2.1. Conservation equations

Eilmer is formulated around the integral form of the conservation equations, which can be written as

(2.1)\begin{equation} \frac{\partial}{\partial t} \int_{V} U \,{\rm d}V ={-} \oint_{S} \left ( \bar{F}_{c} - \bar{F}_{v} \right )\, {\cdot}\, \hat{n}\,{\rm d}A + \int_{V} Q\,{\rm d}V , \end{equation}

where $S$ is the bounding surface and $\hat {n}$ is the outward-facing unit normal of the control surface. The symbol $V$ in (2.1) is the volume of a cell while $A$ is the area of the cell boundary. For a gas in thermal and chemical non-equilibrium governed by two temperatures, the vector of conserved quantities carried in a simulation is

(2.2) \begin{equation} U = \left [ \begin{array}{@{}c@{}} \rho \\ \rho v_{x} \\ \rho v_{y} \\ \rho v_{z} \\ \rho E \\ \vdots \\ \rho_{s} \\ \vdots \\ \rho e_{v} \end{array} \right ], \end{equation}

where the conserved quantities are respectively density, $x$-, $y$- and $z$-momentum per volume, total energy per volume, followed by $n_{sp}$ individual densities for the chemical species and the mixture vibroelectronic energy per volume.

The flux vectors are divided into convective and viscous-transport contributions. With the specific internal energy of the gas being $e$ and the total specific energy being $E = e + v^2/2$, the convective component is

(2.3) \begin{equation} \bar{F}_{c} = \begin{bmatrix} \rho v_{x} \\ \rho v_{x}^{2} + p \\ \rho v_{x} v_{y} \\ \rho v_{x} v_{z} \\ \rho E v_{x} + p v_{x} \\ \vdots \\ \rho_{s} v_{x} \\ \vdots \\ \rho e_{v} v_{x} \\ \end{bmatrix} \hat{i} + \begin{bmatrix} \rho v_{y} \\ \rho v_{y} v_{x} \\ \rho v_{y}^{2} + p \\ \rho v_{y} v_{z} \\ \rho E v_{y} + p v_{y} \\ \vdots \\ \rho_{s} v_{y} \\ \vdots \\ \rho e_{v} v_{y} \\ \end{bmatrix} \hat{j} + \begin{bmatrix} \rho v_{z} \\ \rho v_{z} v_{x} \\ \rho v_{z} v_{y} \\ \rho v_{z}^{2} + p \\ \rho E v_{z} + p v_{z} \\ \vdots \\ \rho_{s} v_{z} \\ \vdots \\ \rho e_{v} v_{z} \\ \end{bmatrix} \hat{k} . \end{equation}

Although the viscous components, $F_{v}$, are available in the Eilmer code, they are not a major player in the present study.

The vector of source terms includes the species rates of reaction and thermal energy exchanges, which are discussed in the following section

(2.4) \begin{equation} Q = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ \vdots \\ M_s \dfrac{{\rm d}[X_s]}{{\rm d}t} \\ \vdots \\ Q^{{vib\text{-}trans}} + Q^{{vib\text{-}chem}}\end{bmatrix}. \end{equation}

2.2. Equations of state and thermodynamic properties

The governing equations are closed using equations of state. In this work, a two-temperature model is used to account for thermochemical non-equilibrium. The two temperatures are a transrotational temperature, $T$, and a vibroelectronic temperature, $T_v$. The thermal equation of state for the pressure of the gas mixture is obtained from Dalton's law of partial pressures. Each component in the mixture is assumed to obey the ideal-gas law (perfectly elastic collisions of point particles). The pressure for a mixture of $n_{sp}$ species is

(2.5)\begin{equation} p = \sum_{s=1}^{n_{sp}} \rho_s R_s T, \end{equation}

where $\rho _s$ is the partial density of species $s$, $R_s$ is the specific gas constant of species $s$ and $T$ is the transrotational temperature.

The caloric equation of state relates the internal energy of the mixture to the temperatures in the gas. The mixture energy is expressed as a mass fraction weighted contribution from the internal energy of each component species as

(2.6)\begin{equation} e = \sum_{s=1}^{n_{sp}} c_s e_s(T, T_v). \end{equation}

The calculation of $e_i$ closely follows the approach described in the two-temperature model presented in Gnoffo, Gupta & Shinn (Reference Gnoffo, Gupta and Shinn1989). That approach is described here. It is convenient to work with specific heat, $C_p$, and enthalpy, $h$, since thermodynamic curve fits are commonly available in these forms.

The two-temperature gas model used in this work assumes one Boltzmann distribution of translation and rotational energy levels described by $T$, and a second Boltzmann distribution of vibrational and electronic energy levels described by $T_v$. Under this assumption, the specific heat at constant pressure for species $s$ is

(2.7)\begin{equation} C_{p,s}(T, T_v) = C_{p_t,s}(T) + C_{p_r,s}(T) + C_{p_{v e},s}(T_v) . \end{equation}

For atomic species, there is no appreciable energy storage in rotational modes, so $C_{p_r,{atoms}} = 0$. The vibroelectronic energy storage has been grouped in one term, $C_{p_{v e}}$. Note that storage in the electronic mode is small enough to be negligible at the temperatures of interest in this work. However, it is included here because it is faithful to the implementation used for the simulations.

Curve fits are available for $C_p$ for individual species as a function of a single temperature. They are valid under an assumption of thermal equilibrium. However, these curve fits are convenient to use to compute the vibrational and electronic contributions to $C_p$ because the translational and rotational contributions are constant. These contributions are constant because these modes are assumed to be fully excited at the temperatures of interest for this work. Here, we use the equilibrium curve fits and data from the Chemical Equilibrium Applications program of Gordon & McBride (Reference Gordon and McBride1994), which has the form

(2.8)\begin{equation} C_{p,s}(T) = R_s (a_0 T^{{-}2} + a_1 T^{{-}1} + a_2 + a_3 T + a_4 T^2 + a_5 T^3 + a_6 T^4). \end{equation}

For all particles, under the assumption of fully excited translation

(2.9)\begin{equation} C_{p_t,s} = \frac{5}{2} R_s. \end{equation}

For linear molecules (N$_2$, O$_2$, CO) with fully excited rotation

(2.10)\begin{equation} C_{p_r,s} = R_s \end{equation}

and for nonlinear molecules (here: CO$_2$)

(2.11)\begin{equation} C_{p_r,s} = \frac{3}{2} R_s. \end{equation}

These constant contributions can be subtracted from (2.8) to compute the vibroelectronic specific heat as

(2.12)\begin{equation} C_{p_{iv e},s}(T_v) = C_{p,s}(T_v) - C_{p_t,s} - C_{p_r,s}. \end{equation}

The calculation of enthalpy for each species follows the same idea as $C_p$, that is, use a curve fit for enthalpy assuming thermal equilibrium and then subtract out the fully excited modes. The idea is shown here in equation form. The curve fits for enthalpy at thermal equilibrium following Gordon & McBride (Reference Gordon and McBride1994) are

(2.13)\begin{align} h_s(T) = (R_sT) \left({-}a_0 T^{{-}2} + a_1 T^{{-}1} \log{T} + a_2 + a_3 \frac{T}{2} + a_4 \frac{T^2}{3} + a_5 \frac{T^3}{4} + a_6 \frac{T^4}{5} + \frac{a_7}{T} \right). \end{align}

The constant $a_7$ in (2.13) represents a reference enthalpy, $h_{{ref.}}$, which for the Gordon & McBride (Reference Gordon and McBride1994) curves is taken as the enthalpy of formation at 298.15 K. Using these curve fits, the vibroelectonic enthalpy is computed as

(2.14)\begin{equation} h_{v e,s}(T_v) = h_s(T_v) - (C_{p_t,s} + C_{p_r,s})(T_v - T_{{ref.}}) - h_{{ref.},s}. \end{equation}

The enthalpy in species $s$ including all energy mode contributions is computed as

(2.15)\begin{equation} h_s(T, T_v) = (C_{p_t,s} + C_{p_r,s})(T - T_{{ref.}}) + h_{v e,s}(T_v) + h_{{ref.},s}. \end{equation}

As a final step, the energy in species $s$ is (assuming the ideal-gas law)

(2.16)\begin{equation} e_s(T, T_v) = h_s - R_sT. \end{equation}

2.3. Chemistry modelling in thermal non-equilibrium

The chemistry source term is computed using the law of mass action. A collection of simple reversible reactions can be represented by

(2.17)\begin{equation} \sum_{s=1}^{n_f} \alpha_s X_s \rightleftharpoons \sum_{s=1}^{n_b} \beta_s X_s, \end{equation}

where $\alpha _s$ and $\beta _s$ represent the stoichiometric coefficients for the reactants and products respectively, and $X_s$ represents gaseous species $s$. For a given reaction $r$, the rate of concentration change of species $s$ is given as,

(2.18)\begin{equation} \left( \frac{{\rm d}[X_s]}{{\rm d}t} \right)_r = \nu_s \left\{ k_f \prod_i [X_i]^{\alpha_i} - k_b \prod_i [X_i]^{\beta_i} \right\}, \end{equation}

where $\nu _s = \beta _s - \alpha _s$. By summation over all reactions, $n_r$, the total rate of concentration change is

(2.19)\begin{equation} \frac{{\rm d}[X_s]}{{\rm d}t} = \sum_{r=1}^{N_r} \left( \frac{{\rm d}[X_s]}{{\rm d}t} \right)_r. \end{equation}

The chemical rate equations require rate constants, $k_{f,r}$ and $k_{b,r}$, for the forward and backward rates of each reaction. These are computed using the generalized Arrhenius form based on a rate-controlling temperature, $T_a$,

(2.20)$$\begin{gather} k_{f,r} = A T_a^{n} \exp{\left( \frac{-C}{T_a}\right)}, \end{gather}$$

(2.21)$$\begin{gather}k_{b,r} = A T_a^{n} \exp{\left(\frac{-C}{T_a}\right)}, \end{gather}$$

or when $k_{b,r}$ rate constant data are unavailable, it is computed using the forward rate constant and the concentration equilibrium constant as

(2.22)\begin{equation} k_{b,r} = \frac{k_{f,r}}{K_{c,r}(T)}. \end{equation}

To account for the effect of thermal non-equilibrium – in this case, we mean specifically vibrational non-equilibrium – a model is used to modify the chemical reaction rates. In this work, the two-temperature model of Park (Reference Park1988) is used. In this model, Park proposes that a weighted average of the translation and vibrational temperatures be used for the rate-controlling temperature in computing forward rate constants involving molecular species. The expression for rate-controlling temperature $T_a$ is

(2.23)\begin{equation} T_a = T^s T_v^{1-s}, \end{equation}

where $s$ is a parameter of the model and is taken as a value between 0.5 and 0.7. The parameter $s$ is usually calibrated against experiment. For a time, it was accepted practice to set $s = 0.5$ in expanding flows and $s = 0.7$ in compressive flows. However, more recently in Park (Reference Park2010), the conclusion is that there has been no clear demonstration of any advantage of using 0.7 over 0.5. In the simulations of this work, we use $s = 0.5$. Thus the rate-controlling temperature is a geometric average of the translational and vibrational temperatures, $\sqrt {TT_v}$.

In the simulations involving nitrogen, presented later, chemical rate constant parameters have been taken from two sources: Kewley & Hornung (Reference Kewley and Hornung1974) and Park (Reference Park1993). For the simulation with a carbon dioxide mixture, the reaction rates from Ebrahim & Hornung (Reference Ebrahim and Hornung1973) are used. All of the reaction rate parameters are given in the Appendix in tables 4–6.

2.4. Energy exchange models

The energy exchange source terms in (2.4) account for the exchange of energy between the vibroelectronic and transrotational modes, $Q^{{vib\text {-}trans}}$, and the energy change in the vibroelectronic mode due to chemistry effects, $Q^{{vib\text {-}chem}}$.

The Landau–Teller model is used to compute the energy exchange between the vibroelectronic and transrotational modes. The model uses the assumption that only one level of jump in vibrational quantum number is allowed. Since the vibroelectronic energy of all molecular species are lumped together in the two-temperature model, the tally for energy exchange between vibroelectronic and transrotational modes occurs over all vibrators

(2.24)\begin{equation} Q^{{vib\text{-}trans}} = \sum_p^{{vib.}} \rho_p \sum_{q=1}^{N_s} \frac{e_{v_p}^{*} - e_{v_p}}{\tau_{{VT}}^{pq}}, \end{equation}

where $p$ represents vibrators (i.e. molecules), $q$ represents all species in the mixture and $\tau _{{VT}}^{pq}$ is the vibrational relaxation time between species $p$ in a bath of species $q$.

The vibrational relaxation time, introduced by Millikan & White (Reference Millikan and White1963), is computed in the form following Park et al. (Reference Park, Howe, Jaffe and Candler1994)

(2.25)\begin{equation} \tau_{{VT}}^{pq} = \frac{A}{p_q} \exp{(a (T^{{-}1/3} - b) - 18.42)}, \end{equation}

where $A = 1$ atm s, $p_q$ is the partial pressure of species $q$ in atmospheres and $a$ and $b$ are parameters of the model. For some collisions, the parameters $a$ and $b$ are chosen to give a best fit to experimental data. In lieu of experiments, $a$ and $b$ can be estimated from molecular constants for the colliders $p$ and $q$ as

(2.26)$$\begin{gather} a = c_1 \mu^{1/2} \varTheta_{p}^{4/3}, \end{gather}$$

(2.27)$$\begin{gather}b = c_2 \mu^{1/4}, \end{gather}$$

with

(2.28)$$\begin{gather} c_1 = 0.00116\,\text{K}^{{-}1}, \end{gather}$$

(2.29)$$\begin{gather}c_2 = 0.015\,\text{K}^{{-}1/3}. \end{gather}$$

Here, $\mu$ (in g mol$^{-1}$) is the reduced molecular weight of the colliders, computed as the product of the molecular weights divided by their sum. The symbol $\varTheta _p$ is the characteristic vibrational temperature for $p$, and the relaxation time parameters for the CO$_2$ mixture are taken from table 1 in Park et al. (Reference Park, Howe, Jaffe and Candler1994).

When molecules dissociate, this process may have a tendency to change the average vibrational energy of the mixture if there is a preference for high-lying molecules to dissociate. In a similar manner, during recombination, the molecules that form may have a vibrational energy content that differs from the bulk average. These effects of chemistry on the vibrational energy content are accounted for using the model of Knab, Frühauf & Messerschmid (Reference Knab, Frühauf and Messerschmid1995). The total change in vibroelectronic energy due to chemistry is a summation over all reactions involving molecular dissociation

(2.30)\begin{equation} Q^{{vib\text{-}chem}} = \sum_{r}^{{dissociations}} Q^{{vib\text{-}chem}}_r, \end{equation}

where

(2.31)\begin{align} Q^{{vib\text{-}chem}}_{r} &= k_{f,r} \varPi_n [X_n]^{\nu_{f,r}} \times \left[ \frac{R_u \varTheta_p}{\exp (\varTheta_p/T^{*}) -1} - \frac{D_r}{\exp (D_r/(R_u T^{*})) - 1} \right]\nonumber\\ &\quad - k_{b,r} \varPi_n [X_n]^{\nu_{b,r}} \frac{1}{2} \left[D_r - R_u \varTheta_p \right]. \end{align}

In (2.31), $T^*$ is a pseudo-temperature computed as $T^* = (T T_v)/(T - T_v)$, $D_r$ is the energy of dissociation for reaction $r$ and $\varTheta _p$ is the characteristic vibrational temperature of molecule $p$ that undergoes dissociation in reaction $r$.

2.5. Finite-volume mechanics

The conservation equations are applied to each finite-volume cell. In a three-dimensional simulation, the cells are hexahedral with quadrilateral boundary interfaces. In a two-dimensional simulation, the cells are assumed to be 1 unit deep (in the $z$-direction) and the boundary interfaces, projected onto the ($x,y$)-plane, consist of four straight lines. Flux values are estimated at midpoints of the cell interfaces and the integral conservation equation (2.1) is approximated as the algebraic expression

(2.32)\begin{equation} \frac{{\rm d}U}{{\rm d}t} ={-} \frac{1}{V} \sum_{cell\text{-}surface} \left ( \bar{F}_{c} - \bar{F}_{v} \right )\, {\cdot}\, \hat{n} \, {\rm d}A + Q ,\end{equation}

where $U$ and $Q$ now represent cell-average values.

The full flow domain is subdivided into blocks of finite-volume cells, arranged into structured grids. During a simulation, a halo of ghost cells around each block of cells facilitates the exchange of data between adjacent blocks and the application of boundary conditions. The exchange of data for adjacent blocks is a simple copy of the cell data, prior to the evaluation of the fluxes of conserved quantities across the cell interfaces. Solid-wall boundary conditions are approximated by reflecting the data for cells that are interior to the block, into the corresponding ghost cells. For the convective fluxes, components of velocity in the ghost cells are adjusted to be mirror images of the interior velocities. For solid-wall boundaries, the flux calculation ensures that normal velocity at the boundary face is zero. At the supersonic inflow boundary, constant flow data are copied into the ghost cells while, at the simple outflow boundary, interior-cell flow data are copied into the downstream ghost cells.

Calculation of the fluxes at each cell interface is preceded by an interpolation phase where cell-average quantities are reconstructed index direction by index direction. Having flow data available in the ghost cells allows all interfaces, including those on a boundary, to be treated by the flux calculator as internal interfaces. For each flow variable, $w$, left and right values ($w_L$ and $w_R$ respectively) at a cell interface are evaluated as the corresponding cell-average value plus a limited higher-order interpolated increment. Given an array of cell centres $[L1,L0,R0,R1]$ with the interface of interest located between $L0$ and $R0$, we fit separate quadratic interpolants to subranges $[L1,L0,R0]$ and $[L0,R0,R1]$ and then evaluate $w_L$ and $w_R$ as

(2.33)\begin{equation} \left. \begin{gathered} w_L = w_{L0} + \alpha_{L0} \left[ \varDelta_{L+} \times \left( 2 h_{L0} + h_{L1} \right) + \varDelta_{L-} \times h_{R0} \right] s_L , \\ w_R = w_{R0} - \alpha_{R0} \left[ \varDelta_{R+} \times h_{L0} + \varDelta_{R-} \times \left( 2 h_{R0} + h_{R1} \right) \right] s_R,\\ \varDelta_{L-} = \frac{w_{L0} - w_{L1}}{\frac{1}{2} \left( h_{L0} + h_{L1}\right)},\\ \varDelta_{L+} = \frac{w_{R0} - w_{L0}}{\frac{1}{2} \left( h_{R0} + h_{L0} \right)} = \varDelta_{R-}, \\ \varDelta_{R+} = \frac{w_{R1} - w_{R0}}{\frac{1}{2} \left( h_{R0} + h_{R1} \right)},\\ \alpha_{L0} = \frac{h_{L0} / 2}{h_{L1} + 2 h_{L0} + h_{R0}} , \\ \alpha_{R0} = \frac{h_{R0} / 2}{h_{L0} + 2 h_{R0} + h_{R1}} , \end{gathered} \right\} \end{equation}

where $h$ represents the width of a cell. The high-order increment is limited by $s_L$ and $s_R$ and, as described by van Albada, van Leer & Roberts (Reference van Albada, van Leer and Roberts1981), these limiter values are evaluated as

(2.34)\begin{equation} \left. \begin{gathered} s_L = \frac{\varDelta_{L-} \varDelta_{L+} + |\varDelta_{L-} \varDelta_{L+}| + \epsilon} {\varDelta_{L-}^2 + \varDelta_{L+}^2 + \epsilon} ,\\ s_R = \frac{\varDelta_{R-} \varDelta_{R+} + |\varDelta_{R-} \varDelta_{R+}| + \epsilon} {\varDelta_{R-}^2 + \varDelta_{R+}^2 + \epsilon} , \\ \epsilon = 1.0 \times 10^{{-}12}. \end{gathered} \right\} \end{equation}

Finally, minimum and maximum limits are also applied, so that the newly interpolated values lie within the range of the original cell-centred values. Unlimited, this reconstruction scheme has third-order truncation error.

With local flow data on the left and right of each face, fluxes are calculated with an adaptive calculator that selects the AUSMDV scheme (the version by Wada & Liou (Reference Wada and Liou1994) of the Advection Upwind Splitting Methods) away from shocks and Hänel's method (Wada & Liou Reference Wada and Liou1997) near shocks. The switching between the two flux calculators is governed by a shock detector that is a simple measure of the relative change in normal velocity at interfaces. Specifically, we indicate a strong compression at the interface when

(2.35)\begin{equation} \frac{v_{R0} - v_{L0}}{\min(a_{R0}, a_{L0})} < \textrm{Tol} , \end{equation}

where Tol is the compression tolerance and is typically set at $-0.3$. This measure is applied to all interfaces in a block and then a second pass propagates the information to nearby interfaces. If a first cell interface is identified as having a strong compression, the equilibrium flux method is used for all interfaces attached to the cell containing that first interface.

Evaluation of the spatial derivatives of temperature and velocity, needed for the viscous fluxes, is performed via the use of the divergence theorem on secondary cells that are temporarily constructed around the midpoint of each interface.

Time integration of the discrete equations is performed with a second-order predictor–corrector method that updates the conserved quantities over time step $\Delta t$ in stages

(2.36)\begin{equation} \left. \begin{gathered} U_1 = U_0 + \Delta t \left.\frac{{\rm d}U}{{\rm d}t}\right|_0 ,\\ U_2 = U_0 + \Delta t \frac{1}{2} \left( \left.\frac{{\rm d}U}{{\rm d}t}\right|_1 + \left.\frac{{\rm d}U}{{\rm d}t}\right|_0 \right). \end{gathered} \right\} \end{equation}

After each stage of the update, the new conserved quantities are used to compute the other flow quantities such as pressure, temperature and sound speed.

With the update process described above, the remaining components of the software set up an initial flow state for all cells across the domain, compute values for the conserved quantities in each cell and then repeatedly step in time, allowing the flow to evolve according to the conservation equations and the applied boundary conditions. This stepping is performed synchronously, with blocks of cells distributed across many message-passing tasks running in parallel on a cluster computer.

5.1. Nominal free-stream conditions

The free-stream conditions reported in Hornung & Smith (Reference Hornung and Smith1979) were determined by first determining the reservoir conditions from the measured initial shock tube pressure and temperature, the measured shock speed and the measured reservoir pressure, assuming equilibrium processes. The nozzle exit conditions were then found by a one-dimensional non-equilibrium nozzle-flow integration in which vibration was assumed to be in equilibrium. In order to improve on this, the nozzle flow was recomputed with viscous axisymmetric flow and with the same two-temperature reacting gas model as in § 3. The nozzle exit conditions obtained from this computation are shown in figure 6.

Figure 6. Profiles of flow variables across the exit plane of the T3 nozzle as computed from the data of Hornung & Smith (Reference Hornung and Smith1979) in axisymmetric flow with the reaction model of Park (Reference Park1993). Here, $r$ is the radial coordinate in the nozzle exit plane.

The most important difference between these conditions and those given by Hornung & Smith (Reference Hornung and Smith1979) is that the vibrational temperature is much higher than the translational temperature. Although there are other minor differences between the two, the nominal conditions for our computations are taken to be as given in table 2.

Table 2. Nominal free-stream conditions for the nitrogen computations.

5.2. The effect of the boundary layer

In order to compare computational and experimental results in this case it is important to determine how important the viscous boundary layer is in the shock detachment distance. The resolution requirements for viscous flows are much more stringent than for inviscid flows and it would save substantial computational time if only the inviscid case needed to be computed. In the following we therefore test the effect of the boundary layer in a particular representative case.

To examine this question, a viscous and an inviscid flow were computed with $\theta =60^\circ$. Figure 7 shows a comparison of the profiles of velocity magnitude $q$ and density $\rho$ as well as their product $\rho q$ plotted against distance normal to the wall $n$ at a point 30 mm from the wedge tip in the symmetry plane ($w=51$ mm). The distance of the grid point nearest to the wall in these computations was 2.5 $\mathrm {\mu }$m from the wall, which is of order 1 viscous length. The integral of the difference between the viscous and inviscid values of $\rho q$ over $n$ is a measure of the displacement effect of the boundary layer. Although the viscous curve for $\rho q$ is somewhat noisy near the wall, and an accurate displacement thickness cannot be determined from the data, it is clear from the comparison that the displacement effect is very small.

Figure 7. Profiles of density, velocity magnitude and their product in the boundary layer at 30 mm from the wedge tip in the $z$-normal symmetry plane. Two-temperature reacting nitrogen, $\theta =60^\circ$, $\varLambda =3$. The fat lines are for viscous flow and they are compared with thin lines for inviscid flow.

This may be tested further by a comparison of the temperature distributions in the immediate vicinity of the point where the stagnation streamline crosses the shock, see figure 8. It is clear from this figure that the effect of viscosity on $\varDelta$ is small enough to be ignored, amounting to less than 1 %. We therefore consider that inviscid computations suffice for the comparison with the experiment.

Figure 8. Enlarged views of the temperature distribution in the region where the stagnation streamline crosses the shock wave in the two flows of figure 7. (a) Inviscid flow, (b) viscous flow. The two green lines are rulers measuring the distance from the stagnation point. These are separated in the $y$-direction by a distance of 30 $\mathrm {\mu }$m and measure 4.96 and 4.99 mm.

5.3. Computational results and comparison with experiment

As for the non-relaxing flows in § 4, three-dimensional computations were made with the two-temperature reacting flow model used in § 3, for a wedge with $\varLambda =3$, $w=51$ mm, at the free-stream conditions of table 2. The results are shown in figure 9 in comparison with the experiments of Hornung & Smith (Reference Hornung and Smith1979). Also shown in the figure are computations of frozen and equilibrium flow at the same free-stream conditions. The striking feature about the figure is that the computational results, shown as filled squares, are quite precisely displaced to the right from the experiments (faster reaction) by $4^\circ$. This is a large discrepancy. It is interesting to note that the reaction rates measured by Hanson & Baganoff (Reference Hanson and Baganoff1972), using pressure measurements in a shock tube, and those measured by Kewley & Hornung (Reference Kewley and Hornung1974), using optical interferometry, agree well with each other and are slower than those of Park (Reference Park1993) in the high-temperature range (10 000–14 000 K) by as much as an order of magnitude. For this reason, additional computations were made with the Kewley & Hornung (Reference Kewley and Hornung1974) rates. The results are shown as filled circles in figure 9. However, as may be seen, the discrepancy is only reduced by a little less than $1^\circ$.

Figure 9. Comparison of three-dimensional computations with the experiments of Hornung & Smith (Reference Hornung and Smith1979) (HS), $w=51$ mm, $\varLambda =3$, free stream: nitrogen, as table 2. Open squares: experiment. Filled squares: rates of Park (Reference Park1993), filled circles: rates of Kewley & Hornung (Reference Kewley and Hornung1974) (KH), filled black triangles: frozen flow, filled red triangles: equilibrium flow. The blue line is a smoothed fit to the experimental data. The thin red lines are (4.6) with $\varepsilon$ determined from the density ratios at these conditions, in very good agreement with the computations.

At this point it is important to relate this result to earlier experiments on dissociating nitrogen flow over circular cylinders performed in the same facility but with a conical nozzle by Hornung (Reference Hornung1972), in which good agreement with two-dimensional computations was observed, specifically between an experimental and computed interferogram of cylinder flow. Cylinder flows are, of course, much less sensitive to relaxation effects than shock detachment from a wedge. However, both source-flow (conical nozzle) and finite-span effects were shown in Hornung & Smith (Reference Hornung and Smith1979) to have the same effect on detachment distance as a faster reaction. Taking proper account of them in computing the cylinder flows would therefore cause a discrepancy of the same sign to be observed there too.

5.4. Tests of possible causes of the discrepancy

In order to examine possible reasons for the discrepancy between experiment and computation that is evident from figure 9, a number of additional computations were made by changing one parameter at a time.

By deliberately reducing all the reaction rates by a factor of 10, the computed detachment distance is brought into agreement with the experiment, see the filled squares in figure 10. This is a much greater change than is credible, since the uncertainty in the reaction rates is more like a factor of 2.

Figure 10. The effect of changing parameters one at a time on the discrepancy. For reference, the frozen and equilibrium results are repeated as in figure 9. Filled black squares: reactions reduced by a factor of 10. Filled red circles: free-stream speed reduced to 5 km s$^{-1}$. Filled green squares: free-stream atomic nitrogen mass fraction increased to 27 %. Filled blue squares: helium contamination mass fraction set to 3.4 %. HS denotes Hornung & Smith (Reference Hornung and Smith1979).

Next, the free-stream speed was reduced from 5.5 to 5.0 km s$^{-1}$. This yielded the results shown as filled red circles in figure 10. Again this step brings the results into line with the experiment, and again, the change is more than twice the uncertainty in $u_\infty$.

Another change was to increase the free-stream atomic nitrogen mass fraction to 27 %, with the results shown as filled green squares in figure 10. This change again brings the results into line with the experiment, but it is much larger than a credible uncertainty.

Finally, a gas model was constructed in which helium was added as a passive component. This was done in order to examine the effect that contamination of the test gas with helium driver gas would have on the measurements. The experiments of Sudani & Hornung (Reference Sudani and Hornung1998) showed that driver-gas contamination occurs in two stages, an early stage in which the concentration of driver gas is limited, followed by a massive increase at a later time. It is possible that the time of the Hornung & Smith (Reference Hornung and Smith1979) experiments occurred during the first stage of contamination. Results obtained with a helium mass fraction of 3.4 % are shown in figure 10 as filled blue squares. This effect reduces the discrepancy by a factor of approximately two.

One might argue that a combination of these four effects, each within a credible limit of uncertainty, could add up to explain the discrepancy. However, we would consider this to be too much of a coincidence. The problem can only be effectively resolved by a repeat of the experiment with more accurately characterized conditions, e.g. using spectroscopic methods such as have recently been applied at Caltech's T5 facility by Girard et al. (Reference Girard, Finch, Strand, Hanson, Yu, Austin and Hornung2021), giving free-stream velocity, temperature and species concentration.