MMVN10 Fluid Mechanics — questions on theory (March 12, 2020)

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. For a simple shear flow, V = u(y)ˆi, show that the shear stress τ is proportional to du/dy. (b) Discuss and illustrate with diagrams some possible characteristics of non-newtonian fluids (liquids). 1.4 State the SI-dimensions of dynamic and kinematic , respectively. Also provide approx- imate values (with two significant digits) for water and air at standard room conditions (20◦C, 1 atm). At around standard room conditions, what are the separate effects of changing the pressure and temperature, respectively? 1.5 Define or explain shortly: (a) incompressible flow, (b) , (c) streamline, (d) streakline.

Chapter 2 — Pressure Distribution in a Fluid

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

2.2 Derive the manometer formula ∆p = (ρm −ρ)g h , where h is the manometer reading (height), ρm the density of the manometer liquid and ∆p the pressure difference across two horizontal points in 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. Illustrate with a figure. (b) Show that the mean (average) velocity for laminar fully developed flow in a pipe of circular cross 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 assumed one-dimensional, the Reynolds transport theorem, at steady flow conditions, reads:

d B = (b ρ V A) − (b ρ V A) dt sys 2 1 State without proof the Reynolds transport theorem for an arbitrarily moving but still non- deforming control volume, at non-steady (time-dependent) conditions. Illustrate the convective part in a figure. 3.3 Assuming steady one-dimensional incompressible flow, state without proof the momentum inte- gral equation applicable to a control volume with one inlet and one outlet. Give a full account for different terms and properties. How can this equation be modified to include velocity variations across the inlet and outlet? Calculate the correction coefficient (β) for the case of fully developed laminar flow in a straight pipe (circular cross section). 3.4 Assuming steady one-dimensional incompressible flow, state without proof the energy integral equation applicable to a control volume with one inlet and one outlet (the extended Bernoulli equation). Give a full account of different terms and properties. How can this equation be modified to include velocity variations across the inlet and outlet? Calculate the correction coefficient (α) for the case of fully developed laminar flow in a straight pipe (circular cross section).

1 Chapter 4 — Differential Analysis

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

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

4.3 Show that Ma2 ≪ 1 is a necessary condition for incompressible flow. Provide an engineering estimate of Ma for which the flow has to be treated as compressible. Hint: Local mass balance, steady flow → ∂(ρu)/∂x + ∂(ρv)/∂y + ∂(ρw)/∂z = 0.

4.4 (a) Define the stress tensor σij in a flowing fluid, expressed in terms of the viscous stress tensor τij 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 the momentum equation for incompressible flow of a newtonian fluid with constant fluid properties.

4.6 Write out the divergence and rotation (curl), respectively, applied to the velocity field V = (u, v, w) in cartesian coordinates. What do they represent physically?

4.7 Consider, at zero time, a cubical fluid element in a flow field. Kinematically, the instantaneous effects on this element from the flow field can be sub-divided into four elementary 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 does it describe physically? (b) For two-dimensional flow in a plane, show that the vorticity vector only has one component. (c) Show that the only component of the vorticity vector in plane two-dimensional flow can be interpreted as twice the instantaneous anti-clockwise rotational rate of the diagonal in a infinitesimal fluid element, initially being quadratic. (d) How is it possible for an initially irrotational flow to become locally rotational, i.e., to possess vorticity? State at least three 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(x, y) and v(x, y). Also show that ψ satisfies the Laplace equation ∇2ψ = 0 when the flow is irrotational. (b) Show that ψ = const. represents streamlines and that the difference in ψ between two streamlines 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 the MLT Θ-dimensions for dynamic (absolute) viscosity µ, heat conductivity k and power P (work per unit time)? (c) State the Π-theorem, in words.

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

2 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 principle of dynamic similarity. Explain why.

5.5 For incompressible steady flow conditions, the drag D on a smooth spherical object depends only 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 CD only depends on the . (b) Suggest an alternative drag coefficient that does not include the flow velocity. Explain why this coefficient could be more appropriate to use, in some cases.

5.6 Show that the Reynolds, Froude and , respectively, each is a rough measure of 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 a rough measure of the ratio between frictional (viscous) and Coriolis (fictional) forces; correspondingly between inertial and Coriolis forces. Define these two numbers if Ω is a characteristic angular rate of rotation (characteristic velocity U; characteristic length L). Coriolis acceleration: 2 Ω × V .

Chapter 6 — Viscous Flows in Ducts

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

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

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

6.4 Consider steady, fully developed laminar flow in a horizontal and straight pipe of circular cross section. How does the pressure drop per unit length ∆p/L vary with the volume flow rate Q, the diameter D, and the fluid density ρ and viscosity µ, respectively? The answer can be given through exponents a − d in the expression ∆p/L ∝ QaDbρcµd. How does the influence of the respective quantity change if the flow is instead turbulent at sufficiently high Reynolds number? Used relationships should be justified.

6.5 Make an account for the recommended scheme of obtaining the pressure loss over a certain length in a duct of constant but non-circular cross section, assuming fully developed turbulent flow. The expression for the corresponding pressure loss in fully developed laminar flow is known.

6.6 (a) Make a sketch of mean velocity profiles in fully developed laminar and turbulent flow in a pipe (circular cross section). For turbulent flow, what is the influence of Reynolds number? (b) State an approximate power-law expression for the mean velocity profile in fully developed turbulent pipe flow. Use this expression to determine the ratio between the average velocity and the 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.

3 6.7 Make a sketch in a semi-logarithmic diagram of the time-mean velocity profile for pipe flow or a fully 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 is the 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 the velocity (incompressible flow). State an approximate condition for the expression to be valid and discuss advantages and disadvantages of this method for measuring local velocity. Illustrate.

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 based on 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. Illustrate. (b) How does δ vary along the surface of 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 a flat-plate boundary layer assuming two-dimensional incompressible flow. Explain why δ∗ is called the displacement thickness.

7.3 It can be shown from momentum-integral analysis that the skin-friction coefficient for the flat- 2 dθ plate boundary layer is cf = 2τw/(ρU ) = 2 dx . The velocity outside the boundary layer is constant (= U). For a laminar boundary layer of thickness θ the velocity profile can be roughly approximated 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 dependent on the pressure gradient. Explain why boundary-layer separation only can occur in regions with decelerating outer velocity (along the wall).

7.5 Define the drag coefficient CD in flow around a sphere. Describe in detail and in a diagram how CD for a smooth sphere varies with the Reynolds number Re (in detail means that certain values on the axes should be marked; logarithmic axes are preferred). What is the asymptotic expression for CD when Re ≪ 1? What is the influence of surface roughness? 7.6 Derive the integral condition D(x) = ρ b U 2 θ(x), valid for a constant-pressure flat-plate boundary layer (θ is the momentum thickness, x the distance from the leading edge, b is the plate width and U is the velocity outside the boundary layer).

7.7 The momentum equations for a two-dimensional boundary layer can be simplified drastically by using order-of-magnitude arguments. Carry out these steps and show in particular the con- sequences for the pressure field. The equations of motion for steady incompressible flow with negligible effects of gravitation reads

∂u ∂v + = 0 ∂x ∂y ( ) ∂u ∂u ∂p ∂2u ∂2u u + v = −ρ−1 + ν + ∂x ∂y ∂x ∂x2 ∂y2 ( ) ∂v ∂v ∂p ∂2v ∂2v u + v = −ρ−1 + ν + ∂x ∂y ∂y ∂x2 ∂y2

4 The boundary layer thickness δ is much smaller than some characteristic dimension of the body L and the body surface radius of curvature, respectively. The Reynolds number based on the oncoming constant velocity U and L is very high (Re = UL/ν ≫ 1).

Introduction to Turbulence

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

T.2 Introduce the Reynolds decomposition of turbulent fluid motion (velocity and pressure) and carry out the time-averaging procedure for the continuity equation in cartesian coordinates assuming a time-mean steady incompressible turbulent flow.

T.3 Consider a two-dimensional boundary layer of thickness δ over a smooth surface in time-mean steady incompressible flow. Suppose that this boundary layer can be divided into two partially overlapping 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 the whole boundary layer. (b) State relevant length scales in the two regions and provide the basic criterion that motivates the assumption of a sub-division of the boundary layer. (c) At distances sufficiently far from the wall, viscous effects can be neglected. Together with the above (+ some further reasoning) it can be inferred that the mean velocity gradient is proportional to the ratio between the characteristic (turbulent) velocity scale and the prevalent length scale. Show that this leads to that the velocity variation within the overlapping region must 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 a constant-pressure turbulent boundary layer. The velocities should be scaled using the friction velocity 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 is this 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. Determine the 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 placed at the origin, along the x-axis. Sketch some streamlines. Hint: Stream function for a line source of strength m placed at x = −a, y = 0: y ψ = m tan−1 k x + a

tan α − tan β Trigonometric identity: tan(α − β) = 1 + tan α tan β

5 8.4 The two dimensional potential flow around a circular cylinder is governed by superposition of a −1 doublet at the origin (ψ1 = −λr sin θ) and a parallel flow (ψ2 = U∞r sin θ). Determine −1 ∂ψ − ∂ψ (a) the velocity field (vr = r ∂θ and vθ = ∂r ), (b) the radius a, (c) the pressure distribution p−p∞ along the surface of the cylinder, expressed as a pressure coefficient Cp = 1 2 , where p∞ is the 2 ρ U∞ 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 a line vortex (steady, incompressible flow in a plane). Hint: 1 ∂ 1 ∂v ζ = (rv ) − r z r ∂r θ r ∂θ 1 ∂ 1 ∂v ∇ · V = (rv ) + θ r ∂r r r ∂θ

(b) The flow field in the outer parts of a tornado can be approximated as a line vortex, the inner parts as like a solid-body rotation. Use these approximations to derive simplified analytical equations for the velocity and pressure field within a tornado. Sketch the variations of velocity and pressure with the radial distance r from the center of the tornado. Radial momentum balance ⇒ 2 ρvθ /r = dp/dr. 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 polar coordinates is Γ v = −2 U∞ sin θ + θ 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 −ρ U∞Γ. ∫ 2π 2 Note, 0 (sin θ) dθ = π. (b) Describe, using streamlines, how the velocity field around the cylinder looks like for different values 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 planar flow. Define quantities and illustrate with a simple sketch. 8.8 A two-dimensional (AR → ∞), symmetric and thin wing is suddenly brought into a constant velocity. The angle of attack is small but not zero. Describe the flow development with special emphasis on the flow at the trailing edge and the generation of a circulation and thus a lift force on the wing. (See Fig. 7.22). 8.9 (a) State the Kutta condition for an airfoil. How can this condition be motivated physically?

(b) Write out the mathematical expression of the lift coefficient CL valid for a slender and slightly cambered two-dimensional wing profile at small angles of attack. The maximum camber is h ≪ C, where C is the wing chord. Illustrate schematically. 8.10 (a) Discuss shortly and illustrate with figures how the lift force on a finite-span wing gives rise to an induced drag.

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

(c) Illustrate the influence of AR on both CD and CL as a function of the angle of attack α. 8.11 Describe what is meant by the concept of added mass (hydrodynamic mass). How can the added mass be calculated with a known potential flow around the body? State without derivation the added mass for potential flow around a sphere.

6 Chapter 9 —

9.1 Use conservation laws of mass and momentum to derive an expression for the velocity C of a finite 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 approximations are applicable. After this, specialize to a vanishing small amplitude wave amplitude assuming adiabatic conditions (velocity of sound).

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

(b) Assuming a perfect gas, derive a relationship between stagnation temperature T0, static temperature T , k = cp/cv and Ma.

(c) Again assuming a perfect gas, derive a relationship between stagnation pressure p0, static pressure p, k = cp/cv and Mach number Ma. 9.3 (a) Assuming one-dimensional adiabatic flow of a perfect gas, write out a complete set of equations which governs the thermodynamic state immediately downstream of a stationary normal-shock wave in a nozzle. (b) How do the stagnation temperature, Mach number, pressure, temperature, density, entropy and 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 constant pressure 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 as adiabatic and frictionless. (a) Sketch the variation of pressure through the nozzle (and just outside of it) for different values of 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 flow strives towards sonic conditions, Ma = 1, regardless of the upstream (inlet) Mach number Ma1. ∗ There is a certain pipe length L (Ma1) for which the exit Mach number will be exactly unity. 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 conditions might be altered.

9.6 (a) Describe geometrically the deflection process associated with an oblique shock in a plane. In particular, 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 conserved after the passage of the oblique shock. (c) Make a sketch on the relation between the deflection angle and the shock wave angle with Mach number Ma1 as parameter. Mark out regions for weak and strong shocks, normal shock waves, and Mach waves. How does the Mach number affect the maximum deflection angle? Also mark out the line where Ma2 = 1. 9.7 (a) Describe the flow pattern around a thin and pointy wing profile in supersonic two-dimensional flow at a small angle of attack. The profile has a prismatic cross-section (planar surfaces). (b) The following formula applies to supersonic two-dimensional flows at small angles θ: p − p kMa2 2 1 = √ 1 θ, p1 2 − Ma1 1

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 flat plate at small angles of attack. The upstream Mach number is Ma∞. The width of the plate (b) is much larger than its chord C.

7 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 open rectangular channel of width b. The undisturbed shallow-water depth is y. After this, specialize to δy/y → 0.

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

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

(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 cross section (and depth) as well as the inner surface condition are constant. √ ≃ 1/6 According to Manning (1891), 8g/f Rh /n, 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. Starting point: the extended Bernoulli equation (the energy equation). Velocity variations across the cross section can be neglected.

10.4 For open-channel flow in a rectangular channel, derive an expression for the critical depth yc at 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 on the 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 and frictionless. 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 Froude numbers (Fr1 < 1 and Fr1 > 1). Illustrate with a diagram (depth y vs. specific energy E).

Christoffer Norberg, tel. 046-2228606

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