Peube J-L. Fundamentals of fluid mechanics and transport phenomena

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Measurement, Representation and Analysis of Temporal Signals 343

7.2.3.2. Lagrangian anemometry

In general we want to know the velocity at an observation point which we define

with respect to the laboratory reference frame (Euler variables). Fluid particles are

not individually visible and it is necessary in all cases to introduce solid elements,

which are entrained by the fluid. The problem of solid particle entrainment by a

fluid is therefore essential.

In the case of cup anemometers (Figure 7.2a), we use the dissymmetry of a

rotating device, which presents a resistance to the fluid movement in a given

direction (cup C

) and very low resistance in the other direction (cup C

). The result

is that the cup situated at position C

tends to follow the flow, whereas the other, C

induces a very low torque on the moveable system. Alternatively a propeller can be

used (Figure 7.2b). As the elements of these devices are not entrained by the fluid

with the fluid velocity, it is necessary to calibrate them.

(a) (b)

Figure 7.2. (a) Cup anemometer or (b) propeller anemometer

Laser Doppler Anemometry (LDA) involves the creation of a system of

interference fringes by means of a laser whose beam is split into two beams focused

by a lens (Figure 7.3a). Very tiny particles crossing the fringes are alternatively

illuminated as they cross the fringes. It is possible to measure the scattered light by

means of a photomultiplier. The frequency value n of the light signal measured

allows a calculation of the particle velocity, equal to i.n, where i is the inter-fringe

spacing, easily obtained from the optical characteristics of the device. The

denomination Doppler comes from the fact that the procedure can also be explained

by considering the variation of the optical frequency which occurs when the particle

scatters the light provided by each of the coherent light beams.

344 Fundamentals of Fluid Mechanics and Transport Phenomena

(a) (b)

L S

Figure 7.3.

(a) Principle of anemometric measurement by ALD (L, laser; S, beam-splitter and

focusing lens; Z, fringes in the measurement zone; P, photodetector centered on the zone Z);

(b) bias in ALD measurement in a non-homogenous flow seeding

The principle of anemometric measurement assumes the presence of small

particles, which are in suspension in the fluid [DUR 81]. These nearly always exist

in water, but it is necessary to “seed” gaseous flows by means of droplets obtained

by the pulverization of some suitable oil, or by very light smoke obtained by

combustion of some suitable substance (incense for example) so as to obtain

particles whose dimensions are in the order of 1 Pm The introduction of particles

needs to be effected so as to produce as uniform a distribution as possible in the

flow. Figure 7.3b shows a region of flow between a stream of velocity V and a

region of flow at rest; the frontier f between the two oscillates unpredictably; only

fluid particles coming from the flow containing seeding will provide a measurement

signal: we thus see a systematic bias in the measurement of the velocity which will

“favor” the same kinds of flow structures.

The LDA procedure provides a local and instantaneous measurement in most

conditions. It is well suited to the laboratory study of small flow structures and it

even allows us to obtain acoustic velocities (from 1 mm/s to 1 cm/s) [PEU 92]. Cup

anemometers only allow measurement in a volume in the order of some

considerable fraction of a dm

, and therefore these are only suitable for quite global

measurements (wind velocity in meteorology or air-conditioning units).

Let us finally recall anemometric measurements at the atmospheric scale, which

are obtained by releasing balloons that move at constant altitude and whose position

can be tracked by satellite. This method provides trajectories (Lagrangian

viewpoint) and not streamlines (Eulerian representation of meteorological charts).

Measurement, Representation and Analysis of Temporal Signals 345

7.2.3.3. Eulerian anemometry

We here use a device, which is fixed in space, on which the flow acts by creating

a phenomenon whose intensity (overpressure, temperature, etc.) depends on the

velocity of the flow.

In the steady flow of an ideal incompressible fluid for which Bernoulli’s theorem

is valid, the difference in driving pressure between two points whose velocities are

respectively equal to V and 0 is equal to:

The measurement of velocity is thus realized by means of a measurement of a

pressure difference. A zero velocity is obtained when a flow is stopped by an

obstacle, on the upstream edge of which a stagnation point A occurs (Figure 7.4a); it

suffices to measure the total driving pressure (or the total pressure for a gas) at this

point A by connecting the orifice to a pressure probe. This orifice needs to be of

very small size in order to obtain a well-defined pressure value. However, if the

flow is not correctly aligned with respect to the obstacle, the orifice will not be at

the stagnation point and will not therefore measure the total pressure (Figure 7.4b).

The effect of the angle of incidence can be reduced by using a tube with a divergent

opening (Prandtl tube, Figure 7.4c).

(a) (b)

(c)

Prandtl tube

(d)

Pitot tube

Figure 7.4.

Measurement of stagnation pressure in a flow

By placing the total pressure tube inside a streamlined cylindrical tube (Figure

2.4d), the flow quickly returns to a uniform state (after 5 or 6 external diameters);

lateral orifices L placed on the external wall will measure the static driving pressure

(or static pressure in a gas) and the difference in pressure between the two tubes

provides a direct estimate of the dynamic pressure

VU . This device, known as a

346 Fundamentals of Fluid Mechanics and Transport Phenomena

Pitot tube, constitutes a somewhat cumbersome obstacle (it is at least a few

centimeters in length); it assumes that the flow velocity is defined and uniform at

this scale.

Bernoulli’s theorem is also applicable for a compressible fluid in subsonic flow,

provided of course that the compressible form of Bernoulli’s equation is used. The

problem is more difficult in supersonic flows because of the existence of a shock

wave upstream of the tube, across which specific laws need to be applied.

Hot wire anemometry is based on a very different principle: the thermal power

evacuated by a heated body is an increasing function of velocity. If the body is

electrically heated, this power can be known by means of a measurement of

intensity. It can be used in a flow in which velocity fluctuations exist with very

small hotwire diameters (1 to 5 Pm) whose thermal inertia is compensated by a

suitable supply (constant temperature anemometer). It is thus possible to measure

velocity fluctuations whose frequencies may be as large as 1,000 to 2,000 Hz. With

two or three differently inclined hotwires it is possible to measure the instantaneous

velocity vector. We therefore have in the hotwire anemometer a powerful means of

knowing the velocity fluctuations provided these remain smaller than the mean

velocity, as it is clearly not possible to distinguish the velocity direction from a

thermal power measurement alone.

7.2.4. Temperature measurements

The local measurement of temperature at the heart of a continuous medium can

be achieved by optical methods (measurements at a distance) or by means of probes

introduced into the flow. Optical procedures are based on emission phenomena,

which are associated with absorption phenomena, the medium considered being thus

semi-transparent for the wavelengths used. This leads to calculation methods which

are often complex and we direct the reader to specialized texts for more detail (see

for example ([BRU 95], [JOH 98], [MAR 99], [MCG 88]).

In a manner analogous to the introduction of pressure or velocity measurement

probes, the introduction of temperature measurement probes provokes a

modification of the flow and associated thermal phenomena only of importance at

high velocities. As with the pressure, the temperature of a fluid will depend on the

position of the measurement element on the wall of the probe; these measurements

are generally performed either at a stagnation point, or on a wall which is parallel to

the flow. For an ideal gas in adiabatic flow, we have conservation of total enthalpy

(section 4.3.2.3.2), or, with T

the stagnation temperature:

Measurement, Representation and Analysis of Temporal Signals 347

app

TC 

However, the measurement of temperature presents additional problems, as the

temperature of the sensible part of the probe is the result of a thermal balance

between the heat flux transmitted by the flow, the heat flux in the metallic electric

wires and in the support structure, and the thermal radiation caused by the

surrounding walls. As the last two quantities are independent of the flow, their

influence is less important on the temperature measurements at higher velocities.

We will note that for high enough velocities, thermal dissipation in the vicinity of

the obstacle, which is constituted by the temperature probe, may be a non-negligible

factor ([SCH 99]). Temperature measurements in flows are always difficult and

should not be attempted without a careful discussion of the different thermal

phenomena which may be present.

7.2.5. Measurements of concentration

The measurement of the concentration of a mixture is quite difficult. Very often

a fluid sample is taken via local suction by a tubular probe. The probe geometry is

not important here, but the suction velocity needs to be suitably chosen so as not to

alter the fluid trajectories in the extraction zone (isokinetic sampling) ([BRU 95],

[JOH 98], [CHE 88], [LIP 05]). An analysis of the fluid sample is then performed

using physico-chemical methods adapted to the mixture under study.

In certain cases, it is possible to perform measurements from a distance by

means of optical procedures associated with radiation emission. It is also possible to

use physico-chemical reactions on a surface placed in the flow, but the specific

procedure employed is always dependent on the particular mixture studied, and

often requires modeling of the heat and mass transfer phenomena involved. For

example, we have already said that evaporation phenomena induce a variation of

temperature: a wet thermometer does not measure temperature of air (section 2.3.2).

In a conducting liquid medium, the use of electrodes allows the measurement of

concentration in certain circumstances (diffusion phenomena have to be negligible).

7.2.6. Fields of quantities and global measurements

7.2.6.1.

Introduction to field quantities

In practice, local quantities are of limited interest, unless they are representative

of the device being studied. The dimensioning of a device or the control of its

operation requires knowledge of global quantities (mass flux, thermal power, etc.)

348 Fundamentals of Fluid Mechanics and Transport Phenomena

or of certain local quantities, which may be associated with possible damage

(temperature at a hot spot in thermal systems). These two kinds of data require

knowledge of the associated field quantities, either for integration (computation of a

flow rate for example) or for identifying the point where critical values may be

attained. In many situations, knowledge of field quantities can only be obtained by

exploring the domain by means of local measurement probes. This is clearly only of

interest if the concerned experiment is reproducible, which can only be the case if

the flow is perfectly defined. It is then possible to obtain velocity, pressure,

temperature fields, etc. This situation does not include poorly defined flows

regardless of their origin (instabilities, turbulence, etc.).

Steady flows do not present particular difficulties, since the instant of

measurement is unimportant. For unsteady flows, this is not the case however, as the

fields of a quantity g(x

t) now depend on time. The installation of an ensemble of

probes, which instantaneously and simultaneously measure the quantity g over the

entire domain considered, is not generally realistic. Only methods which allow the

obtaining of full instantaneous fields by optical means are possible; such techniques

have seen significant progress in recent years.

7.2.6.2.

Direct obtaining of field quantities of a flow

7.2.6.2.1.

Visualizations

The visualization of a flow can consist of “photographing” visible material

elements which have been placed into the flow without disturbing it at fixed

locations with respect to the observer. For instance, it is possible to place pieces of

light wires on a grid inside a flow; these will then be oriented depending on the

direction of the local velocity. The same procedure can be used on a wall in order to

obtain the direction of streamlines and to visualize separated zones. The visible

effect is here a Eulerian representation (observation of streamlines).

It is also possible to introduce streams of smoke into the air or to inject colored

dyes in a liquid medium. This kind of injection requires certain precautions (suitable

injection velocity, density of colored dye equal to that of the liquid medium, etc.).

Diverse particles can also be introduced and entrained by the flow (which has not at

some point stopped to observe the flow structures visible in a river, a gutter, etc.).

However, it is important to remember that this kind of visualization is Lagrangian,

and it shows streaklines which may be very different from the streamlines and

trajectories (section 3.3.2).

Finally, Eulerian visualizations can be performed on walls by means of physical

processes (entrainment of a coating comprising particles for example) or physico-

chemical processes producing a differential deposit on wall streamlines (evaporation

Measurement, Representation and Analysis of Temporal Signals 349

of the coating solvent, or electrochemical deposit under particular conditions). For a

more complete and practical information, see ([HAN 05], [MER 87], [YAN 01]).

7.2.6.2.2. Obtaining instantaneous fields

The preceding visualization methods are rather qualitative. We can make these

quantitative by introducing tracer particles and photographing these with suitable

lighting, either over a period short enough for visualization of the distance covered

during the lighting-time or by means of two successive light pulses. It then remains

to measure the segment covered by each particle in order to ascertain the velocity. It

is obviously necessary to know the position of the particles in space, and this is done

by generating a light sheet of small thickness, particles external to this light sheet

not being visible. This method requires a seeding of particles with very small

diameter (1 to 3 Pm) which is not biased (section 7.2.3.2). Such approaches are

known as particle image velocimetry (PIV). Modern techniques for the generation

of two close light pulses of short duration (5 to 10 ns) by a suitable laser and the

processing of images have made this technique less fastidious than in the past ([RAF

98], [STA 00], [STA 00]). However, in turbulent flows, the obtaining of a mean

velocity field requires statistics to be gathered from a large number of identical

experiments (sections 7.2.6.3.2 and 7.2.6.3.3).

Optical methods of temperature and concentration measurement provide field

quantities; as mentioned earlier, the semi-transparent aspect of the medium

considered (which is both emissive and absorbent) leads to rather global

measurements which depend on thickness, and a deconvolution of signals acquired

in a band of wavelengths is necessary in order to reconstruct the value of the

quantity in each slice. These optical procedures are used in observations by

meteorological satellites.

7.2.6.3.

Application in unsteady flows

7.2.6.3.1.

Unsteady repeatable flows

We here assume that the flow can be reproduced at will, in other words there

exists an initial instant with identical initial and boundary conditions. This is true for

transitional regimes, which precede the established regimes, and also that of

periodic flows.

We repeat the flow regime which is to be studied as many times as necessary,

and measure the value of the quantity g(x, t) each time (Figure 7.5a) at different

points (x

,…, x

,…). We then represent the fields (Figure 7.5b) at successive

instants (tx

,…, t

,…) which we have chosen. These operations, which are in

principle quite simple, clearly require the use of numerical calculations ([PEU 79],

[PEU 89], [PEU 91]).

350 Fundamentals of Fluid Mechanics and Transport Phenomena

(

, t )

(

, t )

(

,t)

(

)

(

, t

)

(

, t

)

(a) time representation of g(x,t) (b) space representation of g(x,t)

Figure 7.5. Obtaining instantaneous fields of quantity g from

point measurements as a function of time

7.2.6.3.2. “Steady” turbulence flows

A turbulent flow presents random fluctuations. We will here call a “steady”

turbulent flow any flow whose mean value of the measured quantity, over a large

enough time, does not depend on the initial instant chosen to begin the

measurement. We see immediately that an analogous procedure to that described in

the last section allows the obtention of quantities derived from the curves g(x, t) by

means of appropriate operations (mean values, fluctuations, variance of the quantity

g, etc.) used in the study of turbulent flows ([SCH 99], [YIH 77]).

7.2.6.3.3. Unsteady “reproducible” turbulent flows

This category of flow is obtained by imposing fixed initial and boundary

conditions that are defined functions of time. As before, we repeat successive

measurements at different points by reproducing the experiment exactly. As the

flow is maintained in a turbulent state, in other words it fluctuates from one

experiment to the next, it is necessary to perform the measurement a sufficient

number of times at each point, for the associated statistical quantities to be

calculated; these statistical quantities are the only ones which are meaningful for this

ensemble of experiments.

Measurement, Representation and Analysis of Temporal Signals 351

7.2.6.4.

Measurement of global quantities

Global quantities such as mass or volume flux, internal energy or enthalpy flux

are very important in industry. Precise knowledge of these can be obtained by

integrating local measurements performed on a grid that covers the region of

interest. This method of integration is used for establishing norms for precise

measurement. However, these methods are time-consuming, expensive and often

cumbersome. It is thus necessary to implement more global procedures based on

laws of fluid mechanics and transfer. The corresponding devices require calibration

in laboratory conditions. Each of these measurement procedures is based on a

particular physical phenomenon, for example, the flow rate in a duct can be obtained

from the mean spatial velocity deduced from the frequency of vortices emission

behind a cylinder, after a calibration of the used device.

7.2.7. Errors and uncertainties of measurements

7.2.7.1.

Introduction

Errors, which occur in the measurement of physical quantities, arise as a result

of diverse factors. It is first of all necessary to appreciate the experimental

conditions, which lead to an experiment always being of an approximate nature. The

control of experimental conditions (temperature and velocity, etc.) are factors which

can significantly diminish these errors, although without eliminating them. In other

words, the data of a problem are only known within the bounds of some uncertainty.

Measurements are always accompanied by errors of various kinds, depending on the

kind of methods that are used:

with analog devices errors come from the sensitivity of the components which

are used, parasite phenomena (influence of temperature, etc.), dry friction in devices

with moving parts, aging of components, etc.;

the numerical treatment of an electric signal introduces no error (with the

exception of rounding errors); it does however introduce errors in the signal

acquisition to be treated (sampling and digitization mainly: for example, the

digitization on 8 bits leads to an additional uncertainty equal to ½

in absolute

value).

Because they retain many decimal places, numerical devices are too often

considered to be capable of a precision that they do not necessarily provide. They

constitute most often only the last visible phase of a complex process of

measurement, the precision of which is not representative: the precision of the

numerical presentation is not directly related to the uncertainty on the quantity

measured and the error in the measurement of a voltage by means of a measurement

system having a numerical display is not simply the digit of the last decimal place.

352 Fundamentals of Fluid Mechanics and Transport Phenomena

7.2.7.2.

General properties

A measurement error is not always exactly known, otherwise it would be

possible to eliminate it. It is systematic when due to a particular flaw in the

measurement device; this flaw is generally not known, otherwise a correction would

be possible. It is random, when it is associated with parasite phenomena (noise from

electronic circuits, diverse fluctuations, etc.). Regardless of their origin, errors will

always exist, and can never be exactly evaluated. The only quantity that is really

available is the uncertainty of a measurement or numerical value, in other words an

upper bound on the absolute value of the error. Uncertainties moreover verify rules

that correspond to properties of distances as they are defined in mathematics

(triangular inequality, etc.).

The evaluation of uncertainty of a measurement should always be made after

error calculation, which is performed in accordance with the usual rules for

calculating small variations in the variables of the problem. For example, we

calculate a quantity ),( bas using measurements of signals a and b. The error

can

immediately be obtained as a function of the errors

G

a and

G

GGG

sass 



The formulae obtained in the calculation of errors allow us to obtain the

uncertainty of the quantity considered by bound estimation for the errors. For the

quantity s, it is possible to obtain the uncertainty 's as a function of the uncertainties

'a and 'b:

bsass

'' '

[7.1]



We draw the reader’s attention to this last point, which is very important in

practice. It is indeed regrettable that these elementary ideas are so often completely

forgotten in the beginning of higher level education.

Let us note that depending on the content of the mathematical expressions, the

uncertainties may be partially compensated or notably worsened, as we will see in

the following example, where we evaluate a quantity ),(

bas

from measurement of

signals

and

each of which has the same uncertainty

. We will leave it to the

reader to calculate the relative uncertainty which results in

in the two following

cases:





(answer:



baab



''

' )