Mechanics of Fluids

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Those particles next to the forward side of a blade have no choice but to be hustled forward by it, but particles further from that side tend to deviate backwards from the prescribed path. As a result of these two effects – the more ready separation along the back of a blade and the inertia of the fluid particles – the mean direction of the fluid leaving the impeller is modified as shown in Fig. 13.33. The velocity vector diagram that the outlet blade angle φ2 leads us to expect is shown in full: the figure actually obtained is indicated by dotted lines. Conditions may vary somewhat round the impeller circumference, but the height of the ‘average’ diagram corresponds to the average value of vr2 (i.e. Q/A 2 ) and therefore, for a given rate of flow, is unchanged. The backward deviation of R2 results in a backward shift of the vertex, a reduction in the whirl component vw2 and a consequent reduction in the head produced by the pump. Methods have suggested for estimating the deviation angle (φ2 − β2) – which may be as high as 15 ◦ in some designs – but these are still semiempirical. The greater the number of blades in the impeller the more uniform is the flow direction, and for an infinite number of blades the deviation of R2 and hence the whirl slip (see Fig. 13.33) would be zero. Frictional losses, however, become greater as the number of blades is increased, and the number of blades used – from six to twelve if they are backward-facing – is a compromise between these two opposing considerations. Forward-facing blades, having a convex real surface, greatly encourage the tendency of the flow to Rotodynamic pumps 633 Fig. 13.32 Fig. 13.33
634 Fluid machines Fig. 13.34 break away from that side. This type of impeller therefore usually has a much greater number of blades (sometimes as many as 60 in large fans). Deviation of the flow also occurs in turbine runners, but to a smaller degree. 13.4.4 Axial- and mixed-flow pumps The axial-flow or propeller pump is the converse of the propeller turbine and is very similar in appearance to it. The impeller consists of a central boss on which are mounted a number of blades. It rotates within a cylindrical casing, ideally of sufficient length to allow uniform flow on each side of the pump, and the clearance between blades and casing is as small as practicable. On the outlet side of the impeller a set of stationary guide vanes is usually fitted: these vanes are designed to remove the whirl component of velocity which the fluid receives from the impeller (see Fig. 13.34). Guide vanes on the inlet side also are sometimes provided but, except in applications where appreciable tangential motion exists in the inlet pipe, they do not significantly improve the pump’s performance and so are usually omitted. Modern axialflow pumps are often provided with impeller blades whose angle may be altered during running so that a high efficiency is maintained over a wide range of discharge. The impeller of a mixed-flow pump is so shaped that the fluid enters axially but leaves with a substantial radial component. The general formulae 13.7–13.10 apply, with reservations, no less to axialflow machines than to centrifugal ones. In an axial machine a particle of fluid, in general, enters and leaves the impeller at the same radius. That is, u1 = u2 and, if we assume that vw does not vary in the circumferential direction, the equations reduce to Work done on fluid divided by mass = u(vw2 − vw1) = 1 2 �� v 2 2 − v2 1 � − � R 2 2 − R2 �� 1 (13.21)
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Mechanics of Fluids
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Mechanics of Fluids Eighth edition
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Contents Preface to the eighth edit
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8 Boundary Layers, Wakes and Other
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Preface to the eighth edition In th
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Fundamental concepts 1 The aim of C
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1.1.1 Molecular structure The diffe
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therefore, to have in place systems
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has been a strong movement in favou
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Notation, dimensions, units and rel
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is nearer 2 m, rather than 1 m. Her
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Properties of fluids 13 The relativ
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that is, p1 − p3 = 1 2BC ϱax (1.
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that there is an almost perfect vac
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calorically perfect. (Some writers
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As a liquid is compressed its molec
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or τ = µ ∂u ∂y (1.9) where µ
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shared among the occupants of bb, a
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the origin with slope equal to µ (
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The interplay of these various forc
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non-uniformity is always encountere
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z direction and therefore the same
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Although Reynolds used water in his
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cylinder. For a certain range of Re
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The roles of experimentation and th
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A flow model which assumes steady,
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2.1 INTRODUCTION Fluid statics 2 Fl
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3. dp/dz =−ϱg. (Since the pressu
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that is, dp p =−g dz R T Variatio
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atmosphere it is termed vacuum or s
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at its centreline be p. Then, provi
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possible, and a sloping manometer m
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The amount of liquid B on each side
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In a pressure transducer the pressu
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where the suffixes C and Oy indicat
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surface (where the pressure is atmo
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which has different depths of water
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Hydrostatic thrusts on submerged su
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plane and may then be combined into
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Since all the elemental thrusts are
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neglected when a body is weighed on
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couple acting on the body in its di
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centre of buoyancy on to the transv
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move, but also the centre of gravit
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that is, W(OG) = (W + F)(OB + BM) =
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is given by ∂p ∂x =−ϱax Equi
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Then ∂p/∂ξ = 0 by definition o
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2.5 Two small vessels are connected
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2.18 In the vertical end of an oil
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The principles governing fluids in
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planes B and C (Fig. 3.2), δs bein
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an actual fluid is thus often remar
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density are small. Then eqn 3.9 has
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General energy equation for steady
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− Energy loss to friction/time Ma
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a decrease of pressure (provided th
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in the neighbourhood of B results i
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it is difficult to measure the heig
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The diagram illustrates an orifice
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� depth h below the free surface
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Solution (a) We consider first the
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of the same liquid. A vena contract
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To ensure that the pressure measure
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Since �h = p1 − p2 ϱwg the flo
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For consistency with the mathematic
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showing the essential form of the r
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With the same assumptions used in d
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∴ Velocity of approach = 0.0660 m
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of the base of the notch. Assume th
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Our aim now is to derive a relation
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The velocity, in general, varies fr
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the control volume is � ϱ1u1ux1
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Applications of the momentum equati
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that the resultant force on the flu
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The existence of the reaction may b
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our reference axes attached to the
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of increase of x-momentum is � D
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This is a simplified picture of wha
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since u4 −u1 is the velocity of t
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(1) u 1 (2) (3) (4) Assume a sea le
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and frictional effects to be neglig
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Physical similarity and dimensional
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A well-known example of kinematic s
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polygon. Now a polygon can be compl
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The condition for dynamic similarit
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equality of the Mach numbers is not
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Consider the incompressible flow al
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moving in a straight line with cons
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of the analysis, and its correct im
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Note: a suffix has been attached to
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For example, the parameter Fϱ/µ 2
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The application of dynamic similari
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atmospheric air, the dynamic viscos
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gives rise to waves on the surface.
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To be tested, the model must theref
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it follows that Hence V2 V1 = P2 P1
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5.5 A torpedo-shaped object 900 mm
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Laminar flow between solid boundari
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the place of y: τ = µ ∂u ∂r A
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compensate for the reduction in vel
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The expression for the total discha
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etween the resisting viscous forces
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must be zero so as to meet the requ
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Example 6.2 Two stationary, paralle
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Thus for any value of y the velocit
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Steady laminar flow between moving
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in eqn 6.24 we obtain � D Vp 2
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and the passage of the liquid level
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this book, it is important to give
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6.6.4 Rotary viscometers A simple m
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The torque T on the cylinder of rad
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stationary when the outer cylinder
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Fundamentals of the theory of hydro
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part of the bearing then gives when
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e determined by our analysis.) From
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where a = 6µ�R h3 dh pc dx Funda
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Integrating this between θ = 0 and
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Rearrangement of eqn 6.71 and subst
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6.4 A cylindrical drum of length l
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7.1 INTRODUCTION Flow and losses in
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a maximum value of 2.0. The line CD
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America the friction factor commonl
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course, quite different from the ro
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Variation of friction factor 253 Fi
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straightforward manner without iter
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Distribution of shear stress in a c
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7.5 FRICTION IN NON-CIRCULAR CONDUI
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estimated by simple theory. The pip
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the junction, the curvature of the
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particular, the performance of a di
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A pipe bend thus causes a loss of h
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cross-sectional area available in t
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correlating experimental data on th
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to do this. But sub-atmospheric pre
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For systems comprising only short r
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7.8 PIPES IN COMBINATION 7.8.1 Pipe
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2. There can be only one value of h
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So For two pipes in parallel, and s
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7.9 CONDITIONS NEAR THE PIPE ENTRY
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where u denotes the mean velocity i
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Assuming that entry and exit losses
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of one-per-revolution electrical co
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of the piping system, this meter is
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losses, calculate (a) the total pow
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How are the running costs altered i
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main. The depth of water in the tan
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the features of boundary layer flow
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the surface other considerations ar
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and θ = = = � δ u 0 um � δ 0
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The momentum equation applied to th
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1 = y η δ The laminar boundary la
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Table 8.1 Comparison of results for
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and � � ∂f (η) B = ∂η A =
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From equation 8.17, the frictional
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The turbulent boundary layer on a s
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Hence F = 0.037ϱu 2 m l(Re l) −1
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From eqn 8.29 Hence and xt − x0 x
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from it. This breakaway before the
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this expression into eqn 8.31 we ge
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When, for example, a ship moves thr
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separate vortices in an inviscid fl
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8.8.4 Profile drag of two-dimension
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are negligible. With reduction in t
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Fig. 8.14 Drag coefficients of smoo
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sphere is falling through the atmos
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At high Reynolds numbers, the varia
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this does not prevent the continual
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Eddy viscosity and the mixing lengt
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Eddy viscosity and the mixing lengt
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Velocity distribution in turbulent
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then, although at very small values
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as η → 1 and (1 − η) is then
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When f −1/2 − 4 log 10 (d/k) is
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comparable with the velocity of sou
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The x, y and z components of the mo
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y y x i-1, j+1 i, j+1 i+1, j+1 i-1,
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e completely covered by a turbulent
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9.1 INTRODUCTION The flow of an inv
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Now consider another point P ′′
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number of smaller ones of which M a
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distortion, the element is thus not
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Any function φ that satisfies Lapl
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ecome perfect squares - except wher
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fluid, and the flow net indicates t
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9.6.2 Flow from a line source A sou
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higher orders of small magnitudes b
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Combining eqns 9.16 and 9.17 we obt
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9.6.5 Forced (rotational) vortex Th
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For a free vortex qR = constant = K
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source is unable to move to the lef
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Also we note that the source is pos
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as shown in Fig. 9.23, encloses all
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eqn 9.23 becomes ψ =−U � r −
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Thus the magnitude of the velocity
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At a stagnation point, qt = 0 and t
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function of the combined flow is gi
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or, more approximately, that betwee
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Then dw dz =−U =−u + iv Hence u
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An introduction to elementary aerof
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ound the aerofoil, its magnitude be
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and lower layers. Since the vortex
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where Ɣ0 represents the circulatio
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9.7 An open cylindrical vessel, hav
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steadily south-east at 6 m · s −
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10.2 TYPES OF FLOW IN OPEN CHANNELS
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The steady-flow energy equation for
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where the hs represent vertical dep
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to the turbulent rough flow regime
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selection of an appropriate value r
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10.6 OPTIMUM SHAPE OF CROSS-SECTION
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oughness coefficient n is independe
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thrust divided by width of the chan
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which the waves follow one another
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The conditions for the critical dep
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y downstream conditions. In these c
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may be obtained by reducing eqn 10.
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this depth is greater than the crit
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The dissipation of energy is by no
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friction, the steady-flow momentum
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Fig. 10.28 Notice that, for a chann
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calculated, and then H + u2 1 /2g m
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Fig. 10.32 If the depth of flow ove
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(Section 10.11.1). Rapid flow can n
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where 0 ≤ θ ≤ 90 ◦ and sin
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(a) The continuity equation is The
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oundaries. It can occur when the fl
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into eqn 10.39 we obtain whence dh
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Table 10.2 Gradually varied flow 46
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The slope of a given channel may, o
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direction. The effect of any bounda
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This momentum term, however, is pro
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With surface tension effects ignore
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are for waves of small amplitude) w
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when the wave was formed. The lengt
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At a particular instant the crests
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Hence h = 2.65λ 2π = 2.65 × 225
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As the ratio of these velocity comp
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water is impulsively displaced and
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is relative movement of one layer o
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the channel bed a drop of 150 mm in
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11.1 INTRODUCTION Compressible flow
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energy ÷ mass, so that the dimensi
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Energy equation with variable densi
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Solution (a) From the equation of s
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whence (c − u)δρ = (ρ + δρ)
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the sonic velocity as supersonic (M
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velocity into the undisturbed fluid
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whence p2 p1 = 1 + γ M2 1 1 + γ M
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of eqns 11.2 and 11.20 T0 T u2 = 1
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the temperature rises greatly (say
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Fig. 11.10 Oblique shock relations
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however, that for a given upstream
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The deflection through the first wa
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As δθ is very small, cos δθ →
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eqn 11.41. Substituting M 2 − 1 =
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oundaries, both of which are curved
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For supersonic flow eqn 11.45 is no
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Some general relations for one-dime
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of minimum cross-section the veloci
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downstream of the minimum cross-sec
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‘telegraphed’ upstream and a pr
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the external pressure. This compres
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Compressible flow in pipes of const
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Table 11.2 Changes with distance al
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From the Fanno Tables at pc/pB = 0.
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Therefore and M1 = u1 a 24.5 m · s
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drag. The abrupt pressure rise thro
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Analogy between compressible flow a
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is, on ∂2n ∂y2 + ∂2 � n ∂
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and the mass flow rate in the duct.
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11.18 Air flows isothermally at 15
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12.2 INERTIA PRESSURE Any volume of
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Equation 12.3 indicates that u beco
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would come to rest later. Although
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a pressure wave in a gas; here we r
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value in eqn 12.7 gives that is, wh
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the moment disregarded. An increase
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at this point remains constant for
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In addition to the reflection of wa
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is closed so that the area coeffici
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The pressure diagrams show that for
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12.3.4 The method of characteristic
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Pressure transients 577 Multiplying
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and velocities are known, then for
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For pipe 2, and hence K ′ = c2 =
Page 594 and 595: (instantaneously, we will assume) a
Page 596 and 597: The one-dimensional continuity rela
Page 598 and 599: to accommodate instantaneous comple
Page 600 and 601: 12.2 A turbine which normally opera
Page 602 and 603: 13.1 INTRODUCTION Fluid machines13
Page 604 and 605: flow. By way of illustration we sha
Page 606 and 607: would obviate the changes of pressu
Page 608 and 609: demand for electric power - for exa
Page 610 and 611: Turbines 599 Fig. 13.7 Runner of Pe
Page 612 and 613: determined, multiplication by the r
Page 614 and 615: Substituting for �vw from eqn 13.
Page 616 and 617: ut the larger its value the more bu
Page 618 and 619: ate of flow possible is achieved wh
Page 620 and 621: the turbine. For the high heads nor
Page 622 and 623: The integrals in eqns 13.4 and 13.5
Page 624 and 625: diagram of Fig. 13.16 we have Simil
Page 626 and 627: would be sufficient to describe the
Page 628 and 629: Table 13.1 Type of turbine Approxim
Page 630 and 631: ∴ Hydraulic efficiency = Overall
Page 632 and 633: If either z (the height of the turb
Page 634 and 635: ∴ Hydraulic efficiency = u1vw1/gH
Page 636 and 637: shows the general effect of change
Page 638 and 639: an unmixed blessing and, except for
Page 640 and 641: orbiting space-craft, for example,
Page 642 and 643: Losses are also possible at the inl
Page 646 and 647: It should be noted that the values
Page 648 and 649: ∴ (β1)A = arctan(va/u) = arctan(
Page 650 and 651: so much that the aerofoil stalls (a
Page 652 and 653: Use these data to deduce the pump c
Page 654 and 655: Manometric efficiency = 0.75 = gH/u
Page 656 and 657: and CP = P/ϱω 3 D 5 in place of Q
Page 658 and 659: yields h 1 = h f = 4fl d ≈ (100Q)
Page 660 and 661: (iii) the power dissipated by pipe
Page 662 and 663: Large numbers of test data for pump
Page 664 and 665: educed by rounding the inlet edges
Page 666 and 667: further: the turbine part then remo
Page 668 and 669: for reaction turbines: 1 − η = 1
Page 670 and 671: 75 mm wide at outlet. The blades oc
Page 672 and 673: Losses at valves, etc. are estimate
Page 674 and 675: APPENDIX Units and conversion 1 fac
Page 676 and 677: Table A1.4 (contd.) Force 32.17 pdl
Page 678 and 679: APPENDIX Physical constants and 2 p
Page 680 and 681: Kinematic viscosity mm 2 ·s -1 App
Page 682 and 683: Range II: 11 000 m ≤ z ≤ 20 000
Page 684 and 685: Table A3.1 (contd.) M 1 M 2 p 2/p 1
Page 686 and 687: Table A3.2 Isentropic flow M p/po
Page 688 and 689: Table A3.3 Adiabatic flow with fric
Page 690 and 691: APPENDIX 4 Algebraic symbols Table
Page 692 and 693: Table A4.1 (contd.) Symbol Definiti
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Table A4.1 (contd.) Symbol Definiti
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Answers to problems 1.1 56.2 m 3 1.
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8.14 0.002955, 2.866 m · s −1 ,
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Absolute pressure 46 Absolute visco
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Eccentricity 230 Eccentricity ratio
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Mach angle 498 intersection 510-1 r
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Slip in fluid couplings 652 in reci
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