Mechanics of Fluids

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Table 8.1 Comparison of results for various approximate velocity distributions in a laminar boundary layer with the Blasius’s exact solution u um δ √ (Rex) x δ ∗√ (Rex) x θ √ (Rex) x √ = cf (Rex) The laminar boundary layer on a flat plate 309 � τ 0 ϱu 2 m � √(Rex) 2η − η 2 5.48 1.826 0.730 0.365 3 2 η − 1 2 η3 4.64 1.740 0.646 0.323 2η − 2η 3 + η 4 5.84 1.751 0.685 0.343 sin(πη/2) 4.80 1.743 0.655 0.328 Blasius’s — 1.721 0.664 0.332 exact solution Example 8.2 Which of the following expressions describes better the velocity distribution for a laminar boundary layer on a flat plate in the absence of a streamwise pressure gradient? (a) u = um 3 � y � − 2 δ 1 � y �3 2 δ (b) u = um 3 � y � − 2 δ 1 � y �2 2 δ Give the reasons for your decision. Solution Write η = y/δ. (a) u um u um u um = 3 1 η − 2 2 η3 = 0 when η = 0 = 1 when η = 1 d(u/um) dη d(u/um) dη d(u/um) dη = 3 3 − 2 2 η2 = 3 2 when η = 0 = 0 when η = 1 (b) u um u um u um = 3 1 η − 2 2 η2 = 0 when η = 0 = 1 when η = 1 d(u/um) dη d(u/um) dη d(u/um) dη = 3 − η 2 = 3 2 = 1 2 when η = 0 when η = 1 Velocity profile (a) is the better of the two, as it satisfies all four of the required boundary conditions, whereas profile (b) fails condition III. ✷
310 Boundary layers, wakes and other shear layers Example 8.3 An approximate relation for the velocity profile in the laminar boundary layer subject to zero pressure gradient is u = a1η + a2η 2 um (a) Determine the values of the constants a1 and a2. (b) Evaluate the constants A and B. (c) Derive relations for the development of δ, δ ∗ and θ with x. Solution (a) Conditions I and II can be tested by substituting for η in the velocity profile equation u = a1η + a2η 2 um Condition I: When η = 0, u um = 0 Thus the no-slip condition is satisfied automatically by the expression. Condition II: When η = 1, u = a1 + a2 = 1 Differentiating d(u/um) = a1 + 2a2η dη Condition III: When η = 1, d(u/um) = a1 + 2a2 = 0 dη Hence we require and um a1 + a2 = 1 a1 + 2a2 = 0 These simultaneous equations are readily solved to yield so a1 = 2 and a2 =−1 f (η) = u um = 2η − η 2 We can confirm that this relation also satisfies condition IV. When η = 0, d(u/um) dη (b) A = = a1 = 2 so � 1 0 d(u/um) dη {1 − f (η)}f (η)dη �= 0 QED.
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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
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 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 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
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,
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e completely covered by a turbulent
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9.1 INTRODUCTION The flow of an inv
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Now consider another point P ′′
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number of smaller ones of which M a
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distortion, the element is thus not
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Any function φ that satisfies Lapl
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ecome perfect squares - except wher
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fluid, and the flow net indicates t
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9.6.2 Flow from a line source A sou
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higher orders of small magnitudes b
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Combining eqns 9.16 and 9.17 we obt
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9.6.5 Forced (rotational) vortex Th
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For a free vortex qR = constant = K
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source is unable to move to the lef
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Also we note that the source is pos
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as shown in Fig. 9.23, encloses all
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eqn 9.23 becomes ψ =−U � r −
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Thus the magnitude of the velocity
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At a stagnation point, qt = 0 and t
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function of the combined flow is gi
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or, more approximately, that betwee
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Then dw dz =−U =−u + iv Hence u
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An introduction to elementary aerof
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ound the aerofoil, its magnitude be
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and lower layers. Since the vortex
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where Ɣ0 represents the circulatio
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9.7 An open cylindrical vessel, hav
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steadily south-east at 6 m · s −
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10.2 TYPES OF FLOW IN OPEN CHANNELS
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The steady-flow energy equation for
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where the hs represent vertical dep
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to the turbulent rough flow regime
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selection of an appropriate value r
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10.6 OPTIMUM SHAPE OF CROSS-SECTION
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oughness coefficient n is independe
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thrust divided by width of the chan
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which the waves follow one another
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The conditions for the critical dep
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y downstream conditions. In these c
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may be obtained by reducing eqn 10.
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this depth is greater than the crit
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The dissipation of energy is by no
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friction, the steady-flow momentum
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Fig. 10.28 Notice that, for a chann
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calculated, and then H + u2 1 /2g m
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Fig. 10.32 If the depth of flow ove
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(Section 10.11.1). Rapid flow can n
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where 0 ≤ θ ≤ 90 ◦ and sin
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(a) The continuity equation is The
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oundaries. It can occur when the fl
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into eqn 10.39 we obtain whence dh
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Table 10.2 Gradually varied flow 46
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The slope of a given channel may, o
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direction. The effect of any bounda
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This momentum term, however, is pro
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With surface tension effects ignore
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are for waves of small amplitude) w
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when the wave was formed. The lengt
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At a particular instant the crests
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Hence h = 2.65λ 2π = 2.65 × 225
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As the ratio of these velocity comp
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water is impulsively displaced and
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is relative movement of one layer o
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the channel bed a drop of 150 mm in
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11.1 INTRODUCTION Compressible flow
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energy ÷ mass, so that the dimensi
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Energy equation with variable densi
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Solution (a) From the equation of s
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whence (c − u)δρ = (ρ + δρ)
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the sonic velocity as supersonic (M
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velocity into the undisturbed fluid
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whence p2 p1 = 1 + γ M2 1 1 + γ M
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of eqns 11.2 and 11.20 T0 T u2 = 1
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the temperature rises greatly (say
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Fig. 11.10 Oblique shock relations
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however, that for a given upstream
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The deflection through the first wa
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As δθ is very small, cos δθ →
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eqn 11.41. Substituting M 2 − 1 =
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oundaries, both of which are curved
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For supersonic flow eqn 11.45 is no
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Some general relations for one-dime
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of minimum cross-section the veloci
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downstream of the minimum cross-sec
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‘telegraphed’ upstream and a pr
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the external pressure. This compres
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Compressible flow in pipes of const
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Table 11.2 Changes with distance al
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From the Fanno Tables at pc/pB = 0.
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Therefore and M1 = u1 a 24.5 m · s
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drag. The abrupt pressure rise thro
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
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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
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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