The experiment demonstrates the use of a Pitot-static tube, and investigates the application of Bernoulli’s theorem to flow along a convergent-divergent passage.

A duct of rectangular section is fitted to the exit of the contraction which leads from the air box, and liners placed along the inside walls of the duct produce a passage which contracts to a parallel throat and then expands to the original width. The convergent portion is shorter than the divergent one. Air is blown through the passage, and a probe may be traversed along the center line to measure the distribution of total pressure P and static pressure p. This probe is a Pitot-static probe. Pressure tappings are connected from the air box and from the Pitot-static probe to a multitube manometer.

The aim of the experiment is to measure the distribution of the difference between the total pressure P and static pressure p along the duct and, using Bernoulli’s equation, determine the absolute velocity values, u, along the same channel.

Consider how the equation is applied to the present case.

According to Bernoulli’s equation the total pressure P, defined by

should be constant along this tube, provided the flow is steady and that the air is incompressible and inviscid. If P₀ denotes the total pressure in the air box, then we should expect the measured value of P along the passage to be everywhere the same as P₀, if Bernoulli’s theorem is valid for this motion.

Now the total pressure P is measured with comparative ease by an opened tube facing the flow. A streamline starts from the air box, passes along the duct, and arrives at the mouth of the Pitot tube. The motion is arrested at this point, so that in equation (1) the local value of u is zero. The pressure recorded by the Pitot tube is therefore the local value of total pressure P. If Bernoulli’s equation applies along the whole length of the streamline from the air box, then P should everywhere be the same as the initial total pressure P₀. The value of P₀ may be found easily from a pressure tapping in the wall of the air box, since the air velocity in the box is so slight as to make the difference between total pressure and static pressure quite negligible.

The variation of static pressure p may be measured by the static pressure tube. Consider a further streamline emanating from the air box and flowing close to the surface of the probe. Provided that the holes in the surface of the probe are placed far enough from the tip of the tube as to be unaffected by the disturbance in this locality (which means in practice about 6 tube diameters away from the tip) then the flow is undisturbed by the holes, which therefore measure the undisturbed pressure, viz. static pressure p. To compare the measured values of p with the result of calculations we must use the continuity equation as well as the Bernoulli equation. Taking the flow as one-dimensional, viz. assuming the velocity over any chosen cross-section to be uniform over that section, then the continuity equation for incompressible flow gives the volume flow rate as

(The suffix t indicates conditions in the throat). The velocity distribution along the duct may be thus be written in the form of the ratio

and since the depth of the duct is constant, cross-sectional area is proportional to width, so

The velocity ratio following from continuity may therefore be calculated simply from the dimensions of the convergent-divergent passage. This now may be compared with the velocity ratio inferred from pressure distribution using Bernoulli’s theorem. For equation (1) gives the local velocity as

and in particular the velocity u_t at the throat is

so from equations (5) and (6)

The right-hand side of this equation may be evaluated from the measured pressure distribution and compared with the values from equation (4).

In fact, the Pitot tube, provided for the experiment, measures the difference between the total pressure and the static pressure, i.e., P – p. The two tappings of the Pitot tube are connected to the multitube manometer.

You may put your results in the form of the following table:

In order to make your results more representative, plot three figures. The first one is pressure difference, P - p (N/m²), versus X, the distance from start of contraction (mm). (Plot P₀ on the same figure). The second figure should be u/u_t versus X. In order to be able to compare your experimental data with the Bernoulli’s theorem predictions, plot the measured values of u/u_t and those given by equation (4) on the same figure. The third figure should represent a dependence u(X). Uncertainty analysis must be performed for the absolute velocity values, i.e., the equation for Du/u must be derived.

1. What is the Mach number at the throat of the duct? For approximate calculation, you may assume that the static pressure and temperature there are approximately the same as in the air box. The air velocity at the throat may be found from the Pitot-static reading, and the acoustic velocity a may be estimated from the equation

(8)

where

g is the ratio of specific heats = 1.4 for air

R is the gas constant = 287.2 J/kg K

T is the absolute temperature in K

2. What difference to the results would you expect if the flow direction were reversed?