Durst F. Fluid Mechanics: An Introduction to the Theory of Fluid Flows

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21.1 Introductory Considerations 655

contamination occurs inherent for all components exposed to the ﬂuid. Re-

liable measurements with hot-wire and hot-ﬁlm anemometers are therefore

often possible only under laboratory conditions. In practical ﬂow studies, one

accepts that dirt is deposited on the measuring sensors continuously with

time. The dirt layer developing on measuring sensors forms a heat insulation

that was not taken into account when the measuring sensor was calibrated.

In order to avoid measuring errors, caused in this way, recalibration of the

measuring probe has to be carried out at short time intervals, which can be

very time consuming.

Most measuring methods that require the insertion of measuring probes in

ﬂows measure the ﬂow velocity of interest only indirectly, i.e. with most ﬂow-

measuring instruments physical quantities are measured which are functions

of the ﬂow velocity. Direct measurements of the local ﬂow velocity are often

not carried out. Unfortunately, the measured quantities, through which the

ﬂow velocity is determined, are mostly also functions of the thermodynamic

state properties of the ﬂuid medium. These ﬂuid-property inﬂuences have to

be known and have to be taken into account in the calibration of the sensor,

in order to make the interpretation of the ﬁnal measured data possible, i.e.

to yield the ﬂow velocity through measurements. When ﬂuctuations of the

state parameters of the ﬂuid medium occur during measurements, e.g. in two-

phase ﬂows, ﬂows with chemical reactions, etc., these have to be known in

order to determine accurately the local velocity with hot-wire anemometers.

However, in practical measurements it is often not possible to know the ﬂuid

properties at all measuring times, and they can therefore not be employed in

the interpretation of the measured velocity signals.

The above-mentioned diﬃculties in the employment of indirect measur-

ing techniques for ﬂow velocities have led to the development of the laser

Doppler anemometry, which allows, almost directly, the local ﬂow velocity

to be measured. By measuring the time which a particle needs for passing

through a well-deﬁned interference pattern, the ﬂow velocity of the parti-

cle is determined. Such measurements do not depend on the often unknown

properties of the ﬂow ﬂuid. Measurements are possible in one- and two-phase

ﬂows, and also in combustion systems and in the atmosphere. The measuring

technique can moreover be employed in ﬂuid media ﬁlled with particles, as

they often occur in practice. However, its employment requires optical access

to the measuring point and thus a suﬃcient transparency of the ﬂow medium.

In this respect, the employment of laser Doppler anemometry is also limited.

Its application allows, nevertheless, the determination of ﬂow velocities in a

number of ﬂows that are important in practice, but cannot be investigated

by other measuring methods.

In addition to the above-mentioned measuring techniques for determining

local, time-resolved ﬂuid velocities, the determination of pressure distribu-

tions is also very important in experimental ﬂuid mechanics. This is given

consideration in the next section which treats the measurement of static pres-

sures. Introductory presentations of the measurement of dynamic pressures

656 21 Introduction to Fluid-Flow Measurement

follow, in order to show that stagnation-pressure probes can be employed for

the determination of local velocities.

In addition to measurements of local velocities and pressures, measure-

ments of wall-shear stresses are often required in ﬂuid-mechanical investiga-

tions. Although reference to measurements of these quantities occasionally

is made, they are not at the center of the considerations in this chapter.

The following considerations have rather to be understood as an introduc-

tion to ﬂow-measuring techniques as a subﬁeld of ﬂuid measurements. The

introduced practice-oriented basics of ﬂow and pressure measurements should

help to round oﬀ or complete the education of students in ﬂuid mechanics.

21.2 Measurements of Static Pressures

In the discussion of ﬂows with boundary-layer behavior (see Chap. 16), it

was shown that the momentum equation in the cross ﬂow direction reduces

to ∂P/∂y = 0. This important property of the pressure ﬁeld of boundary-

layer ﬂows is often employed for the measurement of wall pressures, as it

permits one to determine easily the total pressure distribution in boundary-

layer ﬂows without probes having to be inserted into the ﬂow. In order to

determine now the pressure distribution P (x), with x being the ﬂow direction,

it is suﬃcient to drill holes into the walls of the test rig, such as channels and

pipes, as shown in Fig. 21.2. However, for precise wall pressure measurements,

it is very important that the bore diameter is kept small, about the order of

Too large a hole

for wall pressure

measurements

≥

0.25 d 1.5 mm

Correct hole design

for wall pressure

measurements

Fig. 21.2 Measurements of wall pressures through holes in the walls

21.2 Measurements of Static Pressures 657

magnitude of 0.5 mm diameter. Too large bores lead to recirculation ﬂows as

indicated in Fig. 21.2. These lead to measuring errors that yield wall pressures

that do not correspond to the true values. Similar errors can be caused by

disturbances introduced by burrs, edges, etc., that remain after drilling of

the pressure-measurement holes. Further information on making out pressure

holes can be found in [21.1] and [21.2].

A somewhat diﬃcult task is the measurement of static pressures in ﬂows,

where the pressure in the ﬂow is not obtainable through wall-pressure mea-

surements as described above. In this case, probes have to be inserted into

the ﬂow of the kind shown in Fig. 21.3. These static pressure-measuring

probes have a streamline design and are provided with one or more holes

in their walls. These holes are connected through pressure tubes of small di-

ameter, located in the interior of the probe, to pressure-measuring sensors

through which the desired pressure information is obtained. As indicated

in Fig. 21.3, it is very important that the appropriate location of the holes

is determined experimentally in such a way, that actually the local static

pressure is measured by the probe. Often, with suﬃcient precision, potential-

theory computations are suited to determine the best location for the holes

in the ﬂow direction. The probe design shown in Fig. 21.4 proved valid for

the employment of static pressure measurements in ﬂows.

In order to measure static pressures, it is necessary to employ cor-

responding pressure-measuring instruments. These are brieﬂy described

below.

The simplest of these measuring instruments is the U-tube manometer

shown in Fig. 21.5. If a pressure diﬀerence develops between points A and B,

the liquid levels in the two tubes will adjust themselves in such a way, that

the existing pressure diﬀerence is compensated by gρ

h,withg being the

Pressure

too high

Influence of

probe holde

Position of

hole for static

pressure measurements

Influence of profile

Pressure

to o low

Streamlines

of the flow

1.0

stat,p

-P

stat,1

V /

Fig. 21.3 Measurements of wall pressures with pressure probes

658 21 Introduction to Fluid-Flow Measurement

6 D

15 D

Suggested dimensions

Fig. 21.4 Pressure-measuring probe and location of static pressure-measuring hole

= P

P = P

- P

g h

Pressure measuring probe

= Atmospheric

pressure

= measure of the

pressure difference

Stat. pressure

Fig. 21.5 U-tube manometer for the measurement of pressure diﬀerences

gravitation constant, ρ

the density of the ﬂuid in the U-tube and h the

diﬀerence in the height of the surface layers.

The distance between the layers of the two ﬂuid levels h is therefore in di-

rect relation to the desired pressure diﬀerence and can be read by means of an

installed scale. It can easily be seen that the measurable pressure range of

the instrument can be adapted to each measuring problem by the selection of

diﬀerent ﬂuids (ρ

). The ﬂuids most commonly used in practice are alcohol,

water and mercury, allowing h to be adapted to the measuring problems due

to the chosen ﬂuid density.

When placing a large ﬂuid reservoir at a location in the U-tube manome-

ter and inclining the actual measuring tube at an angle of (ρ

)towardsthe

horizontal line, as shown in Fig. 21.6, one obtains a considerable improve-

ment in the sensitivity of the manometer. A water column h develops, with

21.2 Measurements of Static Pressures 659

Micro-manometer

screw

Fluid

reservoir

∆

Microscope

Zero-marker

∆

Fig. 21.6 Inclined-tube manometer with zero adjustment

Floating element

Transparent

scale

Illuminated scale

and projection system

Zero level setting

Fig. 21.7 Betz manometer with optical reading system for low pressure diﬀerence

measurements

this set-up, i.e. a displacement of the ﬂuid over a distance of h/sin θ in the

inclined arm of the instrument. The zero position, i.e. “zero adjustment” of

the instrument, is obtained by adjusting the height of the ﬂuid reservoir.

In the Betz manometer, as shown in Fig. 21.7, a ﬂoating element with an

attached measuring scale is used to measure the change in the ﬂuid level. The

scale, which is made of glass, is projected by an optical enlarging system on

to a screen, where another interpolation scale is installed. By this means the

reading precision of the measuring instrument is considerably improved.

In the employment of all manometers, a stable and vibration-free base is

of importance, in order to avoid vibrational inﬂuences on displays. Mobile

pipes between the pressure probe and the manometer employed have to be

avoided also, in order to avoid ﬂuctuations of the ﬂuid levels caused by pipe

movements, or by volume changes of the pipes. In particular, however, the

room temperature during a measurement has to be kept constant, as the den-

sity of the ﬂuid (ρ

) is temperature-dependent, so that temperature changes

during the measurements are not measured as pressure changes in the ﬂow.

660 21 Introduction to Fluid-Flow Measurement

In today’s experimental ﬂuid mechanics, the employment of U-tubes as

manometers is less common. Developments of pressure sensors have led

to measuring instruments that nowadays can be employed successfully to

measure pressures in ﬂuid-ﬂow experiments.

21.3 Measurements of Dynamic Pressures

In ﬂuid mechanics, mechanical probes are employed for measuring the total

pressure P

∞

+(ρ/2)U

∞

. For these probes, diﬀerent designs have been de-

veloped over the years. Particularly simple designs of total-pressure probes,

which are named Pitot probes, after the French physicist Henri Pitot (1695–

1771), are sketched in Fig. 21.8. Probes of this kind are very suited for

total-pressure measurements; however, they require precise adjustment, ori-

entating the probe in the ﬂow direction, so that the angle inﬂuence of the

pressure distribution is eliminated. Only if this precaution is taken, one can

measure the desired total pressure correctly.

In some measurements, Pitot tubes of the kind sketched in Fig. 21.9 are

used that possess a nozzle around the actual Pitot tube. These probes are

less direction-sensitive. The nozzle serves a correcting device, guiding the ﬂuid

ﬂow, so that it passes the entrance of the Pitot tube in a more parallel way.

On combining a Pitot probe and a probe for measuring the static pressure,

one obtains the so-called Prandtl probe, that permits the following quantity

Flow direction

ca.

0.2D

P = P +

/ 2 U

tot

Fig. 21.8 Designs of Pitot probes to measure the total pressure P

tot

Flow direction

max. 45°

Fig. 21.9 Pitot probes mounted in a Venturi nozzle

21.3 Measurements of Dynamic Pressures 661

/2 hole

0.052"

I.D

0.16"

I.D.

0.175"

1.3"

0.5"

0.4"

1.6"

0.13"

0.6"

1.83"(5.9

)

2.52"(8.13

)

Fig. 21.10 Designs of Prandtl probes, shown with original dimensions, according to

data from the N.P.L. in England

to be found:

∆P = P

ges

− P =

∞

(21.1)

In Fig. 21.10, practical probe designs for Prandtl probes are shown. Prandtl

probes measure the total pressure at a corresponding probe tip and, through

the pressure holes placed downstream in the probe walls, the static pressure is

measured. Using pressure-measuring instruments to obtain the total pressure

and the static pressure at the same time, results in the displayed pressure

diﬀerence ∆P = P

tot

− P

stat

. This leads to the determination of the ﬂow

velocity:

∞

∆P

(21.2)

Disregarding the perturbations introduced into a ﬂow by the probe,

ﬂow-velocity measurements with Prandtl probes can be categorized as un-

problematic when they are employed in laminar ﬂows. Employment in

turbulent ﬂows, on the other hand, is not free of problems. The reason for

this can be found in the continuous change of the ﬂow direction and the con-

tinuous change in velocity magnitude which occur in turbulent ﬂows. This

leads to integral pressure measurements which can only be interpreted with

diﬃculty, as far as the local mean velocity of the ﬂows is concerned, as it ex-

its at the measuring location. It is important to emphasize that this problem

cannot be removed, even when using pressure-measuring instruments with

large time constants, despite the often held opinion in this respect. Due to

the hose connections from the probe to the measuring instrument, large time

constants already exist for most pressure measurements in ﬂuid mechanics,

so that even the employment of fast pressure-measuring devices is not suited

to resolve the fast time-dependent pressure.

662 21 Introduction to Fluid-Flow Measurement

Hence, pressure probes are not suited to study turbulent ﬂows. In this

respect, it is important to point out that the Prandtl tube measures the

dynamic pressure, but it is not suited to investigate the dynamics of turbulent

ﬂow ﬁelds. Such investigations are better carried out with hot-wire and laser

Doppler anemometers.

When special measuring probes are employed, in spite of the above men-

tioned limitations, for investigations of turbulent velocity ﬂuctuations, the

frequency characteristic of the probe has to be determined before the employ-

ment in turbulent ﬂows ﬁelds. The probe has to be calibrated dynamically.

Such calibrations are diﬃcult. Yet, eﬀorts of this kind are undertaken again

and again in various research eﬀorts in ﬂuid mechanics. This indicates that

in spite of major problems with pressure-measuring probes, the measure-

ment of fast varying pressures is of importance in ﬂuid mechanics. Because

of this, newly developed pressure sensors are employed, utilizing diﬀerent

mechanisms to measure pressure and to convert the pressure into suitable

electronic signals. A pressure probe consists, in general, of the actual sensor

element having low mechanical inertia, a surface as large as possible and very

short transmission elements which connect the sensor to the location where

the electrical signal is recorded. Spring-plunger devices and also capacitive

and inductive transducers are employed in this respect. All these devices have

reached an advanced state of development and nowadays can be adapted to

support pressure measurements in ﬂuid mechanics research.

21.4 Applications of Stagnation-Pressure Probes

In the preceding section, it was shown that stagnation pressure probes, in

connection with pressure-measuring instruments, can be employed in order

to measure velocities of ﬂows. Stagnation probes are in principle open tubes,

one side of which is exposed to the arriving ﬂow, while the other side is

connected to a pressure element. The pressure measured in this way is, in

principle, except for measurements at very low static pressures, proportional

to the mean force per unit surface of a ﬂuid particle located in the stagnation

point of the ﬂow. Measurement of the stagnation pressure and the static

pressure allow one to obtain the sought velocity information, if it is possible to

derive simple relationships between the velocity and the measured pressures.

This is in general possible only for non-viscous media. In viscous media, one

has to take the inﬂuence of the viscosity of the ﬂowing ﬂuid into account. This

was not taken into account when employing the simple evaluation equations

(see the equations in Sect. 21.2). In the following, it is shown how great the

inﬂuences of viscosity on measurement with pressure probes can be.

By appropriate analytical derivations, the total pressure (P

∞

+(ρ/2)U

∞

)

can be set in relation to certain properties of ﬂuid ﬂows, e.g. to the local ﬂow

velocity and the locally existing static pressure. Corresponding analytical

21.4 Applications of Stagnation-Pressure Probes 663

considerations start, however, from assumptions that often do not exist in

experimental investigations. As an example, the measured pressure P

ges

is not

in agreement with the pressure that would be measured if one permitted the

locally existing ﬂow velocity to stagnate in an isotropic manner, i.e. P

ges,i

ges

. Considerations taking geometric and dynamic inﬂuences into account,

allow one to express the pressure relation P

ges,i

stat

, existing for real probes,

by multiplying the measured pressure by a correcting function. The latter

depends on the following parameters:

1. The Reynolds number of the ﬂow, formed with the local ﬂow velocity, the

kinematic viscosity of the ﬂuid and the probe diameter, Re =(U

∞

D)/ν

2. The Mach number of the ﬂow, Ma

3. The Prandtl number of the ﬂow, Pr

4. The ratio of the heat capacitances, κ = c

5. The intensity of the ﬂow turbulence, k

6. The Knudsen number, i.e. the ratio of the mean free path length of the

molecules to the probe radius as a characteristic quantity for a slip velocity,

λ/R

7. The ratio of the relaxation time of the molecules as an assembly to a

characteristic macroscopic time interval, t

/(R/U)

8. The angle of the inﬂow, i.e. the angle between inﬂow direction and probe

axis, α

In the subsequent paragraphs, some experimentally found dependencies of

the measured stagnation pressure on some of the above-mentioned inﬂuencing

quantities are given and discussed.

Provided that the Reynolds number Re =(UD)/ν is larger than approx-

imately 100, the above treated simple description of the ﬂow around the

total-pressure measurements is suﬃciently accurate, i.e. when treated by the

theory of ideal ﬂuid ﬂows. At smaller values of Re, however, the stagnation

pressure increases as a function of the Reynolds number and also depends on

the probe geometry. The measured total pressure can be expressed as

ges

= P +

∞

+2µβ (21.3)

From this pressure coeﬃcient, C

results:

ges

− P

∞

2µβ

∞

(21.4)

The inﬂuence of viscosity on the pressure coeﬃcient (C

)isshownin

Fig. 21.11, indicating the expected deviation at low Re-number.

When compressibility inﬂuences occur, the deviations between the

measured total pressure and the ideal value can be stated as follows:

ges

ges,i

=1+

∞

2κC

1 − C

∞

√



κ − 1

∞



−κ

κ−1

(21.5)

664 21 Introduction to Fluid-Flow Measurement

020

U R

020

100 120 140

U R

(Cylinder)

ges

ges, i

1/2

ges

ges, i

1/2

(Sphere)

Fig. 21.11 Viscosity inﬂuences on total-pressure measurements

All of the above explanations, relating to inﬂuences that exist for the practical

application of pressure probes, indicate that the application of stagnation-

pressure probes for ﬂow investigations is linked to a multitude of practical

problems. Nevertheless, owing to the simplicity of application of pressure

probes in practical measurements and owing to their robustness, they are

still employed today in practical ﬂuid mechanics.

21.5 Basics of Hot-Wire Anemometry

21.5.1 Measuring Principle and Physical Principles

Hot-wire anemometry is a measuring technique that permits electrical mea-

surement of local ﬂow velocities, and it has been employed successfully for a

long time in experimental ﬂuid mechanics and also in other ﬁelds. With hot-

wire and hot-ﬁlm probes, local, rapidly changing ﬂuid ﬂows, requiring high

resolutions of measurements in terms of space and time, can be investigated.

Hot-wire and hot-ﬁlm measurements are based on the measurement of the

release of heat from hot sensors to the medium ﬂowing around them. Hot-

wire anemometry is therefore an indirect measurement technique, as it is not

the ﬂow velocity which is measured, but the velocity-dependent heat release

from a thin, heated wire to the ﬂuid medium surrounding it. In this respect,

the technique takes advantage of this velocity-dependent heat transfer of hot

wires and hot ﬁlms for measuring the local ﬂow velocity.

The heat release of the probe depends not only on the ﬂow velocity,

however, but also on other quantities:

(a) The temperature diﬀerence between the sensor and the ﬂuid medium

(b) The physical properties of the ﬂuid medium and the sensors

When keeping the inﬂuences (a), (b) and (c) constant, the sensor only

reacts to the ﬂow velocity, and the heat, withdrawn from the sensor, is then