F2 Flow of incompressible fluid and gases assumed to be incompressible

Presentation of different flow patters by stream lines

So far, we have only been concerned with the pressure in fluid flow. In order to obtain a clear impression of the flow itself, you would have to follow fluid particles along their trajectories or take a look at what happens at a given point in space in the course of time. The question regarding the velocity and pressure in a flow at a given instant belongs to the most complicated mathematical tasks. The major advances in this area are linked to the names of Euler, Lagrange, Helmholtz, Lord Kelvin. Our elementary description is limited to a few flows. Fig. 249 achieves this through stream lines, which can be made visible by means of dye particles, small aluminium foil, etc.

Irrotational flow (Potential flow)

Fig. 249 a shows a motion which arises through a change in pressure. Due to a definite mathematical property, it is referred to as potential flow¹. Its stream lines surround the body symmetrically and close up again behind it; the pressure and velocity is the same on both sides. For another form of flow is characteristic that the stream lines are closed, as in Fig. 249 b. Completely relevant is the pattern of potential flow only for ideal (frictionless) fluids. However, real fluids like water and air have only very little friction, so that in many cases a statement about the mode of motion of an ideal fluid is a good approximation to that of a real fluid.

¹. For those readers, who know differential calculus, this statement means: There exists a function f(x,y,z), the first partial derivatives of which with respect to the variables yield the velocity components in the directions of the axes X, Y, Z.

The behaviour of a body in a flow, in which the patterns a/b of Fig. 249 are superimposed as in c, are of special interest. In regions where the crowding of the stream lines indicates the presence of largest velocities (the velocities of the flows are added), the pressure is smallest, where they are widest apart, the velocity has a minimum and therefore the pressure is largest. Hence the body experiences from one side a higher pressure than from the other. This occurs in the cases of tennis balls, base balls, etc., whenever you rotate them fast enough about a vertical axis at a right angle to their trajectory; they will during their flight divert to one side. A ball, appropriately hit, can through its lift overcome gravity over a considerable distance. The Magnus effect (Magnus) is the result of superposition of the flow forms of Fig. 249 a/b.

Aeroplane wings

If you give a kite a velocity relative to the moving air by running with it against the wind, it will rise. If its surface is large enough, it can then lift a sizeable weight. Otto Lilienthal 1848-1896, the founder of gliding activities, was the first person who really flew. He discovered with a kite, running down a hill, that approximately 14 m² were sufficient for lifting a medium size person. In order to make his gadget more convenient and more stable, he decomposed

the planes into two planes one above the other, that is, the biplane, which especially through the brothers Orville Wright and Wilbur Wright became a flying machine (Fig. 251). It is driven forwards through propellers which are rotated by engines.

Fig. 250 shows the process which generates the lift of aircraft wings. At the start, you have the flow a, but soon enough a vortex forms at the trailing edge (b) which converts the initially lift free flow of a into that of c. (After its formation, the vortex departs from the wing with the flow and leaves behind the flow of c.)

The force acts vertically upwards, approximately perpendicularly to the wing plane. It can be decomposed into two perpendicular to each other components, one in the direction of the flow, representing resistance - drag -, the other perpendicular to the direction of the flow - lift. If the profile of the wing section has a large lift and small drag, it is suitable for use in aircraft. Theory and practice have resulted in profiles for which the drag is about 4% of the lift. The lift and drag of a wing depend also greatly on the angle which the chord of the profile makes with the direction of the wind (at horizontal flight between 1½ and 4º).

The air ahead of the wing and on its sides is more or less unaffected: Only a lane through which the wing travelled has a downwards velocity. Fig. 252 presents a realistic picture of the motion; you imagine a board of the size of the span of the wing which extends backwards and grows in time in the forward direction, so that it always extends to the advancing wing. If you now move this model downwards, the flow around the board represents in good approximation what happens around a wing (Betz). By computation of the kinetic energy which in this manner stays behind in the air, you arrive at a value for the smallest drag, which every flying object, independently of frictional resistance and such like, must have, in order to fulfil the task of carrying a given weight.

If v is the flight velocity, b the span and r the density of the air, this resistance is given by W_min=2G/prv²b². Every flying creature must also generate per second at least W_min·v = 2G/prvb². In the aerodynamic theory this part of the drag is called induced drag,

Friction of fluids. The Law of Poiseuille (1846)

The friction of fluid particles against each other (internal friction) and against walls (external friction) greatly influences fluid motion; it may become so large that neglecting it impedes the union of experience and theory. You can conceive internal friction as follows: Imagine a fluid moving between two horizontal bounding planes and the fluid subdivided parallel to them in layers (laminar motion, in Latin: Lamina = layer, in contrast to turbulent motion: turbulence). Now assume that all the fluid particles in the same layer have the same motion. An entire layer of fluid particles can then only displace relative to the layers above and below. Let three layers a, a₁, a₂ have at time t₁ the position 1 (Fig. 253) and move from the left to the right at unequal velocities until at time t₂ they have the position 2. The layer a slows down a₁, the layer a₂ accelerates it, both due to friction. For the sake of simplicity, let the magnitudes of these actions be proportional to the velocity differences and to the areas of the layers sliding with respect to each other, an assumption which is justified by experience. If du if the difference of the velocities of two layers at distance dy from each other, then the velocity changes at du/dy per unit length. According to our assumption, we set the force t proportional to this velocity gradient and the area s of the layers sliding with respect to each other - it is a shear which the layers transfer to each other - , that is, we write t = m · s · du/dy, where m is the factor of proportionality, called the coefficient of internal friction or viscosity coefficient. It is equal to the force which acts oppositely to the motion of a fluid layer of unit area, if the layer moves parallel at the steady velocity 1 at the distance 1 from a layer at rest . It dimension is

m = t /(s ·du/dy) = Force/(Area x Velocity/Length) = mlt^-2/(l²·lt^-1/ l) = l^-1mt^-1, that is [cm^-1g sec^-1].

The coefficient of friction decreases strongly with rising temperature and, in general, increases somewhat with rising pressure. At 18 º

The work of Jean Louis Marie Poiseuille 1799-1869 1826 about the motion of fluids in long, narrow, cylindrical tubes (up to 0.6 mm diameter) is decisive. He discovered that the volume Q passing through such tubes in unit time is proportional to the drop of pressure per unit length (p₁ - p₂)/l and the fourth power of the tube radius r, that is, Q = (p r⁴/8m)·(p₁ - p₂)/l. This equation yields the most accurate values for the coefficient of friction. The motion in the tube is laminar, the fluid sticks to the wall, that is, is at rest there, a fact which can be considered to have been proven with great accuracy.

However, Poinseuille's law is valid only for flow at all realizable velocities in narrow tubes; it changes its form suddenly at a certain velocity (the appearance of the jet which was hitherto clear becomes milky) and the assumption that the fluid particles move only parallel to the wall of the tube loses its validity. The flow ceases being a laminar flow; it changes into a turbulent flow.

This change can also be seen when a fluid flows through a tube one section of which has been warmed up. At the instant when turbulence develops, you also have a strong extension of the heat from the wall into the inside of the fluid, and therefore a strengthening of the heat transfer; down stream, passed the heated part of the tube, a thermometer rises strongly (in laminar flow, there occurs only heat conduction!). The critical velocity for the onset of turbulence in cylindrical tubes (of radius r, average velocity v) is characterized by a definite magnitude of Reynolds Number. What is the significance of this number?

Reynolds Number

Imagine that you have a working model of a submarine, which has been tested In a model basin, and you want to construct a full scale submarine. Can you apply without hesitation, for example, to the boat the observations and measurements concerning energy consumption and attained velocity? No! Even if you have enlarged the geometrical dimensions of the model at the prescribed scale and if the model and boat have been constructed out of the same material, nevertheless the physical conditions under which the model moves in the model basin and the boat in the sea differ fundamentally.

The velocity, the depth below the surface, the temperature and density of the water, the waves generated, etc. differ. All of them act together to influence the inertial and frictional forces, which act on the model and the boat. There arises the question: The boat and its model are geometrically similar; under what conditions are the physical processes acting on them mechanically similar?

The answer is: The processes are similar mechanically, when the the ratios of the inertial forces to the frictional forces for the model and the boat are the same, that is, inertial forces/frictional forces = const for both the model and the boat. A consideration, which is linked to the dimensional formulae for the inertial and frictional forces, yields a conclusion as to how the deciding factors of the two groups of forces, that is, the coefficient of friction m, the density r, the velocity v and the length l (any characteristic lengths of the model and boat) must be combined, in order to yield an unnamed , that is, a dimensionless number. Using m, r, v and l, you can form such a dimensionless, unique number; it is rvl/m,and called after its discoverer Reynolds Number. Its dimension is ml^-3·lt^-1·l/l^-1mt^-1.

Note that Reynolds' Law only applies for motions inside a fluid. If you move on the surface of fluid, gravity enters as another determining physical quantity and other laws apply (R. E. Froude). During motion in air and also completely immersed in water, the gravitational force is balanced by the lift, which every fluid particle experiences from its neighbour, and therefore it does not influence the motion.

We can now characterize the index briefly: You can draw conclusions from an experiment under definite hydrodynamical conditions with respect to a mechanically similar case the index if and only if the index is the same in both cases. Applied to the above model experiment, this means: You can only draw conclusions for the submarine, if either the model moves at a velocity which is that faster as corresponds to the reduction in size or if the tests occur in a fluid with a small kinematic friction coefficient, that is, m/r - the ratio viscosity/friction, where a smaller velocity is sufficient for complete adjustment to the actual conditions of the motion.

The following remarks may be added: It is also the same if you execute a certain experiment in air or in water or in any other fluid, provided you change the dimensions and velocity so that the Reynolds Number is the same. The only material constant involved is the kinematic coefficient of friction m/r=n. For air, it about 14 times that for water. An experiment in water gives therefore a strict right lead concerning the process in air, provided the body dimensions in air is double, the velocity seven times that used in water.

We return now to the turbulent flow in a narrow cylindrical tube.

Turbulent flow

At a critical velocity, a laminar flow becomes turbulent. If for a tube with a circular cross-section the the expression vd/n (v average velocity, d tube diameter) serves as critical number R, the validity of Poiseuille's Law ends at R = 2000, when turbulence begins. In a tube with a diameter of 1 cm, through passes water at 10ºC, R = 2000 yields a mean velocity of about 26 cm/sec. Through this tube, a slower flow, as every flow with a smaller value of R will be laminar (layered). If the velocity increases to this value of if you obtain it by enlargement of the tube diameter or heating of the water (decrease of m), there will occur the change over. The amount of fluid for a given pressure drop will then be smaller than the value, given by Poiseuille's formula, that is, the resistance experienced by the flow increases. The causes of turbulence were not yet known in 1935.

Obstruction of rigid bodies by fluid friction

If the value of Reynolds Number is very small, this means that the frictional forces are much larger than the inertial forces. This is what happens, when a very small heavy sphere drops into a fluid. The accelerating action of gravity is soon compensated for by friction and after that the sphere falls with the constant velocity v = 2/9·[(r₁ - r)/m]·r²/g (Stokes), where v is the velocity, r the radius, r₁ the density of the sphere, r the density of the fluid,m the coefficient of friction. This formula is only valid for Reynolds Numbers which are small compared with 1. For drops of water in air, v = 1.3·10^-6·r², where r is in cm; this formula applies to droplets with radii smaller than 0.02 mm.

The frictional forces also are much larger than the inertial forces in the motion of lubricants between a rotating axle and its bearings. The lubricant is in a Poiseuille type motion, because it takes with it the lubricant layer, the bearings force its layer to be at rest. The friction in the lubricant transfers a certain shear from the bearings to the axle. While the internal friction of the lubricant acts against the motion, it is much smaller than would be the friction of the axle in the immediate neighbourhood of the bearings. The action of lubricants depends of the reduction of this friction.

In contrast, a very large Reynolds number means that the frictional forces disappear. But they do this only where the fluid does not touch a wall. At a wall, the fluid adheres to it. Then a boundary layer forms under the influence of the friction - which is the thinner the smaller the friction. Under certain conditions, parts of the boundary layer detach from the wall, enter as separation layers into the free fluid and cause the detachment of the fluid from the wall and the generation of vortices. Boundary layers are of fundamental importance for all of Hydrodynamics. (Prandtl 1904), whence it must be considered in great detail.

Prandtl's Boundary Layer Theory

If you move a fluid by a pressure difference in a certain direction, nowhere inside it is lost mechanical energy; everywhere the energy of motion converts into pressure and vice versa. Bat at the boundaries of the fluid, especially where it moves along a wall, the condition for a lossless flow are no longer fulfilled. Along a wall, the particles near it are retarded by friction. Actually, this layer in which this happens (Fig. 254) is often so thin that the flow outside the boundary layer is hardly changed (this is the name of the thin layer in which the flow velocity drops rapidly to zero without any effect of fluid friction). Also the pressure inside the layer hardly differs from that outside. It is therefore determined everywhere along the wall by the flow outside the boundary layer.

If a fluid flows inside a tube which widens or narrows, the velocity in the direction of flow decreases or increases, because the product of velocity and area is the amount of fluid which passes a section in unit time and this is the same for all sections; at the same time, the pressure changes (in accordance with Bernoulli's Equation p + r··v²/2 = const). Hence in the wider sections the velocity is smaller and the pressure larger than in the narrower parts. What then is the behaviour of a fluid particle in the boundary layer during a change of pressure? During the transition from regions of higher pressure to those of lower pressure, there is no essential difference compared with the undisturbed flow; only the velocities are smaller, since continuously kinetic energy at the wall is lost through friction. It is different when the pressure in the direction of motion increases!

In fact, as a result of its decreased kinetic energy, the particles in the boundary layer cannot any more penetrate the region of increased pressure. They are carried somewhat along by the faster outer flow through friction, but, since during the increase in pressure also the velocity of the unperturbed flow may decrease strongly, this assistance often is not sufficient, in order to take the particles of the boundary layer to the location of highest pressure. Hence the boundary layer flows during rapidly increasing pressure only a short distance along the wall and is then forced by consumption of its kinetic energy to return; while the outer flow with its greater energy continues to flow forwards, there develops near the wall a return flow. The continuously added material of the boundary layer slows down, the boundary later becomes thicker, so that the backward flow spreads rapidly and the outer flow deflects more and more from the wall; it detaches. The separation layer thus generated becomes a vortex and eventually more and more vortices form and cause a reformation of the entire flow. There arises an area which is filled with vortices and in which much mechanical energy is lost.

If you want to convert the kinetic energy of a flow into pressure, you can let if flow through a slowly widening tube (diffuser), in which the velocity decreases and in the process (depending on the losses involved) converts itself more or less into pressure. If you make the widening slow enough, so that the pressure rises only gradually, the boundary layer is taken away by the outer flow, it does not detach from the wall, but becomes thicker. If the angles at which the tube widens exceeds a certain value (about 10º), boundary layer detachment takes place and with it a loss of energy. By suction of the boundary layer, you can improve the flow significantly. Figs. 255-257 demonstrate the effect of suction in an experimental set-up.

As a result, the structure of the boundary layer becomes complicated and frequently turbulence develops. For Reynolds Numbers below 2320, the flow is always laminar unless it is artificially disturbed. However, once it has become turbulent, it remains so; the larger Reynolds Number, the smaller are the perturbations which suffice, in order to make a flow turbulent. In practice, there are always enough perturbations about that a flow at R > about 3000 will almost always be turbulent.

With the type of flow (laminar and turbulent) also changes the law obeyed by the pressure loss in a tube (tube resistance). While during laminar flow the pressure drop In a smooth, straight tube of constant cross-section is proportional to the mean flow velocity, it is during turbulent flow proportional to v^3/2, where v is the flow velocity. In addition, the distribution of the velocity in the tube changes. If you plot the velocities at different points of a diameter as ordinate over the diameter, you obtain during laminar flow a parabola (Fig. 254). During turbulent flow, the velocity at each point oscillates.

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