1. Introduction
The conversion of bulk kinetic energy into molecular thermal motion via shock-induced compression and viscous dissipation activates a number of temperature-dependent thermochemical processes within the shock layers of high-speed flight vehicles (Anderson Reference Anderson2006; Bertin & Cummings Reference Bertin and Cummings2006; Urzay Reference Urzay2018; Urzay & Di Renzo Reference Urzay and Di Renzo2021). These high-enthalpy effects include the activation of additional internal degrees of freedom in the form of vibrational and electronic excitation, together with the initiation of chemical reactions among the atomic and molecular components of high-temperature air (Vincenti & Krüger Reference Vincenti and Krüger1965; Park Reference Park1990; Zel’Dovich & Raizer Reference Zel’Dovich and Raizer2002). Owing to the comparable characteristic time scales of hydrodynamic motion and thermochemical relaxation in hypersonic flows, the composition within high-Mach boundary layers cannot generally be equated to the minimum-free-energy chemical configuration (Lee Reference Lee1984; Gnoffo, Gupta & Shinn Reference Gnoffo, Gupta and Shinn1989; Candler Reference Candler2019). Due to this thermochemical non-equilibrium, the local chemical composition intrinsically depends on the relative rates of advective transport and chemical production, even in the case of laminar hypersonic boundary layers (Inger Reference Inger1964; Liñán & Da Riva Reference Liñán and Da Riva1962; Seror et al. Reference Seror, Zeitoun, Brazier and Schall1997; Williams et al. Reference Williams, Di Renzo, Moin and Urzay2021). With the introduction of boundary-layer disturbances to high-enthalpy flows, exchange of chemical and sensible enthalpy by chemical heat release, as well as internal-energy exchange via vibrational relaxation, modifies the stability characteristics of high-Mach flows in terms of both growth rates and frequencies associated with instabilities (Stuckert & Reed Reference Stuckert and Reed1994; Hudson, Chokani & Candler Reference Hudson, Chokani and Candler1997; Fujii & Hornung Reference Fujii and Hornung2003; Lyttle & Reed Reference Lyttle and Reed2005; Germain & Hornung Reference Germain and Hornung1997; Marxen, Iaccarino & Magin Reference Marxen, Iaccarino and Magin2014; Miró et al. Reference Miró, Fernando, Ethan, Pinna and Reed2019; Passiatore et al. Reference Passiatore, Gloerfelt, Sciacovelli, Pascazio and Cinnella2024). In addition to distorting the base flow relative to an equivalent calorically perfect boundary layer, high-enthalpy processes likewise have been shown through numerical experiments to impact disturbance amplification itself (Johnson, Seipp & Candler Reference Johnson, Seipp and Candler1998; Knisely & Zhong Reference Knisely and Zhong2019). Ultimately, the complex interplay between thermochemical effects and the growth of unstable modes are sensitive to both the particular base flow and instability itself, with exchanges among chemical and sensible enthalpies able to either stabilise or destabilise high-enthalpy boundary layers (Bertolotti Reference Bertolotti1998; Johnson et al. Reference Johnson, Seipp and Candler1998; Kline, Chang & Li Reference Kline, Chang and Li2018; Mortensen Reference Mortensen2018).
The coupling between the hydrodynamic and thermochemical fields is further complicated by the introduction of a multiplicity of hydrodynamic spatial and temporal scales upon breakdown to turbulence, with flow-induced fluctuations in thermodynamic-state variables directly modulating the mean rates of non-equilibrium thermochemical processes (Duan, Beekman & Martin Reference Duan, Beekman and Martin2011; Di Renzo & Urzay Reference Di Renzo and Urzay2021; Urzay & Di Renzo Reference Urzay and Di Renzo2021). As in isotropic turbulence (Martin & Candler Reference Martin and Candler1998, Reference Martin and Candler1999), thermal fluctuations within high-Mach boundary layers give rise to enhancement of dissociation processes (Duan & Martin Reference Duan and Martin2009; Duan & Martín Reference Duan and Martín2011), which correspondingly modifies both the mean-flow thermodynamics and fluctuation statistics. Correspondingly, the chemical heat release associated with exothermic processes has been shown to amplify turbulence kinetic energy and thermal variance in high-Mach turbulence, while both internal-energy excitation and chemical dissociation effectively damp temperature fluctuations owing to their endothermicity (Martin & Candler Reference Martin and Candler2000, Reference Martin and Candler2001; Duan & Martin Reference Duan and Martin2009; Di Renzo, Williams & Pirozzoli Reference Di Renzo, Williams and Pirozzoli2024). In addition to this mutual interaction between dissociation/recombination chemistry and macroscale hydrodynamic fluctuations, the production of radical species by high-temperature reactions and subsequent mixing by boundary-layer turbulence ultimately imparts significant distortions to the mean reaction rates of the Zel’dovich mechanism. The present manuscript characterises this turbulence–chemistry interaction associated with the shuffle reactions, while also assessing the relative importance of compositional and thermal mixing for all chemical-kinetic processes.
Together with the emergence of finite-rate thermochemical processes, significant wall-normal thermal variation itself, as arising from viscous dissipation, represents a distinguishing characteristic of high-Mach turbulence. While this aerodynamic heating effect ultimately distorts the turbulence statistics relative to an incompressible boundary layer (Guarini et al. Reference Guarini, Moser, Shariff and Wray2000; Pirozzoli, Grasso & Gatski Reference Pirozzoli, Grasso and Gatski2004; Duan, Beekman & Martin Reference Duan, Beekman and Martin2010; Duan et al. Reference Duan, Beekman and Martin2011; Pirozzoli & Bernardini Reference Pirozzoli and Bernardini2011), for adiabatic boundary layers or weakly diabatic flows at moderate Mach numbers, the distortions in the mean-flow velocity are largely attributable to mean variations in density and molecular transport properties alone (van Driest Reference van Driest1956; Zhang et al. Reference Zhang, Bi, Hussain, Li and She2012; Patel, Boersma & Pecnik Reference Patel, Boersma and Pecnik2016; Trettel & Larsson Reference Trettel and Larsson2016; Griffin, Fu & Moin Reference Griffin, Fu and Moin2021). For diabatic high-Mach boundary layers, however, intrinsic-compressibility effects (Lele Reference Lele1994; Smits & Dussauge Reference Smits and Dussauge2006; Yu, Xu & Pirozzoli Reference Yu, Xu and Pirozzoli2019) have been shown to emerge in violation of Morkovin’s hypothesis (Morkovin Reference Morkovin1962; Bradshaw Reference Bradshaw1977), associated not only with modulation of pressure fluctuations in the viscous sublayer (Yu, Xu & Pirozzoli Reference Yu, Xu and Pirozzoli2020) but the mean-velocity profile itself (Hasan et al. Reference Hasan, Larsson, Pirozzoli and Pecnik2023, Reference Hasan, Costa, Larsson, Pirozzoli and Pecnik2024). With the activation of chemical reactions in high-Mach wall-bounded flows, pressure perturbations directly modulate the rate of dissociation/recombination processes, and yet, the significance of this intrinsic compressibility as it pertains to turbulence-aerothermochemistry interactions remains uncharacterised.
The objective of this paper is to systematically characterise turbulence–aerothermochemistry interactions in a spatially evolving turbulent reacting hypersonic boundary layer, with particular focus on analysing the effects of species mixing and temperature fluctuations on the Zel’dovich mechanism. To that end, direct numerical simulation results of a high-enthalpy Mach-7 turbulent hypersonic boundary layer, with significant chemical production of radical species, are presented. In order to disambiguate the effects of thermal and compositional fluctuations in high-Mach turbulence–chemistry interaction, a computational approach for statistical decomposition of the expected rates of chemical processes with respect to specific thermodynamic fluctuations is applied to the boundary-layer simulation and, finally, leveraged to assess the impact of pressure perturbations on species production in wall-bounded turbulent hypersonic flows.
The remainder of this manuscript is structured as follows. First, the overall physical configuration for the direct numerical simulation is presented in § 2. The relevant conservation equations, treatment of molecular transport and chemical kinetics, as well as additional details of the computational approach utilised for the direct numerical simulation, are presented in § 3. The corresponding mean-flow results for the turbulent boundary-layer simulation, together with the statistical decompositions of turbulence–chemistry interaction in terms of both chemical production and reaction rates, are presented in § 4. Finally, concluding remarks are provided § 5, while further description of the laminar base flow (Appendix A), verification of the turbulent state (Appendix B), assessment of grid convergence (Appendix C) and additional species-specific decompositions for turbulence–chemistry interaction (Appendix D) are deferred to the appendices.
2. Physical configuration
The direct numerical simulation is performed in a Cartesian computational domain with the incoming laminar boundary-layer profile obtained from an auxiliary two-dimensional simulation of the reacting hypersonic flow over a 16
$^\circ$
wedge at a free-stream Mach number of 25; a schematic of this overall configuration is provided in figure 1. This approach enables leading-edge effects to be fully captured, producing heightened levels of atomic nitrogen in the boundary layer relative to a locally self-similar approximation (Lees Reference Lees1956; Di Renzo & Urzay Reference Di Renzo and Urzay2021). Compression from the nose shock wave produces an inviscid shock layer with a Mach number of 7.0 and a temperature of 2686 K, corresponding to the edge conditions for the boundary layer itself. The wall in both the hypersonic wedge and boundary-layer calculations is treated consistently as isothermal and non-catalytic with a temperature of 3000 K. Although not uncharacteristic of high-Mach atmospheric re-entry, this elevated surface temperature would likely give rise to gas–surface interaction for a realistic thermal protection system. Modification of gas-phase turbulence–chemistry interaction in hypersonic boundary layers by surface chemical effects will therefore be addressed therefore in a subsequent study. Further discussion of the auxiliary wedge calculation and presentation of the corresponding laminar boundary-layer profiles are provided in Appendix A. Following breakdown to turbulence, the friction Knudsen number (McMullen et al. Reference McMullen, Krygier, Torczynski and Gallis2023) for the turbulent reacting boundary layer remains below
$1.5\times 10^{-3}$
throughout the entire simulation domain, evidencing the self-consistency of the continuum formulation utilised in the present study. The vibrational Damköhler numbers based on the Landau–Teller relaxation constant (Landau & Teller Reference Landau and Teller1936; Millikan & White Reference Millikan and White1963; Park Reference Park1990, Reference Park1993), evaluated with the mean thermodynamic variables at the peak-temperature location, and the eddy-turnover time scale remain above
$2.9\times 10^{1}$
,
$9.8\times 10^{2}$
, and
$6.3\times 10^{2}$
for molecular nitrogen, nitric oxide and molecular oxygen, respectively, consistent with the thermal-equilibrium formulation utilised for this investigation.
Figure 1. Composite schematic of the overall physical configuration consisting of the auxiliary inflow calculation together with the Cartesian boundary-layer simulation, reflected across the wedge midplane. Six spanwise periods of the primary simulation domain depict the breakdown to turbulence.
3. Formulation
3.1. Conservation equations
On account of the finite-rate evolution of the chemical state variables in hypersonic flows, determination of the local composition entails integration of a partial-density conservation equation for each of the components, given by
\begin{equation} \frac {\partial {(\rho }{Y_i})}{\partial {t}} + \boldsymbol{\nabla }\,\boldsymbol{\cdot}\,{{(\rho }{Y_i}\boldsymbol {u})} = -\boldsymbol{\nabla }\,\boldsymbol{\cdot}\,\left ({\rho {Y_i}{\boldsymbol{V_i}}}\right ) + \dot {w}_i \hspace {5mm} i = 1, \ldots , N_s, \end{equation}
where
$\rho$
and
$\boldsymbol {u} = [u,\ v,\ w]^{\rm T}$
are the overall density and mass-averaged velocity of the reacting mixture, respectively. The mass fraction for a given species
$i$
is denoted by
$Y_i$
, whereas
$\boldsymbol{V_i}$
is defined as the corresponding diffusion velocity, and
$\dot {w}_i$
is taken to be the net chemical production rate per unit volume. The diffusion velocities are determined as (Curtiss & Hirschfelder Reference Curtiss and Hirschfelder1949; Coffee & Heimerl Reference Coffee and Heimerl1981; Ern & Giovangigli Reference Ern and Giovangigli1994)
\begin{equation} \boldsymbol{V_i} = -D_i\boldsymbol{\nabla }{\ln {X_i}} + \sum _{j=1}^{N_s}{Y_jD_j\boldsymbol{\nabla }{\ln {X_j}}}, \end{equation}
where
$X_i = {\mathcal{M}}Y_i/\mathcal{M}_i$
and
$D_i$
, respectively, correspond to the molar fraction of, and to the mixture-averaged mass diffusivity (Bird, Stewart & Lightfoot Reference Bird, Stewart and Lightfoot1960) for, species
$i$
. In this formulation, the molar mass of species
$i$
is given by
$\mathcal{M}_i$
, whereas
${\mathcal{M}} = [\sum _{i=1}^{N_s}{Y_i/\mathcal{M}_i}]^{-1}$
denotes the effective molar mass of the mixture. The present study utilises the Park (Reference Park1990) chemical mechanism for air dissociation given by
\begin{align}&\,\,\,\, \textrm{N}_2 + \textrm{O} \rightleftharpoons \textrm{NO} + \textrm{N}, \end{align}
\begin{align}&\,\,\,\, \textrm{NO} + \textrm{O} \rightleftharpoons \textrm{O}_2 + \textrm{N}, \end{align}
\begin{align}&\,\,\,\, \textrm{O}_2 + \textrm{M} \rightleftharpoons 2\textrm{O} + \textrm{M}, \end{align}
\begin{align}& \textrm{NO} + \textrm{M} \rightleftharpoons \textrm{N} + \textrm{O} + \textrm{M}, \end{align}
\begin{align}&\,\,\,\, \textrm{N}_2 + \textrm{M} \rightleftharpoons 2\textrm{N} + \textrm{M}, \end{align}
comprising the Zel’dovich exchange reactions, (R1) and (R2), together with the dissociation/recombination reactions for each of the diatomic species species given by (R3)–(R5). The reactive mixture is composed of five species, i.e.
$N_s=5$
, with the symbolic species
$\textrm{M}$
representing the third-body collision partners entering into the dissociation/recombination reactions. As such, accounting for all collision partners, the total number of reactions considered is
$N_r=17$
. With a peak mean temperature of approximately five thousand Kelvin in the turbulent portion of the boundary layer, the consideration of neutral chemistry alone is warranted for the present study, although accurate characterisation of turbulence–chemistry interaction in higher temperature and less collisional flows may, however, demand state-resolved modelling of kinetic processes (Capitelli, Armenise & Gorse Reference Capitelli, Armenise and Gorse1997; Panesi et al. Reference Panesi, Magin, Bourdon, Bultel and Chazot2009, Reference Panesi, Munafò, Magin and Jaffe2014).
In terms of the reaction rates,
$\mathcal{R}_i$
, the chemical production rates merely represent a linear combination with
$\dot {w}_i = \mathcal{M}_i\sum _{j=R_1}^{R_5}(\nu _{\textit{ij}}^{\prime \prime }-{\nu _{\textit{ij}}^\prime })\mathcal{R}_j$
. Following from the law of mass action, then, the net chemical production term for species
$i$
can then be evaluated as
\begin{equation} \dot {w}_i = \mathcal{M}_i\sum _{j=R_1}^{R_5}\big (\nu _{\textit{ij}}^{\prime \prime }-{\nu _{\textit{ij}}^\prime }\big )\sum _{l=1}^{N_s}{F_{lj}}\left [{k_{f,j}\prod _{k=1}^{N_s}\left (\frac {\rho Y_i}{\mathcal{M}_i}\right )^{\nu _{kj}^\prime }-\frac {k_{f,j}}{K_{\textit{eq},j}}\prod _{k=1}^{N_s}\left (\frac {\rho Y_i}{\mathcal{M}_i}\right )^{\nu _{kj}^{\prime \prime }}}\right ]\!, \end{equation}
where
$F_{lj}$
is a chaperon efficiency function of the third-body collision partner
$l$
in reaction
$j$
, while
$\nu _{kj}^{\prime }, \nu _{kj}^{\prime \prime } \in \mathbb{N}$
, respectively, correspond to the forward and reverse stoichiometric coefficients for species
$k$
in elementary reaction
$j$
. The modified-Arrhenius rate constants for each of the reactions in the forward direction are given by
\begin{equation} k_{f,j}= A_jT^{m_{\textit{j}}}\textrm{exp}\left (-\frac {E_{a,j}}{{R}^0{T}}\right ), \quad j = 1, \ldots , N_r, \end{equation}
where
${R}^{0}$
denotes the universal gas constant, and
$A_j$
,
$m_j$
and
$E_{a,j}$
are the Arrhenius parameters provided by Park (Reference Park1989). The chemical equilibrium constants
$K_{\textit{eq},j}$
are evaluated with
\begin{equation} K_{\textit{eq},j} = \exp \left (-\sum _{k=1}^{N_s}(\nu _{kj}^{\prime \prime } - \nu _{kj}^\prime )\frac {\mathcal{G}_k}{{R}^0T}\right )\left (\frac {P_0}{{R}^0T}\right )^{\sum _{i=1}^{N_s}{\big(\nu _{\textit{ij}}^{\prime \prime } - \nu _{\textit{ij}}^\prime\big)}}\!, \end{equation}
where
$\mathcal{G}_k$
is the Gibbs free energy of species
$k$
, evaluated at the reference pressure
$P_0$
. For the present study, the Gibbs free energy for each species is evaluated with the parameterisation of McBride, Zehe & Gordon (Reference McBride, Zehe and Gordon2002); therefore,
$P_0$
is taken to be
$100 \ \textrm{kPa}$
. In this continuum formulation, the conservation of momentum implies the compressible Navier–Stokes equations as given by
\begin{equation} \frac {\partial {(\rho }{\boldsymbol {u}})}{\partial {t}} + \boldsymbol{\nabla }\,\boldsymbol{\cdot}\,{{(\rho }\boldsymbol {u}{\boldsymbol {u}})} = -\boldsymbol{\nabla }{P} + {\boldsymbol{\nabla }}\,\boldsymbol{\cdot}\,{\overline {\overline {{\boldsymbol{\tau }}}}}, \end{equation}
where the ideal-gas equation of state relates the pressure
$P$
to the thermodynamic temperature
$T$
as
\begin{equation} P = \rho {{R}^0}T/{\mathcal{M}}. \end{equation}
Consistent with Stokes’ hypothesis (Valentini, Grover & Bisek Reference Valentini, Grover and Bisek2024), the viscous stress tensor may be expressed in terms of the strain-rate tensor
$\overline {\overline {{\boldsymbol{S}}}} = (\boldsymbol{\nabla }\boldsymbol{u}+\boldsymbol{\nabla }\boldsymbol{u}^T )/2$
and identity tensor
$\overline {\overline {{\boldsymbol{I}}}}$
as
\begin{equation} {\overline {\overline {{\boldsymbol{\tau }}}}} = 2\mu {\overline {\overline {{\boldsymbol{S}}}}} - \frac {2\mu }{3}\left (\boldsymbol{\nabla }\,\boldsymbol{\cdot}\,\boldsymbol {u}\right ){\overline {\overline {{\boldsymbol{I}}}}}, \end{equation}
where
$\mu$
is the dynamic viscosity of the mixture. For the present study,
$\mu$
is determined from Wilke’s mixture rule (Wilke Reference Wilke1950) for which the elementary viscosity of each species is evaluated following the approach of Curtiss & Hirschfelder (Reference Curtiss and Hirschfelder1949). The corresponding conservation equation for the total energy is
\begin{equation} \frac {\partial {(\rho }{E})}{\partial {t}} + \boldsymbol{\nabla }\,\boldsymbol{\cdot}\,{{(\rho }{E}\boldsymbol {u})} = \boldsymbol{\nabla }\,\boldsymbol{\cdot}\,\left ({-\boldsymbol {u}P+{\overline {\overline {{\boldsymbol{\tau }}}}}\boldsymbol {u}+ {\lambda }\boldsymbol{\nabla }{T}} - \rho \sum _{i = 1}^{N_s}{{{Y_i}}{\boldsymbol{V_i}}{h_{i}}}\right )\!, \end{equation}
where
$\lambda$
is the effective thermal conductivity determined from the mixing rule of Mathur, Tondon & Saxena (Reference Mathur, Tondon and Saxena1967), with the thermal conductivity of each component evaluated following the approach of Peters & Warnatz (Reference Peters and Warnatz2013). The stagnation energy
$E$
comprises contributions from specific kinetic energy and internal energy as
$E = (\boldsymbol{u}\,\boldsymbol{\cdot}\,{\boldsymbol{u}})/2+e$
. The internal energy itself accounts for both excitation of thermal motion and chemical energy stored in molecular bonds with
\begin{equation} e=\sum _{i=1}^{N_s}{Y_i{h_i}}-P/\rho , \end{equation}
where
$h_i$
is the specific enthalpy of species
$i$
, evaluated with the polynomial description of McBride et al. (Reference McBride, Zehe and Gordon2002). This set of conservation equations as formulated above is integrated numerically with the HTR solver (Di Renzo, Fu & Urzay Reference Di Renzo, Fu and Urzay2020) for direct numerical simulation of the Mach-7 turbulent reacting hypersonic boundary layer. The inviscid fluxes are evaluated with a low-dissipation, sixth-order hybrid skew–symmetric/targeted essentially non-oscillatory scheme (Pirozzoli Reference Pirozzoli2010; Fu, Hu & Adams Reference Fu, Hu and Adams2016; Williams, Di Renzo & Moin Reference Williams, Di Renzo and Moin2022), while explicit time advancement is performed using a third-order strong-stability-preserving Runge–Kutta method (Gottlieb, Shu & Tadmor Reference Gottlieb, Shu and Tadmor2001).
3.2. Computational approach
The primary simulation domain for the direct numerical simulation extends a length of
$800\delta _0^*, 14\pi \delta _0^*$
and
$40\delta _0^*$
in the streamwise, spanwise and wall-normal directions, respectively, where
$\delta _0^*$
is the displacement thickness of the incoming laminar boundary layer. The streamwise coordinate
$x$
is measured from the leading edge of the wedge and hence
$x \in [x_0, x_0+800\delta _0^*]$
, where
$x_0 = 117\delta _0^*$
is the distance along the wedge surface from the leading edge at which the laminar boundary layer is extracted. The computational domain is discretised with 11 868, 1184 and 464 points in the streamwise, spanwise and wall-normal directions, respectively, with a uniform distribution of points employed in the streamwise and spanwise coordinates. Along the wall-normal coordinate, the points are stretched as
$\hat {y}_j = 40\sinh (s\xi _j)/\sinh (s)$
, with
$\widehat{(\boldsymbol{\cdot})}$
denoting normalisation by
$\delta _0^*$
. The stretching factor,
$s$
, is set equal to 5.0, and
$\xi _j \in [0, 1]$
is a uniformly spaced computational coordinate.
Transition to turbulence is induced by unsteady suction and blowing at the wall, which is applied a distance of
$15\delta ^*_0$
downstream of the inflow boundary, across a strip with a streamwise extent
$5\delta ^*_0$
. The forcing employed for the present study has the same functional form as that of Franko & Lele (Reference Franko and Lele2013) and Di Renzo & Urzay (Reference Di Renzo and Urzay2021), for which a non-zero vertical velocity is imposed at the wall, namely,
$v = f(x)g(z)\sum _{i=1}^2{A_i sin(\omega _\textrm{i}{t}-\beta _i{z})}$
, where
$A_i$
,
$\omega _i$
and
$\beta _i$
naturally correspond to the amplitude, frequency and spanwise wavenumber of the
$i$
th mode. The functions
$f(x) = \exp {[-(x/\delta _0^* - x_s/\delta _0^*)^2/1.125]}$
and
$g(z) = 1.0 + 0.1\{\exp \{-[(z - z_c - z_l)/z_l]^2\} + \exp \{-[(z - z_c + z_l)/z_l]^2\}\}$
provide streamwise localisation and spanwise asymmetry within the forcing strip, where
$x_s = x_0 + 17.5\delta _0^*$
,
$z_c = 7\pi \delta _0^*$
and
$z_l = 1.4\pi \delta _0^*$
. For the present study, two modes are introduced, each with an amplitude set to 5.0 % of the edge streamwise velocity,
$U_e$
. The spanwise wavenumbers for the forcing modes are
$\pm \,2/(7\delta ^*_0)$
, each with a temporal frequency of
$9{a_e}/(10\delta ^*_0)$
, where
$a_e$
is the speed of sound at the edge of the boundary. Finally, periodicity is enforced in the spanwise direction, while characteristic boundary conditions are applied along the outflows located at
$\hat {x} = x_0/\delta _0^* + 800$
and
$\hat {y} = 40$
. Statistical convergence is achieved by averaging in time and along the homogeneous direction over the course of 10 eddy turnovers, sampling the solution fields once every 10 time steps. In terms of notation, this Reynolds-averaging operator is denoted by
$\overline{(\boldsymbol{\cdot})}$
, while the Favre/density-weighted averaging operator is represented with
$\widetilde{(\boldsymbol{\cdot})} = \overline {\rho (\boldsymbol{\cdot})}/\overline {\rho }$
; fluctuations relative to Reynolds-averaged and Favre-averaged variables will meanwhile be denoted with
$(\boldsymbol{\cdot})^{\prime }$
and
$(\boldsymbol{\cdot})^{\prime \prime }$
, respectively.
Figure 2. (a) Isosurface of the Q-criterion coloured by the molar fraction of molecular oxygen, with the side panel depicting contours of the density-gradient magnitude. (b) Instantaneous contours of the nitric oxide molar fraction at the three off-wall locations corresponding to
$(\textrm{i})\ \hat {y} = \ 3.0$
,
$(\textrm{ii})\ \hat {y} = \ 2.0$
and
$(\textrm{iii})\ \hat {y} = \ 1.0$
.
Figure 3. Molar-fraction contours along the streamwise-normal plane at
$\hat {x} = 916.$
The panels correspond to the molar fractions of (a) molecular nitrogen, (b) molecular oxygen, (c) nitric oxide, (d) atomic nitrogen and (e) atomic oxygen, respectively.
4. Results
Breakdown to turbulence in the Mach-7 reacting hypersonic boundary layer strongly modulates the rate of chemical reactions and compositional structure of the boundary layer. Figure 2(a) depicts the Q-criterion (Hunt, Wray & Moin Reference Hunt, Wray and Moin1988) isosurfaces coloured by the local molecular-oxygen molar fraction during the laminar-to-turbulent transition, together with contours of the normalised density gradient to visualise the radiated acoustic waves. Figure 2(b) evidences the strong variation in nitric oxide concentration through the transition process, owing to the enhanced turbulent mixing of thermodynamic states between the lower-temperature largely undissociated air at the boundary-layer edge and the strongly reacting near-wall mixture in the aerodynamic heating layer. Streamwise planes of the molar fractions for each species in the fully turbulent region are depicted in figure 3 reflecting this significant compositional non-uniformity and turbulence-induced species mixing. The mean molar fractions of dissociated species at the non-catalytic wall itself are provided in table 1 for a few streamwise locations, together with the normalised grid spacing in viscous units and the set of dimensionless parameters characterising the hydrodynamics of the turbulent reacting boundary layer.
Table 1. Dimensionless parameters at select streamwise locations based on the averaged primitive variables. Normalised by the inflow displacement thickness,
$\hat {\delta }$
is the height at which the Favre-averaged velocity recovers 99 % of the edge streamwise velocity. The mean wall-to-recovery enthalpy ratio
$\overline {h}_w/\overline {h}_r$
is evaluated as
$h_r = h_e+rU_e^2/2$
with a recovery factor of
$r = 0.9$
(Gibis et al. Reference Gibis, Sciacovelli, Kloker and Wenzel2024). The skin-friction and heat-flux coefficients are given by
$C_f = 2\tau _w/\rho _eU_e^2$
and
$C_q = q_w/\rho _eU_e^3$
, where
$\tau _w$
and
$q_w$
are the average wall stress and heat flux, respectively. The friction Reynolds and Mach numbers are defined as
${Re}_\tau = \overline {\rho }_w{u_\tau }\delta /\overline {\mu }_w$
and
$Ma_\tau = u_\tau /\overline {a}_w$
, respectively, where
$\overline {a}_w$
is the mean sound speed at the wall. The Reynolds numbers based on the momentum thickness
$\theta$
are likewise defined as
${{Re}}_{\delta _2} = \rho _e{U_e}\theta /\mu _w$
and
${{Re}}_{\theta } = \rho _e{U_e}\theta /\mu _e$
. The Eckert number is given by
$Ec = U_e^2/(\overline {h}_w-\overline {h}_r)$
, while the Favre-averaged mole fractions of reaction products at the wall are denoted as
$\widetilde {X}_{i,w}$
. Finally, the grid-spacing dimensions in friction units for the streamwise and spanwise directions, together with the wall-normal spacing as evaluated at the wall and at
${y}=\delta$
, are denoted by
$\Delta {x^+}$
,
$\Delta {z^+}$
,
$\Delta {{y_w^+}}$
and
$\Delta {{y_{\delta }^+}}$
, respectively.
Figure 4. (a) Transformed mean-velocity profiles utilising intrinsic-compressibility transformation of Hasan et al. (Reference Hasan, Larsson, Pirozzoli and Pecnik2023),
$u_{\textit{IC}},$
together with wall-normal variation in the (b) turbulent Mach number, diagonal elements of the Reynolds-stress tensor corresponding to the (c) streamwise, (d) wall-normal and (e) spanwise directions and ( f) Reynolds shear stress. Further characterisations of the van Driest (Reference van Driest1956) and Griffin et al. (Reference Griffin, Fu and Moin2021) mean-velocity transformations are provided in Williams, Di Renzo & Moin (Reference Williams, Di Renzo and Moin2023).
Figure 5. Wall-normal profiles of the (a) Reynolds-averaged density, (b) r.m.s. of density fluctuations, (c) Favre-averaged temperature and (d) r.m.s. of density-weighted temperature fluctuations.
4.1. Velocity, density and temperature statistics
The transformed mean-velocity profiles as a function of the semi-local coordinate (Huang, Coleman & Bradshaw Reference Huang, Coleman and Bradshaw1995; Trettel & Larsson Reference Trettel and Larsson2016; Griffin et al. Reference Griffin, Fu and Moin2021)
$y^* = y(\sqrt {\tau _w\overline \rho })/\overline \mu$
are depicted in figure 4, together with the second-order velocity-fluctuation statistics. The mean-velocity data from the present calculation exhibit a close collapse onto the incompressible scaling under the intrinsic-compressibility transformation of Hasan et al. (Reference Hasan, Larsson, Pirozzoli and Pecnik2023). Consistent with prior numerical simulations of calorically perfect (Zhang, Duan & Choudhari Reference Zhang, Duan and Choudhari2018; Cogo et al. Reference Cogo, Salvadore, Picano and Bernardini2022; Huang, Duan & Choudhari Reference Huang, Duan and Choudhari2022; Huang et al. Reference Huang, Duan and Choudhari2022) and weakly reactive (Duan & Martín Reference Duan and Martín2011; Di Renzo & Urzay Reference Di Renzo and Urzay2021; Passiatore et al. Reference Passiatore, Sciacovelli, Cinnella and Pascazio2022) high-speed turbulent boundary layers, both the turbulent Mach number,
$Ma_t=\sqrt {(\boldsymbol{u}\,\boldsymbol{\cdot}\,{\boldsymbol{u}})/3}/\overline {a}$
, and streamwise-normal component of the Reynolds-stress tensor attain their maxima just outside the peak-mean-temperature location and within the buffer layer at
$y^{*}\simeq 20$
, corresponding to a peak turbulent Mach number of
$Ma_t\simeq 0.7$
. The wall-normal and spanwise normal components of the Reynolds stresses, as well as the Reynolds shear, in contrast reach their maxima further from the wall at semi-local distances of the order of
$y^*\simeq 100$
. Viscous dissipation of kinetic energy in the boundary layer ultimately gives rise to an aerodynamic heating layer with a peak temperature
$\simeq$
5300 K at an off-wall location of
$y^*\simeq 10$
, activating significant dissociation/recombination phenomena and production of nitric oxide via the Zel’dovich mechanism. The presence of significant aerodynamic heating is apparent in both the density-weighted average temperature and Reynolds-averaged density fields, which are characterised in figure 5, with the peak-temperature location naturally corresponding to a global minimum for the mean density. The root-mean-squares (r.m.s.) of density fluctuations and density-weighted temperature fluctuations, as normalised by the local mean value, are likewise represented in figure 5, each realising a local minimum in the vicinity of the temperature peak itself. Within the peak-temperature location, whereas the normalised r.m.s. of density fluctuations increases monotonically with decreasing wall-normal displacement, the normalised r.m.s. of the density-weighted thermal fluctuations exhibits a local maximum in the viscous sublayer at
$y^*\simeq 4$
. The emergence of this near-wall thermal-variance peak is consistent with a turbulent-transport mechanism associated with the combined presence of strong near-wall temperature gradients and inviscid transport of temperature fluctuations away from the temperature peak (Fan, Li & Pirozzoli Reference Fan, Li and Pirozzoli2022; Cogo et al. Reference Cogo, Baù, Chinappi, Bernardini and Picano2023a
), although owing to the presence of chemical non-equilibrium, the balance of thermal variance is mediated not only by convective and conductive processes but also by chemical-heat-release and species-diffusion effects (Martin & Candler Reference Martin and Candler2001).
4.2. Species-production-rate and chemical-composition statistics
The high temperatures in the aerodynamic heating layer ultimately yield significant variations in the boundary layer’s chemical composition, depleting essentially all molecular oxygen within the peak-temperature location and producing measurable concentrations of nitric oxide, atomic oxygen and atomic nitrogen. Characterising the spatial evolution of the compositional statistics, the Favre-averaged molar fractions and their corresponding r.m.s. of density-weighted fluctuations are depicted in figures 6 and 7. While the concentrations of dissociation products are negligible at the edge of the boundary layer, the proportion of undissociated nitrogen and oxygen falls significantly in the aerodynamic heating layer, with
$\widetilde {X}_{N_2}\simeq 0.65$
and
$\widetilde {X}_{{O}_2}\simeq 0.02$
near the peak-temperature location. This almost-complete dissociation of oxygen produces a peak Favre-averaged molar fraction of
$0.27$
for atomic oxygen in the vicinity of the non-catalytic wall. Non-negligible densities of atomic nitrogen and nitric oxide also arise in the boundary layer, with maximum molar fractions of approximately 0.06 and 0.01, respectively. Whereas the Favre-averaged molecular-oxygen and atomic-oxygen molar fractions vary monotonically throughout the boundary layer, molecular and atomic nitrogen in contrast exhibit non-monotonicity with local extrema at
$y^*\simeq 10$
as a consequence of a sign change in their respective chemical production rates within the peak-temperature location. Mediated by both molecular dissociation and the Zel’dovich mechanism, the average nitric-oxide molar fraction reaches its maximum at an off-wall location embedded within the buffer layer of the turbulent boundary layer. Turbulent mixing of the molar fractions ultimately produces maxima in the r.m.s. of density-weighted molar-fraction fluctuations for all molecular species, as well as for atomic oxygen, at semi-local distances of the order of
$y^*\simeq 10^2$
due to transport of reaction products away from the aerodynamic heating layer. For atomic nitrogen, however, the location of maximum variance remains within the high-temperature buffer layer, where in conjunction with species turbulent mixing, the atomic-nitrogen fluctuations are influenced more significantly by fluctuations in the chemical production rate itself.
Figure 6. Favre average and r.m.s. of density-weighted fluctuations for the molar fractions of (a,b) molecular nitrogen, and (c, d) molecular oxygen, respectively.
Figure 7. Favre average and r.m.s. of density-weighted fluctuations for the molar fractions of (a,b) nitric oxide, (c, d) atomic nitrogen and (e, f) atomic oxygen, respectively.
As implied by the spatial evolution of the molar fractions, significant chemical production of atomic species and nitric oxide transpires in the vicinity of the temperature peak, proceeding on temporal scales comparable to that of eddy turnover, as reflected by the characteristic turbulent Damköhler numbers for the chemical production of each species listed in table 2. As depicted in figure 8, the average chemical production rates for all species exhibit local extrema near the peak-temperature location owing to their strong thermal dependence. In particular, the high temperatures in the vicinity of
$y^*\simeq 10$
give rise to significant net positive production of all radical species,
$\textrm{NO}$
,
$\textrm{N}$
and
$\textrm{O}$
, via the consumption of molecular nitrogen and oxygen. Whereas the chemical production of atomic oxygen remains uniformly positive throughout the boundary layer, the chemical production rates of atomic nitrogen and nitric oxide change sign within the temperature peak due to the strong near-wall spatial variation in thermodynamic conditions. The production of nitric oxide and atomic nitrogen becomes negative just within the viscous sublayer at
$y^*\simeq 5$
primarily as a result of the Zel’dovich shuffle reactions. This heightened near-wall activity for the shuffle reactions likewise results in the net production rate of molecular nitrogen becoming positive within the viscous sublayer. Finally, while the overall production of atomic oxygen remains positive within the temperature peak due to the Zel’dovich mechanism, the near-wall production of molecular oxygen ultimately becomes positive at sufficiently high Reynolds numbers due to the predominance of its recombination reaction.
4.3. Thermodynamic decomposition of turbulence/chemistry interaction
Owing to the nonlinearity of the chemical reaction rates with respect to the fluctuating thermodynamic variables, the Reynolds-averaged chemical production rates
\begin{equation} \overline {{{\dot {w}}}}_i=\int _{\mathbb{R}^{N_s+1}}{{\rm d}\mathcal{V}\ {\dot {w}}_i({\{\rho _j\},{T})\mathcal{P}}({\{\rho _j\},{T}})}, \end{equation}
may depart significantly from the mean-field approximation given by
\begin{equation} {\dot {w}}_i^{mf} = \int _{\mathbb{R}^{N_s+1}}{{\rm d}\mathcal{V}\ {\dot {w}}_i{(\{\rho _j\},{T})}\delta (T-\widetilde {T})\prod _{\kappa =1}^{N_s}\delta \left(\rho _\kappa -\overline {\rho }_\kappa \right)} = {\dot {w}}_i\left(\{\overline {\rho }_j\},{\widetilde {T}}\right)\!, \end{equation}
where
${\rm d}\mathcal{V} = {\rm d}T\prod _{\kappa =1}^{N_s}{\rm d}\rho _\kappa$
is the differential volume element in thermodynamic-state space, and
$\mathcal{P}(\boldsymbol{\cdot})$
is the single-point (joint) probability density function of the thermodynamic variables taken as arguments. As reflected in figure 9,
${\dot {w}}_i^{mf}$
differ significantly from
$\overline {{{\dot {w}}}}_i$
for all species apart from nitric oxide, evidencing the strong interactions between turbulent fluctuations and chemical reactions. In particular, the mean-field approximation considerably overpredicts the net formation rate of both molecular nitrogen and atomic oxygen outside the peak-temperature location. Whereas
${\dot {w}}_{N_2}^{mf}$
predicts positive production of
$\textrm{N}_2$
beyond
$y^*\simeq 20$
, the actual mean production remains negative, corresponding to net depletion of
$\textrm{N}_2$
throughout the entire boundary layer outside the viscous sublayer. Meanwhile, for atomic oxygen, the departures of
${\dot {w}}_{O}^{mf}$
from
$\overline {{{\dot {w}}}}_O$
indicate that turbulence-induced thermodynamic fluctuations serve to both reduce the peak production of atomic oxygen by approximately 30 % and shift the location of maximum reactivity inward from
$y^*\simeq 25$
to
$y^*\simeq 16$
. In the case of molecular oxygen and atomic nitrogen, in contrast, the mean-field approximation significantly overestimates the rate at which these species are depleted in the logarithmic layer of the boundary layer. In particular for atomic nitrogen, whereas the actual mean chemical production rate becomes essentially inactive beyond
$y^* \simeq 100$
, the mean-field approximation instead predicts considerable net depletion between
$y^*\simeq 15$
and
$y^*\simeq 1400$
.
Table 2. Turbulent Damköhler numbers,
$Da_{t,i} = \max {(|\overline {\dot {w}}_i/{\overline {\rho }}}|)\delta /{u_\tau }$
, characterising the relative rate of large-scale eddy turnover to chemical production for each species
$i = 1, 2, \ldots , N_s$
at
${Re}_{\delta _2} \simeq 2900$
.
Figure 8. Reynolds-averaged net chemical production for (a) molecular nitrogen, (b) molecular oxygen, (c) nitric oxide, (d) atomic nitrogen and (e) atomic oxygen.
Figure 9. Reynolds-averaged chemical production variables for (a) molecular nitrogen, (b) molecular oxygen, (c) nitric oxide, (d) atomic nitrogen and (e) atomic oxygen.
In order to assess the relative significance of specific thermodynamic fluctuations to this modulation of the mean chemical production rates, the auxiliary chemical reaction variables are required as
\begin{equation} \overline {\mathcal{W}}_i^{\rho _j} =\int _{\mathbb{R}^{N_s+1}}{{\rm d}\mathcal{V}\,{\dot {w}}_i({\{\rho _j\},{{T}})\delta \left(T-\widetilde {T}\right)\mathcal{P}}}(\{\rho _j\}), \end{equation}
and
\begin{equation} \overline {\mathcal{W}}_i^{\theta } = \int _{\mathbb{R}^{N_s+1}}{{\rm d}\mathcal{V}\,{\dot {w}}_i({\{\rho _j\},{T})}\mathcal{P}(T)\prod _{\kappa =1}^{N_s}\delta \left(\rho _\kappa -\overline {\rho }_\kappa \right)}, \end{equation}
which correspond to the expected chemical production rates absent partial-density and temperature fluctuations, respectively. Numerical evaluation of the integrals in (4.1), (4.3) and (4.4), as well as all other auxiliary production-rate or reaction-rate variables to be introduced in this manuscript, are evaluated with a Monte Carlo method, utilising approximately 70 000 temporal samples of the fluctuations at
$\hat {x}=915, {Re}_{\delta _2}\simeq 2900$
over the course of 6.2 eddy turnovers. On the basis of the significant departures of both
$\overline {\mathcal{W}}_i^{\theta }$
and
$\overline {\mathcal{W}}_i^{\rho _j}$
from the mean chemical production rates, as reflected in figure 9, it is apparent that both thermal and compositional fluctuations give rise to the turbulence–chemistry interaction observed in the boundary layer. Particularly for the oxygenic species, neglecting partial-density fluctuations, as in the estimate given by
$\overline {\mathcal{W}}_i^{\theta }$
, proves a worse approximation of the mean chemical production rates than
$\dot {w}_i^{mf}$
itself outside the peak-temperature location. In order to more precisely isolate the effects of specific thermodynamic-state fluctuations, the Reynolds-averaged chemical production rates are decomposed as
\begin{equation} \overline {{\dot {w}}}_i = {\dot {w}}_i^{mf} + \overline {\mathcal{I}}_{{\mathcal{W}}_i}^{\rho _j} + \overline {\mathcal{I}}_{{\mathcal{W}}_i}^{\kern0.5pt\theta } + \overline {\mathcal{I}}_{{\mathcal{W}}_i}^{\rho _j,\theta } ,\end{equation}
where
$\overline {\mathcal{I}}_{{\mathcal{W}}_i}^{\rho _j}$
,
$\overline {\mathcal{I}}_{{\mathcal{W}}_i}^{\kern0.5pt\theta }$
correspond to the distortion of mean chemical production rates arising solely from partial-density and thermal fluctuations, respectively. In contrast, the final term in the decomposition,
$\overline {\mathcal{I}}_{{\mathcal{W}}_i}^{\rho _j,\theta }$
, represents the net effect of the statistical co-moments of temperature and partial-density fluctuations on the mean chemical production rates. Utilising the auxiliary production variables introduced above, the first two terms in the decomposition can be directly evaluated as
\begin{align}& \overline {\mathcal{I}}_{{\mathcal{W}}_i}^{\rho _j} =\overline {\mathcal{W}}^{\rho _j}-\dot {w}_i^{mf}, \end{align}
\begin{align}& \overline {\mathcal{I}}_{{\mathcal{W}}_i}^{\kern0.5pt\theta } = \overline {\mathcal{W}}_i^{\theta }-\dot {w}_i^{mf}, \end{align}
Figure 10. Reynolds-averaged thermodynamic decomposition of turbulence–chemistry interaction for net chemical production rates, corresponding to (a) molecular nitrogen, (b) molecular oxygen, (c) nitric oxide, (d) atomic nitrogen and (e) atomic oxygen.
while the final interaction term immediately follows as
\begin{equation} \overline {\mathcal{I}}_{{\mathcal{W}}_i}^{\rho _j,\theta } = \overline {\dot {w}}_i+\dot {w}_i^{mf}-\overline {\mathcal{W}}_i^{\theta }-\overline {\mathcal{W}}_i^{\rho _j}, \end{equation}
where the summation of (4.6) through (4.8) recovers by construction the overall effect of turbulence–chemistry interaction on the production rates,
$\overline {\dot {w}}_i-\dot {w}_i^{mf}$
. As reflected in figure 10, all contributions to the turbulence–chemistry interaction decomposition generally prove consequential for the net chemical production rate of each species, with the maximum absolute distortion of the production rates attained in the vicinity of
$y^*\simeq 40$
for all species apart from nitric oxide, with its corresponding production rate most increased by temperature fluctuations in the viscous sublayer. Indeed, while the contribution of temperature fluctuations alone to turbulence–chemistry interaction, as represented by
$\overline {\mathcal{I}}_{{\mathcal{W}}_i}^{\kern0.5pt\theta }$
, accounts for much of the overall distortion of the production rates for all species in the near-wall region bounded by the peak-temperature location, the effects of species mixing becomes significant outside
$y^*\simeq 10$
. For the chemical production of molecular and atomic oxygen, in particular, the impact of partial-density fluctuations combines constructively with the joint effect of partial-density/temperature fluctuations to produce significant turbulence–chemistry interactions outside the temperature peak, decreasing the consumption rate of the molecular species and correspondingly inhibiting the production of atomic oxygen. Likewise, for atomic nitrogen, the separate effects of temperature and partial-density fluctuations, both serve to increase the chemical production rate beyond
$y^* \simeq 10$
, giving rise to a global maximum in the turbulence–chemistry interaction effect at the edge of the buffer layer and essentially eliminating consumption of atomic nitrogen outside the temperature peak. Correspondingly, the positive production of molecular nitrogen is largely suppressed by turbulence–chemistry interaction in the buffer and logarithmic layers owing primarily to temperature fluctuations. Finally, for the production of nitric oxide outside the buffer layer, the effect of partial-density fluctuations alone proves more consequential than the thermal fluctuations with respect to the mean distortion of the chemical production rate. As implied by the relatively close alignment of
$\dot {w}_{\textit{NO}}^{mf}$
and
$\overline {\dot {w}}_{\textit{NO}}$
in figure 9(c), however, the modulation of the nitric-oxide production rate by partial-density fluctuations alone is cancelled by the combined effect of thermal fluctuations and non-zero temperature/partial-density co-moments. As such, the modulation of the net nitric-oxide production rate proves most significant within peak-temperature location, where near-wall thermal fluctuations serve to amplify its net production.
4.4. Modulation of chemical reaction rates by thermodynamic fluctuations
To characterise the effect of thermodynamic fluctuations on specific reactions, an analogous decomposition to (4.5) can be invoked for the chemical reaction rates themselves, namely
\begin{equation} \overline {\mathcal{{R}}}_i = \mathcal{R}_i^{mf} + \overline {\mathcal{I}}_{\mathcal{R}_i}^{\rho _j} + \overline {\mathcal{I}}_{\mathcal{R}_i}^{\theta } + \overline {\mathcal{I}}_{\mathcal{R}_i}^{\rho _j,\theta } ,\end{equation}
where
$\overline {\mathcal{{R}}}_i$
denotes the Reynolds-averaged reaction rate and
$\mathcal{R}_i^{mf} = \mathcal{R}_i(\{\overline {\rho }_j\},\widetilde {T})$
corresponds to its mean-field approximation based on the averaged thermodynamic fields. The corresponding expressions for the turbulence–chemistry interaction terms in (4.9), as well as auxiliary reaction-rate variables, can be evaluated precisely as in (4.3) through (4.8), merely exchanging
$\mathcal{W}_i$
with
$\mathcal{R}_i$
. The averaged auxiliary reaction-rate variables are plotted as a function of the semi-local wall-normal coordinate in figure 11, confirming that the mean-field approximation of each reaction rate departs significantly from the Reynolds average itself, with the exception of the dissociation/recombination of molecular oxygen, which exhibits only modest net turbulence/chemistry interaction. The location of maximum distortion of the mean reaction rates by turbulent fluctuations naturally depends on the parameters of the reaction itself: whereas the shuffle reactions in the Zel’dovich mechanism, characterised by relatively lower activation energies, are primarily affected in the buffer and logarithmic layers, the dissociation/recombination processes naturally exhibit the most significant net turbulence–chemistry interaction near the peak-temperature location. In particular, for the Zel’dovich mechanism, the thermodynamic fluctuations serve to reduce the predominance of the reverse reaction, most significantly in the buffer layer. In light of the relative magnitudes of the turbulence–chemistry interaction terms for each reaction, it becomes apparent that the modulation of the net chemical production rates by turbulent fluctuations outside the peak-temperature location can largely be attributed to turbulence–chemistry interactions specific to the Zel’dovich shuffle reactions and not the dissociation/recombination processes. Instead, the mean dissociation rate of all molecular species is naturally maximised in the vicinity of the peak-temperature location, approximately aligned with the location of maximum turbulence–chemistry interaction for the dissociation of
$\textrm{N}_2$
and
$\textrm{O}_2$
. Due to wall-cooling effects, a near-wall region characterised by net recombination of the atomic constituents ultimately emerges, the extent of which is contracted due to fluctuations. Whereas the mean-field approximation implies a recombination layer extending from the wall to
$y^*\simeq 4$
for each molecular species, the actual recombination region based on the Reynolds-averaged reaction rates extends only to
$y^*\simeq 3$
. The close alignment between
$\overline {\mathcal{R}}_i^\theta$
and
$\overline {\mathcal{R}}_i$
within the temperature peak confirms that the introduction of turbulence-induced thermal fluctuations alone accounts for the expansion of the dissociation layer centred at the temperature peak, thereby confining the recombination region closer to the non-catalytic surface.
Figure 11. Reynolds-averaged chemical reaction-rate variables for (a,b) Zel’dovich exchange reactions, and dissociation/recombination of (c) molecular oxygen, (d) nitric oxide and (e) molecular nitrogen.
Figure 12. Reynolds-averaged thermodynamic decomposition of turbulence–chemistry interaction for chemical reaction rates, corresponding to (a,b) Zel’dovich exchange reactions, and dissociation/recombination of (c) molecular oxygen, (d) nitric oxide and (e) molecular nitrogen.
As observed in the statistics of the auxiliary chemical production variables, the chemical reaction rates for both the Zel’dovich mechanism and the molecular dissociation/recombination processes are most significantly modulated by temperature fluctuations alone inside the peak-temperature location, although
$\overline {I}_{\mathcal{R}_i}^\theta$
necessarily vanishes at the wall itself due to isothermal wall condition. As reflected in figure 12, the near-wall temperature fluctuations increase the forward bias of each reaction, giving rise to local maxima in the turbulence–chemistry interactions between the edge of the viscous sublayer and the peak-temperature location. In particular, the turbulence–chemistry interaction observed for the first Zel’dovich reaction qualitatively parallels the turbulence–chemistry interaction observed in the chemical production of its reactant
$\textrm{N}_2$
, in that the distortion of the mean reaction rate is characterised by two local maxima adjacent to the temperature peak. While the inner maxima in the rate of (R1) is largely due to temperature fluctuations alone, the outer peak in turbulence–chemistry interaction is influenced more significantly by partial-density and joint partial-density/temperature fluctuations, increasing the generation rate of
$\textrm{NO}$
and
$\textrm{N}$
radicals. In contrast, the second Zel’dovich shuffle reaction proves far less sensitive to thermal fluctuations, although temperature fluctuations account for most of the forward biasing of the reaction rate within the peak-temperature location. Instead, the significant increase in the net forward rate of the reaction outside
$y^*\simeq 10$
arises primarily due to partial-density fluctuations alone, ultimately contributing to heightened production rates for molecular oxygen and atomic nitrogen as observed in figure 9. For the dissociation/recombination of molecular oxygen, the global maximum in turbulence/chemistry interaction is located just within the peak-temperature location, corresponding to amplified dissociation due almost entirely to temperature fluctuations. For semi-local off-wall displacements exceeding
$y^* \simeq 20$
, however, the overall distortion of the mean
${\rm O}_2$
dissociation/recombination rate proves quite limited. While turbulent fluctuations marginally decrease the effective dissociation rate between
$y^* \simeq 20$
and
$y^*\simeq 100$
, temperature fluctuations alone give rise to an incremental augmentation in the net dissociation rate in the logarithmic layer of the boundary layer. This relatively negligible net turbulence–chemistry interaction, however, represents an approximate balance between the far more significant effects of temperature and joint partial-density/temperature fluctuations. That is, while the transport of temperature fluctuations alone from the peak-temperature location to the buffer and logarithmic layers serves to increase local dissociation rate significantly, the simultaneous transport of the pre-dissociated mixture from the inner part of the aerodynamic heating layer correspondingly displaces the necessary undissociated reactants, such that
$\overline {\mathcal{I}}_{\mathcal{R}_i}^\theta \approx -\overline {\mathcal{I}}_{\mathcal{R}_i}^{\rho _j,\theta }$
. For molecular nitrogen and nitric oxide dissociation/recombination, in contrast, the most significant contribution to the net turbulence–chemistry interaction remains
$\overline {I}_{\mathcal{R}_i}^\theta$
throughout the boundary layer, owing to their relatively higher activation temperatures. To a far lesser extent, and localised near the peak-temperature location, the joint effect of temperature/partial-density fluctuations, as measured by
$\overline {I}_{\mathcal{R}_i}^{\rho _j,\theta }$
, does modestly alter the mean reaction rates for
$\textrm{N}_2$
and
$\textrm{NO}$
. Deferred to Appendix D for the sake of brevity, the impact of partial-density fluctuations on the chemical reaction rates can be further decomposed in terms of each species, confirming that joint temperature/reactant-partial-density fluctuations, as well as temperature/radical-partial-density fluctuations, are the most significant contributions to
$\overline {\mathcal{I}}_{\mathcal{R}_i}^{\rho _j,\theta }$
for the dissociation/recombination processes. In the particular case of molecular oxygen dissociation/recombination, joint
$\rho _{{O}_2}$
/temperature fluctuations almost fully comprise contributions to the overall turbulence–chemistry interaction captured by
$\overline {I}_{\mathcal{R}_i}^{\rho _j,\theta }$
.
4.5. Impact of density fluctuations on turbulence/aerothermochemistry interactions
To further separate the effect of overall density variation from compositional fluctuations on the mean reaction rates, the set of variables defining the thermodynamic state is now chosen to be
$\rho ,Y_{1},Y_{2}, \ldots , Y_{N_s-1},T$
. With this alternative choice of thermodynamic variables, the analogous decomposition to (4.9) is given by
\begin{equation} \overline {\mathcal{{R}}}_i = \mathcal{R}_i^{mf} + \overline {\mathcal{I}}_{\mathcal{R}_i}^{\rho } + \overline {\mathcal{I}}_{\mathcal{R}_i}^{Y_j} + \overline {\mathcal{I}}_{\mathcal{R}_i}^{\theta } + \overline {\mathcal{I}}_{\mathcal{R}_i}^{\rho ,Y_j} + \overline {\mathcal{I}}_{\mathcal{R}_i}^{\rho ,\theta } + \overline {\mathcal{I}}_{\mathcal{R}_i}^{Y_j,\theta } + \overline {\mathcal{I}}_{\mathcal{R}_i}^{\rho ,Y_j,\theta } ,\end{equation}
where
$\overline {\mathcal{I}}_{{\mathcal{R}}_i}^{\rho }$
and
$\overline {\mathcal{I}}_{{\mathcal{R}}_i}^{Y_j}$
represent the mean effect of density and mass-fraction fluctuations on the
$i$
th reaction rate, respectively. In this formulation, the mean distortion of the chemical reaction rates due intrinsically to joint fluctuations in the thermodynamic state are given by
$\overline {\mathcal{I}}_{\mathcal{R}_i}^{\rho ,Y_j}$
,
$\overline {\mathcal{I}}_{\mathcal{R}_i}^{\rho ,\theta }$
,
$\overline {\mathcal{I}}_{\mathcal{R}_i}^{Y_j,\theta }$
and
$\overline {\mathcal{I}}_{\mathcal{R}_i}^{\rho ,Y_j,\theta }$
, with the comma-separated variables appearing in the superscript denoting the relevant fluctuating variables for each term in the decomposition. The effects of density and composition fluctuations on the mean reaction rates are quantified as
\begin{equation} \overline {\mathcal{I}}_{{\mathcal{R}}_i}^{\rho } = \int _{\mathbb{R}^{N_s+1}}{{\rm d}\mathcal{Y}\,{\mathcal{R}}_i({\{\rho {Y}_j\},{{T}})\mathcal{P}}({\rho })}\prod _{\substack {\kappa =1}}^{N_s-1}\delta (Y_\kappa -\widetilde {Y}_\kappa )\delta (T-\widetilde {T}) - \mathcal{R}_i^{mf}, \end{equation}
and
\begin{equation} \overline {\mathcal{I}}_{{\mathcal{R}}_i}^{Y_j} = \int _{\mathbb{R}^{N_s+1}}{{\rm d}\mathcal{Y}\,{\mathcal{R}}_i({\{\rho {Y}_j\},{{T}})\delta (\rho -\overline {\rho })\mathcal{P}}(\{Y_j\})}\delta ({T-\widetilde {T}}) - \mathcal{R}_i^{mf}, \end{equation}
respectively, where
${\rm d}\mathcal{Y} = {{\rm d}\rho }{{\rm d}T}\prod _{\kappa =1}^{N_s-1}{\rm d}Y_\kappa$
is the differential volume element for the given set of thermodynamic variables. The joint interaction terms involving two thermodynamic quantities can likewise be evaluated as
\begin{align}&\,\,\,\, \overline {\mathcal{I}}_{{\mathcal{R}}_i}^{\rho ,Y_j} = \int _{\mathbb{R}^{N_s+1}}{{\rm d}\mathcal{Y}\,{\mathcal{R}}_i({\{{\rho }{Y}_j\},{{T}})\mathcal{P}}(\rho ,\{Y_j\})}\delta (T-\widetilde {T}) - \overline {\mathcal{I}}^\rho _{\mathcal{R}_i}-\overline {\mathcal{I}}^{Y_j}_{\mathcal{R}_i}- \mathcal{R}_i^{mf}, \end{align}
\begin{align}& \overline {\mathcal{I}}_{{\mathcal{R}}_i}^{\rho ,\theta } = \int _{\mathbb{R}^{N_s+1}}{{\rm d}\mathcal{Y}\,{\mathcal{R}}_i({\{\rho {Y}_j\},{{T}})\mathcal{P}}(\rho ,T)}\prod _{\substack {\kappa =1}}^{N_s-1}\delta (Y_\kappa -\widetilde {Y}_\kappa ) - \overline {\mathcal{I}}^\rho _{\mathcal{R}_i}-\overline {\mathcal{I}}^{\theta }_{\mathcal{R}_i}- \mathcal{R}_i^{mf}, \end{align}
\begin{align}&\,\,\,\, \overline {\mathcal{I}}_{{\mathcal{R}}_i}^{Y_j,\theta } = \int _{\mathbb{R}^{N_s+1}}{{\rm d}\mathcal{Y}\,{\mathcal{R}}_i({\{\rho {Y}_j\},{{T}})\delta (\rho -\overline {\rho })\mathcal{P}}(\{Y_j\},T)} -\overline {\mathcal{I}}^{Y_j}_{\mathcal{R}_i} - \overline {\mathcal{I}}^{\kern0.5pt\theta}_{\mathcal{R}_i}- \mathcal{R}_i^{mf}, \end{align}
Figure 13. Reynolds-averaged decomposition, with the thermodynamic state as defined by density, temperature and composition, for the impact of turbulence–chemistry interaction on chemical reaction rates. The panels correspond to (a,b) the Zel’dovich exchange reactions, and dissociation/recombination of (c) molecular oxygen, (d) nitric oxide and (e) molecular nitrogen, respectively.
while the contribution to turbulence/chemistry interaction associated with fluctuations in all variables, i.e.
$\overline {\mathcal{I}}_{\mathcal{R}_i}^{\rho ,Y_j,\theta }$
, then follows immediately from (4.10). Each term in this more fine-grained decomposition is represented as a function of the semi-local coordinate in figure 13, disambiguating the relative impact of density and compositional variations. Direct effects from density fluctuations alone, as measured by
$\overline {\mathcal{I}}_{{\mathcal{R}}_i}^{\rho }$
, are found to be relatively unimportant, apart from the recombination layer where
$\overline {\mathcal{I}}_{\mathcal{R}_i}^\rho$
accounts for additional reverse-reaction activity, representing fractionally significant contributions to the overall turbulence–chemistry interaction for the first Zel’dovich shuffle reaction and oxygen recombination in this very near-wall region. Apart from this relative importance in the recombination layer where the net turbulence–chemistry interaction itself remains quite modest, however, the direct effect of density fluctuations proves largely negligible for all reactions in the buffer and logarithmic layers. In contrast, the direct effect of compositional fluctuations as measured by
$\overline {\mathcal{I}}^{Y_j}_{\mathcal{R}_i}$
represents a major contribution to the net turbulence–chemistry interaction for both shuffle reactions, and to a lesser extent, the dissociation of molecular oxygen. In particular, for the second Zel’dovich reaction, outside the peak-temperature location, the compositional fluctuations alone are primarily responsible for the overall distortion of the mean reaction rates. Hence the significant reduction in the net backwards-direction reactivity previously associated with the partial-density fluctuations can be more precisely attributed to mass-fraction fluctuations. For (R1) and (R3), compositional fluctuations favour the backward reaction rates outside the peak-temperature location, although separately attaining their maximum absolute values in the logarithmic and buffer layers, respectively.
In the case of the first Zel’dovich reaction, all of the two-variable joint interaction terms between the thermodynamic variables prove significant, with the joint effects of compositional/temperature and composition/density fluctuations effectively increasing the net positivity of the reaction rate, while the net impact from joint density/temperature fluctuations is to correspondingly increase the backward reaction activity. For oxygen dissociation, however, the effect of joint fluctuations in composition and temperature emerges as the most significant, effectively negating the impact of temperature fluctuations. Hence, the effective suppression of the forward reaction rate by
$\overline {\mathcal{I}}_{\mathcal{R}_{3}}^{\rho _j,\theta }$
observed previously can be largely attributed to the coupling of mass-fraction/temperature fluctuations consistent with the simultaneous transport of high-temperature fluid, depleted of molecular oxygen, outwards from the aerodynamic heating layer.
4.6. Intrinsic-compressibility effects on finite-rate chemical processes
The density variations present in the boundary layer arise from a combination of pressure, entropic and compositional perturbations (Kovasznay Reference Kovasznay1953; Griffond Reference Griffond2005; Livescu Reference Livescu2020). With Morkovin’s hypothesis relying on the relative insignificance of intrinsic-compressibility effects and hence pressure fluctuations (Morkovin Reference Morkovin1962; Lele Reference Lele1994), its validity for mean reaction-rate statistics can be assessed in part by evaluating the departure of the isobaric reaction rates
\begin{equation} {\overline {\mathcal{R}}_i^{{p}}} = \int _{\mathbb{R}^{N_s+1}}{{\rm d}\mathcal{Y}\,{\mathcal{R}}_i({\{\rho {Y}_j\},{{T}})\delta \left (\rho -\frac {\overline {P}\mathcal{M}}{{R}^0 T}\right )\mathcal{P}}(\{Y_j\},T)}, \end{equation}
for which the density fluctuations are modelled as entirely entropic, from the corresponding Reynolds-averaged counterparts. Alternatively, in approximating the density fluctuations as arising from isentropic pressure waves (Ristorcelli & Blaisdell Reference Ristorcelli and Blaisdell1997; Griffond Reference Griffond2005; Donzis & Jagannathan Reference Donzis and Jagannathan2013), propagative or otherwise, the implied isentropic reaction rates are given by
\begin{equation} {\overline {\mathcal{R}}_i^{{s}}} = \int _{\mathbb{R}^{N_s+1}}{{\rm d}\mathcal{Y}\,{\mathcal{R}}_i({\{\rho {Y}_j\},{{T}})\delta \left (\rho -\frac {\overline {P}\mathcal{M}}{{R}^0 T}\left (1+\frac {(\gamma -1)T'}{\gamma \overline {T}}\right )\right )\mathcal{P}}(\{Y_j\},T)}. \end{equation}
Finally, to characterise the impact of compositional fluctuations, while retaining both entropic and pressure disturbances, the mean isoconcentration reaction rates are defined as
\begin{equation} {\overline {\mathcal{R}}_i^{{c}}} = \int _{\mathbb{R}^{2}}{{\rm d}P{\rm d}T\,{\mathcal{R}}_i\left ({{\frac {{P}\mathcal{M}|_{\overline {Y}_j}}{{R}^0 T}}}{\{\overline {Y}}_j\},{{T}}\right )\mathcal{P}}(P,T), \end{equation}
Figure 14. Reynolds-averaged chemical reaction rates juxtaposed with the mean auxiliary reaction-rate variables neglecting pressure, entropic and compositional fluctuations, for the (a,b) Zel’dovich mechanism, and dissociation/recombination of (c) molecular oxygen, (d) nitric oxide and (e) molecular nitrogen.
where
$\mathcal{M}|_{\overline {Y}_j}$
denotes the mixture’s molar mass based on the Reynolds-averaged mass fractions. Given the presence of near-wall viscous transport and chemical reactions, the entropic, compositional and pressure waves, which evolve independently in the small-disturbance limit, couple in the turbulent boundary layer (Livescu Reference Livescu2020). Nevertheless, as reflected in figure 14, the isobaric approximation for the turbulent reaction rates exhibits close agreement with the corresponding Reynolds averages, particularly with respect to the Zel’dovich reactions. While pressure fluctuations in the buffer layer give rise to slight discrepancies between
$\overline {\mathcal{R}}^p_i$
and
$\overline {\mathcal{R}}_i$
near
$y^* \simeq 10$
, this difference effectively vanishes in both the viscous sublayer and logarithmic region. In contrast, the isentropic approximation, in implicitly correlating temperature fluctuations with compression of the flow, give rises to significant overprediction of the net dissociation rates, in a region extending from the viscous sublayer through the logarithmic layer. The isoconcentration reaction-rate variables likewise depart significantly from the Reynolds averages outside the viscous sublayer, for all reactions apart from nitrogen and nitric-oxide dissociation. Following from the law of mass action, the isobaric, isentropic and isoconcentration chemical production rates are evaluated in terms of the auxiliary reaction-rate variables as
$\overline {\mathcal{W}}_i^{m} = \mathcal{M}_i\sum _{j=R_1}^{R_5}(\nu _{\textit{ij}}^{\prime \prime }-{\nu _{\textit{ij}}^\prime })\overline {\mathcal{R}}_j^{m}$
, where
$m$
represents the given mode neglected, i.e.
$m \in p, s, c$
. In terms of the net production rates of each species and owing to the behaviour observed in the individual chemical reactions, the pressure perturbations generally prove far less significant than the counterpart fluctuations in entropy and composition throughout virtually the entire boundary layer, as shown in figure 15. While this comparison of the Reynolds-averaged and auxiliary chemical-production variables implicitly neglects the impact of pressure fluctuations on establishing the turbulent base flow and corresponding thermodynamic fluctuations themselves, it nevertheless provides strong evidence for a consistent extension of Morkovin’s hypothesis, namely, that turbulence–chemistry interaction in high-speed boundary layers arises primarily from a combination of entropic and compositional fluctuations, with pressure perturbations effecting only marginal impact within the buffer layer.
Figure 15. Reynolds-averaged chemical production rates juxtaposed with the mean auxiliary production-rate variables neglecting pressure, entropic and compositional fluctuations, for (a) molecular nitrogen, (b) molecular oxygen, (c) nitric oxide, (d) atomic nitrogen and (e) atomic oxygen.
5. Conclusions
In order to characterise turbulence–aerothermochemistry interactions in high-Mach external flows, this manuscript presents numerical simulations and statistical analyses of a turbulent hypersonic boundary layer, overriding a non-catalytic surface at 3000 K. With the flow remaining in chemical non-equilibrium, numerical integration of the compressible, reacting Navier–Stokes equations is performed to enable a finite-rate evolution of the composition fields corresponding to
$\textrm{N}_2$
,
$\textrm{O}_2$
,
$\textrm{NO}$
,
$\textrm{N}$
and
$\textrm{O}$
. An auxiliary calculation, comprising Mach-25 flow over a 16-degree sharp-nosed wedge yields an inflow laminar boundary layer with an edge Mach number of 7 and an edge temperature of 2686 K. This laminar base flow is forced to transition in the primary computational domain with spatially localised transpiration comprised of two oscillatory modes. Following breakdown to turbulence, the boundary layer ultimately attains a friction Reynolds and Mach numbers of
${Re}_\tau \simeq 1200$
and
$Ma_\tau \simeq 0.20$
, respectively, with the turbulence integral time scales approximately equal to the characteristic chemical time scales. Chemical activity within the boundary layer, comprising dissociation/recombination reactions for each of the molecular species together with the Zel’dovich mechanism, gives rise to significant compositional variation throughout the boundary layer. Activated by a peak Favre-averaged temperature of approximately 5300 K, the chemical reactions produce considerable concentrations of radical species in the near-wall region. In particular, with the molecular oxygen almost entirely depleted within the peak-temperature location of
$y^*\simeq 10$
, the mean near-wall molar fraction of atomic oxygen reaches 0.27, whereas nitric oxide and atomic nitrogen attain their maximum concentrations in the buffer layer, with peak molar fractions of 0.06 and 0.01, respectively.
As a consequence of the near-unity turbulent Damköhler numbers, the chemical production rates within the aerodynamic heating layer are strongly modulated by turbulence-induced thermodynamic fluctuations, primarily through the exchange processes of the Zel’dovich reactions. In particular, the Reynolds-averaged production rates are shown to differ significantly from their respective mean-field approximations, for all species apart from nitric oxide. While the net depletion rates of both molecular oxygen and atomic nitrogen are severely overpredicted in the buffer and logarithmic layers based on the mean thermodynamic-state variables, the production rates of molecular nitrogen and atomic oxygen are correspondingly overestimated. In order to further characterise this turbulence–chemistry interaction, a computational approach for decomposing the impact of turbulence-induced thermodynamic fluctuations on chemical reactions and production rates is introduced. In general, both thermal and partial-density fluctuations, as well as the impact of their statistical co-moments, are shown to contribute significantly to the net chemical production rate of each species, with the maximum absolute distortion of the production rates attained outside the buffer layer. Within the near-wall region bounded by the peak-temperature location, however, the statistical decomposition reveals that temperature fluctuations alone account for much of the overall distortion of the production rates, with the effects of partial-density fluctuations generally becoming more significant outside
$y^*\simeq 10$
. Further aspects pertaining to the impact of turbulence-chemistry interaction on mean-flow predictions with reduced-order near-wall models are provided in Cogo et al. (Reference Cogo, Williams, Griffin, Picano and Moin2023b
).
In terms of specific reactions, a comparison of the mean-field approximations and Reynolds averages reveals that while the shuffle reactions are primarily affected by thermodynamic fluctuations in the buffer and logarithmic layers, the dissociation/recombination processes in contrast exhibit the most significant distortion in the vicinity of the peak-temperature location, owing to their relatively higher activation energies. Moreover, inside the peak-temperature location, the chemical reaction rates for both the Zel’dovich mechanism and the molecular dissociation/recombination processes are most significantly modulated by temperature fluctuations alone. For dissociation/recombination processes in particular, these near-wall thermal fluctuations manifest in an augmentation of the molecular dissociation rate, thereby further confining the off-wall extent of the recombination layer. Compositional fluctuations are likewise shown to contribute significantly to the net turbulence–chemistry interaction for both shuffle reactions, and to a lesser extent, the dissociation of molecular oxygen. In the case of the second Zel’dovich reaction, the statistical decomposition introduced herein reveals that compositional fluctuations alone are primarily responsible for the overall distortion of its mean reaction rate and hence chemical production rates for all major species apart from molecular nitrogen. To isolate the impact of pressure perturbations on the mean chemical reaction rates, relative to the contributions from compositional and entropic modes, a final set of auxiliary reaction-rate variables is introduced, providing evidence for a consistent extension of Morkovin’s hypothesis, that turbulence–chemistry interaction in high-Mach boundary layers primarily arises from a combination of entropic and compositional fluctuations, with pressure perturbations introducing only marginal reaction-rate distortions in the buffer layer.
Funding
This investigation was funded by the Advanced Simulation and Computing (ASC) program of the US Department of Energy’s National Nuclear Security Administration (NNSA) via the PSAAP-III Center at Stanford, Grant No. DE-NA0002373. C.W. acknowledges support by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-2146755.
Declaration of interests
The authors report no conflict of interest.
Appendix A. Incoming laminar hypersonic boundary layer
The laminar hypersonic boundary layer imposed at the inflow boundary of the Cartesian computational domain is extracted from an auxiliary calculation of the Mach-25 flow field around a 16-degree semi-angle hypersonic wedge, with a leading-edge radius of 0.2 mm. The free-stream pressure and temperature for the wedge simulation were taken to be 2549 Pa and 221 K, respectively, consistent with atmospheric conditions for hypersonic flight in the stratosphere at 25 km. This precursor simulation entailed direct integration of the conservation equations as formulated in Section 3 utilising a single-block curvilinear grid topology. Contours of the local Mach number from the numerical solution are presented in figure 16, in which the shock-induced heating and deceleration of the flow produces a shock layer with a nominal Mach number of 7 and pressure of 203.2 kPa. The laminar boundary-layer profiles are extracted approximately at 117
$\delta _0^*$
downstream of the leading edge, corresponding to a displacement-thickness Reynolds number of 6000. The temperature, molar fractions, and streamwise-velocity profiles for this boundary layer are provided in figure 17, for which the boundary-layer edge conditions have been artificially extended to exclude the oblique shock wave from the Cartesian turbulent boundary-layer simulation. These regularised boundary-layer profiles are then applied as the inflow boundary conditions for the turbulent boundary-layer calculation together with the shock-layer pressure.
Figure 16. Mach-number contours from the numerical simulation of reacting hypersonic flow overriding a 16-degree wedge at an altitude of 25 km, from which the laminar boundary layer is extracted to provide the inflow boundary conditions.
Figure 17. Laminar-boundary-layer profiles from reacting hypersonic wedge calculation utilised as inflow boundary conditions, corresponding to (a) temperature and streamwise velocity, (b) molecular-nitrogen molar fraction, (c) molecular-oxygen molar fraction and (d) molar fractions of radical species.
Appendix B. Confirmation of turbulent state
As depicted in figure 18(a),the breakdown to turbulence in the boundary layer is evidenced by a rapid increase in and subsequent relaxation of the skin-friction and heat-flux coefficients near
$\hat {x} \simeq 600$
. Despite the addition of chemical reactivity and a relative increase in the edge Mach number, the present numerical simulation’s skin-friction coefficient collapses onto the correlation of Ceci et al. (Reference Ceci, Palumbo, Larsson and Pirozzoli2022) to within
$\simeq 5$
%, as reflected in figure 18(b). The normalised power spectral density of velocity and concentration fluctuations in the fully turbulent regime at
$\hat {x} = 915$
are provided in figure 19, which confirms that dominance of the forcing wavenumber does not persist through transition, and instead, the fluctuations exhibit the expected multi-scale behaviour. Likewise, no pileup in spectral density is observed at high wavenumbers, which in conjunction with the grid-sensitivity study presented in the following appendix, verifies the sufficiency of the numerical resolution. Finally, the corresponding two-point autocorrelation functions in figure 20 confirm that the velocity and molar-fraction fluctuations exhibit spatial decorrelation within separation distances less than 20 % of the spanwise extent.
Figure 18. (a) Spatial evolution of the skin-friction and heat-flux coefficients through transition; inset depicts the same data localised near the forcing strip, denoted by the shaded region, (b) collapse of the skin-friction coefficient onto the correlation of Ceci et al. (Reference Ceci, Palumbo, Larsson and Pirozzoli2022), from which we utilise
$\mathcal{A}_f = 0.0131$
and
$\mathcal{B}_f = 0.268$
. The dashed lines demarcate a
$\pm \, 5$
% interval relative to the nominal scaling.
Figure 19. Normalised spanwise spectra for (a) streamwise velocity, (b) wall-normal velocity, (c)
$X_{\textit{NO}}$
and (d)
$X_O$
fluctuations at
${Re}_{\delta _2} \simeq 2900$
. Normalisation of the power spectral density is performed with the semi-local length scale and the total variance given by
$\int {{\rm d}k_z^*}\mathcal{E}_\phi (k_z^*)$
. The range of
$k_z^*$
associated with the numerical tripping in the laminar boundary layer is indicated by the dashed vertical lines.
Figure 20. Two-point autocorrelation functions for (a) streamwise velocity, (b) wall-normal velocity, (c)
$X_{\textit{NO}}$
and (d)
$X_O$
fluctuations as a function of the separation distance,
$z-z'$
, normalised by the total spanwise domain extent
$l_z$
.
Appendix C. Verification of grid convergence
In order to confirm the insensitivity of the relevant direct numerical simulation statistics to grid refinement, a secondary calculation of the Mach-7 reacting hypersonic boundary layer has been performed, discretised with approximately 1.5 times fewer points in each direction, yielding a total of 7920, 310 and 788 points across the streamwise, spanwise and wall-normal directions, respectively. In order to maintain the first off-wall grid point’s location under coarsening, the hyperbolic-sine stretching factor in the wall-normal direction has been increased from 5.0 to 5.5 for the grid-convergence study. The transformed mean-flow streamwise velocity and Reynolds-stress components, as well as the Reynolds-averaged partial densities and chemical production rates, are provided in figure 21 for the coarse calculation. The very close alignment of its statistics with those of the fine grid confirms the grid insensitivity of the numerical solution results analysed in § 2 of this manuscript.
Figure 21. Mean wall-normal profiles of (a) transformed streamwise velocity, (b) normal Reynolds-stress components, (c) species partial densities and (d) chemical production rates. Solid lines correspond to the fine-grid numerical solution included in § 2, whereas the dash-dotted lines denote the coarse-grid results.
Appendix D. Species-specific decomposition of reaction-rate modulation by partial-density fluctuations
In order to isolate species-specific effects, further decomposition of the turbulence–chemistry interaction terms involving partial-density fluctuations, i.e.
$\overline {\mathcal{I}}_{\mathcal{R}_i}^{\rho _j}\textrm{and}\,\overline {\mathcal{I}}_{\mathcal{R}_i}^{\rho _j,\theta },$
yields a finer-grained decomposition
\begin{equation} \overline {{\mathcal{R}}}_i = {\mathcal{R}}_i^{mf} + \overline {\mathcal{I}}_{{\mathcal{R}}_i}^{\theta } + \sum _{\alpha =1}^{N_s}\left (\overline {\mathcal{I}}_{{\mathcal{R}}_i}^{\rho _\alpha } + \overline {\mathcal{I}}_{{\mathcal{R}}_i}^{\rho _\alpha ,\theta }\right )+ \sum _{\alpha =1}^{N_s}\sum _{\gamma =\alpha +1}^{N_s} \overline {\mathcal{I}}_{{\mathcal{R}}_i}^{\rho _\alpha ,\rho _\gamma } + \overline {\epsilon }_{\mathcal{R}_i}^{{\rho _j}} + \overline {\epsilon }_{\mathcal{R}_i}^{\rho _j,\theta } ,\end{equation}
where
$\overline {\mathcal{I}}_{{\mathcal{R}}_i}^{\rho _\alpha }$
,
$\overline {\mathcal{I}}_{{\mathcal{R}}_i}^{\rho _\alpha ,\theta }$
and
$\overline {\mathcal{I}}_{{\mathcal{R}}_i}^{\rho _\alpha ,\rho _\gamma }$
correspond to the mean distortion of the
$i$
th reaction rate arising from fluctuations in the partial density of species
$\alpha$
alone, from fluctuations in both temperature and species-
$\alpha$
partial density, and from fluctuations in the partial densities of species
$\alpha$
and
$\gamma$
, respectively. The final two terms appearing in the decomposition,
$\overline {{\epsilon }}_{\mathcal{R}_i}^{{\rho _j}}$
, and
$\overline {\epsilon }_{\mathcal{R}_i}^{\rho _j,\theta }$
, capture the modulation of the mean chemical reaction rates stemming from fluctuations in the partial densities of three species, and from fluctuations in the partial densities of at least two species and temperature, respectively. These measures of turbulence–chemistry interaction can be expressed can in terms of the probability density of the thermodynamic fluctuations as
\begin{align}&\qquad\quad \overline {\mathcal{I}}_{{\mathcal{R}}_i}^{\rho _\alpha } = \int _{\mathbb{R}^{N_s+1}}{{\rm d}\mathcal{V}\,{\mathcal{R}}_i({\{\rho _j\},{T})}\delta (T-\widetilde {T})\mathcal{P}(\rho _\alpha )\prod _{\substack {\kappa =1,\\\kappa \neq \alpha }}^{N_s}\delta (\rho _\kappa -\overline {\rho }_\kappa )}-\mathcal{R}_i^{mf}, \end{align}
\begin{align}& \overline {\mathcal{I}}_{{\mathcal{R}}_i}^{\rho _\alpha ,\rho _\gamma } \!= \int _{\mathbb{R}^{N_s+1}}\!{{\rm d}\mathcal{V}\,{\mathcal{R}}_i({\{\rho _j\},{T})}\delta (T-\widetilde {T})\mathcal{P}(\rho _\alpha ,\rho _\gamma )\!\prod _{\substack {\kappa =1,\\\kappa \neq \alpha ,\gamma }}^{N_s}\!\delta (\rho _\kappa -\overline {\rho }_\kappa )}-\overline {\mathcal{I}}_{{\mathcal{R}}_i}^{\rho _\alpha }\!-\overline {\mathcal{I}}_{{\mathcal{R}}_i}^{\rho _\gamma }\!-\mathcal{R}_i^{mf}\!, \end{align}
\begin{align}&\qquad\qquad\qquad\qquad\quad \overline {\epsilon }_{\mathcal{R}_i}^{{\rho _j}} = \overline {\mathcal{I}}_{\mathcal{R}_i}^{\rho _j} - \sum _{\alpha =1}^{N_s}\overline {\mathcal{I}}_{{\mathcal{R}}_i}^{\rho _\alpha }-\sum _{\alpha =1}^{N_s}\sum _{\gamma =\alpha +1}^{N_s} \overline {\mathcal{I}}_{{\mathcal{R}}_i}^{\rho _\alpha ,\rho _\gamma }, \end{align}
Figure 23. Fine-grained decomposition of turbulence–chemistry interaction arising jointly from temperature and partial-density fluctuations for the (a) first Zel’dovich reaction,
$R_1$
, as well as the dissociation/recombination of (b) molecular oxygen, (c) nitric oxide and (d) molecular nitrogen.
where, given the bimolecularity of the Zel’dovich mechanism,
$\overline {\epsilon }_{\mathcal{R}_i}^{\rho _j}$
is precisely zero for the nitric-oxide exchange reactions. With the impact of partial-density fluctuations shown to be most significant for the exchange reactions, the further decomposed reaction-rate distortion due to partial-density fluctuations is presented in figure 22 for the Zel’dovich mechanism, as a function of the semi-local wall-normal distance. For the first of the shuffle reactions, while joint fluctuations in the reactant species
$\textrm{N}_2$
and
$\textrm{O}$
serve to increase the net forward reaction rate, particularly near the peak-temperature location, joint fluctuations in the product species correspondingly increase the backwards component of the reaction, apart from the region between the peak-temperature location and
$y^* \simeq 200$
, where the anticorrelation in the product partial densities serves to effectively increase the significance of the forward reaction rate. For the second of the Zel’dovich reactions, joint partial-density fluctuations in
$\textrm{O}_2$
and
$\textrm{N}$
account almost entirely for the distortion in the mean reaction rate associated with
$\overline {\mathcal{I}}_{\mathcal{R}_i}^{\rho _j}$
, significantly increasing the effective forward reaction rate in the buffer and logarithmic layers, associated with anticorrelation between the product-species partial-density fluctuations.
The species-specific terms arising from the further decomposition of
$\overline {\mathcal{I}}_{{\mathcal{W}}_i}^{\rho _j,\theta }$
can be directly evaluated as
\begin{align} \overline {\mathcal{I}}_{{\mathcal{R}}_i}^{\rho _\alpha ,\theta } = \int _{\mathbb{R}^{N_s+1}}{{\rm d}\mathcal{V}\,{\mathcal{R}}_i({\{\rho _j\},{T})}\mathcal{P}(\rho _\alpha ,T)\prod _{\substack {\kappa =1,\\\kappa \neq \alpha }}^{N_s}\delta (\rho _\kappa -\overline {\rho }_\kappa )}-\overline {\mathcal{I}}_{{\mathcal{R}}_i}^{\rho _\alpha }-\overline {\mathcal{I}}_{{\mathcal{R}}_i}^{\theta }-\mathcal{R}_i^{mf}, \end{align}
\begin{align} \overline {\epsilon }_{\mathcal{R}_i}^{\rho _j,\theta } = \overline {\mathcal{I}}_{{\mathcal{W}}_i}^{\rho _j,\theta } - \sum _{\alpha =1}^{N_s} \overline {\mathcal{I}}_{{\mathcal{R}}_i}^{\rho _\alpha ,\theta }. \end{align}
For the first Zel’dovich reaction, joint fluctuations in temperature and both the reactant and product partial densities are shown to modulate the net reaction rate in figure 23. Whereas joint fluctuations in temperature and the atomic species’ partial densities serve to increase the net forward reaction rate, the coupling between thermal and
$\rho _{N_2}$
fluctuations correspondingly serve to decrease the forward bias in the reaction rate. Overall, however, it is the joint fluctuations in temperature with
$\rho _{O}$
and
$\rho _{N}$
that ultimately prove more consequential, uniformly increasing the net forward bias of the reaction throughout the boundary layer. For the molecular-oxygen dissociation/recombination reaction, however, joint
$\rho _{O_2}$
/temperature fluctuations account almost entirely for the overall reaction-rate distortion due to the coupling between partial-density and temperature fluctuations, effectively suppressing the net forward reaction rate. Finally, for the dissociation/recombination of molecular nitrogen and nitric oxide, joint fluctuations in temperature and their respective reactant molecules again prove the most consequential, manifesting in an increased net forward/dissociation reaction rate within the peak-temperature location. For these latter dissociation/recombination reactions, the coupling between temperature and radical-partial-density fluctuations likewise contributes significantly to
$\overline {\mathcal{I}}_{\mathcal{R}_i}^{\rho _j,\theta }$
, particularly outside the peak-temperature location, where it effectively increases the forward bias of the respective reactions and largely balances the effect of joint fluctuations in temperature and the relevant reactant partial density.
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