Mechanics of Fluids

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showing the essential form of the relation between depth and discharge even though the formula so obtained will have to be modified by an experimentally determined coefficient. The sheet of liquid escaping over the notch or weir is known as the nappe. If the pressure underneath it is atmospheric, the nappe (except at very low rates of flow) springs clear of the notch plate. For a notch extending across the entire width of a channel, atmospheric air may not be able to get under the nappe, and the liquid then clings to the downstream side of the notch plate and the discharge is unpredictable. So to obtain flow of a predictable and consistent nature, the space underneath the nappe must be ventilated, if necessary by providing an air vent as shown in Fig. 3.23. Consider a sharp-edged, rectangular notch as shown in Fig. 3.24. The crest is horizontal and normal to the general direction of flow. The classical analysis, usually ascribed to the German engineer Julius Weisbach (1806–71), requires these assumptions: 1. Upstream of the notch, the velocities of particles in the stream are uniform and parallel; thus the pressure there varies according to the hydrostatic equation p = ϱgh. (In practice it is often necessary to install baffles to achieve reasonably steady and uniform conditions.) 2. The free surface remains horizontal as far as the plane of the notch, and all particles passing through the notch move horizontally, and perpendicular to its plane. 3. The pressure throughout the nappe is atmospheric. 4. The effects of viscosity and surface tension are negligible. These assumptions give the idealized pattern of flow shown in Fig. 3.24. At section 1, (p1/ϱg) + z1 = height H of the free surface. So for a typical streamline Bernoulli’s equation gives H + u2 1 /2g = 0 + u2 2 /2g + z2 ∴ u2 ={2g(H − z2 + u2 1 /2g)}1/2 This shows that u2 varies with z2. In the plane of the notch the discharge through a horizontal element of depth δz2 is u2bδz2 and so the idealized Simple applications of Bernoulli’s equation 127 Fig. 3.24
128 The principles governing fluids in motion Fig. 3.25 Suppressed weir. total discharge Q ideal = b � H 0 u2dz2 (the datum of z2 being taken at the crest of the notch). Hence Q ideal = b � (2g) � H 0 =− 2 3 b� (2g) = 2 3 b� (2g) (H − z2 + u 2 1 /2g)1/2 dz2 � (H − z2 + u 2 1 /2g)3/2� H � (H + u 2 1 /2g)3/2 − (u 2 1 /2g)3/2� 0 (3.25) Since u1 depends on Q, solution of eqn 3.25 is troublesome except by trial and error. However, u2 1 /2g is frequently negligible in comparison with H and then the equation becomes Q ideal = 2 3 b� (2g)H 3/2 An experimentally determined coefficient of discharge now has to be inserted to account for the simplifying assumptions used in the mathematical model. The contraction of the nappe as it passes through the notch is a significant factor and the coefficient is considerably less than unity. Its value depends primarily on H and H/Z where Z is the height of the crest above the bed of the approach channel. The effects of viscosity and surface tension are appreciable only when H is small. A suppressed weir is one for which the breadth b is the same as the width of the approach channel. The nappe then contracts in the vertical direction only and not horizontally (see Fig. 3.25). When the notch is symmetrically placed in the width of a channel and has the proportions shown in Fig. 3.26, full contraction of the nappe takes place in the horizontal direction. The contraction at each side of the nappe is about H/10; thus the width of the nappe and consequently the coefficient of discharge vary with H. The V notch scores over the rectangular notch in producing a nappe with the same shape whatever the value of H, and thus, it has a less variable discharge coefficient.
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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
Page 88 and 89: move, but also the centre of gravit
Page 90 and 91: that is, W(OG) = (W + F)(OB + BM) =
Page 92 and 93: is given by ∂p ∂x =−ϱax Equi
Page 94 and 95: Then ∂p/∂ξ = 0 by definition o
Page 96 and 97: 2.5 Two small vessels are connected
Page 98 and 99: 2.18 In the vertical end of an oil
Page 100 and 101: The principles governing fluids in
Page 102 and 103: planes B and C (Fig. 3.2), δs bein
Page 104 and 105: an actual fluid is thus often remar
Page 106 and 107: density are small. Then eqn 3.9 has
Page 108 and 109: General energy equation for steady
Page 116 and 117: − Energy loss to friction/time Ma
Page 118 and 119: a decrease of pressure (provided th
Page 120 and 121: in the neighbourhood of B results i
Page 122 and 123: it is difficult to measure the heig
Page 124 and 125: The diagram illustrates an orifice
Page 126 and 127: � depth h below the free surface
Page 128 and 129: Solution (a) We consider first the
Page 130 and 131: of the same liquid. A vena contract
Page 132 and 133: To ensure that the pressure measure
Page 134 and 135: Since �h = p1 − p2 ϱwg the flo
Page 136 and 137: For consistency with the mathematic
Page 140 and 141: With the same assumptions used in d
Page 142 and 143: ∴ Velocity of approach = 0.0660 m
Page 144 and 145: of the base of the notch. Assume th
Page 146 and 147: Our aim now is to derive a relation
Page 148 and 149: The velocity, in general, varies fr
Page 150 and 151: the control volume is � ϱ1u1ux1
Page 152 and 153: Applications of the momentum equati
Page 154 and 155: that the resultant force on the flu
Page 156 and 157: The existence of the reaction may b
Page 158 and 159: our reference axes attached to the
Page 160 and 161: of increase of x-momentum is � D
Page 162 and 163: This is a simplified picture of wha
Page 164 and 165: since u4 −u1 is the velocity of t
Page 166 and 167: (1) u 1 (2) (3) (4) Assume a sea le
Page 168 and 169: and frictional effects to be neglig
Page 170 and 171: Physical similarity and dimensional
Page 172 and 173: A well-known example of kinematic s
Page 174 and 175: polygon. Now a polygon can be compl
Page 176 and 177: The condition for dynamic similarit
Page 178 and 179: equality of the Mach numbers is not
Page 180 and 181: Consider the incompressible flow al
Page 182 and 183: moving in a straight line with cons
Page 184 and 185: of the analysis, and its correct im
Page 186 and 187: 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
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and CP = P/ϱω 3 D 5 in place of Q
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yields h 1 = h f = 4fl d ≈ (100Q)
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(iii) the power dissipated by pipe
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Large numbers of test data for pump
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educed by rounding the inlet edges
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further: the turbine part then remo
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for reaction turbines: 1 − η = 1
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75 mm wide at outlet. The blades oc
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Losses at valves, etc. are estimate
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APPENDIX Units and conversion 1 fac
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Table A1.4 (contd.) Force 32.17 pdl
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APPENDIX Physical constants and 2 p
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Kinematic viscosity mm 2 ·s -1 App
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Range II: 11 000 m ≤ z ≤ 20 000
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Table A3.1 (contd.) M 1 M 2 p 2/p 1
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Table A3.2 Isentropic flow M p/po
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Table A3.3 Adiabatic flow with fric
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APPENDIX 4 Algebraic symbols Table
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Table A4.1 (contd.) Symbol Definiti
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Table A4.1 (contd.) Symbol Definiti
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Answers to problems 1.1 56.2 m 3 1.
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8.14 0.002955, 2.866 m · s −1 ,
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Absolute pressure 46 Absolute visco
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Eccentricity 230 Eccentricity ratio
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Mach angle 498 intersection 510-1 r
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Slip in fluid couplings 652 in reci
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