# Heat transfer augmentation by recombination reactions in turbulent
reacting boundary layers at elevated pressures

Nikolaos Perakis  Chair of Space Propulsion, Technical University of Munich,
85748 Garching, Germany  Department of Mechanical Engineering, Stanford
University, Stanford, CA 94305, USA  Oskar Haidn  Chair of Space Propulsion,
Technical University of Munich, 85748 Garching, Germany  Matthias Ihme
Department of Mechanical Engineering, Stanford University, Stanford, CA 94305,
USA

###### Abstract

A study of a reacting boundary layer flow with heat transfer at conditions
typical for configurations at elevated pressures has been performed using a
set of direct numerical simulations. Effects of wall temperatures are
investigated, representative for cooled walls of gas turbines and sub-scale
rocket engines operating with hydrocarbon as fuels. The results show that
exothermic chemical reactions induced by the low-enthalpy in the boundary
layer take place predominantly in the logarthimic sub-layer. The majority of
the heat release is attributed to the exothermic recombination of OH and CO to
produce CO2 and H2O. The recombination reactions result in an increase of the
wall heat loads by up to 20% compared to the inert flow. The gas composition
experiences strong deviations from the chemical equilibrium conditions. In
fact, a quenching of the major species is observed within the viscous sub-
layer and the transition region. Analysis of chemical time-scales shows that
the location of quenched composition coincides with the region where the
Damköhler number decreases below unity. Within the viscous sub-layer, a
secondary reaction zone is detected, involving the production of formyl and
formaldehyde radicals that provide an additional source of energy release. The
analysis of the reaction paths showed that reactions with zero activation
energy are responsible for this change in gas composition, which also account
for the initial branching of hydrocarbon fuels decomposition according to
previous auto-ignition studies. The effect of the secondary recombination
reactions is more prominent for the lower wall temperature case. Finally, the
role of turbulent fluctuations on the species net chemical production rates is
evaluated, showing a strong correlation between species and temperature
fluctuations. This leads to a pronounced deviation of the mean reaction rates
$\overline{\dot{\omega}(Y_{k},T)}$ from the reaction rates obtained under the
assumption of laminar finite rate.

††preprint: APS/123-QED

##  I Introduction

Accurate predictions of the flame-wall interaction are crucial in the design
of combustion systems. In most industrial applications that operate in
confined geometries including rocket thrust chambers, gas-turbine combustors
and automotive engines, flame-wall interaction strongly affects fuel
consumption and pollutant formation [1, 2], while temporal and spatial
fluctuations induced by it can influence the thermal loads and engine
lifetime. Classical turbulent combustion models [3] do not account for the
wall effects. Hence a better understanding of the flame-wall interaction is
required for the development of predictive combustion models.

A major physical phenomenon that introduces substantial uncertainties on the
prediction of heat transfer is the effect of exothermic recombination
reactions within the reacting boundary layer. In particular for rocket
applications, modeling these reactions is especially critical in hydrocarbon
combustion, where the chemical time-scales are lower than in the case of
hydrogen/oxygen chemistry. Efforts to model the influence of non-adiabatic
effects on the gas composition, the associated heat release and expected wall
heat loads have been carried out using flamelet methods [4, 5, 6, 7, 8].
Studies aiming at quantifying the importance of the recombination reactions on
heat loads augmentation have been carried out by considering methane/oxygen
mixtures in rocket combustion chambers [9, 10, 11, 12]. At the same time, wall
model extensions have been proposed in an effort to capture the near-wall
chemical kinetics, but so far are limited to hydrogen/oxygen studies, which
can be safely assumed to be in chemical equilibrium even in the cold boundary
layer [13].

Despite ongoing modeling efforts, the physical mechanisms controlling
turbulent flame-wall interaction, recombination reactions and the related wall
heat fluxes have not been completely understood. Because of the difficulties
in obtaining accurate near-wall experimental measurements, Direct Numerical
Simulation (DNS) represents an alternative for studying the processes taking
place in the near-wal region.

As far as high-fidelity simulations of flame-wall interaction are concerned,
head-on quenching configurations have often been chosen, where premixed
laminar hydrocarbon flames are propagated perpendicular to the wall. One-
dimensional simulations of head-on quenching are reported in the works of
Westbrook et al. [14], Hocks et al. [15], Popp et al. [16, 17]. These studies
concur that low-activation energy recombination reactions of chemical radicals
enhance the wall heat loads and require consideration for the flame-wall
interaction processes. Based on these results it is also evident that detailed
chemical kinetic mechanisms are required for capturing these effects. The
importance of radical recombination reactions at the wall is also emphasized
in simulations of H2/O2 flames, as in the work of Dabireau et al. [18].

Due to the large computational costs of multi-dimensional DNS of turbulent
flame-wall interaction, only few investigations have been reported. Two-
dimensional DNS of head-on quenching within a reactive boundary layer was
performed by Poinsot et al. [19] while Bruneaux et al. [20] studied a three-
dimensional configuration of a premixed flame propagating in constant density
turbulent channel flow. Alshaalan et al. [21] investigated sidewall quenching
of a three-dimensional V-shaped premixed flame with a single-step chemistry
approximation. More recently Gruber et al. [22] simulated the same
configuration with H2/air with a detailed chemical mechanism.

As far as reacting turbulent boundary layer simulations are concerned, an even
smaller number of studies can be found in literature, especially for
configurations that are relevant to rocket thrust chambers. Martin et al. [23,
24] performed DNS of hypersonic boundary layers with a single-step reaction
scheme examining the feedback mechanisms between chemistry and turbulence,
showing an increase in temperature fluctuations, induced by exothermic
chemical reactions. Cabrit et al. [25] performed DNS and wall-resolved LES of
multi-component mixtures in rectangular isothermal channels. Configurations
with large temperature gradients and small Mach numbers were simulated,
resembling rocket-like applications. However, the gas mixture that was
introduced consisted to a large degree of inert nitrogen and a simplified
chemical model was utilized, rendering the simulations less applicable to the
thermochemical states experiences in modern methane/oxygen rocket engine
hardware.

In the case of combustion devices at elevated pressures such as gas turbines
and rocket combustion chambers, it is recognized that the effect of the
reacting boundary layer can have a leading order effect on the performance and
wall heat transfer. However, there is still insufficient understanding of the
processes controlling the evolution of chemical reactions within the boundary
layer, which leads to the absence of suitable models able to describe them.
The present work addresses this knowledge gap in the context of recombinations
in low-enthalpy environments using DNS of a reacting turbulent boundary layer.
A canonical configuration is chosen for the study, with the selected operating
conditions being representative for rocket combustion chamber applications. A
reacting methane/oxygen gas mixture is chosen due to the relevance of methane
as a fuel in the design of future space transportation vehicles.

##  II Computational setup

In order to investigate the occurrence of recombination reactions in reacting
boundary layers, direct numerical simulations of spatially evolving turbulent
boundary layer over an isothermal flat plate are performed. The computational
setup consists of a three-dimensional, spanwise periodic domain, which is
schematically illustrated in Fig. 1.

Figure 1:  Computational setup for the DNS.

For the thermodynamic state at the inlet, the chemical equilibrium composition
and temperature for a methane/oxygen mixture with mass mixture ratio equal to
3.0 is chosen, whereas the Mach number is 0.25 and the pressure is 20 bar.
This represents the typical composition and velocity conditions found
downstream of the main reaction zone in sub-scale rocket engines [26]. A
boundary layer thickness of $\delta=1$ mm is chosen, leading to a Reynolds
number $Re_{\delta}=\rho_{\infty}u_{\infty}\delta/\mu_{\infty}=4678$. The
inlet velocity $u_{\infty}$ is 327 m/s, while $\rho_{\infty}$ and
$\mu_{\infty}$ are calculated from the chemical equilibrium thermochemical
state and are equal to 1.41 kg/m3 and 9.87$\times 10^{-5}\,\mathrm{Pas}$
respectively.

As described in Xu et al. [27], the specification of appropriate methods to
generate inflow boundary conditions for LES and DNS of compressible boundary
layers is important for the development of the turbulence in the domain. In
spatially evolving boundary layers, the most convenient procedure involves the
definition of a laminar profile with disturbances far upstream, allowing for a
transition to turbulence. This approach is typically used to investigate
transition [28, 29, 30] and is not generally applicable to turbulence
simulations as it is very costly especially when coupled to the resolution of
chemical reactions. It was hence not considered. Instead, more cost-efficient
methods include the recycling of time series of instantaneous velocity planes
from an auxiliary simulation similar to the work of Li et al. [31], the
specification of the inflow by superposition of random fluctuations on mean
flows as done by Lee et al. [32] and the parallel-flow boundary layer method
used by Lund et al. [33, 34]. In practice, the inflow boundary typically has
to be displaced upstream of the region of interest in order to allow for
relaxation of the errors made in approximating the inflow conditions. The
inclusion of such a ”development section” adds to the overall cost of the
simulation and therefore one would like to minimize its extent while at the
same time, trying to minimize the cost associated with generating the inflow
data themselves.

In the present work, the velocity inflow profile is prescribed by a synthetic
turbulent flow, of which the mean flow obeys the law of the wall [35] with
boundary-layer thickness of $\delta=1\,$mm. The turbulent perturbations with
turbulence intensity of 0.2 in the freestream are generated using the method
of Klein et al. [36]. This method is able to generate pseudoturbulent inflow
conditions based on digital filtering of random data and is able to reproduce
prescribed second order statistics as well as auto-correlation functions. It
hence provides advantages over the classical approach of using random
fluctuations, by reducing the long development section which is induced due to
the lack of proper phase information and non-linear energy transfer in random
methods.

Based on these considerations and on typical development lengths reported for
inflow turbulence [37, 38], a development region of 10 boundary layer
thicknesses is considered. Also, a sponge layer is applied before the outlet
to suppress any numerical wave propagation. In the following analysis, $x=0$
represents the start of the domain after the development region and not the
inflow boundary.

The mesh in wall-normal distance is discretized using 236 grid points
following a geometric growth rate with a first-cell height corresponding to
$y^{+}\approx 0.3$ and adequate points to resolve the viscous sublayer
following the recommendations in Moser et al. [39]. The mesh in the streamwise
and spanwise directions consists of 1400 and 80 equidistant points,
respectively. This corresponds to grid resolutions of $\Delta x^{+}\approx 11$
and $\Delta z^{+}\approx 12$ in the area of interest (after the development
region).

For locations further away from the wall, the flow is dominated by energy-
transferring motions and there, the grid resolution is better evaluated in
terms of the Kolmogorov length scale
$\eta_{K}=((\mu/\rho)^{3}\rho/\epsilon)^{1/4}$, where $\epsilon$ is the
dissipation rate. Due to the presence of heat and mass transfer, apart from
the Kolmogorov scale, the thermal Batchelor scale
$\eta_{T}=\eta_{K}/\sqrt{Pr}$ [40] and scalar Batchelor scale
$\eta_{S}=\eta_{K}/\sqrt{Sc}$ [41] have to be resolved as well. As the Prandtl
$Pr$ and Schmidt $Sc$ numbers of the examined mixture are close to unity for
the entire temperature range, $\eta_{K}$, $\eta_{T}$ and $\eta_{S}$ are all on
the same order of magnitude. The variations of the non-dimensional numbers for
the mixture are shown in Fig. 2 for the two extreme cases of frozen chemistry
and equilibrium conditions. The Schmidt number of CO2 is chosen as it has the
lowest diffusivity among the dominant reaction species. The values are within
the resolution requirements of $\Delta x<12\eta_{T}$, $\Delta y<2\eta_{T}$ and
$\Delta z<6\eta_{T}$, as also reported in other DNS studies [42, 43, 44].

Figure 2:  Prandtl and Schmidt numbers as a function of temperature for the
mixture at 20 bar and O/F = 3.0.

The spanwise extent of the domain was chosen to ensure that it exceeds the
minimal size of $l_{z}^{+}=100$, reported in Jimenez et al. [45]. For a
periodic boundary condition to be reasonably accurate the two-point
correlations are required to be close to zero at a distance of half the domain
size. In Fig. 3, we show the two-point correlations for density
($R_{\rho\rho}$), streamwise velocity ($R_{uu}$) and CO mass fractions
($R_{Y_{CO}Y_{CO}}$) at wall-normal positions $y^{+}=10$ and $y^{+}=1000$ and
streamwise location $x/\delta=5$.

Figure 3:  Two-point correlation for downstream position $x/\delta=5$.

For the simulations presented in the following sections, the governing
equations for continuity, momentum, energy and transported scalars are solved
as follows:

$\partial_{t}\rho+\nabla\cdot(\rho\bm{u})=0$ (1)

$\partial_{t}(\rho\bm{u})+\nabla\cdot(\rho\bm{u}\bm{u}+p\bm{I})=\nabla\cdot\bm{\tau}$
(2)

$\partial_{t}(\rho E)+\nabla\cdot[\bm{u}(\rho
E+p)]=\nabla\cdot(\bm{\tau}\cdot\bm{u})-\nabla\cdot\dot{\bf{q}}$ (3)

$\partial_{t}(\rho Y_{k})+\nabla\cdot(\bm{u}\rho
Y_{k})=-\nabla\cdot\bm{j}_{k}+\dot{\omega}_{k}M_{k}$ (4)

where $\rho$ is the density, $\bm{u}$ is the velocity vector, $p$ is the
pressure, $\bm{\tau}$ is the viscous stress tensor, $\bm{\dot{q}}$ is the heat
flux vector and $E$ is the specific total energy defined as the sum of the
specific internal energy $e$ and the kinetic energy
$\left|\bm{u}\right|^{2}/2$. The ideal gas equation of state is used as
closure for the system of equations.

The viscous stress tensor, heat flux and species diffusion terms are given by:

$\bm{\tau}=\mu\left[\nabla\bm{u}+(\nabla\bm{u})^{T}\right]-\frac{2}{3}\mu(\nabla\cdot\bm{u})\bm{I}\;,$
(5)

$\bm{q}=-\lambda\nabla T\;,$ (6)

$\bm{j}_{k}=-\rho\left(D_{k}\frac{M_{k}}{M}\nabla
X_{k}-Y_{k}\sum_{l=1}^{N}D_{l}\frac{M_{l}}{M}\nabla X_{l}\right)\;,$ (7)

To account for the chemical reactions and the production term
$\dot{\omega}_{k}$ the present DNS study considers the detailed GRI 3.0
chemical mechanism [46]. In the case of the inert simulations, the chemical
source term is set to zero.

The system of equations is discretized based on a finite-volume approach and a
high-order non-dissipative scheme is used for the convective flux
discretization [47], which is fourth-order accurate on uniform meshes. A
strong stability-preserving third-order Runge-Kutta scheme is used for time
advancement. [48]. A Strang-splitting scheme [49, 50] is employed to separate
the convection operator from the remaining operators of the system. A sensor-
based hybrid central-ENO scheme is used to capture flows with large density
gradients and to minimize the numerical dissipation while stabilizing the
simulation. For regions where the density ratio between the reconstructed face
value and the neighboring cells exceeds 25%, a second-order ENO reconstruction
is used on the left- and right-biased face values, followed by an HLLC Riemann
flux evaluation. The flow is well resolved and the ENO scheme is active on
less than 0.01% of all cell faces.

##  III Results

The results of the DNS are presented in this section for four different
computational setups. Reacting and inert simulations are carried out for two
different values of the wall temperature (500 K and 1000 K) resulting in a
total of four cases. In this section will mainly focus on the results of the
case with 500 K wall temperature and we only consider the 1000 K case to show
qualitative differences.

###  III.1 Velocity scaling

For constant-property wall-bounded turbulent flows, the near-wall time-
averaged velocity follows the law of the wall [35] with the wall-normalized
velocity given by $u^{+}=y^{+}$ in the viscous sublayer ($y^{+}<5$) and by the
logarithimic relationship $u^{+}=\frac{1}{\kappa}\log\left(y^{+}\right)+B$ in
the logarithimic layer ($30\leq y^{+}\leq 0.1\delta$).

For variable-property flows, both the fluid density and the dynamic viscosity
are functions of temperature and pressure, leading to additional complexities.
One commonly used velocity transformation that allows the scaled velocity to
collapse with the incompressible law of the wall is the van Driest
transformation [51]:

$u_{\mathrm{VD}}^{+}=\int_{0}^{u^{+}}\left(\frac{\bar{\rho}}{\bar{\rho}_{w}}\right)^{1/2}du^{+}\;.$
(8)

Figure 4:  Normalized velocity profile of the inert and reacting DNS using the
van Driest transformation.

A comparison of the velocity profiles from the DNS is shown in Fig. 4, where
the data support the validity of the van Driest transformation for
compressible reacting flows. This can be attributed to the density
fluctuations in compressible low-Mach number flows, arising from mean fluid-
property variations. This confirms the Morkovin hypothesis [52], which has
been used in the past to investigate the compressible turbulent boundary layer
in the same line as for the incompressible one by accounting for the effects
of mean density variations [53]. In the range of Mach numbers explored in the
present study, our DNS results support previous experimental and numerical
studies that showed the validity of the van Driest transformation in scaling
the velocity profile with wall heat transfer and even chemical reactions [54,
55, 56, 57, 25]. In the presence of higher Mach-number flows and real-fluid
effects [58], it is expected that additional effects require consideration,
related to the compressible nature of the fluid.

###  III.2 Temperature and species mass fractions

The mean temperature profiles (normalized by the wall temperature) are shown
as a function of wall-normal distance in Fig. 5. The effect of chemical
reactions in the boundary layer is evident for both wall temperatures.
Specifically, the temperature profiles for the reacting cases show higher
values in the range $10\leq y^{+}\leq 1000$. The increased temperature is a
result of the exothermic reactions that are induced in the low-enthalpy
boundary-layer region.

Figure 5: Temperature profile within the turbulent boundary layer.

(a)  (b)

Figure 6: Profiles of the (a) transformed temperature $T^{+}$ and (b) van
Driest scaled temperature $T^{+}_{VD}$ within the turbulent boundary layer.

Fig. 6(a) shows the profile of the mean transformed temperature $T^{+}$, which
is defined as the temperature difference $\theta=\overline{T}-T_{w}$
normalized by the friction temperature
$T_{\tau}=\overline{q_{w}}/\left(\overline{\rho_{w}}\,\overline{c_{p}}_{,w}u_{\tau}\right)$.

$T^{+}=\frac{\overline{T}-T_{w}}{T_{\tau}}\;.$ (9)

The discrepancies observed between the reacting and inert cases indicate that
the wall heat flux is sensitive to chemistry. Specifically, the values for the
inert case are higher than the reacting case, indicating the presence of an
appreciable heat transfer augmentation due to the chemical reactions.

Using the van Driest transformed temperature profile [44]

$T_{\mathrm{VD}}^{+}=\int_{0}^{T^{+}}\left(\frac{\bar{\rho}}{\bar{\rho}_{w}}\right)^{1/2}dT^{+}\;,$
(10)

as shown in Fig. 6(b), does not lead to a collapse of the inert and reacting
profiles. Compared to the findings in Fig. 4, where the velocity profiles for
the reacting and inert cases showed similar properties, it is clear that the
chemical reactions have an important effect on the fluxes at the wall, even if
they do not influence the mean velocity profiles. This finding aligns with the
results reported by Carbit et al. [25] and justifies the need of taking care
of fluid heterogeneity in wall models.

Figure 7: Instantaneous species mass fractions and temperature field for the
reacting case with wall temperature equal to 500 K.

To illustrate the effect of the chemical reactions, the instantaneous fields
of major species and temperature are shown in Fig. 7. The formation of a
distinctive species boundary layer in the vicinity of the cooled wall is
evident. Specifically, the recombination of CO to CO2 and that of OH to H2O is
prominent throughout the entire domain. Although only major species are shown
here, the recombination reactions influence all radicals that also react and
form stable species. The locations with increased CO2 mass fraction and hence
reduced CO mass fraction are directly correlated to regions with lower
temperature.

In order to obtain a better understanding of where those reactions occur and
to quantify the correlation between species mass fractions and enthalpy, we
examine Fig. 8, which illustrates the distribution of the major species as a
function of the normalized enthalpy loss. $\mathcal{H}$, which is defined as:

$\mathcal{H}=\frac{h-h_{ad}}{h_{wall}-h_{ad}}\;,$ (11)

and lower values of $\mathcal{H}$ correspond to conditions closer to the wall.
$h_{ad}$ represents the adiabatic enthalpy of the mixture, i.e. the enthalpy
of the hot products at the inlet and $h_{wall}$ the enthalpy that the gas
would have when cooled down to the wall temperature without changing its
composition.

(a)  (b)

Figure 8:  Instantaneous enthalpy conditioned mass fractions for wall
temperature equal to (a) 500 K and (b) 1000 K. Dotted lines correspond to
equilibrium mass fractions and dashed lines correspond to frozen mass
fractions.

The scatter plots in Fig. 8 show a clear correlation between the normalized
enthalpy loss and the species concentration. Compared to the adiabatic values
($\mathcal{H}=0$), a decrease in enthalpy appears to promote the recombination
of OH to H2O and of CO to CO2. A noteworthy remark is the small dispersion of
the instantaneous data in the enthalpy space, as a very small variation is
observed with the point cloud being concentrated around a well established
average for each species. We compare the change in concentration with the
theoretical chemical equilibrium for each enthalpy level, which is illustrated
by the dotted lines in Fig. 8. For enthalpy losses $\mathcal{H}\geq-0.8$ the
results follow the chemical equilibrium concentration, indicating that the
reaction rates are faster than the competing physical phenomena. In lower
enthalpy environment, however, the deviation increases, which implies that the
rate of change of the gas composition is now dictated by larger time-scales.
Finally, the shift towards the recombined products is not continuously
increasing with lower enthalpy but rather reaches a plateau, indicating a
freezing of the reactions directly at the wall.

It is noteworthy that the minimum value of $\mathcal{H}$ in the domain is
smaller than -1, which corresponds to a quenched (non-reacting) mixture that
is cooled to the wall temperature. This arises from the fact that the major
combustion products H2O and CO2 have a lower formation enthalpy and a lower
specific heat capacity than OH and CO. As the gas approaches the wall, the
heat loss favors the formation of stable products and results in a lower
enthalpy compared to its frozen equivalent.

The spatial distribution of the species composition is illustrated in Fig. 9,
where the average mass fraction of the major species is plotted as a function
of the wall-normal distance for the reacting cases with $T_{w}=500\,$K and
$T_{w}=1000\,$K. Distinct regions can be identified when examining the species
mass fraction profiles. Starting from locations away from the wall ($y^{+}\geq
3000$), it appears that no significant change occurs in the composition of the
gas, leading to non-varying mass fractions. However at $y^{+}\approx 3000$, a
species boundary layer starts to develop. The conversion of CO to CO2 is
observed, along with the recombination of OH to H2O. The mass fraction of H2
on the other hand is unaltered. For both wall temperatures, the normalized
wall-distance at which the chemical reactions take effect are comparable. In
accordance with the enthalpy conditioned diagram in Fig. 8 which provides
information about a ”freezing” of the reactions in enthalpy space, a similar
termination of the chemical recombinations is evident in physical space as
well. For wall-normal distances with $y^{+}<10$, all major species have
reached a constant value, which infers a suppression of the reaction rates
because of the low temperature.

Figure 9:  Mass fractions of the major species as a function of the wall-
normal distance for the reacting cases with $T_{w}=500\,$K (solid line) and
$T_{w}=1000\,$K (dashed line).

(a)  (b)

Figure 10:  Minor and radical species mass fractions for the reacting cases as
a function of the wall-normal distance for (a) $T_{w}=500\,$K and (b)
$T_{w}=1000\,$K.

Apart from the major species, shown in Fig. 9, the mass fraction profiles of
the radical species are depicted in Fig. 10. Similar to the behavior of major
species, the radical species concentrations are nearly constant further away
from the wall, where the gas is in a state of chemical equilibrium
($y^{+}>2000$). At locations closer to the isothermal wall, the exothermic
recombination reactions result in a reduction in the mass fractions of all
radicals. This remains for most of the species until a quenching of the
reactions is reached at $y^{+}<10$. For the formyl radical (HCO) and
formaldehyde (CH2O), however, a distinct increase in mass fraction is observed
starting at $y^{+}\approx 100$. A second reaction zone is hence present, in
which the formation of HCO and CH2O is favored.

The results of the species profiles both in conditional and physical space are
qualitatively similar for the tow wall temperatures. For that reason, only the
case with $T_{w}=500\,$K is examined in the remaining sections.

###  III.3 Reaction path analysis

In order to understand the origin of the recombination reactions, the mean net
species molar reaction rates $\dot{\omega}_{k}$ are plotted in Fig. 11(a). In
agreement with the species plots in Fig. 9, the reaction rates for
$y^{+}>1000$ are exactly zero, whereas the main reaction zone extends from
$y^{+}\approx 10$ until $y^{+}\approx 1000$. Starting at locations further
from the wall, it is evident that the hydrogen chemistry is activated first,
with the reaction rates of H2O, H2 and OH increasing in magnitude already at
$y^{+}\approx 1000$. The carbon chemistry on the other hand (represented here
by the reaction rates of CO and CO2) is more localized, with non-negligible
values for the reaction rates beginning at $y^{+}\approx 100$. As far as the
net production and destruction of the species is concerned, in the area where
the bulk of the reactions takes place (between $y^{+}\approx 1000$ and
$y^{+}\approx 10$), there are regions with positive net production of water
and regions with net consumption of water, with the transition occurring at
$y^{+}\approx 500$. For OH, on the other hand, only net consumption rates are
shown. Moreover, as expected based on the scatter results (Fig. 8), CO2 is
predominantly being created in regions where CO has a net consumption rate.

(a)  (b)

Figure 11:  Profiles of (a) mean net chemical production rates of the major
species and (b) mean net rates of progress for the most dominant reactions for
the reacting case with $T_{w}=500\,$K.

Although the mass fraction of CO appears to be unaffected in regions closer to
the wall due to the freezing of the reactions that was established in Fig. 9,
a sharp increase in the consumption rate of CO is observed for distances
$y^{+}<10$, with a simultaneous increase in the H2 net creation rate. This
effect, which is most prominent for the case with lower wall temperature,
occurs at wall-distances that coincide with the regions of increased HCO and
CH2O abundance (from Fig. 10). This secondary reaction zone is hence activated
at low temperatures and is confined within the viscous sub-layer.

In order to investigate the nature of this secondary recombination layer, the
reactions with the highest mean net rates of progress
$\overline{\mathcal{R}_{j}}$ are explored in Fig. 11(b). The instantaneous
rate of progress of reaction $j$ is defined as:

$\mathcal{R}_{j}=\Gamma_{j}\cdot\left(\mathcal{K}_{fj}\prod_{k=1}^{N_{sp}}\mathcal{C}_{k}^{\nu_{kj}^{\prime}}-\mathcal{K}_{rj}\prod_{k=1}^{N_{sp}}\mathcal{C}_{k}^{\nu_{kj}^{\prime\prime}}\right)$
(12)

where $\Gamma_{j}$ is the third-body coefficient, $\mathcal{K}_{fj}$ and
$\mathcal{K}_{rj}$ are the forward and reverse reaction coefficients,
$\mathcal{C}_{k}$ is the molar concentration of species $k$, while
$\nu_{kj}^{\prime}$ and $\nu_{kj}^{\prime\prime}$ are the forward and reverse
stoichiometric coefficients.

Within the viscous sub-layer, it is apparent that reactions involving HCO are
dominant. Specifically, the recombination of CO and H to form HCO using H2O as
collision partner as well as the recombination of HCO and H to CO and H2
demonstrate the largest rates of progress. The net effect is a net production
rate for HCO and a subsequent increase in its mass fraction as shown in Fig.
10. At the same time, the recombination of some of the excess HCO with H to
form formaldehyde is also activated in this low temperature region.

The reason for the sharp increase in the mass fractions of HCO and CH2O is the
absence of activation energy for the aforementioned reactions. The formation
of formaldehyde is driven by a negative activation energy, which promotes the
formation of the molecules in low enthalpy environments. As the temperature
drops in the vicinity of the wall, the other reaction rates with positive
activation energy start decaying, leading to a freezing of the reactions. For
the case of HCO and CH2O production reactions however, the rates of progress
keep increasing as the density and hence radical concentration increase. This
agrees with the results reported by Westbrook et al. [14], who investigated
the importance of radical recombination reactions with low activation energy
and found that radical recombination reactions become the dominant means of
consumption of radical species when the temperatures drop below 700-800 K and
the rates of radical-fuel reactions fall to very low values.

The paths leading to the production of formyl and formaldehyde in the reacting
DNS are illustrated in the reaction flux diagram of Figs. 13. Reaction path
diagrams at different wall-normal distances for a representative instantaneous
snapshot are included. The forward and backward rates of the most dominant
reactions are also used as labels in the diagrams. Going from left to right in
Fig. 13, the normalized distances $y^{+}=2$, $y^{+}=10$, $y^{+}=75$ and
$y^{+}=200$ are shown. Those distinct locations are also indicated with red
markers in Fig. 12. In the regions further from the wall ($y^{+}=200$), the
production of H2O remains most prominent, as the temperature is still high and
the reaction rates of the hydrogen chemistry dominate. Approaching the wall,
in the lower temperature environment ($y^{+}=75$ and $y^{+}=10$), the
recombination of OH and CO to form CO2 is established as the main reaction
path. Directly within the viscous sub-layer nevertheless ($y^{+}=2$), the
reaction rates of the CO-to-CO2 recombination die out and CO feeds the
production of HCO via the CO + H and CO + H + M paths. Further absorption of a
H-atom leads to the production of formaldehyde.

Figure 12: Instantaneous CO2 species mass fraction for the reacting case with
wall temperature equal to 500 K. The wall-normal distances used in Fig. 13 are
indicated with the red markers.

Figure 13:  Reaction flux path diagram for the oxygen atom at different wall-
distances in the reacting case with $T_{w}=500\,$K.

Given the presence of two distinct reaction zones (one where OH and CO
recombine to form carbon dioxide and water and the other where CO recombines
to formyl and formaldehyde), the individual contributions of these two modes
to the total heat release rate are quantified in Fig. 14. The mean local heat
release rate $\overline{\dot{\mathcal{Q}}}$ as well as the integrated
cumulative heat release $\overline{\dot{\mathcal{Q}}}_{cumul}$ are shown for
the two reacting cases. The cumulative heat release is defined as the
percentage of the total energy release starting from the wall and moving
towards the free stream:

$\overline{\dot{\mathcal{Q}}}_{cumul}=\cdot\frac{\int_{0}^{y}\overline{\dot{\mathcal{Q}}}(y)dy}{\int_{0}^{\infty}\overline{\dot{\mathcal{Q}}}(y)dy}$
(13)

As expected from the results of the species production rates in Fig. 11, the
total heat release rate has a local maximum close to $y^{+}\approx 60$ and a
further maximum with a larger value directly at the wall ($y^{+}=1$). For the
case with $T_{w}=500\,$K, the maximum corresponding to the HCO recombination
is more prominent. However, looking at the distribution of
$\overline{\dot{\mathcal{Q}}}_{cumul}$ in Fig. 14, it becomes evident that the
HCO production reactions do not have a significant contribution to the total
heat release. Within the viscous sub-layer ($y^{+}<10$), approximately 6% of
the total energy is set free. The bulk majority of the chemical energy is
released between $y^{+}\approx 10$ and $y^{+}\approx 1000$, with a subsequent
ceasing of the energy production in the regions with high temperature. The
low-temperature recombination reactions with zero activation energy have hence
a minimal impact on the total energy release, as they are only activated in a
region with restricted volume in the direct vicinity of the wall.

By comparing the total heat release rate with the net rate of progress for the
most dominant reaction in Fig. 11(b), it can be inferred that the two
reactions responsible for the majority of the energy being released in the
boundary layer are the production of OH following the path H + H2O
$\rightarrow$ H2 \+ OH and the subsequent recombination of OH and CO following
the path CO + OH $\rightarrow$ CO2 \+ H. The latter reaction path has also
been identified as the most dominant reaction in the boundary layer of rocket
thrust chambers in prior work [9].

Figure 14:  Mean heat release rate and cumulative energy release for the
reacting cases with $T_{w}=500\,$K and $T_{w}=1000\,$K.

###  III.4 Chemical time-scales

The exothermic CO recombination reaction that is activated in the low-
temperature environment can be explained by Le Chatelier’s principle, as the
initially equilibrated gas is subjected to a change in enthalpy and reacts by
undergoing exothermic reactions that aim at counteracting the applied change.
However, with decreasing temperature near the wall, the reaction rates also
approach zero which gives rise to two competing effects: the low temperature,
which is the initiator for the reactions is also the limiting factor that does
not allow the composition to reach its new equilibrium. This results in a
significant increase in the chemical time-scales and is expressed by an
apparent ”quenching” of the chemical reactions.

To understand the competing effects that dominate the change in chemical
composition, we choose to describe the chemical time-scale along the wall
normal direction using the Computational Singular Perturbation (CSP) technique
by Lam et al. [59] which is used in the Chemical Explosive Mode Analysis
(CEMA) as introduced by Lu et al. [60]. The time-scale is then defined as:

$\tau_{c}=\frac{\|\dot{\bm{\omega}}\|}{\|\bm{J}\cdot\dot{\bm{\omega}}\|}\;,$
(14)

where the species Jacobi matrix is given by the sensitivity of the net species
reaction rates to a change in the species mass fractions.

The mean chemical time-scale $\tau_{c}$ for the reacting cases is shown in
Fig. 15 along with the local Damköhler number. Two different definitions for
the Dahmköhler number are included, one based on the Kolmogorov time-scale
$\tau_{K}$ and one based on the integral turbulent time-scale $\tau_{t}$. The
respective time-scales and Damköhler numbers are given as:

$\tau_{K}=\sqrt{\frac{\mu}{\rho\epsilon}}\;\;,\;\;Da_{K}=\tau_{K}/\tau_{c}$
(15)

$\tau_{t}=\frac{k}{\epsilon}\;\;,\;\;Da_{t}=\tau_{t}/\tau_{c}$ (16)

Fig. 15 shows that, the chemical time-scale is small in regions away from the
wall, indicating that the gas mixture is close to its equilibrium
concentration. Moving towards the wall however, as the temperature decreases,
a slowing-down of the reactions takes place. This trend persists up until
$y^{+}\approx 30$, where the chemical time-scale has a local maximum and then
subsequently reduces further. This secondary reduction in the time-scale
represents the zone where the formyl and formaldehyde production become
dominant. At the same time, both the turbulent time-scale and the Kolmogorov
time-scale are monotonically reduced for locations closer to the wall.

This leads to comparable results for $Da_{K}$ and $Da_{t}$. Both of them have
large values between $4\times 10^{4}$ and $9\times 10^{5}$ outside the
boundary layer and reach values between 0.1 and 5 within the viscous sub-
layer. In the region where the reaction rates of the main exothermic
recombination reactions of CO and OH start diminishing ($y^{+}<10$), the
Damköhler number reaches values close to unity. This implies that after this
point the turbulent and Kolmogorov time-scales dictate the evolution of the
species which confirms the assumption that the freezing of the chemical
reactions is due to the large chemical time-scales in the low-temperature
environment.

Figure 15:  Chemical time-scale and Damköhler number for the reacting case
with $T_{w}=1000\,$K.

###  III.5 Wall heat transfer amplification

Understanding the conditions that lead to the enhancement of the exothermic
recombination reactions in the cooled boundary layer is of major importance
for the prediction of wall heat loads in rocket thrust chambers and other
propulsion systems. Using the comparison between the inert and reacting cases,
the effect of the additional energy release in the wall vicinity can be
isolated and quantified. The results for the mean wall heat loads for the
different DNS cases examined here are summarized in Table 1.

Table 1:  Mean wall heat flux for the different cases. Case | Mean heat flux
---|---
Reacting, $T_{w}=500\,$K | 11.93 $\mathrm{MW/m^{2}}$
Reacting, $T_{w}=1000\,$K | 9.87 $\mathrm{MW/m^{2}}$
Inert, $T_{w}=500\,$K | 10.01 $\mathrm{MW/m^{2}}$
Inert, $T_{w}=1000\,$K | 8.27 $\mathrm{MW/m^{2}}$

These results show that, lower wall temperatures lead to a higher heat flux,
as this is proportional to the temperature difference between the wall and the
free stream temperature (which is equal to the adiabatic equilibrium
temperature). The same is true when comparing the two reacting cases to each
other, with the lower wall temperature leading to a higher wall heat transfer
rate. For a given wall temperature, an increase of approximately 20% is
obtained from the effect of the chemical recombinations. This degree of under-
prediction of the heat loads when using models that do not account for
recombination reactions is comparable with the values reported by Betti et al.
[10] for low-pressure methane/oxygen rocket engines. It was also reported in
the same work that the difference between inert and reacting treatment for the
boundary layer reduces with increasing operating pressures.

Figure 16:  Instantaneous wall heat flux. From top to bottom: reacting case
with $T_{w}=500\,$K, reacting case with $T_{w}=1000\,$K, inert case with
$T_{w}=500\,$K, inert case with $T_{w}=1000\,$K Figure 17: PDFs for the wall
heat flux distribution. Figure 18:  PDFs of species mass fractions and
temperature at different wall distances for the reacting case with
$T_{w}=500\,$ K.

Apart from the averaged values for the heat loads reported in Table 1, the
local variation of the heat flux is plotted in Fig. 16 for representative
instantaneous snapshots. Localized ”hot islands” with heat flux values two
times larger than the mean average are visible. This is also illustrated in
Fig. 17, where the Probability Density Function (PDF) of the instantaneous
local heat flux values is plotted for each of the four simulations. A
consequence of the overall higher heat flux level, the reacting cases have a
wider distribution compared to the respective inert ones. The presence of
strong fluctuations in the experienced wall heat loads can lead to significant
deviations from the mean values which in turn could lead to strong temperature
gradients and higher thermal stresses on the walls. However, the time-scales
of the turbulent fluctuations which are responsible for the spatial variation,
shown in Fig. 16 and 17, are much shorter than typical conduction time-scales
in the material and hence do not pose an issue for the design of cooling
systems.

###  III.6 Turbulent fluctuations

Figure 19:  Probability density function of species molar reaction rates at
different wall distances for the reacting case with $T_{w}=500\,$ K.

In an effort to examine the sensitivity of reaction rates to compositional
variations, Figs. 18 and 19 show PDFs of $Y_{k}$, $T$ and $\dot{\omega}_{k}$
at three distinct distances from the wall. The case with $T_{w}=500\,$K is
shown here and the results obtained with the higher wall temperature are
qualitatively similar. In contrast to the enthalpy-conditioned distribution
from Fig. 8, which clearly showed a minimal variation of the species mass
fractions around its average, the PDFs at different spatial locations show a
much larger variability. Wider distributions for both the temperature as well
as major species mass fractions are observed within the logarithmic region of
the boundary layer ($y^{+}=75$). Approaching the wall and entering the viscous
sub-layer where turbulent effects are less prominent, the width of the
distributions quickly shrinks. For OH in fact, the PDF of the mass fraction
converges to a Dirac function around zero, as most of the OH is consumed
before the begin of the viscous sub-layer.

The net molar production rates at $y^{+}=75$ demonstrate a clear correlation
between the consumption of CO and the production of CO2 and exhibit a large
variation which extends in the range from $-$150 to 150
$\mathrm{kmol/(m^{3}s)}$, while the average value lies at $\pm$ 35
$\mathrm{kmol/(m^{3}s)}$ for CO2 and CO, respectively. Closer to the wall, the
correlation between the CO and CO2 reaction rates ceases to be dominant, as
carbon dioxide no longer undergoes any further chemical reactions in alignment
with the results in Fig. 11. Instead, as the reaction path including HCO is
activated, a correlation between the reaction rates of formyl and carbon
monoxide is visible. It is also noteworthy that the reaction rate of HCO was
close to zero for locations within the log-layer, whereas it exhibits a large
variability in the vicinity of the wall owing to non-negligible variations in
temperature and CO mass fraction.

To further examine the non-linearity of mass fractions and reaction rates, a
scatter plot of instantaneous and averaged net species production rates are
given in Fig. 20. The grey points represent instantaneous results, mapped onto
the conditional $Y_{k}-T$ space, while the dashed line corresponds to the
projection of the averaged quantities, i.e. to the manifold
$\overline{Y}_{k}-\overline{T}$.

Figure 20:  Scatter plot of the instantaneous and averaged chemical source
terms for (a) CO2 and (b) CO in conditional space for $T_{w}=500\,$K. The
dashed lines correspond to the manifold correlating mean temperature and mean
mass fractions.

It is evident that the estimation of the reactions rates using the averaged
fields for mass fractions and temperature
$\dot{\omega}_{k}(\overline{Y}_{i},\overline{T})$ leads to significant
overestimation compared to the time-averaged production rates
$\overline{\dot{\omega}_{k}(Y_{i},T)}$ (which correspond to the results from
Fig. 11) both for CO and CO2. The two results for the production rates deliver
similar values in the hot regions (where the gas is near its chemical
equilibrium) as well as in the direct vicinity of the wall, where a quenching
of the reactions occur. For the intermediate positions however, differences
larger than one order of magnitude are observed. This is an indication for
strong turbulence-chemistry interaction, which leads to the confirmation that
the closure of the chemical source term is inadequately described by a first-
order approximation of the reaction rate:

$\overline{\dot{\omega}_{k}(Y_{i},T)}\neq\dot{\omega}_{k}(\overline{Y}_{i},\overline{T})$
(17)

The deviation between the two quantities can be attributed to the correlation
between mass fraction fluctuations and temperature fluctuations. This is shown
in Fig. 21, which clearly illustrates that a positive correlation holds in the
reacting zone for CO, while a negative correlation can be found for CO2. No
significant difference can be found between the compressible and
incompressible definitions ($\widetilde{Y^{\prime\prime}T^{\prime\prime}}$ and
$\overline{Y^{\prime}T^{\prime}}$), as they both give qualitatively identical
results.

Figure 21:  Correlation of turbulent fluctuations for species mass fractions
and temperature for the reacting case with $T_{w}=500\,$K. The solid line
represents the Reynolds averaged quantity $\overline{Y^{\prime}T^{\prime}}$,
while the dashed line represents the Favre averaged
$\widetilde{Y^{\prime\prime}T^{\prime\prime}}$.

Finally, we investigate whether the turbulent fluctuations and the feedback
produced by the chemical reactions give rise to an enhancement of the
turbulent heat transfer. To quantify this, the turbulent Prandtl number is
calculated for all four DNS cases. The turbulent Prandtl number is defined as
the ratio of the turbulent thermal diffusivity $\alpha_{T}$ and the turbulent
viscosity $\alpha_{M}$. The strong Reynolds analogy assumes that the turbulent
heat transfer and the turbulent momentum transfer are similar, resulting in a
turbulent Prandtl number equal to unity. Based on experimental and numerical
studies, it has been found that the Reynolds analogy is valid for most
boundary layer flows, although departures from $Pr_{t}=1.0$ have been reported
[61].

Common practice in most turbulence models for RANS, is to approximate $Pr_{t}$
by a constant value, which in some cases, can lead to incorrect heat transfer
predictions. This is typically the case in supercritical flows, where strong
gradients in the molecular Prandtl number give rise to spatial variations for
$Pr_{t}$ [62]. For gases at high-temperature conditions, with nearly constant
molecular Prandtl number (as in the present operating point) and high
turbulent Peclet number, the Prandtl number is nearly constant within the log
region with values around 0.85 and 1, while in the ”wake” region of external
turbulent boundary layers it was found to decrease in the neighborhood of
0.5-0.7 [61]. Previous studies of reacting turbulent boundary layers performed
by Cabrit et al. [25] found that the turbulent Prandtl number varies from
around 0.5 in the middle of their turbulent channel up to values close to 1.2
in the wall vicinity. No qualitative differences were obtained in the $Pr_{t}$
profiles between inert and reacting cases.

Similar findings are reported in Fig. 22 by analyzing the results of the
present DNS. The compressible definition was employed for the calculation:

$Pr_{t}=\frac{\widetilde{u^{\prime\prime}v^{\prime\prime}}}{\widetilde{v^{\prime\prime}h^{\prime\prime}}}\frac{\partial\widetilde{h}/\partial
y}{\partial\widetilde{u}/\partial y}$ (18)

For all cases, the turbulent Prandtl number appears to be nearly constant
throughout the logarithmic and viscous sub-layer regions. Differences between
the individual cases are small, with the reacting cases having the tendency of
producing higher turbulent Prandtl numbers. Values between 0.9 and 1.2 are
found within the region where the bulk energy release takes place
($y^{+}\approx 10$ to $y^{+}\approx 100$). Further from the wall, a drop of
the turbulent Prandtl number to values as low as 0.5 is observed.

Figure 22:  Turbulent Prandtl number for the four DNS cases as a function of
the wall-normal distance.

##  IV Conclusions

In the present study, DNS of a reacting boundary layer have been carried out.
Mixtures representative of the conditions within rocket engines and high-
pressure propulsion systems have been chosen, corresponding to an equilibrated
mixture of methane and oxygen at 20 bar. In order to isolate effects of
chemical reactions resulting from the low-enthalpy environment in the vicinity
of the isothermal wall, inert and reacting simulations were carried out for
the same operating conditions. A detailed chemical mechanism was employed to
account for the role of chemical radical recombinations in the vicinity of the
wall.

Evaluation of the temperature profile showed that deviations between the inert
and reacting cases were prominent, resulting from the energy release within
the boundary layer. The energy released from the recombination of CO and OH to
form H2O and CO2 was identified as the main effect for the temperature
increase. This effect was found to be qualitatively similar for both wall
conditions.

The analysis of reaction rates showed strong non-equilibrium effects. Although
a clear correlation between the degree of recombination of CO and the local
enthalpy value was found, the species mass fractions profiles deviated from
the theoretical equilibrium values. Moreover, for regions up until the begin
of the viscous sub-layer ($y^{+}\approx 10$), a quenching of the major species
mass fractions was observed. Further examination of the chemical reaction
rates confirmed that the occurrence of this layer with chemically quenched
composition arises from the low temperature and resulting long chemical time-
scales. In fact, the chemical time-scales at the locations where the
recombination rates start diminishing reach values that are larger than the
turbulent and Komlmogorov time-scales and hence chemistry is no longer the
rate-defining process.

Closer to the wall, despite the termination of the CO recombination, the
recombination of CO to HCO and the subsequent formation of CH2O is activated.
This reaction path was detected using reaction flux diagrams and showed that
the production of formyl and formaldehyde is favored by low wall temperatures.
Due to the restricted extent of the region in which the temperatures are low
enough to activate the aforementioned reactions, the total energy released
from this reaction path amounted to approximately 5% of the total heat release
in the boundary layer.

As far as the wall heat transfer is concerned, the exothermic reactions
contributed to an additional 20% in heat flux compared to the equivalent inert
cases. No qualitative difference was observed between the results with low
(500 K) and high (1000 K) wall temperature. The non-negligible augmentation of
the heat transfer puts additional emphasis on the importance of the
recombination reactions in modeling efforts of flame-wall interaction, like in
the case of non-adiabatic flamelet manifolds.

A strong coupling between turbulence and chemistry was also inferred from the
results. Strong variations in mass fractions and temperature within the
reacting zone resulted in a large variance of the resulting reaction rates and
broad PDFs for the source terms. The assumption of first-order representation
of the reaction rate was hence assessed as inadequate to describe the
turbulence-chemistry interaction processes in the reaction zone. Despite the
strong correlation of species and temperature fluctuations and the additional
energy released via the chemical reactions, no significant enhancement of the
turbulent heat transfer was observed. Evaluation of the turbulent number for
both the reacting and inert cases delivered values for $Pr_{t}$ close to unity
with fairly constant profiles along the entirety of the logarithmic and
viscous sub-layers.

The strong presence of non-equilibrium effects, dictated by the competition
between turbulent and chemical time-scales as well as the intensity of the TCI
throughout the turbulent boundary layer, highlight the difficulty of modeling
the near-wall region in the presence of chemically reacting flows and strong
temperature gradients. Examination of further operating points and fuels is of
interest in order to understand the processes occurring within the reacting
boundary layer of rocket engines and gas turbines for a broad spectrum of
applications. Based on the representative load point chosen for this analysis
however, the importance of the recombination effects on the wall heat flux
augmentation is illustrated. For that reason, future models developed for the
treatment of near-wall effects in reacting simulations should account for the
quenching of the reactions within the buffer and viscous sub-layers.

##  V Acknowledgements

###### Acknowledgements.

Resources supporting the DNS study were provided by the NASA High-End
Computing Program through the NASA Advanced Supercomputing Division at Ames
Research Center (Award No. NNX15AV04A) and by the GCS Supercomputer SuperMUC
at Leibniz Supercomputing Centre. Financial support has been provided by the
German Research Foundation (Deutsche Forschungsgemeinschaft DFG) in the
framework of the Sonderforschungsbereich Transregio 40\.

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