Department of Energy Sciences, LTH

C. Norberg, 20/01/2012

MMV211

Fluid Mechanics

LABORATION 1 Flow around

Bodies

OBJECTIVES

(1) To understand how body shape and surface finish influence the flow-related forces

(2) To understand how scaling experiments can be used to determine the forces on a full-

scale, real body

(3) To understand the basic mechanisms of flow separation

(4) To investigate pressure distributions around a circular cylinder and a wing profile

SUMMARY

The flow-related force vector acting on an immersed body can be divided into three named

components, a drag (drag force), which acts in the flow direction, a lift (lift force) and a side

force, all perpendicular to each other. The lift usually is in the direction so that it does a useful

job, for instance upwards for an airplane in horizontal flight or downwards for inverted wings

on race cars. In many cases the (time-mean) side force is zero, for instance when there is flow

symmetry about the plane of lift and drag, as for an airplane flying in still air. Further, the

components can be divided up with respect to their origin, wall surface pressure and wall

friction. The pressure component of the drag, the pressure drag, is often referred to as the form

drag since it is strongly dependent on the body form (shape). The remaining part is the friction

drag, which is due to shearing viscous forces along the body surface. Flow similarity laws are

crucial for model testing experiments. For instance, the Reynolds similarity law says that for

incompressible flow about two geometrically similar bodies, without any effects of free

surfaces, the flow itself is similar, if tested at the same Reynolds number.

The lab session is based on three tests, in air flow.

Test 1: Drag measurements

Determine the drag for some different axisymmetric bodies and two circular cylinders.

Test 2: Measurements of forces on an airfoil

Determine lift and drag for an airfoil model at different angles of attack.

Test 3: Pressure measurements

Measure the distribution of static wall pressure around a circular cylinder in cross flow and an

airfoil model, respectively. Also determine the form drag of the cylinder.

PREREQUISITES

Read this PM and Ch. 7.1, 7.5, and 7.6 in F. M. White, Fluid Mechanics1.

REPORT

Each student should account for all measurement data and results in the handed-out lab

protocol, the report, which is to be finalized during the lab session.

Time in lab: approx. 4 hours

1 Further references are to 7th edition in SI Units (2011).

2

Forces on an immersed body

A solid body that is exposed to flow of a viscous fluid is also affected by a flow-induced

force. This force is the resultant of the normal and shear forces that acts locally on the body

surface. The normal surface force is completely dominated by pressure action, the shear forces

are due to viscous stresses only, and are normally referred to as friction forces. Consider a

wing-like body of large width in relation to its chord, see Fig. 1. The flow around this body

can be considered to be essentially two-dimensional.

Figure 1: Forces on an immersed body, two-dimensional flow.

The force vector F is divided into two components, the drag D, which acts in the flow

direction, and the lift L, which acts normal to the flow. The drag itself is divided into pressure

or form drag and friction drag. Form drag is due to the pressure forces acting on the surface

(static pressure p); the friction drag is due the surface viscous shearing (frictional) forces (wall

shear stress τ). The absolute value of F is normally expressed using a dimensionless

coefficient C, defined as

AV

CF2

2 (1)

V - oncoming, free stream velocity

A - characteristic area

- density of fluid

For geometrically similar bodies and for incompressible flows without influences of free

surface effects we have the Reynolds similarity law:

Flows around geometrically similar bodies are similar for equal Reynolds numbers

The Reynolds number, Re, is dimensionless and is defined as

Re

VV (2)

V - characteristic velocity, e.g. as above

- characteristic length

- dynamic (or absolute) viscosity, - kinematic viscosity ( / )

For a certain geometrical flow situation, the Reynolds similarity law (or rule) means that the

dimensionless coefficient C only is a function of the Reynolds number and a dimensionless

3

time (e.g. /tV ), but only if the prerequisites are fulfilled. Thus, the following applies for the

time-averaged force F in a stationary flow situation:

(Re)fC (3)

If the a flow force on a body is to be determined from experiments, the Reynolds similarity

law means that it is not necessary to use the same velocity, body size, fluid medium, fluid

temperature, etc. as in the actual flow situation. If only the Reynolds number is the same, or if

the Reynolds number is within an interval where C is a constant, the force from the scaled

(model) situation can be transferred directly to determine the force on the actual body (full-

scale, prototype). For instance, it is possible to transfer results from experiments in air to full-

scale water conditions, but only if Reynolds number are equal.

Drag

The drag component caused by normal forces on the body, the form drag p

D , and the friction

drag due to tangential forces f

D , sum up to the total drag,

fp DDD (4)

Consider Fig. 1 where is the angle between the surface normal n, at the surface element dA,

and the oncoming free stream velocity (of magnitude V). The pressure drag pD then is

cosA

p dApD A

dAzgp cos)(

*

cos*Re)*,*,*,(* 22

A

dAzyxpV

*

cos*Re)*,*,*,(*22

A

dAzyxpV (Re)122 fV

Dimensionless quantities are denoted with a *, e.g. /* xx . Now introduce the projected

area of the body in the free stream direction, or any other characteristic area of the body, and

denote it by A. Since this area is proportional to 2 , it follows that

2

2

,

VACD pDp

, where pDC , , the pressure drag coefficient, only depends on Re.

The total friction drag becomes

sindAD f

sindA

n

Vt

sin*

*2 dA

n

VV t

(Re)(Re)(Re) 322

222

2 fVfV

VfV

4

Thus, 2

2

,

VACD fDf

, where the friction drag coefficient fDC , only depends on Re.

Since pDfDD CCC ,, we then obtain for the total drag:

2

2VACD D

where the (total) drag coefficient DC only depends on Re, (Re)DC .

Lift

The total lift force is determined in a similar fashion,

2

2VACL L

where the lift coefficient LC only depends on Re, (Re)LC .

Force components measured in a model scale experiment can thus be transferred to any other

scale, if there is full geometrical similarity and equal Reynolds numbers (and of course,

stationary, incompressible flow without any influence of free surface, e.g., wave interactions).

The choice of characteristic area and length

For geometrical similar bodies the characteristic length and the characteristic area A can be

chosen arbitrarily. However, as a general rule for “bluff bodies”, bodies that under normal

conditions give rise to large separated wake regions (for instance a sphere, ordinary cars, etc.),

the characteristic area is normally chosen as the projected area of the body when viewing it

from the upstream, along the free stream velocity; the so-called frontal area. As a

characteristic length it is customary to choose a typical cross-stream dimension within the

frontal area. For cylinders of circular cross-section the characteristic length is chosen as the

diameter d, the frontal, characteristic area then is dbA , where b is the cylinder length. Also

for spheres the characteristic length is chosen as the diameter ( 4/2dA ). For slender

bodies of the wing type (“wing profiles”) the characteristic length is chosen as the mean

chord, ( c ), see Fig. 2. The so-called planform area (projected wing area) is used as

characteristic area, cbAA p , where b is the wing span.

Figure 2: Wing profile at angle of attack .

5

Symmetrical bodies

In the mean sense, a fully symmetric body, e.g. a sphere, only has a force component in the

flow direction, the drag D. According to eq. (1) this drag corresponds to a drag coefficient

DC , which under the conditions of Reynolds similarity law only is a function of the Reynolds

number (Re). The variations of DC with Re, for the flow around a sphere and a long circular

cylinder in cross-flow, are shown in Fig. 3 (smooth surfaces).2

At very low Reynolds numbers for the sphere the drag coefficient in Fig. 3a approaches a

straight line. For low enough Reynolds numbers, see eq. (7.64) in White, the drag can be

determined from the Stokes formula,

VdD 3 (5)

According to eq. (1) the drag coefficient becomes

Re

24DC

which means a straight line in logarithmic diagram (dotted line in Fig. 3a).

Within 52 103Re105 the drag coefficient for a sphere is approximately constant

( 07.044.0 DC ). Thus the drag within this interval is approximately proportional to the

velocity squared, 2VD . At low Re (approx. Re < 1), from Stokes formula, it is

proportional to the velocity, VD .

Figure 3a: Drag coefficient for a smooth sphere (Fox & McDonald 1994).

2 At high Re the surface roughness has a significant influence on CD, see Fig. 5.3 and D5.2 in White.

6

Figur 3b: Drag coefficient for a long and smooth circular cylinder in cross-flow.

We recall that the total drag can be divided into form (pressure) drag and friction (viscous

shear) drag. The Reynolds number is a measure of the ratio between inertia forces (mass times

acceleration) and viscous forces in the flow field. This means that the viscous forces at large

Re become very small in relation to the pressure forces, with pressure forces approximately

balancing the inertial forces. Consequently, at high enough Re the form drag dominates, and it

can be determined solely from the pressure field acting on the body surface. Fig. 3b shows that

the form drag dominates the total drag for the circular cylinder as from about Re = 3000. For

the cylinder, the friction drag coefficient fDC , can be approximated as,

Re/5.3, fDC (6)

For the circular cylinder and within 52 103Re103 , DC = 1.08 ± 0.15. At Re = 300 the

friction part is about 16% of the total drag, at 5103Re it is only about 0.5%.

At 5103Re , for both the sphere and the cylinder and with increasing Re, there is a

dramatic decrease in the drag coefficient. In fact, over a short interval, the total drag actually

diminishes with increasing velocity. The phenomenon is called drag crisis, and it depends on

a change of character for the boundary layer, which influences the separation process; see

pages 497-499 in White. At higher Re, for 6104Re (approx.), the drag coefficient for the

cylinder again settles to an approximate constant level, 1.06.0 DC , see Fig. 3b.

7

Figure 4: Pressure distribution round a circular cylinder in cross-flow (410Re ).

Fig. 4 shows a schematic of the pressure distribution along the stagnation streamline, for the

circular cylinder in cross-flow at relatively high Reynolds numbers (where the drag coefficient

is approximately constant, 1DC ). The pressure difference p is referred to the undisturbed

pressure far upstream, in this case the atmospheric pressure, atmppp . Due to

deceleration, there is a pressure rise in front of the cylinder. At high Re, the viscous forces in

this region are negligible; the pressure rise at the stagnation point then is equal to the dynamic

pressure 2/2V . Due to acceleration the pressure decreases on the frontal side. Boundary-

layer separation occurs at 80 ; the pressure in the separated (wake) region is

approximately constant. Downstream of the cylinder the pressure eventually recovers to the

undisturbed value. The pressure distribution around the cylinder surface will be studied

further during the laboratory exercise.

Wing profiles (aerofoils)

Under normal circumstances a wing is subjected not only to a drag but also a lift. Consider the

flow around a wide-span wing (a wing profile or aerofoil) exposed to a constant velocity

(constant Reynolds number, assumed high), at varying angles of attack , see Fig. 2. Since

is part of the geometry both DC and LC will depend on . Starting at small , the drag

coefficient is approximately constant but then starts to increase; the lift coefficient increases

rapidly, almost linear with . The lift coefficient reaches a maximum at a critical angle of

attack, c ; beyond this angle there is a rapid decrease in LC while there is a continuing

increase of DC . The occurrence of lift can be explained as follows. At small angles of attack

the flow is attached to wing almost to the trailing edge. Due to this viscous adhesive capacity

there is a net deflection of the flow downwards3, which according to the laws of Newton

means that there is an opposing force upwards on the wing. The wing profile acts like a

turning vane.

On the frontal part of the upper side of the wing there is a compression of streamlines, the

velocity increases and this means, at high enough Re, that the pressure is lowered (cf.

Bernoulli equation). The upper side is thus often called the suction side. On the lower side the

streamlines diverge somewhat and the pressure increases slightly (pressure side). At high Re

the occurrence of lift is thus due mainly to the pressure difference between the upper and

lower side. On the upper side, however, the velocity eventually decreases and the pressure

3 Also the upstream flow is affected, at the wing tip flow is deflected upwards and this also contributes to lift.

8

thus starts to increase. This is an adverse pressure gradient and it means that there is a

potential for flow separation, and for high enough angles of attack this certainly will happen,

starting close to the trailing edge. This is about the point of maximum lift ( c ). A slight

increase in then will move the position of flow separation rapidly upstream, the flow

deflection diminishes significantly and there is a rapid drop in lift. Eventually the separation is

so massive that the flow is barely deflected at all, the force is then dominated by (pressure)

drag; see Fig. 5 and Fig. 7.24 in White. Beyond the critical angle c the wing is said to be

stalled; c is often called the stall angle. The phenomenon of stall is of great importance

when landing an airplane, as it is then crucial to have high enough lift with much drag, at a

velocity as low as possible. If stall occurs the loss in lift might result in a too steep approach

of the landing-ground.

Figure 5: Polar diagram for a wing with 5/ cb (Finnemore & Franzini 2002).

The angle of attack for which the ratio between lift and drag is a maximum is called the (best)

gliding angle ( g ). The angle of attack at cruising conditions is normally set at around this

value. For a glider in still air, maximum sailing distance will occur at this angle of attack. For

the cambered wing in Fig. 5, 1g .

End effects, wing-tip vortices

All real wings (and cylinders) have limited width (length). Naturally, there will be flow

distortions near the ends, end effects. If the width is high enough the end effects will be of

minor importance, sometimes even neglected. At limited widths, the end effects can be

reduced by using end plates, thin plates mounted at the ends, along the flow and of special

design. For real finite-width wings there will be an overflow from the pressure (lower) side of

9

the wing to the suction (upper) side. At the wing tips the pressure is equalized (zero lift). The

overflow results in so-called wing-tip vortices, which reduces the lift. It also increases the

drag. Since this drag is due to lift it is usually called lift-induced drag. For more details, see

Ch. 8.7 in White.

Experimental equipment

Wind tunnel

During scientific measurements it is required to have a flow that is highly uniform, uni-

directional and non-turbulent. To achieve this in air it is usually recommended to use a closed-

return wind tunnel, with a plenum chamber with several gauze screens followed by a well-

designed contraction before the test section, driven by low-noise, highly effective fan unit.

However, such wind tunnels are very expensive and require much space. During the lab

exercise we will instead use two small (and a bit noisy!), open wind tunnels (fan units), which

are quite sufficient for our purposes. The wind tunnels used in this lab basically consist of an

axial fan within a circular duct followed by a short contraction (nozzle). One unit has a

guiding vane ring that will dampen the flow distortions due to the rotating fan blades. This

unit also has exchangeable exit nozzles (of different outlet diameter).

Prandtl tube (Pitot-static tube)

The free stream velocity V is measured by using a so-called Prandtl tube, also called Pitot-

static tube, see Ch. 6.12 in White. Along a streamline in stationary, incompressible and

frictionless flow the Bernoulli equation applies. Along a horizontal streamline or a streamline

where effects of gravitation are negligible, it reads

.2

0

2

constpV

p

(7)

where p is static pressure, the fluid density and V is the local velocity. The combination

2/2V is called the dynamic pressure, the difference between the pressure in a zero-velocity

state (stagnation pressure 0p ) and the flowing state along the same streamline. At the tip of

the Prandtl tube there is a pressure hole; it also has pressure holes along the perimeter further

downstream. When pointing towards the flow, the frontal tip will become a stagnation point

( 0V , tpp ). At the position for the perimeter holes, and by careful design, the pressure

has recovered exactly to the undisturbed value upstream, the free stream pressure p . If

friction effects can be neglected, as they indeed can for high enough Reynolds numbers, the

pressure difference between these two positions is equal to the dynamic pressure, since

0t pp by eq. (7). The undisturbed velocity along the stagnation streamline then is

)(2 0 ppV

(8)

The air density can be calculated from the ideal gas law,

RT

p , where K)J/(kg287R (9)

10

Pressure and force measurements

Measurements of pressure differences will be carried out using a differential liquid U-tube

manometer, with readings directly in pascal (Pa). A more detailed description and how to use

it will be given by the instructor. Drag and lift components are measured using a two-

component force balance, to be described by the instructor.

Outline

The laboratory session consists of three parts:

1. Measurements of Drag (axisymmetric bodies)

(a) Measure the drag D on a set of axisymmetric bodies (Fig. 6), at a certain air velocity.

Figure 6: Axisymmetric bodies.

(b) Measure the drag D on two circular cylinders in cross-flow, for two or three velocities.

Analysis

(1) Compute Re and CD for all cases

(2) Compare the results with Fig. 3 in this PM, and Table 7.3 in White

2. Measurements of drag and lift (wing profile)

For a certain velocity, measure drag D and lift L for a wing profile at different angles of

attack, e.g., = -5o, -2

o, 0

o, 2

o, 5

o, 10

o, 15

o, 20

o, 25

o.

Analysis

(1) Compute the Reynolds number, CD, CL and the lift-to-drag ratio DL CCDL // . Why is

DL / called the glide ratio? (think about unpowered descent of a plane in still air) What

is the gliding angle g ?

(2) Plot CD and CL as a function (same diagram). What is the approximate stall angle?

What happens at stall?

11

3. Pressure measurements (circular cylinder and wing profile)

(a) Pressure distribution around a circular cylinder

The surface pressure is measured through a small drilled hole in the surface. Measure the

pressure difference p between the surface pressure (at mid-span) and the ambient pressure

at various angles from the scale, e.g., -20o, -10

o, 0

o, 10

o, 20

o, 40

o, .. 80

o, 90

o, 100

o, 120

o, ... ,

180o, -20

o. The angular position for the actual stagnation point (where 0 ) may not be

identical to the scale value. Actual angles can be determined from the (assumed) symmetry

condition. The pressure difference p is measured using a differential micro-manometer.

Analysis

The component that contributes to the pressure drag is cosp . Since 0 is the

stagnation point, 2/20 Vp (Bernoulli equation, high Re).

The form drag pD per unit width is determined from the integration around the perimeter,

2

0

cos dRpDp

Because of symmetry it is enough to integrate over half the perimeter and multiply with 2,

d

p

pRVdpRDp

0 0 0

2 coscos2 (10)

The form drag coefficient is

d

p

p

RV

DC

ppD

0 02,

cos

22

(11)

(b) Pressure measurements along a wing profile

Pressure differences xp between the wall static pressure at different fixed positions and the

ambient pressure are measured; see Fig. 7. The pressure tap at 0x is assumed to be at the

stagnation point (limited angles of attack).

Figure 7: Pressure holes on the wing profile.

12

Analysis

Compute Reynolds numbers for both the cylinder and the wing profile.

(a) Circular cylinder

(1) Plot 0/)( ppCp and coscos)/()( 0 pCppf in the same

diagram. Which part of the cylinder contributes most to the drag, frontal side or rear side?

(2) Determine the pressure drag coefficient, pDC , , graphically.

(3) Estimate the total drag coefficient, fDpDD CCC ,, , and compare with results from

previous total drag measurements and Fig. 3b in this PM.

(b) Wing profile

(1) Plot 10 //)( ppppxf xxx , where x is the abscissa for the projected holes; see

Fig. 7. Which side contributes most to the lift, the upper side or the lower side? On which

surface is there a risk for flow separation? Why?

(2) Estimate the lift coefficient LC

REFERENCES

Finnemore, E. J. & Franzini, J. B. (2002), Fluid Mechanics (with Engineering

Applications), Tenth Edition, McGraw-Hill.

Fox, R. W. & McDonald, A. T. (1994), Introduction to Fluid Mechanics, Fourth

Edition, John Wiley & Sons, Inc.

White, F. M. (2011), Fluid Mechanics, Seventh Edition in SI Units, McGraw-Hill.