2.1 Mathematical formulation

The usual approach of stability analysis involves the linearized version of non-dimensional Euler or Navier–Stokes equations, under some simplifying assumptions regarding the base flow (Sipp & Lebedev Reference Sipp and Lebedev2007; Taira et al. Reference Taira, Brunton, Dawson, Rowley, Colonius, McKeon, Schmidt, Gordeyev, Theofilis and Ukeiley2017). Such a linearized and non-dimensional (by the jet diameter $D$ , ambient sound speed $a_{\infty }$ and density $\unicode[STIX]{x1D70C}_{\infty }$ ) set of Euler equations for the disturbances $\boldsymbol{q}(\boldsymbol{x},t)=[u_{x},u_{r},u_{\unicode[STIX]{x1D703}},\unicode[STIX]{x1D70C},p]^{\text{T}}$ , in cylindrical coordinates ( $\boldsymbol{x}\equiv (x,r,\unicode[STIX]{x1D703})$ ), can be written schematically as

(2.1) $$\begin{eqnarray}\displaystyle {\mathcal{L}}_{\bar{\boldsymbol{q}}}\{\boldsymbol{q}(\boldsymbol{x},t)\}=0. & & \displaystyle\end{eqnarray}$$

Here ${\mathcal{L}}_{\bar{\boldsymbol{q}}}\{\cdot \}$ is the linearized operator, which has the form

(2.2) $$\begin{eqnarray}\displaystyle {\mathcal{L}}_{\bar{\boldsymbol{q}}}\{\cdot \}=\frac{\unicode[STIX]{x2202}\{\cdot \}}{\unicode[STIX]{x2202}t}+\boldsymbol{A}(\bar{\boldsymbol{q}})\frac{\unicode[STIX]{x2202}\{\cdot \}}{\unicode[STIX]{x2202}r}+\boldsymbol{B}(\bar{\boldsymbol{q}})\frac{\unicode[STIX]{x2202}\{\cdot \}}{\unicode[STIX]{x2202}x}+\boldsymbol{C}(\bar{\boldsymbol{q}})\frac{\unicode[STIX]{x2202}\{\cdot \}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}+\boldsymbol{D}(\bar{\boldsymbol{q}})\{\cdot \}. & & \displaystyle\end{eqnarray}$$

The coefficients $\boldsymbol{A}$ through $\boldsymbol{D}$ are only dependent on the mean flow $\bar{\boldsymbol{q}}(\boldsymbol{x})$ ,

(2.3) $$\begin{eqnarray}\displaystyle \bar{\boldsymbol{q}}(\boldsymbol{x})=[\bar{u}_{x}(r,\unicode[STIX]{x1D703}),\bar{u}_{r}(r,\unicode[STIX]{x1D703}),\bar{u}_{\unicode[STIX]{x1D703}}(r,\unicode[STIX]{x1D703}),\bar{\unicode[STIX]{x1D70C}}(r,\unicode[STIX]{x1D703}),\bar{p}(r,\unicode[STIX]{x1D703})]^{\text{T}}, & & \displaystyle\end{eqnarray}$$

which in turn does not depend on time or axial coordinate (with the parallel-flow assumption). It follows that the solutions for $\boldsymbol{q}(\boldsymbol{x},t)$ become separable in these directions using normal modes

(2.4) $$\begin{eqnarray}\displaystyle \boldsymbol{q}(\boldsymbol{x},t)=\widetilde{\boldsymbol{q}}(r,\unicode[STIX]{x1D703})\text{e}^{\text{i}\unicode[STIX]{x1D6FC}x}\text{e}^{-\text{i}\unicode[STIX]{x1D714}t}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D714}$ are the axial wavenumber and circular frequency, respectively. In the case of spatial stability formulation, the interest lies in finding (possibly complex) solutions for $\unicode[STIX]{x1D6FC}$ for a given real-valued $\unicode[STIX]{x1D714}$ . When considering strictly parallel ( $\bar{u}_{r}=0$ , $\bar{u}_{\unicode[STIX]{x1D703}}=0$ ) flows, algebraic manipulations to eliminate all variables other than pressure $\widetilde{p}$ lead to a generalized form of the compressible Rayleigh equation (Koshigoe et al. Reference Koshigoe, Gutmark, Schadow and Tubis1988; Gudmundsson & Colonius Reference Gudmundsson and Colonius2007)

(2.5) $$\begin{eqnarray}\displaystyle \frac{1}{r}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}\left(r\frac{\unicode[STIX]{x2202}\widetilde{p}}{\unicode[STIX]{x2202}r}\right)+\frac{1}{r^{2}}\frac{\unicode[STIX]{x2202}^{2}\widetilde{p}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}^{2}}-f\frac{\unicode[STIX]{x2202}\widetilde{p}}{\unicode[STIX]{x2202}r}-g\frac{1}{r^{2}}\frac{\unicode[STIX]{x2202}\widetilde{p}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}-h\widetilde{p}=0, & & \displaystyle\end{eqnarray}$$

where $f$ , $g$ and $h$ are coefficients related to the mean flow

(2.6) $$\begin{eqnarray}\displaystyle & \displaystyle f(r,\unicode[STIX]{x1D703})=\frac{2\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D6FC}\bar{u}_{x}-\unicode[STIX]{x1D714}}\frac{\unicode[STIX]{x2202}\bar{u}_{x}}{\unicode[STIX]{x2202}r}+\frac{1}{\bar{\unicode[STIX]{x1D70C}}}\frac{\unicode[STIX]{x2202}\bar{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x2202}r}, & \displaystyle\end{eqnarray}$$

(2.7) $$\begin{eqnarray}\displaystyle & \displaystyle g(r,\unicode[STIX]{x1D703})=\frac{2\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D6FC}\bar{u}_{x}-\unicode[STIX]{x1D714}}\frac{\unicode[STIX]{x2202}\bar{u}_{x}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}+\frac{1}{\bar{\unicode[STIX]{x1D70C}}}\frac{\unicode[STIX]{x2202}\bar{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}, & \displaystyle\end{eqnarray}$$

(2.8) $$\begin{eqnarray}\displaystyle & \displaystyle h(r,\unicode[STIX]{x1D703})=\unicode[STIX]{x1D6FC}^{2}-\frac{\bar{\unicode[STIX]{x1D70C}}(\unicode[STIX]{x1D6FC}\bar{u}_{x}-\unicode[STIX]{x1D714})^{2}}{\unicode[STIX]{x1D6FE}\bar{p}}. & \displaystyle\end{eqnarray}$$

Previous derivations of this equation assumed implicitly that $\bar{p}=1/\unicode[STIX]{x1D6FE}$ , with $\unicode[STIX]{x1D6FE}$ being the specific heat ratio, which may not be valid for supersonic conditions having shock waves in the pressure mean flow. Thus, a more general formulation is presented herein, where we invoke the ideal gas law for the mean flow, which, under the prevailing non-dimensionalization, is given by $\bar{p}=(\unicode[STIX]{x1D6FE}-1/\unicode[STIX]{x1D6FE})\bar{\unicode[STIX]{x1D70C}}\bar{T}$ .

It is possible to apply a further treatment to this Rayleigh equation using the base-flow periodicity in the azimuthal direction. This follows the Floquet theory (Floquet Reference Floquet1883; Bender & Orszag Reference Bender and Orszag2013) of ordinary differential equations with periodic coefficients, sometimes used to investigate secondary instability in boundary layers (Herbert Reference Herbert1988). If we consider the base flow consisting of $N$ identical corrugations in $\unicode[STIX]{x1D703}$

(2.9) $$\begin{eqnarray}\displaystyle \bar{u}_{x}(r,\unicode[STIX]{x1D703})=\bar{u}_{x}(r,\unicode[STIX]{x1D703}+2\unicode[STIX]{x03C0}/N), & & \displaystyle\end{eqnarray}$$

$\widetilde{p}$ can be written as

(2.10) $$\begin{eqnarray}\displaystyle \widetilde{p}(r,\unicode[STIX]{x1D703})=\widehat{p}(r,\unicode[STIX]{x1D703})e^{\unicode[STIX]{x1D706}\unicode[STIX]{x1D703}}, & & \displaystyle\end{eqnarray}$$

where $\widehat{p}$ has the same periodicity in $\unicode[STIX]{x1D703}$ as in (2.9) and the Floquet exponents ( $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}_{r}+i\unicode[STIX]{x1D707}$ ) could be in principle any complex number. In order to ensure azimuthal periodicity of $\widetilde{p}$ , however, we are restricted to $\unicode[STIX]{x1D706}_{r}=0$ and integer values of $\unicode[STIX]{x1D707}$ . The Rayleigh equation then becomes

(2.11) $$\begin{eqnarray}\displaystyle \frac{1}{r}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}\left(r\frac{\unicode[STIX]{x2202}\widehat{p}}{\unicode[STIX]{x2202}r}\right)+\frac{1}{r^{2}}\frac{\unicode[STIX]{x2202}^{2}\widehat{p}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}^{2}}-\hat{f}\frac{\unicode[STIX]{x2202}\widehat{p}}{\unicode[STIX]{x2202}r}-{\hat{g}}\frac{1}{r^{2}}\frac{\unicode[STIX]{x2202}\widehat{p}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}-{\hat{h}}\widehat{p}=0, & & \displaystyle\end{eqnarray}$$

which has the same form as (2.5), with a slight change in the coefficients,

(2.12) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{f}(r,\unicode[STIX]{x1D703})=\frac{2\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D6FC}\bar{u}_{x}-\unicode[STIX]{x1D714}}\frac{\unicode[STIX]{x2202}\bar{u}_{x}}{\unicode[STIX]{x2202}r}+\frac{1}{\bar{\unicode[STIX]{x1D70C}}}\frac{\unicode[STIX]{x2202}\bar{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x2202}r}, & \displaystyle\end{eqnarray}$$

(2.13) $$\begin{eqnarray}\displaystyle & \displaystyle {\hat{g}}(r,\unicode[STIX]{x1D703})=\frac{2\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D6FC}\bar{u}_{x}-\unicode[STIX]{x1D714}}\frac{\unicode[STIX]{x2202}\bar{u}_{x}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}+\frac{1}{\bar{\unicode[STIX]{x1D70C}}}\frac{\unicode[STIX]{x2202}\bar{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}-2\text{i}\unicode[STIX]{x1D707}, & \displaystyle\end{eqnarray}$$

(2.14) $$\begin{eqnarray}\displaystyle & \displaystyle {\hat{h}}(r,\unicode[STIX]{x1D703})=\unicode[STIX]{x1D6FC}^{2}-\frac{\bar{\unicode[STIX]{x1D70C}}(\unicode[STIX]{x1D6FC}\bar{u}_{x}-\unicode[STIX]{x1D714})^{2}}{\unicode[STIX]{x1D6FE}\bar{p}}+\frac{\unicode[STIX]{x1D707}^{2}}{r^{2}}+\text{i}\left(\frac{2\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x1D6FC}\bar{u}_{x}-\unicode[STIX]{x1D714}}\frac{\unicode[STIX]{x2202}\bar{u}_{x}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}+\frac{1}{\bar{\unicode[STIX]{x1D70C}}}\frac{\unicode[STIX]{x2202}\bar{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}\right)\frac{\unicode[STIX]{x1D707}}{r^{2}}. & \displaystyle\end{eqnarray}$$

The Floquet ansatz allows us to obtain, using (2.11), direct solutions of (2.5) confined to only one of the $N$ identical sectors of the base flow, thus reducing the computational cost.

If the base flow is axisymmetric (i.e. if $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D703}\equiv 0$ ) equation (2.11) readily reduces to its usual axisymmetric form, well known from studies in circular jets (Michalke Reference Michalke1984). In this case, the Floquet exponents ( $\unicode[STIX]{x1D707}$ ) reduce to azimuthal wavenumbers ( $m$ ), which appear when the equation is separable. If the base flow is azimuthally inhomogeneous (i.e.  $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D703}\neq 0$ ), one can still decompose $\widehat{p}$ in a Fourier series owing to its $2\unicode[STIX]{x03C0}/N$ periodicity

(2.15) $$\begin{eqnarray}\displaystyle \widehat{p}(r,\unicode[STIX]{x1D703})=\mathop{\sum }_{j=-\infty }^{\infty }\check{p}_{j}(r)\text{e}^{-\text{i}jN\unicode[STIX]{x1D703}}. & & \displaystyle\end{eqnarray}$$

Substitution in (2.10) yields

(2.16) $$\begin{eqnarray}\displaystyle \widetilde{p}(r,\unicode[STIX]{x1D703})=\mathop{\sum }_{j=-\infty }^{\infty }\check{p}_{j}(r)\text{e}^{\text{i}(\unicode[STIX]{x1D707}-jN)\unicode[STIX]{x1D703}}. & & \displaystyle\end{eqnarray}$$

The above Fourier decomposition shows the correspondence of each Floquet coefficient $\unicode[STIX]{x1D707}$ with the central azimuthal wavenumber $m$ in the flow solutions. For instance, if we consider $N=6$ lobes, the solution for $\unicode[STIX]{x1D707}=0$ comprises the azimuthal wavenumbers $m=(\ldots ,-18,-12,-6,0,6,12,18,\ldots )$ ; for $\unicode[STIX]{x1D707}=1$ the solution is formed by $m=(\ldots ,-17,-11,-5,1,7,13,19,\ldots )$ , and so on. The entire set of unique solutions can then be recovered when we reach $\unicode[STIX]{x1D707}=3$ . This observation is consistent with previous works (Kopiev et al. Reference Kopiev, Ostrikov, Chernyshev and Elliott2004; Gudmundsson Reference Gudmundsson2010; Sinha & Colonius Reference Sinha and Colonius2015; Sinha et al. Reference Sinha, Gudmundsson, Xia and Colonius2016a ). Given that these solutions are formed by a sum of azimuthal modes coupled by the base-flow periodicity, their classification is no longer straightforward. What is usual, at least regarding most non-circular jets studies, is resorting to the symmetries of the eigenfunction shape to classify the solutions. Thus, centre and corner modes appear for rectangular jets (Tam & Thies Reference Tam and Thies1993), modes that peak at flats or vertices are found for triangular configurations (Koshigoe et al. Reference Koshigoe, Gutmark, Schadow and Tubis1989) and solutions with peaks at the flats or tips can be found in corrugated sheets (Kawahara et al. Reference Kawahara, Jiménez, Uhlmann and Pinelli2003) and jets with chevrons (Gudmundsson & Colonius Reference Gudmundsson and Colonius2007; Sinha et al. Reference Sinha, Gudmundsson, Xia and Colonius2016a ). In this work we will restrict our attention to $\unicode[STIX]{x1D707}=0$ , since, as the only case involving the axisymmetric mode, it is likely the most relevant one for the peak jet noise (Michalke Reference Michalke1970; Cavalieri et al. Reference Cavalieri, Jordan, Colonius and Gervais2012). We expect the conclusions reached in this work to extend to other $\unicode[STIX]{x1D707}$ cases (Kopiev et al. Reference Kopiev, Ostrikov, Chernyshev and Elliott2004; Sinha et al. Reference Sinha, Gudmundsson, Xia and Colonius2016a ).

In our present formulation, solutions are eigenfunctions $\widehat{p}(r,\unicode[STIX]{x1D703})$ defined in an azimuthal slice of the jet, that can be transformed back to the whole domain by application of (2.10). Invoking the linear governing equations, velocity components can be obtained as

(2.17) $$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{u}_{r}(r,\unicode[STIX]{x1D703})=-\frac{1}{\text{i}\bar{\unicode[STIX]{x1D70C}}(\unicode[STIX]{x1D6FC}\bar{u}_{x}-\unicode[STIX]{x1D714})}\frac{\unicode[STIX]{x2202}\widetilde{p}}{\unicode[STIX]{x2202}r}, & \displaystyle\end{eqnarray}$$

(2.18) $$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{u}_{\unicode[STIX]{x1D703}}(r,\unicode[STIX]{x1D703})=-\frac{1}{\text{i}\bar{\unicode[STIX]{x1D70C}}(\unicode[STIX]{x1D6FC}\bar{u}_{x}-\unicode[STIX]{x1D714})}\frac{1}{r}\frac{\unicode[STIX]{x2202}\widetilde{p}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}, & \displaystyle\end{eqnarray}$$

(2.19) $$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{u}_{x}(r,\unicode[STIX]{x1D703})=-\left(\frac{1}{r^{2}}\frac{\unicode[STIX]{x2202}\bar{u}_{x}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}\frac{\unicode[STIX]{x2202}\widetilde{p}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}+\frac{\unicode[STIX]{x2202}\bar{u}_{x}}{\unicode[STIX]{x2202}r}\frac{\unicode[STIX]{x2202}\widetilde{p}}{\unicode[STIX]{x2202}r}+(\unicode[STIX]{x1D6FC}\bar{u}_{x}-\unicode[STIX]{x1D714})\unicode[STIX]{x1D6FC}\widetilde{p}\right)/\bar{\unicode[STIX]{x1D70C}}(\unicode[STIX]{x1D6FC}\bar{u}_{x}-\unicode[STIX]{x1D714})^{2}, & \displaystyle\end{eqnarray}$$

and density fluctuations are given by

(2.20) $$\begin{eqnarray}\displaystyle \widetilde{\unicode[STIX]{x1D70C}}(r,\unicode[STIX]{x1D703})=-\frac{\bar{\unicode[STIX]{x1D70C}}}{\text{i}(\unicode[STIX]{x1D6FC}\bar{u}_{x}-\unicode[STIX]{x1D714})}\left(\text{i}\unicode[STIX]{x1D6FC}\widetilde{u}_{x}+\frac{1}{r}\frac{\unicode[STIX]{x2202}(r\widetilde{u}_{r})}{\unicode[STIX]{x2202}r}+\frac{1}{r}\frac{\unicode[STIX]{x2202}\widetilde{u}_{\unicode[STIX]{x1D703}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}\right). & & \displaystyle\end{eqnarray}$$

2.2 Numerical solution

To obtain the direct solution for the spatial stability problem, equation (2.11) is rearranged into a generalized eigenvalue problem that is cubic in eigenvalues $\unicode[STIX]{x1D6FC}$ . Collocation methods, with Chebyshev polynomials along the radius and trigonometric polynomials in the azimuth (Trefethen Reference Trefethen2000; Boyd Reference Boyd2001), lead to a discrete version of the problem

(2.21) $$\begin{eqnarray}\displaystyle (\unicode[STIX]{x1D641}_{3}\unicode[STIX]{x1D6FC}^{3}+\unicode[STIX]{x1D641}_{2}\unicode[STIX]{x1D6FC}^{2}+\unicode[STIX]{x1D641}_{1}\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D641}_{0})\widehat{p}=0, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D641}_{j}$ are matrices and $\widehat{p}$ are values at collocation points. A standard linear generalized eigenvalue problem is obtained rearranging the system,

(2.22) $$\begin{eqnarray}\displaystyle \left[\begin{array}{@{}ccc@{}}0 & I & 0\\ 0 & 0 & I\\ -\unicode[STIX]{x1D641}_{0} & -\unicode[STIX]{x1D641}_{1} & -\unicode[STIX]{x1D641}_{2}\\ \end{array}\right]\left[\begin{array}{@{}c@{}}\widehat{p}\\ \unicode[STIX]{x1D6FC}\widehat{p}\\ \unicode[STIX]{x1D6FC}^{2}\widehat{p}\end{array}\right]=\unicode[STIX]{x1D6FC}\left[\begin{array}{@{}ccc@{}}I & 0 & 0\\ 0 & I & 0\\ 0 & 0 & \unicode[STIX]{x1D641}_{3}\\ \end{array}\right]\left[\begin{array}{@{}c@{}}\widehat{p}\\ \unicode[STIX]{x1D6FC}\widehat{p}\\ \unicode[STIX]{x1D6FC}^{2}\widehat{p}\end{array}\right]. & & \displaystyle\end{eqnarray}$$

The boundary conditions (in the $r$ direction) normally consider asymptotic solutions at $r\rightarrow \infty$ (with Bessel or Hankel equations), which allow numerical solutions via shooting procedures (Michalke Reference Michalke1964; Morris & Bhat Reference Morris and Bhat1995; Gudmundsson Reference Gudmundsson2010). However, here we employ a Dirichlet boundary condition at a large but finite distance from the centreline, $r_{M}$ (Batchelor & Gill Reference Batchelor and Gill1962). The Chebyshev polynomials are mapped from $\unicode[STIX]{x1D709}$ space, $[-1,1]$ , to $(0,r_{M}]$ , where most collocation points can be concentrated within the shear-layer region by a two-parameter transformation (Lesshafft & Huerre Reference Lesshafft and Huerre2007),

(2.23) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D709}(r)=\frac{r_{c}}{r}-\sqrt{1+\frac{r_{c}^{2}}{4r^{2}}+\frac{2r_{c}}{r_{M}}-\frac{r_{c}}{r}}, & \displaystyle\end{eqnarray}$$

(2.24) $$\begin{eqnarray}\displaystyle & \displaystyle r(\unicode[STIX]{x1D709})=r_{c}\frac{1-\unicode[STIX]{x1D709}}{1-\unicode[STIX]{x1D709}^{2}+2r/r_{M}}, & \displaystyle\end{eqnarray}$$

so that approximately half of the points $r(\unicode[STIX]{x1D709})$ are placed in the interval $0<r\leqslant r_{c}$ , concentrated around $r=r_{c}/2$ . Values of $r_{c}=0.9$ and $r_{M}=50$ have been used in all calculations. Close to the centreline, $r=10^{-6}$ , we apply Neumann boundary conditions.

Periodic boundary conditions (in the $\unicode[STIX]{x1D703}$ direction) are inherited by the application of evenly spaced periodic grids (Trefethen Reference Trefethen2000). These are simply scaled to the $[0,2\unicode[STIX]{x03C0}/N]$ domain. Every calculation throughout this work is repeated with a coarser grid, in order to check discretization convergence. We verify that approximately $240$ collocation points in the radial direction and 20 points along the azimuthal slice are sufficient to obtain converged numerical solutions for Strouhal numbers, $St=\unicode[STIX]{x1D714}D/(2\unicode[STIX]{x03C0}U_{j})$ , up to $3.0$ , with $U_{j}$ the jet exit velocity. A validation of the numerical approach is carried out by comparisons with Morris (Reference Morris1988) and Gudmundsson (Reference Gudmundsson2010), who have studied elliptic and serrated jets, respectively. Results are shown in figure 2 and figure 3 where we see a close match in growth rates and phase speeds of the most unstable mode found for each configuration.

Figure 2.

Growth rate (a,c) and phase velocity (b,d) comparisons, for $ce_{0}$ (a,b) and $se_{1}$ (c,d) solutions, from $A/B=1.001$ (solid line) and $2.0$ (dashed) elliptic profiles given by Morris (Reference Morris1988). Here, $ce_{0}$ and $se_{1}$ stand for cosine elliptic and sine elliptic solutions related to the Mathieu functions, and $A/B$ is the ratio between the major and minor axes in the considered elliptic mean velocity profile. This notation is the same as that used by Morris (Reference Morris1988). Circles indicate the present results, and lines show results by Morris (Reference Morris1988).

Figure 3.

Comparisons with growth rates (a) and phase velocities (b), at $x/D=2.0$ , for the SMC000 nozzle (dashed) and SMC001 nozzle (solid). Present numerical results are represented by the circles, and lines show results by Gudmundsson (Reference Gudmundsson2010).

4.1 Classification of modes based on continuation from circular jets

We first explore solutions of the linear stability problem for corrugated jets based on the analysis of the jet issuing from the SMC001 nozzle. This choice also allows direct comparison with corresponding results presented by Gudmundsson & Colonius (Reference Gudmundsson and Colonius2007) and Sinha et al. (Reference Sinha, Gudmundsson, Xia and Colonius2016a ). Growth rates $\unicode[STIX]{x1D6FC}_{i}$ and phase speeds $c_{ph}$ obtained for $\unicode[STIX]{x1D707}=0$ at $St=0.35$ are presented in figure 5. The continuous branch of critical-layer modes, with $\unicode[STIX]{x1D6FC}_{i}=0$ , which results from a singularity of the Rayleigh equation (2.11) whenever $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D6FC}\bar{u}_{x}$ (Case Reference Case1960), can be seen along the real axis. However, since the behaviour of large-scale turbulent structures near the nozzle is expected to be dominated by unstable discrete modes, we will focus on these in what follows.

Figure 5.

Eigenspectra of SMC001 at (a) $x/D=0.5$ , (b) $x/D=1.0$ and (c) $x/D=2.0$ , for $St=0.35$ . Pressure eigenfunctions are shown by their normalized absolute values, and are ordered by their magnitude of $-\unicode[STIX]{x1D6FC}_{i}$ . Thick red curves represent the base-flow axial velocity contour at $\bar{u}_{x}=0.5$ .

Multiple unstable solutions (discrete) are found near the nozzle region ( $x/D=0.5$ and $1.0$ ), but only one unstable mode is obtained further downstream ( $x/D=2.0$ ). These are mostly in good agreement with previous studies (Gudmundsson & Colonius Reference Gudmundsson and Colonius2011; Sinha et al. Reference Sinha, Gudmundsson, Xia and Colonius2016a ), especially modes (ii) and (iii). The associated eigenfunctions are also consistent with the results of the cited works, with some peaking at the lobes of the base flow and others at the flats. Mode (i) that appears at the first two stations are not reported by previous studies.

Our goal is to trace the origin of these multiple modes, by exploring the features of our base-flow parametrization. By continuously varying the fitted $A_{j}$ and $B_{j}$ (simultaneously) with a perturbation factor ( $P$ ) between $0$ and $1.3$ , such that

(4.1) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{R}(\unicode[STIX]{x1D703})=R_{0}\left(1+P\mathop{\sum }_{j=1}^{\infty }A_{j}\cos (jN\unicode[STIX]{x1D703})\right), & \displaystyle\end{eqnarray}$$

(4.2) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D723}(\unicode[STIX]{x1D703})=\unicode[STIX]{x1D6E9}_{0}\left(1+P\mathop{\sum }_{j=1}^{\infty }B_{j}\cos (jN\unicode[STIX]{x1D703})\right), & \displaystyle\end{eqnarray}$$

we can track the solution behaviour as a continuation of corresponding ones for circular jets. Note that $P=1$ stands for the original base flow of the SMC001 jet with chevrons, but smaller $P$ values are also found to approximate the base-flow coefficients of the jet with micro-jets (from table 1). Thus, $P$ can be seen as a metric for corrugation amplitude, which can be continuously varied from a circular (azimuthally averaged) jet to increased corrugated profiles. Results are shown in figure 6, for $x/D=0.5$ , $1.0$ and $2.0$ .

Figure 6.

Eigensolution branches traced with gradual increase of azimuthal corrugation parameter $P$ of (4.1)–(4.2) at $x/D=0.5$ (a), $1.0$ (b) and $2.0$ (c), for $St=0.35$ . Open circles are results for $P=0$ , whereas end dots are results for $P=1.3$ . Arrows indicate directions of increasing $P$ . Presentation of eigenfunctions follows the scheme of figure 5.

The starting points are solutions obtained for $P=0$ . For such circular jets, azimuthal wavenumbers $m=0$ and $m=\pm 6$ possess unstable Kelvin–Helmholtz modes for sufficiently thin shear layers. Such modes are contained in the $\unicode[STIX]{x1D707}=0$ Floquet ansatz, and are represented in figure 6 with open circles; notice that $m=-6$ and $6$ have the same eigenvalue (in $x/D=0.5$ ), and are thus degenerate eigenfunctions. As small azimuthal inhomogeneities are included ( $P\ll 1$ ), we verify the break up of the $m=\pm 6$ degeneracy. Such behaviour was observed analytically by Kopiev et al. (Reference Kopiev, Ostrikov, Chernyshev and Elliott2004), and we expect it to be a common feature in both corrugated jets considered here. Since eigenfunctions for $P>0$ can no longer be associated with a single azimuthal wavenumber $m$ , we identify these solutions according to the azimuthal mode from which they emerge (now labelled as $\tilde{m}$ ) and whether the (pressure) eigenfunction has peaks aligned with the lobes (referred to with superscript ‘ $c$ ’) or flats (superscript ‘ $s$ ’), making reference to their cosinusoidal and sinusoidal character (Morris & Miller Reference Morris and Miller1984; Koshigoe & Tubis Reference Koshigoe and Tubis1986). Thus the $\tilde{m}=0$ mode refers to the continuation of the axisymmetric mode of circular jets ( $P=0$ ) for corrugated ones ( $P>0$ ), and $\tilde{m}=6^{c}$ and $6^{s}$ refer to the continuations of the $m=\pm 6$ modes. The solutions for $m=\pm 6$ are not visible at $x/D=1.0$ and $2.0$ for $P=0$ , but their continuations become apparent after some level of perturbation ( $P=0.5$ and $P=1.1$ , respectively), forming similar paths to those found in $x/D=0.5$ . We thus conclude that these modes are stable for $P=0$ at these stations, and cannot be found as solutions of the Rayleigh equation, unless a deformation of the integration contour is taken to avoid the critical-layer singularity (Tam & Morris Reference Tam and Morris1980; Boyd Reference Boyd1985).

These results highlight that the multiple unstable modes obtained for non-circular jets, seen in previous studies (Gudmundsson & Colonius Reference Gudmundsson and Colonius2007; Sinha et al. Reference Sinha, Gudmundsson, Xia and Colonius2016a ), are all related to the Kelvin–Helmholtz mechanism, as they can be tracked from the standard Kelvin–Helmholtz modes of circular jets. The multiple unstable modes are simple continuations of circular jet modes, keeping in mind that all eigenfunctions that are consistent with the Floquet ansatz for a jet with $N$ corrugations should be tracked. For the present base flow, with $N=6$ corrugations, three unstable modes were seen to be continuations of $m=0$ and $\pm 6$ Kelvin–Helmholtz modes of the corresponding azimuthally averaged jet.

In general, these gradual base-flow deformations are observed to be more significant at stations $x/D=0.5$ and $1.0$ , promoting higher growth rates for modes $\tilde{m}=6^{s}$ , but with minor variations of their phase velocity. Conversely, modes $\tilde{m}=6^{c}$ are found to increase their phase speeds for higher $P$ , but without much variation in growth rates. Solutions for $\tilde{m}=0$ are found to become more unstable with increasing $P$ , presenting minor variations in their phase velocities. The exceptions here are small values of $P$ at $x/D=1.0$ and the results further downstream, at $x/D=2.0$

Regarding the eigenfunction shape, an increase in the base-flow deformation has a significant effect on the eigenfunctions for the $\tilde{m}=0$ and $\tilde{m}=6^{c}$ modes, with the former becoming less axisymmetric and the latter becoming more so. Such redistribution of azimuthal coherence suggests that, in principle, both $\tilde{m}=0$ and $\tilde{m}=N^{c}$ should be considered as acoustically relevant structures in corrugated jet profiles. The streamwise evolution of the pressure fluctuations is illustrated in figure 7, from solutions obtained at $x/D=0.5$ applied to (2.4). We have presented the magnitudes of the pressure eigenfunction in all other figures; here we depict isocontours of their real part to better represent the wavepacket shape in the near-nozzle region.

Figure 7.

Reconstructed pressure disturbances for (a,d,g) $\tilde{m}=0$ , (b,e,h) $\tilde{m}=6^{c}$ and (c,f,i) $\tilde{m}=6^{s}$ . Representative positive and negative of the real part of the pressure ( $\pm 0.5$ , normalized values). Results obtained for $P=0.3$ (a–c), $0.6$ (d–f) and $1.3$ (g–i). The base flow is indicated by the $\bar{u}_{x}=0.5$ contour.

Another feature observed in these solutions is that at $x/D=0.5$ , the modes forming the $\tilde{m}=0$ branch peak at the base-flow flats, while those for $\tilde{m}=6^{c}$ appear to evolve from something similar to the original $m=\pm 6$ shape, at small $P$ , to modes that are more axisymmetric with peaks occurring at the base-flow peaks. Solutions at $x/D=2.0$ behave in the opposite manner, with the increase of $P$ modifying the eigenfunction shape of $\tilde{m}=0$ to those with peaks aligned with the base-flow peaks. This will be further investigated in what follows.

Figure 8.

Saddles for perturbed profiles at (a) $x/D=0.5$ and (b) $x/D=1.0$ , formed by $\tilde{m}=0$ (blue) and $\tilde{m}=6^{c}$ (red) defined at $P\ll 1$ . Arrows indicate direction of increased $P$ , with equally spaced steps of $0.1$ , from $0$ to $1.3$ . For fixed $P$ , the grey lines show the trajectory of the modes as frequency is increased from $St=0.15$ to $0.35$ (from bottom to top).

4.2 Axis switching of eigenfunctions of unstable modes

The complex behaviour of the eigenfunctions for the $\tilde{m}=0$ and $\tilde{m}=6^{c}$ modes can be understood as an axis switching that occurs when these two modes form a saddle in the complex $\unicode[STIX]{x1D6FC}$ plane as the (real) frequency is varied. Figure 8 shows that at both $x/D=0.5$ and $x/D=1.0$ , a saddle is formed and the two modes exchange their identities at the saddle. This is similar to what has been found as the temperature ratio in a round jet is varied (Michalke Reference Michalke1970), which was later found to be related to the occurrence of absolute instability in hot jets (Huerre & Monkewitz Reference Huerre and Monkewitz1985). In that case, however, the saddle is formed by two solutions whose group velocities are in the opposite directions, which is not expected to be the case here. Indeed, by application of the Briggs criterion (Briggs Reference Briggs1964; Huerre et al. Reference Huerre, Batchelor, Moffatt and Worster2000), we confirmed that all these discrete unstable solutions comprise downstream propagating waves, and thus are not related to absolute instability. Saddles formed by two downstream travelling waves also appear in high supersonic cold jets, where the Kelvin–Helmholtz mode blends itself with supersonic instability wave modes (Tam & Hu Reference Tam and Hu1989).

A peculiarity related to eigenfunctions of the two saddle modes is shown in figure 9. At very low $St$ values, $\tilde{m}=0$ closely follows the base-flow shape, while $\tilde{m}=6^{c}$ tends to present peaks at the flats. For situations where $P<1.1$ in $x/D=0.5$ (and $P<1.0$ in $x/D=1.0$ ), the increase of $St$ is followed by the switch of behaviour between these two solutions, which alters the azimuthal location of their peaks; this can be thought of an axis switching of eigenfunctions. For higher $P$ this is not observed, but then an exchange of the solutions trajectories occurs, with $\tilde{m}=6^{c}$ taking similar paths to previous $\tilde{m}=0$ solutions, and vice versa. One should observe that the gradual perturbations at a fixed $St=0.35$ , shown in figure 6, are therefore not restricted to a single mode continuation along $\tilde{m}=0$ and $\tilde{m}=6^{c}$ branches. Our mode classification thus follows the azimuthal mode continuation found at lower $St$ . Please note the supplemental material that is made available in the online version of the paper at https://doi.org/10.1017/jfm.2019.573.

Figure 9.

Axis switching of solutions near the saddle point in $x/D=0.5$ , for $\tilde{m}=6^{c}$ (a) and $\tilde{m}=0$ (b). Branches of $P=0.7$ (red) and $1.3$ (blue) are emphasized, showing the eigenfunctions for $St=0.15$ , $0.2$ , $0.25$ and $0.3$ from bottom to top.

5.1 Azimuthal inhomogeneities of $\boldsymbol{R}$ and $\unicode[STIX]{x1D723}$

The variation in growth rate and phase speed as $P$ is varied are shown in figure 11. Recall that varying $P$ alters the magnitude of the azimuthal inhomogeneity, and thus, for instance, mimics the effect of chevron penetration (Bridges & Brown Reference Bridges and Brown2004); however, one should bear in mind that increasing penetration is, in a real jet, invariably accompanied by higher azimuthally averaged shear-layer thickness $\unicode[STIX]{x1D6E9}_{0}$ . In the present analysis we are able to isolate these two effects. We concentrate on the azimuthal variation aspect of penetration only, as the effect of increasing shear-layer thickness is well understood to be stabilizing (Michalke Reference Michalke1984).

In the near-nozzle region (refer to figure 11 a), we notice that the growth rates and the range of unstable frequencies change more significantly for $\tilde{m}=0$ , and that increasing $P$ is destabilizing for this mode; for $\tilde{m}=6^{c}$ there are only mild changes in growth rate induced by azimuthal inhomogeneities in the base flow. The tendency is mostly monotonic with $P$ , until an exchange of $\tilde{m}=0$ and $\tilde{m}=6^{c}$ (in $P\geqslant 1.1$ ) due to a saddle around $St\approx 0.2$ . On the other hand, regarding phase speeds, $\tilde{m}=6^{c}$ is observed to be the most affected mode. Dominant unstable solutions (those with highest growth rate) are found to closely follow phase velocities obtained from the zeroth-order approximations. Downstream (see figure 11 b), growth rates of mode $\tilde{m}\,=\,0$ are slightly reduced, which is opposite to what is seen near the nozzle. Phase speeds are nearly insensitive to the increase of $P$ .

Further investigation is carried out to ascertain the effect of the applied azimuthal inhomogeneities in a more decoupled manner. By imposing $B_{j}=0$ in (3.3), the perturbations are isolated in the $\boldsymbol{R}$ term. Conversely, by enforcing $A_{j}=0$ , azimuthal variations are confined to $\unicode[STIX]{x1D723}$ . The results for the individual influence of $\boldsymbol{R}$ and $\unicode[STIX]{x1D723}$ are shown in figures 12 and 13, respectively.

Figure 11.

Top: base-flow deformation, iso-lines for $M=0.9$ and $0.1$ . Growth rates and phase speeds for perturbed $P$ (a) near the nozzle, $x/D=0.5$ and (b) downstream, $x/D=2.0$ . Solid lines are $\tilde{m}=0$ and dashed $\tilde{m}=6^{c}$ , coloured by increasing $P$ within the range: $0.25$ , $0.5$ , $0.75$ , $1.0$ and $1.1$ (lightest grey $P=0.25$ , black $P=1.1$ ). Zeroth-order results are solid blue for $m=0$ and dashed blue for $m=\pm 6$ .

Figure 12.

Growth rates and phase speeds for the $\boldsymbol{R}$ effect (a) near the nozzle and (b) downstream. Solid lines are $\tilde{m}=0$ and dashed $\tilde{m}=6^{c}$ . Coloured by increasing $P$ , within the range: $0.25$ , $0.5$ , $0.75$ , $1.0$ and $1.1$ (lightest grey $P=0.25$ , black $P=1.1$ ). Zeroth-order results are shown in solid blue lines for $m=0$ and dashed for $m=\pm 6$ .

Figure 13.

Growth rates and phase speeds for the $\unicode[STIX]{x1D723}$ effect (a) near the nozzle and (b) downstream. Solid lines are $\tilde{m}=0$ and dashed $\tilde{m}=6^{c}$ . Coloured by increasing $P$ , within the range: $0.25$ , $0.5$ , $0.75$ , $1.0$ and $1.1$ (lightest grey $P=0.25$ , black $P=1.1$ ). Zeroth-order results are shown in solid blue lines for $m=0$ and dashed for $m=\pm 6$ .

Interestingly, the $\boldsymbol{R}$ effect seems to control the change in phase speed observed in the combined $\boldsymbol{R}/\unicode[STIX]{x1D723}$ effect, but with little change in growth rate, whereas $\unicode[STIX]{x1D723}$ variation has the opposite effect. In both cases the changes are pronounced at $x/D=0.5$ , but by $x/D=2.0$ the phase speeds and growth rates are again insensitive to $\boldsymbol{R}$ and $\unicode[STIX]{x1D723}$ modulations applied independently, as they were when applied in unison.

5.2 Flow periodicity $N$

Similar to penetration, the effect of increasing the number of corrugations in real jets may be accompanied by a total increase of the azimuthally averaged mixing-layer thickness $\unicode[STIX]{x1D6E9}_{0}$ (Bridges & Brown Reference Bridges and Brown2004). Again, our focus is restricted to the azimuthal variations. In order to investigate the influence of base-flow periodicity, all fit results for the SMC001 nozzle from table 1 are maintained fixed, while the number of identical domain corrugations $N$ is varied. This has the outcome that more modes appear together, especially when larger pieces of the flow domain are considered. Our attention is restricted to $\tilde{m}=0$ and $N^{c}$ (here only considering $8^{c}$ , as it overtakes $\tilde{m}=0$ ). Results are shown in figure 14.

Figure 14.

Growth rates and phase speeds for varying $N$ (a) near the nozzle region and (b) downstream. Solid lines are $\tilde{m}=0$ and dashed $\tilde{m}=N^{c}$ . Colours are for increasing $N$ , within the range: $1$ , $2$ , $3$ , $4$ , $5$ , $6$ , $7$ and $8$ (lightest grey $N=1$ , black $N=8$ ). Zeroth-order results are shown in solid blue lines for $m=0$ .

The increase of $N$ is found to mostly reduce growth rates and phase speeds of the dominant unstable solutions in the near-nozzle region. There is an apparent trade-off in growth rates, increasing with $N$ (when $St<0.7$ ) and decreasing (when $St>0.7$ ). Such behaviour is related to a second saddle, between $\tilde{m}=0$ and $2N^{c}$ (not shown here), that appears with further increase of $N$ (around $N\geqslant 7$ ). The behaviour of these modes around the saddle mostly follows the findings from § 4.2, and for this reason is not further discussed. For the downstream region, solutions are again found to be quite robust, depending only weakly on the number of corrugations.