2. Mathematical model

We consider a flow that is single component, ideal and calorically perfect. The effect of body forces and diffusion are assumed to be negligible. The flow evolves subsonically (up to choking condition) in an adiabatic nozzle, whose cutoff frequency is sufficiently high for the acoustics to be one-dimensional. The conservation of mass, momentum and energy are, respectively (e.g. Shapiro Reference Shapiro1954),

(2.1)$$\begin{gather} \frac{{\rm D}\rho}{{\rm D}t} + \rho\frac{\partial u}{\partial x} + \frac{\rho u}{A}\frac{{\rm d}A}{{\rm d}\kern0.06em x} = {\dot{S}_m}, \end{gather}$$

(2.2)$$\begin{gather}\frac{{\rm D}u}{{\rm D}t} + \frac{1}{\rho}\frac{\partial p}{\partial x} = {\dot{S}_M}, \end{gather}$$

(2.3)$$\begin{gather}T\frac{{\rm D}s}{{\rm D}t} = {\dot{S}_s}, \end{gather}$$

where $x$ is the longitudinal coordinate of the nozzle, $\rho$ is the density, $u$ is the velocity, $p$ is the pressure, $T$ is the temperature, $s$ is the entropy and $A$ is the cross-sectional area of the duct. The right-hand side terms, ${\dot {S_j}}$, are the source terms of mass, momentum and entropy, respectively. The thermodynamic variables satisfy the Gibbs equation

(2.4)\begin{equation} {{\rm d}s} = c_p\frac{{\rm d}T}{T} - R\frac{{\rm d}p}{p}, \end{equation}

where $R$ is the gas constant and $c_p$ is the specific heat at constant pressure. We assume that the mass source is zero, $\dot {S}_m = 0$, in (2.1).

In the energy equation, the dissipation is modelled as an entropy source. Introducing the compressibility factor

(2.5)\begin{equation} \varLambda\equiv {1 + \frac{\gamma - 1}{2}M^2}, \end{equation}

and substituting the thermodynamic relations $T = {T_0}/\varLambda$ and $p = {p_0}/{\varLambda }^{{\gamma }/({\gamma - 1)}}$ in (2.4), where $\gamma$ is the heat-capacity ratio, yield

(2.6)\begin{equation} {\rm d}s = c_p \frac{{\rm d}T_0}{T_0} - R\frac{{\rm d}p_0}{p_0}, \end{equation}

which, under the adiabatic assumption, ${\rm d}T_0 = 0$, simplifies to

(2.7)\begin{equation} \dot{S}_s ={-}RT\frac{{\rm D}\log p_0}{{\rm D}t}. \end{equation}

Physically, a reduction in the stagnation pressure increases the entropy production proportionally to the temperature of the flow. If the flow is isentropic, the stagnation pressure is constant, and vice versa. In the momentum equation, the dissipation is modelled as a momentum source

(2.8)\begin{equation} \dot{S}_M ={-} \frac{4f}{D}\frac{u^2}{2}, \end{equation}

where $f = \tau _w / ({\rho u^2}/{2})$ is the Fanning friction factor, in which $\tau _w$ is the shear stress (Shapiro Reference Shapiro1954). The friction decreases the stagnation pressure as

(2.9)\begin{equation} {{\rm d}\log p_0} ={-}\frac{\gamma M^2}{2} \frac{4f}{D} {{\rm d}\kern0.06em x}, \end{equation}

where $D$ is the nozzle diameter. In quasi-one-dimensional models, the flow variables at $x$ are averages across the cross-section $A(x)$. This means that the friction factor can account for wall friction and two-dimensional (or three-dimensional) effects, such as flow recirculation, in a cross-averaged sense. (These effects can be accurately captured in ducts with a slowing-varying cross-sectional area, for which the quasi-one-dimensional assumption holds.) The Mach number, $M$, is governed by (Shapiro Reference Shapiro1953)

(2.10)\begin{equation} \frac{{\rm d}M^2}{M^2} = \frac{\varLambda}{1 - M^2}\left[{-}2\frac{{\rm d}A}{A} + \gamma M^2 \frac{4f}{D}{{\rm d}\kern0.06em x}\right]. \end{equation}

Equation (2.10) is the extension of the Fanno flow to variable-area ducts. We start with an arbitrary nozzle geometry as shown in figure 1. Using the nozzle profile, at any location $x$, $4\,{\rm d}x/D = {\rm d}A/(A \tan \alpha )$, where $D$ is the diameter of the cross-section and $\tan \alpha$ is the spatial derivative of the nozzle profile at that location, which reduces (2.11) to

(2.11)\begin{equation} \frac{{\rm d}M^2}{M^2} = \left(\frac{\varLambda}{1-M^2}\right)\frac{\zeta}{\tan\alpha}\frac{{\rm d}A}{A} \end{equation}

where, for brevity, we define the competition factor $\zeta \equiv f\gamma M^2 - 2 \tan \alpha$, which quantifies the competition between the dynamics dictated by mass conservation (2.1), through the nozzle geometry, $\alpha$, and the dynamics dictated by momentum conservation (2.2), through the friction, $f$. If the two mechanisms are in balance, $\zeta =0$, the flow evolves at a constant Mach number. The stagnation pressure is

(2.12)\begin{equation} \frac{{\rm D}\log p_0}{{\rm D}t} ={-}\frac{f}{\zeta}\left(\frac{\gamma(1 - M^2)}{2\varLambda}\right) \frac{{\rm D}M^2}{{\rm D}t}. \end{equation}

(The condition $\zeta =0$ is not a singularity because (2.12)$\rightarrow$(2.9) as $\zeta \rightarrow 0$.) To physically interpret (2.12), we consider a symmetric linear geometry nozzle in which $\alpha _c=-\alpha _d$, where $c$ and $d$ denote the convergent and divergent sections, respectively. As defined in figure 1, $\alpha _c<0$. From (2.11) and (2.12), changes in the stagnation pressure for the same Mach number, $M$, are related by

(2.13)\begin{equation} \frac{({\rm d}\log p_0)_d}{({\rm d}\log p_0)_c} \sim \frac{\zeta_c}{\zeta_d} > 1. \end{equation}

This means that the flow experiences greater pressure stagnation losses in the divergent section of the nozzle for the same Mach number, which is physically consistent with experiments (De Domenico et al. Reference De Domenico, Rolland and Hochgreb2019). Finally, the entropy source term (2.7) can be expressed as a function of the friction factor by substituting (2.12) into (2.7). In the limit of constant-area ducts, $\alpha \rightarrow 0$, the equations tend to the Fanno flow. In the limit of frictionless nozzles, $f \rightarrow 0$, the equations tend to the isentropic model.

Figure 1. Schematic of a generic nozzle with nomenclature.

2.1. Linearisation

We model the acoustics as linear perturbations to a mean flow. For this, we decompose a generic flow variable, $v$, as $v\rightarrow {v}(x) + v^{\prime }(x,t)$, where ${v}(x)$ is the steady mean flow component, and $v^{\prime }(x,t)$ is the first-order perturbation.

Assuming a constant friction factor, $f$, the analytical integration of (2.11) and (2.9), provides the nonlinear algebraic equations for the mean-flow Mach number, stagnation pressure and entropy, respectively,

(2.14)$$\begin{gather} \frac{A_2}{A_1} = \frac{M_1}{M_2} \left(\frac{\varLambda_2}{\varLambda_1}\right)^{(\gamma + 1) \tan \alpha/{2\kappa}} \left(\frac{\zeta_1}{\zeta_2}\right)^{(f\gamma - 2 \tan\alpha)/{2\kappa}}, \end{gather}$$

(2.15)$$\begin{gather}\frac{p_{02}}{p_{01}} = \left(\frac{\varLambda_2}{\varLambda_1}\right)^{f\gamma(\gamma + 1)/2(\gamma - 1)\kappa} \left(\frac{\zeta_1}{\zeta_2}\right)^{(f\gamma - 2 \tan\alpha)/{2\kappa}}, \end{gather}$$

(2.16)$$\begin{gather}\frac{\Delta s}{c_p} = \log\left(\frac{\varLambda_1}{\varLambda_2}\right)^{f(\gamma + 1)/2\kappa} \left(\frac{\zeta_2}{\zeta_1}\right)^{((\gamma - 1)/{\gamma})((f\gamma - 2 \tan\alpha)/{2\kappa})}, \end{gather}$$

where, $\kappa = f\gamma + (\gamma - 1)\tan \alpha$. (Solutions (2.14)–(2.16) can be used for curved nozzle geometries by discretising the nozzle shape piecewise linearly.) After some algebra, it can be shown that the linear perturbations are governed by

(2.17)$$\begin{gather} \frac{ {\rm D}}{{\rm D}\tau}\left(\frac{p^{\prime}}{\gamma p}\right) + \tilde{u}\frac{\partial}{\partial \eta}\left(\frac{u^{\prime}}{ u}\right) - \frac{ {\rm D}}{{\rm D}\tau}\left(\frac{s^{\prime}}{ c_p}\right) = 0, \end{gather}$$

(2.18)$$\begin{gather}\frac{ {\rm D}}{{\rm D}\tau}\left(\frac{u^{\prime}}{ u}\right) + \frac{\tilde{u}}{ M^2}\frac{\partial}{\partial \eta} \frac{p^{\prime}}{\gamma p} + \left(2\frac{u^{\prime}}{ u} + (1 - \gamma)\frac{p^{\prime}}{\gamma p} - \frac{s^{\prime}}{ c_p}\right)\left(\frac{\partial\tilde{u}}{\partial \eta} + \frac{4f}{D}\frac{\tilde{u}}{2} \right) = 0, \end{gather}$$

(2.19)$$\begin{gather}\frac{{\rm D}}{{\rm D}\tau}\left(\frac{s^\prime}{ c_p}\right) = g(f,f^2,f^3, {o}(f^3)), \end{gather}$$

where the linearised entropy source term, $g(\cdot )$, is provided in the Appendix (A), and $o$ is the little-o Landau symbol. The linearised Gibbs equation is (Marble & Candel Reference Marble and Candel1977)

(2.20)\begin{equation} \frac{\rho^{\prime}}{\rho} = \frac{p^{\prime}}{\gamma p} - \frac{s^{\prime}}{ c_p}. \end{equation}

The variables are non-dimensionalised as $\tau = tf_a$, $\eta = x/L$, $M = u/ c$ and $\tilde {u} = u / c_{ref}$, where $f_a$ is the frequency of the advected perturbations entering the nozzle, $L$ is the axial length of the nozzle and $c_{ref}$ is the reference speed of sound. Consequently, the non-dimensionalised material derivative becomes ${\rm D}/{\rm D}{\tau } = He {\partial }/{\partial \tau } + \tilde {u}{\partial }/{\partial \eta }$, where the Helmholtz number is defined as $He = {f_a L}/{c_{ref}}$. Physically, the Helmholtz number is the ratio between advected perturbations and acoustic wavelengths. If the length of the nozzle is small compared with the wavelength of the acoustic and the entropy perturbations, the nozzle is compact ($He = 0$). Equation (2.19) is derived by linearising (2.3) and substituting (2.17)–(2.18).

In the limit of isentropic flow, $f = 0$, (2.17), (2.18) and (2.19) tend to the linearised Euler equations of Duran & Moreau (Reference Duran and Moreau2013). On the one hand, in a frictionless flow, the sound is generated by the advected inhomogeneities through the acceleration ${\partial \tilde {u}}/{\partial \eta }$ in (2.18). On the other hand, in non-isentropic flows, the friction term, $4f\tilde {u}/2D$, is a negative acceleration, which competes with the advection of the mean flow. In § 2.1.1, we recast the equations in Riemann invariants, which are necessary for the numerical procedure proposed.

2.1.1. Riemann invariants

First, (2.17)–(2.19) are Fourier transformed with the decomposition $\boldsymbol {q}(\tau,\eta ) = \hat {\boldsymbol {{q}}}({\eta })\exp (2{\rm \pi} {\rm i} \tau )$. Second, the primitive variables are decomposed in travelling waves, where the downstream (superscript $+$) and upstream (superscript $-$) propagating acoustic waves are ${\rm \pi} ^\pm = 0.5[{p^\prime }/{(\gamma p)} \pm {u^\prime }/{ c} ]$; and the advected entropy wave is $\sigma = {s^\prime }/{c_p}$. Third, the equations are solved as a boundary value problem with boundary conditions specified for waves according to the transfer functions being evaluated. This provides a system of four linear equations in the gradients of the four primitive variables, which is solved by inversion. The gradients of the primitive variables provide the gradients of the Riemann invariants at each axial location, which, in turn, are used to update the values of Riemann invariants.

The process is repeated until the boundary conditions are matched (figure 1). As a measure of direct noise, we use the acoustic–acoustic reflection coefficient, ${\rm \pi} _1^- / {\rm \pi}_1^+$ and the acoustic–acoustic transmission coefficient, ${\rm \pi} _2^+ / {\rm \pi}_1^+$. As a measure of indirect noise, we use the entropic–acoustic reflection coefficient, $S_R={{\rm \pi} _1^-}/{\sigma _1}$, and the entropic–acoustic transmission coefficient, $S_T={{\rm \pi} _2^+}/{\sigma _1}$.

3. Physical interpretation of semiempirical non-isentropic models

In the literature, non-isentropic models for indirect noise are semiempirical and valid for compact nozzles (De Domenico et al. Reference De Domenico, Rolland and Hochgreb2019). The dissipation in the divergent section, which is caused by flow separation due to adverse pressure gradients, was modelled as an orifice plate. To do this, the non-isentropicity was embedded in one semiempirical parameter, $\beta = {A_j}/{A_2}$, where $j$ denotes the location at which non-isentropicity is assumed to begin (equivalent orifice area), and $2$ denotes the nozzle exit (inset in figure 2). The non-isentropicity parameter, $\beta$, was assumed to be related to the loss in stagnation pressure using the Borda–Carnot equation, which describes losses in mechanical energy due to sudden flow expansion. As a function of the pressure loss coefficient, De Domenico et al. (Reference De Domenico, Rolland and Hochgreb2019) showed that $C_{p0} = {(p_{0,j} - p_{0,2})}/(0.5\rho _j u_j^2)=(1 - \beta )^2$, where $p_{0,2}$ is the stagnation pressure at the outlet, and $p_{0,j} = p_{0,1}$. The semiempirical parameter, $\beta$, was calibrated experimentally by measuring the pressure loss as a function of the mass flow rate (De Domenico et al. Reference De Domenico, Rolland and Hochgreb2019).

Figure 2. Relation between the semiempirical parameter, $\beta$, in the literature (De Domenico et al. Reference De Domenico, Rolland and Hochgreb2019) (inset), and physical friction, $f$, proposed in this paper.

We propose a physical interpretation of the semiempirical parameter, $\beta$, and a physical equation for it. Consistently with De Domenico et al. (Reference De Domenico, Rolland and Hochgreb2019), we assume the nozzle to be divided into an isentropic part ($1 - j$), and a non-isentropic part ($j - 2$) (figure 2). Integrating the Gibbs equation and the energy equation in $j - 2$, i.e.

(3.1)\begin{equation} \int_{j}^{2}\frac{{\rm d}s}{R} = \int_{j}^{2}\frac{(1 - M^2)}{\varLambda} \frac{{\rm d} M}{M} + \int_{j}^{2}\frac{{\rm d} A}{A}, \end{equation}

yields

(3.2)\begin{equation} \beta = \left[\frac{M_2}{M_j}\left({\frac{\varLambda_j}{\varLambda_2}}\right)^{({\gamma + 1)}/{2(\gamma - 1)}}\right] \exp{\left(-\frac{\Delta s}{R}\right)}. \end{equation}

From (2.6), the net change of entropy is ${\Delta s}/{R} = - \log ({p_{0,2}}/{p_{0,j}})$, which, in terms of the pressure coefficient, is

(3.3)\begin{equation} \frac{\Delta s}{R} ={-} \log\left[1 - \left(\frac{C_{p0}\dfrac{1}{2}\gamma M_j^2 }{\varLambda_j}\right) \right]. \end{equation}

The properties at section $j$ are isentropic. For a fully isentropic nozzle $C_{p0} = 0$ ($\Delta s=0$), therefore, section $j$ coincides with $2$ with $\beta$ becoming unity.

To connect the semiempirical non-isentropicity factor, $\beta$, with the physical friction parameter, $f$, from (2.16) and (3.2), it can be shown that

(3.4)\begin{equation} \beta = \frac{M_2}{M_j}\left({\frac{\varLambda_j}{\varLambda_2}}\right)^{({\gamma + 1)}/{2(\gamma - 1)}}\left(\frac{\varLambda_1}{\varLambda_2}\right)^{{-}f\gamma(\gamma + 1)/2(\gamma - 1)\kappa}\left(\frac{\zeta_1}{\zeta_2}\right)^{(f\gamma - 2\tan\alpha)/{2\kappa}}. \end{equation}

As shown in figure 2, larger values of $\beta$ correspond to lower non-isentropicity, or, equivalently, lower values of the friction factor. For the nozzle geometry of De Domenico et al. (Reference De Domenico, Rolland and Hochgreb2019), which is used in this study, the lowest value is $\beta = 0.024$, which is attained when the throat area is the equivalent orifice plate area. As the effect of non-isentropicity decreases, i.e. $\beta$ increases, the friction factor, $f$, becomes negligible. From (3.4), for an isentropic nozzle, $\beta = 1$, the friction factor, $f$, is zero. The proposed model captures the variation of the friction factor with the throat Mach number. Physically, a flow with a higher throat Mach number needs less friction to generate the same entropy of a flow with a smaller throat Mach number. It can be shown from (2.4), (2.9) and (3.3) that

(3.5)\begin{equation} \frac{{\rm d} s}{{\rm d}M^2} \sim{-} \frac{{\rm d}f}{{\rm d}M^2} \sim (1-\beta)^2, \end{equation}

which means that, for a given non-isentropicity parameter, $\beta$, friction needs to decrease as the Mach number increases. Similarly, the sensitivity of the friction factor to the throat Mach number quadratically decreases with $\beta$, which becomes negligible for $\beta \gtrsim 0.1$ (figure 2). Equation (3.4) shows that the friction factor models (in a cross-averaged sense) the flow recirculation dissipation captured by the orifice-place parameter, $\beta$.

4. Indirect-noise transfer functions in a subsonic-choked regime

We calculate the transmission and reflection coefficients in a linear geometry nozzle in a subsonic (up to choked) regime. Both compact nozzles and non-compact nozzles are analysed. The predictions on the compact nozzle are validated against experimental data available in the literature (De Domenico et al. Reference De Domenico, Rolland and Hochgreb2019). The ambient temperature is 293.15 K, the exit pressure is $10^5$ Pa, and $\gamma =1.4$. The nozzle has inlet and outlet diameters of $46.2$ mm, throat diameter of $6.6$ mm, length of converging section of $24$ mm and length of divergent section of $230$ mm. The angles of the convergent and divergent sections are $\alpha _c = -40^{\circ }$ and $\alpha _d = 4^{\circ }$, respectively (figure 1). The vena contracta factor is $\varGamma = 0.89$, which is defined as the ratio between the cross-sectional area of the stream and the throat area (De Domenico et al. Reference De Domenico, Rolland and Hochgreb2019).

4.1. Mean flow

Figure 3 shows the mean flow parameterised with the friction. As shown in figure 3(a), because the outlet pressure $p_2$ is fixed, to compensate for the pressure loss, the inlet pressure $p_1$ increases as the friction increases. The effect becomes more evident as the inlet Mach number increases. As intuitively expected, higher friction results in larger pressure losses. As shown in figure 3(b), as the friction increases, the outlet Mach number becomes larger than the inlet Mach number (inlet and outlet areas are equal). This is consistent with a Fanno flow-like behaviour. As shown in figure 3(c), entropy is mostly generated near the throat, where the nozzle geometry switches from a converging (the flow is attached) to a diverging (the flow partly separates) regime. (Mathematically, the sign of the angle, $\alpha$, in (2.16) changes.) Physically, in the converging section, the only force opposing the flow is friction, but, in the divergent section, the force from the adverse pressure gradient also opposes the flow motion, which makes it susceptible to separation. At the throat, the overall opposing force changes abruptly, which makes the losses large. This can also be seen from the stagnation pressure equation

(4.1)\begin{equation} \frac{{\rm d}p_0}{p_0} = \frac{{\rm d}p}{p} + \frac{\gamma M^2/2}{\varLambda}\frac{{\rm d}M^2}{M^2}. \end{equation}

On the one hand, in an isentropic flow, the change in pressure term is in equilibrium with the inertia term (Mach number), thus, the stagnation pressure is constant. On the other hand, in a subsonic non-isentropic flow, the pressure gradient becomes adverse right after the throat (${\rm d}p>0$). This change of sign generates maximum entropy production (figure 3c). Figure 4(a) shows the effect of of the nozzle geometry. As the location of the throat moves downstream, the highest entropy generation location also changes. The limiting curves correspond to a diffuser ($L_c/L_{tot} = 0$) and a converging nozzle ($L_c/L_{tot} = 1$), in which the maximum increase in entropy takes place at the inlet and outlet, respectively. Finally, as shown in figure 4(b), as the throat-to-exit-area ratio increases, the entropy production decreases because the flow undergoes a reduced separation. In conclusion, the location of the throat determines the location of maximum entropy generation, whereas the throat-to-exit area ratio determines the amount of entropy generated. Although quasi-one-dimensional, the proposed model is able to capture the two-dimensional dissipation effects in the divergent section, which are averaged across the section, through the friction factor.

Figure 3. Mean flow: (a) ratio of inlet and exit pressures, (b) Mach number and (c) entropy. The arrows indicate increasing friction, $f$. The normalised throat location is $\eta _t = 0.095$.

Figure 4. Effect of geometry on the mean flow. (a) Converging-section-to-total-length ratio ($L_c/L_{tot}$) with $L_{tot}$ fixed. (b) Throat-to-exit area ratio ($A_t/A_e$) with $A_e$ fixed. (In the geometry considered in this paper, the exit area is equal to the inlet area.)

4.2. Indirect noise in compact nozzles

Figure 5 shows (i) the entropic–acoustic reflection, $S_R$, and transmission, $S_T$, coefficients; and (ii) the acoustic–acoustic reflection, ${\rm \pi} _1^-/{\rm \pi} _1^+$, and transmission, ${\rm \pi} _1^-/{\rm \pi} _1^+$, coefficients in a nearly compact nozzle ($He=0.0074$), which is the Helmholtz number of the experiment in De Domenico et al. (Reference De Domenico, Rolland and Hochgreb2019). On the one hand, in isentropic flows, the magnitudes of $S_R$ and $S_T$ are zero throughout. On the other hand, when friction is modelled, both $S_R$ and $S_T$ increase in magnitude. In a nearly compact nozzle, there is a negligible phase difference between the reflected and transmitted waves. The model predictions on entropic–acoustic reflection and transmission coefficients compare favourably with the theoretical results presented in figure 6 of De Domenico et al. (Reference De Domenico, Rolland and Hochgreb2019) where $\beta$ (discussed in § 3) was used to measure the degree of non-isentropicity. For a frictionless compact nozzle with equal inlet and exit areas, the acoustic–acoustic transmission coefficient is unity and the reflection coefficient is zero. In contrast, as the friction, $f$, increases, the reflection coefficient approaches unity for choked conditions, whereas the transmission coefficient approaches zero. This means that modelling friction, thereby relaxing the isentropic assumption, is key to the accurate prediction of indirect noise in subsonic nozzles. Physically, for higher values of friction, all the impinging acoustic waves tend to reflect back. Figure 5(c,d) shows the comparison of the model prediction on acoustic-to-acoustic transfer functions with the experimental data of De Domenico et al. (Reference De Domenico, Rolland and Hochgreb2019). The error bars are reported in De Domenico et al. (Reference De Domenico, Rolland, Rodrigues, Magri and Hochgreb2021). A favourable fit is found for $f = 0.07 - 0.08$, which is a physical range of friction in ducts (Shapiro Reference Shapiro1954).

Figure 5. Compact nozzle. Entropic–acoustic (a) reflection and (b) transmission coefficients for the (nearly) compact nozzle ($He = 0.0074$) of the experimental set-up in De Domenico et al. (Reference De Domenico, Rolland and Hochgreb2019).

4.3. Indirect noise in non-compact nozzles

Figure 6 shows the variation of the gain/phase of the indirect-noise transfer functions with the Helmholtz number. The magnitude of the reflection coefficient, $S_R$, has a non-monotonic behaviour, which increases up to $He \approx 0.1$ and, then, decreases for higher $He$. The magnitude of the transmission coefficient, $S_T$, however, increases monotonically with $He$. (In the limit of a frictionless nozzle, the predictions tend to those of the isentropic model of Duran & Moreau (Reference Duran and Moreau2013).) For compact nozzles, $S_R$ and $S_T$ are zero in frictionless nozzles of equal inlet and exit areas. This is because the velocity gradient is equal in the convergent and divergent sections, but with different signs. The sound waves, thus, cancel each other. However, in a non-compact nozzle, the spatial extent of the velocity gradient is not negligible, which causes sound waves not to cancel each other. Therefore, the sound is generated even in a nozzle with equal inlet and exit areas. As the friction increases, the amplitudes of the reflected and transmitted wave increase. Figure 6(c,d) shows a linear change in phase as the $He$ increases. The peak at small values of $He$ in frictionless nozzles is consistent with experimental and analytical investigation (Bake, Michel & Roehle Reference Bake, Michel and Roehle2006; Lourier et al. Reference Lourier, Huber, Noll and Aigner2014). The phase of reflected waves becomes a key parameter for the prediction of thermoacoustic instabilities, as explained in § 6.

Figure 6. Non-compact nozzle. Entropic–acoustic (a,c) reflection and (b,d) transmission coefficients. Gain (a,b) and phase (c,d). Throat Mach number $M_t = 0.8$.

5. Indirect-noise transfer functions in a supersonic regime

In the previous section, we show the analysis for a subsonic choked regime, which is relevant to nozzle guide vanes of realistic aircraft engines (Giusti, Magri & Zedda Reference Giusti, Magri and Zedda2019). In this section, we extend the study to a supersonic flow with and without a normal shock. A linear steady velocity choked profile is considered in the analysis to compare the results with Duran & Moreau (Reference Duran and Moreau2013). The Mach numbers at the inlet and exit are fixed to $0.29$ and $1.5$, respectively. The area is calculated at each location using (2.14) for different friction factors. The nozzle profile for the isentropic case ($f = 0$) is shown in figure 7.

Figure 7. (a) Nozzle profile for the study in the supersonic regime. (b) Mach number in the nozzle with steady linear velocity profile, with and without shock in the divergent section; $(M_1 = 0.29, M_2 = 1.5)$.

5.1. Supersonic flow without a shock wave

In a choked nozzle, the mass flow rate is maximum, equivalently, the choking condition is derived by imposing zero fluctuations of Mach number at the throat (Marble & Candel Reference Marble and Candel1977), $M^\prime /\bar M = 0$,

(5.1)\begin{equation} 2\frac{u^\prime}{u} + \frac{p^\prime}{\gamma p} (1 - \gamma) - \frac{s^\prime}{c}_p = 0. \end{equation}

Friction affects the choking condition through the mean-flow quantities, but it does not affect the form of the equation. Because the upstream acoustic wave changes direction at the throat, which gives rise to a singularity, the nozzle is divided into two sections on both sides of the throat (Duran & Moreau Reference Duran and Moreau2013). First, the flow is solved in the converging section. Second, the choking condition is imposed at the nozzle throat. Third, the flow is solved in the divergent section.

Figure 8 shows the effect that friction has on the entropic–acoustic reflected and transmitted waves. The results for the isentropic case ($f = 0$) match the results of Duran & Moreau (Reference Duran and Moreau2013) for the same nozzle profile. (The magnitudes of $S_R$ and $S_T$ are half of those of Duran & Moreau (Reference Duran and Moreau2013) because they defined the Riemann invariants without the $1/2$ factor.) The effect of non-isentropicity is significant in the diverging section. Because the nozzle is choked, the effect of non-isentropicity in the divergent section cannot travel upstream, which means that, in contrast to the subsonic case, the reflection coefficient is almost insensitive to friction (figure 8a,b). For small Helmholtz numbers, the magnitudes of the entropic–acoustic transfer functions are insensitive to friction. As the Helmholtz number increases, friction increases the magnitude of the transmission transfer function. In a supersonic nozzle, (4.1) and table 1 show that the sign of ${\rm d}p$ and ${\rm d}M$ remains constant. Therefore, as expected from (4.1), the change in stagnation pressure is smaller, which results in less entropy generation (relative to the subsonic-choked case).

Figure 8. Non-compact nozzle. Entropic–acoustic (a,c) reflection and (b,d) transmission coefficients. Gain (a,b) and phase (c,d). Supersonic nozzle with a linear velocity profile without a shock wave; $(M_a = 0.29, M_b = 1.5)$.

Table 1. Change in pressure and Mach number for supersonic and subsonic nozzles. Here $\eta$ is the nozzle spatial distance and $\eta _t$ is the location of the throat.

In conclusion, the effect of friction in the supersonic case without a shock wave is negligible on the reflected wave for any Helmholtz number, and it is negligible on the transmission wave for small Helmholtz numbers.

5.2. Supersonic flow with a shock wave

A normal shock wave is assumed to occur in the divergent section with a linear velocity profile (figure 7). To solve the flow, the nozzle is divided into two parts: the flow upstream of the shock, which is calculated using the supersonic flow of § 5.1; and downstream of the shock, which is calculated with the subsonic flow conditions of § 4. The shock wave is assumed to oscillate with an infinitesimal amplitude about a mean position. Therefore, the jump conditions have the same form as the linearised Rankine–Hugoniot for frictionless nozzles, which are (e.g. Marble & Candel Reference Marble and Candel1977; Stow, Dowling & Hynes Reference Stow, Dowling and Hynes2002; Moase, Brear & Manzie Reference Moase, Brear and Manzie2007; Goh & Morgans Reference Goh and Morgans2011; Leyko et al. Reference Leyko, Moreau, Nicoud and Poinsot2011)

(5.2)\begin{equation} \left.\begin{gathered} M_{s_2}^2 = \frac{1 + \dfrac{\gamma - 1}{2} M_{s_1}^2}{\gamma M_{s_1}^2 - \dfrac{\gamma - 1}{2}}, \\ {\rm \pi}_{s_2}^+{=} \frac{1 + M_{s_2}^2 M_{s_1} + M_{s_1}^2}{1 + M_{s_1}^2 M_{s_2} + M_{s_1}^2} {\rm \pi}_{s_1}^+{+} \frac{1 - M_{s_2}^2 M_{s_1} + M_{s_1}^2}{1 + M_{s_1}^2 M_{s_2} + M_{s_1}^2} {\rm \pi}_{s_1}^-, \\ \sigma_{s_2} = \sigma_{s_1} + \left( \frac{(\gamma - 1)(M_{s_1} - 1)^2}{M_{s_1}^2(2 + (\gamma - 1)M_{s_1}^2)}\right)({\rm \pi}_{s_2}^+{+} {\rm \pi}_{s_2}^-{-} {\rm \pi}_{s_1}^+{-} {\rm \pi}_{s_1}^-), \end{gathered}\right\} \end{equation}

where the nomenclature is defined in figure 7. Friction affects the mean flow quantities that appear in the jump conditions (5.2). (In this analysis, we assume that ${\rm \pi} _{s_2}^-$ is zero for brevity.) The nozzle response has common features with both the shockless supersonic and subsonic cases (figure 9). On the one hand, because the nozzle is choked, the reflection coefficient remains virtually unaffected by friction, as in the supersonic case without a shock (figure 9a,c). On the other hand, because the flow becomes subsonic after the shock wave, the transmission coefficient is affected by friction starting for small Helmholtz numbers, as in the subsonic case and in contrast to the shockless supersonic case (figure 7b,d). Friction tends to increase the magnitude of the transmitted wave, similarly to the observations in subsonic and supersonic cases 9(c,d). In the limit of zero friction, the trends of figure 7 match qualitatively the results of Goh & Morgans (Reference Goh and Morgans2011). (We use a slightly different nozzle geometry, therefore we do not expect the results to quantitatively match.)

Figure 9. Non-compact nozzle. Entropic–acoustic (a,c) reflection and (b,d) transmission coefficients. Gain (a,b) and phase (c,d). Supersonic nozzle with a linear velocity profile with a shock wave in the divergent section; $(M_a = 0.29, M_b = 1.5)$.

6. Thermoacoustic stability

We investigate the effect that dissipation in the nozzle guide vane has on thermoacoustic feedback. Geometrically, we consider a straight duct that models a laboratory combustor, also known as the Rijke tube (figure 10). The duct is characterised by a left-hand boundary condition with the acoustic-to-acoustic reflection coefficient $R_0$, and a right-hand boundary condition with the acoustic-to-acoustic reflection coefficient $R_1={\rm \pi} ^{-}_1/{\rm \pi} ^+_{1}$ and entropic-to-acoustic reflection coefficient $S_R={\rm \pi} ^{-}_1/\sigma _1$. The reflection coefficient $R_0=-1$ models an open end, which is fixed, whereas the reflection coefficient $R_1$ and $S_R$ are calculated from the proposed nozzle model (§ 2.1.1), which vary with the Helmholtz number, $He$, and the nozzle friction, $f$. The straight duct contains a pointwise heat source, which models the heat released by a flame that responds to acoustic perturbations as ${q^\prime } = n {u^\prime (t-\tau )}$, where $\tau$ is the flame time delay, $n$ is the flame interaction index (e.g. Dowling & Mahmoudi Reference Dowling and Mahmoudi2015). The conservation of mass, momentum and energy are enforced across the flame as jump conditions (Bloxsidge, Dowling & Langhorne Reference Bloxsidge, Dowling and Langhorne1988). After Laplace transformation of the jump conditions, the thermoacoustic stability is governed by a nonlinear eigenvalue problem

(6.1)\begin{equation} \boldsymbol{\mathsf{L}}(\lambda, n, \tau, M, R_0, R_1, S_R)\boldsymbol{q} = 0, \end{equation}

where $\boldsymbol{\mathsf{L}}$ is the scattering matrix, $\lambda$ is the complex eigenvalue, which is the solution of the dispersion relation $\det (\boldsymbol{\mathsf{L}})=0$, and $\boldsymbol {q}$ is the eigenvector that contains the outgoing waves. (The exact scattering matrix $\boldsymbol{\mathsf{L}}$ is reported in Appendix B of Aguilar, Magri & Juniper (Reference Aguilar, Magri and Juniper2017), in which all the details can be found.) If the growth rate of the eigenvalue (real part) is positive, $\mathcal {R}(\lambda )>0$, the thermoacoustic system is linearly unstable. As in Aguilar et al. (Reference Aguilar, Magri and Juniper2017), we fix $n=1$ and $\tau =10^{-3}\ \textrm {s}$. By numerically solving (6.1), we compute the eigenvalue loci $\lambda (R_1(He, f), S_R(He, f))=\lambda (He, f)$ to investigate the effect that the nozzle geometry and dissipation have on thermoacoustic stability. The nozzle throat Mach number is fixed to $M_t=0.8$ as in § 4.3.

Figure 10. Rijke tube with an open left-hand end. The nozzle with dissipation is attached downstream to provide the right-hand boundary conditions through $S_R$, which is the acoustic–entropic transfer function, and $R_1$, which is the acoustic–acoustic transfer function.

Figure 11 shows the trajectory of the dominant eigenvalues with respect to the nozzle guide vane friction and Helmholtz number. In isentropic nozzles, $f=0$, the modes become unstable as $He$ increases. For a small friction, $f = 0.005$, the mode with $He = 0$ (compact nozzle) is stable. As the friction factor increases, all modes become unstable regardless of the Helmholtz number. Figure 11(b) shows that, for higher values of friction factors, the dependence of the growth rate on $He$ decreases. Crucially, thermoacoustic stability can switch from stable to unstable. Physically, the nozzle Helmholtz number and friction factor change the phase of the waves that are reflected off the nozzle. When these waves are sufficiently in phase with the heat released by the flame, according to the Rayleigh criterion (Rayleigh Reference Rayleigh1896), interpreted in the frequency domain (Magri, Juniper & Moeck Reference Magri, Juniper and Moeck2020), the thermoacoustic system is linearly unstable. In conclusion, the analysis shows that the friction and the spatial extent of the nozzle guide vane can have a key effect on thermoacoustic stability.

Figure 11. Thermoacoustic stability (eigenvalue loci). (a) The three dominant eigenvalues (modes) for $He=0 - 0.5$ and friction factor $f = 0 - 0.04$. (b) Close-up on the first thermoacoustic mode. The axes are scaled by a factor of $2c_1/L_n=1553/1=1553\ \textrm {s}^{-1}$.