The structural sensitivity of open shear flows calculated with a local stability analysis

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The structural sensitivity of open shear flows calculated with a local stability analysis

Matthew Juniper, Benoît Pier

To cite this version:

Matthew Juniper, Benoît Pier. The structural sensitivity of open shear flows calculated with a lo- cal stability analysis. European Journal of Mechanics - B/Fluids, Elsevier, 2015, 49, pp.426-437.

�10.1016/j.euromechflu.2014.05.011�. �hal-01084708�

Contents lists available atScienceDirect

European Journal of Mechanics B/Fluids

journal homepage:www.elsevier.com/locate/ejmflu

The structural sensitivity of open shear flows calculated with a local stability analysis

Matthew P. Juniper

^a,∗

, Benoit Pier

^b

aDepartment of Engineering, University of Cambridge, Trumpington Street, Cambridge, CB2 1PZ, UK

bLaboratoire de mécanique des fluides et d’acoustique, École centrale de Lyon - CNRS - Université Claude-Bernard Lyon 1 - INSA Lyon, 36 avenue Guy-de-Collongue, 69134, Écully, France

a r t i c l e i n f o

Article history:

Available online 5 June 2014

Keywords:

Stability

Structural sensitivity Adjoint

Global modes

a b s t r a c t

The structural sensitivity shows where an instability of a fluid flow is most sensitive to changes in internal feedback mechanisms. It is formed from the overlap of the flow’s direct and adjoint global modes.

These global modes are usually calculated with 2D or 3D global stability analyses, which can be very computationally expensive. For weakly non-parallel flows the direct global mode can also be calculated with a local stability analysis, which is orders of magnitude cheaper. In this theoretical paper we show that, if the direct global mode has been calculated with a local analysis, then the adjoint global mode follows at little extra cost. We also show that the maximum of the structural sensitivity is the location at which the localk⁺andk⁻branches have the same imaginary value. Finally, we use the local analysis to derive the structural sensitivity of two flows: a confined co-flow wake atRe=400, for which it works very well, and the flow behind a cylinder atRe=50, for which it works reasonably well. As expected, we find that the local analysis becomes less accurate when the flow becomes less parallel.

©2014 Elsevier Masson SAS. All rights reserved.

1. Introduction

Many open flows have a steady solution to the Navier–Stokes equations that becomes unstable above a critical Reynolds number.

Usually this instability is driven by one region of the flow, which is called the wavemaker region. The rest of the flow merely re- sponds to forcing from this region. The shape, linear growth rate, and frequency of the instability can be calculated by considering the evolution of small perturbations about the steady solution.

This is known as the direct global mode. The direct global mode emanates from the wavemaker region and grows spatially down- stream, reaching a maximum at the streamwise location where the spatial growth rate is zero. For example, in the case of the flow be- hind a cylinder, this direct global mode is a sinuous flapping mo- tion, whose nonlinear development is the familiar Kármán vortex street [1].

The receptivity of the direct global mode to harmonic open loop forcing is given by the last term in Eq. (9) of Ref. [2] and Eq. (7) of Ref. [3]. This term is proportional to the adjoint global mode, which is calculated in the same way as the direct global mode,

∗Corresponding author. Tel.: +44 1223 332 585.

E-mail address:[email protected](M.P. Juniper).

but from the adjoint (rather than direct) linearized Navier–Stokes equations. If the perturbation magnitude is measured by the per- turbation kinetic energy, which is the conventional approach, then there are only two significant differences between the direct and adjoint equations [2,4]. The first is the sign of the convection term, V_j

∂v

i

/∂

^xj, and is calledconvective non-normality. The second is the appearance of a transconjugate operator,

v

j

∂

^Vj

/∂

^xi, and is called component-typenon-normality, For the flows in this paper, the non-normality is almost entirely convective [4]. In a manner anal- ogous to the direct global mode, the adjoint global mode emanates from the wavemaker region but grows spatially upstream, reach- ing a maximum at the streamwise location where the adjoint spa- tial growth rate is zero, or when it meets the upstream boundary.

Physically, this reflects the fact that an open loop forcing signal will have most influence on the flow if it impinges on the wavemaker region, and if it is amplified by the flow before it does so.

The sensitivity of the direct global mode to changes in the lin- earized Navier–Stokes (LNS) equations is given by the penultimate term in Eq. (9) of Ref. [2]. This term is proportional to the over- lap between the direct and adjoint global modes and is known as the structural sensitivity. It is equivalent to the sensitivity of the direct global mode to closed-loop feedback between the pertur- bation and the governing equations in the special case where the sensor and actuator are co-located. For example, in the case of the

http://dx.doi.org/10.1016/j.euromechflu.2014.05.011 0997-7546/©2014 Elsevier Masson SAS. All rights reserved.

flow behind a cylinder, it can quantify the sensitivity of the flow to the presence of a small control cylinder that produces a small force on the flow in the opposite direction to the velocity pertur- bation [5,6]. Given that the direct global mode grows downstream of the wavemaker region and that the adjoint global mode grows upstream, the structural sensitivity is clearly maximal in the wave- maker region itself. Indeed, the wavemaker region is often defined as the position of maximum structural sensitivity, although alter- native definitions exist [4, Section 4.2.1]. Physically, this reflects the fact that, for a closed loop feedback mechanism to be effective, it requires firstly that the perturbation has significant amplitude at that point, which is quantified by the direct global mode, and sec- ondly that the flow has significant receptivity at that point, which is quantified by the adjoint global mode.

The above concepts were first introduced for the flow behind a cylinder atRe

=

50 by Hill [5] and Giannetti and Luchini [6,7] and have been extended to include the sensitivity to steady forcing and modifications to the base flow [8–10]. They have also been applied to recirculation bubbles [11] bluff bodies, both incompressible [12]

and compressible [13], backward-facing steps [14], forward-facing steps [15], confined wakes [16,17], and a recirculation bubble in a swirling flow [18].

The direct global mode is usually found with a global stabil- ity analysis. This typically proceeds in three steps: (i) the Navier–

Stokes (N–S) equations are linearized around a steady laminar flow, which is called the base flow and which is usually unstable;

(ii) the equations are discretized and expressed as a 2D or 3D ma- trix eigenvalue problem; (iii) the most unstable eigenmodes are calculated with an iterative technique, such as an Arnoldi algo- rithm or power iteration. Each eigenmode consists of a complex eigenvalue, which describes the frequency and growth rate, and an eigenfunction, which describes the 2D or 3D shape that grows on top of the base flow until nonlinear effects become significant.

As more elaborate configurations are examined, the number of de- grees of freedom rapidly approaches millions, so global stability analyses can be extremely computationally expensive [4].

If the base flow varies slowly in the streamwise direction then the global stability analysis can be replaced with a local stability analysis [19]. The WKBJ approximation reduces the LNS equations over the entire domain into a series of local LNS or Orr–Sommerfeld (O–S) equations at each streamwise location. Each local equation can be discretized and expressed as a small matrix eigenvalue problem, which represents the dispersion relation between the complex frequency,

ω

, and the complex wavenumber,k. At each streamwise location, the value of

ω

is found for which the disper- sion relation is satisfied and for which d

ω/

^dk

=

0. This is known as the absolute complex frequency,

ω

0and its imaginary part,

ω

0i, is the absolute growth rate. The flow is absolutely unstable in re- gions in which

ω

0i is positive. These regions exist in every flow that is globally unstable due to hydrodynamic feedback. The fre- quency and growth rate of the linear global mode can be derived from the streamwise distribution of

ω

0. This also gives a specific spatial position for the region of the flow that, in the context of the local analysis, is known as the wavemaker [20]. Local stability analyses are much quicker and require much less computer mem- ory than global stability analyses because they convert one large matrix eigenvalue problem into several small independent matrix eigenvalue problems. This is why they have been used so widely in the past and why they are still used for flows that are beyond the range of global analyses [21–23].

In all existing papers, the adjoint global mode is calculated with a global stability analysis. The purpose of this paper is to show that, if a local stability analysis is used to calculate the direct global mode, then the adjoint global mode follows at almost no extra cost.

This means that, for weakly nonparallel flows, adjoint global modes and structural sensitivities can be estimated quickly and cheaply,

without deriving the adjoint equations. After defining the form of the direct and adjoint equations in Section2, we derive this result rigorously in Section3for the Ginzburg–Landau equation (G–L), which is often used as a simple model for slowly-developing flows.

We then apply this to the linearized N–S equations in Section4and demonstrate this on two flows in Section5: a slowly-developing confined wake, and the flow behind a cylinder atRe

=

50.

2. General form of the direct and adjoint equations

Many different conventions are used to describe direct and ad- joint global modes. The convention used here is similar to that used for local stability analysis, so that it is easy to compare the local and global approaches. It differs from that used in Hill [5,24] and Gian- netti and Luchini [6] in three ways. The direct and adjoint govern- ing equations(1)and(2)have the same form so that theirk⁺and k⁻branches in the local analysis have the same physical meaning.

The adjoint variables are denoted with_Ď, rather than

+

or

∗

, so that they are not confused with thek⁺branch or with the complex conjugate. The inner product contains a complex conjugate so that the inner product of a complex state variable with itself is a real number.

The linearized governing equations are expressed in terms of the direct state variable,

ψ(

^x

,

^t

)

, the adjoint state variable,

ψ

^Ď

(

^x

,

^t

)

^, the direct linear spatial operator L, and the adjoint linear spatial operator L^Ď:

∂ψ

∂

^t

−

L

ψ =

0

,

⁽¹⁾

∂ψ

^Ď

∂

^t

−

L^Ď

ψ

^Ď

=

0

.

⁽²⁾

(The relationship between the direct and adjoint quantities will be specified in(8), after the inner product(7)has been defined.) So- lutions to the initial value problems defined by(1)and(2)can be expressed fort

∈ [

0

, ∞ )

as the sum of the direct and adjoint global modes:

ψ(

^x

,

^t

) =



m

ψ ˆ

m

(

^x

)

^exp

( −

i

ω

mt

),

⁽³⁾

ψ

^Ď

(

^x

,

^t

) =



n

ψ ˆ

n^Ď

(

^x

)

^exp

( −

i

ω

nt

).

⁽⁴⁾ Substituting(3)into(1)and(4)into(2)gives, for each mode,

−

i

ω

m

ψ ˆ

m

−

L

ψ ˆ

m

=

0

,

⁽⁵⁾

−

i

ω

n

ψ ˆ

n^Ď

−

L^Ď

ψ ˆ

n^Ď

=

0

.

⁽⁶⁾ An inner product between state variablesfandgis defined as

⟨

f

,

^g

⟩ ≡

 +∞

−∞

f^∗g dx

.

⁽⁷⁾

If boundary terms are assumed to be zero, as in Giannetti and Lu- chini [6], Hill [5], then the relationship between the direct operator, L, and its adjoint, L^Ď, is given by

⟨

L

ψ ˆ

m

, ψ ˆ

n^Ď

⟩ = ⟨ ˆ ψ

m

,

^LĎ

ψ ˆ

n^Ď

⟩ .

⁽⁸⁾ These definitions determine the relationship between

ω

mand

ω

n:

⟨

L

ψ ˆ

m

, ψ ˆ

n^Ď

⟩ = ⟨ ˆ ψ

m

,

^LĎ

ψ ˆ

n^Ď

⟩ ,

⁽⁹⁾

⟨−

i

ω

m

ψ ˆ

m

, ψ ˆ

n^Ď

⟩ = ⟨ ˆ ψ

m

, −

i

ω

n

ψ ˆ

n^Ď

⟩ ,

⁽¹⁰⁾ i

ω

m^∗

⟨ ˆ ψ

m

, ψ ˆ

n^Ď

⟩ = −

i

ω

n

⟨ ˆ ψ

m

, ψ ˆ

n^Ď

⟩ ,

⁽¹¹⁾

(ω

^∗m

+ ω

n

) ⟨ ˆ ψ

m

, ψ ˆ

n^Ď

⟩ =

0

.

⁽¹²⁾ This is the bi-orthogonality condition: every adjoint mode is or- thogonal to every direct mode, except for the pairs that satisfy

ω

n

= − ω

^∗m.

2.1. Structural sensitivity

We would like to find the change in the direct eigenvalue,

δω

m, when there is a small change,

δ

L, in the direct linear operator, L:

δω

m

=

^lim

ϵ →

0



ω

m

(

^L

+ ϵδ

^L

) − ω

m

(

^L

) ϵ



.

⁽¹³⁾

This perturbation causes perturbed eigenvalues,

ω

m

+ ϵδω

m, per- turbed direct eigenmodes,

ψ ˆ

m

+ ϵδ ψ ˆ

m, and perturbed adjoint eigenmodes,

ψ ˆ

n^Ď

+ ϵδ ψ ˆ

n^Ď. We premultiply(5)by

ψ ˆ

n^Ďand substitute in the perturbed variables:

⟨ ( ψ ˆ

n^Ď

+ ϵδ ψ ˆ

n^Ď

), (

ⁱ

ω

m

+

i

ϵδω

m

)( ψ ˆ

m

+ ϵδ ψ ˆ

m

) ⟩

+ ⟨ ( ψ ˆ

n^Ď

+ ϵδ ψ ˆ

n^Ď

), (

^L

+ ϵδ

^L

)( ψ ˆ

m

+ ϵδ ψ ˆ

m

) ⟩ =

0

.

⁽¹⁴⁾ Retaining terms at order

ϵ

and making use of(5),(6), and the bi- orthogonality condition(12)leads to

δω

m

=

i

⟨ ˆ ψ

m^Ď

, δ

^L

ψ ˆ

m

⟩

⟨ ˆ ψ

m^Ď

, ψ ˆ

m

⟩ .

⁽¹⁵⁾ This is the penultimate term in Eq. (9) of [2], but expressed in the notation of this paper. The operator

δ

L describes a generic pertur- bation to the operator, L. If one considers a perturbation that is localized in space then the structural sensitivity [6, Section 8] is defined as:

∇

_L

ω

m

≡

i

ψ ˆ

m^∗

ψ ˆ

m^Ď

⟨ ˆ ψ

m^Ď

, ψ ˆ

m

⟩ ,

⁽¹⁶⁾ where the numerator is a function ofxand the eigenfunctions are usually normalized such that the denominator is 1. This is shown graphically in [2, Fig 5 a,b].

3. Local analysis of the direct and adjoint Ginzburg–Landau equations

For the Ginzburg–Landau (G–L) equation, the operator L acting on

ψ(

^x

,

^t

)

^in(1)is:

∂ψ

∂

^t

=

L

ψ ≡

a₀

(

^x

)ψ +

a₁

(

^x

) ∂ψ

∂

^x

+

a₂

(

^x

) ∂

²

ψ

∂

^x2

,

⁽¹⁷⁾

wherea₀

,

^a1anda₂are complex coefficients that depend on the spatial coordinate,x. The aim of this section is to perform WKBJ analysis on the direct and adjoint G–L equations in order to de- termine

ω

^Ďnin terms of

ω

mandk^Ď_nin terms ofk_m, and to confirm that higher-order terms in the WKBJ analysis do not need to be considered. In this section, the subscriptsmandnwill be dropped because the adjoint mode constructed in Section3.6is always the bi-orthogonal counterpart of the direct mode constructed in Sec- tion3.5.

3.1. Local dispersion relation of the direct G–L equation

In slowly-evolving flows, the coefficientsa₀

,

^a1anda₂in(17) depend only on a slow spatial coordinate X

= ϵ

x. The small parameter

ϵ ≪

1 measures the ratio between typical instability and typical inhomogeneity length scales. Implementing a WKBJ analysis, a global-mode solution of(17)is sought in the form

ψ ∼

A

(

^X

)

^exp

i

ϵ

 X

k

(

^u

)

^du

−

i

ω

^t



,

⁽¹⁸⁾

where the local complex wavenumberk

(

^X

)

is a solution of the local dispersion relation:

ω =

_Ω

(

^k

,

^X

) ≡

ia₀

(

^X

) −

a₁

(

^X

)

^k

−

ia₂

(

^X

)

^k2

.

⁽¹⁹⁾

The dispersion relation can also be written in terms of the local absolute frequency,

ω

0

(

^X

)

, the local absolute wavenumber,k₀

(

^X

)

^, and the local curvature,

ω

kk

(

^X

)

^:

Ω

(

^k

,

^X

) = ω

0

(

^X

) +

¹ 2

ω

kk

(

^X

)



k

−

k₀

(

^X

)

2

,

⁽²⁰⁾

where

ω

0

=

ia₀

−

ia²₁

/

^4a2

,

^k0

=

ia₁

/

^2a2, and

ω

kk

= −

2ia₂. (Equiv- alently,a₀

= −

i

ω

0

−

i

ω

kkk²₀

/

²

,

^a1

= ω

kkk₀, anda₂

=

i

ω

kk

/

^{2.) This} shows how the coefficients of the G–L equation can be derived from the dispersion relation associated with a given weakly developing shear flow by taking a Taylor expansion around the saddle point, which is at

(ω

0

,

^k0

)

and by definition has d

ω/

^dk

=

0. Eq.(20)can be rearranged to givekas an explicit function of

ω

^:

k^±

(

^X

, ω) =

k₀

(

^X

) ±



2

ω − ω

0

(

^X

)

ω

kk

(

^X

) .

⁽²¹⁾

Here, branch cuts of(21)are taken along positive real values of the argument of the square root. This choice of branch cut ensures that, in stable or convectively unstable regions of the complexX-plane, the above definition coincides with the usual labelling of spatial branches based on causality considerations, for which ak⁺-branch corresponds to a downstream response to localized harmonic forcing, and ak⁻-branch corresponds to an upstream response.

3.2. Calculation of the adjoint of the G–L equation

For the G–L equation, the adjoint operator, L^Ď, is found by ex- panding

⟨ ˆ ψ

^Ď

,

^L

ψ ˆ ⟩

, using(7), and then integrating by parts:

⟨ ˆ ψ

^Ď

,

^L

ψ ˆ ⟩ =

 +∞

−∞

ψ ˆ

^Ď∗



a₀

ψ ˆ +

a₁

∂ ψ ˆ

∂

^x

+

a₂

∂

²

ψ ˆ

∂

^x2



dx (22)

=

 +∞

−∞



a₀

ψ ˆ

^Ď∗

− ∂

∂

^x

(

^a1

ψ ˆ

^Ď∗

) + ∂

²

∂

^x2

(

^a2

ψ ˆ

^Ď∗

)



× ˆ ψ

^dx

,

⁽²³⁾

in which the boundary terms have been set to zero with appropri- ate boundary conditions. The adjoint operator is found by noting that, from(8),

⟨ ˆ ψ

^Ď

,

^L

ψ ˆ ⟩ = ⟨

L^Ď

ψ ˆ

^Ď

, ψ ˆ ⟩

, and therefore that

L^Ď

ψ ˆ

^Ď

=

a^∗₀

ψ ˆ

^Ď

− ∂

∂

^x

(

^a∗1

ψ ˆ

^Ď

) + ∂

²

∂

^x2

(

^a∗2

ψ ˆ

^Ď

)

⁽²⁴⁾

=

a^Ď₀

ψ ˆ

^Ď

+

a^Ď₁

∂ ψ ˆ

^Ď

∂

^x

+

a^Ď₂

∂

²

ψ ˆ

^Ď

∂

^x2

,

⁽²⁵⁾

wherea^Ď₀

≡

_a^∗₀

− ∂

^a∗1

/∂

^x

+ ∂

^2a∗2

/∂

^x2

,

^aĎ1

≡ −

_a^∗₁

+

₂

∂

^a∗2

/∂

^{x, and} a^Ď₂

≡

a^∗₂. These expressions are general and do not necessarily as- sume weak spatial inhomogeneities.

3.3. Local dispersion relation of the adjoint problem

Under the quasi-parallel-flow assumption, the coefficients of the direct G–L equations depend only on the slow spatial coordi- nateX

= ϵ

^{x. Eq.(24)becomes:}

L^Ď

ψ ˆ

^Ď

=

a^∗₀

ψ ˆ

^Ď

−

a^∗₁

∂ ψ ˆ

^Ď

∂

^x

− ˆ ψ

^Ď

ϵ ∂

^a∗1

∂

^X

+

a^∗₂

∂

²

ψ ˆ

^Ď

∂

^x2

+

2

ϵ ∂

^a∗2

∂

^X

∂ ψ ˆ

^Ď

∂

^x

+ ˆ ψ

^Ď

ϵ

²

∂

^2a∗2

∂

^X2

.

⁽²⁶⁾

When performing a WKBJ analysis of the adjoint G–L equation(2), the adjoint operator(24)must be expanded in powers of

ϵ

^as L^Ď



∂

∂

^X

;

X



=

L^Ď₀



∂

∂

^X

;

X



+ ϵ

^LĎ1



∂

∂

^X

;

X



+ ϵ

^2LĎ2



∂

∂

^X

;

X



,

⁽²⁷⁾

By inspection of(26), L^Ď₀

,

^LĎ1, and L^Ď₂are:

L^Ď₀



∂

∂

^X

;

X



=

a^∗₀

(

^X

) −

a^∗₁

(

^X

) ∂

∂

^x

+

a^∗₂

(

^X

) ∂

²

∂

^x2

,

⁽²⁸⁾

L^Ď₁



∂

∂

^X

;

X



= − ∂

^a∗1

(

^X

)

∂

^X

+

2

∂

^a∗2

(

^X

)

∂

^X

∂

∂

^x

,

⁽²⁹⁾

L^Ď₂



∂

∂

^X

;

X



= ∂

^2a∗2

(

^X

)

∂

^X2

.

⁽³⁰⁾

A solution of the adjoint problem is then sought in the form

ψ

^Ď

∼



A^Ď₀

(

^X

) + ϵ

^AĎ1

(

^X

) + ϵ

^2AĎ2

(

^X

) + · · ·



×

exp

i

ϵ

 X

k

(

^u

)

^du

−

i

ω

^Ďt



.

⁽³¹⁾

Substituting (28)–(31) into the governing adjoint equation (2) gives, at leading-order,

−

i

ω

^Ď

=

a^∗₀

(

^X

) −

a^∗₁

(

^X

)

^ik

(

^X

) −

a^∗₂

(

^X

)

^k2

(

^X

) =

L^Ď₀

(

^ik

;

X

).

⁽³²⁾ In a manner similar to the direct problem, the adjoint dispersion relation can be rewritten as

ω

^Ď

=

_Ω0^Ď

(

^k

,

^X

) ≡

ia^∗₀

(

^X

) +

a^∗₁

(

^X

)

^k

(

^X

) −

ia^∗₂

(

^X

)

^k2

(

^X

)

⁽³³⁾

= ω

^Ď0

(

^X

) +

¹ 2

ω

kk^Ď

(

^X

)



k

−

k^Ď₀

(

^X

)

2

,

⁽³⁴⁾

where

ω

^Ď0

(

^X

) =

ia^∗₀

(

^X

) −

ⁱ

4a^∗₁²

(

^X

)/

^a∗2

(

^X

) = − ω

^∗0

(

^X

),

⁽³⁵⁾ k^Ď₀

(

^X

) = −

ⁱ

2a^∗₁

(

^X

)/

^a∗2

(

^X

) =

k^∗₀

(

^X

),

⁽³⁶⁾

ω

^Ďkk

(

^X

) = −

2ia^∗₂

(

^X

) = − ω

^∗kk

(

^X

).

⁽³⁷⁾ The higher-order terms L^Ď₁ and L^Ď₂ do not appear in this adjoint dispersion relation, because it is obtained at leading order in the WKBJ analysis. The L^Ď₁component enters only when working out, atO

(ϵ

¹

)

, the solvability condition that governs the leading-order amplitude termA^Ď₀

(

^X

)

ⁱⁿ(31). This amplitude equation is

Ω₀^Ď_,k

k

(

^X

),

^XdA^Ď₀ dX

+

¹

2Ω₀^Ď_,kk

k

(

^X

),

^Xdk dXA^Ď₀

(

^X

) +

iΩ1^Ď

k

(

^X

),

^X

A^Ď₀

(

^X

) =

0

,

⁽³⁸⁾

whereΩ1^Ď

(

^k

,

^X

) ≡

iL^Ď₁

(

^ik

,

^X

)

. Higher-order expansions will not be derived further, however, because the results of this paper re- quire only the local dispersion relations. Turning points, where

∂

Ω₀^Ď

/∂

^k

=

0, are not affected by the higher-order expansions.

The key point of this section is that, at leading order, the disper- sion relation of the adjoint G–L equation is the same as that of the direct G–L equation but with the substitutions(35)–(37).

3.4. Adjoint of a generic polynomial PDE

The development in Sections3.1–3.3is for a parabolic PDE but holds for any polynomial PDE in one spatial dimension, as shown in this section. For a generic polynomial PDE, the direct operator (17)can be written as

∂ψ

∂

^t

=

L

ψ ≡



j

a_j

(

^x

) ∂

^j

ψ

∂

^xj

,

⁽³⁹⁾

and, after integration by parts, the adjoint operator can be written as

∂ψ

^Ď

∂

^t

=

L^Ď

ψ

^Ď

≡



j

( −

1

)

^j

∂

^j

∂

^xj

a^∗_j

(

^x

)ψ

^Ď

.

⁽⁴⁰⁾

If the coefficientsa^∗_j do not depend onxthen L^Ď

ψ

^Ď

=



j

( −

1

)

^ja∗j

∂

x^j

ψ

^{Ďand(35)–(37)}follow immediately. If the coefficientsa^∗_j de- pend onx, then thex-derivatives ofa^∗_j

(

^x

)ψ

^Ďproduce extra terms:

L^Ď

ψ

^Ď

=



a^∗₀

(

^x

) − ∂

^a∗1

(

^x

)

∂

^x

+ ∂

^2a∗2

(

^x

)

∂

^x2

− ∂

^3a∗3

(

^x

)

∂

^x3

+ · · ·



ψ

^{Ď (41)}

+



−

a^∗₁

(

^x

) +

2

∂

^a∗2

(

^x

)

∂

^x

−

3

∂

^2a∗3

(

^x

)

∂

^x2

+ · · ·



∂ψ

^Ď

∂

^{x (42)}

+



a^∗₂

(

^x

) −

3

∂

^a∗3

(

^x

)

∂

^x

+ · · ·



∂

²

ψ

^Ď

∂

^{x2 (43)}

+



−

a^∗₃

(

^x

) + · · ·



∂

³

ψ

^Ď

∂

^x3

+ · · ·

(44) However, under the assumption of slow spatial development, the n^thderivatives of the coefficientsa_jare of order

ϵ

ⁿ, so the local dis- persion relation that is obtained at leading order is the same as that obtained for constant coefficients. This proves that the rela- tions

ω

0^Ď

(

^X

) = − ω

^⋆0

(

^X

),

^kĎ0

(

^X

) =

k^⋆₀

(

^X

)

^and

ω

^Ďkk

(

^X

) = − ω

^⋆kk

(

^X

)

in(35)–(37)hold for systems governed by any dispersion relation that is polynomial ink. We therefore expect this result to remain generally valid in the case of dispersion relations that are analytic inkover large parts of the complexk-plane. We assume that dis- persion relations derived from the linearized Navier–Stokes equa- tions in slowly-varying flows fall into this category.

3.5. Global mode of the direct G–L equation with a local analysis A linear global mode is a global solution of the governing equa- tion (1) with the form

ψ(

^x

,

^t

) ∼

exp

( −

i

ω

gt

)

for a complex global frequency

ω

g. Assuming that the slowly-varying coefficients

ω

0

(

^X

),

^k0

(

^X

)

^and

ω

kk

(

^X

)

are known along the realX-axis, a WKBJ approximation of the global mode can be sought as in(18)with

ω = ω

g. This integral is most easily evaluated in the complex X-plane, as shown in the top half ofFig. 1a. (The bottom half is for the adjoint mode.) The point X_s is a saddle point of

ω

0

(

^X

)

and the diagonal lines have the same value of

ω

0i as the saddle point. Huerre and Monkewitz [19] have shown that the frequency of the dominant global mode,

ω

g, is equal to

ω

s

+

O

(ϵ)

^{, where}

ω

s

= ω

0

(

^Xs

)

^.

At a given

ω

g, there are two valid solutions tok, known as thek⁺ andk⁻spatial branches, and there are therefore two independent WKBJ approximations

ψ

⁺

∼

A⁺

(

^X

)

^exp

i

ϵ

 X Xs

k⁺

(

^u

, ω

g

)

^du

−

i

ω

gt



(45) and

ψ

⁻

∼

A⁻

(

^X

)

^exp

i

ϵ

 X Xs

k⁻

(

^u

, ω

g

)

^du

−

i

ω

gt



.

⁽⁴⁶⁾

These two WKBJ approximations are singular at the saddle point X_s, which is a double turning point of the dispersion relation. From this double turning point, four Stokes lines emerge, defined by Im

 X Xs

[

k⁺

(

^u

, ω

s

) −

k⁻

(

^u

, ω

s

) ]

du

=

0

.

⁽⁴⁷⁾ Along these Stokes lines both WKBJ approximations remain of the same order of magnitude, while inside the sectors delimited by

Fig. 1. (a) Left frame: integration paths in the complexX-plane for the direct (top) and adjoint (bottom) cases. The diagonal lines represent the boundaries between valleys (left and right of the lines) and hills (above and below the lines). The integration paths must pass through the valleys in order to obey causality. (b) Right frame: branch cuts (dashed lines) in the complexX-plane.

the Stokes lines one approximation is exponentially larger than the other. Following classical WKBJ theory [25] the global mode must be sought as a linear combination of the two independent solutions,

ψ =

C⁺

ψ

⁺

+

C⁻

ψ

⁻, within each sector delimited by these Stokes lines.

WhenX

→ +∞

, the solution must be dominated by ak⁺ branch and is therefore made up of the subdominant

ψ

^+approxi- mation. The global mode is therefore of the form

ψ =

C⁺

ψ

^+(with C⁻

=

0) in the region starting from the Stokes lines issuing fromX_s and extending toX

= +∞

. (See Pier [26] for a detailed analysis of a similar case.) For similar reasons, the global mode is of the form

ψ =

C⁻

ψ

^−(withC+

=

0) in the region starting from the Stokes lines issuing fromX_sand extending toX

= −∞

.

Consequently, the global mode is approximated by the WKBJ approximationC⁻

ψ

⁻along the semi-infinite path fromX_sto

−∞

, and byC⁺

ψ

⁺along the semi-infinite path fromX_sto

+∞

. Since the global mode must be continuous atX_s, the coefficientC⁺on the path fromX_sto

+∞

must equalC⁻on the path fromX_sto

−∞

(this includes higher-order terms; asymptotic matching of the two WKBJ-expansions prevailing on each side of the saddle point can be rigorously carried out via an inner layer). After rescaling the solu- tion so thatC⁺

=

C⁻

=

1, the direct global mode is approximated by

ψ

⁺along the semi-infinite path fromX_stoX

= +∞

and by

ψ

⁻ along the semi-infinite path fromX_stoX

= −∞

.

Finally, the approximations of the direct global mode obtained along the path from X

= −∞

toX

= +∞

, passing through the saddle pointX_s, must be continued onto the realX-axis. When crossing a Stokes line, a subdominant WKBJ solution becomes dominant but remains a valid asymptotic approximation. There- fore the global mode is approximated by

ψ

⁻in the sectors adja- cent to the sector extending toX

= −∞

, and by

ψ

⁺in the sectors adjacent to the sector extending toX

= +∞

. Since there are four Stokes lines and two branch cuts emanating from the saddle point, X_s, one may safely assume that one branch cut crosses the real axis atX_cand that no more than one Stokes line crosses the real axis on either side ofX_c. It follows that the global mode is approximated by

ψ

^−forX

<

^Xcalong the real axis and by

ψ

^+forX

>

^Xc. At X_cthere is a smooth relabelling of thek-branches, but otherwise nothing special happens across the branch cut. This division of the integration path becomes important in Section3.7.

3.6. Global mode of the adjoint G–L equation with a local analysis Following the same development as Section3.5, the adjoint global mode is sought as

ψ

^Ď

∼

A^Ď

(

^X

)

^exp

i

ϵ

 X

k^Ď

(

^u

; ω

^Ďg

)

^du

−

i

ω

^Ďgt



.

⁽⁴⁸⁾

We again assume that the coefficients

ω

0

(

^X

),

^k0

(

^X

)

^and

ω

kk

(

^X

)

can be continued analytically into the complex plane and use the relationships(35)–(37). This is represented in the bottom half of Fig. 1(a). We obtain the result that

ω

^Ďg

= ω

^Ďs

+

O

(ϵ)

^{, where}

ω

s^Ď

=

ω

^Ď0

(

^Xs^Ď

)

^{with d}

ω

^Ď0

/

^dX

|

Xs^Ď

=

0. The local wavenumber in(48)follows thek^Ď−branch forX

→ −∞

and thek^Ď+branch forX

→ +∞

. Here, these branches are obtained from the local adjoint dispersion relation(36)as

k^ϱ

(

^X

, ω

^Ďg

) =

k^Ď₀

(

^X

) ±



2

ω

^Ďg

− ω

^Ď0

(

^X

)

ω

^Ďkk

(

^X

) .

⁽⁴⁹⁾

For real values ofX, substituting(35)–(37)into(49)leads to the following relationship between the local branches of the adjoint and the direct global modes:

k^ϱ

(

^X

; ω

g^Ď

) =



k^∓

(

^X

; ω

g

)

∗

.

⁽⁵⁰⁾

This relationship guarantees that a branch cut of the adjointk^ϱ crosses the realX-axis at the same location,X_c, as a branch cut of the directk^±.

The two adjoint spatial branchesk^ϱlead to two independent WKBJ approximations

ψ

^Ď+

∼

A^Ď+

(

^X

)

^exp

i

ϵ

 X Xs^Ď

k^Ď+

(

^u

, ω

g^Ď

)

^du

−

i

ω

^Ďgt



(51) and

ψ

^Ď−

∼

A^Ď−

(

^X

)

^exp

i

ϵ

 X Xs^Ď

k^Ď−

(

^u

, ω

g^Ď

)

^du

−

i

ω

^Ďgt



.

⁽⁵²⁾

Following similar arguments to those in the previous section, it can be shown that the adjoint global mode is approximated along the real axis by

ψ

^Ď−forX

<

^Xcand by

ψ

^Ď+forX

>

^Xc. ForX

<

^Xc, the direct global mode followsk⁻and the adjoint global mode follows k^Ď−, which is

(

^k+

)

^{∗. ForX}

>

^Xc, the direct global mode follows k⁺and the adjoint global mode followsk^Ď+, which is

(

^k−

)

^{∗. AtX}c, there is a smooth re-labelling of thek-branches.

The final result, that the adjoint mode follows

(

^k+

)

^∗upstream of the wavemaker and

(

^k−

)

^∗downstream, is simple and may seem trivial. However, we are not aware of this result being stated or used before in stability analysis, despite its potential usefulness.

3.7. Calculating the structural sensitivity of the G–L equation with a local analysis

The structural sensitivity(16)is the product of the direct and adjoint global modes. ForXalong the real axis, the direct global mode found from the local analysis takes the form

ψ ∼











A⁺

(

^X

)

^exp

i

ϵ

 X Xs

k⁺du



forX

>

^Xc

,

A⁻

(

^X

)

^exp

i

ϵ

 X Xs

k⁻du



forX

<

^Xc

.

(53)

After splitting the integrals fromX_stoXinto two integrals fromX_s toX_cand fromX_ctoX, and using the fact that thek⁺-branch on the