Reflection and transmission of a Kelvin–Helmholtz wave incident on a shock in a jet

Abstract

Screech tones in supersonic jets are underpinned by resonance between downstream-travelling Kelvin–Helmholtz waves and upstream-travelling acoustic waves. Specifically, recent works suggest that the relevant acoustic waves are guided within the jet and are described by a discrete mode of the linearised Euler equations. However, the reflection mechanism that converts downstream-travelling waves into upstream-travelling waves, and vice versa, has not been thoroughly addressed, leading to missing physics within most resonance models. In this work, we investigate the reflection and transmission of waves generated by the interaction between a Kelvin–Helmholtz wave and a normal shock in an under-expanded jet using a mode-matching approach. Both vortex-sheet and finite-thickness shear-layer models are explored, quantifying the impact of the shear layer in the reflection process. This approach could enable more quantitative predictions of resonance phenomena in jets and other fluid systems.

1. Introduction

Shock-containing shear flows involve a rich variety of phenomena including shock–turbulence interaction (STI). In free shear layers, STI leads to an increase in turbulence levels and mixing downstream of the shock (Génin & Menon 2010). In wall-bounded flows, in addition to the enhancement of turbulence, STI may also be accompanied by boundary layer separation and the formation of a separation bubble (Delery 1983; Dolling 2001; Clemens & Narayanaswamy 2014). The STI is also an important feature of supersonic combustion in scramjets (Yang, Kubota & Zukoski 1993). In imperfectly expanded propulsive jets, STI underpins the generation of broad-band shock associated noise (Tanna 1977; Tam & Tanna 1982) and screech (Powell 1953; Tam, Seiner & Yu 1986; Raman 1999; Edgington-Mitchell 2019).

Linear theory has been widely used to study the interaction between disturbance fields and shocks. Ribner (1954) considered the interaction between a vorticity wave and a normal shock. The analysis was later extended to consider STI, where a homogeneous turbulence was modelled as a superposition of Fourier vorticity waves (Ribner 1955). Moore (1954) considered the interaction between sound waves and an oblique shock, and this work was extended by Mahesh et al. (1995) to study an isotropic field of acoustic disturbances interacting with a shock. Later, Mahesh, Lele & Moin (1997) considered the influence of entropy fluctuations on STI as well and Buttsworth (1996) derived expressions for shock-induced vorticity, useful for the estimation of mixing enhancement. The foregoing studies were all based on solution of the Rankine–Hugoniot relations. The unsteady STI was converted into an equivalent steady-flow problem which did not consider the reflection process associated with the incident turbulent disturbance but only the transmission mechanism through the shock wave. A review of these studies and others has been compiled by Andreopoulos, Agui & Briassulis (2000). More recently, Kitamura et al. (2016) used rapid distortion theory to study the interaction between homogeneous isotropic turbulence and a shock wave, and Chen & Donzis (2019) considered STI at high turbulence intensities. Similarly to the works reported above, the authors mainly focused on the turbulence amplification and modification of the turbulence length scales downstream of the shock.

The reflection and transmission of acoustic, vorticity and entropy waves within a convergent–divergent nozzle with and without a shock was studied by Marble & Candel (1977). The study focused on compact disturbances, that is, with wavelengths larger than the nozzle length, thus limiting the application to low frequencies. The inclusion of non-compactness effects was considered by Stow, Dowling & Hynes (2002), who provided a first-order correction for the phase of the reflection coefficient for higher frequencies. The correction was extended to the transmission coefficient also by Goh & Morgans (2011) and a further development was carried out by Duran & Moreau (2013) and Duran & Morgans (2015), who extended the high-frequency correction to the amplitude of the reflection and transmission coefficients in the case of planar and circumferential incident waves, respectively. All of these works were motivated by the problem of combustion noise and the role the reflected waves play on the onset of thermo-acoustic instability in the combustion chamber of the burner–turbine–nozzle configuration of an aero-engine.

The problem we consider is motivated by the sound generated by imperfectly expanded, supersonic jets and, in particular, the phenomenon known as screech, a mechanistic explanation for which was first provided by Powell (1953). The mechanism involves turbulent structures that are convected through the shock-cell structure; this STI results in the generation of upstream-travelling sound waves. According to Powell's phenomenological description, when the phases of the upstream-travelling sound waves and downstream-travelling turbulent structures are suitably matched, at the jet exit plane and at the STI locations, resonance may occur. The downstream-travelling turbulent structures considered important for screech are what are often referred to as coherent structures.

A large body of recent work has shown how coherent structures in turbulent jets, and the sound they produce, can be modelled using linear theory (Jordan & Colonius 2013; Schmidt et al. 2017; Towne, Schmidt & Colonius 2018; Cavalieri, Jordan & Lesshafft 2019; Lesshafft et al. 2019; Nogueira et al. 2019; Edgington-Mitchell et al. 2021a). As shown in these studies, downstream-travelling coherent structures are largely underpinned by Kelvin–Helmholtz (K–H) instability. Powell (1953) assumed that the upstream-travelling waves responsible for the feedback mechanism in screech generation were free-stream acoustic waves, but this has been recently questioned. Shen & Tam (2002) suggested that the upstream-travelling disturbance might comprise a family of guided jet modes, first discussed by Tam & Hu (1989). This hypothesis has been recently confirmed in studies by Gojon, Bogey & Mihaescu (2018) and Edgington-Mitchell et al. (2018), and a simplified screech-tone prediction model based on this idea has been developed and validated by Mancinelli et al. (2019a). In the simplest formulation of the screech-tone model, the spatial growth of the K–H mode is ignored, and a phase-matching criterion is sufficient to provide a reasonable prediction of screech-tone frequencies. A similar resonant mechanism was proposed for subsonic compressible jets (Towne et al. 2017), cavity flows (Rossiter 1964; Rowley, Colonius & Basu 2002) and impinging jets (Tam & Ahuja 1990; Bogey & Gojon 2017). The reflection of waves is implicitly considered in all these mechanisms, but it is rarely studied in detail. In more complete screech-frequency prediction models (Mancinelli et al. 2019b, 2021), where the spatial growth rates of the upstream- and downstream-travelling waves are included, knowledge of the reflection coefficients in the jet exit plane and at the location of STI is required.

In this paper, we investigate the interaction between a downstream-travelling K–H wave and a normal shock and compute the amplitude and phase of the reflected upstream-travelling guided wave active in the screech loop using a mode-matching approach. We consider vortex-sheet (V-S) and finite-thickness (F-T) flow models, which elucidate the role of shear in the reflection and transmission processes. The efficiency of the mode-matching technique in the presence of a discontinuity, as is the shock in the flow we consider herein, has been already shown by Gabard & Astley (2008) for the estimation of the sound attenuation in a lined duct. More recently, a mode-matching approach has been used by Dai (2020, 2021) to calculate the reflection and transmission coefficients in a duct flow in the presence of a cavity. Consistent with these works, we use linear theory to describe the flow dynamics upstream and downstream of the shock and then match the solutions across the discontinuity.

The paper is organised as follows. The general modelling framework, including the jet models adopted and the mode-matching approach used to calculate the reflection and transmission coefficients, is presented in § 2. Results involving the reflection-coefficient calculation, its dependence on the frequency and jet-flow conditions and the reflected and transmitted pressure fields are presented and discussed in § 3. The paper closes with concluding remarks in § 4.

2. Modelling framework

We here present the shock and jet-dynamics modelling and the procedure adopted to calculate the reflection and transmission coefficients. We consider an axisymmetric, shock-containing supersonic jet. It is known that the organised structure of the jet plume of imperfectly expanded supersonic jets is shaped by oblique shocks and expansion waves (see the many flow visualisations presented in Powell 1953; Powell, Umeda & Ishii 1992; Panda 1999; Mercier, Castelain & Bailly 2017). Normal shocks are, however, found in the form of Mach disks for highly imperfectly expanded jets and often encountered in the jet plume of convergent–divergent nozzles and impinging jets (Edgington-Mitchell 2019). Despite many years of research activity (see the many works by Powell 1953; Tam & Tanna 1982; Suzuki & Lele 2003; Lele 2005; Edgington-Mitchell et al. 2021b), a clear picture of the way by which instability waves interact with shock cells to generate screech is still far from being reached. With the aim of keeping the model as simple as possible and in the absence of a clear and unambiguous description of the interaction between instability and shock waves, the shock is herein assumed to be normal, thus allowing the use of the locally parallel-flow assumption both upstream and downstream of the shock. This assumption implies that the linear response to shock oscillations induced by the incoming instability wave, which is a typical non-parallel feature, is not considered. A sketch of the shock-containing jet and the cylindrical reference system used in this paper are depicted in figure 1. We consider a K–H wave with unitary amplitude, $I=1$, incident to a shock. The interaction of the incoming wave with the shock generates a collection of reflected and transmitted modes upstream and downstream of the shock, respectively. The sections upstream and downstream of the shock are hereinafter denoted 1 and 2 and the reflection and transmission coefficients of each wave moving away from the shock are indicated with $R_{n_R}$ and $T_{n_T}$, respectively. The state vector is $\boldsymbol {q}^*=\lbrace \rho ^*, u_x^*, u_r^*, u_\theta ^*, T^*, p^*\rbrace$, where $\rho$ is the flow density, u the velocity, T the temperature and p the pressure. The flow variables are normalised by the nozzle diameter $D$ and the ambient density and speed of sound $\rho _\infty$ and $c_\infty$, respectively, thus leading to a non-dimensional state vector $\boldsymbol {q}$.

Figure 1. Schematic representation of the jet model: (a) sketch of the shock-containing jet, (b) control volume representation with identification of the normal to the inlet and outlet surfaces.

2.1. Shock model

The flow regions upstream and downstream of the shock are well described by a locally parallel model. In order to connect these two regions, we impose the conservation laws of mass, momentum and energy through the shock. This is done by dividing the shock into infinitesimal control volumes $\mathrm {d}V$ of length ${\rm \Delta} x\to 0$ such that the flux terms through the top and bottom surfaces are zero (see figure 1) and enforcing mass, momentum and energy conservation for the control volume, leading to the system of equations

(2.1a–c)\begin{gather}\left.\begin{aligned} &\int_{S_1}\rho^*\boldsymbol{u}^*\boldsymbol{\cdot}\boldsymbol{n}\,\mathrm{d}S + \int_{S_2}\rho^*\boldsymbol{u}^*\boldsymbol{\cdot}\boldsymbol{n}\,\mathrm{d}S=0,\\ &\int_{S_1}\rho^*\boldsymbol{u}^*\left(\boldsymbol{u}^*\boldsymbol{\cdot}\boldsymbol{n}\right)\mathrm{d}S + \int_{S_1} p^*\boldsymbol{n}\,\mathrm{d}S + \int_{S_2}\rho^*\boldsymbol{u}^* \left(\boldsymbol{u}^*\boldsymbol{\cdot}\boldsymbol{n}\right)\mathrm{d}S - \int_{S_2} p^*\boldsymbol{n}\,\mathrm{d}S=0,\\ &\int_{S_1}\rho^* \,{\rm e}^*\boldsymbol{u}^*\boldsymbol{\cdot}\boldsymbol{n}\,\mathrm{d}S + \int_{S_2}\rho^* \,{\rm e}^*\boldsymbol{u}^*\boldsymbol{\cdot}\boldsymbol{n}\,\mathrm{d}S=0, \end{aligned}\right\} \end{gather}

where ${\rm e}^*=h^*+0.5(u_x^{*2}+u_r^{*2}+u_\theta ^{*2})$ is the total specific energy with the enthalpy expressed as $h^*=c_pT^*$, $c_p$ is the specific heat capacity, S is the control volume surface and n the normal to the surface. Normalising the flow variables, performing the Reynolds decomposition,

(2.2)\begin{equation} \boldsymbol{q}\left(x,r,\theta,t\right)=\bar{\boldsymbol{q}}\left(r\right)+ \boldsymbol{q}'\left(x,r,\theta,t\right), \end{equation}

and substituting into (2.1a), removing the mean and linearising, the linearised jump equations for the shock become

(2.3a–e)\begin{gather}\left.\begin{aligned} &\bar{u}_{1x}\rho_1+\bar{\rho}_1u_{1x}=\bar{u}_{2x} \rho_2+\bar{\rho}_2u_{2x},\\ &p_1+2\bar{\rho}_1\bar{u}_{1x}u_{1x}+\bar{u}_{1x}^2 \rho_1=p_2+2\bar{\rho}_2\bar{u}_{2x}u_{2x}+\bar{u}_{2x}^2\rho_2,\\ &u_{1r}=u_{2r},\\ &u_{1\theta}=u_{2\theta},\\ &T_1+\bar{u}_{1x}u_{1x}=T_2+\bar{u}_{2x}u_{2x}, \end{aligned}\right\} \end{gather}

where we removed the primes from the fluctuating variables for notational simplicity. We note that the adiabatic relation between the thermodynamic variables is implicit in the linearised operator. The perturbations upstream and downstream of the shock are modelled using the normal mode ansatz

(2.4)\begin{equation} \boldsymbol{q}'\left(x,r,\theta,t\right)=\hat{\boldsymbol{q}} \left(r\right)\exp({{\rm i}\left(kx+m\theta -\omega t\right)}), \end{equation}

where $k$ is the wavenumber along the axial direction, $m$ is the azimuthal order and $\omega =2{\rm \pi} St M_a$ is a non-dimensional frequency, with $St=fD/U_j$ the nozzle-diameter-based Strouhal number, $M_a=U_j/c_\infty$ the acoustic Mach number, f the frequency, D the nozzle diameter and $U_j$ the fully expanded jet velocity. Considering that the only incident wave is the K–H wave, the system of (2.3a) can be written in a compact form as follows:

(2.5) \begin{equation} \boldsymbol{\mathsf{A}}_1\left(I\hat{\boldsymbol{q}}_{1I}+ \sum_{n_R=1}^\infty R_{n_R}\hat{\boldsymbol{q}}_{1R,n_R}\right)= \boldsymbol{\mathsf{A}}_2\sum_{n_T=1}^\infty T_{n_T}\hat{\boldsymbol{q}}_{2T,n_T}, \end{equation}

where $\hat {\boldsymbol {q}}_{1I}$, $\hat {\boldsymbol {q}}_{1R,n_R}$ and $\hat {\boldsymbol {q}}_{2T,n_T}$ are the incident, reflected and transmitted waves moving upstream and downstream of the shock, respectively, and

(2.6a,b)\begin{equation} \boldsymbol{\mathsf{A}}_1=\begin{bmatrix} \bar{u}_{1x} & \bar{\rho}_1 & 0 & 0 & 0 & 0\\ \bar{u}_{1x}^2 & 2\bar{\rho}_1\bar{u}_{1x} & 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0\\ 0 & \bar{u}_{1x} & 0 & 0 & 1 & 0 \end{bmatrix},\quad \boldsymbol{\mathsf{A}}_2= \begin{bmatrix} \bar{u}_{2x} & \bar{\rho}_2 & 0 & 0 & 0 & 0\\ \bar{u}_{2x}^2 & 2\bar{\rho}_2\bar{u}_{2x} & 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0\\ 0 & \bar{u}_{2x} & 0 & 0 & 1 & 0 \end{bmatrix} \end{equation}

are matrices containing information about the mean flow upstream and downstream of the shock. The vector of eigenfunctions $\hat {\boldsymbol {q}}_{1I}$, $\hat {\boldsymbol {q}}_{1R,n_R}$ and $\hat {\boldsymbol {q}}_{2T,n_T}$ are computed using either a V-S or F-T model (see § 2.3). The procedure used to ascertain whether a wave is reflected or transmitted is described in § 2.3.3.

2.2. Reflection- and transmission-coefficient calculation

Equation (2.5) is exact if a complete basis of jet modes is considered. In order to estimate the reflection- and transmission-coefficient values, we truncate the sum to a finite number of modes $N_R$ and $N_T$ and we introduce an error density $\epsilon (r)$ for each conservation equation. Equation (2.5) can then be written as

(2.7)\begin{equation} \boldsymbol{\mathsf{A}}_1\left(I\hat{\boldsymbol{q}}_{1I}+\sum_{n_R=1}^{N_R} R_{n_R} \hat{\boldsymbol{q}}_{1R,n_R}\right)-\boldsymbol{\mathsf{A}}_2\sum_{n_T=1}^{N_T} T_{n_T} \hat{\boldsymbol{q}}_{2T,n_T}=\boldsymbol{\epsilon}, \end{equation}

where $\boldsymbol {\epsilon }(r)$ is the error density vector and $\boldsymbol {\epsilon }\to 0$ if the number of modes $N_R$ and $N_T\to \infty$. The reflection and transmission coefficients $R_{n_R}$ and $T_{n_T}$ associated with each mode are estimated by a least-mean-square minimisation of the error densities, formalised as

(2.8)\begin{equation} \left[R_{opt,n_R},T_{opt,n_T}\right]=\underset{R_{n_R},T_{n_T} \in \mathcal{C}}{\text{min}}\underbrace{\int_0^\infty\vert\boldsymbol{\epsilon}^2\left(r\right)\vert\,\mathrm{d}r}_{F}. \end{equation}

The objective function $F$ corresponds to the sum of the absolute value of the squared error densities associated with each conservation equation integrated along the radial direction. To solve the minimisation problem in (2.8), we write (2.7) in a matrix form

(2.9)\begin{equation} \begin{bmatrix} \boldsymbol{\mathsf{A}}_1\hat{\boldsymbol{\mathsf{Q}}}_{1R}\quad -\boldsymbol{\mathsf{A}}_2\hat{\boldsymbol{\mathsf{Q}}}_{2T} \end{bmatrix} \begin{Bmatrix} \boldsymbol{R}\\ \boldsymbol{T} \end{Bmatrix} ={-}I\boldsymbol{\mathsf{A}}_1\hat{\boldsymbol{q}}_{1I}+\boldsymbol{\epsilon}, \end{equation}

where $\hat {\boldsymbol{\mathsf{Q}}}_{1R}=[\hat {\boldsymbol {q}}_{1R,1},\ldots,\hat {\boldsymbol {q}}_{1R,N_R}]$ and $\hat {\boldsymbol{\mathsf{Q}}}_{2T}=[\hat {\boldsymbol {q}}_{2T,1},\ldots,\hat {\boldsymbol {q}}_{2T,N_T}]$ are the matrices of the eigenfunctions and $\boldsymbol {R}$ and $\boldsymbol {T}$ are the vectors of the reflection and transmission coefficients, respectively. We then define

(2.10)\begin{equation} \begin{Bmatrix} \boldsymbol{R}\\ \boldsymbol{T} \end{Bmatrix} = \boldsymbol{X},\quad \begin{bmatrix} \boldsymbol{\mathsf{A}}_1\hat{\boldsymbol{\mathsf{Q}}}_{1R} \quad -\boldsymbol{\mathsf{A}}_2\hat{\boldsymbol{\mathsf{Q}}}_{2T} \end{bmatrix} =\boldsymbol{\mathsf{B}},\quad \boldsymbol{y}=\boldsymbol{\mathsf{A}}_1\hat{\boldsymbol{q}}_{1I}I, \end{equation}

so that (2.9) can be written in the compact form $\boldsymbol {\epsilon }=\boldsymbol {B}\boldsymbol {X}+\boldsymbol {y}$. The solution of (2.8) is obtained by finding the stationary point of $F$ by setting its gradient to zero

(2.11)\begin{equation} \frac{{\rm d} F}{{\rm d}\kern0.06em\boldsymbol{X}}=\frac{{\rm d}\left(\boldsymbol{\epsilon^T} \boldsymbol{W}\boldsymbol{\epsilon}\right)}{{\rm d}\kern0.06em\boldsymbol{X}} = \boldsymbol{B}^{\rm T}\boldsymbol{W}\boldsymbol{B}\boldsymbol{X}+ \boldsymbol{B}^{\rm T}\boldsymbol{\mathsf{W}}\boldsymbol{y}=0, \end{equation}

which leads to

(2.12)\begin{equation} \boldsymbol{X}={-}\left(\boldsymbol{B}^{\rm T}\boldsymbol{W}\boldsymbol{B}\right)^{{-}1} \boldsymbol{B}^{\rm T}\boldsymbol{W}\boldsymbol{y}, \end{equation}

where $\boldsymbol {B}\vert _W^+=(\boldsymbol {B}^{\rm T}\boldsymbol {W}\boldsymbol {B})^{-1}\boldsymbol {B}^{\rm T}\boldsymbol {W}$ is the weighted pseudo-inverse matrix and $\boldsymbol {W}$ is a diagonal matrix of elements $dr$.

2.3. Jet models

We here present a local description of the jet dynamics using the parallel-flow linear stability theory. This theory is applied to two different models: a F-T flow model and a simplified cylindrical vortex sheet. Both are governed by the linearised Euler equations (LEE) (see Appendix A).

2.3.1. Finite-thickness model

Writing the LEE exclusively in terms of pressure, the compressible Rayleigh equation (Schmid & Henningson 2001)

(2.13)\begin{align} & \frac{\partial^2\hat{p}}{\partial r^2}+\left(\frac{1}{r}-\frac{2k}{\bar{u}_xk-\omega} \frac{\partial\bar{u}_x}{\partial r}-\frac{\gamma -1}{\gamma\bar{\rho}} \frac{\partial\bar{\rho}}{\partial r}+\frac{1}{\gamma\bar{T}} \frac{\partial\bar{T}}{\partial r}\right) \frac{\partial\hat{p}}{\partial r} \nonumber\\ &\quad -\left(k^2+\frac{m^2}{r^2}- \frac{\left(\bar{u}_xk-\omega\right)^2}{\left(\gamma -1\right)\bar{T}}\right)\hat{p}=0, \end{align}

is obtained, where $\gamma$ is the specific heat ratio for a perfect gas. The solution of the linear stability problem is obtained by specifying a real or complex frequency $\omega$ and solving the resulting augmented eigenvalue problem $k=k(\omega )$, with $\hat {p}(r)$ the associated pressure eigenfunction. The eigenvalue problem is solved numerically by discretising (2.13) in the radial direction using the Chebyshev polynomials and by imposing Dirichlet boundary conditions at the domain boundaries. A mapping function proposed by Lesshafft & Huerre (2007) is used to non-uniformly distribute the 500 grid points to efficiently resolve the shear layer of the jet and to ensure convergence of the computed eigenmodes. The eigenfunctions $\hat {u}_i$, $\hat {\rho }$ and $\hat {T}$ are calculated from the knowledge of $\hat {p}$ (see Appendix A). The eigenfunctions are normalised such that $\angle \hat {p}(r)=0$ for $r=0$ and to have unitary energy norm, which, following Chu (1965) and Hanifi, Schmid & Henningson (1996), is defined as

(2.14)\begin{equation} E=\frac{1}{2}\int_0^{2{\rm \pi}}\int_0^\infty \left(\bar{\rho}\left(\vert \widehat{u_x}\vert^2+\vert \widehat{u_r}\vert^2+\vert \widehat{u_\theta}\vert^2\right) + \frac{\gamma -1}{\gamma} \frac{\bar{T}}{\bar{\rho}}\vert\hat{\rho}\vert^2 + \frac{\bar{\rho}}{\gamma\bar{T}}\vert \hat{T}\vert^2\right)r\mathrm{d}r\,\mathrm{d}\theta. \end{equation}

The derivation of the energy norm is provided in Appendix B.

2.3.2. Vortex-sheet model

The V-S model is an inviscid idealisation of the jet where the infinitely thin V-S separates the interior flow and the outer quiescent fluid, resulting in a jet with a mean top-hat profile. The V-S was used by Lessen, Fox & Zien (1965) and Michalke (1970) to study the stability properties of a compressible jet. We recently showed that the standard V-S model for free jets, which was used in the previous studies, does not support free-stream acoustic waves as discrete modes, as required for the mode-matching procedure. To obtain a discrete representation of free-stream acoustic modes, we use the dispersion relation of a confined jet with the radial distance of the boundary ($r_{MAX}$) sufficiently distant from the jet in order to recover the same dynamical properties of a free jet. The analysis of this surrogate problem allowed us to include the free-stream acoustic modes in the description of the jet dynamics (for more details the reader can refer to Mancinelli et al. 2022). We herein use this confined version of the V-S, whose dispersion relation $D(k,\omega ;M_a,T,m,r_{MAX})=0$ is,

(2.15) \begin{equation} \begin{gathered} \frac{1}{\left(1-\dfrac{kM_a}{\omega}\right)^2} + \frac{1}{T}\frac{I_m\left(\dfrac{\gamma_i}{2}\right)}{K_m\left(\dfrac{\gamma_o}{2}\right)-zI_m \left(\dfrac{\gamma_o}{2}\right)}\\ \frac{\dfrac{\gamma_o}{2}K_{m-1}\left(\dfrac{\gamma_o}{2}\right) + mK_m\left(\dfrac{\gamma_o}{2}\right)+z\left(\dfrac{\gamma_o}{2}I_{m-1} \left(\dfrac{\gamma_o}{2}\right)-mI_m\left(\dfrac{\gamma_o}{2}\right)\right)}{\dfrac{\gamma_i}{2}I_{m-1} \left(\dfrac{\gamma_i}{2}\right) - mI_m\left(\dfrac{\gamma_i}{2}\right)}=0, \end{gathered} \end{equation}

with

(2.16a)\begin{equation} \left.\begin{gathered} \gamma_i = \sqrt{k^2-\frac{1}{T}\left(\omega-M_ak\right)^2},\\ \gamma_o = \sqrt{k^2-\omega^2}, \end{gathered}\right\} \end{equation}

where $I$ and $K$ are modified Bessel functions of the first and second kinds, respectively, $T=T_j/T_\infty$ is the jet-to-ambient temperature ratio such that the relation between the jet and the acoustic Mach numbers is $M_a=M_j\sqrt {T}$ and $z=K_m(\gamma _or_{MAX})/I_m(\gamma _or_{MAX})$. The branch cut in the square root of (2.16) is chosen such that $-{\rm \pi} /2\leq \rm {arg}(\gamma _{i,o})<{\rm \pi} /2$. The dispersion relation in (2.15) for a confined jet differs from the unconfined one for a free jet in the additional terms containing $z(r_{MAX})$. Following Mancinelli et al. (2022), we herein use $r_{MAX}=100$ in order to avoid any effect of the boundary on the eigenmodes. Frequency/wavenumber pairs $(\omega, k)$ that satisfy (2.15) define eigenmodes of the vortex sheet for given values of $m$, $M_a$, $T$ and $r_{MAX}$. To find these pairs, similarly to the F-T model, we specify a frequency $\omega$ (real or complex) and compute the associated eigenvalues $k$. Eigenvalues are computed using a root finder based on the Levenberg–Marquardt method (Levenberg 1944; Marquardt 1963).

After imposing bounded solution for $r=0$ and a soft-wall boundary condition at $r=r_{MAX}$, the solution for the pressure in the inner and outer flows is

(2.17a–b) \begin{gather}\left.\begin{aligned} &\hat{p}_i\left(r\right)=B_iI_m\left(\gamma_ir\right)\quad r\leq 0.5\\ &\hat{p}_o\left(r\right)=C_o\left({-}zI_m\left(\gamma_or\right) + K_m\left(\gamma_or\right)\right) \quad r> 0.5, \end{aligned}\right\} \end{gather}

where $B_i$ and $C_o$ are constants fixed in order to ensure pressure continuity at the V-S location $r=0.5$. The eigenfunctions of the other flow variables are calculated from the knowledge of $\hat {p}_{i,o}(r)$ by exploiting the Fourier-transformed LEE (A4f). The same eigenfunction normalisation procedure described for the F-T model is used for the V-S model as well.

2.3.3. Identification of reflected and transmitted waves

There are two types of waves which appear due to the scattering of the K–H wave at the shock: upstream-travelling reflected waves in region 1, and downstream-travelling transmitted waves in region 2 (see figure 1). Following Towne et al. (2017), we use the terms downstream and upstream travelling to designate the direction of the energy transfer. This property can be characterised using the Briggs–Bers criterion by looking at the asymptotic behaviour of $k(\omega )$ at large $\omega _i$ (Briggs 1964; Bers 1983). The wave is downstream travelling if

(2.18a)\begin{equation} \lim_{\omega_i\to+\infty}k_i={+}\infty, \end{equation}

and upstream travelling if

(2.18b)\begin{equation} \lim_{\omega_i\to+\infty}k_i={-}\infty, \end{equation}

where the subscript $i$ stands for the imaginary part of the variable. The downstream- and upstream-travelling waves are denoted hereinafter with the superscript $+$ and $-$, respectively.

2.3.4. Mean flow

The conditions upstream of the shock in the case of the V-S are provided by

(2.19a)

where $\rho =\rho _j/\rho _\infty$ is the jet-to-ambient density ratio.

In the case of finite thickness, we use the hyperbolic tangent function reported in Lesshafft & Huerre (2007) for the velocity profile upstream of the shock,

(2.20)\begin{equation} \bar{u}_{1x}=\frac{1}{2}M_{a_1}\left(1+\tanh\left(\frac{R}{4\delta} \left(\frac{R}{r}-\frac{r}{R}\right)\right)\right), \end{equation}

where $\delta$ is the shear-layer momentum thickness and $R=0.5$ is the nozzle radius. Consistent with particle image velocimetry results presented in Mancinelli et al. (2021) for an under-expanded supersonic jet with jet Mach number $M_{j_1}=1.1$ and temperature ratio $T\approx 0.81$, we choose a shear-layer thickness $R/\delta =10$. Denoting the dimensional variables with the superscript $*$ and the fully expanded variables with the subscript $j$, the mean density profile is calculated as the inverse of the mean temperature profile, $\rho ^*(r)/\rho _j=(T^*(r)/T_j)^{-1}$, where the mean temperature is calculated using the Crocco–Busemann relation (Michalke 1984)

(2.21)\begin{equation} \frac{T^*\left(r\right)}{T_j}=\frac{T_\infty}{T_j} + \left(1-\frac{T_\infty}{T_j}\right)\frac{u_x^*\left(r\right)}{U_j} + \left(\gamma -1\right)M_j^2\frac{u_x^*\left(r\right)}{U_j}\frac{1}{2} \left(1-\frac{u_x^*\left(r\right)}{U_j}\right). \end{equation}

The mean flow downstream of the shock can then be determined from the upstream conditions using the jump equations of normal shocks

(2.22a–d) \begin{gather}\left.\begin{aligned} M_{j_2}&=\sqrt{\frac{M_{j_1}^2\left(\gamma-1\right)+2}{2\gamma M_{j_1}^2-\left(\gamma-1\right)}},\\ \frac{\bar{p}_2}{\bar{p}_1}&=\frac{2\gamma M_{j_1}^2-\left(\gamma-1\right)}{\gamma+1},\\ \frac{\bar{\rho}_2}{\bar{\rho}_1}&=\dfrac{\left(\gamma+1\right)M_{j_1}^2}{\left(\gamma-1\right)M_{j_1}^2+2},\\ \frac{\bar{T}_2}{\bar{T}_1}&=\frac{\left(1+\dfrac{\gamma-1}{2}M_{j_1}^2\right) \left(\dfrac{2\gamma}{\gamma-1}M_{j_1}^2-1\right)}{M_{j_1}^2\left(\dfrac{2\gamma}{\gamma-1}+ \dfrac{\gamma-1}{2}\right)}. \end{aligned}\right\} \end{gather}

The mean-flow profiles upstream and downstream of the shock in the case of V-S and F-T models for the flow conditions listed above are represented in figure 2. The presence of the shock wave generates a mean pressure gradient along the radial direction downstream of the shock. This $\partial \bar {p}/\partial r$ induces a mean radial velocity $\bar {u}_r$, thus making the flow slowly diverging. The evaluation of this induced radial velocity in the case of a F-T model is reported in Appendix C, where we show that the induced mean radial velocity is small compared with the axial velocity component. We also point out that the transmission and reflection mechanisms occur locally and hence they are not affected by the flow evolution far away from the shock.

Figure 2. Mean flow upstream and downstream of the shock for the V-S and F-T models for $M_{j_1}=1.1$: solid blue lines refer to upstream conditions, dashed red lines to downstream ones. (a) The V-S model, (b) F-T model: (i) axial velocity, (ii) temperature, (iii) density, (iv) pressure.

3. Results

In this section we present the results of the reflection-coefficient calculation obtained by modelling the jet dynamics with both the V-S and F-T models. We first consider a Strouhal number $St=0.68$, a jet Mach number $M_{j_1}=1.1$ and a temperature ratio $T\approx 0.81$ (corresponding to an acoustic Mach number $M_{a_1}\approx 0.987$), which results in a jet Mach number $M_{j_2}=0.91$ and an acoustic Mach number $M_{a_2}\approx 0.84$ downstream of the shock. These upstream jet conditions and Strouhal number are selected to match conditions for which screech has been experimentally observed by Mancinelli et al. (2019a, 2021). Due to the axisymmetric nature of the resonance mode, we here study the azimuthal mode $m=0$. Furthermore, we focus on the reflection coefficient of the upstream-travelling guided mode of the second radial order given that this mode has been proven to be the closure mechanism for axisymmetric screech modes (see Edgington-Mitchell et al. 2018; Mancinelli et al. 2019a, 2021). Finally, for the F-T model, we explore the variation of the reflection coefficient as a function of both $St$ and $M_j$.

3.1. Incident, reflected and transmitted waves

First, we identify the reflected and transmitted waves generated by the interaction of the incident K–H wave with the shock discontinuity for both the V-S and F-T models upstream and downstream of the shock, respectively. Figure 3 shows the V-S eigenspectrum in the complex-$k$ plane upstream of the shock. For the sake of brevity and clarity of the figure, in the present manuscript we show eigenvalues only for real $\omega$ (see Mancinelli et al. (2022) for the corresponding eigenspectrum for $\omega \in \mathcal {C}$). Several distinct families of modes can be identified. The V-S model supports one convectively unstable mode, the K–H mode, which is denoted hereinafter $k_{KH}$. The unstable $k_{KH}$ wave has a complex conjugate and both eigenvalues have positive phase and group velocities according to the criteria (2.18). The V-S model also supports guided modes, i.e. modes that use the jet as a wave guide. These modes, hereinafter denoted $k_p$, belong to a hierarchical family of waves identified by their azimuthal and radial orders $m$ and $n_r$, respectively. According to Towne et al. (2017), these modes are guided or completely trapped inside the jet depending on the $St$ and the radial order considered. Specifically, we observe evanescent $k_p$ waves for $n_r=1$ with supersonic negative phase speed. The wave associated with $k_i>0$ is a downstream-travelling wave, whereas the wave with $k_i<0$ is upstream travelling. The $k_p$ mode for $n_r = 2$ is propagative and upstream travelling and has a slightly subsonic negative phase speed. All the $k_p^\pm$ modes with $n_r\leq 2$ have support both inside and outside of the jet for the $St$ analysed. The $k_p^+$ modes for $n_r> 2$ represent acoustic waves trapped inside the jet due to total reflection at the V-S, which effectively behaves as a soft-walled duct (Towne et al. 2017; Martini, Cavalieri & Jordan 2019). Finally, we find propagative and evanescent acoustic modes, which are hereinafter denoted $k_a$. Among them, modes lying on the real and imaginary axes with $k_r<0$ and $k_i<0$, respectively, are upstream-travelling modes.

Figure 3. Eigenspectrum upstream of the shock obtained using the V-S model for azimuthal mode $m=0$, $T\approx 0.81$, $M_{j_1}=1.1$ and $\omega \in \mathcal {R}$. Blue $\diamond$ represent $k_{KH}^+$ and $k_{KH}^{*+}$ waves, red $\circ$ represents propagative $k_p^-$ mode with $n_r=2$, black $\bigtriangleup$ represent evanescent $k_p^\pm$ modes with $n_r=1$, green $\Box$ represent $k_p^+$ modes with $n_r\geq 2$, magenta $\ast$ represent $k_a^\pm$ waves. Dashed lines refer to the sonic speed $\pm c_\infty$. Incident and reflected waves are indicated with arrows and labelled.

Figure 4 shows the eigenspectrum upstream of the shock computed using linear stability theory for a shear layer with finite thickness $R/\delta =10$. In addition to the mode families supported by the V-S, the F-T model supports critical-layer modes, denoted hereinafter $k_{cr}$. These modes have positive, subsonic phase and group velocities and lie on the real axis. Their spatial support is concentrated in the critical layer of the jet, i.e. the region of the jet where the phase speed equals the local mean-flow velocity (Tissot et al. 2017). Critical-layer modes with small wavenumbers are characterised by a spatial support mainly concentrated in the core of the jet and possess a phase speed close to the mean jet velocity, whereas $k_{cr}$ modes with larger wavenumbers are mostly concentrated in the shear layer and have a phase-speed value which decreases as the spatial support of the mode moves more and more outside of the jet. To summarise, both the V-S and F-T models support two families of reflected waves upstream of the shock: (i) guided jet modes and (ii) propagative and evanescent acoustic modes. Among the guided modes, we may distinguish the evanescent $k_p^-$ mode with $n_r=1$ and the propagative $k_p^-$ mode with $n_r=2$. In the remainder of this paper, we focus our attention on the reflection coefficient between the K–H wave and the propagative $k_p^-$ mode with $n_r = 2$, since this interaction is responsible for the screech resonance at this frequency and Mach number (Edgington-Mitchell et al. 2018; Mancinelli et al. 2019a, 2021; Nogueira et al. 2022).

Figure 4. Eigenspectrum upstream of the shock obtained using the F-T model for azimuthal mode $m=0$, $T\approx 0.81$, $M_{j_1}=1.1$ and $\omega \in \mathcal {R}$. Markers and colours to identify the modes are the same used in figure 3 in the case of the V-S. The modes that are only supported by the F-T model, that is the $k_{cr}^+$ modes, are here indicated by cyan $\times$. Dashed lines refer to the sonic speed $\pm c_\infty$. Incident and reflected waves are indicated with arrows and labelled.

We now consider the eigenspectrum downstream of the shock with the aim of identifying the transmitted modes for both the V-S and F-T models. Figure 5 shows the eigenspectrum obtained using the V-S. According to Towne et al. (2017), in the subsonic regime for $0.82\leq M< 1$ the guided modes are characterised by two upstream-travelling branches delimited by two saddle points in the $k_r$-$St$ plane: one branch characterised by a larger negative phase speed close to the ambient speed of sound, the above-defined $k_p^-$, and one with a lower phase-speed absolute value, herein denoted $k_d^-$. Similar to the eigenspectrum upstream of the shock, we may identify the $k_{KH}^+$ wave and its complex conjugate $k_{KH}^{*+}$, the downstream- and upstream-travelling $k_p$ waves with $n_r=1$, which are evanescent and have supersonic negative phase speed at this frequency, and the $k_a^\pm$ modes. As outlined in Mancinelli et al. (2022), propagative, downstream-travelling acoustic modes are not found in the vicinity of the sonic line due to numerical issues in the root-finder algorithm. We show in Appendix D that these modes are not relevant for the determination of the reflection coefficient. We then locate the upstream-travelling propagative $k_d^-$ wave for $n_r=1$, which has a duct-like behaviour (Towne et al. 2017), and the evanescent $k_p^\pm$ waves with $n_r\geq 2$, which behave like modes in a soft duct as well at this frequency. In this regard, we note that the $k_p$ eigenvalues with $k_i>0$ are associated with $k^+$ waves, whereas the eigenvalues with $k_i<0$ are associated with $k^-$ modes.

Figure 5. Eigenspectrum downstream of the shock obtained using the V-S model for $\omega \in \mathcal {R}$, azimuthal mode $m=0$, $T\approx 0.85$ and $M_{j_2}=0.91$ (which corresponds to $M_{a_2}=0.84$): (a) global view, (b) zoom around the origin. Blue $\diamond$ represent $k_{KH}^+$ and $k_{KH}^{*+}$ waves, red $\circ$ represents propagative $k_d^-$ mode with $n_r=1$, black $\bigtriangleup$ represent evanescent $k_p^\pm$ modes with $n_r=1$, green $\Box$ represent $k_p^\pm$ modes with $n_r\geq 2$, magenta $\ast$ represent $k_a^\pm$ waves. Dashed lines refer to the sonic speed $\pm c_\infty$. Transmitted waves are indicated with arrows and labelled.

Figure 6 shows the eigenspectrum downstream of the shock computed using a F-T model. Unlike those computed by the V-S, the evanescent guided modes for $n_r>1$ bend towards the supersonic phase-speed region as $n_r$ increases and eventually merge with the evanescent free-stream acoustic modes for $n_r>4$. Additionally, we observe the $k_{cr}^+$ waves that are not supported by the V-S. To summarise, within the V-S model, possible downstream-travelling transmitted modes are: (i) the K–H mode and its complex conjugate, (ii) the evanescent guided mode of first radial order with supersonic phase speed, (iii) the evanescent trapped modes of higher radial order with subsonic phase speed and (iv) the acoustic modes. For the F-T model, in addition to the transmitted modes listed above for the V-S, we include the critical-layer modes. For this flow model, unlike the V-S model and consistent with the eigenspectrum shown in figure 6, we use evanescent $k_p^+$ waves on the transmitted side up to the radial order $n_r=4$, that is, before this mode branch merges with the evanescent $k_a^+$ modes and become no longer distinguishable. A summary of the modes involved in the reflection coefficient computation is reported in table 1.

Figure 6. Eigenspectrum downstream of the shock obtained using the F-T model for $\omega \in \mathcal {R}$, azimuthal mode $m=0$, $T\approx 0.85$ and $M_{j_2}=0.91$ (which corresponds to $M_{a_2}=0.84$): (a) global view, (b) zoom around the origin. Markers and colours are the same as used in figure 5 to identify the modes in the case of the V-S. The modes that are only supported by the F-T model, that is the $k_{cr}^+$ modes, are here indicated by cyan $\times$. Dashed lines refer to the sonic speed $\pm c_\infty$. Transmitted waves are indicated with arrows and labelled.

Table 1. Summary of the eigenmodes supported by the V-S and F-T models involved in the reflection coefficient computation.

Examples of the normalised pressure eigenfunctions of the waves upstream and downstream of the shock are reported in figure 7. For the V-S (figure 7a), the incident and transmitted K–H waves show a peak at the V-S location and the reflected waves, that is $k_p^-$ and $k_a^-$ modes, have support both inside and outside the jet. We note that, although there is no energy loss at $r=r_{MAX}$ as a consequence of the imposition of a soft-wall boundary condition, the energy of $k_a$ modes reflected back cannot be transferred to the other discrete modes since the flow is locally parallel. On the transmitted side downstream of the shock, while the evanescent $k_p^+$ wave with $n_r=1$ has a support both in the inner and outer part of the jet, the transmitted $k_p^+$ modes with $n_r>1$ show a spatial support concentrated inside the jet, consistent with their identity as acoustic waves trapped within the jet core. Examples of the pressure eigenfunctions of the waves upstream and downstream of the shock computed using the F-T model are shown in figure 7(b). The incident K–H mode and the reflected $k_p^-$ and $k_a^-$ have a similar shape to that found using the V-S model. As mentioned above, $k_{cr}^+$ modes have eigenfunctions with a spatial support concentrated either inside of the jet or in the shear layer depending on the wavenumber value considered. The spatial support of the critical-layer modes moves to larger $r$ and the phase velocity $U_{\phi }$ decreases as $\vert k_{cr}\vert$ increases.

Figure 7. Pressure eigenfunctions for $m=0$ and $St=0.68$ computed using (a) the V-S model and (b) the F-T model. The colours are the same as those used in figures 3, 4, 5 and 6 to identify the different mode families upstream and downstream of the shock, respectively. (i) Incident and reflected waves upstream of the shock for $M_j=1.1$ and $T\approx 0.81$: solid blue line refers to the incident $k_{KH}^+$ wave, dashed red line to the propagative $k_p^-$ wave with $n_r=2$, dotted black line to the evanescent $k_p^-$ mode with $n_r=1$, dash-dotted magenta line to the propagative $k_a^-$ wave. (ii) Transmitted waves downstream of the shock for $M_{j_2}=0.91$ and $T\approx 0.85$: solid blue line refers to the transmitted $k_{KH}^+$ mode, dotted black line to the evanescent $k_p^+$ with $n_r=1$, dashed green line to the evanescent $k_p^+$ with $n_r=2$, dash-dotted magenta line to the propagative $k_a^+$ wave, solid and bold cyan lines to $k_{cr}^+$ modes.

3.2. Reflection coefficient

Second, we present the reflection-coefficient values calculated by minimising the error objective function $F$ in (2.8) via the pseudo-inverse solution (2.12). Figure 8 shows the evolution of the objective function $F$ as a function of the number of reflected and transmitted modes $n=n_R+n_T$ for both the V-S and F-T models. Here, $F$ is normalised so as to have unitary value when the sole incident K–H wave is considered in the minimisation algorithm, thus allowing us to quantify the relevance of the modes added in the computation in terms of error reduction. We here make a convergence analysis showing how the objective function changes by adding reflected and transmitted modes. In the absence of any unquestionable criterion allowing us to know a priori the relevance of each mode in the reflection/transmission dynamics, we choose to add modes as a function of the family they belong to. For each mode family, waves are added by increasing $\vert k\vert$. Specifically, we first add in the calculation the reflected waves, that is the $k_p^-$ and $k_a^-$ mode families, and then we add the transmitted modes downstream of the shock, that is $k_{KH}^+$ and its complex conjugate, the $k_p^+$ modes for $n_r\geq 1$, the $k_a^+$ modes and the $k_{cr}^+$ waves in the case of the F-T model, for a total number of modes $N=N_R+N_T=800$. The final results are independent of the mode arrangement. The objective function remains approximately constant when reflected waves are added. The transmitted K–H mode provides a significant decay in the cost function, which indicates that it has an important role in the reflection/transmission mechanism, as the impinging and transmitted K–H modes have a similar structure. For the V-S, the addition of the other transmitted modes has a small impact on the cost functional, which saturates at a value of ${\approx }10^{-2}$. On the contrary, for the F-T model, the addition of critical-layer modes produces an important reduction of the cost function, which saturates at a value of $5\times 10^{-3}$, lower than that observed in the case of the V-S.

Figure 8. Evolution of the normalised objective function as a function of the number of modes: (a) V-S, (b) F-T model. Markers and colours are the same as those used to identify the modes in the eigenspectra in figures 3 and 5 for the V-S and figures 4 and 6 for the F-T model.

Figure 9 shows the evolution of the amplitude and phase of the reflection coefficient associated with the upstream-travelling guided mode of second radial order as a function of the number of modes $n$ obtained using the V-S model. On the reflected side, the significant drop of $\vert R\vert$ observed when $k_a^-$ modes are added is due to propagative acoustic modes. Similar to the objective function in figure 8(a), both the amplitude and phase of the reflection coefficient undergo a significant jump when modes on the transmitted side are included in the calculation. Specifically, the solution does not drastically change after the inclusion of the $k_{KH}^+$ mode and remains approximately constant. The solution converges to $5.8\times 10^{-3}$ and $-0.38$ for the amplitude and phase, respectively.

Figure 9. Evolution of the reflection coefficient computed using the V-S as a function of the modes considered in the calculation (markers and colours are the same as those used to identify the modes in the eigenspectra in figures 3 and 5): (a) amplitude, (b) phase. Blue dotted lines indicate $\pm {\rm \pi}$ in the phase plot.

The evolution of the amplitude and phase of the reflection coefficient as a function of $n$ in the case of the F-T model is shown in figure 10. Similar to the V-S model, we see a significant change in the value of both amplitude and phase when modes on the transmitted side are included in the calculation. Consistent with what we observed in the objective function trend, the addition of the $k_{cr}^+$ modes plays an important role in the determination of the reflection coefficient. Specifically, a significant drop in amplitude occurs for critical-layer modes with $5.5\leq \vert k\vert \leq 21$, after which both the amplitude and phase values remain approximately constant. These are modes with wavelength in the range $\approx [0.3D,D]$ and, hence, 6 to 20 times the shear-layer thickness. Their phase speed is $\approx$ $0.2U_j\leq U_{\phi }\leq 0.75U_j$, and they are mostly concentrated in the centre of the shear layer, as shown in figure 11.

Figure 10. Evolution of the amplitude and phase of the reflection coefficient computed using the F-T model as a function of the number of modes considered. Markers and colours are the same as those used in figures 4 and 6 to identify the mode families. (a) amplitude, (b) phase. Blue dotted lines indicate $\pm {\rm \pi}$ in the phase plot.

Figure 11. Normalised eigenfunctions of $k_{cr}^+$ modes computed using the F-T model as a function of wavenumber $k$ and radial distance $r$: (a) pressure, (b) axial velocity, (c) radial velocity.

A summary of the reflection-coefficient values obtained for both flow models is reported in table 2. A large discrepancy between the V-S and F-T model is found for both the amplitude and phase of the reflection coefficient. This result suggests that the presence of a finite thickness and the associated shear-layer dynamics, which cannot be described using a vortex sheet, plays an important role in the reflection/transmission mechanism and needs to be taken into account to perform a reliable estimation of the reflection coefficient.

Table 2. Summary of the reflection-coefficient values associated with the upstream-travelling guided mode of second radial order generated by the interaction between an incident K–H wave on a shock. Results obtained using both the V-S and F-T models to describe the jet dynamics are reported.

3.2.1. Reflected and transmitted pressure fields

The reflected and transmitted pressure fields for the flow condition investigated above using a F-T model are here reconstructed in the $x$–$r$ plane. Using the normal-mode ansatz (2.4), the reflected and transmitted fields are given by

(3.1a)$$\begin{gather} p_R\left(x,r,\theta\right)=\sum_{n_R=1}^{N_R}R_{n_R} \hat{p}_{1R,n_R}\exp({{\rm i}\left(k_{nR}x+m\theta\right)}), \end{gather}$$

(3.1b)$$\begin{gather}p_T\left(x,r,\theta\right)=\sum_{n_T=1}^{N_T}T_{n_T} \hat{p}_{2T,n_T}\exp({{\rm i}\left(k_{nT}x+m\theta\right)}). \end{gather}$$

We set $x=0$ to be the location of the shock discontinuity, so the reflected upstream-travelling and transmitted downstream-travelling waves evolve along negative and positive $x$ directions, respectively. Figure 12 shows the entire reflected field as well as the reflected fields for each type of the upstream-travelling modes considered in the reflection-coefficient calculation, that is the propagative $k_p^-$ mode with $n_r=2$, the evanescent $k_p^-$ mode with $n_r=1$ and an example of the propagative and evanescent $k_a^-$ waves with $\vert k\vert \approx 2.79$ and $2.1$, respectively. The fields reconstructed using the $k_p^-$ modes show a spatial support both inside and outside the jet with a prescribed radial decay. Regarding the axial evolution, the evanescent $k_p^-$ mode with $n_r=1$ exhibits a decay starting from the shock position $x=0$, whereas the $k_p^-$ mode with $n_r=2$ has a fixed axial structure consistent with its neutrally stable nature. A similar behaviour is found for the pressure fields reconstructed using propagative and evanescent free-stream acoustic waves. The entire reflected pressure field, which results from the linear superposition of all the reflected $k^-$ waves, shows an axially and radially decaying structure starting from $x=0$ and $r=0$, respectively.

Figure 12. Reconstruction of the reflected fields in the $x-r$ plane for a shock discontinuity located at $x=0$: (a) $k_p^-$ mode with $n_r=2$, (b) $k_p^-$ mode with $n_r=1$, (c) propagative $k_a^-$ wave, (d) evanescent $k_a^-$ wave, (e) total field.

The entire transmitted field as well as the transmitted fields for each mode family considered in the reflection-coefficient calculation are shown in figure 13. We here report the fields obtained using the $k_{KH}^+$ wave, the evanescent supersonic $k_p^+$ mode with $n_r=1$, the evanescent subsonic $k_p^+$ mode with $n_r=2$, the propagative and evanescent $k_a^+$ waves with $\vert k\vert \approx 2.7$ and $2.8$, respectively, and the $k_{cr}^+$ mode for $\vert k\vert \approx 11.5$. Consistent with its unstable nature, the shape and intensity of the total transmitted field is dominated by the K–H mode. The shape of the transmitted fields reconstructed using the individual mode family is consistent with the neutral/evanescent nature and radial structure described above.

Figure 13. Reconstruction of the transmitted fields in the $x$–$r$ plane for a shock discontinuity located at $x=0$: (a) $k_{KH}^+$ wave, (b) supersonic $k_p^-$ mode with $n_r=1$, (c) subsonic $k_p^+$ with $n_r=2$, (d) propagative $k_a^+$ wave, (e) evanescent $k_a^+$ wave, ( f) $k_{cr}^+$ mode, (g) total field.

3.2.2. Frequency–Mach-number dependence of the reflection coefficient

We here explore the frequency–Mach-number dependence of the reflection coefficient associated with the upstream-travelling guided mode with $n_r=2$ obtained using the F-T model. For this purpose, we let the jet Mach number vary in the range $M_j=[1,1.7]$ and explore the Strouhal-number band for which the guided mode is propagative for such flow conditions, that is the $St$-number range delimited by the branch and saddle points (see Mancinelli et al. 2019a, 2021). The resolutions for ${\rm \Delta} M_j$ and ${\rm \Delta} St$ were set equal to $10^{-2}$ and $5\times 10^{-3}$, respectively. Figure 14 shows the evolution of the normalised objective function $F$ as a function of $M_j$ and $St$. The errors are very small for low $M_j$ and gradually rise as the jet Mach number increases but never exceed 13 %. The amplitude and the phase modulus of the reflection coefficient as a function of $St$ and $M_j$ are shown in figure 15. The reflection-coefficient amplitude rises with increasing $M_j$ for a given $St$ and appears to be larger in the proximity of the saddle point for all $M_j$, with the jet-flow region $M_j=[1.17,1.37]$ showing the highest amplitude. Overall, the phase gradually increases for larger $M_j$ and lower $St$. We note that a jump from in-phase to out-of-phase conditions is found for $M_j=1.58$ for most frequencies.

Figure 14. Normalised error objective function as a function of Strouhal and jet Mach numbers for the F-T model. Dashed and dash-dotted lines refer to the branch- and saddle-point locations, respectively.

Figure 15. Evolution of the reflection coefficient as a function of Strouhal and jet Mach numbers for the F-T model. Dashed and dash-dotted lines refer to the branch- and saddle-point locations, respectively. (a) Amplitude, (b) phase.

4. Conclusions

The scattering of a K–H wave into a discrete, upstream-travelling guided mode due to a normal shock in a jet flow was computed in the present manuscript. Jets with zero-thickness and F-T shear layers were modelled with an analytical V-S and a numerical solution of the Rayleigh equation, respectively. The reflection coefficient associated with the scattered, upstream-travelling wave was calculated via a mode-matching technique enforcing conservation equations through the shock discontinuity. Assuming small disturbances, the shock equations were linearised about the mean flow and the possible scattered modes, i.e. upstream-travelling reflected waves and downstream-travelling transmitted waves, were identified by assessing the corresponding group velocity using the Briggs–Bers criteria. Solutions based on a truncated set of modes were obtained via the minimisation of the error in the conservation equations, which corresponds to a weighted pseudo-inverse solution. The F-T model showed that critical-layer modes, characterised by eigenfunctions with radial support mostly concentrated in the centre of the shear layer, by a wavelength 6 to 20 times the shear-layer thickness and by a phase velocity between 0.2 and 0.75 of the jet velocity, play an important role in the reflection-coefficient calculation and, thus, most likely in the physical mechanisms underpinning the process. The reconstructed reflected and transmitted pressure fields in the $x$–$r$ plane showed an organised structure consistent with the axial and radial evolution of the modes calculated by the F-T model. The frequency–Mach-number dependence of the reflection coefficient has been explored, revealing that the reflection-coefficient amplitude is larger for frequencies in the proximity of the saddle point and for jet Mach-number values in between 1.17 and 1.37. The phase exhibited a gradual increase as the Strouhal and jet Mach numbers decreased and increased, respectively.

The mode-matching approach described in this paper appears to be a relevant and promising tool for better describing and predicting resonant dynamics found in jets, such as screech in supersonic jets. A similar approach may be developed to evaluate the reflection coefficient at the nozzle exit, thus providing all the elements to make screech-frequency predictions without any input from data using the model presented in Mancinelli et al. (2021).