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

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

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

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

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

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

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:

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

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

where the complex vector

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

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).

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:

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:

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 $\unicode[STIX]{x1D740}$ :

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

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 $\unicode[STIX]{x1D6FF}\boldsymbol{U}_{0}$ :

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

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

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.

5.1 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:

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:

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 $\unicode[STIX]{x1D743}$ :

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,

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):

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:

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.