kind of force enters every problem, but a list of the possible types is usefully made at the outset of the discussion.
1. Forces due to differences of piezometric pressure between different points in the fluid. The phrase ‘in the fluid’ is worth emphasis. Dynamic similarity of flow does not necessarily require similarity of thrusts on corresponding parts of the boundary surfaces and so the magnitude of piezometric pressure relative to the surrounding is not important. For the sake of brevity, the forces due to differences of piezometric pressure will be termed pressure forces.
2. Forces resulting form the action of viscosity.
3. Forces acting from outside the fluid – gravity, for example.
4. Forces due to surface tension.
5. Elastic forces, that is, those due to the compressibility of the fluid.
Now any of these forces, acting in combination on a particle of fluid, in general have a resultant, which, in accordance with Newton’s Second Law, F = ma, causes an acceleration of the particle in the same direction as the force. And the accelerations of individual particles together determ-ine the pattern of the flow. Let us therefore examdeterm-ine a little further these accelerations and the forces causing them.
If, in addition to the resultant force F, an extra force (−ma) were applied to the particle in the same direction as F, the net force on the particle would then be F− ma, that is, zero. With zero net force on it, the particle would, of course, have zero acceleration. This hypothetical force (−ma), required to bring the acceleration of the particle to zero, is termed the inertia force: it represents the reluctance of the particle to be accelerated. The inertia force is, however, in no way independent of the other forces on the particle; it is, as we have seen, equal and opposite to their resultant, F. Nevertheless, since our concern is primarily with the pattern of the flow, it is useful to add inertia forces to our list as a separate item:
(6. Inertia forces.)
If the forces on any particle are represented by the sides of a force poly-gon, then the inertia force corresponds to the side required to close the
Ratios of forces arising in dynamic similarity 163 polygon. Now a polygon can be completely specified by the magnitude and
direction of all the sides except one. The remaining, unspecified, side is fixed automatically by the condition that it must just close the polygon.
Consequently, if for any particular particle this hypothetical inertia force is specified, one of the other forces may remain unspecified; it is fixed by the condition that the force polygon must be completely closed, in other words, that the addition of the inertia force would give zero resultant force.
For dynamic similarity between two systems, the forces on any fluid particle in one system must bear the same ratios of magnitude to one another as the forces on the corresponding particle in the other system. In most cases several ratios of pairs of forces could be selected for consideration. But, because the accelerations of particles (and hence the inertia forces) play an important part in practically every problem of fluid motion, it has become conventional to select for consideration the ratios between the magnitude of the inertia force and that of each of the other forces in turn. For example, in a problem such as that studied in Section 1.9.2, the only relevant forces are inertia forces, viscous forces and the forces due to differences of pres-sure. The ratio chosen for consideration in this instance is that of|Inertia force| to |Net viscous force| and this ratio must be the same for correspond-ing particles in the two systems if dynamic similarity between the systems is to be realized. (There is no need to consider separately the ratio of|Inertia force| to |Pressure force| since, once the inertia force and net viscous force are fixed, the pressure force is determined automatically by the condition that the resultant of all three must be zero.) In a case where the forces involved are weight, pressure force and inertia force, the ratio chosen is|Inertia force|
to|Weight|.
We shall now consider the various force ratios in turn.
5.3.1 Dynamic similarity of flow with viscous forces acting
There are many instances of flow that is affected only by viscous, pressure and inertia forces. If the fluid is in a full, completely closed conduit, gravity cannot affect the flow pattern; surface tension has no effect since there is no free surface, and if the velocity is well below the speed of sound in the fluid the compressibility is of no consequence. These conditions are met also in the flow of air past a low-speed aircraft and the flow of water past a submarine deeply enough submerged to produce no waves on the surface.
Now for dynamic similarity between two systems, the magnitude ratio of any two forces must be the same at corresponding points of the two systems (and, if the flow is unsteady, at corresponding times also). There are three possible pairs of forces of different kinds but, by convention, the ratio of
|Inertia force| to |Net viscous force| is chosen to be the same in each case.
The inertia force acting on a particle of fluid is equal in magnitude to the mass of the particle multiplied by its acceleration. The mass is equal to the density times the volume (and the latter may be taken as proportional to the cube of some length l which is characteristic of the geometry of the system). The mass, then, is proportional to l3. The acceleration of the particle is the rate at which its velocity in that direction changes with time and
so is proportional in magnitude to some particular velocity divided by some particular interval of time, that is, to u/t, say. The time interval, however, may be taken as proportional to the chosen characteristic length l divided by the characteristic velocity, so that finally the acceleration may be set proportional to u÷ (l/u) = u2/l. The magnitude of the inertia force is thus proportional tol3u2/l = u2l2.
The shear stress resulting from viscosity is given by the product of viscosity µ and the rate of shear; this product is proportional to µu/l. The magnitude of the area over which the stress acts is proportional to l2 and thus the magnitude of viscous force is proportional to(µu/l) × l2= µul.
Consequently, the ratio
|Inertia force|
|Net viscous force| is proportional tol2u2 µul = ul
µ
The ratioul/µ is known as the Reynolds number. For dynamic similarity of Reynolds number
two flows past geometrically similar boundaries and affected only by viscous, pressure and inertia forces, the magnitude ratio of inertia and viscous forces at corresponding points must be the same. Since this ratio is proportional to Reynolds number, the condition for dynamic similarity is satisfied when the Reynolds numbers based on corresponding characteristic lengths and velocities are identical for the two flows.
The length l in the expression for Reynolds number may be any length that is significant in determining the pattern of flow. For a circular pipe completely full of the fluid the diameter is now invariably used – at least in Great Britain and North America. (Except near the inlet and outlet of the pipe the length along its axis is not relevant in determining the pattern of flow. Provided that the cross-sectional area of the pipe is constant and that the effects of compressibility are negligible, the flow pattern does not change along the direction of flow – except near the ends as will be discussed in Section 7.9.) Also by convention the mean velocity over the pipe cross-section is chosen as the characteristic velocity u.
For flow past a flat plate, the length taken as characteristic of the flow pattern is that measured along the plate from its leading edge, and the char-acteristic velocity is that well upstream of the plate. The essential point is that, in all comparisons between two systems, lengths similarly defined and velocities similarly defined must be used.
5.3.2 Dynamic similarity of flow with gravity forces acting
We now consider flow in which the significant forces are gravity forces, pressure forces and inertia forces. Motion of this type is found when a free surface is present or when there is an interface between two immiscible fluids.
One example is the flow of a liquid in an open channel; another is the wave motion caused by the passage of a ship through water. Other instances are the flow over weirs and spillways and the flow of jets from orifices into the atmosphere.
Ratios of forces arising in dynamic similarity 165 The condition for dynamic similarity of flows of this type is that the
magnitude ratio of inertia to gravity forces should be the same at corres-ponding points in the systems being compared. The pressure forces, as in the previous case where viscous forces were involved, are taken care of by the requirement that the force polygon must be closed. The magnitude of the inertia force on a fluid particle is, as shown in Section 5.3.1, proportional tou2l2where represents the density of the fluid, l a characteristic length and u a characteristic velocity. The gravity force on the particle is its weight, that is, (volume) g which is proportional to l3g where g represents the acceleration due to gravity. Consequently the ratio
|Inertia force|
|Gravity force| is proportional tol2u2
l3g =u2 lg
In practice it is often more convenient to use the square root of this ratio so as to have the first power of the velocity. This is quite permissible: equality of u/(lg)1/2implies equality of u2/lg.
The ratio u/(lg)1/2 is known as the Froude number after William Froude Froude number (1810–79), a pioneer in the study of naval architecture, who first introduced
it. Some writers have termed the square of this the Froude number, but the definition Froude number= u/(lg)1/2is now more usual.
Dynamic similarity between flows of this type is therefore obtained by having values of Froude number (based on corresponding velocities and cor-responding lengths) the same in each case. The boundaries for the flows must, of course, be geometrically similar, and the geometric scale factor should be applied also to depths of corresponding points below the free surface.
Gravity forces are important in any flow with a free surface. Since the pressure at the surface is constant (usually atmospheric) only gravity forces can under steady conditions cause flow. Moreover, any disturbance of the free surface, such as wave motion, involves gravity forces because work must be done in raising the liquid against its weight. The Froude number is thus a significant parameter in determining that part of a ship’s resistance which is due to the formation of surface waves.
5.3.3 Dynamic similarity of flow with surface tension forces acting In most examples of flow occurring in engineering work, surface tension forces are negligible compared with other forces present, and the engineer is not often concerned with dynamic similarity in respect to surface tension.
However, surface tension forces are important in certain problems such as those in which capillary waves appear, in the behaviour of small jets formed under low heads, and in flow of a thin sheet of liquid over a solid surface.
Here the significant force ratio is that of|Inertia force| to |Surface tension force|. Again, pressure forces, although present, need not be separately con-sidered. The force due to surface tension is tangential to the surface and has the same magnitude perpendicular to any line element along the surface.
If the line element is of length l then the surface tension force is γ ( l)
whereγ represents the surface tension. Since inertia force is proportional to
u2l2(Section 5.3.1) and l is proportional to the characteristic length l, the ratio
|Inertia force|
|Surface tension force| is proportional to l2u2 γ l = lu2
γ
The square root of this ratio, u(l/γ )1/2, is now usually known as the Weber number
Weber number after the German naval architect Moritz Weber (1871–1951) who first suggested the use of the ratio as a relevant parameter. Sometimes, however, the ratio(lu2/γ ) and even its reciprocal are also given this name.
5.3.4 Dynamic similarity of flow with elastic forces acting
Where the compressibility of the fluid is important the elastic forces must be considered along with the inertia and pressure forces, and the magnitude ratio of inertia force to elastic force is the one considered for dynamic simil-arity. Equation 1.7 shows that for a given degree of compression the increase of pressure is proportional to the bulk modulus of elasticity, K. Therefore, if l again represents a characteristic length of the system, the pressure increase acts over an area of magnitude proportional to l2, and the magnitude of the force is proportional to Kl2. Hence the ratio
|Inertia force|
|Elastic force| is proportional tol2u2 Kl2 = u2
K
The parameter u2/K is known as the Cauchy number, after the French Cauchy number
mathematician A. L. Cauchy (1789–1857). However, as we shall see in Chapter 11, the velocity with which a sound wave is propagated through the fluid (whether liquid or gas) is a =
(Ks/) where Ks represents the isentropic bulk modulus. If we assume for the moment that the flow under consideration is isentropic, the expressionu2/K becomes u2/a2.
In other words, dynamic similarity of two isentropic flows is achieved if, Mach number
along with the prerequisite of geometric similarity of the boundaries, u2/a2 is the same for corresponding points in the two flows. This condition is equi-valent to the simpler one that u/a must be the same at corresponding points.
This latter ratio is known as the Mach number in honour of Ernst Mach (1838–1916), the Austrian physicist and philosopher. It is very important in the study of the flow of compressible fluids. It should be remembered that a represents the local velocity of sound, which, for a given fluid, is determ-ined by the values of absolute pressure and density at the point where u is measured.
If the change of density is not small compared with the mean density, then thermodynamic considerations arise. In particular, the ratio of principal specific heat capacitiesγ must be the same in the two cases considered. Where appreciable changes of temperature occur, the ways in which viscosity and thermal conductivity vary with temperature may also be important. These matters are outside the scope of this book, but it is well to remember that
Other dimensionless groups 167 equality of the Mach numbers is not in every case a sufficient criterion for