Mechanics of Fluids

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Slip in fluid couplings 652 in reciprocating pumps 596 Slip coupling 652 Slip surface 511 Slope 461–3 Smooth zone of flow 251 Solidity of axial runner 628 of wire gauze 270 Sommerfeld boundary condition 232–3 Sommerfeld condition, half 237 Sommerfeld number 234 Sonic velocity 493–6 Source 375–6, 401 Source and sink of numerically equal strength 387–9 Span (of aerofoil) 403 Specific-energy curve in dimensionless form 436–7 Specific energy in open channels 431–5 Specific entropy 489 Specific heat capacity 489 Specific gravity 13 Specific speed power 616 pumps 639–40 turbines 616 Speed of sound 493–6 Speed ratio of Pelton wheel 603 Spiral vortex 398–9 Stability of atmosphere 79 of bodies in fluids 71–8 of body subject to additional force 78 of floating bodies 72–8 of fluid itself 79–80 of submerged bodies 71–2 Stable equilibrium 71 Stagnation enthalpy 492 Stagnation hypothesis 405 Stagnation point 110, 384, 517 Stagnation pressure 110, 503 Stagnation temperature 492, 502 Stall 337 Stalled flow 337 Stalling angle 337 Stall torque 654 Standing wave 478–9 Stanton diagram 250 Starting vortex 405 Steady flow 36 Steady-flow energy equation derivation 97–100 for open channels 416–9 practical application 103–4 Steady-flow momentum equation 134, 428 Steady uniform flow 415, 419–23 Steep slope 435, 462–3 Stokes (unit) 26 Stokes’s Law 212, 391 Straight-line closure 570 Streak-line 32 Stream filament 31 Stream function 362–4 Streamlined body 325 Streamlines 31, 362–3, 370 pressure variation perpendicular to 107–9 Stream-tube 31 Strength of source 375 Strength of vortex 377 Stress 1, 9, 15 Strickler’s formula 422 Strouhal number 328 Submerged bodies, stability of 71–2 Subsonic flow 522, 524, 526–7 Subsonic velocity 496 Substantial acceleration 89 Super-cavitating machines 644 Supersonic flow 502, 527 between two boundaries 516–7 over concave boundary 516 round corners 512–6 Supersonic velocity 497 Surface profiles 457, 459 classification 461–3 Surface tension 28–30, 469 Surface tension forces 165–6 Surface waves 464–9 Surge tanks 583–6 Surges in open channels 427–31 Système International d’Unités (SI units) 6 Index 695 Tail race 606 Temperature 9 Temperature difference 10 Temperature lapse rate 47 Temporal acceleration 89 Terminal velocity 213, 333 Thermally perfect gas 17–8, 489 Thermodynamic concepts 487–91 Thermodynamic effects 487 Thixotropic liquids 27 Thoma’s cavitation parameter 620 Thomson’s theorem 407 Three-dimensional flow 32, 331–5 Thrust coefficient 153 Thwaites’s method 322–3 Töpler system 549 Torque coefficient 654 Torque converter 654–6 Torricellian vacuum 49 Torricelli’s formula 114 Total energy line 106, 271–5 Total head 96 Total head line 106, 271–5 Tranquil flow 416, 435 Transition region of boundary layer 299 Transition zone of flow 251 Transpiration methods 210–1 Tsunamis 480–2 Turbines 591, 596–625 performance characteristics 623–5 types 596–7 Turbulence 35 free 352–3 Turbulent boundary layer, on smooth flat plate with zero pressure gradient 313–6 Turbulent flow 35 in pipes 246–8 velocity distribution in 344–52 Two-dimensional flow 32–3 Undular jump 441–2 Uniform flow 30 Uniform rectilinear flow 374, 384 Unit flow 624
696 Index Unit power 624 Units 4–10 prefixes 8 Unit speed 624 Universal gas constant 18 Unstable equilibrium 71 Unsteady flow 36–7 Upper (or higher) critical velocity 247 U-tube manometers 50–5 Vacuum 48–9 Valve closure 569–74 Valve opening factor 570 Vapour pressure 16 Varied flow 416 Velocity of flow 630 of sound 496 of whirl 601 Velocity defect 345 Velocity defect law 345 Velocity diagrams 609–10, 612 Velocity distribution in rough pipes 349–50 in smooth pipes and over smooth plates 345–8 in turbulent flow 344–52 Velocity gradient 22 Velocity head 96 Velocity measurement 288–90 Velocity potential 367–9 Velocity profile 22 Vena contracta 113 Venturi flume 451–4 Venturi-meter 119–22 Virtual mass 392–3 Viscoelastic materials 27 Viscometer 211 Engler 211 Ostwald 211 Redwood 211 rotary 215–8 Saybolt 211 Searle 231 Viscometry 210 Viscosimeter see Viscometer Viscosity 21–8 absolute 23 basic SI unit 25 causes of 24–5 dynamic 23, 668 eddy 341–2 influence on flow 37 kinematic 26, 669 measurement of 210 quantitative definition 21–4 variation with temperature 668–9 Viscous forces 163–4 Viscous resistance 191–2 Viscous stresses 199 Viscous sub-layer 251–2, 347 Voidage 240 Volute 605, 626 Vortex forced 381–2 free 376–9 spiral 398–9 starting (on aerofoil) 405 Vortex drag 407–9 Vortex pair 396–8 Vortex shedding 328 Vortex sheet 511–2 Vortex street (or trail) 326–8 Vortex strength 377 Vorticity 365–6 Wake definition 325 flow pattern 326 width 328 Water hammer 558 Wave drag 340 Wave energy 472–3 Wave formation 480 Wave-making resistance 182–3 Wave propagation, finite waves 498–9 Waves Airy 467–8 capillary 469 deep water 468 dispersive 468 elastic 493–7 gravity 468 moving into shallow water 477–8 in open channels 427–31 oscillatory 464–79 reflection 564–9 standing 478–9 Weber number 166–7 Weir broad-crested 444–7 drowned 447–8 rapid flow approaching 449–51 sharp-crested 126–30 suppressed 128 Whirl 609 Whirl slip 633 Wicket gates 606 Windage 603 Wind turbine, momentum theory 154 Wings, aerodynamics of 403–9, 544–6 Yaw meter 291 Yield stress 198 Zone of action 498 Zone of silence 498
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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
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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
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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
Page 656 and 657: and CP = P/ϱω 3 D 5 in place of Q
Page 658 and 659: yields h 1 = h f = 4fl d ≈ (100Q)
Page 660 and 661: (iii) the power dissipated by pipe
Page 662 and 663: Large numbers of test data for pump
Page 664 and 665: educed by rounding the inlet edges
Page 666 and 667: further: the turbine part then remo
Page 668 and 669: for reaction turbines: 1 − η = 1
Page 670 and 671: 75 mm wide at outlet. The blades oc
Page 672 and 673: Losses at valves, etc. are estimate
Page 674 and 675: APPENDIX Units and conversion 1 fac
Page 676 and 677: Table A1.4 (contd.) Force 32.17 pdl
Page 678 and 679: APPENDIX Physical constants and 2 p
Page 680 and 681: Kinematic viscosity mm 2 ·s -1 App
Page 682 and 683: Range II: 11 000 m ≤ z ≤ 20 000
Page 684 and 685: Table A3.1 (contd.) M 1 M 2 p 2/p 1
Page 686 and 687: Table A3.2 Isentropic flow M p/po
Page 688 and 689: Table A3.3 Adiabatic flow with fric
Page 690 and 691: APPENDIX 4 Algebraic symbols Table
Page 692 and 693: Table A4.1 (contd.) Symbol Definiti
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 708: eBooks - at www.eBookstore.tandf.co
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