Mechanics of Fluids

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In addition to the reflection of waves that takes place at open or closed ends, partial reflections occur at changes of section in the pipe line, and at junctions. Our discussion so far has all been based on the concept of instantaneous closure of the valve. An absolutely instantaneous closure is physically impossible, but if the valve movement is completed in a time less than 2l/c the results are not essentially different. The pressure at the valve is gradually built up as the valve is closed; the maximum pressure reached, however, is the same as with instantaneous closure, because the conversion of kinetic energy of the fluid to strain energy is completed before any reflected waves have had time to reach the valve. If, on the other hand, the valve movement is not finished within a time 2l/c, then not all of the kinetic energy has been converted to strain energy before a reflected wave arrives to reduce the pressure again. The maximum pressure rise thus depends on whether the time during which valve movement occurs is greater or less than 2l/c. Movements taking less than 2l/c are said to be rapid; those taking longer than 2l/c are said to be slow. (When pressure changes are considered for a point between the valve and the reservoir – at a distance x, say, from the reservoir – the relevant time interval is 2x/c.) Similar effects follow from a sudden opening of the valve, although the primary result is a reduction of pressure behind the valve instead of a rise. 12.3.3 Slow closure of the valve If the valve is closed in a time longer than 2l/c account must be taken of waves returning to the valve from the reservoir. The subsequent changes of pressure, as the waves travel to and fro in the pipe, may be very complex, and determining the pressure fluctuations requires a detailed step-by-step analysis of the situation. In this section we shall outline a simple, arithmetical, method of solution so as to focus attention on the physical phenomena without the distraction of too much mathematics. In Section 12.3.4 we shall look at the mathematical basis of a more general technique. The essential feature of the arithmetical method is the assumption that the movement of the valve takes place, not continuously, but in a series of steps occurring at equal intervals of 2l/c or a sub-multiple of 2l/c. The partial closure represented by each step is equal in magnitude to that achieved by the actual, continuous, movement during the time interval following the step, and the valve movement for each step is assumed to be instantaneous. Between the steps the valve is assumed stationary. Each of the instantaneous partial closures generates its own particular wave – similar in form to those depicted in Fig. 12.5 – which travels to and fro in the pipe until it is completely dissipated by friction. The total effect of all these individual waves up to a particular moment approximates fairly closely to the effect produced by the actual, continuous, valve movement. (During any interval while the valve is being closed, the pressure at the valve builds up to a maximum which occurs at the end of the interval. If each interval is so chosen that a wave sent off at the beginning does not return before the end of the interval, then Pressure transients 569
570 Unsteady flow Fig. 12.7 Gate valve. a knowledge of the precise way in which the continuous valve movement occurs is unnecessary: eqn 12.9 gives h2 − h1 for a reduction of velocity from u1 to u2, regardless of the manner in which u changes with time.) Let us suppose that the steady flow conditions before the valve closure begins are represented by a head h0 at the valve and a velocity u0 in the pipe. If at the end of the first chosen interval the head and velocity are respectively h1 and u1 then these must be related by eqn 12.9: h1 − h0 = c g (u0 − u1) (12.10) Another relation between h1 and u1 is required if either is to be calculated. For the simple case in which the valve discharges to atmosphere, the valve may be regarded as similar to an orifice, and so Au = Q = C dAv � (2gh) (12.11) where A represents the cross-sectional area of the pipe in which the velocity is u, Av the area of the valve opening and C d the corresponding coefficient of discharge. Putting C d(Av/A) � (2g) = B in eqn 12.11 we have u = B √ h (12.12) The factor B is usually known as the valve opening factor or area coefficient. It should be noted that Cd is not necessarily constant, and the variation of B with the valve setting usually has to be determined � by experiment for each design of valve. From eqn 12.12 u1 = B1 h1 and simultaneous solution of this equation with eqn 12.10 gives corresponding values of h1 and u1. Such a closure in which B varies linearly with time t, that is, B = B0(1 − t/T), is frequently termed straight-line closure. It does not imply uniform motion of the valve. With a gate valve (Fig. 12.7), for example, in which the gate is moved uniformly, the rate of reduction of area is initially small, but very much greater towards the end of the closure. It is therefore common to arrange for such valves on long pipe-lines to close slowly during the final stages of the movement. A wave returning from the reservoir is negative in sign and it thus offsets – partially at least – rises in pressure caused by further reductions of velocity. A numerical example will show how allowance is made for the reflected waves. Example 12.2 A pipe carries water at a (mean) velocity of 2 m · s −1 , and discharges to atmosphere through a valve. The head at the valve under steady flow conditions is 100 m (pipe friction being neglected). Length of pipe = 2400 m; wave celerity c = 1200 m · s −1 . The valve
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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
Page 530 and 531: For supersonic flow eqn 11.45 is no
Page 532 and 533: Some general relations for one-dime
Page 534 and 535: of minimum cross-section the veloci
Page 536 and 537: downstream of the minimum cross-sec
Page 538 and 539: ‘telegraphed’ upstream and a pr
Page 540 and 541: the external pressure. This compres
Page 542 and 543: Compressible flow in pipes of const
Page 544 and 545: Table 11.2 Changes with distance al
Page 550 and 551: From the Fanno Tables at pc/pB = 0.
Page 554 and 555: Therefore and M1 = u1 a 24.5 m · s
Page 556 and 557: drag. The abrupt pressure rise thro
Page 558 and 559: Analogy between compressible flow a
Page 560 and 561: is, on ∂2n ∂y2 + ∂2 � n ∂
Page 562 and 563: and the mass flow rate in the duct.
Page 564 and 565: 11.18 Air flows isothermally at 15
Page 566 and 567: 12.2 INERTIA PRESSURE Any volume of
Page 568 and 569: Equation 12.3 indicates that u beco
Page 570 and 571: would come to rest later. Although
Page 572 and 573: a pressure wave in a gas; here we r
Page 574 and 575: value in eqn 12.7 gives that is, wh
Page 576 and 577: the moment disregarded. An increase
Page 578 and 579: at this point remains constant for
Page 582 and 583: is closed so that the area coeffici
Page 584 and 585: The pressure diagrams show that for
Page 586 and 587: 12.3.4 The method of characteristic
Page 588 and 589: Pressure transients 577 Multiplying
Page 590 and 591: and velocities are known, then for
Page 592 and 593: 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
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∴ Hydraulic efficiency = Overall
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If either z (the height of the turb
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∴ Hydraulic efficiency = u1vw1/gH
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shows the general effect of change
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an unmixed blessing and, except for
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orbiting space-craft, for example,
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Losses are also possible at the inl
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Those particles next to the forward
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It should be noted that the values
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∴ (β1)A = arctan(va/u) = arctan(
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so much that the aerofoil stalls (a
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Use these data to deduce the pump c
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Manometric efficiency = 0.75 = gH/u
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and CP = P/ϱω 3 D 5 in place of Q
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yields h 1 = h f = 4fl d ≈ (100Q)
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(iii) the power dissipated by pipe
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Large numbers of test data for pump
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educed by rounding the inlet edges
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further: the turbine part then remo
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for reaction turbines: 1 − η = 1
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75 mm wide at outlet. The blades oc
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Losses at valves, etc. are estimate
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APPENDIX Units and conversion 1 fac
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Table A1.4 (contd.) Force 32.17 pdl
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APPENDIX Physical constants and 2 p
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Kinematic viscosity mm 2 ·s -1 App
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Range II: 11 000 m ≤ z ≤ 20 000
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Table A3.1 (contd.) M 1 M 2 p 2/p 1
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Table A3.2 Isentropic flow M p/po
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Table A3.3 Adiabatic flow with fric
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APPENDIX 4 Algebraic symbols Table
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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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