Low Mach Number Flow Noise:

Flow as the Quadrupole Source

 

 

Hongbin Ju

Department of Mathematics

Florida State University, Tallahassee, FL.32306

www.aeroacoustics.info

Please send comments to: hju@math.fsu.edu

 

 

In a uniform flow, linear waves of vorticity, sound and heat are independent. They interact with each other only at boundaries. When the flow is not uniform, or disturbances in the flow are not small, different waves will interact by scattering and/or nonlinearity: one type of wave can grow, be generated or dampened by other types of waves.

 

It is difficult to analyze wave interactions in a nonlinear nonuniform flow. In this chapter we will discuss a simple case of nonlinear interaction: the sound generation by a compact vortical flow with low Mach number M, shown by Fig.1. The flow moves at typical speed  which is much smaller than sound speed a. During one cycle of the typical oscillation with period T, the ratio of distance l traveled by the flow to sound wavelength  is . Because Mach number M is small, the two lengths are disparate: . On flow length scale l, the flow in its low order M expansion is vortical and incompressible. On sound wavelength scale , the flow is irrotational in its low order M expansion. That means the short and very long lengths are only weakly coupled in the flow and sound regions. The physical explanation of this weak coupling is, during one cycle of the flow parameter, the sound wave oscillates many cycles so its net effect is almost cancelled; on the other hand, the slowly varying flow is almost constant over one cycle of the sound oscillation. (Crighton, et.al.1992, p.209) Therefore, one may only need to solve an incompressible flow in the flow region and solve the sound wave equation in the sound wave region. The two solutions are then coupled in an overlap region. This coupling is fulfilled only mathematically. There is no physical solution in the overlap region. In the final solution the incompressible flow acts as a quadrupole source to the far sound field, which is the same as in the Lighthillıs Analogy Theory.

 

The mathematic tool for this analysis is the perturbation method. We first assume the flow is incompressible (). An incompressible flow is localized. Then a small compressibility is admitted. Admission of compressibility has two effects. First, the flow energy can propagate away in the form of compressible waves (sound) from the local flow region. Second, there is time lag between the flow and its generated sound. Mach number M is the expansion parameter. It is the ratio of two lengths. The disparity of the two length scales implies that the low Mach number flow sound is a multiple scale problem, and the perturbation will be singular instead of regular (Dyke1975, p.80: a perturbation solution is uniformly valid in the space and time coordinates unless the perturbation quantity is the ratio of two lengths or two times).

 

 

 

Fig.1, Sound generation by low Mach number compact flow.

 

 

There are two different singular perturbation methods. Landau and Lifshitz first used the Matched Asymptotic Expansion (MAE) to compute sound from a small pulsating and oscillating body. Crow1970, Obermeier1985 extended this MAE method to low Mach number flow sound problem. Different length scales are used in the near field flow region and the far field sound region. Regular expansions are developed based on their local length scales in different regions. The two expansions are then matched in the overlap region to form a composite solution.

 

The other method is Multiple Scale Method. One may choose one time scale and two length scales (l and ) in the analysis as in Fortenbach&Munz2003. Since for the same distance, the time spent by the sound wave is much shorter than that by the flow, one may also use one length scale and two time scales in the Multiple Scale method as in Müller1999.

 

MAE method similar to Crow1970 will be employed here. The following are the equations for inviscid, non-heat-conducting flow used in this chapter:

 

,                                                     (1)

,                                             (2)

.                                                        (3)

 

 

Inner Expansion

 

Success of MAE depends on choosing right scales in the region analyzed. Different choice of scales gives different results. In the flow region, the flow is fully characterized by typical eddy speed  (velocity scale), typical eddy size l (length scale), and time scale . From continuity equation (1), density scale has no significance. Generally we use density in the far field  as the density scale. From momentum equation (2), the relative scale of pressure to that of density is critical for the analysis. However, the appropriate pressure scale can't be found from the momentum equation. It can be found from isentropic equation (3). It is straightforward to choose pressure in the far field  as the scale. Just for convenience, we usually choose   (sound speed ) as the pressure scale. Then the dimensionless equations are:

 

,                                                    (4)

, ,                          (5)

.                                                        (6)

 

All variables in (4)~(6) and hereafter are dimensionless quantities unless specified.

 

Let's try the regular expansion series:

 

,                                             (7)

,                                             (8)

.                                            (9)

 

Substituting them into (4)~(6) and equating the terms of like powers of M, we have:

 

 equations:

 

, , .                           (10)

 

The general solution of  is , which should be determined by matching it to the outer expansion in the overlap region. From physics intuition, it should be the atmospheric pressure in the far field . Therefore,

 

 , , and .                                    (11)

 

 equations:

 

, , .                        (12)

 

The same argument as in  equations indicates:

 

, , .                                         (13)

 

(11) and (13) have similar properties. There is no advantage to separate these two orders of equations (Crow1970, Müller1999, Eldredge2002). Therefore,  and  order variables can be combined. The proper expansion series should be:

 

,                                                 (14)

,                                                 (15)

.                                                (16)

 

Now , , and  are variables of  and  orders, instead of variables only of  order as in (7)~(8).

 

Substituting (14)~(16) into (4)~(6), we have:

 

+ equations:

 

, , .                            (17)

 

The same argument as for (11) brings to:

 

, , .                                         (18)

 

In this order the flow is solenoidal (dilation free).

 

 equations:

 

,                                       (19)

,                                        (20)

.                                               (21)

 

From (18) we know in the limit of low compressibility, pressure converges to a constant thermodynamic background pressure. When small compressibility is admitted, from equations (18) and (20), we can form the following complete system of equations:

 

,                                                       (22)

.                                          (23)

 

These are the well known incompressible flow equations. They are actually the low Mach number flow equations. Velocity field is solenoidal. We will use  instead of  in the equations. Taking divergence on both sides of (23), we have the Poisson equation about pressure:

 

.                                              (24)

 

Its solution in three dimensions if  is known is:

 

,                                   (25)

 

where ,  with  fixed. Any harmonic solution J (solution of ) can be added to  to form another solution. But J is analytical and will grow algebraically in the far field (Crighton, et.al.1992). Thus J should not be included in the solution.

 

From (18) and (25), to second order , the formal solution of pressure in the flow region is:

 

.                     (26)

 

Solutions (25) and (26) are only formal since  has to be solved together with  from incompressible equations (22) and (23).  is called hydrodynamic pressure. Hydrodynamic pressure appears in incompressible equation (23) to establish the divergence-free velocity . Associated with  is hydrodynamic density  [Eq.(21)] and second order velocity  [Eq.(19)].  is not necessarily dilation free. That means  may compress the fluid element, although in this low order [] there is no sound, Eq.(24).

 

An acoustic/viscous splitting numerical scheme can be established based on this asymptotic analysis. One may use the next asymptotic series:

 

,                                                 (27)

,                                                 (28)

.                                                (29)

 

, ,  and  have small length scale l. They are solved from incompressible equations (22)&(23) and equations (21)&(19) on a fine grid system. Substitute expansions (27)~(29) into the N.S. equations and subtract the incompressible equations from them to obtain a set of equations for , , and . These equations are acoustic equations. Since only the long length scale, sound wavelength, exists in the acoustic equations, they can be solved on a coarse grid system. By splitting the incompressible flow and the sound waves, the singularity due to length scale disparity is removed. A similar acoustic/viscous splitting numerical scheme was proposed by Hardin&Pope1994, in which  is not solved in the incompressible flow.

 

 

Outer Expansion

 

In the sound region, the appropriate length scale is sound wavelength . Use  as the new length scale to define the dimensionless coordinate:

 

.                                             (30)

 

Eq.(30) is a stretching transformation of the coordinate. The sound is generated by the fluid flow, therefore the inner and outer regions have the same time scale: .  is the density scale and  the pressure scale. We may still use  as the velocity scale. However we may have a better choice. From the  momentum equation of the inner expansion,  (on the inner expansion scales), then the pressure fluctuation in dimensional form is . As we know for sound waves,  ( is the acoustic velocity). Which means the acoustic velocity is in the order of , which seems to be a better choice for velocity scale in the outer region: . Then the dimensionless equations in the outer region are:

 

,                                              (31)

,                                  (32)

.                                                      (33)

 

The next regular expansion series are used:

 

,                                      (34)

,                                        (35)

.                                    (36)

 

We have:

 

+ equations:

 

, , .                                     (37)

 

The  solutions are:

 

, .                                                 (38)

 

 equations:

 

, , .                      (39)

 

From the momentum equation in (39), one has:

 

.

 

In the far field there is no vorticity initially, therefore the velocity field is vorticity free in this order:

 

.                                                       (40)

 

The wave equation is obtained from equations in (39):

 

.                                                  (41)

 

 equations:

 

, , .                                         (42)

 

The solutions are:

 

.                                                 (43)

 

 equations:

 

,                                   (44)

,                        (45)

.                                      (46)

 

In this order vorticity  is not necessarily zero.

 

 equations:

 

,                                              (47)

,                                                (48)

.                                   (49)

 

From (48), .

 

The wave equation is:

 

.                                                  (50)

 

Here we will only discuss the solution to  wave equation (50) because it is the term which will match the inner expansion. The general solution to this wave equation is composed of monopoles, dipoles, quadrupoles, and even higher order components. But only the quadrupole is needed for matching to the inner solution:

 

, where .                             (51)

 

Therefore, pressure in the sound field is:

 

.                        (52)

 

 

Matching

 

Both the inner and outer expansions are the approximations to the same function, but just in different regions. They have to match with each other in the overlap region. There are two matching methods: Intermediate Matching Principle and Asymptotic Matching Principle (Crighton, et.al. 1992, Van Dyke1975, Holmes1995). We will use Intermediate Matching Principle here. In this method, the two expansions match in an overlap region. As , the overlap region is the far field for the inner region, but a near field for the outer expansion. To describe this region mathematically, a new coordinate system is introduced:.  is chosen so that as  when keeping  fixed,

 

, .                                         (53)

 

One may choose , . In the new coordinate system,

 

,                        (54)

 

.

 

Inner solution (26) can be written in the new coordinate system as:

 

.           (55)

 

(Note that the scale of  is still l.)

 

Outer solution (52) is rewritten as:

 

.                    (56)

 

Matching the terms with  in (55) and (56), we have:

 

.                                         (57)

 

Final Solutions

 

Now we summarize the final solutions with scales: length l, velocity , time , density , and pressure .

 

Inner solution:

 

,                                           (58)

,                                              (59)

;                                            (60)

,                                                          (61)

,                                         (62)

,                                                         (63)

.                                       (64)

 

 

Outer solution:

,                                             (65)

,                                           (66)

;                                              (67)

,                          (68)

,                                                      (69)

.                                                   (70)

 

 

Acoustic Analogy of the Low Speed Flow Sound

 

From sound wave equation (50) and solution (68), the equivalent inhomogeneous wave equation in the sound field is:

 

, or                                   (71)

 

.                                   (72)

 

In dimensional form, inhomogeneous wave equation (72) is (the same variable names are used):

 

.                                      (73)

 

The nonlinear incompressible flow in the near field acts as a quadrupole source driving the far field acoustic field. Strength of the quadrupole source is , which is the Lighthill stress tensor in Lighthillıs Analogy Theory. From Eq.(24),

 

.                                          (74)

 

 is the hydrodynamic pressure in the near field.  represents the relative value of  at one point compared to the average pressure around this point.  means  is the local minimum, otherwise it is the local maximum. Therefore, the sound source is the pressure at one point relative to the averaged pressure of its neighbor caused by the turbulent eddies.  is called jetlets by Ribner.

 

The same result can also be obtained by weakly nonlinear analysis.