dimensional analysis 5
5.5 OTHER DIMENSIONLESS GROUPS
We have already dealt in Section 5.4 with a small but important collection of independent dimensionless groups, including Reynolds number, Mach number, Froude number and Weber number. Besides these dimensionless groups, there are numerous others that have their place in the study of fluid mechanics. A number of these fall into one or other of three important sub-sets expressing in a dimensionless form (i) differences in pressure (or head); (ii) forces on bodies; or (iii) surface shear stresses (friction) resulting
Table 5.1 Principal dimensionless groups in fluid dynamics Dimensionless
group
Name Represents
magnitude ratio of these forces
Recommended symbol
ul/µ Reynolds number |Inertia force|
|Viscous force| Re u/(lg)1/2 Froude number |Inertia force|
|Gravity force| Fr u/(l/γ )1/2 Weber number |Inertia force|
|Surface tension force| We u/a Mach number |Inertia force|
|Elastic force| M
from fluid motion. These can be broadly classified as pressure coefficients, force coefficients and friction coefficients, respectively. Amongst the force coefficients are the lift and drag coefficients, of fundamental importance in aircraft aerodynamics. The friction coefficients include amongst their number the friction factor associated with pipe flow, and the skin friction coefficient used to describe in a dimensionless form the surface shear stresses of external fluid flow. Examples of dimensionless groups which fall outside these named categories are the Strouhal number, which arises in the treat-ment of cyclical phenomena, and the mass flow parameter, which is a useful concept when dealing with compressible flows in pipes. Apart from the pres-sure coefficients, which will be discussed explicitly, albeit briefly, all other dimensionless groups will be considered as they arise.
5.5.1 The pressure coefficient and related coefficients
Pressure forces are always present and are therefore represented in any complete description of fluid flow. When expressed in dimensionless form, the ratio of pressure forces to other types of forces appears. For example, the ratio
|Pressure force|
|Inertia force| is proportional to p∗l2
l2u2 = p∗
u2
where p∗ represents the difference in piezometric pressure between two points in the flow. In fluid dynamics, it has become normal practice to use the ratio p∗/12u2, the12being inserted so that the denominator represents kinetic energy divided by volume or, for an incompressible fluid, the dynamic pressure of the stream (see Section 3.7.1). This latter form is usually known as the pressure coefficient, denoted by the symbol Cp.
Several other similar coefficients are, in essence, variants of the pressure coefficient. Amongst these are the static and total pressure loss coefficients, widely used to describe the dissipation of mechanical energy that occurs in internal fluid flows, and the pressure recovery coefficient more specifically used to describe the properties of diffusers. Also in this category are the corresponding head loss coefficients.
5.5.2 The discharge coefficient
The discharge coefficient is an important dimensionless parameter which relates the flow rate through a differential-pressure flow-metering device, such as an orifice plate, nozzle or venturi tube, to the pressure distribu-tion the flow generates. It was first introduced in Chapter 3, where it was used to adjust theoretical values of mass flow rate (or volumetric flow rate), derived from simplistic mathematical models of fluid motion which ignored the effects of viscosity, to yield improved comparisons with the behaviour of real flows. The method of dimensional analysis provides a more rigorous justification for the use of the discharge coefficient, since, from the outset, the analysis takes full account of the viscous nature of real fluids.
Other dimensionless groups 169 Consider the incompressible flow along a straight section of circular pipe
in which there is a constriction due to the presence of an orifice plate, nozzle or venturi tube. Denote the diameter of the pipe by D and the diameter at the minimum cross-sectional area by d. The pressure varies along the pipe partly as a result of viscous effects but mainly as a consequence of the geometry of the constriction, which causes the flow to accelerate. The difference in piezometric pressure p∗measured between two arbitrary points along the pipe, denoted by suffices a and b, depends on the positions at which the pressures are measured. In the most general case, the position of each tapping can be specified in terms of cylindrical polar coordinates x, r,θ. In practice, for flow-metering, the pressure tappings are located at the circumference of the pipe and efforts are made to ensure that the flow approaching the device is axisymmetric. Hence the positions of the pressure tappings are fully specified by the distance along the pipe axis, xa and xb, measured from some arbitrary datum. The positions xaand xbare chosen so that the measured pressure difference is a maximum, or close to a maximum, for a given flow rate. In practice this requires an upstream tapping to be located in the pipe upstream of the constriction, and the downstream tapping to be at, or close to, the plane of minimum cross-section. Besides the effects of D, d, xa and xb, the magnitude of p∗is determined by the mean velocity of the flow in the pipe, u, and the fluid density,, and dynamic viscosity, µ.
Hence, for a device of specified geometry, we may write p∗= f (D, d, xa, xb, u,, µ)
Dimensional analysis (see Section 5.6) yields the relation for the dimen-sionless pressure coefficient
p∗
1 2u2
= f1(d/D, xa/D, xb/D, uD/µ) (5.1) Denote the cross-sectional area of the pipe by A1 and the minimum cross-sectional area at the constriction by A2. The continuity equation, evaluated in the upstream pipe, yields m= A1u, which when substituted in equation 1 gives after rearrangement
m
A1(2 p∗)1/2
= f2(d/D, xa/D, xb/D, uD/µ) (5.2) It has become established practice in flow-metering to use the throat area A2 rather than A1 as the reference cross-sectional area. Hence eqn 5.2 can be replaced by
m
A2(2 p∗)1/2
= f3(d/D, xa/D, xb/D, uD/µ) (5.3) The dimensionless group m/"
A2(2 p∗)1/2#
is sometimes referred to as the flow coefficient.
A universal practice in flow-metering is to use the discharge coefficient Cd rather than the flow coefficient, although the two are closely related.
The discharge coefficient Cdis defined by the equation Cd= m(1 − λ2)1/2
A2(2 p∗)1/2 (5.4)
whereλ = A2/A1= (d/D)2and the dimensionless quantity(1 − λ2)−1/2is known as the velocity of approach factor. A comparison of equations 5.3 and 5.4 shows that Cdcan be expressed by the relation
Cd= f4(d/D, xa/D, xb/D, uD/µ) (5.5) In summary Cdis shown to depend on the area ratio of the constriction (as well as on the basic geometry of the flow-metering device), the positions of the two pressure tappings, and the Reynolds number.
The given derivation of eqn 5.5 applies to the flow through differential-pressure flow-metering devices used in internal flow systems. Weirs and notches are devices used for flow-rate measurement for fluid motion where a free surface exists. An approach, starting with the appropriate independent variables and again based on the methods of dimensional analysis, can be used to derive relationships for the discharge coefficients for these devices.
5.5.3 Cavitation number
In some instances of liquid flow the pressure at certain points may become so low that vapour cavities form – this is the phenomenon of cavitation (see Section 13.3.6). Pressures are then usefully expressed relative to pv, the vapour pressure of the liquid at the temperature in question. A signi-ficant dimensionless parameter is the cavitation number, (p − pv)/12 u2 (which may be regarded as a special case of the pressure coefficient). For fluid machines a special definition due to D. Thoma is more often used (see Section 13.3.6).
Ratios involving electrical and magnetic forces may arise if the fluid is per-meable to electrical and magnetic fields. These topics, however, are outside the scope of this book.