Mechanics of Fluids

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sphere is falling through the atmosphere. This is of practical importance for some meteorological balloons and large hailstones. As we have seen, for very small values of Re (say less than 0.1) Stokes’s Law applies to the motion of a sphere. Then CD = 24/Re and eqn 8.36 yields uT = d2 (�ϱ)g 18µ from which either uT or d may be obtained algebraically. It should be remembered that Fig. 8.15 applies only to spheres well away from other particles or solid boundaries. The sphere also is assumed rigid: if the sphere itself consists of fluid (e.g. a raindrop in the atmosphere or a gas bubble in a liquid) its motion may be modified by flow inside it. The effect of such internal flow is, however, not usually significant for small spheres. Results for some other bodies of revolution are shown in Fig. 8.14, and it will be noted that the decrease in CD when the boundary layer becomes turbulent is somewhat more gradual for the more streamlined bodies than for the sphere. This is because of the smaller contribution made by pressure drag to the total drag of the streamlined bodies. For the circular disc, the body of maximum ‘bluffness’, there is no reduction in CD because, except at low Reynolds numbers, CD is independent of viscous effects, and the position of separation is fixed at the sharp edge of the disc. (It may be recalled that losses due to separation at the abrupt changes of section in pipe-lines (Section 7.6) are also practically independent of Reynolds number.) 8.8.6 Separation from an aerofoil If fluid flowing past a body produces a lift force on it (i.e. a force perpendicular to the direction of the oncoming fluid), then any separation of the flow from the surface of the body not only increases the drag but also affects the lift. We here consider the two-dimensional flow about a section of an aircraft wing (see Section 9.10 for definitions of aeronautical terms). The angle of attack is defined as the angle between the direction of the oncoming fluid and a longitudinal reference line in the crosssection. As we shall show in Section 9.10.2, an increase in the angle of attack causes the lift force on the aerofoil to increase – at least in the absence of boundary-layer effects. The adverse pressure gradient along the rear part of the upper (lower pressure) surface is thereby intensified until, at a particular value of the angle of attack, the boundary layer separates from the upper surface, and a turbulent wake is formed. At fairly small angles of attack the position of separation may be quite close to the trailing (rear) edge of the aerofoil. As the angle increases further, the position of separation moves forward, the wake becomes wider, and there is a marked increase of drag. In addition the main flow round the aerofoil is disrupted, and the lift no longer increases (Fig. 8.16). Figure 8.17 shows these variations of lift and drag – for a particular wing section Effect of pressure gradient 335
336 Boundary layers, wakes and other shear layers Fig. 8.16 Stalled flow above an aerofoil. Fig. 8.17 at a given Reynolds number – in terms of the respective dimensionless coefficients CL and CD. Each coefficient is defined as the corresponding force divided by the product of 1 2ϱu2∞ and the plan area of the wing. (Note that they are plotted to different scales; CL is much larger than CD.) As we have mentioned, the angle of attack α is measured from some reference line in the aerofoil section, but, as this is chosen simply for geometrical convenience, the position α = 0 has, in general, no particular significance.
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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
Page 296 and 297: where u denotes the mean velocity i
Page 298 and 299: Assuming that entry and exit losses
Page 300 and 301: of one-per-revolution electrical co
Page 302 and 303: of the piping system, this meter is
Page 304 and 305: losses, calculate (a) the total pow
Page 306 and 307: How are the running costs altered i
Page 308 and 309: main. The depth of water in the tan
Page 310 and 311: the features of boundary layer flow
Page 312 and 313: the surface other considerations ar
Page 314 and 315: and θ = = = � δ u 0 um � δ 0
Page 316 and 317: The momentum equation applied to th
Page 318 and 319: 1 = y η δ The laminar boundary la
Page 320 and 321: Table 8.1 Comparison of results for
Page 322 and 323: and � � ∂f (η) B = ∂η A =
Page 324 and 325: From equation 8.17, the frictional
Page 326 and 327: The turbulent boundary layer on a s
Page 328 and 329: Hence F = 0.037ϱu 2 m l(Re l) −1
Page 330 and 331: From eqn 8.29 Hence and xt − x0 x
Page 332 and 333: from it. This breakaway before the
Page 334 and 335: this expression into eqn 8.31 we ge
Page 336 and 337: When, for example, a ship moves thr
Page 338 and 339: separate vortices in an inviscid fl
Page 340 and 341: 8.8.4 Profile drag of two-dimension
Page 342 and 343: are negligible. With reduction in t
Page 344 and 345: Fig. 8.14 Drag coefficients of smoo
Page 348 and 349: At high Reynolds numbers, the varia
Page 350 and 351: this does not prevent the continual
Page 352 and 353: Eddy viscosity and the mixing lengt
Page 354 and 355: Eddy viscosity and the mixing lengt
Page 356 and 357: Velocity distribution in turbulent
Page 358 and 359: then, although at very small values
Page 360 and 361: as η → 1 and (1 − η) is then
Page 362 and 363: When f −1/2 − 4 log 10 (d/k) is
Page 364 and 365: comparable with the velocity of sou
Page 366 and 367: The x, y and z components of the mo
Page 368 and 369: y y x i-1, j+1 i, j+1 i+1, j+1 i-1,
Page 370 and 371: e completely covered by a turbulent
Page 372 and 373: 9.1 INTRODUCTION The flow of an inv
Page 374 and 375: Now consider another point P ′′
Page 376 and 377: number of smaller ones of which M a
Page 378 and 379: distortion, the element is thus not
Page 380 and 381: Any function φ that satisfies Lapl
Page 382 and 383: ecome perfect squares - except wher
Page 384 and 385: fluid, and the flow net indicates t
Page 386 and 387: 9.6.2 Flow from a line source A sou
Page 388 and 389: higher orders of small magnitudes b
Page 390 and 391: Combining eqns 9.16 and 9.17 we obt
Page 392 and 393: 9.6.5 Forced (rotational) vortex Th
Page 394 and 395: 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
Page 498 and 499:
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
Page 516 and 517:
the temperature rises greatly (say
Page 518 and 519:
Fig. 11.10 Oblique shock relations
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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 →
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eqn 11.41. Substituting M 2 − 1 =
Page 528 and 529:
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
Page 544 and 545:
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
Page 558 and 559:
Analogy between compressible flow a
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is, on ∂2n ∂y2 + ∂2 � n ∂
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and the mass flow rate in the duct.
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11.18 Air flows isothermally at 15
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12.2 INERTIA PRESSURE Any volume of
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Equation 12.3 indicates that u beco
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would come to rest later. Although
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a pressure wave in a gas; here we r
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value in eqn 12.7 gives that is, wh
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the moment disregarded. An increase
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at this point remains constant for
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In addition to the reflection of wa
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is closed so that the area coeffici
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The pressure diagrams show that for
Page 586 and 587:
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
Page 610 and 611:
Turbines 599 Fig. 13.7 Runner of Pe
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determined, multiplication by the r
Page 614 and 615:
Substituting for �vw from eqn 13.
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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
Page 622 and 623:
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
Page 632 and 633:
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
Page 638 and 639:
an unmixed blessing and, except for
Page 640 and 641:
orbiting space-craft, for example,
Page 642 and 643:
Losses are also possible at the inl
Page 644 and 645:
Those particles next to the forward
Page 646 and 647:
It should be noted that the values
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∴ (β1)A = arctan(va/u) = arctan(
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so much that the aerofoil stalls (a
Page 652 and 653:
Use these data to deduce the pump c
Page 654 and 655:
Manometric efficiency = 0.75 = gH/u
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
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further: the turbine part then remo
Page 668 and 669:
for reaction turbines: 1 − η = 1
Page 670 and 671:
75 mm wide at outlet. The blades oc
Page 672 and 673:
Losses at valves, etc. are estimate
Page 674 and 675:
APPENDIX Units and conversion 1 fac
Page 676 and 677:
Table A1.4 (contd.) Force 32.17 pdl
Page 678 and 679:
APPENDIX Physical constants and 2 p
Page 680 and 681:
Kinematic viscosity mm 2 ·s -1 App
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Range II: 11 000 m ≤ z ≤ 20 000
Page 684 and 685:
Table A3.1 (contd.) M 1 M 2 p 2/p 1
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
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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
Page 704 and 705:
Mach angle 498 intersection 510-1 r
Page 706 and 707:
Slip in fluid couplings 652 in reci
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