2.1 Compressible version

The Euler equations are the governing equations for inviscid compressible flow. Assuming isentropic flow, they are

(2.2a ) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C} & = & \displaystyle -\unicode[STIX]{x1D70C}\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u},\end{eqnarray}$$

(2.2b ) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{u}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u} & = & \displaystyle -\frac{1}{\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D735}p,\end{eqnarray}$$

$\unicode[STIX]{x1D70C}$

$\boldsymbol{u}$

$p$

$r_{i}\leqslant r\leqslant r_{o}$

(2.3) $$\begin{eqnarray}\displaystyle \bar{\boldsymbol{u}}=[\bar{u}_{z},0,Kr], & & \displaystyle\end{eqnarray}$$

where $\bar{u}_{z}$ indicates the uniform axial velocity. The flow is in solid body rotation, such that the mean azimuthal velocity component $\bar{u}_{\unicode[STIX]{x1D703}}=Kr$ increases linearly with radius, where the circulation strength $K$ is – up to a factor $2\unicode[STIX]{x03C0}r_{o}^{2}$ – equal to the circulation of the flow. The mean pressure and density can be determined via the ideal gas law and the radial momentum balance, which are, respectively,

(2.4a ) $$\begin{eqnarray}\displaystyle \bar{p} & = & \displaystyle \frac{c^{2}}{\unicode[STIX]{x1D6FE}}\bar{\unicode[STIX]{x1D70C}},\end{eqnarray}$$

(2.4b ) $$\begin{eqnarray}\displaystyle \bar{\unicode[STIX]{x1D70C}}\frac{\bar{u}_{\unicode[STIX]{x1D703}}^{2}}{r} & = & \displaystyle \frac{\unicode[STIX]{x2202}\bar{p}}{\unicode[STIX]{x2202}r},\end{eqnarray}$$

$c$

$\unicode[STIX]{x1D6FE}$

(2.5) $$\begin{eqnarray}\displaystyle \bar{p}(r)=p_{0}\exp \left(\frac{\unicode[STIX]{x1D6FE}K^{2}r^{2}}{2c^{2}}\right) & & \displaystyle\end{eqnarray}$$

with reference pressure $p_{0}$ . The mean density profile can be trivially retrieved by substituting this relation back into the ideal gas law. Given the above mean fields, equation (2.2) is linearized as

(2.6a ) $$\begin{eqnarray}\displaystyle \frac{1}{c^{2}}\left(\frac{\unicode[STIX]{x2202}p^{\prime }}{\unicode[STIX]{x2202}t}+\bar{u}_{z}\frac{\unicode[STIX]{x2202}p^{\prime }}{\unicode[STIX]{x2202}z}\right) & = & \displaystyle -\displaystyle u_{r}^{\prime }\frac{\unicode[STIX]{x2202}\bar{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x2202}r}-\bar{\unicode[STIX]{x1D70C}}\left(\frac{\unicode[STIX]{x2202}u_{z}^{\prime }}{\unicode[STIX]{x2202}z}+\frac{\unicode[STIX]{x2202}u_{r}^{\prime }}{\unicode[STIX]{x2202}r}+\frac{u_{r}^{\prime }}{r}\right),\end{eqnarray}$$

(2.6b ) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}u_{z}^{\prime }}{\unicode[STIX]{x2202}t}+\bar{u}_{z}\frac{\unicode[STIX]{x2202}u_{z}^{\prime }}{\unicode[STIX]{x2202}z} & = & \displaystyle -\frac{1}{\bar{\unicode[STIX]{x1D70C}}}\frac{\unicode[STIX]{x2202}p^{\prime }}{\unicode[STIX]{x2202}z},\end{eqnarray}$$

(2.6c ) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}u_{r}^{\prime }}{\unicode[STIX]{x2202}t}+\bar{u}_{z}\frac{\unicode[STIX]{x2202}u_{r}^{\prime }}{\unicode[STIX]{x2202}z} & = & \displaystyle -\frac{1}{\bar{\unicode[STIX]{x1D70C}}}\frac{\unicode[STIX]{x2202}p^{\prime }}{\unicode[STIX]{x2202}r}+\displaystyle \frac{1}{c^{2}\bar{\unicode[STIX]{x1D70C}}^{2}}\frac{\unicode[STIX]{x2202}\bar{p}}{\unicode[STIX]{x2202}r}p^{\prime }+\frac{2\bar{u}_{\unicode[STIX]{x1D703}}u_{\unicode[STIX]{x1D703}}^{\prime }}{r},\end{eqnarray}$$

(2.6d ) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}u_{\unicode[STIX]{x1D703}}^{\prime }}{\unicode[STIX]{x2202}t}+\bar{u}_{z}\frac{\unicode[STIX]{x2202}u_{\unicode[STIX]{x1D703}}^{\prime }}{\unicode[STIX]{x2202}z} & = & \displaystyle -u_{r}^{\prime }\left(\frac{\text{d}\bar{u}_{\unicode[STIX]{x1D703}}}{\text{d}r}+\frac{\bar{u}_{\unicode[STIX]{x1D703}}}{r}\right),\end{eqnarray}$$

$p^{\prime }=c^{2}\unicode[STIX]{x1D70C}^{\prime }$

$\unicode[STIX]{x1D70C}^{\prime }$

$p^{\prime }$

$r_{o}$

$c$

(2.7) $$\begin{eqnarray}\displaystyle Ma_{s}=\frac{Kr_{o}}{c}. & & \displaystyle\end{eqnarray}$$

In the limit $Ma_{s}\ll 1$ , the impact of circulation on the mean pressure in (2.5) is negligible and the governing equations can be further simplified: uniform radial profiles of pressure and density may be assumed and the boxed terms in (2.6) may be neglected. Nevertheless, note that the set of equations resulting from the assumption $Ma_{s}\ll 1$ retains compressibility.

In the following, the condition of small swirling Mach number $Ma_{s}\ll 1$ is considered for the compressible flow case. In order to retrieve incompressible equations an additional assumption of infinitely fast speed of sound $c\rightarrow \infty$ is required. This step, which allows further simplifications, is performed in the next subsection.

Modal decomposition as defined in (2.1) simplifies the partial differential equations (2.6) to the following second-order ordinary differential equation for $\hat{u} _{r}$ as

(2.8) $$\begin{eqnarray}\displaystyle \frac{\text{d}^{2}\hat{u} _{r}}{\text{d}r^{2}}+\frac{1}{r}\frac{\text{d}\hat{u} _{r}}{\text{d}r}+\hat{u} _{r}\left(A^{2}-\frac{1}{r^{2}}\right)=0. & & \displaystyle\end{eqnarray}$$

The factor $A$ , which is introduced here for better readability, is defined as

(2.9) $$\begin{eqnarray}\displaystyle A=\sqrt{(C^{2}-4K^{2})\left(\frac{1}{c^{2}}-\frac{k^{2}}{C^{2}}\right)}, & & \displaystyle\end{eqnarray}$$

with modal convective operator $C$ ,

(2.10) $$\begin{eqnarray}\displaystyle C=\unicode[STIX]{x1D714}-\bar{u}_{z}k. & & \displaystyle\end{eqnarray}$$

With impermeability boundary conditions at the inner and outer walls of the annulus,

(2.11a ) $$\begin{eqnarray}\displaystyle \hat{u} _{r}(r_{i}) & = & \displaystyle 0,\end{eqnarray}$$

(2.11b ) $$\begin{eqnarray}\displaystyle \hat{u} _{r}(r_{o}) & = & \displaystyle 0.\end{eqnarray}$$

(2.12) $$\begin{eqnarray}\displaystyle \hat{u} _{r}(r)=\unicode[STIX]{x1D705}_{1}[J_{1}(Ar)+\unicode[STIX]{x1D705}_{2}Y_{1}(Ar)], & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D705}_{2}$ and $A$ are found by applying the boundary conditions, while $\unicode[STIX]{x1D705}_{1}$ is to be determined via an initial condition. The boundary conditions inserted in the solution require

(2.13) $$\begin{eqnarray}\displaystyle J_{1}(Ar_{o})Y_{1}(Ar_{i})-J_{1}(Ar_{i})Y_{1}(Ar_{o})=0, & & \displaystyle\end{eqnarray}$$

where $J_{m}(r)$ and $Y_{m}(r)$ are the Bessel functions of first kind and second kind, respectively. As a characteristic property of the Sturm–Liouville theorem, this equation has infinitely many solutions with eigenvalues $\unicode[STIX]{x1D706}_{n}$ for the variable $A$ . The eigenvalues are ordered as

(2.14) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}_{0}<\unicode[STIX]{x1D706}_{1}<\cdots <\unicode[STIX]{x1D706}_{n}<\cdots \rightarrow \infty . & & \displaystyle\end{eqnarray}$$

Substitution into (2.9) yields dispersion relations, which indicate how the frequencies $\unicode[STIX]{x1D714}_{n}$ of the corresponding eigenmodes depend on the wavenumber $k$ and the parameters $K,c$ that govern the mean flow,

(2.15a ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}_{n,i} & = & \displaystyle k\left(\bar{u}_{z}\pm \frac{c}{\sqrt{2}}\sqrt{L_{n}^{2}-\sqrt{L_{n}^{4}-\frac{16K^{2}}{k^{2}c^{2}}}}\right),\end{eqnarray}$$

(2.15b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}_{n,a} & = & \displaystyle k\left(\bar{u}_{z}\pm \frac{c}{\sqrt{2}}\sqrt{L_{n}^{2}+\sqrt{L_{n}^{4}-\frac{16K^{2}}{k^{2}c^{2}}}}\right).\end{eqnarray}$$

$i$

$a$

$L_{n}$

(2.16) $$\begin{eqnarray}\displaystyle L_{n}=\sqrt{1+\left(\frac{\unicode[STIX]{x1D706}_{n}}{k}\right)^{2}+\left(\frac{2K}{ck}\right)^{2}}. & & \displaystyle\end{eqnarray}$$

Note that (2.15) is valid for $n\neq 0$ . For $n=0$ , i.e. the trivial solution $\unicode[STIX]{x1D706}_{0}=0$ , inertial waves do not exist and the well-known plane acoustic wave dispersion relation is recovered,

(2.17) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}_{0,a}=k(\bar{u}_{z}\pm c). & & \displaystyle\end{eqnarray}$$

For both inertial and acoustic modes there exist two sets of solutions indicated by ‘ $\pm$ ’. The corresponding eigenmodes are determined by substituting the eigenvalues, $\unicode[STIX]{x1D706}_{n}$ , in (2.6) and written

(2.18a ) $$\begin{eqnarray}\displaystyle \hat{u} _{r,n}(r) & = & \displaystyle \unicode[STIX]{x1D705}_{1}M_{1}(\unicode[STIX]{x1D706}_{n}r),\end{eqnarray}$$

(2.18b ) $$\begin{eqnarray}\displaystyle \hat{u} _{\unicode[STIX]{x1D703},n}(r) & = & \displaystyle -\unicode[STIX]{x1D705}_{1}\frac{2K\text{i}}{C_{n}}M_{1}(\unicode[STIX]{x1D706}_{n}r),\end{eqnarray}$$

(2.18c ) $$\begin{eqnarray}\displaystyle \hat{u} _{z,n}(r) & = & \displaystyle \unicode[STIX]{x1D705}_{1}\frac{\text{i}kc^{2}\unicode[STIX]{x1D706}_{n}}{c^{2}k^{2}-C_{n}^{2}}M_{0}(\unicode[STIX]{x1D706}_{n}r),\end{eqnarray}$$

(2.18d ) $$\begin{eqnarray}\displaystyle \hat{p}_{n}(r) & = & \displaystyle \unicode[STIX]{x1D705}_{1}\frac{\text{i}C_{n}\bar{\unicode[STIX]{x1D70C}}c^{2}\unicode[STIX]{x1D706}_{n}}{c^{2}k^{2}-C_{n}^{2}}M_{0}(\unicode[STIX]{x1D706}_{n}r),\end{eqnarray}$$

(2.19) $$\begin{eqnarray}\displaystyle M_{m}(\unicode[STIX]{x1D706}_{n}r)=J_{m}(\unicode[STIX]{x1D706}_{n}r)-\frac{J_{1}(\unicode[STIX]{x1D706}_{n}r_{i})}{Y_{1}(\unicode[STIX]{x1D706}_{n}r_{i})}Y_{m}(\unicode[STIX]{x1D706}_{n}r). & & \displaystyle\end{eqnarray}$$

The modal convective operator is trivially extended for the $n$ th eigenmode as

(2.20) $$\begin{eqnarray}\displaystyle C_{n}=\unicode[STIX]{x1D714}_{n}-\bar{u}_{z}k. & & \displaystyle\end{eqnarray}$$

The interpretation of these results is presented in § 2.3 after the introduction of the incompressible modal analysis.

2.2 Incompressible version

Since the modal analysis of incompressible inertial waves is readily found in the literature (Greenspan Reference Greenspan1968), we will not repeat it in its traditional form. Instead, we demonstrate results as an asymptotic approach from the above compressible solution. This is accomplished by taking the limit of infinite speed of sound for each variable that is defined for the compressible case. From now on the superscript $\dagger$ is employed to indicate incompressible variables, i.e.

(2.21) $$\begin{eqnarray}\displaystyle q^{\dagger }=\lim _{c\rightarrow \infty }q. & & \displaystyle\end{eqnarray}$$

In the incompressibility limit, the linearized compressible equations as in (2.6) reduce to

(2.22a ) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}u_{z}^{\prime }}{\unicode[STIX]{x2202}z}+\frac{\unicode[STIX]{x2202}u_{r}^{\prime }}{\unicode[STIX]{x2202}r}+\frac{u_{r}^{\prime }}{r} & = & \displaystyle 0,\end{eqnarray}$$

(2.22b ) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}u_{z}^{\prime }}{\unicode[STIX]{x2202}t}+\bar{u}_{z}\frac{\unicode[STIX]{x2202}u_{z}^{\prime }}{\unicode[STIX]{x2202}z} & = & \displaystyle -\frac{1}{\bar{\unicode[STIX]{x1D70C}}}\frac{\unicode[STIX]{x2202}p^{\prime }}{\unicode[STIX]{x2202}z},\end{eqnarray}$$

(2.22c ) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}u_{r}^{\prime }}{\unicode[STIX]{x2202}t}+\bar{u}_{z}\frac{\unicode[STIX]{x2202}u_{r}^{\prime }}{\unicode[STIX]{x2202}z} & = & \displaystyle -\frac{1}{\bar{\unicode[STIX]{x1D70C}}}\frac{\unicode[STIX]{x2202}p^{\prime }}{\unicode[STIX]{x2202}r}+\frac{2\bar{u}_{\unicode[STIX]{x1D703}}u_{\unicode[STIX]{x1D703}}^{\prime }}{r},\end{eqnarray}$$

(2.22d ) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}u_{\unicode[STIX]{x1D703}}^{\prime }}{\unicode[STIX]{x2202}t}+\bar{u}_{z}\frac{\unicode[STIX]{x2202}u_{\unicode[STIX]{x1D703}}^{\prime }}{\unicode[STIX]{x2202}z} & = & \displaystyle -u_{r}^{\prime }\left(\frac{\text{d}\bar{u}_{\unicode[STIX]{x1D703}}}{\text{d}r}+\frac{\bar{u}_{\unicode[STIX]{x1D703}}}{r}\right).\end{eqnarray}$$

$u_{r}^{\prime }$

(2.23) $$\begin{eqnarray}\displaystyle \frac{\text{d}^{2}\hat{u} _{r}}{\text{d}r^{2}}+\frac{1}{r}\frac{\text{d}\hat{u} _{r}}{\text{d}r}+\hat{u} _{r}\left((A^{\dagger })^{2}-\frac{1}{r^{2}}\right)=0, & & \displaystyle\end{eqnarray}$$

but the factor $A^{\dagger }$ reduces to

(2.24) $$\begin{eqnarray}\displaystyle A^{\dagger }=\frac{\text{i}k}{C}\sqrt{(C^{2}-4K^{2})}. & & \displaystyle\end{eqnarray}$$

In the incompressible limit, the propagation of the acoustic waves becomes non-local and thus its dispersion relation drops out and the inertial wave dispersion relation simplifies to

(2.25) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}_{n,i}^{\dagger }=k\left(\bar{u}_{z}\pm \frac{2K}{kL_{n}^{\dagger }}\right). & & \displaystyle\end{eqnarray}$$

Now the non-dimensional parameter $L_{n}^{\dagger }$ is

(2.26) $$\begin{eqnarray}\displaystyle L_{n}^{\dagger }=\sqrt{1+\left(\frac{\unicode[STIX]{x1D706}_{n}}{k}\right)^{2}}. & & \displaystyle\end{eqnarray}$$

Corresponding eigenmodes are in the incompressible limit simplified to

(2.27a ) $$\begin{eqnarray}\displaystyle \hat{u} _{r,n}^{\dagger }(r) & = & \displaystyle \unicode[STIX]{x1D705}_{1}M_{1}(\unicode[STIX]{x1D706}_{n}r),\end{eqnarray}$$

(2.27b ) $$\begin{eqnarray}\displaystyle \hat{u} _{\unicode[STIX]{x1D703},n}^{\dagger }(r) & = & \displaystyle -\unicode[STIX]{x1D705}_{1}\frac{2K\text{i}}{C_{n}}M_{1}(\unicode[STIX]{x1D706}_{n}r),\end{eqnarray}$$

(2.27c ) $$\begin{eqnarray}\displaystyle \hat{u} _{z,n}^{\dagger }(r) & = & \displaystyle \unicode[STIX]{x1D705}_{1}\frac{\text{i}\unicode[STIX]{x1D706}_{n}}{k}M_{0}(\unicode[STIX]{x1D706}_{n}r),\end{eqnarray}$$

(2.27d ) $$\begin{eqnarray}\displaystyle \hat{p}_{n}^{\dagger }(r) & = & \displaystyle \unicode[STIX]{x1D705}_{1}\frac{\text{i}C_{n}\bar{\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D706}_{n}}{k^{2}}M_{0}(\unicode[STIX]{x1D706}_{n}r).\end{eqnarray}$$

$\unicode[STIX]{x1D706}_{n}$

2.3 Interpretation of modal results

In this section, we interpret the results of modal analysis for both compressible and incompressible cases. In the literature, several publications present a similar analysis, but each of them focuses on different aspects. It is instructive to briefly discuss these studies and juxtapose our work.

For turbomachinery applications, Kerrebrock (Reference Kerrebrock1977) employed modal analysis to demonstrate wave modes in inviscid, compressible, ducted swirling flows. The author distinguished acoustic and ‘nearly convective’ waves and discussed the stability of modes. Golubev & Atassi (Reference Golubev and Atassi1998) extended this work by analysing a variety of azimuthal velocity profiles, i.e. free vortex $\bar{u}_{\unicode[STIX]{x1D703}}=\unicode[STIX]{x1D6E4}/r$ and solid body rotation $\bar{u}_{\unicode[STIX]{x1D703}}=Kr$ . They confirmed the existence of acoustic and nearly convective waves by means of asymptotic analysis and numerical solution of the modal equations with spectral methods. Kousen (Reference Kousen1996) followed a similar numerical approach. These studies mentioned also a third branch of ‘purely convective’ wave solutions for the free vortex case. Employing an initial value formulation, Tam & Auriault (Reference Tam and Auriault1998) confirmed the existence of the acoustic and nearly convected waves, but showed that the purely convective mode is a spurious mode, i.e. a numerical artefact.

Note that all of these early studies speak of ‘nearly convective’ waves. This term indicates that wave propagation is of dispersive character, with speeds close to those of convection by the mean flow – an order of magnitude slower than acoustic propagation. However, there were no further attempts to quantify the wave propagation velocities. The Coriolis force was correctly identified as the driving mechanism of these waves, however, there was no mention of the literature on inertial waves.

A modal analysis framework was also employed by Rusak & Lee (Reference Rusak and Lee2002) for a numerical investigation of vortex breakdown in compressible flows. Based on the concept of critical swirl introduced by Benjamin (Reference Benjamin1962), it was shown that vortex breakdown occurs at lower swirl strength as Mach number increases. Renac et al. (Reference Renac, Sipp and Jacquin2007) confirmed this numerical result by analytically deriving the inertial wave dispersion relation. By setting $k=0$ and $\unicode[STIX]{x1D714}=0$ in the dispersion relation, the impact of compressibility on the critical swirl strength was assessed. Interestingly, the result for the dispersion relation agrees with (2.15) in non-dimensional form, although different assumptions were employed: Renac et al. (Reference Renac, Sipp and Jacquin2007) assumed that the product of wavenumber $k$ and Mach number $Ma$ is small $k\,Ma\ll 1$ , whereas the present derivation is based on the assumption of small swirling Mach number, i.e. $Ma_{s}\ll 1$ .

In contrast to prior publications, the present study employs modal analysis in order to characterize the eigenmode structures and quantify propagation speeds. Our main objective is to derive a time–space domain solution of an initial value problem (see § 3), where an initial perturbation of swirling flow results from an acoustic wave that is incident on a swirler. The dispersion relation is especially important in this context because propagation speeds of perturbations can be deduced from the group speed, $c_{g}=\unicode[STIX]{x2202}\unicode[STIX]{x1D714}/\unicode[STIX]{x2202}k$ . The modal analysis is a crucial step before proceeding to the initial value problem, as initial perturbations are decomposed into eigenmodes. In this regard, the present analysis follows Arendt et al. (Reference Arendt, Fritts and Andreassen1997), who represent an initial vortical perturbation of an incompressible vortex tube as a superposition of the corresponding inertial wave eigenmodes. An extensive discussion with respect to this publication is given in § 3.

Figure 2.

Structure of first three eigenmodes $n=1,2,3$ (solid black, dashed red, dotted blue lines). (a) Profiles of $\hat{u} _{\unicode[STIX]{x1D703}}$ and $\hat{u} _{r}$ as a function of radius. (b) Profiles of $\hat{u} _{z}$ and $\hat{p}$ .

Figure 3.

Inertial wave group speeds $c_{g}$ versus normalized axial wavenumber $k^{\ast }$ . (a) Incompressible case, reduced group speed $c_{g}^{\dagger }-\bar{u}_{z}$ of the first three eigenmodes $n=1,2,3$ (solid black, dashed red, dotted blue lines). (b) Comparison of incompressible (solid black line) and compressible normalized reduced group speeds $c_{g}^{\ast }$ for swirling Mach numbers $Ma_{s}=0.3$ (dashed red) and $Ma_{s}=0.8$ (dotted blue). Only first fast propagating group speed is considered.

Figure 2 shows radial profiles of the first three eigenmodes $n=1,2,3$ as defined in (2.18) and (2.27). Panel (a) shows profiles of radial and azimuthal velocities $\hat{u} _{r}$ and $\hat{u} _{\unicode[STIX]{x1D703}}$ , while panel (b) shows axial velocity $\hat{u} _{z}$ and pressure $\hat{p}$ . The radial coordinate is normalized as $r^{\ast }=(r-r_{i})/(r_{o}-r_{i})$ . The profiles are normalized with the respective maximum values, i.e. $M_{n}^{\ast }(\unicode[STIX]{x1D706}_{n}r^{\ast })=M_{n}(\unicode[STIX]{x1D706}_{n}r^{\ast })/\max (M_{n}(\unicode[STIX]{x1D706}_{n}r^{\ast }))$ . Note that the respective normalized profiles of the incompressible and the compressible cases agree with each other, thus the figure represents both cases. Obviously, the eigenmodes become more oscillatory with increasing order $n$ .

Figure 3 shows the dependence of inertial wave group speed on axial wavenumber. In these plots, reduced group speeds $c_{g}-\bar{u}_{z}$ are shown, i.e. the speed of convection is not taken into account. The wavenumber is normalized by the first eigenvalue, $k^{\ast }=k/\unicode[STIX]{x1D706}_{1}$ . Panel (a) represents the first three eigenmodes $n=1,2,3$ of the incompressible case. For each eigenmode, two values of the group speed are found, corresponding to the $\pm$ sign in the dispersion relations (2.15) and (2.25) for the compressible and the incompressible case, respectively. The opposite signs of the reduced group speeds $c_{g}-\bar{u}_{z}$ imply that inertial waves propagate at speeds above as well as below that of base flow convection. The deviation of the group speeds from the convective velocity increases with circulation strength $K$ , but diminishes with increasing order $n$ and with increasing axial wavenumber $k^{\ast }$ , indicative of dispersive wave propagation. In the low axial wavenumber limit, group speeds are the highest and are written

(2.28) $$\begin{eqnarray}\displaystyle \lim _{k\rightarrow 0}c_{g}^{\dagger }-\bar{u}_{z}=\pm \frac{2K}{\unicode[STIX]{x1D706}_{n}}. & & \displaystyle\end{eqnarray}$$

These values are shown on the ordinate of the plot.

These results provide an explanation for the unexpected experimental results on the propagation speed of azimuthal velocity perturbations that were discussed in the introduction. The 40 %–50 % deviation in the propagation speed from convection is attributed to dispersive inertial wave propagation. In § 3, an initial value problem is proposed for a detailed analysis and quantification of the inertial wave propagation.

Figure 3(b) shows the fast propagating group speed of the first eigenmode $n=1$ for values of the swirling Mach number $Ma_{s}=0,0.3,0.8$ , where the value of zero Mach number is represented by the solution for the incompressible case. The reduced group speeds are normalized $c_{g}^{\ast }=(c_{g}-\bar{u}_{z})\unicode[STIX]{x1D706}_{1}/K$ , such that the maximum group speed in the incompressible limit is $\lim _{c\rightarrow \infty }c_{g}^{\ast }=2$ . Obviously, the impact of compressibility on inertial wave propagation is marginal even for $Ma_{s}=0.8$ , which shows only a slight decrease in the group speed at small wavenumbers. This decrease can be quantified in the low wavenumber limit as

(2.29) $$\begin{eqnarray}\displaystyle \lim _{k\rightarrow 0}c_{g}-\bar{u}_{z}=\frac{2Kc}{\sqrt{\unicode[STIX]{x1D706}_{n}^{2}c^{2}+4K^{2}}}. & & \displaystyle\end{eqnarray}$$

Note that this solution implicitly confirms the compressibility dependency of the critical swirl state discussed by Rusak & Lee (Reference Rusak and Lee2002). The subcritical swirl state requires upstream propagating waves. By observing the above solution, this is satisfied for the following critical circulation strength as

(2.30) $$\begin{eqnarray}\displaystyle K_{c}=\frac{\bar{u}_{z}\unicode[STIX]{x1D706}_{n}c}{2\sqrt{c^{2}-\bar{u}_{z}^{2}}}. & & \displaystyle\end{eqnarray}$$

This condition agrees with the result of Renac et al. (Reference Renac, Sipp and Jacquin2007), i.e. $1/\sqrt{1-Ma^{2}}$ in non-dimensional form. The value $Ma_{s}=0.8$ seems incompatible with the assumption of small swirl Mach number, but note that this large value was selected to demonstrate that the change in the group speed is negligible. Also note that Renac et al. (Reference Renac, Sipp and Jacquin2007) provide a comparison against numerical solutions where the low Mach number assumption is relaxed. A good match is achieved, which confirms the validity of the results obtained with $Ma_{s}=0.8$ .

As mentioned in the introduction, Palies et al. (Reference Palies, Ilak and Cheng2017) employed linearized Euler equations to show that the propagation of azimuthal velocity perturbations is entirely convective. However, instead of using the complete set of Euler equations (2.6), only the radial and azimuthal momentum equations are taken into account (see equations (A 1) and (A 2) in Palies et al. (Reference Palies, Ilak and Cheng2017)), while mass conservation and the axial momentum equation are neglected. In the light of the present analysis, we argue that this incomplete set of equations is not capable of correctly describing inertial wave propagation, and that conclusions based upon these equations are invalid.

2.4 Comparison of inviscid theory against modal linearized Navier–Stokes equations

In this section, analytical results for the inviscid case are compared against numerical modal analysis based on the linearized incompressible Navier–Stokes equations. This comparison serves not only as validation, but also allows us to investigate the effects of viscosity on inertial waves. We employ an approach used previously in stability/sensitivity analysis (see Juniper (Reference Juniper2012) and Juniper, Hanifi & Theofilis (Reference Juniper, Hanifi and Theofilis2014)). In a flow that is not absolutely unstable, there is a convenient and reasonably accurate mapping between the temporal and spatial analyses at a given wavenumber or angular frequency (see Gaster (Reference Gaster1962)). Here, we perform a temporal analysis. The Chebyshev spectral collocation technique proposed by Khorrami, Malik & Ash (Reference Khorrami, Malik and Ash1989) is employed. A total of $50$ grid points are used for the discretization. The simplified flow field described in (2.3) is also employed for this numerical study. Slip wall boundary conditions are used to match the analytical inviscid approach accurately,

(2.31) $$\begin{eqnarray}\displaystyle \hat{u} _{r}=\hat{u} _{\unicode[STIX]{x1D703}}=\frac{\unicode[STIX]{x2202}\hat{u} _{z}}{\unicode[STIX]{x2202}r}=\frac{\unicode[STIX]{x2202}\hat{p}}{\unicode[STIX]{x2202}r}=0,\quad \text{for}~r=r_{i},r_{o}. & & \displaystyle\end{eqnarray}$$

In figure 4, the eigenfrequencies $\unicode[STIX]{x1D714}_{n}$ of the first three modes $n=1,2,3$ are represented in terms of the respective phase speeds and growth rates for three values of axial wavenumber $k$ . The Reynolds number is selected as $Re=\bar{u}_{z}D_{H}/\unicode[STIX]{x1D708}=18\,000$ to match with the experimental configuration of Komarek & Polifke (Reference Komarek and Polifke2010), where $D_{H}$ is the hydraulic diameter and $\unicode[STIX]{x1D708}$ is the kinematic viscosity. Similarly, the swirl number is chosen as $S=KD_{H}/\bar{u}_{z}=1.6$ . However, owing to the non-dimensionalization described below, the results in figure 4 are independent of both these parameters.

Figure 4.

Normalized temporal decay rate $\unicode[STIX]{x1D714}_{i}^{\ast }$ of the first six eigenvalues as a function of the normalized phase speed $c_{p}^{\ast }$ . Dots represent the temporal analysis for the linearized Navier–Stokes equation for three normalized axial wavenumbers, $k^{\ast }=[0.1,0.75,1.25]$ , which are connected by lines. Squares indicate the eigenvalues from the inviscid analysis as in (2.25) for $k^{\ast }=0.1$ . The corresponding radial profiles of azimuthal velocity perturbation are shown on the right.

The phase speed $c_{p}=\unicode[STIX]{x1D714}_{r}/k$ depends on the real part $\unicode[STIX]{x1D714}_{r}$ of the angular frequency. The abscissa of the plot corresponds to a normalized reduced phase speed, i.e. $c_{p}^{\ast }=(c_{p}-\bar{u}_{z})\unicode[STIX]{x1D706}_{1}/K$ , i.e. again, the plot represents the deviation from the speed of convection. The ordinate corresponds to the normalized decay rate, $\unicode[STIX]{x1D714}_{i}^{\ast }=\unicode[STIX]{x1D714}_{i}ReD_{H}/\bar{u}_{z}$ , i.e. the imaginary part of the angular frequency. Negative values indicate that the mode amplitude decays in time. Round markers represent eigenvalues calculated by the linearized Navier–Stokes equations. Solid lines are used to track eigenvalues for varying axial wavenumber. Squares represent analytical eigenvalues determined from the incompressible dispersion relation via (2.25). For a better visualization, the analytical eigenvalues are plotted only for $k^{\ast }=0.1$ . In figure 2(b), azimuthal velocity profiles of the eigenmodes are shown. Since analytical inviscid eigenmode profiles, which are shown in figure 2, are in perfect agreement with the numerical approach for the linearized Navier–Stokes equations, only a single mode profile is plotted. The arrows indicate which eigenmode corresponds to which set of eigenvalues. For each eigenmode, two eigenvalues exist, one being faster and the other being slower than the convection speed. Therefore, the eigenvalues are numbered as pairs. Inviscid phase speeds are also in perfect agreement with the numerical viscous approach. The obvious mismatch in decay rates is expected, since the inviscid analysis does not account for viscous dissipation.

The first pair of eigenvalues, i.e. $n=1$ , decays slowest and deviates most from the convective speed. With increasing mode order $n$ or wavenumber $k$ , the phase speeds approach the convection speed. These statements were also confirmed in the previous section for the inviscid approach. More importantly, modes are more strongly damped with increasing $n$ and $k$ . This trend in decay rates suggest that the low wavenumber region is more relevant for the propagation of a perturbation, with the first mode as the leading term, while high-order modes decay more rapidly. We shall use these findings to interpret the inertial wave propagation in thermo-acoustic systems, see § 4.

3.1 Mode conversion: actuator disk theory

The mode conversion process across a swirler from incident plane acoustic waves to azimuthal velocity fluctuations was modelled via actuator disk theory by Cumpsty & Marble (Reference Cumpsty and Marble1977) and revised by Palies et al. (Reference Palies, Durox, Schuller and Candel2011a ) for thermo-acoustic applications. Here, we briefly introduce actuator disk theory and the set-up that we use for the initial value problem. The swirler is modelled as a thin disk that generates azimuthal momentum instantly. The jump condition across the swirler is schematically illustrated in figure 5. Solid black and dashed red arrows indicate the base and perturbed flow quantities, respectively. Grey planes represent upstream perturbations in downstream propagating plane acoustic waves.

Figure 5.

Schematic illustration of the swirler inside an annular mixing duct. Grey planes indicate initial perturbations of downstream propagating acoustic waves as denoted by $f$ -waves. The base flow vectors are shown with solid black vectors. Right after the swirler, the mode conversion process is illustrated. Corresponding dashed red flow vectors indicate perturbed velocities resulting from actuator disk theory. The dashed red box indicates the domain for the initial value problem.

To derive jump conditions for small fluctuating quantities, Palies et al. (Reference Palies, Durox, Schuller and Candel2011a ) introduced several simplifications, which are also employed here: the flow is assumed to be isentropic, while the pressure drop across the swirler is neglected. Moreover, the swirler is assumed to be acoustically transparent, which means that the incident plane acoustic waves are transmitted in a lossless manner across the swirler. Under these assumptions, jump conditions are written as

(3.6a ) $$\begin{eqnarray}\displaystyle f|_{down} & = & \displaystyle f|_{up},\end{eqnarray}$$

(3.6b ) $$\begin{eqnarray}\displaystyle u_{\unicode[STIX]{x1D703}}^{\prime }|_{down} & = & \displaystyle u_{z}^{\prime }|_{up}\tan (\unicode[STIX]{x1D719}),\end{eqnarray}$$

$f$

$\bar{\boldsymbol{u}}$

In summary, plane acoustic waves incident on a swirler generate azimuthal velocity perturbations with a strength that depends on the swirler blade angle. From this point onward, mere convective transport of azimuthal velocity perturbations (also referred to as ‘vorticity waves’) has been taken for granted (Cumpsty & Marble Reference Cumpsty and Marble1977; Richards & Yip Reference Richards and Yip1995; Komarek & Polifke Reference Komarek and Polifke2010; Palies et al. Reference Palies, Durox, Schuller and Candel2011a ). Signal propagation times observed in experiments and simulations do not support this assumption, however, as discussed in the introduction, and no further attempts have been made until now to characterize the subsequent propagation of azimuthal velocity perturbations.

Another important point is that the Coriolis force has not been taken into consideration. Indeed, Palies et al. (Reference Palies, Durox, Schuller and Candel2011a ) simply used the one-dimensional plane acoustic wave solution, i.e.

(3.7a ) $$\begin{eqnarray}\displaystyle f & = & \displaystyle 0.5\left(\frac{p^{\prime }}{\bar{\unicode[STIX]{x1D70C}}c}+u_{z}^{\prime }\right),\end{eqnarray}$$

(3.7b ) $$\begin{eqnarray}\displaystyle g & = & \displaystyle 0.5\left(\frac{p^{\prime }}{\bar{\unicode[STIX]{x1D70C}}c}-u_{z}^{\prime }\right).\end{eqnarray}$$

$g$

$g=0$

(3.8a ) $$\begin{eqnarray}\displaystyle u_{z}^{\prime }|_{down} & = & \displaystyle f|_{up},\end{eqnarray}$$

(3.8b ) $$\begin{eqnarray}\displaystyle p^{\prime }|_{down} & = & \displaystyle \bar{\unicode[STIX]{x1D70C}}cf|_{up}.\end{eqnarray}$$

(3.9) $$\begin{eqnarray}\displaystyle p^{\prime }|_{down}=\bar{\unicode[STIX]{x1D70C}}cf|_{up}+2K\bar{\unicode[STIX]{x1D70C}}\int u_{\unicode[STIX]{x1D703}}^{\prime }|_{down}\,\text{d}r. & & \displaystyle\end{eqnarray}$$

This additional term is crucial for the compressible initial value problem (see (3.3)), but does not play a role in the incompressible case (see (3.5)). These assertions will be explained further in the next section.

3.2 Initial perturbation

To pose an initial value problem for inertial wave propagation, a step perturbation is introduced, as discussed in the previous subsection. This condition is written mathematically as

(3.10a ) $$\begin{eqnarray}\displaystyle u_{z}^{\prime }(z,r,t=0)|_{up} & = & \displaystyle \unicode[STIX]{x1D700}_{a}\text{H}(-z),\end{eqnarray}$$

(3.10b ) $$\begin{eqnarray}\displaystyle p^{\prime }(z,r,t=0)|_{up} & = & \displaystyle \unicode[STIX]{x1D700}_{a}\bar{\unicode[STIX]{x1D70C}}c\text{H}(-z),\end{eqnarray}$$

$\unicode[STIX]{x1D700}_{a}$

$\text{H}(-z)$

$z<0$

(3.11a ) $$\begin{eqnarray}\displaystyle u_{z}^{\prime }(z,r,t=0)|_{down} & = & \displaystyle \unicode[STIX]{x1D700}_{a}\text{H}(-z),\end{eqnarray}$$

(3.11b ) $$\begin{eqnarray}\displaystyle u_{r}^{\prime }(z,r,t=0)|_{down} & = & \displaystyle 0,\end{eqnarray}$$

(3.11c ) $$\begin{eqnarray}\displaystyle u_{\unicode[STIX]{x1D703}}^{\prime }(z,r,t=0)|_{down} & = & \displaystyle \left(\mathop{\sum }_{n=1}^{\infty }\unicode[STIX]{x1D6FC}_{n}M_{1}(\unicode[STIX]{x1D706}_{n}r)\right)\text{H}(-z),\end{eqnarray}$$

(3.11d ) $$\begin{eqnarray}\displaystyle p^{\prime }(z,r,t=0)|_{down} & = & \displaystyle \left(\unicode[STIX]{x1D700}_{a}\bar{\unicode[STIX]{x1D70C}}c-2K\bar{\unicode[STIX]{x1D70C}}\mathop{\sum }_{n=1}^{\infty }\frac{\unicode[STIX]{x1D6FC}_{n}}{\unicode[STIX]{x1D706}_{n}}M_{0}(\unicode[STIX]{x1D706}_{n}r)\right)\text{H}(-z),\end{eqnarray}$$

(3.12) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FC}_{n}=\frac{\displaystyle \int _{r_{i}}^{r_{o}}f_{\unicode[STIX]{x1D703}}(r)M_{1}(\unicode[STIX]{x1D706}_{n}r)\,\text{d}r}{\displaystyle \int _{r_{i}}^{r_{o}}M_{1}^{2}(\unicode[STIX]{x1D706}_{n}r)\,\text{d}r}, & & \displaystyle\end{eqnarray}$$

with an arbitrary radial profile of azimuthal velocity fluctuations $f_{\unicode[STIX]{x1D703}}(r)$ . Here we employ the Kutta condition result, c.f. (3.6a ). Since the upstream acoustic axial velocity condition is uniform, the corresponding azimuthal velocity perturbation is also assumed to be uniform:

(3.13) $$\begin{eqnarray}\displaystyle f_{\unicode[STIX]{x1D703}}(r)=\unicode[STIX]{x1D700}_{a}\tan (\unicode[STIX]{x1D719}). & & \displaystyle\end{eqnarray}$$

This also confirms that only the axisymmetric inertial wave modes are generated.

The substitution of these initial conditions into the ordinary differential equation (3.3) for the compressible case yields

(3.14) $$\begin{eqnarray}\displaystyle \frac{\text{d}^{2}\hat{u} _{r}}{\text{d}r^{2}}+\frac{1}{r}\frac{\text{d}\hat{u} _{r}}{\text{d}r}+\hat{u} _{r}\left(A^{2}-\frac{1}{r^{2}}\right)=\frac{k^{2}}{C^{2}}\left(2K\mathop{\sum }_{n=1}^{\infty }\unicode[STIX]{x1D6FC}_{n}M_{1}(\unicode[STIX]{x1D706}_{n}r)\right)\left(\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}(k)+\frac{\text{i}}{k}\right). & & \displaystyle\end{eqnarray}$$

Note that the direct impact of plane acoustic waves does not enter the equation for $u_{r}^{\prime }$ , since first radial derivatives of $p^{\prime }$ and $u_{z}^{\prime }$ are zero. This is an important result, which indicates that plane acoustic waves do not introduce a direct force in the radial velocity component.

In the incompressible limit (3.14) reduces to

(3.15) $$\begin{eqnarray}\displaystyle \frac{\text{d}^{2}\hat{u} _{r}}{\text{d}r^{2}}+\frac{1}{r}\frac{\text{d}\hat{u} _{r}}{\text{d}r}+\hat{u} _{r}\left((A^{\dagger })^{2}-\frac{1}{r^{2}}\right)=\frac{k^{2}}{C^{2}}\left(2K\mathop{\sum }_{n=1}^{\infty }\unicode[STIX]{x1D6FC}_{n}M_{1}(\unicode[STIX]{x1D706}_{n}r)\right)\left(\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}(k)+\frac{\text{i}}{k}\right). & & \displaystyle\end{eqnarray}$$

It is remarkable that the inhomogeneous part is the same as for the compressible case. This is achieved by cancellation of two terms in (3.3), i.e. the coefficient of the initial value for azimuthal velocity perturbation reduces and, at the same time, the initial condition in the pressure perturbation drops out. In combination, these two effects yield the same inhomogeneous part.

3.3 Solution

Equation (3.14) can be solved via the method of undetermined coefficients, which is a standard method to solve inhomogeneous ordinary differential equations. The resulting expression, $\hat{u} _{r}$ , is then substituted in (3.2) to obtain other components, which are written as

(3.16a ) $$\begin{eqnarray}\displaystyle \hspace{-27.0pt}\hat{u} _{r}(k,r,s)=2Kc^{2}k^{2}\left(\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}(k)+\frac{\text{i}}{k}\right)\mathop{\sum }_{n=1}^{\infty }\frac{\unicode[STIX]{x1D6FC}_{n}M_{1}(\unicode[STIX]{x1D706}_{n}r)}{C^{4}-(c^{2}(\unicode[STIX]{x1D706}_{n}^{2}+k^{2})+4K^{2})C^{2}+4K^{2}c^{2}k^{2}}, & & \displaystyle\end{eqnarray}$$

(3.16b ) $$\begin{eqnarray}\displaystyle \hat{u} _{\unicode[STIX]{x1D703}}(k,r,s)=\text{i}C\left(\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}(k)+\frac{\text{i}}{k}\right)\mathop{\sum }_{n=1}^{\infty }\frac{(C^{2}-c^{2}(\unicode[STIX]{x1D706}_{n}^{2}+k^{2})-4K^{2})\unicode[STIX]{x1D6FC}_{n}M_{1}(\unicode[STIX]{x1D706}_{n}r)}{C^{4}-(c^{2}(\unicode[STIX]{x1D706}_{n}^{2}+k^{2})+4K^{2})C^{2}+4K^{2}c^{2}k^{2}}, & & \displaystyle\end{eqnarray}$$

(3.16c ) $$\begin{eqnarray}\displaystyle \hat{u} _{z}(k,r,s) & = & \displaystyle \left(\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}(k)+\frac{\text{i}}{k}\right)\left[\frac{\text{i}\unicode[STIX]{x1D700}_{a}}{C-ck}\right.\nonumber\\ \displaystyle & & \displaystyle -\left.2K\text{i}k\mathop{\sum }_{n=1}^{\infty }\frac{(C^{2}-c^{2}\unicode[STIX]{x1D706}_{n}^{2}-4K^{2})\unicode[STIX]{x1D6FC}_{n}M_{0}(\unicode[STIX]{x1D706}_{n}r)}{\unicode[STIX]{x1D706}_{n}(C^{4}-(c^{2}(\unicode[STIX]{x1D706}_{n}^{2}+k^{2})+4K^{2})C^{2}+4K^{2}c^{2}k^{2})}\right]\end{eqnarray}$$

(3.16d ) $$\begin{eqnarray}\displaystyle \hat{p}(k,r,s) & = & \displaystyle \left(\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}(k)+\frac{\text{i}}{k}\right)\left[\frac{\text{i}c\bar{\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D700}_{a}}{C-ck}\right.\nonumber\\ \displaystyle & & \displaystyle -\left.2K\text{i}C\bar{\unicode[STIX]{x1D70C}}\mathop{\sum }_{n=1}^{\infty }\frac{(C^{2}-c^{2}\unicode[STIX]{x1D706}_{n}^{2}-4K^{2})\unicode[STIX]{x1D6FC}_{n}M_{0}(\unicode[STIX]{x1D706}_{n}r)}{\unicode[STIX]{x1D706}_{n}(C^{4}-(c^{2}(\unicode[STIX]{x1D706}_{n}^{2}+k^{2})+4K^{2})C^{2}+4K^{2}c^{2}k^{2})}\right].\quad\end{eqnarray}$$

Before performing the inverse transformations to retrieve the solution in the time–space domain, some important features of the solution can be analysed in the frequency and axial wavenumber domain. Similar to initial conditions, the inertial wave solution is also decomposed into eigenmodes. Moreover, plane acoustic waves are not influenced by inertial waves, i.e. the first terms in square brackets of $\hat{u} _{z}$ and $\hat{p}$ correspond to the one-dimensional plane acoustic wave solution. Hence, the solution of inertial waves is separated from that of acoustic waves. Note that roots of the denominator in the above equation correspond to the dispersion relation as defined in (2.15). The dispersive propagation nature of inertial waves is therefore indicated implicitly in these equations.

To the authors’ knowledge, an inverse transform in both time and space of the above equations cannot be found in an analytical manner. One possible simplification is to assume a low frequency and low axial wavenumber, $s\rightarrow 0$ and $k\rightarrow 0$ . This limit illustrates characteristic features of the inertial wave propagation, because, as indicated in figure 3, group speeds deviate most from the convective speed in the low wavenumber region. Moreover, decay rates calculated from the linearized Navier–Stokes equations as shown in figure 4 also suggest that mode damping increases with increasing wavenumber. Although no mode damping can be captured in the inviscid solution, this numerical observation justifies the low wavenumber simplification. This assumption is also relevant for thermo-acoustic applications, because the flame response is by nature a low pass filter. This means that flames barely respond to perturbations at high frequencies. Under the low wavenumber and low frequency assumption, both inverse Laplace and Fourier transformations can be performed analytically. The simplified time–space domain solution is written

(3.17a ) $$\begin{eqnarray}\displaystyle u_{r}^{\prime }(z,r,t) & {\approx} & \displaystyle -0.5\mathop{\sum }_{n=1}^{\infty }\frac{c}{\sqrt{c^{2}\unicode[STIX]{x1D706}_{n}^{2}+4K^{2}}}\unicode[STIX]{x1D6FC}_{n}M_{1}(\unicode[STIX]{x1D706}_{n}r)\nonumber\\ \displaystyle & & \displaystyle \times \left[\unicode[STIX]{x1D6FF}\left(z-\frac{2Kct}{\sqrt{c^{2}\unicode[STIX]{x1D706}_{n}^{2}+4K^{2}}}\right)-\unicode[STIX]{x1D6FF}\left(z+\frac{2Kct}{\sqrt{c^{2}\unicode[STIX]{x1D706}_{n}^{2}+4K^{2}}}\right)\right],\end{eqnarray}$$

(3.17b ) $$\begin{eqnarray}\displaystyle u_{\unicode[STIX]{x1D703}}^{\prime }(z,r,t) & {\approx} & \displaystyle 0.5\mathop{\sum }_{n=1}^{\infty }\unicode[STIX]{x1D6FC}_{n}M_{1}(\unicode[STIX]{x1D706}_{n}r)\nonumber\\ \displaystyle & & \displaystyle \times \left[2-\text{H}\left(z-\frac{2Kct}{\sqrt{c^{2}\unicode[STIX]{x1D706}_{n}^{2}+4K^{2}}}\right)-\text{H}\left(z+\frac{2Kct}{\sqrt{c^{2}\unicode[STIX]{x1D706}_{n}^{2}+4K^{2}}}\right)\right],\qquad\end{eqnarray}$$

(3.17c ) $$\begin{eqnarray}\displaystyle u_{z}^{\prime }(z,r,t) & {\approx} & \displaystyle \unicode[STIX]{x1D700}_{a}\left[1-\text{H}(z-ct)\right]-0.5\mathop{\sum }_{n=1}^{\infty }\frac{\sqrt{c^{2}\unicode[STIX]{x1D706}_{n}^{2}+4K^{2}}}{c\unicode[STIX]{x1D706}_{n}}\unicode[STIX]{x1D6FC}_{n}M_{0}(\unicode[STIX]{x1D706}_{n}r)\nonumber\\ \displaystyle & & \displaystyle \times \left[\text{H}\left(z-\frac{2Kct}{\sqrt{c^{2}\unicode[STIX]{x1D706}_{n}^{2}+4K^{2}}}\right)-\text{H}\left(z+\frac{2Kct}{\sqrt{c^{2}\unicode[STIX]{x1D706}_{n}^{2}+4K^{2}}}\right)\right],\end{eqnarray}$$

(3.17d ) $$\begin{eqnarray}\displaystyle p^{\prime }(z,r,t) & {\approx} & \displaystyle \unicode[STIX]{x1D700}_{a}\bar{\unicode[STIX]{x1D70C}}c\left[1-\text{H}(z-ct)\right]-\bar{\unicode[STIX]{x1D70C}}K\mathop{\sum }_{n=1}^{\infty }\frac{\unicode[STIX]{x1D6FC}_{n}M_{0}(\unicode[STIX]{x1D706}_{n}r)}{\unicode[STIX]{x1D706}_{n}}\nonumber\\ \displaystyle & & \displaystyle \times \left[2-\text{H}\left(z-\frac{2Kct}{\sqrt{c^{2}\unicode[STIX]{x1D706}_{n}^{2}+4K^{2}}}\right)-\text{H}\left(z+\frac{2Kct}{\sqrt{c^{2}\unicode[STIX]{x1D706}_{n}^{2}+4K^{2}}}\right)\right].\qquad\end{eqnarray}$$

(3.18) $$\begin{eqnarray}\displaystyle \lim _{k\rightarrow 0}\unicode[STIX]{x1D714}_{n,i}=k\left(\bar{u}_{z}\pm \frac{2cK}{\sqrt{c^{2}\unicode[STIX]{x1D706}_{n}^{2}+4K^{2}}}\right). & & \displaystyle\end{eqnarray}$$

(This relation was also introduced as the low wavenumber limit of the group speed, see (2.29) in § 2.1.)

In most combustion applications, low Mach number flow is considered. At the same time, for thermo-acoustic applications, acoustic waves are also relevant. Therefore, an accurate and reasonable approximation is to keep the acoustic wave solution as it is and to neglect compressibility effects on inertial waves. This assumption further simplifies the solution to

(3.19a ) $$\begin{eqnarray}\displaystyle u_{r}^{\prime }(t,z,r) & {\approx} & \displaystyle -0.5\mathop{\sum }_{n=1}^{\infty }\frac{\unicode[STIX]{x1D6FC}_{n}M_{1}(\unicode[STIX]{x1D706}_{n}r)}{\unicode[STIX]{x1D706}_{n}}\nonumber\\ \displaystyle & & \displaystyle \times \left[\unicode[STIX]{x1D6FF}\left(z-\bar{u}_{z}t-\frac{2Kt}{\unicode[STIX]{x1D706}_{n}}\right)-\unicode[STIX]{x1D6FF}\left(z-\bar{u}_{z}t+\frac{2Kt}{\unicode[STIX]{x1D706}_{n}}\right)\right],\end{eqnarray}$$

(3.19b ) $$\begin{eqnarray}\displaystyle u_{\unicode[STIX]{x1D703}}^{\prime }(t,z,r) & {\approx} & \displaystyle 0.5\mathop{\sum }_{n=1}^{\infty }\unicode[STIX]{x1D6FC}_{n}M_{1}(\unicode[STIX]{x1D706}_{n}r)\nonumber\\ \displaystyle & & \displaystyle \times \left[2-\text{H}\left(z-\bar{u}_{z}t-\frac{2Kt}{\unicode[STIX]{x1D706}_{n}}\right)-\text{H}\left(z-\bar{u}_{z}t+\frac{2Kt}{\unicode[STIX]{x1D706}_{n}}\right)\right],\end{eqnarray}$$

(3.19c ) $$\begin{eqnarray}\displaystyle u_{z}^{\prime }(t,z,r) & {\approx} & \displaystyle \unicode[STIX]{x1D700}_{a}[1-\text{H}(z-\bar{u}_{z}t-ct)]+0.5\mathop{\sum }_{n=1}^{\infty }\unicode[STIX]{x1D6FC}_{n}M_{0}(\unicode[STIX]{x1D706}_{n}r)\nonumber\\ \displaystyle & & \displaystyle \times \left[\text{H}\left(z-\bar{u}_{z}t-\frac{2Kt}{\unicode[STIX]{x1D706}_{n}}\right)-\text{H}\left(z-\bar{u}_{z}t+\frac{2Kt}{\unicode[STIX]{x1D706}_{n}}\right)\right],\end{eqnarray}$$

(3.19d ) $$\begin{eqnarray}\displaystyle p^{\prime }(t,z,r) & {\approx} & \displaystyle \unicode[STIX]{x1D700}_{a}\bar{\unicode[STIX]{x1D70C}}c[1-\text{H}(z-\bar{u}_{z}t-ct)]-K\bar{\unicode[STIX]{x1D70C}}\mathop{\sum }_{n=1}^{\infty }\frac{\unicode[STIX]{x1D6FC}_{n}M_{0}(\unicode[STIX]{x1D706}_{n}r)}{\unicode[STIX]{x1D706}_{n}}\nonumber\\ \displaystyle & & \displaystyle \times \left[2-\text{H}\left(z-\bar{u}_{z}t-\frac{2Kt}{\unicode[STIX]{x1D706}_{n}}\right)-\text{H}\left(z-\bar{u}_{z}t+\frac{2Kt}{\unicode[STIX]{x1D706}_{n}}\right)\right].\end{eqnarray}$$

$c\rightarrow \infty$

(3.20) $$\begin{eqnarray}\displaystyle \lim _{c\rightarrow \infty }\frac{2Kc}{\sqrt{c^{2}\unicode[STIX]{x1D706}_{n}^{2}+4K^{2}}}=\frac{2K}{\unicode[STIX]{x1D706}_{n}}. & & \displaystyle\end{eqnarray}$$

The other difference is that the inertial wave amplitude for axial velocity component, $u_{z}^{\prime }$ , decreases in the incompressible limit. Similarly, this can be shown by taking the limit for the amplitude as

(3.21) $$\begin{eqnarray}\displaystyle \lim _{c\rightarrow \infty }\frac{\sqrt{c^{2}\unicode[STIX]{x1D706}_{n}^{2}+4K^{2}}}{c\unicode[STIX]{x1D706}_{n}}=1. & & \displaystyle\end{eqnarray}$$

Note that these differences depend on Mach number and are therefore negligible for subsonic flows.

In terms of flame dynamics, the solution in (3.19) reveals the importance of inertial waves for flame front kinematics. Fleifil et al. (Reference Fleifil, Annaswamy, Ghoneim and Ghoniem1996) demonstrated that any flow perturbation normal to the flame surface disturbs the kinematic balance between flow and flame speeds and hence causes unsteady heat release rate. The flow perturbation due to plane acoustic waves at the flame front is one of the major contributions to thermo-acoustic instabilities and is investigated extensively in the literature (see Schuller, Durox & Candel (Reference Schuller, Durox and Candel2003), Blumenthal et al. (Reference Blumenthal, Subramanian, Sujith and Polifke2013) and Steinbacher et al. (Reference Steinbacher, Albayrak, Ghani and Polifke2019)). Similarly, axial and radial velocity perturbations resulting from inertial waves, respectively $u_{z}^{\prime }$ and $u_{r}^{\prime }$ in (3.19), play an important role in flame dynamics by modulating the flame surface. This is further discussed in the authors’ recent publication (Albayrak, Bezgin & Polifke Reference Albayrak, Bezgin and Polifke2018a ).

As discussed above, the time–space domain solution results from a simplified approach. The inertial wave dispersion effects are not present, but important properties such as propagation speeds and eigenmode structures are revealed. For comparison purposes, we also repeat the step response solution that results from the convective assumption for azimuthal velocity perturbations. This solution is equivalent to the solution proposed by Palies et al. (Reference Palies, Durox, Schuller and Candel2011a ) and is written as

(3.22a ) $$\begin{eqnarray}\displaystyle u_{r}^{\prime }(t,z,r) & = & \displaystyle 0,\end{eqnarray}$$

(3.22b ) $$\begin{eqnarray}\displaystyle u_{\unicode[STIX]{x1D703}}^{\prime }(t,z,r) & = & \displaystyle \unicode[STIX]{x1D700}_{a}\tan (\unicode[STIX]{x1D719})[1-\text{H}(z-\bar{u}_{z}t)],\end{eqnarray}$$

(3.22c ) $$\begin{eqnarray}\displaystyle u_{z}^{\prime }(t,z,r) & = & \displaystyle \unicode[STIX]{x1D700}_{a}[1-\text{H}(z-\bar{u}_{z}t)],\end{eqnarray}$$

(3.22d ) $$\begin{eqnarray}\displaystyle p^{\prime }(t,z,r) & = & \displaystyle \unicode[STIX]{x1D700}_{a}\bar{\unicode[STIX]{x1D70C}}c^{2}[1-\text{H}(z-\bar{u}_{z}t)].\end{eqnarray}$$

3.3.1 Asymptotic solution: method of steepest descent

In this section, we derive the asymptotic time–space domain solution by the method of steepest descent. For details of this method, the reader may consult Bender & Orszag (Reference Bender and Orszag1999). The asymptotic analysis here merely focuses on the incompressible inertial wave solution. Note that the same analysis can also be performed for the compressible case. However, as discussed in the previous section, plane acoustic waves are decoupled from the inertial wave solution at low Mach numbers. Moreover, the impact of compressibility on the inertial wave solution is negligible. This means that the analysis with compressible equations does not introduce additional aspects to the solution. In the incompressible limit, equation (3.16) reduces to

(3.23a ) $$\begin{eqnarray}\displaystyle \hat{u} _{r}(k,r,s) & = & \displaystyle -2Kk^{2}\left(\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}(k)+\frac{\text{i}}{k}\right)\mathop{\sum }_{n=1}^{\infty }\frac{\unicode[STIX]{x1D6FC}_{n}M_{1}(\unicode[STIX]{x1D706}_{n}r)}{C^{2}(\unicode[STIX]{x1D706}_{n}^{2}+k^{2})-4K^{2}k^{2}},\end{eqnarray}$$

(3.23b ) $$\begin{eqnarray}\displaystyle \hat{u} _{\unicode[STIX]{x1D703}}(k,r,s) & = & \displaystyle \text{i}C\left(\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}(k)+\frac{\text{i}}{k}\right)\mathop{\sum }_{n=1}^{\infty }\frac{(\unicode[STIX]{x1D706}_{n}^{2}+k^{2})\unicode[STIX]{x1D6FC}_{n}M_{1}(\unicode[STIX]{x1D706}_{n}r)}{C^{2}(\unicode[STIX]{x1D706}_{n}^{2}+k^{2})-4K^{2}k^{2}},\end{eqnarray}$$

(3.23c ) $$\begin{eqnarray}\displaystyle \hat{u} _{z}(k,r,s) & = & \displaystyle -2K\text{i}k\left(\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}(k)+\frac{\text{i}}{k}\right)\mathop{\sum }_{n=1}^{\infty }\frac{\unicode[STIX]{x1D706}_{n}\unicode[STIX]{x1D6FC}_{n}M_{0}(\unicode[STIX]{x1D706}_{n}r)}{C^{2}(\unicode[STIX]{x1D706}_{n}^{2}+k^{2})-4K^{2}k^{2}},\end{eqnarray}$$

(3.23d ) $$\begin{eqnarray}\displaystyle \hat{p}(k,r,s) & = & \displaystyle -2K\text{i}C\bar{\unicode[STIX]{x1D70C}}\left(\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}(k)+\frac{\text{i}}{k}\right)\mathop{\sum }_{n=1}^{\infty }\frac{\unicode[STIX]{x1D706}_{n}\unicode[STIX]{x1D6FC}_{n}M_{0}(\unicode[STIX]{x1D706}_{n}r)}{C^{2}(\unicode[STIX]{x1D706}_{n}^{2}+k^{2})-4K^{2}k^{2}}.\end{eqnarray}$$

Note that the plane acoustic wave solution can be superposed into the above equation as they are decoupled. However, it is neglected here for ease of presentation. By performing the inverse Laplace transformation, the time domain solution is recovered as

(3.24a ) $$\begin{eqnarray}\displaystyle \hat{u} _{r}(k,r,t) & = & \displaystyle \frac{k}{2}\text{e}^{-\text{i}kt\bar{u}_{z}}\left(\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}(k)+\frac{\text{i}}{k}\right)\mathop{\sum }_{n=1}^{\infty }\unicode[STIX]{x1D6FC}_{n}M_{1}(\unicode[STIX]{x1D706}_{n}r)\nonumber\\ \displaystyle & & \displaystyle \times \,\frac{\exp \left(-{\displaystyle \frac{2K\text{i}kt}{\sqrt{\unicode[STIX]{x1D706}_{n}^{2}+k^{2}}}}\right)-\exp \left({\displaystyle \frac{2K\text{i}kt}{\sqrt{\unicode[STIX]{x1D706}_{n}^{2}+k^{2}}}}\right)}{\sqrt{\unicode[STIX]{x1D706}_{n}^{2}+k^{2}}},\end{eqnarray}$$

(3.24b ) $$\begin{eqnarray}\displaystyle \hat{u} _{\unicode[STIX]{x1D703}}(k,r,t) & = & \displaystyle \frac{1}{2}\text{e}^{-\text{i}kt\bar{u}_{z}}\left(\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}(k)+\frac{\text{i}}{k}\right)\mathop{\sum }_{n=1}^{\infty }\unicode[STIX]{x1D6FC}_{n}M_{1}(\unicode[STIX]{x1D706}_{n}r)\nonumber\\ \displaystyle & & \displaystyle \times \,\left[\exp \left(-{\displaystyle \frac{2K\text{i}kt}{\sqrt{\unicode[STIX]{x1D706}_{n}^{2}+k^{2}}}}\right)+\exp \left({\displaystyle \frac{2K\text{i}kt}{\sqrt{\unicode[STIX]{x1D706}_{n}^{2}+k^{2}}}}\right)\right],\end{eqnarray}$$

(3.24c ) $$\begin{eqnarray}\displaystyle \hat{u} _{z}(k,r,t) & = & \displaystyle \frac{\text{i}}{2}\text{e}^{-\text{i}kt\bar{u}_{z}}\left(\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}(k)+\frac{\text{i}}{k}\right)\mathop{\sum }_{n=1}^{\infty }\unicode[STIX]{x1D706}_{n}\unicode[STIX]{x1D6FC}_{n}M_{0}(\unicode[STIX]{x1D706}_{n}r)\nonumber\\ \displaystyle & & \displaystyle \times \,\frac{\exp \left(-{\displaystyle \frac{2K\text{i}kt}{\sqrt{\unicode[STIX]{x1D706}_{n}^{2}+k^{2}}}}\right)-\exp \left({\displaystyle \frac{2K\text{i}kt}{\sqrt{\unicode[STIX]{x1D706}_{n}^{2}+k^{2}}}}\right)}{\sqrt{\unicode[STIX]{x1D706}_{n}^{2}+k^{2}}},\end{eqnarray}$$

(3.24d ) $$\begin{eqnarray}\displaystyle \hat{p}(k,r,t) & = & \displaystyle -K\bar{\unicode[STIX]{x1D70C}}\text{e}^{-\text{i}kt\bar{u}_{z}}\left(\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}(k)+\frac{\text{i}}{k}\right)\mathop{\sum }_{n=1}^{\infty }\unicode[STIX]{x1D706}_{n}\unicode[STIX]{x1D6FC}_{n}M_{0}(\unicode[STIX]{x1D706}_{n}r)\nonumber\\ \displaystyle & & \displaystyle \times \,\frac{\exp \left(-{\displaystyle \frac{2K\text{i}kt}{\sqrt{\unicode[STIX]{x1D706}_{n}^{2}+k^{2}}}}\right)+\exp \left({\displaystyle \frac{2K\text{i}kt}{\sqrt{\unicode[STIX]{x1D706}_{n}^{2}+k^{2}}}}\right)}{\unicode[STIX]{x1D706}_{n}^{2}+k^{2}}.\end{eqnarray}$$

The pure convection process acts as a Doppler shift with the exponential term $\text{e}^{-\text{i}kt\bar{u}_{z}}$ . The inertial wave propagation is governed by exponential functions inside the summation, where fast and slow propagating waves are indicated by the negative and positive sign in the exponential function, respectively. The denominator of the exponential function, $\sqrt{\unicode[STIX]{x1D706}_{n}^{2}+k^{2}}$ , is the reason for the dispersive propagation. At this point, no closed form solution for the inverse Fourier transform exists. Similar to the previous section, one can seek the solution at low wavenumbers, i.e. $k\rightarrow 0$ . This leads exactly to the inertial wave solution as described in (3.19). Instead, we continue with the method of steepest descent. Here, for simplicity only the fast propagating solution of the azimuthal velocity $\hat{u} _{\unicode[STIX]{x1D703}}$ is presented. Other components may be derived in a similar manner. Moreover, only the $n$ th mode is considered, so that the summation is not required. The convection process is neglected and terms that do not depend on $k$ are not included. However, in the final expression, equation (3.28), the most general form of the solution is presented. Under these conditions, the integral for the inverse Fourier transformation is written

(3.25) $$\begin{eqnarray}\displaystyle I(z)=\int _{-\infty }^{\infty }f(k)\text{e}^{zg(k)}=\int _{-\infty }^{\infty }\left(\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}(k)+\frac{\text{i}}{k}\right)\exp \left(z\left[\text{i}k-{\displaystyle \frac{2\text{i}Kk}{\unicode[STIX]{x1D702}\sqrt{\unicode[STIX]{x1D706}_{n}^{2}+k^{2}}}}\right]\right)\,\text{d}k, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D702}=z/t$ . Real and imaginary parts of the exponent function, $g(k)$ , are shown in figure 6 as contour plots. Columns correspond to the real and imaginary parts, respectively. Darker colours indicate lower values. The singularities are marked with red circles at $k=\pm \unicode[STIX]{x1D706}_{n}$ . Corresponding branch cuts are on the imaginary axis, i.e. $k>\text{i}\unicode[STIX]{x1D706}_{n}$ and $k<-\text{i}\unicode[STIX]{x1D706}_{n}$ . Dashed blue contours indicate constant levels of imaginary part, $\text{Im}(g(k))$ .

Figure 6.

Contour plots of real (a) and imaginary (b) parts of $g(k)$ . From (a–f), three branches are plotted, $\unicode[STIX]{x1D702}>2K/\unicode[STIX]{x1D706}_{n}$ , $\unicode[STIX]{x1D702}=2K/\unicode[STIX]{x1D706}_{n}$ and $\unicode[STIX]{x1D702}<2K/\unicode[STIX]{x1D706}_{n}$ . Circles and crosses indicate poles of $g(k)$ and zeros of $g^{\prime }(k)$ , respectively. Dashed blue and solid green (with arrows) lines indicate constant levels of contours and closed integral paths for the steepest descent path, respectively.

Saddle points, $\unicode[STIX]{x2202}g/\unicode[STIX]{x2202}k|_{k=k_{0}}=0$ , are determined as

(3.26) $$\begin{eqnarray}\displaystyle k_{0}=\pm \unicode[STIX]{x1D706}_{n}\sqrt{\left(\frac{2K}{\unicode[STIX]{x1D702}\unicode[STIX]{x1D706}_{n}}\right)^{2/3}-1}. & & \displaystyle\end{eqnarray}$$

Observing the form of saddle points, three different branches are recognized as $\unicode[STIX]{x1D702}>2K/\unicode[STIX]{x1D706}_{n}$ , $\unicode[STIX]{x1D702}=2K/\unicode[STIX]{x1D706}_{n}$ and $\unicode[STIX]{x1D702}<2K/\unicode[STIX]{x1D706}_{n}$ . The first branch is shown in figure 6(a,b). Two non-degenerate saddle points marked with red crosses are located on the imaginary axis. Similarly, there exists two saddle points for the branch, $\unicode[STIX]{x1D702}<2K/\unicode[STIX]{x1D706}_{n}$ , which is shown in the last row. The saddle points lie on the real axis. For $\unicode[STIX]{x1D702}=2K/\unicode[STIX]{x1D706}_{n}$ , there exist a single degenerate saddle point at the origin.

For each branch, the initial integral path, $k=(-\infty ,\infty )$ is constructed by additional paths from the steepest descent method. Cauchy’s integral theorem is employed, which states that the integral along a closed path with no singularities is zero. The closed integral paths are indicated with the green colour (solid lines with arrows). As an example, only the first branch is discussed in detail. The closed integral path is written

(3.27) $$\begin{eqnarray}\displaystyle \underbrace{\int _{\unicode[STIX]{x1D6E4}_{1}}+\int _{\unicode[STIX]{x1D6E4}_{2}}}_{I(z)}+\underbrace{\int _{\unicode[STIX]{x1D6E4}_{3}}+\int _{\unicode[STIX]{x1D6E4}_{6}}}_{C(z)}+\underbrace{\int _{\unicode[STIX]{x1D6E4}_{4}}+\int _{\unicode[STIX]{x1D6E4}_{5}}}_{S(z)}=0, & & \displaystyle\end{eqnarray}$$

where $S(z)$ is the contribution by steepest descent path, which follows the constant level of $\text{Im}(g(k))$ . Note that the saddle point in the negative imaginary axis is a local minimum and does not define a steepest descent path. Thus, only the saddle point in the positive imaginary axis is employed. The circular paths, $C(z)$ , enclose the contour so that Cauchy’s integral theorem can be employed. As the radius of the circular path grows to infinity, its contribution vanishes, $C(z)=0$ . Hence, the integral, $I(z)$ , can be related to the steepest descent integral, $I(z)=-S(z)$ . Similar arguments apply for the other two branches of $\unicode[STIX]{x1D702}$ . The leading term for steepest descent integral results in the general expression for azimuthal velocity perturbations as

(3.28) $$\begin{eqnarray}\displaystyle & & \displaystyle \hspace{-35.00005pt}u_{\unicode[STIX]{x1D703}}^{\prime }(t,z,r)\approx \mathop{\sum }_{n=1}^{\infty }\unicode[STIX]{x1D6FC}_{n}M_{r,n}\nonumber\\ \displaystyle & & \displaystyle \hspace{-35.00005pt}\quad \times \,\left\{\begin{array}{@{}ll@{}}\displaystyle 1-\left(\frac{2Kt}{\unicode[STIX]{x1D706}_{n}(\bar{u}_{z}t-z)}\right)^{1/3}\frac{\exp (-\unicode[STIX]{x1D706}_{n}(-c_{m})^{3/2})}{2\sqrt{6\unicode[STIX]{x03C0}\unicode[STIX]{x1D706}_{n}}(-c_{m})^{3/4}}\quad & \text{if }\displaystyle \unicode[STIX]{x1D702}<\bar{u}_{z}-\frac{2K}{\unicode[STIX]{x1D706}_{n}},\\ \displaystyle \frac{5}{6}\quad & \text{if }\displaystyle \unicode[STIX]{x1D702}=\bar{u}_{z}-\frac{2K}{\unicode[STIX]{x1D706}_{n}},\\ \displaystyle \frac{1}{2}+\left(\frac{2Kt}{\unicode[STIX]{x1D706}_{n}(\bar{u}_{z}t-z)}\right)^{1/3}\frac{\cos (\unicode[STIX]{x1D706}_{n}c_{m}^{3/2})-\sin (\unicode[STIX]{x1D706}_{n}c_{m}^{3/2})}{2\sqrt{3\unicode[STIX]{x03C0}\unicode[STIX]{x1D706}_{n}}c_{m}^{3/4}}\quad & \text{if }\displaystyle \bar{u}_{z}-\frac{2K}{\unicode[STIX]{x1D706}_{n}}<\unicode[STIX]{x1D702}<\bar{u}_{z},\\ \displaystyle \frac{1}{2}-\left(\frac{2Kt}{\unicode[STIX]{x1D706}_{n}(z-\bar{u}_{z}t)}\right)^{1/3}\frac{\cos (\unicode[STIX]{x1D706}_{n}c_{p}^{3/2})-\sin (\unicode[STIX]{x1D706}_{n}c_{p}^{3/2})}{2\sqrt{3\unicode[STIX]{x03C0}\unicode[STIX]{x1D706}_{n}}c_{p}^{3/4}}\quad & \text{if }\displaystyle \bar{u}_{z}<\unicode[STIX]{x1D702}<\bar{u}_{z}+\frac{2K}{\unicode[STIX]{x1D706}_{n}},\\ \displaystyle \frac{1}{6}\quad & \text{if }\displaystyle \unicode[STIX]{x1D702}=\bar{u}_{z}+\frac{2K}{\unicode[STIX]{x1D706}_{n}},\\ \displaystyle \left(\frac{2Kt}{\unicode[STIX]{x1D706}_{n}(z-\bar{u}_{z}t)}\right)^{1/3}\frac{\exp (-\unicode[STIX]{x1D706}_{n}(-c_{m})^{3/2})}{2\sqrt{6\unicode[STIX]{x03C0}\unicode[STIX]{x1D706}_{n}}(-c_{m})^{3/4}}\quad & \text{if }\displaystyle \unicode[STIX]{x1D702}>\bar{u}_{z}+\frac{2K}{\unicode[STIX]{x1D706}_{n}},\end{array}\right.\end{eqnarray}$$

where

(3.29) $$\begin{eqnarray}\displaystyle c_{p} & = & \displaystyle \left(\frac{2Kt}{\unicode[STIX]{x1D706}_{n}}\right)^{2/3}-(z-\bar{u}_{z}t)^{2/3},\end{eqnarray}$$

(3.30) $$\begin{eqnarray}\displaystyle c_{m} & = & \displaystyle \left(\frac{2Kt}{\unicode[STIX]{x1D706}_{n}}\right)^{2/3}-(\bar{u}_{z}t-z)^{2/3}.\end{eqnarray}$$

In figure 7, three snapshots of the spatio-temporal development of azimuthal velocity fluctuations after a step perturbation are presented. The colour raster plots show velocity values ranging from 0 (black) to 1 (white) in the $z$ – $r$ plane, i.e. the annulus between inner radius $r_{i}$ and outer radius $r_{o}$ . For simplicity, only the first mode $n=1$ with a radial profile corresponding to the first eigenmode of azimuthal velocity $M_{1}(\unicode[STIX]{x1D706}_{1}r)$ is presented here. Corresponding results for other modes or multiple modes can be generated without difficulty. Below each surface plot, axial profiles of perturbation amplitude are shown. Solid black lines indicate the numerical inverse fast Fourier transformation of (3.24). Dotted blue lines indicate the simplified non-dispersive solution (see (3.19)) developed in the previous section. Dashed red lines indicate the asymptotic solution in (3.28). By inspecting this equation, the solution can be divided into regions: The first region, $\unicode[STIX]{x1D702}<\bar{u}_{z}-2K/\unicode[STIX]{x1D706}_{n}$ , converges exponentially to 1. For the non-dispersive solution, this region is estimated as 1 without an exponential term. Similarly, the last region, $\unicode[STIX]{x1D702}>\bar{u}_{z}-2K/\unicode[STIX]{x1D706}_{n}$ , decays exponentially to 0. Again, the non-dispersive solution estimates only a constant value of 0. The second region and third regions, $\bar{u}_{z}-2K/\unicode[STIX]{x1D706}_{n}>\unicode[STIX]{x1D702}>\bar{u}_{z}+2K/\unicode[STIX]{x1D706}_{n}$ , correspond to the dispersive region of inertial propagation, where oscillations around $0.5$ are observed. The asymptotic solution captures the oscillations accurately via the combination of trigonometric functions. Like a chirp function, oscillations become squeezed as they approach the convective line $\unicode[STIX]{x1D702}=\bar{u}_{z}$ . In other works, near the convective region, high axial wavenumbers are observed. This is also confirmed in figure 3, where the group speed is shown as a function of the axial wavenumber. As the axial wavenumber increases, the propagation of inertial waves converges to the convection speed. The non-dispersive solution, as its name suggests, cannot capture these oscillations and estimates a constant value of 0.5. Note that the asymptotic solution diverges close to the boundaries $\unicode[STIX]{x1D702}=\bar{u}_{z}-2K/\unicode[STIX]{x1D706}_{n}$ and $\unicode[STIX]{x1D702}=\bar{u}_{z}+2K/\unicode[STIX]{x1D706}_{n}$ , since saddle points move in the complex plane abruptly (see regions in figure 6). Exactly at these boundaries, the asymptotic solution yields $5/6$ and $1/6$ , respectively. These points are indicated in figure 7 by circles. The non-dispersive solution estimates step changes at these locations.

Figure 7.

Three snapshots of the azimuthal velocity fluctuation field developing after a step perturbation by a downstream propagating acoustic wave. Only the first eigenmode is considered, thus the radial profile $u_{\unicode[STIX]{x1D703}}^{\prime }(r)\propto M_{1}(\unicode[STIX]{x1D706}_{n}r)$ . Colour raster plots were generated via the inverse fast Fourier transformation of (3.24). Axial profile plots compare that solution (solid black line) with the non-dispersive solution (3.19) (dotted blue line) and the asymptotic solution (3.28) obtained with the steepest descent method (dashed red line).

Figure 8.

Three snapshots of the axial velocity fluctuation field developing after a step perturbation by a downstream propagating acoustic wave. Only the first eigenmode is considered, thus the radial profile $u_{z}^{\prime }(r)\propto M_{0}(\unicode[STIX]{x1D706}_{n}r)$ . Colour raster plots were generated via the inverse fast Fourier transformation. Axial profile plots compare that solution (solid black line) with the non-dispersive solution (3.19) (dotted blue line).

Figure 9.

Three snapshots of the radial velocity fluctuation field developing after a step perturbation by a downstream propagating acoustic wave. Only the first eigenmode with radial profile $u_{r}^{\prime }(r)\propto M_{1}(\unicode[STIX]{x1D706}_{n}r)$ is considered. Colour raster plots were generated via the inverse fast Fourier transformation. Axial profile plots compare that solution (black line) with the non-dispersive solution (3.19) (blue line). Note that the Dirac delta function is represented by the circles at $\pm 0.5$ .

Figure 10.

Three snapshots of the pressure fluctuation field developing after a step perturbation by a downstream propagating acoustic wave. Only the first eigenmode radial profile $p^{\prime }(r)\propto M_{0}(\unicode[STIX]{x1D706}_{n}r)$ is considered. Colour raster plots were generated via the inverse fast Fourier transformation. Axial profile plots compare that solution (solid black line) with the non-dispersive solution (3.19) (dotted blue line).

The other components of the fluctuation field are shown in subsequent figures. Figure 8 presents axial velocity. Perturbations decay to zero for both the first and the last regions. This means that the axial velocity perturbations are constrained to the second and third region, where dispersive oscillations around the value of $-0.5$ are present. The non-dispersive solution estimates a constant value of $-0.5$ for the oscillatory region. The mode shape corresponds to $M_{0}(\unicode[STIX]{x1D706}_{1}r)$ .

In figure 9 the radial velocity fluctuation is shown. No step changes are observed, and the solution oscillates around zero for the second and third region. Note that the non-dispersive solution estimates only Dirac delta functions for at $\unicode[STIX]{x1D702}=\bar{u}_{z}-2K/\unicode[STIX]{x1D706}_{n}$ and $\unicode[STIX]{x1D702}=\bar{u}_{z}+2K/\unicode[STIX]{x1D706}_{n}$ , indicated at $\pm 0.5$ by blue circles.

Figure 10 presents pressure fluctuations, normalized by $2\bar{\unicode[STIX]{x1D70C}}K$ . An analogous argument to that for the azimuthal velocity applies but the amplitude is negative, i.e. the first region converges to $-1$ . As discussed previously, the initial perturbation in pressure does not show up as an initial condition for the incompressible equations due to the elliptic continuity equation. However, the initial azimuthal velocity perturbation induces also pressure fluctuations as seen in the solution due to the Coriolis force in the radial momentum balance. Although it does not play a role in the incompressible equations, it should be included for the compressible equations because they show up in the equations as an initial condition. Therefore, this term is introduced in the actuator disk theory (see (3.11)) for the consistency of inertial waves.