Mechanics of Fluids

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velocity into the undisturbed fluid. A second tiny wave follows immediately, but the fluid into which this second wave moves has already been traversed by the first wave. That fluid is therefore at a slightly higher pressure, density and temperature than formerly. If for simplicity a perfect gas is assumed so that a = � (γ RT) (although the conclusion may be shown to apply to any fluid) it is clear that the second wave travels slightly faster than the first. Similarly the third wave is propagated with a slightly higher velocity than the second and so on. The pressure distribution at a certain time might be as shown in Fig. 11.5a, and a little while later would become like that shown at b. In practice, separate small waves would not be distinguishable but, instead, a gradual rise of pressure which becomes steeper as it advances. Before long the wave front becomes infinitely steep and a sharp discontinuity of pressure results. This is known as a shock wave, and it will be studied in the next section. A similar argument applied to a rarefaction wave shows that this becomes less steep as shown in Figs. 11.5c and d. The foremost edge of the wave proceeds into the undisturbed fluid with the sonic velocity in that fluid, but all other parts of the wave have a lower velocity. Consequently no effect of a pressure decrease, no matter how large or how sudden it is initially, is propagated at more than the sonic velocity in the undisturbed fluid. A pressure increase, on the other hand, may be propagated at a velocity greater than that of sound. 11.5 SHOCK WAVES Whereas in Section 11.4 we considered a pressure change of infinitesimal size, we now turn attention to an abrupt finite pressure change known as a shock. The possibility of such an abrupt change in a compressible fluid was envisaged by the German mathematician G. F. Bernhard Riemann (1826–66) Fig. 11.5 Shock waves 499
500 Compressible flow of gases Fig. 11.6 Normal shock. and the theory was developed in detail by W. J. M. Rankine (1820–72) and the French physicist Henri Hugoniot (1851–87). In practice a shock is not absolutely abrupt, but the distance over which it occurs is of the order of only a few times the mean free path of the molecules (about 0.3 µm in atmospheric air). For most purposes, therefore, the changes in flow properties (pressure, density, velocity and so on) may be supposed abrupt and discontinuous and to take place across a surface termed the shock wave. (In photographs a thickness apparently greater than 0.3 µm may be observed because the wave is seldom exactly plane or exactly parallel to the camera axis.) We shall not concern ourselves here with what happens within the very narrow region of the shock itself, for such a study is very complex and involves non-equilibrium thermodynamics. Moreover, the analysis that follows will be restricted to a perfect gas because a general solution is algebraically complicated and explicit results cannot usually be obtained. Qualitatively, however, the phenomena discussed apply to any gas. 11.5.1 Normal shock waves We consider first a normal shock wave, that is one perpendicular to the direction of flow. Such shocks may occur in the diverging section of a convergent-divergent nozzle or in front of a blunt-nosed body. We shall see that in every case the flow upstream of the shock is supersonic, while that downstream is subsonic and at higher pressure. We shall see too that the changes occurring in a shock are not reversible and so not isentropic. Our first objective is to determine the relations between quantities upstream and downstream of the shock. As shown in Fig. 11.6, quantities upstream are denoted by suffix 1 and those downstream by suffix 2. To obtain steady flow we consider the shock stationary and so the velocities u1 and u2 are relative to it. As the shock region is so thin, any change in the cross-sectional area of a stream-tube from one side of the shock to the other is negligible, and so the continuity relation is simply ρ1u1 = ρ2u2 (11.22) If effects of boundary friction are negligible, the momentum equation for a stream-tube of cross-sectional area A is (p1 − p2)A = m(u2 − u1) where m = ρAu is the mass flow rate. Thus p1 − p2 = ρ2u 2 2 − ρ1u 2 1 (11.23) Since a 2 = pγ/ρ and Mach number M = u/a, eqn 11.23 may be written p1 − p2 = p2γ M 2 2 − p1γ M 2 1
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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
Page 460 and 461: Fig. 10.32 If the depth of flow ove
Page 462 and 463: (Section 10.11.1). Rapid flow can n
Page 464 and 465: where 0 ≤ θ ≤ 90 ◦ and sin
Page 466 and 467: (a) The continuity equation is The
Page 468 and 469: oundaries. It can occur when the fl
Page 470 and 471: into eqn 10.39 we obtain whence dh
Page 472 and 473: Table 10.2 Gradually varied flow 46
Page 474 and 475: The slope of a given channel may, o
Page 476 and 477: direction. The effect of any bounda
Page 478 and 479: This momentum term, however, is pro
Page 480 and 481: With surface tension effects ignore
Page 482 and 483: are for waves of small amplitude) w
Page 484 and 485: when the wave was formed. The lengt
Page 486 and 487: At a particular instant the crests
Page 488 and 489: Hence h = 2.65λ 2π = 2.65 × 225
Page 490 and 491: As the ratio of these velocity comp
Page 492 and 493: water is impulsively displaced and
Page 494 and 495: is relative movement of one layer o
Page 496 and 497: the channel bed a drop of 150 mm in
Page 498 and 499: 11.1 INTRODUCTION Compressible flow
Page 500 and 501: energy ÷ mass, so that the dimensi
Page 502 and 503: Energy equation with variable densi
Page 504 and 505: Solution (a) From the equation of s
Page 506 and 507: whence (c − u)δρ = (ρ + δρ)
Page 508 and 509: the sonic velocity as supersonic (M
Page 512 and 513: whence p2 p1 = 1 + γ M2 1 1 + γ M
Page 514 and 515: of eqns 11.2 and 11.20 T0 T u2 = 1
Page 516 and 517: the temperature rises greatly (say
Page 518 and 519: Fig. 11.10 Oblique shock relations
Page 520 and 521: however, that for a given upstream
Page 522 and 523: The deflection through the first wa
Page 524 and 525: As δθ is very small, cos δθ →
Page 526 and 527: eqn 11.41. Substituting M 2 − 1 =
Page 528 and 529: 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
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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
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Substituting for �vw from eqn 13.
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ut the larger its value the more bu
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ate of flow possible is achieved wh
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the turbine. For the high heads nor
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The integrals in eqns 13.4 and 13.5
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diagram of Fig. 13.16 we have Simil
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would be sufficient to describe the
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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
Page 644 and 645:
Those particles next to the forward
Page 646 and 647:
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
Page 652 and 653:
Use these data to deduce the pump c
Page 654 and 655:
Manometric efficiency = 0.75 = gH/u
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and CP = P/ϱω 3 D 5 in place of Q
Page 658 and 659:
yields h 1 = h f = 4fl d ≈ (100Q)
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(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
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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
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Range II: 11 000 m ≤ z ≤ 20 000
Page 684 and 685:
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
Page 690 and 691:
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
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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