3.1. Three-degrees-of-freedom gas

The first case examined is $f = 3$ that corresponds to $\gamma = 5/3$ . These degrees of freedom are all related to translational motion. This case typically applies to monatomic gases such as the helium, argon and other noble gases (Anderson Reference Anderson2019). These gases are often employed where chemical inertness is desired, such as shock tubes (Würmel et al. Reference Würmel, Silke, Curran, Ó Conaire and Simmie2007; Collen et al. Reference Collen, Satchell, Di Mare and McGilvray2022). The case of a monatomic gas represents a physical lower bound where only translational motion contributes to the molecular internal energy (Boyd & Schwartzentruber Reference Boyd and Schwartzentruber2017). While mathematically other solutions exist where $f \lt 3$ , which may be useful for an interpolating function or other purposes, they are not physically valid and are ignored here.

The mass conservation equation for a three-degrees-of-freedom gas is found by substituting $f=3$ into the unsquared form of the mass conservation relation, shown in (3.5), which yields a fourth-order polynomial in $M_2$ . Expanding, rearranging by powers of $M_2$ and normalising by the leading coefficient gives the monic form

(3.12) \begin{equation} M_{2}^{4} + 6 M_{2}^{2} - 9 \sqrt {C} M_{2} + 9 = 0. \end{equation}

The polynomial shown in (3.12) is a fourth-order polynomial in $M_2$ and, therefore, has an exact closed-form solution (Irving Reference Irving2013; Zwillinger Reference Zwillinger2018). This indicates that, for any three-degrees-of-freedom gas, an exact closed-form solution is guaranteed to exist and does not require any iteration. This, however, does not guarantee that one of the resulting roots is physically valid given the boundary conditions applied. The existence and identification of physically valid solutions is discussed later.

3.2. Five-degrees-of-freedom gas

The second case examined is $f = 5$ that corresponds to $\gamma = 7/5$ . This case is typical for a linear diatomic gas that has three translational and two rotational degrees of freedom when the static temperature is small compared with the characteristic vibrational temperature of the species (Anderson Reference Anderson2019). These gases are of particular interest since Earth’s atmosphere is primarily composed of diatomic nitrogen and oxygen, which are both linear diatomic molecules, i.e. $\textrm {N}_2$ and $\textrm {O}_2$ .

The mass conservation equation for a five-degrees-of-freedom gas is found by substituting $f=5$ into the unsquared form of the mass conservation equation, shown in (3.5), which yields a sixth-order polynomial in $M_2$ . Expanding, rearranging by powers of $M_2$ and normalising by the leading coefficient gives the monic form

(3.13) \begin{equation} M_{2}^{6} + 15 M_2^4 + 75 M_2^2 -125 \sqrt {C} M_2 + 125 = 0. \end{equation}

The function shown in (3.13) is a sixth-order polynomial in $M_2$ . Therefore, no exact closed-form solution exists in terms of radicals. The solution for the five-degrees-of-freedom gas must be found numerically (Irving Reference Irving2013). Two example methods that are capable of solving the roots of this polynomial are discussed in Aurentz et al. (Reference Aurentz, Mach, Vandebril and Watkins2015) and Oliva & Jemcov (Reference Oliva and Jemcov2024). The existence and identification of physically valid solutions is discussed later.

3.3. Seven-degrees-of-freedom gas

The case of $f=7$ is used here only as a constant- $\gamma$ surrogate for a diatomic gas with an additional active mode. A full treatment of vibrational excitation would require temperature and state-dependent specific heats and, therefore, a thermally perfect gas model. However, such a model lies outside of the calorically perfect gas formulation developed in the present work (Vincenti Reference Vincenti1965; Anderson Reference Anderson2019).

The mass conservation equation for a seven-degrees-of-freedom gas is found by substituting $f=7$ into the unsquared form of the mass conservation, shown in (3.5), which yields an eighth-order polynomial in $M_2$ . Expanding, rearranging by powers of $M_2$ and normalising by the leading coefficient gives the monic form

(3.14) \begin{equation} M_{2}^{8} + 28 M_{2}^{6} + 294 M_{2}^{4} + 1372 M_2^2 -2401 \sqrt {C} M_{2} + 2401 = 0. \end{equation}

The function shown in (3.14) is an eighth-order polynomial in $M_2$ . Therefore, no general, exact, closed-form solution exists in terms of radicals. The solution for the seven-degrees-of-freedom gas must be found numerically (Irving Reference Irving2013). Two example methods that are capable of solving the roots of this polynomial are discussed in Aurentz et al. (Reference Aurentz, Mach, Vandebril and Watkins2015) and Oliva & Jemcov (Reference Oliva and Jemcov2024). The existence and identification of physically valid solutions is discussed later.

3.4. Infinite-degrees-of-freedom gas

The final case examined is the infinite-degrees-of-freedom gas, i.e. $f\to \infty$ , which corresponds to $\gamma \to 1$ . This limit can approximate large, complex polyatomic molecules using an effective constant $\gamma$ . In this limit $(c_{\!P},c_V)\to \infty$ at a fixed specific gas constant $R$ , so for finite dimensional heat transfer $q_w$ , the temperature change becomes small and the flow approaches an isothermal behaviour. Furthermore, the case $f\to \infty$ represents a mathematical upper bound for solutions within the present calorically perfect, constant $\gamma$ , ideal gas model.

The mass conservation equation for an infinite-degrees-of-freedom gas is found by taking the limit of (3.6) as $f \to \infty$ . The resulting equation is

(3.15) \begin{equation} \exp (M_2^2) - C_{\infty } M_2^2 = 0, \end{equation}

where $C_{\infty }$ is defined as

(3.16) \begin{equation} C_{\infty } \equiv \lim _{f\to \infty } C = \exp (M_1^2) \left ( \frac {{P}_{o2}}{{P}_{o1}} \sqrt {\frac {T_{o1}}{T_{o2}}} \frac {A_2}{M_1 A_1}\right )^2. \end{equation}

The function shown in (3.15) differs from (3.12), (3.13) and (3.14) as it is not polynomial due to the exponential function being an infinite series. The solution of (3.15) in terms of $M_2$ is

(3.17) \begin{equation} M_2 = \pm \sqrt {-W_k \left ( -\frac {1}{C_{\infty }} \right )}, \end{equation}

where $W_k$ is the $k$ th branch of the Lambert $W$ function (Zwillinger Reference Zwillinger2018; Mezo Reference Mezo2022). The Lambert $W$ function has infinite branches; choosing the correct branch to produce a physically valid result will be shown in the next section.

A summary of degrees of freedom, resulting function orders, variable and solution to the cases discussed is shown in table 1.

Table 1.

Resulting function order, variable and solution method for gases with molecular degrees of freedom $f=3$ , $5$ , $7$ and $\infty$ .

4.1. Finite-degrees-of-freedom gases

All finite-degrees-of-freedom gases produce functions whose highest-order term is a finite order but the function is not necessarily a polynomial if $f \in \mathbb{R}_{\gt 0}$ . In special cases when (3.6) results in a polynomial, directly evaluating the roots may be done for polynomials up to order 4. Polynomials of order higher than 4 require numerical approaches, an example of such an algorithm is shown in Aurentz et al. (Reference Aurentz, Mach, Vandebril and Watkins2015). However, for the more general case of $f \in \mathbb{R}_{\gt 0}$ , some definitive statements on the existence of real, positive solutions and bounds on these solutions may be made. Mathematically, the following existence conditions and bounds hold for $f\gt0$ ; the physical ideal gases considered here use $f \geq 3$ .

The existence of real, positive solutions requires understanding specific properties of a governing equation being used for the given input information of the geometric area ratio $A_2/A_1$ , stagnation pressure ratio $P_{o2}/P_{o1}$ , non-dimensional heat transfer $q_w/h_{o1}$ , degrees of freedom $f$ , and inlet Mach number $M_1$ . First, taking $M_2^2$ as the unknown, the convex part of (3.6) is

(4.2) \begin{equation} \left ( 1 + \frac {1}{f} M_2^2 \right )^{f+1}, \end{equation}

and a simple proof is shown in Appendix A. The remaining part of (3.6) is linear in the unknown $M_2^2$ , i.e. $C M_2^2$ . Thus, for a convex and linear function to intersect, there must exist some minimum critical slope on the linear function to intercept the convex function (Boyd & Vandenberghe Reference Boyd and Vandenberghe2004). The critical slope for this problem is found by locating the zero of the derivative of (3.6). The resulting minimum slope is defined as

(4.3) \begin{equation} C^* \equiv \left ( 1 + \frac {1}{f}\right )^{f+1}, \end{equation}

and the critical point where the convex and minimum-slope linear functions intersect occurs at $M_2=1$ . Additionally, $C$ encapsulates the boundary conditions and sets the slope of the linear term. If $C = C^*$ then a double root occurs at $M_2=1$ . Moreover, if $C\gt C^*$ then two positive real solutions exist. No real solutions exist if $C \lt C^*$ . These conditions are summarised in table 2. The form of these conditions is independent of $f$ , although the values of $C$ and $C^*$ depend on $f$ . Further details on the geometric interpretation and graphic representation of these terms is now discussed.

Table 2.

Conditions for the existence of solutions to (3.6) given $C$ and $C^*$ .

Graphically presenting the conditions described in table 2 aids in understanding the existence of real and positive solutions to (3.6). Figure 2 shows graphically how $C^*$ varies as a function of the degrees of freedom $f$ . The abscissa is the degrees of freedom $f$ and the ordinate is the critical slope $C^*$ . Here $C^*$ (shown as a solid black line) is shown to decrease with increasing $f$ . The asymptotic limit of $C^*$ defines $C_{\infty }^*$ as

(4.4) \begin{equation} C_{\infty }^* \equiv \lim _{f\to \infty } C^* = \mathrm{e} , \end{equation}

where $\mathrm{e}$ is Euler’s number and is shown as a black dotted line. This $C^*$ value sets the minimum value of $C$ for two real, positive solutions to (3.6) to exist.

Figure 2.

Plot of $C^*$ derived from (3.6) (solid black line) versus degrees of freedom $f$ with the asymptotic limit as $f\to \infty$ (black dotted line).

Further geometric interpretation is found in figure 3. The abscissa is the outlet Mach number squared $M_2^2$ and the ordinate is either (4.2) or $C^* M_2^2$ . Equation (4.2) is shown as a blue solid line for $f=3$ and a dark yellow solid line for $f\to \infty$ . These two values of $f$ physically bound all physically realisable calorically perfect ideal gases. The linear function $C^* M_2^2$ is shown as a blue dashed line for $f=3$ and as a dark yellow dashed line for $f\to \infty$ . The convex function for both $f$ values starts at unity and increases nonlinearly with increasing outlet Mach number. The linear functions for both $f$ values increase linearly with outlet Mach number. However, the linear functions have different slopes that are set by $C^*$ . The respective $C^*$ curves represent the slope of a linear function required to intersect the respective convex functions at exactly one point (Boyd & Vandenberghe Reference Boyd and Vandenberghe2004). Should the slope of the linear function be any lower, then no intersection would be present. Moreover, should the slope of the linear function be any higher, then two distinct, real, positive roots would be present, indicated by the intersection of the linear function and the convex function at two separate points. The $f \to \infty$ case is included in figure 3 for completeness as identical observations are found to the $f=3$ case. Therefore, if $C = C^*$ then a double root occurs at $M_2=1$ . Moreover, if $C \lt C^*$ then no real solution exists. Only if the slope of the linear function is sufficiently large will two intersections be present corresponding to two real positive solutions to (3.6), i.e. $C\gt C^*$ . These conditions are summarised in table 2.

Figure 3.

Plot of $( 1 + ({1}/{f}) M_2^2)^{f+1}$ (shown as solid lines) and $C^* M_2^2$ (shown as dashed lines) for both $f=3$ (blue lines) and $f \to \infty$ (dark yellow lines) versus the outlet Mach number squared $M_2^2$ .

Gases of arbitrary degrees of freedom, i.e. $f \in \mathbb{R}_{\gt 0}$ , cannot be solved directly using the approach discussed in previous sections. However, assuming that two, real, positive solutions exist to (3.6) for a gas with arbitrary degrees of freedom, then purely mathematical bounds may be placed on both the subsonic and supersonic roots. These bounds are determined by exploiting the convex nature of (3.6). The lower bound for the subsonic solution is found by intersecting $\textit{CM}_2^2$ with the two-term Taylor series of (4.2) around $M_2=0$ . The upper bound on the subsonic solution is found by the intersection of $\textit{CM}_2^2$ with the secant line between the critical point at $M_2=1$ and (4.2) evaluated at the lower bound. The true subsonic solution must obey the inequality

(4.5) \begin{equation} M_{L}^2 \leqslant M_2^2 \leqslant \frac {(1-M_L^2)\,C^*- \left (C^* - \left ( 1 + \frac {1}{f} M_L^2 \right )^{f+1}\right )}{(1-M_L^2)\,C - \left (C^*-\left ( 1 + \frac {1}{f} M_L^2 \right )^{f+1}\right )} \quad \textrm { if } M_2 \lt 1, \end{equation}

where $M_{L}$ is defined as

(4.6) \begin{equation} M_{L} \equiv \sqrt {\frac {1}{C-\frac {f+1}{f}}}. \end{equation}

The upper bound on the supersonic solution is found using a two-step process. First, the intersection of the horizontal line $\textit{Cf}$ with $ (1+({M_2^2}/{f} ))^{f}$ is found since

(4.7) \begin{equation} \left (1+\frac {1}{f} M_2^2\right )^{f} \lt \left (1+\frac {1}{f} M_2^2\right )^{f+1}, \end{equation}

so the $M_2^2$ at the intersection point is larger than the true solution given that $f \in \mathbb{R}_{\gt 0}$ . Then an analytical Newton step is taken from this intersection point, which further reduces the upper bound due to the function being convex. The lower bound on the supersonic solution is then found from the intersection of $\textit{CM}_2^2$ with the secant line between (4.2) evaluated at the supersonic upper bound and the critical point where $M_2=1$ . The bounds on the supersonic root are

(4.8) \begin{equation} \frac {(M_U^2-1)\,C^* + \left (C^* - \left (1 + \frac {1}{f} M_U^2\right )^{f+1} \right )}{(M_U^2-1)\,C + \left (C^* - \left (1 + \frac {1}{f} M_U^2\right )^{f+1} \right )} \leqslant M_2^2 \leqslant M_U^2\quad \textrm { if } M_2 \gt 1, \end{equation}

where $M_U$ is the upper bound and is defined as

(4.9) \begin{equation} M_U \equiv \sqrt {f \left ((C f)^{1/f} -1\right )-1}. \end{equation}

The $f ((C f)^{1/f} -1 )$ part of (4.9) is from the first step in finding the supersonic upper bound. The remaining subtraction of $1$ in (4.9) outside the parentheses is a result of the analytical Newton step. The subsonic lower bound and the supersonic upper bound can be further refined by use of further analytical Newton steps. Additional bounds using the Lambert $W$ function are also discussed in Appendix B.

4.2. Infinite-degrees-of-freedom gas

Similar to the finite-degrees-of-freedom gas, statements may be made regarding the existence of physically valid solutions for the infinite-degrees-of-freedom gas. The Lambert $W$ function produces real output for only the principal and secondary branches, i.e. $k=0$ and $k=-1$ (Corless et al. Reference Corless, Gonnet, Hare, Jeffrey and Knuth1996; Zwillinger Reference Zwillinger2018; Mezo Reference Mezo2022). Furthermore, these branches produce purely real output only when the following condition is met:

(4.10) \begin{equation} C_{\infty } \geqslant C_{\infty }^*. \end{equation}

The principal branch $W_0$ does have a purely real output for positive arguments, however, the argument of $W_k$ , shown in (3.17), is bounded from above as

(4.11) \begin{equation} -\frac {1}{C_{\infty }} \leqslant 0. \end{equation}

Therefore, there is no physical solution from the positive part of the principal branch, i.e. $W_0^+$ . All solutions are present in either the negative part of the principal branch $W_0^-$ or the secondary branch $W_{-1}$ . These conditions are seen graphically in figure 4. The abscissa is the argument of the Lambert $W$ function shown in (3.17) and the ordinate is the Lambert $W$ -function output. The positive principal branch $W_0^+$ is shown as a red dotted line, the negative principal branch $W_0^-$ is shown as a blue dash-dot line and the secondary branch $W_{-1}$ is shown as a green dashed line. Due to the requirement that $M_2 \geqslant 0$ , $W_0^+$ must be non-physical, which necessitates that only $W_0^-$ and $W_{-1}$ are possible solutions. The negative principal branch has a purely real output in the range $-1 \leqslant W_0^- (z) \leqslant 0$ , where $z$ is the argument of the Lambert $W$ function, for the input range $-1/\mathrm{e} \leqslant z \leqslant 0$ . Similarly, the secondary branch has a purely real output in the range $-1 \geqslant W_{-1}(z)$ over the input range $-1/\mathrm{e} \leqslant z \leqslant 0$ . Therefore, the subsonic solution is contained in the negative part of the principal branch $W_0^{-}$ while the supersonic solution is associated with the secondary branch $W_{-1}$ , i.e.

(4.12) \begin{equation} M_2 = \begin{cases} \sqrt { -W_{0}\left ( - \frac {1}{C_{\infty }}\right ) }\quad\textrm {if}\, M_2 \leqslant 1,\\ \\ \sqrt { -W_{-1}\left ( - \frac {1}{C_{\infty }}\right ) }\quad\textrm {if}\, M_2 \geqslant 1. \end{cases} \end{equation}

For the case where $M_2=1$ , there is no discontinuity and the principal and secondary branches of the Lambert $W$ function meet at the same point such that

(4.13) \begin{equation} \lim _{z\to -\frac {1}{\mathrm{e}}} W_0^-(z) = \lim _{z\to -\frac {1}{\mathrm{e}}} W_{-1}(z) = -1, \end{equation}

where $z$ is the argument of the Lambert $W$ function.

Figure 4.

Real output from the Lambert $W$ function positive principal branch $W_0^+$ , negative principal branch $W_0^-$ and the secondary branch $W_{-1}$ versus $-1/C_{\infty }$ .

The Lambert $W$ function is well defined but is a transcendental equation. Therefore, it can be advantageous to use approximations of the Lambert $W$ function for either estimates or as a seed for iterative algorithms. Examples of these approximations are

(4.14) \begin{equation} W_0(z) = \sum _{n=1}^\infty \frac {(-n)^{n-1}}{n!} z^n \approx z - z^2 + \frac {3}{2} z^3 - \frac {8}{3} z^4\quad \textrm { for } \quad |z|\ll 1, \end{equation}

which is appropriate for small $|z|$ and, for $z$ approaching zero from the negative side,

(4.15) \begin{equation} W_{-1}(z) \approx \ln (-z) - \ln (-\ln (-z))\,\textrm { for } \quad z\to 0^-. \end{equation}

These approximations and others are discussed in the literature (Corless et al. Reference Corless, Gonnet, Hare, Jeffrey and Knuth1996; Zwillinger Reference Zwillinger2018; Mezo Reference Mezo2022).

4.3. Algorithm work flow

Both the finite- and infinite-degrees-of-freedom cases have now been explored sufficiently to identify both the existence and identification of physically valid solutions. The algorithmic procedure for determining the output state is shown graphically in figure 5. The problem begins with a given inlet state $M_1$ , geometry $A_2/A_1$ , non-dimensional heat transfer $q_w/h_{o1}$ and gas degrees of freedom $f$ (or $\gamma$ ). The entropy constraint is prescribed either by $\Delta s$ or $P_{o2}/P_{o1}$ , either directly or in coefficient form. From the input values, the stagnation temperature ratio $T_{o2}/T_{o1}$ is determined from the energy balance shown in (3.11). Should $\Delta s$ have been specified then $P_{o2}/P_{o1}$ should now be evaluated using Gibbs relation in (2.2).

The problem set-up must now be confirmed to be able to produce physically valid solutions. This confirmation requires satisfying two primary requirements. The first is satisfying the Clausius–Duhem inequality in the context of Q1D flows using (4.1). Compliance with the Clausius–Duhem inequality requires selecting a wall temperature $T_w$ should the non-dimensional heat transfer $q_w/h_{o1}$ be non-zero. However, if $q_w = 0$ then satisfying the second law simplifies to $s_2 - s_1 \geqslant 0$ . If the Clausius–Duhem inequality is violated then no admissible solution exists for the conditions provided.

Provided the given conditions satisfy the Clausius–Duhem inequality then the solution must be checked for real and positive solutions. The values of $C$ and $C^*$ must be evaluated using (3.7) and (4.3), respectively. If $C\lt C^*$ then no real and positive solutions exist and there are no admissible solutions. If $C=C^*$ then $M_2=1$ and the solution algorithm is complete. If $C\gt C^*$ then two real and positive solutions exist for $M_2$ and may be found by solving (3.6) for finite $f$ or (3.15) for $f \to \infty$ . In special cases, (3.6) may reduce to a polynomial such as for $f=3$ , 5 or 7. Should (3.6) require iteration, then bounds have been provided in (4.5) and (4.8) for subsonic and supersonic solutions, respectively. The subsonic or supersonic solutions are both valid for the supplied input conditions and, thus, the user must understand what solution is appropriate (Oliva & Morris Reference Oliva and Morris2023). From the final solution of $M_2$ , the non-dimensional property ratios may be computed, e.g. static pressure, static temperature, static density, velocity ratios, etc. Once the ratios are computed with the solution of $M_2$ , the output state has been fully defined and the algorithm is complete.

Figure 5.

Computation procedure for the entropy-constrained Q1D formulation.

5.1. Isentropic flow

Isentropic flow with area change is now examined. The literature provides a typical expression for $A/A^*$ that represents the ratio of the geometric cross-sectional area at a specific location to the geometric cross-sectional area at the sonic state where $M=M^*=1$ , which is denoted as $A^*$ (Shapiro Reference Shapiro1953; John & Keith Reference John and Keith2006). Isentropic flow requires adiabatic and reversible flow so that $\Delta s = 0$ ; hence, Gibbs relation requires that $P_{o2}/P_{o1} = T_{o2}/T_{o1}=1$ . The $A^*/A$ equation, for the gases being considered, is

(5.1) \begin{equation} \frac {A^*}{A} = \begin{cases} M \exp \left(\dfrac {1}{2}(1-M^2)\right)&\,\textrm {if } f \to \infty, \\ M \left ( \frac {2}{\gamma +1} \left ( 1 + \frac {\gamma -1}{2} M^2 \right ) \right )^{\frac {-(\gamma +1)}{2(\gamma -1)}}&\,\textrm {otherwise}. \end{cases} \end{equation}

The first expression in (5.1) represents the case of an infinite-degrees-of-freedom gas while the second expression is valid for all other finite-degrees-of-freedom ideal gases. Solutions were obtained by using $M_1 = M^* = 1$ (i.e. the geometric throat is used for normalisation) and setting the geometric area ratio $A^*/A$ , where $A_2 = A$ so that $M_2 = M$ for use in (3.12), (3.13), (3.14) and (4.12).

Figure 6.

Isentropic flow prediction of Mach number $M$ taking $M_1=M^*=1$ versus $A^*/A$ for (a) subsonic and (b) supersonic solutions using (3.12) for $f=3$ (solid blue line), (3.13) for $f=5$ (green dash-dot line), (3.14) for $f=7$ (dark yellow dashed line) and (4.12) for $f \to \infty$ (magenta dotted line) compared with the exact results obtained from (5.1) shown as ‘+’ symbols.

Figure 6 shows the comparison between (5.1) and the solutions from (3.12), (3.13), (3.14) and (4.12). More specifically, figure 6(a) shows subsonic and figure 6(b) shows supersonic flow solutions. The abscissa and ordinate, for both subfigures, are $A^*/A$ and Mach number $M$ , respectively. Additionally, $A^*/A$ was chosen for the abscissa because the function is bounded between $0$ and $1$ for all $M$ . Solutions were obtained by using $M_1=M^*=1$ and setting the geometric area ratio $A^*/A$ , where $A_2=A$ so that $M_2=M$ for use in (3.12), (3.13), (3.14) and (4.12). The solution from (3.12) for $f=3$ is shown as a blue solid line, the solution from (3.13) for $f = 5$ is shown as a green dash–dot line, the solution from (3.14) for $f = 7$ is shown as a dark yellow dashed line and the solution from (4.12) is shown as a magenta dotted line. The black plus symbols ‘ $+$ ’ represent the results from (5.1). The results show that $A^*/A$ increases toward Mach 1 in both the subsonic and supersonic regimes. At a given Mach number, the value of $A^*/A$ for the $f = 3$ case is the largest while $f \to \infty$ is the smallest value of $A^*/A$ in both the subsonic and supersonic regimes. The subsonic results, observed in figure 6(a), show a relatively small difference between the bounding cases of $f=3$ and $f\to \infty$ at either constant $M$ or constant $A^*/A$ , indicating that there is no significant difference for gases in this range. However, for the supersonic flow results, shown in figure 6(b), there is a relatively large difference, at constant $A^*/A$ or constant $M$ , between the bounding cases of $f=3$ and $f\to \infty$ , especially as $M$ becomes larger.

Figure 7.

Static-to-stagnation pressure ratio $P/P_o$ predictions of siloxane MDM versus non-dimensional nozzle location using (4.12), shown as lines, for (a) the Mach 2.0 nozzle and (b) the Mach 1.5 nozzle compared with data, shown as symbols, from Spinelli et al. (Reference Spinelli, Cammi, Gallarini, Zocca, Cozzi, Gaetani, Dossena and Guardone2018).

All results from (3.12), (3.13), (3.14) and (4.12) are identical to the analytical expression shown in (5.1) to numerical precision. Furthermore, the $f=3$ and $f \to \infty$ solutions represent physical bounds between which all other ideal gas solutions must lie in the context of steady Q1D flow. The differences between the results for the gases, shown in figure 6, are explained by considering the underlying physics.

The differences between the results for the gases, shown in figure 6, may also be interpreted using the general differential Q1D relation introduced earlier in (2.4). For the present isentropic case, wall friction and heat transfer are absent, so the local behaviour reduces to the contribution from area change alone. The effect of ${\rm d}A/A$ on ${\rm d}M/M$ is amplified for $\gamma \gt 1$ relative to the limiting case $\gamma \to 1$ . Therefore, for a given geometric area change, larger changes in $M$ are expected for gases with larger $\gamma$ , i.e. monatomic gases exhibit the largest changes in $M$ , whereas gases with $f \to \infty$ exhibit the smallest changes. This is consistent with the trends observed in figure 6. Additional examination of the influence coefficients for various quantities is discussed in Appendix C.

The infinite-degrees-of-freedom gas is an approximation for larger, more complex molecules that will now be shown. Spinelli et al. (Reference Spinelli, Cammi, Gallarini, Zocca, Cozzi, Gaetani, Dossena and Guardone2018) uses siloxane MDM (octamethyltrisiloxane) vapour in non-ideal compressible gas experiments of interest for organic Rankine cycles. The ratio of specific heats of siloxane MDM was approximately 1.02 as calculated with real gas properties for the subset of experimental data used in the analysis below (Bell et al. Reference Bell, Wronski, Quoilin and Lemort2014). Figures 7(a) and 7(b) show experimental data from converging–diverging nozzles designed for Mach 2.0 and 1.5 flow, respectively. The abscissas are the streamwise location with each respective nozzle normalised by the half-height $H$ at the geometric throat. The ordinate is the static-to-stagnation pressure ratio. The Mach 2.0 nozzle had a small backward facing step immediately at the geometric throat that created a shock train. Therefore, figure 7(a) was only plotted up to the geometric throat so the isentropic assumption remains valid. The Mach 1.5 nozzle had the same geometric throat but did not have the small backward facing step and so all available data was utilised. The data used from the Mach 2.0 and 1.5 experiments correspond to compressibility factors at the stagnation state of 0.98 and 0.9688, respectively. These data were chosen to limit the deviations from the ideal gas law that was utilised in the derivation shown for (4.12). For predictions, the Mach number was taken to be unity at the geometric throat location. Furthermore, (4.12) was used to obtain either the subsonic or supersonic Mach number using the area ratio defined by the nozzle geometry and using the geometric throat for normalisation. From the Mach number, the static-to-stagnation pressure ratio may be evaluated for a $f \rightarrow \infty$ gas as

(5.2) \begin{equation} \frac {P}{P_o} = \exp \left (-\frac {1}{2} M^2\right )\!. \end{equation}

Figure 7(a) shows the subsonic portion of the Mach 2.0 nozzle that is upstream of the throat (Spinelli et al. Reference Spinelli, Cammi, Gallarini, Zocca, Cozzi, Gaetani, Dossena and Guardone2018). The data, shown as symbols with error bars representing $95\,\%$ uncertainty, show a decreasing static-to-stagnation pressure ratio that is consistent with an increasing Mach number. The prediction, shown as a dashed blue line, also shows a decreasing static-to-stagnation pressure ratio consistent with the data. Moreover, the static-to-stagnation pressure ratio prediction is within the respective uncertainty bars at each location. This agreement between data and prediction indicates that the $f \rightarrow \infty$ approximation is well justified for siloxane MDM.

Figure 7(b) shows both the subsonic and supersonic portions of the Mach 1.5 nozzle measurements from Spinelli et al. (Reference Spinelli, Cammi, Gallarini, Zocca, Cozzi, Gaetani, Dossena and Guardone2018). The data, shown as symbols with error bars representing $95\,\%$ uncertainty, shows a decreasing static-to-stagnation pressure ratio that is consistent with an increasing Mach number into the diverging section of the nozzle resulting in supersonic flow. The predictions are shown as two separate lines. The subsonic prediction is shown as a dashed blue line and the supersonic prediction is shown as a dotted olive coloured line. The predictions are consistent with the experimental data. The predicted static-to-stagnation pressure ratio $P/P_o$ agrees with measurements over the range shown. This agreement between data and prediction indicates that the $f \rightarrow \infty$ approximation is, again, well justified for siloxane MDM.

5.2. Fanno flow

Fanno flow is the subsequent problem to be examined. Fanno flow considers a constant-area flow with a frictional force, which opposes the velocity vector of the oncoming flow. The entropy change between the inlet state normalised by $R$ for finite- and infinite-degrees-of-freedom gases is

(5.3) \begin{equation} \frac {s-s^*}{R} = \begin{cases} \ln M + \dfrac {1}{2} (1 - M^2)& \, \textrm {if }f \to \infty, \\[7pt] \ln M - {\frac {\gamma +1}{2(\gamma -1)}} \ln \left ( \frac {2}{\gamma +1} \left ( 1 + \frac {\gamma -1}{2} M^2 \right ) \right ) &\,\textrm {otherwise}, \end{cases} \end{equation}

where the superscript $^*$ refers to the sonic state where $M=1$ . The first expression in (5.3) represents the case of an infinite-degrees-of-freedom gas while the second expression is valid for all other finite-degrees-of-freedom ideal gases. The derivation of the first expression for $f \to \infty$ in (5.3) is shown in Appendix D. The second expression in (5.3) and more information on Fanno flow may be found in Shapiro (Reference Shapiro1953); John & Keith (Reference John and Keith2006). In the Fanno and Rayleigh comparisons below, the sonic state is used only as a thermodynamic reference state for inverting the classical star relations; it is not a physical process initialized at station 1, and admissibility for any actual inlet and outlet pair is still governed by the 2nd law of thermodynamics.

Figure 8 shows the comparison between (5.3) and solutions from (3.12), (3.13), (3.14) and (4.12). Subsonic and supersonic results are shown in figures 8(a) and 8(b), respectively. The ordinate of each of these subfigures is the Mach number and the abscissa is the entropy change normalised by the specific gas constant, i.e. $(s-s^*)/R$ . Since $s^*$ refers to the sonic state where $M=1$ , which is the maximum entropy state for Fanno flow, the normalised entropy change satisfies $(s-s^*)/R \leqslant 0$ , with equality only at the sonic condition. Solutions were obtained by using $M_1=1$ and selecting a value of $(s-s^*)/R$ to use with the Gibbs relation to evaluate $P_{o2}/P_{o1}$ for use in (3.12), (3.13), (3.14) and (4.12). The solution from (3.12) for $f=3$ is shown as a blue solid line, the solution from (3.13) for $f = 5$ is shown as a green dash-dot line, the solution from (3.14) for $f = 7$ is shown as a dark yellow dashed line and the solution from (4.12) is shown as a magenta dotted line. The black symbols ‘ $+$ ’ represent the results from (5.3). The resulting Mach number calculated corresponds precisely to the exact result given in (5.3). Additionally, for a given Mach number $M$ , the resulting value for the normalised entropy change is the largest for $f = 3$ and smallest for $f \to \infty$ in both the subsonic and supersonic regimes. The subsonic results in figure 8(a) show a relatively small difference compared with supersonic flow between the bounding cases of $f=3$ and $f\to \infty$ at either constant $M$ or constant $(s-s^*)/R$ , indicating that there is no significant difference for gases in the subsonic range. However, for supersonic flow, shown in figure 8(b), there is a significantly larger difference at constant $M$ or constant $(s-s^*)/R$ between the bounding cases of $f=3$ and $f\to \infty$ , especially as $M$ becomes larger.

Figure 8.

Fanno flow prediction of Mach number $M$ taking $M_1=M^*=1$ versus entropy change normalised by $R$ using (3.12) for $f=3$ (solid blue line), (3.13) for $f=5$ (green dash-dot line), (3.14) for $f=7$ (dark yellow dashed line) and (4.12) for $f \to \infty$ (magenta dotted line) compared with the exact results obtained from (5.3), shown as ‘+’ symbols, over (a) subsonic and (b) supersonic flow regimes.

All results from (3.12), (3.13), (3.14) and (4.12) are identical to the analytical expression shown in (5.3) to numerical precision. Within the present calorically perfect, constant $\gamma$ ideal gas model, the cases $f=3$ and $f\to \infty$ bound the solutions obtained for intermediate $f$ in steady Q1D flow. Additionally, along a line of constant $(s-s^*)/R$ from $M=1$ , the $f\to \infty$ result is encountered first and the $f=3$ result is encountered last. This trend is also consistent with the general differential Q1D relation introduced earlier in (2.4). In the Fanno flow limit, area change and heat transfer are absent, so the local variation of Mach number is governed by wall friction alone. Since the friction contribution to ${\rm d}M/M$ is amplified for $\gamma \gt 1$ relative to $\gamma \to 1$ , larger changes in $M$ are expected for gases with larger $\gamma$ . Thus, monatomic gases exhibit the largest changes in $M$ , whereas gases with $f\to \infty$ exhibit the smallest changes. This is consistent with the trends shown in figure 8 and with the ordering of the solutions discussed above. Additional examination of the influence coefficients for various quantities is discussed in Appendix C.

5.3. Rayleigh flow

Rayleigh flow is the following problem to be examined. Rayleigh flow considers a constant-area flow with heat transfer through the walls. The entropy change between the inlet state normalised by $c_{\!P}$ for a finite- and infinite-degrees-of-freedom gas is

(5.4) \begin{equation} \frac {s-s^*}{c_{\!P}} = \begin{cases} 2 \ln \left ( \frac {2M}{1+M^2}\right )&\, \textrm {if }f \to \infty, \\[7pt] \ln M^2 + \frac {\gamma +1}{\gamma } \ln \left ( \frac {1+\gamma }{1+\gamma M^2} \right )&\,\textrm {otherwise}, \end{cases} \end{equation}

where $^*$ indicates the sonic state where $M=M^*=1$ . The first expression in (5.4) represents the case of an infinite-degrees-of-freedom gas while the second expression is valid for all other finite-degrees-of-freedom ideal gases. The derivation of the first expression for $f \to \infty$ in (5.4) is shown in Appendix D. The second expression in (5.4) and more information on Rayleigh flow may be found in Shapiro (Reference Shapiro1953); John & Keith (Reference John and Keith2006).

Figure 9 shows the comparison between (5.4) and solutions from (3.12), (3.13), (3.14) and (4.12). Subsonic and supersonic Mach number results are shown in figures 9(a) and 9(b), respectively. The ordinate of each of these subfigures is the Mach number and the abscissa is the entropy change normalised by the specific heat at constant pressure, i.e. $(s-s^*)/c_{\!P}$ . This normalisation by $c_{\!P}$ , rather than $R$ , is used to keep the $f\to \infty$ limit finite and to facilitate comparisons across a range of $f$ , including $f \to \infty$ . Since $s^*$ refers to the sonic state where $M=1$ , which is the maximum entropy state for Rayleigh flow, the normalised entropy change satisfies $(s-s^*)/c_{\!P} \leqslant 0$ , with equality only at the sonic condition. Solutions were obtained by using $M_1=1$ and selecting a value of $(s-s^*)/c_{\!P}$ to use with the Gibbs relation to evaluate $P_{o2}/P_{o1}$ for use in (3.12), (3.13), (3.14) and (4.12). The abscissa $(s-s^*)/c_{\!P}$ maps directly to the stagnation temperature ratio for Rayleigh flow used in the Gibbs relation as

(5.5) \begin{equation} \frac {T_o}{T_o^*} = \frac {2(\gamma+1)M^2}{(1+\gamma M^2)^2} \left( 1 + \frac{\gamma-1}{2} M^2\right), \end{equation}

which is the standard form found in various texts (Shapiro Reference Shapiro1953; John & Keith Reference John and Keith2006). The solution from (3.12) for $f = 3$ is shown as a blue solid line, the solution from (3.13) for $f = 5$ is shown as a green dash-dot line, the solution from (3.14) for $f = 7$ is shown as a dark yellow dashed line and the solution from (4.12) for $f \to \infty$ is shown as a magenta dotted line. The black ‘+’ symbols represent the results from (5.4). The resulting Mach number calculated corresponds precisely to the exact result given in (5.4). The subsonic results, shown in figure 9(a), show a relatively small difference between the bounding cases of $f=3$ and $f\to \infty$ at either constant $M$ or constant $(s-s^*)/c_{\!P}$ compared with the supersonic flow results in figure 9(b). This observation indicates that there is no significant difference for physically possible gases over the subsonic range. However, for supersonic flow shown in figure 9(b), there is a significantly larger difference at constant $M$ or constant $(s-s^*)/c_{\!P}$ between the bounding cases of $f=3$ and $f\to \infty$ , especially as $M$ becomes larger.

Figure 9.

Rayleigh flow prediction of Mach number $M_2$ taking $M_1=M^*=1$ versus entropy change normalised by $c_{\!P}$ using (3.12) for $f=3$ (solid blue line), (3.13) for $f=5$ (green dash-dot line), (3.14) for $f=7$ (dark yellow dashed line) and (4.12) for $f \to \infty$ (magenta dotted line) compared with the exact results obtained from (5.4), shown as ‘+’ symbols, over (a) subsonic and (b) supersonic flow regimes.

All results from (3.12), (3.13), (3.14) and (4.12) are identical to the analytical expression shown in (5.4) to numerical precision. Within the present calorically perfect, constant $\gamma$ ideal gas model, the cases $f=3$ and $f\to \infty$ bound the solutions obtained for intermediate $f$ in Q1D flow. Additionally, along a line of constant $(s-s^*)/c_{\!P}$ from $M=1$ , the $f\to \infty$ result is encountered first and the $f=3$ result is encountered last. This behaviour is also consistent with the general differential Q1D relation introduced earlier in (2.4). For Rayleigh flow, area change and wall friction are absent, so the local behaviour reduces to the contribution from heat transfer alone. The effect of heat transfer on ${\rm d}M/M$ is amplified for $\gamma \gt 1$ relative to the limiting case $\gamma \to 1$ . Therefore, for a given thermal forcing, larger changes in $M$ are expected for gases with larger $\gamma$ , i.e. monatomic gases exhibit the largest changes in $M$ , whereas gases with $f\to \infty$ exhibit the smallest changes. This is consistent with the trends shown in figure 9 and with the ordering of the solutions discussed above. Additional examination of the influence coefficients for various quantities is discussed in Appendix C.

5.4. Flow through a normal shock

The problem of a normal shock is the last of the classical problems to be examined. Flow through a normal shock occurs at constant area with no heat transfer and, for the physical shock solution, requires $M_1\gt 1$ along with a selection of the gas, i.e. $f$ . The outlet Mach number $M_2$ for a normal shock in terms of the inlet Mach number $M_1$ and the ratio of specific heats is

(5.6) \begin{equation} M_2 = \sqrt {\frac {\frac {\gamma -1}{2} M_1^2 + 1}{\gamma M_1^2 - \frac {\gamma -1}{2}}}. \end{equation}

The expression shown in (5.6) is valid for all degrees of freedom discussed in this work. The change in entropy normalised by $R$ is

(5.7) \begin{equation} \frac {s_2-s_1}{R} = \begin{cases} \frac {M_1^4-1}{2M_1^2} - \ln (M_1^2)\,&\textrm {if } f\to \infty ,\\[9pt] \ln \left [ \left ( \frac {(\gamma +1) M_1^2}{2+(\gamma -1)M_1^2} \right )^{\frac {-\gamma }{\gamma -1}} \left ( \frac {2\gamma }{\gamma +1} M_1^2- \frac {\gamma -1}{\gamma +1} \right )^{\frac {1}{\gamma -1}} \right ]\,&\textrm {otherwise.} \end{cases} \end{equation}

The first expression in (5.7) represents the case of an infinite-degrees-of-freedom gas while the second expression is valid for all other finite-degrees-of-freedom ideal gases. The derivation of the first expression for $f \to \infty$ in (5.7) is shown in Appendix D. The second expression in (5.7) and more information on flow through a normal shock may be found in Shapiro (Reference Shapiro1953); John & Keith (Reference John and Keith2006).

Figure 10.

Normal shock prediction of (a) the outlet Mach number and (b) the entropy change normalised by $R$ versus the inlet Mach number $M_1$ using (3.12) for $f=3$ (solid blue line), (3.13) for $f=5$ (green dash-dot line), (3.14) for $f=7$ (dark yellow dashed line) and (4.12) for $f \to \infty$ (magenta dotted line) compared with the exact results, shown as ‘+’ symbols, obtained from (5.6) and (5.7), respectively.

Figure 10(a) shows the comparison between (5.6) and the solutions from (3.12), (3.13), (3.14) and (4.12). The abscissa is the inlet Mach number $M_1$ and the ordinate is the outlet Mach number $M_2$ . The solution from (3.12) for $f = 3$ is shown as a blue solid line, the solution from (3.13) for $f = 5$ is shown as a green dash-dot line, the solution from (3.14) for $f = 7$ is shown as a dark yellow dashed line and the solution from (4.12) for $f \to \infty$ is shown as a magenta dotted line. The black ‘+’ symbols represent the results from (5.6) and (5.7) for figures 10(a) and 10(b), respectively. All results in figure 10(a) from (3.12), (3.13), (3.14) and (4.12) are identical to the analytical results from (5.6) to numerical precision. Within the present calorically perfect, constant $\gamma$ ideal gas model, the cases $f=3$ and $f\to \infty$ bound the solutions obtained for intermediate $f$ . The results in figure 10(a) show a change in $M_2$ for a given $M_1$ from $f =3$ to $f \to \infty$ when $(s_2-s_1)/R \gt 0$ . The infinite-degrees-of-freedom gas has the smallest $M_2$ for a given $M_1$ across all gases examined.

Figure 10(b) shows the inlet Mach number $M_1$ on the abscissa and the normalised entropy change on the ordinate. Furthermore, figure 10(b) has the same legend scheme as figure 10(a). For $M_1\lt 1$ , the analytic continuation of (5.7) yields $\Delta s\lt 0$ and is therefore non-admissible. The physical normal shock solution corresponds to $M_1\gt 1$ , for which (5.7) gives $s_2 - s_1\gt 0$ and $M_2\lt 1$ . This admissibility restriction is consistent with the literature (Shapiro Reference Shapiro1953; John & Keith Reference John and Keith2006). Interestingly, in figure 10(b) an appreciable change is observed in the normalised entropy change $(s_2-s_1)/R$ for a given $M_1$ for $f =3$ to $f \to \infty$ over the range of $M_1$ shown. More specifically, the infinite-degrees-of-freedom gas shows significantly larger entropy change across the normal shock through the range of $M_1$ shown as compared with the $f=3$ , $5$ or $7$ gases.

Figure 11.

Control volume for the sudden expansion example. Adapted from Oliva & Morris (Reference Oliva and Morris2023), CC BY 4.0.

6.1. Sudden expansion

Flow through a sudden expansion in a compressible internal flow is a standard benchmark with extensive measurements and modelling available in the literature (Hall & Orme Reference Hall and Orme1955; Marjanovic & Djordjevic Reference Marjanovic and Djordjevic1994; Greitzer et al. Reference Greitzer, Tan and Graf2004; Oliva & Morris Reference Oliva and Morris2023; Oliva et al. Reference Oliva, Szczudlak, Jemcov and Morris2024). A control volume for the sudden expansion problem is shown in figure 11. The control volume comprises one inflow, one outflow and impermeable wall control surfaces. Wall friction and heat transfer are neglected. Many treatments approximate the wall pressure on control surface 3 using the upstream static pressure. The present closure does not require prescribing a wall pressure. Instead, an inlet-based stagnation pressure loss coefficient is used to explicitly determine the stagnation pressure ratio. The inlet-based stagnation pressure loss coefficient is defined by

(6.1) \begin{equation} \omega _1 \equiv \frac {P_{o1} - P_{o2}}{P_{o1} - P_1} = \frac {1 - \frac {P_{o2}}{P_{o1}}}{1 - (1 + \frac {\gamma -1}{2} M_1^2 )^{\frac {-\gamma }{\gamma -1}}}, \end{equation}

which is the stagnation pressure decrease normalised by the inlet compressible dynamic pressure (Dixon & Hall Reference Dixon and Hall2010). In the case of incompressible flow, it can be shown that the approximation

(6.2) \begin{equation} \omega _1 \approx \left ( 1 - \frac {A_1}{A_2} \right )^2 \end{equation}

holds, which is shown in Marjanovic & Djordjevic (Reference Marjanovic and Djordjevic1994); Greitzer et al. (Reference Greitzer, Tan and Graf2004). Thus, in the limit of an infinite sudden expansion, the inlet-based stagnation pressure loss coefficient approaches unity, which is consistent with other previous results (Oliva et al. Reference Oliva, Szczudlak, Jemcov and Morris2024). Using (6.1) and (6.2), the stagnation pressure ratio may be explicitly determined a priori as

(6.3) \begin{equation} \frac {P_{o2}}{P_{o1}} = 1 - \left ( 1 - \frac {A_1}{A_2}\right )^2 \left ( 1 - \left (1 + \frac {\gamma -1}{2} M_1^2 \right )^{\frac {-\gamma }{\gamma -1}} \right )\!. \end{equation}

Figure 12 compares the predicted outlet Mach number to the measurements of Hall & Orme (Reference Hall and Orme1955) for air using $f=5$ . The abscissa is the inlet Mach number $M_1$ and the ordinate is the outlet Mach number $M_2$ . The lines are predictions from (3.13) and the various symbols are the corresponding experimental data for various area ratios. All predictions use (6.1) and (6.2) to determine the stagnation pressure ratio. For increasing inlet Mach number $M_1$ , there is a corresponding increase in the outlet Mach number $M_2$ . For larger area ratios, the outlet Mach number is smaller compared with smaller area ratios. There is excellent agreement between the experimental data and the prediction over the full range of area ratios and inlet Mach numbers.

Figure 12.

Sudden expansion predictions of the outlet Mach number $M_2$ versus the inlet Mach number $M_1$ using (3.13) given $A_2/A_1$ , $\omega _1 = (1 - ({A_1}/{A_2}))^2$ , $f = 5$ , which are displayed using lines, compared with experimental data from Hall & Orme (Reference Hall and Orme1955), which are displayed using symbols.

6.2. Sudden contraction

Sudden contraction provides a second benchmark that utilises loss modelling under a discontinuous area reduction. A control volume for the sudden contraction problem is shown in figure 13. There is a single inlet and single outlet for this problem along with an impermeable wall, which is consistent with figure 1. The outflow station is chosen sufficiently downstream of the contraction so that a uniform, one-dimensional outlet flow state is a reasonable approximation. Similar to sudden expansion, wall friction and heat transfer are neglected. Classical closures approximate the wall pressure on surface 3 using the inlet stagnation pressure. However, here the wall pressure is not directly specified a priori. Instead, an outlet-based stagnation pressure loss coefficient is used to explicitly determine the stagnation pressure ratio. The outlet-based stagnation pressure loss coefficient is defined as

(6.4) \begin{equation} \omega _{2} \equiv \frac {P_{o1} - P_{o2}}{P_{o2} - P_2} = \frac {1 - \frac {P_{o2}}{P_{o1}}}{\frac {P_{o2}}{P_{o1}}\left ( 1 - \left(1+\frac {\gamma -1}{2} M_2^2\right)^{\frac {-\gamma }{\gamma -1}} \right )}, \end{equation}

which is the stagnation pressure decrease normalised by the outlet compressible dynamic pressure (Denton Reference Denton1993; Aungier Reference Aungier2005). It may be shown that the outlet-based stagnation pressure loss coefficient can be approximated as

(6.5) \begin{equation} \omega _{2} \approx 1 - \frac {A_2}{A_1}, \end{equation}

which assumes incompressible flow. The derivation of (6.5) is shown in Marjanovic & Djordjevic (Reference Marjanovic and Djordjevic1994); Oliva et al. (Reference Oliva, Szczudlak, Jemcov and Morris2024). Thus, in the limit of an infinite sudden contraction, the outlet-based stagnation pressure loss coefficient approaches unity. The presence of the outlet conditions in (6.4) introduces another factor of the outlet Mach number to a non-integer power into (3.6), thereby breaking the current polynomial structure used thus far. However, if $M_{2}\approx M_{2s}$ , where $M_{2s}$ is the isentropic outlet Mach number, then the stagnation pressure ratio may be estimated a priori as

(6.6) \begin{equation} \frac {P_{o2}}{P_{o1}} = \left [1 + \left (1 - \frac {A_2}{A_1} \right ) \left ( 1 - \left ( 1 + \frac {\gamma -1}{2} M_{2s}^2 \right )^{\frac {-\gamma }{\gamma -1}} \right )\right ]^{-1} \!, \end{equation}

which allows the polynomial structure in (3.6) to remain unchanged (Farokhi Reference Farokhi2014). Similarly, for certain problems, one may choose to use the isentropic outlet-based stagnation pressure loss coefficient $\omega _{2s}$ rather than the outlet-based stagnation pressure loss coefficient (Horlock Reference Horlock1966; Dixon & Hall Reference Dixon and Hall2010).

Figure 13.

Control volume used for the sudden contraction example. Adapted from Oliva & Morris (Reference Oliva and Morris2023), CC BY 4.0.

Using (6.6) requires that the sudden contraction first be solved assuming isentropic flow, which provides $M_{2s}$ . This solution provides $M_{2s}$ directly. Then the value of $M_{2s}$ may be used to evaluate $P_{o2}/P_{o1}$ with (6.6). Lastly, the resulting stagnation pressure ratio estimate is input into (3.6) to calculate $M_{2}$ without iteration.

Figure 14 shows a comparison of experimental data for air flow through a sudden contraction from Benedict, Carlucci & Swetz (Reference Benedict, Carlucci and Swetz1966) and the results from (3.13) for a $f = 5$ gas. The abscissa is the outlet static-to-stagnation pressure ratio $P_2/P_{o2}$ and the ordinate is the outlet-to-inlet stagnation pressure ratio $P_{o2}/P_{o1}$ . Experimental data are shown as the various symbols that correspond to different area ratios. The black dotted line is a reference line for $\omega _2=1$ , i.e. an infinite contraction. The remaining coloured lines are predictions from (3.13) for the respective area ratios. For a decreasing outlet static-to-stagnation pressure ratio, the stagnation pressure ratio is shown to decrease. Moreover, for smaller area ratios (i.e. larger contraction ratios), the outlet-to-inlet stagnation pressure ratio is shown to decrease more rapidly with smaller area ratios. These trends are consistent in both the predictions and the experimental data.

Overall, the experimental data show good agreement with the predictions, especially at larger area ratios; the largest discrepancy is approximately $0.025$ and is concentrated at small area ratios and low $P_2/P_{o2}$ (high $M_2$ ). However, there is some discrepancy between the predictions and the experimental data especially at small area ratios and small $P_2/P_{o2}$ , which corresponds to a high outlet Mach number $M_2$ . The largest discrepancy is of the order of approximately $0.025$ . This discrepancy likely reflects the incompressible scaling used to estimate $\omega _2$ and the approximation $M_2 \approx M_{2s}$ . An improved estimate of $\omega _2$ or an alternative loss-coefficient definition that closes the system without this approximation may reduce the error.

Figure 14.

Sudden contraction predictions of outlet-to-inlet stagnation pressure ratio $P_{o2}/P_{o1}$ versus outlet static-to-stagnation pressure ratio $P_2/P_{o2}$ using (3.13) given $A_2/A_1$ , $\omega _{2} = (1 - ({A_2}/{A_1}))$ , $f = 5$ , which are displayed using lines, compared with experimental data from Benedict et al. (Reference Benedict, Carlucci and Swetz1966) which are displayed using symbols.

6.3. Simultaneous friction and heat transfer

Combined wall friction and heat transfer in a constant-area duct is a convenient test case for coupling dissipation and heat addition in a Q1D setting. The configuration, shown in figure 15, has one inlet and one outlet flow control surface. The wall contributes a friction force and heat transfer to the working fluid. Ferrari (Reference Ferrari2021a ) derived an implicit algebraic relation for this problem that can be solved iteratively and is used as a reference solution.

Figure 15.

Constant-area duct control volume with wall friction and heat transfer for the combined effects example. Adapted from Oliva & Morris (Reference Oliva and Morris2023), CC BY 4.0.

Figure 16.

Combined wall friction and heat transfer solutions for Mach number $M$ versus non-dimensional streamwise distance $c_{\!f} ( {x}/{D})$ using (3.12) for $f=3$ , (3.13) for $f=5$ , (3.14) for $f=7$ and (4.12) for $f \rightarrow \infty$ compared with the exact results from Ferrari (Reference Ferrari2021a ) for (a) subsonic inlet flow with $M_1=0.4$ and $\varGamma =0.01$ , and (b) supersonic inlet flow with $M_1=2.5$ and $\varGamma =1.0$ .

Figure 16 shows two results for simultaneous friction and heat transfer. More specifically, figure 16(a) shows a result for subsonic flow and figure 16(b) for supersonic flow. The abscissa for both is a non-dimensional length along the constant-area duct. The ordinate is the Mach number in the duct at that particular $x$ location. The exact solution from Ferrari (Reference Ferrari2021a ) is shown as black ‘+’ symbols. The lines are predictions from the $f = 3, 5, 7$ and $f \to \infty$ solutions, which are shown as a blue solid line, green dash-dot line, dark yellow dashed line and magenta dotted line, respectively.

The subsonic problem shown in figure 16(a) is now described in detail. The inlet Mach number is fixed at $0.4$ . The non-dimensional heat-transfer-to-friction ratio $\varGamma$ is $0.01$ , where $\varGamma$ is defined as

(6.7) \begin{equation} \varGamma \equiv \frac {q_w/h_{o1}}{c_{\!f} L /D}. \end{equation}

Here $L$ is the length of the constant-area duct, $D$ is the diameter of the pipe and $c_{\!f}$ is the friction coefficient. All gases start at the same prescribed inlet Mach number and are shown to increase in Mach number as the non-dimensional length increases. The Mach numbers increase for each gas until the constant-area duct becomes sonic, indicated by the respective curves having an infinite slope at $M=1$ . The solution from Ferrari (Reference Ferrari2021a ) is identical, within numerical precision, to the solutions from (3.12) for $f = 3$ , (3.13) for $f = 5$ , (3.14) for $f = 7$ and (4.12) for $f \rightarrow \infty$ . Gases with lower degrees of freedom are observed to become sonic at lower non-dimensional lengths along the constant-area duct compared with gases with higher degrees of freedom. Therefore, gases with a lower number of degrees of freedom are more sensitive to the combined effects of heat transfer and friction. For finite dimensional heat input $q_w$ , the $f \to \infty$ limit has vanishing thermal sensitivity because $q_w / h_{o1} \to 0$ . Under that limiting convention, choking in the $f \to \infty$ case is due to friction. For the prescribed finite values of $\Gamma$ used here, the $f \to \infty$ curve should instead be interpreted as the weakest thermal response among the cases plotted. These observations are also consistent with the previous discussion on the influence coefficients of ${\rm d}M/M$ shown in (2.4).

The supersonic problem shown in figure 16(b) is now described in detail. The inlet Mach number is fixed at $2.5$ . The non-dimensional heat-transfer-to-friction ratio $\varGamma$ is $1.0$ . All gases start at the same prescribed inlet Mach number and are shown to decrease in Mach number as the non-dimensional length increases. The Mach numbers decrease for each gas until the supersonic constant-area flow becomes sonic, indicated by the respective curves having an infinite slope at $M=1$ . The solution of Ferrari (Reference Ferrari2021a ) agrees, to numerical precision, with the solutions from (3.12) for $f=3$ , (3.13) for $f=5$ , (3.14) for $f=7$ and (4.12) for $f\to \infty$ . Gases with fewer degrees of freedom are observed to become sonic at smaller non-dimensional lengths along the constant-area duct than gases with more degrees of freedom. Therefore, gases with fewer degrees of freedom are more sensitive to the combined effects of heat transfer and friction, whereas the $f\to \infty$ limit shows the weakest thermal response for the prescribed $\Gamma$ values.