Mechanics of Fluids

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where Ɣ0 represents the circulation in the centre of the aerofoil of span b. At a distance x from the centre the circulation Ɣ = Ɣ0{1 − (x/ 1 2 b)2 } 1/2 . ∴ Le = � b/2 −b/2 −ϱUeƔdx =−ϱUeƔ0 � b/2 −b/2 An introduction to elementary aerofoil theory 409 � � 1 − x/ 1 2b � � 2 1/2 dx =−ϱUeƔ0bπ/4 (9.44) Combining eqns 9.42, 9.43 and 9.44 we obtain Di = viLe Ue = −Ɣ0Le 2bUe = 4Le ϱUebπ Le 2bUe = 2 ϱπb 2 � �2 Le Ue = 2 ϱπb 2 � �2 L U Division by 1 2ϱU2 S gives the result in terms of the dimensionless coefficients: � � � CDi = C 2 L πb 2 S = C 2 L /(πAR) (9.45) The vortex drag is shown to decrease as the aspect ration AR of the wing increases. It may also be shown that the elliptical distribution of lift here assumed is the condition for the vortex drag to be a minimum for a given value of the lift. Equation 9.45 enables the effect of vortex drag to be separated from other components of drag, and thus data obtained at one aspect ratio can be converted for wings with the same aerofoil section but a different aspect ratio. The change in the effective angle of attack is � � CL arctan(vi/U) = arctan(Di/L) = arctan(CDi/CL) = arctan πAR Phenomena associated with the viscous flow round wings are considered in Section 8.8.6 and the effect of compressibility is briefly treated in Section 11.11. Example 9.7 A wing of span 10 m and mean chord 2 m has a lift coefficient of 0.914 and a drag coefficient of 0.0588 for an angle of attack of 6.5 ◦ . If the distribution of lift over the span is elliptical, what are the corresponding lift and drag coefficients for an aerofoil of the same profile and effective angle of attack, but aspect ratio 8.0? Solution Aspect ratio = 10 m/2 m= 5 CDi = C2 L πAR = 0.9142 /5π = 0.0532
410 The flow of an inviscid fluid ✷ The effective angle of attack is less than the nominal value by arctan CL 0.914 = arctan = 3.33◦ πAR 5π that is, effective angle is 6.5 ◦ − 3.33 ◦ = 3.17 ◦ . For AR =∞andsono vortex drag, CD = 0.0588 − 0.0532 = 0.0056. For AR = 8.0, L = Le cos ε ≈ Le since ε is small. Therefore if the effective angle of attack is unchanged, the lift is not appreciably different and CL = 0.914 again. CDi = 0.9142 8π = 0.0332 and CD = 0.0056 + 0.0332 = 0.0388 The value of ε is now arctan(CL/πAR) = arctan(0.914/8π) = 2.08 ◦ , and so, if the effective angle of attack is to be unchanged, the nominal angle must be 3.17 ◦ + 2.08 ◦ = 5.25 ◦ . PROBLEMS 9.1 Is a flow for which u = 3m· s−1 , v = 8x s−1 possible in an incompressible fluid? Can a potential function exist? 9.2 Which of the following functions could represent the velocity potential for the two-dimensional flow of an inviscid fluid? (a) x + 5y; (b) 3x2 − 4y2 ; (c) cos(x − y); (d) ln(x + y); (e) arctan(x/y); (f) arccosec (x/y). 9.3 Show that the two-dimensional flow described (in metre-second units) by the equation ψ = x + 2x2 − 2y2 is irrotational. What is the velocity potential of the flow? If the density of the fluid is 1.12 kg · m−3 and the piezometric pressure at the point (1, −2) is 4.8 kPa, what is the piezometric pressure at the point (9, 6)? 9.4 Determine the two-dimensional stream function corresponding to φ = A ln(r/r0) where A is a constant. What is the flow pattern? 9.5 A two-dimensional source at the origin has a strength (3π/2) m2 · s−1 . If the density of the fluid is 800 kg · m−3 9.6 calculate the velocity, the piezometric pressure gradient and the acceleration at the point (1.5 m, 2 m). An enclosed square duct of side s has a horizontal axis and vertical sides. It runs full of water and at one position there is a curved right-angled bend where the axis of the duct has radius r. If the flow in the bend is assumed frictionless so that the velocity distribution is that of a free vortex, show that the volume rate of flow is related to �h, the difference of static head between the inner and outer sides of the duct, by the expression � Q = r 2 − s2 � (sg�h/r) 4 1/2 � � 2r + s ln 2r − s
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Mechanics of Fluids
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Mechanics of Fluids Eighth edition
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Contents Preface to the eighth edit
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8 Boundary Layers, Wakes and Other
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Preface to the eighth edition In th
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Fundamental concepts 1 The aim of C
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1.1.1 Molecular structure The diffe
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therefore, to have in place systems
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has been a strong movement in favou
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Notation, dimensions, units and rel
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is nearer 2 m, rather than 1 m. Her
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Properties of fluids 13 The relativ
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that is, p1 − p3 = 1 2BC ϱax (1.
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that there is an almost perfect vac
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calorically perfect. (Some writers
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As a liquid is compressed its molec
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or τ = µ ∂u ∂y (1.9) where µ
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shared among the occupants of bb, a
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the origin with slope equal to µ (
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The interplay of these various forc
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non-uniformity is always encountere
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z direction and therefore the same
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Although Reynolds used water in his
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cylinder. For a certain range of Re
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The roles of experimentation and th
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A flow model which assumes steady,
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2.1 INTRODUCTION Fluid statics 2 Fl
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3. dp/dz =−ϱg. (Since the pressu
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that is, dp p =−g dz R T Variatio
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atmosphere it is termed vacuum or s
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at its centreline be p. Then, provi
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possible, and a sloping manometer m
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The amount of liquid B on each side
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In a pressure transducer the pressu
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where the suffixes C and Oy indicat
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surface (where the pressure is atmo
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which has different depths of water
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Hydrostatic thrusts on submerged su
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plane and may then be combined into
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Since all the elemental thrusts are
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neglected when a body is weighed on
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couple acting on the body in its di
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centre of buoyancy on to the transv
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move, but also the centre of gravit
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that is, W(OG) = (W + F)(OB + BM) =
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is given by ∂p ∂x =−ϱax Equi
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Then ∂p/∂ξ = 0 by definition o
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2.5 Two small vessels are connected
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2.18 In the vertical end of an oil
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The principles governing fluids in
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planes B and C (Fig. 3.2), δs bein
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an actual fluid is thus often remar
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density are small. Then eqn 3.9 has
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General energy equation for steady
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− Energy loss to friction/time Ma
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a decrease of pressure (provided th
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in the neighbourhood of B results i
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it is difficult to measure the heig
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The diagram illustrates an orifice
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� depth h below the free surface
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Solution (a) We consider first the
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of the same liquid. A vena contract
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To ensure that the pressure measure
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Since �h = p1 − p2 ϱwg the flo
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For consistency with the mathematic
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showing the essential form of the r
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With the same assumptions used in d
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∴ Velocity of approach = 0.0660 m
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of the base of the notch. Assume th
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Our aim now is to derive a relation
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The velocity, in general, varies fr
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the control volume is � ϱ1u1ux1
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Applications of the momentum equati
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that the resultant force on the flu
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The existence of the reaction may b
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our reference axes attached to the
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of increase of x-momentum is � D
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This is a simplified picture of wha
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since u4 −u1 is the velocity of t
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(1) u 1 (2) (3) (4) Assume a sea le
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and frictional effects to be neglig
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Physical similarity and dimensional
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A well-known example of kinematic s
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polygon. Now a polygon can be compl
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The condition for dynamic similarit
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equality of the Mach numbers is not
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Consider the incompressible flow al
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moving in a straight line with cons
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of the analysis, and its correct im
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Note: a suffix has been attached to
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For example, the parameter Fϱ/µ 2
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The application of dynamic similari
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atmospheric air, the dynamic viscos
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gives rise to waves on the surface.
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To be tested, the model must theref
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it follows that Hence V2 V1 = P2 P1
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5.5 A torpedo-shaped object 900 mm
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Laminar flow between solid boundari
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the place of y: τ = µ ∂u ∂r A
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compensate for the reduction in vel
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The expression for the total discha
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etween the resisting viscous forces
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must be zero so as to meet the requ
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Example 6.2 Two stationary, paralle
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Thus for any value of y the velocit
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Steady laminar flow between moving
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in eqn 6.24 we obtain � D Vp 2
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and the passage of the liquid level
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this book, it is important to give
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6.6.4 Rotary viscometers A simple m
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The torque T on the cylinder of rad
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stationary when the outer cylinder
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Fundamentals of the theory of hydro
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part of the bearing then gives when
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e determined by our analysis.) From
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where a = 6µ�R h3 dh pc dx Funda
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Integrating this between θ = 0 and
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Rearrangement of eqn 6.71 and subst
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6.4 A cylindrical drum of length l
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7.1 INTRODUCTION Flow and losses in
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a maximum value of 2.0. The line CD
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America the friction factor commonl
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course, quite different from the ro
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Variation of friction factor 253 Fi
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straightforward manner without iter
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Distribution of shear stress in a c
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7.5 FRICTION IN NON-CIRCULAR CONDUI
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estimated by simple theory. The pip
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the junction, the curvature of the
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particular, the performance of a di
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A pipe bend thus causes a loss of h
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cross-sectional area available in t
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correlating experimental data on th
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to do this. But sub-atmospheric pre
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For systems comprising only short r
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7.8 PIPES IN COMBINATION 7.8.1 Pipe
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2. There can be only one value of h
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So For two pipes in parallel, and s
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7.9 CONDITIONS NEAR THE PIPE ENTRY
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where u denotes the mean velocity i
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Assuming that entry and exit losses
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of one-per-revolution electrical co
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of the piping system, this meter is
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losses, calculate (a) the total pow
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How are the running costs altered i
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main. The depth of water in the tan
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the features of boundary layer flow
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the surface other considerations ar
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and θ = = = � δ u 0 um � δ 0
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The momentum equation applied to th
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1 = y η δ The laminar boundary la
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Table 8.1 Comparison of results for
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and � � ∂f (η) B = ∂η A =
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From equation 8.17, the frictional
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The turbulent boundary layer on a s
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Hence F = 0.037ϱu 2 m l(Re l) −1
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From eqn 8.29 Hence and xt − x0 x
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from it. This breakaway before the
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this expression into eqn 8.31 we ge
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When, for example, a ship moves thr
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separate vortices in an inviscid fl
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8.8.4 Profile drag of two-dimension
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are negligible. With reduction in t
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Fig. 8.14 Drag coefficients of smoo
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sphere is falling through the atmos
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At high Reynolds numbers, the varia
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this does not prevent the continual
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Eddy viscosity and the mixing lengt
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Eddy viscosity and the mixing lengt
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Velocity distribution in turbulent
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then, although at very small values
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as η → 1 and (1 − η) is then
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When f −1/2 − 4 log 10 (d/k) is
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comparable with the velocity of sou
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The x, y and z components of the mo
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y y x i-1, j+1 i, j+1 i+1, j+1 i-1,
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
Page 396 and 397: source is unable to move to the lef
Page 398 and 399: Also we note that the source is pos
Page 400 and 401: as shown in Fig. 9.23, encloses all
Page 402 and 403: eqn 9.23 becomes ψ =−U � r −
Page 404 and 405: Thus the magnitude of the velocity
Page 406 and 407: At a stagnation point, qt = 0 and t
Page 408 and 409: function of the combined flow is gi
Page 410 and 411: or, more approximately, that betwee
Page 412 and 413: Then dw dz =−U =−u + iv Hence u
Page 414 and 415: An introduction to elementary aerof
Page 416 and 417: ound the aerofoil, its magnitude be
Page 418 and 419: and lower layers. Since the vortex
Page 422 and 423: 9.7 An open cylindrical vessel, hav
Page 424 and 425: steadily south-east at 6 m · s −
Page 426 and 427: 10.2 TYPES OF FLOW IN OPEN CHANNELS
Page 428 and 429: The steady-flow energy equation for
Page 430 and 431: where the hs represent vertical dep
Page 432 and 433: to the turbulent rough flow regime
Page 434 and 435: selection of an appropriate value r
Page 436 and 437: 10.6 OPTIMUM SHAPE OF CROSS-SECTION
Page 438 and 439: oughness coefficient n is independe
Page 440 and 441: thrust divided by width of the chan
Page 442 and 443: which the waves follow one another
Page 444 and 445: The conditions for the critical dep
Page 446 and 447: y downstream conditions. In these c
Page 448 and 449: may be obtained by reducing eqn 10.
Page 450 and 451: this depth is greater than the crit
Page 452 and 453: The dissipation of energy is by no
Page 454 and 455: friction, the steady-flow momentum
Page 456 and 457: Fig. 10.28 Notice that, for a chann
Page 458 and 459: calculated, and then H + u2 1 /2g m
Page 460 and 461: Fig. 10.32 If the depth of flow ove
Page 462 and 463: (Section 10.11.1). Rapid flow can n
Page 464 and 465: where 0 ≤ θ ≤ 90 ◦ and sin
Page 466 and 467: (a) The continuity equation is The
Page 468 and 469: oundaries. It can occur when the fl
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into eqn 10.39 we obtain whence dh
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Table 10.2 Gradually varied flow 46
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The slope of a given channel may, o
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direction. The effect of any bounda
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This momentum term, however, is pro
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With surface tension effects ignore
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are for waves of small amplitude) w
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when the wave was formed. The lengt
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At a particular instant the crests
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Hence h = 2.65λ 2π = 2.65 × 225
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As the ratio of these velocity comp
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water is impulsively displaced and
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is relative movement of one layer o
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the channel bed a drop of 150 mm in
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11.1 INTRODUCTION Compressible flow
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energy ÷ mass, so that the dimensi
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Energy equation with variable densi
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Solution (a) From the equation of s
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whence (c − u)δρ = (ρ + δρ)
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the sonic velocity as supersonic (M
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velocity into the undisturbed fluid
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whence p2 p1 = 1 + γ M2 1 1 + γ M
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of eqns 11.2 and 11.20 T0 T u2 = 1
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the temperature rises greatly (say
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Fig. 11.10 Oblique shock relations
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however, that for a given upstream
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The deflection through the first wa
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As δθ is very small, cos δθ →
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eqn 11.41. Substituting M 2 − 1 =
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oundaries, both of which are curved
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For supersonic flow eqn 11.45 is no
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Some general relations for one-dime
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of minimum cross-section the veloci
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downstream of the minimum cross-sec
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‘telegraphed’ upstream and a pr
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the external pressure. This compres
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Compressible flow in pipes of const
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Table 11.2 Changes with distance al
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From the Fanno Tables at pc/pB = 0.
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Therefore and M1 = u1 a 24.5 m · s
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drag. The abrupt pressure rise thro
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Analogy between compressible flow a
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is, on ∂2n ∂y2 + ∂2 � n ∂
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and the mass flow rate in the duct.
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11.18 Air flows isothermally at 15
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12.2 INERTIA PRESSURE Any volume of
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Equation 12.3 indicates that u beco
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would come to rest later. Although
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a pressure wave in a gas; here we r
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value in eqn 12.7 gives that is, wh
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the moment disregarded. An increase
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at this point remains constant for
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In addition to the reflection of wa
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is closed so that the area coeffici
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The pressure diagrams show that for
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12.3.4 The method of characteristic
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Pressure transients 577 Multiplying
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and velocities are known, then for
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For pipe 2, and hence K ′ = c2 =
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(instantaneously, we will assume) a
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The one-dimensional continuity rela
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to accommodate instantaneous comple
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12.2 A turbine which normally opera
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13.1 INTRODUCTION Fluid machines13
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flow. By way of illustration we sha
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would obviate the changes of pressu
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demand for electric power - for exa
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Turbines 599 Fig. 13.7 Runner of Pe
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determined, multiplication by the r
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Substituting for �vw from eqn 13.
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ut the larger its value the more bu
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ate of flow possible is achieved wh
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the turbine. For the high heads nor
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The integrals in eqns 13.4 and 13.5
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diagram of Fig. 13.16 we have Simil
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would be sufficient to describe the
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Table 13.1 Type of turbine Approxim
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∴ Hydraulic efficiency = Overall
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If either z (the height of the turb
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∴ Hydraulic efficiency = u1vw1/gH
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shows the general effect of change
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an unmixed blessing and, except for
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orbiting space-craft, for example,
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Losses are also possible at the inl
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Those particles next to the forward
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It should be noted that the values
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∴ (β1)A = arctan(va/u) = arctan(
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so much that the aerofoil stalls (a
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Use these data to deduce the pump c
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Manometric efficiency = 0.75 = gH/u
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and CP = P/ϱω 3 D 5 in place of Q
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yields h 1 = h f = 4fl d ≈ (100Q)
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(iii) the power dissipated by pipe
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Large numbers of test data for pump
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educed by rounding the inlet edges
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further: the turbine part then remo
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for reaction turbines: 1 − η = 1
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75 mm wide at outlet. The blades oc
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Losses at valves, etc. are estimate
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APPENDIX Units and conversion 1 fac
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Table A1.4 (contd.) Force 32.17 pdl
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APPENDIX Physical constants and 2 p
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Kinematic viscosity mm 2 ·s -1 App
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Range II: 11 000 m ≤ z ≤ 20 000
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Table A3.1 (contd.) M 1 M 2 p 2/p 1
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Table A3.2 Isentropic flow M p/po
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Table A3.3 Adiabatic flow with fric
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APPENDIX 4 Algebraic symbols Table
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Table A4.1 (contd.) Symbol Definiti
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Table A4.1 (contd.) Symbol Definiti
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Answers to problems 1.1 56.2 m 3 1.
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8.14 0.002955, 2.866 m · s −1 ,
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Absolute pressure 46 Absolute visco
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Eccentricity 230 Eccentricity ratio
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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