Eindhoven University of Technology MASTER Aero · PDF file2.1 Orifice at the end of a pipe .. 2.1.1 Incompressible theory ... C Model for flow through pipe with friction; ... pressure

Eindhoven University of Technology

MASTER

Aero-acoustics of a bend : quasi-stationary models

Huijnen, J.H.

Award date:1999

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Technische Universiteit t(Ï) Eindhoven

Titel:

Auteur:

V erslagnurnrner:

Datum:

Werkeenheid:

Begeleider(s):

Aero-acoustics of a bend:

quasi-stationary models .

. J.H. Huijnen

R-1476-A

.December 1998

. Gasdynamica

.TUE: A. Hirschberg

LAUM: Y. Auregan

Vakgroep Transportfysica Faculteit Technische Natuurkunde Gebouw W&S Postbus 513 5600 MB Eindhoven

Summary

Within the frameworkof a st.rudy of flow instahilities in gas transport systems (Gasunie) and the European project on duet acoustics FLODAC we considered the aero-acoustical response of 90° bends. For engineering applications the response is often described by assuming a quasi-stationary flow.

The validity of those models has been considered for 90° bends with a sharp inner corner and two bends with larger ratios of curvature to pipe diameter (respectively one and three).

The theory describes quite well the behaviour of the sharp bend up to M = 0.23. The response of the smooth bends is difficult to measure and may not correspond to a quasi-stationary behaviour.

The bend has generally shown a dissipating behaviour. However, measurements on the smooth bends suggest that these bends are able to produce sound when flow is added to the system. Further research is recommended, in particular a measurement of the scattering matrix at higher flow veloeities and reflection coefficient measurements as a function of the acoustic velocity amplitude.

Contents

Introduetion

1 General concepts 1.1 Basics of fluid dynamics

1.1.1 Conservation laws . . . 1.1. 2 Constitutive equations 1.1.3 The Bernoulli equation .

1. 2 Acoustics . . . . . . 1.2.1 Introduetion ...... .

Plane waves . . . . . . . 1.2.2 1.2.3 1.2.4 1.2.5 1.2.6 1.2.7 1.2.8

Approximations; non-dimensional numbers Damping in the boundary layers of the flow The reflection coefficient . . . . . . The transfer and scattering matrix Abrupt cross-section change .... The bend and other pipe segments

2 Quasi-stationary behaviour of the sharp bend 2.1 Orifice at the end of a pipe ..

2.1.1 Incompressible theory 2.1.2 Compressible theory .

2.2 Flow through a sharp bend .. 2.2.1 Incompressible model . 2.2.2 Compressible model ..

2.3 The hodograph methad . . . . 2.3.1 Two-dimensional potential flow 2.3.2 Principlesof the hodograph methad . 2.3.3 The hodograph methad used for the sharp bend

2.4 The vena-contracta ..... . 2.4.1 Loss coefficient Cd ........ . 2.4.2 The Borda mouthpiece . . . . . . . 2.4.3 Application of multiple corrections

3 Smooth bends 3.1 Loss coefficient Cd 3.2 Wall roughness .. 3.3 Quasi-stationary models

11

1

3 3 3 4 5 5 5 6 8

10 11 12 13 15

16 16 16 17 19 19 21 22 23 25 25 28 29 30 31

32 32 32 32

4 Experimentalset-up 4.1 Measurement of the reflection coefficient; two microphones method .

4.1.1 Theory ......................... . 4.1.2 The frequency range of the two microphones method

4.2 Measurement of the scattering matrix; two souree method 4.2.1 Translation of the scattering matrix element

4.3 Calibration method 4.4 The set-up at TUE . . . . . . . . . . .

4.4.1 The siren . . . . . . . . . . . . 4.4.2 Positioning of the microphones. 4.4.3 Measurement of the Mach number . 4.4.4 Accuracy analysis .

4.5 The set-up at LAUM . . . . . . 4.5.1 The souree ....... . 4.5.2 Microphone positioning . 4.5.3 Determination of the Mach number 4.5.4 Anechoic termination . . . . . . . . 4.5.5 Determination of the wave number

5 Results of measurements at the sharp bend, R = D /2 5.1 The reflection coefficient (TUE) ...

5.1.1 Measurement at an open end 5.1.2 Measurement at a sharp bend

5.2 The transfer matrix (LAUM) 5.2.1 Measurement without mean flow 5.2.2 Measurement with mean flow . .

6 Results of measurements on smooth bends 6.1 The reflection coefficient (TUE) 6.2 The scattering matrix (LAUM)

7 general conclusions

A A sharp bend in a tube; compressible equations

B Measurement of the scattering matrix elements

C Model for flow through pipe with friction; Fanno theory

lll

36 36 36 37 37 39 39 40 41 42 43 44 44 46 47 47 48 48

50 50 50 51 53 53 56

59 59 61

66

69

72

74

List of symbols

Roman symbols lowercase

a m, m s-1 radius, velocity b m s-1 velocity c m s-1 velocity of sound e m2 s-2 internal energy per unit of mass f s-1 frequency f F Fanning friction factor fc s-1 cut-off frequency 2 v=r k n p q r s t

tij

u

V

w

x

y

iJ z z

Pa J m-2 s-1

m J kg-1 K-1

s

m s-1

m s-1

m s-1

m s-1

m

m

m

wave number natural number pressure heat flux radial space coordinate vector entropy time transfer matrix component x-component of the velocity, real part of the complex variabie w y-component of the velocity, imaginary part of the complex variabie w

complex coordinate in the velocity plane velocity vector spacial coordinate, real part of the complex variabie z imaginary part of the complex variabie z two-port output state vector complex coordinate in the physical plane two-port input state vector

Roman symbols uppercase

J kg-1 K-1 J kg-1 K-1

m

constant transfer matrix constant constant irreversible loss coefficient specific heat at constant pressure specific heat at constant volume diameter

1V

F(z) m2 s-1 stream funetion He Helmholtz number L m charaeteristic length M Mach number M matrix Pr Prandtl number Q rad m2 s-1 souree strength R m, 0 radius, pressure reileetion coefficient

Rb enthalpy reileetion coefficient Re Reynolds number

RE energy reileetion coefficient

Rg J kg-1 K specifîc gas constant s m2 cross-section s scattering matrix Sr Strouhal number T K, 0 absolute temperature, transmission coefficient T transfer matrix

V m s-1 velocity (V2 = (iJ· iJ))

Greek symbols

aeff m-1 effective damping coefficient

a ij cross-section ratio

av1 contraction ratio, vena-contracta

ao m-1 damping coefficient in absence of a mean flow

/3ij density ratio

"( Poisson's constant 6 m end correction

ÓL m viseaus laminar sublayer thickness

br m thermal boundary layer

Óv m viseaus boundary layer

Tl kg m-1 s-1 dynamic viscosity

T/ij calibration coefficient

"" J m-1 s-1 K-1 coefficient of thermal conductivity

À m wave length V m2 s-1 kinematic viscosity ç m complex variabie p kg m-3 density (J kg m-1 s-2 entropy wave T non-dimensional time parameter

Tw kg m-1 s-2 shear stress

c/Jx rad amplitude of the phase shift

x non-dimensionallength parameter

1/J friction coefficient w rad s-1 radial frequency m2 s-1 velocity potential j m3 s-1 volume flux

V

complex velocity potential

Other symbols

A 0( ... )

amplification factor order of magnitude of ...

Functions

:F( ... ) 9( ... ) I( ... ) n( ... )

function of .. . function of .. . imaginary part of ... re al part of ...

Subscripts

a radius b enthalpy c cut-off d diaphragm exp experimental f flow meter h heart of the flow z index

J index k index l index, length ref reference t turbulent th theoretica! tot tot al u end V vena-contract a w wall x at position x B anechoic, Borda mouth piece I first test state IJ second test state 0 refers to the attached flow 1 upstream 2 downstream

Superscripts

+ in downstream direction in u pstream direction

± in down- and upstream direction, respectively refers to the acoustic flow

Vl

amplitude conjugate complex

Vll

Introduetion

Low frequency acoustic pulsations in gas transport systems can cause severe noise and vibration problems. Resonators along a pipe are delimited by two reflectors such as two resonating side branches. In complex manifolds a large number of potential resonators exists.

Gasunie and the section Gasdynamica at the University of Technology in Eindhoven (TUE) study the stability of the flow in gas transport systems. Research at TUE has determined the resonance conditions of resonators formed by closed side branches along a straight pipe. According to the results of this research, pulsations should occur much more often than found in practice by Gasunie. A logical explanation for this is the presence of a damping element.

The bend is generally expected to have a strong influence on such acoustic pulsations and to be such a damping element. Snel [1] showed that the sharp edge has a strong damping effect when placed in between two closed side branches. That a sharp bend has a strong acoustic influence on pulsations is also shown by Ziada [2], who measured at a sharp bend which was placed upstream of two closed side branches.

In addition an important question is whether or not this damping effect of bends can be described by a so-called quasi-stationary model. This model is commonly used in engineering practice. From research by Ter Riet [4] it has been concluded that this should be investigated by measuring the acoustic response of a bend to imposed incident waves: reflection coefficients and scattering matrices. In the presence of a main flow this response involves interaction of the acoustic field with the vortical field ( vorticity).

In this report we present the results of an experimental study of the aero-acoustic response of a 90° bend, see figure 1. The experiments in the first part of this study have been carried out at TUE. The reflection coefficient is measured as a function of the

Q.) b)

Figure 1: The three bends which were investigated: a) sharp bend (R = D /2) with rounded outer corner, b) smooth bend (R = D), c) smooth bend (R = 3D).

1

Mach number M which is the ratio of the mean flow velocity to the speed of sound c0

(0 < M < 0.25). The second part of this study was carried out at the Laboratoire d' Acoustique de

l'Université du Maine in Le Mans (LAUM). It is a study within the frame work of the EC project FLODAC. We have carried out measurements of the influence of the main flow (0 < M < 0.07) on the acoustical response of bends (scattering matrix) in a broad frequency range (50Hz < M < 1300Hz).

This report is divided into four main parts. In the first chapter we will discuss general theoretica! concepts, such as the conservation laws in fluid mechanics and elementary acoustics.

Chapter two and three focus on the aero-acoustical influence and quasi-stationary model of, respectively, sharp and smooth bends.

Chapter four describes the measurement methods and the set-ups of TUE and LAUM. In chapter five and six, present results from measurements at respectively sharp en

smooth bends will be presented and compared with the quasi-stationary models. In the last chapter, chapter seven, the conclusions of this research are summarized.

2

Chapter 1

General concepts

This chapter is an introduetion to the chapters two, three and four contains the general concepts needed to understand the measurement methods. Section 1.1 introduces the conservation laws, constitutive equations and the Bernoulli equation.

Paraghraph 1.2 deals with duet acoustics and introduces the pressure reflection coefficient and the transfer matrix.

A more complete introduetion in fluid dynamics and basic acoustics can be found in Batchelor [3].

1.1 Bas i es of fi uid dynamics

1.1.1 Conservation laws

The conservation laws of mass, momenturn and energy are the starting point in the derivation of the basic fluid dynamica! and acoustical equations. We consider a fluid partiele in a frictionless flow on which no external force acts and to which no external heat is supplied. The conservation laws in differential vector notation for a fluid partiele are, respectively,

in which: p p v d dt e ij K,

T

density pressure velocity

1 dp

p dt dv

p dt

d(e + ~v2 ) p dt

-(~. v)

-(~. p)

- (~ . if) - P(~ . v)

substantial derivative, -ft = gt + ( v · ~) internal energy per unit of mass heat flux, ij= -K,~T coefficient of thermal conductivity temperature.

3

(1.1)

(1.2)

(1.3)

1.1.2 Constitutive equations

The heat conductivity coefficient K, is empirically specified. This specification is among others given in table 1.1 for dry air. The five conservation laws of mass, momenturn and energy form, tagether with the following constitutive equations and thermadynamie identities, a closed system of eight equations with eight unknowns (p, v, p, e, T and the entropy s):

e e(s, p) (1.4)

T ( äe) äs P

(1.5)

p = 2 ( äe) p ä .

p s (1.6)

Some important concepts which will be used are the specific heats Cv and CP at constant volume and pressure, respectively, and Poisson's constant r:

Cv (:), (1.7)

CP (:), (1.8)

I CP Cv·

(1.9)

The speed of sound c is defined by

c2 = ( ~~) s = I ( ~~) T . (1.10)

Applying equations 1.4, 1.5 and 1.6 for an ideal gas gives

(1.11)

in which R9 is the specific gas constant (R9 =CP- Cv). With the definition of the speed of sound we find

(1.12)

We call an ideal gas "perfect" when CP and Cv are temperature independent. The thermadynamie identity of a perfect gas is

e =CvT rv p1-

1 exp ( dv) ,

which in the isentropic case gives the Poison's equations,

p (

p )' ( T ) 0 ( c ) ~ Pref = Tref = Cref Pref

4

(1.13)

(1.14)

Table 1.1: Specifications of dry air at T = 273K and p = 1bar, unless mentioned otherwise.

Coefficient of heat conductivity "' 24. 10 3 Wm 1K 1

Dynamic viscosity T] 1.71. 10-5 kg m-1s-1

Specific heat at constant pressure CP 1.00. 103 J kg 1K 1

Specific gas constant Rg 287 J kg 1K 1

Poisson's constant I 1.40 Density p 1.293 kgm-3

Speed of sound (T = 20°C) c 342.20 ms-1

1.1.3 The Bernoulli equation

The Bernoulli equation can be derived from the momenturn conservation law. Taking the inner product of v with the momenturn conservation law for a flow without external forces (equation 1.2) gives

.... dv .... (1~ .... ) V · dt - V · p V • V = 0. (1.15)

Substitution of ft= gt + (v· ~) and v· ~v= ~~lvl 2 +((~x v) x v), in which the term

( (~ x v) x v) will drop off since it stands perpendicular to v, gives

(1.16)

in which V 2 = (v · v). For an incompressible flow ~~P = ~~ and equation 1.16 simplifies to

.... ov .... ~ [ 1 y2 pl v·-+v· v - +- =0 at 2 P '

which gives us the stationary incompressible Bernoulli equation:

Incompressible : ~ V2 + 'E = constant along a streamline. 2 p

(1.17)

(1.18)

When the fluid is a compressible ideal gas, we find from equation 1.16 the stationary Bernoulli equation:

Stationary : ~ V2 + _I_?_ = constant along a streamline. 2 1-1 p

(1.19)

1.2 Acoustics

1.2.1 Introduetion

The ambient state describes the medium through which sound propagates. Since sound waves, or acoustic waves, are small-amplitude compression waves which can be regarcled as small-amplitude perturbations to the steady ambient state, it is useful to write the pressure p, the density p and the velocity u as

5

p(r, t) p(f, t) iJ(f, t)

Po(f') + p'(r, t) Po(f') + p'(r, t) ilo (i) + iJ' ( r, t) ,

(1.20)

where the stationary ambient state is designed by p0 (i), p0 (i) and iJ0 (i) and p', p' and u' represent the acoustic contributions. These perturbations are assumed to be small compared to the ambient state (p' /Po « 1, etc.). We will further neglect spatial variations of p0 and p0 and assume a uniform main flow velocity v0 = (u0 , 0, 0).

We will now briefly explain why it is justified to use one dimensionallinearized inviscid equations in the pipe systems which we consider. This approximation is of course not valid at a bend but only in the straight pipe segments between bends.

Since the acoustical disturbances are very small in comparison with the ambient states, second and higher order terms in these perturbations can be neglected. In other words, we linearize the equations. It can be shown that for our conditions viscous and thermal dissipation is limited to boundary layers at the walls of the duet. Furthermore, these acoustical viscous and thermal boundary layers are very thin. These boundary layers Öv and Ör respectively, have a characteristic thickness given by

(1.21)

(1.22)

in which v is the kinematic viscosity of the gas and w the radial frequency of the wave. In our applications the ratio of Öv to the pipe diameterDis Ov/ D = 0(10-2

). This justifies the inviscid approximation.

The last simplification, the one dimensional representation of the basic equations, can be explained in the following way :

It can be proven that below a certain cut-off frequency fc the only propagating acoustical waves in pipes are plane waves which propagate in axial direction (Pierce [5]). This cut-off frequency applies to, in case of a cylindrical pipe without main flow,

co 271" Je= 1.8412-,

a (1.23)

in which a the radius of the pipe. The pipe which was used in the experiment has a radius a = 15mm. The cut-off frequency becomes fc = 6. 7kHz. The highest frequency at which we have measured is 1300Hz. This is much lower than the cut-off frequency and therefore it is justified to use the one dimensional approximation to describe the acoustic wave propagation. In the next section a description for plane waves will be derived.

1.2.2 Plane waves

With use of the definition of the speed of sound (equation 1.10), the mass and momenturn conservation laws can be written as follows in the one dimensional case:

6

1 (op op) au pc2 ot + u ox + ox 0 (1.24)

(ou ou) op p -+u- +-ot ax ax = 0, (1.25)

in which u is the velocity in the x-direction. The viscous forces are neglected to obtain better insight. These forces would lead to souree terms in the wave equations.

Multiplying equation 1.24 with c and dividing equation 1.25 by p and adding and subtracting these results give

au au 1 op 1 op -+(u±c)-+--+(u±c)-- 0 ot ox pc at pc ox

(1.26)

Ç? [! +(u±c)!] (u±~~~) = 0. (1.27)

This means that r± =u± f ~~ is invariant along c± lines in the x,t-plane. These characteristics c± are described by

( ~~) c± = u ± c. (1.28)

It shows that the propagation velocity of acoustical fiuctuations contains a convective translation term due to the flow velocity. The acoustical waves propagate with the velocity c with regard toanobserver moving with the fiuid. An observer in the laboratory coordinate system sees them propagating with a velocity u± c in respectively down- ( +) and upstream (-) directions. With use of p'_ « 1, lL « 1 and 3!!_ « 1 equation 1.27 can be linearized in

Po Po uo a characteristic form:

[a a](' p') at+ (uo ±co) ox u ±PoCo = O. (1.29)

The corresponding wave equation for p' is

(1.30)

The general solution is

x x p' (x' t) = :F ( t - ) + g ( t + )

co+ uo co- uo (1.31)

which corresponds with traveling waves propagating in respectively the c+ and c- directions in the x, t-plane.

We only consider harmonica! solutions and use the complex notation:

(1.32)

7

---~- ~~---

in which: p+, p- complex amplitudes of the traveling waves in the different directions at time t = 0 and position x = 0

w radial frequency, w = 21r f k0 wave number of the traveling wave without a main flow, k0 = wjc0

M Mach number of the main flow, M = u0 jc0 .

From here R( ... ) will be left out of the equations and assumed implicitly. By defining two different wave numbers k+ and k- for the waves traveling in the

downstream and upstream direction, respectively,

we write for the acoustic pressure

ko 1±M'

(1.33)

(1.34)

in which the hat denotes the complex notation. The complex notation for the acoustical velocity follows from substitution of equation 1.34 in the conservation of mass ( equation 1.24):

u'(x, t) = û(x)eiwt = ( p+ e-ik+x _ P- eik-x) eiwt. PoCo PoCo

(1.35)

From now we will use the amplitudes p+ and p- instead of the pressure and velocity to describe the acoustic field in a pipe segment.

1.2.3 Approximations; non-dimensional numbers

We start the discussion of the validity of some approximations by introducing a few nondimensional numbers:

uo =M:

co Mach number (1.36)

wL =He: Helmholtz number (1.37)

co wL

=Sr: Strouhal number with respect to the main flow. (1.38) uo

(1.39)

L is the characteristic lenght.

Incompressible; M 2 « 1

We first consider a stationary flow. The Mach number M provides a measure for the density variations in this flow. New variables, the total variables, follow from the stationary Bernoulli equation ( equation 1.19):

8

~ v2 + _"!_!!_ "( Ptot (1.40) ---

2 "(-lp "( - 1 Ptot

1 1 1 2 -V2+ --c2 "( _ 1 Ctot (1.41) 2 "(-1

~V2 + CpT CpTtot· (1.42)

The variables Ttot, Ct0 t, Ptot and Ptot are the total temperature, total speed of sound, total pressure and the total density, respectively. They are the state variables which are found when a given flow with velocity V, pressure p and density p is stopped isentropically. When writing the second equation with the Mach number M = V I c we get

"! - 1 M2 + l = ctot = Ptot 2 ( )"(-1 2 c2 p

(1.43)

Compressibility effects appear to be a second order effect in the Mach number:

(1.44)

The flow can be considered incompressible when M2 « 1. In this research Mach numbers of the measured main flow varied from M = 0 to M = 0.25. We expect only small departmes from the incompressible flow approximation.

Quasi-stationary; Sr « 1

The Strouhal number Sra = waluo gives a measure of the ratio of the residence time aluo of a partiele and the acoustical oscillation period. The momenturn conservation law 1.2 can be written in non-dimensional coordinates as follows:

aa - -Sra OT + il'\lil = -('\7 · p), (1.45)

in which T = wt, V= a V, il = vluo and j5 = ~. puo

At low Strouhal numbers Sra « 1 the partial derivative with respect to the time can be neglected and the flow may considered to be quasi-stationary. Changes in pressure ( or other variables) of a partiele due to the unsteadyness of acoustic waves are much smaller than the change due to the flow through a pipe discontinuity. The lowest Strouhal numbers in this research are about Sra = 2 · 10-1 . The quasi-stationary approximation is expected to be reasonable.

Compact; He « 1

The Helmholtz number He = w L I c0 = 27r L I À gives the ratio of a characteristic length L and the acoustical wavelength À. The wave propagation can be neglected in case the length scale Lis much smaller than the wavelength À, exp(ikL) = 1 + O(kL) = 1 + O(He). The flow is then called compact. Because He, Sr and Marenotindependent (He= SrM), a quasi-stationary incompressible flow is always compact. But an instationary flow with Sr= 0(1) can be compact in case M « 1.

9

1.2.4 Damping in the boundary layers of the flow

Large velocity and temperature gradients arise in the thin boundary layers at the pipe wall due to the boundary conditions at the wall for the velocity ( v = 0) and the temperature (T = constant). The velocity gradients in these layers cause viscous dissipation, and a heat flux towards the wall arises due to the temperature gradients. Because the density fluctuations in the boundary layer may not be considered isentropic, as in the bulk of the flow, a mass flux towards or from the boundary layer will arise. The component of this flux which is in phase with the acoustical pressure fluctuations corresponds to acoustical sound absorption. Both processes (viscous and thermal) cause energy losses, due to which the acoustical energy of the plane waves decreases. The souree terms of the wave equations may not be neglected as was assumed in equation 1.27. In first approximation this effect can bedescribed by introducing a complex wave number k.

A derivation of the dispersion relation in absence of mean flow ( u0 = 0) is given by Boot [6]. The result of Kirchhoff fora cylindrical pipeis

ko + (1- i)ao

ko_!!_ 1 + -- , 6 [ 1'- 1] 2a JPr

(1.46)

(1.47)

in which a0 is called the damping coefficient. We find for the used set-up in which the pipe radius equals a = 1.5cm and a frequency stays below f = 1300Hz

(1.48)

The damping has a dominant influence on the amplitude of the pressure wave only when the lengthof the system becomes of the order of 1/a0 , which is about 14 meters. In pipes of the order of lm as considered, this will be a small but significant correction.

Inft uence of a main flow

The damping of acoustical waves in pipes can increase due to the presence of a mean flow. The flow in a pipe with radius a is among others characterized by the Reynolds number Re = 2a;o. Above a certain flow velocity the flow will be turbulent. The turning point for pipes is at Re ::::::: 2300. In this research the Reynolds numbers are of the order 104 - 105

and so the flow is always turbulent. Because the velocity approaches zero at the wall a viscous sub-layer with thickness ÓL ::::::: 12.5vJ p0 /Tw arises in which the degree of turbulence is negligible, see Peters [7]. The stationary wall shear stress Tw can be found with Blasius' empirical friction law which can be applied for Re < 105 and gives

(1.49)

(1.50)

in which 1/J is the non-dimensional friction coe:fficient, see Bchlichting [8]. The boundary layer will become thinner when the flow velocity increases. Outside this viscous sub-layer the turbulent viscosity 'T}t will be much larger than the molecular viscosity 'TJ·

10

Wh en the viseaus sub-layer is larger than the acoustical sub-layer ( c5 L > Óv), the shear wave and the heat conduction wave will be damped away completely within the viseaus sub-layer. We expect that the thermal wave salution and the shear wave salution in this situation will be equal to the situation without flow and the wave number is given by

k± = ko + (1 - i)a0

1±M ' (1.51)

This formula fails at low frequencies or high flow veloeities (Sr « 1). This is due to the fact that the shear waves will not be strongly damped within the viseaus sub-layer when Óv > c5L. Particularly the shear wave will be strongly influenced by the interaction with the turbulent heart of the flow.

The damping coefficient can bedescribed by an effective damping coefficient aeff similar to that proposed by Peters [7] in the limit ÓL « Óv:

in which r0 is the fluctuating part of the shear stress.

~ 12.5v !Pi_, V Tw

(1.52)

The damping factor increases considerably when the flow velocity increases. Figure 1.1 shows the relationship between aeff and ódc = c5v/c5L. A detailed discussion about the dependency of the damping factor to the flow velocity is given by Peters [7], Reijnen [9] and Howe [10].

4~~========~----~ t' x, I

\

105 [ -x ~ (a) ~

~ ''x I 0 I

3 1\ 0~ ~ I 0.

95 n B 10 12 14

Re (Z) 2

0

,.,.~

!0 20

" 0

'b

JO 40 50

:~[,-.-!i' A-.-~--------(h-)---,

I a A

""f ' lm (Z) IJ.n t ~o o

0.4l ·. 0

: •

0

0 2

~~~~-~~-~~J

0

•

10 20 30 .\j) 50

Figure 1.1: Averaged wall impedance Z obtained from the averaged wavenumber k0 . a) R(Z) = aettfao as a function of ó;tc. b) I(Z) = [ko- (w/co)/(1- M2 )]/(-iao) as a function of ó;t;. Measurements by Peters [7] at variable Mach number: 6., ka= 0.0371; o, ka= 0.0121. Variable Helmholtz number: x, M = 0.011, 0.042 and 0.107.

1.2.5 The reflection coefficient

In general the plane wave salution (see section 1.2.2) is written as a summation of a wave propagating in positive x-direction and a wave propagating in negative x-direction,

11

p+e-ik+xeiwt and p-eik-xeiwt (p' = p(x)eiwt), respectively. The complex pressure amplitude reflection coefficient R is defined as the ratio between the complex amplitudes of the in negative and positive x-direction traveling pressure waves,

p-eik-x R(x) = .k+ . p+e-2 x (1.53)

In the presence of a mean flow, the absolute value of this pressure reflection coefficient can become larger than 1 for a passive acoustical system. This is not in conflict with the energy conservation law because the energy reflection coefficient is given by

(1.54)

which becomes zero for each finite pressure reflection coefficient when M = 1. When M = 1 the reflected waves cannot get upstream anymore. In this report we will give some results in terms of the total enthalpy reflection coefficient Rb = ~-

1.2.6 The transfer and scattering matrix

When linear theory is valid, a pipe discontinuity can be modeled as an acoustical twoport. A two-port can be defined as a linear, time-invariant physical system. Such a system describes the relationship between an inputsignaland an output signal (velocity, pressure), which are independent of each other. Figure 1.2 shows the input and output states of a two-port system. The subscripts 1 and 2 denote the states at, respectively, the upstream and downstream side of the discontinuity. The discontinuities mentioned in this report

{wo -por~

+--P;

----------------------------------------~X-~lS

Figure 1.2: The in- and output signals of a two-port system. In this report the bend is regarcled as a two-port system. The positive direction of the x-axis corresponds to the direction of the main flow.

(the bend, orifice plate and open pipe termination) are regarcled as passive elements. Now, the relationship between the input and output states of a passive two-port can, in the frequency domain, be written as

Y-= Mi -' (1.55)

in which i is the state vector at the input, ff the state vector at the output and M the matrix which relates ff to i.

12

When we choose z to be the amplitudes of the waves downstream of the two-port and iJ the amplitudes of the waves upstream of the two-port, we write the definition of the transfer matrix T:

(1.56)

When choosing the state variable vector z to be the amplitudes of the waves incident on the two-port while iJ represents the waves leaving the two-port, like in figure 1.2, this gives us the definition of the scattering matrix 5 which contains four elements r+, r-, R+ and R-:

(1.57)

In section 4.2 we will see how these scattering matrix elements can be determined from measurements.

1.2. 7 Abrupt cross-section change

s,

\.V

--------------------~--------------------.x-ax's

Figure 1.3: A cross-section change in a tube at position x = 0 from 51 to 52 .

As an example of a discontinuity we will look at the reileetion at an abrupt cross-section change from 5 1 for x < 0 to 52 for x > 0, as shown in figure 1.3, without a main flow. We describe the acoustic field in terms of traveling plane waves by

p~ (x, t) = Pl (x)eiwt = (vt e-ik+x + P} eik-x) eiwt for x < 0

p;(x, t) = P2(x)eiwt = (v;e-ik+x + p:;eik-x) eiwt for x> 0.

(1.58)

(1.59)

The pressure at the cross-section change is continuous and in the compact case the terms of order k0a are neglected. This gives

( + + -) iwt + ( + + - ) iwt P1o + P1 P1 e = P2o P2 P2 e , (1.60)

13

in which p10 and p2a are the stationary pressures on the left and right hand side of the widening, respectively. Because this equation has to hold for any time t we can separate the equation in a time dependent and time independent part and which after dividing by eiwt gives

(1.61)

A second relation between the acoustical pressures of the left and right hand side follows from the mass flow continuity equation. With the definitions p = p' I eiwt and û = u' I eiwt

this becomes

S1p1u1

:::::} sl (PlO + Pl eiwt)ûl eiwt

=? S1p10û1 =? SI(Pt- pi)

S2p2u2

s2 (P2a + P2eiwt)û2eiwt

S2p10û2 (1.62)

S2(Pt- P2)

where we used equation 1.35 to calculate û as a function of p+ and p-. From equations 1.61 and 1.62 we get a transfer matrix representation for the acoustic response of the cross-section change:

[p+ l [ 1+.!.~ _.!_82-81 l [p+ l 1 _ 2 81 2 81 2

P- - _.!. 82-81 1 +.!. 82-81 p- · 1 2 8 1 2 81 2

(1.63)

When S1 = S2 this is the unit matrix. With. the condition for an infinite long tube in region 2, P2 = 0, the reflection coefficient at position x = 0 becomes

Ra = p} = S1 - S2

Pt s1 + s2' (1.64)

which agrees with typicallimit cases. Fora closed tube end S2 is zero and we find Ra = 1. In case S1 = S2 , Ra is zero which we call anechoic.

Open pipe termination

For an ideal open pipe termination, which corresponds to SI/ S2 --+ 0, we find Ra = -1. An i deal open pipe termination is perfectly reflecting but induces a phase shift of 1r.

At a distance of the order of the pipe radius a (downstream of the pipe exit) the acoustical flow is already close to a spherical symmetrical flow. In the absence of a mean flow, the inertia of this acoustical flow just outside the pipe induces a difference in Ra which is aften represented in terms of an end correction ó:

Ra= -1 + 2ikó. (1.65)

When a mean flow is added to the open pipe termination an extra correction in Ra is needed for the Mach number:

Ra = (2ikó- 1) (1 +AM), (1.66)

in which A is an amplification factor which is described by Cargill's theory, see Peters [7].

14

1.2.8 The bend and other pipe segments

In order to deduce the reileetion coefficient of a system consisting of a few pipe segments and a bend, we need the transfer matrix representation of a pipe segment. This matrix representation A for traveling waves through a smooth pipe with length l follows from the wave equation for p',

[ PT l = [ Pt l = [ eik+l 0 _ l [ Pt l - A 1 - o -tk l - · P1 - P2 e P2 (1.67)

We now look at the configuration in which there are a pipe segment, a bend and another pipe segment downstream of the position x = 0 where we want to calculate the reileetion coefficient. It can be deduced when the reileetion coefficient at the end Ru is known. The configuration is described by consiclering every segment as a black-box which is described by one of the given matrix representations:

(1.68)

15

Chapter 2

Quasi-stationary behaviour of the sharp bend

Plane acoustical waves in pipes interact with each pipe discontinuity. The interaction between geometry changes and the acoustical waves is described in a so-called transfer matrix which provides a relationship between the complex pressures on both sides of the geometry change.

In section 2.1 we will look at a simple example of an interaction between flow and acoustics. An incompressible and compressible theory for an orifice at the end of a pipe is given. In section 2.2 an incompressible and compressible model for the sharp bend is derived, by analogy with the diaphragm treated in section 2.1.

In section 2.3 the hodograph methad is described and used to discuss the flow through a sharp bend in more detail.

Section 2.4 deals with the vena-contracta and the introduetion of a Mach and Reynolds dependency in the contraction ratio.

2.1 Orifice at the end of a pipe

2.1.1 lncompressible theory

Pct= o

-----+------------~----~r,---------~X-OXIS )( : 0 ;t.J

Figure 2.1: A diaphragm at the end of a pipe.

Bechert [11] presented a simple quasi-stationary incompressible theory for the reflection coefficient of a flow through a diaphragm at an end of a pipe. We will look at the configuration as shown in figure 2.1. A main flow in the pipe with cross-section 5 1 passes a

16

diaphragm with cross-section sd and forms a free jet with cross-section sd into the free space by tearing itself away from the sharp edge of the diaphragm. The flow is considered to be incompressible and without friction till in the free jet. For simplicity we assume a quasi-stationary behaviour, which is reasonable for Srd = 1r fd/ud = 0(10-2). Changes in velocity ( or in other variables) of a partiele due to the acoustical wave propagating are much smaller than the changes due to passing of the discontinuity in geometry. Along a streamline we can use the incompressible Bernoulli equation (1.18). Tagether with the incompressible continuity equation this gives two equations which relate the pressure and velocity upstream of the diaphragm to those in the jet:

continuity : S1u1 = Sdud Bernoulli: 1 2 1 2

P1 + 2PoU1 = Pd + 2 PoUd

When we eliminate the velocity from the farmer two equations we get

(2.1)

(2.2)

(2.3)

At low acoustical amplitudes (u~ « u10 ) the quadratic term uî linearizes to uî0 + 2u10u~. The time dependent part of equation 2.3 gives

Pt + P! + M1 ( 1 - ~~) (Pt - pl) = Pd· (2.4)

Because of the free field conditions all pressure fluctuations are negligible in the free jet, Pd = 0. So, the reileetion coefficient becomes

Ru P! MB -M1 (2.5) Pt MB +M1'

s2 (2.6) MB d

- S2- s2· 1 d

The reileetion coefficient is real and vanishes at M = MB. The pipe termination is not reflecting, in other words, it is then anechoic. Ru is negative for M1 < MB and positive for M1 > MB)· So the phase of R jumps from 1r to 0 at MB. Figure 2.2 shows the absolute value of R. The energy reileetion coefficient equals

(2.7)

2.1.2 Compressible theory

Ronneberger [12] has presented a compressible theory for the reileetion at a discontinuous increase in pipe diameter with the introduetion of aso-called entropy wave a. The entropy wave a stands for the non-isentropie part of the density fluctuations. The equations for

17

0.8

0.6 0.6 a: UJ

a: 0.4 0.4

0.2 0.2

0 0 0 Me

M, 2Me 3Me 0 Me

M, 2Me 3Me

Figure 2.2: Becherts theory for a perforated end plate, SI/ Sd = 2 or M 8 = ~- It shows that there is no pressure or energy refiection for a certain Mach number M1 = M 8 .

the diaphragm at the end of pipeend are now complemented with p = p(p, s). This gives after linearisation the density p

p =Po+ peiwt ; p = 12 (p +a). (2.8) Co

From the mass continuity equation the isentropic Bernoulli equation (1.19) and Poisson's equations for a perfect gas we find

continuity : S1p1u1 = Sbpdud (2.9)

Bernoulli: lu2 + _î_Pl 2 1 -y-1 Pl

lu2 + _î_Esl. 2 d -y-1pd (2.10)

Poisson' s equation : Pl - (~)'Y. (2.11) Pd

By eliminating ud with the continuity equation and linearizing for small pressure fiuctuations and for small acoustical veloeities compared to the main flow velocity we get

U Û + _î_Plü ( 'Îll fh) 10 1 -y-1 Plü Plü - Plü

Plü ( 'Îll fh ) Pi_o Plü - T PlO

(2.12)

(2.13)

in which a 1b = t. With the boundary condition in the free jet, Pd = 0, the former two

equations reduce in terms of Pt, p} and a 1 to

in which /31d = E.!Q. In case we have no incoming entropy wave, a 1 = 0, the refiection PdO

coefficient becomes

R = - (1 + M1)(1- aîdf3fdMt) (1- M1)(l + aîdf3îdM1).

18

(2.15)

To obtain the reflection coefficient we need to know the reference situation (stationary solution) at position x1 and xd· Both f31d and M1 are functions of the pressure difference over the diaphragm. Therefore the Mach number is as well a function of f3Id· From the continuity equation, the compressible Bernoulli equation and with help of the isentropic

gas identities p rv p'"'~ rv cf!-ï and c2 = lP/ p we can write in terms of the density and the Mach number

::r.±l S1M1P1

2

(I ; 1 MÎ + 1) p{-1

::r.±l SdMdpd 2

(I; 1 M~ + 1) Prl.

(2.16)

(2.17)

Eliminating Md gives the following expression for M1 as a function of the density ratio f31d over the diaphragm:

1 - /31-'"Y ld (2.18)

Figure 2.3 shows the reflection coefficient as a function of the Mach number for both the incompressible and the compressible approximation. The Mach number at which the anechoic behaviour is achieved is smaller than the incompressible theory predicts. When the flow becomes critical in the free jet, the pressure reflection coefficient goes to 1 but the

energy reflection coefficient is a factor ( ~~~~) 2

lower.

a: 0.4

0.2

0 0

-lncompressible

Compressible

~=2.5

Ms M,

2M8

~-5 s,-

3Ms

UJ

a: 0.4

0.2

0 0

- Compressible

lncompressible

0.1 0.2 M,

~=5

0.3

Figure 2.3: The reflection coefficient as a function of the Mach number, calculated with both the incompressible and the compressible theory at different cross-sections (a1d =

SI/ Sd)· At M1 =MB the incompressible model becomes anechoic.

2.2 Flow through a sharp bend

2.2.1 Incompressible model

In this section a model for the acoustical behaviour of the sharp bend, as shown in figure 2.4 will be introduced. As the flow goes through the sharp bend, flow separation occurs

19

Uv- .._/")../

s" ul. '""- G I

• J) -:; 1i Po- s2 V ( (..(, \.. "-

x=o Xv- x2. x-axi.s

R u.., x,

S,

Figure 2.4: A flow passes a sharp bend. In area 1 upstream a uniform flow is considered which tears itself away from the sharp edge of the corner and forms a free jet which contracts further ( the vena-contracta effect). After a turbulent mixing zone the flow gets again uniform over the cross-section of area 2.

at the sharp inner corner which results in the formation of a free jet. The free-jet narrows downstream to a typical cross-section Sv due to the so-called vena-contracta effect. The ratio Sv/ 5 1 we call the vena-contracta Ctvl· To simplify the model we first consider a constant vena-contracta, independent of the Mach number. The validity of this assumption is discussed in section 2.4.

Up till the free jet the flow is similar to the flow through an orifice as discussed in section 2 .1. Ho wever, the boundary condition p~ = 0 is not val id. A pressure increase will take place in the turbulent mixing zone between xv and x2 . In this mixing zone friction is essential and the flow is adiabatic but not anymore isentropic. The integral conservation laws of mass and momenturn are used in order to describe this area. At position Xv the pressure is uniform over the cross-section but the mass flux outside the free jet is zero. At position x2 both pressure and velocity are uniform over the cross-section. In the quasistationary incompressible approximation we find

mass: S1u1 = Svuv = S2u2 (2.19) 2

2 Po Uv Bernoulli: Pl + Po2ul =pv + -- (2.20) 2

momenturn: PvSv + Pou~S2 = P2S2 + Pou~S1, (2.21)

from which Pv and Uv can be easily eliminated. We find similar equations as for Becherts end plate with only a different factor in front of the quadratic velocity term:

2 (s )2 PoU2 1 P2+-- --1

2 Sv

20

(2.22)

(2.23)

We linearize again for small acoustical amplitudes in the velocity compared to the main flow velocity,

Pt +Pi Pi+ P2 + M, (~: -1 )'(Pi- P2)

Pt- P2·

(2.24)

(2.25)

This results in two equations which describe the effect of a flow through a sharp bend on the acoustical field up- and downstream of the bend. In the form of a transfer matrix

multiplication with the constant C = ( t - 1) 2

this becomes

[ Pt l = T [ P~ l = [ 1 + iCM2 - iCM2] [ PP22~ l· (2.26) p1 - p2 2CM2 1 - 2CM2

In the form of a scattering matrix we get

2 2+CM2

CMz 2+CM2

(2.27)

Note that in this incompressible model M1 = M2 because of mass conservation and equal cross-sections in region 1 and 2.

2.2.2 Compressible model

When we take the compressibility effects into account in a quasi-stationary approximation, then the following six equations give the relations between the pressures, densities and veloeities at different positions:

mass: PlU! avlPvUv

mass: avlPvUv P2U2 Bernoulli: lu2 + ...:LPl lu2 + ...î_Pv

2 1 -y-1 Pl 2 V 1-lpv (2.28) energy: lu2 + ...:LPl lu2 + ...:LP2

2 1 1-1 Pl 2 2 1-l P2

momenturn: Pv + avlPvU; 2

P2 + P2U2

Poisson' s equation : Pl (~)'. Pv

From equations 2.28 we can eliminate the velocity inthefree jet Uv with the first equation. Fr om the first order terms ( zeroth order in acoustics) remains

{h u1o + P10Û1 = P2U2o + P2oÛ2

U Û + ...î_PlO (Pl _ .h_)- PÎoU~o (.h_ + .fu_ _ Pv) + ...î_PvO (Pv _ Pv) lO l ')'-1 PlO PlO PlO - P~oavl PlO UlO Pvo ')'-1 Pvo Pvo Pvo

u û + ...î_PlO ( Pl - .h_) -u û + ...î_P20 ( P2 - ..h...) lO l 1-1 PlO Plo Plo - 20 20 ')'-1 P20 P20 P2o

Pv + Pv P~ouio + 2Pvo P~ouio (.h_ + .fu_ - _b_) = P2 + P2U~o + 2P2oU2oÛ2 Pvoavl Pvoavl PlO UlO PvO

~~~ (:110 - :VVO) = ry (~) ')' (~- :vvO) •

21

Substitution of

+ -P~l = p+l +p-1 û - pl -pl

1 - PlOC!O + -

P~2 = p+l + P2- û - P2 -p2 2 - P20C20

gives five equations with eight unknowns, Pt, P1, Pv, Pt, P2, a1, av and a2. Again we consider that there is no incoming entropy wave, a 1 = 0. After elimination of the term Pv + av and a 2 (see appendix A for a more complete derivation) two equations remain which can be written as

[ ~f l = [ ~~~ ~~~ l [ ~f l· (2.29)

In case we know two of the pressure waves the two unknown waves follow from equation 2.29 after determination of the reference variables M1, M2, f31v and (312 = p10 / p20 . At a given M1 , f31v can be determined with equation 2.18. Equation 2.16 gives the Mach

number inthefree jet Mv = M1f3?v('y+l) /o:vl· The momenturn conservation law applied for the turbulent mixing zone can be written as

(2.30)

With help of P2/Pv = (P2/P2)(P2/PvHPv/Pv) and P2/Pv = O:vluv/u2 this equation gives an expression for the ratio c2/ Cv as a function of the Mach numbers:

:r=.!. M + M2

f3 2')' - c2 - _2 O:lv I V

2v - - M 1 M2 ' Cv v +1 2

(2.31)

in which f32v = p20 / Pvo· Together with the energy equation,

(2.32)

this gives a quadratic equation for M~. M2 and f32v can be determined at given Mv. In appendix A this is worked out.

In this report the incoming entropy wave a 1 is considered to be zero. Figure 2.5 shows the reflection coefficient calculated with the compressible and incom

pressible model as a function of the Mach number in the pipe when we assume o:v1 = 0.5.

2.3 The hodograph method

This section will discuss the hodograph method. It uses some features of the two dimensional potential theory and uses the fact that from physical considerations some general statements can be made regarding the shape of streamlines and about some velocities. A more complete overview of the potential theory and the hodograph method can be found in Prandtl et al. [13].

22

0.8 0.8

0.6 0.6 0: w

0:

0.4 0.4

0.2 - lncompressible 0.2 -lncompressit>

Compressible Compressible 0 0 0 0.1 0.2

M, 0.3 0 0.1 0.2

M, 0.3

Figure 2.5: The calculation of the reileetion coefficient at l.OOlm from an ideal open pipe termination (Ru = -1) with the sharp bend at 0.519m from the open pipe termination. The frequency f is 343Hz (! / fc = 0.0509) and avl is 0.5.

Some principlesof the two-dimensional motion theory will be explained and some useful formulas will be derived. Then the use of the hodograph methad will be explained and will be used to discuss the flow through a right-angled bend.

2.3.1 Two-dimensional potential flow

The motion of a two-dimensional fiuid of which the veloeities at the different points in space are independent of time is described by the vector iJ (representing the velocity (u, v) of a partiele) given as a function of the position x or the two corresponding coordinates x, y. In order to find out what is happening to the individual fiuid particles we have a set of equations for each particle:

dx

dt dy

dt

u (2.33)

V. (2.34)

If we assume that the velocity field iJ( x) is everywhere free from rotation this velocity field can be descri bed by a potential (x). This means that there is a scalar function (x, y) so that

u

V

(2.35)

(2.36)

The particular importance of the two-dimensional problem lies in the fact that it is particularly amenable to mathematica! analysis. The simplification is connected with the fact that, as soon as the analysis depends on the two Cartesian coordinates (x, y), the real and imaginary partsof any analytic function of the complex argument (x+ iy) satisfy the

23

Laplace differential equation of potential theory. The analytic function F(x + iy) = F(z) of the complex argument (x+ iy) = z can always be split up in a real and an imaginary part: F(z) = F(x+iy) = (x,y)+iW(x,y). F(z) is called the stream function. By taking the first partial derivatives of F(x + iy) with respect to x and y we see that

oF dFoz dF --

ox dz ox dz oF dFoz .dF oy

--=z-dz oy dz l

thus giving

.oF oF z ox = oy ·

But, sirree

oF o .ow ox

-+z-ox ox

oF o .ow oy oy + z oy,

we derive the Cauchy-Riemann differential equations:

o ox o oy

o'll

oy ow ox.

(2.37)

(2.38)

(2.39)

(2.40)

(2.41)

(2.42)

(2.43)

When we differentiate these equations partially with respect to x and y we arive at Laplace's equation:

(2.44)

We also see that 'll implies that we automatically satisfy the continuity equation ( ~~ + ~~ =

O)valid for an incompressible flow. The total differential dF of F(x + iy) is

dF = oF oF -dx+-dy ox oy

( o + iow) dx + (o + iow) dy ox ox oy oy

(u- iv)(dx + idy) wdz,

where w is the complex conjugate to w =u+ iv. Therefore we have

w = d~~z) = F'(z).

24

(2.45)

(2.46)

(2.47)

(2.48)

(2.49)

2.3.2 Principlesof the hodograph method

The hodograph method may be considered as a special case of the methods in which conforma! transformations are used. By a conformal transformation we mean a transformation from one plane toanother (e.g. from the x,y-plane to the ,w-plane) of such a nature that angles in one plane are transformed into equal angles in the other plane, and that the ratio of any two contiguous smalllengths in one plane is equal to the ratio of the conesponding lengths in the other plane.

Since w = F'(z) is an analytic function of z, the transformation from the w-plane to the z-plane is a conformal one. Since on the other hand the transformation from the z-plane to the F-plane is also conformal, the w-plane is mapped on the F-plane with equal angles, so that w = F(F) is also an analytic function.

The hodograph method uses some properties of complex function theory, such as that an analytic complex function is determined by its singular points. When we consider an analytic function in the z-plane of which we can assume its singularities in the w-plane, than we look at the w-plane as if it is the physical plane. By suggesting a flow in the w-plane, which applies to the same boundary conditions as the original problem in the z-plane we find a solution in the w-plane. After transformation back into the z-plane, the only thing remairring is that we should verify if the solution satisfies the boundary conditions of our original problem.

The relationship between the stream function and z is usually much too complicated to be written down a priori, but the relation between w and F is often so simple that it is possible to find its analytic expression from a simple physical inspection. If we know the function w = F(F), we can always obtain the desired relationship between z and F by integration, because w = F'(z), or dz = dFjw. Hence

z f dF w +constant (2.50)

f 1 oF(w) w ow dw +constant. (2.51)

If the boundary conditions of the velocity field in the z-plane are known, it is possible to construct a w-plane, and this roundabout way through the w-plane is often useful for first finding the function F(w) and then, by integration, for finding F(z). In this process we first have to make a diagram in which velocity veetors of all points on some chosen streamlines are drawn.

2.3.3 The hodograph method used for the sharp bend

We will now apply this method to the sharp bend. Without knowing the stream function we can indicate the probable flow of the streamlines from physical principles (figure 2.6).

The fluid flows from the left towards the bend and the velocity has a constant value a provided that the points are sufficiently far away from the corner; then u = a and v = 0. The walls from A1 to C and A2 to D are streamlines. The fluid does, however, not flow completely around the sharp corner at point C; hence the velocity does not become infinite here. The fluid tears itself away from this point and forms a jet around the corner. Finally, sufficiently far away from the corner, it will flow vertically and in between B1 and B2 the

25

Az D

S, A,

(

~LJI Bl..

Figure 2.6: Flow through a sharp bend.

velocity will again have a constant value, b; then u = 0 and v = -b. The pressure along the free surface must be equal to that of the surrounding fiuid, which can be considered uniform. Hence from Bernoulli's equation, ~piP must also be constant, or, in other words, the velocity lvl on the free surface of the jet must have a constant value equal to the final velocity b. The wall from D to B 1 will be a streamline as well. Because point D is a stagnation point the velocity will be equal to zero; v = u = 0.

As aresult of these observations we shall now build up the w-plane (figure 2.7). Since all streamlines come from sufficiently far from the left with the samevelocity a towards the bend, all of the points in between A1 and A2 of the z-plane are represented by one point A on the real axis where u = a. All streamlines in the w-plane will start from this point. In other words, the souree in the z-plane which is located in z = -oo is in the w-plane represented by a souree in a finite point, w = a.

Figure 2.7: Hodograph of flow of figure 2.6.

Because all streamlines end in between B 1 and B 2 with the same direction and value b (a sink in z = -ioo), they will also end in one point B, on the imaginary axis where v = -b. So until now we found a souree in A and a conesponding sinkin B.

26

Let us follow the streamline from A2 to D, of which, since its direction remains constant, the velocity decreases from a to zero. In the w-plane, therefore, this streamline is a straight line from w =a to w = 0. When we follow the streamline from D to B2 we see again that the direction remains constant but the velocity increases from zero to b. So in the w-plane this streamline is a straight line from w = 0 to w = -ib. Because of mass conservation we see that b > a.

Now we will follow the streamline at the free surface of the jet. Because the velocity has a constant value b and the direction changes from horizontally in C to vertically in B 1 , this streamline is an are of a circle in the w-plane from w = b to w = -ib. The last streamline to consider is the one from A 1 to C, of which the direction is constant and the velocity increases from a to b. In the w-plane, therefore, this streamline is a straight line from w =a to w = b.

Now we obtained the w-plane in which we found a souree in w = a, a sink with the same strength in w = -ib and three walls, the positive real axis, the negative imaginary axis and an are of a circle with radius b. In order to find an expression for the function F ( w) we will use the methad of mirror imaging.

Firstly we need to find an expression F1 ( w) which describes the situation as if there were only the souree in w = a and the sink in w = -ib. To do so we will have a short look at how a souree in the origin of the z-plane would be described.

The motion of the fl.uid is purely radial and because of mass conservation, the same amount of mass Q fl.ows through every circle with radius lzl. So Q = 27rlzllwl gives us the velocity Iw I = ~

1

;

1

and F(z) becomes F(z) = ~ ln(z). Q we call the souree strength. Since the souree and the sink in our z-plane are not located in the origin but are of a

very special kind due to which they are represented as a souree and a sink in finite points in the w-plane (in w =a and w = -ib, respectively), we find

F1(w) = .2_ln(w- a)- .2_ln(w + ib). 27f 27f

(2.52)

Secondly we add the two walls on the axes by using the mirror images of F1 ( w) in the real and imaginary axis and we find

Q {ln(w- a)+ ln(w +a) -ln(w + ib) -ln(w- ib)} 7f

Q {ln(w2- a2

)- ln(w2 + b2)}.

7f

(2.53)

(2.54)

The last thing remaining is to include the circle wall by using the circle-theorem of Milne-Thomson which says that

(2.55)

in which R is the radius of the circle, :F1 the expression as if there wouldn't be a circle wall and :F the expression which includes the wall by adding the mirror image of :F1. Using this theorem results in

27

F( w) = ~ { ln ( w2 - a

2) - ln ( w2 + b2

) + ln ( !: -a2

) - ln ( !: +b2

) } (2.56)

Now we are ready to use equation 2.51, z = J 4- 8~~w) dw +constant, to find the relation between z and F. So by substituting equation 2.56 we find

(2.57)

and integration gives

z x+ iy = ~ { ln (:~~) - 4% arctan (~) + (%) 2ln (~:~~~)}+constant (2.58)

!:l. {In (u-iv-a) - 4Q. arctan (u-iv) + (Q.) 2 ln (a(u-iv)-b2)} +constant. (2.59)

1ra u-w+a b b b a(u-zv)+b2

Since the constant in this expression simply means a change of origin in the x, y-plane, we can put it equal to -~(1 + (%) 2)ln(-1) (thus making the origin coincide with point D).

Until now we did notmention the dimensions of the bend, the diameters 51 , 5v and 82 of the tubes and the free jet (see figure 2.6). By choosing equal diameters before and after the bend (51 = 52) or, in other words, choosing a square sharp bend, the ratio a/b will get a value which is directly related to the vena contracta av1 via the mass conservation law (p5vb = p81a):

5v a av1 = 51 = t;· (2.60)

We find the value av1 by looking at z in point C where u = b, v = 0 and x = y ( or the

real and imaginary part of z are equal). This gives with use of ln( -1) def i1r

2 (1- av1) ( 1 + av 1) ln - av 1 7f + 7f = 0. 1 + avl

(2.61)

Solving equation 2.61 gives av1 = 0.5255.

2.4 The vena-contracta

In the preceding section we used the hodograph method to find a theoretica! value for the vena-contracta. We found av1 = 0.5255 which is independent of the main flow velocity. However, in reality the vena-contracta has a Reynolds (Re = u0D / v) and a Mach number dependency. We will now introduce the loss coefficient Cd to discuss the Reynolds dependency and then consider the vena-contracta of the Borda mouthpiece to discuss the Mach dependency.

28

2.4.1 Loss coefficient Cd

The pressure drop !:1p over the bend can be expressed in terms of a non-dimensional loss coefficient cd,

(2.62)

which linearizes in the incompressible approximation to

(2.63)

where we assume Cd to be constant. With equation 2.24 from the incompressible theory of section 2.2.1 we see that

( s, ) 2

Cd= Sv -1 (2.64)

or

1 (2.65) avl = ~+ 1 .

Blevins [14] as well as Idelchik [15] give an overview of the loss coefficient Cd for a number of geometries as a function the Reynolds number. The loss coefficient for the sharp bends with a rounded and right-angled outer corner are shown as a function of Re in figure 2.8, as well as the contraction ratio of the vena-contracta calculated from the data on cd by means of equation 2.65 forthese two geometries.

1.4

1.2

0.48

0 :;::; ~0.46

Figure 2.8: The loss coefficient Cd and contraction ratio of the vena-contracta av1 as a function of Re for the sharp bends with a rounded outer corner (-- and · · · · · ·) and a right-angled outer corner (- · - ). The data shown is taken from Blevins [14] (--) and Idelchik [15] (· · · · · · and - ·-). av1 is calculated with equation 2.65.

29

r -----------

I p_,. I I I

L------------

Figure 2.9: Borda mouthpiece

2.4.2 The Borda mouthpiece

Hofmans [16] gives a derivation of the contraction ratio (avds in the Borda mouthpiece shown in figure 2.9. Assuming a steady isentropic flow, momenturn conservation, the stationary Bernoulli equation and Poisson's equation give

and one obtains:

momenturn: Bernoulli:

Poisson's equation :

2 Pv + avlPvUv .!.u2 + ___:j_Pv 2 V 1-1 Pv

Pv Po

~

(1 + 1~ 1 M;) •- 1 - 1

(avl)B = M2 . "'( V

(2.66)

(2.67)

In figure 2.10 is (av1)s shown as a function of Mv. Because the behaviour of the flow in the Borda mouthpiece is similar to the flow through a sharp bend we will use equation 2.67 as a correction for the Mach dependency of the vena contracta.

g0.65 ~ c:: 0 ~ 0.6 ~ E 0 0

0.55

0.2 0.4 0.6 Mv

0.8

Figure 2.10: Contraction ratio (av1)s as a function of jet Mach number Mv for the Borda mouthpiece shown in figure 2.9. The salution is given by equation 2.67.

30

2.4.3 Application of multiple corrections

When the corrections for the Reynolds and Mach number dependency are small compared to the reference contraction ratio, we assume that we can take along multiple corrections by multiplying relative contraction ratios. When because of a certain geometry and certain Reynolds number in the incompressible limit the vena-contracta equals (av1)ref, the compressibility correction is applied as follows:

(2.68)

In figure 2.11 is the absolute value of the pressure refiection coefficient shown as function of the Mach number M1 , for the incompressible model, the compressible model with a constant vena-contracta, and the compressible model with a Mach and Reynolds number dependent vena-contracta.

0.8

0.6 a:

0.4

0.2

0 0 0.1 0.2

M, 0.3

Figure 2.11: The absolute value of the pressure refiection coefficient at 1.001m of an ideal open pipe termination with the sharp bendat 0.519m from the open pipe termination. The four lines represent the incompressible model and the compressible model with constant av1 = 0.5 (-- and --, respectively), and the compressible model with a correction for only the Mach number dependency as well as both the Mach and Reynolds number dependency of the vena-contracta (- · - and · · · · · ·, respectively). The frequency f is 343Hz (! / fc = 0.051).

31

Chapter 3

Smooth bends

The models described in section 2.2 are in principle not valid for smooth bends because these havenosharp inner corners. However, those modelscan beseen as a way of descrihing the pressure drop over the bend.

Section 3.1 discusses the loss coefficient Cd of the R = D and R = 3D bends ( see figure 1).

Section 3.2 shows the infiuence of the wall roughness of the bend on their acoustical behaviour.

Section 3.3 discusses the quasi-stationary models for smooth bends and introduces the Fanno model.

3.1 Loss coefficient Cd

As mentioned in section 2.4.1, Blevins [14] and Idelchik [15] give an overview of the loss coefficient Cd for a number of geometries as a function the Reynolds number. The loss coefficient for the three different geometries (of figure 1) are shown as a function of Re in figure 3.1, as well as the contraction ratio of the vena-contracta calculated from the data on cd by means of equation 2.65 forthese three geometries.

3.2 Wall roughness

Apart from the R/ D value, the behaviour of bends may differ from each other in a number of ways. One of them is the roughness of the inner wall. It appears that the behaviour of the loss coefficient Cd with an increasing Reynolds number is highly dependent on this roughness. Figure 3.2 shows a graph, which sketches the roughness dependency of the loss coefficient (Idelchik [15]). The ratio of the roughness amplitude and the pipe diameter in our experimentsis of the order 0.0003. From figure 3.2 we see astrong dependency in the range of our measurements (104 < Re< 2 · 105

).

3.3 Quasi-stationary models

As mentioned before, we can consider the models from section 2.2 as a way to describe the pressure drop over the bend. With equation 2.64, Cd= (a~l- 1) 2

, we can 'translate'

32

2.5

0.7 AID=3 2 0

1.5 ~ ~ 0.6

t3 0 AID= 1 - Blevins :;:::

- Blevins (.) I delehik ct! .... -

AID= 1 I delehik § 0.5 0.5 (_)

AID=3 0 10 4

10 5

Re 10 6 10 5

Re 10 6

Figure 3.1: The loss coefficient Cd and contraction ratio of the vena-contracta av1 as a function of Re for the sharp bend with rounded outer corner (R/ D = 0.5), the R/ D = 1 bend and the R/ D = 3 bend. The data shown is taken from Blevins [14] (~-) and Idelchik [15] (· · · · · ·). av1 is calculated with equation 2.65.

Figure 3.2: Resistance curves of a bend with different relative roughnesses of the surface of the entire inner walls. (tot is a parameter for the resistance. The ratio of the roughness amplitude and the pipe diameter is LS.. (Idelchik [15]).

the loss coefficient into terms of a vena-contracta effect and substitute this in the quasistationary models of the sharp bend.

We will campare these models to results from measurements at the R = D and R = 3D ben ds.

In a different quasi-stationary model we will use the Fanno theory. The Fanno theory describes the pressure drop over a straight pipe segment in which friction at the wall is taken into acount. Due to the production of entropy a (subsonic) flow will speed up as it moves through the pipe.

Thompson [17] describes the Fanno theory. After assuming that the pipe has a constant area, that the one-dimensional theory is applicable and that the walls are perfectly insulated (adiabatic walls), we can derive the following equation:

1- M2 dM2 4f

1M4 (1 + 1 ;1 M2) dx D'

(3.1)

in which D is the diameter of the pipe and f the Fanning friction factor.

33

Appendix C works out a model in which compressibility effects are taken into acount in a quasi-stationary approximation.

The pressure drop, described by the loss coefficient Cd, over a straight pipe segment of length L is in the Fanno theory

(3.2)

in which fp is the Fanning friction factor which is non-dimensional and depends upon the Mach and Reynolds numbers and the roughness of the wall. f F is the friction factor averagedover the length L. The Mach dependency of fp is weak and the Reynolds number is almost independent of the position x. Therefore we can choose a constant value 1 F·

With equation 3.2 we can 'translate' the loss coefficient Cd in termsof the friction factor 1 F· We will substitute cd values from literature (Blevins [14]) fortheR = D and R = 3D bend in the Fanno model and compare this model to experimental results.

Figure 3.3 shows the scattering matrix elements from the compressible and incompressible model (from section 2.2) and the Fanno model, in which the same value of Cd = 1 (independent of Reynolds or Mach number) is chosen.

34

0.5

0.9 0.4

' 0.8 ' '\ 0.3

+ \ ó:: I-0.7 0.2 I

0.6 0.1

0.50 0.1 0.2 0.3 0.1 0.2 0.3 M1 M1

0.5 1

0.4 0.9

' 0.3 I 0.8 '\

I '\ + ~ cc I \

0.2 / 0.7 \ /

\

0.1 0.6 \

0.5 0.1 0.2 0.3 0 0.1 0.2 0.3

M1 M1

Figure 3.3: The scattering matrix elements from the incompressible (--) and compressible model (- - -, see section 2.2) and the Fanno model (- · · · · ·, see appendix C) for the same value of Cd= 1 as a function of the Mach number.

35

Chapter 4

Experimentalset-up

The pressure refiection coefficient was measured as a function of the Mach number M =

u0 jc0 (0:::; M < 0.25) for three different bends (see figure 1) at TUE. The scattering matrix elements were measured as a function of M (0 :::; M < 0.07) for the same bends at LAUM. The two microphones methad which is explained insection 4.1 was used todetermine the pressure refiection coefficient and the two sourees methad which is explained insection 4.2 was used to determine the scattering matrix elements.

The set-ups at TUE and LAUM are described in sections 4.4 and 4.5, respectively. A global overview of the set-ups is shown, the individual instruments are discussed and the equations used for the determination of the steady main flow velocity, pressure and temperature are given.

4.1 Measurement of the reflection coefficient; two microphones method

4.1.1 Theory

The pressure signal at an arbitrary axial position x can be described by the composition of a wave p+ traveling in positive direction and a wave p- travelling in negative direction (see equation 1.31):

p(x, t) = p+(x- (u0 + c0 )t) + p-(x- (uo- c0 )t) (4.1)

When we consider harmonie waves of radial frequency w, the pressure p(xi, w) at Xi is related to that at position x by:

(4.2)

in which p±(w) is the complex amplitude of the acoustical wave. The complex transfer function Hij ( w) of a pressure signal at position x j to a pressure signal at position x i is the ratio of the complex pressure amplitudes:

(4.3)

36

By means of a F astF ourier T ransform-system we can measure the transfer function between the two microphones at any frequency w.

The pressure reflection coefficient Rx at a position x is defined as the ratio of the complex amplitudes of the wave propagating in the negative x-direction (the reflected wave) and the wave propagating in the positive x-direction ( the incoming wave):

( 4.4)

where R0 = p- ( w) I p+ ( w). This pressure reflection coefficient is a complex variabie with the amplitude iRx(w) I and a phase <Px(w) which represent respectively the ratio of the amplitudes and the phase shift of the reflected and the incoming wave. Combination of equation 4.3 and 4.4 gives

e-ik+(xi-X) + Rxeik-(xi-X)

Hij = e-ik+(xj-X) + Rxeik-(xj-X).

Solving this equation for Rx gives the following important relation:

In this report We take x = 0 as a reference point.

4.1.2 The frequency range of the two microphones method

(4.5)

(4.6)

A condition for the validity of equation 4.3 is that !Hij I =/= 1. Without flow k+ = k- = k0

which gives for the condition validity

e2iko(Xi -x) =!= e2iko(Xj -x) {:::} e2iko(Xi -Xj) =!= 1

{:::} lxi- Xji =/= n2>., n = 0, 1, 2, ... , (4.7)

in which À = 21r I k0 is the wavelength of the sound. From this equation we can determine a frequency interval for which the results of the two microphones method is valid:

(4.8)

in which f 9 is the frequency for which studies by Ábom and Bodén [18, 19] and Reijnen [9] show that the inaccuracy drastically increases for higher frequencies. The optimal accuracy is obtained for lxi-xjl = -XI4 and Xj ~ 0. However it is necessary to place the microphones a few diameters upstream from the discontinuity which we want to study, because closer to the discontinuity the flow might not be uniform over the cross-section.

4.2 Measurernent of the scattering matrix; two souree rnethod

To determine the scattering matrix S defined by equation 1.57 two independent acoustic tests are needed, denoted as I and I I. These can be obtained by using the two souree method. This method uses one souree upstream and one souree downstream of the bend,

37

<::.ourc.e I

X i x· J

Souree 1[

~ource

I

X; x· J

Figure 4.1: Description of the two souree method. For convenience the bend is drawn as a box.

see figure 4.1, of which only one is active in each test state. If the input and output state veetors for these test cases are measured, we obtain the following matrix equation:

[ Pti Ptn l = [ T: R= l [ Pti Ptn l· (4.9) Pu Pui R T P2I P2 n

If the input states [Pi IP2 I JT and [pi IiP2 n JT are linearly independent, S can be solved from this equation. We find

+ - - + y+ P2IP2II - P2 IP2 n ( 4.10) + - - +

P1 IP2 IJ - P2 IP1 n

R+ P1 IP2 IJ - P2 IP1 n (4.11) + - - + PuP2 n- P2 IPui

+ + + + R- PliP2II- P2IP1II (4.12) + - - +

P1 IP2 IJ - P2 IP1 n + - - +

r- P1 IPl IJ - P1 IP1 IJ (4.13) - + - - + P1 IP2 IJ - P2 IP1 n

When we have two microphones at each side of the bend we can determine r+, R+, R- and r- using the definition of the transfer function ( equation 4.3). In appendix B this is worked out. However, it is more convenient when anechoic terminations are used on both sicles of the bend. This would imply that when souree I is active, P2 I = 0, and when souree I I is active, Pi IJ = 0. This yields for the scattering matrix elements:

+ y+ P2I (4.14) + Pu

R+ P!I (4.15) + Pu +

R- P2 n (4.16) P2n

y- Pui ( 4.17) P2n

38

This implies that the elements of the scattering matrix have simple physical interpretations as the reflection and transmission coefficients measured under anechoic conditions either up- or downstream of the bend.

In practice the terminations are not perfectly anechoic so that we have to use the general form given in appendix B. However when the anechoic conditions are approached, the experimental inaccuracy in the scattering matrix is minimized.

4.2.1 Translation of the scattering matrix element

In principle the scattering matrix elements are obtained for all pipe segments in between the two micropbones dosest tothebend (at positions Xj and xk, see figure 4.1. In order to correct for the different pipe segments we will translate the scattering matrix components from the microphone positions Xj and xk towards x 0 = 0. The x-axis is chosen along the centerline of the bend with x 0 in the middle of bend. In these calculations we apply the translation equations for an ideal straight pipe. Therefore, the scattering matrix elements will be multiplied with the following exponential factors:

(4.18)

in which the subscript t stands for the translated matrix element. From bere we willleave the subscript t out, sirree we will always translate the components to the middle of the bend ( except when mentioned otherwise).

4.3 Calibration method

The calibration procedure takes care that the measured transfer functions can be corrected for difference in response of the channels of the data acquisition system, the charge amplifiers and the microphones. The calibration was dorre with the configuration as given in figure 4.2.

a)

X-ClX\> --------rl---t~rl----~•

X: x_\ X=O

Figure 4.2: Configuration for the calibration in a) TUE, b) LAUM.

The pipe is closed at the end and the combination of micropbones i and j which have to be calibrated is placed as close as possible to this closed end. The theoretical transfer function (Hij)th between both channels can theoretically be found with equation 4.5 and R0 = p- / p+ = 1. Comparison of the theoretical transfer function (Hij )th with the measured value (Hij )exp gives the following calibration factor 'J)ij:

39

(p(xi)) (p(xi)) -;:--( ·) = -;:--( ·) 'TJij = (Hij)exp'TJij P X J th P X J exp

(Hij)th

(Hij)exp. ::::} 'TJij =

From the definition of the calibration factor follows

1 'TJij

'TJji

'TJik 'TJij'TJjk·

( 4.19)

( 4.20)

( 4.21)

( 4.22)

In figure 4.3 one of the calibration factors from the set-up at LAUM is shown as a function of the frequency.

1.012 ,.---,.---,---,.---,---,.---,----,

1.010

<? (i) en 1.ooa .0 Cl!

1.006

400 f (Hz) 800 1200

0.4 ,.---,.---,---,.---,---,.---,----,

-0.2'-----'--~-~-~-~-~-'---'

0 400 f (Hz) 800 1200

Figure 4.3: The calibration factor TJ for one of the used microphones in the set-up of the LAUM, with respect to the reference mieropharre (--), and the fit which is used in the processing of the data (--).

4.4 The set-up at TUE

The global overview of the set-up at TUE is given in figure 4.4. The high pressure control system of the laboratory (A) provides air with an adjustable pressure Pr of 1 to 15bar and a dew point of -40°C. So the relative humidity is negligible. This pressure can further be adjusted by means of a valve (B). Flow meter (C) measures the volume flux in the supply pipe and sends 12156 pulses per cubic meter to the counter (D).

The settling chamber (E), internally covered with damping material, reduces the acoustical disturbances from the valves. A siren (F) modulates the flow periodically with an adjustable frequency up to 1000Hz. Also the ratio between the acoustical velocity amplitude u' and the mean flow velocity u0 is adjustable by means of a bypass.

A flexible pipe (G) forms the conneetion between the siren and the rest of the pipe (H). This is dorre to reduce the transfer of mechanica! vibrations to the pipe wall. The pipe (H)

40

A

E.

Figure 4.4: The set-up at TUE: (A) high pressure control system, (B) valve, (C) flow meter, (D) counter, (E) settling chamber, (F) siren, (G) flexible connection, (H) pipe end, (I) sand box, ( J) piezo-electric microphones, (K) charge amplifiers, (L) data-acquisition system, (M) personal computer, (N) thermometer, (0) barometer.

in which the acoustical measurements are performed is internally smooth (wall roughness is of the order 0.1f.lm). The wall thickness is 0.5cm. A heavy metal frame supports the pipe at arbitrary chosen positions. Rubber is placed between the frame and the floor and the pipe is lead through a sand box (I) to damp mechanica! vibrations from the siren.

The pipe consists of several components in between which elements like microphones and bends can be placed. Piezo-electric microphones (PCB, model 116A) ( J) placed in the wall of the pipe measure the pressure signal in the pipe. Via charge amplifiers (Kistler, type 5011, with a frequency bandwith from 1Hz till 3kHz) (K) the measured signals are lead to a data-acquisition system (HP-3566A) (L) after which they are further processes with a computer (M). Thermometers (Pt 100) (N) are placed close to the flow meter and in the measurements section. A mechanica! barometer ( 0) gives the static pressure in the flow meter which is used for the determination of the volume flux in the pipe. The pressure in the test section is assumed to be equal to the atmospheric pressure which is measured within 20Pa by means of a mercury manometer.

The characteristic parameters of the set-up are given in table 4.1.

4.4.1 The siren

The souree in the experiment is the siren which is schematically shown in figure 4.5. The air is pressed through a hollow cylinder and is periodically supplied into the pipe by an electrically driven wheel with three circular wholes (diameter = 10mm). In this way a

41

Table 4.1: Characteristic parameters of the set-up at TUE.

Pressure 1 bar < Pr < 15bar Main flow M < 0.3

Re< 2 · 106

Flow meter 8dm3s 1 < <P f < 90dm3s 1

Siren lHz < f < 1000Hz u'< 2.6 · uo

Pipe Radius a = 15.013mm Wall roughness O.lp,m

_j(~\1 ... ( . )"

J

j ~~a) b)

Figure 4.5: The siren; a) flow, b) without flow, c) diversion.

fluctuating flow at the siren is imposed which is at most twice the volume flux. the ratio between the acoustical amplitude and the mean velocity amplitude can be reduced with help of an adjustable diversion.

4.4.2 Positioning of the microphones.

The piezo-electric microphones are mounted in the pipe wall as sketched in tigure 4.6. Due to their large diameter of 10.3mm it is impossible to mount them flush at the pipe

b) Figure 4.6: a) Mounting of the microphones in the pipe wall. b) Two of the three pipe segments in which the microphones are mounted have a helmholtz resonator just accross the microphone. L1 = 2 · 10-3m, L2 = 2 · 10-2m and SI/ S2 = 3

16 .

wall. Therefore the microphone is mounted in a chamber which is connected with the pipe through a small hole in the pipe wallof 3.5mm in diameter. The volume in this chamber is critical because a too large volume causes reflections in the pipe while a to small volume causes undesirable viseaus effects which influence the measurement of the pressure. When

42

we assume that the latter affect is independent of the amplitude it is eliminated by a calibration procedure of the microphones. The effect of the undesirable reflections has been investigated by Van de Konijnenberg [20]. The relative error in the measurement of the reileetion coefficient due to these reflections appear to be much smaller than 0.01 %, which is much smaller than the accuracy of the measurement.

Recently we found that two of the pipe segmentsin which the microphones were mounted at TUE, happened to have cavities just across the microphones (see figure 4.6). These act as efficient Helmholtz resonators and cause some additional systematic errors in the measurements.

The calibrations were done with the Helmholtz resonators at positions in the standing wave pattern different from their positions during the measurements. Because the influence of the Helmholtz resonators depend strongly on their position in the standing wave patterns it is not easy to correct the measurements for the systematic errors. However, because the measurements are reproductive within the errors of the set-up, we can use the measurements to qualitatively discuss the results of and differences between the three measured bends. Comparison between measurements of the reileetion at an open p1pe termination will be used to estimate the experimental accuracy.

4.4.3 Measurement of the Mach number

The Mach number M is equal to the ratio of the mean flow velocity u 0 in the pipe and the local speed of sound c. The mean velocity can be computed with the integral mass conservation law:

in which: PJ

if?f

Po So

uo

P!if? f = poSouo,

density of the air at the flow meter volume flux at the flow meter (1m3

rv 12156 pulses) density of the air in the pipe cross-section of the pipe S0 = na2

mean flow velocity in the pipe.

From equation 4.23, using the ideal gas law p = pR9T, it follows:

( 4.23)

( 4.24)

in which: PJ, Po static pressure in respectively the flow meter and the measurement section T1, T0 temperature in respectively the flow meter and the measurement section.

Because for an ideal gas c6 = 1R9T the speed of sound in dry air will be computed with

( 4.25)

See for the specifications of dry air table 1.1.

43

4.4.4 Accuracy analysis

The accuracy of M is determined by the accuracy of the measurements of the variables p f, p0 , Tf, T0 , f and a. The high pressure control system stahilizes Pr around a certain value. This involves pressure fluctuations in Pr and thus also in PJ· These fluctuations must remain as small as possible. In practize it appears that a high reservoir pressure upstream of the valve combined with a strong pressure reduction by means of this valve minimizes the fluctuations in the flow.

However, because measurements take place over large time scales ("-' 102s), the low frequent fluctuations caused by the pressure fluctuations in the control system are significant.

The relative error 8::/ as a result of the pressure measurement equals ( ~)

2

+ ( ~0°) 2

and is a bout 2 · 10-3 .

The temperatures T1 and T0 are measured by means of Pt-100 thermometers which are calibrated with respect to a mercury thermometer in between 15 and 30°C. The resulting accuracy is 0.2°C and the relative error becomes 0.07%. The temperature in the heart of the flow differs from the wall temperature. For a turbulent boundary layer at a adiabatic (isolated) plane wall we find a wall temperature Tw,

( 4.26)

in which Th is the temperature of the mean flow (see Shapiro [21]). We correct for this effect assuming a uniform flow temperature Th. This implies an additional error which at high Mach numbers can reach values of 0.2% in M.

The pipe has a radius of 1.5013cm with an accuracy of 0.01 %. This causes a relative error in M of 0.02%.

The flow meter (Insomet 13057) was calibrated by the Nederlands Meetinstituut in 1984. Van Ballegooien [22] and Reijnen [9] checked the calibration in 1992. Their results are given in figure 4. 7. The relative error due to the flow meter is estimated to be 1% on account of figure 4. 7 and of 1% due to the unsteadyness of f during the experiment.

Concluding we can suppose that the Mach number M is determined with an accuracy of the order of 1.5% inaccuracy which is mainly caused by the error in the flow meter.

4.5 The set-up at LAUM

The global overview of the set-up at LAUM is given in figure 4.8. A ventilator (type ABB Solyvent-Ventee FHK VB ED 500) (A) which has an adjustable rotation frequency, provides a constant flow of air with veloeities from M = 0.015 up to M = 0.075. Variations in flow are at most 1% for the highest Mach number. Flow meter (type ITT Barton 7445) (B) measures the main volume flux and calculates the main flow velocity. It uses the information from the thermometer ( C) and barometer (D) to correct the main flow velocity for temperature and pressure.

The measurement section consists of two anechoic pipe terminations (E), two acoustical sourees (F) which can be separated from the measurement section by valves (G), and several components like microphones and pipe segments.

44

"I:! -& s:: ..... ..... ::s

c.8

0.01

+

0 +

+ x•

XX

x. x •

(].)

:> x •

-~ -0.01 ~ ...... ~

• x . "' x • x

x x

• Prandtlbuis

x Meetflens

+ Debietmeter (ROTA)

- IJkcurve N.M.I. -0.02 l___...J....__....J.._ _ _..L. _ ___j_ _ _....jL-------.J

0 0.01 0.02 0.03 0.04 0.05 0.06

<Pd (m3s-l)

Figure 4. 7: Independent measurements of the volume flux through the pipe, obtained by Van Ballegooien [22] and Reijnen [9] with three different methods. The line is the calibration curve of the Nederlands Meetinstituut.

F A

F

Figure 4.8: The set-up at LAUM: (A) ventilator, (B) flow meter, (C) thermometer, (D) barometer, (E)anechoic termination, (F) acoustical source, (G) valve, (H) microphones, (I) bend, (J) amplifier, (K) data-acquisition system, (L) personal computer, (M) thermometer.

45

Eight BK microphones (H) with integrated pre-amplifiers are placed in the wall of the pipe, four upstream and four downstream ofthe bend (I). They measure the pressure signal in the pipe and lead the signals to the amplifier (type NEXUS 2690) (J) which sends the signal further to a data-acquisition system (HP-3566A) (K) after which they are processed with a computer (L).

Thermometers (Pt100) (M) are placed in the measurement section. The characteristic parameters of both the set-up of LAUM and TUE are given in table 4.2.

Table 4.2: The characteristic parameters of the set-ups of LAUM and TUE.

11 I TUE I LAUM 11

Pressure lbar < Pr < 15bar 1bar Main flow M < 0.3 M < 0.07

Re< 2 · 106 Re< 7 ·105

Flow meter 8dm::ss 1 < 1 < 90dm::ss 1

Souree Sirene: speakers: 1Hz < f < 1000Hz 40Hz < f < 1300Hz u'< 2.6 · u0 max. 130dB SPL

Pipe Radius a = 15.013mm idem Wall roughness 0.1J.Lm idem

4.5.1 The souree

The two identical sourees consist of boxes in which six speakers are mounted around a cylindrical pipe with diameter of 100mm, see figure 4.9. A fl.exible ring is used to con-

b) :l

----------·~~·--------~ { l 0)}

neet the box to a diffusor in order to minimize the transmission of mechanica! vibrations. The diffusor is connected to a T-joint in which a valve can separate the souree from the measurement section.

The souree is originally designed to give a constant acoustical level up to 130dB SPL in between 40Hz and 1000Hz. We used the souree in the range 50Hz to 1300Hz.

46

4.5.2 Microphone positioning

The eight microphones are mounted in the pipe wall in the same way as in the set-up of TUE, see figure 4.6. Figure 4.10 shows their positions in the measurement section. They are chosen in such a way that measurements can be done in between 50Hz and 1300Hz.

1,o12."" \. I, Oi:Z.. 0...

I' ,/ I' <> '<lS-.

, o,<c"1~""' ... ...

I' 7 1,. 0 *""- o s-s~-.. I' L 0,100"" ..... pe-Y)d. !"' I o,ulD~

.r 7

~ l •• •• •• • -I. ...

Figure 4.10: Details of the acoustic measurement section.

Wh en L is the distance in between two microphones, then sin2 (27r f L / c0 ) gives an indication of the accuracy. Summation for these accuracies of the different microphone pairs gives an idea of how the set-up performs as a function of the frequency. Figure 4.11 shows the summation of the different accuracies as a function of f. We see that around 330Hz the measurement will be the least accurate due to the positions of the microphone.

1~------~--------~------~--------~------~--------~--.

0.8 -c: ~0.6 ~ :3 0.4 ..... co

200 400 600 800 1000 1200 f

Figure 4.11: Influence of frequency on the accuracy of the set-up: i I:~=l I:~=i+l sin

2(27rf(xj - xi)/c0 ).

4.5.3 Determination of the Mach number

The main flow velocity is measured before the flow enters the measurement section. In order to calculate the Mach number in the up- and downstream part of the measurement section, the speed of sound is corrected for the temperature. The wall temperature is measured in

47

the up- and downstream part of the measurement section. These temperatures are used as an first indication of the temperatures inside the flow.

Because the measurement is dorre with eight microphones, we have more equations than unknowns. In other words, the set of equations to obtain the scattering matrix is over-determined and the extra information can be used to determine scattering matrix, the main flow veloeities and temperatures in the up- and downstream part more accurately by applying a least square salution method.

This method of over determination is described by Ajello [23].

4.5.4 Anechoic termination

In the set-up two anechoic terminations are used. In principle they are not necessary to measure the scattering matrix. However, when anechoic ends are used, pressure nodes won't appear at the microphones(, because no standing wave patterns will exist) . This means that if the souree level is large compared to the background noise, this will remain so at all positions and for all frequencies.

The anechoic pipe terminations are perforated pipes covered with cloth. The perforation fraction is increasing in the direction away from the measurement section.

Ajello [23] measured the quality of the used anechoic terminations. Figure 4.12 shows the absolute values of the reflection coefficients of the anechoic terminations as a function of the frequency at M = 0.043.

0.9

o.a

O.i

0.6

IRI 0.5

0.3

0.1

0.1

0o 100 200 :lXl 400 500 600 700 ll10

Fréquence (Hz)

Figure 4.12: The absolute values of the reflection coefficients of the anechoic terminations upstream ( +) and downstream ( o) as a function of the frequency at M = 0.043.

4.5.5 Determination of the wave number

To determine the wave numbers we use equation 1.46 which becomes in the situation with flow:

k± = ko + (1 - i)aref 1±M '

(4.27)

48

in which, in case Ó11 < 8L,

ko ~ ( ~-1) aef f = ao = 2a V 7r f 1 + VPr ( 4.28)

and, in case Ó11 > lh,

Óv a =a~----~--

eff 0 (1.2 + 0.4i)6L. ( 4.29)

With help of the isentropic gas identities (1.14) we find the cinematic viscosity at temperature T, and TJ(T) !'V T 0

·7 with

~ = (Tref) 7.:_

1 (___!]__) = (Tref) 7.:_1

+0.7

Vref T TJref T (4.30)

In the measurements of the scattering matrix, experimental values for k from measurements done by Ronneberger [12] were used, see Ajello [23].

49

Chapter 5

Results of measurements at the sharp bend, R == D /2

In this chapter we will show the results of the measurements at the sharp bend. Section 5.1 presents the reileetion coefficient measurements clone at TUE. Section 5.2 presents the scattering matrix measurements clone at LAUM. The results are compared to the quasistationary modeland to numerical calculations clone by Dequand [24].

For a sharp bend with right-angled outer corner in a pipe with square cross-sectional area, Miles has proposed an analytica! approximation procedure to calculate the scattering matrix (Miles [25]). This theory is only valid without flow. We will campare these results to our results for a pipe with circular cross section. The model has already been verified experimentally by Lippert [26] for pipes with square cross sections.

5.1 The reflection coefficient (TUE)

5.1.1 Measurement at an open end

Figure 5.1 shows the absolute value of the enthalpy reileetion coefficient I Rb I as a function of the Mach number M of an open pipe termination. Also I Rb I is shown according to the theory for the open pipe termination ( equation 1.66) with the wavenumber taken from measurements by Peters [7]. Since Peters' measurements are well understood, camparing these with our results show the quality of our measurements. The enthalpy reileetion coefficient is measured 1.001m from the open pipe termination. The frequency was f =

343Hz (f / fc = 0.0509).

For Strouhal numbers Sra = ka/M > 2, the amplification factor A in equation 1.66 is not well described by CArgull's theory. Experimental values for A are not available for our configuration. Therefore and because A is of the order 1 we substitute A = 1 in the theory of the open pipe termination. This gives rise to a defiection between the experiments and the theory for M < 0.05, see figure 5.1.

Defiections between the experiment and theory for higher Mach numbers are caused by systematic errors due to the preserree of Helmholtz resonators as mentioned in section 4.4.2.

50

0.8

0.7 rf

0.6

0.5

0.4

0.3 0 0.1 0.2

M, 0.3

Figure 5.1: The absolute value of the enthalpy reflection coefficient as a function of M at 1.001m from an open pipe termination (x and +) as well as the theoretica! reflection coefficient (--). Measurement is done at J = 343Hz (f I Je= 0.0509).

5.1.2 Measurement at a sharp bend

The measured sharp bend is placed 51.9cm upstream of an open pipe termination. The pressure reflection coefficient is measured and calculated with the incompressible and compressible models (see section 2.2) at 48.2cm upstream of the bend. The frequency at which is measured is f =343Hz (f I Je= 0.0509).

Figure 5.2 shows the measured pressure reflection coefficient as a function of the Mach number as wellas the incompressible and compressible model for three constant contraction ratios (avl = 0.44, avl = 0.50 and avl = 0.56).

The incompressible model is reasanabie up to M = 0.12 for av1 = 0.50. For larger Mach numbers, deflections between model and experiment rapidly become larger. The flow may not anymore assumed to be incompressible for M > 0.12.

In the compressible model, shown in figure 5.2, acoustic variations in av1 and Mach or Reynolds dependencies of the vena-contracta are not taken into acount. That causes the anechoic behaviour to be predicted at too small Mach numbers.

Figure 5.3 shows the same experimental results compared to two times the compressible model: Once with the vena-contracta corrected for the Mach number and once with the vena-contracta corrected for both Mach and Reynolds numbers. The corrections are carried out as suggested in section 2.4.3.

We see that the compressible model which is corrected for only the Mach dependency, well describes the experiment up to M = 0.2 for (avl)ref = 0.52. (avdref = 0.52 in this case means that in the limit M -+ 0, av1 -+ 0.52.

The compressible model corrected for both Mach and Reynolds dependencies describes the measurement well in between M =0.1 and M = 0.23 for (avl)ref = 0.55. (avdref means now that in the limmit Re-+ oo and M-+ 0, av1 -+ 0.55. In the correction for the Reynolds dependency an empirica! formula given by Blevins [14] for the sharp bend with rounded outer corner is used. This formula is not suitable at low Reynolds numbers. This explains why the deflection between the measurement and the model suddenly increases for M < 0.03.

51

0.8

a: 0.6

0.4

0.1 0.2 M,

0.3

0.8

a: 0.6

0.4 ' '

0.1 0.2 M,

!

I

' I

' . I 't, I

0.3

Figure 5.2: The pressure refiection coefficient of the sharp bend as a function of the Mach number measured with two microphone pairs (x and +). The bend is placed 51.9cm upstream of an open pipe termination and the refiection coefficient is measured at 46.5cm upstream of the bend, f = 343Hz (!I fc = 0.0509). The incompressible model from section 2.2.1 and the compressible model from section 2.2.2 are given on, respectively, the leftand right-hand side, at three contraction ratios independent of M or Re. ( av1 = 0.44 --, avl = 0.50--- and avl = 0.56- · -).

0.9 0.9 • • --""'· ',~ ..

0.8 0.8 '~ '"-

"' a: 0.7 a: 0.7

"\( ... "'"' •, .

0.6 0.6 • ',. ',.

' " ' 0.5 0.5

,. .-- -

0.4 0.4 / 0 0.1 0.2 0.3 0 0.1 0.2 0.3

M, M,

Figure 5.3: The pressure refiection coefficient of the sharp bend as a function of the Mach number measured with two microphone paires (x and + ). The bend is placed 51.9cm upstream of an open pipe termination and the refiection coefficient is measured at 48.2cm upstream of the bend, f =343Hz (!I fc = 0 .. 0509). The compressible model from section 2.2.2 is given with the vena-contracta dependent on only M on the lefthand side and both Mand Re on the righthand side. ((avl)ref = 0.52 --, (avl)ref = 0.55 ---).

52

We notice that the best fitting value of av1 is around 0.52. From the hodograph method (see section 2.3) we found a constant (i.e. independent of M or Re) value of av1 = 0.5255 for a sharp bend with a right-angled outer corner and square cross-section. In order to have a more complete overview we give the contraction ratio's used in the models as a function of the Mach number in figure 5.4.

:2 0.6 ~ c 0

:;::::: (.)

~ ë 0.5 0 () ' I

I I I

I

0.4[._1 -~~-~~----'----~-~___)

0 0.1 0.3 0.4

Figure 5.4: The contraction ratio av1 as a function of the M, corrected for only the Machnumber ((avdref = 0.52, --), as wellas corrected for both M and Re ((av1)ref = 0.55, -- -).

5.2 The transfer matrix (LAUM)

5.2.1 Measurement without mean flow

Miles [25] has presented an analytica! approximation for the acoustical response of the sharp bend with square cross-section. The scattering matrix elements calculated by Miles are shown in figure 5.5. Because M = 0 the bend acts symmetrical and therefore the scattering matrix is symmetrical: r+ = r- and R+ = R-. Furthermore, conservation of energy implies IR±I2 + IT±I2 = 1 at all frequencies below fc·

The scattering matrix is a function of the ration of the frequency f and the cut-off frequency fc for propagation of non-planar modes in the pipe. In practice, when the frequency approaches zero, the bend is transparent to acoustic waves and it acts as a straight pipe: T± = 1 and R± = 0.

When the frequency approaches the cut-off frequency, the dimension of the bend region is of the same order of magnitude as the wavelength of the acoustic wave. For plane waves the bend acts as a straight pipe with a closed end. The waves will refiect completely: T± = 0 and R± = 1.

As mentioned before, Lippert [26] has done a series of measurements on a sharp bend with a right-angled and rounded outer corner, see figure 5.6. The results for the transmission and refiection coefficients r± and R± are plotted in figure 5.5. As can be seen in figure 5.5 the scattering matrix elements change more slowly with increasing frequency in case of a rounded outer corner. When the frequency approaches the cut-off frequency a part of the acoustic waves will still be transmitted. Intuitively this is expected since a

53

--"0 2 1 nl .... 0. -[?0.8 [?1.5 - -U) Q) .c U)

as 0.6 as ..c 0 "0 1 c: 0.

as "0 i='0.4 c:

0 as -U) i='0.5 .c as 0.2 oCO -Q)

0

0 1 0 0.5

f/fc

U)

as ..c 0. 0.5

f/fc 1

Figure 5.5: The scattering matrix elements y± and R± from the model of Miles (--), as well as of Lippert's verification of Miles' theory ( +) and his experiment on the bend with a rounded outer corner ( o).

Figure 5.6: The bends which Lippert has used for his measurements: a rectangular pipe with a square bend and the same geometry with an added rounded plate in the outer corner (see Lippert [26]).

rounded part 'guides' (or reflects) the wave better through the pipe and the bendis a less effective reflector (backwards).

Miles and Lippert worked with pipes with square cross-sections. However, our pipes and bends have circular cross-sections. We assume that the reflection and transmission patterns of a sharp bend in a pipe, whether this pipe has a square or a circular crosssection, are similar for equal values of the relative frequency f I fc· With this assumption some differences like for example the difference in acoustic flow behaviour at the bend-edge are not taken into account.

Dequand recently performed numerical calculations fora sharp bend with rounded outer corner (but square cross-section), with the Euler-code. We will compare these calculations to our experimental results.

Figure 5. 7 shows the absolute value of the scattering matrix elements for a sharp bend as a function of f I fc· The scattering matrix elements are in this case calculated outside the bend.

Surprisingly, we see that ITI according to the calculation does not go to 1 for the limit case f I fc -+ 0. This is caused by a numerical error. We also notice a difference between the calculated and measured phase of the transmission coefficient. The angle between the

54

1.01 0.5

1 0.4 -I-- * * Có0.3 I-

:2'0.99 * en

*** ro .c:

ro P-0.2 ï5 ..

0.98 0 0.1

0

0.970 0.05 0.1 0.15 0.2 0.05 0.1 0.15 0.2

fffc f/fc

1

0.8 0

0.1 -a: - (i)0.6 a: ........ en en ro .0 ..c ro p-0.4

0.05 ·c. 0.2

0 . 0 0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2

fffc fffc

Figure 5. 7: Measured scattering matrix elements y± and R± of the sharp bend (RI D = 0.5) (T+, R+ --; r-, R- · · · · · ·), as well as numerical calculations by Dequand [24] ( *) and the experimental verification of Mil es' theory by Lippert [26] ( o).

curve which represents the measurement of the phase and the f I fc-axis can be seen as a translation of the transmision coeficient from the center of the bend towards a position outside the bend. In this way the difference between the measurement and the calculations would be equivalent with a translation of 1.2cm of the measured transmission coefficient. In other words, when the measured transmission coefficient would be translated so that it describes the bend and a piece of pipe of 1.2cm, than the phase of the calculated and experimental transmission coefficient would be equal.

Differences in behaviour of the measured sharp bend and the calculations can be explained by the fact that we measured at pipes with circular cross-sections and the calculations were clone for pipes with square cross-sections.

When we have a look at the energy conservation, IR±I2 + IT±I2 = 1 (see figure 5.8), we notice that energy is not conserved for relative frequencies larger than 0.1. We measure deviations of the order of 0.01 for f I fc = 0.17 at higher frequencies a deviation up to 2.5%.

55

N

1-+0.99 N

a:

0.98

0.04 0.08 f/fc 0.12 0.16 0.2

Figure 5.8: Energy conservation for the up- and downstream scattering matrix elements of the sharp bend ( upstream: --, downstream: · · · · · ·). The theoretical value is one.

5.2.2 Measurement with mean flow

If we add flow to the system it willlose its symmetry. The up- and downstream scattering matrix elements are now expected to be different from each other. Furthermore, turbulent noise is brought into the system which reduces the coherence of our measurements.

In section 2.2 we have derived an incompressible and compressible model for the flow through a sharp bend. The incompressible model implies a symmetrie scattering matrix (T+ = r- and R+ = R-). In the compressible model subtle differences can be made by taking a constant vena-contracta or a vena-contracta which is dependent on either M or Re or both. Because the measurements were done up to M = 0.067 and the incompressible and compressible model show little difference from each other, we will only compare the experiments to the incompressible model.

Figure 5.9 shows the enthalpy reflection coefficient IRbl from the measurement at the TUE as well as according to the measurement at the LAUM. In order to compare both experiments we had to use the theory of an open pipe termination for the measurement in such a way, that the measurement of the LAUM represented the reflection coefficient l.OOlm from an open pipe termination with a bendat 0.519m from an open pipe termination. As we have pointed out in section 5.1.1 , this theory does not apply for M < 0.05. From figure 5.9 we see that, within the errors of the set-ups, the measurements agree with each other for 0.05 < M < 0.07.

Figure 5.10 shows the absolute value and phase of the scattering matrix elements as a function of the Mach number at three different frequencies (! = 100Hz (!I ie = 0.0148), i = 200Hz (!I ie = 0.0297) and i = 340Hz (!I ie = 504)), as well as the incompressible model with av1 = 0.52 independent of M or Re. As mentioned befare the scattering matrix loses its symmetry. The incompressible model, however, is symmetrie. Because the measurements were done at low mean flow veloeities (0 < M < 0.07) and the models show only very little difference from each other in that region, we will not discuss the differences between the models. However, we can see that since the models are valid for the limit case i I ie --+ 0, they better fit to the measurements at low frequency.

56

0.9

0 • 0 0. o• ri 0.8 ~ .

0

•o •o

0.7

0.6 0 0.02 0.04 M 0.06 0.08 0.1

Figure 5.9: The enthalpy reileetion coefficient IRbl as a function of the Mach number M from measurements at the TUE ( + and x) and the LAUM ( o). At the TUE the reileetion coefficient is measured l.OOlm from an open pipe termination with a sharp bend at0.519m from the open pipe termination. At the LAUM the scattering matrix elements are measured and used to calculate the reileetion coefficient in the same configuration as at the TUE. The frequencies were J = 343Hz (f I Je = 0.0509) at the TUE and J = 340Hz (f I Je = 0.0504) at the LAUM.

57

tJ~

-ó::: -en .0 ctl

0 0.02 0.04 0.06 M1

0.05.----~--~----.-------,

x x x x

0.04 0.06 M1

l1

I~ 0.95~--~--~--~---'

0 0.02 0.04 0.06 M1

1.:30L._ __ ~~I!l!!-j*î_~__._~-~-~~~ -~~1 0.02 0.04 0.06

-~ 0.----~--~----,------,

- 0 0 0 0 0 C/lQ) -0.2 0 0 ° + d, co >~ëxslëxÀ -3.-0.4 * * x )!( ~ ~-~~--~--~--~

·a. 0 0.02 0.04 0.06 M1

-ó::: o.----~--~----,------. - oooooooo ~ -0.2 co 0 x+~:.:**~)f::.: "3.-0.4 :.: ~ ~---~---~---~-~

·a. 0 0.02 0.04 0.06 M1

Figure 5.10: Measurement values of the scattering matrix elements of the sharp bend as a function of M at different frequencies: f = 100Hz (f I fc = 0.0148) ( o ), f = 200Hz (f I fc = 0.0297) ( +) and f = 340Hz (f I fc = 0.0504) (x); as well as the incompressible model (--) with av1 = 0.5255 independent of M or Re.

58

Chapter 6

Results of measurements on smooth bends

In this chapter we will show some results of the smooth bends with R = D and R = 3D and campare these results to the results of measurements done at the sharp bend and to quasi-stationary models.

6.1 The reflection coefficient (TUE)

We expect the influence of a geometry change to be the largest when it is placed in an acoustic velocity anti-node. Measurements done at the bends (with R = 0.5D, R = D and R = 3D) in an acoustic velocity node show no significant difference with measurements at straight pipes of the same lengths.

Sirree we measured at frequencies around f = 343Hz, the wavelength was about 1m and the hearts of the bends were placed at about 0.5m from an open end. Figure 6.1 shows the absolute value of the pressure reflection coefficients of the bends with R = 0.5D, R = D and R = 3D as a function of M at frequencies f = 343Hz, f = 344.5Hz and f = 343Hz, respectively. The hearts of the bends are placed at, respectively, 51.9cm, 50.5cm and 44.1cm from an open pipe termination and the reflection coefficients are measured l.Oülm from the open pipe terminations for all three bends (in which the centerline of the bends are taken to measure their lengths). The bends are not placed at exactly the same position in the standing wave pattern. This causes the amplitude of the acoustic velocity in the hearts of the bends to be at most 0.002% smaller than the amplitude in the maximum of the acoustic velocity node.

Figure 6.1 shows the contrast of the behaviour of the sharp bend to the behaviour of the other bends. For M < 0.1 the R = D and R = 3D bend act like a straight pipe. For higher Mach numbers the reflection coefficients of the two bends become smaller than of a straight pipe. Surprisingly for M > 0.18 the R = D bend has a bigger reflection coefficient than the straight pipe. This suggests that the R = D bend produces sound. However, because the measurement conditions of the different bends and the pipe were not equal, such conclusions cannot be drawn definitely.

Figure 6.2 shows the same results of the R = D and R = 3D bends, compared to the incompressible and compressible model from section 2.2 in which Cd = 0.25

59

.. ... 0.7

0.6

0.5

0.05 0.1

.. . . . .

0.2 0.25 0.3

Figure 6.1: The absolute value of the pressure reileetion coefficient as a funetion of the Mach number for an open pipe termination ( +) and the bends with R = 0.5D ( *), R = D (x) and R = 3D ( o). The measurements are clone at f = 343Hz, except for R = 3D f = 344.5Hz. The reileetion coefficients are measured at 1.001m from an open pipe termination. The bends were placed at 51.9cm, 50.5cm and 44.1cm from the open pipe termination,

respectively.

0.9

0.85 a:

0.8

0.75

0.7L--~-~-~-~-~--

0.05 0.1 0.15 0.2 0.25 0.3 M, O

0.9

0.85 a:

0.8

0.75

0.05 0.1

x

0.15 0.2 0.25 0.3 M,

Figure 6.2: The absolute value of the pressure reileetion coefficient as a function of the Mach number for the bends with R = D (x) and R = 3D ( o). The measurements are clone at f = 343Hz andf = 344.5Hz, respectively. The reileetion coefficients are measured at 1.001m from an open pipe termination. The bends were placed at 50.5cm and 44.1cm from the open pipe termination, respectively. The incompressible and compressible model of section 2.2 are respectively shown on the left- and righthand side for Cd = 0.25 (--)

and cd= 0.18 (-- -).

60

1.1 r---~-~--,-------r--~-----,

1.05

0.95

a: 0.9

0.85

0.8

0.75

0.70 0.05 0.1 0.15 0.2 0.25 0.3

M,

Figure 6.3: The absolute value of the pressure refiection coefficient as a function of the Mach number for the bends with R = D (x) and R = 3D ( o). The measurements are done at f = 343Hz and f = 344.5Hz, respectively. The refiection coefficients are measured at 1.001m from an open pipe termination. The bends were placed at 50.5cm and 44.1cm from the open pipe termination, respectively. The Fanno model (see section 3.3) is shown for cd= 0.25 (-) and cd = 0.18 (- - - ).

and cd= 0.18 are substituted. Figure 6.3 shows the same results of the R = D and R = 3D bends, compared to the

Fanno model in which Cd = 0.25 and Cd = 0.18 is substituted. The chosen Cd values for the R = D and R = 3D bends are taken from Blevins [14].

Both figures 6.2 and 6.3 show large discrepancies with the experimental results and the models. Although the same Cd values are chosen in the different models we notice that the Fanno model gives larger refiection coefficients and the model of section 2.2 gives smaller refiection coefficients than the experiments. For the compressible model of section 2.2 we could introduce a Mach dependency in the 'vena-contracta' effect. This does not give significant changes for M < 0.25.

Since the Cd values apply according Blevins in the limit case Re --+ oo, they would be smaller for M --+ 0. However, since the acoustical behaviour for the R = D and R = 3D bends are small, correct values of Cd at low Mach number are not easy to find. Moreover, the behaviour of the R = D bend cannot bedescribed in termsof the modelsof section 2.2 because it shows larger refiection coefficients than the open pipe termination which would mean cd< 0.

Apparently, smooth bends are not easy to describe with quasi-stationary models, moreover because the R = D bend shows a behaviour which suggests production of sound. Production of sound cannot be explained by a quasi-stationary theory.

6.2 The scattering matrix {LAUM)

Figure 6.4 shows the absolute value of the experimental scattering matrix elements of the three different bends as a function of the relative frequency f / fc at M = 0. First of all we notice that the scatter in the measurements of the R = D and R = 3Dbend are of the order of ±0.01 where the scatter in the measurement of the sharp bend is ±0.003.

61

1.02

1.01

-+ 1-._ 1 (/) .0 co

0.99

0.98 0 0.05 0.1 0.15

f/fc

0.1 r-----..----..-------,-,-,

0.08

[0.06 ._ (/)

~ 0.04

0.02

0.05 0.1 0.15 f/fc

0.1

0.08

;t0.06 ._ (/)

~ 0.04

0.02

0 0

1.02

1.01

-~ ._ 1 (/)

.0 co

0.99

0.98 0

" I :\ i I j\ t I \ I \ J

. 'I " \ I I /" \ . "· . I \;·I'J i··"-1 .,.I_.. ·.\c·.

·t.>.._r.fl" .r:··· .. .-r .\ ... ·.· . ".

0.05 0.1 0.15

0.05

f/fc

0.1 f/fc

0.15

Figure 6.4: The absolute value of the scattering matrix elements of the bends R = 0.5D (--), R = D (- - -) and R = 3D ( · · · · · ·) as a function of the relative frequency f I fc at M = 0.

Furthermore we see that the infiuence of the geometry on the transmission coefficients is smaller than the scatter in the measurement. The refiection coefficients from the R = D and R = 3D bends drop with about 75% and 90%, respectively, compared totheR = 0.5D bend.

Figure 6.5 shows the energy conservation for the different bends. Within the scatter of the data, all three bends show the same kind of behaviour. Though, the R = 3D shows a slighty larger defiection than the other two bends.

Figure 6.6 shows the absolute values of the scattering matrix elements of the three bends as a function of f I fc at M = 0.067. We notice that adding a flow to the system gives a different kind of behaviour of the R = D and R = 3D bends than of the R = 0.5D bend. Where the sharp showed an decrease in the transmission coefficients the two other bends show a small increase, they even become bigger than one which can be interpretedas a production of sound.

The refiection coefficients of the R = D and R = 3D bends remain almast unchanged

62

1.02 1.02

1.01 1.01 IJ•I, .. ,,

~ + "' f- 0.99 ~0.99 + + ~ 0.98 ' ~0.98 a: '" ' a: /, ,,

0.97 :::: 0.97 IJ liJ

tjl

0.96 1·11

0.96 I

' 0.95 0.95 0 0.04 0.08 f 0.12 0.16 0.2 0 0.04 0.08 flfc 0.12 0.16 0.2 f/ c

Figure 6.5: Energy conservation for the up- and downstream scattering matrix elements of the bends with R = 0.5D (solid line), R = D (dashed line) and R = 3D (dotted line) ( upstream: lefthandside, downstream: righthandside).

after adding the flow. However, the scatter in the measurement became smaller which might imply that after adding a flow the position where interactions take place, are better defined.

Figure 6. 7 shows the absolute values of the scattering matrix elements as of the three bends as a function of M at f = 90Hz (! / fc = 0.0134). The behaviour shown by the R = D and R = 3D bend cannot bedescribed by any of the suggested models.

63

1.02 r------,...---~----.----,

0 0.05 0.1 0.15 flfc

0. 1 r------,...---~--"~--.----,

0.08

;f0.06 -en .g 0.04

0.02 • I\ I" " " r r .. A 1 rl \

1 •

I I 0 ~ \ "' . ' - r \ ,r i 1/ " "\ \ .

. ··~.·'···· ·.· 00

0.05 0.1 0.15 flfc

0.08

~0.06 -en .a co 0.04

0.02

0.05 0.1 flfc

0.15

1.02 r-----r------,----,----,

E en .g 0.98

0.96

0 0.05 0.1 flfc

0.15

Figure 6.6: The absolute value of the scattering matrix elements of the bends R = 0.5D (--), R = D (- - -) and R = 3D ( · · · · · ·) as a function of the relative frequency i/ ie at M=0.067.

64

1.01 .----~--~--~----.., 0.03.----~--~--~----,

,. 0.02 , .. _...., ... . . . _....,

+ ' ó::: 1-._ ._ Cl) Cl)

.0 .0

<tl 0.99 <tl

0.01

<.

0.98 0 ----0 0.02 0.04 0.06 0 0.02 0.04 0.06

M1 M1

0.03 1.01

0.02 1 - --·....:....· .-.·.:_· --_...., _...., + t:. a: ._

Cl) Cl)

.0 .0

<tl 0.01 <tl

0.99

. ---0 0 0.02 0.04 0.06

0.980 0.02 0.04 0.06

M1 M1

Figure 6. 7: The absolute value of the scattering matrix elements of the bends R = 0.5D (-), R = D (---) and R = 3D (······)as a function of Mat f =90Hz (f / fc = 0.0134).

65

Chapter 7

general conclusions

The goal of this study was to find a theoretical model, such as the quasi-stationary model, which describes the aero-acoustical response of a 90° bend to acoustic pulsations. Measurements have beendoneon a sharp bend with a rounded outer corner as wellas on bends with R = D and R = 3D.

Although systematic errors were introduced by the presence of Helmholtz resonators in the set-up of TUE, we were able to compare the results with the quasi-stationary models. The measurements at TUE should still be corrected for the presence of the Helmholtz resonators.

It appears that the behaviour of the reEleetion coefficient of the sharp bend is well described by the quasi-stationary compressible model of section 2.2.2 up to M = 0.23 provided that the contraction ratio of the vena-contracta is well chosen and dependent on the Mach and Reynolds numbers. An analytica! salution for this contraction ratio from a compressible theory of the Borda mouth piece gives good results.

For smaller Mach numbers, up to M = 0.12, the compressible model of section 2.2.2 well describes the reEleetion coefficient of the sharp bend when the contraction ratio is corrected for both the Mach and Reynolds numbers.

Although scattering matrix measurements show that the sharp bend behaves as predicted by the numerical calculations for a sharp bend with a square cross-section, these calculations could still be improved. Scattering matrix measurements at larger Mach uurnbers should be performed in order to compare the scattering matrix to the quasi-stationary models

The R = D and R = 3D bends show small differences compaired to a straight piece of pipe. The acoustic effects are small and therefore hard to measure. The refiection coefficients of the R = D and R = 3D bends may not correspond with either the quasistationary models which describe the sharp bend or the Fanno-model.

ReEleetion coefficient measurements suggest that the R = D bend can produce sound when flow is added to the system. Sound production cannot be understood in terms of quasi-stationary models. It is a phenomenon which is caused by complex interactions between vortices and the geometry of the bend. These interactions depend strongly on the acoustic velocity amplitude. Measurements of the reEleetion coefficient as a function of the acoustic velocity amplitude should be performed to get better insight in the production of sound by the R = D bend.

66

Bibliography

[1] J .K. Snel, The effects of a sharp 90° bend on the selfsustained aero-acoustical pulsations in gas transport systems: the two side-branch system, Report (TUE, 1998).

[2] S. Ziada, Flow excited resonances of piping systems containing side-branches: excitation, counter measures and design guidelines, (Seminar on Acoustic Pulsations in Rotating Machinery, Toronto, 1993).

[3] G.K. Batchelor, An Introduetion toFluid Dynamics, (Cambridge, U.K. 1967).

[4] R. Ter Riet, The influence of a bendon pulsations in gas transport systems, R-1469-A (TUE, 1998).

[5] A.D. Pierce, Acoustics: an introduetion to its physical principles and applications, (Acoustical Society of America, New York, 1989).

[6] R.J.J. Boot, Het aeroakoestisch gedrag van diafragma's, R-1370-A (TUE, 1995).

[7] M.C.A.M. Peters, Aeroacoustic sourees in internal flows, (TUE, 1993).

[8] H. Schlichting Boundary-layer theory (Me. Graw-Hill Inc., New York, 1979).

[9] A.J.Reijnen, Demping en reflectie van akoestische golven in een buis, R-1183-A (TUE, 1992).

[10] M.S.Howe, J. Acoust. Soc. Am., 98 (1995) 1723.

[11] D. Bechert, Journalof sound and vibration, 70 (1980) 389.

[12] D. Ronneberger, Acustica, 19 (1967) 222.

[13] L. Prandtl & O.G. Tietjens, Fundamentals of hydro- and earomechanics, (Dover Publications Inc., New York, 1934).

[14] R.D. Blevins, Applied Fluid Dynamics Handbook, (Van Nostrand Reinhold Company Inc., New York, 1984).

[15] I.E. Idelchik, Handbook of hydrolic resistance: coeflicients of local resistance and of friction, (Israel program for scientific translations, J erusalem, 1966).

[16] G.C.J. Hofmans, Vortex Sound in Coniined Flows, (TUE, 1998).

[17] P.A. Thompson Compressible-fluid dynamics (Me. Graw-Hill Inc., New York, 1972).

67

[18] M. Ábom, H. Bodén, J. Acoust. Soc. Am., 83 (1988) 2429.

[19] H. Bodén, M. Ábom, J. Acoust. Soc. Am., 79 (1986) 541.

[20] J.A. van de Konijnenberg, An experimental study on the acoustic reflection coefficient of open pipes at low Helmholtz numbers, R-1119-A (TUE, 1991).

[21] A.H. Shapiro, The dynamics and thermodynamics of compressible Huid flow, volume I, (New York, 1953).

[22] W.G.E. van Ballegooien, Demping van akoestische golven in pijpen: ijking van de debietmeting, meting van de demping, R-1145-S (TUE, 1992).

[23] G. Ajello, Mesures acoustiques dans les guides d'ondes en présence d'écoulement: Mise au point d'un banc de mesure Application à des discontinuités, (Université du Maine, 1997).

[24] S. Dequand Private communication, (1998).

[25] J.W. Miles J. Acoust. Soc. Am., 19 (1947) 572.

[26] W.K.R. Lippert Acustica, 5 (1955) 274.

68

Appendix A

A sharp bend in a tube; compressible equations

From equations 2.28 we can eliminate the velocity inthefree jet Uv with the first equation. Fr om the first order terms ( zeroth order in acoustics) re rnains

U Û + I PlO (n .h._) 10 1 1-1 PlO PlO - PIO

u û + I PlO ( ih PI ) 10 1 1-1 PIO PlO - PIO

Substitution of

= a1vP~ou10 1!1.. + ..l!l._ _ 1!.3!_ + _l_Pvo .h__ _ 1!.3!_ 222( •• ') (' ') Pvo PIO u1o Pvo 1-1 Pvo Pvo Pvo

_ U Û + _l_ P2o ( fh _ h._) - 20 20 1-1 P2o P2o P20

+ -Û _ P1 -pi 1 - PIOClQ

+ -û - p2 -p2 2 - P20C20

gives five equations with eight unknowns, Pi, p]", Pv, Pi, P2, 0"1, O"v and 0"2 . These can be written in the following matrix representation:

0<12 -0<12 a12M10 0 0 (1 + Mw)p{ 1- D<Ivi3fvM10 1 + aiv/JivMw I 2 !32 M2 -!3Iv + aL!3J,:t-

2MÎo ~/Jiv - -y-1 - Qlv lv 10

(1- Mw)PÏ 1 1 1 0 0 --y-1 a1 2al2D<Ivi31vM10 -20<J20<Ivi3lvM10 2al2D<Ivi3lvMÎo 1- O<J20<Jvi3?,:t-

1MÎo -1 Pv + av

0 0 !3--y 0 -1 av lv :r=l :r=l :r=l

!3122

-!3122 !312

2 M2o (1 + M2o)pt 0 0 0

[ ] !312 !312 --1-!312 (1- M2o)P2 -y-1 1 + M2o 1 - M2o M~o a2

0 0 0

69

Sweeping one time with the fifth row gives

[

0[2

1- oîv,6ÎvM10

2o1201:,61vM10

-012

1 + oîv,B?vMw 1

-20[20!v,6lvM10

Sweeping another time with the second row results in

[A5 -~A1 A,

-1 M10 ] [ {1 + M10)pj l

1 - 1~1 (1 - Mw)P! -As- A2A4 A6- A3 A4 a1

[ ~ A7 A7 (A.1) ::t.=l

a,1f3\fMzo ] [ {1 + M,o)P! ] Ci21f3122 -a21(3122

(312 !312 --(312 (1 - M2o)P2 'Y-1 1 + M2o 1- M2o M~o 0"2

in which

These equations can be rearranged so that the outgoing waves are on the left and the incoming waves are on the right:

[ ~ a 21{3rM20 ] [ {1 + M,o)P! l Ci21 (3122 1 (312 -1 -

1_ 1/312 (1- Mw)P1

1 + M20 As+ A2~~ M2 a2 20 (A.2) ::t.=l

[ {1 + M10)pj l [A

5 _:A

1 A,

Ci21f3122

M10 ] -(312 __ 1_ (1 - M2o)P2 'Y-1 -1 + M2o A6- A3 A4 0"2

A7 A7

The scattering matrix equation is then the result of the multiplication of the inverse of the left-hand side matrix and the right-hand side matrix.

In case that there is no incoming entropy wave the matrices can be reduced to dimensions 2 x 2. The resulting scattering matrix has the form:

70

(A.3)

where

y+ -2(ca- db) d-e+ b-a '

R+ d+c-b-a d-e+ b-a '

R- d-c-b+a) d-c+b-a '

y- -2

d-c+b-a '

and

a

b

c

d

The quadratic equation which has to be solved to determine the reference variabie (see equation 2.32) is

which gives for b2 + 4a :::; 0

-b + vb2 + 4a { a 2a b

(A.4)

71

Appendix B

Measurement of the scattering matrix elements

From two independent test states we find the matrix equation

Solving this gives

This can be rewritten as

in which

+ - P2 r =-P2

r = P~ P2

(B.l)

and the subscripts I and I I denote the first and secoud test state respectively. With two microphones upstream (at position xi and xj) and two downstream (at position Xk and x 1)

72

of the bend (see figure 4.1), t+, r+, r- and r can be determined by using the definition of the transfer function ( equation 4.3):

Hij p+(xj)eik+(xj-x;) + p-(xj)e-ik-(xj-x;)

p+(xj) + p-(xj) eik+(xj-x;) + r+e-ik-(xj-x;)

1 + r+ ik+(x -x) H

::::} r+ e 1 • - ij H·. _ e-ik (xj -x;)

~J

Hkl p+(xk) + p-(xk)

p+(xk)e-ik+(xz-xk) + p-(xk)eik-(x1-xk)

r- + 1 r-e-ik+(xz-xk) + eik-(xz-xk)

::::} 1 _ Hkleik-(xz-xk)

r Hije-ik+(xz-xk) _ 1

Hjk p+(xj) + p-(xj)

p+(xk) + p- (xk) r-(l+r+)

t+(1+r-)

r(1+r+) r+(1+r-)

::::} t+ r-(1 + r+)

Hjk(1 + r-)

::::} c Hjkr+(1 + r-) -

1 + r+

The scattering matrix which can now be calculated gives the relation between the incoming and outgoing pressure waves at positions Xj and xk·

73

Appendix C

Model for flow through pipe with friction; Fanno theory

Thompson [17] works out this theory and derives the following equation:

1- M2 dM2 4f 1M4 (1 + ";1M2) dx D'

(C.1)

in which D is the diameter of the pipe and f the Fanning friction factor and which after integration from position x1 to x 2 gives the machnumber M 2 when M 1 is known:

(C.2)

in which f = - 1- Jx 2 fdx. We will now use this model to derive the scattering matrix X2-Xl X!

for a piece of pipe from x1 to x 2 where f is assumed to be constant (independent of M, Re).

We take the compressibility effects into account in a quasi-stationary approximation, the following seven equations give the relations among the pressures, densities, velocities, speeds of sound and Mach numbers at positions x 1 and x2 :

Mass: P1U1 P2U2

Energy: c2+1-1u2 1 2 1 c2 + 1.=..!. u 2 2 2 2

Fanno: 4f(x2-xl) 1-M12 _ 1-Mi2 + 1+1ln ( MHl+?MO) D 1Mi "M2 2, MHl+?Mn

Perfect gas : c2 IP.l. (C.3) 1 Pl

Perfect gas : c2 lP]_ 2 P2

Definition : Mr Uj C)

Definition : M2 U2 C2

Fr om the first order terms ( zeroth order in acoustics) remains

74

_h_ + JlL = fJ2 + ~ PlO UIQ P20 U20

2c10ê1 + ( 1 - 1 )u10û1 = 2c2oê2 + ( 1 - 1 )u2oÛ2 AloMl = A2oM2 A-o- _2_ + 2(1-MJo) + 1+1 { (1-l)M;o - _2_}

t - 1M;o 1M~0 21 I+yM?0

M;o

(C.4)

After elimination of ê1, ê2 , M1 and M2 from the second and third equation with the last four equations, and after substitution of

+ -P~l = P+l + P-1 û - PI -pi

l - PIOCJO + -

P~2 = P2+ + P2- û - P2 -p2 2 - P20C20

(C.5)

three equations are left with six unknowns, Pt, p]", a-1 , Pt, P2 and a-2 . They can be given in a matrix equation with the incoming waves on the right hand side and the outgoing waves on the left hand side:

in which a. = Plo. Pl2 P2o

1- M10 -(1- 1)(1- M1o)

2+(r-1)MIO

We consider the incoming entropy wave a-1 to be zero. By multiplying with the inverse of the three times three matrix on the left hand side and by eliminating a-2 we can write the matrix equation in the form of a scattering matrix equation:

The scattering matrix elements are

in which

r+=~ eg-dJ

R+ =be-ad eg-dJ

R- = cg-dJ er;-dJ r- = d(e-c) eg-dJ'

A1o _ti! a 1 + M1o- (2- (1- 1)Mlo)-A !312

2

20

b (r- 1)(1 + M10) + (2- (r- 1)Mlo)M A~ f3Î;1

20 20

75

(C.6)

(C.7)

c

d

e

f

2::..!. - ('y + 1) ,8122

( ~20 + ('y- 1 )M2o) ,812

2::..!. ('y - 1) ,8122

A10 - .:r.±l 1- M10- (2 + ('y- 1)M10)-A /312

2 20

g = -('y- 1)(1- M1o) + (2 + ('y- 1)M10) MA~ /3i2' 20 20

In case we know two of the pressure waves the two unknown waves follow from the former equation after determination of the reference variables M1, M1 and ,812 . At a known friction factor and M1 given, M2 can be calculated with equation C.2. From mass

2

conservation (p10u10 = p20u20 ) and Poisson's equation ( E!J.Q = (fl.) -y-l) we can derive P20 C2

(-1 = (M2 ) 1'~ 1 1-'12 Ml .

76

Eindhoven University of Technology MASTER Aero · PDF file2.1 Orifice at the end of a pipe .. 2.1.1 Incompressible theory ... C Model for flow through pipe with friction; ... pressure

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