1 Introduction
In the framework of linear stability analysis, open flows showing self-sustained oscillations, such as spiral vortex breakdown (Ruith et al.
Reference Ruith, Chen, Meiburg and Maxworthy2003; Qadri, Mistry & Juniper Reference Qadri, Mistry and Juniper2013), are globally unstable and are called oscillators. In contrast, stable flows which exhibit a strong response to external disturbances are named amplifiers. Trailing vortices generated by a lifting wing are a classical example of amplifier flows in aeronautics (Spalart Reference Spalart1998) and are commonly modelled by using the self-similar solution proposed by Batchelor (Reference Batchelor1964). In its parallel approximation, the stability properties of the Batchelor vortex have been well characterized in the literature. For instance, Lessen, Singh & Paillet (Reference Lessen, Singh and Paillet1974) and Mayer & Powell (Reference Mayer and Powell1992) found that inviscid helical modes are temporally unstable and higher-wavenumber modes are destabilized when the swirl increases (Duck & Foster Reference Duck and Foster1980). These helical modes reach their maximum growth rate at a certain swirl number (Leibovich & Stewartson Reference Leibovich and Stewartson1983) before becoming stable when the azimuthal velocity of the vortex further increases. The mechanism underlying this destabilization is a generalized centrifugal instability unravelled by Ludwieg (Reference Ludwieg1962), Leibovich & Stewartson (Reference Leibovich and Stewartson1983) and Eckhoff (Reference Eckhoff1984). On the other hand, viscous modes have been investigated numerically and asymptotically by Khorrami (Reference Khorrami1991), Fabre & Jacquin (Reference Fabre and Jacquin2004), Fabre, Sipp & Jacquin (Reference Fabre, Sipp and Jacquin2006) and Heaton (Reference Heaton2007), whereas Delbende, Chomaz & Huerre (Reference Delbende, Chomaz and Huerre1998), Olendraru et al. (Reference Olendraru, Sellier, Rossi and Huerre1999) and Olendraru & Sellier (Reference Olendraru and Sellier2002) carried out a spatio-temporal analysis showing that, in the case of strong advection and moderate wake deficit (less than approximately
$80\,\%$
of the external flow), the flow is convectively unstable. More recently, Heaton, Nichols & Schmid (Reference Heaton, Nichols and Schmid2009) carried out a global stability analysis of a non-parallel Batchelor vortex with an inflow wake deficit of
$90\,\%$
that is locally absolutely unstable close to the inlet and globally unstable. In the framework of amplifier flows, Viola, Arratia & Gallaire (Reference Viola, Arratia and Gallaire2016) investigated the response to harmonic inlet and body forcing of a globally stable non-parallel vortex by using a local WKB (Wentzel–Kramers–Brillouin) approach and a global resolvent analysis.
These studies suggest that the stability properties of swirling flows depend in a complex way on multiple factors, such as the advection rate, the swirl number, the presence of localized forces, turbulence level, fluid properties, domain geometry and boundary conditions. Thus, a modification of one of these factors yields a variation of the linearized operators of the stability analysis, which turns into a modification of the eigenvalues in the stability spectrum. The dependence of the stability properties on one or more of these factors is commonly called sensitivity. Chomaz (Reference Chomaz2005) explained that the sensitivity may become large when the linear stability operator is non-normal, which is usually the case in hydrodynamics. For this reason, several studies have been devoted over the years to sensitivity analyses. Giannetti & Luchini (Reference Giannetti and Luchini2007) defined the so-called structural sensitivity, which is the sensitivity to a spatially localized feedback, and showed that the combined analysis of the global mode and its adjoint mode allows the identification of the wavemaker of the stability, i.e. the core region where the self-excited instability mechanism takes place. Bottaro, Corbett & Luchini (Reference Bottaro, Corbett and Luchini2003) were the first to study the sensitivity to a base-flow modification, and they identified the regions of the parallel Couette flow where a base-flow modification has the most significative effect on temporal stability. Such base-flow variations can be the result of a passive control acting on the flow. Using a variational technique, Marquet, Sipp & Jacquin (Reference Marquet, Sipp and Jacquin2008) derived a general expression of the base-flow sensitivity valid for oscillator flows.
In the case of amplifiers, the flow response to an external forcing is given by the resolvent analysis (Chomaz Reference Chomaz2005), and its sensitivity with respect to a base-flow modification was derived analytically by Brandt et al. (Reference Brandt, Sipp, Pralits and Marquet2011). They focused on the variation of the resolvent norm in the case of the Blasius boundary layer subjected to a harmonic body force where both componentwise and convective non-normalities are present. This analysis was applied by Boujo & Gallaire (Reference Boujo and Gallaire2015) to the backward-facing step flow and extended to the case of inlet forcing and time-stochastic perturbation. However, when the base flow evolves slowly in the advection direction, these global approaches do not exploit the physical amplification mechanism, which is the convective growth of perturbation that can be related to the local stability properties of the flow. Indeed, by using both a local WKB asymptotic analysis (Crighton & Gaster Reference Crighton and Gaster1976) and a global resolvent approach (Åkervik et al. Reference Åkervik, Ehrenstein, Gallaire and Henningson2008) for a non-parallel Batchelor vortex, Viola et al. (Reference Viola, Arratia and Gallaire2016) showed that the linear global amplification of perturbations in amplifier flows is well captured by the spatial stability branches. This raises the question whether it is possible to determine a systematic way to control the spatial amplification of a given mode, i.e. its global amplification, resulting from the integration of the local spatial growth along the streamwise direction.
The objective of this paper is to provide a general framework so as to solve the above-mentioned control problem. The method proposed here generalizes to the case of weakly non-parallel amplifier flows the adjoint-based strategies usually employed for controlling oscillators (see Camarri (Reference Camarri2015) for a review). Noise amplifiers comprise many fundamental flows, such as boundary layers, convectively unstable wakes and trailing vortices, and thus the proposed method is general and has many direct engineering applications. In this paper we present only a particular prototypical application, which consists in controlling the spatial instability of the non-parallel Batchelor vortex using a perturbation of the velocity profile generating the vortex.
The particular selected test case, which is aimed at showing an example of flow control, is loosely applicable to disturbance suppression in trailing vortices (Spalart Reference Spalart1998) and in wind turbine wakes (Iungo et al. Reference Iungo, Viola, Camarri and Gallaire2013). To this end, we need first to address in § 2 the sensitivity of the local spatial stability problem in the parallel-flow framework. The non-parallel Batchelor vortex flow and its local stability properties are presented in § 3. Successively, on the basis of the previous result, the sensitivity of the integral amplification factor of a single helical disturbance to a modification of the inflow vortex velocity is derived in § 4. Furthermore, a strategy to design the passive inlet control to stabilize all helical modes amplified over the vortex flow is proposed in § 5. Lastly, conclusions and possible applications are discussed in § 6.
2 Sensitivity of the local spatial stability spectrum
2.1 Parallel flow and spatial stability analysis
Figure 1. (a) Velocity components of the parallel Batchelor vortex with wake and swirl parameters equal to
$\unicode[STIX]{x1D6FC}=0.667$
and
$S=0.333$
, respectively. (b) Temporal growth rate maximized over the streamwise wavenumber,
$\max _{k}(\unicode[STIX]{x1D714}_{i})$
, for increasing swirl number at
$Re=100$
,
$\unicode[STIX]{x1D6FC}=0.667$
and azimuthal modes
$m=1,2,3$
. The vertical dashed line depicts the swirl number used in the paper (
$S=0.333$
). (c) Local spatial stability properties of the flow, where the spatial growth rate,
$-k_{i}$
, of the unstable helical perturbations
$m\in \{-1,1,2,3\}$
is reported as a function of the frequency
$\unicode[STIX]{x1D714}$
. The red circle refers to the maximum
$-k_{i}$
, which corresponds to
$m=1$
and
$\unicode[STIX]{x1D714}=0.6$
.
Although the formalism used here applies to all kinds of convectively unstable shear flows, we consider now a Batchelor vortex profile with dimensionless velocity components
$$\begin{eqnarray}\displaystyle U_{r}=0,\quad U_{\unicode[STIX]{x1D703}}=S(1-\text{e}^{-r^{2}})/r,\quad U_{x}=1-\unicode[STIX]{x1D6FC}\text{e}^{-r^{2}}, & & \displaystyle\end{eqnarray}$$
which are shown in figure 1(a). Here,
$r$
,
$\unicode[STIX]{x1D703}$
and
$x$
are the cylindrical coordinates,
$\unicode[STIX]{x1D6FC}$
and
$S$
are the wake and swirl parameters, and
$Re$
designates the Reynolds number, which is based on the vortex core radius and on the free-stream axial velocity. The behaviour of infinitesimal three-dimensional disturbances,
$\boldsymbol{q}(x,r,\unicode[STIX]{x1D703};t)=(\boldsymbol{u},p)$
, superimposed on top of the base flow is described by the linearized Navier–Stokes equations
$$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{u}}{\unicode[STIX]{x2202}t}+\boldsymbol{U}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{U}=-\unicode[STIX]{x1D735}p+\frac{1}{Re}\unicode[STIX]{x0394}\boldsymbol{u},\quad \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0, & & \displaystyle\end{eqnarray}$$
where
$\boldsymbol{u}$
and
$p$
are the disturbance velocity and pressure. In the case of parallel axisymmetric base flow, these perturbations can be decomposed in the standard form
$$\begin{eqnarray}\displaystyle \boldsymbol{q}(x,r,\unicode[STIX]{x1D703};t)=\hat{\boldsymbol{q}}(r)\text{e}^{\text{i}(kx+m\unicode[STIX]{x1D703}-\unicode[STIX]{x1D714}t)}, & & \displaystyle\end{eqnarray}$$
where
$m$
is the azimuthal wavenumber and
$\hat{\boldsymbol{q}}(r)=(\hat{\boldsymbol{u}},\hat{p})$
.
The wake parameter and Reynolds number are here set to
$\unicode[STIX]{x1D6FC}=0.667$
and
$Re=100$
respectively, and the swirl parameter
$S$
is chosen in such a way as to ensure the simultaneous amplification of different helical modes in the swirling wake. By recalling that in the temporal analysis framework
$k$
is real and
$\unicode[STIX]{x1D714}$
is taken complex, figure 1(b) displays the temporal growth rate,
$\unicode[STIX]{x1D714}_{i}$
, maximized over the streamwise wavenumber,
$k$
, for
$m=1,2,3$
and increasing swirl parameter. It shows strong competition between the first and second helical modes at
$S=0.333$
, which is further considered in this study. Such a competition is representative of several swirling jet/wake experiments (Gallaire & Chomaz Reference Gallaire and Chomaz2003; Iungo et al.
Reference Iungo, Viola, Camarri and Gallaire2013). The considered test-case vortex was shown to be convectively unstable by Viola et al. (Reference Viola, Arratia and Gallaire2016), calling for a spatial stability analysis, where the frequency
$\unicode[STIX]{x1D714}$
is real and wavenumber
$k$
complex. The real part of
$k$
,
$k_{r}$
, corresponds to the streamwise wavenumber of the travelling perturbation and the imaginary part of
$k$
, in particular
$-k_{i}$
, is the spatial amplification rate. The equations governing the linearized dynamics of
$\hat{\boldsymbol{q}}$
, obtained by substituting the normal-mode expansion (2.3) in (2.2), are reported in appendix A. They can be recast in a quadratic eigenvalue problem of the type
$$\begin{eqnarray}\displaystyle k^{2}{\mathcal{C}}_{2}(\boldsymbol{U},\unicode[STIX]{x1D714})\hat{\boldsymbol{q}}+k{\mathcal{C}}_{1}(\boldsymbol{U},\unicode[STIX]{x1D714})\hat{\boldsymbol{q}}+{\mathcal{C}}_{0}(\boldsymbol{U},\unicode[STIX]{x1D714})\hat{\boldsymbol{q}}=0, & & \displaystyle\end{eqnarray}$$
where the linear operators
${\mathcal{C}}_{0},{\mathcal{C}}_{1},{\mathcal{C}}_{2}$
, together with boundary conditions are also reported in appendix A. The discretization is ensured through a Chebyshev spectral collocation method including an algebraic mapping of the domain, as detailed in Viola et al. (Reference Viola, Iungo, Camarri, Porté-Agel and Gallaire2014), where the influence of the radial extension of the domain is discussed in appendix B. In figure 1(c) the spatial growth rates are shown as a function of the frequency
$\unicode[STIX]{x1D714}$
, and the integer numbers at the curve peaks depict their azimuthal wavenumber mode,
$m$
. In particular, four helical modes are seen to be amplified in space, and
$m=1$
at a frequency approximately equal to
$0.6$
is the dominant mode. For this particular case
$(m=1,~\unicode[STIX]{x1D714}=0.6)$
, the eigenvalue spectrum that is obtained by solving (2.4) numerically is reported in figure 2(a): unstable perturbations propagating downstream are characterized by a positive phase velocity,
$\unicode[STIX]{x1D714}/k_{r}$
, and a negative growth rate,
$k_{i}$
. Thus, the eigenvalues of interest are located in the fourth quadrant, i.e.
$k_{r}>0\text{ and }k_{i}<0$
. The red circle depicts the non-spurious eigenvalue associated with the largest spatial growth rate and whose imaginary part corresponds to the red circle in figure 1(c). This eigenvalue was distinguished among many spurious eigenvalues (outside of the figure’s visualization axis) by using the Gaster transformation of the temporal analysis, which has been used to obtain a target for the complex
$k^{+}$
spatial branches as explained in more detail in Iungo et al. (Reference Iungo, Viola, Camarri and Gallaire2013).
Figure 2. (a) The spatial stability spectrum (circles) of the parallel Batchelor vortex with
$\unicode[STIX]{x1D6FC}=0.667$
,
$S=0.333$
,
$Re=100$
,
$m=1$
and
$\unicode[STIX]{x1D714}=0.6$
. The red circle indicates the most unstable physically meaningful eigenvalue. The blue asterisks mark the variation of the dominant eigenvalue (red circle) due to a modification of the base-flow velocity components (see text). The red dashed line in (a) is the linear prediction of the eigenvalue variation based on the sensitivity map, whose components are reported in (b).
2.2 Generic sensitivity analysis of the local stability problem
The sensitivity of a given eigenvalue
$k$
to generic perturbations of the linear operators in (2.4) is here derived by a standard expansion procedure. A small variation of the linear operators
${\mathcal{C}}_{0}^{\prime }={\mathcal{C}}_{0}+\unicode[STIX]{x1D6FF}{\mathcal{C}}_{0},~{\mathcal{C}}_{1}^{\prime }={\mathcal{C}}_{1}+\unicode[STIX]{x1D6FF}{\mathcal{C}}_{1},~{\mathcal{C}}_{2}^{\prime }={\mathcal{C}}_{2}+\unicode[STIX]{x1D6FF}{\mathcal{C}}_{2}$
yields a small variation of the eigenvector
$\hat{\boldsymbol{q}}^{\prime }\sim \hat{\boldsymbol{q}}+\unicode[STIX]{x1D6FF}\hat{\boldsymbol{q}}$
together with a small variation in the eigenvalue
$k^{\prime }\sim k+\unicode[STIX]{x1D6FF}k$
. At first order we have
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}k=-\frac{\langle \boldsymbol{q}^{\dagger },[\unicode[STIX]{x1D6FF}{\mathcal{C}}_{0}+k\unicode[STIX]{x1D6FF}{\mathcal{C}}_{1}+k^{2}\unicode[STIX]{x1D6FF}{\mathcal{C}}_{2}]\hat{\boldsymbol{q}}\rangle _{p}}{\langle \boldsymbol{q}^{\dagger },[{\mathcal{C}}_{1}+2k{\mathcal{C}}_{2}]\hat{\boldsymbol{q}}\rangle _{p}}, & & \displaystyle\end{eqnarray}$$
where
$\langle \cdot \rangle _{p}$
designates a suitable scalar product in the framework of parallel flows and
$\boldsymbol{q}^{\dagger }=(\boldsymbol{u}^{\dagger },p^{\dagger })$
is the adjoint vector of the stability eigenproblem (2.4) associated with the defined scalar product. From now on we will use the Hermitian inner product defined as
$$\begin{eqnarray}\displaystyle \langle \boldsymbol{q}_{A},\boldsymbol{q}_{B}\rangle _{p}=\int _{0}^{\infty }\boldsymbol{q}_{A}^{H}\boldsymbol{q}_{B}r\,\text{d}r, & & \displaystyle\end{eqnarray}$$
where the symbol
$^{H}$
indicates the trans-conjugate and
$\boldsymbol{q}_{A}$
and
$\boldsymbol{q}_{B}$
are two complex vector fields. In this case, the adjoint vector satisfies the following adjoint eigenvalue problem:
$$\begin{eqnarray}\displaystyle k^{\ast \,2}{\mathcal{C}}_{2}^{\dagger }\boldsymbol{q}^{\dagger }+k^{\ast }{\mathcal{C}}_{1}^{\dagger }\boldsymbol{q}^{\dagger }+{\mathcal{C}}_{0}^{\dagger }\boldsymbol{q}^{\dagger }=0, & & \displaystyle\end{eqnarray}$$
where the superscript
$^{\ast }$
designates the complex conjugate and the adjoint operators
${\mathcal{C}}_{0}^{\dagger }$
,
${\mathcal{C}}_{1}^{\dagger }$
and
${\mathcal{C}}_{2}^{\dagger }$
are reported in appendix A. The adjoint vector
$\boldsymbol{q}^{\dagger }$
is defined up to a multiplicative factor and from now the chosen normalization is such that the denominator of the expression (2.5) is equal to
$1$
. Equation (2.5) is the generalization of the sensitivity of a standard eigenvalue problem described in Chomaz (Reference Chomaz2005), which would correspond to the case
${\mathcal{C}}_{2}=0$
.
2.3 Sensitivity of the local spatial stability properties to base-flow modifications
We are now interested in determining the sensitivity of
$k$
to a generic base-flow modification
$\unicode[STIX]{x1D6FF}\boldsymbol{U}=(\unicode[STIX]{x1D6FF}U_{r}=0,\unicode[STIX]{x1D6FF}U_{\unicode[STIX]{x1D703}},\unicode[STIX]{x1D6FF}U_{x})$
, where the restriction
$\unicode[STIX]{x1D6FF}U_{r}=0$
is dictated by the local analysis setting. A perturbation at the base-flow level in the linearized equation (2.2) yields a perturbation of the linear operators
$\unicode[STIX]{x1D6FF}{\mathcal{C}}_{0},~\unicode[STIX]{x1D6FF}{\mathcal{C}}_{1},~\unicode[STIX]{x1D6FF}{\mathcal{C}}_{2}$
such that
$\unicode[STIX]{x1D6FF}{\mathcal{C}}_{2}=0$
and
$$\begin{eqnarray}\displaystyle [\unicode[STIX]{x1D6FF}{\mathcal{C}}_{0}+k\unicode[STIX]{x1D6FF}{\mathcal{C}}_{1}]\hat{\boldsymbol{q}}=\left(\begin{array}{@{}c@{}}\text{i}m\hat{u} _{r}\unicode[STIX]{x1D6FF}U_{\unicode[STIX]{x1D703}}/r+\text{i}k\hat{u} _{r}\unicode[STIX]{x1D6FF}U_{x}-2\hat{u} _{\unicode[STIX]{x1D703}}\unicode[STIX]{x1D6FF}U_{\unicode[STIX]{x1D703}}/r\\ \hat{u} _{r}\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}U_{\unicode[STIX]{x1D703}}/\unicode[STIX]{x2202}r+\text{i}m\hat{u} _{\unicode[STIX]{x1D703}}\unicode[STIX]{x1D6FF}U_{\unicode[STIX]{x1D703}}/r+\text{i}k\hat{u} _{r}\unicode[STIX]{x1D6FF}U_{x}+\hat{u} _{r}\unicode[STIX]{x1D6FF}U_{\unicode[STIX]{x1D703}}/r\\ \hat{u} _{r}\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}U_{x}/\unicode[STIX]{x2202}r+\text{i}m\hat{u} _{x}\unicode[STIX]{x1D6FF}U_{\unicode[STIX]{x1D703}}/r+\text{i}k\hat{u} _{x}\unicode[STIX]{x1D6FF}U_{x}\\ 0\end{array}\right). & & \displaystyle\end{eqnarray}$$
Substituting (2.8) in (2.5) and integrating by parts, we can transfer the differential operators from the vector
$\unicode[STIX]{x1D6FF}\boldsymbol{U}$
to the adjoint vector
$\boldsymbol{u}^{\dagger }$
. If the base-flow modification does not occur at the domain boundaries, the boundary terms coming from the integration by parts are null and (2.5) reads
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}k=\{\unicode[STIX]{x1D735}_{\boldsymbol{U}}k,\unicode[STIX]{x1D6FF}\boldsymbol{U}\}, & & \displaystyle\end{eqnarray}$$
where the complex vector
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}_{\boldsymbol{U}}k=\left(\begin{array}{@{}c@{}}0\\ \text{i}m(u_{x}^{\dagger }\hat{u} _{x}^{H}+u_{r}^{\dagger }\hat{u} _{r}^{H}+u_{\unicode[STIX]{x1D703}}^{\dagger }\hat{u} _{\unicode[STIX]{x1D703}}^{H})+\unicode[STIX]{x2202}u_{\unicode[STIX]{x1D703}}^{\dagger }/\unicode[STIX]{x2202}r\hat{u} _{r}^{H}+u_{\unicode[STIX]{x1D703}}^{\dagger }\unicode[STIX]{x2202}\hat{u} _{r}^{H}/\unicode[STIX]{x2202}r+2u_{r}^{\dagger }\hat{u} _{\unicode[STIX]{x1D703}}^{H}/r\\ \text{i}k(u_{x}^{\dagger }\hat{u} _{x}^{H}+u_{r}^{\dagger }\hat{u} _{r}^{H}+u_{\unicode[STIX]{x1D703}}^{\dagger }\hat{u} _{\unicode[STIX]{x1D703}}^{H})+\unicode[STIX]{x2202}u_{x}^{\dagger }/\unicode[STIX]{x2202}r\hat{u} _{r}^{H}+u_{x}^{\dagger }\unicode[STIX]{x2202}\hat{u} _{r}^{H}/\unicode[STIX]{x2202}r+u_{x}^{\dagger }\hat{u} _{r}^{H}/r\end{array}\right) & & \displaystyle\end{eqnarray}$$
is the sensitivity of the considered spatial eigenvalue,
$k$
, to a base-flow modification. The azimuthal and streamwise components of
$\unicode[STIX]{x1D735}_{\boldsymbol{U}}k$
are the sensitivities of the complex eigenvalue
$k$
to an azimuthal and streamwise base-flow modification. Their real (dashed line) and imaginary (full line) parts are reported in figure 2(b) for the most unstable spatial eigenvalue of the parallel Batchelor vortex in the case of
$\unicode[STIX]{x1D6FC}=0.667$
and
$S=0.333$
, i.e.
$m=1$
and
$\unicode[STIX]{x1D714}=0.6$
. Since
$\unicode[STIX]{x1D735}_{\boldsymbol{U}}k$
is a complex vector and
$\unicode[STIX]{x1D6FF}\boldsymbol{U}$
is a real quantity, the real parts of
$\unicode[STIX]{x1D735}_{U_{\unicode[STIX]{x1D703}}}k$
and
$\unicode[STIX]{x1D735}_{U_{x}}k$
are the sensitivities of
$k_{r}$
to a generic base-flow modification. On the other hand, the sensitivity of the spatial growth rate is given by the imaginary part of expression (2.10). The radial component of the sensitivity to base flow is null consistently within the parallel-flow assumption of the local analysis.
From the sensitivity maps in figure 2(b), it is seen that a base-flow modification with positive azimuthal and streamwise components superimposed onto the reference base flow (2.1) leads
$k_{r}$
to increase. Moreover, a swirl increase and a decrease of the wake deficit yield a lower spatial growth rate,
$-k_{i}$
. Hence, (2.9) allows us to determine the direction of displacement of the spatial eigenvalues in the complex
$k$
-plane corresponding to a given base-flow modification. In order to verify the sensitivity maps, we study the effect of a specific base-flow modification, i.e.
$\unicode[STIX]{x1D6FF}U_{x}=c\text{e}^{-r^{2}}$
, on the most unstable spatial eigenvalue in figure 2(a). The considered base-flow modification mimics a variation or uncertainty on the wake parameter
$\unicode[STIX]{x1D6FC}$
in (2.1), and its effect on the spatial stability is determined by solving the spatial stability analysis equation (2.4) with the modified base-flow component
$U_{x}+c\text{e}^{-r^{2}}$
. Results are shown in figure 2(a) for
$c$
ranging from
$3\,\%$
to
$30\,\%$
of
$\unicode[STIX]{x1D6FC}$
. The true path followed by the selected eigenvalue in the complex plane as the parameter
$c$
is varied is indicated with blue asterisks. The same path is approximated by the red straight line using the sensitivity analysis carried out on the unperturbed flow. It is seen that the perturbed eigenvalue positions, which migrate towards the stability region as
$c$
is increased, are well aligned with the linear prediction provided by sensitivity analysis, especially when
$c$
is small, thus validating the method and its implementation. For larger base-flow modifications, nonlinearities neglected in the linearized relation (2.5) become more important and lead to a progressively increasing discrepancy between the predicted and the true eigenvalue trajectories.
Equation (2.10) is an extension to the spatial analysis of a concept that was originally developed by Bottaro et al. (Reference Bottaro, Corbett and Luchini2003) for temporal analysis. It should also be noted that the two non-null components of
$\unicode[STIX]{x1D735}_{U}k$
are equal to the azimuthal and axial components of the vector
$$\begin{eqnarray}\displaystyle -\boldsymbol{u}^{\dagger }\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\hat{\boldsymbol{u}})^{H}+\hat{\boldsymbol{u}}^{\ast }\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}^{\dagger }, & & \displaystyle\end{eqnarray}$$
which is formally similar to the sensitivity to base-flow modifications of a global mode found by Marquet et al. (Reference Marquet, Sipp and Jacquin2008). The use of (2.11) in the framework of a local stability analysis requires
$\hat{\boldsymbol{u}}$
to be the direct spatial mode, while
$\boldsymbol{u}^{\dagger }$
is the spatial adjoint vector, which satisfies the spatial adjoint problem (2.7). Moreover, the streamwise and azimuthal derivatives have to be replaced by the Fourier derivatives, according to the modal expansion (2.3). Finally, the radial component of (2.11), which would give the sensitivity to a radial velocity modification in a global framework, does not bear any meaning in the local spatial analysis.
Figure 3. (a) Radial, (b) azimuthal and (c) streamwise velocity components of the non-parallel Batchelor vortex at
$Re=100$
with inlet condition given by the velocity profile in (2.1).
3 Uncontrolled trailing vortex prototype
Rather than parallel flows, we now consider weakly non-parallel amplifier flows which slowly evolve in the streamwise direction, resulting in variations of the local stability properties. The non-parallel Batchelor vortex represents a suitable prototype flow, where its parallel approximation (2.1) considered in the previous section is here used as inlet condition together with free-stress condition at the outlet and radial far-field boundaries. Its velocity components and pressure
$\boldsymbol{Q}=(\boldsymbol{U},P)$
depend on both the radius
$r$
and the streamwise distance
$x$
from the inlet and are found by solving the discretized Navier–Stokes equation (3.1) in cylindrical coordinates:
$$\begin{eqnarray}\displaystyle \boldsymbol{N}(\boldsymbol{Q},\boldsymbol{U}_{0})=\left(\begin{array}{@{}c@{}}\boldsymbol{U}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{U}+\unicode[STIX]{x1D735}P-\displaystyle \frac{1}{Re}\unicode[STIX]{x0394}\boldsymbol{U}\\ \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{ U}\end{array}\right)=0, & & \displaystyle\end{eqnarray}$$
where the dependence on the inlet condition
$\boldsymbol{U}_{0}$
is emphasized in the notation. For the numerical solution of (3.1), we have used a Newton–Raphson method based on a staggered pseudospectral Chebyshev–Chebyshev collocation method (Viola et al.
Reference Viola, Arratia and Gallaire2016). The computational domain is
$0<x<20$
and
$0<r<10$
and it has been discretized by using
$80$
and
$40$
nodes in the streamwise and radial directions. The resulting velocity field at
$Re=100$
is reported in figure 3(a–c). Note that in figure 3(a) the radial velocity is significantly smaller than the other two velocity components, thus validating the assumption of weakly non-parallel flow. Proceeding downstream the vortex core gets slightly diffused, as shown in figure 3(b), and the axial velocity deficit in figure 3(c) gradually recovers. The flow is seen to be locally convectively unstable and the resulting spatial growth rates,
$-k_{i}$
, are reported in figure 4 as a function of the frequency,
$\unicode[STIX]{x1D714}$
, and the streamwise position,
$x$
. Consistently with the spatial stability properties of the inlet section (see figure 1
c), four helical modes are amplified in the trailing vortex, namely
$m\in \{-1,1,2,3\}$
. The largest amplification is observed in the inlet region where the wake deficit is more pronounced. In particular, the single helical mode,
$m=1$
, has the largest spatial growth rate and is intense in the low-frequency band, whereas the double helical mode,
$m=2$
, dominates for
$0.8<\unicode[STIX]{x1D714}<1.3$
. In addition,
$m=3$
is the most amplified mode at high frequency and
$m=-1$
is weakly amplified at low frequency.
Figure 4. Contours of the spatial growth rate
$-k_{i}$
in the
$(\unicode[STIX]{x1D714},x)$
plane for the convectively unstable helical modes, namely (a)
$m=1$
, (b)
$m=2$
, (c)
$m=3$
and (d)
$m=-1$
. Each colour bar ranges from
$0$
(neutral amplification) to
$\max (-k_{i}(\unicode[STIX]{x1D714},x,m))$
. The horizontal dashed line in (a) corresponds to the spatial branch examined in § 4.
Asymptotic theory for a weakly non-parallel base flow (Huerre & Rossi Reference Huerre and Rossi1998) prescribes that the global disturbance with frequency
$\unicode[STIX]{x1D714}$
and azimuthal wavenumber mode
$m$
takes the following modulated wave form at leading order:
$$\begin{eqnarray}\displaystyle \boldsymbol{q}(r,\unicode[STIX]{x1D703},x;t)\sim \hat{\boldsymbol{q}}(r,x)\exp \left[\text{i}\left(\int _{0}^{x}k(x^{\prime },\unicode[STIX]{x1D714})\,\text{d}x^{\prime }+m\unicode[STIX]{x1D703}-\unicode[STIX]{x1D714}t\right)\right], & & \displaystyle\end{eqnarray}$$
where
$k(x,\unicode[STIX]{x1D714})$
is the local spatial eigenvalue at the flow location
$x$
and
$\hat{\boldsymbol{q}}(r,x)=(\hat{\boldsymbol{u}},\hat{p})$
is the corresponding spatial eigenvector. In particular, the perturbation (3.2) is amplified or damped in space according to the local spatial growth rate
$-k_{i}(x)$
, and the integral amplification factor
$$\begin{eqnarray}\displaystyle G(\unicode[STIX]{x1D714},m)=\exp \left(\int _{0}^{x_{f}}-k_{i}(x^{\prime },\unicode[STIX]{x1D714},m)\,\text{d}x^{\prime }\right), & & \displaystyle\end{eqnarray}$$
is a measure of the ratio between the final,
$x_{f}$
, and the initial,
$x=0$
, disturbance amplitudes, where
$x_{f}$
has been set equal to
$x_{f}=10$
. The factor
$G(\unicode[STIX]{x1D714},m)$
, which is obtained by integrating in
$x$
the spatial growth rates in figure 4, is reported in figure 9(b) by a black line: the azimuthal geometry of the most amplified mode depends on the disturbance frequency, with
$m=1$
well resonating around
$\unicode[STIX]{x1D714}=0.6$
and
$m=2$
around
$\unicode[STIX]{x1D714}=1.1$
.
Furthermore, we define the total amplification factor as
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}=\mathop{\sum }_{m\in {\mathcal{M}}}\int _{\unicode[STIX]{x1D714}_{min}}^{\unicode[STIX]{x1D714}_{max}}G(\unicode[STIX]{x1D714},m)\,\text{d}\unicode[STIX]{x1D714}, & & \displaystyle\end{eqnarray}$$
where
$[\unicode[STIX]{x1D714}_{min},\unicode[STIX]{x1D714}_{max}]$
is the frequency band of noise amplification (here equal to
$[0,2.1]$
; see figure 9
b) and
${\mathcal{M}}$
is the set of convectively unstable helical modes, namely
${\mathcal{M}}=\{-1,1,2,3\}$
. The factor
$\unicode[STIX]{x1D6E4}$
represents the overall amplification of all modes at all frequencies after they have undergone their entire amplification process in space between
$0$
and
$x_{f}$
.
In what follows, the sensitivity of the integral (3.3) and total (3.4) amplification factors to an inflow modification is derived by using the results presented in § 2.3, thus providing information for their control.
4 Sensitivity of the integral amplification factor in weakly non-parallel flows
4.1 Lagrangian formulation
In this section, specifically for control purposes, we are interested in determining the sensitivity of the integral amplification factor (3.3) of a specific spatial mode. As a reference case, we consider the streamwise evolution of the single helical mode
$m=1$
at frequency
$\unicode[STIX]{x1D714}=0.6$
in the evolving Batchelor vortex (see § 3), which is the most amplified mode in the domain and which corresponds to the horizontal dashed line in figure 4(a). By carrying out the local spatial analysis at different positions in the
$x$
direction, it is seen that both the axial wavenumber,
$k_{r}$
, and the spatial growth rate,
$-k_{i}$
, decrease while moving downstream due to the reduction of the wake deficit; see figure 5.
This control problem can be formulated as an optimization problem where the objective function is the integral amplification itself. Rather than a continuous approach where the continuous optimization problem is defined and then discretized, we directly apply the Lagrangian variational technique to the discretized governing equations:
$$\begin{eqnarray}\displaystyle \mathscr{L}_{G}(\boldsymbol{Q},\boldsymbol{U}_{0})=G(\boldsymbol{Q})+\unicode[STIX]{x1D740}^{\text{T}}\boldsymbol{N}(\boldsymbol{Q},\boldsymbol{U}_{0}), & & \displaystyle\end{eqnarray}$$
where the superscript
$^{\text{T}}$
stands for the transpose. The constraint is given by the discretized steady Navier–Stokes equations,
$\boldsymbol{N}(\boldsymbol{Q},\boldsymbol{U}_{0})$
, and
$\unicode[STIX]{x1D740}$
is the vector of Lagrange multipliers or discrete adjoint vector. Thus, the integral amplification factor (3.3) is written in discretized form as follows:
$$\begin{eqnarray}\displaystyle G(\unicode[STIX]{x1D714},m)=\exp \left(-\mathop{\sum }_{n=1}^{N_{x}}w_{n}k_{i}(x_{n},\unicode[STIX]{x1D714},m)\right). & & \displaystyle\end{eqnarray}$$
In (4.2) the coefficients
$w_{n}$
are the integration weights of the standard trapezoidal formula and
$N_{x}$
is the number of locations used to discretize the streamwise direction.
Figure 5. Streamwise evolution of (a) the axial wavenumber,
$k_{r}$
, and (b) the spatial growth rate,
$-k_{i}$
, for the single-helix spatial branch at
$\unicode[STIX]{x1D714}=0.6$
.
The gradient of the integral amplification factor
$G$
with respect to a base-flow inlet modification is derived by considering variations of the Lagrangian
$\mathscr{L}_{G}$
. Imposing the stationarity of
$\mathscr{L}_{G}$
with respect to
$\unicode[STIX]{x1D740}$
reduces to the discrete state (3.1), which is satisfied by the uncontrolled base flow
$\boldsymbol{Q}$
. Then, the stationarity of the Lagrangian with respect to the state variable,
$\boldsymbol{Q}$
, yields an equation that has to be satisfied by the adjoint variable
$\unicode[STIX]{x1D740}$
:
$$\begin{eqnarray}\displaystyle -\unicode[STIX]{x1D740}^{\text{T}}\unicode[STIX]{x1D735}_{\boldsymbol{Q}}\unicode[STIX]{x1D649}=\unicode[STIX]{x1D735}_{\boldsymbol{Q}}G, & & \displaystyle\end{eqnarray}$$
where
$\unicode[STIX]{x1D735}_{\boldsymbol{Q}}\unicode[STIX]{x1D649}$
is the Jacobian matrix of the discrete Navier–Stokes operator. Recalling that the local growth rates
$-k_{i}(x_{n})$
depend only on the streamwise and azimuthal velocity at the location
$x_{n}$
, the right-hand side of (4.3) can be rewritten as
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}_{\boldsymbol{Q}}G=-G\mathop{\sum }_{n=1}^{N_{x}}w_{n}[\unicode[STIX]{x1D735}_{U_{x}}k_{i}(x_{n})\unicode[STIX]{x1D735}_{\boldsymbol{Q}}U_{x}(x_{n})+\unicode[STIX]{x1D735}_{U_{\unicode[STIX]{x1D703}}}k_{i}(x_{n})\unicode[STIX]{x1D735}_{\boldsymbol{Q}}U_{\unicode[STIX]{x1D703}}(x_{n})], & & \displaystyle\end{eqnarray}$$
where the terms
$\unicode[STIX]{x1D735}_{U_{x}}k_{i}(x_{n})$
and
$\unicode[STIX]{x1D735}_{U_{\unicode[STIX]{x1D703}}}k_{i}(x_{n})$
are the local spatial sensitivities of
$k_{i}$
to a local base-flow modification and the related expressions have already been derived in § 2.3. Specifically,
$\unicode[STIX]{x1D735}_{U_{x}}k_{i}(x_{n})$
and
$\unicode[STIX]{x1D735}_{U_{\unicode[STIX]{x1D703}}}k_{i}(x_{n})$
have to be evaluated by injecting the local direct
$\hat{\boldsymbol{u}}(x_{n})$
and adjoint
$\boldsymbol{u}^{\dagger }(x_{n})$
modes in (2.10). The terms
$\unicode[STIX]{x1D735}_{\boldsymbol{Q}}U_{x}(x_{n})$
and
$\unicode[STIX]{x1D735}_{\boldsymbol{Q}}U_{\unicode[STIX]{x1D703}}(x_{n})$
are the interpolation matrices which extract from a vector
$\boldsymbol{Q}$
the streamwise and azimuthal velocity components at the streamwise position
$x_{n}$
. Hence, given the local spatial sensitivity analyses, the adjoint vector
$\unicode[STIX]{x1D740}$
can be determined by solving the linear system (4.3).
Finally, the derivative of the Lagrangian
$\mathscr{L}_{G}$
with respect to the control variable gives the variation of the integral amplification factor with respect to a small-amplitude generic inlet modification
$\unicode[STIX]{x1D6FF}\boldsymbol{U}_{0}$
:
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}G=\frac{\text{d}G(\boldsymbol{Q})}{\text{d}\boldsymbol{U}_{0}}\unicode[STIX]{x1D6FF}\boldsymbol{U}_{0}=\unicode[STIX]{x1D740}^{\text{T}}\unicode[STIX]{x1D735}_{\boldsymbol{U}_{0}}\unicode[STIX]{x1D649}\unicode[STIX]{x1D6FF}\boldsymbol{U}_{0}, & & \displaystyle\end{eqnarray}$$
where
$\unicode[STIX]{x1D735}_{\boldsymbol{U}_{0}}\unicode[STIX]{x1D649}$
is a restriction matrix, which extracts the velocity components at the inlet
$x=0$
from a velocity–pressure vector
$\boldsymbol{Q}$
. Equation (4.5) can be formally rewritten as
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}G=\langle \unicode[STIX]{x1D735}_{\boldsymbol{U}_{0}}G,\unicode[STIX]{x1D6FF}\boldsymbol{U}_{0}\rangle , & & \displaystyle\end{eqnarray}$$
where
$\langle \boldsymbol{u}_{A},\boldsymbol{u}_{B}\rangle =\boldsymbol{u}_{A}^{\text{T}}\unicode[STIX]{x1D648}\boldsymbol{u}_{B}$
is the inner product in the discrete setting, and
$\unicode[STIX]{x1D648}$
is the integration matrix containing the integration weights on its diagonal. The real vector
$\unicode[STIX]{x1D735}_{\boldsymbol{U}_{0}}G$
is the integral amplification factor sensitivity with respect to an inlet modification, and is defined as
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}_{\boldsymbol{U}_{0}}G=\unicode[STIX]{x1D648}^{-1}(\unicode[STIX]{x1D735}_{\boldsymbol{U}_{0}}\unicode[STIX]{x1D649})^{\text{T}}\unicode[STIX]{x1D740}. & & \displaystyle\end{eqnarray}$$
For the reference case of
$\unicode[STIX]{x1D714}=0.6$
and
$m=1$
, the three components of
$\unicode[STIX]{x1D735}_{\boldsymbol{U}_{0}}G$
, which correspond to the sensitivity maps to a radial, azimuthal and axial inlet velocity modification, are reported in figure 6(a). The sensitivity with respect to
$U_{\unicode[STIX]{x1D703}_{0}}$
reveals that, in order to increase the amplification, the azimuthal velocity, and therefore the swirl parameter, has to be decreased. Similarly,
$\unicode[STIX]{x1D735}_{U_{x_{0}}}G$
shows that an increase of the wake deficit and a higher streamwise velocity at the vortex periphery lead to a more intense response to noise. Owing to the non-parallelism of the base flow
$\boldsymbol{U}$
, the spatial amplification is also sensitive to a variation of the radial velocity. This velocity component acts on the local spatial stability properties of the flow solely through the coupling with the streamwise and azimuthal velocity components in the base-flow governing equations (3.1). In particular, it is seen that a positive inlet radial velocity, which increases the vortex core diameter and consequently favours the axial deceleration, has a destabilizing effect.
Figure 6. (a) The sensitivity maps of
$G(\unicode[STIX]{x1D714}=0.6,m=1)$
to radial (green), azimuthal (red) and axial (black) inlet velocity modifications. (b) The true amplification factor modification (blue asterisks) as a function of the magnitude of the inlet control
$c$
. The red dashed line shows the linear amplification modification according to the inlet sensitivity map.
Similar results on the sensitivity with respect to radial and axial velocity components are observed for
$m=2$
and
$m=3$
at the frequency corresponding to the largest amplification factor,
$\unicode[STIX]{x1D714}=1.1$
and
$\unicode[STIX]{x1D714}=1.6$
respectively; see figure 7. In contrast to
$m=1$
(figure 6
a), however, the sensitivity with respect to the azimuthal velocity is positive in the core region for both
$m=2$
and
$m=3$
. This signifies that an increase of the swirl has a destabilizing effect on the double and triple helical modes. This result can be rationalized by recalling that, starting from a pure axial flow, helical modes are first progressively destabilized by an increasing amount of swirl till
$S$
exceeds
$S^{\ast }(m)$
, beyond which a further increase of swirl starts to have a damping effect, until complete stabilization of the flow for
$S\approx 1.5\unicode[STIX]{x1D6FC}$
(Mayer & Powell Reference Mayer and Powell1992; Delbende et al.
Reference Delbende, Chomaz and Huerre1998). As illustrated in figure 1(b) for the flow pertaining at the inlet section, the swirl value corresponding to the maximum growth rate,
$S^{\ast }(m)$
, depends on the helical mode and it is seen to increase with
$m$
for the chosen flow parameters, thus yielding a different sensitivity to a modification in the inlet azimuthal velocity.
Figure 7. Sensitivity maps of (a)
$G(\unicode[STIX]{x1D714}=1.1,m=2)$
and (b)
$G(\unicode[STIX]{x1D714}=1.6,m=3)$
to a radial (green), azimuthal (red) and axial (black) inlet velocity modifications.
4.2 Optimal inlet control
The sensitivity
$\unicode[STIX]{x1D735}_{\boldsymbol{U}_{0}}G$
not only predicts the linear modification of the integral amplification factor due to an inlet modification through (4.6), but also corresponds to the most destabilizing infinitesimal inlet modification (Camarri Reference Camarri2015). For this reason,
$\unicode[STIX]{x1D735}_{\boldsymbol{U}_{0}}G$
can be used to design an inlet control which stabilizes the global spatial branch. Figure 6(b) shows the amplification variation of the single-helix spatial branch at frequency
$\unicode[STIX]{x1D714}=0.6$
due to an inlet modification of the type
$\unicode[STIX]{x1D6FF}\boldsymbol{U}_{0}=-c\unicode[STIX]{x1D735}_{\boldsymbol{U}_{0}}G$
, with
$c$
the magnitude of the control. The blue line in figure 6(b) represents the real amplification variation with respect to the uncontrolled condition. More precisely, at a given
$c$
, the real amplification variation (blue asterisk) is computed by (i) solving the nonlinear base flow equation (3.1) with the inlet condition
$\boldsymbol{U}_{0}-c\unicode[STIX]{x1D735}_{\boldsymbol{U}_{0}}G$
, and (ii) carrying out the spatial stability analysis on the obtained base flow. Then, (iii) the associated integral amplification factor
$G(c)$
is computed according to the definition (4.2) and its variation
$\unicode[STIX]{x0394}G=1-G(c)/G(c=0)$
is obtained, where
$G(c=0)$
is the amplification factor of the uncontrolled condition. It is seen that the linear amplification variation predicted by (4.6), which is depicted by a red line in figure 6(b), correctly captures the amplification variation for small values of
$c$
. In contrast, owing to the nonlinear dependence of the amplification factor on the magnitude of the perturbation, the error increases as
$c$
is progressively increased. Still, the linearization leading to the sensitivity analysis allows the prediction of even large variations of amplification with reasonable accuracy.
As an application, figure 8(a) shows the uncontrolled and the controlled velocity profiles at the inlet for
$c=7\times 10^{-4}$
(see figure 6
a). Although the inlet condition differs very slightly from the uncontrolled case, the effect of this variation on the considered instability is substantial, as shown in figure 8(b,c), where the axial vorticity of the spatial branch given by (3.2) with
$\unicode[STIX]{x1D714}=0.6$
and
$m=1$
is reported. In the first case the helical perturbation is convectively unstable in all the flow domain and is continuously amplified while propagating. When the control is applied, the spatial branch is damped for
$x>7$
and the integral amplification factor experiences a significative reduction of
$35\,\%$
with respect to the uncontrolled flow.
Figure 8. (a) The velocity components of the uncontrolled inlet condition (full lines), which correspond to the Batchelor vortex profile (2.1) with
$\unicode[STIX]{x1D6FC}=0.667$
and
$S=0.333$
. The velocity components perturbed by the application of the control
$\unicode[STIX]{x1D6FF}\boldsymbol{U}_{0}=7\times 10^{-4}\unicode[STIX]{x1D735}_{\boldsymbol{U}_{0}}G(\unicode[STIX]{x1D714}=0.6,m=1)$
are shown with dashed lines. (b,c) Isosurfaces of
$\pm 95\,\%$
of the maximum axial vorticity of the global spatial mode (3.2) in the (b) uncontrolled and (c) controlled cases.
5 Sensitivity of the total growth factor in weakly non-parallel flows
5.1 Lagrangian formulation
The control strategy illustrated in the previous section is based on the inlet sensitivity of the integral amplification factor of a single helical mode at a given frequency. Hence, a control is designed to reduce the amplification of the single spatial branch considered, but it could dangerously increase the spatial growth of other helical modes and of other frequencies. Therefore, we now turn to assess the sensitivity of the total amplification factor
$\unicode[STIX]{x1D6E4}$
(see (3.4)) to a modification of the inflow conditions. Indeed, the control of
$\unicode[STIX]{x1D6E4}$
requires one to account for the local growth rates of all helical modes in their entire frequency band of amplification.
Similarly to the analysis in § 4.1, the problem can be tackled by considering the following Lagrangian:
$$\begin{eqnarray}\displaystyle \mathscr{L}_{\unicode[STIX]{x1D6E4}}(\boldsymbol{Q},\boldsymbol{U}_{0})=\unicode[STIX]{x1D6E4}(\boldsymbol{Q})+\unicode[STIX]{x1D743}^{\text{T}}\boldsymbol{N}(\boldsymbol{Q},\boldsymbol{U}_{0}). & & \displaystyle\end{eqnarray}$$
The steady Navier–Stokes equations
$\boldsymbol{N}(\boldsymbol{Q},\boldsymbol{U}_{0})$
are the constraint, the Lagrange multiplier
$\unicode[STIX]{x1D743}$
is the adjoint vector and the quantity to minimize is now the total amplification factor
$\unicode[STIX]{x1D6E4}(\boldsymbol{Q})$
, which in discrete form reads:
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}=\mathop{\sum }_{m\in {\mathcal{M}}}\mathop{\sum }_{l=1}^{N_{\unicode[STIX]{x1D714}}}\unicode[STIX]{x1D6FC}_{l}G(\unicode[STIX]{x1D714}_{l},m)=\mathop{\sum }_{m\in {\mathcal{M}}}\mathop{\sum }_{l=1}^{N_{\unicode[STIX]{x1D714}}}\unicode[STIX]{x1D6FC}_{l}\exp \left(-\mathop{\sum }_{n=1}^{N_{x}}w_{n}k_{i}(x_{n},\unicode[STIX]{x1D714}_{l},m)\right). & & \displaystyle\end{eqnarray}$$
As in (4.2), the coefficients
$w_{n}$
are the integration weights of the standard trapezoidal formula and
$N_{x}$
is the number of locations used to discretize the control window in the streamwise direction. In addition, the coefficients
$\unicode[STIX]{x1D6FC}_{l}$
are the integration weights corresponding to the integral in the frequency domain, which is discretized using
$N_{\unicode[STIX]{x1D714}}$
points, and
${\mathcal{M}}=\{-1,1,2,3\}$
is the set of convectively unstable helical modes.
When imposing the stationarity of
$\mathscr{L}_{\unicode[STIX]{x1D6E4}}$
with respect to
$\unicode[STIX]{x1D743}$
, the problem reduces to the discrete state (3.1), which is satisfied by the uncontrolled base flow
$\boldsymbol{Q}$
that we have determined previously. Then, the stationarity of the Lagrangian with respect to the state variable,
$\boldsymbol{Q}$
, yields a governing equation for the adjoint variable
$\unicode[STIX]{x1D743}$
:
$$\begin{eqnarray}\displaystyle -\unicode[STIX]{x1D743}^{\text{T}}\unicode[STIX]{x1D735}_{\boldsymbol{Q}}\unicode[STIX]{x1D649}=\unicode[STIX]{x1D735}_{\boldsymbol{Q}}\unicode[STIX]{x1D6E4}, & & \displaystyle\end{eqnarray}$$
which is now forced at the right-hand side by the variation of the total amplification factor,
$\unicode[STIX]{x1D6E4}$
, with respect to a variation of the base flow,
$\boldsymbol{Q}$
. This term can be rewritten as a function of the sensitivities to a base-flow variation of the helical modes’ amplification factors at different frequencies,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}_{\boldsymbol{Q}}\unicode[STIX]{x1D6E4}=\mathop{\sum }_{m\in {\mathcal{M}}}\mathop{\sum }_{l=1}^{N_{\unicode[STIX]{x1D714}}}\unicode[STIX]{x1D6FC}_{l}\unicode[STIX]{x1D735}_{\boldsymbol{Q}}G(\unicode[STIX]{x1D714},m), & & \displaystyle\end{eqnarray}$$
which reduces, by using (4.4), to a weighted sum of the local spatial sensitivities to a generic local base-flow modification (see § 2.3):
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}_{\boldsymbol{Q}}\unicode[STIX]{x1D6E4}=\mathop{\sum }_{m\in {\mathcal{M}}}\mathop{\sum }_{l=1}^{N_{\unicode[STIX]{x1D714}}}-\unicode[STIX]{x1D6FC}_{l}G(\unicode[STIX]{x1D714},m)\mathop{\sum }_{n=1}^{N_{x}}w_{n}[\unicode[STIX]{x1D735}_{U_{x}}k_{i}(x_{n})\unicode[STIX]{x1D735}_{\boldsymbol{Q}}U_{x}(x_{n})+\unicode[STIX]{x1D735}_{U_{\unicode[STIX]{x1D703}}}k_{i}(x_{n})\unicode[STIX]{x1D735}_{\boldsymbol{Q}}U_{\unicode[STIX]{x1D703}}(x_{n})].\quad & & \displaystyle\end{eqnarray}$$
Note that the local sensitivities
$\unicode[STIX]{x1D735}_{U_{\unicode[STIX]{x1D703}}}k_{i}(x_{n})$
and
$\unicode[STIX]{x1D735}_{U_{x}}k_{i}(x_{n})$
in (5.5) are multiplied by the integral amplification factor of the given spatial branch
$G(\unicode[STIX]{x1D714},m)$
. Hence, as reasonable, most amplified helical modes correspond to a higher contribution to the forcing term
$\unicode[STIX]{x1D735}_{\boldsymbol{Q}}\unicode[STIX]{x1D6E4}$
. Then, by solving the linear system (5.3) for the adjoint field
$\unicode[STIX]{x1D743}$
, the sensitivity of the total amplification factor with respect to a small-amplitude generic inlet modification
$\unicode[STIX]{x1D6FF}\boldsymbol{U}_{0}$
reads:
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}_{\boldsymbol{U}_{0}}\unicode[STIX]{x1D6E4}=\unicode[STIX]{x1D648}^{-1}(\unicode[STIX]{x1D735}_{\boldsymbol{U}_{0}}\unicode[STIX]{x1D649})^{\text{T}}\unicode[STIX]{x1D743}. & & \displaystyle\end{eqnarray}$$
The three components of the sensitivity
$\unicode[STIX]{x1D735}_{\boldsymbol{U}_{0}}\unicode[STIX]{x1D6E4}$
are shown in figure 9(a) and they correspond to the sensitivity maps to a radial (green line), azimuthal (red line) and axial (black line) inlet velocity modification.
Similarly to the sensitivity maps in figures 6(a) and 7, where only one helical mode at a given frequency was considered, an increase of the wake deficit and a higher streamwise velocity at the vortex periphery lead to a more intense amplification of disturbances. Moreover, a positive inlet radial velocity is positively correlated with
$\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E4}$
. In contrast, the sensitivity with respect to a modification of the azimuthal velocity does not have a monotonic behaviour since it is the result of a compromise between the sensitivities of different helical modes. As previously discussed in § 4.1, double and triple helical modes are destabilized by an increase of the azimuthal velocity close to the centreline; see figure 7. On the other hand, the single helical mode requires a decrease in swirl to enhance the amplitude of the response, as depicted in figure 6(a).
5.2 Optimal inlet control
Figure 9. (a) Sensitivity maps (full lines) of the total amplification factor
$\unicode[STIX]{x1D6E4}$
(see (5.6)) to radial (green), azimuthal (red) and streamwise (black) inlet velocity modifications. The
$+$
symbols correspond to the sensitivity maps computed with a finer mesh (see appendix B), showing convergence with respect to grid resolution. (b) The variation of the integral amplification factor
$G(\unicode[STIX]{x1D714},m)$
obtained by using the sensitivity map as inlet control
$\unicode[STIX]{x1D6FF}\boldsymbol{U}_{0}=c\unicode[STIX]{x1D735}_{\boldsymbol{U}_{0}}\unicode[STIX]{x1D6E4}$
. Grey lines correspond to several forcing amplitudes
$c$
ranging from
$c=0$
(uncontrolled condition, black line) to
$c=2\times 10^{-3}$
(red line) in equal intervals.
The sensitivity map (5.6) is now used as an inlet control to reduce the integral amplification factor
$G(\unicode[STIX]{x1D714},m)$
of all helical modes over the base flow. Figure 9(b) shows the variation of
$G(\unicode[STIX]{x1D714},m)$
due to an inlet modification of the type
$\unicode[STIX]{x1D6FF}\boldsymbol{U}_{0}=-c\unicode[STIX]{x1D735}_{\boldsymbol{U}_{0}}\unicode[STIX]{x1D6E4}$
, with
$c$
the magnitude of the control. Grey lines depict the dependence of
$G(\unicode[STIX]{x1D714},m)$
on the control amplitude, ranging from the uncontrolled condition
$c=0$
(black line) to
$c=2\times 10^{-3}$
in equal intervals. At a given
$c$
, the integral amplification factor is obtained by (i) solving the nonlinear base flow equation (3.1) with the inlet condition
$\boldsymbol{U}_{0}-c\unicode[STIX]{x1D735}_{\boldsymbol{U}_{0}}\unicode[STIX]{x1D6E4}$
, and (ii) carrying out the spatial stability analysis on the obtained base flow.
Figure 9 demonstrates how, with a single linearization of the governing equation, it is possible to design a control strategy able to reduce the spatial amplification of all convectively unstable disturbances. In particular, with an amplitude of the control of
$c=2\times 10^{-3}$
(red line) the integral amplification factor is reduced by
$50\,\%$
for
$m=1$
and by
$40\,\%$
for
$m=2$
, and higher azimuthal wavenumber modes are seen not to be destabilized by the control. Moreover, the high-frequency mode
$m=3$
is completely stabilized and
$m=-1$
, which was already weakly amplified, is slightly stabilized.
Note that the control is more effective on
$m=1$
rather than
$m=2$
, which progressively becomes the most amplified mode while the control amplitude
$c$
is increased. This behaviour can be explained by recalling that the control is designed on the uncontrolled base state where
$m=1$
has the largest
$G(\unicode[STIX]{x1D714},m)$
and therefore has the highest contribution to the forcing term (5.5) and on the sensitivity map. However, the sensitivity maps defined here can be used in an iterative shaping of the control as proposed by Camarri & Iollo (Reference Camarri and Iollo2010). This technique, which is based on successive linearizations of the governing equations, allows one to properly modify the control according to the new stability property of the controlled base flow.
6 Conclusions
In this work a general formulation is proposed which allows the application of systematic adjoint-based techniques for the control of the integral amplification factor in weakly non-parallel amplifier flows. The method is based on the sensitivity analysis of the local stability properties of the flow, which was here derived by adjoint methods.
Firstly, the sensitivity of the spatial stability spectrum of a locally parallel flow to a generic modification of the stability operators and to a base-flow modification is found. In the case of a parallel Batchelor vortex, the local sensitivity map is seen to correctly predict the displacement of the spatial eigenvalues in the complex
$k$
-plane due to a specific base-flow modification.
Then, for the purpose of control in slowly evolving amplifier flows, these local results are used to determine the sensitivity of the spatial growth of disturbances to a modification of the inflow conditions. Although the method can be easily extended to cover very general cases, here we presented its application to the control of the spatial instability in an evolving Batchelor vortex, which is traditionally used to model trailing vortices in aeronautics. The base flow is obtained by axisymmetric direct numerical simulation starting from the velocity profile of a parallel Batchelor vortex as inflow boundary condition, and is seen to be locally convectively unstable to four helical modes,
$m\in \{-1,1,2,3\}$
. The largest amplification is observed in the inlet region where the wake deficit is more pronounced and the single- and double-helical modes are the most amplified modes.
The sensitivity of the integral amplification factor,
$G(\unicode[STIX]{x1D714},m)$
, of a single spatial branch is formulated as an optimization problem, where
$G(\unicode[STIX]{x1D714},m)$
is the objective function and the Navier–Stokes equations are included in the Lagrangian formulation as a constraint. Rather than a continuous approach where the continuous optimization problem is defined and then discretized, we directly apply the Lagrangian variational technique to the discretized governing equations. In this framework, we show that the local spatial sensitivities are the building blocks of the sensitivity of the integral amplification factor to generic modifications of the inlet velocity profile. The resulting sensitivity map indicates the optimal modification of the inlet velocity profile so as to stabilize the considered helical spatial mode of a non-parallel Batchelor vortex, which is locally convectively unstable. It is shown that the control, formulated using only one linearization of the flow dynamics carried out on the uncontrolled configuration, successfully reduces the global spatial amplification of a given spatial branch.
This analysis is then extended to determine the sensitivity to an inflow modification of the total amplification factor, which is a measure of the overall amplification of all the convectively unstable spatial branches. Hence, the corresponding sensitivity map accounts for all the amplified helical modes in the frequency band of amplification. It is seen that, in order to increase the total amplification factor, the streamwise velocity has to decrease close to the centre of the wake and to increase at the vortex periphery, thus enhancing the wake deficit. A positive variation of the azimuthal velocity has a destabilizing effect close to the vortex centreline whereas it stabilizes the flow in the outer region. This non-monotonic behaviour is seen to be the result of the different sensitivities to the base-flow swirl of different helical modes. Furthermore, owing to the non-parallelism of the base flow, also the radial velocity has an effect on the spatial amplification: a positive inlet radial velocity increases the vortex core diameter, favouring the axial deceleration and destabilizing the flow. This map is subsequently used as an inlet control to effectively reduce the integral amplification factor of all helical modes over the base flow. We observe that with a control of size
$c=2\,\unicode[STIX]{x2030}$
the spatial amplification of all modes is significantly reduced, and the amplification factor of
$m=1$
and
$m=2$
is almost halved.
The information provided by the control map is valid only for small control amplitudes perturbing the uncontrolled configuration. However, the control design method proposed here can be easily extended to include larger control amplitudes, by employing the sensitivity map in a standard iterative gradient method, as done for instance by Bottaro et al. (Reference Bottaro, Corbett and Luchini2003) for temporal analysis on parallel flow. We conclude by underlining that the systematic framework proposed here to control the integral amplification factor in amplifier flows has potential impact in many cases of interest, such as the control of the hub vortex in wind turbines (Iungo et al. Reference Iungo, Viola, Camarri and Gallaire2013), which has inspired this work. As a final note, in the spirit of weakly non-parallel flows, the full Navier–Stokes equations in the Lagrangian formulation (4.1) can be replaced by the parabolized equations in the streamwise direction, allowing a marching technique for their numerical solution as for the boundary layer equations (see Zuccher, Bottaro & Luchini Reference Zuccher, Bottaro and Luchini2006). In this case, both the base-flow and stability problems are carried out in a weakly non-parallel setting.
Appendix A. Local stability analysis of swirling flows
The linear evolution of helical disturbances of the type
$(\hat{\boldsymbol{u}},\hat{p})\text{e}^{\text{i}(kx+m\unicode[STIX]{x1D703}-\unicode[STIX]{x1D714}t)}$
over a parallel swirling flow
$U_{x}(r),U_{\unicode[STIX]{x1D703}}(r)$
is governed by
$$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle -\text{i}\unicode[STIX]{x1D714}\hat{u} _{r}+\unicode[STIX]{x1D6EC}_{m,k}\hat{u} _{r}-2\frac{U_{\unicode[STIX]{x1D703}}\hat{u} _{\unicode[STIX]{x1D703}}}{r}=-\frac{\unicode[STIX]{x2202}\hat{p}}{\unicode[STIX]{x2202}r}+\frac{1}{Re}\left[\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}\left(\frac{1}{r}\frac{\unicode[STIX]{x2202}r\hat{u} _{r}}{\unicode[STIX]{x2202}r}\right)-(m^{2}/r^{2}+k^{2})\hat{u} _{r}-2\text{i}m\frac{\hat{u} _{\unicode[STIX]{x1D703}}}{r^{2}}\right]\!,\\ \displaystyle -\text{i}\unicode[STIX]{x1D714}\hat{u} _{\unicode[STIX]{x1D703}}+\unicode[STIX]{x1D6EC}_{m,k}\hat{u} _{\unicode[STIX]{x1D703}}+\frac{\hat{u} _{r}}{r}\frac{\unicode[STIX]{x2202}rU_{\unicode[STIX]{x1D703}}}{\unicode[STIX]{x2202}r}=-\frac{\text{i}m}{r}\hat{p}+\frac{1}{Re}\left[\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}\left(\frac{1}{r}\frac{\unicode[STIX]{x2202}r\hat{u} _{\unicode[STIX]{x1D703}}}{\unicode[STIX]{x2202}r}\right)-(m^{2}/r^{2}+k^{2})\hat{u} _{\unicode[STIX]{x1D703}}+2\text{i}m\frac{\hat{u} _{r}}{r^{2}}\right]\!,\\ \displaystyle -\text{i}\unicode[STIX]{x1D714}\hat{u} _{x}+\unicode[STIX]{x1D6EC}_{m,k}\hat{u} _{x}+\hat{u} _{r}\frac{\unicode[STIX]{x2202}U_{x}}{\unicode[STIX]{x2202}r}=-\text{i}k\hat{p}+\frac{1}{Re}\left[\frac{1}{r}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}\left(r\frac{\unicode[STIX]{x2202}\hat{u} _{x}}{\unicode[STIX]{x2202}r}\right)-(m^{2}/r^{2}+k^{2})\hat{u} _{x}\right]\!,\\ \displaystyle \frac{1}{r}\frac{\unicode[STIX]{x2202}r\hat{u} _{r}}{\unicode[STIX]{x2202}r}+\frac{\text{i}m}{r}\hat{u} _{\unicode[STIX]{x1D703}}+\text{i}k\hat{u} _{x}=0,\end{array}\right\} & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
where
$\unicode[STIX]{x1D6EC}_{m,k}=U_{\unicode[STIX]{x1D703}}\text{i}m/r+U_{x}\text{i}k$
is the base-flow advection operator. Homogeneous Neumann boundary conditions are imposed at the lateral boundary
$r_{max}$
together with the regularity conditions on the axis proposed by Batchelor & Gill (Reference Batchelor and Gill1962):
$$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \hat{u} _{r}=\hat{u} _{\unicode[STIX]{x1D703}}=\frac{\unicode[STIX]{x2202}\hat{u} _{x}}{\unicode[STIX]{x2202}r}=0\quad \text{for }m=0,\\ \displaystyle \frac{\unicode[STIX]{x2202}\hat{u} _{r}}{\unicode[STIX]{x2202}r}=\frac{\unicode[STIX]{x2202}\hat{u} _{\unicode[STIX]{x1D703}}}{\unicode[STIX]{x2202}r}=\hat{u} _{x}=0\quad \text{for }|m|=1,\\ \displaystyle \hat{u} _{r}=\hat{u} _{\unicode[STIX]{x1D703}}=\hat{u} _{x}=0\quad \text{for }|m|>1.\end{array}\right\} & & \displaystyle\end{eqnarray}$$
Temporal stability analysis allows the investigation of the temporal evolution of the helical disturbances (see Schmid & Henningson Reference Schmid and Henningson2012). In this framework, the axial wavenumber
$k$
is real and
$\unicode[STIX]{x1D714}$
is complex. Its real part,
$\unicode[STIX]{x1D714}_{r}$
, is the frequency of the perturbation and its imaginary part,
$\unicode[STIX]{x1D714}_{i}$
, is the temporal growth rate. Given the azimuthal,
$m$
, and streamwise,
$k$
, wavenumbers, equations (A 1) reduce to a generalized eigenvalue problem,
$$\begin{eqnarray}\displaystyle {\mathcal{A}}\hat{\boldsymbol{q}}=\unicode[STIX]{x1D714}{\mathcal{B}}\hat{\boldsymbol{q}}, & & \displaystyle\end{eqnarray}$$
where
$\unicode[STIX]{x1D714}$
is the eigenvalue and the linear operators are defined as follows:
$$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{A}}=\left[\begin{array}{@{}cccc@{}}\unicode[STIX]{x1D6EC}_{m,k}-\displaystyle \frac{1}{Re}\unicode[STIX]{x1D6E5}_{m,_{r\unicode[STIX]{x1D703}x}}^{\ast } & -2U_{\unicode[STIX]{x1D703}}/r+\displaystyle \frac{1}{Re}\displaystyle \frac{2\text{i}m}{r^{2}} & 0 & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}\\ \displaystyle \frac{\unicode[STIX]{x2202}U_{\unicode[STIX]{x1D703}}}{\unicode[STIX]{x2202}r}+U_{\unicode[STIX]{x1D703}}/r-\displaystyle \frac{1}{Re}\displaystyle \frac{2\text{i}m}{r^{2}} & \unicode[STIX]{x1D6EC}_{m,k}-\displaystyle \frac{1}{Re}\unicode[STIX]{x1D6E5}_{m,_{r\unicode[STIX]{x1D703}x}}^{\ast } & 0 & \displaystyle \frac{\text{i}m}{r}\\ \displaystyle \frac{\unicode[STIX]{x2202}U_{x}}{\unicode[STIX]{x2202}r} & 0 & \unicode[STIX]{x1D6EC}_{m,k}-\displaystyle \frac{1}{Re}\unicode[STIX]{x1D6E5}_{m,_{r\unicode[STIX]{x1D703}x}} & \text{i}k\\ \displaystyle \frac{1}{r}+\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r} & \displaystyle \frac{\text{i}m}{r} & \text{i}k & 0\end{array}\right]\!,\quad & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{B}}=\left[\begin{array}{@{}cccc@{}}\text{i} & 0 & 0 & 0\\ 0 & \text{i} & 0 & 0\\ 0 & 0 & \text{i} & 0\\ 0 & 0 & 0 & 0\end{array}\right]\!, & \displaystyle\end{eqnarray}$$
with
$\unicode[STIX]{x1D6E5}_{m,_{r\unicode[STIX]{x1D703}x}}=((1/r)(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}r)+\unicode[STIX]{x2202}^{2}/\unicode[STIX]{x2202}r^{2}-k^{2}-m^{2}/r^{2})$
and
$\unicode[STIX]{x1D6E5}_{m,_{r\unicode[STIX]{x1D703}}x}^{\ast }=\unicode[STIX]{x1D6E5}_{m,_{r\unicode[STIX]{x1D703}}}-1/r^{2}$
.
On the other hand, the spatial stability analysis allows the investigation of the spatial evolution of the helical disturbances (Schmid & Henningson Reference Schmid and Henningson2012) and the frequency
$\unicode[STIX]{x1D714}$
is a real number whereas the wavenumber
$k$
is complex. In particular, its real part,
$k_{r}$
, corresponds to the axial wavenumber and its imaginary part,
$-k_{i}$
, is the spatial growth rate. Therefore equations (A 1) can be recast as the quadratic eigenvalue problem (2.4), where the linear operators are defined as follows:
$$\begin{eqnarray}\displaystyle {\mathcal{C}}_{0}=\left[\begin{array}{@{}cccc@{}}\text{i}mU_{\unicode[STIX]{x1D703}}/r-\displaystyle \frac{1}{Re}\unicode[STIX]{x1D6E5}_{m,_{r\unicode[STIX]{x1D703}}}^{\ast }-\text{i}\unicode[STIX]{x1D714} & -2U_{\unicode[STIX]{x1D703}}/r+\displaystyle \frac{1}{Re}\displaystyle \frac{2\text{i}m}{r^{2}} & 0 & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}\\ \displaystyle \frac{\unicode[STIX]{x2202}U_{\unicode[STIX]{x1D703}}}{\unicode[STIX]{x2202}r}+U_{\unicode[STIX]{x1D703}}/r-\displaystyle \frac{1}{Re}\displaystyle \frac{2\text{i}m}{r^{2}} & \text{i}mU_{\unicode[STIX]{x1D703}}/r-\displaystyle \frac{1}{Re}\unicode[STIX]{x1D6E5}_{m,_{r\unicode[STIX]{x1D703}}}^{\ast }-\text{i}\unicode[STIX]{x1D714} & 0 & \displaystyle \frac{\text{i}m}{r}\\ \displaystyle \frac{\unicode[STIX]{x2202}U_{x}}{\unicode[STIX]{x2202}r} & 0 & \text{i}mU_{\unicode[STIX]{x1D703}}/r-\displaystyle \frac{1}{Re}\unicode[STIX]{x1D6E5}_{m,_{r\unicode[STIX]{x1D703}}}-\text{i}\unicode[STIX]{x1D714} & 0\\ \displaystyle \frac{1}{r}+\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r} & \displaystyle \frac{\text{i}m}{r} & 0 & 0\end{array}\right]\!, & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle {\mathcal{C}}_{1}=\left[\begin{array}{@{}cccc@{}}\text{i}U_{x} & 0 & 0 & 0\\ 0 & \text{i}U_{x} & 0 & 0\\ 0 & 0 & \text{i}U_{x} & \text{i}\\ 0 & 0 & \text{i} & 0\end{array}\right],\quad {\mathcal{C}}_{2}=\left[\begin{array}{@{}cccc@{}}\displaystyle \frac{1}{Re} & 0 & 0 & 0\\ 0 & \displaystyle \frac{1}{Re} & 0 & 0\\ 0 & 0 & \displaystyle \frac{1}{Re} & 0\\ 0 & 0 & 0 & 0\end{array}\right], & & \displaystyle\end{eqnarray}$$
with
$\unicode[STIX]{x1D6E5}_{m,_{r\unicode[STIX]{x1D703}}}=((1/r)(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}r)+\unicode[STIX]{x2202}^{2}/\unicode[STIX]{x2202}r^{2}-m^{2}/r^{2})$
and
$\unicode[STIX]{x1D6E5}_{m,_{r\unicode[STIX]{x1D703}}}^{\ast }=\unicode[STIX]{x1D6E5}_{m,_{r\unicode[STIX]{x1D703}}}-1/r^{2}$
.
Integration by parts of the spatial stability problem using the hermitian scalar product (2.6) yields the adjoint spatial stability problem (2.7), where
$$\begin{eqnarray}\displaystyle {\mathcal{C}}_{0}^{\dagger }=\left[\begin{array}{@{}cccc@{}}-\text{i}mU_{\unicode[STIX]{x1D703}}/r-\displaystyle \frac{1}{Re}\unicode[STIX]{x1D6E5}_{m,_{r\unicode[STIX]{x1D703}}}^{\ast }+\text{i}\unicode[STIX]{x1D714} & \displaystyle \frac{\unicode[STIX]{x2202}U_{\unicode[STIX]{x1D703}}}{\unicode[STIX]{x2202}r}+U_{\unicode[STIX]{x1D703}}/r+\displaystyle \frac{1}{Re}\displaystyle \frac{2\text{i}m}{r^{2}} & \displaystyle \frac{\unicode[STIX]{x2202}U_{x}}{\unicode[STIX]{x2202}r} & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}\\ -2U_{\unicode[STIX]{x1D703}}/r-\displaystyle \frac{1}{Re}\displaystyle \frac{2\text{i}m}{r^{2}} & -\text{i}mU_{\unicode[STIX]{x1D703}}/r-\displaystyle \frac{1}{Re}\unicode[STIX]{x1D6E5}_{m,_{r\unicode[STIX]{x1D703}}}^{\ast }+\text{i}\unicode[STIX]{x1D714} & 0 & \displaystyle \frac{\text{i}m}{r}\\ 0 & 0 & -\text{i}mU_{\unicode[STIX]{x1D703}}/r-\displaystyle \frac{1}{Re}\unicode[STIX]{x1D6E5}_{m}+\text{i}\unicode[STIX]{x1D714} & 0\\ \displaystyle \frac{1}{r}+\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r} & \displaystyle \frac{\text{i}m}{r} & 0 & 0\end{array}\right]\!, & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle {\mathcal{C}}_{1}^{\dagger }=\left[\begin{array}{@{}cccc@{}}-\text{i}U_{x} & 0 & 0 & 0\\ 0 & -\text{i}U_{x} & 0 & 0\\ 0 & 0 & -\text{i}U_{x} & \text{i}\\ 0 & 0 & \text{i} & 0\end{array}\right],\quad {\mathcal{C}}_{2}^{\dagger }={\mathcal{C}}_{2}, & & \displaystyle\end{eqnarray}$$
which satisfies the same boundary conditions as the direct problem.
Appendix B. Convergence of the results to grid and computational domain
We show here the numerical independence of the results from the size of the domain and grid refinement.
B.1 Local stability analysis
Figure 10. (a) Temporal and (b) spatial growth rates computed with radial extension of the domain equal to
$r_{max}=10$
and
$N=40$
(black line) and
$r_{max}=15$
and
$N=60$
(red line).
Figure 10 shows the insensitivity of the (a) temporal and (b) spatial stability analysis of the parallel Batchelor vortex (2.1) on the radial extension of the domain,
$r_{max}$
, and on the number of Gauss–Lobatto–Chebyshev (GLC) nodes,
$N$
. Black line stands for
$r_{max}=10,N=40$
whereas red line indicates
$r_{max}=15,N=60$
.
B.2 Direct numerical simulations (DNS)
Figure 11. Streamwise velocity field
$U_{x}$
computed using the computational domain (a)
${\mathcal{D}}_{1}$
and (b)
${\mathcal{D}}_{2}$
(see text). The corresponding integral amplification factor
$G(\unicode[STIX]{x1D714},m)$
is shown in (c) by black line for
${\mathcal{D}}_{1}$
and red line for
${\mathcal{D}}_{2}$
.
Similarly to figure 3(c), figure 11(a) reports the streamwise velocity obtained by using the domain
${\mathcal{D}}_{1}$
in the DNS, which consists of:
$r_{max}=10$
, axial extension of the domain
$x_{max}=20$
and
$80\times 40$
GLC nodes for the velocity components. In figure 11(b) the same quantity is obtained by using the larger domain
${\mathcal{D}}_{2}$
(
$r_{max}=15,~x_{max}=25$
,
$100\times 60$
GLC nodes), thus revealing the lack of influence of the radial and axial extents of the domain and of the free-stress constraint on the base flow in the region of interest for this study.
B.3 Amplification factor and sensitivity map
Similar independence of the radial and streamwise extents of the domain is found for the integral amplification factor that is shown in figure 11(c) by a black line in the case of
$r_{max}=10$
,
$N=40$
GLC nodes,
$N_{x}=30$
and using the
${\mathcal{D}}_{1}$
base flow (in the range
$x\in [0,10]$
). There the red line stands for
$G(\unicode[STIX]{x1D714},m)$
computed with
$r_{max}=15$
,
$N=60$
GLC nodes,
$N_{x}=40$
and using the
${\mathcal{D}}_{2}$
base flow (for
$x\in [0,10]$
).
In addition, the sensitivity map of the total amplification factor with respect to an inflow base-flow modification is shown in figure 9(a) by a full line when using
$r_{max}=10$
,
$N=40$
GLC nodes,
$N_{x}=30$
,
${\mathcal{D}}_{1}$
base flow and
$N_{\unicode[STIX]{x1D714}}=210$
, and by symbols in the case of
$r_{max}=15$
,
$N=60$
GLC nodes,
$N_{x}=40$
,
${\mathcal{D}}_{2}$
base flow and
$N_{\unicode[STIX]{x1D714}}=420$
. The very good agreement between the sensitivity maps computed with different domain sizes and grids represents a significant convergence test because all the numerical tools of the analysis are involved: the local spatial analysis, the DNS, integration in
$x$
and
$\unicode[STIX]{x1D714}$
.
