2.1 The governing equation for non-axisymmetric vortex-sheet flows

Following the routine procedure of stability analysis, we decompose the flow into base and fluctuation parts. Note that the time-average mean flow is often taken as the base flow, and hence in this paper we use the mean flow and base flow interchangeably. We start with the incompressible Navier–Stokes equations since installed jet noise is relevant primarily at low Mach numbers. At this stage, we write equations in a vector form to avoid the introduction of coordinate systems. The momentum equation can be written as

(2.1) $$\begin{eqnarray}\frac{\text{D}\boldsymbol{v}}{\text{D}t}=-\frac{1}{\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D735}p+\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{v},\end{eqnarray}$$

where $t$ denotes time, $\boldsymbol{v}$ denotes the fluid velocity, $\unicode[STIX]{x1D70C}$ denotes the flow density, $p$ denotes the pressure and $\unicode[STIX]{x1D708}$ denotes the dynamic viscosity. We assume that base flow is steady, with flow density, velocity and pressure to be $\unicode[STIX]{x1D70C}$ , $\boldsymbol{U}$ and $\bar{p}$ , respectively, and that the base flow satisfies

(2.2) $$\begin{eqnarray}\frac{\text{D}\boldsymbol{U}}{\text{D}t}=-\frac{1}{\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D735}\bar{p}+\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{U}.\end{eqnarray}$$

The total flow field is given by the sum of the base flow and the small perturbation. After substituting the total flow into (2.1) and ignoring second-order quantities, we have the following linearized equation:

(2.3) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\boldsymbol{v}^{\prime }}{\unicode[STIX]{x2202}t}+\boldsymbol{U}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{v}^{\prime }+\boldsymbol{v}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{U}=-\frac{1}{\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D735}p^{\prime }+\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{v}^{\prime },\end{eqnarray}$$

where the prime symbols denote the corresponding fluctuation quantities. When the Reynolds number is high, we expect that the viscous term plays a negligible role. Hence the term $\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{v}^{\prime }$ can be neglected, i.e.

(2.4) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\boldsymbol{v}^{\prime }}{\unicode[STIX]{x2202}t}+\boldsymbol{U}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{v}^{\prime }+\boldsymbol{v}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{U}=-\frac{1}{\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D735}p^{\prime }.\end{eqnarray}$$

Equation (2.4), together with the incompressible continuity equation

(2.5) $$\begin{eqnarray}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{v}^{\prime }=0,\end{eqnarray}$$

governs the small-amplitude inviscid perturbations over a steady base flow.

The perturbation field is generally rotational for general shear base flows. Therefore, it cannot be described using a potential function. However, for parallel flows of a vortex-sheet type, the base flows inside and outside the vortex sheet are both irrotational. Hence, the perturbation should also be irrotational. This suggests the existence of a potential function $\unicode[STIX]{x1D713}$ for the velocity perturbations, i.e.

(2.6) $$\begin{eqnarray}\boldsymbol{v}^{\prime }=\unicode[STIX]{x1D735}\unicode[STIX]{x1D713},\end{eqnarray}$$

on either side of the vortex sheet. The function $\unicode[STIX]{x1D713}$ will be discontinuous across the shear layer. From the (2.5), one can see that the velocity potential satisfies the Laplace equation, i.e.

(2.7) $$\begin{eqnarray}\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D713}=0.\end{eqnarray}$$

The pressure perturbation is effectively decoupled from the velocity potential and can be easily obtained from (2.4).

The vortex-sheet simplification was used extensively in stability analysis, in large due to the fact that an analytical dispersion relation can be generally found (Batchelor & Gill Reference Batchelor and Gill1962; Crighton Reference Crighton1973; Kawahara et al. Reference Kawahara, Jiménez, Uhlmann and Pinelli2003). From these dispersion relation, one can gain more insight than numerical simulations can offer. In addition, the vortex-sheet simplification is often permissible, particularly for analysing low-frequency instability waves. Though a realistic jet mean flow spreads slowly and has an increasingly thick mixing layer towards downstream, the vortex-sheet simplification should serve as a good approximation to the realistic flow close to the jet nozzle. Therefore, in this paper we assume the base flow to be parallel and of a vortex-sheet type.

Since the velocity potentials exist for the vortex-sheet problem, we let $\unicode[STIX]{x1D713}^{+}$ and $\unicode[STIX]{x1D713}^{-}$ denote the potentials outside and inside of the vortex sheet, respectively, i.e.

(2.8) $$\begin{eqnarray}{\boldsymbol{v}^{\prime }}^{\pm }=\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}^{\pm },\end{eqnarray}$$

where ${\boldsymbol{v}^{\prime }}^{+}$ and ${\boldsymbol{v}^{\prime }}^{-}$ denote the velocity perturbations outside and inside the vortex sheet, respectively. Considering the parallel-flow assumption, we now introduce the cylindrical coordinates $\unicode[STIX]{x1D70E}$ , $\unicode[STIX]{x1D719}$ and $z$ , as shown in figure 1. In this coordinate frame, the velocity potentials $\unicode[STIX]{x1D713}^{\pm }(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719},z,t)$ satisfy the following Laplace equation:

(2.9) $$\begin{eqnarray}\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D713}^{\pm }(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719},z,t)=0.\end{eqnarray}$$

Note that we have not yet restricted the profiles of the vortex sheet. It therefore can be of arbitrary geometry, such as rectangular, elliptic or lobed. One can therefore let a general function ${\mathcal{R}}(\unicode[STIX]{x1D719})$ denote the radius of the vortex sheet at polar angle $\unicode[STIX]{x1D719}$ . Consequently, the profile of the vortex sheet can be specified as

(2.10) $$\begin{eqnarray}{\mathcal{F}}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})=\unicode[STIX]{x1D70E}-{\mathcal{R}}(\unicode[STIX]{x1D719})=0.\end{eqnarray}$$

2.2 The eigenvalue problem

Without losing generality, one may assume

(2.11) $$\begin{eqnarray}\unicode[STIX]{x1D713}^{\pm }=\mathop{\sum }_{m=-\infty }^{\infty }A_{m}E_{m}^{\pm }(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})\text{e}^{\text{i}\unicode[STIX]{x1D6FC}z}\text{e}^{-\text{i}\unicode[STIX]{x1D714}t},\end{eqnarray}$$

where $A_{m}$ are complex constants, $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D714}$ are the streamwise wavenumber and frequency, respectively. The functions $E_{m}^{\pm }(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})$ are linearly independent of each other and each pair of them at a given $m$ satisfies both the governing equations and appropriate boundary conditions. The functions $E_{m}^{+}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})$ and $E_{m}^{-}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})$ are therefore defined in the two-dimensional regions outside ( $\unicode[STIX]{x1D70E}>{\mathcal{R}}(\unicode[STIX]{x1D719})$ ) and inside ( $\unicode[STIX]{x1D70E}\leqslant {\mathcal{R}}(\unicode[STIX]{x1D719})$ ) the vortex-sheet profile, respectively, as shown in figure 2.

Figure 1. The cylindrical coordinate system: the $z$ axis is parallel to the vortex sheet, the origin is at the centre of the lobed profile and $\unicode[STIX]{x1D719}$ denotes the anticlockwise angle between the radial axis and the horizontal dashed line.

Figure 2. Schematic illustration of the boundary conditions of the stability problem of a parallel lobed vortex sheet.

We choose to normalize the functions $E_{m}^{-}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})$ such that for any integer $m$

(2.12) $$\begin{eqnarray}\frac{1}{2\unicode[STIX]{x03C0}}\int _{0}^{2\unicode[STIX]{x03C0}}E_{m}^{-}(a,\unicode[STIX]{x1D719}){E_{m}^{-}}^{\ast }(a,\unicode[STIX]{x1D719})\,\text{d}\unicode[STIX]{x1D719}=1,\end{eqnarray}$$

where $a$ , as defined in § 1 and shown in figure 2, is the mean radius of the vortex sheet, which is defined by

(2.13) $$\begin{eqnarray}a=\frac{1}{2\unicode[STIX]{x03C0}}\int _{0}^{2\unicode[STIX]{x03C0}}{\mathcal{R}}(\unicode[STIX]{x1D719})\,\text{d}\unicode[STIX]{x1D719}.\end{eqnarray}$$

We cannot normalize $E_{m}^{+}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})$ according to (2.12), because $E_{m}^{-}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})$ and $E_{m}^{+}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})$ are not independent. In fact, either the dynamic or the kinematic boundary condition is sufficient to determine $E_{m}^{+}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})$ from a given $E_{m}^{-}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})$ . Therefore, $E_{m}^{+}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})$ has to be determined such that for the same mode number $m$ , the functions $E_{m}^{+}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})$ and $E_{m}^{-}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})$ satisfy the boundary conditions. One should note that $E_{m}^{-}(a,\unicode[STIX]{x1D719})$ are not properly defined within some ranges of $\unicode[STIX]{x1D719}$ (because $E_{m}^{-}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})$ is defined for $\unicode[STIX]{x1D70E}\leqslant {\mathcal{R}}(\unicode[STIX]{x1D719})$ and $a>{\mathcal{R}}(\unicode[STIX]{x1D719})$ in some ranges of $\unicode[STIX]{x1D719}$ , see figure 2 for example). However, they can be naturally defined using analytical continuation, which will become clear at a later stage. Hence, the normalization defined by (2.12) is valid.

Substituting $E_{m}^{\pm }(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})$ into (2.9) yields the governing equation

(2.14) $$\begin{eqnarray}\left[\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}^{2}}+\unicode[STIX]{x1D70E}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}}\left(\unicode[STIX]{x1D70E}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}}\right)-\unicode[STIX]{x1D6FC}^{2}\unicode[STIX]{x1D70E}^{2}\right]E_{m}^{\pm }(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})=0.\end{eqnarray}$$

Equation (2.14) is to be solved subject to appropriate boundary conditions. These are a finite value of $E_{m}^{-}(0,\unicode[STIX]{x1D719})$ , a decay behaviour for $E_{m}^{+}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})$ as $\unicode[STIX]{x1D70E}\rightarrow \infty$ , and the kinematic and dynamic boundary conditions across the vortex sheet (Batchelor & Gill Reference Batchelor and Gill1962; Crighton Reference Crighton1973). Because all the function pairs $E_{m}^{\pm }(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})$ satisfy both (2.14) and their relevant boundary conditions, they are referred to as the eigenfunctions of this system. The functions $\{E_{m}^{\pm }(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})\}$ should form a complete set of basis for the Hilbert space determined by (2.14) and appropriate boundary conditions. The aim of the next section is to calculate these eigenfunctions analytically (without discretizing (2.14) and then solving it numerically).

2.3 The calculation of eigenfunctions

Due to the coupling of $\unicode[STIX]{x1D70E}$ and $\unicode[STIX]{x1D719}$ of the boundary conditions, $E_{m}^{\pm }(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})$ are not generally expected to be of a separable form. However, in the case of a cylindrical vortex sheet (round jet), the eigenfunctions $E_{m}^{\pm }(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})$ are known to be separable. Before proceeding to the case of more general vortex sheets, it is instructive to review the characteristics of $E_{m}^{\pm }(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})$ for a cylindrical vortex sheet.

2.3.1 The solution for a cylindrical vortex sheet

For the cylindrical vortex-sheet flow, the solutions were derived by Batchelor & Gill (Reference Batchelor and Gill1962) and we review them to introduce the notation and, more importantly, these solutions form the basis for analysing more complicated geometries in the following section. The potentials $\unicode[STIX]{x1D713}^{\pm }$ are known to be able to be expanded as

(2.15) $$\begin{eqnarray}\unicode[STIX]{x1D713}^{\pm }=\mathop{\sum }_{n=-\infty }^{\infty }\unicode[STIX]{x1D6F9}_{n}^{\pm }(\unicode[STIX]{x1D70E})\text{e}^{\text{i}n\unicode[STIX]{x1D719}}\text{e}^{\text{i}\unicode[STIX]{x1D6FC}z}\text{e}^{-\text{i}\unicode[STIX]{x1D714}t},\end{eqnarray}$$

where the functions $\unicode[STIX]{x1D6F9}_{n}^{\pm }(\unicode[STIX]{x1D70E})$ satisfy the modified Bessel equation

(2.16) $$\begin{eqnarray}\unicode[STIX]{x1D70E}^{2}\frac{\text{d}^{2}\unicode[STIX]{x1D6F9}_{n}^{\pm }}{\text{d}\unicode[STIX]{x1D70E}^{2}}+\unicode[STIX]{x1D70E}\frac{\text{d}\unicode[STIX]{x1D6F9}_{n}^{\pm }}{\text{d}\unicode[STIX]{x1D70E}}-(\unicode[STIX]{x1D6FC}^{2}\unicode[STIX]{x1D70E}^{2}+n^{2})\unicode[STIX]{x1D6F9}_{n}^{\pm }=0.\end{eqnarray}$$

Considering the boundary condition at the centre of the mean flow and at infinity, one can show that

(2.17) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D6F9}_{n}^{-}(\unicode[STIX]{x1D70E})=C_{n}^{-}\frac{1}{I_{n}(\unicode[STIX]{x1D6FC}a)}I_{n}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70E}),\\ \displaystyle \unicode[STIX]{x1D6F9}_{n}^{+}(\unicode[STIX]{x1D70E})=C_{n}^{+}\frac{1}{K_{n}(\unicode[STIX]{x1D6FC}a)}K_{n}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70E}),\end{array}\right\}\end{eqnarray}$$

where $C_{n}^{-}$ and $C_{n}^{+}$ are arbitrary complex constants and $I_{n}$ and $K_{n}$ are the modified Bessel functions of the first and second kinds, respectively.

Applying the kinematic and dynamic boundary condition on the vortex sheet, one obtains

(2.18a ) $$\begin{eqnarray}\displaystyle & \displaystyle (\unicode[STIX]{x1D714}-\unicode[STIX]{x1D6FC}U)\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F9}_{n}^{+}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}}=\unicode[STIX]{x1D714}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F9}_{n}^{-}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70E}}, & \displaystyle\end{eqnarray}$$

(2.18b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D714}\unicode[STIX]{x1D6F9}_{n}^{+}=(\unicode[STIX]{x1D714}-\unicode[STIX]{x1D6FC}U)\unicode[STIX]{x1D6F9}_{n}^{-}, & \displaystyle\end{eqnarray}$$

where use is made of the fact that the function set $\{\text{e}^{\text{i}n\unicode[STIX]{x1D719}}\}$ is orthogonal. For a non-trivial pair of solutions to exist ( $C_{n}^{-}C_{n}^{+}\neq 0$ ), equations (2.18a ) and (2.18b ) can be rearranged to yield the dispersion relation

(2.19) $$\begin{eqnarray}\left(\frac{\unicode[STIX]{x1D6FC}U}{\unicode[STIX]{x1D714}}-1\right)^{2}=\frac{I_{n}^{\prime }(\unicode[STIX]{x1D6FC}a)K_{n}(\unicode[STIX]{x1D6FC}a)}{I_{n}(\unicode[STIX]{x1D6FC}a)K_{n}^{\prime }(\unicode[STIX]{x1D6FC}a)}.\end{eqnarray}$$

In this special case, one can verify that the eigenfunctions, $E_{n}^{+}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})$ and $E_{n}^{-}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})$ , take the separable form of

(2.20) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle E_{n}^{-}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})=\frac{1}{I_{n}(\unicode[STIX]{x1D6FC}a)}I_{n}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70E})\text{e}^{\text{i}n\unicode[STIX]{x1D719}},\\ \displaystyle E_{n}^{+}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})=\left(1-\frac{\unicode[STIX]{x1D6FC}U}{\unicode[STIX]{x1D714}}\right)\frac{1}{K_{n}(\unicode[STIX]{x1D6FC}a)}K_{n}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70E})\text{e}^{\text{i}n\unicode[STIX]{x1D719}},\end{array}\right\}\end{eqnarray}$$

where $(1-\unicode[STIX]{x1D6FC}U/\unicode[STIX]{x1D714})$ is obtained from (2.19). Here, $E_{n}^{\pm }(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})$ take these simple forms because the boundary condition involves no coupling between $\unicode[STIX]{x1D70E}$ and $\unicode[STIX]{x1D719}$ (separable), and therefore these separable solutions are just the eigenfunction of the mathematical problem.

2.3.2 The solution for a vortex sheet of arbitrary geometry

For the case of a non-axisymmetric vortex sheet, the perturbations outside and inside the vortex sheet still remain determined by the Laplace equation. However, the boundary condition is now more complicated. If we were to find an orthogonal coordinate system in which the vortex-sheet profile can be represented by one of the constant coordinate axes, we may be able to find a separable solution. For elliptic profiles, such a coordinate system exists and eigenfunctions of separable form can be obtained (Crighton Reference Crighton1973). However, it seems rather unlikely to find such a coordinate system for a general profile.

However, in light of the completeness of the orthogonal function set $\{\text{e}^{\text{i}n\unicode[STIX]{x1D719}}\}$ , we are still able to write the solutions inside and outside the vortex sheet as

(2.21) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D713}^{-}=\mathop{\sum }_{-\infty }^{\infty }C_{n}^{-}\frac{1}{I_{n}(\unicode[STIX]{x1D6FC}a)}I_{n}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70E})\text{e}^{\text{i}n\unicode[STIX]{x1D719}}\text{e}^{\text{i}\unicode[STIX]{x1D6FC}z}\text{e}^{-\text{i}\unicode[STIX]{x1D714}t},\\ \displaystyle \unicode[STIX]{x1D713}^{+}=\mathop{\sum }_{-\infty }^{\infty }C_{n}^{+}\frac{1}{K_{n}(\unicode[STIX]{x1D6FC}a)}K_{n}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70E})\text{e}^{\text{i}n\unicode[STIX]{x1D719}}\text{e}^{\text{i}\unicode[STIX]{x1D6FC}z}\text{e}^{-\text{i}\unicode[STIX]{x1D714}t},\end{array}\right\}\end{eqnarray}$$

respectively. We must emphasize here that neither the solution $I_{n}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70E})\text{e}^{\text{i}n\unicode[STIX]{x1D719}}/I_{n}(\unicode[STIX]{x1D6FC}a)$ nor that $K_{n}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70E})\text{e}^{\text{i}n\unicode[STIX]{x1D719}}/K_{n}(\unicode[STIX]{x1D6FC}a)$ is the eigenfunction for this problem, since they do not satisfy the boundary conditions on the vortex sheet (although they satisfy (2.14)). However, since they form a complete set, a suitable combination of these separable solutions which satisfies the boundary conditions will be the eigenfunction that we aim to obtain in this section. There may be multiple such combinations, corresponding to eigenfunctions of different orders. For a more compact presentation of the rest of the derivation, let

(2.22) $$\begin{eqnarray}\displaystyle & \displaystyle \bar{C}_{n}^{-}=C_{n}^{-}\frac{1}{I_{n}(\unicode[STIX]{x1D6FC}a)}, & \displaystyle\end{eqnarray}$$

(2.23) $$\begin{eqnarray}\displaystyle & \displaystyle \bar{C}_{n}^{+}=C_{n}^{+}\frac{1}{K_{n}(\unicode[STIX]{x1D6FC}a)}. & \displaystyle\end{eqnarray}$$

Hence, equation (2.21) becomes

(2.24) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D713}^{-}=\mathop{\sum }_{-\infty }^{\infty }\bar{C}_{n}^{-}I_{n}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70E})\text{e}^{\text{i}n\unicode[STIX]{x1D719}}\text{e}^{\text{i}\unicode[STIX]{x1D6FC}z}\text{e}^{-\text{i}\unicode[STIX]{x1D714}t},\\ \displaystyle \unicode[STIX]{x1D713}^{+}=\mathop{\sum }_{-\infty }^{\infty }\bar{C}_{n}^{+}K_{n}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70E})\text{e}^{\text{i}n\unicode[STIX]{x1D719}}\text{e}^{\text{i}\unicode[STIX]{x1D6FC}z}\text{e}^{-\text{i}\unicode[STIX]{x1D714}t}.\end{array}\right\}\end{eqnarray}$$

Equation (2.24) needs to satisfy both the kinematic and dynamic boundary conditions on the vortex sheet, i.e. when $\unicode[STIX]{x1D70E}={\mathcal{R}}(\unicode[STIX]{x1D719})$ . The two boundary conditions can be shown to be (see more details in appendix A),

(2.25a ) $$\begin{eqnarray}\displaystyle & \displaystyle (\unicode[STIX]{x1D714}-\unicode[STIX]{x1D6FC}U)\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}^{+}\boldsymbol{\cdot }\boldsymbol{n}=\unicode[STIX]{x1D714}\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}^{-}\boldsymbol{\cdot }\boldsymbol{n}, & \displaystyle\end{eqnarray}$$

(2.25b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D714}\unicode[STIX]{x1D713}^{+}=(\unicode[STIX]{x1D714}-\unicode[STIX]{x1D6FC}U)\unicode[STIX]{x1D713}^{-}, & \displaystyle\end{eqnarray}$$

where $\boldsymbol{n}$ denotes the unit vector normal to the surface of the vortex sheet.

Figure 3. (a) The weakly lobed profiles when $\unicode[STIX]{x1D716}=0.1$ . (b) Strongly lobed profiles when $\unicode[STIX]{x1D716}=0.2$ .

3.1 Weakly lobed profile

For weakly lobed profile, $\unicode[STIX]{x1D716}\ll 1$ , therefore we may expand both left- and right-hand sides of (3.4) around $\unicode[STIX]{x1D716}=0$ and keep only the first order without causing too much error. In doing so, the first-order equations can be obtained and, after collecting the terms of the same $\text{e}^{\text{i}n\unicode[STIX]{x1D719}}$ , written as

(3.5) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\begin{array}{@{}l@{}}\displaystyle \mathop{\sum }_{n=-\infty }^{\infty }\left[\bar{C}_{n}^{+}K_{n}^{\prime }(\unicode[STIX]{x1D6FC}a)+\bar{C}_{n-N}^{+}\left(K_{n-N}^{\prime \prime }(\unicode[STIX]{x1D6FC}a)\frac{\unicode[STIX]{x1D6FC}a}{2}+K_{n-N}(\unicode[STIX]{x1D6FC}a)\frac{(n-N)N}{2\unicode[STIX]{x1D6FC}a}\right)\unicode[STIX]{x1D716}\right.\\ \displaystyle \qquad \left.+\,C_{n+N}^{+}\left(K_{n+N}^{\prime \prime }(\unicode[STIX]{x1D6FC}a)\frac{\unicode[STIX]{x1D6FC}a}{2}-K_{n+N}(\unicode[STIX]{x1D6FC}a)\frac{(n+N)N}{2\unicode[STIX]{x1D6FC}a}\right)\unicode[STIX]{x1D716}\right]\text{e}^{\text{i}n\unicode[STIX]{x1D719}}\\ \displaystyle \quad =\left(\frac{1}{1-\frac{\unicode[STIX]{x1D6FC}U}{\unicode[STIX]{x1D714}}}\right)\mathop{\sum }_{n=-\infty }^{\infty }\left[\bar{C}_{n}^{-}I_{n}^{\prime }(\unicode[STIX]{x1D6FC}a)+\bar{C}_{n-N}^{-}\left(I_{n-N}^{\prime \prime }(\unicode[STIX]{x1D6FC}a)\frac{\unicode[STIX]{x1D6FC}a}{2}+I_{n-N}(\unicode[STIX]{x1D6FC}a)\frac{(n-N)N}{2\unicode[STIX]{x1D6FC}a}\right)\unicode[STIX]{x1D716}\right.\\ \displaystyle \qquad \left.+\,\bar{C}_{n+N}^{-}\left(I_{n+N}^{\prime \prime }(\unicode[STIX]{x1D6FC}a)\frac{\unicode[STIX]{x1D6FC}a}{2}-I_{n+N}(\unicode[STIX]{x1D6FC}a)\frac{(n+N)N}{2\unicode[STIX]{x1D6FC}a}\right)\unicode[STIX]{x1D716}\right]\text{e}^{\text{i}n\unicode[STIX]{x1D719}},\end{array}\\ \begin{array}{@{}l@{}}\displaystyle \mathop{\sum }_{n=-\infty }^{\infty }\left(\bar{C}_{n}^{+}K_{n}(\unicode[STIX]{x1D6FC}a)+\bar{C}_{n-N}^{+}K_{n-N}^{\prime }(\unicode[STIX]{x1D6FC}a)\frac{\unicode[STIX]{x1D6FC}a}{2}\unicode[STIX]{x1D716}+\bar{C}_{n+N}^{+}K_{n+N}^{\prime }(\unicode[STIX]{x1D6FC}a)\frac{\unicode[STIX]{x1D6FC}a}{2}\unicode[STIX]{x1D716}\right)\text{e}^{\text{i}n\unicode[STIX]{x1D719}}\\ \displaystyle \quad =\left(1-\frac{\unicode[STIX]{x1D6FC}U}{\unicode[STIX]{x1D714}}\right)\mathop{\sum }_{n=-\infty }^{\infty }\!\left(\bar{C}_{n}^{-}I_{n}(\unicode[STIX]{x1D6FC}a)+\bar{C}_{n-N}^{-}I_{n-N}^{\prime }(\unicode[STIX]{x1D6FC}a)\frac{\unicode[STIX]{x1D6FC}a}{2}\unicode[STIX]{x1D716}+\bar{C}_{n+N}^{-}I_{n+N}^{\prime }(\unicode[STIX]{x1D6FC}a)\frac{\unicode[STIX]{x1D6FC}a}{2}\unicode[STIX]{x1D716}\right)\!\text{e}^{\text{i}n\unicode[STIX]{x1D719}}.\end{array}\end{array}\!\!\!\right\} & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The above equations can be written in a more compact matrix form, from which the effects of lobed jets can be seen more clearly. Let $\unicode[STIX]{x1D646}(\unicode[STIX]{x1D6FC}a)$ denote the diagonal matrix

(3.6) $$\begin{eqnarray}\unicode[STIX]{x1D646}(\unicode[STIX]{x1D6FC}a)=\text{diag}(\ldots K_{-1}(\unicode[STIX]{x1D6FC}a),K_{0}(\unicode[STIX]{x1D6FC}a),\,K_{1}(\unicode[STIX]{x1D6FC}a)\ldots ),\end{eqnarray}$$

and let $\unicode[STIX]{x1D646}_{c}(\unicode[STIX]{x1D6FC}a)$ and $\unicode[STIX]{x1D646}_{s}(\unicode[STIX]{x1D6FC}a)$ be

(3.7) $$\begin{eqnarray}\unicode[STIX]{x1D646}_{c}(\unicode[STIX]{x1D6FC}a)=\left[\begin{array}{@{}cc@{}}\ldots & \ldots \\ \ldots K_{-N-1}(\unicode[STIX]{x1D6FC}a) & \ldots K_{N-1}(\unicode[STIX]{x1D6FC}a)\ldots \ldots \ldots \\ \ldots \ldots K_{-N}(\unicode[STIX]{x1D6FC}a) & \ldots \ldots K_{N}(\unicode[STIX]{x1D6FC}a)\ldots \ldots \\ \ldots \ldots \ldots K_{-N+1}(\unicode[STIX]{x1D6FC}a) & \ldots \ldots \ldots K_{N+1}(\unicode[STIX]{x1D6FC}a)\ldots \\ \ldots \ldots \ldots \ldots & \ldots \ldots \ldots \ldots \\ \end{array}\right],\end{eqnarray}$$

and

(3.8) $$\begin{eqnarray}\unicode[STIX]{x1D646}_{s}(\unicode[STIX]{x1D6FC}a)=\left[\begin{array}{@{}cc@{}}\ldots & \ldots \\ \ldots K_{-N-1}(\unicode[STIX]{x1D6FC}a)(-N-1) & \ldots -K_{N-1}(\unicode[STIX]{x1D6FC}a)(N-1)\ldots \ldots \ldots \\ \ldots \ldots K_{-N}(\unicode[STIX]{x1D6FC}a)(-N) & \ldots \ldots -K_{N}(\unicode[STIX]{x1D6FC}a)N\ldots \ldots \\ \ldots \ldots \ldots K_{-N+1}(\unicode[STIX]{x1D6FC}a)(-N+1) & \ldots \ldots \ldots -K_{N+1}(\unicode[STIX]{x1D6FC}a)(N+1)\ldots \\ \ldots \ldots \ldots \ldots & \ldots \ldots \ldots \ldots \\ \end{array}\right].\end{eqnarray}$$

Clearly, each element of the matrices $\unicode[STIX]{x1D646}(\unicode[STIX]{x1D6FC}a)$ , $\unicode[STIX]{x1D646}_{c}(\unicode[STIX]{x1D6FC}a)$ and $\unicode[STIX]{x1D646}_{s}(\unicode[STIX]{x1D6FC}a)$ is a function of $\unicode[STIX]{x1D6FC}a$ and, consequently, these matrices are essentially function matrices with an argument $\unicode[STIX]{x1D6FC}a$ . We can therefore define the $n$ th derivative of a matrix to be the matrix formed by the $n$ th derivative of each element function. For example, the first derivative of $\unicode[STIX]{x1D646}_{c}(\unicode[STIX]{x1D6FC}a)$ is

(3.9) $$\begin{eqnarray}\unicode[STIX]{x1D646}_{c}^{\prime }(\unicode[STIX]{x1D6FC}a)=\left[\begin{array}{@{}cc@{}}\ldots & \ldots \\ \ldots K_{-N-1}^{\prime }(\unicode[STIX]{x1D6FC}a) & \ldots K_{N-1}^{\prime }(\unicode[STIX]{x1D6FC}a)\ldots \ldots \ldots \\ \ldots \ldots K_{-N}^{\prime }(\unicode[STIX]{x1D6FC}a) & \ldots \ldots K_{N}^{\prime }(\unicode[STIX]{x1D6FC}a)\ldots \ldots \\ \ldots \ldots \ldots K_{-N+1}^{\prime }(\unicode[STIX]{x1D6FC}a) & \ldots \ldots \ldots K_{N+1}^{\prime }(\unicode[STIX]{x1D6FC}a)\ldots \\ \ldots \ldots \ldots \ldots & \ldots \ldots \ldots \ldots \\ \end{array}\right].\end{eqnarray}$$

If one replaces the modified Bessel function of the second kind $K_{n}(\unicode[STIX]{x1D6FC}a)$ , in matrices $\unicode[STIX]{x1D646}(\unicode[STIX]{x1D6FC}a)$ , $\unicode[STIX]{x1D646}_{c}(\unicode[STIX]{x1D6FC}a)$ and $\unicode[STIX]{x1D646}_{s}(\unicode[STIX]{x1D6FC}a)$ , with function $\unicode[STIX]{x1D644}_{n}(\unicode[STIX]{x1D6FC}a)$ , the matrices $\unicode[STIX]{x1D644}(\unicode[STIX]{x1D6FC}a)$ , $\unicode[STIX]{x1D644}_{c}(\unicode[STIX]{x1D6FC}a)$ and $\unicode[STIX]{x1D644}_{s}(\unicode[STIX]{x1D6FC}a)$ can be similarly defined. Upon defining the vector

(3.10) $$\begin{eqnarray}\bar{\boldsymbol{C}}^{\boldsymbol{\pm }}=\left[\ldots ,\bar{C}_{-n}^{\pm },\ldots ,\bar{C}_{0}^{\pm },\ldots ,\bar{C}_{n}^{\pm },\ldots \right]^{\text{T}},\end{eqnarray}$$

where $[~\,]^{\text{T}}$ denotes the transpose of matrix $[~\,]$ , equation (3.5) can be readily written as

(3.11) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\begin{array}{@{}l@{}}\displaystyle \left(1-\frac{\unicode[STIX]{x1D6FC}U}{\unicode[STIX]{x1D714}}\right)\left[\unicode[STIX]{x1D646}^{\prime }(\unicode[STIX]{x1D6FC}a)+\unicode[STIX]{x1D716}\left(\unicode[STIX]{x1D646}_{c}^{\prime \prime }(\unicode[STIX]{x1D6FC}a)\frac{\unicode[STIX]{x1D6FC}a}{2}+\unicode[STIX]{x1D646}_{s}(\unicode[STIX]{x1D6FC}a)\frac{N}{2\unicode[STIX]{x1D6FC}a}\right)\right]\bar{\boldsymbol{C}}^{+}\\ \displaystyle \quad =\left[\unicode[STIX]{x1D644}^{\prime }(\unicode[STIX]{x1D6FC}a)+\unicode[STIX]{x1D716}\left(\unicode[STIX]{x1D644}_{c}^{\prime \prime }(\unicode[STIX]{x1D6FC}a)\frac{\unicode[STIX]{x1D6FC}a}{2}+\unicode[STIX]{x1D644}_{s}(\unicode[STIX]{x1D6FC}a)\frac{N}{2\unicode[STIX]{x1D6FC}a}\right)\right]\bar{\boldsymbol{C}}^{-},\end{array}\\ \displaystyle \left[\unicode[STIX]{x1D646}(\unicode[STIX]{x1D6FC}a)+\unicode[STIX]{x1D716}\unicode[STIX]{x1D646}_{c}^{\prime }(\unicode[STIX]{x1D6FC}a)\frac{\unicode[STIX]{x1D6FC}a}{2}\right]\bar{\boldsymbol{C}}^{+}=\left(1-\frac{\unicode[STIX]{x1D6FC}U}{\unicode[STIX]{x1D714}}\right)\left[\unicode[STIX]{x1D644}(\unicode[STIX]{x1D6FC}a)+\unicode[STIX]{x1D716}\unicode[STIX]{x1D644}_{c}^{\prime }(\unicode[STIX]{x1D6FC}a)\frac{\unicode[STIX]{x1D6FC}a}{2}\right]\bar{\boldsymbol{C}}^{-}.\end{array}\right\}\end{eqnarray}$$

Equation (3.11) represents the dispersion relation for the considered lobed jet to the first-order accuracy. It is worth noting that both $\unicode[STIX]{x1D646}(\unicode[STIX]{x1D6FC}a)$ and $\unicode[STIX]{x1D644}(\unicode[STIX]{x1D6FC}a)$ are diagonal matrices. Therefore, in the case of $\unicode[STIX]{x1D716}=0$ , i.e. the axisymmetric vortex sheet, equation (3.11) represents a set of decoupled dispersion relation equations. Consequently, stability analysis can be performed for each mode individually and the results are identical to those obtained by Batchelor & Gill (Reference Batchelor and Gill1962), as shown above. When $\unicode[STIX]{x1D716}\neq 0$ , the equations governing the dispersion relations are coupled equations, hence the equations must be solved together.

Before proceeding to solve these equations, it is informative to examine how lobed vortex-sheet profiles affect the characteristic matrices. First, one can see that the coupling only occurs between modes $n$ , $n-N$ and $n+N$ . Had we included higher-order terms ( $\unicode[STIX]{x1D716}^{2},\unicode[STIX]{x1D716}^{3},\ldots$ ), the coupling would involve modes $n\pm kN$ ( $k=0,\pm 1,\pm 2,\ldots$ ). This shows that the lobed profile affects the instability waves by modulating them with its own periodicity. Second, the modulating effects occur in two ways: modifying the radial length scales and changing the normal directions of vortex sheet. The effects of modifying the radial length scales are represented by the $\unicode[STIX]{x1D646}_{c}^{(i)}(\unicode[STIX]{x1D6FC}a)$ and $\unicode[STIX]{x1D644}_{c}^{(i)}(\unicode[STIX]{x1D6FC}a)$ matrices ( $i$ here denotes the $i$ th derivative). Take the strongly-lobed profile of $N=2$ , as shown in figure 3(b), as an example. At such a large value of $\unicode[STIX]{x1D716}$ , the lobe profile resembles that of an elliptic vortex sheet. Hence, the radial length scales of the major and the minor axes are different. It is known that this causes different behaviour for instability waves orientated with different axes (Crighton Reference Crighton1973). In the dynamic boundary conditions shown in (3.11), the $\unicode[STIX]{x1D646}_{c}^{\prime }(\unicode[STIX]{x1D6FC}a)$ and $\unicode[STIX]{x1D644}_{c}^{\prime }(\unicode[STIX]{x1D6FC}a)$ terms account for the different length scales (to the first-order accuracy) and ensure pressure is continuous across the vortex sheet. The $\unicode[STIX]{x1D646}_{s}(\unicode[STIX]{x1D6FC}a)$ and $\unicode[STIX]{x1D644}_{s}(\unicode[STIX]{x1D6FC}a)$ matrices, on the other hand, account for changing of the restrictions on the normal perturbation velocities across the vortex sheet. Equation (2.25a ) shows that it is the normal (to the vortex sheet) perturbation velocities that have to satisfy the jump condition. From (3.3) it is evident that the use of lobed nozzles can significantly change the local normal directions of the vortex sheet and, hence, the instability characteristics. In addition, it is clear from (3.3) that the normal direction changes more pronouncedly as $N$ increases.

To solve (3.11), we write the two matrix equations in a more compact form as

(3.12a ) $$\begin{eqnarray}\displaystyle \displaystyle \left(1-\frac{\unicode[STIX]{x1D6FC}U}{\unicode[STIX]{x1D714}}\right)\unicode[STIX]{x1D646}_{k}\bar{\boldsymbol{C}}^{+}=\unicode[STIX]{x1D644}_{k}\bar{\boldsymbol{C}}^{-}, & & \displaystyle\end{eqnarray}$$

(3.12b ) $$\begin{eqnarray}\displaystyle \displaystyle \unicode[STIX]{x1D646}_{d}\bar{\boldsymbol{C}}^{+}=\left(1-\frac{\unicode[STIX]{x1D6FC}U}{\unicode[STIX]{x1D714}}\right)\unicode[STIX]{x1D644}_{d}\bar{\boldsymbol{C}}^{-}. & & \displaystyle\end{eqnarray}$$

$\unicode[STIX]{x1D646}_{k}$

$\unicode[STIX]{x1D644}_{k}$

$\unicode[STIX]{x1D646}_{d}$

$\unicode[STIX]{x1D644}_{k}$

$\unicode[STIX]{x1D6FC}a$

$\bar{\boldsymbol{C}}^{\boldsymbol{\pm }}$

$\boldsymbol{C}^{\boldsymbol{\pm }}$

$\boldsymbol{C}^{\boldsymbol{\pm }}$

$C_{n}^{\pm }$

$\bar{C}_{n}^{\pm }$

$C_{n}^{\pm }$

$\bar{C}_{n}^{\pm }$

$I_{n}(\unicode[STIX]{x1D6FC}a)$

$\unicode[STIX]{x1D6FC}a$

$n$

$\bar{C}_{n}^{-}$

$n$

$C^{\pm }$

$\boldsymbol{C}^{+}=\unicode[STIX]{x1D646}\bar{\boldsymbol{C}}^{+}$

$\boldsymbol{C}^{-}=\unicode[STIX]{x1D644}\bar{\boldsymbol{C}}^{-}$

$\unicode[STIX]{x1D644}$

$\unicode[STIX]{x1D646}$

$\unicode[STIX]{x1D644}^{-1}$

$\unicode[STIX]{x1D646}^{-1}$

(3.13a ) $$\begin{eqnarray}\displaystyle & \displaystyle \left(1-\frac{\unicode[STIX]{x1D6FC}U}{\unicode[STIX]{x1D714}}\right)\widetilde{\unicode[STIX]{x1D646}}_{k}\boldsymbol{C}^{+}=\widetilde{\unicode[STIX]{x1D644}}_{k}\boldsymbol{C}^{-}, & \displaystyle\end{eqnarray}$$

(3.13b ) $$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{\unicode[STIX]{x1D646}}_{d}\boldsymbol{C}^{+}=\left(1-\frac{\unicode[STIX]{x1D6FC}U}{\unicode[STIX]{x1D714}}\right)\widetilde{\unicode[STIX]{x1D644}}_{d}\boldsymbol{C}^{-}, & \displaystyle\end{eqnarray}$$

$\widetilde{\unicode[STIX]{x1D646}}_{k}=\unicode[STIX]{x1D646}_{k}\unicode[STIX]{x1D646}^{-1}$

$\widetilde{\unicode[STIX]{x1D646}}_{d}=\unicode[STIX]{x1D646}_{d}\unicode[STIX]{x1D646}^{-1}$

$\widetilde{\unicode[STIX]{x1D644}}_{k}=\unicode[STIX]{x1D644}_{k}\unicode[STIX]{x1D644}^{-1}$

$\widetilde{\unicode[STIX]{x1D644}}_{d}=\unicode[STIX]{x1D644}_{d}\unicode[STIX]{x1D644}^{-1}$

$\unicode[STIX]{x1D644}^{-1}$

$\unicode[STIX]{x1D646}^{-1}$

From (3.13b ), we see that

(3.14) $$\begin{eqnarray}\boldsymbol{C}^{+}=\left(1-\frac{\unicode[STIX]{x1D6FC}U}{\unicode[STIX]{x1D714}}\right)\widetilde{\unicode[STIX]{x1D646}}_{d}^{-1}\widetilde{\unicode[STIX]{x1D644}}_{d}\boldsymbol{C}^{-}.\end{eqnarray}$$

Substituting (3.14) into (3.13a ), we have

(3.15) $$\begin{eqnarray}\left(1-\frac{\unicode[STIX]{x1D6FC}U}{\unicode[STIX]{x1D714}}\right)^{2}\widetilde{\unicode[STIX]{x1D646}}_{k}\widetilde{\unicode[STIX]{x1D646}}_{d}^{-1}\widetilde{\unicode[STIX]{x1D644}}_{d}\boldsymbol{C}^{-}=\widetilde{\unicode[STIX]{x1D644}}_{k}\boldsymbol{C}^{-}.\end{eqnarray}$$

Upon multiplying $\widetilde{\unicode[STIX]{x1D644}}_{k}^{-1}$ on both sides of (3.15) and defining $\unicode[STIX]{x1D63C}=\widetilde{\unicode[STIX]{x1D644}}_{k}^{\,-1}\widetilde{\unicode[STIX]{x1D646}}_{k}\widetilde{\unicode[STIX]{x1D646}}_{d}^{-1}\widetilde{\unicode[STIX]{x1D644}}_{d}$ , we obtain the following eigenvalue problem

(3.16) $$\begin{eqnarray}\unicode[STIX]{x1D63C}\boldsymbol{C}^{-}=\unicode[STIX]{x1D706}\boldsymbol{C}^{-},\end{eqnarray}$$

where

(3.17) $$\begin{eqnarray}\unicode[STIX]{x1D706}=\left(1-\frac{\unicode[STIX]{x1D6FC}U}{\unicode[STIX]{x1D714}}\right)^{-2}.\end{eqnarray}$$

The matrix $\unicode[STIX]{x1D63C}$ is of an infinite dimension. To calculate its eigenvalues in practical cases, we may drop all the modes higher than $M$ (and less than $-M$ ). Though we expect results to become inaccurate for large modes close to $M$ , it may yield satisfactory results for relatively low-order modes when $M$ is taken to be adequately large. These low-order modes are of our primary interest in this study, since high-order modes vanish sufficiently quickly according to experimental results (Tinney & Jordan Reference Tinney and Jordan2008). In addition, the vortex-sheet assumption would fail for high-order modes anyway. By truncating high-order terms, we obtain a matrix of $(2M+1)\times (2M+1)$ , and there are $2M+1$ eigenvalues (degenerate eigenvalues are counted more than once) and their corresponding eigenvectors. For each obtained eigenvector $\boldsymbol{C}^{-}$ , we can obtain the corresponding $\boldsymbol{C}^{+}$ easily from (3.14). The fact that the non-zero eigenvector $\boldsymbol{C}^{-}$ satisfies (3.16) entails that the non-trivial velocity potential $\unicode[STIX]{x1D713}^{-}$ , determined by $\boldsymbol{C}^{-}$ , and the corresponding $\unicode[STIX]{x1D713}^{+}$ , determined by $\boldsymbol{C}^{+}$ , satisfy both the kinematic and dynamic boundary conditions on the vortex sheet. Therefore, each eigenvector represents an eigenfunction of the lobed problem, i.e.

(3.18) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle E_{n}^{-}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})=\mathop{\sum }_{n=-\infty }^{\infty }C_{n}^{-}\frac{1}{I_{n}(\unicode[STIX]{x1D6FC}a)}I_{n}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70E})\text{e}^{\text{i}n\unicode[STIX]{x1D719}},\\ \displaystyle E_{n}^{+}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})=\mathop{\sum }_{n=-\infty }^{\infty }C_{n}^{+}\frac{1}{K_{n}(\unicode[STIX]{x1D6FC}a)}K_{n}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D70E})\text{e}^{\text{i}n\unicode[STIX]{x1D719}}.\end{array}\right\}\end{eqnarray}$$

One can readily verify that, when $\boldsymbol{C}^{-}$ is normalized such that $(\boldsymbol{C}^{-})^{\ast \text{T}}\boldsymbol{C}^{-}=1$ and $\boldsymbol{C}^{+}$ is obtained from the dynamic boundary conditions shown in (3.14), both $E_{n}^{-}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})$ and $E_{n}^{+}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})$ are normalized as described in § 2.2.

3.2 The mode labelling strategy

When $\unicode[STIX]{x1D716}=0$ , there is a well-defined mode number for each mode. For example, mode $n$ can be defined as the eigenvector

(3.19) $$\begin{eqnarray}\boldsymbol{C}^{-}=\left[\ldots 0,\ldots ,0,\ldots ,C_{n}^{-}=1,\ldots \right]^{\text{T}}.\end{eqnarray}$$

When $\unicode[STIX]{x1D716}\neq 0$ , however, the eigenvector does not posses this simple property. Instead, the resulting eigenvector has other non-vanishing elements in addition to $C_{n}^{-}$ . We need to develop an unambiguous strategy to label the eigenmodes.

We can show that the eigenvector $\boldsymbol{C}^{-}$ can be always defined as either symmetric or antisymmetric (with respect to the element of index $0$ ). This is due to the rotationally symmetric property possessed by the matrix $\unicode[STIX]{x1D63C}$ , and for brevity we have placed detailed derivation in appendix B.

Based on this property, we define an eigenvector to have a mode number $n$ if it is symmetric and

(3.20) $$\begin{eqnarray}\Vert \boldsymbol{C}^{-}-\boldsymbol{G}\Vert =\sqrt{\mathop{\sum }_{j=-M}^{M}(|C_{j}^{-}|-G_{j})^{2}}\end{eqnarray}$$

yields a minimum value when $n$ varies from $-M$ to $M$ and

(3.21) $$\begin{eqnarray}\boldsymbol{G}=[\ldots ,g_{-n}=\sqrt{2}/2,\ldots ,0,\ldots ,g_{n}=\sqrt{2}/2,\ldots ]^{\text{T}},\end{eqnarray}$$

where $g_{n}$ is the element of the gauge vector $\boldsymbol{G}$ . For antisymmetric eigenvectors we label it as $-n$ in a similar manner. For $n=0$ , it is trivial to label its mode number and it can be shown that it is a symmetric vector. By labelling the eigenvectors in this way, equation (3.18) implies that all nonnegative eigenfunctions are even functions of $\unicode[STIX]{x1D719}$ and negative ones odd.

In the following analysis, the mode number for the obtained eigenfunction is designated according to the above conventions. Then for each mode $n$ , we can calculate its corresponding eigenvalue $\unicode[STIX]{x1D706}_{n}$ at a given value of $\unicode[STIX]{x1D6FC}a$ . The complex frequency $\unicode[STIX]{x1D714}$ can be directly obtained from $\unicode[STIX]{x1D706}_{n}$ according to (3.17). This complex number determines both the growth rate and the convection velocity of its corresponding instability wave. Therefore, by varying the values of $\unicode[STIX]{x1D716}$ and $N$ , one can easily examine how different lobed geometry changes both the growth rate and convection velocity of instability waves of different mode numbers.

3.3 Strongly lobed nozzle

For strongly lobed nozzles, e.g.  $\unicode[STIX]{x1D716}\sim 0.2$ , it is necessary to include high-order terms $\unicode[STIX]{x1D716}^{n}$ . Luckily this is not a difficult extension, and all the aforementioned procedures used to solve the eigenvalue problem remain the same. It suffices to find the high-order coefficient matrices and add them into (3.11), i.e.

(3.22) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D646}_{k}=\unicode[STIX]{x1D646}^{\prime }(\unicode[STIX]{x1D6FC}a)+\unicode[STIX]{x1D716}\left(\unicode[STIX]{x1D646}_{c}^{\prime \prime }(\unicode[STIX]{x1D6FC}a)\frac{\unicode[STIX]{x1D6FC}a}{2}+\unicode[STIX]{x1D646}_{s}(\unicode[STIX]{x1D6FC}a)\frac{N}{2\unicode[STIX]{x1D6FC}a}\right)+\unicode[STIX]{x1D716}^{2}(\cdots \,)+\cdots \,,\\ \displaystyle \unicode[STIX]{x1D644}_{k}=\unicode[STIX]{x1D644}^{\prime }(\unicode[STIX]{x1D6FC}a)+\unicode[STIX]{x1D716}\left(\unicode[STIX]{x1D644}_{c}^{\prime \prime }(\unicode[STIX]{x1D6FC}a)\frac{\unicode[STIX]{x1D6FC}a}{2}+\unicode[STIX]{x1D644}_{s}(\unicode[STIX]{x1D6FC}a)\frac{N}{2\unicode[STIX]{x1D6FC}a}\right)+\unicode[STIX]{x1D716}^{2}(\cdots \,)+\cdots \,,\\ \displaystyle \unicode[STIX]{x1D646}_{d}=\unicode[STIX]{x1D646}(\unicode[STIX]{x1D6FC}a)+\unicode[STIX]{x1D716}\unicode[STIX]{x1D646}_{c}^{\prime }(\unicode[STIX]{x1D6FC}a)\frac{\unicode[STIX]{x1D6FC}a}{2}+\unicode[STIX]{x1D716}^{2}(\cdots \,)+\cdots \,,\\ \displaystyle \unicode[STIX]{x1D644}_{d}=\unicode[STIX]{x1D644}(\unicode[STIX]{x1D6FC}a)+\unicode[STIX]{x1D716}\unicode[STIX]{x1D644}_{c}^{\prime }(\unicode[STIX]{x1D6FC}a)\frac{\unicode[STIX]{x1D6FC}a}{2}+\unicode[STIX]{x1D716}^{2}(\cdots \,)+\cdots \,.\end{array}\right\}\end{eqnarray}$$

It is worth noting that incorporating higher-order terms does not invalidate the matrix $\unicode[STIX]{x1D63C}$ being rotationally symmetric, hence all the previous conclusions about its eigenvectors still remain valid. Owing to the nature of higher-order modified Bessel functions, expanding them around $\unicode[STIX]{x1D6FC}a$ results in a slow convergence when $\unicode[STIX]{x1D716}$ is large. Therefore, the number of high-order terms needed increases quickly as $\unicode[STIX]{x1D716}$ increases. It also increases when we increase the value of $M$ . However, this problem can be overcome by expanding properly scaled modified Bessel functions. Since in this study we only need a relatively small $M$ , and the extension of incorporating more higher-order terms can be promptly automated using computer programming, it is not strictly necessary to expand the scaled modified Bessel functions instead. For example, a MATLAB code has been developed that can automatically incorporate as many orders of terms as needed. Owing to its analytical nature, the computation is very fast. For example, a comprehensive eigenvalue analysis of order $10$ with $M=20$ takes less than 50 ms. All the results shown in the following sections are obtained by incorporating higher-order terms to the order of $10$ ( $\unicode[STIX]{x1D716}^{10}$ ) and with $M=20$ . A convergence analysis, to be shown in § 4.1, shows that this is much more than necessary.

4.1 Convergence analysis

We first examine the convergence characteristics of this new method when the order of accuracy $m$ ( $\unicode[STIX]{x1D716}^{m}$ ) and $M$ change. To separate the effects of $m$ and $M$ , we can fix $M$ and vary the values of $m$ and then vice versa. For brevity, we choose to present the convection velocity and temporal growth rate of mode $1$ instability wave and fix the number of lobes $N=2$ .

Figure 4. The convergence of the temporal growth rate and convection velocity for the mode $1$ instability wave when $N=2$ and $\unicode[STIX]{x1D716}=0.1$ : (a) the order of accuracy is fixed at $\unicode[STIX]{x1D716}^{10}$ while $M$ varies from $2$ to $8$ ; (b) $M$ is fixed to be $20$ while the order of accuracy varies from $\unicode[STIX]{x1D716}^{2}$ to $\unicode[STIX]{x1D716}^{7}$ .

We first show the results for a relatively small penetration ratio, i.e.  $\unicode[STIX]{x1D716}=0.1$ . The results are shown in figure 4. For a compact presentation, we define a complex number $U_{c}\equiv \unicode[STIX]{x1D714}/\unicode[STIX]{x1D6FC}$ . Therefore, the real part of $U_{c}$ denotes the convection velocity while the imaginary part represents the temporal growth rate. Figure 4 shows the real and imaginary parts of $U_{c}$ .

Figure 4(a) shows the results when $m=10$ and $M$ varies from $2$ to $8$ . One can see that both the growth rates and convection velocities are hardly distinguishable. This shows that, for a small value of $\unicode[STIX]{x1D716}\approx 0.1$ , $M=2$ is essentially sufficient, at least for the mode number $1$ . Of course, we would need a slightly larger value of $M$ if we were to consider higher-order modes. However, this increase in $M$ is likely to be on a small scale, because we are only interested in low-order modes as only those are physically relevant in experiments. Figure 4(b) shows the results when $M$ is fixed at $20$ and $m$ varies from $2$ to $7$ . Similar to figure 4(a), the lines are nearly on top of each other. This shows that for $\unicode[STIX]{x1D716}\approx 0.1$ , a second-order accuracy is sufficient for a good convergence.

As mentioned in § 3.3, owing to the nature of the modified Bessel functions, the number of high-order $\unicode[STIX]{x1D716}^{n}$ terms needed increases quickly as $\unicode[STIX]{x1D716}$ increases. To show that $M=20$ and $m=10$ is also sufficient for a large value of $\unicode[STIX]{x1D716}$ , we present the convergence characteristics of this analysis when $\unicode[STIX]{x1D716}=0.2$ in figure 5.

Figure 5(a) shows the results when $m=10$ and $M$ varies from $2$ to $8$ . We can see that there is an observable difference between the convection velocities calculated using $M=2$ and $M=4$ . However, there is little change between the results for $M=4$ and $M=8$ . This shows that for strongly lobed geometry at $\unicode[STIX]{x1D716}\approx 0.2$ , $M$ must be at least $4$ . On the other hand, it is interesting to see that the temporal growth rate is much less sensitive to the change in $M$ than the convection velocity. Figure 5(b) similarly shows the results for a fixed number $M=20$ and a varying $m$ . The observation is similar to figure 5(a), and it shows that for $\unicode[STIX]{x1D716}\approx 0.2$ , a high-order accuracy up to $\unicode[STIX]{x1D716}^{4}$ or $\unicode[STIX]{x1D716}^{6}$ is recommended. In summary, the results in this section show that $M=20$ and $n=10$ can ensure that the obtained eigensolutions are well converged.

Figure 5. The convergence of the temporal growth rate and convection velocity for the mode $1$ instability wave when $N=2$ and $\unicode[STIX]{x1D716}=0.2$ : (a) the order of accuracy is fixed at $\unicode[STIX]{x1D716}^{10}$ while $M$ varies from $2$ to $8$ ; (b) $M$ is fixed to be $20$ while the order of accuracy varies from $\unicode[STIX]{x1D716}^{2}$ to $\unicode[STIX]{x1D716}^{7}$ .

4.2 Validation

Section 4.1 merely shows that $M=20$ and $n=10$ are sufficient for a good convergence. A converged solution, however, does not always imply a correct one. Consequently, before presenting any results, it is necessary to validate this new analysis framework. Luckily, it is very straightforward to do so. Since the entire analysis is devoted to calculating the eigenfunctions that satisfy both the kinematic and dynamic boundary conditions on the vortex sheet, we can examine the obtained eigenfunctions to ensure that they do indeed satisfy the two boundary conditions. More precisely, we can show that for each pair of eigenfunctions obtained above, we have, on the vortex sheet,

(4.1a ) $$\begin{eqnarray}\displaystyle & \displaystyle (1-\unicode[STIX]{x1D6FC}U/\unicode[STIX]{x1D714})\unicode[STIX]{x1D735}E_{m}^{+}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})\boldsymbol{\cdot }\boldsymbol{n}=\unicode[STIX]{x1D735}E_{m}^{-}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})\boldsymbol{\cdot }\boldsymbol{n}, & \displaystyle\end{eqnarray}$$

(4.1b ) $$\begin{eqnarray}\displaystyle & \displaystyle 1/(1-\unicode[STIX]{x1D6FC}U/\unicode[STIX]{x1D714})E_{m}^{+}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})=E_{m}^{-}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719}). & \displaystyle\end{eqnarray}$$

$n$

$n\geqslant 0$

$E_{n}^{-}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})$

$E_{-n}^{-}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})$

$-n$

$n$

$n\geqslant 0$

$-n$

$n>0$

$-2$

$2$

Figure 6. The match validation of the dynamic (a,c,e) and kinematic (b,d,f) boundary conditions for the eigenfunctions of modes $\pm 2$ when the number of lobes $N$ are different: (a,b) $N=1$ ; (c,d) $N=2$ ; (e,f) $N=3$ . The normalized wavenumber is $\unicode[STIX]{x1D6FC}a=1$ and the lobed profile has a penetration ratio $\unicode[STIX]{x1D716}=0.1$ .

Figure 6 shows how the boundary conditions are satisfied by the eigenfunctions of modes $\pm 2$ . These results are obtained with $\unicode[STIX]{x1D6FC}a=1$ and $\unicode[STIX]{x1D716}=0.1$ . Figures 6(a) and 6(b) show the matches of the left- and right-hand sides of (4.1b ) and (4.1a ), respectively, when the number of lobes is $N=1$ . From figure 3(a), one can see that when $N=1$ , the lobed profile is not very significantly different from the axisymmetric one. The single lobe merely causes a displacement of the profile centre of the circular vortex sheet. Therefore, one expects that the dynamics of instability waves shall remain largely unchanged. Figure 6(a,b) indeed confirms this. Modes $2$ and $-2$ resemble those of an axisymmetric vortex sheet in every aspect. When $N$ increases to $2$ , the effects of lobes are more pronounced, as shown in figure 6(c,d). In particular, the symmetric (with respect to $\unicode[STIX]{x1D719}=0$ ) eigenfunction shown in figure 6(d) starts to deviate from the normal $-\text{cos}\,2\unicode[STIX]{x1D719}$ function, with its peak somewhat flattened. The mode $-2$ , on the other hand, remains largely similar to $\sin 2\unicode[STIX]{x1D719}$ . When $N$ increases to $3$ , both modes $2$ and $-2$ respond to the change of vortex-sheet geometry and change their shapes significantly.

It is, however, worth noting that, for $N=1$ and $N=2$ , the eigenvalues $\unicode[STIX]{x1D706}_{2}\neq \unicode[STIX]{x1D706}_{-2}$ . Therefore, both $E_{2}^{-}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})$ and $E_{-2}^{-}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D719})$ have a standing wave pattern with respect to $\unicode[STIX]{x1D719}$ and they cannot be combined to produce a travelling wave. However, for $N=3$ we find that $\unicode[STIX]{x1D706}_{2}=\unicode[STIX]{x1D706}_{-2}$ . Therefore, a travelling-wave pattern can be obtained. The important observation is that, for all the six figures, exceptionally good matches of the pressure and normal velocity across the layer of vortex sheet are achieved. The excellent agreement seen from figure 6 shows that the analytical framework developed in this section works exceptionally well, at least for the low-order modes such as those we have just shown.

4.3 The effects of lobed profiles on the convection velocity and growth rate of instability waves

Having validated the analytical framework, we are now in a position to examine the effects of lobed profiles on the convection velocity and growth rate of instability waves. In the rest of this section, we plot both quantities versus the normalized frequency $\unicode[STIX]{x1D6FC}a$ , for lobed profiles of different geometry.

We start from showing results for mode $0$ . These are shown in figure 7. Figure 7(a) shows the convection velocity and growth rate for a single-lobe profile. To facilitate a direct comparison with the results of a cylindrical vortex sheet, both quantities are plotted when $\unicode[STIX]{x1D716}=0$ first. As can be seen, when the frequency $\unicode[STIX]{x1D6FC}a$ increases, the normalized convection velocity ( $\text{Re}\,U_{c}/U$ ) decreases from unity to around $0.5$ , whereas the normalized growth rate ( $\text{Im}\,U_{c}/U$ ) increases from $0$ to the same limit value. However, it shows that a single lobe does not cause any observable changes to the characteristics of mode $0$ at all frequencies, no matter what value of $\unicode[STIX]{x1D716}$ is used. The same conclusion can be reached for $N=2$ , which is shown in figure 7(b). Increasing the number of lobes to $3$ , however, starts to cause a slightly larger convection velocity and a marginally lower growth rate. These results are shown in figure 7(c), from which we see that the changes are only observable at high frequencies. However, they become more pronounced when the penetration ratio $\unicode[STIX]{x1D716}$ increases. The increase of the convection velocity and the reduction of the temporal growth rate, caused by lobed vortex-sheet profiles, are more evident when the number of lobes is increased to $5$ . As shown in figure 7(d), at high frequencies ( $\unicode[STIX]{x1D6FC}a>1.5$ ), a large penetration ratio results in an effective rise of the convection velocity and a less effective drop of the growth rate. However, one should note that although these changes are observable, they are not in any way significant.

Figure 7. The convection velocity and growth rate of the mode $0$ jet instability waves for vortex sheets of different lobed geometry: (a) $N=1$ ; (b) $N=2$ ; (c) $N=3$ ; (d) $N=5$ .

The fact that serrations increase the convection velocity and reduce growth rates of mode $0$ is in accord with the findings of Lajús Jr et al. (Reference Lajús, Cavalieri and Deschamps2015) and Sinha et al. (Reference Sinha, Gudmundsson, Xia and Colonius2016). However, it should be noted that in the work of Sinha et al. (Reference Sinha, Gudmundsson, Xia and Colonius2016), there also exists a low-frequency band where the spatial convection velocity is slightly reduced. Such a difference might be caused by the difference between temporal and spatial analysis, the difference in jet mean flow profiles or the difference in the shear layer thickness.

Figure 8. The convection velocity and growth rate of the mode $1$ jet instability waves for vortex sheets of different lobed geometry: (a) $N=1$ ; (b) $N=2$ ; (c)  $N=3$ (d) $N=5$ .

Figure 7 shows that instability waves of mode $0$ are not very sensitive to either the number of lobes or the penetration ratio. Figure 8, however, shows a different story for mode $1$ . The eigenfunctions corresponding to mode $1$ are even functions of $\unicode[STIX]{x1D719}$ . Figure 8(a) still indicates that a single lobe does not noticeably change the characteristics of the mode $1$ instability waves. This is somewhat expected. Because, as we observed before, one single lobe merely causes a displacement of the profile’s geometrical centre. Therefore, the physics should more or less stay the same as that of an axisymmetric vortex sheet. This is consistent with the results shown in figure 8(a). However, figure 8(b) shows that the use of two lobes leads to a pronounced increase of the convection velocity, and a slight decrease of the temporal growth rate. In contrast to those shown in figure 7, the increase of the convection velocity is more marked at low frequencies. Similarly, increasing $\unicode[STIX]{x1D716}$ results in a stronger rise of the convection velocity. It is very interesting to note that the rise of the convection velocity is nearly linear with respect to $\unicode[STIX]{x1D716}$ . The decrease of the temporal growth rate, however, is most notable in the intermediate frequency range, and the maximum decrease is very small. Figure 8(c) shows the results for $N=3$ . Compared with figure 8(b), the effects of lobes on both the convection velocity and the growth rate appear to be more effective. The most pronounced change, however, occurs when $N$ increases to $5$ , as shown in figure 8(d). Again, these results are consistent with the findings of Sinha et al. (Reference Sinha, Gudmundsson, Xia and Colonius2016). The dependence of both the convection velocity and the temporal growth rate on $\unicode[STIX]{x1D716}$ appears to linear. However, the relative change of the former is significantly larger than that of the latter.

Figure 9. The convection velocity and growth rate of the mode $-1$ jet instability waves for vortex sheets of different lobed geometry: (a) $N=1$ ; (b) $N=2$ ; (c)  $N=3$ ; (d) $N=5$ .

Figure 9 presents the results for the mode $-1$ . The eigenfunctions are odd functions of $\unicode[STIX]{x1D719}$ . Still, figure 9(a) does not show observable changes when $\unicode[STIX]{x1D716}$ increases. However, zooming in this figure, one can see that the convection velocity is weakly increasing as $\unicode[STIX]{x1D716}$ increases, and this is opposite to that observed in figure 8(a). This different behaviour is because $\unicode[STIX]{x1D706}_{1}$ is no longer identical to $\unicode[STIX]{x1D706}_{-1}$ . Therefore, the odd and even eigenfunctions are now independent of each other and they change in opposite ways as $\unicode[STIX]{x1D716}$ increases. If figures 9(a) and 8(a) are too similar to each other to make this trend clear, figure 9(b) makes it much more evident. The number of lobes is now two, and the lobed profile is approximately elliptic. Instead of obtaining higher convection velocities, increasing $\unicode[STIX]{x1D716}$ from $0$ now results in increasingly smaller convection velocities. The temporal growth rate, on the other hand, starts to drop at low frequencies ( $\unicode[STIX]{x1D6FC}a<0.5$ ), but gradually changes to increase at high frequencies, although both are on a small scale. The different eigenvalues of modes $\pm 1$ , hence distinctive characteristics of instability waves of modes $\pm 1$ , are consistent with the analytical results obtained by Crighton (Reference Crighton1973) for elliptic vortex sheets and also agree with the findings in the spatial stability analysis carried out by Kopiev et al. (Reference Kopiev, Ostrikov, Chernyshev and Elliot2004) (because the mode number $1$ is a half of the number of lobes $N=2$ ). Figure 9(c,d) shows the results for $N=3$ and $N=5$ , respectively. One can easily verify that they are identical to those shown in figure 8(c,d). This is because, for both $N=3$ and $N=5$ , the eigenvalues $\unicode[STIX]{x1D706}_{n}$ remain identical to $\unicode[STIX]{x1D706}_{-n}$ . As discussed in § 4.2, this also implies that azimuthally travelling waves can exist, in contrast to the case of $\unicode[STIX]{x1D706}_{n}\neq \unicode[STIX]{x1D706}_{-n}$ , where only azimuthally standing waves are allowed.

Figure 10. The convection velocity and growth rate of the mode $2$ jet instability waves for vortex sheets of different lobed geometry: (a) $N=1$ ; (b) $N=2$ ; (c)  $N=3$ ; (d) $N=5$ .

Figure 10 shows results for the instability waves of mode $2$ . Figure 10(a) is for $N=1$ . We expect little change caused by one single lobe, and this is demonstrated clearly by the figure. The results for $N=2$ are shown in figure 10(b). The mode $2$ instability waves have increasingly lower convection velocities when $\unicode[STIX]{x1D716}$ increases. However, the changes are very small. The changes of the temporal growth rate are nearly unobservable. However, using three lobes can still effectively reduce the convection velocity while marginally altering the temporal growth rate of the mode $2$ instability waves. Like all the results we have reported, the use of five lobes is the most effective way of increasing the convection velocity and decreasing the temporal growth rate. Figure 11 shows the results for the mode $-2$ instability waves. Figure 11(a) exhibits expected behaviour for a lobed profile of $N=1$ . Figure 11(b) shows a very slight increase of the convection velocity and no change to the temporal growth rate. We emphasize again that this is due to $\unicode[STIX]{x1D706}_{n}\neq \unicode[STIX]{x1D706}_{-n}$ . Figures 11(c) and 11(d) are identical to figures 10(c) and 10(d) respectively because of identical eigenvalues.

Figure 11. The convection velocity and growth rate of the mode $-2$ jet instability waves for vortex sheets of different lobed geometry: (a) $N=1$ ; (b) $N=2$ ; (c)  $N=3$ ; (d) $N=5$ .

In summary, the stability characteristics of base flows of a lobed vortex-sheet type are different from those of axisymmetric ones. The differences consist of changes to both the convection velocity and the temporal growth rate of instability waves. The changes become more pronounced as the number of lobes $N$ and the penetration ratio $\unicode[STIX]{x1D716}$ increase. However, instability waves with different mode numbers are affected differently by the lobed geometry. In particular, little change occurs for mode $0$ , no matter how large both $N$ and $\unicode[STIX]{x1D716}$ are. On the other hand, an evident alteration of the characteristics of jet instability waves with large mode numbers occurs when $N>1$ . For $N=3$ and $N=5$ , azimuthally even and odd instability waves demonstrate the same characteristics. However, for $N=2$ and $N=1$ , even and odd instability waves of lobed jets exhibit two different types of behaviour, with one having favourable effects on installed jet noise reduction and the other having adverse. Therefore, for the sake of suppressing instability waves, or achieving installed jet noise reduction, it is desired to use a lobed profile of large $N$ , such as $N=5$ , with a large penetration ratio. Because this results in a larger reduction of the temporal growth rates, and hence may lead to weaker instability waves.

Note that the results discussed above are obtained from a temporal analysis, from which the results of a spatial analysis may be recovered from Gaster’s transformation (Gaster Reference Gaster1962). Gaster’s transformation provides a straightforward way to connect the temporal and spatial stability analyses. One of the key assumptions used in Gaster’s transformation is that both the spatial and temporal growth rates have to be small, and under this assumption Gaster showed that, to the first-order accuracy, the temporal angular frequency, $\text{Re}(\unicode[STIX]{x1D714})$ , rather than the convection velocity $\text{Re}(\unicode[STIX]{x1D714})/\unicode[STIX]{x1D6FC}$ , is the same as the spatial angular frequency, and the temporal growth rate is equal to the product of the spatial growth rate and the temporal group velocity. With the temporal results available, we can therefore obtain the spatial growth rate by using Gaster’s transformation, provided the aforementioned assumption is not violated. Care must be taken for the spatial convection velocity as the $\text{Re}(\unicode[STIX]{x1D714})$ being identical does not imply the same thing for $\text{Re}(\unicode[STIX]{x1D714})/\text{Re}(\unicode[STIX]{x1D6FC})$ (see the spatial convection velocity of a round vortex sheet given by Michalke (Reference Michalke1970) for example), in which case high-order terms need to be incorporated to improve accuracy.

However, the aim of this study is to investigate the effects of lobes on the growth rates of jet instability waves so as to examine the feasibility of controlling installed jet noise using lobed jets, and these effects are unlikely to be significantly different between the temporal or spatial frameworks. For example, if the lobed geometry slightly reduces the temporal growth rate, we would expect similar behavior for the spatial growth rate. Therefore, in this paper, we focus on the temporal results only. Also note that previous studies mainly focused on chevron jets, and we do not expect their results to be identical to the stability results for the special type of lobed base flow used in this study.

4.4 The change to mode shapes

We have studied the effects of lobed geometries on the convection velocity and temporal growth rate of jet instability waves. In this section, we show their effects on the mode shapes. In the rest of this section, we present the contour plot of the pressure field corresponding to each eigenmode. The pressure is normalized such that it is equal to the right-hand side of (4.1b ) inside the vortex sheet. We choose to shown the pressure fields for modes $0$ , $-1$ and $2$ , respectively. When doing so, we fix $\unicode[STIX]{x1D716}=0.15$ but let $N$ vary between $1$ , $2$ , $3$ and $5$ , respectively. For each mode at each value of $N$ , we show the pressure distribution at a low frequency $\unicode[STIX]{x1D6FC}a\approx 0.25$ and then at a high frequency $\unicode[STIX]{x1D6FC}a\approx 2.5$ . These results are shown from figures 12 to 17. In each figure, the lobed profile is represented by a thick solid black line.

Figure 12. The shapes of mode 0 for some strongly lobed geometries when $\unicode[STIX]{x1D716}=0.15$ at a low frequency of $\unicode[STIX]{x1D6FC}a\approx 0.25$ : (a) $N=1$ ; (b) $N=2$ ; (c) $N=3$ ; (d)  $N=5$ . The thick black solid lines show the lobed vortex sheets.

Figure 13. The shapes of mode 0 for some strongly lobed geometries when $\unicode[STIX]{x1D716}=0.15$ at a high frequency of $\unicode[STIX]{x1D6FC}a\approx 2.5$ : (a) $N=1$ ; (b) $N=2$ ; (c) $N=3$ ; (d) $N=5$ . The thick black solid lines show the lobed vortex sheets.

Figure 12 shows the pressure distributions for mode $0$ at $\unicode[STIX]{x1D6FC}a\approx 0.25$ . Figure 12(a) is for $N=1$ . We have mentioned that when $N=1$ the lobed profile is more or less the same as a displaced circle and, hence, its stability characteristics should be nearly the same as a round jet. Figure 12(a) proves this by showing a nearly axisymmetric pressure distribution. The pressure inside the vortex sheet is nearly uniform. This is because the frequency is very low and the modified Bessel function of the first kind approaches to a constant value for a small argument. The pressure outside gradually decays to zero as the distance to the centre of the vortex sheet increases. Note how the pressure is matched continuously across the vortex sheet: another indication for a well-converged eigensolution. Figure 12(b) shows the result for $N=2$ . In this case the lobed profile resembles an ellipse. The effect of the geometry change is to stretch the pressure field inside the vortex sheet to have a distribution similar to the shape of the vortex sheet itself. The pressure outside gradually becomes axisymmetric and decays to zero at a large distance. The behaviours for $N=3$ and $5$ , as shown in figure 12(c,d), respectively, are very similar to $N=2$ .

Figure 14. The shapes of mode $-$ 1 for some strongly lobed geometries when $\unicode[STIX]{x1D716}=0.15$ at a low frequency of $\unicode[STIX]{x1D6FC}a\approx 0.25$ : (a) $N=1$ ; (b) $N=2$ ; (c) $N=3$ ; (d)  $N=5$ . The thick black solid lines show the lobed vortex sheets.

Figure 13 shows the pressure fields for mode $0$ at a high frequency $\unicode[STIX]{x1D6FC}a\approx 2.5$ . A striking difference is the much smaller size of contour regions in each sub-figure. This is due to the fact that the pressure decays more quickly outside the vortex sheet at high frequencies. This is in accord with the fact that installed jet noise is only significant at low frequencies. Figure 13(a) again shows the result for $N=1$ . One difference from figure 12(a) is that the pressure variation inside the vortex sheet is clear at this high frequency, which is what we would expect. When $N=2$ , the lobed geometry has the same stretching effects as those shown in figure 12(b), but with a more marked pressure variation. The same tendencies can be seen from figure 12(c,d). The relatively insignificant change to the shape of the mode $0$ instability wave, as shown in figures 12 and 13, is consistent with the fact that both the convection velocity and temporal growth rate remain roughly the same as those for an axisymmetric jet.

Figure 15. The shapes of mode $-$ 1 for some strongly lobed geometries when $\unicode[STIX]{x1D716}=0.15$ at a high frequency of $\unicode[STIX]{x1D6FC}a\approx 2.5$ : (a) $N=1$ ; (b) $N=2$ ; (c) $N=3$ ; (d) $N=5$ . The thick black solid lines show the lobed vortex sheets.

Figure 14 shows the results for mode $-1$ at $\unicode[STIX]{x1D6FC}a\approx 0.25$ . From § 3.2, we know that for negative mode numbers the pressure distribution is antisymmetric with respect to $\unicode[STIX]{x1D719}=0$ . This is reflected in figure 14. Figure 14(a) very much resembles to that of an axisymmetric jet. When $N=2$ , the pressure field still largely resembles that in figure 14(a). A clear change in mode shape occurs when $N=3$ , as shown in figure 14(c). We can see that a three-lobe profile causes the two lobes of the pressure field to tilt towards two lobes of the vortex sheet. This signifies a potentially large change in the convection velocity and temporal growth rate. The mode shape at $N=5$ shows similar characteristics to that at $N=3$ . Moreover, there are some small and local changes in response to the appearance of local geometry corrugations. We wish to point out, however, that although changes are observed, the mode shapes are still bearing the signature of conventional $\sin \unicode[STIX]{x1D719}$ behaviour, suggesting the appropriateness of our earlier mode labelling strategy.

Figure 15 is similar to figure 14 in almost every aspect, apart from the smaller size of the contour regions owing to the quicker decay of the instability waves outside the vortex sheet at high frequencies. We therefore omit a repetitive description. Instead, we try to understand the behaviour of non-identical eigenvalues, as discussed in § 4.3, from the perspective of mode shapes.

We have observed that for $N=1$ and $N=2$ , the eigenvalue $\unicode[STIX]{x1D706}_{-1}$ is not degenerate any more, and consequently, only standing modes are allowed, whereas travelling modes (in the azimuthal direction, similar to a behaviour of $\text{e}^{\text{i}(m\unicode[STIX]{x1D719}-\unicode[STIX]{x1D714}t)}$ ) are allowed for $N=3$ and $5$ . To understand the difference we can compare figures 15(b) and 15(c). We conclude that, if an eigenvalue is not degenerate, then each of the symmetric plane of the lobe profile must also be a symmetric (or antisymmetric) plane of the eigenmode. For example, in figure 15(b), the lobed profile has two symmetric planes, namely the horizontal and vertical planes, each of which is also a symmetric or antisymmetric plane of the pressure field. This is consistent with the fact that $\unicode[STIX]{x1D706}_{1}$ is not degenerate. On the other hand, in figure 15(c), the lobed profile has three symmetric planes, but only one of them is the antisymmetric plane of the pressure field. Hence, $\unicode[STIX]{x1D706}_{-1}$ must be degenerate. Figure 15(d) is very similar to figure 15(c). For an axisymmetric jet, the vortex sheet profile has infinitely many symmetric planes, however only two of them are the symmetric and antisymmetric planes of mode $-1$ pressure field. Therefore, $\unicode[STIX]{x1D706}_{-1}$ is degenerate for a round jet. Note that the vortex sheet profile in figure 15(a) is close to a displaced circle, but is not strictly one. It has a slight eccentricity and, therefore, has a non-degenerate eigenvalue.

The connection between the degeneracy and the mode shape can be understood as follows. Take figure 15(c) as an example. We mentioned that only one of three symmetric planes, which are angled by $120^{\circ }$ from each other, is the antisymmetric plane of the pressure field. Now we can rotate the pressure field by $120^{\circ }$ , then the resulting pressure field must also be an eigensolution of the stability problem, with the same eigenvalue. Therefore, the corresponding eigenvalue must be degenerate. The combination of the eigenvectors corresponding to the same eigenvalue creates infinitely many eigensolutions, and two of them are those obtained by rotating figure 15(c) by $120^{\circ }$ and $240^{\circ }$ , respectively.

Figure 16. The shapes of mode 2 for some strongly lobed geometries when $\unicode[STIX]{x1D716}=0.15$ at a low frequency of $\unicode[STIX]{x1D6FC}a\approx 0.25$ : (a) $N=1$ ; (b) $N=2$ ; (c) $N=3$ ; (d)  $N=5$ . The thick black solid lines show the lobed vortex sheets.

We expect that the non-degeneracy is likely to occur when the mode number $n$ is equal to $\pm N/2$ , $\pm N$ , etc. This is because, for mode $n$ , the pressure field normally has $2|n|$ lobes, and the symmetric and antisymmetric planes of such a mode can be aligned with all the symmetric planes of the vortex sheet profile. Comparing with the figures shown in § 4.3, we find that this is indeed the case. For example, we see $\unicode[STIX]{x1D706}_{-1}\neq \unicode[STIX]{x1D706}_{1}$ when $N=2$ and $\unicode[STIX]{x1D706}_{-2}\neq \unicode[STIX]{x1D706}_{2}$ when $N=2$ .

Figure 17. The shapes of mode 2 for some strongly lobed geometries when $\unicode[STIX]{x1D716}=0.15$ at a high frequency of $\unicode[STIX]{x1D6FC}a\approx 2.5$ : (a) $N=1$ ; (b) $N=2$ ; (c) $N=3$ ; (d) $N=5$ . The thick black solid lines show the lobed vortex sheets.

Figure 16 shows the distributions of the pressure field for mode $2$ at the low frequency $\unicode[STIX]{x1D6FC}a\approx 0.25$ . Figure 16(a) shows its similarity to that for an axisymmetric vortex sheet while figure 16(b) for an elliptic vortex sheet. Note how the earlier conclusion about non-degeneracy remains valid in figure 16(b–d). Figure 17 shows qualitatively similar results at a higher frequency so we avoid an unnecessary repetition.