3.1 Classical Myers boundary condition

Assuming the continuity of the acoustic normal displacement through an infinitely thin boundary layer leads to the classical Ingard–Myers boundary condition (Ingard Reference Ingard1959; Myers Reference Myers1980). It reads with the present conventions:

(3.2a,b ) $$\begin{eqnarray}\displaystyle \hat{u} (h)={\displaystyle \frac{kU_{x}(h)+mU_{\unicode[STIX]{x1D703}}(h)/h-\unicode[STIX]{x1D714}}{\unicode[STIX]{x1D714}Z_{h}}}\hat{p}(h),\quad \text{and}\quad \hat{u} (1)=-{\displaystyle \frac{kU_{x}(1)+mU_{\unicode[STIX]{x1D703}}(1)-\unicode[STIX]{x1D714}}{\unicode[STIX]{x1D714}Z_{1}}}\hat{p}(1), & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

at the hub and at the tip respectively. In the following, it will be shown that (3.2) is not the correct limit when the boundary-layer thickness tends to zero in the presence of swirl.

3.2 Assumptions

The aim of this section is to substitute the effect of a flow including a boundary layer with the boundary condition in (3.1) by an equivalent boundary condition in a flow where the boundary layer is not taken into account. In order to address such a problem, Eversman & Beckemeyer (Reference Eversman and Beckemeyer1972) proposed an approach based on an asymptotic matching between an inner solution in the boundary layer, and an outer solution outside the boundary layer which would exist if there was no boundary layer. This method has been reused by several authors such as Myers & Chuang (Reference Myers and Chuang1984), Brambley (Reference Brambley2011) and Khamis & Brambley (Reference Khamis and Brambley2016) for uniform axial flow. It is extended here to the case of a swirling flow. Considering a thin boundary layer at the wall ( $r=1$ or $r=h$ ) of typical width (in dimensional terms, $\unicode[STIX]{x1D6FF}^{\ast }$ should be smaller than any other length scale, including not only the mid-span radius $r_{m}^{\ast }$ but also axial and radial wavelengths. This will be noted where pertinent)  $\unicode[STIX]{x1D6FF}$ , $\unicode[STIX]{x1D6FF}\ll 1$ , the distance to the wall $\unicode[STIX]{x1D6FF}y$ is introduced to develop the asymptotic expansion. As in Brambley (Reference Brambley2011), it is scaled by the boundary-layer thickness $\unicode[STIX]{x1D6FF}$ such that $r=1-\unicode[STIX]{x1D6FF}y$ at the outer wall and $r=h+\unicode[STIX]{x1D6FF}y$ at the inner wall. Let $U_{x}^{nbl}(r)$ and $U_{\unicode[STIX]{x1D703}}^{nbl}(r)$ be the outer mean flow velocity profiles without the boundary layer.

In the vicinity of walls, a Taylor expansion to first order applied on the outer mean flow velocity profiles reads:

(3.3a,b ) $$\begin{eqnarray}\displaystyle U_{x}^{nbl}(1-\unicode[STIX]{x1D6FF}y)=U_{x}^{nbl}(1)+O(\unicode[STIX]{x1D6FF}),\quad U_{\unicode[STIX]{x1D703}}^{nbl}(1-\unicode[STIX]{x1D6FF}y)=U_{\unicode[STIX]{x1D703}}^{nbl}(1)+O(\unicode[STIX]{x1D6FF}), & & \displaystyle\end{eqnarray}$$

(3.4a,b ) $$\begin{eqnarray}\displaystyle U_{x}^{nbl}(h+\unicode[STIX]{x1D6FF}y)=U_{x}^{nbl}(h)+O(\unicode[STIX]{x1D6FF}),\quad U_{\unicode[STIX]{x1D703}}^{nbl}(h+\unicode[STIX]{x1D6FF}y)=U_{\unicode[STIX]{x1D703}}^{nbl}(h)+O(\unicode[STIX]{x1D6FF}). & & \displaystyle\end{eqnarray}$$

Next, the outer mean flow is assumed constant up to leading order in the vicinity of the walls. For the sake of conciseness, the constants $M_{x,i}=U_{x}^{nbl}(i)$ and $M_{\unicode[STIX]{x1D703},i}=U_{\unicode[STIX]{x1D703}}^{nbl}(i)$ are introduced for $i=h,1$ . This assumption will be relaxed in § 3.5.

3.3 Derivation of the corrected boundary condition at the outer wall

The boundary condition is developed at the outer wall first.

3.3.1 Outer solution

The outer solution corresponds to the fluctuations propagating in the outer flow. The inner expansions of the outer solutions are obtained by writing a Taylor expansion of $\hat{u}$ and $\hat{p}$ in the vicinity of the outer wall. At leading order, it reads:

(3.5) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{u} _{o}(1-\unicode[STIX]{x1D6FF}y)=\hat{u} _{o}(1)+O\left(\unicode[STIX]{x1D6FF}\right), & \displaystyle\end{eqnarray}$$

(3.6) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{p}_{o}(1-\unicode[STIX]{x1D6FF}y)=\hat{p}_{o}(1)+O\left(\unicode[STIX]{x1D6FF}\right), & \displaystyle\end{eqnarray}$$

where the subscript $o$ stands for the outer solution. The eigenfunctions $\hat{p}_{o}$ and $\hat{u} _{o}$ represent the pressure and the radial velocity respectively if the boundary layer did not exist. Even if there is no analytic expression for them, they can be determined by a pseudo-spectral method for example, as done in § 4. They are assumed to be known, in particular $\hat{p}_{\infty }^{1}$ and $\hat{u} _{\infty }^{1}$ are introduced with

(3.7a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{p}_{o}(1)=\hat{p}_{\infty }^{1},\quad \hat{u} _{o}(1)=\hat{u} _{\infty }^{1}. & \displaystyle\end{eqnarray}$$

In the following, the inner solution for $\hat{u}$ and $\hat{p}$ will be matched to (3.5) and (3.6).

3.3.2 Inner solution

To compute the inner solution, equations (2.11) and (2.12) are expanded by replacing $r$ by $1-\unicode[STIX]{x1D6FF}y$ . It is also possible to start the matching method from (2.9d ). To leading order, equation (2.12) becomes

(3.8) $$\begin{eqnarray}\displaystyle \hat{p}_{y}(y)={\displaystyle \frac{2\text{i}\unicode[STIX]{x1D70C}_{0}U_{\unicode[STIX]{x1D703}}U_{\unicode[STIX]{x1D703},y}}{kU_{x}+mU_{\unicode[STIX]{x1D703}}-\unicode[STIX]{x1D714}}}\hat{u} (y)+O(\unicode[STIX]{x1D6FF}), & & \displaystyle\end{eqnarray}$$

where the $y$ -dependence has been kept implicitly. The subscript $y$ represents the derivative with respect to $y$ and for every variable $\unicode[STIX]{x1D6F7}$ ; $\unicode[STIX]{x1D6F7}(y)$ improperly stands for $\unicode[STIX]{x1D6F7}(r=1-\unicode[STIX]{x1D6FF}y)$ . Similarly, equation (2.11) is rewritten in terms of the inner variable $y$

(3.9) $$\begin{eqnarray}\displaystyle & \displaystyle \left({\displaystyle \frac{\hat{u} (y)}{kU_{x}+mU_{\unicode[STIX]{x1D703}}-\unicode[STIX]{x1D714}}}\right)_{y}=O\left(\unicode[STIX]{x1D6FF}\right). & \displaystyle\end{eqnarray}$$

The solution to these equations must be determined to be matched with the outer solution. The acoustic pressure and normal velocity are now considered as perturbation series, although only the leading-order term is required for our analysis:

(3.10) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{p}(y)=\hat{p}_{0}(y)+O(\unicode[STIX]{x1D6FF}), & \displaystyle\end{eqnarray}$$

(3.11) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{u} (y)=\hat{u} _{0}(y)+O(\unicode[STIX]{x1D6FF}). & \displaystyle\end{eqnarray}$$

The form of equation (3.9) allows determining of $\hat{u} _{0}$ . At zeroth order, it reduces to:

(3.12) $$\begin{eqnarray}\displaystyle & \displaystyle \left({\displaystyle \frac{\hat{u} _{0}(y)}{kU_{x}(y)+mU_{\unicode[STIX]{x1D703}}(y)-\unicode[STIX]{x1D714}}}\right)_{y}=0. & \displaystyle\end{eqnarray}$$

Then, by posing $\unicode[STIX]{x1D6EC}_{0}(y)=kU_{x}(y)+mU_{\unicode[STIX]{x1D703}}(y)-\unicode[STIX]{x1D714}$ ,

(3.13) $$\begin{eqnarray}\displaystyle \hat{u} _{0}(y)=B_{0}\unicode[STIX]{x1D6EC}_{0}(y). & & \displaystyle\end{eqnarray}$$

Similarly, only leading-order terms in $\unicode[STIX]{x1D6FF}$ are kept in (3.8). This reads:

(3.14) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{p}_{0,y}(y)={\displaystyle \frac{2\text{i}\unicode[STIX]{x1D70C}_{0}U_{\unicode[STIX]{x1D703}}(y)U_{\unicode[STIX]{x1D703},y}(y)}{\unicode[STIX]{x1D6EC}_{0}(y)}}\hat{u} _{0}(y), & \displaystyle\end{eqnarray}$$

and hence

(3.15) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{p}_{0}(y)=A_{0}+\text{i}B_{0}\int _{0}^{y}\unicode[STIX]{x1D70C}_{0}(y^{\prime })\left(U_{\unicode[STIX]{x1D703}}^{2}(y^{\prime })\right)_{y^{\prime }}\,\text{d}y^{\prime }. & \displaystyle\end{eqnarray}$$

The integral term is bounded as $y\rightarrow \infty$ since $U_{\unicode[STIX]{x1D703}}(y)=M_{\unicode[STIX]{x1D703}}$ is assumed constant outside the boundary layer, and hence its derivative is zero. The constants $B_{0}$ and $A_{0}$ can be determined by matching the leading-order inner solutions with the outer ones.

3.3.3 Matching

Matching equation (3.13) with the outer solution (3.5) for large $y$ gives the value of the constant $B_{0}$ :

(3.16) $$\begin{eqnarray}\displaystyle B_{0}={\displaystyle \frac{\hat{u} _{\infty }^{1}}{\unicode[STIX]{x1D6EC}_{\infty }^{1}}}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6EC}_{\infty }^{1}=kM_{x,1}+mM_{\unicode[STIX]{x1D703},1}-\unicode[STIX]{x1D714}$ has been defined for conciseness. Matching $\hat{p}_{0}$ defined in (3.15) with $\hat{p}_{o}$ in (3.6) for large $y$ gives:

(3.17) $$\begin{eqnarray}\displaystyle & \displaystyle A_{0}=\hat{p}_{\infty }^{1}-{\displaystyle \frac{\text{i}\hat{u} _{\infty }^{1}I_{0}}{\unicode[STIX]{x1D6EC}_{\infty }^{1}}}, & \displaystyle\end{eqnarray}$$

where

(3.18) $$\begin{eqnarray}\displaystyle & \displaystyle I_{0}=\int _{0}^{\infty }\unicode[STIX]{x1D70C}_{0}(y)\left(U_{\unicode[STIX]{x1D703}}^{2}(y)\right)_{y}\text{d}y=-\int _{r_{m}}^{1}\unicode[STIX]{x1D70C}_{0}(r){\displaystyle \frac{\text{d}}{\text{d}r}}\left(U_{\unicode[STIX]{x1D703}}^{2}(r)\right)\text{d}r. & \displaystyle\end{eqnarray}$$

3.3.4 Boundary condition

At $r=1$ (or $y=0$ ), the surface impedance of the liner at the outer wall is defined by $Z_{1}=\hat{p}(1)/\hat{u} (1)$ . Furthermore, we note that $\unicode[STIX]{x1D6EC}_{0}(1)=-\unicode[STIX]{x1D714}$ since $U_{x}(1)=U_{\unicode[STIX]{x1D703}}(1)=0$ . As in Khamis & Brambley (Reference Khamis and Brambley2016), the boundary condition is expressed as an equivalent impedance relating $\hat{u} _{\infty }^{1}$ and $\hat{p}_{\infty }^{1}$ . At leading order, it reads:

(3.19a ) $$\begin{eqnarray}\displaystyle Z_{eff}^{\dagger }(1)={\displaystyle \frac{\hat{p}_{\infty }^{1}}{\hat{u} _{\infty }^{1}}}=-{\displaystyle \frac{\unicode[STIX]{x1D714}}{\unicode[STIX]{x1D6EC}_{\infty }^{1}}}Z_{1}^{\dagger }, & & \displaystyle\end{eqnarray}$$

(3.19b ) $$\begin{eqnarray}\displaystyle \text{with}~Z_{1}^{\dagger }=Z_{1}+{\displaystyle \frac{\text{i}}{\unicode[STIX]{x1D714}}}\int _{r_{m}}^{1}\unicode[STIX]{x1D70C}_{0}{\displaystyle \frac{\text{d}}{\text{d}r}}\left(U_{\unicode[STIX]{x1D703}}^{2}\right)\text{d}r, & & \displaystyle\end{eqnarray}$$

which shows an additional term in comparison with the classical Myers boundary condition, (3.2). Equation (3.19) is the correct boundary condition for a vanishing boundary layer in the presence of a swirling flow.

3.4 Extension to the inner wall

At the inner wall, the surface impedance is defined by $Z_{h}=-\hat{p}(h)/\hat{u} (h)$ . Applying the same method leads to the corrected Myers boundary condition at the inner wall:

(3.20a ) $$\begin{eqnarray}\displaystyle Z_{eff}^{\dagger }(h)=-{\displaystyle \frac{\hat{p}_{\infty }^{h}}{\hat{u} _{\infty }^{h}}}=-{\displaystyle \frac{\unicode[STIX]{x1D714}}{\unicode[STIX]{x1D6EC}_{\infty }^{h}}}Z_{h}^{\dagger }, & & \displaystyle\end{eqnarray}$$

(3.20b ) $$\begin{eqnarray}\displaystyle \text{with}~Z_{h}^{\dagger }=Z_{h}+{\displaystyle \frac{\text{i}}{h\unicode[STIX]{x1D714}}}\int _{h}^{r_{m}}\unicode[STIX]{x1D70C}_{0}{\displaystyle \frac{\text{d}}{\text{d}r}}\left(U_{\unicode[STIX]{x1D703}}^{2}\right)\text{d}r, & & \displaystyle\end{eqnarray}$$

$\hat{p}_{\infty }^{h}=\hat{p}_{o}(h)$

$\hat{u} _{\infty }^{h}=\hat{u} _{o}(h)$

$\unicode[STIX]{x1D6EC}_{\infty }^{h}=kM_{x,h}+mM_{\unicode[STIX]{x1D703},h}/h-\unicode[STIX]{x1D714}$

3.5 Extension to a varying outer mean flow away from the walls

The corrected Myers boundary condition has been developed so far assuming constant axial and azimuthal velocities. This assumption allows application of the matching between the inner and the outer solutions for large $y$ (see § 3.3.3), since the outer solution has a well-defined limit when $y\rightarrow \infty$ . The method is extended here to velocity profiles which may vary through the duct section. In this case, $U_{x}$ and $U_{\unicode[STIX]{x1D703}}$ no longer have a limit when $y\rightarrow \infty$ and the matching must be handled differently. To do so, the parameters $\widehat{U_{x}}$ and $\widehat{U_{\unicode[STIX]{x1D703}}}$ are introduced such that

(3.21) $$\begin{eqnarray}\displaystyle \widehat{U_{x}}(y)={\displaystyle \frac{M_{x,i}U_{x}(y)}{U_{x}^{nbl}(y)}}=U_{x}(y)+O(\unicode[STIX]{x1D6FF}),\quad i=h,1, & & \displaystyle\end{eqnarray}$$

and

(3.22) $$\begin{eqnarray}\displaystyle \widehat{U_{\unicode[STIX]{x1D703}}}(y)={\displaystyle \frac{M_{\unicode[STIX]{x1D703},i}U_{\unicode[STIX]{x1D703}}(y)}{U_{\unicode[STIX]{x1D703}}^{nbl}(y)}}=U_{\unicode[STIX]{x1D703}}(y)+O(\unicode[STIX]{x1D6FF}),\quad i=h,1, & & \displaystyle\end{eqnarray}$$

where $M_{x,i}$ and $M_{\unicode[STIX]{x1D703},i}$ are defined in § 3.2, and where $U_{x}^{nbl}$ and $U_{\unicode[STIX]{x1D703}}^{nbl}$ now depend on $y$ . Then, $U_{\unicode[STIX]{x1D703}}$ can be simply replaced by $\widehat{U_{\unicode[STIX]{x1D703}}}$ in (3.8) and (3.9) since the equations are unchanged at leading order. Unlike $U_{x}$ and $U_{\unicode[STIX]{x1D703}}$ , $\widehat{U_{x}}$ and $\widehat{U_{\unicode[STIX]{x1D703}}}$ have a well-defined limit as $y\rightarrow \infty$ . In particular,

(3.23a,b ) $$\begin{eqnarray}\displaystyle \widehat{U_{\unicode[STIX]{x1D703}}}(y)=M_{\unicode[STIX]{x1D703},i},\quad \widehat{U_{x}}(y)=M_{x,i} & & \displaystyle\end{eqnarray}$$

outside the boundary layer and the integral $I_{0}$ remains bounded. Since $\widehat{U_{x}}(0)=U_{x}(0)$ and $\widehat{U_{\unicode[STIX]{x1D703}}}(0)=U_{\unicode[STIX]{x1D703}}(0)$ , the boundary condition becomes:

(3.24) $$\begin{eqnarray}\displaystyle Z_{eff}(h)=-{\displaystyle \frac{\hat{p}_{\infty }^{h}}{\hat{u} _{\infty }^{h}}}=-{\displaystyle \frac{\unicode[STIX]{x1D714}}{\unicode[STIX]{x1D6EC}_{\infty }^{h}}}\left(Z_{h}+{\displaystyle \frac{\text{i}}{h\unicode[STIX]{x1D714}}}\int _{h}^{r_{m}}\unicode[STIX]{x1D70C}_{0}{\displaystyle \frac{\text{d}}{\text{d}r}}\left(\widehat{U_{\unicode[STIX]{x1D703}}}^{2}\right)\text{d}r\right), & & \displaystyle\end{eqnarray}$$

and

(3.25) $$\begin{eqnarray}\displaystyle Z_{eff}(1)={\displaystyle \frac{\hat{p}_{\infty }^{1}}{\hat{u} _{\infty }^{1}}}=-{\displaystyle \frac{\unicode[STIX]{x1D714}}{\unicode[STIX]{x1D6EC}_{\infty }^{1}}}\left(Z_{1}+{\displaystyle \frac{\text{i}}{\unicode[STIX]{x1D714}}}\int _{r_{m}}^{1}\unicode[STIX]{x1D70C}_{0}{\displaystyle \frac{\text{d}}{\text{d}r}}\left(\widehat{U_{\unicode[STIX]{x1D703}}}^{2}\right)\text{d}r\right). & & \displaystyle\end{eqnarray}$$

This extension has been implemented in Mathews et al. (Reference Mathews, Masson, Moreau and Posson2018), and results can be found in figure 4 therein. In the next section we discuss an approximation to this boundary condition, which will be used in this paper instead.

3.6 Approximating the boundary condition

For thin boundary layers, the density hardly varies and the corrected boundary conditions could be approximated by:

(3.26a ) $$\begin{eqnarray}\displaystyle Z_{eff}^{\ddagger }(h)=-{\displaystyle \frac{\unicode[STIX]{x1D714}}{\unicode[STIX]{x1D6EC}_{\infty }^{h}}}Z_{h}^{\ddagger }, & & \displaystyle\end{eqnarray}$$

(3.26b ) $$\begin{eqnarray}\displaystyle \text{with}~Z_{h}^{\ddagger }=Z_{h}+{\displaystyle \frac{\text{i}}{h\unicode[STIX]{x1D714}}}\unicode[STIX]{x1D70C}_{0}^{nbl}(h)M_{\unicode[STIX]{x1D703},h}^{2}, & & \displaystyle\end{eqnarray}$$

(3.27a ) $$\begin{eqnarray}\displaystyle Z_{eff}^{\ddagger }(1)=-{\displaystyle \frac{\unicode[STIX]{x1D714}}{\unicode[STIX]{x1D6EC}_{\infty }^{1}}}Z_{1}^{\ddagger }, & & \displaystyle\end{eqnarray}$$

(3.27b ) $$\begin{eqnarray}\displaystyle \text{with}~Z_{1}^{\ddagger }=Z_{1}-{\displaystyle \frac{\text{i}}{\unicode[STIX]{x1D714}}}\unicode[STIX]{x1D70C}_{0}^{nbl}(1)M_{\unicode[STIX]{x1D703},1}^{2}, & & \displaystyle\end{eqnarray}$$

3.7 Physical interpretation of the corrected boundary condition

In equations (3.19) and (3.20), there is an extra integral term that does not appear in the classical Myers boundary condition, equation (3.2). This integral term scales with the square of the mean swirl, and scales inversely with the frequency $\unicode[STIX]{x1D714}$ . In a common mass–spring–damper impedance model, $Z(\unicode[STIX]{x1D714})=R_{Z}+\text{i}(K_{Z}/\unicode[STIX]{x1D714}-M_{Z}\unicode[STIX]{x1D714})$ , where $K_{Z}$ is the spring constant of the impedance boundary, and therefore, owing to the $1/\unicode[STIX]{x1D714}$ frequency dependence, the extra integral term may be seen as providing an extra spring-like term. The extra integral term is seen from (3.26b ) and (3.27b ) to change sign between the inner wall and the outer wall. Physically, therefore, it is suggested that this extra integral term is due to the mismatch in centrifugal force acting on the perturbation between one side and the other of the infinitely thin boundary layer over the wall, increasing the perceived springiness of the inner wall and decreasing the perceived springiness of the outer wall. Even at the high frequencies typical in aeroacoustics, it is common to retain the $K_{Z}$ term in the wall impedance, especially near the tuned resonance when $K_{Z}/\unicode[STIX]{x1D714}=M_{Z}\unicode[STIX]{x1D714}$ , and hence the extra integral term may prove to be important even at high frequencies, and especially near resonances of the impedance walls. When the mean swirl is zero, the additional term reduces to zero and equations (3.19) and (3.20) reduce to Myers’ boundary condition for an axial mean flow. In the case of hard walls ( $Z=\infty$ ), it can be observed that the Myers, the corrected Myers and the simple approximation all reduce to the same boundary condition:

(3.28) $$\begin{eqnarray}\displaystyle \hat{u} (h)=\hat{u} (1)=0. & & \displaystyle\end{eqnarray}$$

4.1 Classical Myers boundary condition

Since Myers’ boundary condition is $k$ -linear, it can be easily included in the eigenmode formulation (see 3.2). This is what has been done so far in the literature (see Guan et al. Reference Guan, Luo and Wang2008; Posson & Peake Reference Posson and Peake2013b ; Maldonado et al. Reference Maldonado, Astley, Coupland, Gabard and Sutliff2015; Gabard Reference Gabard2016).

4.2 Corrected Myers boundary condition

The corrected Myers boundary condition is also $k$ -linear. Using the impedances $Z_{h}^{\dagger }$ and $Z_{1}^{\dagger }$ introduced in (3.26b ) and (3.27b ) respectively, the corrected Myers boundary condition to leading order reads:

(4.2a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{u} (h)={\displaystyle \frac{kU_{x}(h)+mU_{\unicode[STIX]{x1D703}}(h)/h-\unicode[STIX]{x1D714}}{\unicode[STIX]{x1D714}Z_{h}^{\dagger }}}\hat{p}(h)\quad \text{and}\quad \hat{u} (1)=-{\displaystyle \frac{kU_{x}(1)+mU_{\unicode[STIX]{x1D703}}(1)-\unicode[STIX]{x1D714}}{\unicode[STIX]{x1D714}Z_{1}^{\dagger }}}\hat{p}(1), & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

which is the same formulation as the classical Myers boundary condition, where only the impedances $Z$ have been replaced by $Z^{\dagger }$ .

4.3 Boundary-layer treatment

In order to challenge the original Myers and the corrected Myers boundary conditions, a boundary-layer profile can be included in the mean flow, such that the flow is zero at the interface. Given the outer flow (varying or not), the exponential envelope $L_{\unicode[STIX]{x1D6FC}}$ ,

(4.3) $$\begin{eqnarray}\displaystyle & \displaystyle L_{\unicode[STIX]{x1D6FC}}(r)=1-\text{e}^{-\unicode[STIX]{x1D6FC}(r-h)}-\text{e}^{-\unicode[STIX]{x1D6FC}(1-r)}, & \displaystyle\end{eqnarray}$$

is introduced. A realistic profile with a boundary layer is defined by multiplying the outer flow by $L_{\unicode[STIX]{x1D6FC}}$ . The boundary-layer thickness is controlled by the parameter $\unicode[STIX]{x1D6FC}$ , with $\unicode[STIX]{x1D6FF}\propto 1/\unicode[STIX]{x1D6FC}$ . The displacement thickness is defined by:

(4.4a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D700}_{h}=2\int _{h}^{r_{m}}\left(1-{\displaystyle \frac{U_{x}(r)}{U_{x}^{nbl}(r)}}\right)\text{d}r,\quad \unicode[STIX]{x1D700}_{1}=2\int _{r_{m}}^{1}\left(1-{\displaystyle \frac{U_{x}(r)}{U_{x}^{nbl}(r)}}\right), & & \displaystyle\end{eqnarray}$$

at the inner and outer wall respectively. Gabard (Reference Gabard2013) applied several boundary-layer profiles in the case of a non-swirling flow and showed that the shape of the boundary layer had a weak impact on the result. The same conclusion is expected for a swirling mean flow. However, the influence of the boundary-layer shape will not be addressed in the framework of the present study. For a mean flow which includes a boundary layer, the boundary condition reduces to equation (3.1).

5.1 Validation of the approximation for thin boundary layers

Figure 1.

(a) Magnitude of the additional term in (3.19b ),  $\text{log}_{10}(|Z_{1}^{\dagger }-Z_{1}|)$ . (b) Difference between the corrected boundary condition (3.19b ) and its thin boundary-layer approximation (3.27b ),  $\text{log}_{10}(|Z_{1}^{\dagger }-Z_{1}^{\ddagger }|)$ . Both are calculated at the outer wall for $U_{\unicode[STIX]{x1D703}}^{nbl}=0.5$ .

The magnitude of the new term in equation (3.19b ) is presented in figure 1(a) in the $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D714})$ -plane, where $\unicode[STIX]{x1D6FC}$ from (4.3) sets the boundary-layer thickness and $\unicode[STIX]{x1D714}$ is the frequency, for a typical case of $U_{\unicode[STIX]{x1D703}}^{nbl}=0.5$ . As expected, this term decreases with the frequency because of the $1/\unicode[STIX]{x1D714}$ term. However, it appears to be largely independent of the $\unicode[STIX]{x1D6FC}$ parameter. The additional term due to the correction is very weak almost everywhere in the $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D714})$ -plane and may become significant only below $\unicode[STIX]{x1D714}=3$ (with $\text{log}_{10}(3)\approx 0.48$ ), which is lower than the typical frequency range of turbofan applications. For this angular frequency $|Z_{1}^{\dagger }-Z_{1}|\approx 0.1$ , which must be compared with the amplitude of a typical liner impedance ( $|Z_{1}|\approx 1$ ). Close to the liner resonance defined by $\operatorname{Im}(Z_{1})=0$ , this additional term could become significant however.

In order to validate the thin boundary-layer approximated form of the corrected Myers boundary condition introduced in § 3.6, $Z_{1}^{\dagger }$ and $Z_{1}^{\ddagger }$ defined in (3.19b ) and (3.27b ) respectively are also compared in the $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D714})$ -plane. The results are presented in  figure 1(b), again for $U_{\unicode[STIX]{x1D703}}^{nbl}=0.5$ . As expected, the approximate solution $Z_{1}^{\ddagger }$ gets closer to the reference $Z_{1}^{\dagger }$ when the boundary-layer thickness tends to zero (e.g. $\unicode[STIX]{x1D6FC}\rightarrow \infty$ ) and when the frequency increases. The difference between the two terms is at least 3 orders of magnitudes lower than the magnitude of the additional term for all the considered values of $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D714})$ . Because of this, the simpler approximated form of the boundary condition ((3.26b ) and (3.19b )) will be used for the numerical results that follow. This notably allows application of the boundary condition without knowing the details of the flow parameters inside the boundary layer. In both figures 1(a) and 1(b), $Z_{1}$ need not be specified even though it appears in the legend. Indeed, a glance at (3.19b ) and (3.27b ) will show that $\left|Z_{1}^{\dagger }-Z_{1}\right|$ and $\left|Z_{1}^{\dagger }-Z_{1}^{\ddagger }\right|$ do not depend on $Z_{1}$ . Besides, it is worth noting that these terms scale with $M_{\unicode[STIX]{x1D703},1}^{2}$ .

5.2 Validation of the corrected Myers boundary condition

First, it is proposed to assess the corrected boundary condition for a simple case. The canonical case chosen for the study is the constant swirl and constant axial flow such that $M_{x}=M_{\unicode[STIX]{x1D703}}=0.5$ . The duct section is characterised by the dimensionless inner radius $h=0.5$ . The dimensionless frequency is $\unicode[STIX]{x1D714}=4$ and the azimuthal mode order $m=-1$ is considered. The surface impedance at the inner and outer walls are defined by $Z_{1}=Z_{h}=1-\text{i}$ . For this case, the approximated boundary condition gives $Z_{h}^{\ddagger }=1-0.887\text{i}$ and $Z_{1}^{\ddagger }=1-1.067\text{i}$ .

Table 1.

Effect of the boundary condition on the position of the most cut-on eigenmodes for the canonical case $M_{x}=M_{\unicode[STIX]{x1D703}}=0.5$ , $h=0.5$ , $\unicode[STIX]{x1D714}=4$ , $Z_{h}=Z_{1}=1-\text{i}$ , with $|m|=1$ . U1: first upstream mode, U2: second upstream mode, D1: first downstream mode, D2: second downstream mode.

Eigenmodes obtained with the original Myers and the corrected Myers boundary conditions are compared with the ones obtained with a realistic boundary layer, for several boundary-layer thicknesses. As the boundary-layer thickness tends to zero, the eigenmodes become harder to calculate numerically when simulating the boundary layer. In table 1, the two most ‘cut-on’ upstream and downstream eigenmodes are considered.

The additional centrifugal term in the corrected Myers boundary condition has an effect on the eigenmodes, since they differ from the eigenmodes of the classical Myers boundary condition at the second decimal in this case. Furthermore, when the boundary-layer thickness tends to zero, the eigenmodes indeed converge towards the eigenmodes from the corrected Myers boundary condition.

For a given eigenmode, the corresponding attenuation rate in dB is given by $\pm 20\operatorname{Im}(k)/\text{log}(10)$ , the sign depending on the direction of propagation. The attenuation rates related to the eigenmodes of table 1 are presented in table 2. As for the eigenmodes, the attenuation rates converge towards the corrected Myers boundary condition when the boundary-layer thickness tends to zero.

These observations suggest that (3.26) and (3.27) are the correct limits for a vanishing boundary layer.

Table 2.

Attenuation rates in dB per radius for the different boundary conditions with the parameters as in table 1.

5.3 Centrifugal effect on the eigenmodes at lower frequency

Because of the $\unicode[STIX]{x1D714}^{-1}$ factor contained in the centrifugal term in (3.26) and (3.27), larger differences are expected between the original Myers boundary condition and the corrected Myers boundary condition in the low frequency range. To illustrate this, the above test case is considered again, but with a frequency ten times lower ( $\unicode[STIX]{x1D714}=0.4$ ), all the other parameters being unchanged. The eigenmodes obtained with both boundary conditions are plotted in figure 2. They are compared to the reference eigenmodes, which are also obtained by solving numerically the set of (2.9) according to the method described at the beginning of § 4. These reference eigenmodes are computed for a realistic flow profile defined by $U_{x}(r)=U_{\unicode[STIX]{x1D703}}(r)=0.5L_{2500}(r)$ , where $L_{2500}$ represents a thin boundary layer (displacement thickness $\unicode[STIX]{x1D700}=4\times 10^{-4}$ ), together with the boundary conditions $Z_{eff}(h)=Z_{h}$ and $Z_{eff}(1)=Z_{1}$ since the mean flow cancels at both walls. The values of some of the eigenmodes are given in  table 3 for the three boundary conditions.

Figure 2.

Eigenmode positions in the complex $k$ -plane at low frequency ( $\unicode[STIX]{x1D714}=0.4$ ). The case is defined by $M_{x}=M_{\unicode[STIX]{x1D703}}=0.5$ , $h=0.5$ , $m=-1$ , $Z_{h}=Z_{1}=1-\text{i}$ . A total of 300 points were used for the radial discretisation.

Table 3.

Effect of the boundary condition on the position of some eigenmodes for the low frequency case ( $\unicode[STIX]{x1D714}=0.4$ ).  $M_{x}=M_{\unicode[STIX]{x1D703}}=0.5$ , $h=0.5$ , $m=-1$ , $Z_{h}=Z_{1}=1-\text{i}$ . The relative errors to the reference eigenvalues are shown in brackets.

As expected, the centrifugal effects in the boundary condition are much stronger than for the previous frequency of observation. Indeed, for this specific case, $Z_{h}^{\ddagger }=1+0.13\text{i}$ and $Z_{1}^{\ddagger }=1-1.67\text{i}$ . The original Myers boundary condition inaccurately predicts the eigenmodes while the new corrected boundary condition gives eigenmodes much closer to the reference.

5.4 Illustration of the correct behaviour for vanishing boundary layers

In this section, the exact value of $Z_{eff}^{1}=\hat{p}_{\infty }^{1}/\hat{u} _{\infty }^{1}$ , which would be obtained if there was no boundary layer, is compared with the effective impedance arising from the different boundary conditions. To isolate the boundary condition being investigated at $r=1$ , a slipping mean flow with no boundary layer is assumed at $r=h$ and two artificial boundary conditions are introduced there, given by $\hat{p}(h)=1-\text{i}$ and $\hat{u} (h)=-1$ . For any given values of $\unicode[STIX]{x1D714}$ , $m$ and $k$ , the governing (2.11) and (2.12) may be integrated to give $\hat{p}(r)$ and $\hat{u} (r)$ . Hence, the exact value of $\hat{p}_{\infty }^{1}$ and $\hat{v}_{\infty }^{1}$ (and hence $Z_{eff}^{1}=\hat{p}_{\infty }^{1}/\hat{u} _{\infty }^{1}$ ) can be computed by taking no boundary layer near $r=1$ . By introducing an exponential boundary layer similar to $L_{\unicode[STIX]{x1D6FC}}$ at $r=1$ , the exact value of $Z_{BL}^{1}=\hat{p}_{\unicode[STIX]{x1D6FC}}(1)/\hat{u} _{\unicode[STIX]{x1D6FC}}(1)$ can be computed in the same way.

Thus, the accuracy of the boundary conditions can be assessed by comparing the exact value $Z_{eff}^{1}=\hat{p}_{\infty }^{1}/\hat{u} _{\infty }^{1}$ with the ones which would be obtained by inserting $Z_{BL}^{1}$ in the boundary conditions. The new boundary condition, the approximated boundary condition and the classical Myers boundary condition are written in terms of an effective impedance. They read

(5.1) $$\begin{eqnarray}\displaystyle & \displaystyle Z_{eff}^{\dagger ,1}=-{\displaystyle \frac{\unicode[STIX]{x1D714}}{\unicode[STIX]{x1D6EC}_{\infty }^{1}}}\left(Z_{BL}^{1}+{\displaystyle \frac{\text{i}}{\unicode[STIX]{x1D714}}}\int _{r_{m}}^{1}\unicode[STIX]{x1D70C}_{0}{\displaystyle \frac{\text{d}}{\text{d}r}}\left(U_{\unicode[STIX]{x1D703}}^{2}\right)\text{d}r\right), & \displaystyle\end{eqnarray}$$

(5.2) $$\begin{eqnarray}\displaystyle & \displaystyle Z_{eff}^{\ddagger ,1}=-{\displaystyle \frac{\unicode[STIX]{x1D714}}{\unicode[STIX]{x1D6EC}_{\infty }^{1}}}\left(Z_{BL}^{1}-{\displaystyle \frac{\text{i}}{\unicode[STIX]{x1D714}}}\unicode[STIX]{x1D70C}_{0}^{nbl}(1)M_{\unicode[STIX]{x1D703},1}^{2}\right), & \displaystyle\end{eqnarray}$$

(5.3) $$\begin{eqnarray}\displaystyle & \displaystyle Z_{eff}^{M,1}=-{\displaystyle \frac{\unicode[STIX]{x1D714}Z_{BL}^{1}}{\unicode[STIX]{x1D6EC}_{\infty }^{1}}}, & \displaystyle\end{eqnarray}$$

respectively. For these three boundary conditions, the equivalent impedance is compared with $Z_{eff}^{1}$ . In figure 3, the evolution of the errors $|Z_{eff}^{\dagger ,1}/Z_{eff}^{1}-1|$ , $|Z_{eff}^{\ddagger ,1}/Z_{eff}^{1}-1|$ and $|Z_{eff}^{M,1}/Z_{eff}^{1}-1|$ as a function of the boundary-layer thickness are shown for four values of $k$ : $1$ , $-1$ , $1+\text{i}$ and $-1-\text{i}$ . The parameters used for the study are $h=0.5$ , $\unicode[STIX]{x1D714}=5$ , $m=1$ and $M_{x}=M_{\unicode[STIX]{x1D703}}=0.5$ .

Figure 3.

Plot of the relative error $|Z_{eff}^{\dagger ,1}/Z_{eff}^{1}-1|$ (plusses), $|Z_{eff}^{\ddagger ,1}/Z_{eff}^{1}-1|$ (circles) and $|Z_{eff}^{M,1}/Z_{eff}^{1}-1|$ (crosses) at the outer wall. The other parameters are $h=0.5$ , $\unicode[STIX]{x1D714}=5$ , $m=1$ and $M_{x}=M_{\unicode[STIX]{x1D703}}=0.5$ .

Since the corrected Myers boundary condition has been developed to leading order with error $O(\unicode[STIX]{x1D6FF})$ , it is expected that the error decreases linearly with the boundary-layer thickness. The linear relationship between the error and the boundary-layer thickness is drawn in yellow. The corrected Myers boundary condition as well as its simple approximation match very well the expected behaviour while the classical Myers condition is clearly wrong.

Applying the same procedure at the inner wall leads to the same conclusions. Comparisons have not been shown here for the sake of conciseness.

5.5 A realistic turbofan engine configuration: the SDT test case

Finally, the corrected Myers boundary condition is tested on a realistic fan configuration. The mean flow has been extracted from the core flow of a Reynolds-averaged Navier–Stokes (RANS) simulation performed on the NASA SDT test case (see Hughes et al. Reference Hughes, Jeracki, Woodward and Miller2002; Woodward et al. Reference Woodward, Hughes, Jeracki and Miller2002) so that the flow is representative of a bypass interstage with no boundary layer. After interpolating the mean flow on a set of rational functions, a boundary layer can be added by multiplying the mean flow by the exponential envelope defined in (4.3). The value of $\unicode[STIX]{x1D6FC}=200$ was chosen since it corresponds to a displacement thickness of $\unicode[STIX]{x1D700}=0.01$ , which is representative of a boundary-layer thickness in the interstage. The mean flow profiles with and without the boundary layer are shown in figures 4(a) and 4(b).

Figure 4.

Interpolated profiles from the SDT test case, with or without a boundary layer.

The hub-to-tip ratio of the SDT test case is $h=0.5$ . The frequency of the study is chosen so that it fits with the maximum of the rotor–stator interaction broadband noise spectra at approach condition. The value $f_{\text{max}}^{\star }=3000~\text{Hz}$ has been taken from Masson et al. (Reference Masson, Posson, Sanjose, Moreau and Roger2016), which corresponds to $\unicode[STIX]{x1D714}=15$ . Even though there is no acoustic treatment in the NASA SDT, the walls are assumed to be lined with typical acoustic treatments. The impedance is set to $Z_{h}=Z_{1}=2.55+1.5\text{i}$ . The imaginary part is above zero which corresponds to a resonance at a higher frequency. The effect of the boundary condition on the eigenmodes and the eigenfunctions will be assessed for $m=1$ and $m=15$ . Since the azimuthal flow is negative, these values correspond to contra-rotating modes according to the conventions of the paper.

5.5.1 Position of the eigenmodes

The azimuthal mode $m=1$ is considered first. The eigenmodes of (4.1) are plotted in figure 5 for the three boundary conditions. The original Myers boundary condition ((3.2), violet triangles) and the corrected Myers boundary condition ((3.27), red crosses) are compared with the reference ((3.1), blue circles) which is obtained by considering the flow with a boundary layer. For these three cases, the same boundary condition is applied both at the inner and outer walls. Since the frequency is quite high, results obtained from the original Myers boundary condition and from the corrected Myers boundary condition are very similar, the eigenmodes being almost indistinguishable. The ‘most’ cut-on modes, which are the most relevant when studying in-duct transmission, are pretty well predicted, especially in the downstream direction, while the accuracy decreases as the modes move away from the real axis.

Figure 5.

Eigenmode positions in the complex $k$ -plane for a realistic test case based on the SDT mean flow. The case is defined by $\unicode[STIX]{x1D714}=15$ , $h=0.5$ , $m=1$ , $Z_{h}=Z_{1}=2.55+1.5\text{i}$ .

Figure 6.

Eigenmode positions in the complex $k$ -plane for a realistic test case based on the SDT mean flow. The case is defined by $\unicode[STIX]{x1D714}=15$ , $h=0.5$ , $m=15$ , $Z_{h}=Z_{1}=2.55+1.5\text{i}$ .

The same methodology is then applied to the azimuthal mode order $m=15$ . The results are presented in figure 6. Once again, the original Myers boundary condition gives very similar results as the corrected one. However, significant error is observed with respect to the reference case, even for the eigenmodes located close to the real axis. This is expected to alter the prediction when acoustic transmission is considered. To improve the accuracy of the corrected Myers boundary condition with swirl, it is possible to extend the method presented in § 3 to the first order in $\unicode[STIX]{x1D6FF}$ , as done by Brambley (Reference Brambley2011) in the case of no swirl.

6.1 The surface-wave dispersion relation with swirl

The surface-wave dispersion relation is considered first at the outer wall. The surface waves located near the outer wall are characterised by an exponential decay away from the outer wall, so that the acoustic pressure and normal velocity $\hat{p}$ and $\hat{u}$ are of the form

(6.1a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{p}(r)=\unicode[STIX]{x1D713}_{\hat{p}}(r)\text{e}^{-\unicode[STIX]{x1D707}_{1}(1-r)},\quad \hat{u} (r)=\unicode[STIX]{x1D713}_{\hat{u} }(r)\text{e}^{-\unicode[STIX]{x1D707}_{1}(1-r)} & \displaystyle\end{eqnarray}$$

respectively, where $\unicode[STIX]{x1D707}_{1}$ is the radial wavenumber near the outer wall and $\text{d}\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D711}}/\text{d}r=O(1)$ for $\unicode[STIX]{x1D711}=\hat{p},\hat{u}$ . For surface waves to decay exponentially from the wall, it is required that $\text{Re}(\unicode[STIX]{x1D707}_{1})>0$ and $|\text{Re}(\unicode[STIX]{x1D707}_{1})|\gg 1$ . Note that, for the infinitely thin boundary-layer assumption $\unicode[STIX]{x1D6FF}\ll 1$ to still hold for such surface waves, $\unicode[STIX]{x1D6FF}\ll 1/|\unicode[STIX]{x1D707}_{1}|\ll 1$ is required. From these assumptions, the derivative of $\hat{p}$ and $\hat{u}$ can be evaluated near the outer wall

(6.2a,b ) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{\text{d}\hat{p}}{\text{d}r}}(1)=\unicode[STIX]{x1D707}_{1}\hat{p}(1)+O(1),\quad {\displaystyle \frac{\text{d}\hat{u} }{\text{d}r}}(1)=\unicode[STIX]{x1D707}_{1}\hat{u} (1)+O(1). & & \displaystyle\end{eqnarray}$$

The boundary conditions at the inner wall impose conditions on $\unicode[STIX]{x1D713}_{\hat{p}}(h)$ and $\unicode[STIX]{x1D713}_{\hat{u} }(h)$ , but since both $\hat{p}$ and $\hat{u}$ are exponentially small away from $r=1$ this inner boundary condition does not affect the dominant behaviour near $r=1$ . This decoupling of the boundary conditions for the surface waves was also shown by Rienstra (Reference Rienstra2003, § 9) in the case of uniform flow in an annular duct, and can be verified explicitly from the high frequency approximation as in Heaton & Peake (Reference Heaton and Peake2005), Mathews & Peake (Reference Mathews and Peake2017) for arbitrary swirling flows.

Equations (2.10) and (2.12) can be rewritten at the outer wall for the specific case of surface waves propagating in a homentropic mean flow. To leading order the pressure and normal velocity satisfy

(6.3) $$\begin{eqnarray}\displaystyle \displaystyle & & \displaystyle \left({\displaystyle \frac{\unicode[STIX]{x1D6EC}^{2}(1)}{c_{0}^{2}(1)}}-m^{2}-k^{2}\right)\hat{p}+\left[-\unicode[STIX]{x1D707}_{1}\unicode[STIX]{x1D6EC}(1)-\unicode[STIX]{x1D6EC}(1)-{\displaystyle \frac{U_{\unicode[STIX]{x1D703}}^{2}(1)}{c_{0}^{2}(1)}}\unicode[STIX]{x1D6EC}(1)\right.\nonumber\\ \displaystyle & & \displaystyle \qquad \left.+\,k{\displaystyle \frac{\text{d}U_{x}}{\text{d}r}}(1)+m\left({\displaystyle \frac{\text{d}U_{\unicode[STIX]{x1D703}}}{\text{d}r}}(1)+U_{\unicode[STIX]{x1D703}}(1)\right)\right]\text{i}\unicode[STIX]{x1D70C}_{0}(1)\hat{u} =0,\end{eqnarray}$$

(6.4) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle \left(\unicode[STIX]{x1D707}_{1}\unicode[STIX]{x1D6EC}(1)+2mU_{\unicode[STIX]{x1D703}}(1)-{\displaystyle \frac{U_{\unicode[STIX]{x1D703}}^{2}(1)\unicode[STIX]{x1D6EC}(1)}{c_{0}^{2}(1)}}\right)\hat{p}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\left(\unicode[STIX]{x1D6EC}^{2}(1)-2U_{\unicode[STIX]{x1D703}}(1){\displaystyle \frac{\text{d}U_{\unicode[STIX]{x1D703}}}{\text{d}r}}(1)-2U_{\unicode[STIX]{x1D703}}^{2}(1)\right)\text{i}\unicode[STIX]{x1D70C}_{0}(1)\hat{u} =0.\end{eqnarray}$$

At the outer wall, the approximate corrected Myers boundary condition reads:

(6.5) $$\begin{eqnarray}\displaystyle & \displaystyle \text{i}\unicode[STIX]{x1D714}\unicode[STIX]{x1D70C}_{0}(1)Z_{1}^{\ddagger }\hat{u} (1)=-\text{i}\unicode[STIX]{x1D70C}_{0}(1)\unicode[STIX]{x1D6EC}(1)\hat{p}(1), & \displaystyle\end{eqnarray}$$

where $Z_{1}^{\ddagger }$ is defined in (3.27b ). This is substituted in (6.3) and (6.4), and after eliminating $\hat{p}$ the coupled velocity and pressure equations read

(6.6) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle {\displaystyle \frac{\unicode[STIX]{x1D6EC}^{2}(1)}{c_{0}^{2}(1)}}-m^{2}-k^{2}+\left[-\unicode[STIX]{x1D707}_{1}\unicode[STIX]{x1D6EC}(1)-\unicode[STIX]{x1D6EC}(1)-{\displaystyle \frac{U_{\unicode[STIX]{x1D703}}^{2}(1)}{c_{0}^{2}(1)}}\unicode[STIX]{x1D6EC}(1)\right.\nonumber\\ \displaystyle & & \displaystyle \quad \left.+\,k{\displaystyle \frac{\text{d}U_{x}}{\text{d}r}}(1)+m\left({\displaystyle \frac{\text{d}U_{\unicode[STIX]{x1D703}}}{\text{d}r}}(1)+U_{\unicode[STIX]{x1D703}}(1)\right)\right]\left(-\text{i}{\displaystyle \frac{\unicode[STIX]{x1D70C}_{0}(1)\unicode[STIX]{x1D6EC}(1)}{\unicode[STIX]{x1D714}Z_{1}^{\ddagger }}}\right)=0,\end{eqnarray}$$

and

(6.7) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D707}_{1}\unicode[STIX]{x1D6EC}(1)+2mU_{\unicode[STIX]{x1D703}}(1)-{\displaystyle \frac{U_{\unicode[STIX]{x1D703}}^{2}(1)\unicode[STIX]{x1D6EC}(1)}{c_{0}^{2}(1)}}\nonumber\\ \displaystyle & & \displaystyle \quad +\,\left(\unicode[STIX]{x1D6EC}^{2}(1)-2U_{\unicode[STIX]{x1D703}}(1){\displaystyle \frac{\text{d}U_{\unicode[STIX]{x1D703}}}{\text{d}r}}(1)-2U_{\unicode[STIX]{x1D703}}^{2}(1)\right)\left(-\text{i}{\displaystyle \frac{\unicode[STIX]{x1D70C}_{0}(1)\unicode[STIX]{x1D6EC}(1)}{\unicode[STIX]{x1D714}Z_{1}^{\ddagger }}}\right)=0.\end{eqnarray}$$

Finally, the dispersion relation for the surface waves at the outer wall is obtained by eliminating $\unicode[STIX]{x1D707}_{1}$ in (6.6) and (6.7). It reads:

(6.8) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle \text{i}{\displaystyle \frac{\unicode[STIX]{x1D714}Z_{1}^{\ddagger }}{\unicode[STIX]{x1D70C}_{0}}}\left[\text{i}{\displaystyle \frac{\unicode[STIX]{x1D714}Z_{1}^{\ddagger }}{\unicode[STIX]{x1D70C}_{0}}}\left({\displaystyle \frac{\unicode[STIX]{x1D6EC}^{2}}{c_{0}^{2}}}-m^{2}-k^{2}\right)-\unicode[STIX]{x1D6EC}^{2}\left(1+{\displaystyle \frac{2U_{\unicode[STIX]{x1D703}}^{2}}{c_{0}^{2}}}\right)\right.\nonumber\\ \displaystyle & & \displaystyle \quad \left.+\,\unicode[STIX]{x1D6EC}\left(k{\displaystyle \frac{\text{d}U_{x}}{\text{d}r}}+3mU_{\unicode[STIX]{x1D703}}+m{\displaystyle \frac{\text{d}U_{\unicode[STIX]{x1D703}}}{\text{d}r}}\right)\vphantom{{\displaystyle \frac{\unicode[STIX]{x1D714}Z_{1}^{\ddagger }}{\unicode[STIX]{x1D70C}_{0}}}}\right]+\unicode[STIX]{x1D6EC}^{2}\left(\unicode[STIX]{x1D6EC}^{2}-2U_{\unicode[STIX]{x1D703}}{\displaystyle \frac{\text{d}U_{\unicode[STIX]{x1D703}}}{\text{d}r}}-2U_{\unicode[STIX]{x1D703}}^{2}\right)=0,\end{eqnarray}$$

where all the quantities are evaluated at $r=1$ .

Similarly, the dispersion relation for surface waves located near the inner wall can be derived. The acoustic variables are now proportional to $\exp \{-\unicode[STIX]{x1D707}_{h}(r-h)\}$ , where $\text{Re}(\unicode[STIX]{x1D707}_{h})>0$ and $|\text{Re}(\unicode[STIX]{x1D707}_{h})|\gg 1$ . Following the same approach as for the outer wall yields:

(6.9) $$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle -\text{i}{\displaystyle \frac{\unicode[STIX]{x1D714}Z_{h}^{\ddagger }}{\unicode[STIX]{x1D70C}_{0}}}\left[-\text{i}{\displaystyle \frac{\unicode[STIX]{x1D714}Z_{h}^{\ddagger }}{\unicode[STIX]{x1D70C}_{0}}}\left({\displaystyle \frac{\unicode[STIX]{x1D6EC}^{2}}{c_{0}^{2}}}-{\displaystyle \frac{m^{2}}{h^{2}}}-k^{2}\right)-\unicode[STIX]{x1D6EC}^{2}\left({\displaystyle \frac{1}{h}}+{\displaystyle \frac{2U_{\unicode[STIX]{x1D703}}^{2}}{hc_{0}^{2}}}\right)\right.\nonumber\\ \displaystyle & & \displaystyle \quad \left.+\,\unicode[STIX]{x1D6EC}\left(k{\displaystyle \frac{\text{d}U_{x}}{\text{d}r}}+{\displaystyle \frac{3mU_{\unicode[STIX]{x1D703}}}{h^{2}}}+{\displaystyle \frac{m}{h}}{\displaystyle \frac{\text{d}U_{\unicode[STIX]{x1D703}}}{\text{d}r}}\right)\vphantom{{\displaystyle \frac{\unicode[STIX]{x1D714}Z_{1}^{\ddagger }}{\unicode[STIX]{x1D70C}_{0}}}}\right]+\unicode[STIX]{x1D6EC}^{2}\left(\unicode[STIX]{x1D6EC}^{2}-{\displaystyle \frac{2U_{\unicode[STIX]{x1D703}}}{h}}{\displaystyle \frac{\text{d}U_{\unicode[STIX]{x1D703}}}{\text{d}r}}-{\displaystyle \frac{2U_{\unicode[STIX]{x1D703}}^{2}}{h^{2}}}\right)=0,\end{eqnarray}$$

where all the quantities are evaluated at $r=h$ . Equations (6.8) and (6.9) are polynomials in $k$ of degree four. Therefore there are at most four surface-wave modes at each wall, although some of the solutions may not satisfy $\text{Re}(\unicode[STIX]{x1D707}_{r_{0}})>0,r_{0}=\{h,1\}$ where $\unicode[STIX]{x1D707}_{1}$ is given by (6.7) for instance.

6.2 No-swirl case

If the swirl is set to zero, equations (6.8) and (6.9) reduce to

(6.10) $$\begin{eqnarray}\displaystyle & \displaystyle \pm \text{i}{\displaystyle \frac{\unicode[STIX]{x1D714}Z_{r_{0}}}{\unicode[STIX]{x1D70C}_{0}}}\left[\pm \text{i}{\displaystyle \frac{\unicode[STIX]{x1D714}Z_{r_{0}}}{\unicode[STIX]{x1D70C}_{0}}}\left({\displaystyle \frac{\unicode[STIX]{x1D6EC}^{2}}{c_{0}^{2}}}-{\displaystyle \frac{m^{2}}{r_{0}^{2}}}-k^{2}\right)-{\displaystyle \frac{\unicode[STIX]{x1D6EC}^{2}}{r_{0}}}+\unicode[STIX]{x1D6EC}k{\displaystyle \frac{\text{d}U_{x}}{\text{d}r}}\right]+\unicode[STIX]{x1D6EC}^{4}=0, & \displaystyle\end{eqnarray}$$

(where $\pm =+$ at the outer wall and $\pm =-$ at the inner wall), which can also be written in the form

(6.11a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}_{r_{0}}=\text{i}{\displaystyle \frac{\unicode[STIX]{x1D70C}_{0}\unicode[STIX]{x1D6EC}^{2}}{\unicode[STIX]{x1D714}Z_{1}}},\quad \unicode[STIX]{x1D707}_{r_{0}}^{2}={\displaystyle \frac{m^{2}}{r_{0}^{2}}}+k^{2}-{\displaystyle \frac{\unicode[STIX]{x1D6EC}^{2}}{c_{0}^{2}}}\pm \unicode[STIX]{x1D707}_{r_{0}}\left({\displaystyle \frac{k}{\unicode[STIX]{x1D6EC}}}{\displaystyle \frac{\text{d}U_{x}}{\text{d}r}}-{\displaystyle \frac{1}{r_{0}}}\right). & & \displaystyle\end{eqnarray}$$

Equation (6.11) is equivalent at leading order in $1/\unicode[STIX]{x1D707}_{r_{0}}$ to the surface-wave dispersion relation of Brambley & Peake (Reference Brambley and Peake2006), as used by Brambley (Reference Brambley2013). Since both equation (6.11) and the surface-wave dispersion relation of Brambley & Peake (Reference Brambley and Peake2006) were derived only to leading order in $1/\unicode[STIX]{x1D707}_{r_{0}}$ by different methods, it is unsurprising that they differ at high orders of $1/\unicode[STIX]{x1D707}_{r_{0}}$ , and such differences are not significant. The difference between the two is due to the presence of the $1/r_{0}$ and $kU_{x}^{\prime }/\unicode[STIX]{x1D6EC}$ terms. These terms physically correspond to geometric effects of the curved cylindrical surface and mean flow shear effects at the wall respectively.

6.4 Stability of surface waves

Figure 10.

Briggs–Bers contours as the frequency is increased from $\unicode[STIX]{x1D714}=10$ to $\unicode[STIX]{x1D714}=10+8\text{i}$ . The blue/red circles indicate the position at $\unicode[STIX]{x1D714}=10$ , green/orange crosses at $\unicode[STIX]{x1D714}=10+8\text{i}$ , and the black contour are the paths between. The other parameters are $M_{x}=0.5$ , $M_{\unicode[STIX]{x1D703}}=0.5$ , $h=0.5$ , $m=1$ and $Z(\unicode[STIX]{x1D714})=1.6-0.08\text{i}\unicode[STIX]{x1D714}+6\text{i}/\unicode[STIX]{x1D714}$ .

The stability of the surface waves is now considered. To begin with, the Briggs–Bers (Briggs Reference Briggs1964; Bers Reference Bers1983; Brambley Reference Brambley2009) contours are plotted for figure 7(a), which show the stability of the modes. These are traces of the eigenmodes as the imaginary part of frequency is increased from zero (in Brambley (Reference Brambley2009) it is decreased from zero due to the different sign of $\text{e}^{-\text{i}\unicode[STIX]{x1D714}t}$ ). In  figure 10 these traces are shown in black as the frequency is increased from $\unicode[STIX]{x1D714}=10$ to $\unicode[STIX]{x1D714}=10+8\text{i}$ , with the blue circles (acoustic modes) and red circles (surface-wave modes) showing the initial positions at $\unicode[STIX]{x1D714}=10$ , and green crosses (acoustic modes) and orange crosses (surface-wave modes) showing the position at $\unicode[STIX]{x1D714}=10+8\text{i}$ . The other parameters are the same as in  figure 7(a).

The acoustic modes are seen not to move that much as the imaginary part of $\unicode[STIX]{x1D714}$ is varied. For each frequency exactly one surface wave per wall is found. These were obtained by using the dispersion relations given by (6.8) and (6.9) as starting guesses for calculating these surface-wave modes numerically. As the imaginary part of the frequency is increased, the surface-wave modes move to the right and down, and do not cross the real axis. Therefore these surface-wave modes are stable.

Figure 11.

Briggs–Bers contours as the frequency is increased from $\unicode[STIX]{x1D714}=10$ to $\unicode[STIX]{x1D714}=10+20\text{i}$ . The blue/red circles indicate the position at $\unicode[STIX]{x1D714}=10$ , green/orange crosses at $\unicode[STIX]{x1D714}=10+20\text{i}$ , and the black contours are the paths between. The other parameters are $M_{x}=0.8$ , $M_{\unicode[STIX]{x1D703}}=0.5$ , $h=0.5$ , $m=1$ and $Z(\unicode[STIX]{x1D714})=1.6-0.08\text{i}\unicode[STIX]{x1D714}+6\text{i}/\unicode[STIX]{x1D714}$ .

For a more interesting case concerning unstable surface-wave modes, a flow with $U_{x}(r)=0.8$ is considered instead, with the other parameters unchanged. The Briggs–Bers contours are plotted in figure 11 for $\unicode[STIX]{x1D714}=10$ to $\unicode[STIX]{x1D714}=10+20\text{i}$ . The downstream acoustic modes are again seen not to move that much as the imaginary part of $\unicode[STIX]{x1D714}$ is varied, although the upstream acoustic modes move more than in the previous case. We also note that the two downstream acoustic modes with the largest imaginary part in figure 11 have more interesting behaviour than their counterparts, with the trajectory of one of them forming a loop.

Initially, there is one surface wave per wall, labelled $S_{1}$ and $S_{h}$ , and both have negative imaginary axial wavenumbers $k$ . They move considerably as the imaginary part of frequency is varied, with both crossing the real axis and moving into the upper half-plane. The surface mode $S_{1}$ crosses the real axis for $16<\operatorname{Im}(\unicode[STIX]{x1D714})<17$ , while the surface mode $S_{h}$ crosses shortly afterwards when $18<\operatorname{Im}(\unicode[STIX]{x1D714})<19$ . If the imaginary part of frequency was further increased then these surface-wave modes would continue to move upwards, with both crossing the line $\operatorname{Im}(k)=100$ at around $\operatorname{Im}(\unicode[STIX]{x1D714})\approx 58$ . Owing to their presence in the upper half-plane for sufficiently large $\operatorname{Im}(\unicode[STIX]{x1D714})$ , these surface waves are found to be convectively unstable.

Thus, each wall is found to support up to one convectively unstable surface wave for swirling flow. Furthermore, at each wall this unstable mode travels downstream. This is the same situation as in the non-swirling case. It is entirely possible and reasonable that we get a stable surface mode at one wall, and an unstable surface mode at the other wall. Whilst we cannot rule out the possibility that more than one surface mode per wall is unstable for some set of parameters, for all the realistic parameters we have calculated at most one instability per wall was present.

The stability of the SDT flow from  figure 4 is finally considered but now with $\unicode[STIX]{x1D714}=10$ , $m=-5$ , $h=0.5$ and the mass–spring–damper impedance. Two surface modes are found that have negative axial wavenumber for $\unicode[STIX]{x1D714}=10$ , and as the imaginary part of frequency is increased they move further from the real line, with the negative imaginary part of the axial wavenumber increasing. The Briggs-Bers contours would look very similar to figure 10, and the SDT flow would be stable. If a frequency of $\unicode[STIX]{x1D714}=3$ was considered instead for the SDT flow with the other parameters unchanged then two surface modes would be found to cross the real axis as the imaginary part of frequency is increased and hence the flow is unstable.

6.4.1 Temporal stability of the surface waves

Finally, the temporal stability of the surface waves is considered. Khamis & Brambley (Reference Khamis and Brambley2016) considered the temporal stability of the unstable surface wave for uniform axial non-swirling flow using three different boundary conditions: the Myers, modified Myers and second-order boundary condition. It was shown that using the Myers boundary condition gives an unbounded temporal growth rate and makes the problem ill posed, while using the modified Myers or second-order boundary condition gives a bounded growth rate, with the second-order boundary condition more closely replicating the numerics.

The growth rate of the corrected Myers boundary condition in the presence of swirl is of interest here. The simple swirling flow from  figure 11 is considered, and the frequency $\unicode[STIX]{x1D714}$ of the unstable surface wave is solved for as $k$ is varied with $k$ real. The unstable surface wave $S_{h}$ lies on the real line at $k=109.89$ , $\unicode[STIX]{x1D714}=10+18.42\text{i}$ , and the surface wave $S_{1}$ lies on the real line at $k=89.48$ , $\unicode[STIX]{x1D714}=10+16.75\text{i}$ . Instead of solving the system in (4.1) for the axial wavenumber $k$ with $\unicode[STIX]{x1D714}$ given, the inverse problem is considered with wavenumber $k$ given and the eigenmode $\unicode[STIX]{x1D714}$ to be found. The resulting $k$ versus growth rate, $\operatorname{Im}(\unicode[STIX]{x1D714})$ , is plotted in figure 12(a), along with a plot of $\operatorname{Im}(\unicode[STIX]{x1D714})$ against $\text{Re}(\unicode[STIX]{x1D714})$ in figure 12(b). These lines correspond to surface wave transitioning from spatially stable to spatially unstable, as their related axial wavenumber crosses the real axis in the complex $k$ -plane. As an example, the points where the surface-wave modes cross the real axis in figure 11 are marked with circles in figure 12, and correspond to surface waves becoming unstable for this specific set of parameters. Note also that as $k$ or $\text{Re}(\unicode[STIX]{x1D714})$ are decreased, the surface-wave modes get closer together as seen in figure 12(b).

Figure 12.

Trajectory of the two surface-wave modes $S_{h}$ and $S_{1}$ from figure 11 as $k$ is varied along the real line. Plot of (a) growth rate $\operatorname{Im}(\unicode[STIX]{x1D714})$ versus $k$ and (b) $\text{Re}(\unicode[STIX]{x1D714})$ versus $\operatorname{Im}(\unicode[STIX]{x1D714})$ .

It appears that, for both surface waves, the growth rate $\operatorname{Im}(\unicode[STIX]{x1D714})$ is unbounded as both $k$ and $\text{Re}(\unicode[STIX]{x1D714})$ increase, suggesting that the corrected Myers boundary condition is ill posed in swirling flow, as in the axial flow case. Therefore, the present stability analysis is not strictly correct and a well-posed boundary condition should be considered instead. Such a model is developed in Mathews et al. (Reference Mathews, Masson, Moreau and Posson2018).