2.1 Problem definition

The situation considered here is the flow of a viscous fluid of density $\unicode[STIX]{x1D70C}$ and viscosity $\unicode[STIX]{x1D708}$ through a circular hole or radius $R_{h}$ and area $A_{h}=\unicode[STIX]{x03C0}R_{h}^{2}$ inside a planar thin plate of a thickness that is negligible with respect to the radius, connecting an inner and an outer open domain, as shown in figure 1. We note by $Q$ the mean volumetric flow rate across the aperture, and from that latter quantity we classically define the mean velocity as $U_{M}=Q/A_{h}$ . Thus the Reynolds number of the flow is defined as:

(2.1) $$\begin{eqnarray}Re=\frac{2R_{h}U_{M}}{\unicode[STIX]{x1D708}}\equiv \frac{2Q}{\unicode[STIX]{x03C0}R_{h}\unicode[STIX]{x1D708}}.\end{eqnarray}$$

When subjected to harmonic forcing with frequency $\unicode[STIX]{x1D714}$ , a second dimensionless parameter naturally emerges: the dimensionless frequency $\unicode[STIX]{x1D6FA}$ (or Strouhal number) defined as:

(2.2) $$\begin{eqnarray}\unicode[STIX]{x1D6FA}=\frac{\unicode[STIX]{x1D714}R_{h}}{U_{M}}.\end{eqnarray}$$

The final goal of our study is to characterize the interaction of the jet with acoustic waves, and in the general case this calls for a description using compressible equations. However, in many situations, it is justified to consider that the flow is locally incompressible, and hence to assume a uniform density $\unicode[STIX]{x1D70C}$ of the fluid. This simplification is justified under two hypotheses. First, the Mach number $Ma=U_{M}/c_{0}$ based on the mean velocity of the jet must be small enough. Secondly, all length scales charactering the aperture (here, the only relevant one is the radius $R_{h}$ since the thickness is assumed to be zero) have to be small compared to the acoustic wavelength $\unicode[STIX]{x1D706}_{ac}=2\unicode[STIX]{x03C0}c_{0}/\unicode[STIX]{x1D714}$ . This latter hypothesis is often referred to in acoustic textbooks as the acoustic compactness hypothesis. In dimensionless terms, the hypothesis can be formulated as $\unicode[STIX]{x1D706}_{ac}/R_{h}=2\unicode[STIX]{x03C0}/(Ma\unicode[STIX]{x1D6FA})\gtrsim 10$ , so for the largest frequencies considered here ( $\unicode[STIX]{x1D6FA}\approx 6$ ) it is valid up to $Ma\approx 0.1$ . For a flow of air through a typical hole of radius $R_{h}=1~\text{mm}$ , the condition $Ma\approx 0.1$ corresponds to $Re\approx 4500$ , confirming the relevance of the range of parameters investigated in the present paper.

Figure 1. Sketch of the oscillating flow through a circular aperture in a thin plate.

Under these two hypotheses, it is justified to assume that, far away from the hole, the pressure levels in the upstream and downstream regions tend to uniform values noted respectively by $p_{in}(t)$ and $p_{out}(t)$ . In the harmonic regime, the upstream and downstream pressure levels as well as the flow rate will be expanded as:

(2.3) $$\begin{eqnarray}\left(\begin{array}{@{}c@{}}p_{in}(t)\\ p_{out}(t)\\ q(t)\end{array}\right)=\left(\begin{array}{@{}c@{}}P_{in}\\ P_{out}\\ Q\end{array}\right)+\unicode[STIX]{x1D700}\left(\begin{array}{@{}c@{}}p_{in}^{\prime }\\ p_{out}^{\prime }\\ q^{\prime }\end{array}\right)\text{e}^{-\text{i}\unicode[STIX]{x1D714}t}+\text{c.c.},\end{eqnarray}$$

where $\unicode[STIX]{x1D700}$ is a given amplitude of the harmonic perturbation and $\unicode[STIX]{x1D714}\in \mathbb{R}$ is the oscillation rate. The interaction of the jet with external systems can thus be characterized by the sole relationship between the pressure drop $[\,p_{in}^{\prime }-p_{out}^{\prime }]$ and the flow rate $q^{\prime }$ of the harmonic part.

Ultimately, if one wants to introduce the jet in the description of an acoustic system of much larger dimensions, the description (2.3) can be matched with an external solution derived from the equations of acoustics. Such a matching is not conducted here but examples will be given in a forthcoming paper.

2.2 Steady flow

The steady flow corresponding to the present situation is globally characterized by the mean pressure drop $[P_{in}-P_{out}]$ and the mean flow rate $Q$ . In the inviscid case, a classical model to relate these quantities was proposed by Levi-Civita and Prandtl. The model consists of a vortex sheet formed at the hole and surrounding the jet (see figure 1). After several diameters, the jet becomes parallel, but with a radius $R_{J}$ smaller than that of the hole. We classically call the ratio of surfaces $\unicode[STIX]{x1D6FC}=(\unicode[STIX]{x03C0}R_{J}^{2})/(\unicode[STIX]{x03C0}R_{h}^{2})$ the vena contracta coefficient. This coefficient is classically associated with the pressure loss across the aperture. Assuming a constant velocity $U_{J}$ inside the jet (see figure 1), the conservation of flux through the hole leads to $Q=\unicode[STIX]{x03C0}R_{J}^{2}U_{J}=\unicode[STIX]{x03C0}R_{h}^{2}U_{M}^{2}$ . Applying the Bernoulli theorem along streamlines passing through the hole thus leads to

(2.4) $$\begin{eqnarray}[P_{in}-P_{out}]=\frac{\unicode[STIX]{x1D70C}U_{J}^{2}}{2}=\frac{\unicode[STIX]{x1D70C}U_{M}^{2}}{2\unicode[STIX]{x1D6FC}^{2}},\end{eqnarray}$$

that links the pressure jump across the hole and the mean velocity (or flow rate) inside it. Theoretical inviscid calculations by Prandtl and Levi-Civita provided the value $\unicode[STIX]{x1D6FC}=0.5$ , that represents also the lower limit for this coefficient. Smith & Walker (Reference Smith and Walker1923), instead, estimated the vena contracta coefficient $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x03C0}/(2+\unicode[STIX]{x03C0})\approx 0.611$ for round inviscid jets discharging in open spaces. This value has been found to agree very well with experiments (Cummings & Eversman Reference Cummings and Eversman1983) and numerical calculations (Scarpato, Ducruix & Schuller Reference Scarpato, Ducruix and Schuller2011) at very high Reynolds number.

2.3 Unsteady flow: conductivity and impedance concepts

We now consider the relationship between the pressure jump and the flow rate in the unsteady case, under the hypothesis of harmonic perturbations (2.3). As explained in the introduction, the Rayleigh conductivity ( $K_{R}$ ) is defined as the proportionality coefficient between the acceleration of the fluid particles located within the hole and the pressure jump across the hole. More specifically,

(2.5) $$\begin{eqnarray}K_{R}=\frac{-\text{i}\unicode[STIX]{x1D714}\unicode[STIX]{x1D70C}q^{\prime }}{(p_{in}^{\prime }-p_{out}^{\prime })}.\end{eqnarray}$$

The conductivity is, in the general case, a complex quantity, and has the dimension of a length. Following Howe, it is classically noted $K_{R}=2R_{h}(\unicode[STIX]{x1D6FE}-\text{i}\unicode[STIX]{x1D6FF})$ . The real part $\unicode[STIX]{x1D6FE}$ represents the inertia of the system, while the imaginary part $\unicode[STIX]{x1D6FF}$ is directly related to the average value of the power absorbed by the hole. In effect, for harmonic perturbations described with the convention (2.2), the power is given by

(2.6) $$\begin{eqnarray}\langle \unicode[STIX]{x1D6F1}\rangle =\langle ([p_{in}^{\prime }-p_{out}^{\prime }]\text{e}^{-\text{i}\unicode[STIX]{x1D714}t}+\text{c.c.})(q^{\prime }\text{e}^{-\text{i}\unicode[STIX]{x1D714}t}+\text{c.c.})\rangle =2\text{Re}([p_{in}^{\prime }-p_{out}^{\prime }]\bar{q^{\prime }}),\end{eqnarray}$$

where the brackets $\langle \cdot \rangle$ represent the averaging over a complete period of oscillation $2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D714}$ , $\text{Re}$ means the real part and the overbar denotes the complex conjugate. Using the definition of the conductivity, this formula directly leads to

(2.7) $$\begin{eqnarray}\langle \unicode[STIX]{x1D6F1}\rangle =\frac{4R_{h}\unicode[STIX]{x1D6FF}}{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}}|p_{in}^{\prime }-p_{out}^{\prime }|^{2}.\end{eqnarray}$$

So, when $\unicode[STIX]{x1D6FF}>0$ , this term represents a resistance (or the ability to absorb acoustic energy), meaning that exciting the jet at a given frequency necessitates the provision of energy by an outer system.

As an alternative to the Rayleigh conductivity, we can also define the impedance of the aperture ( $Z_{h}$ ) as the ratio between the pressure jump and the flow rate:

(2.8) $$\begin{eqnarray}Z_{h}=\frac{(p_{in}^{\prime }-p_{out}^{\prime })}{q^{\prime }}\quad \left(=\frac{-\text{i}\unicode[STIX]{x1D714}\unicode[STIX]{x1D70C}}{K_{R}}\right)\!.\end{eqnarray}$$

The impedance is also a complex quantity, with physical dimension $\text{Mass}\cdot \text{Length}^{-2}\cdot \text{Time}^{-1}$ . In the following we decompose it as

(2.9) $$\begin{eqnarray}Z_{h}=\frac{\unicode[STIX]{x1D70C}U_{M}}{R_{h}^{2}}(Z_{R}+\text{i}Z_{I}),\end{eqnarray}$$

where $Z_{R}$ is the dimensionless resistance and $Z_{I}$ is the dimensionless reactance. It is easy to verify that the equation (2.6) for the power absorbed by the hole can be written as function of $Z_{R}$ as follows:

(2.10) $$\begin{eqnarray}\langle \unicode[STIX]{x1D6F1}\rangle =2\frac{\unicode[STIX]{x1D70C}U_{M}}{R_{h}^{2}}Z_{R}|q^{\prime }|^{2}.\end{eqnarray}$$

The Rayleigh conductivity and the impedance are conceptually and practically interchangeable quantities, and both have been used in the literature to characterize the interaction of a jet flow with acoustic fields. In the case of thin holes acting as a sound attenuators, most authors have used the conductivity as initially introduced by Howe. On the other hand, in cases where the jet can act as an energy source for external acoustic systems, leading to instabilities, it proves to be more convenient to employ the impedance (Fabre et al. Reference Fabre, Bonnefis, Charru, Russo, Citro, Giannetti and Luchini2014; Yang & Morgans Reference Yang and Morgans2016). In the present paper, we will use both concepts. A more detailed discussion of impedance-based instability criteria and a parametric study of the impedance of long holes will be given in a forthcoming paper.

2.4 The classical Rayleigh solution in the absence of mean flow

The problem initially solved by Rayleigh (Reference Rayleigh1945) is the simplest situation corresponding to the absence of mean flow. In this case, the problem admits an analytical solution under the framework of potential flow theory. This solution yields a direct proportionality between the flow acceleration and the pressure jump, namely

(2.11) $$\begin{eqnarray}(p_{in}^{\prime }-p_{out}^{\prime })=-\frac{\text{i}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}}{2R_{h}}q^{\prime }.\end{eqnarray}$$

The classical interpretation of this result is that the fluid in the vicinity of the hole behaves as a simple solid plug with mass $m=\unicode[STIX]{x1D70C}\unicode[STIX]{x03C0}R_{h}^{2}\ell _{eff}$ oscillating across the hole, where $\ell _{eff}$ is the equivalent length of the plug given by $\ell _{eff}=\unicode[STIX]{x03C0}R_{h}/2$ .

When reformulated in terms of conductivity (respectively impedance) and using the non-dimensionalization choices introduced in the previous section, the Rayleigh solution thus corresponds to $\unicode[STIX]{x1D6FE}=1;\unicode[STIX]{x1D6FF}=0$ (respectively $Z_{R}=0;Z_{I}=-\text{i}\unicode[STIX]{x1D6FA}/2$ ). An obvious consequence is that, under this model, the power absorbed by the hole predicted by (2.10) is exactly zero.

2.5 Review and criticism of Howe’s inviscid model

We now review and discuss in more detail the classical model of Howe already mentioned in the introduction. Howe models the jet as a cylindrical vortex sheet of constant radius $R_{h}$ formed at the rim of the aperture. He subsequently assumes a vorticity perturbation of this vortex sheet with the form

(2.12) $$\begin{eqnarray}\unicode[STIX]{x1D709}^{\prime }=\unicode[STIX]{x1D70E}H(x)\unicode[STIX]{x1D6FF}(r-R_{h})\exp [-\text{i}\unicode[STIX]{x1D714}(t-x/U_{c})],\end{eqnarray}$$

where $\unicode[STIX]{x1D6FF}$ and $H$ are respectively the Dirac and Heaviside functions, $U_{c}$ the assumed convection velocity of vorticity structures and $\unicode[STIX]{x1D70E}$ the amplitude of the vorticity perturbation. This later parameter is determined by imposing a Kutta condition (Crighton Reference Crighton1985), requiring finite velocity and pressure fluctuations at the rim of the hole. Starting from this point, and going through a series of very technical mathematical transformations, Howe was eventually able to predict the Rayleigh conductivity under the following analytical form:

(2.13) $$\begin{eqnarray}K_{R}=2R_{h}(\unicode[STIX]{x1D6FE}-\text{i}\unicode[STIX]{x1D6FF})=2R_{h}\left\{1+\frac{(\unicode[STIX]{x03C0}/2)I_{1}(\unicode[STIX]{x1D6FA}^{H})\text{e}^{-\unicode[STIX]{x1D6FA}^{H}}-\text{i}K_{1}(\unicode[STIX]{x1D6FA}^{H})\sinh (\unicode[STIX]{x1D6FA}^{H})}{\unicode[STIX]{x1D6FA}^{H}[(\unicode[STIX]{x03C0}/2)I_{1}(\unicode[STIX]{x1D6FA}^{H})\text{e}^{-\unicode[STIX]{x1D6FA}^{H}}+\text{i}K_{1}(\unicode[STIX]{x1D6FA}^{H})\cosh (\unicode[STIX]{x1D6FA}^{H})]}\right\}\!,\end{eqnarray}$$

where $I_{1}$ and $K_{1}$ are the order-one modified Bessel functions of respectively first and second kind and $\unicode[STIX]{x1D6FA}^{H}=\unicode[STIX]{x1D714}R_{h}/U_{c}$ is the Strouhal number.

Despite its mathematical rigor, a number of starting hypotheses of Howe’s model are questionable. The main caveats of the model can be summarized in four points:

1. (i) First, the study models the mean flow as a cylindrical vortex sheet with radius $R_{h}$ , and constant velocity $U_{M}$ , hence completely overlooks the vena contracta phenomenon discussed above. In a subsequent step of his analysis (p. 215 of his paper), Howe intended to incorporate partially this effect in his model, but this a posteriori modification remains imperfect.

2. (ii) Secondly, Howe’s model assumes that the perturbation affects only the strength of the vortex sheet but not its location, so that the perturbed vortex sheet is assumed to remain perfectly cylindrical. A better starting point would be to assume a vortex sheet with location given by (see figure 1):

   (2.14) $$\begin{eqnarray}r_{J}(r)=R_{J}+\unicode[STIX]{x1D700}\unicode[STIX]{x1D702}(x,r,t)=R_{J}+\unicode[STIX]{x1D700}\unicode[STIX]{x1D702}^{\prime }(r)\exp [\text{i}k(\unicode[STIX]{x1D714})x-\text{i}\unicode[STIX]{x1D714}t],\end{eqnarray}$$
   where $k(\unicode[STIX]{x1D714})=k_{r}+\text{i}k_{i}$ is a complex wavenumber which has to be determined as a function of the frequency $\unicode[STIX]{x1D714}$ . The inviscid stability analysis of this model is a classical problem whose solution can be found, for instance, in Batchelor & Gill (Reference Batchelor and Gill1962) or in Abid, Brachet & Huerre (Reference Abid, Brachet and Huerre1993). For completeness, this problem is reviewed in appendix A.

3. (iii) Thirdly, the starting point of Howe’s analysis (2.12) assumes that the perturbations are convected at a constant velocity $U_{c}$ which is assumed to be half the velocity of the jet. This choice is justified by analogy with the classical result for the Kelvin–Helmholtz instability of a planar vortex sheet. This choice is a questionable simplification and it would seem more rigorous to predict $U_{c}$ using spatial stability analysis of the cylindrical vortex sheet model, namely $U_{c}=k_{r}/\unicode[STIX]{x1D714}$ . This analysis shows that for small frequencies the convection velocity is actually closer to the velocity of the jet than half its value (see appendix A).

4. (iv) Finally, Howe completely ignores the fact that perturbations of the vortex sheet are spatially amplified in addition to being convected.

According to the two last criticisms, it would thus seem more appropriate to replace the starting point (2.12) by the following ansatz:

(2.15) $$\begin{eqnarray}\unicode[STIX]{x1D709}^{\prime }=\unicode[STIX]{x1D70E}H(x)\unicode[STIX]{x1D6FF}(r_{J}(r)-R_{J}-\unicode[STIX]{x1D700}\unicode[STIX]{x1D702}^{\prime }(r)\exp [\text{i}k(\unicode[STIX]{x1D714})x-\text{i}\unicode[STIX]{x1D714}t])\exp [\text{i}k(\unicode[STIX]{x1D714})x-\text{i}\unicode[STIX]{x1D714}t].\end{eqnarray}$$

We have not intended to reconstruct the whole analysis from this modified starting point, an option which would anyway not address the first criticism discussed above (vena contracta effect) and would remain limited to the high Reynolds number range. Instead, our chosen approach to address the problem is to compute the impedance (or alternatively the conductivity) through a global resolution of the linearized Navier–Stokes equations (LNSE) for given values of the Reynolds number.

3.1 General equations

Taking the diameter of the hole $D_{h}=2R_{h}$ as a length scale and the mean velocity $U_{M}$ as a velocity scale, the problem is governed by the axial–symmetric incompressible dimensionless Navier–Stokes equations:

(3.1) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0,\\ \displaystyle \unicode[STIX]{x2202}_{t}\boldsymbol{u}+(\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{u}+\unicode[STIX]{x1D735}p-\displaystyle \frac{1}{Re}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}=0,\end{array}\right\}\end{eqnarray}$$

where $\boldsymbol{u}(x,r,t)=(u_{x},u_{r})$ and $p$ is the reduced pressure. The variables $x$ and $r$ are respectively the axial and radial coordinate while $u_{x}$ and $u_{r}$ represent the axial and radial velocity components.

The flow is further decomposed into a base flow $(\boldsymbol{U},P)$ associated with the mean flux $Q$ and a harmonic perturbation $\unicode[STIX]{x1D700}(\boldsymbol{u}^{\prime },p^{\prime })\text{e}^{-\text{i}\unicode[STIX]{x1D714}t}$ associated with the oscillating flow rate $q^{\prime }\text{e}^{-\text{i}\unicode[STIX]{x1D714}t}$ . A crucial hypothesis in this treatment is that the amplitude of the harmonic perturbation is small, namely $\unicode[STIX]{x1D700}\ll 1$ . Inserting this decomposition into the Navier–Stokes equations (3.1) and linearizing, two different sets of partial differential equations are obtained:

1. (i) First, the leading order yields the base-flow equations:

   (3.2) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{U}=0,\\ \displaystyle (\boldsymbol{U}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{U}+\unicode[STIX]{x1D735}P-\frac{1}{Re}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{U}=0.\end{array}\right\}\end{eqnarray}$$
   The link between the base flow $(\boldsymbol{U},P)$ and the quantities $P_{in}$ , $P_{out}$ , $Q$ introduced in § 2 is given by the asymptotic matching conditions and flow rate definition as follows:
   (3.3) $$\begin{eqnarray}\displaystyle & \displaystyle P(x,r)\rightarrow P_{in}\quad \text{for }\sqrt{x^{2}+r^{2}}\rightarrow \infty \text{and }x<0, & \displaystyle\end{eqnarray}$$
   (3.4) $$\begin{eqnarray}\displaystyle & \displaystyle P(x,r)\rightarrow P_{out}\quad \text{for }\sqrt{x^{2}+r^{2}}\rightarrow \infty \text{and }x>0, & \displaystyle\end{eqnarray}$$
   (3.5) $$\begin{eqnarray}\displaystyle & \displaystyle \int _{{\mathcal{S}}}\boldsymbol{U}\boldsymbol{\cdot }\boldsymbol{n}\,\text{d}S=Q, & \displaystyle\end{eqnarray}$$
   where ${\mathcal{S}}$ is any surface traversed by the flow and $\boldsymbol{n}$ a normal unitary vector oriented in the direction of the flow.

2. (ii) Secondly, the $\unicode[STIX]{x1D700}$ -order yields the linearized Navier–Stokes equations (LNSE) governing the perturbation:

   (3.6) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}^{\prime }=0,\\ \displaystyle -\text{i}\unicode[STIX]{x1D714}\boldsymbol{u}^{\prime }+(\boldsymbol{U}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{u}^{\prime }+(\boldsymbol{u}^{\prime }\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{U}+\unicode[STIX]{x1D735}p^{\prime }-\frac{1}{Re}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}^{\prime }=0.\end{array}\right\}\end{eqnarray}$$
   The link with the quantities $p_{in}^{\prime }$ , $p_{out}^{\prime }$ , $q^{\prime }$ introduced in § 2 and allowing to define the impedance/conductivity is:
   (3.7) $$\begin{eqnarray}\displaystyle & \displaystyle p^{\prime }(x,r)\rightarrow p_{in}^{\prime }\quad \text{for }\sqrt{x^{2}+r^{2}}\rightarrow \infty \text{and }x<0, & \displaystyle\end{eqnarray}$$
   (3.8) $$\begin{eqnarray}\displaystyle & \displaystyle p^{\prime }(x,r)\rightarrow p_{out}^{\prime }\quad \text{for }\sqrt{x^{2}+r^{2}}\rightarrow \infty \text{and }x>0, & \displaystyle\end{eqnarray}$$
   (3.9) $$\begin{eqnarray}\displaystyle & \displaystyle \int _{{\mathcal{S}}}\boldsymbol{u}^{\prime }\boldsymbol{\cdot }\boldsymbol{n}\,\text{d}S=q^{\prime }. & \displaystyle\end{eqnarray}$$

Note that, as is customary when dealing with incompressible flows, the pressure is defined up to an arbitrary constant. We can choose this constant by setting $P_{out}=0$ and $p_{out}^{\prime }=0$ in (3.4) and (3.8), so that the mean pressure drop and fluctuating pressure drop are actually given by $[P_{in}-P_{out}]=P_{in}$ , $[p_{in}^{\prime }-p_{out}^{\prime }]=p_{in}^{\prime }$ .

With the addition of no-slip conditions $\boldsymbol{U}=\boldsymbol{u}^{\prime }=\mathbf{0}$ on the upstream and downstream surfaces of the plate (noted $\unicode[STIX]{x1D6E4}_{w}$ ) and symmetry conditions $U_{r}=u_{r}^{\prime }=0$ ; $\unicode[STIX]{x2202}U_{x}/\unicode[STIX]{x2202}r=\unicode[STIX]{x2202}u_{x}^{\prime }/\unicode[STIX]{x2202}r=0$ at the axis (noted $\unicode[STIX]{x1D6E4}_{axis}$ ), the set of (3.2)–(3.9) completely defines the nonlinear problem allowing us to compute the vena contracta coefficient $\unicode[STIX]{x1D6FC}$ and the linear problem allowing us to compute the impedance/conductivity.

In practice, the boundary conditions at $\sqrt{x^{2}+r^{2}}$ have to be imposed at the boundaries of a finite computational domain, both upstream and downstream. Treatment of these boundary conditions requires special attention and is detailed in the next sections.

Figure 2. Structure of the mesh $M_{1}$ obtained at the end of the mesh adaptation process, and nomenclature of boundaries. The mesh is adapted to both the base flow for $Re=1000$ and the harmonic perturbation for $\unicode[STIX]{x1D714}=2$ . The inset shows a zoom of the mesh structure in the range $X\in [-0.5;0.8]R_{h}$ and $R\in [0.5;1.3]R_{h}$ . Note that owing to the coordinate mapping, the actual dimension of the outlet domain is $[x_{max},r_{max}]=[1022+306i,337]$ .

3.2 Upstream domain

As sketched in figure 1, the upstream domain is expected to originate from an upstream container of large dimension, and sufficiently far away from the hole. Moreover, the flow is assumed to be radially convergent. However, in the numerical implementation, it is necessary to specify a given geometry for this upstream domain. Here, we chose to assume that the upstream region is a closed cavity of cylindrical section, with radius $R_{in}$ and length $L_{in}$ . The volumetric flux conditions (3.5) and (3.9) are imposed by assuming that both the base flow and the perturbation velocities are constant along the bottom of the cavity, noted $\unicode[STIX]{x1D6E4}_{in}$ (see figure 2), i.e.

(3.10) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\boldsymbol{U}=Q/S_{in}\boldsymbol{n}\\ \boldsymbol{u}^{\prime }=q^{\prime }/S_{in}\boldsymbol{n}\end{array}\right\}\quad \text{on }\unicode[STIX]{x1D6E4}_{in},\end{eqnarray}$$

where $S_{in}=\unicode[STIX]{x03C0}R_{in}^{2}$ is the area of the bottom wall. The values of $Q$ and $q^{\prime }$ have been selected in order to have a mean velocity equal to one into the hole, for both the base flow and the perturbation. The pressure levels $P_{in}$ and $p_{in}^{\prime }$ , which are required for the calculation of the mean pressure loss (and the vena contracta coefficient) and the impedance (or conductivity), are extracted by averaging along the inlet boundary:

(3.11) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle P_{in}=2\unicode[STIX]{x03C0}/S_{in}\int _{0}^{R_{in}}P(r)r\,\text{d}r\\ \displaystyle {p^{\prime }}_{in}=2\unicode[STIX]{x03C0}/S_{in}\int _{0}^{R_{in}}p^{\prime }(r)r\,\text{d}r\end{array}\right\}\quad \text{on }\unicode[STIX]{x1D6E4}_{in}.\end{eqnarray}$$

Since the upstream cavity used in our mesh definition is expected to represent an upper domain of infinite extend, its precise geometry has no real importance, but is dimension has to be large enough so that the results are independent of this geometry. In practice we verified that the choice $L_{in}=R_{in}=10R_{h}$ fulfils this condition. Finally, at the lateral wall of the cavity for $r=R_{in}$ (noted $\unicode[STIX]{x1D6E4}_{lat}$ ), we simply choose no-penetration ( $u_{r}=0$ ) and no-stress ( $\unicode[STIX]{x2202}_{r}u_{x}=0$ ) conditions for both base flow and perturbation. This condition ensures that the volumetric flux imposed at the bottom of the cavity effectively corresponds to the one traversing the hole, preventing the occurrence of an unphysical boundary layer that would be obtained using a no-slip condition.

3.3 Downstream domain: boundary conditions and change of coordinates

The treatment of the outlet boundary conditions is a delicate point here, as the structure of the perturbation leads to some difficulties, especially when the Reynolds number becomes large. In effect, due to the strongly spatially unstable nature of the jet, all perturbations are strongly amplified along the axial direction. In particular, the pressure field $p^{\prime }(x,r)$ can be reach huge levels (reaching $10^{15}$ or even more for $Re\approx 3000$ ) along the axis ( $r=0$ ) for large $x$ , and this conflicts with the necessity of imposing the boundary condition $p_{out}^{\prime }=0$ at a finite distance $x_{max}$ corresponding to the boundary of the computational domain.

To detail the origin of the problem and introduce the idea used to overcome it, let us review the classical modelling of the Kelvin–Helmholtz instability for a planar shear layer of zero thickness in the inviscid case. The formal derivation can be found in any classical textbook on hydrodynamical stability (see for example Drazin & Reid (Reference Drazin and Reid2004) or Charru (Reference Charru2011)). Consider as base flow a shear layer separating two regions of constant axial velocity, namely $u_{x}=U$ for $r<0$ and $u_{x}=0$ for $r>0$ . Now assume that the perturbation consists of a displacement of the shear layer with the form

(3.12) $$\begin{eqnarray}\unicode[STIX]{x1D702}(x,r,t)\propto \text{e}^{\text{i}kx-\text{i}\unicode[STIX]{x1D714}t},\end{eqnarray}$$

and assume a similar modal expansion for the velocity potential in the upper and lower regions. Matching the two regions at the interface leads to the classical dispersion relation:

(3.13) $$\begin{eqnarray}c\equiv \frac{\unicode[STIX]{x1D714}}{k}=\frac{1\pm \text{i}}{2}U.\end{eqnarray}$$

In a temporal stability framework, this means that a perturbation with a real wavenumber $k$ is convected downstream with a phase velocity $U/2$ and temporally amplified with a growth rate $Uk/2$ . On the other hand, in a spatial stability framework which is more relevant here, a perturbation with real frequency $\unicode[STIX]{x1D714}$ will be spatially amplified downstream with a complex wavenumber $k$ and will diverge at $x\rightarrow +\infty$ . This divergence forbids a global resolution of the function $\unicode[STIX]{x1D702}(x,t)$ when the variable $x$ is real. However, the problem disappears if we consider an analytical continuation of the function $\unicode[STIX]{x1D702}(x,t)$ with a complex variable $x$ . More specifically, as $\arg (k)=-\unicode[STIX]{x03C0}/4$ , the function $\unicode[STIX]{x1D702}(x,t)$ becomes convergent as soon as $|x|\rightarrow \infty$ in a direction of the complex plane verifying $\unicode[STIX]{x03C0}/4<\arg (x)<5\unicode[STIX]{x03C0}/4$ . These considerations suggest a possible way to overcome the problem, namely using a complex coordinate change $x={\mathcal{G}}_{x}(X)$ which maps a (real) numerical coordinate $X$ defined over a finite-size computational downstream domain $X\in [-L_{in};L_{out}]$ , onto the physical coordinate $x$ in such a way that it enters the complex plane and follows a direction where the perturbation is spatially damped.

The coordinate mapping effectively transforms the outlet location $X=L_{out}$ into a location $x=x_{max}={\mathcal{G}}_{x}(L_{out})$ located in the complex plane. In order for the boundary conditions at the outlet $X=L_{out}$ of the computational domain to best represent the physical boundary condition at $|x|\rightarrow \infty$ , it is desirable for $x_{max}={\mathcal{G}}_{x}(L_{out})$ to be as large as possible. This can be achieved using coordinate stretching in order to have short numerical domains and large physical ones.

Combining both ideas, namely stretching and complex mapping, we designed the following mapping function from numerical coordinate $X$ to physical coordinate $x$ :

(3.14) $$\begin{eqnarray}\displaystyle x={\mathcal{G}}_{x}(X) & = & \displaystyle \frac{X}{\displaystyle \left[1-\left(\frac{X}{L_{A}}\right)^{2}\right]^{2}}\left[1+\text{i}\unicode[STIX]{x1D6FE}_{c}\tanh \left(\frac{X}{2L_{C}}\right)^{2}\right]\quad \text{for }X>0,\nonumber\\ \displaystyle & = & \displaystyle X\quad \text{for }X<0.\end{eqnarray}$$

This function is characterized by three parameters which have the following interpretation. First, the parameter $L_{C}$ controls the transition range from real coordinate to complex coordinate. For $X\ll L_{C}$ the mapping is almost identity $({\mathcal{G}}_{x}(X)\approx X)$ so that the transition with the upstream, unmapped domain is as smooth as possible. For $X\approx L_{c}$ the imaginary part of the corresponding physical coordinate $x$ gradually increases. For $X\gg L_{c}$ the argument of $x$ asymptotes to a constant value, namely $\arg (x)\approx \tan ^{-1}(\unicode[STIX]{x1D6FE}_{c})$ . The third parameter $L_{A}$ controls the stretching effect associated with the coordinate mapping. This parameter has to be chosen so that $L_{A}>L_{out}$ . $L_{A}\rightarrow \infty$ means no coordinate stretching, so that the real part of $x_{max}$ is the same as the dimension $L_{out}$ of the computational domain, while if $L_{A}-L_{out}$ is small the corresponding $x_{max}$ is rejected very far away in the complex plane.

Finally, although the issue is less crucial with respect to the axial coordinate, we also used a mapping $r={\mathcal{G}}_{r}(R)$ to stretch the radial coordinate from $R\in [0,R_{out}]$ to $r\in [0,r_{out}]$ in order to enlarge the effective radial dimension of the physical domain. Here there is no point in using a complex deformation, so we used the following mapping function:

(3.15) $$\begin{eqnarray}\displaystyle r={\mathcal{G}}_{r}(R) & = & \displaystyle R_{M}+\frac{R-R_{M}}{\displaystyle \left[1-\left(\frac{R-R_{M}}{R_{A}-R_{M}}\right)^{2}\right]^{2}}\quad \text{for }X>0\text{ and }R>R_{A},\nonumber\\ \displaystyle & = & \displaystyle R\quad \text{otherwise}.\end{eqnarray}$$

This function leaves the radial coordinate unchanged in the region $r<R_{M}$ where the jet develops, but it stretches the limit of the domain from $R_{out}$ to $r_{out}={\mathcal{G}}_{r}(R_{out})$ which is very large as soon as $R_{A}$ is close to $R_{out}$ (with the constraint $R_{A}>R_{out}$ ).

Having explained this change of coordinates, it remains to specify the numerical boundary conditions effectively used at the boundaries of the numerical domain $R=R_{out}$ (corresponding to $r=r_{out}$ ) and $X=L_{out}$ (corresponding to $x=x_{max}$ ). In the framework of finite elements, the usual way to impose outlet conditions is to take advantage of integration by parts leading to the weak formulation. The most natural condition emerging in this way is the zero-traction condition, namely $-p\boldsymbol{n}+Re^{-1}\unicode[STIX]{x1D735}\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{n}=0$ . In the present case, we used the zero-traction condition as an approximation of the physical condition $p=0$ for both the base-flow and perturbation computations. This choice is justified if the viscous stresses are negligible in the vicinity of the boundaries of the domain, which turns out to be the case here.

We stress that using the present method, outflow boundary conditions are effectively applied at a location $x_{max}$ located in the complex plane. The validity of the method is not justified by rigorous mathematical argument, but only by the fact that it effectively works. Detailed validations are given in § B.1 of this paper. In particular, we show that at low Reynolds numbers results obtained with and without complex mapping are identical, and are independent of the precise choice of the parameters ( $\unicode[STIX]{x1D6FE}_{c},L_{C},L_{A}$ ) for the mapping function.

Note that the idea of using a complex coordinate mapping is not completely new. Indeed, the method is conceptually similar to the perfectly matching layer (PML) method, which is a numerical approach largely used in electromagnetics and acoustics to impose non-reflection boundary conditions in wave propagation problems (see Colonius (Reference Colonius2004) for a complete review). In stability studies of incompressible flows, complex coordinate mappings have also been used in linear problems involving a single spatial coordinate and characterized by a critical layer singularity (see for instance Fabre, Sipp & Jacquin Reference Fabre, Sipp and Jacquin2006) and mathematical theorems are available to justify how to choose the mapping as a function of the singularities of the problem (see for example Bender & Orszag (Reference Bender and Orszag2013)). However, we are not aware of any usage of such methods to get rid of convective amplifications (which is a different issue compared to reflexion of waves along boundaries and critical layer singularities). The usage of complex mappings for solving a nonlinear problem (i.e. computation of the base flow) involving two spatial coordinates is also totally new to our knowledge.

3.4 Numerical implementation

The numerical resolution of the problem was performed with a finite element method, using the FreeFem++ (http://www.freefem.org) open source code (Hecht (Reference Hecht2012)). The procedure follows the classical approach initially introduced by Sipp & Lebedev (Reference Sipp and Lebedev2007). The only notable originalities are the introduction of the complex mapping into the weak formulation, and the use of mesh adaptation using the adaptmesh command provided by the Freefem++ (see Fabre et al. Reference Fabre, Sabino, Citro, Bonnefis and Giannetti2018 for a demonstration of the efficiency of this method for solving linear and nonlinear problems arising from stability analysis).

In order to solve the problem, the base flow and perturbation equations have first to be expressed in terms of the mapped coordinates. The treatment of both sets of equations in very similar so we document only the base-flow equations. First, the spatial derivatives have to be modified as follows:

(3.16) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x2202}_{x}\equiv {\mathcal{H}}_{x}(X)\unicode[STIX]{x2202}_{X}\quad \text{with }{\mathcal{H}}_{x}=\frac{1}{\unicode[STIX]{x2202}_{X}{\mathcal{G}}_{x}(X)},\\ \displaystyle \unicode[STIX]{x2202}_{r}\equiv {\mathcal{H}}_{r}(R)\unicode[STIX]{x2202}_{R}\quad \text{with }{\mathcal{H}}_{r}=\frac{1}{\unicode[STIX]{x2202}_{R}{\mathcal{G}}_{r}(R)}.\end{array}\right\}\end{eqnarray}$$

The steady incompressible Navier–Stokes equations (3.2) thus take the following form:

(3.17) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle {\mathcal{H}}_{x}\unicode[STIX]{x2202}_{X}U_{x}+{\mathcal{H}}_{r}/r\unicode[STIX]{x2202}_{R}(rU_{r})=0,\\ \displaystyle \mathscr{C}\{U_{x}\}+{\mathcal{H}}_{x}\unicode[STIX]{x2202}_{X}P-\frac{1}{Re}[{\mathcal{H}}_{x}\unicode[STIX]{x2202}_{X}({\mathcal{H}}_{x}\unicode[STIX]{x2202}_{X}U_{x})+{\mathcal{H}}_{r}/r\unicode[STIX]{x2202}_{R}(r{\mathcal{H}}_{r}\unicode[STIX]{x2202}_{r}U_{x})]=0,\\ \displaystyle \mathscr{C}\{U_{r}\}+{\mathcal{H}}_{r}\unicode[STIX]{x2202}_{R}P-\frac{1}{Re}[{\mathcal{H}}_{x}\unicode[STIX]{x2202}_{X}({\mathcal{H}}_{x}\unicode[STIX]{x2202}_{X}U_{r})-U_{r}/r^{2}+{\mathcal{H}}_{r}/r\unicode[STIX]{x2202}_{R}(r{\mathcal{H}}_{r}\unicode[STIX]{x2202}_{r}U_{r})]=0,\end{array}\right\}\end{eqnarray}$$

where, from (3.15), $r={\mathcal{G}}_{r}(R)$ and $\mathscr{C}\{\cdot \}=U_{x}{\mathcal{H}}_{x}\unicode[STIX]{x2202}_{X}(\cdot )+U_{r}{\mathcal{H}}_{r}\unicode[STIX]{x2202}_{R}(\cdot )$ . The weak formulation is classically obtained by multiplying by test functions $[U_{x}^{+},U_{r}^{+},P^{+}]$ and integrating over the domain. Note that this integration has to be done over the physical domain, so in terms of the numerical variables the elementary volume of integration is $\text{d}V=2\unicode[STIX]{x03C0}r\,\text{d}r\,\text{d}x=2\unicode[STIX]{x03C0}({\mathcal{H}}_{x}{\mathcal{H}}_{r})^{-1}r\,\text{d}R\,\text{d}X\equiv 2\unicode[STIX]{x03C0}({\mathcal{H}}_{x}{\mathcal{H}}_{r})^{-1}{\mathcal{G}}_{r}(R)\,\text{d}R\,\text{d}X$ . After integration by parts of the pressure gradient and Laplacian terms of equation (3.17), we are thus lead to the following weak formulation of the mapped Navier–Stokes equations:

(3.18) $$\begin{eqnarray}\displaystyle & & \displaystyle -\int [U_{x}^{+}(U_{x}{\mathcal{H}}_{x}\unicode[STIX]{x2202}_{X}U_{x}+U_{r}{\mathcal{H}}_{r}\unicode[STIX]{x2202}_{R}U_{x})+U_{r}^{+}(U_{x}{\mathcal{H}}_{x}\unicode[STIX]{x2202}_{X}U_{r}+U_{r}{\mathcal{H}}_{r}\unicode[STIX]{x2202}_{R}U_{r})]\,\text{d}V\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\int [P({\mathcal{H}}_{x}\unicode[STIX]{x2202}_{X}U_{x}^{+}+{\mathcal{H}}_{r}\unicode[STIX]{x2202}_{R}U_{r}^{+}+U_{r}^{+}/r)-P^{+}({\mathcal{H}}_{x}\unicode[STIX]{x2202}_{X}U_{x}+{\mathcal{H}}_{r}\unicode[STIX]{x2202}_{R}U_{r}+U_{r}/r)]\,\text{d}V\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\frac{1}{Re}\int [{\mathcal{H}}_{x}^{2}\unicode[STIX]{x2202}_{X}U_{x}\unicode[STIX]{x2202}_{X}U_{x}^{+}+{\mathcal{H}}_{r}^{2}\unicode[STIX]{x2202}_{R}U_{x}\unicode[STIX]{x2202}_{R}U_{x}^{+}]\,\text{d}V\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\frac{1}{Re}\int [{\mathcal{H}}_{x}^{2}\unicode[STIX]{x2202}_{X}U_{r}\unicode[STIX]{x2202}_{X}U_{r}^{+}+{\mathcal{H}}_{r}^{2}\unicode[STIX]{x2202}_{R}U_{r}\unicode[STIX]{x2202}_{R}U_{r}^{+}+U_{r}U_{r}^{+}/r^{2}]\,\text{d}V\nonumber\\ \displaystyle & & \displaystyle \quad =0.\end{eqnarray}$$

Note that with this formulation, the no-traction boundary conditions at the outlet boundary, as well as the symmetry condition at the axis and the zero tangential stress condition at the lateral wall of the cavity are automatically satisfied thanks to the integration by parts. The other boundary conditions are imposed by penalization.

The LNSE (3.6) is treated in a similar way, leading to the following weak formulation:

(3.19) $$\begin{eqnarray}\displaystyle & & \displaystyle \text{i}\unicode[STIX]{x1D714}\int [U_{x}^{+}u_{x}^{\prime }+U_{r}u_{r}^{\prime }]\,\text{d}V\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\int [U_{x}^{+}(U_{x}{\mathcal{H}}_{x}\unicode[STIX]{x2202}_{X}u_{x}^{\prime }+u_{x}^{\prime }{\mathcal{H}}_{x}\unicode[STIX]{x2202}_{X}U_{x}+u_{r}^{\prime }{\mathcal{H}}_{r}\unicode[STIX]{x2202}_{R}U_{x}+U_{r}{\mathcal{H}}_{r}\unicode[STIX]{x2202}_{R}u_{x}^{\prime })]\,\text{d}V\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\int [+U_{r}^{+}(U_{x}{\mathcal{H}}_{x}\unicode[STIX]{x2202}_{X}u_{r}+u_{x}^{\prime }{\mathcal{H}}_{x}\unicode[STIX]{x2202}_{X}U_{r}+u_{r}^{\prime }{\mathcal{H}}_{r}\unicode[STIX]{x2202}_{R}U_{r}+U_{r}{\mathcal{H}}_{r}\unicode[STIX]{x2202}_{R}u_{r}^{\prime })]\,\text{d}V\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\int [p^{\prime }({\mathcal{H}}_{x}\unicode[STIX]{x2202}_{X}U_{x}^{+}+{\mathcal{H}}_{r}\unicode[STIX]{x2202}_{R}U_{r}^{+}+U_{r}^{+}/r)+P^{+}({\mathcal{H}}_{x}\unicode[STIX]{x2202}_{X}u_{x}^{\prime }+{\mathcal{H}}_{r}\unicode[STIX]{x2202}_{R}u_{r}^{\prime }+u_{r}^{\prime }/r)]\,\text{d}V\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{1}{Re}\int [{\mathcal{H}}_{x}^{2}\unicode[STIX]{x2202}_{X}u_{x}^{\prime }\unicode[STIX]{x2202}_{X}U_{x}^{+}+{\mathcal{H}}_{r}^{2}\unicode[STIX]{x2202}_{R}u_{x}^{\prime }\unicode[STIX]{x2202}_{R}U_{x}^{+}]\,\text{d}V\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{1}{Re}\int [{\mathcal{H}}_{x}^{2}\unicode[STIX]{x2202}_{X}u_{r}^{\prime }\unicode[STIX]{x2202}_{X}U_{r}^{+}+{\mathcal{H}}_{r}^{2}\unicode[STIX]{x2202}_{R}u_{r}^{\prime }\unicode[STIX]{x2202}_{R}U_{r}^{+}+u_{r}^{\prime }U_{r}^{+}/r^{2}]\,\text{d}V\nonumber\\ \displaystyle & & \displaystyle \quad =0.\end{eqnarray}$$

Once the weak formulation is written, the discrete matrices are assembled using classical Taylor–Hood $(P_{2},P_{2},P_{1})$ finite elements for the spatial discretization. The use of mesh adaptation to generate a efficient mesh is done in a way very similar to that explained in Fabre et al. (Reference Fabre, Sabino, Citro, Bonnefis and Giannetti2018). The procedure is as follows:

1. (i) we generate an initial coarse mesh using the Delaunay–Voronoi triangulation of the domain;

2. (ii) we use Newton iteration to compute a base flow at a moderate value of the Reynolds (for instance $Re=10$ );

3. (iii) we adapt the mesh to the base-flow solution of the previous step and recompute the base flow on the resulting mesh;

4. (iv) we repeat points (ii) and (iii) for gradually increasing values the Reynolds number up $Re=1000$ .

   After this stage, we are guaranteed to have a mesh yielding converged results for base-flow characteristics;

5. (v) we solve the linear problem for a value of $\unicode[STIX]{x1D714}$ in the range of interest, adapt the mesh to fit with the corresponding structure and recompute the base flow on the resulting mesh.

After this stage, we are ensured to have a mesh yielding converged results for both the base flow and the perturbation for a given $\unicode[STIX]{x1D714}$ . For even better efficiency, it is also possible to do the last mesh adaptation $(v)$ for two values of $\unicode[STIX]{x1D714}$ spanning the range of parameters in which converged results are expected.

To obtain the results presented in the next sections, two different meshes were designed in this way. The first mesh, noted $\mathbb{M}_{0}$ , is generated without the use of complex mapping, with a large domain corresponding to $L_{out}=x_{max}=80$ . This mesh was used to compute impedances at low Reynolds number (up to 1000) and to plot the base-flow characteristics. The second, noted $\mathbb{M}_{1}$ , uses complex mapping and was used for most results at larger Reynolds number values. The structure of this mesh $\mathbb{M}_{1}$ is illustrated in figure 2.

Additional meshes were designed for convergence tests and for demonstrations of the robustness of the complex mapping technique. Details are given in appendix B. The full characteristics of all meshes designed in this study (including $\mathbb{M}_{0}$ and $\mathbb{M}_{1}$ ) are given in table 3, in § B.2. We mention that the number of grid points $n_{t}$ in $\mathbb{M}_{1}$ is approximately half the value compared to mesh $\mathbb{M}_{0}$ .

5.1 Structure of the unsteady flow for $Re=500$

Let us now investigate the structure of the flow perturbation due to harmonic forcing. Figure 7 displays this structure for a moderate value of the Reynolds number, namely $Re=500$ , and for $\unicode[STIX]{x1D6FA}=3$ , computed in physical coordinates without mapping (mesh $\mathbb{M}_{0}$ ). As can be observed, the effect of a periodic forcing is to generate vortical structures in the jet which are amplified and convected in the downstream direction. In the case plotted, the maximum level is reached at approximately $x\approx 8$ . Progressing further downstream, the perturbations are no longer amplified but begin to slowly decrease, until they vanish for $x\gtrsim 20$ . This is consistent with the fact that for $x\gtrsim 8$ the shear layer bounding the jet has diffused (as documented in figure 3 b) and is not steep enough to sustain a spatial instability.

Figure 7. Harmonic perturbation for $\text{Re}=500,\unicode[STIX]{x1D6FA}=3$ computed in physical coordinates $(x,r)$ (mesh $\mathbb{M}_{0}$ ). Real part of the axial velocity component $u_{x}^{\prime }$ (upper) and vorticity $\unicode[STIX]{x1D709}^{\prime }$ (lower).

Figure 8. Harmonic perturbation for $\text{Re}=500,\unicode[STIX]{x1D6FA}=3$ (in physical coordinates $(x,r)$ ; mesh $\mathbb{M}_{0}$ ) on the axis of symmetry. Real (——) and imaginary (– –) part of the axial velocity component $u_{x}^{\prime }$ (thin lines) and pressure $p^{\prime }$ (thick lines).

In figure 8 we display the values taken along the axis of the jet of the axial velocity $u_{x}^{\prime }(x,0)$ and the pressure $p^{\prime }(x,0)$ associated with the harmonic perturbation previously described. One can clearly observe that the pressure perturbation asymptotes to different limit values in the upstream and downstream domains, allowing us to deduce the pressure jump $[p_{in}^{\prime }-p_{out}^{\prime }]$ which is the key parameter allowing us to define the impedance and/or the conductivity. In the upstream region the asymptote is reached rapidly and the pressure is almost constant with $p^{\prime }(x,0)\approx p_{in}^{\prime }=1.8-3i$ for $x\lesssim -3$ . On the other hand, in the downstream region, the asymptotic limit (which amounts to $p_{out}^{\prime }=0$ owing to the way the boundary conditions are introduced in the problem, see § 3) is only reached for $x\gtrsim 25$ , after the spatial growth and subsequent decay.

Figure 9. Harmonic perturbation for $\text{Re}=500,\unicode[STIX]{x1D6FA}=3$ (in numerical coordinates $(X,R)$ with complex mapping; mesh $\mathbb{M}_{1}$ ). Real part of the axial velocity component $u_{x}^{\prime }$ (upper) and vorticity (lower).

5.2 Efficiency of the complex mapping technique

Figure 9 displays the structure of the perturbation for the same parameters as in figure 7, but using the complex mapping technique (with mesh $\mathbb{M}_{1}$ ). Accordingly, the structure is plotted as function of the (real) numerical coordinates $[X,R]$ . As one can observe, the complex mapping technique completely fulfils the goal of getting rid of the strong spatial amplification in the downstream direction.

Of course, the spatial structure displayed in figure 9 has no physical meaning as soon as $X>0$ , because a point $(X,R)$ in the numerical domain corresponds to a point $(x,r)$ in the physical domain for some complex $x$ defined by $x=G(X)$ , and there is no easy way to access the structure of the perturbation for some real $x$ . However, since our focus is on the impedance of the jets, we are not interested in a full characterization of the perturbation field but only by a determination of the associated pressure jump.

Figure 10. (a) Pressure contours of the perturbation $p^{\prime }$ computed in physical coordinates (mesh $\mathbb{M}_{0}$ ; upper part) and with the complex mapping (mesh $\mathbb{M}_{1}$ ; lower part). (b) Pressure of the perturbation on the symmetry axis $p^{\prime }(X,0)$ with (——) and without (– –) the complex mapping ( $Re=500$ ; $\unicode[STIX]{x1D6FA}=3$ ).

Figure 11. Same as figure 10(a,b) but for $Re=1000$ . The inset (c) displays a zoom in the near hole region in order to catch the pressure jump across the hole.

Figure 10 compares the results obtained with and without complex mapping (again for $\text{Re}=500$ and $\unicode[STIX]{x1D6FA}=3$ ), focussing on the pressure component $p^{\prime }$ (real part) of the harmonic disturbance. In figure 10(a) the same iso-levels are used for both results without complex mapping (upper half) and with complex mapping (lower half). The comparison shows again that the structure computed without complex mapping quickly grows to reach large values, saturating the iso-levels, while the structure computed with complex mapping nicely decays to rapidly reach zero. Figure 10(b) complements the comparison with plots of the pressure field along the axis. The comparison confirms that in the inlet region ( $X<0$ ) the results exactly coincide. For $0<X\lesssim 1.25$ the results with complex mapping remain qualitatively similar to the ones without mapping while for $X\gtrsim 1.25$ they become completely different and rapidly decay to zero. This not surprising, as our definition the mapping function defining our mesh contains a parameter $L_{C}=1.25$ such that for $X<L_{C}$ the corresponding physical variable $x=G_{x}(X)$ is almost real while for $X>L_{C}$ it is fully complex.

Let us now consider the case $\text{Re}=1000$ , $\unicode[STIX]{x1D6FA}=3$ . The pressure $p^{\prime }$ of the harmonic disturbance is plotted in figure 11(a,b), with the same conventions as in figure 10. Inspection shows that the results without complex mapping lead to the same difficulties as for $Re=500$ but the spatial growth is much more pronounced. In this case, the pressure levels reach an amplitude of order $10^{3}$ for $x\approx 15$ and the asymptotic value $p_{out}^{\prime }=0$ is only reached for $x\gtrsim 70$ . This justifies that fact that, to correctly resolve this mode, we had to design the dimension of the mesh $\mathbb{M}_{0}$ to be as large as $L_{out}=80$ . On the other hand, results obtained with complex mapping behave very similarly as for $Re=500$ , and the asymptotic limit $p_{out}^{\prime }=0$ is reached quite rapidly, for $X\gtrsim 8$ .

In figure 11(b), the pressure levels of the results with and without complex mapping are so different that it is impossible to check that the pressure $p^{\prime }$ effectively asymptotes to the same limits $p_{in}^{\prime }$ and $p_{out}^{\prime }$ in both cases. To remedy this, figure 11(c) shows a zoom of the results in figure 11(b), in the region close to the hole. This representation confirms that the two computations lead to identical results in the upstream region where no mapping is used (and in particular that the upstream limit $p_{in}^{\prime }$ is the same), and that for the case using complex mapping the downstream limit is reached after only a few oscillations with amplitudes of order one.

Figures 10 and 11 thus confirm that the use of complex mapping is a convenient and efficient way to access to the pressure jump $p_{in}^{\prime }-p_{out}^{\prime }$ associated with the harmonic perturbation without having to deal with the strong spatial amplifications. It is worth pointing out that in this method is also computationally economical, as the number of point in mesh $\mathbb{M}_{1}$ is approximately half that of the unmapped case $\mathbb{M}_{0}$ . The figures also indicate that the difficulties encountered when solving the problem in physical coordinates without mapping become worse as the Reynolds number becomes large. In practice, as soon as $Re\gtrsim 1500$ , the huge levels reached by the perturbations in the far-field region lead to round-off errors and it becomes impossible to accurately resolve the near-field region. We verified that enlarging the domain of dimension $L_{out}$ to even larger than 80 does not improve the results. Using ‘sponge’ regions with artificially large viscosity was also tried as an alternative idea to get rid of the problems associated with huge spatial amplifications, but this idea proved to be unsuccessful. In the end the only efficient way we found to obtain reliable results for $Re\gtrsim 1500$ was to use the complex mapping technique. An illustration of the failure of the resolution in physical variables for large Reynolds is given in § B.1 (figures 20 and 21).

Note that in order to be consistent, the base flow also needs to be computed with the same numerical coordinates. The structure of the base flow in mapped coordinates has no physical meaning for the same reasons as the harmonic perturbation, but we verified that the pressure jump and the associated vena contracta coefficient are identical to the results in physical coordinates. We detail this in appendix C, and display an example of base flow obtained in such a way in figure 25.

5.3 Impedance and conductivity

Having illustrated the structure of the perturbation due to a harmonic forcing and justified the validity of the numerical method, we now come to the most important result of this work, namely prediction of the impedance as function of $Re$ and $\unicode[STIX]{x1D6FA}$ .

Figure 12. (a) Resistance $Z_{R}$ and (b) reactance $-Z_{I}/\unicode[STIX]{x1D6FA}$ for $Re=100$ (——), $Re=500$ (– –), $Re=1500$ (— ⋅ —), $Re=2000$ (— —) and $Re=3000$ (— ⋅ ⋅ —).

Figure 12 displays the real and imaginary parts of the impedance, calculated according to equation (2.8), as a function of the forcing frequency $\unicode[STIX]{x1D6FA}$ at various Reynolds numbers.

As for the resistance (panel $a$ ), only the case $Re=100$ is notably different from the other ones. On the other hand, the results obtained for $Re>1500$ seem to collapse in a unique curve, indicating that a large Reynolds number asymptotic regime is attained after this value. The resistance is maximum in the low-frequency limit $\unicode[STIX]{x1D6FA}\approx 0$ , with a value $Z_{R}\approx 0.85$ which will be explained in the next section. As $\unicode[STIX]{x1D6FA}$ is increased, $Z_{R}$ first decreases to reach a minimum for $\unicode[STIX]{x1D6FA}\approx 3.5$ and then it reaches a quasi-constant value equal to $0.53$ . Moreover, one can observe that the resistance increases with the Reynolds number for all values of the frequency $\unicode[STIX]{x1D6FA}$ . One can note that the resistance is always positive, meaning that, according to equation (2.10), the jet is an energy sink and so, in order to excite the jet at a given frequency, we need to provide energy from an outer system, as recently observed by Howe (Reference Howe1979) for an inviscid flow.

The reactance $Z_{I}$ is documented in panel ( $b$ ). As this quantity turns out to be a negative and approximately linear function of $\unicode[STIX]{x1D6FA}$ , it is more practical to plot $-Z_{I}/\unicode[STIX]{x1D6FA}$ as a function of $\unicode[STIX]{x1D6FA}$ . Under this representation, we can make the same observation as for the resistance, regarding the existence of a high Reynolds number asymptotic regime for $Re\gtrsim 1500$ , and the notable difference of the case $Re=100$ . Note that in the high-frequency range, the curves indicate an asymptotic trend given by $Z_{I}/\unicode[STIX]{x1D6FA}\approx 0.5$ . This value matches with that predicted by the simple Rayleigh model (§ 2.3), indicating that, for large frequencies, the impedance of the hole is at leading order the same as in the absence of a mean flow.

We now document the results using the equivalent concept of conductivity (see § 2.2), and compare these with the predictions of Howe’s model. Just as for the impedance, the results for $Re\gtrsim 1500$ collapse onto a single curve characterizing a high Reynolds number asymptotic regime. In figure 13 we plot with a thick line the results obtained for $Re=3000$ which are representative of this regime.

Figure 13. (a) Real part $\unicode[STIX]{x1D6FE}$ and (b) imaginary part $\unicode[STIX]{x1D6FF}$ of the Rayleigh conductivity. Plain lines: LNSE results for $\text{Re}=3000$ . Dash-dotted lines: Howe’s original model. Dotted lines: Howe’s modified model.

As explained in § 2.5, Howe’s expression for the conductivity (2.13) is expressed in terms of $\unicode[STIX]{x1D6FA}^{H}=\unicode[STIX]{x1D714}R_{h}/U_{c}$ where $U_{c}$ is the convection velocity of the structures along the vortex sheet, which choice is one of the most questionable points of the analysis. In the initial model, Howe disregarded the vena contracta phenomenon and assumed a jet with radius $R_{h}$ and velocity $U_{J}=U_{M}$ . Then, assuming $U_{c}=U_{M}/2$ leads to $\unicode[STIX]{x1D6FA}^{H}=2\unicode[STIX]{x1D6FA}$ . The values for $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D6FF}$ obtained with this choice are plotted in the figure with dash-dotted lines. As can be seen, in this initial version, Howe’s model rather badly agrees with our LNSE results. In a subsequent step of his analysis, Howe argued that the effect of the vena contracta can be partly accounted for by using the more appropriate choice $U_{J}=2U_{M}$ . This choice leads to $U_{c}=U_{M}$ and hence $\unicode[STIX]{x1D6FA}^{H}=\unicode[STIX]{x1D6FA}$ . The predictions of this modified model are plotted by the dashed lines in the figure. As can be seen, the agreement with LNSE results is improved but the curves still differ notably, especially as for the imaginary part $\unicode[STIX]{x1D6FF}$ (panel $b$ ) in the range $\unicode[STIX]{x1D6FA}\approx 2$ where Howe’s modified model underestimates the numerically computed one by approximately $30\,\%$ . On the other hand, the model overestimates the real part $\unicode[STIX]{x1D6FE}$ for $\unicode[STIX]{x1D6FA}\lesssim 2$ by approximately $10\,\%$ and underestimates it for $\unicode[STIX]{x1D6FA}\gtrsim 2$ by the same amount.

As discussed in § 2.5, the result of Howe is expressed in terms of a non-dimensional frequency $\unicode[STIX]{x1D6FA}^{H}=\unicode[STIX]{x1D6FA}R_{h}/U_{c}$ based on the convection velocity of vorticity structures along the vortex sheet $U_{c}$ , whose precise value is questionable. In figure 13 we followed the original choice of Howe $U_{c}=U_{M}$ which leads to $\unicode[STIX]{x1D6FA}^{H}=\unicode[STIX]{x1D6FA}$ . We also tried to compare the results using improved modellings of $U_{c}$ , leading only to mild ameliorations of the agreement.

Figure 14. Argument $\unicode[STIX]{x1D719}$ of the impedance, for $Re=100$ (——), $Re=500$ (– –), $Re=1500$ (— ⋅ —), $Re=2000$ (— —) and $Re=3000$ (— ⋅ ⋅ —), and from Howe’s modified model $(\cdots )$ .

Finally, a useful quantity which can be extracted from the impedance is the delay angle of the pressure with respect to the velocity:

(5.1) $$\begin{eqnarray}\unicode[STIX]{x1D719}=\text{arg}(Z_{h})=\tan ^{-1}\left(\frac{Z_{I}}{Z_{R}}\right)\!.\end{eqnarray}$$

This quantity has been used in a number of experiments, as it allows us to discriminate the cases where the impedance is mainly resistive ( $\unicode[STIX]{x1D719}\approx 0$ ) from the ones where it is mainly reactive ( $\unicode[STIX]{x1D719}\approx -\unicode[STIX]{x03C0}/2$ ). This quantity is plotted in figure 14, confirming that the behaviour switches from purely resistive to purely reactive as the frequency is increased. We also observe in this plot a collapse of the curves obtained in the high Reynolds number asymptotic regime $Re\gtrsim 1500$ . The angle $\unicode[STIX]{x1D719}$ extracted from Howe’s modified model is also plotted in the figure (note that in terms of conductivity, the definition of $\unicode[STIX]{x1D719}$ translates into $\unicode[STIX]{x1D719}=\unicode[STIX]{x03C0}/2-\text{arg}(K_{R})=-\text{tan}^{-1}(\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D6FF})$ ). Again, a substantial deviation is observed, especially in the range of intermediate frequencies $\unicode[STIX]{x1D6FA}\approx 2$ where the deviation can be as large as $\unicode[STIX]{x03C0}/12\equiv 15^{\circ }$ . Oddly, the inviscid Howe model turns out to give better predictions for the case $Re=100$ than for the high Reynolds number regime.

5.4 The quasi-static limit for $\unicode[STIX]{x1D6FA}\rightarrow 0$

We have observed that in the limit of small frequencies ( $\unicode[STIX]{x1D6FA}\rightarrow 0$ ), the impedance becomes purely real and tends to a constant value. This limit value can be predicted using a quasi-static approximation, and this property will be used to verify the consistency of our impedance calculations. As explained in § 2.3 for a steady flow, the pressure jump and the mean velocity across the hole are related through the Bernoulli equation which can be written in the form (2.4)

(5.2) $$\begin{eqnarray}\unicode[STIX]{x0394}p=\frac{\unicode[STIX]{x1D70C}u_{M}^{2}}{2\unicode[STIX]{x1D6FC}^{2}(Re_{M})}.\end{eqnarray}$$

Assuming $\unicode[STIX]{x0394}p=\unicode[STIX]{x0394}P+\unicode[STIX]{x0394}p^{\prime }$ and $u_{M}=U_{M}+u_{M}^{\prime }$ , inserting into (5.2) with $Re_{M}=(U_{M}+u_{M}^{\prime })R_{h}/\unicode[STIX]{x1D708}=Re(1+u_{M}^{\prime }/U_{M})$ and linearizing leads to

(5.3) $$\begin{eqnarray}\unicode[STIX]{x0394}P+\unicode[STIX]{x0394}p^{\prime }\approx \frac{\unicode[STIX]{x1D70C}U_{M}^{2}}{2\unicode[STIX]{x1D6FC}^{2}}+\frac{\unicode[STIX]{x1D70C}u_{M}^{\prime }U_{M}}{\unicode[STIX]{x1D6FC}^{2}}\left(1-\frac{1}{\unicode[STIX]{x1D6FC}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x2202}Re}\right)\!.\end{eqnarray}$$

Remembering now that $\unicode[STIX]{x0394}P=(\unicode[STIX]{x1D70C}U_{M}^{2})/(2\unicode[STIX]{x1D6FC}^{2})$ , this equation allows us to obtain a prediction for the impedance which is assumed to be valid in the quasi-static limit ( $\unicode[STIX]{x1D6FA}\rightarrow 0$ ):

(5.4) $$\begin{eqnarray}Z_{QS}=\frac{\unicode[STIX]{x0394}p^{\prime }}{\unicode[STIX]{x03C0}R_{h}^{2}u_{M}^{\prime }}=\frac{\unicode[STIX]{x1D70C}U_{M}}{\unicode[STIX]{x1D6FC}^{2}\unicode[STIX]{x03C0}R_{h}^{2}}\left(1-\frac{1}{\unicode[STIX]{x1D6FC}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}{\unicode[STIX]{x2202}Re}Re\right)\!.\end{eqnarray}$$

Table 1 compares the impedance computed using the method of the previous section for a small value of the frequency, namely $\unicode[STIX]{x1D6FA}=10^{-6}$ , to the quasi-static prediction (5.4) obtained using the base-flow characteristics computed in § 4. One can note that the results agree with a less than $1\,\%$ error. Finally, we can note that the term  $(1/\unicode[STIX]{x1D6FC})(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}/\unicode[STIX]{x2202}Re)Re$ in (5.4) is small because $\unicode[STIX]{x1D6FC}$ is a slowly varying function of $Re$ . The fourth column of table 1 gives the prediction of the quasi-static impedance obtained when neglecting this term. The comparison shows that this simplified prediction is still an excellent approximation, and slightly overestimates the actual value, except for the case $Re=100$ , where it underestimates it. This is consistent with the fact that the $\unicode[STIX]{x1D6FC}-Re$ curve reaches a maximum for $Re\approx 120$ (see figure 6).

The low-frequency limit was also addressed by Howe in the framework of his model. A Taylor series of expression (2.13) leads to $\unicode[STIX]{x1D6FF}\approx \unicode[STIX]{x03C0}\unicode[STIX]{x1D6FA}^{H}/4$ (equation 3.15(b) of Howe’s paper), which, when expressed in terms of impedance, translates into $Z_{R}\approx (2/\unicode[STIX]{x03C0})(U_{c}/U_{M})\approx 0.637(U_{c}/U_{M})$ . Thus, the choice $U_{c}/U_{M}=1$ made by Howe actually yields a prediction for $Z_{R}$ which underestimates the high Reynolds number value by approximatively $37\,\%$ . Note that this mismatch can also be observed in figure 13(b) regarding the initial slope of the curve $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D6FA})$ . This error in the quasi-static limit may be cancelled using an ad hoc choice of $U_{c}/U_{M}$ , but as previously explained, such a modification does not improve substantially the agreement in other ranges of  $\unicode[STIX]{x1D6FA}$ .

Table 1. Values of the impedance in the low-frequency range. Comparison of values obtained numerically with a very small $\unicode[STIX]{x1D6FA}$ , quasi-static approximation (5.4) and simplified approximation obtained assuming $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}/\unicode[STIX]{x2202}Re=0$ .