2. Non-isentropic linearised Euler equations with wall friction

The nozzle with varying area and wall friction is shown in figure 1. The gas inside the nozzle is assumed to be an ideal, calorically perfect gas, and the flow is considered inviscid and compressible. The present analysis adopts a quasi-one-dimensional formulation, in which the nozzle is assumed to be sufficiently long compared with its characteristic transverse dimension, such that the mean flow varies predominantly in the axial direction. In this framework, the flow variables at each axial location are taken as cross-sectional averages. In addition, the wavelengths of the acoustic and entropy perturbations of interest are assumed to be large compared with the transverse length scale of the nozzle, so that transverse variations of the perturbation fields remain weak (Duran & Moreau Reference Duran and Moreau2013).

Figure 1.

Sketch of the acoustic and entropy waves in a converging–diverging nozzle with skin friction factor, $f$ , along all nozzle walls.

Although the effects of wall friction are, in essence, not strictly one-dimensional, they are incorporated here through an equivalent axial momentum loss within the quasi-one-dimensional framework for modelling purposes. The Fanning friction factor $f(x)$ , which can vary axially along the nozzle, is used to express the wall-friction effects. Here, the volumetric forces, mass diffusion and heat transfer are not taken into account, and the continuity and momentum equations can be given as

(2.1) \begin{align} A\frac {{\partial \rho }}{{\partial t}} + \frac {{\partial (\rho Au)}}{{\partial x}} &= 0, \end{align}

(2.2) \begin{align} \frac {{\partial u}}{{\partial t}} + u\frac {{\partial u}}{{\partial x}} + \frac {1}{\rho }\frac {{\partial p}}{{\partial x}} &= - \frac {{4f}}{D}\frac {{{u^2}}}{2}, \end{align}

where $u$ , $\rho$ and $p$ denote the flow axial velocity, density and pressure, respectively. Here, the Fanning friction factor $f$ is defined via the wall shear stress $\tau _w=f\rho u^2/2$ , and is equal to one fourth the Darcy–Weisbach friction factor. Also, $A$ stands for the cross-sectional area of the nozzle, and $D$ denotes the diameter of the nozzle. Equation (2.2) represents the quasi-one-dimensional momentum equation for compressible flow with wall friction. It is obtained by applying an axial momentum balance to a control volume of variable cross-sectional area under the quasi-one-dimensional approximation, in which transverse velocity components are neglected. The effect of wall friction enters through the streamwise wall shear stress, resulting in a momentum loss term proportional to the friction factor and the wetted perimeter. Related derivations and formulations of the quasi-one-dimensional momentum equation with friction can be found in classical treatments of compressible internal flows (Shapiro Reference Shapiro1953).

Following the first law of thermodynamics, the conservation of energy for adiabatic nozzle flow with varying area and wall friction can be given as

(2.3) \begin{equation} \frac {{\partial \left ( {\rho {e_t}A\text{d}x} \right )}}{{\partial t}} + \frac {{\partial \left ( {\rho {e_t}uA} \right )}}{{\partial x}}\text{d}x + \frac {{\partial \left ( {puA} \right )}}{{\partial x}}\text{d}x = 0\, ,\end{equation}

with $e_t=e+u^2/2$ standing for the total energy per unit mass. Expanding the equation and subtracting the mass continuity (2.1) from it, we obtain

(2.4) \begin{equation} \rho \frac {{\text{D}e}}{{\text{D}t}} + \frac {\rho }{2}\frac {{\text{D}{u^2}}}{{\text{D}t}} + \frac {1}{A}\frac {{\partial \left ( {puA} \right )}}{{\partial x}} = 0\ ,\end{equation}

where $e$ is the internal energy per unit and $D/Dt=(\partial /\partial t)+u(\partial /\partial x)$ represents the total derivative. The entropy change can be evaluated from the thermodynamic relationship

(2.5) \begin{equation} T\text{d}s = \text{d}e - \big ( {p/{\rho ^2}} \big )\text{d}\rho ,\end{equation}

where $T$ and $s$ represent the temperature and entropy, respectively. Substituting the above equation and momentum (2.2) into (2.4) yields the energy conservation equation expressed in terms of entropy. It reads

(2.6) \begin{equation} \frac {{\partial s}}{{\partial t}} + u\frac {{\partial s}}{{\partial x}} = \frac {{4f}}{D}\frac {{{u^3}}}{2}\frac {{\rho R}}{p}\quad ,\end{equation}

where $R$ is the gas constant. Equation (2.6) is derived directly from the unsteady energy conservation with wall friction, in contrast to the formulation given by Jain & Magri (Reference Jain and Magri2022), which relates entropy generation to total-pressure loss and then evaluates the total-pressure change from the steady frictional flow. In the present work, the effect of wall friction is incorporated explicitly and consistently at the level of the unsteady governing equations, which is essential for linearisation and for capturing the unsteady contribution of wall friction to nozzle acoustics.

Together with the state equation $p=\rho RT$ , (2.1), (2.2) and (2.6) form the governing equations for quasi-one-dimensional nozzle flow with wall friction.

Under the linearisation assumption, the thermodynamic and flow variables are decomposed into a steady, time-averaged mean component and a time-dependent fluctuating component, which are denoted by an overbar and a prime, respectively. This decomposition enables the analysis of unsteady, small-amplitude perturbations superimposed on a steady base flow. The governing equations are then linearised with respect to these variables, and the mean components are collected and balanced, yielding the following system:

(2.7) \begin{align} &\frac {1}{{\bar \rho }}\frac {{{\textrm {d}}\bar \rho }}{{{\textrm {d}}x}} + \frac {1}{{\bar u}}\frac {{{\textrm {d}}\bar u}}{{{\textrm {d}}x}} + \frac {1}{A}\frac {{{\textrm {d}}A}}{{{\textrm {d}}x}} = 0, \end{align}

(2.8) \begin{align} &\ \, \bar u\frac {{{\textrm {d}}\bar u}}{{{\textrm {d}}x}} + \frac {1}{{\bar \rho }}\frac {{{\textrm {d}}\bar p}}{{{\textrm {d}}x}} = - \frac {{4f}}{D}\frac {{{{\bar u}^2}}}{2}, \end{align}

(2.9) \begin{align} &\quad \ \bar u\frac {{{\textrm {d}}\bar s}}{{{\textrm {d}}x}} = \frac {{4f}}{D}\frac {{{{\bar u}^3}}}{2}\frac {{\bar \rho R}}{{\bar p}}, \end{align}

(2.10) \begin{align} &\qquad \bar p = \bar \rho R\bar T. \end{align}

Here, we normalise the time-dependent quantities as

(2.11) \begin{equation} \hat p = {{p'}}/({{\gamma \bar p}}), \hat u = {{u'}}/{{\bar u}}, \hat s = {{s'}}/{{{c_p}}} ,\end{equation}

with $c_p$ denoting the specific heat capacity at constant pressure. By subtracting the mean components from the governing equations and retaining only the first-order fluctuating terms, the quasi-one-dimensional linearised Euler equations (LEE) for nozzle flow with wall friction are obtained in dimensionless form as

(2.12) \begin{align}&\quad \frac {{\partial \hat p}}{{\partial t}} + \bar u\frac {{\partial \left ( {\hat p + \hat u} \right )}}{{\partial x}} = \frac {{\gamma - 1}}{\gamma }\frac {{\bar \rho }}{{\bar p}}\frac {{4f}}{D}\frac {{{{\bar u}^3}}}{2}\left ( {2\hat u - \left ( {\gamma - 1} \right )\hat p - \hat s} \right )\!, \end{align}

(2.13) \begin{align}& \frac {{\partial \hat u}}{{\partial t}} + \bar u\frac {{\partial \hat u}}{{\partial x}} + \frac {{{{\bar c}^2}}}{{\bar u}}\frac {{\partial \hat p}}{{\partial x}} + \left ( {\frac {{\partial \bar u}}{{\partial x}} + \frac {{4f}}{D}\frac {{\bar u}}{2}} \right )\left ( {2\hat u - \left ( {\gamma - 1} \right )\hat p - \hat s} \right ) = 0, \end{align}

(2.14) \begin{align}&\qquad\,\, \frac {{\partial \hat s}}{{\partial t}} + \bar u\frac {{\partial \hat s}}{{\partial x}} = \frac {{\gamma - 1}}{\gamma }\frac {{\bar \rho }}{{\bar p}}\frac {{4f}}{D}\frac {{{{\bar u}^3}}}{2}\left ( {2\hat u - \left ( {\gamma - 1} \right )\hat p - \hat s} \right )\!, \end{align}

where $\bar c$ and $\gamma$ stand for the mean adiabatic speed of sound and the adiabatic index or heat capacity ratio, respectively. Equations (2.12)–(2.14) form a self-consistent set of linearised governing equations for non-isentropic nozzle flows with wall friction. They show that wall friction influences the unsteady dynamics through two complementary mechanisms: indirectly via the modified mean flow, and directly through source terms proportional to the friction factor that appear explicitly in the linearised equations. These source terms appear not only in the momentum equation, but also in the pressure and entropy equations, yielding a fully coupled description of acoustic, velocity and entropy fluctuations in the presence of friction.

To obtain the wave transfer functions using a Magnus-expansion approach, the normalised mass flow rate, total temperature and entropy fluctuations that are invariants of the isentropic flow at zero frequency are chosen as the variables for recasting the LEE. They read

(2.15) \begin{equation} {I_A} = {\dot m}'/{\bar {\dot m}}, {I_B} = {T_t}^\prime /{{\bar T}_t}, {I_c} = s'/{c_p}. \end{equation}

These invariants can be expressed in terms of the dimensionless primitive variables $\hat p$ , $\hat u$ and $\hat s$ as follows:

(2.16) \begin{align} {I_A} &= \hat u + \hat p - \hat s, \end{align}

(2.17) \begin{align} {I_B} &= \left ( {\gamma - 1} \right )\frac {{{M^2}\hat u + \hat p + \hat s/\left ( {\gamma - 1} \right )}}{{1 + \left ( {\gamma - 1} \right ){M^2}/2}}, \end{align}

(2.18) \begin{align} {I_C} &= \hat s , \end{align}

with $M={\bar u}/{\bar c}$ denoting the mean local Mach number. Given a vector containing the above invariants $\boldsymbol{I} =[I_A\ I_B\ I_C]^T$ , the LEE can be recast in the matrix form as

(2.19) \begin{equation} \frac {\partial }{{\partial t}}{\boldsymbol{{ I}}} + \bar u{{\boldsymbol{E}}_{{x}}}\frac {\partial }{{\partial x}}{\boldsymbol{I}} + \frac {{4f}}{D}\frac {{\bar u}}{2}{{\boldsymbol{E}}_{{s}}}{\boldsymbol{I}} = 0, \end{equation}

where ${\boldsymbol{E}}_{{x}}$ and ${\boldsymbol{E}}_{{s}}$ are non-constant 3 $\times$ 3 matrices defined by the mean flow. They read

(2.20) \begin{equation} {{\boldsymbol{E}}_{{x}}} = \left [ {\begin{array}{*{20}{c}} 1&{\frac {\varLambda }{{\left ( {\gamma - 1} \right ){M^2}}}}&{ - \frac {1}{{\left ( {\gamma - 1} \right ){M^2}}}}\\[8pt] {\frac {{\gamma - 1}}{\varLambda }}&1&{\frac {{\gamma - 1}}{\varLambda }}\\[8pt] 0&0&1 \end{array}} \right ]\!, \end{equation}

(2.21) \begin{align} & {{\boldsymbol{E}}_{{s}}} = \dfrac {1}{{{M^2} - 1}} \nonumber \\ &\quad \times\left [\!\!\! {\begin{array}{*{20}{c}} { - 2\varLambda }&{\dfrac {{\varLambda \left ( {\gamma + 1} \right )}}{{\left ( {\gamma - 1} \right )}}}&{ - \dfrac {1}{{\left ( {\gamma - 1} \right ){M^2}}}}\\[12pt] { - 2\left ( {\gamma - 1} \right )\!\left ( \!{{M^2} - \dfrac {{\bar \rho }}{{\bar p}}{{\bar u}^2}} \right )}&{\left ( \!{{M^2} - \dfrac {{\bar \rho }}{{\bar p}}{{\bar u}^2}} \right )\!\left ( {\gamma + 1} \right )}&{\dfrac {{\gamma - 1}}{\varLambda }\!\left (\! {{M^2} - \dfrac {{\bar \rho }}{{\bar p}}{{\bar u}^2}} \right )\!\left (\! { - \gamma {M^2} - \dfrac {{2\gamma }}{{\gamma - 1}}} \right )}\\[12pt] {2\varLambda \dfrac {{\gamma - 1}}{\gamma }\dfrac {{\bar \rho }}{{\bar p}}{{\bar u}^2}}&{ - \varLambda \dfrac {{\gamma + 1}}{\gamma }\dfrac {{\bar \rho }}{{\bar p}}{{\bar u}^2}}&{ - \dfrac {{\gamma - 1}}{\gamma }\dfrac {{\bar \rho }}{{\bar p}}{{\bar u}^2}\left ( { - \gamma {M^2} - \dfrac {{2\gamma }}{{\gamma - 1}}} \right )} \end{array}} \!\!\!\!\right ]\!, \end{align}

where $\varLambda =1+(\gamma -1)M^2/2$ .

3. General analytical solution

Assuming the above time-dependent fluctuating quantities to be harmonic, we can take $\hat z=z^*e^{i\omega t}$ , where $\omega$ stands for the angular frequency and $z^*$ is the wave amplitude. Applying the temporal Fourier transform to recast (2.19) in the frequency domain yields

(3.1) \begin{equation} \begin{array}{*{20}{c}} {\frac {\text{d}}{{\text{d}x}}{\boldsymbol{I}} = {\boldsymbol{A}}(x,\omega ,f){\boldsymbol{I}}},& \text{where} \quad {{\boldsymbol{A}} = - {{[{{\boldsymbol{E}}_{{x}}}]}^{ - 1}}\left ( {\frac {{i\omega }}{{\bar u}}{\boldsymbol{J}} + \frac {{4f}}{{2D}}{{\boldsymbol{E}}_{{s}}}} \right )}, \end{array} \end{equation}

with $\boldsymbol{J}$ denoting a third-order identity matrix. Equation (3.1) is similar to the equation obtained by Duran & Moreau (Reference Duran and Moreau2013) but with an extra term, involving the matrix $E_s$ , which accounts for wall-friction effects. The above formulation requires the matrix ${\boldsymbol{E}}_{{x}}$ to be invertible. When the determinant of ${\boldsymbol{E}}_{{x}}$ is non-zero, i.e. $M \ne 1$ , the matrix $\boldsymbol{A}$ is well defined. For choked-nozzle flows, the Mach number reaches unity at the throat, where $\det ({\boldsymbol{E}}_{{x}})=0$ and the matrix system cannot be inverted. This case will be discussed in § 3.2.

The position and frequency are specified in terms of the non-dimensional parameters $\xi =x/L$ (dimensionless axial coordinate) and $\textit{He} = \omega L/\bar c_0$ (axial Helmholtz number), respectively, where $L$ denotes the nozzle length and $\bar c_0$ corresponds to the stagnation speed of sound at the inlet. The first-order system of differential (3.1) is solved using a Magnus-expansion-based method (Blanes et al. Reference Blanes, Casas, Oteo and Ros2009). In this method, the solution is expressed as an infinite expansion around the axial Helmholtz number $\textit{He}$ and the wall-friction factor $f$ , and the infinite series expansion is truncated to a certain order that guarantees reasonable accuracy for the model under given conditions. The final solution is expressed as a relation between the flow invariants

(3.2) \begin{equation} \boldsymbol{I}(\xi , \textit{He}, f) = \left [ \exp \left ( \boldsymbol{B}(\xi , \textit{He}, f) \right ) \right ] \boldsymbol{I_0} \quad \text{with} \quad \boldsymbol{B}(\xi , \textit{He}, f) = \sum _{k=0}^{\infty } \boldsymbol{B}^{(k)}(\xi , \textit{He}, f), \end{equation}

where $\boldsymbol{I_0}$ is the value of the invariant vector at the inlet. Instead of expanding $\boldsymbol{I}$ directly, the asymptotic expansion is performed in terms of $\boldsymbol{B}$ , where $\boldsymbol{B}^{(k)}$ represents the $k$ th-order term of the Magnus expansion, each of which is of order $O(\textit{He},f)^k$ , with $k$ being the expansion order. Following the method discussed by Blanes et al. (Reference Blanes, Casas, Oteo and Ros2009) and Duran & Moreau (Reference Duran and Moreau2013), the matrices $\boldsymbol{B}^{(k)}$ are obtained with a recursive procedure, reducing computational costs.

The exponential of the matrix $\boldsymbol{B}$ in (3.2) can be expanded using Taylor series up to an order of $k$ to obtain the invariant transfer matrix at the same accuracy as the truncated Magnus series. Here, we first introduce a term that captures the effects of non-isentropicity induced by wall friction and non-zero frequency as the expansion parameters in the Magnus framework. The higher-order terms capture the deviations arising from wall friction and non-zero frequency.

Achieving convergence of the series solution sometimes requires dividing the nozzle into multiple axial segments, within which the transfer matrices are computed independently and subsequently combined. In the isentropic subcritical nozzle flow, the series converges rapidly across the entire nozzle length at zero frequency and low frequencies, making segmentation unnecessary. In contrast, at higher frequencies, the series converges more slowly, and the nozzle must be discretised into several axial sections to maintain computational efficiency and accuracy. This segmentation approach achieves fast convergence when the condition given by

(3.3) \begin{equation} \int _0^{\xi _F} \| \boldsymbol{A}(\xi ) \|_2 \, \mathrm{d}\xi \lt \pi ,\end{equation}

with $\xi _F$ denoting that the maximum length of the segments is satisfied (Blanes et al. Reference Blanes, Casas, Oteo and Ros2009), but may diverge when the value of the integral is larger than $\pi$ .

Considering the nozzle flow shown in figure 1, the wave components at a particular location are represented using the wave vector $\boldsymbol{W} =[w^+\ w^-\ w^s]^T$ , where $w^+$ , $w^-$ and $w^s$ stand for the downstream-propagating acoustic wave, the upstream-propagating acoustic wave and the entropy wave, respectively. These waves can be expressed as functions of the primitive variables in a non-dimensional form, as done by Duran & Moreau (Reference Duran and Moreau2013) and Yeddula et al. (Reference Yeddula, Guzmán-Iñigo and Morgans2022). They read

(3.4) \begin{equation} w^+=p^*+Mu^*, w^-=p^*-Mu^*, w^s=s^*, \end{equation}

where $p^*$ , $u^*$ and $s^*$ represent the complex Fourier coefficients for pressure, velocity and entropy, respectively. Therefore, the flow invariants can be related to the wave vector $\boldsymbol{W}$ using the matrix $\boldsymbol{D}$ as follows:

(3.5) \begin{equation} \begin{bmatrix} I_A \\[3pt] I_B \\[3pt] I_C \end{bmatrix} =\boldsymbol{DW}= \begin{bmatrix} \frac {1}{2M} (1 + M) & -\frac {1}{2M} (1 - M) & -1 \\[8pt] \frac {\gamma -1}{2\varLambda } (1 + M) & \frac {\gamma -1}{2\varLambda } (1 - M) & \frac {1}{\varLambda } \\[8pt] 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} w^+ \\ w^- \\ w^s \end{bmatrix}. \end{equation}

A transfer matrix $\boldsymbol{T}$ is therefore defined to relate the wave vector at any location, $\xi$ , to that at the inlet, i.e. $\boldsymbol{W}_0$ , as

(3.6) \begin{equation} \boldsymbol{W}(\xi ) = \boldsymbol{T}(\xi )\, \boldsymbol{W}_0\ \text{with}\ \boldsymbol{T}(\xi )\ = [\boldsymbol{D}(\xi )]^{-1}\boldsymbol{C}(\xi )\boldsymbol{D}_0, \end{equation}

where $\boldsymbol{C}(\xi )=\exp ( \boldsymbol{B}(\xi , \textit{He}, f) )$ .

3.1. Subsonic flow

For a nozzle sustaining subsonic flow Mach numbers, three different incoming waves are imposed that correspond to a downstream-propagating acoustic wave at the inlet ( $w^+_{0,f}$ ), an entropy wave at the inlet ( $w^s_{0,f}$ ) and an upstream-propagating acoustic wave at the outlet ( $w^-_{1,f}$ ) as shown in figure 1. Here, the subscript $'f'$ denotes externally forced waves and should not be confused with the wall-friction factor $f$ . These incoming waves result in three outgoing waves, namely, an upstream-propagating acoustic wave at the inlet ( $w^-_0$ ), a downstream-propagating acoustic wave at the outlet ( $w^+_1$ ) and an entropy wave at the outlet ( $w^s_1$ ).

A transfer function relates the outgoing wave to the incoming externally forced wave, which is defined by an extended scattering matrix $\boldsymbol{S}$ as

(3.7) \begin{equation} \begin{bmatrix} w_1^+ \\[3pt] w_0^- \\[3pt] w_1^s \end{bmatrix} = \boldsymbol{S} \begin{bmatrix} w_{0,f}^+ \\[3pt] w_{1,f}^- \\[3pt] w_{0,f}^s \end{bmatrix} .\end{equation}

Note that, since $\boldsymbol{W}(\xi )$ in (3.6) denotes the wave vector at an arbitrary axial location, the transfer matrix $\boldsymbol{T}$ relates both incoming and outgoing wave components. To construct the scattering matrix $\boldsymbol{S}$ , the reflected wave $w_0^-$ is moved to the left-hand side, while the imposed outlet forcing $w_{1,f}^-$ is moved to the right-hand side. The resulting system is then reordered to express the outgoing waves as linear functions of the externally forced waves, which defines the scattering matrix $\boldsymbol{S}$ .

3.2. Choked flow without shock waves

For a choked flow without any shock waves, the upstream-propagating acoustic wave ( $w^-_{1}$ ) changes the propagation direction at the throat, causing a singularity at the throat of a choked nozzle. There are only two incoming waves, $w^+_{0,f}$ and $w^s_{0,f}$ , which can be applied, with fast and slow-propagating acoustic waves at the outlet ( $w^+_{1}$ and $w^-_{1}$ ), an entropy wave at the outlet ( $w^s_{1}$ ) and an upstream-propagating acoustic wave at the inlet ( $w^-_{0}$ ) as the outgoing waves.

The converging–diverging nozzle is then partitioned into two regions: a subsonic section extending from the inlet to a location infinitesimally upstream of the throat ( $x_u$ ), and a supersonic section extending from a point infinitesimally downstream of the throat ( $x_d$ ) to the outlet. The subscripts ` $u$ ’ and ` $d$ ’ denote positions immediately upstream and downstream of the throat, respectively. For the subsonic section, Mach-number fluctuations cannot occur at the choked throat, as the throat is an infinitesimally wide section (Marble & Candel Reference Marble and Candel1977; Moase, Brear & Manzie 2007). The unsteady response of the subsonic region can be calculated, imposing two waves at the inlet ( $w^+_{0,f}$ and $w^s_{0,f}$ ) and calculating the upstream-propagating acoustic wave at the throat ( $w^-_u$ ) using the boundary condition $ M^\prime /M=u^*-(\gamma -1)p^*/2-s^*/2=0$ at the nozzle throat. Although this boundary condition is not exact at non-zero frequencies, as reported by Olivon et al. (Reference Olivon, Genot, Hirschberg, Moreau and Hirschberg2025), its influence on the resulting acoustic response was shown to be negligible, and it is therefore adopted in the present analysis. Enforcing this condition yields a linear relationship between the upstream-propagating acoustic wave at a location infinitesimally upstream of the throat and the two downstream-propagating waves, namely the acoustic and entropy waves. This leads to

(3.8) \begin{align}&\qquad\quad w_u^- = R_u w_u^+ + R_s w_u^s, \end{align}

(3.9) \begin{align}& \text{with}\quad R_u = \frac {3 - \gamma }{\gamma + 1} \quad \text{and} \quad R_s = \frac {-2}{\gamma + 1}, \end{align}

where $R_u$ and $R_s$ denote the acoustic and entropy reflection coefficients, respectively. The subsonic part of the nozzle is therefore governed by three input conditions: the acoustic and entropy forcing inputs at the inlet and the throat relation given by (3.8). With these conditions, the subsonic region can be solved using the same procedure as in § 3.1.

In the choked flow without any shock waves, the upstream-propagating acoustic wave from the downstream supersonic region at the throat, $w_{u,f}^-$ , is zero, because no upstream-propagating disturbance can be transmitted to the throat (Yeddula et al. Reference Yeddula, Guzmán-Iñigo and Morgans2022). Similar to (3.7) of the subsonic case, collecting the forcing inputs on one side and the unknown wave amplitudes on the other, the system can be recast as

(3.10) \begin{equation} \begin{bmatrix} w_u^+ \\[3pt] w_0^- \\[3pt] w_u^s \end{bmatrix} = \boldsymbol{S_{sub}} \begin{bmatrix} w_{0,f}^+ \\[3pt] w_{u,f}^- \\[3pt] w_{0,f}^s \end{bmatrix} =\boldsymbol{S_{sub}} \begin{bmatrix} w_{0,f}^+ \\[4pt] 0 \\ w_{0,f}^s \end{bmatrix}, \end{equation}

where $\boldsymbol{S_{sub}}$ is defined as the scattering matrix of the subsonic part for the choked flow. Excluding the $w_{0}^-$ in (3.7) yields

(3.11) \begin{equation} \boldsymbol{W_u}= \begin{bmatrix} w_u^+ \\[3pt] 0\\ w_u^s \end{bmatrix} = \boldsymbol{S'} \begin{bmatrix} w_{0,f}^+ \\[3pt] 0 \\ w_{0,f}^s \end{bmatrix}, \end{equation}

where $\boldsymbol{S'}$ can be separated from $\boldsymbol{S_{sub}}$ . The wave vector infinitesimally upstream of the throat propagates unaltered at the throat and hence $\boldsymbol{W_u}=\boldsymbol{W_d}$ . The transfer matrix in the supersonic portion of the nozzle is denoted by $\boldsymbol{T_{su}}$ , and relates the wave vectors as $\boldsymbol{W_1}=\boldsymbol{T_{su}}\boldsymbol{W_d}=\boldsymbol{T_{su}}\boldsymbol{W_u}$ . Here, $\boldsymbol{T_{su}} = [\boldsymbol{D_1}]^{-1}\boldsymbol{CD_d}$ , where matrix $\boldsymbol{D_d}$ relates the wave vector to the flow invariants at the throat and matrix $\boldsymbol{C}$ connects these flow invariants at the throat to the outlet. Combining the subsonic and supersonic parts, we get

(3.12) \begin{equation} \begin{bmatrix} w_1^+ \\[3pt] w_1^- \\[3pt] w_1^s \end{bmatrix} = \boldsymbol{T_{su}S'} \begin{bmatrix} w_{0,f}^+ \\[3pt] 0 \\ w_{0,f}^s \end{bmatrix}. \end{equation}

Equation (3.12) expresses the fast ( $w_1^+$ ) and slow ( $w_1^-$ ) propagating acoustic waves and the entropy wave ( $w_1^s$ ) at the outlet as a function of the acoustic ( $w_0^+$ ) and entropy ( $w_0^s$ ) forcing at the nozzle inlet. The upstream-propagating wave at the inlet ( $w_0^-$ ) in the subsonic portion is given by (3.10).

5. A solution for the nozzle flow with both wall friction and heat transfer

The above sections present a model for the acoustic and entropic transfer functions of non-isentropic nozzle flows in which wall friction is the sole source of non-isentropicity, and the nozzle wall is assumed to be adiabatic. Steady heat transfer is another important factor inducing non-isentropic nozzle flows, and occurs widely when there is heat exchange with surroundings and reactions (Yeddula et al. Reference Yeddula, Guzmán-Iñigo and Morgans2022). To account for this, the formulation is extended to incorporate steady heat transfer alongside wall friction. Such combined effects arise, for example, in actively cooled nozzles or exhaust ducts, as well as in configurations subject to net radiative heat losses to the surroundings. Here, we include a measure of these two critical flow non-isentropic factors as expansion parameters for the first time and develop models for nozzle flow.

When accounting for heat transfer, the energy conservation equation expressed in terms of entropy can be reformulated as follows:

(5.1) \begin{equation} \frac {{\partial s}}{{\partial t}} + u\frac {{\partial s}}{{\partial x}} = \frac {R}{p}\dot Q + \frac {{4f}}{D}\frac {{{u^3}}}{2}\frac {{\rho R}}{p} ,\end{equation}

where $\bar {\dot {Q}}$ stands for the steady volumetric heat source term. This equation governs nozzle flow with wall friction and heat transfer together with (2.1) and (2.2). Here, we focus on the primary effect of mean heat transfer for simplicity, and assume that there are no heat fluctuations inside the nozzle. The time-averaged governing equations with mean components read

(5.2) \begin{align} &\frac {1}{{\bar \rho }}\frac {{{\textrm {d}}\bar \rho }}{{{\textrm {d}}x}} + \frac {1}{{\bar u}}\frac {{{\textrm {d}}\bar u}}{{{\textrm {d}}x}} + \frac {1}{A}\frac {{{\textrm {d}}A}}{{{\textrm {d}}x}} = 0 ,\end{align}

(5.3) \begin{align} &\,\bar u\frac {{{\textrm {d}}\bar u}}{{{\textrm {d}}x}} + \frac {1}{{\bar \rho }}\frac {{{\textrm {d}}\bar p}}{{{\textrm {d}}x}} = - \frac {{4f}}{D}\frac {{{{\bar u}^2}}}{2} , \end{align}

(5.4) \begin{align} &\bar u\frac {{{\textrm {d}}\bar s}}{{{\textrm {d}}x}} = \frac {{{R}}}{{\bar p}}\bar {\dot Q} + \frac {{4f}}{D}\frac {{{{\bar u}^3}}}{2}\frac {{\bar \rho R}}{{\bar p}}. \end{align}

The linearised form of the Euler equations for the nozzle flow with wall friction and steady heat transfer are further written in terms of the normalised fluctuating variables $\hat {p}$ , $\hat {u}$ and $\hat {s}$ . This gives

(5.5) \begin{align}& \frac {{\partial \hat p}}{{\partial t}} + \bar u\frac {{\partial \left ( {\hat p + \hat u} \right )}}{{\partial x}} = \frac {{\gamma - 1}}{\gamma }\frac {{\bar \rho }}{{\bar p}}\frac {{4f}}{D}\frac {{{{\bar u}^3}}}{2}\left ( {2\hat u - \left ( {\gamma - 1} \right )\hat p - \hat s} \right ) - \frac {{\gamma - 1}}{\gamma }\frac {{\bar {\dot Q}}}{{\bar p}}(\hat u + \gamma \hat p) , \end{align}

(5.6) \begin{align}&\qquad\quad \frac {{\partial \hat u}}{{\partial t}} + \bar u\frac {{\partial \hat u}}{{\partial x}} + \frac {{{{\bar c}^2}}}{{\bar u}}\frac {{\partial \hat p}}{{\partial x}} + \left ( {\frac {{\partial \bar u}}{{\partial x}} + \frac {{4f}}{D}\frac {{\bar u}}{2}} \right )\left ( {2\hat u - \left ( {\gamma - 1} \right )\hat p - \hat s} \right ) = 0 , \end{align}

(5.7) \begin{align}&\quad\,\, \frac {{\partial \hat s}}{{\partial t}} + \bar u\frac {{\partial \hat s}}{{\partial x}} = \frac {{\gamma - 1}}{\gamma }\frac {{\bar \rho }}{{\bar p}}\frac {{4f}}{D}\frac {{{{\bar u}^3}}}{2}\left ( {2\hat u - \left ( {\gamma - 1} \right )\hat p - \hat s} \right ) - \frac {{\gamma - 1}}{\gamma }\frac {{\bar {\dot Q}}}{{\bar p}}(\hat u + \gamma \hat p). \end{align}

The linearised Euler equations, (5.5)–(5.7), govern the unsteady flow in a nozzle sustaining mean non-isentropic flow with both wall friction and steady heat transfer. It is worth noting that both (5.5) and (5.7) contain two source terms, arising from wall friction and steady heat transfer, respectively, while the momentum equation features only a source term associated with wall friction, which means heat transfer has no direct influence on the momentum balance. When only steady heat transfer is considered (i.e. $f$ = 0), the system simplifies to the LEE presented by Yeddula et al. (Reference Yeddula, Guzmán-Iñigo and Morgans2022). The Magnus-expansion method described in § 3 is further employed to obtain the analytical solution for the converging–diverging nozzle at all frequencies. Here, we continue to use the converging–diverging nozzle whose area profile is given by (4.1). A fifth-order Magnus expansion is employed to predict the coefficients of the acoustic transfer functions, and the computational set-up outlined in § 4 is adopted to obtain the numerical solutions.

For clarity and brevity, we focus on the results for the subsonic case. Although both wall friction and heat transfer introduce non-isentropic effects, their impacts on the mean nozzle flow differ. The equation for the mean temperature in nozzle flow with wall friction and heat transfer is given as follows:

(5.8) \begin{equation} \frac {{\text{d}\bar T}}{{\text{d}x}} = - \frac {{\gamma - 1}}{{\gamma R}}\bar u\frac {{\text{d}\bar u}}{{\text{d}x}}+ \frac {{\gamma - 1}}{{\gamma R}}\frac {{\bar {\dot Q}}}{{\bar \rho \bar u}}. \end{equation}

This equation is recalled here to facilitate the interpretation of the results discussed below. In particular, the heat-transfer term appears explicitly as a source in the mean temperature equation and therefore directly controls the temperature level, whereas wall friction influences the temperature only indirectly through its effect on the velocity field. With regard to Mach-number variation, Yeddula et al. (Reference Yeddula, Guzmán-Iñigo and Morgans2022) proposed that a steady heat source ( ${\bar {\dot Q}}\gt 0$ ) leads to an increase in Mach number downstream of the throat, while a heat sink ( ${\bar {\dot Q}}\lt 0$ ) results in a decrease. Thus, a heat source acts in a manner analogous to wall friction, while a heat sink produces the opposite effect.

Motivated by this observation, we explore whether negative heat transfer can be used to offset the influence of wall friction. Such conditions arise, for example, in actively cooled nozzles or in nozzles subject to net radiative heat losses, where wall friction and negative heat transfer naturally coexist. We consider a case where both wall friction and negative heat transfer are present. Figure 7 presents the spatial distribution of the mean Mach number and temperature inside the subsonic nozzle with both wall friction and negative heat transfer ( $f=0.015$ , ${\tilde {\dot Q}}=$ −0.35), where ${\tilde {\dot Q}}=\bar {\dot Q}L/{\bar p_0}{\bar c_0}$ , with $\bar p_0$ and $\bar c_0$ denoting the stagnation pressure and stagnation speed of sound at the inlet, respectively. It is observed that the Mach-number distribution closely follows that of the isentropic solution, indicating that negative heat transfer can effectively offset the effect of wall friction on the Mach number in the mean flow. The temperature at the outlet is close to the case with heat transfer only, and is notably reduced compared with the isentropic case, which is consistent with (5.8).

Figure 7.

Mach number, $M$ , (lines in blue) and mean temperature, $\bar T$ , (lines in red) for subsonic case with different levels of wall friction and heat transfer, where ${\tilde {\dot Q}}=\bar {\dot Q}L/{\bar p_0}{\bar c_0}$ , with $\bar p_0$ and $\bar c_0$ denoting the stagnation pressure and stagnation speed of sound. Here, $f=0$ , ${\tilde {\dot Q}}=0$ (solid line); $f=0.015$ , ${\tilde {\dot Q}}=0$ (dotdashed line); $f=0$ , ${\tilde {\dot Q}}=$ −0.35 (dotted line); $f=0.015$ , ${\tilde {\dot Q}}=$ -0.35 (dashed line).

Figure 8.

Transfer functions of the upstream-propagating acoustic wave at the inlet, $w^-_0$ , for the subcritical nozzle flow in the frequency domain ( $\textit{He}$ ) for the isentropic case, case of $f=0.015$ , ${\tilde {\dot Q}}=0$ and case of $f=0.015$ , ${\tilde {\dot Q}}=-0.35$ . Model predictions and numerical solutions are included.

Figure 9.

Transfer functions of the downstream-propagating acoustic wave at the outlet, $w^+_1$ , for the subcritical nozzle flow in the frequency domain ( $\textit{He}$ ) for the isentropic case, case of $f=0.015$ , ${\tilde {\dot Q}}=0$ and case of $f=0.015$ , ${\tilde {\dot Q}}=-0.35$ . Model predictions and numerical solutions are included.

Figure 9. Long description

The image contains six line graphs labeled (a) through (f), each depicting different aspects of the transfer functions of downstream-propagating acoustic waves for subcritical nozzle flow in the frequency domain. Panel A: A line graph showing the magnitude of the transfer function for the isentropic case. The x-axis represents the frequency (He/2π) ranging from 0 to 2.0, and the y-axis represents the magnitude of the transfer function (|w+/w0,f|) ranging from 0.9 to 1.3. Different lines represent model predictions and numerical solutions. Panel B: A line graph showing the magnitude of the transfer function for the case of T0’=0. The x-axis represents the frequency (He/2π) ranging from 0 to 2.0, and the y-axis represents the magnitude of the transfer function (|w+/w1,f|) ranging from 0.0 to 0.20. Different lines represent model predictions and numerical solutions. Panel C: A line graph showing the magnitude of the transfer function for the case of T0’=0.5. The x-axis represents the frequency (He/2π) ranging from 0 to 2.0, and the y-axis represents the magnitude of the transfer function (|w+/w0,f|) ranging from 0.0 to 0.15. Different lines represent model predictions and numerical solutions. Panel D: A line graph showing the phase of the transfer function for the isentropic case. The x-axis represents the frequency (He/2π) ranging from 0 to 2.0, and the y-axis represents the phase of the transfer function (〈w+/πw0,f〉) ranging from −1.0 to 1.0. Different lines represent model predictions and numerical solutions. Panel E: A line graph showing the phase of the transfer function for the case of T0’=0. The x-axis represents the frequency (He/2π) ranging from 0 to 2.0, and the y-axis represents the phase of the transfer function (〈w+/πw1,f〉) ranging from −1.0 to 1.0. Different lines represent model predictions and numerical solutions. Panel F: A line graph showing the phase of the transfer function for the case of T0’=0.5. The x-axis represents the frequency (He/2π) ranging from 0 to 2.0, and the y-axis represents the phase of the transfer function (〈w+/πw0,f〉) ranging from −1.0 to 1.0. Different lines represent model predictions and numerical solutions.

The influence of the wall friction and negative heat transfer on the unsteady response of the nozzle flow is explored further. Figures 8 and 9 show the variation of the magnitude and phase of the acoustic transfer functions for the isentropic and different non-isentropic cases. It is observed again that the model estimates closely match the numerical predictions. When both wall friction and negative heat transfer are included, the magnitudes of $w^-_0/w^+_{0,f}$ , $w^-_0/w^s_{0,f}$ and $w^+_1/w^s_{0, f}$ exhibit significant deviations from the isentropic case at low frequencies, which quickly disappear with increasing frequency. In contrast to the case with wall friction only, the introduction of negative heat transfer reduces the magnitude of the acoustic response to entropy fluctuations at low frequencies. For the response to the upstream-propagating acoustic wave at the outlet, the magnitude is increased compared with the case with wall friction only. Moreover, regarding the transfer function $w^+_1/w^+_{0,f}$ , the presence of wall friction results in a noticeably smaller magnitude compared with the isentropic case. This behaviour is associated with the additional terms introduced by wall friction in (2.12)–(2.14), which modify the coupling between acoustic, velocity and entropy fluctuations as the wave propagates through the nozzle, thereby reducing the effective transmission of the downstream-propagating acoustic wave. However, upon introducing negative heat transfer, the magnitude increases to a level even higher than that of the isentropic case, indicating that negative heat transfer can partially offset this reduction and increase the acoustic gain.

These results highlight that, even when the combined effect of wall friction and negative heat transfer leads to a mean Mach-number distribution close to the isentropic case, the unsteady acoustic and entropy responses can still differ. This is because both wall friction and heat transfer influence the unsteady response not only indirectly through the mean flow, but also directly through additional source terms in the LEE. Although negative heat transfer does not eliminate the influence of wall friction on the Mach-number distribution as in the mean flow, it nevertheless counteracts the effect of wall friction in the unsteady response.