Mechanics of Fluids

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Compressible flow in pipes of constant cross-section 531 To account for the changes of fluid properties we must consider initially an infinitesimal length of the pipe. For adiabatic flow the relation involving the friction force may be integrated over a finite length of the pipe by introducing the continuity equation and the energy equation. For isothermal conditions integration is made possible by inserting the condition T = constant. Applying the steady-flow momentum equation to an element with crosssectional area A, perimeter P and length δl (Fig. 11.26), we have pA − (p + δp)A − τ 0Pδl − ρgAδl sin α = ρAuδu Substituting τ 0 = 1 2 f ρu2 (eqn 7.3), δl sin α = δz and dividing by ρgA we obtain −δz = δp uδu f δl + + ρg g A/P u 2 2g (11.62) The final term corresponds to Darcy’s formula for head lost to friction. Equation 11.62 applies to any steady one-dimensional flow where A is constant, but, for integration, the relation between density and pressure must be known and this depends on the degree of heat transfer. We consider first the case of zero heat transfer. 11.10.1 Adiabatic flow in a pipe: Fanno flow Before embarking on the integration of eqn 11.62 for adiabatic conditions it will be instructive to look at the problem in general thermodynamic terms. When no external work is done and changes of elevation may be neglected, the steady-flow energy equation (11.12) and the continuity equation m = ρAu together give Constant h0 = h + 1 2 u2 = h + 1 � �2 m (11.63) 2 ρA For given values of m, A and the stagnation enthalpy h0, curves of h against ρ could be plotted from eqn 11.63. A more significant relation, however, is that between h and specific entropy s. For any pure substance, s, like h and ρ, is a function of state and so may be determined from values of h and ρ. In particular, for a perfect gas s − s1 = cv ln(h/h1)(ρ1/ρ) γ −1 . Starting from a specified state (point 1 on Fig. 11.27) the curve of h against s traces the Fig. 11.26
532 Compressible flow of gases Fig. 11.27 states through which the substance must pass in an adiabatic process. Such a curve is termed a Fanno curve in honour of Gino Fanno, the Italian engineer who first studied its properties. All Fanno curves show a maximum value of s. At this maximum ds = 0 = 1 T dqrev = 1 T � de + pd Also, from the definition of specific enthalpy, � � 1 dh= d(e + p/ρ) = de + pd ρ Equations 11.64 and 11.65 together give � �� 1 ρ (11.64) + 1 dp (11.65) ρ dh = 1 dp (11.66) ρ Differentiation of eqn 11.63 gives 0 = dh−(m2 /A2 )(dρ/ρ3 ) and substitution for dh from eqn 11.66 then shows that when s is a maximum u 2 = m2 A2 � � ∂p = = a ρ2 ∂ρ 2 (from eqn 11.18) s That is, the specific entropy is a maximum when the Mach number is unity. The upper branch of the curve in Fig. 11.27, which approaches the stagnation enthalpy h0, thus corresponds to subsonic flow, and the lower branch to supersonic flow. Since for adiabatic conditions the entropy cannot decrease, friction acts to increase the Mach number in subsonic flow and to reduce the Mach number in supersonic flow. Changes in other properties with distance along the pipe may be deduced as shown in Table 11.2. Moreover, as friction involves a continual increase of entropy, sonic velocity can be reached only at the exit end of the pipe, if at all. If sonic velocity is to be reached in a particular pipe then, for given inlet conditions and exit pressure, a certain length is necessary. If the actual length is less than this limiting value sonic conditions are not reached. If the length of the pipe is increased beyond the limiting value an initially subsonic flow will be choked; that is, the rate of flow will be reduced so as again to give sonic
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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
Page 492 and 493: water is impulsively displaced and
Page 494 and 495: is relative movement of one layer o
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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 510 and 511: velocity into the undisturbed fluid
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 544 and 545: Table 11.2 Changes with distance al
Page 546 and 547: Compressible flow in pipes of const
Page 550 and 551: From the Fanno Tables at pc/pB = 0.
Page 554 and 555: Therefore and M1 = u1 a 24.5 m · s
Page 556 and 557: drag. The abrupt pressure rise thro
Page 558 and 559: Analogy between compressible flow a
Page 560 and 561: is, on ∂2n ∂y2 + ∂2 � n ∂
Page 562 and 563: and the mass flow rate in the duct.
Page 564 and 565: 11.18 Air flows isothermally at 15
Page 566 and 567: 12.2 INERTIA PRESSURE Any volume of
Page 568 and 569: Equation 12.3 indicates that u beco
Page 570 and 571: would come to rest later. Although
Page 572 and 573: a pressure wave in a gas; here we r
Page 574 and 575: value in eqn 12.7 gives that is, wh
Page 576 and 577: the moment disregarded. An increase
Page 578 and 579: at this point remains constant for
Page 580 and 581: In addition to the reflection of wa
Page 582 and 583: is closed so that the area coeffici
Page 584 and 585: The pressure diagrams show that for
Page 586 and 587: 12.3.4 The method of characteristic
Page 588 and 589: Pressure transients 577 Multiplying
Page 590 and 591: and velocities are known, then for
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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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