TABLE OF CONTENTS
3.1 Bernoulli's Theorem (After D. Bernoulli 17001782)
3.2 Loss of Head due to Friction
3.3 The Continuity Equation
3.4 Application to Bernoulli's Apparatus
3.5 Laminar and Turbulent Flow
4.1 Bernoulli's Apparatus allows two experiments to be conducted
4.2 Equipment Preparation
The flow of a fluid has to conform with a number of scientific principles in particular the conservation of mass and the conservation of energy. The first of these when applied to a liquid flowing through a conduit requires that for steady flow the velocity will be inversely proportional to the flow area. The second requires that if the velocity increases then the pressure must decrease.
Cussons P6231 Bernoulli's Apparatus demonstrates both of these principles and can also be used to examine the onset of turbulence in an accelerating fluid stream.
Both Bernoulli's equation and the Continuity equation are essential analytical tools required for the analysis of most problems in the subject of Mechanics of Fluids.
Bernoulli's Apparatus consists essentially of a two dimensional rectangular section convergent divergent duct designed to fit between Cussons P6103 Constant Head Inlet Tank and P6104 Variable Head Outlet Tank. An eleven tube static pressure manometer bank is attached to the convergent divergent duct. A dye injection system is provided which allows for a single filament of dye to be introduced into the entrance to the convergent section to enable laminar and turbulent flow regimes to be demonstrated. The differential head across the test section can be varied from zero up to a maximum of 450mm. The test section, which is manufactured from acrylic sheet, is illustrated in figure 1 below.
Figure 1  Bernoulli's Apparatus
The convergent divergent duct is symmetrical about the centre line with a flat horizontal upper surface into which the eleven static pressure tappings are drilled. The lower surface is at an angle of 4º 29'. The width of the channel is 6·35 mm. The height of the channel at entry and exit is 19·525 mm and the height at the throat is 6·35 mm. The static tappings are at a pitch of 25 mm distributed about the centre and therefore about the throat. The flow area at each tapping is tabulated below the dimensions which are shown in figure 2.
Figure 2 Duct Dimension
Tapping Number 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
Flow Area 
102 · 56 
90 · 11 
77 · 66 
65 · 22 
52 · 77 
40 · 32 
52 · 77 
65 · 22 
77 · 66 
90 · 11 
102 · 56 
List of Symbols
A 
Cross sectional area of channel 
m^{2} 
D 
Diameter 
m 
D_{e} 
Equivalent diameter 
m 
G 
Acceleration due to gravity (9·807) 
m/s^{2} 
h 
Height of passage 
m 
H 
Head 
m 
H_{f} 
Frictional head 
m 
L 
Length of pipe 
m 
Mass flow rate 
kg/s 

P 
Pressure 
bars 
Flow rate 
m^{3}/s 

R 
Pressure recovery  
S 
Distance 
m 
t 
Time 
secs 
V 
Velocity 
m/s 
w 
Width of passage 
m 
Z 
Vertical height 
m 
r 
Density 
kg/m^{3} 
m 
Viscosity 
Ns/m^{2} 
3.1 Bernoulli's Theorem (After D. Bernoulli 17001782)
Consider steady flow of an inviscid fluid in a small section of a stream tube as illustrated in figure 3 below.
Figure 3 Element in a Streamtube
The length of the element d S is sufficiently small for any curvature of the streamtube to be neglected. The pressure and velocity will vary along the streamtube but since the flow is assumed to be steady these properties will not vary with time at any one fixed point. At the upstream end assume that the cross sectional area is A, the static pressure P, and the velocity V. At the downstream end these properties will have changed by d A, d P and d V respectively. The height of the downstream section be d Z higher than the upstream section. Now consider the forces acting on the fluid element in the direction of travel.
Pressure force on upstream end = P A
Pressure force on sides =
Pressure force on downstream end =
Gravitational force =
There will be no frictional forces due to the assumption of no viscosity
The resultant of these forces is :
expanding and neglecting second orders of small quantities the resultant force reduces to :
Applying Newtons second law along the streamline
Dividing through by r A d S
Now can be replaced by
Hence
Since P, Z and V are all functions of S then the partial derivatives may be replaced by total derivatives, and on rearranging the equation can be written as :
This is known as Euler's equation, it can be integrated with respect to S provided the variation of the density r with distance S along the streamline is known. When dealing with liquid which are essentially incompressible an obvious and reasonable assumption is to assume that the density is constant. Therefore integrating Euler's equation for a constant density gives :
or dividing through by g
This result is known as Bernoulli's equation. It is applicable to the steady flow of an incompressible and inviscid fluid. Bernoulli's equation shows that the sum of the three quantities
are constant. Therefore the three terms must be interchangeable so that, for example, if in a horizontal system the velocity head is increased then the pressure head must decrease.
3.2 Loss of Head due to Friction
If the fluid is not inviscid then there will be a small loss of head due to friction within the fluid and between the fluid and the walls of the passage. Bernoulli's Equation can then be modified by the inclusion of the frictional head loss H_{f}
where Bernoulli's equation has been written in the integrated form and has been applied between the upstream section 1 and the downstream section 2.
3.3 The Continuity Equation
The continuity equation is a statement of the conservation of mass. Consider the steady flow of a fluid through a streamtube of varying cross sectional area as shown in figure 3. For steady flow the mass of fluid entering the streamtube at section 1 must equal the mass of fluid leaving the streamtube at section 2. The mass flow rate of fluid at any section along the streamtube must be constant so that :
For an incompressible fluid the density is constant and the continuity equation can be written as :
For an incompressible fluid flowing in a converging duct it follows that as the area reduces then the velocity must increase, whilst in a diverging duct as the area increases then the velocity must decrease. Applying Bernoulli's equation if the velocity increases then the pressure must decrease whilst as the velocity decreases the pressure must increase. These processes are illustrated in figure 4 below.
A decreases  A increases 
V increases  V decreases 
P decreases  P increases 
This is a NOZZLE  This is a DIFFUSER 
Figure 4 Nozzle and Diffuser
3.4 Application to Cussons P6231 Bernoulli's Apparatus
In Cussons P6231 Bernoulli's Apparatus the passage is two dimensional with a constant width but with a linearly varying height. The flow area of the passage therefore also varies linearly. From the continuity equation

where

For this channel w is constant in size throughout since the channel is formed between two parallel plates, hence the product h V = Constant. The velocity head V^{2}/2g will be proportional to h^{2} and therefore to S^{2}. A graph of static head against distance S will be an inverted parabola.
The effect of loss of head due to friction can be investigated by comparing the static head at positions of equal area for the converging and diverging parts of the duct. Using Bernoulli's equation
Since the passage is horizontal Z_{1} = Z_{2}. At two positions of equal area the two velocities will be equal thus the equation reduces to
Most of the pressure loss in the converging part of the duct is recovered in the diverging part of the duct. The degree of pressure recovery is given by :
Experiments show very clearly that whilst it is possible to change pressure head to velocity head without appreciable loss of energy, it is impossible to change velocity head to pressure head without loss.
3.5 Laminar and Turbulent Flow
Laminar and turbulent flow regimes may be seen by the behaviour of a fine line of dye injected into the centre of the converging passage. If a condition can be set up in which the flow velocity at the start of the converging duct is high but laminar then as the velocity increases in the converging duct the flow will become turbulent and this transition can be observed by the behaviour of the dye line.
The Reynolds number is defined as
For a rectangular duct the equivalent dimension to diameter to include in the calculation of Reynolds number is :
The value of R_{e} at which turbulence commences is indicative of the higher critical Reynolds number of the fluid. Similarly, the value at which turbulence finally subsides is indicative of the lower critical Reynolds number. Careful manipulation of the flow rate will show turbulence at the throat while the flow in the convergent passage is still laminar. One of the important aspects of this experiment is in the facility with which random variation of flow may be achieved and the corresponding flow patterns observed.
4.1 Cussons P6231 Bernoulli's Apparatus allows two experiments to be conducted
Experiment 1 Verification of Bernoulli's Equation
Experiment 2 Demonstration of Laminar and Turbulent Flow
4.2 Equipment Preparation
Position the Inlet Head Tank and the Variable Head Outlet Tank on the mounting studs provided on the Hydraulic Bench working surface and connect the Bernoulli Apparatus between them using the union connections. Connect the Bench Feed hose to the Inlet Head Tank and attach an overflow hose to the overflow outlet of the Inlet Head Tank. Attach the Dye Reservoir to the top of the Inlet Head Tank using the attached mounting clip and ensure that the spring clip is attached to the rubber hose so that ink cannot flow to the injector needle. Fill the ink reservoir with a water miscible dye, washable blue ink is recommended. Make sure that the dye is free to flow through the dye injector needle when the spring clip is adjusted. If a blockage of the ink does occur in the injector needle, this is usually caused by a failure to wash out all the ink when the equipment has been previously used. The blockage can normally be rectified by flushing the dye injection system thoroughly with clean water. Remove the brass blanking plug from the side of the Inlet Head Tank and insert the Dye Injector Needle so that the tip protrudes approximately 20mm into the transparent Bernoulli Test section. In order to record the height of the water level in each of the manometer tubes, a sheet of paper should be positioned as in figure 1.
EXPERIMENT 1  VERIFICATION OF BERNOULLI'S EQUATION
Aim. To verify Bernoulli's equation by demonstrating the relationship between pressure head and kinetic head.
Equipment Preparation. Prepare the equipment to the following specification
Inlet 
P6103 Constant Head Inlet Tank with overflow extension fitted. 
Test Section 
P6231 Bernoulli's Apparatus 
Exit 
P6104 Variable Head Outlet Tank 
Manometer 
Insert a sheet of graph paper 440mm high by 325mm wide behind the manometer tubes to provide an easy method of obtaining a record of the results. 
Experimental Procedure.
Start the pump and initiate a flow of water through the test section. Regulate the flow to the inlet head tank so that there is a small but steady overflow from the reservoir. Adjust the swivel tube of the outlet tank to obtain a differential head.
Measure the height of the water level in each manometer tube by marking the paper positioned behind the tubes and record on the test sheet. Measure the time taken to fill the bench measuring tank from zero to 10 litres and record.
Increase the differential head between the inlet and outlet head tanks by 50mm increments, until the water level in the centre manometer tubes drops off the scale. For each condition record the heights of liquid in the manometer tubes by once again marking the paper positioned behind the tubes and measure the flow rate.
Results and Analysis.
RESULTS SHEET 31 BERNOULLI'S APPARATUS
EXPERIMENT 1  VERIFICATION OF BERNOULLI'S EQUATION
EXPERIMENT 2  DEMONSTRATION OF LAMINAR AND TURBULENT FLOW
Aim. To demonstrate visually laminar (or streamlined) flow and its transition to turbulent flow at a particular velocity.
Equipment Preparation. Prepare the equipment to the following specification
Inlet 
P6103 Constant Head Inlet Tank with overflow extension fitted. 
Test Section 
P6231 Bernoulli's Apparatus 
Exit 
P6104 Variable Head Outlet Tank 
Manometer 
Insert a sheet of graph paper 440mm high by 325mm wide behind the manometer tubes to provide an easy method of obtaining a record of the results. 
Dye injection 
Fitted 
Experimental Procedure.
Start the pump and initiate a flow of water through the test section. Regulate the flow to the inlet head tank so that there is a small but steady overflow from P6103. Adjust the swivel tube of the outlet tank to obtain a differential head of 20mm.
Raise the dye reservoir to the top of its column (the spring loaded bracket can be freed by squeezing the two ends together between the fingers) and open the tube clip.
Open the small cock on the base of the reservoir to permit dye to flow from the nozzle at the entrance to the channel. This will be visible as a coloured stream along the passage. If the dye accumulates around the nozzle, increase the velocity of flow in the passage and/or check the flow from the dye reservoir.
Under laminar flow conditions the stream will be visible along the whole length of the passage. If this is not so, reduce the flow until a continuous stream of dye is visible along the passage.
Steadily increase the flow rate by increasing the total differential head, while carefully observing the condition of the fluid in the channel. When instability occurs leading to the breakup of the dye stream, note the position in the passage and measure the corresponding value of the flow rate.
Continue to maintain close observation of the passage while further increasing the flow rate until the whole system is turbulent with no visible dye stream at any point.
Reduce the flow rate to the point at which turbulence disappears and stable laminar flow conditions are regained along the whole passage. Measure the flow rate and position of the last traces of turbulence.
Continuous manipulation of the flow rate while observing the flow conditions may be conducted as a useful visual aid to the appreciation of laminar and turbulent flow conditions.
Switch off the pump and allow the apparatus to drain back to the main reservoir.
Note : Continuous use of the dye will tint the circulating water. When this becomes severe the water should be changed. Do not use water containing dye in other apparatus
Completely clean all the apparatus of any trace of water containing dye before returning the apparatus to store.
Results and Analysis.