Category Noise Sources in Turbulent Shear Flows

As explained in section 6.5, closed side branches are almost perfect re­flectors at frequencies such that the length corresponds to an odd number of quarter wave-length. Early quantitative research on the self-sustained oscillation of this type of closed side-branch resonators has been carried out by Bruggeman et al. (1991) and Ziada and Biihlmann (1992). A literature review on this subject is provided by Tonon et al. (2011). The simplest configuration is a set of two side branches of equal length L and diame­ter D placed opposite each other, forming a cross with the main pipe of diameter Dp. Figure 22a shows the amplitude of pressure fluctuations p’ measured at the closed side-branch termination as a function of the aver­age flow velocity U along the main pipe. It also shows the corresponding frequency f. With increasing flow velocity U we observe successively the

first three acoustic modes of the system with: HeL = fL/c0 « || and |. For each of these acoustic modes we observe two critical Strouhal numbers (two hydrodynamic modes)for optimal whistling: SrD = fD/U « 0.5 and 1.0. They correspond to a travel time of the vortices across the opening of the side-branches of one period and two periods respectively.

Figure 22b shows flow visualization for these two hydrodynamic modes. In this case, the acoustic velocity is mainly normal to the convection veloc­ity of the vortices. If the edges of the T-junctions between the main pipe and the side-branches are rounded, the acoustic velocity amplitude is, in first approximation, spatially uniform over the path of the vortices. The in­crease in circulation of the vortex during the first oscillation period, whilst traveling from the upstream edge towards the downstream edge explains the net sound production in the system.

It should be noted that for the prediction of the oscillation amplitude we consider self-sustained oscillations. The instability of the shear layer acts

as an amplifier in this feedback loop, transferring energy from the main flow to the acoustic field. The acoustic resonator selects oscillations close to the resonance frequency. It is a filter. The acoustic oscillation induces new vorticity perturbations at the upstream edge, where the flow separates from the wall. If all these elements in the feedback loop were linear the sys­tem would be either stable, neutrally stable or unstable. In stable conditions perturbations decay exponentionally to zero. In unstable conditions the am­plitude increases indefinitely. In neutrally stable conditions, the amplitude is the amplitude imposed at the initial conditions, and can have any value. We cannot predict a stable limit cycle oscillation with finite amplitude on the basis of linear theory [Bruggeman et al. (1991), Tonon et al. (2011)]. The non-linear phenomenon which limits the pulsation amplitude depends on the amplitude of the oscillations. For moderate pulsation amplitudes (u’/U = p’/p0c0U < 0.1), the main non-linearity is the saturation of the shear layer amplification due to the concentration of the vorticity into dis­crete vortices. Once the shear layer has rolled-up to form a discrete vortex we have reached a maximum perturbation. Actually, because the amount of vorticity shed is almost independent of the amplitude dT /dt к – UUc and the path of the vortex almost straight (from the upstream edge to the downstream edge), the product p0(w x v) is mainly determined by the mean flow. It scales with p0U2. The power generated by this source is at a fixed Strouhal number proportional to the acoustic amplitude u’/U. The losses due to friction and radiation are proportional to the square of the amplitude W2/U2. A finite amplitude is found by balancing the power generated by the source and the power related to friction and radiation losses [Tonon et al. (2011)]. When u’/U > 1 vortex shedding and path become dependent on the amplitude. Furthermore, additional vortex shedding from the down­stream edge of the T-junction induces additional losses which are scaling with (u’/U)3. As a consequence u’/U = 0(1) is a kind of maximum of the pulsation amplitude which is reached when friction and radiation losses are negligible.

The exact value of the maximum pulsation amplitude depends on the details of geometry. For a cross configuration with rounded edges one can reach u’/U ~ 2 [Slaton and Zeegers (2005)]. At such large pulsation am­plitudes shock waves are formed in the side branches due to non-linear wave steepening. For sharp edges one finds u’/U < 0.8 [Kriesels et al. (1995)]. In a Helmholtz resonator under grazing flow with rounded edges one finds u’/U < 0.6 [Dequand et al. (2003a)]. In flue instruments such as a recorder flute or a flute one typically finds u’/U < 0.3 [Verge et al. (1997), Dequand et al. (2003b)].

An interesting aspect of this discussion is that in contrast to turbulence noise produced by flows in free field conditions, the acoustic flow due to whistling is not a small perturbation of the flow. This implies that numerical simulation of the flow is not impossible. In following section an example of such a successful numerical simulation is given.

For Reynolds numbers ReD = UD/v above 50 the wake of a cylinder of diameter D placed with its axis normal to a uniform flow (velocity U, kinematic viscosity v) is unstable with periodic vortex shedding occurring. Vortices have alternating vorticity signs, which results in an oscillating lift force (normal to the flow direction). The force of this flow on the cylinder comes from a reaction force F of the cylinder on the flow. This reaction force acts as a source of sound. For low Mach numbers the cylinder is compact so that we can neglect variation of the retarded time over the source region. Equation (94) can be written in the following form:

xi dFj

4^cq|X|2 dt

The lift force scales with pU2DL where L is the length of the cylinder over which the vortex shedding is coherent. The oscillation frequency corre­sponds to a Strouhal number which is somewhat dependent on the Reynolds number SrD = fD/U « 0.2. This was already observed by Strouhal (1878) (Figure 15). The most important conclusions that can be drawn from this experience is:

• the cylinder does not need to vibrate or oscillate in order to generate the whistling tone,

• the vortices shed by the cylinder do not impinge on any surface or edge.

The first statement contradicts our intuition that sound is produced by wall vibrations. Oscillation of the cylinder can occur and can strongly affect the frequency. This occurs when the mechanical oscillation frequency is close to the natural vortex shedding frequency. One can then observe a strong increase in coherence length of the vortex shedding [Blake (1986)].

The second statement contradicts the intuition that sound is produced upon impingement of vortices on edges. In early work [Rockwell (1983)] this assumption is generally accepted, although it is a rather misleading assumption, as the following examples show.

5.5

Human whistling

As we have seen in section (6.4) convective effects induce acoustical en­ergy absorption upon reflection at an open pipe termination with outflow. The ratio of reflected and incoming acoustic intensity is, following equation (178) and with R = —1:

This is a consequence of the losses in total enthalpy AB’ = UU in the free jet formed by flow separation at the pipe exit. This corresponds to the modulation of the kinetic energy in the jet. This kinetic energy is dissipated by turbulence in the jet with negligible pressure recovery (p’ = 0).

We now analyse the same phenomenon by using the energy corollary of Howe (193). As a first step we consider the spatial distribution of the acoustic velocity field U at the pipe exit. A potential flow such as the acous­tic field u’ = Vp’ does not separate from sharp edges. This flow follows
the walls smoothly. The acoustic streamlines around the edges of the open pipe termination are curved. Which implies that there should be a pressure gradient directed towards the inner side of the bend which provides the centripetal force bending the streamlines. Following the Bernoulli equation (36) this decrease in pressure implies an increase of velocity towards the interior of the bend. Actually, this also follows directly from the condition that the acoustic flow should be irrotational Vx и’ = Vx V(f’ = 0. Rota­tion due to the bending of streamlines should be compensated by a gradient in the radial direction of the tangential component of the velocity. In terms of forces the radial pressure gradient balances the centrifugal force. As the velocity increases and the radius of curvature of the streamlines decreases as we approach the interior of the bend, the centrifugal force increases dra­matically. Obviously for a sharp edge we have a singularity in a potential flow [Prandtl (1934), Milne-Thomson (1952), Paterson (1983)]. As we ap­proach the edge, the magnitude of the acoustic velocity becomes infinitely large. However, moving away from the edge in the direction of the pipe axis, causes rapid decrease of the amplitude of the acoustic field. The direction of the acoustic velocity also turns gradually from normal to the pipe axis to­wards the direction of the pipe axis (Figure 16). A harmonically oscillating acoustic field implies that the acoustic flux is directed pipe-outward during half the oscillation period and is directed pipe-inward during the next half period.

The next step in our analysis is to consider the vortex shedding. Vortex shedding is the result of viscous effects in the boundary layers of the flow. In these boundary layers the flow velocity |v| decreases from the bulk flow velocity down to the zero velocity imposed by the no-slip boundary condition v = 0 at the wall. At high Reynolds numbers the boundary layers are thin. The flow in these boundary layers is mainly directed along the wall and this implies that the pressure in the boundary layer is equal to the pressure imposed by the bulk flow at the outer edge of the boundary layer. In the bulk of the flow there is an equilibrium between inertia and pressure gradient (as the viscous forces are negligible for high Reynolds numbers). An increase in pressure is compensated by a reduction of velocity. This allows fluid particles to move against an adverse pressure gradient. In the viscous boundary layer, the fluid has lost much of its kinetic energy and cannot use its inertia to overcome an adverse pressure gradient. Viscous drag of the fluid in the boundary layer by the bulk flow can allow to overcome a small pressure gradient. However, in a steady flow with bulk velocity U there will be back flow along the wall, when the characteristic time for momentum diffusion across the boundary layer ff2 /v (with ff the momentum thickness

of the boundary layer) becomes larger than the characteristic deformation time (dU/dx)-1. As a consequence the flow will separate from the wall. As the flow passes a sharp edge at the end of an unflanged open pipe, the flow will certainly separate from the wall. This implies that the flow continues tangentially to the wall (along the direction of the axis of the pipe) rather than following the bend, as does the potential flow. A shear layer is formed separating the main flow from a dead water region around the free jet. In this shear layer there is vorticity ш. Due to the instability of the shear layer this vorticity concentrates in coherent vortical structures (vortices). Each time the acoustic field turns from pipe inward to pipe outward a new vortex is formed at the edge of the pipe termination. This vortex accumulates most of the vorticty shed at the sharp edge while travelling at almost constant velocity Uc « U/2 in the direction of the pipe axis. The strength of the vortex is measured by the circulation Г = <fC v ■ dx = fS ш ■ dS taken along a contour C enclosing the vortex. The circulation is the flux of the vorticity vector through a surface sustained by the contour. In first approximation, with the acoustic velocity considerably smaller than the bulk flow velocity, we have dr/dt = – UUc, as illustrated in Figure 17 [Nelson et al. (1983)]. After an oscillation period a new vortex is shed from the pipe edges and the
old vortex continues to travel at almost constant speed. Further downstream these vortex rings are eventually dissipated by turbulence.

Figure 17. Vorticity and circulation of a shear layer.

Considering a new vortex shed at the edges of the pipe, we can see that the vector Uxv is directed normal to the pipe axis in the direction away from this axis. In first approximation the convection velocity is v « (Uc, 0, 0). At this point in time the acoustic velocity U is oriented in the same direction and locally very large due to the singularity of the acoustic flow at the edge. Hence the triple product —p0(U x v) ■ U is very large and negative. The formation of a new vortex by acoustic excitation of the shear layer implies sound absorption, which seems quite logical. The less trivial message from the theory of Howe, is that, after half a period, the same vortex will start to generate sound, because the sign of the acoustic velocity changes while those of the rotation ш and of the convection velocity v do not change. The power produced in the second half period is much lower than the initial sound absorption because the growth in the vortex circulation is not able to compensate for the decrease in acoustic velocity amplitude and its rotation in the direction of the pipe axis (Figure 18). We obtained a result similar to the predicted sound absorption found when using the quasi-steady model.

The major gain is the understanding that there is a possibility of net

sound production by the vortex shedding, if we can reduce the initial ab­sorption and enhance production. This is exactly what occurs when we whistle with our lips. Flow separation at our lips occurs actually almost at the neck of the channel formed by our lips. This implies that there is a strong reduction of the singularity of the acoustic velocity, because the lips are rounded rather than sharp and the acoustic velocity is almost parallel to the axis of the flow. Moreover, if we ensure that the vortex travels over the radius of curvature R of our lips within half an oscillation period, it will start producing sound. As the acoustic field has not expanded in free space its amplitude is still large and the direction reasonably normal to the main flow axis. This particularly favourable condition is met when the Strouhal number SrR = fR/U = (fR/Uc)(Uc/U) « 0.25. The frequency f is im­posed by the Helmholtz resonance of our mouth cavity in combination with the neck formed by our lips (Figure 19) [Wilson et al. (1971), Hirschberg et al. (1995)]. Once we have adjusted this geometry we should tune the flow velocity to match the Strouhal number condition. This explains why a child, that is blowing too hard will never be able to whistle by blowing harder and harder.

This simple experiment confirms that we do not need any impingement of vortices on an edge to generate sound. It furthermore indicates that sharp

edges are not necessary for vortices to be shed. Finally, it indicates that sharp edges at the flow separation point actually tend to reduce sound due to vortex shedding. We should note, however, that sharp edges will strongly enhance broadband noise production. This effect is clear when we consider the sound produced by blowing hard trough our lips in comparison with the sound produced by blowing along our teeth (as we do when we generate a fricative sound such as an [s]).

A related configuration is that of a pipe terminated by a diffuser, which is a conical expansion from the pipe cross section Sp to the outlet cross section S0. This allows reducing the loss of energy by dissipation of kinetic energy in the free jet at the outlet. This works only typically if the increase in cross section is not much larger than a factor 2. Furthermore, the opening angle of the diffuser cone should be less than 8 degrees. This would imply very long diffusers. In practice one uses therefore opening angles of about 20 to 25 degrees. In this case the flow partially separates from the wall within the diffuser. Considering the steady flow performance the losses due to this flow separation is rather marginal. However, it has a spectacular consequence on the energy reflection coefficient for acoustic waves travelling in the pipe towards the open end. Measurements of RE = I-/I+ appear to be larger than unity for two ranges of Strouhal numbers (Figure 20). This implies sound generation and potentially whistling of the pipe system too. The lowest Strouhal number corresponds roughly to a travel time of the vortices through the diffuser half that of the oscillation period. The second higher Strouhal number corresponds to a travel time of one and a half periods. This means that there are two co-existing vortices within the diffuser. These two flow conditions are called hydrodynamic modes or stages Howe (1998). The same type of behaviour can be observed with another configuration in the next section.

5.4 Powell/Howe analogy

The quantitative relationship between vortex shedding and sound pro­duction was first established by Powell (1964). His approach was limited

to free field conditions and low Mach numbers. Howe [Howe (1975), Howe (1998), Howe (2002)] proposed a generalization of this approach to arbi­trary Mach numbers, which is valid for confined flows. In its most general form it implies a numerical solution of a complex convective wave equation [Doak (1995), Musafir (1997)] and it is mostly used at low Mach numbers. In most cases it is used for the analysis of the sound production based on an energy corollary, which we are looking at.

In section (5.2) we have seen that the choice of variable is important in an analogy, because it determines the approximations that are intuitively reasonable. In section (6.4) we have seen that in the presence of a frictionless mean flow, the total enthalpy fluctuation is a natural variable. Following equation (31):

dt + VB = — (w + v) + f (190)

we see that the Corriolis acceleration (w x v) acts as a source of sound if we define the acoustic velocity U as the time-dependent part of the potential flow in a Helmholtz decomposition of the flow velocity:

v = V(p0 + ) + Vx ф. (191)

This yields the definition proposed by Howe (1984) for the acoustic velocity:

U = V<f’ . (192)

For low Mach number flows Howe (1984) proposes the use of the following approximation for the time average acoustic power < P > produced by a flow:

< P >= —po < I (w x v) • U dV > . (193)

JV

This corresponds to the use of the energy corollary (64) assuming f = —p0(w x v). This intuitive statement gives an excellent insight into the sound production associated with vortex shedding in low Mach number flows, which is due to the fact that vorticity is a conserved quantity in 2­D frictionless flows. We therefore have an intuition for the dynamics of vortices in such flows [Prandtl (1934), Milne-Thomson (1952), Paterson (1983), Saffman (1992)].

A drawback of the vortex sound theory is that it stresses the dipole character of the sound source: V – (w x v). Unlike the analogy of Lighthill, it does not impose a quadrupole character to the sound field. Thus, in order
to apply this analogy to flows, such as a free jet, one has to use analogies as proposed by Mohring (1978) or Schram and Hirschberg (2003), which do take this aspect into account. In our discussions we limit ourselves to applications with a dominant dipole source term. In such cases the formulation of Howe (1984), as given in equation (193) can be used.

The bottle or Helmholtz resonator is an elementary acoustical resonator [Dowling and Ffowcs Williams (1983), Pierce (1990), Rienstra and Hirschberg (1999)]. It is an acoustical mass-spring system, because the volume of the bottle acts as a spring, while the inertia of the flow (mass) is concentrated in the neck (Figure 6.6). If the neck has a uniform cross section S and a

length L the mass obviously is:
M = p0S(L + 25)	(181)
Ap AV _ S Ax
(182)
where Ax is the acoustic fluid displacement in the neck. The uniform density assumption is actually in agreement with the fact that we neglect inertia in the volume of the bottle, implying a uniform pressure. This is exactly the same assumption as for a massless spring, which implies that the tension is uniform over the spring. Assuming an adiabatic compression we have: Ap = c2Ap. The force acting on the fluid in the neck is therefore:
AF = SAp	(183)

From this we deduce that the spring constant K of the system is:

K 2S2

K = Poco -у

and the resonance frequency of the resonator is given by:

= (185)

It is interesting to note that for an ideal gas poc0 = jp0 where 7 is the Poisson ratio of specific heats at constant pressure and volume respectively. When considering an oven or furnace with an open door, the gas density in the neck of the system is close to that of the surrounding air at room temperature, while the average pressure p0 in the volume is atmospheric. Hence, the resonance frequency depends only weakly on the temperature in the oven.

A bottle of cider or champagne has a neck with a non-uniform cross section S(x). In order to calculate its resonance frequency we need a more sophisticated approach [Cummings (1972)]. We start again by applying the integral mass conservation law on the volume, assuming a uniform density in the volume:

Pou’S (L)

where u’ is the acoustic velocity at the pipe opening x = L. Furthermore, we use the integral of the momentum equation (45) over the neck of the bottle:

where we assumed the flow in the neck to be incompressible. Elimination of u’ yields a second-order harmonic equation corresponding to the resonance frequency:

^0 = C0

with:

Acoustic energy can accumulate in parts of a duct system delimited by strongly reflecting boundaries, an example of this being an expansion cham­ber of length L and cross section Sm along a pipe of cross section Sp << Sm. Such an expansion chamber can also be used as muffler, to reflect waves generated by an engine. Maximum transmission losses are found in cases where the expansion chamber length matches an odd number of quarter wave-length of the incoming waves (figure 13). Other obvious examples of resonators are pipe segments terminated either by open or closed pipe ter­minations.

The flute can be approximated as an open-open pipe with uniform cross section displaying resonances when an integer number of half wave length matches the pipe length. Since the mouth opening of the flute is smaller than the pipe cross section, the end correction of the mouth opening is quite large. This implies an important inertia, which would detune the pipe resonances if it was not combined with a compliance approaching that of a pipe segment of the length of the end correction. For this reason the mouth of the flute is not at the pipe termination. The volume in the dead end between the closed pipe termination and the mouth opening is adjusted by means of a movable piston (cork) so that the first resonances of the pipe are exactly multiples of the fundamental. This strongly enriches the sound produced by the instrument [Chaigne and Kergomard (2008)].

The close-open pipe of a uniform cross-section is a model for the clarinet. It displays resonances when the length of the pipe matches an odd number of quarter wave-length. This promotes odd harmonics of the fundamental, giving the sound a particular character. The use of a conical close-open pipe, such as the oboe or the saxophone, provides a series of resonances at frequencies that are a multiple of half wave length matching the pipe

• Experiments — Theory

Figure 13. Comparison between measurement [Davis 1954] and theory [Dowling 1983] for transmission losses of a simple expansion chamber along a pipe with an anechoic termination. The transmission losses are defined by: TL = 20log(I/T), with I the amplitude of the incident wave and T the amplitude of the transmitted wave. Maxima of transmission losses corre­spond to a length equal to an odd number of quarter wave-length. Minima correspond to an integer number of half wave-length.

length. In conical pipes the acoustic field is dominated by spherical waves rather than plane waves. Consequently the radiation efficiency of the sound source (reed) increases at low frequences linearily with the frequency, as demonstrated by equation (63). Thus, contrary to the clarinet, the low­est resonance frequency does not correspond to the strongest impedance in conical pipe instruments [Chaigne and Kergomard (2008)].

A pipe system can display localized acoustic standing waves, captured between two reflectors. A typical reflector is a closed side branch and the closed end of it imposes a standing wave within the side branch [Bruggeman et al. (1991), Ziada and Buhlmann (1992), Tonon et al. (2011)]. The incom­ing and reflected waves have equal amplitudes as imposed by the closed pipe termination R =1. Whenever the closed side-branch length corresponds to

an odd number of quarter wave-length the standing wave imposes a pres­sure node at the junction of the closed side branch with the main pipe. At low frequencies the pressure is continuous over the junction and this imposes a pressure node in the main pipe (equation 157). Thus, this pres­sure node acts as an ideal open pipe termination with R = —1. When two closed side branches with equal resonance frequency are placed at a distance corresponding to an integer number of half a wave length, we ob­tain an acoustically perfectly closed system [Bruggeman et al. (1991), Ziada and Buhlmann (1992), Tonon et al. (2011)]. Paradoxically enough, this system is open for the flow [Hein and Koch (2008)]. The most spectacular resonances are obtained when considering two opposite closed side-branches forming a cross configuration with the main pipe [Keller (1984), Kriesels et al. (1995), Dequand et al. (2003c), Slaton and Zeegers (2005)]. This will be discussed in more detail in the next chapter.

A series of closed side branches of equal length can display strong acous­tical resonances even if the side-branches are placed at arbitrary distances from each other. A system of deep closed side branches of random depth can also display Anderson localization [Depolier et al. (1986)].

Another example of strong localization of a resonant acoustic field in an apparently open system is the Beta Parker mode in a pipe system with a splitter plate [Welsh and Stokes (1984),Stokes and Welsh (1986)]. When the longitudinal splitter plate (separating the pipe in two equally wide parallel ducts) is longer than the pipe width, there is a resonance for which the half wave length is longer than the pipe width. Hence, at this frequency only plane waves propagate along the main pipe. If the two pipe segments separated by the splitter plate are oscillating in opposite phases, the system will not radiate any plane waves and actually does not radiate at all. This type of resonance has been observed in ventilation ducts (due to guiding vanes at bends), turbines (stator or rotor) [Welsh and Stokes (1984), Stokes and Welsh (1986)] and even protection grid in building (ventilators, roof) [Spruyt (1972)].

Until now it was assumed that the fluid in the pipe is stagnant. Now we will consider the influence of a uniform, steady outgoing flow Up in the pipe on the acoustic response of the pipe termination. For plane waves the convected d’Alembert solution is:

(168)

du’ dp’

dx

we get: , p+ Л (, x u — ——- exp iu t—————- — poco V V co + Up
p —— exp PQCq
lu	(170)

Looking at a pipe of a uniform cross section Sp, terminated by an orifice plate with opening S0. The flow leaves the pipe through the orifice forming a free jet downstream of the pipe, which contracts slightly after leaving the orifice to reach a minimum cross section Sj before mixing with the surrounding air. The pressure at the minimum cross section in the jet is equal to the pressure of the surroundings. Typically, the contraction factor is Sj/Sq — 0.7 for a thin orifice plate with sharp edges. In the low frequency limit we can describe the flow by using the integral mass conservation law:

ppUpSp — pjUjSj .

The integration of the momentum equation in the quasi-static approxima­tion, neglecting friction and heat transfer and assuming an irrotational flow (equations 35 and 37) yields:

We observe that the pipe termination is anechoic (R — 0) for Mp — Up0/cp0 — (Upq/Uj0)2 ~ (Sj/Sp) (figure 12). This particular behaviour

was first observed and explained by Bechert [1980]. It is a consequence of sound absorption by vortex shedding (modulation of the shear layers of the jet). While the model does not explicitly take the effect of viscosity into account, the assumption that pj = 0 can only be explained by the presence of a free jet, which is a consequence of flow separation due to viscosity. Also we assume implicitly that all the kinetic energy in this jet is dissipated by turbulence without any pressure recovery.

Note that in the presence of flow the acoustical intensity is given by [Morfey (1971)]:

Where m’ is the fluctuation in mass flux:

m = pu

and B’ is the fluctuation in the total enthalpy:

(180)

Consequently a reflection coefficient R = —1 indicates energy losses. This is the limit found when Uj0 = Up0.

In the ideal open pipe termination limit discussed in the previous section, the radiation of sound from the open pipe termination was ignored.

This is the very low frequency limit. With increasing the frequency we get deviations. Firstly the inertia of the oscillating acoustic flow outside the pipe, around the open end, which implies that there is a finite pressure at the outlet of the pipe supplying the force needed for the acceleration of the fluid. As discovered by Bernoulli, this effect can be accounted for by assuming that the wave reflection occurs at a small distance 6 outside the pipe. This is called the end correction [Rayleigh (1954), Pierce (1990), Dowling and Ffowcs Williams (1983)]. The exact value of this end correction depends strongly on the geometry of the pipe termination. Whilst we get in the low frequency limit 6 = 0.61a [Levine and Schwinger (1948)] for a pipe with a radius a and infinitely thin walls (unflanged pipe) we get 6 = 0.82a for a flanged pipe (pipe end flush with a flat wall) [Morse and Ingard (1968), Peters et al. (1993)]. The order of magnitude of this end correction can be estimated by considering the pulsation of a sphere of radius a0. The end correction corresponds to the part of the solution (63) for the pressure field which does not carry energy (not in phase with iwa)

We find 6 = a0. Another way to look at this is to consider the incompress­ible part of the acoustic velocity (vr = i^a(a0/r)2 exp(iwt)) associated with the pulsation of the sphere and to integrate the radial component of the

momentum (p0(dv’r/dt = —(dp/dr)) to calculate the associated pressure on the surface of the sphere p(a0) = J Poш2d(a0/r)2dr = p0ui2aa0. It shows that for a 3-D radial flow the end correction is determined by the near field, which is incompressible.

This does not apply to a two-dimensional flow through a slit. Assuming an incompressible flow would result in an infinite large end correction, the near field is essentially compressible [Lesser and Lewis (1972)]. This illus­trates the complexity of two-dimensional acoustic fields, as is discussed in Dowling and Ffowcs Williams (1983). In practice, this means that the use of a two-dimensional model for an unbounded flow can lead to unrealistically large radiation losses.

The fact that we hear music generated by a wind instrument is a clear demonstration that waves are radiated by the open pipe terminations (Fig­ure 11). We now estimate the amplitude of these waves by coupling a plane wave propagation model in the pipe with a spherical wave emerging from the pipe termination. We assume that the frequency is so low that one can neglect compressibility in the region of the transition from plane waves to spherical waves.

In this case the mass conservation law implies:

Spv’x = 4nR2v’r (163)

where Sp is the pipe cross sectional area and R is the distance chosen such that it is small compared to the acoustic wave length, but large compared to the pipe radius (R > RSp). In terms of plane waves amplitude and spherical wave amplitude we have:

Sp(p+ – p-) = 4nR2 r f 1 + kR) exp(-*koR) – – i4n — ■ (164)

The conservation of acoustic energy over the same control volume yields:

sP(p+2 – pi2) -4п Щ – (165)

ko

neglecting the phase of the waves and combined with (164) this implies:

Zr 1 — R (koa)2

Poco 1 + R 4

For a flanged pipe termination we have :

Zr 1 — R (koa)2

Poco 1 + R 2

Reflections on the flange (wall) double the radiation power (Figure 3).

5.2 Pipes modes

We are considering propagation of harmonic waves p’ = pexp(iwt) in a duct with a uniform rectangular cross section, with the duct axis is in the x3 direction. The duct is delimited by rigid walls in the planes: xi = 0,xi = h1,x2 = 0,x2 = h2 (Figure 9). For such harmonic waves the wave equation

Figure 9. Duct with rectangular cross section.

(47) can be written as:

This is the Helmholtz equation.

Seeking a solution by using the method of separation of variables:

p = F(xi)G(x2)H(x3) . (139)

and substituting (139) in (138) we get:

2 1 d2F 1 d2G 1 d2H

0 + F dxf + G dx2 + H dx3

As this equation should be valid for any value of x – in (140) should be constant:

The constants a and в are determined by the boundary conditions of zero normal velocity at the rigid walls. The normal component of the pressure gradient, which is proportional to this normal velocity, vanishes at the walls:

(148)

There are two types of solution, depending on the sign of k2mn. For positive values we have propagating wave modes:

. mn / nn cos ( —id cos I—x2 ) exp(T*|kmn|x3) (149)

pmn

and for negative values we have evanescent modes:

. mn / nn і cos ( ~^~xi ) cos I ^X2) exp(Tkmnx3) (150)

Pmn

with

(151)

The solution we are looking for is a linear superposition of these modes:

where the amplitudes of the modes are determined by the boundary con­ditions at the boundaries of the duct in the x3 direction. For each mode there is a cut off frequency (wmn)c below which the mode is evanescent. For example for the mode pi0 we have:

(wio)c = ПС0 . (153)

hi

The duct width should be larger than half the wave length to allow prop­agation of this first higher-order mode. The mode p00 is the plane wave mode and will always propagate.

Evanescent waves do not propagate energy. They decay exponentionally with the distance along the duct. In the low frequency limit ш ^ (umn)c the pressure perturbation due to an evanescent mode will decay faster than exp ^—mf-x^j. For mode (1,0) a distance hi is sufficient for a decay by a factor exp(n) « 23. All other higher-order modes will decay even faster.

Hence, the acoustic field will be dominated by plane waves in the low fre­quency limit for distances larger than the duct width away from disconti­nuities. This result will be applied in the following sections.

In this discussion we have neglected damping. In ducts damping due to visco-thermal losses at the walls is usually dominant. These effects on acoustical wave propagation are discussed in Pierce (1990), Bruneau (2006) and Rienstra and Hirschberg (1999). Damping, in presence of flow, has been extensively studied by Ronneberger and Ahrens (1977), Peters et al. (1993), Howe (1998) and Allam and Abom (2005).

5.3 Reflection at pipe discontinuities at low frequencies

We are looking at the reflection and transmission of plane acoustic waves at an abrupt transition at x = 0 between a pipe with uniform cross-section Si(x < 0) and another with uniform cross-section S2(x > 0) (Figure 10). Assuming that small perturbations of a stagnant fluid, as described by the

Figure 10. Reflection and transmission of waves at a pipe discontinuity.

linearized mass and momentum equations (44 and 45), the integral formu-

lation of the mass equation can be applied:

to a volume V enclosed by the fixed surface S with outer unit normal vector П. The surface S is chosen to enclose the pipe discontinuity and cut the pipes at a distance of the order of the pipe diameter from the discontinuity. As explained in the previous section, the acoustic field in the pipes is dominated at low frequences by the plane travelling waves. This implies that the acoustic field is given by:

Pi = P+ exp(-ikox) + p – exp(ikox) (155)

where the index i = 1 corresponds to the x < 0 and i = 2 to x > 0. Using equation (53) we have:

S1(P+ – P-) – S2 (P+ – P-) ~ (C^S^ (P+ + P – ^ (156)

At low frequency the volume flux across the pipe discontinuity is constant and the flow is locally incompressible.

Assuming, f = 0 and integrating the momentum equation (45) along the x-axis, we get the linearized Bernoulli equation:

d Ґ2 , ,

Podt J v’xdx = p (xi) – p (x2) . (157)

In terms of plane waves we have:

p+ + p – – (p+ + p-) – (P+ -P-^ (158)

which implies that, at low frequencies cV/(c0S1) << 1, the pressure is continuous across the pipe discontinuity.

Thus, we have:

S1(p+ – p-)= S2(p+ – p-) (159)

and

p+ + p – = p+ + p – . (160)

In the form of a scattering matrix we get:

For an anechoic (non-reflecting) pipe termination of segment 2 (p— = 0) we have a reflection coefficient R = p—/p+ = (Si — S2)/(S1 + S2) and a transmission coefficient T = p+ /p+ = 2Si/(Si + S2). For the limit S2 ^ Si we find the ideal open-end behaviour R = —1 and T = 0. The acoustic flow from pipe segment 1 cannot change the pressure in the much wider pipe segment 2, so that p+ + p– = 0. The outflow corresponding to a positive incoming pressure wave produces an under-pressure at the pipe outlet which propagates into the pipe as a reflected wave with negative amplitude wave. In the opposite limit S2 ^ S1, R =1 because the acoustic flux meets a closed pipe termination causing into a positive wave travelling back p+ — p– = 0, the result being that the pressure at the end of section 1 is twice as high. Hence, the transmission coefficient is T = 2. One can check by using the energy equation that this transmitted wave does not carry any energy in the limit S2/S;|_ ^ 0.

In the previous sections we used the analogy of Lighthill (1952-54) to obtain a scaling law for sound production by subsonic isothermal free jets. One of the choices in this derivation is to express the analogy in terms of fluctuations of density p’ (equation 40). As an alternative, we could have also used the fluctuations of pressure p’ (equation 39). In principle both formulations are equivalent as long as no approximations are involved. However, an analogy is only meaningful if we do use approximations. De­pending on the choice of the aero-acoustic variable some approximations will appear naturally. For example using the pressure formulation, the en­tropy noise source term has the form d2(p’/c2 — p’)/dt2. This is a monopole sound source, to be understood as the time dependent volume expansion due to unsteady combustion. A more detailed analysis of thermal effects is provided by Morfey et al. (1978) and Dowling [in, Crighton et al. (1992)]). Using the density formulation, the entropy sound source term is a spatial derivative d2 (p’ — c2p’)/dx2. We will now explain the physical meaning of this apparently obscure sound source term. For this we consider the sound

produced by a turbulent free jet in a bubbly liquid, as observed by a lis­tener immerged in the pure liquid. In such a case the speed of sound c in the source region is much lower than the speed of sound c0 of the fluid surrounding the listener.

Considering the low frequency limit of the behaviour of a mixture of gas bubbles and a liquid (Figure 8). We find that low frequency implies that gas density pg and fluid density pi are both uniform so that the mixture density p is given by [Crighton et al. (1992)]:

P = вРд + (1 – в)Р1 (133)

where в is the volume fraction of gas in the mixture. Assuming a quasi­steady behaviour, the pressure is uniform. Thus, we can add the compress­ibility of the two phases to obtain the compressibility of the mixture:

JL = _A_ і (1 – в)

pc2 Pg c2 Plcl

where cg is the speed of sound in gas and ci is the speed of sound in liquid.

Eliminating the density by multiplying (133) by (134) yields:

For air/water mixtures at neither too small or too large a value of в we can neglect both the contribution of air to the mass density and the contribution of water to the compressibility. We then get:

For air with (pg = 1.2kg/m, cg = 340m/s) and water with (pi = 1000kg/m, ci = 1500m/s) we get a minimal speed of sound cmin « 20m/s at в = 0.5 . The entropy term in the analogy of Lighthill for an isentropic flow can be written as follows:

d 2 d 2 / c2

■Щ p — c0P’> = щp (l – ^ . (137)

The pressure fluctuations in the source region are of the same order as the fluctuations in the Reynolds stress tensor: p ~ pU2. Hence, compared to a free jet of water surrounded by water, the bubbly liquid turbulence sound is enhanced by a factor |(1 — c2/c2)| = 5 x 103, which is 74 dB. Infact, taking a shower in a bath tub, we observe that the water jet impinging on the water surface is much noisier than the jet immerged in the water, as we can understand qualitatively in terms of the analogy of Lighthill. According to Morfey (1973) and Powell (1990) this entropy term can be understood as the sound produced by the unsteady force exerted on the mixture as a result of the “buoyancy” force due to the difference in density between the two phases undergoing a pressure gradient. This corresponds to a slip between the two phases. Obviously, as there are no net external forces, this sound source must be a quadrupole, the force of the gas on the liquid being balanced by the reaction force of the liquid on the gas.

Similar effects, though much weaker can be found in non-isothermal gas free jets. Contrary to earlier literature predicting a dipole [Morfey (1973), Obermeier (1975)], recent studies indicate that the overall acoustic power level radiated by hot jets is also in line with the height power law of Lighthill [Viswanathan (2009)], which actually confirms that this sound source is also a quadrupole. In the early literature it was also suggested that next to con­vection effects due to density differences, the heat transfer between a hot
gas free jet and its surroundings would generate a monpole sound source. In cases with ideal gasses and a uniform constant Poisson ratio y, this does not occur due to the jet contraction by cooling compensating exactly expansion of the surroundings due to heating [Morfey and Wright (2007)]. Monopole sound sources do occur as a result of combustion or phase transition (mois­ture condensation).

Bubble resonance can induce an even larger amplification of turbulent sound production [Dowling and Ffowcs Williams (1983)]. Yet, it is argued by Crighton (1975) that typical turbulent eddies corresponding to frequencies close to resonance frequencies of bubbles are much smaller than the bubbles and can therefore not excite the bubbles coherently. He therefore uses the low frequency approximation described above.

5.1 Isothermal free jet

Considering the sound production of a turbulent free jet. This is the flow with a velocity U0 at the outlet of a pipe of diameter D. Turbulence is an unsteady chaotic fluid motion which appears when viscous forces are small compared to non-linear convective forces. This corresponds to high Reynolds numbers ReD = UqD/v. We limit ourselves to a low Mach num­ber flow M = Uq/oq ^ 1 of an air jet surrounded by air with the same temperature as its surroundings. The prediction of the scaling rule between the power of this sound source and the Mach number was a major suc­cess of the theory of Lighthill (1952-54). As stressed by Powell (1990), the scaling law was predicted before it was corroborated by experiments. The steps taken by Lighthill were, however, quite intuitive and justification of some of these steps came only long after the original publication [Mor – fey (1973),(1976),(1978), Obermeier (1975)]. We now follow the Lighthill prodecure [1954].

Firstly Lighthill assumes that there are no external forces working on the flow and that the effect of walls can be neglected. In free field conditions equation (99) simplifies to:

This implies that the solution we are seeking for is, at most, a quadrupole field. In fact, we have imposed this by assuming that there are no external forces acting on the fluid and the potential monopole sources were neglected. Please note that in the analogy of Lighthill, p is used as aeroacoustical vari­able. In the next section we will discuss why this choice can be important. Carrying the time integration and using the far field approximation we find:

The sound appears to be produced mainly by large coherent vortex struc­tures with a length scale of the order of the pipe diameter D. For such scales the Reynolds number is large. We therefore expect the Reynolds stress ten­sor pvivj to be much larger than the viscous stress tensor Tij [Morfey (1976)]. Furthermore, at low Mach numbers variations in temperature and density are negligible [Morfey (1973), Morfey et al. (1978)], which implies that we

Figure 7. Overall acoustic sound power level (OAPWL) of the sound radiation from an isothermal free jet as a function of the jet Mach number: comparison of theory with experimental results [Fisher et al. (1973),Viswanathan (2009)].

can use the approximation proposed by Lighthill (1952-54):

Tij & poViVj. (127)

For a circular jet cross section the dominant frequency corresponds to a Strouhal number of unity. Hence the dominating frequency is U0/D and the corresponding acoustic wavelength is D/M = Dc0/U0. The sound source has a volume V of the order of D3. At low Mach numbers the sound source is small compared to the wave length. This implies that we can neglect

Substitution in (126) yields:

In terms of sound source power < P >

<P>

1 p U3 nD2

2 p0U0 4

where we assumed an isotropic radiation pattern. This famous global scaling rule of Lighthill (1952-54) appears to be valid up to Mach numbers of order unity. At these high Mach numbers the radiation pattern has a high forward directivity due to the Doppler effect and, due to refraction of sound by the shear layers, it displays a cone of silence around the axis. The fact that the theory remains valid up to relatively high Mach numbers can be partially explained by the fact that the convection velocity Uc of the vortices in the jet is only a fraction of the main flow velocity [Crighton et al. (1992)]. Typ­ically we have Uc/U0 « 0.3. Recent discussions on jet noise are Morris and Farassat (2002) and Viswanathan (2009) as well as the discussion in Part 2.

Obviously, by increasing the Mach number, the scaling law of Lighthill fails simply because the radiated power would become larger than the avail­able jet power 1 pU0 ^4Р-. Also the sound production mechanism changes drastically. The sound radiation from supersonic jets above M = 3 is largely due to hydrodynamic shear waves which display highly directional radiation patterns. Entropy effects due to temperature differences in the flow also be­come very important. In a supersonic flow the temperature typically varies from the stagnation temperature Tt to the isentropic expansion temperature T = Tt/(1 + (y — 1)M2/2). Starting from room temperature Tt « 300^ in the reservoir, M = 3 implies a main flow temperature T « 100^. Obvi­ously, such a flow is not isothermal and we can use many different definitions of the temperature or Mach numbers to characterize the flow [Viswanathan (2009)].

Finally, most supersonic jet are either over – or underexpanded, and there­fore display standing shock structures, which interact with vortices (turbu­lence) that give strong sound radiation. In some cases, this leads to spec­tacular self-sustained oscillation (jet screetch).

Note that approximation (128) is based on the fact that in a circular jet the characteristic Strouhal number for the sound production is of or­der unity SrD = Df/U = 0(1). In a planar jet of thickness H we find

SrH = Hf/U = 0(10 1), which again stresses that the assumptions are not trivial [Bjprnp et al. (1984)].

Turbulence noise is essential because, when all other sound sources have been suppressed, this will always remain as the minimum remaining noise production which we can achieve. Lighthill’s scaling law indicates that the most efficient way to reduce this noise is to reduce the flow velocity. The result derived for free-field conditions remains valid for confined flow. In the absence of resonances, one finds at low frequencies in a pipe p’ ~ poU0/co and < P >~ M6.

It is important to stress again that the analogy of Lighthill does not impose the quadrupole character of the source. Because we neglected the monopoles (no heat sources and negligible variation in density) and the dipoles (no external force acting on the “free” jet), the source has at most, a quadrupole character. Based on the integral formulation (126) the proce­dure imposes this assumed quadrupole character on the solution. So even if the applied model predicting the stress tensor Tij does involve density fluctuations and external forces, the formulation ensures that these contri­butions are ignored. This explains the success of such analogies [Schram and Hirschberg (2003)]. They filter out spurious sound sources due to errors in the estimation of the stress tensor Tij.