2.1. Governing equations

Consider an internal flow consisting of two parallel streams (cf. figure 1) of the same ideal gas. The theory can be extended to any number of streams, but this generalisation is left for future work. The streams are modelled as uniform 1-D streams, i.e. in terms of cross-stream averaged quantities. The main assumption for compound-compressible flows is that the static pressure is uniform in all cross-sections, hence

(2.1)\begin{equation} p_p(x) = p_s(x) = p(x), \end{equation}

where $p$ denotes the static pressure, $x$ is the axial coordinate and the subscripts $p$ and $s$ refer to the primary and secondary stream. The distribution of the cross-sections $A_i(x)$, with $i \in [p,s]$, is initially unknown and should be calculated. These are subject to a geometrical constraint imposed by the cross-section of the channel $A(x)$:

(2.2)\begin{equation} \sum_{i\in[p,s]} A_i(x) = A(x). \end{equation}

The governing equations consist of the conservation of mass, momentum and energy in each stream. For a steady, quasi-1-D flow, these read as follows:

(2.3)$$\begin{gather} \frac{{\rm d}}{{\rm d}\kern0.7pt x} (\rho_i A_i u_i) = 0, \end{gather}$$

(2.4)$$\begin{gather}\frac{{\rm d}}{{\rm d}\kern0.7pt x} (\rho_i A_i u_i^2) ={-}A_i \frac{{\rm d}p_i}{{\rm d}\kern0.7pt x} + F_i, \end{gather}$$

(2.5)$$\begin{gather}\frac{{\rm d}}{{\rm d}\kern0.7pt x} (\rho_i A_i u_i h_{t,i}) = 0, \end{gather}$$

where $\rho$ denotes the density, $u$ denotes the axial velocity, $F_i$ is the net force per unit length exerted on stream $i$ and $h_t$ denotes the total enthalpy. The force $F_i$ accounts for the friction between the streams and at the wall, and is defined in § 2.2. The equations imply that the streams do not exchange mass and that the separating line between the streams is the dividing streamline. Heat exchange is not considered.

Figure 1.

The considered axisymmetric nozzle has a contour $A(x)$ and contains two streams, which have the same total temperature $T_t$, but can have a different total pressure $p_t$. They exchange momentum through a friction force $F_{ps}$ at the dividing streamline and a wall friction force $F_w$. The static pressure $p$ is assumed to be uniform in each cross-section. The solution consists of the streamwise distribution of the flow variables $p$, $p_{t,p}$, $p_{t,s}$, $T_{t,p}$, $T_{t,s}$ and the individual cross-sections $A_p$ and $A_s$.

This system of equations can be converted to an explicit system of ODEs for numerical integration. The derivation can be found in classic textbooks; for example in Chapter 8 of Shapiro (Reference Shapiro1953). The equations in terms of the static pressure $p$, the total pressure $p_t$ and the total temperature $T_t$ read as follows:

(2.6)$$\begin{gather} \frac{1}{p_i} \frac{{\rm d}p_i}{{\rm d}\kern0.7pt x} = \left[\frac{1 + \left(\gamma - 1\right)\textit{Ma}_i^2}{1-\textit{Ma}_i^2}\right] \frac{F_i}{A_i p_i} + \left[\frac{\gamma \textit{Ma}_i^2}{1-\textit{Ma}_i^2}\right]\frac{1}{A_i}\frac{{\rm d} A_i}{{\rm d}\kern0.7pt x}, \end{gather}$$

(2.7)$$\begin{gather}\frac{1}{p_{t,i}}\frac{{\rm d}p_{t,i}}{{\rm d}\kern0.7pt x} = \frac{F_i}{A_i p_i}, \end{gather}$$

(2.8)$$\begin{gather}\frac{1}{T_{t,i}}\frac{{\rm d}T_{t,i}}{{\rm d}\kern0.7pt x} = 0, \end{gather}$$

having introduced the ratio of specific heats $\gamma$ and the Mach numbers $\textit {Ma}_i$,

(2.9a,b)\begin{equation} \textit{Ma}_i = u_i / a_i \quad \mbox{and} \quad a_i = \sqrt{\gamma R T_i} , \end{equation}

where $R$ denotes the specific gas constant. Equations (2.6)–(2.8) do not form a closed system of equations because the cross-sections $A_i(x)$ are unknown (as opposed to a single stream where $A_i = A$). Nonetheless, their sum should equal the known channel's cross-section $A(x)$ (2.2). The derivative ${\rm d}A/{{\rm d}\kern0.7pt x}$ can be computed from the channel's profile, so the unknown terms ${\rm d}A_i/{{\rm d}\kern0.7pt x}$ can be eliminated from the system by summing over the streams $i$. The gradient of the cross-section $A_i$ is first isolated from (2.6),

(2.10)\begin{equation} \frac{{\rm d}A_i}{{\rm d}\kern0.7pt x} = \left[A_i\frac{1-\textit{Ma}_i^2}{\gamma \textit{Ma}_i^2}\right]\frac{1}{p} \frac{{\rm d}p}{{\rm d}\kern0.7pt x} - \left[\frac{1 + \left(\gamma - 1\right)\textit{Ma}_i^2}{\gamma \textit{Ma}_i^2}\right] \frac{F_i}{p}, \end{equation}

where the index for the pressure is omitted because of (2.1). Note that (2.10) can be integrated to compute the cross-section of a single stream if the pressure gradient is known. Summing over the streams $i$ gives

(2.11)\begin{equation} \sum_{i\in[p,s]} \frac{{\rm d}A_i}{{\rm d}\kern0.7pt x} = \frac{{\rm d}A}{{\rm d}\kern0.7pt x} = \sum_{i\in[p,s]} \left[A_i\frac{1-\textit{Ma}_i^2}{\gamma \textit{Ma}_i^2}\right]\frac{1}{p} \frac{{\rm d}p}{{\rm d}\kern0.7pt x} - \sum_{i\in[p,s]}\left[\frac{1 + \left(\gamma - 1\right)\textit{Ma}_i^2}{\gamma \textit{Ma}_i^2}\right] \frac{F_i}{p}. \end{equation}

The pressure gradient is the only remaining unknown, for which an explicit equation is obtained from (2.11),

(2.12)\begin{equation} \frac{1}{p} \frac{{\rm d}p}{{\rm d}\kern0.7pt x} = \dfrac{1}{\beta}\left( \dfrac{{\rm d}A}{{\rm d}\kern0.7pt x} + \sum_{i\in[p,s]}\left[\dfrac{1 + \left(\gamma - 1\right)\textit{Ma}_i^2}{\gamma \textit{Ma}_i^2}\right] \dfrac{F_i}{p}\right), \end{equation}

where $\beta$ denotes the compound choking indicator,

(2.13)\begin{equation} \beta = \sum_{i\in[p,s]} A_i\dfrac{1-\textit{Ma}_i^2}{\gamma \textit{Ma}_i^2}. \end{equation}

Bernstein et al. (Reference Bernstein, Heiser and Hevenor1967) showed that $\beta > 0$ corresponds to a compound-subsonic flow, $\beta =0$ to a compound-sonic flow and $\beta < 0$ to a compound-supersonic flow. As a result, the pressure gradient and the area gradient of a compound-subsonic flow share the same sign in the absence of forces, leading to an expansion in a convergent channel and a compression in a divergent channel. Recently, Metsue et al. (Reference Metsue, Debroeyer, Poncet and Bartosiewicz2021) extended the compound choking condition to real gases and showed that the condition $\beta =0$ corresponds to the maximal combined mass flow rate.

A convenient variable to identify the compound flow regime is the so-called equivalent Mach number $\textit {Ma}_{eq}$, introduced by Hedges & Hill (Reference Hedges and Hill1974),

(2.14)\begin{equation} \textit{Ma}_{eq} = \left(\gamma \frac{\beta}{A}+1\right)^{-{1}/{2}}. \end{equation}

This equals one for a compound choked flow. Section 2.3 analyses choking in more detail. It is important to note that the forces do not affect $\beta$ and hence the compound choking criterion, as pointed out by Otis (Reference Otis1976). Moreover, as shown by Bernstein et al. (Reference Bernstein, Heiser and Hevenor1967), compound waves can no longer travel upstream of a location $x$ if $\beta (x) \leqslant 0$, which constitutes the fundamental principle of choking as in the case of a single stream.

Equations (2.7), (2.8), (2.10) and (2.12) form a closed system for the (uniform) static pressure $p$ and the total pressure $p_{t,i}$, the total temperature $T_{t,i}$ and the cross-section $A_i$ in each stream. These variables fully define the state of the flow and other variables can easily be calculated from them using classic gas dynamic relations. The explicit nature of the equations facilitates numerical integration with a shooting method. More details on the numerical solution are provided in § 4.

2.2. Model closure

The compound flow is assumed to be axisymmetric, with the primary stream surrounded by the secondary stream. Planar flows are left as technical extensions for future studies. Hence, wall friction acts only on the secondary stream. A friction coefficient $f_w$ is defined such that the friction force per unit length $F_w$ equals

(2.15)\begin{equation} F_w = \tfrac{1}{2} f_w \rho_s u_s^2 l_w = \tfrac{1}{2} f_w \gamma p \textit{Ma}_s^2 l_w, \end{equation}

where $l_w$ denotes the wetted perimeter of the cross-section. In a circular duct, $l_w$ is related to the cross-section $A$ as follows:

(2.16)\begin{equation} l_w = 2 {\rm \pi}r = 2 \sqrt{{\rm \pi} A}, \end{equation}

where $r$ denotes the radius of the total cross-section $A$. Similarly, the friction between the streams is defined as (Papamoschou Reference Papamoschou1993)

(2.17)\begin{equation} F_{ps} = \frac{1}{2} f_{ps} \dfrac{\rho_p+\rho_s}{2} \left(u_p - u_s\right) \left|u_p - u_s\right| l_{ps}, \end{equation}

where $f_{ps}$ denotes a friction coefficient and $l_{ps}$ denotes the perimeter of the primary stream,

(2.18)\begin{equation} l_{ps} = 2 {\rm \pi}r_p = 2 \sqrt{{\rm \pi} A_p}. \end{equation}

The friction force between the streams opposes the stream with the highest velocity and is defined to be positive by (2.17) if $u_p > u_s$. Therefore, $F_{ps}$ opposes the primary stream, while the secondary stream is entrained by the interstream friction and decelerated by the wall friction,

(2.19a,b)\begin{equation} F_p ={-} F_{ps} \quad \mbox{and} \quad F_s = F_{ps} - F_w. \end{equation}

Two approaches were used in this work for both coefficients $f_w$ and $f_{ps}$ and tested to the RANS simulations in § 6.1.

The first approach was to leverage known correlations for similar flow configurations. The formula of Van Driest (Reference Van Driest1951) for the local wall friction coefficient $f_w$ under a turbulent boundary layer in a compressible flow gives

(2.20)\begin{equation} \dfrac{0.242}{\sqrt{f_w}} \sqrt{1 - \lambda^2} \dfrac{\arcsin (\lambda)}{\lambda} = 0.41 + \log_{10} \left(f_w Re_x\right) + \log_{10} \left((1 - \lambda^2)\left(1 - \dfrac{\theta \lambda^2}{1 + \theta}\right)\right),\end{equation}

where

(2.21a–c)\begin{equation} 1-\lambda^2 = \left(1 + \frac{\gamma - 1}{2} \textit{Ma}_s^2\right)^{{-}1}, \quad Re_x = \dfrac{\rho_s u_s x}{\mu_s} \quad \mbox{and}\quad \theta = \dfrac{S}{T_s}, \end{equation}

with the dynamic viscosity $\mu$ computed with the law of Sutherland (Reference Sutherland1893), a reference viscosity $\mu _{ref}$ of $1.716 \times 10^{-5}$ Pa s at $T_{ref} = 273.2$ K, and $S = 110.4$ K. For the interstream friction coefficient $f_{ps}$, the formula of Papamoschou (Reference Papamoschou1993) gives

(2.22)\begin{equation} f_{ps} = 0.013 \dfrac{(1 + u_s/u_p) (1 + \sqrt{\rho_s / \rho_p})}{1 + (u_s/u_p) \sqrt{\rho_s / \rho_p}} \left(0.25 + 0.75 \exp({-}3 \textit{Ma}_c^2)\right), \end{equation}

with all quantities evaluated locally and the local convective Mach number $\textit {Ma}_c$ is defined as follows:

(2.23)\begin{equation} \textit{Ma}_c = \dfrac{u_p - u_s}{a_p + a_s}. \end{equation}

The second approach was to infer these coefficients from the RANS results using an inverse method. In this setting, both coefficients $f_w$ and $f_{ps}$ are assumed to be constant and are determined by an optimiser which seeks to minimise the $l_2$ norm of the discrepancy between the proposed model and the results of RANS simulations. This approach is expected to be more accurate due to the calibration, provided that the assumption of constant coefficient is adequate. On the other hand, these coefficients are not expected to generalise well outside the reference data, while the correlations (2.20) and (2.22) make the model self-sufficient and closed.

2.3. Choking mechanism

Substituting the forces $F_p$ and $F_s$ as defined in the section above in (2.12) leads to a rational equation for the static pressure gradient,

(2.24)\begin{equation} \frac{1}{p} \frac{{\rm d}p}{{\rm d}\kern0.7pt x} = \dfrac{N}{\beta}, \end{equation}

with $\beta$ defined in (2.13) and the numerator,

(2.25)\begin{equation} N = \dfrac{{\rm d}A}{{\rm d}\kern0.7pt x} + N_{ps} + N_w, \end{equation}

where

(2.26)$$\begin{gather} N_{ps} = \dfrac{f_{ps}l_{ps}}{2}\frac{\textit{Ma}_p^2 - \textit{Ma}_s^2}{\gamma \textit{Ma}_p^2 \textit{Ma}_s^2} \dfrac{\rho_p+\rho_s}{2} \left(u_p - u_s\right) \left|u_p - u_s\right| \dfrac{1}{p}, \end{gather}$$

(2.27)$$\begin{gather}N_w ={-} \dfrac{f_{w} l_{w}}{2} \left(1 + \left(\gamma - 1\right)\textit{Ma}_s^2\right). \end{gather}$$

The term $N_w$ stems from wall friction ( $f_w$) and is negative by definition. The term $N_{ps}$ is due to friction between the streams and it is positive unless the terms $\textit {Ma}_p^2-\textit {Ma}_s^2$ and $u_p-u_s$ have a different sign. In the natural case $u_p>u_s$, this is possible only if $T_p/T_s \geqslant u^2_p/u^2_s$. This scenario is not encountered in Lamberts et al. (Reference Lamberts, Chatelain, Bourgeois and Bartosiewicz2018) and Kracik & Dvorak (Reference Kracik and Dvorak2023) and thus not discussed further. Therefore, the interstream friction has a positive contribution to the numerator $N$ in (2.24) in the considered conditions and the wall friction has a negative contribution. This leads to some fundamental implications.

A compound-subsonic flow ($\beta > 0$) expands in a convergent section (${\rm d}A/{{\rm d}\kern0.7pt x} < 0$) and under the action of wall friction (as in a classic Fanno flow). Conversely, it recovers pressure in a divergent section and under the action of interstream friction. This force opposes the stream with the highest velocity, seeking to equalise velocities between streams. Consequently, momentum exchange between the streams promotes flow uniformity and results in a net increase in static pressure. The opposite is true for a compound-supersonic flow.

Equation (2.24) is singular in the sonic section ($\beta = 0$), similarly to a single stream where the denominator equals $1 - \textit {Ma}^2$ (cf. Shapiro Reference Shapiro1953; Restrepo & Simões-Moreira Reference Restrepo and Simões-Moreira2022). Nevertheless, classic nozzle flows and compound flows expand continuously through the sonic section, meaning that their pressure gradient is finite at $\beta =0$. This is possible according to (2.24) only if both the numerator $N$ and the denominator $\beta$ tend to zero, leading to an indeterminate form with a finite value. This observation is corroborated by (2.11), which simplifies to $N=0$ if $\beta =0$. In isentropic conditions ( $f_{ps} = f_w = 0$), a compound flow becomes sonic at the throat (${\rm d}A/{{\rm d}\kern0.7pt x} = 0$), as shown by Bernstein et al. (Reference Bernstein, Heiser and Hevenor1967). In non-isentropic conditions, the relative magnitudes of $N_{ps}$ and $N_w$ in (2.25) determine the position of the sonic section.

Wall friction has a negative contribution to the numerator. Therefore, in the absence of interstream friction ( $f_{ps}=0$), one must have ${\rm d}A/{{\rm d}\kern0.7pt x}>0$ to respect $N=0$. This implies that the sonic section is located in a divergent section, hence downstream of the throat. The wall friction has the same effect on a nozzle flow with a single stream (Shapiro Reference Shapiro1953; Beans Reference Beans1970; Restrepo & Simões-Moreira Reference Restrepo and Simões-Moreira2022). Likewise, in the absence of wall friction ( $f_w=0$), one must have ${\rm d}A/{{\rm d}\kern0.7pt x} < 0$ to respect $N=0$. This implies that the interstream friction pushes the sonic section upstream of the throat. This analysis offers a plausible explanation for the observed displacement of the sonic section by Lamberts et al. (Reference Lamberts, Chatelain, Bourgeois and Bartosiewicz2018) and Kracik & Dvorak (Reference Kracik and Dvorak2023). The interplay between both forces is investigated further in § 6.

The indeterminate pressure gradient can be computed by applying l'Hôpital's rule on (2.24) (cf. Bernstein et al. Reference Bernstein, Heiser and Hevenor1967; Restrepo & Simões-Moreira Reference Restrepo and Simões-Moreira2022),

(2.28)\begin{equation} \left(\frac{1}{p}\frac{{\rm d}p}{{\rm d}\kern0.7pt x}\right)^* = \frac{N^*}{\beta^*} = \frac{0}{0} = \frac{({\rm d}N/{{\rm d}\kern0.7pt x})^*}{({\rm d}\beta / {{\rm d}\kern0.7pt x})^*}, \end{equation}

where the superscript $*$ refers to the sonic section. The derivatives ${\rm d}N/{{\rm d}\kern0.7pt x}$ and ${\rm d}\beta /{{\rm d}\kern0.7pt x}$ give rise to the derivatives of Mach numbers $\textit {Ma}_i$, cross-sections $A_i$, forces $F_i$ and the static pressure $p$ (see (2.25) and (2.13)), and these can be all related to the local pressure gradient. The relevant equations are presented in Appendix A.

For example, the gradient of $\beta$ is given by

(2.29)\begin{equation} \dfrac{{\rm d}\beta}{{\rm d}\kern0.7pt x} = \sum_{i\in[p,s]} \left(\dfrac{{\rm d}A_i}{{\rm d}\kern0.7pt x}\dfrac{1-\textit{Ma}_i^2}{\gamma \textit{Ma}_i^2} - \dfrac{A_i}{\gamma \textit{Ma}_i^4}\dfrac{{\rm d}\textit{Ma}_i^2}{{\rm d}\kern0.7pt x}\right). \end{equation}

Writing the Mach numbers $\textit {Ma}_i$ as a function of the total and the static pressure,

(2.30)\begin{equation} \textit{Ma}_i^2 = \frac{2}{\gamma-1}\left(\left(\frac{p_{t,i}}{p}\right)^{(\gamma-1)/{\gamma}}-1\right), \end{equation}

the derivatives ${\rm d}\textit {Ma}_i^2/{{\rm d}\kern0.7pt x}$ in (2.29) can be substituted by the gradients of the static and total pressures. The resulting expression for the gradient of $\beta$ thus contains first derivatives of the static pressure, the cross-sections $A_i$ and the total pressures $p_{t,i}$. The latter two can be substituted using (2.7) and (2.10), which contain the static pressure gradient. The resulting expression for the gradient of $\beta$ is linear with respect to the static pressure gradient, with nonlinear coefficients depending on the Mach numbers, forces and static pressure,

(2.31)\begin{equation} \dfrac{{\rm d}\beta}{{\rm d}\kern0.7pt x} = c_{\beta0}(\textit{Ma}_p,\textit{Ma}_s)\left(\dfrac{1}{p}\dfrac{{\rm d}p}{{\rm d}\kern0.7pt x}\right) + c_{\beta1}(\textit{Ma}_p,\textit{Ma}_s,F_p,F_s,p). \end{equation}

The analytical expressions of $c_{\beta 0}$ and $c_{\beta 1}$ are given by (A26) and (A27). The same approach leads to a similar linear equation for the derivative of the numerator (see (A21)). These equations are derived for constant friction coefficients, i.e. the correlations (2.20) and (2.22) are not explicitly differentiated. This approximation is deemed acceptable since the distribution of $f_w$ and $f_{ps}$ are relatively flat, as later shown in § 6. The result is an implicit equation for the static pressure gradient in the sonic section,

(2.32)\begin{equation} \left(\frac{1}{p}\frac{{\rm d}p}{{\rm d}\kern0.7pt x}\right)^* = \dfrac{c_{N0}^* \left(\dfrac{1}{p}\dfrac{{\rm d}p}{{\rm d}\kern0.7pt x}\right)^* + c_{N1}^*}{c_{\beta0}^*\left(\dfrac{1}{p}\dfrac{{\rm d}p}{{\rm d}\kern0.7pt x}\right)^* + c_{\beta1}^*}. \end{equation}

Rearranging the equation above leads to the quadratic equation (A28) for the static pressure gradient. The two roots correspond to the subsonic and the supersonic solution downstream of the sonic point (cf. Shapiro Reference Shapiro1953). For an isentropic compound flow, the derivative of the numerator corresponds to the second derivative of the cross-section ${\rm d}^2A/{{\rm d}\kern0.7pt x}^2$ ( $f_{ps}=f_w=0$), leading to the tractable expression reported by Bernstein et al. (Reference Bernstein, Heiser and Hevenor1967). The additional terms introduced by the friction forces preclude such a concise expression, which is why the coefficients and their analytical expressions are reported in Appendix A. Note, however, that these equations are exact and analytical.

5.1. Numerical set-up for the RANS validation

The computational domain is shown in figure 4. It consists of an axisymmetric convergent–divergent nozzle with two distinct inlets. The simulation was carried out with a wedge-shaped domain of five degrees and a single cell in the tangential direction. The contour of the upper wall was defined as a cosine between ${\rm \pi} /2$ and $2{\rm \pi}$ with appropriate scaling to reach the dimensions shown in figure 4:

(5.1)\begin{equation} r(x) = \dfrac{R_o + R_{th}}{2} + \dfrac{R_o - R_{th}}{2} \cos\left(\dfrac{3{\rm \pi}}{2} \dfrac{x}{L} + \dfrac{\rm \pi}{2}\right). \end{equation}

Figure 4.

The computational domain consists of an axisymmetric convergent–divergent nozzle with two distinct inlets. The primary stream (index $p$) is surrounded by the secondary stream (index $s$). The reference geometry is defined by $L = 187.5$ mm, $R_i = 9.5$ mm, $R_{th} = 9.0$ mm, $R_o = 10$ mm and $R_p = 0.5 R_i$. The model validation considered various throat $R_{th}$ and the primary inlet $R_p$ radii, as described in § 5.2, while keeping the same resolution at the wall. The coarse mesh is displayed for visibility.

Hence, the radius at the inlet ($x=0$) follows from the outlet radius $R_o$, the throat radius $R_{th}$ and the length $L$. The throat is located at one-third of the domain. The analysis of the sonic section displacement was focused on a test case with nozzle lengths of 187.5 mm, outlet radius $R_o=10$ mm, inlet radius $R_i = 9.5$ mm, throat radios $R_{th} = 9$ mm and $R_p = 0.5 R_i$. For the model validation, several other geometries were considered as reported in more details in § 5.2.

The mesh was generated using Gmsh (https://gmsh.info/) coupled to Python to allow full control of the number of elements and of the refinement near the walls. The simulations were performed with an adapted version of the compressible transient solver rhoCentralFoam in OpenFOAM v9 (https://openfoam.org/version/9/) with the $k-\omega$ shear stress transport turbulence model of Menter (Reference Menter1993). The adapted solver does not solve for the tangential component of the velocity, which remains equal to zero (no swirl). The law of Sutherland (Reference Sutherland1893) is used to compute the viscosity. The simulations are wall-resolved as shown in § 5.2.

The boundary conditions at the inlets consist of the total pressure and total temperature. The primary inlet is chosen to be supersonic, so static pressure is also imposed. A wave transmissive boundary condition is imposed at the exit as the flow is supersonic. The wall is adiabatic and treated with a slip or no-slip condition, depending on the test case. The ‘wedge’ boundary condition is imposed on the sides of the wedge to inform OpenFOAM of the axisymmetry.

The resulting 2-D flow fields are postprocessed by identifying the dividing streamline between the streams and by averaging over the resulting cross-sections. The radial position $r_{d}$ of the dividing streamline is found at any point $x$ by integrating the axial mass flux,

(5.2a,b)\begin{equation} \dot{m}_p(x) = \int_0^{r_{d}(x)} \rho(x,r) u(x,r) 2 {\rm \pi}r \, {\rm d}r \quad \mbox{and} \quad \dot{m}_s(x) = \int_{r_{d}(x)}^{R} \rho(x,r) u(x,r) 2 {\rm \pi}r \, {\rm d}r, \end{equation}

where $u$ denotes the axial component of the velocity vector. Knowing the radius $r_d(0)$ of the dividing streamline at the primary inlet, the primary mass flow rate is calculated. The evolution of the dividing streamline at any position $x$ follows from (5.2a,b). The second step consists of averaging the relevant flow variables across the ejector's section, i.e.

(5.3)$$\begin{gather} \hat{\rho}_{i} (x) = \frac{1}{A_{i}}\int_{A_{i}}\rho \, {\rm d}A_{i}, \end{gather}$$

(5.4)$$\begin{gather}\hat{u}_{i} (x) = \frac{1}{\hat{\rho}_{i} A_{i}}\int_{A_{i}}\rho u \, {\rm d}A_{i}, \end{gather}$$

(5.5)$$\begin{gather}\hat{e}_{t,i} (x) = \frac{1}{\hat{\rho}_{i} A_{i}}\int_{A_{i}}\rho e_t \, {\rm d}A_{i}, \end{gather}$$

for $i \in [p,s]$ and where $e_t = c_v T + u^2/2$ denotes the total specific internal energy and $c_v$ is the specific heat at constant volume. The density-averaged velocity and internal energy ensure that the 1-D mass flow rate and internal energy match their integrated 2-D counterpart. The averaged static temperature, static pressure and Mach number follow from (5.3)–(5.5) and the specific gas constant $R$:

(5.6a–c)\begin{equation} \hat{T}_{i} = \frac{1}{c_v} \left(\hat{e}_{t,i}-\frac{1}{2}\hat{u}_{i}^2\right), \quad \hat{p}_i = \hat{\rho}_i R \hat{T}_i \quad \mbox{and} \quad \widehat{\textit{Ma}}_{i} = \frac{\hat{u}_{i}}{\sqrt{\gamma R \hat{T}_{i}}}.\end{equation}

5.2. Selected test cases

Three simulations were carried out where the wall and interstream friction forces were numerically (de)-activated. The boundary conditions are listed in table 1. At both inlets, the turbulent mixing length was set equal to 7 % of the hydraulic diameter, as recommended by Versteeg (Reference Versteeg2007). The turbulence intensity was set to 1 %, as this value proved to best match the friction coefficients predicted by (2.22).

Table 1.

Boundary conditions of the simulations presented in § 6. The primary inlet is supersonic with $\textit {Ma}_p=1.22$. They cover (i) an isentropic case, (ii) a case with friction between the streams but without wall friction and (ii) a case with both forces, respectively.

The first simulation is inviscid and thus serves as a reference for the original compound flow theory without momentum exchange. The second simulation is viscous with a slip condition at the wall: only friction forces between the streams are thus active in this case. Finally, wall friction is included in the third case by imposing a no-slip condition along the upper boundary. A mesh convergence study was carried out on the viscous simulation with the no-slip condition. The number of cells was doubled in each direction, leading to meshes with $400 \times 25$, $800 \times 52$ and $1600 \times 106$ cells. The resulting distributions of $y^+$ near the wall and the Mach number at the centreline are shown for the three meshes in figure 5. The medium and fine meshes show good agreement and are both wall-resolved. The wall friction Reynolds number $Re_{\tau } = \rho u_{\tau } D_h / \mu$ equals $1.9\times 10^4$ at the inlet and $5.3 \times 10^3$ at the outlet in case 3. The results on the fine mesh are used in the next section to profit from its higher resolution.

Figure 5.

The coarse, medium and fine mesh contain $400 \times 25$, $800 \times 52$ and $1600 \times 106$ elements, respectively. The medium and fine meshes lead to a wall-resolved simulation and to good agreement for the Mach number at the centreline. The fine mesh is retained for the other simulations and the postprocessing. Here (a) $y^+$ of the first cell near the wall; (b) Mach number at the centreline.

A parametric study was carried out to validate the model in a wider range of conditions. The reference case is the viscous simulation with the no-slip condition at the wall (case 3 in table 1), as it is the most representative for practical applications. Three parameters were varied independently: (i) the total pressure at the secondary inlet between $p_{t,s} = 0.5$ and 3 bar; (ii) the radius of the throat between $R_{th}/R_o = 0.6$ and 0.9; and (iii) the radius of the primary inlet between $R_p/R_i = 0.3$ and 0.7 (see figure 4). The correlations (2.20) and (2.22) were used in all cases, so the model was not tuned further.

The resulting distributions of the static pressure and the Mach numbers depend on the geometry and the inlet conditions, so the static pressure imposed at the supersonic primary inlet changes too. However, the static pressure at the inlet corresponding to a choked flow without a shock train is initially unknown. The 1-D model proved to be a useful tool to compute this pressure, as it provides a cross-stream averaged description of a choked flow with a uniform static pressure profile. The 1-D model was thus run first, to provide the boundary conditions for the RANS simulations afterwards (see table 2). The primary stream is supersonic in most cases, except if $p_{t,s} \geqslant 2$ bar. In that case, only the total pressure and total temperature were imposed. The other boundary conditions remained identical to those in table 1.

Table 2.

Boundary conditions for the parametric study. The primary static pressure $p_p$ was computed a priori with the 1-D model and served a posteriori as a boundary condition for the RANS simulations. The primary stream is subsonic at the inlet for $p_{t,s} \geqslant 2$ bar, so the static pressure is not imposed.