## Abstract

The acoustic-vortical wave equation is derived describing the propagation of sound in (i) a unidirectional shear flow with a linear velocity profile upon which is superimposed (ii) a uniform cross flow; together with an impedance wall boundary condition representing the effect of a locally reacting acoustic liner in the presence of bias and shear flow. This leads to a third-order differential equation in the presence of cross flow, and in its absence simplifies to the Pridmore–Brown equation (second-order); also the singularity of the Pridmore–Brown equation for zero Doppler-shifted frequency is removed by the cross flow. Because the third-order wave equation has no singularities (except at the sonic condition), its general solution is a linear combination of three linearly independent MacLaurin series in powers of the distance from the wall. The acoustic field in the boundary layer is matched through the pressure and horizontal and vertical velocity components to the acoustic field in a uniform free stream consisting of incident and reflected waves. The scattering coefficients are plotted for several values of the five parameters of the problem, namely the angle of incidence, free stream and cross-flow Mach numbers, specific wall impedance and Helmholtz number using the boundary layer thickness.

## 1. Introduction

The air inlets and exhaust nozzles of jet engines make extensive use of liners to absorb or attenuate sound. A locally reacting acoustic liner can be represented by an impedance wall condition. In the case of a duct with mean flow, the boundary layer near the wall leads to interaction between acoustic and vortical modes that also affects the pressure perturbation and energy [1]. In addition, a perforated plate can have a bias flow, thus superimposing a cross flow to the shear flow in the boundary layer. This paper addresses the combination of the three effects, namely (i) a plane flow over a flat impedance wall (figure 1); (ii) a boundary layer with a unidirectional shear flow consisting of a linear velocity profile matched to a uniform stream; and (iii) in addition, a uniform cross flow representing the bias flow out of the perforated liner. The pressure perturbation in the free stream consists of incident and reflected plane waves; it must be matched to the pressure field in the boundary layer (§2) in order to apply the impedance boundary condition at the wall (§3); the latter specifies the reflection coefficient, and thus the wave pressure perturbation in the whole flow, inside and outside the boundary layer.

The wave equation for sound in a unidirectional shear flow with a superimposed cross flow does not appear to have been derived in the literature. In the absence of cross-flow, the acoustic wave equation in a unidirectional shear flow [2,3] has been known for a long time [4,5]. The simplest solution concerns a linear profile equivalently described by four methods: (i) parabolic cylinder functions [6]; (ii) Whittaker functions [7]; (iii) confluent hypergeometric functions [8–10]; and (iv) Frobenius–Fuch series around the critical layer where the Doppler-shifted frequency vanishes [11]. The critical layer also leads to the appearance of a continuous spectrum [12]. The acoustic wave equation in a unidirectional shear flow has been solved for three other velocity profiles: (i) an exponential boundary layer [13]; (ii) an hyperbolic tangent shear layer [14]; and (iii) a parabolic profile in a duct [15]. All the preceding solutions concern sound in an homentropic unidirectional shear flow; the isentropic non-homentropic case allowing for the presence of mean flow temperature gradients has been considered without [16] and with [17] sound sources.

The acoustic-vortical wave equation is at least of third order because (i) the sound is described by a second-order wave equation allowing for propagation in opposite directions; (ii) vorticity transport is specified by a first-order conservation equation along the streamlines (Kelvin’s circulation theorem) in a potential homentropic flow; (iii) excluding entropy modes, the coupling of (i) sound and (ii) vorticity is specified by a third-order wave equation. In the case of sound in a unidirectional shear flow [18,19], the wave equation is a cubic on frequency and horizontal wavenumber and leads to a second-order differential equation for the dependence on the cross-stream coordinate. This differential equation has a critical layer when the Doppler-shifted frequency vanishes; this singularity occurs when the horizontal phase velocity equals the velocity of the mean shear flow, so that the wave cannot propagate further, and must become evanescent, or be reflected, or be absorbed or transformed to another mode. The presence of a cross flow (§2*a*) convects the wave away from this condition and thus the acoustic-vortical wave equation in a unidirectional shear flow with linear velocity profile superimposed on a uniform cross flow is of third order (§2*b*) and has no singularity or critical layer (§2*c*).

The pressure perturbation in a linear unidirectional shear flow with uniform cross flow is a linear combination of three solutions, each specified by Taylor series as a function of the distance from the wall (§3*a*). The three constants are specified by matching the pressure and horizontal as well as vertical velocity perturbations at the edge of the boundary layer to the sound field in the free stream (§3*b*), consisting of an incident downward propagating and reflected upward propagating plane wave. This specifies the pressure perturbation in the boundary layer involving the reflection coefficient which is determined from the impedance boundary condition at the wall (§3*c*). The transmission coefficient is determined as the ratio of the pressure perturbation at the wall and at the junction of the boundary layer with the free stream. Thus, the modulus and phase of the reflection and transmission coefficients (§4*c*) are plotted as function of the angle of incidence at the free stream (figures 12–19) for several combinations of the four parameters (§4*b*) of the problem, namely (i) the free-stream Mach number; (ii) the cross-flow Mach number; (iii) the specific wall impedance; (iv) the dimensionless frequency or Helmholtz number comparing the thickness of the boundary layer to the wavelength. The modulus and phase of the pressure perturbation as a function of the distance from the wall (§4*c*) are plotted (figures 7–11) for several combinations of the preceding four parameters (i–iv) and several angles of incidence. The level curves for the modulus of the pressure (figure 5) compare the zones of silence and propagation in the free stream (figure 4) with the corresponding regions in the boundary layer.

## 2. Sound in superimposed shear and cross flows

The acoustic-vortical wave equation (§2*b*) is obtained in a unidirectional shear flow with a linear velocity profile superimposed with a uniform cross flow (§2*a*). The dependence of the pressure perturbation on the coordinate perpendicular to the wall is specified in the presence (absence) of the cross flow by a third (second)-order differential equation without (with) a singularity, that is a critical layer does not (does) occur (§2*c*).

### (a) Linear unidirectional shear with uniform cross flow

The fundamental equations of fluid mechanics are considered as (i) the continuity equation for the mass conservation:
2.1
where is the mass density and is the velocity; (ii) the inviscid momentum equation:
2.2
where is the pressure; and (iii) the adiabatic equation:
2.3
where is the adiabatic sound speed. In all three equations the material derivative appears:
2.4
The tilde notation is used to denote the total flow variables, the subscript ‘0’ the mean flow variables and neither is used for the perturbations. The mean flow is assumed to be plane and consists (figure 1) of: (i) a uniform bias or cross flow orthogonal to a straight wall with velocity *V* _{0}:
2.5
(ii) a unidirectional shear flow parallel to the wall with a linear velocity profile in a boundary layer of thickness *L*:
2.6*a*,*b*
matched to a uniform stream with velocity .

Denoting with a subscript ‘0’ the mean flow quantities, the momentum equation (2.2) becomes
2.7
Using equation (2.5), the *y*- and *x*-components of the momentum equation for the mean flow (2.2) are respectively
2.8*a*,*b*
From (2.8*a*), it follows that the mean flow pressure does not depend on the distance from the wall, but it does depend on the distance along the wall (2.8*b*) for a shear flow. For a shear flow with a linear velocity profile (2.9*a*) from (2.8*b*) follows (2.9*b*):
2.9*a*,*b*
showing that the mass density does not depend on the distance from the wall (2.10*a*); using (2.10*b*) in the continuity equation (2.1) for the mean flow (2.9*a*) shows that
2.10*a*,*b*
where the mass density is constant (2.11*a*); because *V* _{0} and *dU*_{0}/*dy* are constants (2.5), (2.9*a*) the two components (2.8*a,b*) of the momentum equation show that the mean flow pressure is a linear function of the distance along the wall (2.11*b*):
2.11*a*,*b*
Considering a limited length of the wall satisfying (2.12*a*), the mean flow pressure is approximately constant (2.12*b*):
2.12*a*,*b*
Under the same conditions, the mean flow is adiabatic (2.13*a*), and the sound speed is constant (2.13*b*):
2.13*a*,*b*
These conditions simplify the following (§2*b*) derivation of the acoustic-vortical wave equation in a linear unidirectional shear flow (2.6*a,b*) superimposed on a uniform cross flow (2.5) for a limited wall distance (2.12*a*).

### (b) Third-order acoustic-vortical wave equation

An unsteady, non-uniform perturbation of the mean flow (2.5) is considered
2.14*a*–*d*
and the momentum equation (2.2) is linearized
2.15
using the linearized material derivative:
2.16

The adiabatic (2.3) and continuity (2.1) equations are combined
2.17
and are linearized
2.18
The linearized material derivative (2.13) commutes (2.19*a*) with the *x*-component gradient, but not with the *y*-component gradient (2.19*b*):
2.19*a*,*b*
For example, applying *d*/*dt* to equation (2.15) gives
2.20
and to derive (2.20) properly the commutation relations (2.19*a,b*) were used.

The *x*- and *y*-components of the momentum equation (2.12) are
2.21*a*,*b*
substitution in (2.17) gives
2.22
The l.h.s of (2.19) is the convected wave equation for pressure which is valid when the r.h.s is zero, i.e. in the absence of shear flow. In the presence of shear flow in order to eliminate (*v*,*ρ*) from the r.h.s of (2.19) and obtain a wave equation for the pressure alone, *d*/*dt* is applied once more leading to
2.23
In the case (2.9*a*) of a linear shear flow (2.24*a*) two terms on the r.h.s of (2.20) vanish, and the linearized adiabatic equation (2.24*b*):
2.24*a*,*b*
together with (2.21*b*) substituted in (2.20) give
2.25
The first three terms on the l.h.s of (2.22) specify the acoustic-vortical wave equation in a unidirectional shear flow [2–5,18,19] in the absence of cross flow. The generalization to include a uniform cross flow adds the fourth term on the l.h.s of (2.22) and restricts the shear flow to the linear velocity profile (2.24*a*). It is shown next (§2*c*) that the extra cross-flow term eliminates the critical layer that would otherwise exist.

### (c) The existence or absence of a critical layer

Because the mean flow is steady and uniform in the wall direction, a Fourier integral representation exists
2.26
where *P* is the pressure perturbation spectrum for a wave of frequency *ω* and horizontal wavenumber *k* at the distance *y* from the wall. The linearized material derivative (2.13) leads to (2.27*a*)
2.27*a*,*b*
where *ω*_{*} is the Doppler-shifted frequency calculated for the shear flow alone. Substitution of (2.23) and (2.27*b*) in the acoustic-vortical wave equation (2.22) leads to the dependence of the pressure perturbation spectrum on the distance from the wall:
2.28
Because the acoustic-vortical wave equation (2.22) is of the third order, (2.25) is (i) a cubic dispersion relation in the frequency *ω* and horizontal wavenumber *k*, and retaining (ii) the dependence on the distance from the wall as a third-order differential equation.

In the absence of the cross flow (2.29*a*), the differential equation (2.25) reduces to the second order
2.29*a*,*b*
Equation (2.29*b*) has a singularity when the coefficient of *P*′′, that is the Doppler-shifted frequency (2.27*b*) vanishes. To interpret this physically, consider (figure 2*a*) a wave of frequency *ω* and horizontal wavenumber *k*, hence horizontal phase speed * w*=

*ω*/

*k*propagating against a unidirectional shear flow. At the critical layer when the horizontal phase speed equals the mean flow velocity (2.30

*a*), the Doppler-shifted frequency vanishes (2.30

*b*) 2.30

*a*,

*b*

The wave can propagate no further, and the wave equation (2.29*b*) has a singularity, implying one of the following possibilities: (i) the wave becomes evanescent beyond the critical layer that acts as a total reflector; and (ii) the wave is partly absorbed, partly reflected and partly transmitted as another mode able to propagate beyond the critical layer. In all cases, the critical layer occurs at the point where the wave is ‘stopped’ by the mean flow. The sonic condition in a potential flow is also a singularity of the acoustic wave equation. The difference is that sound waves in a potential flow are non-dispersive and the critical layer occurs at the sonic condition that is the same for all wavenumbers. In the shear flow, the critical layer occurs at different positions for fixed frequency *ω* and varying wavenumber *k* (or vice versa) so that the forbidden values form a continuous spectrum [12]. Applying a cross flow (figure 2*b*), this convects the wave away from the critical layer and removes the singularity. Thus, the acoustic-vortical wave equation with cross flow (2.25) has no singularity, that is the coefficient of the third-order derivative does not vanish, except if the cross flow reaches the sonic condition. The acoustic-vortical wave equation in a unidirectional shear flow is simpler in the presence of cross flow, in the sense that it has no singularity, so it has solution as Taylor series valid in the whole flow region or the radius of convergence of the Taylor series is limited only by another singularity *y*_{*} owing to another feature of the mean flow (figure 3*a*). In the absence of cross flow (figure 3*b*), the singularity at the critical layer would: either require a Frobenious–Fuch series around the critical layer with a radius of convergence limited only by another singularity (ii) or restrict the Taylor series solution around another point to a radius of convergence limited by the critical layer. The presence of the cross flow increases the order of the differential equation (2.25) to three, and can lead to stiff solutions for small cross-flow velocity *V* _{0} that appears as a factor of the highest order derivative *P*′′′.

## 3. Solution of the third-order wave equation

The pressure perturbation spectrum in the boundary layer is a solution of the third-order differential equation (2.25) without singularities, that is a linear combination of three particular integrals specified by MacLaurin series of the distance from the wall (§3*a*); the three constants of integration are determined by three conditions matching to the acoustic field in the free stream (§3*b*). This extends the pressure field up to the wall, so that the reflection coefficient is specified by the impedance boundary condition (§3*c*).

### (a) Pressure perturbation inside the boundary layer

The acoustic-vortical wave equation (2.25) is applied inside the boundary layer (2.6*a*), and the distance from the wall is normalized to its thickness (3.1*a*) leading for the pressure perturbation spectrum (3.1*b*):
3.1*a*,*b*
to the third-order differential equation:
3.2
that involves three independent dimensionless parameters, namely (i) the cross-flow Mach number (3.3*a*); (ii) the free-stream Mach number (3.3*b*); (iii) the dimensionless frequency or Helmholtz number (3.3*c*) comparing the thickness of the boundary layer *L* to the wavelength *λ*_{0} in an homogeneous medium at rest:
3.3*a*–*c*
Thus, *Ω*≫1 for sound rays in the boundary layer, *Ω*≪1 for acoustically thin boundary layer and *Ω*∼1 in the more interesting case of wavelength compared with the thickness of the boundary layer. In (3.2) appear another two dimensionless coefficients, namely (i) the horizontal compactness defined as the product of the horizontal wavenumber (3.4*a*) by the thickness of the boundary layer (3.4*b*) that depends on the angle of incidence.
3.4*a*,*b*
(ii) the dimensionless (3.4*c*) Doppler-shifted frequency (2.27*b*):
3.4*c*
The angle *θ* is measured from the horizontal in the direction of the core flow. The word ‘horizontal’ means parallel to the wall or along the core flow (2.6*a*,*b*); in the presence of cross flow (2.5), the total velocity is oblique to the wall and streamwise is not ‘horizontal’. Similarly, the vertical direction is perpendicular to the wall not perpendicular to the stream. Substituting (3.4*c*) in (3.2), it follows that the pressure perturbation spectrum in the boundary layer satisfies a third-order differential equation with polynomial coefficients:
3.5
The coefficient of the highest order derivative is a non-zero constant except for a bias flow at sonic speed; excluding this case, the differential equation (3.3) has no singularities, and the solution exists as a MacLaurin series (3.6*b*) with infinite radius of convergence (3.6*a*):
3.6*a*,*b*
Substitution of (3.6*b*) in (3.3) leads to the recurrence formula for the coefficients:
3.7
The MacLaurin series (3.6*b*) with coefficients satisfying (3.4) specifies the pressure perturbation spectrum in the boundary layer to be matched to the acoustic field in the free stream.

### (b) Matching to the acoustic field in the free stream

The pressure perturbation spectrum specified by (3.6*b*), (3.4) corresponds to acoustic-vortical waves in the boundary layer (3.8*a*), and because the differential equation (3.2) is linear of third-order its general integral (3.8*b*) is a linear combination of three linearly independent particular integrals or fundamental solutions:
3.8*a*,*b*
where (*B*_{1}, *B*_{2}, *B*_{3}) are arbitrary constants of integration. First are obtained the three linearly independent particular integrals or fundamental solutions *Q*_{1−3}(*z*), and then the arbitrary constants *B*_{1−3}. Setting (3.9*a*) the recurrence formula (3.4) shows (3.9*b*) that the first three coefficients (*a*_{0}, *a*_{1}, *a*_{2}) are arbitrary:
3.9*a*,*b*
The remaining coefficients *a*_{n} with *n*=3,4,5… are determined uniquely from (*a*_{0}, *a*_{1}, *a*_{2}) for example *a*_{3} is given by:
3.10
By choosing one of (*a*_{0}, *a*_{1}, *a*_{2}) to be unity and the others zero, three particular integrals are obtained, namely
3.11*a*
3.12*b*
and
3.13*c*
where the coefficients (*b*_{n},*c*_{n},*d*_{n}) are determined from the recurrence formula (3.4) starting respectively from (3.11*a*), (3.12*b*), (3.13*c*). Because implies that is finite and non-zero, and are infinitesimals of orders respectively one and two [20,21], the three particular integrals are linearly independent, and form a set of three fundamental solutions, so that their linear combination (3.8*b*) specifies the general integral [22,23].

The three arbitrary constants of integration are obtained by matching the acoustic-vortical waves in the boundary layer to acoustic waves in the free stream given by
3.14
with (i) the same frequency *ω* and horizontal wavenumber *k* as (2.23) for the pressure perturbation of acoustic-vortical waves in the boundary layer; (ii) the dependence on the distance from the wall (3.15*a*) consisting of a downward propagating wave with unit amplitude and a upward propagating wave whose amplitude is the reflection coefficient *R*:
3.15*a*,*b*
(iii) the vertical wavenumber is
3.16
where *θ* is the angle of incidence which also appears in the horizontal wavenumber (3.4*a*); and (iv) this confirms that the vertical compactness:
3.17
involves the dimensionless frequency (3.3*c*) and the angle of incidence. The vertical wavenumber (3.7) exists in the free stream where waves are sinusoidal (3.15*b*), but not in the boundary layer where the dependence of the pressure on the distance from the wall is non-sinusoidal, because the differential equation (3.2) has variable coefficients. The acoustic wave field at the free stream (3.15*a*,*b*) can be rewritten (3.18*a*,*b*) using (3.1*a*,*b*) and (3.8):
3.18*a*,*b*
is matched to the acoustical–vortical wave field (3.8*a*,*b*) in the boundary layer by the continuity of the pressure perturbation and its first two derivatives:
3.19*a*-*c*
These conditions are proved next (§3*c*), as a preliminary to determine the reflection and transmission coefficients from the wall impedance.

### (c) Wall impedance and scattering coefficients

The matching conditions at the edge of the boundary layer *y*=*L* or *z*=1 between the acoustic field outside (3.18*a*,*b*) and the acoustic-vortical wave field inside (3.8*a*,*b*) are the continuity of the pressure (3.20*a*):
3.20*a*,*b*
and of the horizontal (3.20*b*) and vertical (3.20*c*) velocity perturbations:
3.21
with dependence on the distance from the wall specified respectively by *U*(*z*) and *V* (*z*); in (3.11) was also included (2.14*d*) the mass density perturbation *ρ* and its dependence *Γ*(*z*) on the dimensionless distance from the wall (3.1*a*). The condition of the continuity of pressure is the same in (3.19*a*) and (3.20*a*), thus it remains to prove that the continuity of the horizontal and vertical velocities (3.20*b*,*c*) is equivalent to the continuity of the first two derivatives of the pressure (3.19*b*,*c*). The proof uses the momentum (2.21*a*,*b*) and the adiabatic (2.24*b*) linearized equations:
3.22*a–c*
where prime denotes derivative regard to *y*. The proof is made first in the simpler case where is no cross flow (3.23*a*), and (3.22*a*,*b*) reduce to (3.23*b*,*c*):
3.23*a*–*c*
It follows from (3.23*b*) that the continuity of (*U*,*V*,*P*) is consistent; then, (3.23*c*) shows that the continuity of *V* implies that of *P*′. Thus, in the absence of cross flow, there are two matching conditions (3.19*a*,*b*) stating the continuity of *P* and *P*′; this agrees with the wave equation being of the second order (2.29*b*) in the absence of the cross flow (2.29*a*), and thus having two linearly independent solutions. By contrast, in the presence of cross flow, the wave equation (2.25) is of the third order, hence has (3.8*b*) three linearly independent solutions (3.11*b*), (3.12*b*), (3.13*b*) and three matching conditions (3.19*a*–*c*) are needed at the edge of the boundary layer. The conditions (3.19*b*,*c*) are proved in the presence of cross flow as follows: (i) the differential equation (3.22*b*) of the first order can be solved expressing *V* as function of *P*′; (ii) hence the continuity of *V* implies the continuity of *P*′, proving (3.19*b*); (iii) the differential equation of the first order (3.22*c*) can be solved for *Γ* as a function of *P* and *P*′; (iv) because (*P*,*P*′) are continuous *Γ* is continuous too; (v) also *Γ*′ depends on (*P*,*P*′,*P*′′); (vi) it follows that the r.h.s. of (3.22*a*) depends on (*P*,*P*′,*P*′′); (vii) solving the first-order differential equation (3.22*a*) specifies *U* as a function of (*P*,*P*′,*P*′′); and (viii) because (*U*,*P*,*P*′) are continuous so is *P*′′, proving (3.19*c*).

Therefore, the continuity of *V* , *U* and *P* implies the continuity of *P*, *P*′, *P*′′ at the edge between the boundary layer and the free stream. Combining (3.15*b*) and (3.19*a*–*c*) the matching conditions are reduced to
3.24*a–c*
The impedance boundary condition at the wall in terms of the admittance (3.25*a*) can be substituted in (3.22*b*) the *y*-component of the momentum equation (3.25*b*):
3.25*a*,*b*
leading to
3.26
thus the boundary condition (3.14) specifies the reflection coefficient *R* by the substitution of (3.24*a*–*c*) in the form (3.27*b*),
3.27*a*,*b*
involving (i) the specific admittance (3.27*a*) obtained dividing by that of a plane wave; and (ii) the cross flow Mach number (3.3*a*) and dimensionless frequency (3.3*c*). The transmission coefficient is defined by
3.28
as the ratio of the pressure perturbations at the wall and in the free stream at the edge of the boundary layer. The modulus of the transmission coefficient is unity if the modulus of the pressure is the same at the wall and at the edge of the boundary layer. If the modulus of the transmission coefficient is greater (smaller) than unity, then there is an amplification (attenuation) of the sound in the boundary layer. In the absence of the boundary layer and cross flow, the wall boundary condition simplifies to (3.29*a*), that applied to the pressure in the free stream (3.18*b*) leads to (3.29*b*),
3.29*a–c*
and the reflection coefficient is given by (3.29*c*). In the same conditions, the transmission coefficient is given by (3.17):
3.30
In the presence of the boundary layer with shear and cross flow, the simple expressions for the reflection (3.29*c*) and the transmission (3.17) coefficients in a uniform stream no longer hold; they are replaced by (3.19*a*–*c*) and (3.27*b*) that must be solved for the reflection coefficient in (3.18*b*), with the transmission coefficient given by (3.15). This completes the determination of the pressure perturbation in the whole flow outside and inside of the boundary layer, and of the scattering coefficients consistent with an impedance wall boundary condition.

## 4. Impedance liner with bias flow

The preceding model of sound propagation in a boundary layer over an impedance wall with a cross flow is applied next with values of the parameters (§4*b*) typical of an acoustic liner with bias flow to plot (§4*c*): (i) the amplitude and phase of the reflection and transmission coefficients as a function of the angle of incidence; (ii) the pressure perturbation as a function of the distance from the wall for fixed angles of incidence. This is preceded by the calculation of the zones of propagation and zones of silence associated with an oblique flow (§4*a*) corresponding to the uniform free stream parallel to the wall and a uniform bias flow orthogonal to it emerging from perforations in the wall.

### (a) Zones of silence and propagation sectors

The vertical wavenumber (3.7) is real in the propagation zone(s) at the free stream, and imaginary corresponding to the zone(s) of silence where the sound waves are evanescent. These zones(s) of silence and propagation alternate and correspond to angular sectors depending on *M*_{0} and . The boundaries between the zone(s) of silence and propagation zone(s) are the roots of the equation *φ*=0 with *φ* defined in (3.8), that is a periodic transcendental equation, that would have to be solved iteratively. Instead, it is possible to obtain an algebraic equation for alone, whose roots can be determined explicitly, by using the following procedure: (i) setting *φ*=0 in (3.8) and taking the square root of the radical leads to:
4.1*a*,*b*
(ii) separating and squaring leads to:
4.2*a*,*b*
that is a polynomial of the second degree in :
4.3*a*,*b*
(iii) the roots of the quadratic equation (4.3*a*,*b*) are
4.4*a–d*
where the second ± applies to *M*_{0} and the first ± to the remaining terms. The validity of the roots obtained by the preceding method (i) to (ii) can be confirmed by checking that substitution in (3.8) gives *φ*=0; the roots are also confirmed by plotting *φ*^{2} as a function of *θ* for fixed values of *M*_{0} and . Of the four roots in (4.4*a*–*d*) for , two are imaginary *θ*_{−±} if ; moreover, real angles require . In the absence of cross flow (4.5*a*), there are only two roots (4.5*b*,*c*):
4.5*a–c*
real values require , that is always met for *θ*_{+}, and is met for *θ*_{−} only if .

As a first example (figure 4*a*) consider in the absence of cross flow *M*_{0}=0; there is only one real angle separating a zone of silence in the downstream arc 0≤*θ*≤*θ*_{+} from a propagation zone *θ*_{+}<*θ*≤180° in the upstream arc. A second example (figure 4*b*) with still without cross flow *M*_{0}=0, there is only one angle *θ*_{+}=70.52° defining a zone of silence 0<*θ*<*θ*_{+} and an upstream propagation zone 180°>*θ*>*θ*_{+}. The cases with cross flow are considered for all combinations of the free stream (4.1a) and cross flow (4.1b) Mach numbers:
4.6a
and
4.6b
The bias flow of acoustic liners usually has a low Mach number corresponding to the smaller values in (4.1b); the larger values in (4.1b) are intended to simulate a higher speed fluidic jet normal to the wall. For the combination of values (4.6*a*,*b*), the roots *θ*_{++} in (4.4*d*) do not satisfy *φ*=0 in (3.8), and the values of the remaining three roots *θ*_{−−}, *θ*_{+−} and *θ*_{−+} are indicated respectively in the tables 1–3; only the real values of the angles corresponding to are indicated. The third example with cross flow *M*_{0}=0.8 and free stream leads in the tables 1–3 to two angles *θ*_{−−}=77.31°<*θ*_{+−}=86.14°; it can be checked that the dimensionless vertical wavenumber (3.8) is imaginary the for the intermediate value *θ*=80° and real for *θ*=60° and *θ*=90°. Thus, there are (figure 4*c*) downstream 0°<*θ*<77.31°=*θ*_{−−} and upstream 180°>*θ*>86.14°=*θ*_{+−} propagation zones, with a zone of silence *θ*_{−−}<77.31°<*θ*<86.14°=*θ*_{+−} in between. A fourth example with higher free stream and intermediate cross flow *M*_{0}=0.3 Mach numbers leads in the tables 1–3 to the angles *θ*_{−−}=33.39° and *θ*_{+−}=76.33°; the vertical wavenumber is imaginary for *θ*=60° in this range and real for *θ*=30° and 90° outside this range. Thus (figure 4*d*), there are downstream 0°<*θ*<33.39°=*θ*_{−−} and upstream *θ*_{+−}=76.33°<*θ*<180° propagation zones, with a zone of silence in between *θ*_{−−}=33.39°<*θ*<76.33°=*θ*_{+−}.

The distinction between propagation zones and zones of silence applies strictly in a uniform stream where the vertical wavenumber is constant, corresponding: (i) to real values to upward and downward waves; (ii) to imaginary values *κ*=*i*|*κ*|, when the divergent solution is discarded, and only the evanescent wave is retained. In a shear flow, the distinction is less clear, because the waveforms are not sinusoidal in the *y*-direction and a vertical wavenumber strictly does not exist. A local varying vertical wavenumber *κ*(*y*) would exist in the ray limit of the weak shear on a wavelength scale, that is *Ω*^{2}≪1 in (3.3*c*); this would break down for *Ω*∼1 or *Ω*<1. Thus, it may be appropriate to plot wave fields into the zones of silence, because waves can penetrate these regions in the boundary layer. The pressure fields may be compared (i) in the uniform free stream where a zone of silence exists; (ii) in the boundary layer where the non-uniform flow excludes the existence of a zone of silence in a strict sense. All of the four cases in figure 4 are illustrated in figure 5 with the free stream in the l.h.s. and the boundary layer on the r.h.s. The contour plots for the modulus for the amplitude of the acoustic pressure are shown: (i) using (4.7*a*) in the free stream (3.18*b*), so that at the boundary between the zone of silence and the upstream and downstream propagation zones *φ*=0 and the value if *S*_{+} is zero:
4.7*a*,*b*
(ii) the boundary layer is used (3.8*b*) with the same normalization (4.7*b*). In both plots, the horizontal *x*- and vertical *y*-coordinates, respectively, along and perpendicular to the wall are made dimensionless by dividing by the thickness of the boundary layer. In the plots in figure 5, polar coordinates are taken, and the pressure is represented by:
4.7c
The r.h.s plots in figure 5 show the pressure amplitude in a large region of the free stream −20≤*x*/*L*≤20 by 1≤*y*/*L*≤20 above the boundary layer. The boundary layer corresponds to an horizontal strip 0≤*y*/*L*≤1. The pressure amplitude in the shear and cross flow in the boundary layer is depicted inside a smaller box −1≤*x*/*L*≤1 by 0≤*y*/*L*≤1.

The choice of the dimensionless horizontal wavenumber *ε*=*kL*=1 implies a spatial period 2*π* in dimensionless form or 2*πL* in dimensional form. This spatial period would not be observed extending the plots on the r.h.s of figure 5 to the whole horizontal strip 0≤*y*/*L*≤1 and for the following reasons: (i) the acoustic pressure is the product of separate functions of *x* and *y*; (ii) the function of *x* is and hence periodic, that is it takes the same value at the points (*x*_{1},*y*_{0}) and (*x*_{2},*y*_{0}) in figure 6 whose horizontal distance is a multiple of one period 2*πL*; (iii) although the *y*-coordinates are the same, the angles *θ*_{1} and *θ*_{2} are different, and appear in the horizontal wavenumber (3.4*b*), thereby changing the function of *y*. Thus, the polar representation chosen to illustrate the zones(s) of propagation (silence) in figure 4*a*–*d*, when applied to the pressure field that is periodic in *x*, destroys this periodicity, because the function of *y* depends on the angle . In the l.h.s of figure 5, the amplitude of the pressure shows a clear distinction between: (*a*,*b*) the zone of silence downstream and propagation zone upstream; (*c*,*d*) the zone of silence between upstream and downstream propagation zones. In all cases (*a*–*d*), the pressure amplitude is much lower in the zone of silence with a sharp transition towards the propagation zone(s). On the r.h.s., the pressure amplitude in the boundary layer with shear and cross flow: (*a*) is fairly smooth in the absence of the bias flow; (*b*–*d*) the bias flow causes some concentration of the pressure. In all cases (*c*,*d*), the acoustic ‘zone of silence’ has been destroyed by the vortical modes in the boundary layer. In figure 5 a dimensionless frequency (3.3*c*) equal to the unity *Ω*=1 was assumed and a mixed specific wall impedance 1+*i*. The preceding values of the flow and acoustic parameters correspond to the ‘baseline case’ that is justified next together with variations around the baseline values.

### (b) Baseline case and variations of parameters

The present model of an acoustic liner with bias flow relies on the following assumptions: (i) a boundary layer of thickness *L* with a linear shear velocity profile matched (2.6*a*,*b*) to a uniform stream of velocity ; (ii) a cross flow with uniform velocity *V* _{0}; (iii) homentropic flow and limited wall distance (2.12*a*), so that mean flow parameters such as the sound speed *c*_{0} are constant; (iv) locally reacting liner with specific (3.27*a*) wall admittance (3.25*a*). The baseline parameters are taken as: (i) the sound speed (4.8*a*) in sea-level atmospheric conditions; (ii,iii) a high subsonic free stream (4.8*b*) with a much smaller bias velocity (4.8*c*); (iv) a boundary layer of moderate thickness (4.8*d*); (v) a wave frequency of 1 kHz in the part of audible range 20 Hz–20 kHz more sensitive to noise (4.8*e*); (vi) a wall impedance (4.8*f*) or inverse of wall admittance combining resistance and inductance:
4.8*a–f*
These are typical values as order of magnitude of conditions in the inlet and exhaust ducts of modern jet engines, where the acoustic liners with bias flow are used to attenuate the sound; for example, precise values of the specific admittance depend on frequency and may not be available for a specific application, but it is known that the real and imaginary parts do not usually exceed unity. The preceding baseline values (4.8*a*–*f*) serve only to calculate the reference values of the four dimensionless parameters, namely the specific impedance (4.8*f*) and: (i,ii) the free-stream (4.9*a*) and cross-flow (4.9*b*) Mach numbers; (iii) the dimensionless frequency (4.9*c*):
4.9*a–c*
These reference dimensionless values (4.9*a*–*c*) will each be varied in turn over a sufficiently wide range to cover most aeronautical applications, so that the choice of starting numbers (4.8*a*–*f*) is not critical.

The dimensionless parameters (4.9*a*–*c*) specify the scattering coefficients, namely the amplitude and the phase of the reflection and transmission coefficients (figures 12–19) as a function of the angle of incidence in the full range of directions (4.10*a*):
4.10*a*,*b*
for the plots of the pressure as a function of the distance from the wall (figures 7–11) the angle of incidence is given by five values (4.10*b*) at 30° intervals symmetric relative to the vertical. The dimensionless parameters in the baseline case (4.8*f*), (4.9*a*–*c*) are varied in turn: (i) the free-stream Mach number is given (figures 7 and 11) by the values (4.1a), noting that the shear flow in the boundary layer is incompressible, hence the Mach number is not restricted: (ii) the bias flow is uniform and the values (4.1b) are considered including larger Mach numbers corresponding to fluidic jets; (iii) the specific impedance is given by the values:
4.11*a–f*
corresponding to a rigid wall, and unit values of the real part and plus or minus the imaginary part and their combinations. The frequency may vary over the more sensitive part of the audible range 100 Hz–5 kHz and the boundary layer thickness *L* = 0.05–0.15 m leading to values of the dimensionless frequency in the range:
4.12*a–e*

### (c) Pressure fields and scattering coefficients

The acoustic pressure normalized to the wall value is plotted as a function of dimensionless distance from the wall within the boundary layer in figures 7–11 with the amplitude at the l.h.s and the phase at the r.h.s. In all figures appear, besides the baseline case (4.9*a*–*c*) with bias flow (4.9*b*) also the comparable case without bias flow in order to assess the importance of the cross flow. The cases without bias flow are different in each figures 7–11 leading to five distinct comparisons with bias flow. Because the solution (3.8*b*) of the wave equation of the third order (3.3) is singular for *M*_{0}=0.0, the case without bias flow was calculated from (2.9*b*) using methods described in the existing literature [11,24]. The baseline concerns *θ*=60° propagation downstream and the cases with and without flow bias flow are compared for a free-stream Mach number in figure 7. The l.h.s shows that the curvature is opposite in the case with (line with blobs) and without (dashed line) bias flow; thus, the bias flow causes a slower decrease of the acoustic pressure away from the wall. In the presence of bias flow, the pressure decreases away from the wall more slowly for larger free-stream Mach numbers (figure 7) while the phase varies more slowly; the cases marked with asterisk corresponding to the zone of silence in the free stream do lie among the others within the boundary layer as a continuous variation. The r.h.s of the figure 7 shows that the phase of the acoustic pressure varies almost linearly with the distance from the wall for lower Mach numbers; for larger Mach numbers the phase is nonlinear indicating strong sound refraction effects. Concerning the comparison of the phase of the acoustic pressure such as the amplitude, it is larger without than with bias flow. It should be borne in mind that the Mach number of the bias flow is small, and thus its main effect is to remove the critical layer at the condition of zero Doppler-shifted frequency for the horizontal core flow. In spite of the smallness of the Mach number of the bias flow is has a detectable effect on the acoustic pressure, both for the amplitude (l.h.s of the figure 7) and for the phase (r.h.s of the figure 7).

To make more visible, the effects of cross flow the corresponding Mach number are given larger values well in excess of a typical bias flow, and more representative of a fluidic jet. This is done only in one figure, figure 8. Increasing the Mach number of the cross flow leads to a slower decay of the modulus of the acoustic pressure away from the wall (figure 8, l.h.s) as was the case increasing the core-flow Mach number (figure 7, l.h.s); the phase is an almost linear function of the distance from the wall both for lower core flow (figure 7, r.h.s) and lower cross flow (figure 8, r.h.s) Mach numbers. Large cross-flow Mach numbers, well in excess of a typical bias flow, and implying a high flow rate for a fluidic jet, lead to strong sound refraction effects illustrated by nonlinear amplitude and phase dependencies on the distance from the wall. The comparison of the cases with and without bias flow is made for the baseline case in figure 8, hence for the free-stream Mach number instead of in figure 7. Both the amplitude (l.h.s) and the phase (r.h.s) of the pressure vary more slowly with the distance from the wall in the absence of the bias flow (dashed line) compared with its presence (solid line). Returning to small cross-flow Mach number (4.9*b*) typical of bias flow, the effect of the wall impedance is shown in figure 9. The absorbing wall *A*=1 leads to a nearly constant pressure amplitude (figure 9, l.h.s) and a rapidly varying phase (figure 9, r.h.s). The amplitude decays away from the wall (figure 9, l.h.s) in all other cases, except for an ‘active’ wall 1/*A*=*i* when it increases rapidly and 1/*A*=1−*i* when it increases more slowly. The phase varies more slowly than for the absorbing wall in all cases (figure 9, r.h.s) with bias flow. The case without bias flow is considered for an absorbing wall *A*=1 to compare with the same wall with bias flow. The amplitude of the pressure is nearly uniform in the presence of bias flow and decays towards the free stream in its absence; the phase variation is greater in the absence of bias flow. The effect of the Helmholtz number or dimensionless frequency (figure 10) is to have larger amplitude and phase variation at high frequencies when the boundary layer is thicker on the scale of a wavelength, and smaller variations are observed for wavelengths larger than the thickness of the boundary layer. The amplitude of the pressure ceases to be a monotonic function of the distance from the wall for higher Helmholtz number *Ω*=2. This case is chosen for the comparison of the presence (line with crosses) and absence (dashed line) of bias flow. The absence of bias flow decreases notably the phase variation; it also changes significantly the modulus of the acoustic pressure by reversing its curvature and hence reducing it near the wall. The angle of incidence (figure 11) leads to larger amplitude and phase variations for propagation upstream 90°<*θ*<180° and smaller for propagation downstream 0°<*θ*<90° with strong refraction effects on the amplitude for almost counter-flow propagation *θ*=150°. The case of no bias flow (dashed line) is considered for the propagation normal to the wall *θ*=90° in comparison with the presence of bias flow (line with blobs). The absence of bias flow reduces the phase variation and more significantly changes the growth of the modulus of the pressure away from the wall to an initial decay with gradual a change towards growth.

Considering the reflection coefficient as a function of the angle of incidence (figures 12–15) again the amplitude and phase are plotted separately, respectively, at the r.h.s and l.h.s. The amplitude and phase are relatively smooth (figure 12) at low free-stream Mach number, whereas at higher Mach numbers the interaction of the sound with the shear and bias flows leads to sharper phase changes and several amplitude peaks. The free-stream Mach number is used for the comparison of the cases without (dashed line) and with (line with circles) bias flow. The variation in the amplitude and phase of the reflection coefficient are smaller and smoother in the absence of bias flow. The increase in the Mach number of the cross flow also leads to sharper changes in the amplitude and phase of the reflection coefficient (figure 13). In figure 13, the baseline case is used to compare the cases of no bias flow (dashed line) and bias flow (solid line); the variations of the amplitude and phase of the reflection coefficient are again smaller and smoother in the absence of bias flow. The reflection coefficient in the zone of silence is larger without bias flow. The wall impedance also affects strongly the directional dependence of the amplitude and phase of the reflection coefficient (figure 14), for example for the baseline case. The absorbing wall 1/*A*=1 is used for the comparison of bias flow absent (dashed line) and present (line with triangles). The bias flow makes the variations of the amplitude and phase of the reflection coefficient smaller and smoother. The effect of the Helmholtz number (figure 15) is a marked attenuation in the zone of silence for higher frequencies close to sound rays *Ω*=2; the lower frequencies are less affected by the ‘zone of silence’. In the propagation zone *θ*>56°, the amplitude changes are more notable for lower frequencies *Ω*=0.1. For all frequencies, there is a large phase transition between the zones of silence and propagation, with more rounded edges at lower frequencies. The highest Helmholtz number *Ω*=2 is used to compare the cases with (line with crosses) and without (dashed line) bias flow. The absence of bias flow eliminates the amplitude decay in the zone of silence without changing much the amplitude of the reflection coefficient in the propagation zone; the absence of bias flow reduces significantly the phase changes replacing the phase jump between the zones of silence and propagation by a smaller and much more gradual change.

Concerning the transmission coefficient, the amplitude and phase are again plotted separately at the r.h.s and l.h.s (figures 16–19) versus the angle of incidence. The transmission coefficient has severally larger amplitude and phase changes for downstream propagation and increasing free-stream (figure 16) and cross-flow (figure 17) Mach numbers. The comparison with the absence of the bias flow (dashed line) is made for the baseline case (solid line) in figure 17 and for (line with blobs) in figure 16. The baseline case in figure 17 shows that the bias flow increases the amplitude in the zone of silence, and increases the phase in the propagation zone. This is consistent with of what was seen in figure 13 for the reflection coefficient in the zone of silence, the bias flow reduces the reflection coefficient and increases the transmission coefficient. Thus, the bias flow spreads the acoustic energy over a wider range of directivities by increasing the pressure field in what would be the ‘zone of silence’ in the absence of boundary layer. The comparison of bias flow present (line with blobs) and absent (dashed line) for a free-stream Mach number in figure 16 shows that the modulus of transmission coefficient is increased by the bias flow in the zone of silence, and is less affected in the propagation zone, and also the phase jump at the transition between the two zones is smoothed by the bias flow. The effect of the wall impedance (figure 18) is more pronounced for the active wall 1/*A*=*i* and for the rigid wall, that lead to peaks respectively in the downstream and upstream arcs. The absorbing wall *A*=1 is used for the comparison of bias flow absent (dashed line) or present (line with triangles). The bias flow mostly eliminates the reduction in the modulus of the transmission coefficient factor in the zone of silence and has less effect on its amplitude in the propagation zone; the changes in the phase of the transmission factor are small for an absorbing wall without bias flow, and the bias flow causes a smooth increase. The higher values of the Helmholtz number (figure 19) lead to larger variations of the transmission coefficient, corresponding to wavelengths smaller than or comparable with the thickness of the boundary layer; the longer wavelengths lead to almost total transmission with small phase change. The highest value of the Helmholtz number *Ω*=2 is used for the comparison of absence (dashed line) and presence (line with crosses) of bias flow. The bias flow increases the amplitude of the transmission coefficient in the zone of silence and decreases it in the propagation zone, leading to a more isotropic radiation pattern and a smoother phase change across the boundary between the two zones. Because the present theory is based on the exact solutions of the wave equation, it holds for unrestricted values of the parameters, in particular for wavelengths large, small or comparable with flow scales in the boundary layer, covering a frequency range from sound rays to compact scattering.

## 5. Discussion

The wave equation was obtained for sound in a unidirectional shear flow with a linear velocity profile superimposed on a uniform cross flow. The acoustical–vortical wave equation is of third order owing to the coupling of second-order sound waves with first-order vorticity transport. Because the mean flow is steady (and uniform in the wall direction), there exists a frequency and a horizontal wavenumber that appear as cubics in the coefficients of the differential equation specifying the dependence of the pressure perturbation on the distance from the wall. The differential equation is of the second (third) order in the absence (presence) of cross flow. In the absence of cross flow, the second-order differential equation has a singularity or critical layer corresponding to the vanishing of the Doppler-shifted frequency and leading to a continuous spectrum besides the discrete spectrum. The cross flow eliminates the critical layer and leads to a third-order differential equation without singularities at finite distance. Thus, the pressure in the presence of cross flow is specified by MacLaurin series convergent for any finite distance from the wall. In the absence of cross flow, the MacLaurin series around the wall will have radius of convergence limited by the critical layer; the solution around the critical layer is specified by a Frobenius–Fuchs series instead of a MacLaurin series, to account for the singularity at the critical layer.

In the absence (presence) of cross flow, there are two (three) linearly independent pressure fields, because the differential equation is of the second (third) order. The matching at the edge of the boundary layer of the pressure perturbation of acoustic-vortical waves inside and sound waves outside requires in both cases the continuity of the pressure and horizontal and vertical velocity perturbations. In the presence (absence) of cross flow, these three conditions are independent (redundant and reduce to two). Thus, the matching conditions lead to the matching of the pressure and first two (only first) derivatives. These three (two) matching conditions specify the three (two) constants of integration in the general solution of the differential equation of order three (two). Because the wave field in the free stream consists of an incident and reflected sound wave, the two (three) acoustic-vortical waves in the boundary layer have amplitudes that depend on the reflection coefficient. The extension of the wave field up to the wall specifies the reflection coefficient using an impedance boundary condition for a locally reacting liner. The transmission coefficient is defined as the ratio of the pressure at the wall to the value in the free stream at the edge of the boundary layer. This completes the determination of the amplitude and phase of the reflection and transmission coefficients and also of the pressure perturbation in the whole flow region inside and outside the boundary layer. The preceding theory of acoustic-vortical waves can be used to model the wave field near an acoustic liner with bias flow.

The comparison of the acoustic liner with and without bias flow shows that the mathematical consequences are (i) the removal of the singularity at the critical layer; (ii) the increase in the order of the differential equation from 2 to 3, implying that there is further decoupling of the acoustic and vortical modes. From the physical point of view, the comparison of the bias flow present and absent demonstrates changes in the modulus and phase of: (i) the acoustic pressure as a function of the distance from the wall; (ii) the reflection and transmission coefficient as a function of the angle of incidence, including different trends in the zones(s) of silence and propagation. The overall conclusion is that a bias flow with small Mach number can change significantly the acoustic pressure in the boundary layer by: (i) broadening the range of directions of propagation into the zone of silence; (ii) changing the ratio of pressure amplitudes in the free stream and at the wall. Both effects (i) and (ii) can have a beneficial effect on noise reduction, depending on the core flow, boundary layer flow and wall impedance characteristics.

Some brief concluding remarks are made on three effects not included in the present theory that may be expected to affect the sound field in the boundary layer over an acoustic liner with bias flow, namely (i) viscosity; (ii) vortex shedding; and (iii) turbulence. The effect of vorticity on the sound is represented in the present model by a boundary layer with a linear velocity profile; the velocity profile would be different in a viscous turbulent boundary layer. The acoustic of shear flows has been considered for the other shear velocity profiles namely exponential [13], hyperbolic tangent [14] and parabolic [15]; there are both fundamental qualitative similarities (existence of critical layer and continuous spectrum) and moderate quantitative changes (pressure fields and scattering coefficients). Viscosity can absorb sound in a boundary layer [25] but this is mostly high-frequency effect. More significant is the effect of the vortex shedding [26] from the edges of the holes in the acoustic liner, in which passes the bias flow; the circulation of the shed vorticity is specified by the Kutta condition at downstream edges [21,27]. Turbulence may be seen as a continuous spectrum of eddies of different scales; it causes random changes in direction and frequency of propagation [28] and hence causes interference effects that change the directivity and spectrum of sound [29]. In the presence of strong turbulence, the amplitude and phase of the acoustic pressure become random [30] and their correlation specifies the power spectrum of sound.

- Received November 4, 2013.
- Accepted December 11, 2013.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.