3.1. Mean thermochemical state

This section first examines the species mass fraction distributions for the LR and HR cases. Figure 1 shows the wall-normal distributions of the Favre-averaged species mass fractions, $\tilde {Y}_s$ , and the Reynolds-averaged mixture molar mass $\overline {M}/M_e$ . In both cases, the dominant chemical process is O $_2$ dissociation, whereas N $_2$ remains largely intact. The dissociation of N $_2$ is negligible, with atomic N reaching only 0.8 % near the mean temperature peak in the HR case. In the LR case, only 7.3 % of the O $_2$ is dissociated, whereas the dissociation level increases to roughly 21 % in the HR case, owing to the higher chemical activity.

Figure 1.

Wall-normal profiles of the mean species mass fractions $\tilde {Y}_s$ ; (a) N $_2$ , (b) O $_2$ , (c) NO, (d) O and (e) N and (f) mean molar mass $\overline {M}/M_e$ for the LR and HR cases.

Figure 1. Long description

The image contains six line graphs labeled (a) through (f), each depicting wall-normal profiles for different species mass fractions and mean molar mass. Panel A: A line graph showing the mean mass fraction of N2. The x-axis is labeled y* and ranges from 10^0 to 10^3. The y-axis is labeled YN2 and ranges from 0.72 to 0.78. Two lines are present: a dashed red line for the low-resolution (LR) case and a solid black line for the high-resolution (HR) case. Panel B: A line graph showing the mean mass fraction of O2. The x-axis is labeled y* and ranges from 10^0 to 10^3. The y-axis is labeled YO2 and ranges from 0 to 0.30. Two lines are present: a dashed red line for the LR case and a solid black line for the HR case. Panel C: A line graph showing the mean mass fraction of NO. The x-axis is labeled y* and ranges from 10^0 to 10^3. The y-axis is labeled YNO and ranges from 0 to 0.08. Two lines are present: a dashed red line for the LR case and a solid black line for the HR case. Panel D: A line graph showing the mean mass fraction of O. The x-axis is labeled y* and ranges from 10^0 to 10^3. The y-axis is labeled YO and ranges from 0 to 0.20. Two lines are present: a dashed red line for the LR case and a solid black line for the HR case. Panel E: A line graph showing the mean mass fraction of N. The x-axis is labeled y* and ranges from 10^0 to 10^3. The y-axis is labeled YN and ranges from 0 to 1.00. Two lines are present: a dashed red line for the LR case and a solid black line for the HR case. Panel F: A line graph showing the mean molar mass. The x-axis is labeled y* and ranges from 10^0 to 10^3. The y-axis is labeled M/M and ranges from 0.80 to 1.00. Two lines are present: a dashed red line for the LR case and a solid black line for the HR case.

In addition to the different levels of O $_2$ dissociation, the HR and LR cases exhibit distinct behaviours in the distributions of N $_2$ and NO, as shown in figure 1(a,c). In the HR case, both species display non-monotonic wall-normal profiles with extrema near the mean temperature peak $y^* \approx 10$ , whereas monotonic distributions are observed in the LR case. This behaviour can be attributed to the enhanced atomic O concentration in the HR case, which alters the balance of the Zel’dovich reactions ( $\mathcal{R}_4$ and $\mathcal{R}_5$ in (2.11d ), (2.11e )), promoting NO formation near the mean temperature peak and its subsequent consumption closer to the wall (Williams et al. Reference Williams, Di Renzo and Moin2025). In contrast, the weaker O $_2$ dissociation in the LR case leads to much lower atomic O levels, limiting the activation of these reactions and resulting in monotonic profiles. The high-enthalpy reactions also reduce the mean molar mass $\overline {M}$ , as shown in figure 1(f). In the LR case, the molar mass decreases monotonically by 4.23 % relative to the free-stream value, whereas in the HR case the reduction reaches 14.42 %.

Given the pronounced changes in species composition in the HR case, we further examine the corresponding influence of high-enthalpy effects on the flow properties. The wall-normal profiles of the mean streamwise velocity $\tilde {u}/u_e$ , mean density $\overline {\rho }/\rho _e$ , mean temperature $\tilde {T}/T_e$ and mean enthalpy $\tilde {h}/h_e$ are shown in figure 2. Although the species composition changes significantly in the HR case, the mean velocity and density profiles remain very close to those of the FC case, as shown in figure 2(a,b). More specifically, the FC case exhibits a slightly steeper velocity profile in the buffer region, leading to marginally higher velocities than in the HR case for $y^* \gtrapprox 10$ . Since the mean density is nearly unchanged in this region shown in figure 2(b), this difference is expected to be attributed to the lower viscosity in the HR case, as illustrated by the inset in figure 2(a). In the near-wall region, the HR case exhibits an increase in mean density relative to the FC case near the non-catalytic wall.

Figure 2.

Wall-normal profiles of the mean (a) velocity $\tilde {u}/u_e$ , (b) density $\overline {\rho }/\rho _e$ , (c) temperature $\tilde {T}/T_e$ and (d) enthalpy $\tilde {h}/h_e$ for the LR, HR and FC cases. The inset of (a) shows the wall-normal profiles of mean viscosity $\overline {\mu }/\mu _e$ for the HR and FC cases. The FC wall-normal profiles of $\overline {\rho }/\rho _e$ , $\tilde {T}/T_e$ and $\tilde {h}/h_e$ , estimated from the mean velocity profiles of the LR and HR cases, are shown in (b–d). The wall-normal profiles of the mean enthalpy $h(\tilde {T}, \tilde {Y}_s)/h_e$ for the LR and HR cases are shown in (d).

According to figure 2(c,d), the HR case exhibits a lower peak mean temperature $\tilde {T}/T_e$ but a higher peak mean enthalpy $\tilde {h}/h_e$ compared with the FC case. This behaviour reflects the dominant energy exchange mechanism in HE-TBLs, whereby endothermic chemical reactions absorb sensible heat and convert it into chemical energy, thereby reducing the temperature. Since the kinetic energy remains nearly unchanged (and is even slightly lower in the HR case), the additional enthalpy is primarily associated with the reduced wall heat loss (see Appendix A). Furthermore, the peak mean temperature $\tilde {T}/T_e$ in the HR case is approximately 18.8 % lower than that in the FC case, a reduction comparable to that in the mixture molar mass $\overline {M}$ . According to the approximate mean equation of state

(3.1) \begin{align} \overline {p} \approx \overline {\rho }\, \frac {\mathscr{R}}{\overline {M}} \, \tilde {T}, \end{align}

one could reasonably argue that the comparable reductions in $\tilde {T}$ and $\overline {M}$ largely offset each other, leading to only minor variations in the mean density $\overline {\rho }$ under nearly constant pressure. As a result, the mean streamwise momentum $\overline {\rho }\tilde {u}$ remains nearly unchanged.

To verify this argument and the approximate mean equation of state (3.1), we assume that the mean streamwise velocity profiles $\tilde {u}/u_e$ are identical in the FC and non-equilibrium-chemistry cases. Under this assumption, the FC mean enthalpy profile $\tilde {h}/h_e$ can be estimated from the non-equilibrium-chemistry mean velocity profile $\tilde {u}/u_e$ using the mean velocity–temperature relation proposed by Duan & Martin (Reference Duan and Martin2011b )

(3.2) \begin{align} \frac {\tilde {h}}{h_e} = \frac {\tilde {h}_w}{h_e} + \frac {h_{r} - \tilde {h}_w}{h_e} f\left ( \frac {\tilde {u}}{u_e} \right ) + \frac {h_e - {h}_r}{h_e} \left (\frac {\tilde {u}}{u_e}\right )^2, \end{align}

where the function $f(\tilde {u}/u_e)$ is given by

(3.3) \begin{align} f\left ( \frac {\tilde {u}}{u_e} \right ) = \alpha \frac {\tilde {u}}{u_e} + (1 - \alpha )\left (\frac {\tilde {u}}{u_e}\right )^2, \end{align}

with $\alpha = 0.8259$ , as determined from DNS data fitting by Duan & Martin (Reference Duan and Martin2011b ). An additional assumption, supported by DNS observations, is that the mean enthalpy $\tilde {h}$ obtained directly from DNS collapses with that evaluated from the mean temperature and mean species mass fractions $h(\tilde {T},\tilde {Y}_s)$ (see figure 2 d)

(3.4) \begin{align} \widetilde {h\left (T,Y_s\right )} - h\big (\tilde {T},\tilde {Y}_s\big ) = \sum _{s=1}^{N_s} \widetilde {Y_{s}^{\prime \prime }h_s^{\prime \prime }} + \sum _{s=1}^{N_s} \tilde {Y}_s \big (\tilde {e}_{{V},s} - {e}_{{{V},s}} (\tilde {T})\big ) \approx 0. \end{align}

Both the species–enthalpy coupling term $\sum _{s=1}^{N_s}\widetilde {Y_s^{\prime \prime }h_s^{\prime \prime }}$ and the nonlinear vibrational energy contribution $\sum _{s=1}^{N_s}\tilde {Y}s(\tilde {e}_{{V},s} - e_{{V},s}(\tilde {T}))$ are found to be negligible compared with $\tilde {h}$ and are therefore not shown in figure 2(d). Consequently, the FC mean temperature $\tilde {T}$ and density $\overline {\rho }$ can be estimated by inverting the enthalpy relation $\tilde {h} \approx h(\tilde {T},\tilde {Y}_s)$ in conjunction with (3.1). The estimated FC profiles of density $\overline {\rho }/\rho _e$ , temperature $\tilde {T}/T_e$ and enthalpy $\tilde {h}/h_e$ for the HR case are in good agreement with the DNS results of the FC case, as shown in figure 2(b–d). In addition, for the LR case, the mean density profile $\overline {\rho }/\rho _e$ closely matches the corresponding estimated FC profile, consistent with the above interpretation. The peak mean temperature $\tilde {T}/T_e$ is reduced by only approximately 3.7 % relative to the estimated FC value, which is also comparable to the reduction in the mixture molar mass $\overline {M}$ . These results further confirm that the decrease in mean molar mass $\overline {M}$ largely offsets the reduction in mean temperature $\tilde {T}$ induced by chemical reactions, resulting in negligible variations in the mean momentum $\overline {\rho }\tilde {u}$ .

According to Morkovin’s hypothesis (Morkovin Reference Morkovin1962), compressibility effects influence turbulence primarily through mean density variations. Since the mean density is only weakly affected by high-enthalpy effects in the present cases, the wall-normal profiles of Reynolds stresses remain largely unchanged (see Appendix A). Consequently, the classical theories for ideal-gas turbulent boundary layers remain largely applicable under high-enthalpy conditions (Duan & Martin Reference Duan and Martin2011b ).

3.2. Role of chemical non-equilibrium

To further elucidate the role of high-enthalpy TCI, the Damköhler number $Da_s^{\textit{LE}} = (\delta ^*/u_{\tau }^*)/(\overline {\rho }/\overline {\dot {\omega }}_s)$ is examined for all species in the LR and HR cases, as shown in figure 3. The Damköhler number $Da_s^{\textit{LE}}$ essentially represents the ratio of the semi-local large-eddy turnover time ( $t^{\textit{LE}} = \delta ^*/u_{\tau }^*$ ) to the chemical reaction time of species $s$ ( $t^{{chem}} = \overline {\rho }/\overline {\dot {\omega }}_s$ ), where $\delta ^*$ denotes the displacement boundary-layer thickness. It is observed that $Da_s^{\textit{LE}}$ in the HR case is larger than that in the LR case, particularly in the near-wall region $y^* \lessapprox 10$ . Furthermore, in both cases, $Da_s^{\textit{LE}}$ remains below $O(10^{-1})$ across most of the boundary layer, indicating that the chemical time scale is much longer than the large-eddy turnover time scale, i.e. $t_{{chem}} \gg t_{{LE}}$ . Consequently, turbulent mixing dominates over high-enthalpy chemical reactions.

Figure 3.

Wall-normal profiles of Damköhler number $Da_s^{\textit{LE}}$ for the (a) LR and (b) HR cases.

This behaviour is further supported by the strong correlations between temperature and species mass fractions, quantified by $C_{T^{\prime \prime }Y_s^{\prime \prime }}=\widetilde {T^{\prime \prime }Y_s^{\prime \prime }}/(T_{\textit{rms}}^{\prime \prime }Y_{s,\textit{rms}}^{\prime \prime })$ , where subscript rms denotes the root-mean-square value, as illustrated in figure 4. Away from the near-wall region $y^* \gtrapprox 50$ , the correlations approach unity for the major species N $_2$ , O $_2$ , NO and O, confirming that the mass fraction fluctuations $Y_s^{\prime \prime }$ are largely governed by turbulent transport. In other words, the fluctuations in the mass fractions of species $Y_s^{\prime \prime }$ can be approximated as passive scalars. The sign of the correlations reflects the underlying thermochemical processes, with dissociation products (e.g. NO, O, N) positively correlated with temperature and reactants (e.g. N $_2$ , O $_2$ ) negatively correlated. In the region below the mean temperature peak ( $y^* \lessapprox 10$ ), the correlations decrease towards the non-catalytic wall, where the mass fraction fluctuations become very small (Di Renzo & Urzay Reference Di Renzo and Urzay2021; Williams et al. Reference Williams, Di Renzo and Moin2025), such that the correlation coefficients lose significance. The linear relation is further supported by the instantaneous fields in figure 5, where the spatial structures of the temperature fluctuation $T^{\prime \prime }/\tilde {T}$ and the mass fraction fluctuation $Y_{\mathrm{O}}^{\prime \prime }/\tilde {Y}_s$ display highly correlated spatial structures in the $x$ – $z$ plane at $y^* \approx 200$ for both the LR and HR cases. Similar results can also be extended to the other species (for economy of space, not shown herein).

Figure 4.

Wall-normal profiles of correlation coefficients $C_{T^{\prime \prime }Y_s^{\prime \prime }}$ for the (a) LR and (b) HR cases.

Figure 5.

Instantaneous fields of the (a, c) temperature fluctuations $T^{\prime \prime }/\tilde {T}$ and (b, d) mass fraction fluctuations of atomic oxygen $Y_{\text{O}}^{\prime \prime }/\tilde {Y}_{\text{O}}$ in wall-parallel slices ( $x$ – $z$ plane) at $y^* \approx 200$ for the (a, b) LR and (c, d) HR cases.

4.1. Extension of SRA

The preceding results indicate that the turbulent momentum field remains largely unaffected in both the LR and HR cases, while the thermochemical fluctuations are strongly coupled with temperature. This motivates a quantitative extension of the SRA to HE-TBLs.

First, the relationship among enthalpy, temperature and species mass fraction fluctuations is examined. The enthalpy fluctuation $h^{\prime \prime }$ is approximated by a first-order expansion around $\tilde {h}$ , under the assumption (3.4)

(4.1) \begin{align} \begin{aligned} h^{\prime \prime } \approx h - h(\tilde {T}, \tilde {Y}_s) &\approx \left (\frac {\partial h}{\partial T}\right )_{\tilde {T}, \tilde {Y}_s} T^{\prime \prime } + \sum _{s=1}^{N_s}\left (\frac {\partial h}{\partial Y_s}\right )_{\tilde {T}, \tilde {Y}_s} Y_s^{\prime \prime }\\& \quad = \hat {c}_p T^{\prime \prime } + \sum _{s=1}^{N_s}\hat {h}_s Y_s^{\prime \prime }, \end{aligned} \end{align}

where $\hat {c}_p = (\partial h/ \partial T)_{\tilde {T}, \tilde {Y}_s}$ is the mixture specific heat at constant pressure evaluated at the mean temperature $\tilde {T}$ and species mass fractions $\tilde {Y}_s$ , and similarly for $\hat {h}_s = (\partial h/ \partial Y_s)_{\tilde {T}, \tilde {Y}_s}$ . Then, the corresponding variance of the enthalpy fluctuation $\widetilde {h^{\prime \prime 2}}$ is

(4.2) \begin{align} \widetilde {{h^{\prime \prime 2}}} \approx \underbrace { \hat {c}_p^{\,\,2} \widetilde {{T^{\prime \prime 2}}} \vphantom {\sum _{s=1}^{N_s}}}_{\mathcal{F}_T} + \underbrace {2\sum _{s=1}^{N_s} \hat {c}_p \hat {h}_s \widetilde {T^{\prime \prime }Y_s^{\prime \prime }}}_{\mathcal{F}_{TY_s}} + \underbrace {\sum _{s=1}^{N_s} \hat {h}_s^{\,\,2} \widetilde {{Y_{s}^{\prime \prime 2}}} + 2\sum _{s\lt r}^{N_s} \hat {h}_s \hat {h}_r \widetilde {Y_s^{\prime \prime }Y_r^{\prime \prime }}}_{\mathcal{F}_{Y_s}}. \end{align}

On the right-hand side of (4.2), the first term $\mathcal{F}_T$ is the temperature fluctuation term, representing the contribution exclusively due to the temperature fluctuation. The second term $\mathcal{F}_{TY_s}$ is the temperature–composition cross-term, accounting for the coupling of the temperature and species composition fluctuation. The sum of the third and fourth terms $\mathcal{F}_{Y_s}$ is collectively referred to as the composition fluctuation term, which represents the contribution exclusively from species composition fluctuations.

The decomposing terms of the enthalpy variance $\widetilde {h^{\prime \prime 2}}$ are plotted in figure 6. The first-order expansion of enthalpy fluctuations, denoted as $\mathcal{F}_{{sum}} = \mathcal{F}_T + \mathcal{F}_{TY_s} + \mathcal{F}_{Y_s}$ , agrees well with the DNS results, indicating that the higher-order terms can be neglected. In the region below the mean temperature peak ( $y^* \lessapprox 10$ ), the non-catalytic wall condition suppresses species mass fraction fluctuations (Di Renzo & Urzay Reference Di Renzo and Urzay2021; Williams et al. Reference Williams, Di Renzo and Moin2025), such that the temperature fluctuation term $\mathcal{F}_T$ dominates and agrees well with the DNS results for both the LR and HR cases. In the region $y^* \gtrapprox 10$ , apparent differences emerge between the LR and HR cases. In the LR case, $\mathcal{F}_T$ remains dominant, and the deviation from the DNS results mainly arises from the temperature–composition cross-term $\mathcal{F}_{TY_s}$ , while the contribution of the composition fluctuation term $\mathcal{F}_{Y_s}$ is negligible. In contrast, in the HR case, the dominance of $\mathcal{F}_T$ is significantly reduced, and the contribution of $\mathcal{F}_{Y_s}$ becomes non-negligible, with all three terms attaining comparable magnitudes. This comparison indicates that, with increasing high-enthalpy effects, species mass fraction fluctuations and their strong coupling with temperature become important contributors to the enthalpy fluctuations.

Figure 6.

Wall-normal profiles of the enthalpy variance $\widetilde {h^{\prime \prime 2}}/\tilde {h}^2$ compared with the first-order expansion (4.2) and its constituent terms for the (a) LR and (b) HR cases.

According to the turbulent mixing length hypothesis, the following relations can be obtained for the enthalpy, temperature and species mass fraction fluctuations:

(4.3a,b,c) \begin{align} h_{\textit{rms}}^{\prime \prime } \approx l_h \left |\frac {\partial \tilde {h}}{\partial y}\right |, \quad T_{\textit{rms}}^{\prime \prime } \approx l_T \left |\frac {\partial \tilde {T}}{\partial y}\right |, \quad Y_{s,\textit{rms}}^{\prime \prime } \approx l_{Y_s} \left |\frac {\partial \tilde {Y}_s}{\partial y}\right |, \end{align}

where $l_h$ , $l_T$ and $l_{Y_s}$ are the turbulent mixing lengths for the corresponding fluctuations. This assumption is supported by the observed near-linear correlations between temperature and species mass fraction fluctuations (see figure 4), indicating that they exhibit passive-scalar-like behaviour under turbulent transport. Considering the thermodynamic relation $h = h(T,Y_s)$ and applying the chain rule at the mean state, one obtains

(4.4) \begin{align} \frac {\partial \tilde {h}}{\partial y} \approx \frac {\partial h\big (\tilde {T}, \tilde {Y}_s\big )}{\partial y} = \hat {c}_p \frac {\partial \tilde {T}}{\partial y} + \sum _{s=1}^{N_s} \hat {h}_s \frac {\partial \tilde {Y}_s}{\partial y}. \end{align}

Combining (4.3a ) and (4.4), the ratio between the enthalpy and temperature fluctuation intensities can be expressed as

(4.5) \begin{align} \frac {h^{\prime \prime }_{\textit{rms}}}{T^{\prime \prime }_{\textit{rms}}} \approx \frac {l_h}{l_T} \frac {\partial \tilde {h}}{\partial \tilde {T}} = \frac {l_h}{l_T} \left ( \hat {c}_p + \sum _{s=1}^{N_s} \frac {\partial \tilde {Y}_s}{\partial \tilde {T}} \hat {h}_s \right )\!. \end{align}

On the other hand, substituting (4.3a) into (4.2) and invoking the strong linear correlations between temperature and species mass fraction fluctuations, yields

(4.6) \begin{align} \begin{aligned} \frac {h^{\prime \prime }_{{rms}}}{T^{\prime \prime }_{{rms}}} &\approx \left \{ \hat {c}_p^{\,2} + 2\sum _{s=1}^{N_s} \hat {c}_p \hat {h}_s \frac {l_{Y_s}}{l_T} \frac {\partial \tilde {Y}_s}{\partial \tilde {T}} + \sum _{s=1}^{N_s}\left ( \hat {h}_s \frac {l_{Y_s}}{l_T} \frac {\partial \tilde {Y}_s}{\partial \tilde {T}} \right )^{2} \right . \\&\quad \left . + 2\sum _{s\lt r}^{N_s} \hat {h}_s \hat {h}_r \frac {l_{Y_s}}{l_T} \frac {l_{Y_r}}{l_T} \frac {\partial \tilde {Y}_s}{\partial \tilde {T}} \frac {\partial \tilde {Y}_r}{\partial \tilde {T}} \right \}^{\frac {1}{2}} \\&\simeq \hat {c}_p + \sum _{s=1}^{N_s} \hat {h}_s \frac {l_{Y_s}}{l_T} \frac {\partial \tilde {Y}_s}{\partial \tilde {T}}. \end{aligned} \end{align}

Combining (4.5) and (4.6) yields

(4.7) \begin{align} \left (\frac {l_h}{l_T} - 1\right ) \hat {c}_p + \sum _{s=1}^{N_s} \left (\frac {l_h}{l_T} - \frac {l_{Y_s}}{l_T}\right ) \frac {\partial \tilde {Y}_s}{\partial \tilde {T}} \hat {h}_s = 0. \end{align}

In regions where temperature and species mass fraction fluctuations are linearly correlated, the coefficients in (4.7) approximately vanish individually, leading to

(4.8) \begin{align} \frac {l_h}{l_T} \approx \frac {l_{Y_s}}{l_T} \approx 1. \end{align}

This result indicates that the turbulent mixing processes of enthalpy, temperature and species mass fractions share a common characteristic length scale under high-enthalpy conditions. Thus, the proportional relations can be obtained

(4.9a,b) \begin{align} \frac {h_{\textit{rms}}^{\prime \prime }}{T_{\textit{rms}}^{\prime \prime }} \approx \left |\frac {\partial \tilde {h}}{\partial \tilde {T}}\right |, \quad \frac {T_{\textit{rms}}^{\prime \prime }}{Y_{s,\textit{rms}}^{\prime \prime }} \approx \left |\frac {\partial \tilde {T}}{\partial \tilde {Y}_s}\right |.\end{align}

Based on (4.9a), it can be further deduced that the enthalpy-based turbulent Prandtl number $Pr_{t,h} = {\widetilde {u^{\prime \prime }v^{\prime \prime }}}/{\widetilde {v^{\prime \prime }h^{\prime \prime }}}{\partial \tilde {h}}/{\partial \tilde {u}}$ , the temperature-based turbulent Prandtl number $Pr_{t,T} = {\widetilde {u^{\prime \prime }v^{\prime \prime }}}/{\widetilde {v^{\prime \prime }T^{\prime \prime }}}{\partial \tilde {T}}/{\partial \tilde {u}}$ and the turbulent Schmidt number $Sc_{t,Ys} = {\widetilde {u^{\prime \prime }v^{\prime \prime }}}/{\widetilde {v^{\prime \prime }Y_s^{\prime \prime }}}{\partial \tilde {Y}_s}/{\partial \tilde {u}}$ are equivalent. This equivalence is confirmed in figure 7 in the region away from the mean temperature peak ( $y^* \gtrapprox 50$ ). The turbulent Schmidt number of N deviates from those of the other species, which can be attributed to its low concentration, as shown in figure 1(e). Consequently, the SRA can be extended to temperature and species mass fraction fluctuations in HE-TBLs. The RSRA, recently proposed by Huang et al. (Reference Huang, Su, Guo, Yuan, Xiong and Zhou2025), is written in terms of enthalpy, temperature and species mass fractions as follows:

(4.10a,b,c) \begin{align} \frac {h_{\textit{rms}}^{\prime \prime }}{u_{\textit{rms}}^{\prime \prime }} \approx \frac {C}{\sqrt {Pr_{t,h}}} \left |\frac {\partial \tilde {h}}{\partial \tilde {u}}\right |, \quad \frac {T_{\textit{rms}}^{\prime \prime }}{u_{\textit{rms}}^{\prime \prime }} \approx \frac {C}{\sqrt {Pr_{t,T}}} \left |\frac {\partial \tilde {T}}{\partial \tilde {u}}\right |, \quad \frac {Y_{s,\textit{rms}}^{\prime \prime }}{u_{\textit{rms}}^{\prime \prime }} \approx \frac {C}{\sqrt {Sc_{t,Y_s}}} \left |\frac {\partial \tilde {Y}_s}{\partial \tilde {u}}\right |,\end{align}

where the constant $C \approx 1.09$ is determined from the nearly constant ratio of the turbulent production time scales associated with velocity and enthalpy fluctuations (Huang et al. Reference Huang, Su, Guo, Yuan, Xiong and Zhou2025). The deviations of the RSRAs from the DNS data are shown in figure 8, where they are evaluated as the ratio of the right-hand side to the corresponding left-hand side of (4.10a–c). The DNS results verify the accuracy of (4.10a) in the region away from the mean temperature peak $y^* \gtrapprox 50$ . Owing to the non-catalytic wall condition, (4.10a) breaks down for the mass fraction fluctuations in the near-wall region ( $y^* \lessapprox 50$ ).

Figure 7.

Wall-normal profiles of the temperature-based turbulent Prandtl number $Pr_{t,T}$ , enthalpy-based turbulent Prandtl number $Pr_{t,h}$ and turbulent Schmidt numbers $Sc_{t,Y_s}$ for the (a) LR and (b) HR cases.

Figure 7. Long description

Two line graphs depict wall-normal profiles of turbulent Prandtl and Schmidt numbers for different cases. Panel A: The graph shows the wall-normal profiles for the LR case. The x-axis represents the wall-normal coordinate (y+) on a logarithmic scale, ranging from 10^0 to 10^3. The y-axis represents the turbulent Prandtl and Schmidt numbers, ranging from 0 to 2. Multiple lines represent different quantities: Prandtl number based on temperature (Pr_t,T), Prandtl number based on enthalpy (Pr_t,h), and Schmidt numbers for various species (Sc_t,YN2, Sc_t,Y0, Sc_t,Y02, Sc_t,YN, Sc_t,YNO). The lines show different trends, with some increasing and others decreasing as y+ increases. Panel B: The graph shows the wall-normal profiles for the HR case. The axes and quantities are the same as in Panel A. The trends in this panel are similar to those in Panel A, with variations in the profiles of different quantities.

Figure 8.

Assessment of the RSRA for enthalpy $h$ , temperature $T$ and species mass fractions $Y_s$ for the (a) LR and (b) HR cases.

4.2. Correlation correction of SRA

The limitation of the RSRA model manifests in its inability to accurately predict enthalpy fluctuation intensities in the vicinity of vanishing mean enthalpy gradient ( $\partial \tilde {h}/\partial y = 0$ ) under wall-cooling conditions. In this region, enthalpy fluctuations are primarily governed by small-scale turbulent motions (Cheng & Fu Reference Cheng and Fu2023). Gerolymos & Vallet (Reference Gerolymos and Vallet2024) investigated the statistical characteristics of total enthalpy fluctuations in strong wall-cooling channel turbulent flows. One important finding is that $C_{u^{\prime \prime }h_t^{\prime \prime }}$ and $C_{u^{\prime \prime }v^{\prime \prime }}/C_{v^{\prime \prime }h_t^{\prime \prime }}$ are unity under strong wall-cooling conditions. To assess the validity of these relations under moderate wall-cooling conditions, figure 9(a) presents the wall-normal distributions of $C_{u^{\prime \prime }h^{\prime \prime }}$ and $C_{u^{\prime \prime }h_t^{\prime \prime }}$ . It is observed that $C_{u^{\prime \prime }h_t^{\prime \prime }}$ remains close to unity in the vicinity of the mean temperature peak ( $3 \lesssim y^* \lesssim 30$ , where $C_{u^{\prime \prime }h_t^{\prime \prime }} \gt 0.98$ ), but deviates from unity outside this region. Besides, $C_{u^{\prime \prime }h^{\prime \prime }}$ vanishes at $y^* \approx 10$ , corresponding to the location of the mean temperature peak. Figure 9(b) shows the wall-normal profiles of $C_{u^{\prime \prime }v^{\prime \prime }}/C_{v^{\prime \prime }h_t^{\prime \prime }}$ and $C_{u^{\prime \prime }v^{\prime \prime }}/C_{v^{\prime \prime }h^{\prime \prime }}$ . At $y^* \approx 10$ , corresponding to the location of the mean temperature peak, $C_{u^{\prime \prime }v^{\prime \prime }}/C_{v^{\prime \prime }h_t^{\prime \prime }} \approx 1$ , while $C_{v^{\prime \prime }h^{\prime \prime }}$ vanishes. These observations indicate that the relations reported by Gerolymos & Vallet (Reference Gerolymos and Vallet2024) do not always hold under moderate wall-cooling conditions, that is, they are only valid in the vicinity of the mean temperature peak and deviate outside this region.

Figure 9.

Wall-normal profiles of the (a) the correlation coefficients $C_{u^{\prime \prime }h^{\prime \prime }}$ and $C_{u^{\prime \prime }h_t^{\prime \prime }}$ , and (b) the ratios $C_{u^{\prime \prime }v^{\prime \prime }}/C_{v^{\prime \prime }h^{\prime \prime }}$ and $C_{u^{\prime \prime }v^{\prime \prime }}/C_{v^{\prime \prime }h_t^{\prime \prime }}$ for the LR, HR and FC cases.

Figure 9. Long description

Two line graphs depict wall-normal profiles of correlation coefficients and their ratios for different cases. Panel A shows the correlation coefficients C_u’’h’’ and C_u’’h’_i’’ plotted against y*. The graph includes three data series: LR (gray solid line), HR (black solid line), and FC (red dashed line). The x-axis is labeled y* on a logarithmic scale, and the y-axis ranges from -1.0 to 1.0. Panel B displays the ratios C_u’’v’’/C_v’’h’’ and C_u’’v’’/C_v’’h’_i’’ against y*. The x-axis is again labeled y* on a logarithmic scale, and the y-axis ranges from -2 to 2. The same three data series are used: LR (gray solid line), HR (black solid line), and FC (red dashed line). The graphs illustrate how these ratios vary under different conditions, highlighting the differences between the LR, HR, and FC cases.

For further analysis, the statistical relations between the total enthalpy and enthalpy fluctuation are presented in (4.11). The statistical relations are derived under the assumption of quasi-parallel flow ( $\tilde {v} = \tilde {w} \approx 0$ and $\tilde {v}, \tilde {w} \ll \tilde {u}$ ), and further simplified by neglecting the higher-order statistical moments, which are given as follows (Gerolymos & Vallet Reference Gerolymos and Vallet2024):

(4.11a) \begin{align} \tilde {h}_t & \approx \tilde {h} + \frac {1}{2}\tilde {u}^2, \\[-12pt] \nonumber \end{align}

(4.11b) \begin{align} \widetilde {v^{\prime \prime }h^{\prime \prime }} & \approx \widetilde {v^{\prime \prime }h_t^{\prime \prime }} - \tilde {u}\widetilde {u^{\prime \prime }v^{\prime \prime }}, \\[-12pt] \nonumber \end{align}

(4.11c) \begin{align} \widetilde {{h^{\prime \prime }}^2} & \approx \widetilde {{h_{t}^{\prime \prime 2}}} + \widetilde {{u^{\prime \prime 2}}}\tilde {u}^{2}-2C_{h_{t}^{\prime \prime }u^{\prime \prime }}\tilde {u}u_{\textit{rms}}^{\prime \prime }h_{t,10rms}^{\prime \prime }. \\[24pt] \nonumber \end{align}

Given the wall-normal turbulent heat flux $\widetilde {v^{\prime \prime }h^{\prime \prime }} = 0$ where $\partial \tilde {h}/ \partial y = 0$ , it is evident that the total-enthalpy-based turbulent Prandtl number $Pr_{t,h_t}$ also reaches unity

(4.12) \begin{align} Pr_{t,h_t} = \frac {\widetilde {u^{\prime \prime }v^{\prime \prime }}}{\widetilde {v^{\prime \prime }h_t^{\prime \prime }}} \frac {\partial \tilde {h}_t}{\partial \tilde {u}} \approx \frac {\widetilde {u^{\prime \prime }v^{\prime \prime }}}{\widetilde {v^{\prime \prime }h^{\prime \prime }} + \tilde {u}\widetilde {u^{\prime \prime }v^{\prime \prime }}} \left ( \frac {\partial \tilde {h}}{\partial \tilde {u}} + \tilde {u} \right ) =1. \end{align}

By combining $Pr_{t,h_t} = 1$ and $C_{u^{\prime \prime }v^{\prime \prime }}/C_{v^{\prime \prime }h_t^{\prime \prime }} = 1$ (as shown in figure 9 b) near the mean temperature peak, the following relation for the total enthalpy fluctuations can be derived:

(4.13) \begin{align} \frac {h_{t,rms}^{\prime \prime }}{u_{\textit{rms}}^{\prime \prime }} \equiv \frac {C_{u^{\prime \prime }v^{\prime \prime }}}{C_{v^{\prime \prime }h_t^{\prime \prime }}} \frac {1}{Pr_{t,h_t}} \frac {\partial \tilde {h}_t}{\partial \tilde {u}} \approx \frac {\partial \tilde {h}_t}{\partial \tilde {u}}. \end{align}

Substituting (4.11a ) and (4.13) into (4.11c ) leads to

(4.14) \begin{align} \frac {h_{\textit{rms}}^{\prime \prime }}{u_{\textit{rms}}^{\prime \prime }} \approx \sqrt {\left (\frac {\partial \tilde {h}_t}{\partial \tilde {u}}\right )^2 + \tilde {u}^2 - 2\alpha _{h_t u} \frac {\partial \tilde {h}_t}{\partial \tilde {u}}\tilde {u}} \approx \sqrt {\left (\frac {\partial \tilde {h}}{\partial \tilde {u}}\right )^2+2\left (1-\alpha _{h_t u} \right )\left (\frac {\partial \tilde {h}}{\partial \tilde {u}} + \tilde {u}\right )\tilde {u}}, \end{align}

where the coefficient $\alpha _{h_t u} = C_{u^{\prime \prime }h_t^{\prime \prime }}$ . This expression, here referred to as the correlation-corrected GSRA (CC-GSRA), reduces to the GSRA proposed by Gaviglio (Reference Gaviglio1987) when $\alpha _{h_t u} = 1$ . Gerolymos & Vallet (Reference Gerolymos and Vallet2024) emphasised that small deviations of the correlation coefficient from unity are essential, as enforcing $\alpha _{h_t u} = 1$ can lead to singular behaviour.

Figure 10(a) presents the results of the CC-GSRA with $\alpha _{h_t u} = C_{u^{\prime \prime }h_t^{\prime \prime }}$ , evaluated as the ratio of the right-hand side to the corresponding left-hand side of (4.14). Compared with figure 8, the results show a noticeable improvement in the region near the mean temperature peak ( $y^* \approx 10$ ), whereas they are adversely affected away from this region. Figure 10(b) illustrates the dependence of the CC-GSRA on the coefficient $\alpha _{h_t u}$ in the range $0.97 \leqslant \alpha _{h_t u} \leqslant 1$ for the LR case. Although all the chosen values of $\alpha _{h_t u}$ are close to unity, significant differences can be observed near the mean temperature peak. In the cases of a strong wall-cooling condition, Gerolymos & Vallet (Reference Gerolymos and Vallet2024) recommended $\alpha _{h_t u} \approx 0.997$ . For the present LR case ( $\varTheta = 0.09$ ), the CC-GSRA with a slightly smaller value, $\alpha _{h_t u} = 0.985$ , yields good agreement with the DNS result at the mean temperature peak $y^* \approx 10$ . The coefficient $\alpha _{h_t u}$ clearly depends on the intensity of wall cooling, and it tends to approach unity as the wall cooling becomes stronger (i.e. as the diabatic parameter $\varTheta$ decreases).

Figure 10.

Assessment of the CC-GSRA (4.14): (a) results with $\alpha _{h_t u} = C_{u^{\prime \prime }h_t^{\prime \prime }}$ for the LR, HR and FC cases; (b) sensitivity to constant $\alpha _{h_t u}$ in the range $0.97 \leqslant \alpha _{h_t u} \leqslant 1$ for the LR case.

Also shown in figure 10(b), the results are virtually unaffected by variations in $\alpha _{h_t u}$ in the near-wall region $y^* \lessapprox 5$ . In the region $y^* \gtrapprox 13$ , the influence of $\alpha _{h_t u}$ gradually diminishes with increasing wall-normal distance. Specifically, for $y^* \gt 170$ ( $y/\delta \gt 0.3$ ), a variation of 0.1 in $\alpha _{h_t u}$ leads to less than 5 % deviation in the results, which indicates that the correlation correction has minimal influence in the outer layer. One may reasonably argue that the correlation correction in (4.14) can be analogously and formally extended to the RSRA. The correlation-corrected RSRA (CC-RSRA) is written as follows:

(4.15) \begin{align} \frac {h_{\textit{rms}}^{\prime \prime }}{u_{\textit{rms}}^{\prime \prime }} \approx \sqrt {\frac {1}{Pr_{t,h}} \left (\frac {\partial \tilde {h}}{\partial \tilde {u}}\right )^2+2\left (1-\alpha _{h_t u}\right ) \left (\frac {\partial \tilde {h}}{\partial \tilde {u}} + \tilde {u}\right ) \tilde {u}}. \end{align}

The coefficient $C$ in the RSRA (4.10a) no longer appears explicitly in (4.15), as its effect is implicitly incorporated into the additional term introduced during the modification.

Since the RSRA performs well in the outer region, $\alpha _{h_t u}$ is determined by enforcing agreement between CC-RSRA and DNS under different diabatic conditions. Specifically, for each $\varTheta$ , $\alpha _{h_t u}$ is adjusted to keep the CC-RSRA estimations within $\pm 5\,\%$ of the DNS in the outer region, and the resulting values are subsequently fitted to obtain an empirical relation between $\alpha _{h_t u}$ and $\varTheta$ . To this end, DNS data from Yu et al. (Reference Yu, Zhao, Yuan and Xu2024) for ideal-gas turbulent boundary layers are incorporated. The resulting dependence of $\alpha _{h_t u}$ on $\varTheta$ is shown in figure 11(a), where an approximately linear relation is observed between $\ln (1-\alpha _{h_t u})$ and $\varTheta$ , i.e. $\ln (1 - \alpha _{h_t u}) = 3.675\varTheta - 4.473$ . The CC-RSRA with the corresponding $\alpha _{h_t u}$ under different wall-cooling conditions is evaluated in figure 11(b) as the ratio of the right-hand side to the corresponding left-hand side of (4.15). It yields a substantial improvement over the original RSRA (shown in figure 8), particularly in the vicinity of the mean temperature peak under wall-cooling conditions ( $\varTheta \lt 1$ ), while maintaining good agreement in the outer region. The effectiveness of the correlation correction decreases as the wall-cooling intensity is reduced. Under adiabatic wall conditions ( $\varTheta = 1$ ), the discrepancies inevitably arise in the near-wall region $y^* \approx 2$ .

Figure 11.

(a) Dependence of $ln(1 - \alpha _{h_t u})$ on the diabatic parameter $\varTheta$ for different cases, with the dashed line indicating the linear fit: $\ln (1 - \alpha _{h_t u})=3.675\varTheta -4.473$ ; (b) assessment of the CC-RSRA (4.15) based on the fitted $\alpha _{h_t u}$ . The Mach 4 and Mach 6 data under different wall-cooling conditions are from DNS by Yu et al. (Reference Yu, Zhao, Yuan and Xu2024).

Figure 11. Long description

Panel A: A scatter plot shows the dependence of ln(1 - α_hru) on the diabatic parameter Θ for different cases. The x-axis represents Θ, and the y-axis represents ln(1 - α_hru). The plot includes data points for various conditions: Ma = 6, Θ = 0.133; Ma = 6, Θ = 0.422; Ma = 6, Θ = 1; and Ma = 4, Θ = 1. A dashed line indicates the linear fit: ln(1 - α_hru) = 3.675Θ - 4.473. Panel B: A scatter plot assesses the CC-RSRA (4.15) based on the fitted α_hru. The x-axis represents y* on a logarithmic scale, and the y-axis represents CC-RSRA. The plot includes multiple lines representing different conditions: LR, HR, FC, Ma = 6, Θ = 0.133; Ma = 6, Θ = 0.422; Ma = 6, Θ = 1; and Ma = 4, Θ = 1. Each line shows how CC-RSRA varies with y* under these conditions.

5.1. Statistical decomposition of high-enthalpy TCI

Due to the nonlinear dependence of $\dot {\omega }_s$ on temperature $T$ and the set of densities $\{\rho _s\}$ , the mean production rate $\overline {\dot {\omega }_s(T,\{\rho _s\})}$ generally differs from the value evaluated at the corresponding mean quantities, i.e. $\overline {\dot {\omega _{s}}(T,\{\rho _s\})} \neq \dot {\omega _{s}}(\tilde {T},\{\overline {\rho _s}\})$ . This discrepancy poses a key challenge in turbulence closure with chemical reactions. By averaging the mass production rate (2.12), the mean reaction rate requiring closure can be decomposed as follows:

(5.1) \begin{eqnarray} \overline {k_r\prod _{s=1}^{N_s}\rho _s^{\nu _{sr}}} \approx \underbrace {\overline {k_r}\prod _{s=1}^{N_s} \overline {\rho _s}^{\nu _{sr}} \vphantom {\sum _{m=1,\atop \nu _{mr}\gt 0}^{N_s}}}_{\mathcal{W}_1} + \underbrace {\sum _{m=1,\atop \nu _{mr}\gt 0}^{N_s} \nu _{mr} \overline {k_r^{\prime }\rho _m^{\prime }} \overline {\rho _m}^{\nu _{mr}-1} \prod _{s\neq m}^{N_s}\overline {\rho _s}^{\nu _{sr}}}_{\mathcal{W}_2} \quad \quad \quad \quad &&\nonumber \\ + \underbrace {\overline {k_r} \left (\sum _{m=1,\atop \nu _{mr}=2}^{N_s}\overline {{\rho _m^{\prime }}^2}\prod _{s\neq m}^{N_s}\overline {\rho _s}^{\nu _{sr}} + \sum _{m \neq n,\atop \nu _{mr},\nu _{nr}\gt 0}^{N_s}\nu _{mr}\nu _{nr}\overline {\rho _m^{\prime }\rho _n^{\prime }}{\overline {\rho _m}}^{\nu _{mr}-1}{\overline {\rho _n}}^{\nu _{nr}-1}\prod _{s\neq m,n}^{N_s}\overline {\rho _s}^{\nu _s} \right )}_{\mathcal{W}_3},&& \end{eqnarray}

with higher-order moments (third order and above) truncated. On the right-hand side of (5.1), the first term $\mathcal{W}_1$ is the base term, representing the first-order mean approximation that neglects the fluctuations in the reaction-rate coefficient and species densities. The second term $\mathcal{W}_2$ is the reaction-rate–density cross-term, which represents the contribution of the correlation between fluctuations in the reaction rate and species density. The third and fourth terms are collectively referred to as the density-fluctuation term $\mathcal{W}_3$ , which represents the contribution of fluctuations in species densities.

To quantitatively assess the impact of turbulent fluctuations on high-enthalpy reactions, figures 12 and 13 compare the mean reaction rates obtained using different approaches with the DNS results. The dissociation reactions $\mathcal{R}_1$ – $\mathcal{R}_3$ are shown in figure 12, while the Zel’dovich reactions $\mathcal{R}_4$ and $\mathcal{R}_5$ are presented in figure 13. Here, $\dot {R}_{\tilde {T},\overline {\rho }_s}$ denotes the reaction rate evaluated using the mean temperature $\tilde {T}$ and densities $\overline {\rho }_s$ , and $\dot {R}_{\mathcal{W}_i}\,(i=1,2,3)$ represents the statistically decomposed contributions based on the corresponding terms in (5.1), e.g. $\dot {R}_{\mathcal{W}_{12}}=\dot {R}_{\mathcal{W}_1}+\dot {R}_{\mathcal{W}_2}$ . Overall, $\dot {R}_{\mathcal{W}_{123}}$ agrees well with the DNS results, confirming the validity of the statistical decomposition (5.1) and indicating that higher-order fluctuation terms are negligible. Among the dissociation reactions, the rate of the reaction $\mathcal{R}_1$ (N $_2$ dissociation) is much smaller than the others and has a negligible impact on the overall chemical activity. In the LR case, the NO concentration remains low, and the rate of the reaction $\mathcal{R}_3$ is more than one order of magnitude smaller than the other reactions (see figure 12 e), whereas it becomes non-negligible in the HR case.

Figure 12.

Wall-normal profiles of the mean reactions rates $\overline {\dot {R}}_r$ of the reactions $\mathcal{{R}}_1$ (a, b), $\mathcal{{R}}_2$ (c, d) and $\mathcal{{R}}_3$ (e, f) for the LR (a, c, e) and HR (b, d, f) cases obtained using different approaches. Here, $\dot {R}_{\tilde {T},\overline {\rho }_s}$ denotes the rate evaluated using mean temperature and species densities, $\dot {R}_{W_i}$ represents the contributions from the different terms in (5.1) and $\dot {R}_{\overline {p}}$ denotes the rate evaluated under the isobaric assumption (5.2).

Figure 13.

Wall-normal profiles of the mean reaction rates $\overline {\dot {R}}_r$ of the reactions $\mathcal{{R}}_4$ (a, b) and $\mathcal{{R}}_5$ (c, d) for the LR (a, c) and HR (b, d) cases obtained using different approaches. Here, $\dot {R}_{\tilde {T},\overline {\rho }_s}$ denotes the rate evaluated using mean temperature and species densities, $\dot {R}_{W_i}$ represents the contributions from the different terms in (5.1) and $\dot {R}_{\overline {p}}$ denotes the rate evaluated under the isobaric assumption (5.2).

Figure 13. Long description

The image contains four line graphs labeled (a), (b), (c), and (d). Each graph shows wall-normal profiles of the mean reaction rates for different reactions and cases. Panel A: The line graph shows the mean reaction rates for the LR case. The x-axis is labeled y* and ranges from 10^0 to 10^3. The y-axis is labeled μM R4/(ρτw) and ranges from -2.0 x 10^-5 to 8.0 x 10^-5. Different lines represent various approaches: black solid line for R, red dashed line for RT,Ps, green dashed line for RW1, blue dotted line for RW12, and purple dash-dotted line for RW13. Panel B: The line graph shows the mean reaction rates for the HR case. The x-axis is labeled y* and ranges from 10^0 to 10^3. The y-axis is labeled μM R4/(ρτw) and ranges from -4.0 x 10^-4 to 4.0 x 10^-4. Different lines represent various approaches: black solid line for R, red dashed line for RT,Ps, green dashed line for RW1, blue dotted line for RW12, orange dash-dotted line for RW13, and purple dash-dotted line for RW123. Panel C: The line graph shows the mean reaction rates for the LR case. The x-axis is labeled y* and ranges from 10^0 to 10^3. The y-axis is labeled μM R5/(ρτw) and ranges from -8.0 x 10^-5 to 0. Different lines represent various approaches: black solid line for R, red dashed line for RT,Ps, green dashed line for RW1, blue dotted line for RW12, and purple dash-dotted line for RW13. Panel D: The line graph shows the mean reaction rates for the HR case. The x-axis is labeled y* and ranges from 10^0 to 10^3. The y-axis is labeled μM R5/(ρτw) and ranges from -3.0 x 10^-4 to 0. Different lines represent various approaches: black solid line for R, red dashed line for RT,Ps, green dashed line for RW1, blue dotted line for RW12, orange dash-dotted line for RW13, and purple dash-dotted line for RW123.

Direct evaluation using mean quantities, i.e. $\dot {R}_{\tilde {T},\overline {\rho }_s}$ , exhibits apparent deviations from the DNS results. In the LR case, except for the reaction $\mathcal{R}_5$ , $\dot {R}_{\tilde {T},\overline {\rho }_s}$ underpredicts the magnitude of all reaction rates. This discrepancy is substantially reduced when $\dot {R}_{\mathcal{W}_1}$ is employed, indicating that the inequality $k_r(\tilde {T}) \neq \overline {k}_r$ is the primary source of high-enthalpy TCI under the LR condition (Duan & Martin Reference Duan and Martin2011c ; Wang & Xu Reference Wang and Xu2026). In contrast, the HR case exhibits a different behaviour: $\dot {R}_{\tilde {T},\overline {\rho }s}$ provides a more accurate prediction than $\dot {R}_{\mathcal{W}_1}$ for the reaction $\mathcal{R}_2$ , yields comparable deviations for the reaction $\mathcal{R}_5$ and performs worse for the reactions $\mathcal{R}_1$ , $\mathcal{R}_3$ and $\mathcal{R}_4$ . The markedly different performance of $\dot {R}_{\mathcal{W}1}$ relative to $\dot {R}_{\tilde {T},\overline {\rho }_s}$ in the LR and HR cases suggests that modelling the temperature fluctuation effects alone via a presumed PDF for $\overline {k}_r$ is adequate for the LR case, but leads to a deterioration in prediction accuracy in the HR case.

Further insight is obtained by comparing $\dot {R}_{\mathcal{W}_1}$ with $\dot {R}_{\mathcal{W}_{12}}$ and $\dot {R}_{\mathcal{W}_{13}}$ . In the LR case, the contributions from the reaction-rate–density cross-term $\mathcal{W}_2$ and density-fluctuation term $\mathcal{W}_3$ are small, and thus the focus is placed on the HR case. For the reactions $\mathcal{R}_1$ , $\mathcal{R}_3$ and $\mathcal{R}_4$ , $\dot {R}_{\mathcal{W}_1}$ already agrees closely with the DNS results. For the reaction $\mathcal{R}_2$ , $\dot {R}_{\mathcal{W}_{13}}$ shows a nearly same deviation to $\dot {R}_{\mathcal{W}_{1}}$ , whereas $\dot {R}_{\mathcal{W}_{12}}$ significantly improves the agreement with the DNS compared with $\dot {R}_{\mathcal{W}_{1}}$ . In contrast, for the reaction $\mathcal{R}_5$ , $\dot {R}_{\mathcal{W}_{12}}$ behaves similarly to $\dot {R}_{\mathcal{W}_{1}}$ , while $\dot {R}_{\mathcal{W}_{13}}$ provides better agreement. These results together indicate that the relative importance of the cross-term $\mathcal{W}_2$ and density-fluctuation term $\mathcal{W}_3$ varies among reactions, and that accounting for only one of them is insufficient for accurate TCI prediction.

Recently, Williams et al. (Reference Williams, Di Renzo and Moin2025) reported that pressure fluctuations have a small effect on high-enthalpy TCI, such that the species densities can be approximated under an isobaric assumption using the instantaneous temperature and composition. Accordingly, the isobaric mean reaction rate can be evaluated as

(5.2) \begin{align} \dot {R}_{\overline {p}} = \overline {\dot {R} \left(T,\, \left\{Y_s \frac {\overline {p}M}{\mathscr{R}T} \right\} \right)}. \end{align}

The corresponding wall-normal profiles of $\dot {R}_{\overline {p}}$ are included in figures 12 and 13. In the LR case, $\dot {R}_{\overline {p}}$ improves upon the estimate based on mean quantities, $\dot {R}_{\tilde {T},\overline {\rho }s}$ , but shows slightly larger deviations from the DNS than $\dot {R}_{\mathcal{W}_1}$ . In contrast, in the HR case, $\dot {R}_{\overline {p}}$ yields a marked improvement for the reactions $\mathcal{R}_2$ and $\mathcal{R}_5$ , where both $\dot {R}_{\tilde {T},\overline {\rho }s}$ and $\dot {R}_{\mathcal{W}_1}$ fail to provide accurate predictions. This behaviour can be attributed to the fact that $\dot {R}_{\overline {p}}$ neglects only pressure fluctuations while largely retaining the effects of temperature and composition fluctuations. Although $\dot {R}_{\overline {p}}$ is not fully closed, the isobaric approximation nevertheless provides a useful guideline for improving presumed-PDF-based modelling strategies in high-enthalpy TCI.

5.2. Isobaric presumed PDF model

In conventional presumed PDF models, the effects of turbulent fluctuations on high-enthalpy chemical reaction rates are accounted for solely through temperature fluctuations (Duan & Martin Reference Duan and Martin2011c ; Wang & Xu Reference Wang and Xu2026), such that the mean mass production rate $\overline {\dot {\omega }}_s$ is evaluated as

(5.3) \begin{align} \dot {\omega }_s^{\textit{PDF}} = \int _{-\infty }^\infty \dot {\omega }\big ( \tilde {T} + T^{\prime \prime }, \{\overline {\rho }_s\} \big ) P\left (T^{\prime \prime }\right ) \mathrm{d}T^{\prime \prime }, \end{align}

where $P(T^{\prime \prime })$ is the PDF of the temperature fluctuation, which is typically approximated by a Gaussian distribution

(5.4) \begin{align} P\left (T^{\prime \prime }\right )=\frac {1}{\sqrt {2\pi {T_{\textit{rms}}^{\prime \prime }}^2}}\exp \left (-\frac {{T^{\prime \prime }}^2}{2{T_{\textit{rms}}^{\prime \prime }}^2}\right ), \end{align}

or alternatively by a more physically consistent log-normal distribution, as proposed by Wang & Xu (Reference Wang and Xu2026). The mean temperature $\tilde {T}$ can be directly obtained from the RANS solutions. To avoid solving an additional transport equation for the temperature fluctuation $T_{\textit{rms}}^{\prime \prime }$ , Duan & Martin (Reference Duan and Martin2011c ) proposed to estimate $T_{\textit{rms}}^{\prime \prime }$ based on the SRA. Here, this model is improved by incorporating the CC-RSRA (4.15), which resolves the deficiency of the conventional SRA in predicting temperature fluctuations near the mean temperature peak. The velocity fluctuation intensity $u_{\textit{rms}}^{\prime \prime }$ required for the CC-RSRA is related to the turbulent kinetic energy $k$ as

(5.5) \begin{align} u_{\textit{rms}}^{\prime \prime } = \frac {\sqrt {2k}}{\sqrt {1+C\left (1-\text{exp}\left ({-D\frac {y}{\delta }}\right )\right )^2}}, \end{align}

where $C=1.06$ and $D=15$ are taken from Duan & Martin (Reference Duan and Martin2011c ). The temperature fluctuation $T_{\textit{rms}}^{\prime \prime }$ is obtained from the enthalpy fluctuation $h_{\textit{rms}}^{\prime \prime }$ using (4.2), based on the assumption of linear correlations

(5.6) \begin{align} \widetilde {T^{\prime \prime }Y_s^{\prime \prime }} \approx \text{sgn}\left (\frac {\partial \tilde {T}}{\partial y} \frac {\partial \tilde {Y}_s}{\partial y}\right ) T_{\textit{rms}}^{\prime \prime }Y_{s,\textit{rms}}^{\prime \prime }, \quad \widetilde {Y_s^{\prime \prime }Y_r^{\prime \prime }} \approx \text{sgn}\left (\frac {\partial \tilde {Y}_s}{\partial y} \frac {\partial \tilde {Y}_r}{\partial y}\right ) Y_{s,\textit{rms}}^{\prime \prime }Y_{r,rms}^{\prime \prime }, \end{align}

where the sgn function is used to determine the sign of the covariance based on the local gradients. The mass fraction fluctuation $Y_{s,\textit{rms}}^{\prime \prime }$ is evaluated using the RSRA (4.10a), with the turbulent Prandtl and Schmidt numbers, $Pr_{t,h}$ and $Sc_{t,Y_s}$ , both set to unity.

Despite these modelling efforts, the degraded performance in the HR case shown in figures 12 and 13 indicates that the conventional presumed PDF model still requires further improvement. As shown in § 3.2, the strong linear correlation between mass fraction and temperature fluctuations suggests that composition variations can be parameterised by temperature fluctuations. By further applying the isobaric approximation proposed by Williams et al. (Reference Williams, Di Renzo and Moin2025), an isobaric presumed PDF model can be formulated as follows:

(5.7) \begin{align} \dot {\omega }_{s, \overline {p}}^{\textit{PDF}} = \int _{-\infty }^\infty \dot {\omega }\big ( \tilde {T} + T^{\prime \prime }, \big\{\rho _{s,\overline {p}} \left(T^{\prime \prime } \right) \big\} \big ) P\left (T^{\prime \prime }\right ) \mathrm{d}T^{\prime \prime }, \end{align}

where

(5.8) \begin{align} \rho _{s,\overline {p}} \left (T^{\prime \prime }\right ) = Y_s\left (T^{\prime \prime }\right ) \frac {\overline {p}}{\mathscr{R}} \frac { M\left (\big\{Y_s\left (T^{\prime \prime }\right )\big\}\right )}{\tilde {T} + T^{\prime \prime }}, \\[-28pt] \nonumber \end{align}

(5.9) \begin{align} Y_s\left (T^{\prime \prime }\right ) = \tilde {Y}_s + \text{sgn}\left (\frac {\partial \tilde {T}}{\partial y} \frac {\partial \tilde {Y}_s}{\partial y}\right ) \frac {T^{\prime \prime }}{T_{\textit{rms}}^{\prime \prime }}Y_{s,\textit{rms}}^{\prime \prime }. \\[0pt] \nonumber \end{align}

The species mass fraction fluctuation $Y_{\textit{rms}}^{\prime \prime }$ in (5.9) is evaluated using the RSRA (4.10a).

Figures 14 and 15 present comparisons of the mean production rates $\overline {\dot {\omega }}_s$ obtained using different presumed PDF models against the DNS results for the LR and HR cases. The results for N $_2$ , O $_2$ and NO are shown in figure 14, while figure 15 presents those for O and N. In the LR case, the isobaric presumed PDF model $\dot {\omega }_{s, \overline {p}}^{\textit{PDF}}$ and conventional presumed PDF model $\dot {\omega }_{s}^{\textit{PDF}}$ yield overall similar predictions. In the region near the temperature peak ( $y^* \approx 10$ ), $\dot {\omega }_{s, \overline {p}}^{\textit{PDF}}$ shows slightly larger deviations from the DNS compared with $\dot {\omega }_{s}^{\textit{PDF}}$ . For $y^* \gtrapprox 20$ , the results of $\dot {\omega }_{s, \overline {p}}^{\textit{PDF}}$ exhibit better agreement with the DNS. In the HR case, $\dot {\omega }_{s, \overline {p}}^{\textit{PDF}}$ demonstrates much clearer advantages. It significantly improves the prediction of the production rates of O $_2$ , NO, O and N compared with $\dot {\omega }_{s}^{\textit{PDF}}$ , which accounts only for temperature fluctuations. Moreover, the comparison between the isobaric presumed PDF results obtained using RSRA and CC-RSRA demonstrates that the incorporation of CC-RSRA leads to a marked improvement.

Figure 14.

Wall-normal profiles of the mean mass production rates $\overline {\dot {\omega }}_s$ of the species N $_2$ (a, b), O $_2$ (c, d) and NO (e, f) for the LR (a, c, e) and HR (b, d, f) cases obtained using different approaches. Here, $\dot {\omega }_{s}^{\textit{PDF}}$ denotes the rate evaluated using the conventional presumed PDF model (5.3), and $\dot {\omega }_{s, \overline {p}}^{\textit{PDF}}$ denotes the rate evaluated using the isobaric presumed PDF model (5.7).

Figure 15.

Wall-normal profiles of the mean mass production rates $\overline {\dot {\omega }}_s$ of the species O (a, b) and N (c, d) for the LR (a, c) and HR (b, d) cases obtained using different approaches. Here, $\dot {\omega }_{s, \overline {p}}^{\textit{PDF}}$ denotes the rate evaluated using the isobaric presumed PDF model (5.7), and $\dot {\omega }_{s}^{\textit{PDF}}$ denotes the rate evaluated using the conventional presumed PDF model (5.3).

Figure 15. Long description

The image contains four line graphs showing wall-normal profiles of the mean mass production rates for species O and N in low (LR) and high (HR) Reynolds number cases. Panel A: This line graph shows the mean mass production rate of species O for the LR case. The x-axis is labeled y* and ranges from 10^0 to 10^3. The y-axis is labeled with a complex unit involving mass production rate and ranges from 0 to 2.0 x 10^-4. Different lines represent various approaches, including the isobaric presumed PDF model and the conventional presumed PDF model. Panel B: This line graph shows the mean mass production rate of species O for the HR case. The x-axis is labeled y* and ranges from 10^0 to 10^3. The y-axis is labeled with the same complex unit and ranges from 0 to 8.0 x 10^-4. Different lines represent various approaches, including the isobaric presumed PDF model and the conventional presumed PDF model. Panel C: This line graph shows the mean mass production rate of species N for the LR case. The x-axis is labeled y* and ranges from 10^0 to 10^3. The y-axis is labeled with a complex unit involving mass production rate and ranges from -1.0 x 10^-5 to 1.0 x 10^-5. Different lines represent various approaches, including the isobaric presumed PDF model and the conventional presumed PDF model. Panel D: This line graph shows the mean mass production rate of species N for the HR case. The x-axis is labeled y* and ranges from 10^0 to 10^3. The y-axis is labeled with the same complex unit and ranges from -3.0 x 10^-4 to 3.0 x 10^-4. Different lines represent various approaches, including the isobaric presumed PDF model and the conventional presumed PDF model.

Table 4 summarises the normalised relative $L_2$ errors of different models. The error for each species is defined as

(5.10) \begin{align} \varepsilon _s = \frac { \left ( \int _0^{\infty } \left ( \dot {\omega }_s^{{model}} - \overline {\dot {\omega }}_s \right )^2 \, {\rm d}y \right )^{1/2} }{ \left ( \int _0^{\infty } \left ( \overline {\dot {\omega }}_s \right )^2 \, {\rm d}y \right )^{1/2} } \times 100\,\%, \end{align}

where the integrals are evaluated numerically over the wall-normal direction. The error (5.10) provides a robust measure of the overall deviation across the entire boundary layer. The CC-RSRA-based $\dot {\omega }_{s, \overline {p}}^{\textit{PDF}}$ consistently outperform their RSRA counterparts by yielding substantially lower errors across most species. It can be seen that both approaches exhibit comparable accuracy in the LR case, the simpler $\dot {\omega }_s^{\textit{PDF}}$ formulation remains adequate under low-reactivity conditions, as shown in figures 14(a,c,e) and 15(a,c). However, in the HR case, $\dot {\omega }_{s,\overline {p}}^{\textit{PDF}}$ provides a markedly improved prediction, further demonstrating the reliability of the isobaric presumed PDF model across different levels of chemical reactivity. Although large deviations are observed for species N, the discrepancies are primarily associated with their relatively small production rates and thus have a limited impact on the overall predictive performance.

Table 4.

Normalised $L_2$ errors (in per cent) of mean production rates predicted by different models for the LR and HR cases.

Table 4. Long description

A table comparing normalized errors of different models for LR and HR cases. The table has 8 rows and 6 columns. The columns are labeled Cases, Model, epsilon N2, epsilon O2, epsilon NO, epsilon O, and epsilon N. The rows are labeled with different cases and models. Row 1: Cases, LR; Model, omega sub s, tilde P; epsilon N2, 39.7; epsilon O2, 21.5; epsilon NO, 13.4; epsilon O, 26.1; epsilon N, 319.2. Row 2: Cases, LR; Model, omega sub s PDF (RSRA); epsilon N2, 18.2; epsilon O2, 15.5; epsilon NO, 8.0; epsilon O, 19.7; epsilon N, 155.3. Row 3: Cases, LR; Model, omega sub s PDF (RSRA); epsilon N2, 36.8; epsilon O2, 14.0; epsilon NO, 9.7; epsilon O, 17.1; epsilon N, 288.0. Row 4: Cases, LR; Model, omega sub s PDF (CC-RSRA); epsilon N2, 8.6; epsilon O2, 18.7; epsilon NO, 7.5; epsilon O, 25.0; epsilon N, 84.4. Row 5: Cases, LR; Model, omega sub s PDF (CC-RSRA); epsilon N2, 16.3; epsilon O2, 9.2; epsilon NO, 7.9; epsilon O, 10.0; epsilon N, 58.1. Row 6: Cases, HR; Model, omega sub s, tilde P; epsilon N2, 98.2; epsilon O2, 80.3; epsilon NO, 65.5; epsilon O, 78.3; epsilon N, 226.8. Row 7: Cases, HR; Model, omega sub s PDF (RSRA); epsilon N2, 42.8; epsilon O2, 111.4; epsilon NO, 91.2; epsilon O, 108.3; epsilon N, 161.8. Row 8: Cases, HR; Model, omega sub s PDF (RSRA); epsilon N2, 124.7; epsilon O2, 95.2; epsilon NO, 42.9; epsilon O, 109.6; epsilon N, 253.7. Row 9: Cases, HR; Model, omega sub s PDF (CC-RSRA); epsilon N2, 26.1; epsilon O2, 120.1; epsilon NO, 94.5; epsilon O, 118.5; epsilon N, 138.8. Row 10: Cases, HR; Model, omega sub s PDF (CC-RSRA); epsilon N2, 31.4; epsilon O2, 18.2; epsilon NO, 24.3; epsilon O, 17.7; epsilon N, 53.3.