Mechanics of Fluids

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educed by rounding the inlet edges of the blades, such a refinement is not worth the considerable extra expense. The principal advantages of the coupling are found in unsteady operation. Torsional vibrations in either the primary or secondary shaft are not transmitted to the other; further, the full torque is developed only at full speed, thus easing the starting load on an internal-combustion engine or an electric motor. A so-called slip coupling may also be used as a slipping clutch, that is, with ω2 much less than ω1: this is achieved by restricting the normal circulation of fluid either by reducing the quantity of fluid in the coupling or by throttling the flow. Although the efficiency suffers, the control of slip in this way is a useful temporary facility. The relations derived for pump and turbine rotors (e.g. 13.7, 13.10) apply to the elements of a coupling – again with the provisos about uniformity of conditions at inlet and outlet of runners. For example, from eqn 13.7, the work done on the secondary runner divided by mass of fluid = u1vw1 − u2vw2. If zero whirl slip is assumed, then, since radial blades are used in both runners, the initial whirl component vw1 of the fluid entering the secondary is identical with the blade velocity of the primary at that radius, that is, ω1r0, where r0 is the relevant radius (see Fig. 13.40). Moreover, vw2 for the secondary is identical with the blade velocity at outlet, u2 = ω2ri. Therefore Work done on the secondary divided by mass of fluid = ω2r0ω1r0 − ω 2 2 r2 i = ω1ω2r 2 0 − ω2 2 r2 i Hydrodynamic transmissions 653 (13.29) Similarly it may be shown that the work done by the primary divided by mass of fluid is ω 2 1 r2 0 − ω1ω2r 2 i (13.30) The difference between the expressions 13.29 and 13.30 is the energy dissipated divided by mass of fluid and, as the flow is highly turbulent, is proportional to Q 2 , where Q represents the volume rate of flow round the circuit. Fig. 13.40
654 Fluid machines In practice, however, there is a significant variation of radius, and therefore of blade velocity, across the inlet and outlet sections of both runners. Moreover, the rate at which fluid passes from one runner to the other varies with the radius in a manner not readily determined. Consequently, the expressions 13.29 and 13.30 can be regarded as no better than first approximations to the truth even if mean values are used for r0 and ri. Dimensional analysis indicates that the torque coefficient CT of a coupling is given by CT = T ϱω2 � ϱω1D2 = φ s, 1D5 µ , V D3 � where V represents the volume of fluid in the coupling and D the diameter. The term V/D 3 is thus proportional to the ratio of the volume of fluid to the total volume, and ϱω1D 2 /µ corresponds to the Reynolds number of the fluid flow. Since the flow is usually highly turbulent, however, the effect of Reynolds number on the torque is small, and that of slip s is far greater. A fluid coupling used in a motor-car transmission is normally incorporated in the engine flywheel: it is then loosely known as a fluid flywheel. The general characteristics of a coupling are illustrated in Fig. 13.40. The stall torque is the torque transmitted when the secondary shaft is locked. The highest speed at which the primary can rotate under these conditions is given by the intersection of the two torque curves. For motor-car engines this speed is of the order of 100 rad · s −1 (16 rev/s). 13.5.2 The torque converter The essential difference between a fluid coupling and a torque converter is that the latter has a set of stationary blades in addition to the primary and secondary runners. If the stationary blades are so designed that they change the angular momentum of the fluid passing through them, a torque is exerted on them: to prevent their rotation an opposing torque must be applied from the housing. With this additional torque on the assembly as a whole, the torque on the secondary runner no longer equals that on the primary. Appropriate design of the stationary (reaction) blades allows the torque on the secondary to be as much as five times the input torque at the primary. Since, however, the reaction blades do not move, no work is done on them and the power output from the secondary runner is equal to the power input at the primary minus the power dissipated in turbulence. Many arrangements of the three elements of a torque converter are possible; one for a single-stage converter is shown in Fig. 13.41. For some purposes, particularly where a large torque multiplication is required, more complicated arrangements are used having two or even three turbine stages. A greater torque on the turbine element than on the pump element requires the fluid to undergo a greater change of angular momentum in the turbine element. The reaction member is therefore so shaped as to increase the angular momentum of the fluid: the pump impeller increases the angular momentum
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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 =
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(instantaneously, we will assume) a
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The one-dimensional continuity rela
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to accommodate instantaneous comple
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12.2 A turbine which normally opera
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13.1 INTRODUCTION Fluid machines13
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flow. By way of illustration we sha
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would obviate the changes of pressu
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demand for electric power - for exa
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Turbines 599 Fig. 13.7 Runner of Pe
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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 644 and 645: Those particles next to the forward
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 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
Page 694 and 695: Table A4.1 (contd.) Symbol Definiti
Page 696 and 697: Answers to problems 1.1 56.2 m 3 1.
Page 698 and 699: 8.14 0.002955, 2.866 m · s −1 ,
Page 700 and 701: Absolute pressure 46 Absolute visco
Page 702 and 703: Eccentricity 230 Eccentricity ratio
Page 704 and 705: Mach angle 498 intersection 510-1 r
Page 706 and 707: Slip in fluid couplings 652 in reci
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