MMV211 Fluid Mechanics — questions on theory(January 19, 2015)

Chapter 1 — Introduction

1.1 What is the fundamental difference between a solid and a fluid?

1.2 Give a practical definition of the density of a fluid. Illustrate with a figure.

1.3 (a) For newtonian fluids, shear stresses are proportional to corresponding shear strain rates. Fora simple shear flow, V = u(y)i, show that the shear stress τ is proportional to du/dy.

(b) Discuss and illustrate with diagrams some possible characteristics of non-newtonian fluids.

1.4 State the SI-dimensions of dynamic and kinematic viscosity, respectively. Also provide approx-imate values (with two significant digits) for water and air at standard room conditions (20C,1 atm). At around standard room conditions, what are the separate effects of changing thepressure and temperature, respectively?

1.5 Explain or define (a) streamline, (b) particle path, (c) streak line.

Chapter 2 — Pressure Distribution in a Fluid

2.1 Derive the hydrostatic balance equation ∇p = ρg, valid for a fluid at rest.

2.2 Derive the manometer formula ∆p = (ρm−ρ)g h , where h is the manometer reading (height), ρmthe density of the manometer liquid and ∆p the pressure difference across two horizontal pointsin the fluid of density ρ (U-tube manometer, see Example 2.3).

Chapter 3 — Integral Analysis

3.1 (a) Provide general definitions of volume flow rate and mass flow rate through a surface. Illustratewith a figure.

(b) Show that the mean (average) velocity for laminar fully developed flow in a pipe of circularcross section is equal to half the velocity at the centerline. The velocity profile is parabolic.

3.2 Let B be an arbitrary extensive property and b the same property but expressed per unit mass.For a fixed and non-deforming control volume with one inlet (1) and one outlet (2), both assumedone-dimensional, the Reynolds transport theorem, at stationary conditions, reads:

d

dtBsys = (b ρ V A)2 − (b ρ V A)1

State without proof the Reynolds transport theorem for an arbitrarily moving but still non-deforming control volume, at non-stationary conditions. Illustrate the convective part in a figure.

3.3 (a) Explain briefly what is meant by the concept of incompressible flow.

(b) Assuming stationary one-dimensional incompressible flow, state without proof the momentumintegral equation applicable to a control volume with one inlet and one outlet. Give a fullaccount for different terms and properties. How can this equation be modified to include velocityvariations across the inlet and outlet? Calculate the correction coefficient β for the case of fullydeveloped laminar flow in a straight pipe of circular cross section.

3.4 Assuming stationary one-dimensional incompressible flow, state without proof the energy integralequation applicable to a control volume with one inlet and one outlet (the extended Bernoulliequation). Give a full account for different terms and properties. How can this equation bemodified to include velocity variations across the inlet and outlet? Calculate the correctioncoefficient α for the case of fully developed laminar flow in a straight pipe of circular crosssection.

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Chapter 4 — Differential Analysis

4.1 A velocity field is described in a cartesian coordinate system xyz. Use the rule of chains toexpress the material (substantial) acceleration in local and convective terms. Give a physicaldescription of each term.

4.2 Derive the differential form of the continuity equation using a cartesian coordinate system. Afterthis, specialize to incompressible flow conditions.

4.3 Show that Ma2 ≪ 1 is a necessary condition for incompressible flow. Provide an engineeringestimate of Ma for which the flow has to be treated as compressible.

4.4 (a) Define the stress tensor σij in a flowing fluid, expressed in terms of the viscous stress tensorτij and the pressure p. What symmetry condition is normally valid for σij?

(b) Illustrate the components σ11, σ22, σ12 and σ21 in a figure (σ12 = σxy, . . . ).

(c) Define τxy for a newtonian fluid using cartesian coordinates.

4.5 Using cartesian coordinates, write out the continuity equation as well as the x-component of themomentum equation for incompressible flow of a newtonian fluid with constant fluid properties.

4.6 Write out the divergence applied to the velocity field V = (u, v, w), in cartesian coordinates.What does this scalar quantity represent physically?

4.7 Consider a cubical infinitesimal fluid element in a flow field at a certain time instant. Kinemat-ically, the instantaneous effects on this element from the flow field can be sub-divided into fourelementary motions. Define and describe these elementary motions. Illustrate with a figure.

4.8 (a) State the general definition of the vorticity vector at a point in a flow field. What doesit represent physically? Write out, in cartesian coordinates, the components of this vector.Verify that the vorticity vector only has one component for two-dimensional flow in a plane;V = (u, v, 0), where u(x, y), v(x, y).

(b) Show that the only component of the vorticity vector for two-dimensional flow in a planecan be interpreted as twice the instantaneous anti-clockwise rotational rate of the diagonal in ainfinitesimal fluid element, initially being quadratic.

(c) How is it possible for an initially irrotational flow to become locally rotational, i.e., to possessvorticity? State at least two mechanisms.

4.9 Consider two-dimensional incompressible flow in a plane, V = (u, v, 0), where u(x, y), v(x, y).

(a) Define the stream function ψ through implicit expressions using u and v. Also show that ψsatisfies the Laplace equation ∇2ψ = 0 when the the flow is irrotational.

(b) Show that ψ = const. represents streamlines and that the difference in ψ between twostreamlines is equal to the volume rate per unit width.

Chapter 5 — Dimensional Analysis and Similarity

5.1 (a) What is the principle of dimensional homogeneity, PDH?

(b) What are theMLTΘ-dimensions for dynamic (absolute) viscosity µ, heat conductivity k andpower P (work per unit time)?

(c) State the Π-theorem, in words.

5.2 Define the Strouhal number (St) for periodic vortex shedding downstream of a circular cylinderof diameter d, exposed to a constant flow velocity U perpendicular to the cylinder axis. Assumingincompressible flow, state an approximate value of the Strouhal number over the interval Re =Ud/ν = 300− 105.

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5.3 Explain briefly

(a) homologous points and homologous times

(b) geometric, kinematic and dynamic similarity

5.4 In practical cases concerning free-surface flows, most model tests have to renounce on the principleof dynamic similarity. Explain why.

5.5 For incompressible flow conditions, the drag D on a smooth spherical (and solid) object dependsonly on the diameter d, fluid viscosity µ, fluid density ρ and flow velocity U .

(a) Define the conventional drag coefficient CD, and show, using dimensional analysis, that CDonly depends on the Reynolds number.

(b) Suggest an alternative drag coefficient that does not include the flow velocity. Explain whythis coefficient could be more appropriate to use, in some cases.

5.6 Show that the Reynolds, Froude and Weber number, respectively, each is a (rough) measureof a ratio between certain fluid forces in a flow field (mass times acceleration = inertia force;[Υ] = N/m).

5.7 Ekman (Ek) and Rossby (Ro) numbers are of importance in rotating flows.1 Ekman number is ameasure of the ratio between frictional (viscous) and Coriolis (fictional) forces on a fluid element;Rossby number correspondingly between inertial and Coriolis forces. Define these two numbersif Ω is a characteristic angular rate of rotation (characteristic velocity U ; length L).

Coriolis acceleration: 2Ω×V.

Chapter 6 — Viscous Flows in Ducts

6.1 Make a sketch on the variation of pressure and the velocity profile development for stationaryflow in a horizontal pipe connected to a large reservoir. The pipe inlet is well rounded. State anapproximate expression for the entrance length at laminar flow conditions.

6.2 Define the hydraulic diameter Dh in pipe and duct flows. Find the hydraulic diameter for acircular and a square cross-section. In engineering calculations, state the commonly acceptedReynolds number above which the flow can be regarded as fully turbulent.

6.3 Derive the Hagen-Poiseuille formula for the volume flow rate in a straight circular pipe with fullydeveloped laminar flow. The suggested starting point is a force balance on a cylindrical (andhorizontal) element.

6.4 Define the Darcy-Weisbach friction factor f for pipe flow. How is the (major) pressure losscalculated? What can be stated generally for f in fully developed laminar duct flow, regardlessof the cross section?

6.5 Make an account for the text-book’s recommended scheme of obtaining the pressure loss overa certain length in a duct of constant but non-circular cross section, assuming fully developedturbulent flow. The expression for the corresponding pressure loss in fully developed laminarflow is known.

6.6 (a) Make a sketch of mean velocity profiles in fully developed laminar and turbulent flow in apipe (circular cross section). In turbulent flow, what is the influence of Reynolds number?

(b) State an approximate power-law expression for the mean velocity profile in fully developedturbulent pipe flow. Use this expression to determine the ratio between the average velocity andthe mean velocity at the center of the pipe. What is the influence of Reynolds number?

1Ekman and Rossby were Swedes; Vagn Walfrid Ekman (1874–1954) professor at Lund University 1910–1939.

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6.7 Make a sketch in a semi-logarithmic diagram of the time-mean velocity profile for pipe flow or afully developed turbulent boundary-layer. Use inner scaling for velocity and distance from wall,u+ and y+, which both should be defined. Mark out different regions in the diagram and state,if possible, pertinent expressions for the velocity in these regions. For boundary-layers, what isthe influence of pressure gradient? What is the influence of wall roughness?

6.8 Explain in detail how a Pitot-static tube (Prandtl tube) works and derive an expression for thevelocity. State an approximate condition for the expression to be valid and discuss advantagesand disadvantages of this method for measuring local velocity.

6.9 Describe briefly the concept of hot-wire anemometry (HWA). State some advantages and disad-vantages with this method of measuring local velocity, as compared with some other method.

6.10 Describe two ways of measuring volume flow rate in a pipe. The two methods should be basedon different basic principles.

Chapter 7 — Boundary Layers and Flow past Immersed Bodies

7.1 (a) Give a practical definition of the boundary layer thickness δ.

(b) How does δ vary along a flat, wide plate when exposed to a tangential oncoming flow?Power-law expressions in different regions should be stated. Illustrate with a figure.

7.2 Define the displacement thickness δ∗, the momentum thickness θ and the shape factor H for aflat-plate boundary layer assuming two-dimensional incompressible flow. Explain why δ∗ is calledthe displacement thickness.

7.3 It can be shown from momentum-integral analysis that the skin-friction coefficient for the flat-plate boundary layer is cf = 2τw/(ρU

2) = 2 dθdx . The velocity outside the boundary layer isconstant (= U). For a laminar boundary layer of thickness θ the velocity profile can be roughlyapproximated as a linear function: u/U = A+Bη, where η = y/δ.

Determine the constants A,B and derive an expression for cf as a function of Rex = Ux/ν.

Hint: θ = δ/C, where δ is a function of x and C is a constant (pure number).

7.4 Describe how the shape of the velocity profile in a boundary layer, close to the wall, is dependenton the pressure gradient. Explain why boundary-layer separation only can occur in regions withdecelerating outer velocity.

7.5 Define the the drag coefficient CD in flow around a sphere. Describe in detail and in a diagramhow CD for a smooth sphere varies with the Reynolds number Re (in detail means that certainvalues on the axes should be marked; logarithmic axes are preferred). What is the asymptoticexpression for CD when Re ≪ 1? What is the influence of surface roughness?

7.6 Derive the integral condition D(x) = ρ bU2 θ(x), valid for a constant-pressure flat-plate boundarylayer (θ is the momentum thickness, x the distance from the leading edge, b is the plate widthand U is the velocity outside the boundary layer).

7.7 The momentum equations for a two-dimensional boundary layer can be simplified drastically byusing order-of-magnitude arguments. Carry out these steps and show in particular the conse-quences for the pressure field. The equations of motion for stationary, incompressible flow withnegligible effects of gravitation reads

∂u

∂x+∂v

∂y= 0

u∂u

∂x+ v

∂u

∂y= −ρ−1 ∂p

∂x+ ν

(∂2u

∂x2+∂2u

∂y2

)

u∂v

∂x+ v

∂v

∂y= −ρ−1 ∂p

∂y+ ν

(∂2v

∂x2+∂2v

∂y2

)

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The boundary layer thickness δ is much smaller than some characteristic dimension of the bodyL and the body surface radius of curvature, respectively. The Reynolds number, based on theoncoming constant velocity at large distances U∞ and L, is very large (Re = U∞L/ν ≫ 1).

Introduction to Turbulence

T.1 Make a brief account of (at least) four characteristics of turbulence (fully developed turbulentflow).

T.2 Introduce the Reynolds decomposition of turbulent fluid motion (velocity and pressure) and carryout the time-averaging procedure for the continuity equation in cartesian coordinates assumingstationary, incompressible turbulent flow.

T.3 Consider a two-dimensional boundary layer of thickness δ over a smooth surface in stationary,incompressible flow. Suppose that this boundary layer can be divided into two partially overlap-ping cross-flow regions: (1) the near-wall region with very high mean velocity gradients and (2)the outer region with small mean velocity gradients.

(a) State and define a velocity scale that is characteristic for the turbulent motions across thewhole boundary layer.

(b) State relevant length scales in the two regions and provide the basic criterion that motivatesthe assumption of a sub-division of the boundary layer.

(c) At distances sufficiently far from the wall, viscous effects can be neglected. Together withthe above (+ some further reasoning) it can be inferred that the mean velocity gradient isproportional to the ratio between the characteristic (turbulent) velocity scale and the prevalentlength scale. Show that this leads to that the velocity variation within the overlapping regionmust be logarithmic, i.e. u = C1 ln y + C2.

T.4 Define the root-mean-square value (r.m.s.-value) of a quantity that has a non-zero time average.Sketch how the r.m.s.-values of the fluctuating components of the velocity vector vary across aconstant-pressure turbulent boundary layer. The velocities should be scaled using the frictionvelocity u∗.

Chapter 8 — Incompressible Potential Flow

8.1 For irrotational flow (∇×V = 0) there exists a velocity potential function ϕ.

(a) Define ϕ and derive its’ differential equation for irrotational incompressible flow. What isthis equation called?

(b) State boundary conditions for ϕ in flows around a body without any effect of free surfaces.

(c) For a given potential field ϕ, how is the pressure field determined?

8.2 (a) Define circulation Γ along a closed curve C.

(b) The following applies for a line vortex placed at the origin: vθ = K/r, vr = vz = 0. Determinethe circulation along a closed curve that encompasses the origin.

8.3 Derive the governing equation for streamlines belonging to a doublet of strength λ = 2 am placedat the origin, along the x-axis. Sketch some streamlines.

Hint: The stream function for a line source with strength m placed at x = −a, y = 0:

ψk = m tan−1 y

x+ a

Trigonometric identity: tan(α− β) =tanα− tanβ

1 + tanα tanβ

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8.4 The two dimensional potential flow around a circular cylinder is governed by superposition of adoublet at the origin (ψ1 = −λr−1 sin θ) and a parallel flow (ψ2 = U∞r sin θ).

Determine

(a) the velocity field (vr = r−1 ∂ψ∂θ and vθ = −∂ψ

∂r ), (b) the radius a, (c) the pressure distributionalong the surface of the cylinder, expressed as a pressure coefficient Cp =

p−p∞12ρU2

∞, where p∞ is the

pressure at large distances upstream along the stagnation line.

8.5 (a) Use the condition of irrotational flow together with vr = 0 to derive the velocity field for aline vortex (stationary, incompressible flow in a plane).

Hint:

ζz =1

r

∂

∂r(rvθ)−

1

r

∂vr∂θ

∇ · V =1

r

∂

∂r(rvr) +

1

r

∂vθ∂θ

(b) The flow field in the outer parts of a tornado can be approximated as a line vortex, theinner parts as like a solid-body rotation. Use these approximations to derive simplified analyticalequations for the velocity and pressure field within a tornado. Sketch the variations of velocityand pressure with the radial distance r from the center of the tornado.

Hint: Momentum balance in radial direction ⇒ ρ v2θ/r =∂p∂r

8.6 (a) Consider incompressible potential flow around a circular cylinder with added circulation Γ.The axis of the cylinder is at the origin. The velocity along the cylinder surface (r = a) in polarcoordinates is

vθ = −2U∞ sin θ +Γ

2πa

where U∞ is the undisturbed velocity upstream of the cylinder.

Show, using integrations, that the drag is zero and that the lift per unit width is −ρU∞Γ.

Note,∫ 2π0 (sin θ)2dθ = π.

(b) Describe, using streamlines, how the velocity field around the cylinder looks like for differentvalues of β = K/(U∞a), where K is the vortex strength. How is K related to Γ?

8.7 State the Joukowsky lift theorem for incompressible, potential plane flow. Define quantities andillustrate with a simple sketch.

8.8 A two-dimensional (AR → ∞), symmetric and thin wing is suddenly brought into a constantvelocity. The angle of attack is small but not zero. Describe the flow development with specialemphasis on the flow at the trailing edge and the generation of a circulation and thus a lift forceon the wing. (See Fig. 7.22).

8.9 (a) State the Kutta condition. How can this condition be motivated physically?

(b) Write out the mathematical expression of the lift coefficient CL valid for a slender butcambered two-dimensional wing profile at small angles of attack. The maximum camber ish≪ C, where C is the wing chord. Also state the position of the resultant lift force (the centerof pressure, CP). Illustrate schematically.

8.10 (a) Discuss shortly and illustrate with figures how the lift force on a finite-span wing gives riseto an induced drag.

(b) State the correct formula for the lift coefficient CL under the assumptions of elliptical planformarea, small angles of attack, high Re, and for a given wing span-to-chord ratio AR.

(c) Sketch how AR influences CD and CL as a function of the angle of attack α.

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8.11 Describe what is meant by the concept of added mass. How can the added mass be estimatedwith a known potential flow around the body? State without derivation the added mass forpotential flow around a sphere.

Chapter 9 — Compressible Flow

9.1 Use conservation laws of mass and momentum to derive an expression for the velocity C of afinite pressure wave which propagates in an otherwise still fluid. The thickness of the wave-front can be assumed to be infinitely small so that so that one-dimensional approximationsare applicable. After this, specialize to a vanishing small amplitude wave amplitude assumingadiabatic conditions (velocity of sound).

9.2 (a) Define the concepts of stagnation temperature and stagnation pressure.

(b) Assuming compressible flow of a perfect gas, derive a relationship between stagnation tem-perature T0, static temperature T , k = cp/cv and Mach number Ma.

(c) Assuming compressible flow of a perfect gas, derive a relationship between stagnation pressure

p0, static pressure p, k = cp/cv and Mach number Ma. Isentropic relation: p/Tk

k−1 = const.

9.3 (a) Assuming one-dimensional adiabatic flow of a perfect gas, write out a complete set of equationswhich governs the thermodynamic state immediately downstream of a stationary normal-shockwave in a nozzle,

(b) How does the stagnation temperature, Mach number, pressure, temperature, density, entropyand stagnation pressure vary across a normal-shock wave?

9.4 A converging-diverging nozzle (a Laval nozzle) is connected to a large reservoir with a constantpressure pr. The pressure outside the nozzle, the back pressure, is pb (< pr). The gas is perfect(ideal gas with constant cp and cv). The flow can be considered adiabatic and frictionless.

(a) Sketch the variation of pressure through the nozzle (and just outside of it) for different valuesof the pressure ratio pb/pr. In particular, mark out the positions of shocks and expansion waves.

(b) How does the mass flow rate vary with the pressure ratio?

9.5 According to the theory of adiabatic frictional flow of a perfect gas in a straight pipe the flowalways strives towards sonic conditions, Ma = 1, regardless of the upstream (inlet) Mach numberMa1. There is a certain pipe length L∗(Ma1) for which the exit Mach number will be exactlyunity. The flow is then choked.

Explain in detail what happens if the pipe length is longer than this critical length, L > L∗. Sub-critical and supercritical inlet flow should be treated separately. Please note that inlet conditionsmight be altered.

9.6 (a) Describe geometrically the deflection process associated with an oblique shock in a plane. Inparticular, mark out the shock wave angle β and the deflection angle θ.

(b) Use control volume analysis to show that the tangential component of the velocity is conservedafter the passage of the oblique shock.

(c) Make a sketch on the relation between the deflection angle and the shock wave angle withMach number Ma1 as parameter. Mark out regions for weak and strong shocks, normal shockwaves, and Mach waves. How does the Mach number affect the maximum deflection angle? Alsomark out the line where Ma2 = 1.

9.7 (a) Describe the flow pattern around a thin and pointy wing profile in supersonic two-dimensionalflow at a small angle of attack. The profile has a prismatic cross-section (plane surfaces).

(b) The following formula applies to supersonic two-dimensional flows at small angles θ:

p2 − p1p1

=kMa21√Ma21 − 1

θ,

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where index 1 refers to the state upstream of the deflection (Ma1 > 1).

Use this formula to derive expressions for CL and CD valid for supersonic flow around a thin flatplate at small angles of attack. The upstream Mach number is Ma∞. The width of the plate (b)is much larger than its chord C.

Chapter 10 — Open-Channel Flow

10.1 Derive an expression for the wave speed c of a surface wave front with amplitude δy in an openrectangular channel of width b. The undisturbed shallow-water depth is y. After this, specializeto δy/y → 0.

10.2 Consider one-dimensional flow in an open channel of constant cross-section (area A, width atsurface b0, velocity V ).

(a) State the wave speed c0 of an infinitesimal shallow-water surface wave. Define the Froudenumber.

(b) State c0 and the specific energy E for the rectangular cross-section (b0 = b, A = by).

10.3 Water flows in an open channel laid on a certain slope, S0 = tan θ (θ ≪ 1). The water crosssection (and depth) as well as the inner surface condition are constant.

According to Manning (1891),√8g/f ≃ n−1R

1/6h , where the hydraulic radius Rh is in meters

and n is a constant. Define Rh and describe how the flow rate Q can be determined. Startingpoint: the extended Bernoulli equation (the energy equation). Velocity variations across thecross section can be neglected.

10.4 For open-channel flow in a rectangular channel, derive an expression for the critical depth ycat a given discharge per unit width (q). Show that the Froude number at this depth is unity(Frc = 1).

10.5 Water flows at vanishing slope along a straight open channel. Perpendicular to the flow and onthe bottom of the channel there is a small ridge, a bump, with maximum height (amplitude) ∆h.The cross section is rectangular. The flow over the bump can be treated as one-dimensional andfrictionless. Assume given upstream velocity, upstream depth and ∆h.

(a) Derive an implicit relationship for the depth at the top of the bump.

(b) Describe the depth variation along the bump, for different ∆h < ∆hmax and upstream Froudenumbers (Fr1 < 1 and Fr1 > 1). Illustrate with a diagram (depth y vs. specific energy E).

Christoffer Norberg, tel. 046-2228606

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