The small disturbance of flow parameters is assumed as a wave travelling along axial
x and azimuthal
\(\varphi\) directions with the amplitude varying along the radial direction
r in the following form:
$$\begin{aligned} \left( u',v',w',p'\right) =\left[ {\hat{u}} \left( r \right) ,{\hat{v}} \left( r \right) ,{\hat{w}}\left( r \right) ,{\hat{p}} \left( r \right) \right] \exp {\left[ i \left( \alpha x +m\varphi -\omega t \right) \right] } \end{aligned}$$
(15)
Here
\(u',v',w'\) and
\(p'\) are axial, radial, azimuthal velocity and pressure disturbances, respectively,
\(\alpha\) is the complex wave number and
\(\omega\) the complex frequency,
m is the azimuthal real wave number. Such disturbances are introduced into the continuity equation and Euler equations in cylindrical coordinates and linearised around the given base flow. As shown by Michalke [
19], after eliminating the
\({\hat{u}}\) and
\({\hat{w}}\) velocities, the stability equations can be written as
$$\begin{aligned} r\sigma \frac{d{\hat{p}}}{dr}&= {} i \left[ 2WZ -r \sigma ^2\right] {\hat{v}}-\frac{2mW}{r}{\hat{p}} \end{aligned}$$
(16)
$$\begin{aligned} \frac{\sigma }{r}\frac{d \left( r{\hat{v}} \right) }{dr}&= {} i \left[ \alpha ^2 +\frac{m^2}{r^2} \right] {\hat{p}} + \left[ \alpha \frac{dU}{dr} +m \frac{Z}{r}\right] {\hat{v}} \end{aligned}$$
(17)
where
$$\begin{aligned} \sigma \left( r \right)&= {} \alpha U - \omega + \frac{W}{r}m \end{aligned}$$
(18)
$$\begin{aligned} Z \left( r\right)&= {} \frac{d W}{dr}+ \frac{W}{r} \end{aligned}$$
(19)
To solve the system of the stability equations (
16) and (
17) the boundary conditions must be formulated on the jet axis
\(r=0\) and for
\(r \rightarrow \infty\) using features of the axial and azimuthal velocity profiles of the base flow. The boundary conditions require that
\({\hat{v}} \left( r \right)\) and
\({\hat{p}} \left( r \right)\) are bounded on the jet axis and both quantities vanish at the jet periphery. The asymptotic behavior for
\(r \rightarrow \infty\) can be obtained if one takes into account that
$$\begin{aligned}&\lim _{r \rightarrow \infty } U, \frac{dU}{dr}, Z =0 \\&\lim _{r \rightarrow \infty } \frac{W}{r}=\frac{A_{\infty }}{r^{N_W}} \;\;\;\; \text {where} \;\;\;\; A_{\infty }=\frac{Ar_{W_{max}}^{N_W-1}}{1-\exp \left( -br_{W_{max}}^{N_W-1} \right) } \end{aligned}$$
(20)
Introducing the notation:
$$\begin{aligned} \sigma _{\infty }=\lim _{r \rightarrow \infty } \sigma \left( r \right) =m \frac{A_{\infty }}{r^{N_W}}-\omega \end{aligned}$$
(21)
the stability equations (
16) and (
17) for
\(r \rightarrow \infty\) take the following forms:
$$\begin{aligned} r \sigma _{\infty }\frac{d {\hat{p}}}{dr}&=-i r\sigma ^2_{\infty } {\hat{v}}-2m \frac{A_{\infty }}{r^{N_W}} {\hat{p}} \end{aligned}$$
(22)
$$\begin{aligned} \frac{\sigma _{\infty }}{r} \frac{d\left( r {\hat{v}}\right) }{dr}&=i \left[ \alpha ^2+ \left( \frac{m}{r} \right) ^2 \right] {\hat{p}} \end{aligned}$$
(23)
Introducing the notation
\(\Phi =r {\hat{v}}\) Eq. (
23) reads as
$$\begin{aligned} \frac{d}{dr} \left[ \frac{r}{m^2+ \left( \alpha r \right) ^2} \frac{d\Phi }{dr} \right] - \frac{\Phi }{r}=0 \end{aligned}$$
(24)
with the decaying solution
$$\begin{aligned} \Phi =rK_m' \left( \alpha r \right) \end{aligned}$$
(25)
where
\(K_m\) is the modified Bessel function of the second kind and order
m. Hence,
\({\hat{v}}\) velocity at the limit
\(r \rightarrow \infty\) is
$$\begin{aligned} {\hat{v}} \left( r \right) =K'_m \left( \alpha r \right) \end{aligned}$$
(26)
The asymptotic limit for a pressure perturbation at the jet periphery, with Eq. (
22), is
$$\begin{aligned} {\hat{p}} \left( r\right) = - \frac{i}{\alpha } \sigma _\infty K_m \left( \alpha r \right) = - \frac{i}{\alpha } \left( m\frac{A_\infty }{r^{N_W}} -\omega \right) K_m \left( \alpha r \right) \end{aligned}$$
(27)
The boundary conditions for the jet axis, despite a different base flow used in the current analysis, are exactly the same as shown by Michalke [
19]. Pressure perturbation at the jet axis is expressed as
$$\begin{aligned} {\hat{p}} \left( r \right) = I_m \left( \beta r \right) \end{aligned}$$
(28)
where
\(I_m\) is the modified Bessel function of the first kind and order
m, and the velocity
$$\begin{aligned} {\hat{v}} \left( r \right) = \frac{i \alpha }{\sigma _0 \sqrt{1-\mu ^2}} \left[ I_m' \left( \beta r \right) + \frac{ m \mu }{\beta r}I_m \left( \beta r \right) \right] \end{aligned}$$
(29)
The eigenvalue problem now is solved numerically by integrating stability equations (
16) and (
17) by means of the Runge–Kutta–Fehlberg procedure [
29] of
\(4{\mathrm{th}}\) order starting with
\(r=10^{-6}\) and boundary conditions (
28) and (
29) up till
\(r=1\) yielding
\({\hat{p}}_L\left( 1 \right)\) and
\({\hat{v}}_L \left( 1 \right)\), and from
\(r=\infty\), where the asymptotic solutions (
26) and (
27) are applied, back to
\(r=1\) yielding
\({\hat{p}}_R \left( 1\right)\) and
\({\hat{v}}_R \left( 1 \right)\). The eigenvalue condition then follows from the matching of these solutions at
\(r=1\), requiring:
$$\begin{aligned} G \left( \omega , \alpha \right) = \frac{{\hat{p}}_L}{{\hat{v}}_L}-\frac{{\hat{p}}_R}{{\hat{v}}_R}=0 \end{aligned}$$
(30)
This condition leads, for a given
\(\omega\), to a relation
\(\alpha \left( \omega \right)\), or, for a given
\(\alpha\) to
\(\omega \left( \alpha \right)\). The eigenvalue problems can be solved by Newton’s method as
$$\begin{aligned} \omega ^{\left( n+1 \right) }=\omega ^{\left( n \right) } -\frac{G \left( \omega ^{\left( n \right) },\alpha \right) }{ \partial G/ \partial \omega \vert _{\omega ^{\left( n \right) }}} \end{aligned}$$
(31)
$$\begin{aligned} \alpha ^{\left( n+1 \right) }=\alpha ^{\left( n \right) } -\frac{G \left( \omega ,\alpha ^{\left( n \right) } \right) }{\partial G/ \partial \alpha \vert _{\alpha ^{\left( n \right) }}} \end{aligned}$$
(32)
where
$$\begin{aligned} \frac{\partial G}{ \partial \omega }=\frac{1}{{\hat{v}}_L} \left( \frac{\partial {\hat{p}}}{\partial \omega } \right) _L-\frac{1}{{\hat{v}}_R} \left( \frac{\partial {\hat{p}}}{\partial \omega }\right) _R -\frac{{\hat{p}}_L}{{\hat{v}}_L^2} \left( \frac{\partial {\hat{v}}}{\partial \omega } \right) _L+ \frac{{\hat{p}}_R}{{\hat{v}}_R^2}\left( \frac{\partial {\hat{v}}}{\partial \omega }\right) _R \end{aligned}$$
(33)
$$\begin{aligned} \frac{\partial G}{ \partial \alpha }=\frac{1}{{\hat{v}}_L} \left( \frac{\partial {\hat{p}}}{\partial \alpha } \right) _L-\frac{1}{{\hat{v}}_R} \left( \frac{\partial {\hat{p}}}{\partial \alpha }\right) _R -\frac{{\hat{p}}_L}{{\hat{v}}_L^2} \left( \frac{\partial {\hat{v}}}{\partial \alpha } \right) _L+ \frac{{\hat{p}}_R}{{\hat{v}}_R^2}\left( \frac{\partial {\hat{v}}}{\partial \alpha }\right) _R \end{aligned}$$
(34)
In order to use Newton’s method to solve eigenvalue problem one needs information on the derivatives of pressure and velocity perturbations with respect to the complex wave number
\(\alpha\) or the complex frequency
\(\omega\) respectively. This information can be obtained solving the differential equation obtained from the stability equations (
16), (
17) and the boundary conditions (
26), (
27), (
28) and (
29) differentiated with respect to wave number
\(\alpha\) or frequency
\(\omega\), respectively.