1. 4.4 DISCUSSION OF DIMENSIONLESS PARAMETERS
The five dimensionless parameters:
pressure coefficient;
Reynolds number;
Froude number;
Weber number;
Mach number
- are of importance in correlating experimental data.

2. Pressure Coefficient
The pressure coefficient △p/(ρV2/2) is the ratio of pressure to dynamic pressure
When multiplied by area, it is the ratio of pressure force to inertial force, as (ρV2/2)A would be the force needed to reduce the velocity to zero.
It may also be written as △h/(V2/2g) by division by γ.
For pipe flow the Darcy-Weisbach equation relates losses h1 to length of pipe L, diameter D, and velocity V by a dimensionless friction factor f
as fL/D is shown to be equal to the pressure coefficient.
In pipe flow, gravity has no influence on losses; therefore, F may be dropped out. Similarly, surface tension has no effect, and W drops out.

3. For steady liquid flow, compressibility is not important, and M is dropped. l may refer to D; l1 to roughness height projection є in the pipe wall; and l2 to their spacing є'; hence,
(4.4.1)
If compressibility is important,
(4.4.2)
With orifice flow,
(4.4.3)
in which l may refer to orifice diameter and l1 and l2 to upstream dimensions.
Viscosity and surface tension are unimportant for large orifices and low-viscosity fluids. Mach number effects may be very important for gas flow with large pressure drops, i.e., Mach numbers approaching unity.

4. In steady, uniform open-channel flow, the Chezy formula relates average velocity V, slope of channel S, and hydraulic radius of cross section R (area or section divided by wetted perimeter) by
(4.4.4)
C is a coefficient depending upon size, shape, and roughness of channel. Then
(4.4.5)
since surface tension and compressible effects are usually unimportant.
The drag F on a body is expressed by F = CDAρV2/2, in which A is a typical area of the body, usually the projection of the body onto a plane normal to the flow. Then F/A is equivalent to △p, and
(4.4.6)
R is related to skin friction drag due to viscous shear as well as to form, or profile, drag resulting from separation of the flow streamlines from the body; F is to wave drag if there is a free surface, for large Mach numbers CD may vary more markedly with M than with the other parameters; the length ratios may refer to shape or roughness of the surface.

5. The Reynolds Number
The Reynolds Number VDρ/μ is the ratio of inertial forces to viscous forces.
A critical Reynolds number distinguishes among flow regimes, such as laminar or turbulent flow in pipes, in the boundary layer, or around immersed objects.
The particular value depends upon the situation.
In compressible flow, the Mach number is generally more significant than the Reynolds number.

6. The Froude Number
The Froude Number , when squared and then multiplied and divided by ρA, is a ratio or dynamic (or inertial) force to weight.
With free liquid-surface flow the nature of the flow (rapid or tranquil) depends upon whether the Froude number is greater or less than unity.
It is useful in calculations of hydraulic jump, in design of hydraulic structures, and in ship design.

7. The Weber Number
The Weber Number V2lρ/σ is the ratio of inertial forces to surface-tension forces (evident when numerator and denominator are multiplied by l)
It is important at gas-liquid or liquid-liquid interfaces and also where these interfaces are in contact with a boundary.
Surface tension causes small (capillary) waves and droplet formation and has an effect on discharge of offices and weirs at very small heads.
Fig. 4.1 shows the effect of surface tension on wave propagation.
To the left of the curve's minimum the wave speed is controlled by surface tension (the waves are called ripples), and to the right of the curve's minimum gravity effects are dominant.

8. Figure 4.1 Wave speed vs. wavelength for surface waves

9. The Mach Number
The speed of sound in a liquid is written if K is the bulk modulus of elasticity or (k is the specific heat ratio and T the absolute temperature for a perfect gas).
V/c or is the Mach number. It is a measure of the ratio of inertial forces to elastic forces.
By squaring V/c and multiplying by ρA/2 in numerator and denominator, the numerator is the dynamic force and the denominator is the dynamic force at sonic flow.
It may also be shown to be a measure of the ratio or kinetic energy or the flow to internal energy of the fluid. It is the most important correlating parameter when velocities are near or above local sonic velocities.

10. 4.5 SIMILITUDE; MODEL STUDIES
Model studies of proposed hydraulic structures and machines: permit visual observation or the flow and make it possible to obtain certain numerical data. e.g., calibrations of weirs and gates, depths of flow, velocity distributions, forces on gates, efficiencies and capacities of pumps and turbines, pressure distributions, and losses.
To obtain accurate quantitative data: there must be dynamic similitude between model and prototype. This similitude requires (1) that there be exact geometric similitude and (2) that the ratio of dynamic pressures at corresponding points be a constant (kinematic similitude, i.e., the streamlines must be geometrically similar).
Geometric similitude extends to the actual surface roughness of model and prototype. For dynamic pressures to be in the same ratio at corresponding points in model and prototype, the ratios of the various types or forces must be the same at corresponding points.
Hence, for strict dynamic similitude, the Mach, Reynolds, Froude, and Weber numbers must be the same in both model and prototype.

11. Wind- and Water-Tunnel Tests
Used to examine the streamlines and the forces that are induced as the fluid flows past a fully submerged body.
The type of test that is being conducted and the availability of the equipment determine which kind of tunnel will be used.
Kinematic viscosity of water is about one-tenth that of air  a water tunnel can be used for model studies at relatively high Reynolds numbers.
At very high air velocities the effects of compressibility, and consequently Mach number, must be taken into consideration, and indeed may be the chief reason for undertaking an investigation.
Figure 4.2 shows a model of an aircraft carrier being tested in a low-speed tunnel to study the flow pattern around the ship's super-structure. The model has been inverted and suspended from the ceiling so that the wool tufts can be used to give an indication of the flow direction. Behind the model there is an apparatus for sensing the air speed and direction at various locations along an aircraft's glide path.

12. Figure 4. 2 Wind tunnel tests on an aircraft carrier superstructure
Figure 4.2 Wind tunnel tests on an aircraft carrier superstructure. Model is inverted and suspended from ceiling.

13. Pipe Flow
In steady flow in a pipe, viscous and inertial forces are the only ones of consequence.
Hence, when geometric similitude is observed, the same Reynolds number in model and prototype provides dynamic similitude.
The various corresponding pressure coefficients are the same
For testing with fluids having the same kinematic viscosity in model and prototype, the product, VD, must be the same.
Frequently this requires very high velocities in small models.

14. Open Hydraulic Structures
Structures such as spillways, stilling pools, channel transitions, and weirs generally have forces due to gravity (from changes in elevation of liquid surfaces ) and inertial forces that are greater than viscous and turbulent shear forces.
In these cases geometric similitude and the same value of Froude's number in model and prototype produce a good approximation to dynamic similitude; thus
Since gravity is the same, the velocity ratio varies as the square root of the scale ratio λ = lp/lm
The corresponding times for events to take place (as time for passage of a particle through a transition) are related; thus

15. Figure 4.3 Model test on a harbor to determine the effect of a breakwater

16. Ship’s Resistance
The resistance to motion of a ship through water is composed of pressure drag, skin friction, and wave resistance. Model studies are complicated by the three types of forces that are important, inertia, viscosity, and gravity. Skin friction studies should be based on equal Reynolds numbers in model and prototype, but wave resistance depends upon the Froude number. To satisfy both requirements, model and prototype must be the same size.
The difficulty is surmounted by using a small model and measuring the total drag on it when towed. The skin friction is then computed for the model and subtracted from the total drag. The remainder is stepped up to prototype size by Froude's law, and the prototype skin friction is computed and added to yield total resistance due to the water.
Figure 4.4 shows the dramatic change in the wave profile which resulted from a redesigned bow. From such tests it is possible to predict through Froude's law the wave formation and drag that would occur on the prototype.

17. Figure 4.4
Model tests showing the influence of a bulbous bow on bow wave

18. Hydraulic Machinery
The moving parts in a hydraulic machine require an extra parameter to ensure that the streamline patterns are similar in model and prototype. This parameter must relate the throughflow (discharge) to the speed of moving parts.
For geometrically similar machines, if the vector diagrams of velocity entering or leaving the moving parts are similar, the units are homologous; i.e., for practical purposes dynamic similitude exists.
The Froude number is unimportant, but the Reynolds number effects (called scale effects because it is impossible to maintain the same Reynolds number in homologous units) may cause a discrepancy of 2 or 3 percent in efficiency between model and prototype.
The Mach number is also of importance in axial-flow compressors and gas turbines.

19. Example 4.4
The valve coefficients K = Δp/(ρV2/2) for a 600-mm-diameter valve are to be determined from tests on a geometrically similar 300-mm-diameter valve using atmospheric air at 27°C. The ranges of tests should be for flow of water at 20°C at 1 to 2.5 m/s. What ranges of airflows are needed?
Solution
The Reynolds number range for the prototype valve is
For testing with air at 27°C

20. Then the ranges of air velocities are