4.1. Point properties

The influence of free-stream conicity on various point properties in the flow over a sphere – including the boundary layer thickness and tangential velocity gradient, which have never been examined before to any extent in the literature – is shown in figure 10. The qualitative trends exhibited by these properties from the influence of free-stream conicity have intuitive physical interpretations. The ‘$y$’ component (see figure 3) of the free-stream velocity immediately upstream of the shock (and not exactly on the axisymmetry axis) becomes more prominent with increasing free-stream conicity. Near the axisymmetry axis, the shock is aligned almost parallel with the $y$-axis, which allows this increasing ‘$y$’ velocity to transfer through the shock and thereby increase the tangential velocity and tangential velocity gradient in the flow behind the shock in this region, as shown in figure 10(c) for the tangential velocity gradient at the boundary layer edge on the axisymmetry axis. This increased tangential velocity gradient duly causes the sonic condition to be reached after a shorter distance and, consequently, shifts the sonic point closer to the axisymmetry axis, as shown in figure 10(b). Also, the increased tangential velocity increases the inertial force (over the viscous force) in the flow making the boundary layer thinner, as shown in figure 10(d). Because the boundary layer edge pressure, density and temperature on the axisymmetry axis are essentially unchanged with free-stream conicity, this thinner boundary layer directly increases the temperature gradient at the wall near the axisymmetry axis, resulting in a larger heat flux, as shown in figure 10(e). Furthermore, as indicated in figure 10(a), the increased tangential velocity forces the shock stand-off distance near the axisymmetry axis to decrease, considering the control volume in figure 9, to maintain $\dot m_{in}=\dot m_{out}$ since both the flow density leaving the control volume and $\dot m_{in}$ are essentially not influenced by free-stream conicity. This statement can be formulated mathematically as follows, assuming the tangential velocity is constant across the shock layer, an idea from Hornung (Reference Hornung2019), and equal to $[({\textrm {d}u}/{\textrm{d}\kern0.06em x})^{0}\,{\textrm {d}y}]$,

(4.1)\begin{equation} \bar{\rho}^{out}_{\infty} \left[\left(\frac{{\rm d}u}{{\rm d}\kern0.06em x}\right)^{0}_{\infty}\,{{\rm d} y}\right] 2{\rm \pi}\, {{\rm d} y} \varDelta^{0}_{\infty} = \bar{\rho}^{out} \left[\left(\frac{{\rm d}u}{{\rm d}\kern0.06em x}\right)^{0}\, {{\rm d} y}\right] 2{\rm \pi}\,{{\rm d} y} \varDelta^{0}, \end{equation}

where $\rho ^{out}$ is the average density leaving the control volume. The left-hand side corresponds to $\dot m_{out}$ in a uniform free stream while the right-hand side corresponds to that in a conical free stream. Assuming $\rho ^{out} = \bar {\rho }^{out}_{\infty }$, one obtains

(4.2)\begin{equation} \frac{\left(\dfrac{{\rm d}u}{{\rm d}\kern0.06em x}\right)^{0}}{\left(\dfrac{{\rm d}u}{{\rm d}\kern0.06em x}\right)^{0}_{\infty}} = \frac{\varDelta^{0}_{\infty}}{\varDelta^{0}}, \end{equation}

which can actually be obtained from (2.8) if one assumes $(R_s + \varDelta ^0)/(R_s + \varDelta ^0_\infty ) \approx 1$, that is, the change in shock stand-off distance caused by free-stream conicity is negligible compared with the distance between the shock and the centre of the sphere (appropriate, since the shock layer is generally thin in hypersonic flows); shown in figure 10(c), this is a fine approximation as (4.2) agrees well with the other results, which also validates the simple model used in its derivation.

Figure 9. Inviscid flow over a sphere in the vicinity of the axisymmetry axis.

Figure 10. The influence of the degree of free-stream conicity, measured by $d$, on the (a) shock stand-off distance on the symmetry axis (‘Hornung’ and ‘Shapiro’ are from (2.4) and (2.1), respectively), (b) sonic point location (‘Shapiro’ is from (2.2)), (c) tangential velocity gradient at the boundary layer edge on the stagnation streamline (‘Shapiro’, ‘Olivier’ and ‘Current work’ are from (2.10), (2.8) and (4.2), respectively), (d) boundary layer thickness at the stagnation point and (e) stagnation point heat flux (‘Eremeitsev & Pilyugin’, ‘Current work 1’ and ‘Current work 2’ are from (2.6), (2.9) and (2.11), respectively).

Examining the different results for the shock stand-off distance on the symmetry axis, figure 10(a), one can see that the theoretical results match the numerical results well, with errors of less than $\pm$0.03 at $d = 4$. The influence of free-stream conicity on the shock stand-off distance is shown to mostly have little sensitivity to the free-stream condition; the PG results at different Mach and Reynolds numbers are essentially identical, differing by less than 0.03 for $d = 4$, consistent with the finding of Golovachov (Reference Golovachov1985). The EQ result is also very similar to the PG results, which is consistent with the finding of Golovachov (Reference Golovachov1985) and Shapiro (Reference Shapiro1975), who suggested that PG and EQ flows have the same influence of the free-stream conicity. On the other hand, the NONEQ results do have a more noticeable difference from the other results. More precisely, the free-stream conicity is shown to have a lesser influence on the NONEQ conditions compared with the other conditions. This can be explained as follows. Because the free-stream conicity causes the shock stand-off distance to decrease, the flow along the stagnation streamline becomes more frozen, which is obvious when examining the Damköhler number for $O_{2}$ dissociation (which is the main reaction occurring in the inviscid flow in the NONEQ condition) written as (following Candler Reference Candler2018)

(4.3)\begin{equation} Da^{0}_{sk} = \frac{\varDelta^{0} k_{D,O_{2}} p_{p}}{\overline{{u}^{0}} T_{p} \mathcal{R}}, \end{equation}

where $\overline {{u}^{0}}$ is the mean post-shock velocity on the stagnation streamline, $\mathcal {R}$ is the universal gas constant, $T_{p}$ and $p_{p}$ are the equilibrium post-shock total temperature and pressure, respectively, and $k_{D,O_{2}}$ is the oxygen dissociation rate constant at $T_{p}$ ($Da^{0}_{sk} = O (0)$ for the NONEQ condition); since the free-stream condition immediately upstream of the shock on the stagnation streamline is unchanged, $\overline {{u}^{0}}$, $T_{p}$, $p_{p}$ and $k_{D,O_{2}}$ are essentially not influenced by free-stream conicity, which means $Da^{0}_{sk}$ decreases due to the smaller shock stand-off distance (e.g. $(Da^{0}_{sk})^{d=4}/(Da^{0}_{sk})^{d=\infty } = \varDelta ^{0}_{d=4}/\varDelta ^{0}_\infty = 0.77$), leading to a more frozen flow along the stagnation streamline. However, such freezing tends to increase the shock stand-off distance, as shown by Wen & Hornung (Reference Wen and Hornung1995). Consequently, this results in the non-equilibrium flow having a resistance to the decrease in shock stand-off distance caused by the free-stream conicity; such resistance is uniquely a non-equilibrium effect and is non-existent in PG and EQ flows.

Equation (2.4) from Hornung (Reference Hornung2019) assumes the average density across the shock layer on the stagnation streamline outside of the boundary layer remains constant, which is true for perfect-gas or equilibrium flows. For non-equilibrium flows, this average density does change with free-stream conicity, as shown in figure 11, which shows the density on the stagnation streamline between the shock and the boundary layer edge (defined as the wall-normal distance where the local total enthalpy is 99 % of the free-stream total enthalpy). In this case, Hornung's equation should be given as

(4.4)\begin{equation} \frac{\varDelta^{0}}{\varDelta^{0}_\infty} = \frac{\bar{\rho}^{0}_\infty}{\bar{\rho}^{0}}\frac{1}{1+\dfrac{\left(R_{c}^{0}\right)_{\infty}}{L_{1}}}, \end{equation}

where $\bar {\rho }^{0}$ and $\bar {\rho }^{0}_\infty$ are the average density across the shock layer on the stagnation streamline outside of the boundary layer in the non-uniform and uniform free streams, respectively. Therefore, although thermodynamics was not explicitly considered in Hornung's derivations, the effect of non-equilibrium flow is allowed to enter through the average density across the shock. Figure 11 indicates that the average density across the shock in the $d = 4$ flow is approximately 4 %–5 % lower than that in the uniform flow which, according to (4.4), means the non-equilibrium value of $\varDelta ^{0}/\varDelta ^{0}_\infty$ at $d = 4$ should be higher than the perfect or equilibrium gas value by the same amount; this is indeed observed in the results shown in figure 10(a) when comparing the NONEQ results with the PG and EQ results from CFD.

Figure 11. The density on the stagnation streamline between the shock and the boundary layer edge for condition 5 (NONEQ) with a non-catalytic wall. Since ‘$n$’ is the normal distance from the wall and $\delta ^{0}$ is the boundary layer thickness at the stagnation point, the $x$-axis shows the normal distance from the boundary layer edge normalized with the shock stand-off distance.

To examine the importance of free-stream conicity, the result in figure 10(a) is compared with the experimental uncertainties for the shock stand-off distance, as shown in table 2. The influence of the free-stream conicity becomes comparable to the total uncertainty when $d \lessapprox 10$. If a unique free-stream estimate for each individual shot is available, then the uncertainty from the test condition repeatability is eliminated and only the measurement uncertainty needs to be considered, in which case the influence of the free-stream conicity becomes relevant when $d \lessapprox 20$. Therefore, because $d$ values as small as around 4 can realistically be encountered, as discussed in § 1, experimental measurements of the shock stand-off distances made in facilities with conical nozzles may be significantly influenced by the divergent free stream and, thus, this should be considered and checked before interpreting the experimental results. If corrections are required, they can be done easily using the analytical expressions that are shown in the current work to be very accurate (within the measurement uncertainty shown in table 2).

It is very interesting to observe that the two theoretical results (Shapiro Reference Shapiro1975; Hornung Reference Hornung2019), which were derived from different methods and have completely different expressions ((2.1) and (2.4)), produce essentially the same curves in figure 10(a). The Shapiro expression requires the calculation of the influence of free-stream conicity on the sonic point location, $\theta ^{s}/\theta _{\infty }^{s}$, a priori; an analytical expression for this is provided (2.2) and its comparison with the numerical results is shown in figure 10(b). One can see that the expression is very accurate, matching with the numerical results which are found at the boundary layer edge defined as the location where the local total enthalpy is 99 % of the free-stream total enthalpy. All the results, including the NONEQ and EQ results, are essentially identical at a given $d$, which means that the results are not condition dependent. The overall excellent prediction of $\theta ^{s}/\theta _{\infty }^{s}$ is a significant result as this value is also, importantly, required a priori for the Shapiro transformation (Shapiro Reference Shapiro1975), introduced in § 2.1, for the property distributions.

For the sonic point and tangential velocity gradient, shown in figures 10(b) and 10(c), respectively, the fact that the NONEQ results are essentially indistinguishable from the other results may be surprising given that the NONEQ shock stand-off distance displays a resistance that may transfer to the tangential velocity gradient, as shown in (2.8) and (4.2), which may, in turn, influence the sonic point location. However, (2.8) is for perfect-gas and equilibrium flows; for non-equilibrium flows, the relationship should, instead, be derived from Olivier (Reference Olivier1995) as

(4.5)\begin{equation} \left(\frac{{\rm d}u}{{\rm d}\kern0.06em x}\right)^{0,e}\propto\frac{R_{s}+\varDelta^{0}}{\varDelta^{0}}\frac{1}{\rho^{0,e}}, \end{equation}

because the density at the boundary layer edge on the stagnation streamline, $\rho ^{0,e}$, depends on the thermochemistry along the stagnation streamline which is influenced by free-stream conicity, which is unlike in perfect-gas and equilibrium flows where $\rho ^{0,e}$ is not influenced in this way. In non-equilibrium flows, the free-stream conicity makes the flow near the stagnation streamline more frozen, which decreases the density (Anderson Reference Anderson2019); this is shown in figure 12 where the non-equilibrium effect reduces $\rho ^{0,e}$ by approximately 4 % at $d = 4$, which is comparable to its effect on the shock stand-off distance where the NONEQ results at $d = 4$ are approximately 4 % greater than the other results, as shown in figure 10(a). Consequently, the effect of the reduction in $\rho ^{0,e}$ on $(\textrm {d}u/{\textrm{d}\kern0.06em x})^{0,e}$ cancels out the effect of the resistance in shock stand-off distance, resulting in $(\textrm {d}u/{\textrm{d}\kern0.06em x})^{0,e}$ effectively having no special non-equilibrium effect, as shown in figure 10(c), where the analytical perfect or equilibrium gas results obtained using (2.8) (along with (2.4) to analytically predict the change in $\varDelta ^{0}$) gives excellent agreement with all the numerical results. Likewise, retaining the density terms in (4.1), one obtains

(4.6)\begin{equation} \frac{\left(\dfrac{{\rm d}u}{{\rm d}\kern0.06em x}\right)^{0,e}}{\left(\dfrac{{\rm d}u}{{\rm d}\kern0.06em x}\right)^{0,e}_{\infty}} = \frac{\bar{\rho}^{out}_{\infty}}{\bar{\rho}^{out}} \frac{\varDelta^{0}_{\infty}}{\varDelta^{0}}, \end{equation}

which allows non-equilibrium effects to enter through the density ratio. As shown in figure 11, non-equilibrium reduces the average density by approximately 4 %–5 %, which is cancelled out by its effect on the shock stand-off distance, resulting in the tangential velocity gradient ratio being effectively not influenced by the non-equilibrium according to (4.6). This cancellation can be expected from theory (conservation of mass) where the product $[\bar {\rho }^{out} \varDelta ^{0}]$ is known to be a constant for a given free stream and sphere size regardless of the thermochemistry involved (Wen & Hornung Reference Wen and Hornung1995).

Figure 12. The influence of the degree of free-stream conicity on the density at the boundary layer edge on the stagnation streamline.

Furthermore, due to the lack of non-equilibrium effects on the sonic point location and tangential velocity gradient, (2.10) from Shapiro (Reference Shapiro1975), which assumes a linear velocity distribution on the boundary layer edge between the axisymmetry axis and the sonic point, also predicts the tangential velocity gradient accurately as, shown in figure 10(c) (basically indistinguishable from Olivier's method). Shapiro mentioned that the error of his equation due to the aforementioned assumption is no larger than $5\,\%$; this is further confirmed in the current work by comparing with the CFD results. From CFD, the velocity distribution is essentially linear – with only a slight concave down curvature – as shown exemplarily for condition 4 (PG $M = 11$) in figure 13. Shapiro's equation actually gives the ratio of the average tangential velocity gradient between the axisymmetry axis and the sonic point. Although the tangential velocity gradient at $\theta = 0$ is slightly higher than the average value due to the slight concave down curvature, this same trend is observed in both the uniform and non-uniform free-stream simulations, as shown in figure 13, allowing the errors to essentially cancel out and resulting in a good prediction of the ratio at $\theta = 0$.

Figure 13. The boundary layer edge velocity, $u^{e}$, distribution between the axisymmetry axis and the sonic point for condition 4 (PG $M = 11$).

The boundary layer thickness at the stagnation point – defined as the wall-normal distance where the local total enthalpy is 99 % of the free-stream total enthalpy – is examined in figure 10(d). One can see that the free-stream conicity decreases the boundary layer thickness at the stagnation point, and the results are rather insensitive to the different flow conditions at any given $d$. Shown together with the Navier–Stokes solutions in this figure are the PG and NONEQ results from the numerical solutions of the self-similar boundary layer at the stagnation point of a sphere, which are denoted as ‘Theory’. This theory (Anderson Reference Anderson2019) does not explicitly account for free-stream conicity. However, as discussed earlier, free-stream conicity influences the tangential velocity gradient at the stagnation point, which can be varied in the aforementioned theory to possibly predict the influence of the free-stream conicity on the boundary layer thickness at the stagnation point. Using the tangential velocity gradient at the stagnation point for a uniform free stream calculated analytically with Newtonian theory, and using (2.4) and (2.8) to calculate the change in tangential velocity gradient with free-stream conicity, $d$, while all other boundary layer edge (stagnation point) properties remain unchanged for each condition (equilibrium stagnation conditions are used as the edge conditions in the NONEQ cases), the theoretical results are produced and excellent agreement with CFD is observed. This indicates, firstly, that the free-stream conicity decreases the boundary layer thickness solely due to the tangential velocity gradient, which increases due to the decreasing shock stand-off distance, and, secondly, the self-similarity of the boundary layer at the stagnation point is not influenced by the free-stream conicity. Furthermore, the results indicate essentially no dependence on the flow condition and gas type. No distinct non-equilibrium effects are observed, which means that the changes to the edge condition caused by non-equilibrium, as mentioned earlier, do not significantly influence the boundary layer thickness.

The result for the stagnation point heat flux is shown in figure 10(e), where the free-stream conicity is found to increase the stagnation point heat flux, which is expected given the non-uniformity decreases the shock stand-off distance which increases the tangential velocity gradient, as discussed earlier in § 2.1. The theoretical results again match the numerical results well; the error is less than $\pm$0.03 at $d = 4$. Also, the three theoretical results, which come from different expressions with different origins ((2.6), (2.9) and (2.11)) are essentially identical. The results for the stagnation point heat flux, as with the shock stand-off distance, are mostly insensitive to the flow condition and gas type. The exception here is the NONEQ result using an NC wall which is a little lower than the other results, as can be clearly seen when examining the $d = 4$ results. Interestingly, the NONEQ result using a SC wall does not exhibit this result. Consequently, it is found that the cause of the NONEQ NC result differing from the other results is due to the non-equilibrium effect in the boundary layer. The thinning of the boundary layer at the stagnation point (which is almost frozen) with increasing conicity, discussed above, allows even less recombination to occur in the boundary layer, as demonstrated in the Navier–Stokes solution of condition 5 with a NC wall shown in figure 14; this same trend is also observed in the solutions of the NONEQ NC self-similar boundary layer. This phenomenon can also be shown through the gas-phase oxygen recombination Damköhler number (also called the recombination rate parameter) for the stagnation point boundary layer given as (Fay & Riddell Reference Fay and Riddell1958; Inger Reference Inger1963)

(4.7)\begin{equation} Da_{BL}^{0} = \frac{k_{r,O_{2}}p_{p}^{2}}{({\rm d}u/{{\rm d}\kern0.06em x})^{0,e}T_{p}^{2}\mathcal{R}^{2}}, \end{equation}

where $k_{r,O_{2}}$ is the oxygen recombination rate constant at $T_{p}$ ($Da_{BL}^{0} = O (-4)$ for the NONEQ condition). Because the free-stream condition immediately upstream of the shock on the stagnation streamline is unchanged, the only parameter in the above equation that changes due to free-stream conicity is $(\textrm {d}u/{\textrm{d}\kern0.06em x})^{0,e}$, which increases with increasing free-stream conicity, as mentioned earlier. Therefore, $Da_{BL}^{0}$ decreases with increasing free-stream conicity due to the increasing $(\textrm {d}u/{\textrm{d}\kern0.06em x})^{0,e}$ (e.g. $(Da_{BL}^{0})^{d=4}/(Da_{BL}^{0})^{d=\infty } = (\textrm {d}u/{\textrm{d}\kern0.06em x})^{0,e}_{\infty }/(\textrm {d}u/{\textrm{d}\kern0.06em x})^{0,e}_{d=4} = 0.75$), which was also shown earlier to decrease the boundary layer thickness, resulting in a more frozen boundary layer, and this is consistent with the CFD results. This phenomenon, consequently, results in less heat release in the boundary layer and a lower heat flux when the wall is non-catalytic (Fay & Riddell Reference Fay and Riddell1958).

Figure 14. The $O_{2}$ mass fraction, $c_{O_{2}}$, distribution in the stagnation point boundary layer of condition 5 (NONEQ) with a non-catalytic wall, where ‘$n$’ is the normal distance from the wall and superscript ‘$e$’ refers to the boundary layer edge.

On the other hand, if the wall is super-catalytic, the non-equilibrium in the boundary layer becomes irrelevant in terms of predicting the heat flux, as shown by Fay & Riddell (Reference Fay and Riddell1958). That is, the heat flux at a super-catalytic wall is essentially the same regardless of the behaviour of the chemical kinetics in the boundary layer. This is further demonstrated in figure 15, which shows the solutions from the non-equilibrium self-similar boundary layer with varying tangential velocity gradient while the other boundary layer edge conditions remain constant and equal to the equilibrium stagnation point condition of condition 5. The results show that the heat flux scales perfectly with $\sqrt {(\textrm {d}u/{\textrm{d}\kern0.06em x})^{0,e}}$ when the wall is super-catalytic, but not when the wall is non-catalytic due to the inhibiting of recombination by boundary layer thinning. Therefore, the NONEQ SC result in figure 10(e) is not affected by the aforementioned phenomenon and, hence, agrees well with the other results. As a corollary, one can suggest that (2.7), which works very well for perfect-gas and equilibrium flows, also works very well for non-equilibrium flows when the wall is super-catalytic, and this is consistent with the results of Fay & Riddell (Reference Fay and Riddell1958).

Figure 15. The solutions of the non-equilibrium self-similar stagnation point boundary layer for condition 5 (NONEQ) with varying tangential velocity gradient.

To examine the importance of free-stream conicity, the result in figure 10(e) is compared with the experimental uncertainties for the surface heat flux, as shown in table 2. The influence of the free-stream conicity becomes comparable to the total uncertainty when $d \lessapprox 3$. In this case, given the context of the experimental uncertainty, the influence of the free-stream divergence may generally be considered insignificant as it is within the experimental uncertainty even when the largest possible test model is used. However, if a unique free-stream estimate for each individual shot is available, then the influence of the free-stream conicity becomes relevant when $d \lessapprox 10$, and, thus, corrections to the experimental results may be necessary in certain cases which can easily be carried out using the analytical expressions given in the current work which are shown to be very accurate (within the measurement uncertainty shown in table 2).

4.2. Distributions

The influence of free-stream conicity on various normalized distributions in the flow over a sphere is shown in figure 16. Although these normalized distributions are insensitive to the free-stream condition in a uniform flow, as discussed in § 3, they are all significantly influenced by the free-stream conicity. From hereon in, all the NONEQ results refer to the NC wall case because the SC wall case produces essentially the same results and no special wall catalycity effects are observed, therefore, it is appropriately omitted for clarity. Looking at the normalized shock stand-off distance distribution in figure 16(a), one can see that the free-stream conicity causes the normalized shock stand-off distance to increase. In other words, the shock angle at any given $\theta$ increases with increasing free-stream conicity, as shown exemplarily in figure 17. This is an expected observation considering that the divergent free stream expands in the $y$ direction, which effectively turns the shock in the anti-clockwise direction about the origin, as seen in expansion fan/shock wave interactions (Nel, Skews & Naidoo Reference Nel, Skews and Naidoo2015). The increase is more severe the larger the angle is away from the stagnation point. At $\theta = 90^{\circ }$ and $d = 4$, the normalized shock stand-off distance is around two times larger than that in the corresponding uniform free stream. For reference, the absolute shock stand-off distance distribution is shown exemplarily in figure 18(a) for condition 4. One can see that the shock stand-off distance on the symmetry axis decreases with decreasing $d$, as expected from the previous section. Decreasing $d$ also increases the gradient ($\textrm {d}\varDelta /\textrm {d}\theta$) throughout, resulting in the shock stand-off distance in the non-uniform flow being eventually greater than that in the uniform flow when $\theta$ becomes large. This is why the normalized distributions have the qualitative trend shown in figure 16(a).

Figure 16. The normalized distributions of the (a) shock stand-off distance, (b) surface heat flux (‘Eremeitsev $\&$ Pilyugin’ is from (2.12)) and (c) surface pressure (‘Lunev & Khramov’ is from (2.13)). ‘Shapiro’ refers to the Shapiro transformation (Shapiro Reference Shapiro1975).

Figure 17. The shock locations for condition 2 (PG $M = 8$). The curves are shifted on the $x$-axis such that they pass through the origin.

Figure 18. The absolute distributions of the (a) shock stand-off distance, and (b) surface heat flux for condition 4 (PG $M = 11$).

Comparing figure 16(a) with the experimental uncertainty shown in table 2, one can see that measurements of the normalized shock stand-off distance should be corrected for the influence of free-stream conicity when $\theta$ is close to $90^{\circ }$ at $d \approx 25$ and when $\theta \gtrapprox 30^{\circ }$ at $d \approx 4$. The Shapiro transformation (Shapiro Reference Shapiro1975), discussed in § 2.1, is found to give a reasonable prediction when $\theta$ is not too large ($\theta \lessapprox 40$ at $d = 4$), as shown in figure 16(a), consistent with the finding by Golovachov (Reference Golovachov1985), which may be used to correct for the free-stream conicity. At large $\theta$, the Shapiro transformation is found to overpredict the normalized shock stand-off distance, and, thus, numerical methods must be used to correct for the free-stream conicity in this case. An alternative transformation may be proposed, as mentioned in § 2.1, in which all the results (non-uniform and uniform) are assumed to coalesce when the distribution is given in terms of $\theta +\omega$ (where $\omega$ is the flow divergence angle at $\theta$, defined earlier in § 2.1) instead of $\theta$. That is, it assumes that the normalized shock stand-off distance at some $\theta = \theta _{1}$ in a uniform flow is equal to that at $\theta = \theta _{1}-\omega$ in a non-uniform flow. This transformation, denoted as ‘Current work’ in figure 16(a), is found to underpredict the normalized shock stand-off distance which, together with the Shapiro transformation, forms the bounds on the more accurate numerical results. Regarding the numerical results, although the PG results for different free-stream conditions show very little difference, the results do show some sensitivity to the thermochemistry as the EQ, NONEQ and PG results differ slightly from each other which can be seen when looking at the $d = 4$ results.

Looking at figure 16(c), one can see that the free-stream conicity causes the normalized surface pressure to decrease. As discussed by Lunev & Khramov (Reference Lunev and Khramov1970), this can simply be explained with the Newtonian theory: in a conical free stream, as $\theta$ increases the free-stream flow angle, $\omega$, increases as well, which effectively makes the body surface more parallel with the free stream (figure 3), compared with the corresponding uniform free stream, and this causes the pressure distribution to decrease more rapidly in a conical free stream. The decrease is more severe the larger the angle away from the stagnation point. Because free-stream conicity does not influence the Pitot pressure (Golovachov Reference Golovachov1985), the normalized and absolute distributions have the same qualitative shape. Comparing figure 16(c) with the experimental uncertainty shown in table 2, one can see that measurements of the normalized surface pressure should be corrected for the influence of free-stream conicity when $\theta \approx 90^{\circ }$ at $d \approx 100$, $\theta \gtrapprox 40^{\circ }$ at $d \approx 25$, and $\theta \gtrapprox 10^{\circ }$ at $d \approx 4$. The Shapiro transformation and the expression of Lunev & Khramov (Reference Lunev and Khramov1970) (2.13) give similar results, and both are found to work reasonably well when $\theta$ is not too large ($\theta \lessapprox 50^{\circ }$ at $d = 4$), allowing analytical corrections for the free-stream conicity. When $\theta$ is too large ($\theta \gtrapprox 60^{\circ }$ at $d = 4$), not only are the analytical methods inaccurate, but also the influence from the free-stream conicity becomes dependent on the flow condition and gas type, consistent with the work of Golovachov (Reference Golovachov1985).

Looking at the normalized heat flux distribution in figure 16(b), one can see that the free-stream conicity causes the normalized heat flux to decrease, which is qualitatively the same trend seen in the normalized surface pressure distribution. This is expected considering the work of Lees (Reference Lees1956), who showed that the normalized heat flux distribution around a sphere is closely related to its normalized surface pressure distribution. The absolute heat flux distribution is shown exemplarily in figure 18(b), and one can see that the stagnation point heat flux increases with decreasing $d$, as expected from the previous section, while the gradient $\textrm {d}q/\textrm {d}\theta$ is decreased (made steeper) throughout, resulting in the normalized distributions having the qualitative trend shown in figure 16(b). Comparing figure 16(b) with the experimental uncertainty shown in table 2, one can see that measurements of the normalized heat flux should be corrected for the influence of free-stream conicity when $\theta \gtrapprox 50^{\circ }$ at $d \approx 25$ and when $\theta \gtrapprox 20^{\circ }$ at $d \approx 4$. The Shapiro transformation (Shapiro Reference Shapiro1975) is found to work reasonably well when $\theta$ is not too large ($\theta \lessapprox 50^{\circ }$ at $d = 4$), as shown in figure 16(a), consistent with the finding of Golovachov (Reference Golovachov1985), as with the normalized shock stand-off distance and pressure. The expression of Eremeitsev & Pilyugin (Reference Eremeitsev and Pilyugin1984) (2.12) gives results that are very similar to the Shapiro transformation in which good agreement with the numerical results is also attained when $\theta$ is not too large. Hence, in the case of $\theta$ not being too large, these two analytical methods are available for the correction of free-stream conicity, while numerical methods are required otherwise. Also, when $\theta$ is not too large, the numerical results show that the influence of the free-stream conicity is essentially independent of the free-stream condition and thermochemistry; only when $\theta$ becomes large ($\theta \gtrapprox 50^{\circ }$ at $d \approx 4$) does the dependence on the flow condition and gas type show up, which is similar to the normalized pressure and is consistent with the work of Golovachov (Reference Golovachov1985).

It should be mentioned that, for the Shapiro transformation results shown in figure 16, the disagreement trend at large values of $\theta$ is not due to poor predictions of the corresponding uniform free-stream distributions, required a priori, obtained using the analytical expressions given by (2.15)–(2.17). This is because this disagreement exists even when the numerically obtained uniform free-stream distributions are used, instead of the analytical expressions, as the inputs for the Shapiro transformation. This is shown exemplarily in figure 19 for condition 2 with $d = 4$. Therefore, the failure of the Shapiro transformation at large values of $\theta$ is inherent to the transformation itself rather than from the inputs. Nevertheless, for the surface pressure, significant quantitative improvements can be achieved at $\theta > 50^{\circ }$ by using a more accurate input, as shown in figure 19(c), indicating (2.15) (from Newtonian theory) is inaccurate at larger values of $\theta$; this makes sense because the shock lies far from the surface at large $\theta$, hence, deviating from an ideal Newtonian flow (Anderson Reference Anderson2019). For the shock stand-off distance and heat flux distributions shown in figures 19(a) and 19(b), respectively, no significant quantitative improvements are observed when using a more accurate input, indicating the analytical expressions are accurate enough.

Figure 19. The normalized distributions of the (a) shock stand-off distance, (b) surface heat flux and (c) surface pressure, for condition 2 with $d = 4$ using Shapiro's transformation with uniform free-stream distributions obtained analytically and numerically.

For completeness, the boundary layer thickness – which has never been examined before in this context to any extent – is examined in figure 20. The free-stream conicity decreases the boundary layer thickness at the stagnation point, as discussed earlier. Downstream of the stagnation point, the free-stream conicity causes the boundary layer thickness to grow rapidly, as shown in figure 20(a), and beyond approximately 30$^{\circ }$ the boundary layer thickness at any given $\theta$ becomes greater than that in the uniform free stream. This thickening of the boundary layer caused by free-stream conicity is consistent with the decrease in heat flux, as shown in figure 16(b). This is also consistent with the unit Reynolds number distribution as shown in figure 21; near the stagnation point, the inertial force relative to the viscous force is greater in a conical free stream due to the higher boundary layer edge velocity, which results in a thinner boundary layer, while further away from the stagnation point the inertial force becomes relatively smaller in a conical free stream due to the lower boundary layer edge density, resulting in a thicker boundary layer.

Figure 20. The influence of free-stream conicity on the (a) absolute boundary layer thickness distribution and (b) normalized boundary layer thickness distribution.

Figure 21. Ratio of the unit Reynolds number distribution using the boundary layer edge properties around a sphere between a conical free stream with $d = 4$ and a uniform free stream for condition 1 (PG $M = 5$).

Regarding the normalized distribution, it is of interest to test if the Shapiro transformation also works with the boundary layer thickness. The result is shown in figure 20(b), where the Shapiro transformation is applied to predict the distributions for $d$ = 4, 25 and 100 using the normalized distributions for the uniform free stream computed from CFD. One can see that, similar to the normalized distributions of the other properties shown above in figure 16, good agreement is observed for most cases when $\theta$ is not too large (e.g. $\theta \lessapprox 50$ at $d = 4$). The exception is the NONEQ result for which the Shapiro transformation does not work, even at small values of $\theta$. The results in figure 20 indicate that free-stream conicity has a quantitatively different (lesser) influence on non-equilibrium flow where the differentiation with the other conditions is noticeable even at small values of $\theta$; among the other conditions, the differentiation only becomes noticeable at large values of $\theta$. This demonstrates another special non-equilibrium effect, non-existent in frozen and equilibrium flows, that is mild and is like the resistance shown by $\varDelta ^0$ and by $q^0$ when the wall is non-catalytic, as demonstrated above in § 4.1.

4.3. Boundary layer transition

Another aspect of the flow around a sphere worth examining is the boundary layer transition, which is observed experimentally. Despite substantial recent work on this topic, a theoretical understanding of the boundary layer transition on a blunt body remains elusive (Paredes, Choudhari & Li Reference Paredes, Choudhari and Li2017, Reference Paredes, Choudhari and Li2018; Hein et al. Reference Hein, Theiss, Di Giovanni, Stemmer, Schilden, Schröder, Paredes, Choudhari, Li and Reshotko2019; Schilden et al. Reference Schilden, Pogorelov, Herff and Schröder2020; Di Giovanni & Stemmer Reference Di Giovanni and Stemmer2018). The boundary layer flow over a blunt body does not support the growth of modal instability waves, and this problem has been termed the ‘blunt-body paradox’. Roughness-induced transient growth has been considered a possible cause; however, transient growth analysis for purely stationary disturbances in weakly non-parallel boundary layers and direct numerical simulations of the flow behind a roughness patch on a spherical forebody only found moderate energy amplification (Paredes et al. Reference Paredes, Choudhari and Li2017, Reference Paredes, Choudhari and Li2018; Hein et al. Reference Hein, Theiss, Di Giovanni, Stemmer, Schilden, Schröder, Paredes, Choudhari, Li and Reshotko2019). Due to the lack of theoretical foundations in this problem, the relevant research relies heavily on experimentation which can, consequently, involve the use of conical nozzles (Lin et al. Reference Lin, Reeves and Siegelman1977).

Currently, the best way to predict the aforementioned transition is using semi-empirical correlations with inputs obtained via laminar CFD simulations. From experimental data, which show that transition always occurs in the subsonic region (upstream of the sonic point $\theta ^s$), the following correlation is given for a sphere (Paredes et al. Reference Paredes, Choudhari and Li2017):

(4.8)\begin{equation} Re_{\varTheta}\left(\frac{k}{{\varTheta}}\frac{T^e}{T^w}\right)^{0.7}\begin{matrix}\geq\\ = \\ \end{matrix} \left\{\begin{array}{@{}ll@{}}255\ \text{at}\ \theta^{s}:&\text{transition onset}\\ 215:& \text{onset location}\end{array}\right., \end{equation}

where $\varTheta$ is the boundary layer momentum thickness, $k$ is the peak-to-valley roughness height, $T^{w}$ is the wall temperature, $T^{e}$ is the boundary layer edge temperature and $Re_{\varTheta }$ is the Reynolds number based on the momentum thickness and flow conditions at the boundary layer edge, $\rho ^{e}u^{e}\varTheta /\mu ^{e}$. The correlation shows that the left-hand side of (4.8) has to exceed a value of 255 at the sonic point for transition to occur at all, and transition occurs at a point where the left-hand side of (4.8) equals 215. To study, for the first time, how free-stream conicity affects the transition location in the flow over a sphere, the influence of free-stream conicity on the left-hand side of (4.8) is shown in figure 22(a), considering that $k$ and $T^{w}$ are not influenced. Examining this figure, one can see that the free-stream conicity increases the value of the left-hand side in the subsonic region, which means that the transition location in the conical free stream, if transition were to occur, would occur closer to the stagnation point than in the uniform free stream. An alternative (and more recent) correlation to predict the onset location is given by Paredes et al. (Reference Paredes, Choudhari and Li2018)

(4.9)\begin{equation} Re_{\varTheta}\left(\frac{k}{{\varTheta}}\right)\left(\frac{T^e}{T^w}\right)^{1.31}= 455, \end{equation}

and the influence of free-stream conicity on the left-hand side of this equation is shown in figure 22(b); the same trend is observed. Also, both results in figure 22 show very little dependence on the flow condition and gas state. Equations (4.8) and (4.9) were derived for perfect-gas flows, and, thus, their validity in reactive flows is unknown. Nevertheless, they are still applied to the non-equilibrium and equilibrium results in the current work due to the lack of any alternatives.

Figure 22. The influence of free-stream conicity on the distribution of the left-hand sides of (a) (4.8) and (b) (4.9), assuming $k$ and $T^{w}$ are constants.

To understand the trend found in figure 22, it is of interest to examine the trends of the flow properties making up the left-hand sides of (4.8) and (4.9); this is shown in figure 23. One can see that the increase in the value of the left-hand side in the subsonic region by free-stream conicity is mainly due to the increase in the boundary layer edge tangential velocity, as shown in figure 23(d), caused by the free-stream conicity which is shown in § 4.1 to increase the tangential velocity gradient. With increasing free-stream conicity, this increase in the edge velocity, together with the decrease in the edge viscosity shown in figure 23(b), overcomes the contributions to decrease the left-hand side caused by the decrease in the edge density and temperature, shown in figures 23(c) and 23(a), respectively, and the decrease in the momentum thickness in the subsonic region, shown in figure 23(e). Downstream of the sonic point, the influence of the edge density and temperature wins over and the left-hand side is shown to decrease with increasing free-stream conicity. However, what happens upstream of the sonic point is more important because current experimental data indicate that transition only occurs in the subsonic region.

Figure 23. The influence of free-stream conicity on the distribution of the (a) edge temperature, (b) edge viscosity (calculated using Sutherland's formula), (c) edge density, (d) edge tangential velocity and(e) momentum thickness.

Regarding the edge temperature (and, consequently, the viscosity) shown in figure 23, an exception to the mainstream trend can be observed in the NONEQ result where the free-stream conicity is shown to cause an increase in the value in the subsonic region; this is the same phenomenon mentioned in § 4.1 where the free-stream conicity is found to make the flow near the stagnation streamline more frozen, which increases the translational temperature because less energy is transferred to the vibrational and chemical modes. This phenomenon is also evident in the edge density results, with the NONEQ flow having its edge density in the subsonic region decreased more by the free-stream conicity compared with the other conditions, as mentioned earlier in § 4.1.

In addition to examining the distribution of the values of the left-hand side of (4.8) in the subsonic region, it is also of interest to examine the value of the left-hand side of (4.8) at the sonic point because the left-hand side has to exceed a certain value at this location for transition to occur. The result is shown in figure 24(a). One can see that the left-hand side at the sonic point decreases very slightly, ${\approx }5\unicode{x2013} 8\,\%$ at $d = 4$, with increasing free-stream conicity for all the conditions. This is because, although free-stream conicity increases the left-hand side at any given $\theta$ in the subsonic region, free-stream conicity also shifts the location of $\theta ^{s}$ closer to the stagnation point where the left-hand side has a lower value, as shown exemplarily in figure 24(b) for condition 1. Ultimately, the shift of $\theta ^{s}$ to a location with a lower value of the left-hand side slightly overcomes the overall increase of the left-hand side in the subsonic region, resulting in a slight decrease of the left-hand side at $\theta ^{s}$. Because this decrease is only very slight, it can be suggested that the free-stream conicity will not influence whether transition occurs. Therefore, if transition occurs in a uniform free stream, it would also occur in a conical free stream, albeit with the transition point shifted upstream closer to the stagnation point, as mentioned earlier in this section. This result shows no significant dependency on the flow condition and gas type.

Figure 24. The (a) influence of free-stream conicity on the value of the left-hand side of (4.8) at the sonic point, and (b) absolute distribution of the left-hand side of (4.8), without $k$ and $T^{w}$ which are constants, for condition 1 (PG $M = 5$).

Finally, to provide some idea of how much the transition point gets shifted upstream due to free-stream conicity, figure 25 is produced. To systematize the comparison, $k$ for each condition is selected such that the left-hand sides of (4.8) and (4.9) are equal to 280 and 500, respectively, at the sonic point in the uniform free-stream case; this value of $k$ remains constant for the same condition at different $d$. The results show that the transition point can get shifted upstream by as much as 15 %–20 % and 20 %–25 % for the different conditions at $d = 4$ using (4.8) and (4.9), respectively, and demonstrate no significant dependency to the flow condition and gas type. Such a shift is significant, and it should be accounted for when interpreting the experimental results if a conical nozzle is used along with a significantly large spherical test model. Note that the results presented in figure 25 (and figure 24a) are only given at discrete points because their calculation involves significant inputs from CFD which can only be obtained for a few values of $d$ ($d = 4, 25, 100$). As indicated in (4.8) and (4.9), parameters such as the boundary layer momentum thickness, edge velocity, edge density and edge temperature in both uniform and non-uniform free streams are required, and these have to be attained using CFD. Consequently, the influence of free-stream conicity on transition, unlike some of the other properties analysed earlier, cannot be predicted purely analytically.

Figure 25. The influence of free-stream conicity on the transition onset point, $\theta ^{tr}$, using equations (a) (4.8) and (b) (4.9). The wall temperature $T^{w}$ is 295 K in all the cases.

4.4. Flow field

For completeness, it is of interest to examine the entire flow field. The results are exemplarily shown in figure 26 for condition 2 (PG $M = 8$), and the same trends are observed in the other conditions. As expected from § 4.1, the shock stand-off distance on the axisymmetry axis is clearly smaller in the conical free stream (bottom half) than in the uniform free stream (top half). Also, as shown in figure 26(a), free-stream conicity makes the entire sonic line move towards the axisymmetry axis, which is consistent with the sonic point results presented in § 4.1. Looking at figure 26(e, f), one can see that the velocity in the $z$-direction does not change much with free-stream conicity but the velocity in the $y$-direction does. Although the conical free stream expands in both directions, the shock is mostly aligned closer to the $y$-axis than the $z$-axis, which allows more of the $y$ component of the free-stream velocity to transfer through the shock, resulting in this observation. Examining figure 26(b–d), free-stream conicity does not influence the pressure, temperature and density near the stagnation region but does decrease these parameters elsewhere, which is consistent with the corresponding distributions along the boundary layer edge, as presented earlier in § 4, due to the expansion in the free stream. Finally, examining figure 26(g,h), in which both panels are indicative of the entropy variations in the flow field, one can see that free-stream conicity does not significantly influence the entropy distribution in the flow field; in both the uniform and non-uniform free-stream cases, the entropy at the boundary layer edge is approximately constant and equal to the entropy around the stagnation region, as expected in the flow over a sphere (Anderson Reference Anderson2019), and an entropy layer forms from the shock wave at $\theta \gtrapprox 40^{\circ }$.

Figure 26. The condition 2 (PG $M = 8$) flow field (a) Mach number, (b) pressure, (c) temperature, (d) density, (e) velocity $z$, ( f) velocity $y$, (g) total pressure and (h) entropy $(\varDelta s = ({\gamma }/({\gamma - 1})) \ln {({T}/{T_{\infty }})} - \ln {({p}/{p_{\infty }})})$. The top half corresponds to a uniform free stream while the bottom half corresponds to $d = 4$.

Further analysis is undertaken for the non-equilibrium condition to examine how free-steam conicity changes the thermochemical non-equilibrium behaviour in the flow field. Thermal non-equilibrium is examined in figure 27 by looking at the difference between the translational–rotational temperature and vibrational temperature; the NC wall results are shown exemplarily, and the same is observed for the SC wall. The flow near the shock front has strong thermal non-equilibrium, with $T_{tr}$ being significantly greater than $T_{v}$, while the flow in the boundary layer near the wall is essentially in thermal equilibrium, and no significant differences are observed between the uniform and conical free-stream cases concerning these observations. On the other hand, the thermal non-equilibrium seen in the inviscid flow near the boundary layer edge at $\theta \gtrapprox 30^{\circ }$, where $T_{v}$ is significantly greater than $T_{tr}$, does exhibit a difference between the two free-stream cases: the conical free stream produces stronger thermal non-equilibrium here. This is expected considering the flow expanding around the sphere from the stagnation region is further assisted by the expansion in the conical free stream resulting in a more rapid expansion due to free-stream conicity leading to a stronger thermal non-equilibrium of this kind ($T_{v} > T_{tr}$). This is also consistent with the results shown above in this section where free-stream conicity is found to increase the velocity and decrease the pressure, temperature (translational–rotational) and density in the flow over a sphere outside of the stagnation region. Consider the vibrational Damköhler number of the inviscid flow travelling around the boundary layer edge of the sphere (following from Passiatore et al. Reference Passiatore, Sciacovelli, Cinnella and Pascazio2022)

(4.10)\begin{equation} Da^{e}_{v} = \frac{R_{s} / u^{e}}{\tau_{v}}, \end{equation}

where $\tau _{v}$ is the vibrational relaxation time ($Da^{e}_{v} = O (0)$ for the current condition). The decrease in pressure and temperature by free-stream conicity increases $\tau _{v}$ (Millikan & White Reference Millikan and White1963) which, together with the increase in $u^{e}$, decreases $Da^{e}_{v}$ making the flow more vibrationally frozen (e.g. $(Da^{e}_{v})^{d=4}/(Da^{e}_{v})^{d=\infty }=0.4$ using conditions at the boundary layer edge at $\theta =45^{\circ }$). Looking at figure 27(b,c), the translational–rotational temperature is lower in the conical free-stream case while the vibrational temperature remains basically the same between the two cases, resulting in the larger thermal non-equilibrium seen in the conical free-stream case.

Figure 27. The condition 5 (NONEQ) NC wall flow field of the (a) difference between the translational–rotational temperature, $T_{tr}$, and vibrational temperature, $T_{v}$, (b) translational–rotational temperature and (c) vibrational temperature.

To examine the finite-rate chemistry, which is dominated by the oxygen dissociation/recombination reaction in this condition, figure 28 is made, which shows the $O_{2}$ mass fraction flow field. Examining the difference between the uniform free-stream and conical free-stream results, the $O_{2}$ mass fraction distribution remains largely the same near the shock front while some differences can be observed in the inviscid region near the boundary layer edge, like with the thermal non-equilibrium. This can be seen more clearly in figure 29(a), which shows the $O_{2}$ mass fraction along $\theta = 0^{\circ }, 30^{\circ }, 60^{\circ }$ rays in the inviscid flow; the results in this figure are for an NC wall, and the same is observed for a SC wall. One can see that the $O_{2}$ mass fraction is always higher in the conical free stream, indicating inhibition of dissociation and the presence of larger chemical non-equilibrium. This is confirmed when examining figure 29(b), which shows the difference between the equilibrium $O_{2}$ mass fraction (calculated using Cantera (Goodwin et al. Reference Goodwin, Moffat, Schoegl, Speth and Weber2023) at the local translational–rotational temperature and pressure) and the actual $O_{2}$ mass fraction in the inviscid flow. In this region, the actual $O_{2}$ mass fraction is always in excess of the local equilibrium value (dissociating non-equilibrium with $[(c_{O_{2}})_{eq}-c_{O_{2}}]<0$), and one can see that free-stream conicity generally increases the degree of this kind of chemical non-equilibrium here because the conical free-stream results (dashed lines) are always lower than the uniform free-stream results (solid lines) at all three $\theta$ values. The result for $\theta = 0^{\circ }$ was already presented in § 4.1; for this case, the observation is caused by the smaller shock stand-off distance, as explained earlier. For the $\theta = 30^{\circ }, 60^{\circ }$ cases, the larger chemical non-equilibrium observed in the conical free stream is due to the same reason explained above for the larger thermal non-equilibrium: the expanding free stream assists the expansion of the flow around the sphere from the stagnation region, resulting in a more rapid expansion which creates a larger non-equilibrium. Examining the $O_2$ dissociation Damköhler number which, for the current discussion, can be written as (following from CandlerReference Candler2018)

(4.11)\begin{equation} Da^{e}_{c} = \frac{R_{s} \rho^{e} k_{D,O_{2}}}{u^{e} \mathcal{M}_{O_{2}}}, \end{equation}

where $\mathcal {M}_{O_{2}}$ is the $O_2$ molar mass, and $k_{D,O_{2}}$ is the $O_2$ dissociation rate constant, which increases exponentially with temperature ($Da^{e}_{c} = O (-1)$ for the current condition), we see that, for the inviscid gas flowing around the sphere, the free-stream conicity causes the velocity to increase, and the density and temperature to decrease, which all contribute to decrease the $Da^{e}_{c}$ and make the flow more frozen (e.g. $(Da^{e}_{c})^{d=4}/(Da^{e}_{c})^{d=\infty }=0.4$ using conditions at the boundary layer edge at $\theta =45^{\circ }$).

Figure 28. The condition 5 (NONEQ) NC wall $O_{2}$ mass fraction, $c_{O_{2}}$, flow field.

Figure 29. The (a) $O_{2}$ mass fraction, $c_{O_{2}}$, and (b) difference between the equilibrium $O_{2}$ mass fraction (at the local translational–rotational temperature and pressure), $(c_{O_{2}})_{eq}$, and the actual $O_{2}$ mass fraction in the inviscid flow along rays of $\theta = 0^{\circ }, 30^{\circ }, 60^{\circ }$ for condition 5 with NC wall. The $x$-axis is normalized to give the distribution between the shock front and boundary layer edge.

Finally, details of the gas-phase reaction in the boundary layer are important for the NC wall (unlike the SC wall) due to its influence on the wall heat flux, as mentioned earlier (Fay & Riddell Reference Fay and Riddell1958). Therefore, to examine this more closely, figure 30 is made, which shows the $O_{2}$ mass fraction along $\theta = 0^{\circ }, 30^{\circ }, 60^{\circ }$ rays in the boundary layer with an NC wall. One can see that, in both the conical and uniform free streams, the mass fraction does not change much through the boundary layer, especially when $\theta$ is not large, with only minor recombination occurring near the wall, indicating the boundary layer is basically frozen. Larger variation of the $O_{2}$ mass fraction is seen through the boundary layer at $\theta = 60$, particularly in the conical free stream, where the $O_{2}$ mass fraction is higher near the boundary layer edge and decreases with decreasing distance from the wall, but this is not due to chemical reactions happening in the boundary layer. This is due to the growing thickness of the boundary layer which swallows the inviscid flow with radially varying $O_{2}$ mass fraction, as seen in figures 28 and 29(a) (similar to the entropy layer swallowing phenomenon Anderson Reference Anderson2019). In other words, at $\theta = 60$, the flow near the wall in the boundary layer originates from the inviscid region with $\theta \approx 0$ while the flow in the boundary layer near the boundary layer edge originates from the inviscid region with $\theta \gg 0$, resulting in the aforementioned $O_{2}$ mass fraction distribution through the boundary layer since the chemistry is essentially frozen in the boundary layer. This description is seen more clearly in figure 31, which shows four streamlines that pass through $\theta = 60$ at $n/\delta = 0.25, 0.5, 0.75, 1.0$. One can see that the mass fraction along the streamlines essentially freezes after entering the boundary layer. Because different streamlines enter the boundary layer with different mass fractions, an obvious mass fraction distribution forms through the boundary layer at larger values of $\theta$ despite the flow being basically frozen in the boundary layer. This distribution is, therefore, related to the $O_{2}$ mass fraction distribution along the boundary layer edge, which is shown in figure 32. Free-stream conicity, in addition to reducing the dissociation in the inviscid flow as it expands around the sphere, also increases the rate of growth of the boundary layer, as mentioned earlier in § 4.2, making it swallow more of the inviscid flow; these factors combine to result in the $O_{2}$ mass fraction distribution along the boundary layer edge being higher and having a larger variation in the conical free stream, as shown in figure 32. This larger mass fraction variation along the boundary layer edge is directly responsible for the corresponding larger mass fraction variation through the boundary layer in the conical free stream seenin figure 30.

Figure 30. The $O_{2}$ mass fraction in the boundary layer along rays of $\theta = 0^{\circ }, 30^{\circ }, 60^{\circ }$ for condition 5 with NC wall. The $x$-axis is normalized to give the distribution between the wall and boundary layer edge.

Figure 31. The condition 5 (NONEQ) NC wall $O_{2}$ mass fraction streamlines overlaid on the inviscid and boundary layer flow domains represented by the dark grey and light grey contours in the background, respectively. The four streamlines pass through $\theta = 60$ at $n/\delta = 0.25, 0.5, 0.75, 1.0$.

Figure 32. The condition 5 (NONEQ) NC wall $O_{2}$ mass fraction along the boundary layer edge.