Pressure Distribution around Circular Cylinder Lab Report

The aims of the investigation is to measure the pressure distribution on the surface of a smooth cylinder placed with its axis perpendicular to the flow and to compare it with the distribution predicted for frictionless flow, and to calculate the drag coefficient of the cylinder.

In the investigation being carried out, a cylinder in a closed circuit wind tunnel will be experimented upon to gather the pressure distribution acting on it at different speeds. When the cylinder is standing vertically to the incoming flow in a wind-tunnel, two experiments will be carried out for the same cylinder, one with smooth (laminar) flow and the other with turbulent flow. The experimental pressure distribution obtained from each experiment will be compared with the theoretical distribution predicted for frictionless flow. The drag coefficient for the cylinder will be calculated together with the tunnel calibration constant for both tests.

The smooth cylinder has got 12 pressure tappings at angular intervals of 30° on its surface; it is also placed with its axis vertical on a turntable on the floor. These tappings are connected to multi-tube (methylated spirits) manometer, which is inclined at an angle of 30° to the horizontal. The multi-tube manometer has got a total of 34 tubes, out of which the first 12 are directly connected to the pressure tappings on the cylinder, so that tube 1 is connected to pressure tapping 1 and so on. Pressure tapping 1 is facing the oncoming flow when the angular position indicator is set at 0°. Since pressure tapping 1 is connected to tube 1 in the multi-tube manometer, the head pressure shown on tube 1 will represent the stagnation pressure. Tube 34 in the multi-tube manometer is connected to the upstream part of the wind-tunnel. The Betz manometer is used to change the incoming flow velocity at the upstream section. Since both the Betz manometer and tube 34 (in the multi-tube manometer) are connected to the upstream section, both will show the same equivalent reading for pressure but in different units.

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Background theory:

Laminar flow is defined when a fluid flows in parallel layers, with no disruption between the layers. In comparison to this Turbulent flow has a much more disorganized pattern, it is characterized by mixing of the fluid by eddies of varying size within the flow.

The Reynolds number (Re), gives the measure for laminar and turbulent flows. Laminar flow takes place when Reynolds number is lower than 104, and for Turbulent flow the Re must be greater than 3Ã-105. Reynolds number has got no units since it is just a ratio. There are many diverse types of equations for deriving the Reynolds number of an actual shape.

Fig. A shows the different types of flow patterns at various angles.

The appropriate equation for the cylinder’s Reynolds number can be acquired from:

Re = [eq. 1]

Where: d = Diameter of cylinder

Vˆž = Velocity of fluid upstream

v = Kinematic viscosity of air (1.46Ã-10-5 m2s-1)

From the equation both values of d and v stay constant for both experiments, therefore the change in Reynolds number depends directly on the upstream velocity of the wind-tunnel.

From the Bernoulli’s equation the relationship between fluid pressure and velocity can be established

[eq.2]

Where: P = pressure of the fluid

Z = Height

air = air density

 = The density of the fluid

G = Gravity

V= the velocity of the fluid

The height in remains constant therefore the flow in the wind tunnel has an equation.

[eq.3]

By dividing both sides with  and taking measurements from the point where the flow of velocity stops, (the stagnation point where V2 = 0). This is for the reason that at the stagnation point on the surface of the tube that is perpendicular to the flow to it, therefore the dynamic pressure is given by:

[eq. 4]

The pressure is measured using the manometer, and then therefore the pressure at the tapping must be the same as the pressure head at gH.

Then the stagnation head and static head measured on the multi-tube manometer Inclined at a specific angle are given by:

[eq.5] [eq.6]

So when using these two equations the pressure differences found using them become.

[eq.7]

The dynamic pressure upstream of the cylinder is gained from:

[eq.8]

In the equation above k = the tunnel calibration constant. The pressure change across the wind tunnel contraction is measured using two different instruments; the Betz manometer and the Multi-tube manometer (tube no. 34). In an ideal system where there are no losses in energy. In procedures with energy losses, the ratio value is below 1. The relation is known as the tunnel calibration constant (k). is the pressure variation across the contraction as displayed by the reading on the Betz micro-manometer.

The tunnel calibration constant can be attained directly from:

[eq.9]

Differences between the pressure at angles from the front of the stagnation point and the free steam pressure P is gained from the following equation:

[eq.10]

h is a reading on tube 1 when pressure tapping number 1 is at an angle from the front of the stagnation point. Therefore the pressure coefficient at an angle can be defined:

[eq.11]

To find the pressure coefficient based on frictionless flow by using:

[eq.12]

The equation above is applied when plotting a graph, of variation of cΦ vs. Φ, by replacing values of angles into the equation above and then finding out the subsequent values for pressure coefficient (cΦ). The consequent graph drawn from the derived equation will only correspond to a theoretical relationship, where the flow is believed to be frictionless.

In the illustration below, the pressure in the wake region is less than pressure upstream; this causes drag, mainly due to flow separation behind the body. The streamline pattern and the pressure distribution are not balanced and a wake of slow-moving air is produced behind the cylinder.

Fig. B shows flow separation taking place behind the cylinder

The drag force, due to the pressure forces on the cylinder can be derived from:

[eq.13]

As the term integrates to zero, the drag coefficient can be simplified to:

[eq.14]

Apparatus

The cylinder being experimented on is placed in the wind tunnel. The part that will be under testing will be of size of 1.000m X 0.760m. The wind tunnel will have a contraction ratio of 5.6

To connect the pressure tappings from upstream and downstream of the tunnel contraction to a Betz micro-manometer (mmH2O).

The velocity of air in the test section is to be fluctuated by adjusting the fan speeds on a controller.

Thirdly a smooth circular cylinder with diameter 114.3 mm to be placed with its axis vertical, on a turntable on the ground of the test section area. It can be seen in the cylinder where halfway along there are pressure tappings at angular intervals of 30ï‚° on its surface, near to the tappings are marked numbers from 1 to 12, these are connected to the upper ends of 1 to 12 tubes on a manometer. This manometer is to be a multi-tube methylated spirits manometer.

The pressure upstream of the cylinder is sensed by a taping on the tunnel wall and is connected to one of the tubes. In this experiment to be tube number 34. The remaining tubes 13 to 33 are open to the atmosphere.

The level of turbulence has to be changed, so in this test section it is small however to be increased by the insertion of a grid. This grid as an array of circular rods upstream of the test section.

Lastly as the cylinder is to be placed on the turntable that is to be rotated. The angular position of pressure tapping number 1 is indicated on a digital counter in degrees and in tenths of degree.

Method:

To be able to do this experiment the cylinder to be already oriented so that the pressure from tapping 1 is facing the oncoming flow and the angular position indicator will have to be set at 0ï‚°. As we know that the pressure P1 tapping is the stagnation pressures and exceeds the pressure Infinity upstream of the cylinder by an amount.

The experiment firstly to test laminar flow – The laminar flow of the velocity of the wind tunnel will be increased gradually until the Betz manometer reads 15mH2O. For this velocity to remain constant adjustments are made.

From the multi-tube manometer, to take readings of fluid heights to show a general idea of the pressure distribution.

The fluid height to be noted from the tube which is connected to the tunnel wall upstream of the cylinder.

To measure the fluid height in tube 1 is measured then the table is to be turned in intervals of 10 degrees, this is to be repeated for every 10 degrees until it has fully rotated around 360 degrees.

The experiment is also to test in a turbulent flow – To have a grid with an array of squares inserted in to the wind tunnel, where the air flow and the velocity increases until the Betz manometer reads 35mmH2O, as the air becomes turbulent.

This whole procedure to be repeated.

Fig. C shows the manometer tube readings at a zero angle for smooth flow.

Fig. D shows the manometer tube readings at a zero angle for turbulent flow.

The readings of the multi-tube manometer were taken before starting to rotate the cylinder (at zero angle). This preliminary data collected is presented visually to show the shape of the pressure distribution around the cylinder. Tube 33 is open to air; hence it shows the atmospheric pressure. And tube 34 as mentioned earlier, shows the head pressure of the upstream section of the wind-tunnel.

Results

Raw Results:

The scale of the manometer used was in inch. Therefore, the results obtained have to be changed to metres. This is done as follows:

1 inch = 0.0254 metres

The heights of the fluids have to be multiplied by 0.0254 to change to metres.

Smooth flow:

α = 30°

hˆž = 11 inch = 11 x 0.0254

= 0.2794m

Turbulent flow:

α = 30°

hˆž = 10.4 inch = 10.4 x 0.0254

= 0.26416m

The pressure coefficient, cϕ, at an angle can be found by using eq 8.

The calculations to find cÏ• will be the same for both laminar and turbulent flows. The only difference would be that the value of hˆž would be different in each case.

The value for h1 is the value obtained when the cylinder is at 0°.

The calculations to find the pressure coefficient for the laminar flow at Φ = 0° is shown below:

= 1

Calculated Results

The following data shows values, which will be used to determine the pressure coefficient, this will be calculated using eq. 11.

Smooth flow Transonic Flow

Frictionless flow

The values for the pressure coefficients will be the same for both laminar and turbulent flows since cϕ only depends on the angle Φ.

The pressure coefficient for a frictionless flow is found using eq. 12.

Theoretical graph:

The graph represents in a theoretical manner in which the experimental values should be able to compare to, whereby the air flowing in the graph, shows constant change at regular angle intervals. Also all peaks and troughs on the graph show relevant pressure coefficients.

The graphs illustrate the pressure coefficient

variation with changes in angle.

The above graph shows the variation of cΦ vs. Φ in laminar flow.

This is then compared to the cΦ vs. Φ in turbulent flow.

Calculations to find the drag coefficient, CD:

The value for the drag coefficient depends on the value of cΦ cosΦ, this relation can also be noticed in eq. 14.

Laminar flow

Discussion:

Throughout this experiment several factors were found out these include:

The pressure distribution in the system

Drag of the cylinder

Drag coefficient

Reynolds number

Errors in the experiment which may have caused anomalies

1. Looking at the graphs it can be seen that the pressure distribution in the system as in both lamina and turbulent flow also in parts off the graph it shows steady correlation, between angles 900 and 3100 in laminar flow. Angles 1300 and 2300 in turbulent flow.
2. The drag on the cylinder in turbulent and lamina conditions show through the results and graphs shown. As there is more drag when there is turbulent flow than lamina, however this easy to understand as, in lamina flow the eddies produced have a small wake so therefore it does not have a large pressure so do not increase drag. From the graphs it can be seen that the pressure coefficient in lamina flow at 90ï‚° is greater than the pressure coefficient at turbulent flow. However in the turbulent flow the motion reduce the pressure and so increases the drag.

The drag coefficient can be found by looking at the results and graphs, which both show that it is less in turbulent flow as the separation point occurs after 90ï‚° resulting in less eddies so less wake and therefore a high pressure with a end of low drag coefficient. Whereas in lamina flow it is greater than in turbulent. This may be due to the fact that the separation point occurs before 90ï‚°, this has a resulting effect of more eddies which induce wakes and low pressure, the end result of this is a high drag coefficient.

Separation point is where the angle flows become steady. The separation point occurs when the velocity of the fluid is reducing, in which the pressure flows induce a positive pressure gradient.

Then once the separation has passed the boundary layers bend over and flow in the opposite direction.

The pressure remains constant after the separation point because eddies are transferred to another energy.

The separation point at lamina flow is at 90ï‚° than for turbulent which is after 90ï‚° as the pressure gradient is greater in laminar flow, which means that the greater the pressure gradient the earlier the separation.

The reason why Reynolds number is greater in turbulent flow than lamina , as the main reason for this is that less pressure and drag coffeicnet and more drag is acting on to the cylinder.

Conclusion:

In this experiment a cylinder was used to find laminar and turbulent flow around it, the main objective was to see if the drag and flow increased or decreased, this was achieved and so this was shown that they increased in turbulent conditions and decreased in laminar conditions. Also in a turbulent condition the separation will increase to 90 degrees and the Reynolds number also increases.

Discussion

Figure G & H shows the head pressure distribution around the cylinder. As it can be seen from the laminar head pressure distribution (figure G), the pressure between tubes 3-11 (i.e. angle 90° up to 330°) is approximately the same. This shows that the pressure is more or less the same around the cylinder (seen from top view), except from its front point facing the oncoming flow, where the pressure is equal to the stagnation pressure.

In the case of turbulent flow (figure H), the head pressure distribution pattern is somewhat different. The head pressure values drop between tube no. 5 to 9, which is the rear part of the cylinder. This proofs that a “low-pressure” region exists at the rear of the cylinder in turbulent flow. This region of “low-pressure” is referring to the wake region. The pressure distribution is also clearly symmetrical about tubes 6 and 7, which proofs that the pressure distribution on one (horizontal) side of the cylinder is the similar to the one on the opposite side of the cylinder.

The percentage error for the coefficient of drag (CD) value in experiment 1 was calculated to be around 2.5%. This is a relative small percentage error, which shows that the experimental errors involved in experiment 1 were not significant. However, the percentage error for the CD value in experiment 2 was a massive 56%, which clearly shows that the experimental errors involved in experiment 2 did play a significant role.

The error due to parallax is one of those errors. The parallax error is human reading error, where the eye needs to be exactly in line with the reading to be measured.

In both experiments, the multi-tube manometer was at an inconvenient position (on the floor). And furthermore the whole manometer system was slanting at an angle of 30° to the horizontal. Both of these factors made it difficult to get the eye level exactly in line with the reading to be measured from the multi-tube manometer. This might have caused inaccuracy in the readings.

The trapezium rule was used to determine the area under the graph of cΦcosΦ, which was used to calculate the coefficient of drag. Since the graph had regions of both “negative” and “positive” areas, the trapezium rule had to be applied separately for each section of the graph. The whole graph was divided into strips, each with a width of Ï€/18 (10 degrees in radians).

When the graph of cΦcosΦ goes from a positive region into a negative one, the experimental data in some cases does not reach exactly zero before the data “switches signs”. This results in some minor areas of the graph being neglected. This would not significantly affect the total area found from the trapezium rule, since the areas neglected are relatively small. But since some areas of the graph are ignored, the total area found would not be the exact area under the graph. This will result in an error in the final values of the coefficient drag.

The tunnel calibration constant for laminar flow (k1) was found to be 1.0959. Clearly this value cannot be accurate because the ratio of the two pressures must be equal to 1 or below, since both are representing the pressure across the same points. The unexpected high value for k1 must have been a result of experimental error. Since most terms are constant in the equation used to find k, the only factors that could have contributed to the error must have been the values of hˆž & h1, which were obtained from the multi-tube manometer.

While taking the readings from the multi-tube manometer, the fluid (methylated spirit) in the tubes was fluctuating. Some of the fluctuations were as large as +/- 0.5 inch. For this reason, many of the readings obtained might have been greatly inaccurate, which eventually could have lead to a significant error in the final values for the coefficient of drag in both experiments, as well as the value of k for laminar flow.

The tunnel calibration constant for turbulent flow (k2) was found to be 0.7632. This value clearly indicates that energy losses did take place since the value is well below 1. Energy losses may have occurred in 2 main forms; as heat and sound energy produced by friction and collisions of air molecules with especially the grid system at the upstream.

Heat energy (and some sound energy) is also produced due to friction of the inner wall of the wind tunnel. The flowing air must do work to overcome this friction, and therefore some kinetic energy of the flowing air is lost as heat. To get turbulent flow, the flow rate was increased and a grid system was introduced.

With the introduction of a grid system at the upstream in experiment 2, more collisions of air molecules took place, hence resulting in increased “loudness” of flow. This increase in loudness (due to the increase in collisions and friction) resulted in some kinetic energy of the flow being converted into heat and more noticeably sound energy, and therefore some of the initial kinetic energy of the flow was “lost”.

Energy in form of heat is also lost due to formation of turbulent eddies. The formation of eddies takes place in turbulent flow. All these energy loss factors mentioned earlier might explain why the k-value for turbulent flow was less than the k-value of laminar flow.

In figure I, three graphs were plotted for the variation of pressure coefficient with angle. Each of the graphs was representing data for a unique condition. In the case of the theoretical data graph (green coloured), which represents the condition of frictionless flow, shows that at zero angle the pressure is a maximum (stagnation pressure). Moreover at an angle of 180° (rear of cylinder), the pressure once again reaches a maximum value. This relationship clearly indicates that the pressure distribution pattern would have been exactly symmetrical around a vertical axis at the centre of the cylinder.

However, in the case of both laminar and turbulent flow the lines if symmetry is not vertical but horizontal about the centre of the cylinder. By looking at the graphs, all graphs show a maximum pressure coefficient at an angle of zero (equivalent to 360°), which shows that whatever flow condition is imposed a maximum pressure will still remain at the stagnation point, which is the point where the fluid is brought to a stop.

For laminar flow, the pressure coefficient remains more or less constant after an angle of approximately 75°. This point is referred as the angle of separation, where the flow starts to “separate” from the cylinder’s surface. This value of 75° is very close to the value of 82° given for angle of separation for laminar flow in figure E.

In the case for turbulent flow, the pressure coefficient stays more or less constant after an angle of approximately 130° (i.e. angle of separation). Once again, even this value for angle of separation is very similar to the value of 120° given for turbulent flow in figure E.

One of the main reasons why there is a difference in shape between the theoretical graph and the experimental graphs for pressure coefficient is due to the fact that the assumption of air being a frictionless flow is invalid, since air is a viscous fluid. Overall the characteristics of the pressure coefficient graphs can be said to be an accurate presentation of actual data, since the three pressure coefficient graphs (between angle 0 to 180°) are highly identical in terms of both the shape and scale of the pressure coefficient graphs shown in figure E.

Conclusion

The value of tunnel calibration constant (k) for experiment 1 was found to be inaccurate (due to experimental errors) since the value is not expected to exceed 1. However, the value of k for experiment 1 shows that very negligible energy losses take place across the contraction of the wind tunnel under laminar flow. The value of k was significantly lower for experiment 2; this clearly shows that the energy losses that take place across the contraction of the wind tunnel are significant. The main reason for significant amount of energy losses is due to the introduction of a grid system at the upstream in experiment 2. The grid system dramatically increases the effects of friction to the oncoming flow. For this reason, some of the initial kinetic energy of the airflow is “lost” as heat and sound energy.

The obtained graphs for variation of pressure coefficient with angle around the cylinder perimeter can be said to be fairly accurate, since both the shape and scale of them is extremely identical to the graphs representing the same data in figure D.

For experiment 1 the coefficient of drag value had a percentage error of only 2.5%, which shows that the experimental errors did not have a significant impact on the final result. However, in the case of experiment 2, the percentage error for the coefficient of drag was a massive 56%, clearly the experimental error did significantly affect the final result in experiment 2.

In future improvements, the percentage error of the CD value in experiment 2 could be further decreased by reducing experimental errors mentioned in the discussion section.