experimental methods in fluid dynamics · experimental methods in fluid dynamics aim this course...

G F Lane-Serff 1 15/9/05

Experimental Methods in Fluid Dynamics Aim This course serves as a practical introduction to fluid flows. The main aim of this module is to enable students to directly experience and observe a range of fluid flows and to be able to make and interpret observations and measurements of these flows. Module structure The module consists of a series of lectures and laboratory sessions. The lectures cover some background information about the underlying fluid dynamics in addition to specific information about the individual laboratory experiments. The laboratory work will be done in small groups, and learning to work in these groups is one of the objects of the module. However, students are expected to make their own records of observations and measurements from the experiments and to produce independent laboratory reports. The module runs over both semesters, concentrated at the start of each semester.

A Introduction to fluid flows A1 Introduction A2 Equations of motion A3 Laboratory experiments

B Basic principles in fluid dynamics B1 Reynolds number and flow types B2 Pressure and energy (Bernoulli's equation) B3 Control volumes and the momentum principle B4 Flow past bluff and streamlined bodies B5 Open channel flows and hydraulic jumps

C Visualisation experiments C1 Smoke tunnel (Experiment 1A) C2 Water channel (Experiment 1B)

D Momentum and drag experiments D1 Wind tunnel (Experiment 2) D2 Hydraulic jump (Experiment 3)

E Flow/structure interactions E1 Natural and forced oscillations E2 Oscillations of a simple structure (Experiment 4)

F Waves F1 Water wave theory F2 Wave tank (Experiment 5)

G Turbulent jets G1 Turbulent jets G2 Turbulent air jet (Experiment 6)

Assessment Students will be expected to submit detailed reports on the laboratory experiments. Students will use a laboratory notebook to make records during the laboratory experiments and this notebook will also need to be submitted. The assessment for this module will be based entirely on the submitted laboratory reports and the laboratory notebook. Module coordinator: Dr G F Lane-Serff (P/B20, extn 64602, [email protected])


A Introduction to fluid flows

A1 Introduction Much of the underlying fluid dynamics relating to the experiments is covered in other modules, so will not be repeated in detail here. In particular, a more formal, mathematical derivation of the equations of motion is given in the module "Basic Fluid Mechanics" (though that module deals with inviscid theory only). In this module we will give a more physical introduction to fluid dynamics (and include the effects of viscosity). Fluid deforms continuously ("flows") under the action of a force (solids deform a finite amount). Gases and liquids are fluids. Gases are more compressible and their thermodynamic behaviour is easier to calculate (kinetic theory). The properties of liquids are generally more complex (especially water). Continuum Real fluids are made up of discrete molecules but it is easier mathematically to treat the fluid as if it had a continuous structure. This allows us to attach a definite meaning to the notion of values and derivatives "at a point." In practice we mean an average over a volume small compared with the scales of the fluid flow but large compared with the molecular separation. Density of a fluid is the mass per unit volume, usually denote by ρ, SI units: kg m-3. The density may vary with position, though we often deal with fluids of uniform density (in which we can regard ρ as a constant). Pressure is a scalar function of position and time, force per unit area, SI units: N m-2 or Pa.

p(x, t) Velocity of a fluid is a vector function of position and time, SI units: m s-1.

u(x, t) = (u, v, w) or (ux, uy, uz) where x = (x, y, z). Typically, velocity is measured by an instrument placed at a particular position x in the flow. For example: hot wire probes, pitot tubes, anemometers, current meters, LDA, acoustic Doppler. This way of thinking of the velocity as a function of position is called Eulerian representation. The disadvantage of the Eulerian representation is that you're not measuring the same piece of fluid. Alternatively, flow can be measured by following the motion of particular fluid elements (e.g. using tracers, dyes, floats, bubbles, or small particles). Thinking of the flow in this way is called the Lagrangian representation. Forces act (of course) on fluid elements and not on positions in space. Thus we need to consider accelerations in a Lagrangian frame when deriving the effect the forces acting on a fluid have on the fluid flow. However, fluid properties (such as velocity, pressure and density) are easier to measure and represent as functions of position (Eulerian description). Path line or particle path The line followed by a particle in the fluid released at some point in the flow (what you would see with a long exposure photograph of the flow). Streakline or dyeline The line of dye resulting by continuously injecting dye at a particular point in the flow.

particle released here (eg at t=0) particle is carried

by the flow

dye continuously injected here


Streamlines Lines everywhere tangent to the flow direction. This gives an overall picture of the flow field (at a particular instant in time). Flow speed increases where streamlines converge. Streamtube Bounded by a set of streamlines, so fluid remains within the tube (a "virtual pipe"). Steady flow Flow that doesn't change in time (though the velocity may be different in different parts of the fluid).

0u=

∂∂

t (at constant x)

For steady flow, streamlines, streaklines and particle-paths are all the same. Note that just because the velocity at a point isn't changing, this doesn't mean that fluid passing that point isn't accelerating.


Examples Streamlines, particles paths and streaklines in unsteady flow. ux = V0cos(2πt/T) uy = V0sin(2πt/T) What do streamlines look like (at a particular value of t)? Release particle at (x,y) = (0,0) at t = 0, what is the path? Release dye at (0,0) from t = 0 to t = T, what is the resulting dyeline? Acceleration An air supply system consists of ducts with rectangular cross-sections. The upstream part of the ducting has a width of 0.4 m and height of 0.3 m while the downstream part has the same width (0.4 m) but a height of 0.2 m. These two sections are connected by a simple contraction where the height changes linearly from 0.3 m to 0.2 m over a distance of 1.0 m (see figure). (a) If the air flow rate along the duct is 0.5 m3 s-1, find an expression for the speed of the flow as a function of the distance from the start of the contraction, x. (Assume the velocity is uniform across the duct cross-section at any given position x, i.e. find u(x).) (b) Find an expression for the acceleration of the air as a function of distance along the contraction.

0.3 m 0.2 m

1.0 m

x


A2 Equations of motion Material (or substantive) derivative Rate of change following a fluid element, usually written as D/Dt. E.g. the acceleration of a fluid element is Du/Dt (here u is taken to be a function of space and time - Eulerian). This derivative can be found in the usual way, looking at the change of a small time step, δt, and taking the limit δt → 0. The changes following a fluid element occur partly because of changes in time and partly because of changes in space (and the fluid element is moving to a different position).

∇•+∂∂

= utDt

D

Continuity (Conservation of Mass) Consider the flow through a small fixed rectangular volume of sides δx, δy, δz (and thus volume δV = δx×δy×δz). The rate of change of the mass of fluid in the volume due to the flow through the two faces perpendicular to the x-axis is (the faces have area δy×δz),

{u(x)ρ(x) - u(x+δx)ρ(x+δx)}(δy×δz) = - δx( )

xu∂

ρ∂ δy×δz = -

( )x

u∂

ρ∂ δV.

With similar expressions for the other two directions, this gives ( )ρ•∇− u δV. But the rate of change of mass must also be equal to the rate of change of density multiplied by the volume, so

( )t∂ρ∂

=ρ•∇− u .

Expanding the divergence of the product uρ, and using the expression for the material derivative we can obtain

01=•∇+

ρρ

uDtD

.

If the density of fluid elements remains (at least approximately) constant, then the continuity equation reduces to ∇.u = 0 (incompressible). A vector field with zero divergence is known as "solenoidal." The assumption that the flow is incompressible is usually good for flow speeds much smaller than the speed of sound (340 m s-1 in air, 1470 m s-1 in water), and we will treat flows as incompressible in this module (see other modules for consideration of high speed flows). Momentum Force is equal to rate of change of momentum (Newton's Second Law). For a fluid this becomes,

DtDuF ρ= ,

where F is the total force per unit volume. We now consider the various forces acting on the fluid contained in our small volume δV.

δx

δz

δy


Pressure By considering the pressure forces acting on the faces of the small volume, we find the pressure force per unit volume is given by,

pzp

yp

xp

p −∇=

∂∂

∂∂

∂∂

−= ,,F .

(Note the pressure acts down pressure gradients, from high to low). Viscosity Consider a series of blocks of fluid on top of one another, all moving in the x-direction but at different speeds for different z. The viscous force between blocks depends on the velocity gradient (strain), with force per unit area (stress) proportional to strain for simple fluids (dynamic viscosity µ),

stress zu

∂∂

µ=τ .

The force on a given block is the sum of the forces due to the movement of blocks above and below it,

Vzuyx

zzuyx

zzzu

δ∂∂

µ=δδ∂

∂µ−δδ

∂δ+∂

µ 2

2)()(.

Developing the result for a general flow is complicated (see e.g. Batchelor). The viscous force per unit volume is

uF 2∇µ=v , where 2

2

2

2

2

22

zyx ∂∂

+∂∂

+∂∂

≡∇ .

Other forces: gravity The gravitational force per unit volume can be expressed as Fg = ρg, where g = (0, 0, -g). Other forces can be included (e.g. electromagnetic forces, important in conducting fluids such as in stars and the earth's interior). Navier-Stokes equations Combining the pressure, viscous and gravitational forces gives,

guu+∇ν+

ρ∇

−= 2pDtD

,

where ν = µ/ρ is the kinematic viscosity. This, together with a continuity equation (e.g. ∇.u = 0), gives a system of four equations in four unknowns (u, ,v, w, p) known as the Navier-Stokes equations. (The system without the viscous term is known as the Euler equations.) It is convenient to use the vector form above but often necessary to expand in more detail when solving particular problems. For example, the third component of the momentum equation is

gzw

yw

xw

zp

zww

ywv

xwu

tw

DtDw

−

∂∂

+∂∂

+∂∂

ν+∂∂

ρ−=

∂∂

+∂∂

+∂∂

+∂∂

≡ 2

2

2

2

2

21.


A3 Laboratory experiments Some questions What are laboratory experiments? Why do we do laboratory experiments? How do we do laboratory experiments? Safety Laboratories are potentially dangerous places and the Departmental and UMIST safety policies must be observed. Risk assessments have been carried out for the laboratory as a whole and for the individual experiments, and these are posted in the laboratory. These risk assessments identify the main hazards involved and you should read them before you begin your experiments. Experimental design Where an experiment is used as part of a scientific investigation, the aim is to construct an experiment that isolates the phenomenon under investigation. This generally results in an experimental design that is as simple as possible, excluding as many extraneous parameters as possible. This is also true of the experiments used in this course to illustrate the basic features of fluid flows. The experiments will largely be prescribed but you will have some decisions to make in terms of setting some flow parameters (e.g. flow rate) and in terms of what to measure and how many measurements to make. In some cases experiments are conducted in order to simulate a fluid flow (especially around a proposed machine or building), in which case the accuracy of the simulation is an important consideration in the experimental design. Recording observations Laboratory notebook Having gone to the trouble of making an experiment, it is essential that you record what happens. A vital tool is the laboratory notebook, traditionally hardbound.

• It is sensible to put your name and contact details at the front of the notebook, so there's a chance it might be returned if lost.

• Give each experiment a title and remember to put the date. Begin with a short description of the experiment. A sketch of the apparatus with important dimensions is always useful.

• You should make notes directly into the handbook during the experiment (you can make further notes/calculations later, but you should date these).

• Note down all the information you can at the time: it's always much harder to find out important details later.

• A useful test of whether you've put down enough information is to ask the question "if I look at this in a year or two's time will I be able to work out what I did?"

Qualitative descriptions The information you record should include visual observations. These can be just as important (often more important) than numerical measurements in interpreting fluid flows. Don't leave things out just because you can't understand or explain them: these are often the most interesting features. Quantitative measurements Record numerical measurements to as many figures as you can. You can decide later what the appropriate level of accuracy is in giving your results, but if you've only recorded measurements to the nearest centimetre instead of millimetre then you can never recover the information. In general you should try to read to 1/10th of most marked scales. Reports You will be expected to produce detailed reports on all your experiments. Your account should contain a reasonably detailed description of the experiment (at the level you might find in a research paper). Suggested format for the report is (though the precise number/arrangement of sections is up to you, and will depend on the experiment):

1) Introduction, including an outline of the fluid dynamics. 2) Apparatus/method: main features of apparatus (dimensions, etc), experimental techniques


3) Experiments: what experiments did you do? Ranges of parameters, etc 4) Results: basic experimental results (raw measurements), processed results (e.g. integrated, etc) 5) Discussion/conclusion: how do the laboratory results relate to the fluid dynamics?

Flow visualisation techniques Streamlines, streaklines and particle paths: recap Streamlines are lines which are everywhere parallel to the flow. They give a picture of the instantaneous flow field. Steaklines are lines made by a fixed source of tracer (e.g. dye, smoke, bubbles) released continuously into the flow. A particle path is the path traced out by a marked fluid element as it moves in the flow. Neutrally buoyant particles will follow the fluid flow to a good approximation. For steady flow, streamlines, streaklines and particle paths are all the same, but this is not true of unsteady flows. Dyes and tracers As well as releasing dye at a point, some region of the flow may be marked by mixing dye throughout the region. The dyed fluid can be used to estimate the flow and also to estimate mixing with undyed fluid. Other tracers of the flow (e.g. temperature, salinity) may also be monitored (e.g. by thermocouples and conductivity probes). Shadowgraphs This technique depends on variations in the refractive index of the fluid caused, for example, by changes in temperature, salinity or alcohol content. Light is shone through the fluid onto a translucent screen (e.g. tracing paper, frosted glass) to form an image. The variations in the refractive index focus and unfocus the light to give dark and light patches on the image. Image processing In addition to providing qualitative information about the flow, visualisation techniques are increasingly used to give quantitative information about the flow. Digitised images of the flow are captured on computers and processed to give information about concentrations (dye intensities) and velocities (various forms of particle-tracking). This allows us to get detailed information about all parts of the flow in a way which was hitherto only available in numerical models. For accurate simulation of high Reynolds number flows in complex geometries, laboratory experiments still have a very important role. Flow visualization experiments These are described in detail below (section C). You will use a smoke tunnel (using streaklines formed by injecting smoke at fixed positions into an air flow) and a water channel (using particle paths by sprinkling the surface with a fine powder). A range of bluff and streamlined bodies (cylinders, spheres, aerofoils, etc.) will be available for you to experiment with.


B Basic principles in fluid dynamics

B1 Reynolds number and flow types Features of the Navier-Stokes equations Recall the Navier-Stokes equations for an incompressible flow are:

guuuuu+∇ν+

ρ∇

−=∇•+∂∂

≡ 2ptDt

D,

0=•∇ u .

The equations are non-linear (because of the advection term in the material derivative) and second order (because of the viscous term). At a rigid boundary, moving with velocity U and having normal n, there are two types of boundary conditions. No normal flow: there can be no flow through the boundary, so u.n = U.n at the boundary. No-slip condition: the fluid in contact with the boundary must have the same tangential component of velocity at the boundary, else the viscous stresses would give an infinite restoring force. Taken with the no normal flow condition this gives u = U at the boundary. The no-slip condition can be relaxed if we are ignoring viscous effects, leaving the no normal flow condition. Ignoring viscous effects reduces the equation to first order. Non-dimensionalisation and Reynolds number It is often useful to transform from the standard units of measurement (metres, seconds, etc) to scales that are "natural" for the problem. For example, for flow past a cylinder the natural length scale would be the diameter, D (or possibly radius, R) of the cylinder while the natural velocity scale would be the speed of the flow, U, far from the cylinder. The natural time scale would then be T = D/U. If the dimensional velocities and positions are u* and x*, then in non-dimensional variables we have, u = u*/U, x = x*/D, …boundary conditions u = 0 on |x| = 1/2, and u → (1, 0, 0) as |x| → ∞. In the non-dimensional variables the boundary conditions are the same for all cylinders. The equation of motion becomes (in non-dimensional variables),

uu 2∇ν

+ρ

∇−=

UDp

DtD

.

If both the boundary conditions and the equation are the same, then the solutions and flow behaviour must be the same. This will be true if the quantity ν/UD is the same. The reciprocal of this quantity R = UD/ ν, is known as the Reynolds number. The Reynolds number is the ratio of the order of magnitude of the inertial and viscous acceleration terms, and thus low Re implies viscously dominated flow while high Re implies inviscid flow (flows where the viscous forces are small). Laminar and turbulent flow In practice we find that for low Re flows (in which viscosity dominates), disturbances in the flow field tend to be damped leading to smooth, stable, laminar flows. At high Re, the flow is likely to be unstable, and may break down into chaotic, turbulent flow. For example, for flow through pipes laminar flow is observed for Re < 2000 (where Re = UD/ν, U average flow speed, D pipe diameter), while turbulent flow is generally observed for flows with Re > 4000. Viscous flow At very low Re, the equation of motion reduces to a balance between pressure and viscous stresses. Both the non-linearity and the explicit time-dependence are removed, so that the flow is reversible and the flow field adjusts instantaneously to applied conditions. Reversibility results in symmetric flows since, for example, the flow past a cylinder from right to left must look identical to flow from left to right with the flow direction reversed but the streamlines remaining the same.


Flow past cylinders At low Reynolds number the flow must have forward/backward symmetry but as Re increases the flow separates at the rear of the obstacle and a recirculating region (vortex pair) is formed. Initially this region is steady and symmetric about an axis through the obstacle but as Re increases further (Re ~ 40) waves develop downstream. For Re > 70 the vortices alternate in size and are shed forming a "vortex street." If the frequency at which the vortices are shed (e.g. number per second) is σ, then we relate this to the flow parameters through the Strouhal number (a non-dimensional frequency) S = σD/U. Typical value for cylinders is S ~ 0.2. The alternating vortex shedding produces an oscillatory force on the cylinder.

B2 Pressure and energy (Bernoulli's equation) Hydrostatic pressure The equations of motion for a fluid at rest reduce to,

yp

xp

∂∂

ρ−=

∂∂

ρ−=

110 , gzp

−∂∂

ρ−=

10 .

Thus the pressure must be constant on horizontal surfaces and can be found by integrating from some reference level (e.g. p = p0 at z = 0),

∫=

ρ−=z

z

gdzpp0

0 .

The pressure at a point in the stationary fluid is simply equal to the weight of fluid above it with the vertical pressure gradient opposing the gravitational force. The pressure in a fluid in equilibrium at rest is known as the hydrostatic pressure. For uniform ρ, the hydrostatic pressure is given by p = p0 - ρgz. Since the hydrostatic pressure plays no dynamic role within a fluid body of uniform density, it is common to work in terms of a pressure from which the hydrostatic pressure has been subtracted,

gzpp ρ+=′ . With this version of the pressure, the gravitational term in the equations of motion is eliminated. Bernoulli equation and energy conservation Steady flow, no friction (inviscid) Fluid element moves along a fixed streamline (like a bead on a wire). Distance measured along streamline denoted by s, velocity along the streamline by q (=ds/dt). Volume of fluid element = dA ds Mass of fluid element m = ρ dA ds Gravitational force = -mg = -ρg dA ds (downwards).

q

s+ds dA s

z


Resolving forces in the s-direction Pressure force (s-direction) = p dA - (p+dp) dA = - dp dA Gravitational force (s-direction) = -mg cosθ = -ρg dA ds cosθ = -ρg dA dz

F = ma, and a = dqdt so,

ρ dA ds dqdt = - dp dA - ρg dA dz.

dqdt = -

1ρ

dpds - g

dzds .

Now, dqdt =

dsdt

dqds = q

dqds =

12

d(q2)ds , so we have,

12

d(q2)ds +

1ρ

dpds + g

dzds = 0.

Integrate ∫ds (along streamline)

Kinetic energy + potential energy = constant (no friction), H = "head" Pitot and Pitot-static tubes A Pitot-static tube is a thin tube with an opening at the tip, facing the flow, and another opening in the side of the tube. The tip of the probe is a stagnation point, so the velocity is zero there. The pressure difference (e.g. measured by a manometer) gives a measurement of the flow. If the manometer shows a height difference of ∆y and the manometer fluid has density ρm, then the difference in head is

∆h = ∆y(ρm - ρ)/ρ = ∆y(ρm/ρ - 1), and so

u2 = 2 g ∆h. A simple pitot tube just has an opening in the tip and the pressure there may be compared with either atmospheric pressure, or the pressure at the side of the channel, wind tunnel or pipe through which the flow is passing. If there is little curvature in the flow then there will be little pressure difference between the side of a pitot-static tube and a side-tapping.

ds

p

p+dp

mg

θ

dz = ds cosθ

12 q2 +

pρ + gz = constant =gH

(along streamlines)Bernoulli's equation.


B3 Control volumes and the momentum principle Control volume

• Identify a region of the flow. • Identify all the forces acting on the fluid in this region. • Calculate the rate of change of momentum of the fluid passing through this region (out - in). • Force = rate of change of momentum

Example: force on an array of obstacles We take a control volume as shown, sufficiently large that the downstream flow has settled back to a simple parallel flow (any distortion past the obstacles has been smoothed out). If the total drag on the obstacles is D (x-direction), then the force by the obstacles on the fluid is -D. If the cross-sectional area is A, then the pressure forces acting on the control volume are (p1 - p2)A. For this flow there is no change in momentum so (assuming there are no viscous stresses at the upper and lower boundary) we have,

-D + (p1 - p2)A = 0, or D = (p1 - p2)A. Note that we have not made any assumptions about energy conservation. Force on a contraction If the force on the contraction is F, then the force by the contraction on the fluid is -F.

-F + p1A1 - p2A2 = ρu2(u2A2) - ρu1(u1A1)

F = p1A1 - p2A2 - ρu2(u2A2) + ρu1(u1A1) Notes: for incompressible flow, the total flow rate Q = u2A2 = u1A1. For inviscid flow, we can use Bernoulli's equation to relate the pressures and velocities. Example Water flowing at 0.05 m3 s-1 out of a nozzle (diameter changing from 100 mm to 50 mm) into the atmosphere (take p2 = 0). What is the force on the nozzle?

Q = 0.05 m3 s-1, A1 = (π/4) 0.12 = 7.85×10-3 m2; A2 = (π/4) 0.052 = 1.96×10-3 m2;

u u

p2

p1

p1

A1

u1

p2

u2

A2

area A


and thus u1 = Q/A1 = 6.37 m s-1; u2 = Q/A2 = 25.5 m s-1; Assuming no energy loss in the contraction,

gp

gu

Hg

pg

uρ

+==ρ

+ 2221

21

22,

100081.90

81.925.25

100081.981.9237.6 2

12

×+

×=

×+

×p

so p1 = 3.05×105 N m-2. Force on the nozzle = p1A1 + ρQ(u1 - u2) = 2393 + 1000×0.05×(6.37 - 25.5) = 1440 N. Flow around a pipe bend a) no change in area (A1 = A2) and horizontal (so no effect of gravity) Assume no energy loss and so no change in pressure (p1 = p2 = p). Rate of change of momentum

(u2 - u1) ρQ = (u2 - u1) ρAu Where u is the speed (u = |u2| = |u1|), and so u1 = (u, 0) and u2 = (u cos45°, u sin45°) So the x-component of the total force on the fluid is:

Fx = (u cos45° - u) ρAu = -(1 - cos45°) ρAu2, and the y-component is

Fy = u sin45° ρAu = sin45° ρA u2. The sum of the pressure forces at the two ends is:

(pA, 0) + (-pAcos45°, -pAsin45°) = pA(1-cos45°, -sin45°)

total force on fluid = pressure force at ends + force by pipe on fluid

force by fluid on pipe = -force by pipe on fluid = pressure force - total force

= pA(1-cos45°, -sin45°) - (Fx, Fy)

= (pA + ρA u2)(1 - cos45°, -sin45°) Example Pipe carrying water, diameter 10 mm, pressure 104 Pa, flow rate Q = 1.20×10-4 m3 s-1.

A = (π/4)(0.01)2 = 7.85×10-5 m2. u = Q/A = 1.53 m s-1.

(pA + ρA u2) = 7.85×10-5 × (104 + 1000×1.532) = 0.968 N. Force on pipe: (0.28, -0.68) N.

45° u1

u2

x

y


B4 Flow past bluff and streamlined bodies Drag on an aerofoil In experiment 2 you will be using the control volume technique to calculate the drag on an aerofoil. In the wake of the aerofoil the flow speed is slower, so calculating the momentum flux leaving the control volume requires an integral across the downstream face of the control volume using measured velocities. The aerofoil used in the experiments is symmetric and mounted horizontally (zero angle of attack) so the lift is zero. Equating the rate of change of momentum to the applied forces gives (taking the width of the wind tunnel and the aerofoil to be b),

dragdybpdybpdybUdybu −−=ρ−ρ ∫∫∫∫ 2122 .

Using continuity,

∫∫ = dyudyU

and so the drag per unit width is ( ) ( )dyppuUubdrag 21/ −+−ρ= ∫ .

The pressure does not change significantly with y, so the pressure difference can be taken as constant. The calculation of this integral from the experimental data is covered below (section C). Drag coefficients The drag coefficient is defined as

AU

dragCd2

21

ρ= ,

where A is an appropriate area. At high Reynolds number, the drag coefficient is approximately constant and depends on the shape of the obstacle. The area used in the definition is usually the cross-sectional area normal to the flow direction (for the aerofoil this would be its thickness×b). However, for the aerofoil it is also useful to compare the measured drag to that you would expect to act on a pair of flat plates each of area A = b×c (see the Boundary Layer module). In terms of the Reynolds number defined by the length of the plate in the flow direction Re = Uc/ν,

Cd = 1.328 Re-1/2 (laminar), Cd = 0.0315 Re-1/7 (turbulent), Retr ~ (0.5 to 3)×106. Drag on a cylinder Again we can measure the drag force. The area used in the drag coefficient in this case is usually the area facing the flow, i.e. 2Rb. for a cylinder of radius R and length b. Note that the flow pattern for viscous flow is the same as that for potential flow, though the (theoretical) potential flow would produce no drag. At high Reynolds Numbers it is the departure of the real flow pattern from potential flow that produces the drag. In pure potential flow the pressure and velocity are related through Bernoulli's equation and a symmetric flow would have the flow speed rise (and pressure fall) and then return to its original value in a symmetric way around the cylinder, leading to no net force. In practice the flow separates and the pressure behind the cylinder (in the wake region) does not increase. It is this asymmetry that accounts

U u(y)

wake

chord = c

lift drag

p1 p2


for the drag. (In addition in practice there will be viscous drag on the cylinder surface, but this is usually a relatively small component.) Thus the drag on a cylinder is (mostly) from the pressure forces acting on the cylinder surface. The horizontal component of the pressure force (per unit width) is given by

θθ∫π

dRp cos2

0

and p will be measured directly as a function of θ in the experiment. (The vertical component should be zero.)

B5 Open channel flows and hydraulic jumps Conservation of energy (Bernoulli's Equation) If the channel bed and the flow is slowly-varying, so that there are no strong vertical velocities or accelerations, then the pressure is approximately hydrostatic:

p = ρgZ = ρg(η - z)

Thus the total head (for steady flow) on a streamline through z is given by,

( )η+=+

ρ−ηρ

+=+ρ

+=g

uzg

zgg

uzgp

guH

222

222.

Note this depends on the height of the free-surface, but not the actual height of the streamline. If the flow is uniform (the same at all depths in the fluid), we simply have

η+=g

uH2

2,

for all positions along the channel. (In practice there are frictional losses along the channel - see below.)

p

θ R

free surface

channel bed

z=0

z

d = η - B Z = η - z

η

B

streamline


Conservation of Momentum: Hydraulic jumps Assume the width, b, is constant.

Q = bu1d1 = wu2d2, and so, u2 = u1d1/d2. Pressure forces: hydrostatic pressure giving a horizontal force at each end. Total force (per unit width),

F = ⌡⌠0

d1

ρgZ dZ - ⌡⌠0

d2

ρgZ dZ = 12 ρg(d1

2 - d22) = u1d1ρ (u2 - u1)

12 g(d1

2 - d22) = u1

2d1(d1/d2 - 1)

u1 = gd1 12r(1+r) ,

where r= d2/d1 is the size of the jump. Note that for a jump (r>1), we need u1> gd1 ("supercritical flow"). Energy loss

H1 = u12/(2g) + d1, H2 = u2

2/(2g) + d2. This gives the head loss:

H1 - H2 = (d1/4r) (r - 1)3. Power loss is given by,

ρg (H1 - H2)Q.

u1

d1 d2

u2

Control volume

p = ρgZ p = ρgZ


Froude number The ratio Fr = u/ gd is known as the Froude number. Since gd is the speed of surface waves, the Froude number can be thought of as analogous to the Mach number, in that it is a ratio of a flow speed to a wave speed.

Fr > 1 Supercritical flow Fr < 1 Subcritical flow

Fr = 1 Critical flow Friction in open channel flows For energy conserving flows we had the head remaining constant:

dBg

ug

uH ++=η+=22

22,

where d is the depth of the fluid and B is the height of the channel bottom. In real flows energy is lost and the head decreases. Denoting the head loss gradient by i and the channel slope by s as follows,

xHi

∂∂

−= and xBs

∂∂

−= ,

we can find an expression for the gradient in the fluid depth,

21 Fris

xd

−−

=∂∂ .

Note that the sign conventions are such that a positive value for i represents energy loss and a positive value for s represents a channel sloping down in the x-direction. The energy gradient may be calculated from Manning's formula

2/13/21 imn

u = , (S I units)

where n is Manning's roughness coefficient and m is the hydraulic mean depth, m = A/P, where P is the wetted perimeter. For a rectangular channel of width b, P = b + 2d, A = d×b. For u in m s-1, m in metres, a typical value for n is 0.015 to 0.018. For a roughness of size k (in metres) n = (1/26) k1/6. (Strictly speaking, n has dimensions, not normally given in these engineering formulae.) Calculating the location of hydraulic jumps If the upstream flow is set to be supercritical while the downstream flow is set to be subcritical, then there must be a hydraulic jump in between. For supercritical flow in a horizontal channel, the depth must increase downstream as shown. At any given position the jump equations (above) can be used to calculate the corresponding subcritical flow if there were a jump at that location ("sequent depth"). The friction in the immediate region of the jump is usually ignored, so the result is obtained by solving the quadratic,

r2 + r - 2Fr2 = 0, where r = d2/d1.

Similarly we can work back from the downstream condition to give a predicted depth of the subcritical flow. We would expect the jump to occur where the two predictions intersect. Note: the forward and backward integrations from the end conditions have to be done numerically, stepping along in the x-direction.

sluice weir

d2

d1


C Visualisation experiments

C1 Smoke tunnel (Experiment 1A) You will use a small smoke tunnel to visualise a range of different flows. Air is drawn upwards through a vertical duct by an adjustable fan at the top of the duct. The working section has a transparent side to allow you to view the flow, with lights either side. WARNING: The lights get hot! The transparent side is removable so that you can change the models in the flow. Two transparent sides are available, one completely clear and one with a set of vertical parallel lines which may help in recording the flow. The flow is visualised by injecting smoke through a set of outlets upstream of (i.e. below) the working section. The smoke outlets can be moved from side to side to adjust the streaklines that you are viewing. The smoke is generated by vapourising paraffin. The smoke supply should be clamped off, and then the fan turned off, before changing the model in the working section. A range of models is available, including a cylinder, a sphere, an orifice and an aerofoil. The sphere has a "seam" running around a circumference, which may be opened up, and it may be mounted so that the seam can be tilted at different angles to the flow. The aerofoil can have smoke introduced at its leading edge. There is no direct way of measuring the flow speed in this experiment but you can record the fan power setting. A plot of fan setting against flow speed is provided, but this should be treated as only a rough guide. Working in small groups you will have approximately one hour to use the apparatus. Particularly look for where flow separates from a body and where the flow is turbulent.

C2 Water channel (Experiment 1B) A shallow water channel will allow you to look at a range of flows past various obstacles and make some more detailed measurements. The flow will be visualised by sprinkling the water surface with powder - please don't use too much! You can measure flow speeds by watching the motion of larger particles (e.g. scraps of paper) on the water surface (a very simple form of particle tracking). You will be provided with rulers and stopwatches. You may also have access to a video camera and recorder which will enable more accurate time measurements to be made. As well as single obstacles you can look at the flow past more than one obstacle to examine more complicated interactions. Each group will have approximately two hours to use the equipment. Report Experiments 1A and 1B should be reported together in a single report. This could have separate sections for the two experiments or be arranged (for example) to treat flow past particular types of obstacle. The report should include sketches (approximately to scale) of the various flows you observe, with as much quantitative information as you can derive.


D Momentum and drag experiments

D1 Wind tunnel (Experiment 2) General You will use a small wind tunnel which has a pair of fans that draw air through the tunnel. The working section is transparent and the tunnel is equipped with various pitot-static tubes, pitot tubes and tappings to allow you to measure the flow. The models are mounted on cradles that allow you to measure the drag force on the body directly by balancing the drag force with weights. (Note that the arrangement of the cradles is such that drag = weight for the aerofoil but drag = (2/3)weight for the cylinder.) In both cases a pitot tube is mounted upstream of the obstacle to allow you to measure the upstream velocity (approximately uniform across the tunnel). The velocity can be found using a manometer to measure the difference between the free-stream and stagnation pressures (see B2). You should move this tube out of the way when making measurements further downstream to avoid disrupting the flow. Air density This can be found using the gas law equation

p = ρRT, with R = 287 J kg-1 K-1. A thermometer and barometer are mounted on the laboratory wall. Aerofoil The aerofoil is of a standard type: NACA 0012. It has chord 0.152 m and 0.3007 m span (the width of the wind tunnel). The thickness of the aerofoil is 12% of its chord. The balance arms in this case are 1:1, so the weight added is equal to the drag. Use the multiple manometer for all pressure measurements. The comb of pitot and static tubes are connected to tubes 1-31 (with 5, 12, 21 and 28 measuring static pressure: best to take an average of these for the static pressure). The tubes on the comb are spaced at intervals of 0.1 inches (2.54 mm) - use the difference between the stagnation and local static pressure to calculate the flow speed u for each position, y. The pitot-static tube (used to measure pressures and velocities upstream) is connected to two of the spare tubes. The manometer should be tilted to its maximum inclination (18° from the horizontal, so the measured readings need to be multiplied by sin 18° to get ∆y). The density of the fluid in the manometer is 0.79 g cm-3. The set-up is not perfectly symmetric, so the wake isn't perfectly symmetric. You should use the obvious wake region in calculating the contributions to the momentum balance. Best to use formulae in versions with velocity differences, which can be taken to be zero outside the wake. I.e. plot the integrand from

( ) ( )dyppuUubdrag 21/ −+−ρ= ∫

and find the area under the curve, ignoring the region outside the wake where the curve is flat. Cylinder The cylinder diameter is 6.35 cm. When measuring the drag using the balance it is necessary to tighten the screws securing the cylinder to the end plates. This stops the plates being sucked against the outside of the viewing section. The screws must be loosened to rotate the cylinder for the later part of the experiment. DO NOT OVER TIGHTEN THE SCREWS (finger tight is quite sufficient). The ratio of the arms of the drag balance is 2:3 so the added weight is 1.5 times the drag. Use the single manometer for all the pressure measurements. The cylinder has a small hole on the surface which allows measurement of the pressure there and the cylinder can rotate so that the pressure distribution around the cylinder can be determined. The flow may not be perfectly horizontal but you should be able to detect the line of symmetry from the pressure distribution. The drag can be calculated by integrating the component of the pressure in the flow direction. (You could look at the pressure readings in the wake, but the wake is too broad and unsteady for useful calculations.)


Report You should prepare a single report covering both the aerofoil and cylinder experiments. The measured and calculated drag can be compared, and you can compare the drag coefficients for the two shapes. The drag on the aerofoil can be compared with the expected drag on (a pair of) flat plates.

D2 Hydraulic jump (Experiment 3) These experiments are performed using water (density 1000 kg m-3) in a clear-sided channel approximately 10 cm wide (check the precise width). The water is introduced under a sluice near one end of the tank and the initial flow is supercritical. The flow undergoes a hydraulic jump before exiting at the downstream end. Flow rates The flow rate is calculated by measuring the volume flowing in a measured time. One of the channels is fitted with a flow meter that records the volume passing while the other uses a measuring tank: the water is diverted into this tank for a measured period of time and the volume is read using a float with a scale. Water depths The water level (and bottom of the channel) are measured using a point gauge mounted on a movable carriage above the channel. This is fitted with a vernier scale to allow measurement to 0.1 mm, though the oscillating surface downstream of the jump is hard to measure accurately (take an average of several readings). 3) So the measurement is 1.23 cm. (Some of the scales are marked in mm rather than cm.) Horizontal channel The height of the downstream end of the channel can be adjusted. With a few centimetres of (stationary) water in the channel, check that the water depth is the same at both ends of the channel. Run the water through the channel and measure the flow rate. Find the water depth at suitable intervals along the channel. Sloping channel Now adjust the channel so it slopes down in the direction of the flow (a slope of about 1%) and repeat the experiment. Analysis You should be able to calculate the flow rate and flow speed, consider the momentum and energy either side of the hydraulic jump and investigate how effective the (frictional) theory is in calculating the position of the hydraulic jump. Suggested values for Manning's coefficient is n = 0.015 to 0.018, though you could attempt to calculate it from the experimental data.

1.0

1.5

2.0

0

5 1) The measurement is between 1.2 and 1.3

2) The third line on the vernier scale lines up with a line on the main scale.


E Flow/structure interactions

E1 Natural and forced oscillations Harmonic motion

tFkxxcxm Fω=++ sin&&& (c damping coefficient, k stiffness, ωF forcing frequency) For undamped, unforced oscillations (c = F = 0), the general solution is

)sin( φ+ω= tAx N , where the natural frequency is mk

N =ω .

For damped, unforced motion, critical damping occurs for NC mcc ω== 2 , and for underdamped motion, the general damped frequency is,

22

12

−ω=

−=ω

CND c

cmc

mk

.

with general solution,

)sin(2 φ+ω=

−

tAex D

tmc

. (*) For forced damped motion, the oscillations are at the forcing frequency and the "magnification factor" is given by,

222

21

1

ωω

+

ωω

−

=

N

F

CN

F

cc

MF .

E2 Oscillations of a simple structure (Experiment 4) At the simplest level, structures may be regarded as simple damped oscillators. If there is a flow such that the natural frequency of the structures oscillations is close to the frequency at which vortices are shed then we can have resonance resulting in a large (and sometimes catastrophic) response from the structure. (See the notes on Strouhal number in §B1.) It is also possible for flow past one structure to influence another, for example eddies shed from one chimney causing a downstream chimney to fail. The case we will examine here is the interaction between a simple flow and a cylinder. The experiments will be conducted in a water flume. The flow speed in the flume can be measured using a flow meter consisting of small impellor and associated electronics. A cylinder will be mounted so that it can move perpendicular to the flow direction (again with some electronics to monitor the motion). The restoring force (stiffness) is provided by springs. You should examine the response of the cylinder under a range of conditions. You should note the frequency and damping of the cylinder in the absence of any flow (both in air and surrounded by stationary water). The damping in air should be very small, so that the frequency of oscillation will be close to the natural frequency.


Analysis No flow (air and water). By fitting equations of the form given in (*), you should be able to determine various parameters such as the natural frequency and the ratio (c/cC). Flow. Estimate the forcing frequency from the Strouhal number (note that this is usually given in terms of frequency rather than angular frequency). Now consider the form of the magnification factor and compare with your results. You should also observe the flow closely to see if you can identify the actual eddy-shedding behaviour, both with the cylinder moving and also with the cylinder fixed. The mathematical results above all assume "small" oscillations and thus an analysis of the linear equations. In practice, if the deflections are large you can expect non-linear effects to be important giving rise to more complicated behaviour.


F Waves

F1 Water wave theory Linear surface waves Consider an infinite body of water of mean depth h over a flat horizontal surface. If the flow is inviscid, incompressible and irrotational then we can write the velocity field in terms of a velocity potential, u = ∇φ. The velocity potential form is substituted into the equations of motion and then we linearise, assuming η << h. The boundary conditions at the free surface (z = η) are moved to conditions at z = 0 (use a Taylor expansion about η = 0 and then linearise). The final, linear equations are

02 =φ∇ in the fluid,

0=∂

φ∂n

at z = -h, (no flow through base),

zg

t ∂φ∂

−=∂

φ∂2

2 at z = 0 ("free surface"),

while the free surface displacement is given by

( )0

1,,=∂

φ∂=η

ztgtyx .

General wave solution Wave frequency ω (period T = 2π/ω), wave number k (wave length λ = 2π/k), amplitude A, phase ψ.

( )( )( ) ( )ψ+ω−

+ω

=φ tkxkh

hzkgA coscosh

cosh ,

( )ψ+ω−=η tkxAsin .

The frequency and period are then given by

ω2 = gk tanh(kh) and thus 1/T2 = (g/2πλ) tanh(kh).

The phase speed c = ω/k, so c2 = (g/k) tanh(kh).

We can also show that particle paths are ellipses of the form

( )( )( )

( )( )( ) 2

22

02

20

02

20

sinhcosh ω=

+

−+

+

− ckhzk

zzhzk

xx.

h

η(x, y, t)

z = 0

z = -h


Shallow and deep water approximations We can use approximate forms of the above results for deep water (kh → ∞) and shallow water (kh → 0) cases ("deep" and "shallow" refers to the depth of the water with respect to the wavelength). For example, for waves on deep water, the phase speed and period are given by,

πλ

==2

/ gkgc and g

T πλ=

2 .

while for shallow water,

ghc = and cgh

T λ=

λ= .

Wave breaking For waves where the wave height is significant compared to the water depth, the linear theory breaks down. The mean flow has a near surface component in the direction of the wave propagation ("Stokes drift"). The wave shape departs from sinusoidal with an increasingly sharp crest as the water depth becomes shallower compared to the wave height (as happens when the wave approaches a beach). Eventually the wave crest has a sharp cusp at the crest (theoretical angle 120°) with vertical accelerations becoming equal to g. Past this point the wave "breaks" though the initial form of this breaking is simply some rippling and air entrainment at the crest, with the complete overturning and plunging breaker forming as the waves reaches even shallower water. If the local mean depth is h, and the wave height is H (note this is twice the amplitude for a sinusoidal wave), then "wave breaking" should occur where H/h = 0.78.

F2 Wave tank (Experiment 5) A wave flume 18.5 m long, 1.2 m wide and 1 m deep is used for this experiment. There is a wave maker (paddle driven by a motor) at one end of the flume and you can adjust the stroke (which affects the amplitude) and frequency of the paddle. There is a beach (slope 0.1) at the other end of the flume. You should check the (stationary) water level in the flume at the start of the experiment. The period of the waves (T) can be determined by counting the oscillations of the paddle or number of crests passing a fixed point in a measured time. The wave height (twice the amplitude) can be found by measuring water levels on the glass walls of the flume. The wavelength is found using the pair of wave probes: start with the probes close to each other and then move one probe along the flume until the waveforms on the monitor are in phase (the wavelength is then the distance between the probes). A few permanganate crystals sprinkled on the surface (they will sink through the water column) can be used to visualise the fluid motion (make quantitative measurements if possible). Try to identify where the waves are breaking. Once you have located this position, turn off the wave maker and measure the stationary depth at this location. You also need to measure the wave height ( this can be done in the flat region before the beach). Analysis By using a range of frequencies (and perhaps two different paddle strokes) you should be able to compare the observed behaviour with the results for linear waves given above (and also consider the shallow and deep approximations). (Periods, wavelengths, particle paths, etc.) You can also examine the location where the waves break and compare with the result given above.

H

h


G Turbulent jets

G1 Turbulent jets Where fluid is forced through an outlet into a body of the same fluid a jet is formed. For a review of turbulent jets (and buoyant plumes) see List (1982: Ann. Rev. Fluid Mech., 14: 189-212). Near the exit there are strong shear layers which are unstable, developing vortices. Further instabilities result in a fully turbulent flow by approximately 40 diameters downstream. The turbulent jet entrains (engulfs) ambient fluid into the flow and the mean velocity (and other speeds) in the jet decay with distance downstream. The entrainment of ambient fluid into the jet increases the total volume flux within the jet. In fully developed flow, the mean velocity distribution is found to fit the Gaussian form,

( )2/)(),( brm ezUrzu −= ,

where b(z) is a measure of the radius of the jet. Here z is distance downstream and r the perpendicular distance from the jet axis. We can integrate the velocity profile across the jet to find useful integral properties (these are functions of z),

volume flow ( ) mUbdrruQ 2

0

2 π=π= ∫∞

and momentum flux ( ) 22

0

2

212 mUbdrruM ρπ=πρ=ρ ∫

∞

.

A simple treatment of the momentum flux (in the absence of viscous stresses) suggests that the momentum flux should remain constant with z. The entrainment into the jet is often assumed to be represented by an entrainment velocity proportional to the local mean velocity, entraining fluid in around the circumference of the jet. Thus we would expect,

( ) mUbdzdQ

πα= 2 (where α is a constant).

G2 Turbulent air jet (Experiment 6) In this experiment we will look at a free jet formed as air flows out of a cylindrical tube. Air is forced through a tube, 38 mm internal diameter and length 1.24 m. The air flow is driven by a compressor which attempts to keep the pressure in a large cylindrical reservoir between an upper and a lower limit (it switches on when the pressure drops below the lower limit and keeps running until the pressure reaches the upper limit). The air flow through the jet-experiment is sufficiently large that the best way to get a steady flow is to allow the flow to run until the compressor starts and runs continuously. You should check the air density as you did for the wind tunnel experiments (the temperature and pressure in the wind tunnel room are usually similar to that in the main lab). The tube is able to swivel about a pivot and to move along its axis (through the pivot). The pivot is 50 cm from the tip of a pitot tube (which is connected to a manometer - mean velocities can be calculated by measuring the pressure in the flow compared to the pressure outside the jet). Thus the flow can be measured at various values of z and r by moving the tube (keeping the measuring point fixed) rather than by moving the measuring point. SIDE VIEW

pitot tube tube

to manometer

from air supply


A scale is mounted on the pivot, and the distance along the scale to a collar on the tube measures the distance of the pitot tube from the mouth of the tube when the tube and pitot-tube are aligned. PLAN VIEWS A scale mounted near the rear of the tube, 50 cm from the pivot, allows measurement of the perpendicular distance of the tube centre-line from the pitot tube. For small angles, the value of z will remain approximately constant as the tube is rotated and the value of x will be approximately equal to r. (You may like to examine how good these approximations are.) It is suggested that you measure the flow at various values of z, from very close to the outlet to the furthest extent the apparatus will allow. At each value of z you should use several values of x (positive and negative to check the flow is symmetric). You will probably need a smaller spacing in x for small values of z (especially very close to the outlet). Analysis You should calculate the velocity profiles at various distances from the outlet, and also numerically calculate the volume and momentum fluxes as functions of z. If you also calculate the quantity,

mbUdruA 2/1

021

π== ∫∞

(the area under a graph of u vs r),

then you can estimate the quantities in the Gaussian profile, since (using standard integrals)

AQb 2/12π

= and QAUm

24= .

Use these values to plot Gaussian profiles to see how well the measured profiles fit, and how the profiles develop downstream. You can also look at how the momentum flux and volume flow varies with distance and whether it fits the forms suggested above.

z z

x

x

pitot tube

scale

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