Hello! You are consulting an examination paper from the archivesat https://�uidmech.ninja/.In the winter semester 2020-2021, the general structure of theexamination will be largely the same as in this archive. Neverthe-less, because the course content progressively changes from yearto year, there are a few di�erences. In former years,

• The course contained a chapter about compressible air �ow(involving tables for air properties) that is no longer part ofthe course now;

• Conversely, several chapters have been added to the exam-inable content over the years;

• The course contained a duct �ow problem involving a ballfountain (“Kugel fountain”) that is no longer part of thecourse now;

• Viscosity values were read in a di�erent diagram, and maynot match values read in the 2020 viscosity diagram;

• Many small updates in the notation had not yet been carriedout.

To obtain precise information about the next examination, consultthe dedicated appendix in the lecture notes. If you have questions,contact me as detailed in the course syllabus. Thanks, and goodluck in your revisions!Olivier CleynenDecember 2020

https://fluidmech.ninja/

Fluid Mechanics exam — May 5, 2016

Fluid Mechanics for Masters Students

Solve exercise 1, plus three other exercises among exercises 2 to 6.

Duration: 2 h – Use of calculator is authorized; documents are not authorized.

Except otherwise indicated, we assume that Wuids are Newtonian, and that:

ρwater = 1 000 kg m−3; patm. = 1 bar; ρatm. = 1,225 kg m−3; Tatm. = 11,3 ◦C;

µatm. = 1,5 · 10−5 N s m−2; д = 9,81 m s−2. Air is modeled as a perfect gas (Rair = 287 J K−1 kg−1;

γair = 1,4; cpair = 1 005 J kg−1 K−1).

Reynolds Transport Theorem:

dBsysdt

=ddt

$CVρb dV +

"CSρb (~Vrel · ~n) dA (1)

Mass conservation:

dmsysdt

= 0 =ddt

$CVρ dV +

"CSρ (~Vrel · ~n) dA (2)

Change in linear momentum:

d(m~Vsys )

dt= ~Fnet =

ddt

$CVρ~V dV +

"CSρ~V (~Vrel · ~n) dA (3)

Change in angular momentum:

d(~rXm ∧m~V )sysdt

= ~Mnet,X =ddt

$CV

~rXm∧ρ~V dV+"

CS~rXm∧ρ (~Vrel ·~n)~V dA (4)

Continuity equation:1ρ

DρDt+ ~∇ · ~V = 0 (5)

Navier-Stokes equation for incompressible Wow:

ρD~VDt= ρ~д − ~∇p + µ~∇2~V (6)

[St]∂~V ∗

∂t∗+ [1] ~V ∗ · ~∇∗~V ∗ = 1

[Fr]2~д∗ − [Eu] ~∇∗p∗ + 1

[Re]~∇∗2~V ∗ (7)

in which [St] ≡ f LV , [Eu] ≡p0−p∞ρ V 2 , [Fr] ≡

V√д L

and [Re] ≡ ρ V Lµ .

1

In a highly-viscous (creeping) steady Wow, the drag FD exerted on a spherical body

of diameter D at by Wow at velocity V∞ is quantiVed as:

FDsphere = 3πµV∞D (8)

In cylindrical pipe Wow, we accept the Wow is always laminar for [Re]D . 2 300, and

always turbulent for [Re]D & 4 000. The Darcy friction factor f is deVned as:

f ≡ |∆p |LD

12ρV

2av.

(9)

A pump or turbine subjected to a pressure diUerence ∆p and a volume Wow V̇ has apower expressed by:

Ẇ = |∆p |V̇ (10)

Figure 1 quantiVes the viscosity of various Wuids as a function of temperature, and

Vgure 2 (Moody diagram) quantiVes losses in cylindrical pipes.

In boundary layer Wow, we accept that transition occurs at [Re]x ≈ 5 · 105.The shear coeXcient c f , a function of distance x , is deVned based on the free-stream

Wow velocity U :

c f (x ) ≡τwall12ρU

2(11)

Exact solutions to the laminar boundary layer along a smooth surface yield:

δ

x=

4,91√[Re]x

δ ∗

x=

1,72√[Re]x

(12)

θ

x=

0,664√[Re]x

c f (x ) =0,664√[Re]x

(13)

Solutions to the turbulent boundary layer along a smooth surface yield the following

time-averaged characteristics:

δ

x≈ 0,16

[Re]17x

δ ∗

x≈ 0,02

[Re]17x

(14)

θ

x≈ 0,016

[Re]17x

c f (x ) ≈0,027

[Re]17x

(15)

2

The speed of sound a in a perfect gas is a local property expressed as:

a =√γRT (16)

The total properties (subscript 0) of a perfect gas are expressed as:

T0 ≡ T +1cp

12V 2

p0ρ0= RT0 (17)

In isentropic, one-dimensional Wow, we accept that the mass Wow ṁ is quantiVed as:

ṁ = ρ VA = A[Ma]p0

√γ

RT0

1 +

(γ − 1)[Ma]2

2

−γ−12(γ−1)

(18)

This expression reaches a maximum ṁmax when the Wow is choked:

ṁmax =

[2

γ + 1

] γ+12(γ−1)

A∗p0

√γ

RT0(19)

The properties of air Wowing through a converging-diverging nozzle are described

in Vgure 3, and Vgure 4 describes air property changes through a perpendicular

shockwave. You may approximate your table readings to those of the line with the

nearest value.

Figure 1 – Viscosity of various Wuids at a pressure of 1 bar (in practice viscosity is almost independent ofpressure).

Figure © White, 2011, Fluid Mechanics, 7th ed. pub. McGraw-Hill

3

Figure 2 – A Moody diagram, which presents values for the friction factor f measured experimentally, asa function of the diameter-based Reynolds number [Re]D , for diUerent relative roughness values. In thisVgure, the pipe diameter is noted d .

Diagram CC-by-sa S Beck and R Collins, University of SheXeld4

https://commons.wikimedia.org/wiki/file:Moody diagram.jpghttps://creativecommons.org/licenses/by-sa/3.0/deed.en

Figure 3 – Properties of air (modeled as a perfect gas) as it expands through a converging-diverging nozzle.In this Vgure, the Mach number is noted “Ma”, and the perfect gas parameter γ is noted k . Data alsoincludes the parameter Ma∗ ≡ V /a∗ (speed non-dimensionalized relative to the speed of sound at thethroat).

Figure © Çengel & Cimbala 2010, Fluid Mechanics, 2nd ed., pub. McGraw-Hill

5

Figure 4 – Properties of air (modeled as a perfect gas) as it passes through a perpendicular shockwave.In this Vgure, the Mach number is noted “Ma”, and the perfect gas parameter γ is noted k .

Figure © Çengel & Cimbala 2010, Fluid Mechanics, 2nd ed., pub. McGraw-Hill

6

Solve problem 1,

and three other problems among problems 2 to 6.

7

1 Governing equations

1.1. Write out the Navier-Stokes equation for incompressible Wow in its fully-developed

form in three Cartesian coordinates.

1.2. In which conditions does the continuity equation (eq. 5) apply?

2 Water canal access door

A water canal used in a laboratory is Vlled with stationary water (Vg. 5). An observation

window is installed on one of the walls of the canal, to enable observation and measure-

ments. The window is hinged on its bottom face.

The hinge stands 1,2 m below the water surface. The window has a length of 0,8 m and

a width of 1,5 m. The walls of the canal are inclined with an angle θ = 60° relative to

horizontal.

Figure 5 – A door installed on the wall of a water canal. The water in the canal is perfectly still.

2.1. Represent graphically the pressure of the water and atmosphere on each side of

the window.

2.2. What is the moment exerted by the pressure of the water about the axis of the

window hinge?

2.3. If the same door was positioned at the same depth, but the angle θ was decreased,

would the moment be modiVed? (brieWy justify your answer)

8

3 Drag measurements in a wind tunnel

A wing is positioned from wall to wall across a wind tunnel in which the air is Wowing at

moderate speed. An experimenter proceeds with velocity measurements. The objective

is to measure the drag applying on the wing proVle.

Figure 6 – Wing proVle positioned across a wind tunnel. The distribution of the horizontal velocity u isrepresented as measured, upstream and downstream of the proVle. The width of the tunnel and proVle(towards the viewer in this Vgure) is 45 cm.

Upstream of the proVle, the air Wow velocity is uniform (u1 = U = 40 m s−1).

Downstream of the proVle, horizontal velocity measurements are made every 3 cm across

the Wow; the following results are obtained:

vertical position (cm) horizontal speed u2 (m s−1)

0 40

3 40

6 39

9 38

12 35

15 31

18 29

21 30

24 34

27 37

30 38

33 40

36 40

9

The span of the wing (width perpendicular to the Wow) is 45 cm. The airWow is incom-

pressible (ρair = 1,225 kg m−3) and the pressure is uniform across the measurement

area.

3.1. What is the drag applying on the proVle?

3.2. What is the mechanical power lost by the air because of the drag force? In which

energy form is this power transformed?

4 Power requirements in a water piping system

A long steel pipe is installed to carry water from one large reservoir to another (Vg. 7).

The total length of the pipe is 19 km, its diameter is 0,6 m, and its surface roughness

is ϵ = 0,5 mm. It must climb over a hill, so that the altitude changes along with distance.

Figure 7 – Layout of the water pipe. For clarity, the vertical scale is greatly exaggerated. The diameter ofthe pipe is also exaggerated.

The pump must be powerful enough to push 0,9 m3 s−1 of water at 20 ◦C.

Figure 1 quantiVes the viscosity of various Wuids, and Vg. 2 quantiVes losses in cylindrical

pipes.

4.1. Will the Wow in the water pipe be turbulent?

4.2. What is the pressure drop generated by the water Wow?

4.3. What is the pumping power required to meet the design requirements?

4.4. What would be the power required to carry the same volume Wow in the other

direction?

10

5 Flight of a dragonWy

A dragonWy (sketched in Vg. 8) has a 9 cm wingspan, a mass of 70 mg, and cruises

at 3 m s−1, beating its four wings 22 times per second.

Figure 8 – Plan view of a dragonWy

A team of researchers wishes to investigate the mechanics of the air Wow around the

wings of the dragonWy in Wight. For this, they wish to build a motorized mechanical

model, and make measurements around it in a wind tunnel. In the wind tunnel, the

atmospheric conditions are ambient: Tatm. = 10 ◦C, ρatm. = 1,225 kg m−3.

Two possible models are being considered for the study: one with wingspan 45 cm, and

one with wingspan 60 cm.

5.1. How much more wing surface area and weight would the larger model have,

relative to the small model?

5.2. Which Wow velocity in the wind tunnel would be required for each of the two

models, so that the boundary layer over the wings of the dragonWy is adequately

reproduced on the model?

After careful consideration, the team opts for the model with wingspan 60 cm.

5.3. At which frequency should the model wings beat so that dynamic similarity is

maintained between reality and experiment?

5.4. What will be the lift force developed by the model during the experiment?

5.5. How much mechanical power will the model require, compared to the real dragonWy?

5.6. If the experiment in the wind tunnel was later run on a very hot summer day,

how would you adjust the experiment parameters? (provide a brief qualitative

explanation, e.g. in 30 words or less).

11

6 Disk viscometer

An instrument designed to measure the viscosity of Wuids consists of a disk rotating a

small distance above a large Wat stationary surface (Vg. 9). A Newtonian Wuid is present

between the disk and the bottom Wat surface; and the moment generated by viscous

eUects is measured to deduce the viscosity of the Wuid.

Figure 9 – Sketch of a disk viscometer. The width of the gap has been exaggerated for clarity.

The disk is rotated slowly, so that the Wow pattern between disk and the bottom surface

is entirely laminar (steady and with a smooth velocity distribution).

The diameter of the disk is 20 cm, and the gap height is 0,5 mm. When the disk rotates

at 3 rpm, the moment about the shaft axis is measured at 1,2 · 10−3 N m.

6.1. What is the viscosity of the Wuid?

6.2. Would an non-Newtonian Wuid induce a higher moment? (brieWy justify your

answer)

12

1 Governing equations2 Water canal access door3 Drag measurements in a wind tunnel4 Power requirements in a water piping system5 Flight of a dragonfly6 Disk viscometer