure 1, there are many different regimes. At really low Reynolds numbers (1), there is a balance between inertial and viscous forces in the Reynolds number expression. This is referred to as Stokes flow which has almost perfectly symmetrical streamlines but as it is viscous dominated, it displays a lot of drag. The viscosity at the surface then starts to affect the flow (2). This causes the flow to separate into two stable vortices. Further increase in Reynolds number (3) will lead to a loss in stability thus giving rise to the phenomenon of the alternative shedding of vortices known as a von Karman street. After a Reynolds number of 1000 (4), a clear wake is formed as the laminar boundary layer starts to separate from the front face of the cylinder and the shear layer starts to transit to turbulent. The drag coefficient is stable at approximately one. At a Reynolds number of approximately 400,000 (5), there is a sharp drop in drag due to a smaller pressure drop caused by the boundary layer transitioning to turbulent and reattaching with the cylinder on the rear face. In the last regime (6), the boundary layer transits to turbulent on the forward face and the point of separation start to creep back across the rear face and back onto the front face. This coupled with the increase in skin friction and wake size lead to a rise in drag coefficient once again.

An explanation of the apparatus used and the experimental procedure can be found in the handout 2.

Results
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Figure 2: Uncorrected CP against angular position.
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Figure 3: CP against angular position for both boundary layer cases.

The anomalous points in the Figures 2 and 3 are circled. They are due to the pressure tapping at 140o being faulty. Due to this, the value of the manometer reading at this tapping has been derived using linear interpolation and this new value is used in further calculations.
Laminar Boundary Layer % Variance with Uncorrected Values Turbulent Boundary Layer % Variance with Uncorrected Values
Uncorrected CD 1.48 N/A 0.68 N/A
Corrected CD – Numerical Integration of Corrected CP 1.21 18.24% 0.59 8.82%
Corrected CD – Direct Roshko Correction 1.06 28.38% 0.54 14.71%
Uncorrected CL -2.59 N/A -2.30 N/A
Corrected CL – Numerical Integration of Corrected CP -1.76 32.05% -1.76 23.49%
Separation Points 80o N/A 105o N/A
Table 1: Results from the circular cylinder experiment.

Although the CL can be obtained from the pressure curves, since the flow around a symmetrical circular cylinder is considered, it is quite clear that the lift force generated will be identically zero as opposed to the values given in Table 1. Pressure readings for the bottom half of the cylinder are not required to make this conclusion and since for the drag, only half the cylinder needs to be analysed, readings for the bottom half of the cylinder are not required in this experiment.

Discussion
From Figure 2, it is clear to see the points of boundary layer separation. For the case of the laminar boundary layer, it separates at 80o to the oncoming flow. This is due to the adverse pressure gradient acting on the boundary layer and so the laminar boundary layer has to separate before the maximum thickness of the cylinder. For the turbulent boundary layer, separation occurs on the rear face at 105o to the flow. This is as the turbulent boundary layer is less susceptible to an adverse pressure gradient. This means that the separation is delayed to the rear surface of the cylinder.
Even though the Reynolds numbers for both cases were the same, the separation occurs at two distinct places due to the boundary layer being affected by the trip wire. This tripping of the boundary layer effectively simulates the increase of the Reynolds number. By comparing the corrected value of the CD for the tripped case, it is evident from prior experimental data 3 that the effective Reynolds number was approximately 400,000.

This effective increase in Reynolds number puts the operating regime in the “drag bucket” region rather than the smooth plateau depicted by region 4 in Figure 1. Using the general equation for drag per unit span, the effect of drag on the cylinder can be quantified. For the laminar case, the drag is determined to be 52.70N whereas, for the turbulent case, the drag experienced on the cylinder is 26.85N. This represents a 49.05% decrease in drag when the boundary layer is tripped, even whilst operating at the same Reynolds number.
This property is very useful in real life situations. By tripping the boundary layer, although the skin friction is increased at higher Reynolds numbers, the pressure drag reductions at the lower Reynolds numbers outweighs that concern. This principle is behind dimples on golf balls, fluff on tennis balls and stitching on cricket balls to enhance their performance. In the field of aeronautical engineering, this principle is applied to aircraft operating in lower Reynolds numbers such as gliders. They make use of vortex generators to delay boundary layer separation by encouraging the boundary layer to transit from laminar to turbulent.

For a flow over a circular cylinder operating at a Reynolds number of around 200,000, Anderson’s source 3 suggests that the CD should be approximately 1.10. This agrees very well with this experiment as the percentage differences to the numerical integration method and the direct correction method are only 10.00% and 3.64% respectively.
For the processing of data, blockage corrections had to be made to take into account the effect of the tunnel walls. The blockage ratio for this experiment was 0.22 requiring significant corrections. Due to the correction, the velocity increases by an average of 9.04%. Applying Bernoulli’s equation between the outer edge of the boundary layer and the adjacent streamline on the front face of the cylinder, it can be noted that the pressure drops (as is evident from the corrections in Figure 3). This reduces the net imbalance in pressure distribution between the front and rear faces, thus leading to a lower value for the drag coefficient. As the flow itself was of low turbulence, a positive correction had to be made to the CP curves 5. Also, a more significant correction has to be made in the wake region; the effect being to reduce the overall CD. This indicates that CD increases as blockage increases.

Error Analysis
There were two major classes of error; human and physical. Amongst human error, the primary one was that caused by reading the multi tubed manometer. Parallax error or that produced by the incorrect reading of the meniscus are the contributories. Another human error was caused by the ineffective method to ensure the cylinder had its zero degree pressure tapping towards the oncoming flow. The cylinder had to be kept still by hand and any misalignment would have led to inaccurate measurements.

One source of physical error was that influenced by blockage effects. An attempt was made to overcome this error but if there were in fact errors in the readings then these would propagate regardless of blockage corrections made. Another error could be caused by the trip wire. If the wire were to come in contact with the pressure tappings then the readings would be disturbed. The presence of the pitot static tube facing the oncoming flow would itself affect the flow. This again creates a source of error in the measurements taken.

Conclusion
This experiment showed that potential flow theory does not apply in real life due to the viscous nature of the flow. The laminar and turbulent boundary layers separate at 80o and 105o respectively, causing an increase in pressure drag. As the Reynolds number increases, and boundary layer transits from laminar to turbulent, the separation occurs towards the rear face leading to a significant drop of 49.05% in the CD. Blockage effects were highly significant due to a blockage ratio of 0.22 and thus large corrections had to be made effectively reducing the CD. Nonetheless, the objectives of this report were met and the flow around a circular cylinder is better understood.

References
1 Eric W. Weisstein. (2007) Cylinder Drag. Online. Available from: http://scienceworld.wolfram.com/physics/CylinderDrag.html Accessed 20th November 2011
2 Morgans, A (2011) Measurement of the pressure distribution on a circular cylinder. Department of Aeronautics, Imperial College London.3 Anderson, J. D, Jr. (2011) Fundamentals of Aerodynamics. 5th edition. Singapore, McGraw Hill.

4 Roshko, A. (1961). Experiments on the flow past a circular cylinder at very high Reynolds number. Journal of Fluid Mechanics, Online 10, 345-356 doi:10.1017/S0022112061000950 Accessed 20th November 2011
5 Cheung, C. K. ; Melbourne, W. H. (1980) Wind tunnel blockage effects on a circular cylinder in turbulent flows. 7th Australian Hydraulics and Fluid Mechanics Conference, 18-22 August 1980, Brisbane. pp. 127-130. Online Available from: http://www.mech.unimelb.edu.au/people/staffresearch/AFMS%20site/7/CheungMelbourne.pdf Accessed 20th November 2011

Appendix
Unprocessed data for case 1: laminar boundary layer
Tapping Number Angular Position/degrees Manometer Reading/inch
1 -30 10
2 0 7
3 5 7.1
4 10 7.4
5 15 7.8
6 20 8.4
7 30 10
8 40 11.9
9 50 13.9
10 55 14.7
11 60 15.5
12 65 15.9
13 70 16
14 76 15.8
15 80 15.5
16 85 15.5
17 90 15.5
18 95 15.5
19 100 15.4
20 105 15.3
21 110 15.3
22 115 15.3
23 120 15.3
24 130 15.4
25 140 12.7
26 150 15.6
27 165 15.9
28 180 16
Start End Average Reading
Absolute Pressure 102000 Pa 102000 Pa 102000 Pa
Digital Manometer Dynamic Pressure 46.91 mmH2O 48.09 mmH2O 47.5 mmH2O
Digital Manometer Temperature 23.42 °C 24.21 °C 296.965 K
Stagnation Pressure Reading 6.9 Inch Static Pressure Reading 10.2 Inch Unprocessed data for case 2: turbulent boundary layer
Tapping Number Angular Position/degrees Manometer Reading/inch
1 -30 10.5
2 0 7.2
3 5 7.3
4 10 7.6
5 15 8.1
6 20 8.7
7 30 10.5
8 40 12.4
9 50 14.9
10 55 15.9
11 60 16.9
12 65 17.6
13 70 18.1
14 76 18.4
15 80 18.35
16 85 17.9
17 90 17
18 95 15.4
19 100 14.4
20 105 13.9
21 110 13.8
22 115 13.8
23 120 13.8
24 130 13.8
25 140 12.5
26 150 13.8
27 165 13.8
28 180 13.7
Start End Average Reading
Absolute Pressure 102000 Pa 102000 Pa 102000 Pa
Digital Manometer Dynamic Pressure 47.91 mmH2O 51.47 mmH2O 49.69 mmH2O
Digital Manometer Temperature 23.02 °C 23.95 °C 296.635 K
Stagnation Pressure Reading 7.1 Inch Static Pressure Reading 10.6 Inch Calculations
Mentioned below is the list of calculations that were performed to allow the results to be derived.

Equation 1
Using the absolute pressure and the average temperature, the density of air was obtained.

Equation 2
The dynamic pressure, q, can be obtained by using the digital manometer pressure reading given in mmH2O, converting it into SI, and multiplying it by the gravitational field strength and the density of water.

Equation 3
By using the relation between equations 2 and 3, the free stream velocity can be determined.

Equation 4
Sutherland’s formula for obtaining dynamic viscosity can be used by inputting the required reference values of, and C = 120.
Equation 5
Thus, by using equations 1 and 4, coupled with the free stream velocity obtained previously, it is possible to calculate the Reynolds number, where the characteristic length is the diameter of the cylinder.

Equation 6
Equation 7
Using the definition of pressure coefficient in equation 6, the CP can be simple derived by the manometer height readings. The theoretical CP in accordance to potential flow theory can be obtained by using equation 7.

Equation 8
Equation 9
The drag and lift coefficients can be evaluated by equations 8 and 9 respectively. They are obtained using the trapezium rule numerical integration method.

Equation 10
The drag force per unit span can be calculated from equation 10 where d is the diameter of the cylinder and CD is the direct correction of the drag coefficient applied by those stated by Roshko 4.

Equation 11
Equation 12
Equation 13
Equations 11, 12 and 13 are those provided by Roshko 4, used to correct for blockage effects. The starred quantities refer to the uncorrected values and the blockage ratio (between the cylinder diameter and the tunnel breadth) is given by the ratio.