y = η h
and the final result is
C h
Equation (5-18) provides a means to calculate CD from the pressure drop along the duct and the velocity distribution in the wake. Note that the derivation does not restrict the result to pressure drag only; the contributions of both pressure and skin friction forces are contained in the force D which comes into the momentum equation.
The skin friction drag on the walls also contributes to the momentum change and is also included in D. It is also worth mentioning that Equation (5-18) applies only to the case of flow along a duct where the flow is confined between parallel walls.
The foregoing analysis shows how drag force may be found from the pressure distribution over the surface of the cylinder and by measurements in the wake. The results obtained from both of these methods may be compared with the drag measured by direct weighing, and this is described in the next section.
Description of Apparatus and Test Procedure
Figure 5.4 shows the three main experiment arrangements for the AF12. Figure 5.4a shows the main pressure tapping points and their notation. These arrangements allow study of the drag of various bodies, the pressure distribution around a cylinder and the wake of different bodies. The balance arm is attached to the side of the main unit and holds the models in place inside the main unit. It is used with adjustable weights in the drag force experiments and as a fixed support in the wake traverse experiments. It is not used in the pressure distribution experiment. For the pressure distribution experiment, a protractor and cylinder model is fitted in the main body. The cylinder is connected to the separate AF10a manometer to display the pressure distribution as the cylinder is rotated. For the wake traverse experiments, a pitot tube assembly is attached to the main unit.
Test model Balance Arm Locking Screw
Cylinder
Quick release coupling
Protractor (located at back of main unit)
Test Model Clamp Screws (x2) Test Model Thumbnuts (x2)
Drag Force Pressure Distribution
Wake Traverse
Figure 5.4 Diagram of Apparatus
The apparatus is supplied with the cylinder and protractor model for pressure distribution experiment and three other models for the wake traverse and drag force experiments. The models include: a circular cylinder, a flat plate (inverted prism) and a symmetrical aerofoil section.
Po
po
p
Pe
p = 0, atmospheric datume Pitot tube for traverse in exit plane
AIR FLOW BENCH AF10
Figure 5.4a Pressure Tappings
The Drag Force Experiments - Procedure
Fit the circular cylinder model in position and adjust the weights to achieve equilibrium. Record the value of the weights. Every 1 mm on the front scale (100 g weight) represents 1 g and every 1 mm on the back scale (10 g weight) represents 0.1 g (use for fine adjustment). Switch on the fan and adjust the wind speed to a low level and readjust the weights to achieve equilibrium. Record the value of the weights and the total pressure (Po) and the static pressure (po) at the inlet. Increase the wind speed in increments up to its maximum level, each time readjust the weights to achieve equilibrium and record the pressures and the value of the weights. Subtract the initial value of the weights (with zero wind speed) from all the readings to give an actual value of equilibrium at each wind speed. Repeat the procedure with the other two models.
The Pressure Distribution Experiments - Procedure
Fit the circular cylinder and protractor model into the AF12. Connect the model to the manometer. Set the wind speed to maximum and the protractor to zero angle. Record the surface pressure (p) and the static pressure po. Rotate the protractor in 5° intervals for readings over the front half (0 to 90°) and 10° intervals for readings over the rear half (90 to 180°). Monitor the total pressure (Po) and static pressure (po) at the inlet to ensure that the wind speed is kept constant throughout the experiment.
The Wake Traverse Experiments - Procedure
Fit the circular cylinder model into the AF12. Fit the pitot traverse assembly. Set up a constant wind speed (a pressure difference of approximately 450 N/m2 is recommended). Traverse the Pitot tube across the section, in increments of 2 mm near the centre, increasing to 5 mm or 10 mm in regions where the total pressure is seen to be substantially constant. Record the values of Pe and pe The readings in the wake are extremely unsteady because of the high level of turbulence. Some users may wish to damp the oscillations by placing a tubing clip on the flexible connection to the manometer. If this method is used, take care to avoid excessive damping that can cause error. It is better to use too little than too much. Repeat the experiment with the other two models.
Results
Table 5.1 gives the drag force measured in units of gram-force by direct weighing at various air speeds. The drag force is written as the product of the drag D per unit length and the length l of the cylinder.
Drag Force Table 5.1 Drag Force on the cylinder measured by Direct Weighing
Air temperature 24°C = 297 K
In Figure 5.5 the force is plotted against dynamic pressure and a good linear relationship is established with a slope
600
CD may now be found by substitution in Equation (5-4)
Value of drag coefficient by direct weighing CD = 1.03
0
Drag Force D1 (gmf)
Figure 5.5 Measured Drag Force on a Circular Cylinder
The measured pressure distribution round the cylinder is given in Table 5.2, and graphs of cp and cp cosθ as functions of angle θ from the front are presented in Figures 5.6 and 5.7. The velocity pressure at inlet during this test was:
Po − po = 588.6 (x2), 598.41, 598.4 (x7) : Mean 596.448 N/m2
∴½ρU2 = 596.448 N/m2
The pressure is seen to be relatively symmetrical about the line θ = 0. The pressure coefficient falls over the front portion to a minimum at θ = 70°, and thereafter rises a little. Over the rear half of the cylinder the pressure is reasonably uniform. The distribution of cp for a cylinder in inviscid, incompressible fluid is shown for comparison, calculated from Equation (5-14).
θ
Table 5.2 Pressure Distribution around a Cylinder
-3.00 -2.00 -1.00 0.00 1.00
0 20 40 60 80 100 120 140 160 180
θ (Degrees)
Cp
0 to 180 0 to -180 Ideal Fluid Theory
Calculated from equation 5-14
Figure 5.6 Pressure Distribution around a Circular Cylinder
-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2
0 20 40 60 80 100 120 140 160 180
θ (Degrees) Cp cos θ
0 to180 0 to -180
Figure 5.7 Distribution of cp cosθ around a Cylinder
CD may be obtained from Figure 5.7 from the area beneath the curve. The area A beneath the mean curve is
∫
A cp
o
= π cosθ θd
which, from Equation (5-10), we recognise as the drag coefficient CD. This area can be evaluated in various ways; Appendix A describes how it is done by use of Simpson’s rule. The result is
A = 1.07
The value of drag coefficient obtained by plotting CD = 1.07
y
Table 5.3 Results of Velocity Traverse in the Wake
In table 5.3 the value of U2 is calculated from the experiment results:
To obtain a value of drag coefficient from the Wake Traverse method, the effects of the duct must be taken into consideration. Because of this the value of free stream velocity (U1) is calculated using the free stream flow of the duct (the area outside the wake of the cylinder. Looking at the results of Pe & pe in table 5.3, taken during the experiment, the free stream flow region can be easily identified as the stable region.
In this case the values obtained between 25 mm & 40 mm.
The average values of Pe & pe are:
Pe = 16.7 Pe = 11
Using these values, U1 is calculated using;
ρ
Figure 5.8 Velocity Traverse in the Wake of a Cylinder
The drag coefficient may be obtained by use of the curve of
5.8. The area beneath the curve is found using the trapezoidal rule (see Appendix A) to be:
Using this Equation gives
d
The values obtained for drag coefficient of the circular cylinder are as follows
By direct weighing CD = 1.03
By pressure plotting CD = 1.07
By wake traverse CD = 1.05
Taking the first of these as being the most reliable, we see that pressure plotting yields a result within the probable limits of accuracy.
There have been very many measurements of drag of circular cylinders, and the variation of CD with Reynolds number Re, where
Re = Ud ν
is well established. In this experiment, the absolute viscosity is, at 24°C,
µ = 2.40 × 10-5kg/m s
so the kinematic viscosity is
ν = µ
The velocity U at the typical value of velocity pressure
½ρU2 = 500 N/m2
So a typical value of Re for these tests is:
10 5 value usually quoted for a cylinder being
CD = 1.20
This is for the case of a long cylinder, for which l
d is so great that the three dimensional flows at the ends have no significant effect on the result. Our experiments are made for a cylinder with l
d = 3.9 for which end effects must be significant. Moreover, the cylinder presents an appreciable blockage to the cross section of the flow - its projected area is about 1/8th of the cross sectional area of the working section, so the results cannot be compared directly with those obtained from cylinders in unconfined flows.
Questions for Further Discussion
1. A yawmeter is an instrument for finding the direction of a fluid stream. One type of yawmeter consists of a circular cylinder with two surface holes at different positions in the same diametral plane. It is placed in the stream and rotated slowly about its axis until a balance is obtained between the observed pressures at the holes. Suggest a suitable angular spacing between the holes to give good sensitivity.
2. Measure the drag coefficients of a flat plate and an aerofoil section by direct weighing, and comment on the results obtained. Could pressure plotting be used to establish the drag coefficients of these sections?
(CD = 2.0 for plate, CD = 0.04 for aerofoil)
3. The circular cylinder presents a substantial blockage to the flow along the working section. Suppose instead of using the approach velocity U, the velocity U1 past the cylinder was used as the velocity on which the reference value of velocity head is based. Show that
Ul = 8 7U
for the results presented here, and find the corresponding value CD1 from
C D
D1 1 U d
2 12
= ρ
(0.76)
6. THE ROUND TURBULENT JET
AF13 Round Turbulent Jet Apparatus
Introduction
The behaviour of a jet as it mixes into the fluid which surrounds it has importance in many engineering applications. The exhaust from a gas turbine is an obvious example. In this experiment we establish the shape of an air jet as it mixes in a turbulent manner with the surrounding air. It is convenient to refer to such a jet as a
‘submerged’ jet to distinguish it from the case of the ‘free’ jet where no mixing with the surrounding medium takes place, as is the case when a smooth water jet passes through the atmosphere.
If the Reynolds number of a submerged jet (based on the initial velocity and diameter of the jet) is sufficiently small, the jet remains laminar for some length - perhaps 100 diameters or more. In this case the mixing with the surrounding fluid is very slight, and the jet retains its identity. Laminar jets are important in certain fluidic applications, where a typical diameter may be 1 mm, but the vast majority of engineering applications occur in the range of Re where turbulent jets are produced.
Figure 6.1 Schematic Representation of a Round Turbulent Jet
The essential features of a round turbulent jet are illustrated in Figure 6.1. The jet starts where fluid emerges uniformly at speed U from the end of a thin-walled tube, of cross-sectional radius R, placed in the body of a large volume of surrounding fluid.
The sharp velocity discontinuity at the edge of the tube gives rise to an annular shear layer which almost immediately becomes turbulent.
The width of the layer increases in the downstream direction as shown in the diagram.
For a short distance from the end of the tube the layer does not extend right across the jet, so that at section 1 there is a core of fluid moving with the undisturbed velocity U, the velocity in the shear layer rising from zero at the outside to U at the inside.
Further downstream the shear layer extends right across the jet and the velocity uo on the jet axis starts to fall as the mixing continues until ultimately the motion is completely dissipated.
There is entrainment from the fluid surrounding the jet by the turbulent mixing process so that the mass flux in the jet increases in the downstream direction. The static pressure is assumed to be constant throughout, so there is no force in the direction of the jet. The momentum of the jet is therefore conserved. The kinetic energy of the jet decreases in the downstream direction due to the turbulent dissipation. It should be emphasised that the velocity profiles indicated in Figure 6.1 are mean velocity distributions, and that the very severe turbulence in the jet will cause instantaneous velocity profiles to vary considerably from these mean ones.
Velocity Distribution and Momentum Flux
Consider the jet of Figure 6.1. If we assume that the flow pattern is independent of Reynolds number, then we might expect the velocity on the jet axis to depend on position in the dimensionless form
u
U f x
R
o =
(6-1)