## Abstract

The evolution of acoustic and vorticity perturbations in a two-dimensional incompressible linear flow is investigated. A weighted decomposition of the flow into a hyperbolic part and a rotation part allows continuous spanning of all linear flows such as hyperbolic flow, plane Couette flow and rigid rotation for instance. Using the Kelvin non-modal approach, the equations governing the time evolution of plane wave perturbations are reduced into a system of three first-order ordinary differential equations. This system is analysed using a WKB method where the small parameter *ε* is the ratio of the shear rate of the flow over the typical frequency of the perturbations. With this method, a basis of three modes naturally appears: two acoustic modes and one vorticity mode. At finite but small *ε*, couplings between the modes appear when the length of the wavenumber is minimal. For hyperbolic flow, incident vorticity mode generates the two acoustic modes, and an incident acoustic mode generates the other acoustic mode. More generally, for all flows, the hyperbolic part of the flow is responsible of the coupling between acoustic and vorticity modes, but also of the coupling between the two acoustic modes. These phenomena are illustrated by displaying wavepacket evolutions.

## 1. Introduction

Propagation of acoustic waves in uniform flows is a well-understood topic [1]. Nevertheless, most flows are not uniform, and shear flows are known to display subtle features as testified by the large range of literature devoted to the study of their stability in the incompressible limit [2–11]. The classical modal analysis based on the study of the eigenvalues of the linearized equations used to be largely used, but led to predictions significantly different from experimental results owing to the existence of transient growth [5–7]. To overcome these discrepancies, non-modal methods, initiated by William Thomson, Lord Kelvin [12], have been developed [2,3,5–7,10,11]. In the case of a linear flow, this method is greatly simplified by using a new system of convected coordinates. One of its interests is to transform the governing equations into a three-dimensional time ordinary differential equation (ODE) with constant coefficients with respect to space, what allows us analytic treatment of the equations.

Acoustic waves in plane Couette flow have been studied by Chagelishvili *et al.* [13] in the continuity of flow stability analysis. Using the non-modal approach, they derived an inhomogeneous equation governing the horizontal velocity disturbance evolution and defined acoustic waves as the solutions of the associated homogeneous equation, and vorticity waves as the particular solution of the complete equation. This frame permitted to show some coupling phenomena between what they defined as acoustic and vorticity effects [14–21]. Among these effects, one is the generation of acoustic waves by vortices, first attributed to a discontinuity by Chagelishvili *et al.* [15], and later to an interaction of the disturbance velocity field with mean shear by George & Sujith [22]. This approach has also been extended to arbitrary linear flow [17,18] (including the Couette flow). The Wentzel–Kramers–Brillouin (WKB) method has been used since the earlier papers of Chagelishvili *et al.* [14,23] while not mentioned explicitly (‘adiabatic evolution’). A more extensive use of it has been performed by Gogoberidze *et al.* [20] to explain the couplings as resulting of ‘non-adiabatic transitions’, first studied in quantum mechanics by Landau and co-workers [24,25], and Zener [26].

In this paper, we propose a systematic study of compressible perturbations in general linear incompressible shear flows. This linear form of flow is meant to describe the local behaviour of a shear flow. We introduce a WKB method for which the two acoustic modes and the vorticity mode naturally come out from the equations as eigenvectors of an eigenvalue problem. An order 1 approximation is used instead of the classical order 0 WKB to improve the description and to take into account the well-known inherent rotational part of the acoustic mode which is due to potential vorticity conservation [27]. The plan of this paper is as follows. In §2, we derive the governing equations following the Kelvin non-modal approach [13,17]. We display the variety of flows that can be described under the assumption of linear flow. We also briefly tackle two limit cases of the model: the no flow, and the incompressible cases. The WKB method is presented in §3. Section 4 exhibits results obtained with the proposed approach. All flows are systematically investigated. Results are then applied to the evolution of wavepackets. Finally, we compare numerically the mode coupling efficiency depending on flow shape and intensity.

## 2. Model

In this section, we derive the equations in the same way as Chagelishvili *et al.* [23] and other papers (in particular Mahajan & Rogava [17]). As mentioned in §1, the proposed method is inspired by the stability analysis which is itself closely linked to the rapid distortion theory (RDT) of homogeneous turbulence [28–30]. Indeed, as emphasized by Cambon *et al.* [31], for example, the equations derived in these two approaches are similar. Hence, the approach presented in this paper is also strongly related to the RDT. The aim of this work is the study of the local effect of the shearing of the steady flow on the coupling between acoustic and vorticity perturbations. We suppose that the variation length scales *L*_{ρ0} and *L*_{s0} of the density *ρ*_{0} and of the entropy *s*_{0} of the steady flow are large compared with the typical length scale *l* of the perturbations *L*_{ρ0}≫*l*, *L*_{s0}≫*l*. For this reason, we neglect the variations of density and entropy of the mean flow. So we consider a bi-dimensional incompressible and homentropic steady flow *u*_{0} of ideal fluid on which we superimpose compressible unsteady perturbations. The steady flow satisfies the incompressible Euler equations. To describe the local behaviour of a shear flow, it is linearized by keeping the two first terms of its Taylor expansion
2.1where is a constant vector, which is responsible for a simple convection, and *A* is the constant shear matrix, which is responsible for shearing. We also suppose that the mean flow has a small Mach number and a small shearing, namely , where would be the adiabatic speed of sound at the origin ( being the pressure at the origin ). It is shown in appendix A, that under these assumptions the variation of the steady flow pressure *p*_{0} can also be neglected, being a small quantity of order |*A*|^{2}. Therefore, *c*_{0} is eventually the speed of sound that is supposed constant in space (within the frame of the local approximation of the general steady flow). It is also shown that the only effect of the constant component on the following development is to add an additional term to the phase of the perturbations. For this reason, we consider in the following that for the sake of clarity. The steady flow is then linear and is fully described by
2.2

We now consider linear perturbations of velocity ** u**=(

*u*,

*v*)

^{T}, density

*ρ*and pressure

*p*. Their evolution is governed by the linearized Euler equations 2.3aand 2.3bin which we neglected the term

*ρ*(

*u*_{0}⋅

**∇**)

*u*_{0}which is of order |

*A*|

^{2}. Assuming that the perturbations are also homentropic, the pressure and density are linked by the adiabatic law . Appendix A provides a more detailed derivation of equation (2.3) and a discussion on the order of magnitude of the neglected terms.

We introduce the convected coordinates coming from the solution of the trajectory equation with ** x**(0)=

**: 2.4where**

*X***corresponds to Lagrangian coordinates. Analytical expression of the exponential is obtained by use of Pauli matrices as is detailed in appendix C (equation (A10)). With these new coordinates (**

*X***,**

*X**T*), equations (2.3) become 2.5aand 2.5bwhere the superscript

^{T}stands for matrix transposition. The coefficients involved in (2.5) are constant with respect to space, which will be of a particular interest to apply Fourier. Indeed, by writing

**(**

*u***,**

*X**T*)=

**(**

*U**T*)

*e*

^{ik0⋅X}and

*p*(

**,**

*X**T*)=

*P*(

*T*)

*e*

^{ik0⋅X}, the equations become 2.6aand 2.6bwhere 2.7Equations (2.6) constitute a system of three first-order ODEs, which allows us to study the governing equations as an evolution problem of dimension three.

Equations are now turned into a dimensionless form. We define the dimensionless shear matrix
2.8where the quantity *ζ* is defined in equation (2.16) of §2*a*. We define *k*_{0}=∥*k*_{0}∥ and use 1/*k*_{0}, 1/*c*_{0}*k*_{0}, *c*_{0} and , as reference values to obtain dimensionless lengths, time, velocities and pressure (note that in the new dimensionless variables *k*_{0}=1). Equations (2.6) reduce to
2.9aand
2.9bwhere appears the dimensionless parameter
2.10That *ε* is the ratio of the characteristic shear rate of the flow over the typical frequency of the perturbations. For instance, for the plane Couette flow, this is the ratio of the vorticity of the mean flow over the typical frequency of perturbations. In the following, we name the elements appearing in (2.9) as follows
2.11and
2.12A consequence of equations (2.7) and (2.9) is the conservation of potential vorticity
2.13This relation plays a very important role in the following developments, as it is responsible of cross dependences between acoustic and vorticity parts.

### (a) Linear flows parametrization

The incompressibility and linearity assumptions allow us to parametrize all flows with two parameters. As is developed in appendix B, without loss of generality, we can write the shear matrix in the form
2.14where
2.15The dimensionless shape parameter *δ*∈[0;1], and the shear rate *ζ* (homogeneous to a frequency) are given by (see appendix B for derivation)
2.16and
2.17The two matrices *B*_{h} and *B*_{r} correspond respectively to a hyperbolic flow and to a clockwise rigid rotation flow. Thus, any flow is decomposed as a weighted sum of a hyperbolic part and of a rotation part, and the whole set of flow shapes is parametrized by *δ* as shown in figure 1. It includes the rigid rotation (*δ*=0), elliptic flows (), the plane Couette flow () and the hyperbolic flow (*δ*=1). Note that with the present decomposition the parameters appearing in (2.11) are *r*=0, *a*=1 and *b*=2*δ*−1∈[−1;1]. The dimensionless shear matrix is then
2.18

### (b) Convected wavevector

We discuss here some points concerning the wavevector ** k**. Because it satisfies

*k*_{0}⋅

**=**

*X***⋅**

*k***,**

*x***is the wavevector in fixed coordinates**

*k***. It is governed by 2.19which is the same equation as the one governing the wavenumber in geometrical acoustics approximation. It means that**

*x***is drifted by effect of the flow [13], and the**

*k***vector follows the streamlines of the steady flow, up to a −**

*k**π*/2 rotation. Equation (2.7) gives an explicit formula to compute analytically

**. The derivation is based on Pauli matrices and is detailed in appendix C. It yields 2.20where**

*k**Id*is the identity matrix.

Figure 2 illustrates for different flows the time evolution of the wavevector and of its norm that we name
2.21In the case of the rigid rotation (*δ*=0), *k* is constant. When (elliptic flows), ** k** has a periodic behaviour with an amplitude that grows and a frequency that decreases as . The Couette flow can be viewed as the limit when this frequency tends to zero. For this flow, there is a minimum of

*k*at

*T*

_{⋆}=

*β*

_{0}/

*εα*

_{0}, which is 0 in figure 2 because

*β*

_{0}=0. Far from

*T*

_{⋆},

*k*evolves as ∼|

*εT*|. When , the

*k*curve is qualitatively the same but the behaviour of

*k*becomes asymptotic to an exponential . The most important point for following developments is the presence of minima of

*k*as time evolve, for all flows except rigid rotation; these minima will correspond to couplings between acoustic and vorticity parts.

### (c) Acoustic waves: the no flow limit

We consider here, as an illustrative example, the no flow limit, where *u*_{0}=**0** (*ε*=0 with finite speed of sound). In this case, the wavevector is constant with respect to time:
2.22Equations (2.9), written as a vectorial ODE, reduce to
2.23which implies the scalar wave equation (more precisely an harmonic oscillator). The matrix in the right-hand side of equation (2.23) has the three eigenvalues
2.24associated to the modes
2.25where *k*^{⊥}_{0}=(*β*_{0},−*α*_{0})^{T}. The first mode is a vorticity mode, and it has a divergence free velocity field, whereas the last ones correspond to acoustic modes with curl-free velocity fields. General solution of the system (2.23) is
2.26where *C*_{H},*C*_{+},*C*_{−} are constant. We recover here classical acoustics with two acoustic waves travelling towards opposite directions. The first term in (2.26) corresponds to trivial vorticity perturbations that are frozen in time.

### (d) Hydrodynamic waves: the incompressible limit

We now present as a second illustrative example the incompressible limit case (with infinite speed of sound and finite *u*_{0}). Equations (2.9) become
2.27aand
2.27bThe dynamics of such systems has been the subject of many studies based on the work of Lord Kelvin [2–12]. For three-dimensional linear flows, Craik & Criminale [2] have given explicit solutions. Using the incompressibility condition (2.27b), the pressure is expressed in terms of the velocity
2.28and the velocity evolution is then governed by
2.29This equation has the two independent solutions
2.30and
2.31Except for elliptic flows (), these solutions are bounded and tend to zero as . However, their components evolve as 1/*k*, and the time dependence of *k* (see the minima in figure 2) allows the presence of a transient growth that can lead to high amplitudes at finite time [5–7,11]. The case of elliptic flows is different. One solution is bounded and periodic, and the other one solution grows as ∼*T*. Figure 3 shows an illustration of these two solutions for different flows. The solution *U*_{1} is unstable in the case of elliptic flows (), whereas *U*_{2} is stable. In the case of the Couette plane flow (), the solution *U*_{1} becomes constant but the other solution *U*_{2} displays a transient growth. This solution tends to zero at infinite positive or negative times but has a strong maximum at *T*=0 (here normalized to 1). Even if the solution finally tends to zero, it is possible to find an initial condition, say of norm 1, which leads to an arbitrary high amplitude depending on how far from *T*=0 is imposed this initial condition. For hyperbolic and nearly hyperbolic flows (), the two solutions *U*_{1} and *U*_{2} are subject to a transient growth. They are even more efficient in the potential hyperbolic flow.

In the following sections of this paper, we focus on situations where both acoustic and hydrodynamic modes coexist (finite *ε*≪1 and compressible perturbations).

## 3. WKB approach

In this section, we use the WKB method [32,33] to construct the base of acoustic and vorticity modes in the presence of flow. For small values of *ε*, which means perturbations of frequency higher than the shear rate of the flow, we introduce the slow time *τ*=*εT*. From (2.9), it leads to
3.1where
3.2and
3.3Here, *H*_{0} is antisymmetric, but the operator *H*≡*H*_{0}+*ε**H*_{1} has no particular symmetry in general; it is antisymmetric only in the case of the rigid rotation with antisymmetric *B* (which is also the only case where we know a simple analytic solution). This lack of symmetry implies the lack of the conservation of the canonical energy of the system defined by
3.4Indeed, according to equation (3.1), its evolution is governed by
3.5where the superscript † refers to the complex conjugation transposition. Perturbations can therefore gain or lose energy over time, and this evolution is due to the hyperbolic part of the flow *B*_{h} defined in equation (2.15).

In previous works [20,23], the definition of acoustic and vorticity parts was based on whether the perturbation corresponds to the general homogeneous solution or to the particular non-homogeneous solution of the second-order, non-homogeneous equation governing the horizontal component of the velocity perturbation. In our work, the WKB method will provide automatically three modes. Two of these modes will correspond to the acoustic part, and the third mode will correspond to the vorticity part.

To apply the WKB method on (3.1), we assume the following ansatz
3.6Inserting equation (3.6) into (3.1), and identifying terms of same order in power of *ε* lead to the classical recursive WKB system to be solved
3.7aand
3.7bwhere the overdot refers to differentiation with respect to the time *τ*.

### (a) Order 0 Wentzel–Kramers–Brillouin modes

The order 0 equation (3.7a) is an eigenvalue problem. Therefore, is one the three eigenvalues of *H*_{0}
3.8and *φ*_{0} is proportional to the associated eigenmode among
3.9There remains to find the proportionality factors whose time dependence is imposed by the *compatibility condition* derived from the order 1 equation, as is usual in the WKB method [34,35]. Taking the hermitian product of equation (3.7b) for *n*=1 with *φ*_{0} shows that
3.10which allows us to obtain the sought proportionality factors. Eventually, we have
3.11and the order 0 WKB modes in (3.6) are
3.12In the same manner as in the case of the no flow limit (2.25), we identify one vorticity mode (under-script *H*) and two acoustic modes, one backward-going (under-script +) and one forward-going (under-script −). To get some insights on these modes, we first focus on the eigenvalues.

Figure 4*a* represents the time evolution of the imaginary part of the eigenvalues, in the widely studied Couette flow (), with *ε*=0.2, *α*_{0}=1 and *β*_{0}=0. This is a typical case of avoided crossing of the eigenvalues with near coalescence near *τ*_{⋆}=*β*_{0}/*α*_{0}=0. This is because *τ*_{⋆} is the closest point on the real *τ* axis to the complex zero *τ*_{c}=*β*_{0}/*α*_{0}+*iα*_{0} of *k*, where all three eigenvalues coalesce. In the two-state Landau–Zener model [24–26], two modes with opposite eigenvalues exchange energy at such a point. This phenomenon, called non-adiabatic transition, is of order ∼*e*^{−C/ε}, and cannot be produced by WKB asymptotics. We expect that some couplings occur at such *τ*_{⋆}.

Concerning the WKB modes themselves, they have a varying wavevector, in opposition to the exact modes obtained in the no flow limit (§2*c*), and their norms depend on time. The components of the acoustic modes evolve as . For flows where , they have a minimum at *τ*_{⋆} and grow indefinitely over time. That phenomenon has been interpreted as an energy transfer phenomenon between disturbances and the mean flow for the plane Couette flow () [23]. The vorticity mode evolves as ∼1/*k*, it has a maximum at *τ*_{⋆} and decay for positive and negative time. It is the compressible analogue of the incompressible solution *U*_{2} in equation (2.31) of §2*d*, which displays one of the transient growths [5–7,11]. It is important to notice that at this order the acoustic modes are curl-free, and the vorticity mode is divergence-free.

We now focus on the evolution of the solution in the base formed by these WKB modes. When *ε*=0 with *u*_{0}=**0**, it has been already seen that these amplitudes are constant (see §2*c*). When *ε*≪1, if the goal of providing analytic solutions is achieved, we would expect constant amplitudes for any solution expressed in this base. Let ** c**(

*τ*)=(

*c*

_{H}(

*τ*),

*c*

_{+}(

*τ*),

*c*

_{−}(

*τ*))

^{T}be the vector representing the solution in this base: 3.13where the coefficients

*c*

_{m}will be referred as the amplitude of the mode

*m*(

*m*∈{

*H*,+,−}). In this base, the energy definition (3.4) becomes 3.14For constant amplitude

*c*

_{m}, acoustic modes see their energy evolving as

*k*, and the vorticity mode as 1/2

*k*

^{2}. This also means that for acoustic mode, the wave action [36,37] 3.15is conserved. Note that far from

*τ*

_{⋆}, the contribution of the vorticity mode to the wave action is very small (∼1/

*k*

^{3}) and that the total wave action is almost conserved.

Figure 4*b* displays the time evolution of mode amplitudes. At *τ*_{i}=−5, the WKB acoustic mode is imposed as an initial condition ** Y**(

*τ*

_{i})=

*Y*_{0,+}(

*τ*

_{i}), or equivalently

*c*

_{+}(

*τ*

_{i})=1,

*c*

_{H}(

*τ*

_{i})=

*c*

_{−}(

*τ*

_{i})=0. It corresponds to an incident acoustic mode. Equation (3.1) is numerically integrated using a Magnus–Möbius scheme [38,39]. As expected, the amplitude of the incident acoustic mode remains close to 1, but the vorticity mode amplitude immediately oscillates. These fluctuations may be understood from the potential vorticity conservation: using (2.13) and assuming

*c*

_{+}≃1,

*c*

_{−}≃0 it comes that 3.16So to ensure potential vorticity conservation, the imposed purely acoustic perturbation (curl-free) acquires a rotational part by driving the vorticity mode; we recover the well-known fact that acoustic perturbation cannot be curl-free in shear flow [27]. From a more technical point of view, it is in fact a non-uniformity of the WKB method with an amplitude 〈

*c*

_{H}〉∼

*ετ*

_{i}. More generally, for any flow, the potential vorticity conservation enforces 3.17The (1−

*δ*) factor in (3.17) shows that the rotation part of the flow is responsible for this effect. This happens in all flows except the hyperbolic flow (

*δ*=1), in which the vorticity mode has a constant amplitude. This comes with no surprise as the hyperbolic flow is potential and it is the only case where acoustic perturbations are curl-free (potential).

Another interesting effect is the emergence of the non-incident acoustic mode (of amplitude *c*_{−}) at the time *τ*_{⋆}=0. This is a non-adiabatic transition owing to the avoided crossing of the eigenvalues. As stated earlier, this is an exponentially small phenomenon of order *e*^{−C/ε}. This cannot be modelled by the standard WKB asymptotic expansion itself even if higher-order approximation was used (all derivatives of the exponential are null at *ε*=0). However, a large number of authors have been investigating this kind of phenomena with extensions of the WKB method. It is particularly popular to study non-adiabatic state transitions in quantum mechanics [40–43].

Figure 4*c* displays the amplitudes of the three modes when vorticity mode is incident at *τ*_{i}=−5. As in the preceding case, the incident mode amplitude stays close to 1. At *τ*_{⋆} a non-adiabatic transition occurs, and the vorticity mode generates the two acoustic modes. After *τ*_{⋆}, we can note that the vorticity mode amplitude starts to oscillate due to the presence of the acoustic modes and to the earlier-mentioned non-uniformity. The acoustics produced by the incident vorticity mode can be estimated analytically. The exact equations governing the evolution of ** c** are derived from equations (3.1), (3.12) and (3.13), and yield
3.18where

*e*

_{±}stands for , and 3.19which is constant thanks to equation (2.19). We recover the fact that for the hyperbolic flow (

*δ*=1),

*c*

_{H}is constant. When the vorticity mode is incident,

*c*

_{H}∼1,

*c*

_{+}is governed at first-order by 3.20and is obtained by quadrature. This estimation of the acoustic amplitude

*c*

_{+}is similar to the Born approximation and it is displayed in figure 5. It provides a good approximation of the acoustic produced by vorticity.

In the following, in order to get rid of the non-uniformity, responsible for the fluctuations of *c*_{H} and owing to the potential vorticity conservation, we propose a slight modification of the WKB base.

### (b) Order 1 WKB modes

To get the WKB modes at next order, we need to determine *φ*_{1}. For this, we use equation (3.7b) with *n*=1 and express *φ*_{1} in terms of the *φ*_{0,m} (the eigenmodes of *H*_{0}). For the acoustic mode, it gives
3.21where *w*_{++}, the component of *φ*_{1,+} along *φ*_{0,+}, is still undetermined because the kernel of in equation (3.7b) is along *φ*_{0,+}. To determine this last component *w*_{++}, the classical method would consist of taking the projection on *φ*_{0,+} of (3.7b) at order *n*=2. In our case, this is the source of non-uniformities and secular terms. To avoid this problem, Smith [44] proposed a method in which both phase and mode shape are expanded in *ε* power series [45]. As we intend only to remain at order 1, we propose a simpler method: we choose to determine the last component *w*_{++} so as to minimize the variation of the invariant potential vorticity . This choice relies on the fact that the non-uniformity is imposed by the potential vorticity conservation. Combining (2.13), (3.12) and (3.21), we obtain
3.22To make constant for the WKB mode, we impose
3.23Eventually, the order 1 acoustic WKB modes are
3.24The acoustic mode includes now the vorticity oscillations (non-uniformities) discussed in §4*a*. The factor (1−*δ*) shows that this correction is due to the rotation part of the flow, and, consequently, does not exist in the case of the hyperbolic flow. Applying the same method to determine the order 1 vorticity mode leads to
3.25Here, the invariant can be kept constant only up to order *ε* by setting *w*_{H,H}=0. Therefore, the order 1 vorticity WKB mode is
3.26Now that the order 1 base is obtained we define ** d** the same manner as

**: 3.27We will focus on the evolution of the exact solution in this order 1 base, which is free of non-uniformities and for which only non-adiabatic transitions at**

*c**τ*=

*τ*

_{⋆}may occur.

## 4. Main results

### (a) Plane Couette flow

Figure 6 shows the time evolution of order 1 WKB mode amplitudes in the plane Couette flow when order 1 WKB modes are incident. It is instructive to compare these results with those obtained with the order 0 WKB modes in figure 4*b,c*. For an incident acoustic mode (figure 6*a*), the vorticity mode fluctuations discussed earlier in figure 4*b* no longer exist because the acoustic mode at order 1 contains the vorticity correction needed to ensure potential vorticity conservation. Indeed, this latter condition becomes
4.1From figure 6*a*,*b*, we see that the order 1 WKB modes give results very near exact solutions except the presence of the non-adiabatic amplitude transitions at *τ*=*τ*_{⋆}. The transition owing to the order 1 acoustic mode is now very small in figure 6*a*. Figure 6*b* shows a transition similar to the one with order 0 modes: the final amplitude is the same, whereas the behaviour during transition is smoother. Note also that |*d*_{h}| remains very close to one without the oscillations owing to the non-uniformity observed in figure 4*c*.

### (b) Hyperbolic flow

As the hyperbolic flow (*δ*=1) is the only one to be potential, we could guess that no couplings would occur. Indeed, the vorticity mode amplitude is constant because equation (4.1) reduces here to . However, the two non-adiabatic transitions mentioned for the plane Couette flow also occur here, and the amplitude transitions are more important (figure 7). Concerning intermediate flows between plane Couette and hyperbolic flows (), mode amplitudes behave qualitatively as in the case of the Couette flow. As it could be expected, the closer the flow is to the hyperbolic flow (*δ* close to 1), the more the amplitude transition is important. Figure 8 corresponds to one of these intermediate flows, with *δ*=0.6.

### (c) Rigid rotation

The rigid rotation (*δ*=0) has two particularities: *k* is constant, and we analytically know the exact solution. This solution is the sum of the three following independent solutions
4.2and
4.3The order 1 WKB vorticity mode coincides with the exact solution *χ*_{H}. The WKB acoustic mode is composed of the first terms of the expansion in *ε* power series of the other solution *χ*_{+}. Figure 9 shows the evolution of the mode amplitudes for a rotation flow. There is no transition, but in the case of an incident acoustic mode, the other acoustic mode has an amplitude that oscillates permanently. The difference between exact solution and WKB approximation can be obtained from equation (4.3)
4.4so that
4.5The oscillations of the non-incident acoustic mode are the result of an order *ε*^{2} error.

Figure 10 shows an example concerning an elliptic flow (). In such flows, ** k** and its components become periodic. This leads to successive avoided crossings, and successive amplitude transitions occur. At each transition, the same transition matrix is involved, but the mode amplitudes are each time different before the transitions explaining the apparent erratic transitions. The closer the flow is to the perfect rigid rotation (

*δ*near to 0) the more important are the abrupt oscillations appearing during transitions, and the more the amplitude transitions become smaller.

### (d) Wavepackets

Hereafter are presented computations based on wavepackets and on the ray method. Narrowband wavepackets have to behave in a way very close to that of spatial Fourier harmonics studied in preceding sections. Figure 11*a* shows the time evolution of an acoustic wavepacket in a Couette flow () with *ε*=0.6. The choice of such a ‘high’ value of *ε* is made so that the acoustic wavepacket generated during the transition can be seen. Note that this simulation is rather qualitative than quantitative as once the Mach number is greater than one the linearity assumption used in the model breaks down. At initial time (*τ*=−4), a packet of order 1 WKB acoustic mode is imposed at the position of the blue diamond in figure 11*a*. The acoustic wave propagates and is refracted by the flow (due to both convection and to the drift of the wavevector). When it reaches the transition time, the centre of the packet is located at the central dot. Then, the acoustic wavepacket generates the other acoustic mode packet, and each acoustic packet goes in opposite direction. Figure 11*b* shows the evolution of a vorticity wavepacket, in an area where the hypothesis of the base equation is respected (for a more moderate value of *ε*=0.2). As the mean flow is null at its starting point, it stands still. At the transition time, it generates two acoustic wavepackets which then propagates in opposites directions. Figure 12 shows trajectories of wavepackets (using the ray method) for Couette and elliptic flows. The ray trajectories are augmented by transitions occurring at *τ*=*τ*_{⋆}. Figure 12*a*,*b* corresponds to figure 6*a*,*b* and figure 12*c*,*d* to figure 10*b*,*c*.

### (e) Wave generation efficiency

We are now interested in the efficiency of different types of flows in transferring energy between two modes. Let be *S* the matrix composed of order 1 WKB modes as columns, which permits the transformation
4.6where ** d** is the vector of amplitudes of acoustic and vorticity WKB modes (equation (3.27)). Let also be

*τ*

_{1}and

*τ*

_{2}two times, respectively, before and after a non-adiabatic transition at

*τ*=

*τ*

_{⋆}, and the resolvent associated to the evolution ODE (3.1) 4.7We define the transmission matrix such that 4.8We can compute this matrix using 4.9Figure 13

*a*,

*b*shows the transmission coefficients between, respectively, vorticity and acoustic order 1 WKB modes, and two different acoustic order 1 WKB modes, for flow of any shape (parametrized by

*δ*) and for values of

*ε*varying between 0 and 0.5. The hyperbolic flow once again appears as the most efficient flow in terms of coupling by non-adiabatic transitions. For any

*δ*, the amount of coupling also shows an exponentially small behaviour

*e*

^{−C/ε}when .

## 5. Concluding remarks

The proposed WKB method predicts good approximations of the evolution of compressible perturbations in all incompressible linear flows for small *ε*. It offers a relevant frame to analyse the involved physics. Moreover, WKB solutions provide a base where the couplings between acoustic and vorticity modes appear as non-adiabatic transitions. All the results obtained in this paper suggest that the hyperbolic part of a flow is responsible of the non-adiabatic transitions. This is a quite surprising result, because as the flow is potential, we could think that acoustic and hydrodynamic effects remain independent. Future developments will consist of deriving analytical asymptotic expressions of the transition amplitudes following the methods developed in quantum mechanics. The asymptotic smoothing phenomenon described by Berry [41] and by Lim & Berry [42] could be investigated. The intensively studied Couette flow finally appears as the middle step point being under the influence of the effects of the two reference flows: the rotation part induces a non-curl free velocity field of the acoustic waves, and the hyperbolic part induces couplings between the modes through non-adiabatic transitions. Meanwhile, for moderate flow shear rates, the major coupling effects remain (exponentially) small correction compared with solutions obtained with the WKB method. Another future work could consists in adapting the method developed in this paper to more general flows which are not linear. As an example, we could consider flows which are locally linear, that is with a shear matrix that varies slowly in space. Provided the variation scale of the shear matrix is sufficiently small compared with the length scale of the perturbations, we could consider the use of a multiple scales method involving these two length scales.

## Appendix A. Derivation of equations (sec2.3)

In this appendix, we derive formally the equations governing the evolution of the unsteady perturbations. The starting point is to consider an incompressible steady flow and then to superimpose unsteady compressible perturbations on it. The study performed in this paper is focused on the local effect of the shearing of the flow on the perturbations. So this steady flow *u*_{0}, which satisfies the incompressible Euler equations, is linearized. We also assume it has constant density and entropy. The linearization of the steady flow yields
A1which corresponds to the sum of a convection part which is a constant vector, and of a shearing part with constant shear matrix *A*. It satisfies the incompressible Euler equations in steady regime:
A2The associated pressure is obtained by integration of (A2) that gives
A3This expression is in fact the Taylor expansion of *p*_{0} at order 2, and represents the generic local description of the pressure of a linearized incompressible flow.

The equations governing the evolution of the compressible linear perturbations are
A4aand
A4bwhere D_{t}=(∂/∂*t*+*u*_{0}⋅**∇**). We also consider the adiabatic ideal gas law which under the assumption of an homentropic flow reduces to
A5

The equations are then turned into a dimensionless form by introducing , , , and , where *l* is a characteristic length scale of the perturbations, and where is the adiabatic speed of sound at the origin. The dimensionless form of the components of the steady flow is and *B*=*A*/*ζ*, where *ζ* is the characteristic order of magnitude of the shear matrix (defined precisely in appendix B). Equations (A4) then become
A6aand
A6bwhere is the Mach number of the constant part of the steady flow, and *ε*=*ζl*/*c*_{0} is defined the same way as in equation (2.10) if we consider *l* to be 1/*k*_{0}. The work presented in this paper focuses on the local effect of the shearing of the steady flow. For this reason, we consider the effects of the constant part of the steady flow to be small, namely . This is in fact a small Mach number and small shearing assumption which is equivalent to . All quantities of order *ε*^{2} are now neglected. The third term in the left-hand side of (A6a), of order , is also neglected as being of order *ε*^{2}. Note also that writing the steady flow pressure (equation (A3)) in its dimensionless form yields
A7The two last terms of the right-hand side are also neglected as order *ε*^{2} quantities, and the pressure is eventually constant. Hence, *c*_{0} is the relevant speed of sound and is constant in space. Equation (A5) then reduces to , and the evolution of the perturbations is finally governed by
A8aand
A8b

The transformations introduced in §2 to derive the ODE (2.6), namely the convected coordinates (2.4) and the spatial Fourier harmonic form of the solution, can be reduced to the introduction of the following form for the solution
A9where ** k** is the same as defined in equation (2.7). The insertion of (A9) into (A8) gives
A10aand
A10bTherefore, and satisfy the same equation as

**and**

*U**P*(equations (2.6)). Eventually, the only effect due to the constant part of the steady flow on the work presented in this paper is to add the term in the phase of the perturbation — (kind of) Doppler effect.

## Appendix B. Flow decomposition

We develop in this appendix how general linear incompressible flows are decomposed. The derivation relies on a decomposition of the shear matrix into symmetric and antisymmetric parts (using the same idea as Craik & Criminale [2]). As any studied flow is written in the form *u*_{0}=*A*** x**, it is fully characterized by its shear matrix

*A*. Because of the incompressibility assumption, the shear matrix is traceless: B1The shear matrix is first decomposed into symmetric and antisymmetric parts B2which can be written as B3with B4and B5The first symmetric matrix in (A3) describes a hyperbolic flow, which has two degrees of freedom, a scalar factor

*ϱ*

_{H}that determines its intensity and an angle

*θ*. The second matrix corresponds to a solid rotation. It also has two degrees of freedom, the intensity

*ϱ*

_{R}and the parameter

*s*defining if the rotation is clockwise (

*s*=+1) or anticlockwise (

*s*=−1). We will show that we can choose arbitrarily both

*θ*=

*π*/2 and

*s*=1, without any loss of generality. This particular choice is explained below.

Let us first focus on the case *s*=+1. We introduce the change of basis where *R*_{ϕ} is the transformation matrix associated to a *ϕ* angle rotation:
B6Then, *A* becomes in this new basis
B7and by choosing *ϕ*=−*θ*/2+*π*/4, we obtain the desired form
B8In the case where *s*=−1, we introduce the change of basis , where *Δ*_{y} is the matrix associated to a symmetry with respect to the *y*-axis:
B9This time, the transformation becomes
B10and, if we take *ϕ*=−*θ*/2−*π*/4, we obtain the same decomposition as in equation (A8).

Now, by defining *ζ*=*ϱ*_{H}+*ϱ*_{R} and *δ*=*ϱ*_{H}/*ζ*, we have the decomposition
B11Eventually, we showed that for any flow matrix *A* we can find a basis in which this matrix has the form we wanted. In the model developed in §2, the side effect of such a change of basis is only to change the angle of the vector *k*_{0}. As a consequence, by applying the corresponding change of basis on *k*_{0}, we can focus our study only on flows defined by equation (A11).

Each flow can be viewed as a step in a continuum axis parametrized by *δ*∈[0 1]. The hyperbolic flow corresponds to a value of *δ*=1, and the rotation flows to *δ*=0. It is worth noticing that the plane Couette flow is the middle point for which contributions of hyperbolic and rotation flows are equal. The reason of our choice concerning *θ* and *s*, is to obtain this plane Couette flow oriented horizontally as it is usually presented.

## Appendix C. Computation of *X* and *k*

In this appendix is described how are computed ** X** and

**, defined in equations (2.4) and (2.7) respectively, which both rely on the exponential of the matrix**

*k**B*. To compute this exponential, we decompose

*B*using the Pauli matrices C1According to appendix A, we write the

*B*matrix in the form C2We define that can be imaginary. Then, using the well-known Pauli matrices properties C3 C4 C5we get the following relations C6which allows us to express the matrix exponential in the form C7In equation (2.7),

**is defined by C8therefore, using (A7), we finally obtain C9which gives explicit formulas to compute**

*k***,**

*k**α*,

*β*,

*k*and also, thanks to (A8), .

The same method is used to compute the transformation matrix between fixed coordinates ** x** and convected coordinates

**. It yields C10**

*X*- Received December 4, 2012.
- Accepted January 29, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.