Mechanics of Fluids

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the features of boundary layer flow are common to the velocity profiles that occur in fully developed flow through pipes, a topic that will also be considered in this chapter. 8.2 DESCRIPTION OF THE BOUNDARY LAYER The simplest boundary layer to study is that formed in the flow along one side of a thin, smooth, flat plate parallel to the direction of the oncoming fluid (Fig. 8.1). No other solid surface is near, and the pressure of the fluid is uniform. If the fluid were inviscid no velocity gradient would, in this instance, arise. The velocity gradients in a real fluid are therefore entirely due to viscous action near the surface. Description of the boundary layer 299 The fluid, originally having velocity u∞ in the direction of the plate, is Transition region retarded in the neighbourhood of the surface, and the boundary layer begins at the leading edge of the plate. As more and more of the fluid is slowed down the thickness of the layer increases. We have already noted that the boundary layer merges into the main flow with no sharp line of demarcation but, for convenience, the thickness of the layer may be taken as that distance from the surface at which the velocity reaches 99% of the velocity of the main stream. The flow in the first part of the boundary layer (close to the leading edge of the plate) is entirely laminar. With increasing thickness, however, the laminar layer becomes unstable, and the motion within it becomes disturbed. The irregularities of the flow develop into turbulence, and the thickness of the layer increases more rapidly. The changes from laminar to turbulent flow take place over a short length known as the transition region. Downstream of the transition region the boundary layer is almost entirely turbulent, and its thickness increases further. For a plane surface over which the pressure is uniform the increase of thickness continues indefinitely. It should be noted that the y scale of Fig. 8.1 is greatly enlarged. At any distance x from the leading edge of the plate the boundary layer thickness δ is very small compared with x. Fig. 8.1 Boundary layer on flat plate (y scale greatly enlarged).
300 Boundary layers, wakes and other shear layers Fig. 8.2 Typical velocity distributions in laminar and turbulent boundary layers on a flat plate. Viscous sub-layer The random fluctuations of turbulent flow must die out very close to the surface and so beneath the turbulent boundary layer an even thinner viscous sub-layer is formed in which the flow is basically laminar. In a turbulent layer there is more intermingling of fluid particles and therefore a more nearly uniform velocity than in a laminar layer (Fig. 8.2). As a result the turbulent layer usually has a greater velocity gradient at the surface. Because of the essentially laminar flow immediately adjacent to the surface the shear stress there, τ0, is given by � � ∂u τ0 = µ ∂y and thus the shear stress associated with a turbulent boundary layer is usually greater than that for a wholly laminar one. Moreover, there is a more ready interchange of fluid particles between a turbulent layer and the main flow, and this explains the more rapid increase in thickness of a turbulent layer. Whereas the thickness of a laminar boundary layer increases as x 0.5 (when the pressure is uniform) a turbulent layer thickens approximately as x 0.8 . The point at which a laminar boundary layer becomes unstable depends on a number of factors. Roughness of the surface hastens the transition to turbulence, as does the intensity of turbulence in the main stream. The predominant factor, however, is the Reynolds number of the flow in the boundary layer. This is usually expressed as u∞x/ν, where u∞ represents the velocity of the oncoming flow far upstream, ν the kinematic viscosity of the fluid and x the distance from the leading edge (where the boundary layer starts) to the point in question. This Reynolds number is given the symbol Rex, the suffix x indicating that it is calculated with the distance x as the characteristic length. For values of Rex = u∞x/ν below about 10 5 the laminar layer is very stable. However, as the value increases, transition is more and more easily induced and when Rex > 2 × 10 6 it is difficult to prevent transition occurring even when the surface is smooth and there is no turbulence in the main stream. If the pressure p is not uniform over y=0
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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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Page 262 and 263: course, quite different from the ro
Page 264 and 265: Variation of friction factor 253 Fi
Page 266 and 267: straightforward manner without iter
Page 268 and 269: Distribution of shear stress in a c
Page 270 and 271: 7.5 FRICTION IN NON-CIRCULAR CONDUI
Page 272 and 273: estimated by simple theory. The pip
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Page 276 and 277: particular, the performance of a di
Page 278 and 279: A pipe bend thus causes a loss of h
Page 280 and 281: cross-sectional area available in t
Page 282 and 283: correlating experimental data on th
Page 284 and 285: to do this. But sub-atmospheric pre
Page 286 and 287: For systems comprising only short r
Page 288 and 289: 7.8 PIPES IN COMBINATION 7.8.1 Pipe
Page 290 and 291: 2. There can be only one value of h
Page 292 and 293: So For two pipes in parallel, and s
Page 294 and 295: 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 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 346 and 347: sphere is falling through the atmos
Page 348 and 349: At high Reynolds numbers, the varia
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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
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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
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
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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
Page 544 and 545:
Table 11.2 Changes with distance al
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From the Fanno Tables at pc/pB = 0.
Page 552 and 553:
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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
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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
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(
Page 650 and 651:
so much that the aerofoil stalls (a
Page 652 and 653:
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
Page 662 and 663:
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
Page 670 and 671:
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
Page 704 and 705:
Mach angle 498 intersection 510-1 r
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Slip in fluid couplings 652 in reci
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