The experiment conducted involves the study of flow past a circular cylinder in a uniform stream.

by maxiphil (student)

Table of Contents

Flow Past A Circular Cylinder

LABORATORY REPORT

Introduction

The experiment conducted involves the study of flow past a circular cylinder in a uniform stream. The objective of this experiment was to compare and evaluate the differences between the experimental and theoretical pressure distributions whilst taking into account the reasons why they are different, its causes and effect on the drag on the cylinder as quantified by the calculated drag coefficients.

The flow pattern on a cylinder, according to theory (D'Alembert's paradox), suggests that the drag force on the cylinder moving with constant velocity relative to the fluid is zero as shown in figure 1. However, when this is experimented, the flow streamlines do not follow the surface shape of the cylinder over much of the geometry but rather, follow the solid surface shape closely over the upstream face but then a separation occurs at a point causing a von Kármán Vortex Street wake to develop as shown in figure 2. Theoretically, the separation is not predicted as the potential flow theory indicates the downstream face is similar to the upstream face.

The occurrence of such a theoretical paradox was due to the negligence of the effects of viscosity on a body such as the cylinder.

By using dimensional analysis, we can show that for low speeds, the flow pattern and viscosity on a cylinder are functions of the Reynolds number (Re) – a dimensionless number which determines the flow pattern. This is defined as:

Where V is a velocity, D is the diameter of the cylinder and nu (ν) is the kinematic viscosity of the fluid.

At reasonably high values of Reynolds number in a real flow, a thin layer known as the boundary layer is formed on the body surface as shown in figure 3.This boundary layer inevitably detaches from the solid body to form a boundary layer separation which increases the drag on the body. The pressure on the other hand remains low and approximately constant over the downstream face of the cylinder. This causes an overall imbalance of pressure forces on the cylinder known as the pressure drag. At the same time, skin friction drag affects the cylinder at the upstream face where the boundary layer is thin.

Theoretical Considerations

According to theory i.e. the inviscid potential flow theory, the pressure distribution around a circular cylinder can be predicted in 2-dimensional, low speed flow. It is stated that . In this case, and are the static pressure and free-stream velocity upstream of the cylinder respectively.

By defining the pressure coefficient Cp as: , a complete equation that is used to predict the pressure coefficient distribution can be formulated.

This expression is given as where θ is the angle measured from the back of the cylinder (forward stagnation point) as shown ...

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Theoretical Considerations

By defining the pressure coefficient Cp as: , a complete equation that is used to predict the pressure coefficient distribution can be formulated.

This expression is given as where θ is the angle measured from the back of the cylinder (forward stagnation point) as shown in figure 4. However, this theory is unrealistic as it suggests that the viscosity/drag on the cylinder is zero as at the forward stagnation point, θ = 0 and Cp = 1.

This is incorrect as drag on the real cylinder is not zero. Due to the fact that there is a boundary layer near the circular cylinder, viscosity cannot be neglected as the presence of this boundary layer increases the effect of even small amount of viscosity.

Equipment Details

The experiment setup was made up of four sections: A wind tunnel, Bertz manometer, multi-tube manometer and a circular cylinder.

The wind tunnel, when switched on, uses the laboratory atmosphere as the working fluid to simulate flow during the experiment. Attached to the tunnel was a Bertz manometer which measures the dynamic pressure/ airspeed of the tunnel.

Mounted inside the working section of the wind tunnel was a cylindrical model with a diameter of 110mm. This was equipped with 36 pressure tappings at 10 degree intervals around its circumference at mid span. Each one of these pressure tappings are connected to one of the tubes of the multi-tube manometer.

The multi-tube manometer works by comparing the laboratory atmospheric pressure to the pressure transmitted by each of the 36 tubes from the surface of the cylinder model to the top of each water column in the manometer. If the atmospheric pressure is higher than the pressure of the cylinder, the water is drawn up the tube. Otherwise, if the cylinder pressure is higher than the atmospheric, the water is depressed.

Experimental Procedure

To begin the experiment, the Bertz manometer was checked to ensure that the value displayed was zero.

When this was done, the multi-tube manometer inclination, αMAN was recorded as well as the specific gravity of the manometer fluid, SGMAN and the calibration factor K1, of the Bertz manometer.

The wind tunnel was then switched on and carefully stepped up to around 15-20 units on the Bertz scale and allowed to stabilise. Due to the fact that the manometer fluid was not lockable, multiple readings had to be carried out to allow for “drift” in fan output during the test.

First, the Bertz manometer was read and the value was recorded. Then two fellow students read the tubes 1-36, 38 and 39 of the multi-tube manometer simultaneously starting at either end of the manometer and reading the tube heights in descending and ascending order.

Once all the tube readings had been read, a second Bertz reading was taken and the fan was shut down.

With the intention of knowing the kind of fluid that was being used for the experiment, the air pressure and temperature were recorded by a barometer and a built-in manometer respectively.

Results

Measurements were recorded in order to calculate results. These measurements recorded were in varying units and these were converted automatically into SI units when inputted into a computer. These converted values were then used to calculate a range of values such as the coefficients. A sample of the results are shown below and the full results are shown on Appendix a.

Values displayed in the square boxes as shown above were the data actually recorded during the experiment. The other data were automatically calculated by the computer using formulas already programmed on the spreadsheet.

On the full table showed in Appendix A, the tube heights on the manometer were inputted and the other data e.g. Cp were calculated by the computer. A pressure coefficient graph was also produced. Graph exhibited in Appendix B.

To compare results stated by the outputted data by the computer, I attempted to calculate the dynamic pressure and the airspeed of the system.

To do this, I used the standard manometer equation to calculate the dynamic pressure:

To find airspeed of the wind tunnel, density of the working fluid in the tunnel was calculated:

Where Pa equals:

Therefore the airspeed, V is determined from q and ρa to give:

To complete calculations, the Reynolds number determined from data above gives:

Where the kinematic viscosity

Discussion

The values calculated for the dynamic pressure and airspeed as shown in the results section have a slight variance to those in Appendix A.

Calculating the dynamic pressure requires readings from tubes 38 and 39 from the multi-tube manometer. The disadvantage of having to read from such equipment was that it was difficult to read the change in fluid heights with much accuracy. By having two fellow students read the manometer simultaneously made the experiment much more accurate but not entirely as everyone is prone to mistakes. By having such an error in the reading means the value calculated for the dynamic pressure would be higher thus causing the airspeed V value also to rise.

By comparing the plots on the graph of “Variation of Cp Around the Cylinder” as shown in Appendix B, the theoretical and real pressure distributions being to differ at θ=±45º.

Initially, the real (viscous) flow plots follow the pattern of the inviscid flow (θ=0º to θ=45º). A reason for this is that the boundary layer separation had not been formed as the points of separation are usually formed at the downstream face. It is unusual to have the points of separation at the upstream face.

As the angle on which the flow hits the surface increases (θ=45º to θ=135º), the coefficient pressure plots do not reach the value of Cp=-3.0 as predicted by the theoretical value. At these angles, the boundary layer separation has formed causing the actual pressure of the flow to drop hence exhibits the fact that viscosity effects are felt.

The value of the pressure coefficients attempts to rise at θ=135º to θ=180º but not by a large amount. A probable reason for such differences in pressure coefficients of the real and inviscid flows would be that the flow past a circular cylinder is assumed to be 2-dimensional. This unexpected form of pressure distribution on the cylinder may be, to a certain extent, explained in 3-dimensions: z-direction. By conducting an experiment and analysing the flow past a circular cylinder is 3 dimensions, it would be observed and concluded that the pressure is also distributed in the z-direction and not only in the x and y directions.

Comparing the real flow curve on the experimental graph and the sub-critical Reynolds number curve as shown in Appendix C, it is noticeable that the graph on Berlin and Smith (1998) has a curve that is closely matched to the theoretical curve as opposed to our experimental curve. Once again, this would be due to misreading the instruments and by having different equipment such as a digital manometer, accurate readings can be taken and used to perform calculations.

On the whole, the experimental data recorded seemed to be of lower value compared to other works. In terms of accuracy, more data could be recorded to make experiments more accurate. For example, the temperature and air pressure of the laboratory was recorded after the experiment was completed. To make these accurate, the temperate should be recorded before and after the laboratory session in order to get an accurate value. Some equipment such as the Bertz and multi-tube manometer could be replaced by digital equipment to further increase the accuracy of the experiment.

Conclusion

Obtaining an experimental curve that has pressure coefficients/degrees values not close to the theoretical, it is plausible to say that when the inviscid potential flow theory is experimented, viscosity actually affects the flow past a circular cylinder. This happens by means of a boundary layer and this is not predicted by the inviscid potential flow theory.

References

Anon., 2009. Flow Around a Cylinder in a Steady Current [online]. Available at: <http://www.worldscibooks.com/etextbook/6248/6248_chap01.pdf> [Accessed 3rd Mar 2011].

Bertin, J.J (2002). Aerodynamics for Engineers. 4th ed. Upper Saddle River: Prentice-Hall, Inc. p79-89.

Devernport, W.J. (2007). Flow Past a Circular Cylinder. Available: http://www.aoe.vt.edu/~devenpor/aoe3054/manual/expt3/. [Accessed 25th Feb 2011].

Edwards, C.D., 2000. Calibration of the Reference Velocity in the Test Section of the Low Speed Wind Tunnel at the Aeronautical and Maritime Research Laboratory. [online] Melbourne, DSTO: Aeronautical and Maritime Research Laboratory. Available at: <http://www.dsto.defence.gov.au/publications/2200/DSTO-TN-0248.pdf> [Accessed 6th Mar 2011].

Fornberg B. (1985). Steady Viscous Flow Past a Circular Cylinder up to Reynolds Number 600*. 61 (2), p297-301.

Stern, F., 2009. Intermediate Fluid Mechanics – Bluff Body. [online]. Available at: <http://css.engineering.uiowa.edu/~me_160/lecture_notes/Bluff%20Body2.pdf> [Accessed 2nd Mar 2011].