# Modeling the effect of flame-wall interaction on the wall heat flux

Karine Truffin, B. Leveugle, Gilles Bruneaux, Y. D'angelo, Julien Reveillon

► To cite this version:

Karine Truffin, B. Leveugle, Gilles Bruneaux, Y. D'angelo, Julien Reveillon. Modeling the effect of flame-wall interaction on the wall heat flux. THMT-12. Seventh International Symposium On Turbulence, Heat and Mass Transfer, Sep 2012, Palermo, Italy. pp.602-615, 10.1615/ICHMT.2012.ProcSevIntSympTurbHeatTransfPal.650. hal-04185517

HAL Id: hal-04185517

https://ifp.hal.science/hal-04185517v1

Submitted on 22 Aug 2023

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
# Modeling the effect of flame-wall interaction on the wall heat flux

K. Truffin<fnref n="1" />, B. Leveugle<fnref n="2" />, G. Bruneaux<fnref n="1" />, Y. D'Angelo<fnref n="2" />, J. Reveillon<fnref n="2" />

**Abstract** — For new generation highly downsized spark ignition engines, the proportion of fuel which is consumed near the wall increases significantly. It is therefore necessary to model accurately the flame / wall interactions. While current models take into account the effect of the wall on the flame, the effect of the turbulent flame-wall interaction on the wall heat flux is still lacking. A model, based on the flame surface density concept, is proposed in the present study in order to take into account this effect. It attempts to evaluate the proportion of the flame in the computational cell that interacts with the wall as the quenching distance is reached. A first validation of the new wall model is presented using DNS data of a 2D V-flame.

## 1. Introduction

In spark ignition engines, flame-wall interaction occurs after the ignition and flame propagation phases. At the beginning of interaction, the amount of fuel to be burnt can reach 20 to 30% of the initial mass [1]. This proportion will tend to increase for new generation SI engines which are currently developed in order to increase the energy efficiency by downsizing and boosted air loop technologies [2].

Flame-wall interaction is characterised by the consumption of the fuel trapped between the flame front and the wall and high wall heat losses due to increasing temperature gradient. These phenomena lead to quenching of the flame at a quenching distance $\delta_Q$ of the order of 0.1 mm in a spark ignition engine. The resulting local wall heat flux can reach several MW/m$^2$ during a few milliseconds. Consequently, the energy efficiency is affected and in some cases thermal fatigue can be expected [3, 4, 5].

Studying flame-wall interaction in engines is a major challenge since the unsteady phenomenon is characterised by small time and length scales, complex chemistry and transport effects [6, 7] in complex geometry with limited optical access near walls. However, appropriate modeling for 3D simulation is a necessary requirement to help engine design. The understanding of the effect of turbulence near walls is a source of numerous DNS studies [8, 9, 10, 11]. They are accounted for in anisothermal flows through the so called "law-of-the-wall" [12, 13, 14] while the effect of the cold wall on the turbulent flame is incorporated in a few combustion models [10, 15, 16]. The flame impact on wall heat transfers in turbulent flow was investigated in a few experimental studies for example in constant volume chamber [3] and rapid compression machine [4]. Simultaneous measurements of local, time resolved wall heat flux and velocity field were conducted by Boust et al. [3]. They demonstrated a direct correlation of the maximum wall heat flux fluctuation during flame-wall interaction with the direction fluctuation of the large scale velocity. However to the knowledge of the authors, the modeling of wall heat transfer taking into account the effect of the flame was rarely addressed. In [13] and [17] wall functions based on steady state assumption were derived including an average chemical heat release. But the modeling of the heat lost by the flame during quenching was not incorporated. Alshaaland and Rutland [8] used DNS results to include the reaction term a posteriori in the law-

<footnote n="1">¹IFP Energies nouvelles, 1-4 av. de Bois-Préau, 92852 Rueil-Malmaison, France, karine.truffin@ifpen.fr</footnote>

<footnote n="2">²UMR6614-CORIA, Technopôle du Madrillet, BP 12, Avenue de l'Université, 76801 Saint-Etienne-du-Rouvray, France, benoit.leveugle@coria.fr</footnote>
of-the-wall scaling of the temperature profile. Choi and Huh [18] proposed a simple model to express the flame induced wall heat flux based on the coherent flamelet model [19, 20, 21, 22]. But this model did not account for the flame brush nor direction and went to zero in laminar flows. Moreover no validation was provided. The objective of this study was to develop a model for the mean wall heat flux that accounts for the interaction of the turbulent flame with cold walls. In the present work, the coherent flame model (CFM) approach was used to model turbulent premixed combustion, which is well justified considering that combustion is mainly taking place in the flamelet regime [15, 23]. This model was modified by Bruneaux et al. [10] to account for the effect of the wall on the flame as briefly discussed in section 2. In section 3, a wall law based on the so called LnKC model [12] is derived taking into account the heat transfered by the flame during its interaction with the wall in a steady-state configuration. A model for the contribution of the wall heat flux due to the presence of the flame is proposed in section 4. First validation of this new model, called the α-model, is provided using DNS data of a turbulent 2D V-flame [24].

## 2. Combustion modeling in turbulent flows

The CFM approach [19, 20, 21, 22] is based on the flame surface density (Σ) concept. In this approach the turbulent flame is considered as an interface between the fresh and burnt gases. Each flamelet is described by its laminar flame speed $S_L^0$ and its flame thickness $δ = ν^u/S_L^0$, where $ν^u$ is the kinematic viscosity in the fresh gases. In the absence of quenching, these local features are related to a mean reaction rate through $Σ$ which characterizes the flame wrinkling:

$$ \bar{\omega}_F = -\bar{\rho}^u \tilde{Y}_F^u \langle \overline{S_L} \rangle_s \bar{\Sigma} \quad (1) $$

where $\bar{\rho}^u$ and $\tilde{Y}_F^u$ are respectively the mass density and the fuel mass fraction in the fresh gases. The operator $\langle \cdot \rangle_s$ denotes a surface average over the $Σ$ interface [10]. To simplify we assume in this study that the mixture is homogeneous. The presence of the wall decreases the turbulence scales. These scales can become smaller than the flame thickness so that flamelet approaches are no longer valid. A proper approach was proposed in [10]: using DNS results, an extension of the flame surface density equation was proposed to account for the decrease of the turbulent scales in the vicinity of the wall and the quenching through heat losses. The resulting model was included in the ECFM combustion model [25, 26] and was used in this study. During the interaction near the wall, part of the interface $Σ$ is quenched so that $\bar{\omega}_F = -\bar{\rho}^u \tilde{Y}_F^u \langle \overline{S_L} \rangle_s \bar{\Sigma}_R$ with $\bar{\Sigma}_R < \bar{\Sigma}$ is the reactive flame surface density that takes into account the effect of the quenching. A model for $Q_R = \bar{\Sigma}_R/\bar{\Sigma}$ was proposed and validated in [10].

## 3. A wall law for reactive flows

The main issues to address concerning wall modeling in spark ignition engines are the presence of strong anisothermal flows due to compression and chemical reactions. The wall law provides analytical solutions relating the wall shear stress $τ_w$ and the wall heat flux $q_w$ to the mean flow quantities. Density variations in the boundary layer due to temperature gradients are accounted for in [12, 13, 14, 17]. Using DNS, Alshaalan and Rutland [8] showed that during flame-wall interaction it should be taken into account in the derivation of a wall model to properly reproduce the temperature profile in the inner layer. For this reason, we consider here a reactive turbulent, steady-state, non-separating flow. The transport equations for mean momentum and enthalpy are written:
$$0 = \frac{d}{dy} \left( (\bar{\mu}_t + \bar{\mu}) \frac{d\tilde{u}}{dy} \right), \qquad (2)$$

$$0 = \frac{d}{dy} \left( (\bar{\lambda}_t + \bar{\lambda}) \frac{d\tilde{T}}{dy} \right) + \tilde{\omega}_T^I, \qquad (3)$$

where $\tilde{\omega}_T^I$ is the mean heat release during flame-wall interaction. The integration of Eqs. (2) and (3) in the boundary layer from the wall provides relations between wall heat and momentum fluxes with mean flow quantities:

$$(\bar{\mu} + \bar{\mu}_t) \frac{d\tilde{u}}{dy} = \bar{\tau}_w = \mu_w \left(\frac{d\tilde{u}}{dy}\right)_{y=0}, \qquad (4)$$

$$\left(\bar{\lambda} + \bar{\lambda}_t\right) \frac{d\tilde{T}}{dy} + \int_0^y \tilde{\omega}_T^I dy = -\bar{q}_w = \lambda_w \left(\frac{d\tilde{T}}{dy}\right)_{y=0}. \qquad (5)$$

The influence distance of the flame during quenching $\delta_I$ should be smaller than the near wall computational cell width so that $\int_0^y \tilde{\omega}_T^I dy = \int_{\delta_I < y} \tilde{\omega}_T^I dy$. To derive the analytical function, Eqs. (4) and (5) are usually written in a dimensionless form. In isothermal flows, the following dimensionless variables are usually used:

$$y^+ = \frac{y u_{\tau}}{\bar{\nu}_w}, \quad u^+ = \frac{\tilde{u}}{u_{\tau}}, \quad T^+ = \frac{\tilde{T} - \tilde{T}_w}{T_{\tau}}, \quad \rho^+ = \frac{\bar{\rho}}{\bar{\rho}_w}, \quad \nu^+ = \frac{\bar{\nu}_t}{\bar{\nu}}, \qquad (6)$$

where the index w designates wall values. The friction velocity and temperature are defined using the wall shear stress $\bar{\tau}_w$ and the wall heat flux $\bar{q}_w$ following:

$$u_{\tau} = \sqrt{\frac{\bar{\tau}_w}{\bar{\rho}_w}} \quad \text{and} \quad T_{\tau} = \frac{\bar{q}_w}{\bar{\rho}_w \bar{c}_p u_{\tau}}. \qquad (7)$$

In non-isothermal flows, the thermodynamic properties of the gas in the boundary layer are not constant. Following [12] we define the following set of non-dimensional variables to account for density variations in the boundary layer:

$$d\eta^+ = \frac{\bar{\nu}_w}{\bar{\nu}} dy^+, \quad d\psi^+ = \rho^+ du^+, \quad d\theta^+ = \rho^+ dT^+, \quad \nu^{++} = \frac{\bar{\nu}_t}{\bar{\nu}}. \qquad (8)$$

Assuming an ideal gas and quasi-constant pressure in the boundary layer, so that $\bar{\rho}/\bar{\rho}_w = \tilde{T}_w/\tilde{T}$, the dimensionless density and temperature are related following [12]:

$$\rho^+ = (1 + \zeta T^+)^{-1} \quad \text{with} \quad \zeta = \frac{-\bar{q}_w}{\bar{\rho}_w \bar{c}_p u_{\tau} \tilde{T}_w}. \qquad (9)$$

Eqs. (4) and (5) then write:

$$1 = (1 + \nu^{++}) \frac{d\psi^+}{d\eta^+}, \qquad (10)$$

$$1 = \left( \frac{1}{Pr} + \frac{\nu^{++}}{Pr_t} \right) \frac{d\theta^+}{d\eta^+} - \frac{1}{\bar{q}_w} \int_0^y \tilde{\omega}_T^I dy. \qquad (11)$$
with the boundary conditions:

$$ \psi^{+}(\eta^{+} = 0) = 0, \quad d\psi^{+}/d\eta^{+}(\eta^{+} = 0) = 1, \qquad (12) $$

$$ \theta^{+}(\eta^{+} = 0) = 0, \quad d\theta^{+}/d\eta^{+}(\eta^{+} = 0) = Pr. \qquad (13) $$

Using the expression $\nu^{++} = (\kappa\eta^{+})^2 d\psi^{+}/d\eta^{+}$ for the dimensionless turbulent viscosity, the integration of Eqs. (10) and (11) yields:

$$ \psi^{+} = \begin{cases} \eta^{+}, & \eta^{+} \le \eta_{c,\psi}^{+} \\ \frac{1}{\kappa} \ln(\eta^{+}) + C_{\psi,0}, & \eta^{+} > \eta_{c,\psi}^{+} \end{cases} \qquad (14) $$

$$ \theta^{+} = \begin{cases} Pr\eta^{+} \left(1 + \frac{1}{q_w} \int_{0}^{y} \tilde{\omega}_{T}^{I} dy\right), & \eta^{+} \le \eta_{c,\theta}^{+} \\ \frac{Pr_t}{\kappa} \ln(\eta^{+}) \left(1 + \frac{1}{q_w} \int_{0}^{y} \tilde{\omega}_{T}^{I} dy\right) + C_{\theta,0}, & \eta^{+} > \eta_{c,\theta}^{+} \end{cases} \qquad (15) $$

where $C_{\psi,0} = 5.0$ and $C_{\theta,0} = 3.9$ [12, 14]. As the influence of the flame on the values $\eta_{c,\psi}^{+}$ and $\eta_{c,\theta}^{+}$ delimiting the viscous and inertial sub-layers respectively is unknown, they are set equal to 10.8 and 13.2 resp. as provided in [12]. Assuming that the relation $u_{\tau} = C_{\mu}^{1/4} \tilde{k}^{1/2}$ is still valid in our case, the wall shear stress is given by:

$$ \bar{\tau}_w = \bar{\rho} C_{\mu}^{1/4} \tilde{k}^{1/2} \frac{\tilde{u}}{\psi^{+}}. \qquad (16) $$

Integrating $d\theta^{+}$ as given by Eq. (8) over the wall distance and using the temperature dependency of the mean mass density as given by Eq. (9) leads to [12]:

$$ \theta^{+} = \frac{1}{\zeta} \ln(\tilde{T}/\tilde{T}_{w}). \qquad (17) $$

The wall heat flux is therefore deduced from Eqs. (15) and (17):

$$ \bar{q}_w = \begin{cases} \frac{-\bar{\rho}_w \bar{c}_p u_{\tau} \tilde{T}_w \ln(\tilde{T}/\tilde{T}_w)}{\frac{Pr}{\kappa}} - \int_{0}^{y} \tilde{\omega}_T^I dy, & \eta^{+} < \eta_{c,\theta}^{+} \quad (\text{or in laminar flows}) \\ -\bar{\rho}_w \bar{c}_p u_{\tau} \tilde{T}_w \ln(\tilde{T}/\tilde{T}_w) - \frac{Pr_t}{\kappa} \ln \eta^{+} \int_{0}^{y} \tilde{\omega}_T^I dy, & \eta^{+} > \eta_{c,\theta}^{+} \end{cases} \qquad (18) $$

This result shows that the wall heat flux is composed of two contributions:

$$ \bar{q}_w = \bar{q}_{LnKC} + f(\eta^{+})\bar{q}_I, \qquad (19) $$

where $\bar{q}_{LnKC}$ is the contribution from the thermal boundary layer as taken into account by the LnKC formulation [12] which depends on the near wall computational cell width and $\bar{q}_I$ is an additional contribution due to the presence of the flame that needs to be modeled. Eq. (18) is derived for a reactive turbulent, steady-state, non separating flow. In the engine, the flow might not correspond to these characteristics, but we suppose here that the wall heat flux in the engine can be expressed as the sum of two contributions as obtained in Eq. (19), and therefore an expression for $\bar{q}_I$ has to be proposed. It is the objective of the next section.
## 4. Proposition of a model for the wall heat flux $\bar{q}_I$ due to the presence of the flame: the $\alpha$-model

The model proposed here to account for the contribution of the flame presence on the wall heat flux is based on the laminar flame-wall interaction process which is extended to the turbulent flame using the flamelet analogy. Therefore a description of the process in a laminar configuration is presented first before proposing an extension to the turbulent flame wall interaction and the modeling of different terms.

### 4.1. Laminar flame-wall interaction

Two main configurations are studied in the literature [27] in the case of laminar interaction: Head on Quenching (HOQ) where the flame propagates in the normal direction towards the wall, and Side Wall Quenching (SWQ) where the flame propagates parallel to the wall. In both configurations the wall heat flux reaches a maximum during the interaction when the distance between the wall and the flame reaction zone reaches a minimum $\delta_Q$ also called the quenching distance. The quenching distance can be related to the diffusive flame thickness $\delta$ through the critical Peclet number following:

$$Pe_Q = \frac{\delta_Q}{\delta} \quad \text{with} \quad \delta = \frac{\lambda^u}{\rho^u c_p S_L^0} \qquad (20)$$

For each flame configuration (HOQ and SWQ), the critical Peclet number is a constant showing that the quenching parameters are correlated with the laminar flame speed and the flame temperature only. It varies approximately between 3.4 for HOQ and 7 for SWQ [27, 28, 29]. In order to relate wall heat flux and the quenching distance, Ferguson and Keck [27] considered that the reaction zone thickness was negligible compared with the preheat zone. Moreover they assumed that the temperature gradient in the quenching zone was linear. Therefore they related the heat released by the flame during interaction to the conductive flux through the quenching area as follows:

$$q_{lam} = \lambda^u \frac{T^{ad} - T_w}{\delta_Q} = \int_0^y \dot{\omega}_T^I dy. \qquad (21)$$

### 4.2. Extension to turbulent flame-wall interaction

In the turbulent case, we propose to base the expression of $\bar{q}_I$, the contribution of the presence of the flame on the wall heat flux, on the laminar behaviour at the flamelet scale. Clearly, $\bar{q}_I$ should also account for the flame brush and wrinkling and it should tend to $q_{lam}$ for laminar flows. A conceptual representation of the flame in the computation cell is presented Fig. 1 (a) to support the derivation of the model:

* the flame is represented as the interface between the fresh and burnt gases where the reaction occurs, its total surface in the cell is defined as $A_{tot}$.

* Part of this surface, $A_R^{NI}$ is contained in a zone too far from the wall for the interaction to occur. It has not been quenched and is therefore reactive.

* Another part of this surface is contained in a zone close to the wall were flamelet wall interaction is likely to occur. It is then either still reactive, at the beginning or during the peak of the interaction, in such case it is called the interactive flame surface $A_R^I$, either it has been quenched by the interaction, in such case it is called $A_{NR}$ the non reactive surface after quenching.
* The delimitation between the interacting and the non interacting flamelet elements is defined by a given distance to the wall $\delta_I$.

<figure><img src="image_2.png" /><figcaption>Figure 1: (a) Schematic of the flame near a wall. The total flame surface $A_{tot}$ is the sum of three contributions: $A_R^I$: interacting flame surface; $A_R^{NI}$: reactive flame surface that does not interact with the wall; $A_{NR}$: non reactive surface after quenching; (b) Principle of the evaluation of $\alpha_I$.</figcaption></figure>

Also, the total surface of the interface $A_{tot}$ can be expressed as $A_{tot} = \int_V \bar{\Sigma} dV = \int_V \Xi |\nabla \bar{c}| dV = \Xi A_{mean}$, where $\Xi$ is the turbulent wrinkling of the flame and $A_{mean}$ is the mean flame surface delimiting the volumes of the fresh and burnt gases. The mean wall heat flux $\bar{q}_I$ is defined by the following expression of the heat transfered $F_I$ to the section of area $A_{mean}$:

$$F_I = \bar{q}_I A_{mean}. \qquad (22)$$

In our flamelet approach, we suppose here that each element of the flame that interacts with the wall leads to a wall heat flux close to the laminar value. As a result, the expression of the heat transfered to the wall can also be expressed as:

$$F'_I = \int_{A_R^I} q_{lam} d\mathbf{A} = A_R^I \bar{q}_{lam}. \qquad (23)$$

Combining expressions (22) and (23) leads to:

$$\bar{q}_I = \int_0^y \tilde{\omega}_T^I dy = \frac{A_R^I}{A_{mean}} \bar{q}_{lam} \quad \text{with} \quad \frac{A_R^I}{A_{mean}} = \underbrace{\frac{A_R^I}{A_R}}_{\alpha_I} \underbrace{\frac{A_R}{A_{tot}}}_{Q_R} \underbrace{\frac{A_{tot}}{A_{mean}}}_{\Xi}. \qquad (24)$$

As a result, the mean wall heat flux due to the presence of the flame is expressed by the contribution of four terms:

$$\bar{q}_I = \alpha_I Q_R \Xi \bar{q}_{lam}, \qquad (25)$$

where $\bar{q}_{lam}$ is the mean maximal wall heat flux, $\alpha_I$ is the fraction of the reactive flame interacting with the wall and $Q_R$ is the fraction of reactive flame surface. An expression of $Q_R$ has been proposed in [10], while the expression of $\Xi$ is modeled by $\Xi = \bar{\Sigma}/|\nabla \bar{c}|$. Therefore a model has to be proposed for $\bar{q}_{lam}$ and $\alpha_I$ in order to complete the expression of the mean wall heat flux due to the presence of the flame.
### 4.3. Modelling of $\bar{q}_{lam}$

In order to determine the mean laminar wall heat flux corresponding to flamelet elements, we consider here that each surface element has its own orientation $\theta$ between the wall and the local flame front. Considering all of these elements, the mean values $\bar{q}_{lam}$ and $\overline{Pe}_Q$ could be obtained using a probability density function $P$:

$$ \bar{q}_{lam} = \int_{\theta=0}^{\theta=\pi} q_{lam}(\theta)P(\theta)d\theta \quad \text{and} \quad \overline{Pe}_Q = \int_{\theta=0}^{\theta=\pi} Pe_Q(\theta)P(\theta)d\theta. \qquad (26) $$

We assume here a dirac function using the mean angle $\bar{\theta}$ between the normal to the wall $\mathbf{n}_w$ and the mean flame propagation direction $\bar{\mathbf{n}}_f$ (Fig. 1(b)):

$$ \bar{q}_{lam} = \bar{q}_{lam}(\bar{\theta}) = \bar{\lambda}^u \frac{\tilde{T}^{ad} - \tilde{T}_w}{\delta_I} \quad \text{with} \quad \delta_I = \delta_Q(\bar{\theta}). \qquad (27) $$

$\bar{\theta}$ is deduced from the normal to the mean flame front $\bar{\mathbf{n}}_f$:

$$ \bar{\theta} = \text{angle}(-\bar{\mathbf{n}}_f, \mathbf{n}_w) \quad \text{with} \quad \bar{\mathbf{n}}_f = -\frac{\nabla \bar{c}}{|\nabla \bar{c}|}, \qquad (28) $$

where $\bar{c}$ is the volume fraction of burnt gases:

$$ \bar{c} = \frac{\bar{\rho}}{\bar{\rho}^b} \tilde{c} \quad \text{with} \quad \tilde{c} = 1 - \frac{\tilde{Y}_F^u}{\tilde{Y}_{TF}^u}. \qquad (29) $$

$\tilde{c}$ is the mass fraction of burnt gases. $\tilde{Y}_{TF}$ is the fuel tracer and corresponds to the fuel mass fraction value before combustion. $\tilde{Y}_F^u$ and $\tilde{Y}_{TF}^u$ are obtained from transport equations [25, 26].

$\bar{q}_{lam}$ can therefore be expressed knowing the quenching distances of the flamelet elements, also related to the critical Peclet number. We propose to express it as a function of the mean flame orientation $\bar{\theta}$ and of its boundary values known from the laminar cases (SWQ and HOQ):

$$ \delta_I = \bar{\delta} Pe_Q(\bar{\theta}) = \bar{\delta} (3.4 |\cos \bar{\theta}| + 7.0(1 - |\cos \bar{\theta}|)) \qquad (30) $$

### 4.4. Modelling of $\alpha_I$

As a first approximation, we assume that the reactive flame surface density is homogeneously distributed (from a statistical point of view) in the near wall computational cell. This assumption can be justified as each local quenching leads to the formation of pockets of burnt gases which are ejected away from the wall by the turbulent flow [28]. In that case, the model for the fraction of reactive flame surface can be applied both to the total flame surface and to the quenching flame surface in the quenching area, so that:

$$ \frac{A_R^I}{A_R} = \frac{A_R^I/Q_R}{A_R/Q_R} = \frac{A^I}{A_{tot}} \qquad (31) $$

where $A^I = A_R^I + A_{NR}^I$ is the total surface in the quenching area. With the assumption of homogeneous flame surface density, the surface ratio $A^I/A_{tot}$ is therefore equivalent to the volume ratio $V^I/V_{tot}$, which is more convenient to estimate. The determination of this ratio requires geometrical considerations that are proposed here. First, we assume here that the mean flame position is a linear function $f_0$ of the wall distance in the hexahedral computation cell
(Fig. 1(b)). In that case, the flame of width $\delta_T$ (the turbulent flame brush thickness) is contained within an hexahedral volume $V_{tot}$ delimited by the computational cell width $\Delta y$ and a critical distance $\delta_{nof}$ under which no chemical reaction occurs. This distance $\delta_{nof}$ can be neglected compared to $\Delta y$. The modeling of the mean flame brush thickness $\delta_T$ is a difficult task which cannot be provided in the absence of DNS data. Also for free propagating flames, the capabilites of most models to correctly describe $\delta_T$ has not yet been proved [30]. Zimont [31] introduced the concept of intermediate steady propagation flames with constant turbulent flame speed and increasing flame brush $\delta_T(t) = 2((2u'l_t t)^2)^{1/2}$. The flame brush does not depend on chemical kinetics but exclusively on the turbulence. S. Richard [32] proposed an analytical expression for fully developed planar flames based on the ECFM approach $\delta_T = 2\Xi\overline{S}_L / (\tilde{K}_T + \tilde{K}_m)$ where $\tilde{K}_m = \langle \nabla \cdot \tilde{u} - \overline{nn} : \nabla \tilde{u} \rangle_s$ and $\tilde{K}_T = \langle \nabla \cdot u'' - \overline{nn} : \nabla u'' \rangle_s$ are the mean and turbulent strain rates acting on the flame surface and induced by the mean flow field and the turbulent motions respectively [33]. But this asymptotic expression is rather appropriate for fully turbulent flames in the flamelet regime. We propose to use here a simple model proposed by Lipatnikov and Chomiak [30] which is in agreement with the present DNS results (not shown here):

$$ \delta_T = \Xi l_t, \qquad (32) $$

where $l_t$ is the integral length scale of turbulence. The evaluation of $\delta_T$ will be presented in future studies. The portion of the flame that interacts is contained in the volume $V^I$ located within the influence area of the wall of width the mean quenching distance $\delta_I$. The evaluation of $V^I$ and $V_{tot}$ requires the knowledge of the mean flame position within the near wall cell. $V^w$ is the volume located between the wall and the mean flame position given by $f_0 = ax + b$ (Fig. 1(b)). The slope $a$ is given by $a = |tan\theta|$. The constant $b$ is unknown. $V^w$ corresponds to the fresh gas volume or to the burnt gas volume whether $cos\theta > 0$ or $cos\theta < 0$. The computation of $b$, $V^I$ and $V_{tot}$ and their analytical solutions are detailed in the Appendix. In the case of head on quenching ($cos\theta = 1$), the fraction $\alpha_I$ is directly deduced from the knowledge of $V^w$, $\delta_T$ and $\delta_I$:

$$ \alpha_I^{HOQ} = \frac{\delta_I - \max(0, V^w/\Delta x - \delta_T/2)}{\min(\Delta y, V^w/\Delta x + \delta_T/2) - \max(0, V^w/\Delta x - \delta_T/2)} \qquad (33) $$

In case of side wall quenching ($cos\theta = 0$), the computation of $\alpha_I$ is trivial:

$$ \alpha_I^{SWQ} = \frac{\delta_I}{\Delta y} \qquad (34) $$

## 5. DNS of the interaction between a turbulent V-flame and a wall

A DNS of the interaction between a turbulent V-flame and a wall was performed with the implicit solver Asphodele [34]. The Navier-Stokes equations were solved using a low-Mach number formulation for variable density flows. Complex chemistry is available but it is computationally expensive therefore only simple chemistry was considered here. Also, the prediction of catalytic reactions and detailed description of radical reactions are beyond the scope of this study. A simplified chemistry [35] is sufficient to account for the oxydation of hydrocarbons on cold wall [29, 36, 37] and to determine the wall heat flux and quenching distance [27, 38], which are the quantities of interest here. The configuration studied was an anchored V shaped flame in a channel [24]. A stoechiometric mixture of propane / air was injected at ambient pressure and temperature with cold wall ($T_w = 300$K). The turbulent 2D case with $Re = U_c * h/\nu = 800$
was used in this study as a first validation of the methodology and model formulation. It was obtained by imposing a turbulent intensity of 27% on the velocity profile at the inlet. The box dimensions were $Lx/h = 16$, $Ly/h = 2$. The combustion regime was characterized by the ratio of the velocity fluctuation to the laminar flame speed $u'/S_L^0 = 3.37$ and of the integral length scale to the diffusive flame thickness $l_t/\delta = 16$.

An instantaneous field of the reaction rate is shown in Fig. 2 while Fig. 3 shows the longitudinal evolutions of the mean angle $\bar{\theta}$, the mean distance of the flame from the wall and the average reaction rate in the boundary layer. The flame position corresponds to the distance between the wall and the maximum reaction rate. As the flame brush approached the wall, flamelets flattened and most of them reached the wall with a normal incidence angle ($\bar{\theta}$ close to 0). The mean Peclet number decreased to a minimum value when total quenching occured at $Pe_Q = 4.3$ (Fig. 3) which was greater than the laminar value for a flame reaching the wall at normal angles ($Pe_Q = 3.4$). This indicates that the fluctuations of the flamelets directions were not negligible and should be taken into account in the model. This point will be investigated in later studies.

The fraction of the flame that interacts $\alpha_I$ was obtained numerically by detection of the flamelets associated with an increase of the wall heat flux during the interaction. The numerical value was compared with the model formulation (section 4.4) obtained from the output of the DNS ($\bar{\theta}, \Xi, l_t, \delta$) in Fig. 4(a). To investigate the effect of the fluctuation of $\theta$ on the model, a simple test was performed by computing the quenching distance following Eq. (26) using a gaussian pdf and various values of the variance $\sqrt{\bar{\theta}^2}$. Globally the modeled results were comparable and in excellent agreement with the DNS data but a fluctuation of 50 degrees in this case was necessary to retrieve the DNS value of the mean quenching distance ($Pe_Q = 4.3$). Fig. 4(b) shows the longitudinal wall heat flux. The DNS value was obtained from the temperature gradient at the wall $\bar{q}_{w,DNS} = \bar{\lambda} * \frac{\partial \bar{T}}{\partial x}$ while the modeled value ($\bar{q}_I$) related to the interaction was obtained from Eq. (24) using the output of the DNS. The location and level of the peak heat flux during quenching was quite well predicted by the model. The total DNS flux varied between 0 before interaction (where only fresh gases were present) and higher values after quenching (where only burnt gases were present near the wall). These boundary values were not retrieved by the LnKC model ($\bar{q}_{lnKC}$) which used the average thermodynamic conditions along the boundary layer width.

In order to evaluate the model validity it was also interesting to analyse the global wall heat flux obtained by integrating $\bar{q}_w$ over the wall, since the entire domain can be viewed as a RANS computational cell. While the LnKC model, commonly used in RANS simulations, was not able to correctly predict the wall heat fluxes since the DNS fluxes gave 27 MW but the LnKC model only gave 14 MW, the $\alpha$-model was in good agreement with the DNS result since it gave a total flux of 26 MW. This result shows that the $\alpha$-model is able to correctly take into account the effect of the flame-wall interaction on the wall heat flux, therefore validating the assumptions used to formulate the model.

# 6. Conclusion

This study was focused on the derivation of a new wall law including the effects of the turbulent flame during its interaction with the wall. To this purpose a model for the heat transfered to
the wall during quenching was proposed. It is based on the flame surface density concept and accounts for the local flame quenching on the mean wall heat flux. A first validation of the model was presented using the DNS results of a 2D turbulent V-flame. In later studies 3D simulations will be performed and used to further validate the model and investigate some modeling issues such as the temperature and velocity profiles in the boundary layers, the flame brush thickness, the statistical distribution of the flamelets orientation.

## 7. Acknowledgements

The authors acknowledge the financial support received from the ANR under the grant ANR-08-VTT-002-01 for the InterMAC project.

## 8. Appendix

This Appendix details the computation of the mean flame position $b$, The volume $V_I$ containing the portion of the flame that interacts and the volume containing the flame brush $V^{tot}$. The fraction of the flame that interacts $\alpha_I$ is defined by $\alpha_I = V_I/V^{tot}$. First the volume $V^w$ located between the wall and the mean flame position $f_0 = ax + b$ is expressed following:

$$V^w = \int_{0}^{\Delta x} (0 < ax + b < \Delta y)dx \qquad (35)$$

Defining $f_0^+$ and $f_0^-$ such as:

$$f_0^+ = \frac{f_0 + |f_0|}{2} = \begin{cases} f & \text{if } f_0 \ge 0 \\ 0 & \text{otherwise} \end{cases} \quad \text{et } f_0^- = \frac{f_0 - |f_0|}{2} = \begin{cases} f & \text{if } f_0 \le 0 \\ 0 & \text{otherwise} \end{cases} \qquad (36)$$

In the opposite reference frame with origin $(\Delta x, \Delta y)$, the mean flame position is defined by the equation $g_0 = ax + \Delta y - b - a\Delta x$. One can write:

$$V^w = \int_{0}^{\Delta x} (f_{0}^{+} + g_{0}^{-}) dx = \frac{1}{2} \int_{0}^{\Delta x} (f_{0} + |f_{0}|) dx + \frac{1}{2} \int_{0}^{\Delta x} (g_{0} - |g_{0}|) dx \qquad (37)$$

The integration of $|f_0|$ along the wall face of the computational cell yields:

$$\int_{0}^{\Delta x} |ax + b|dx = \frac{1}{2a} [(a\Delta x + b) |a\Delta x + b| - b|b|] \qquad (38)$$

Therefore, the following expression for Eq. (37) can be derived:

$$V^w = \frac{\Delta x}{4} \left[ (b + a\Delta x) \left( 1 + \frac{|b+a\Delta x|}{a\Delta x} \right) + b \left( 1 - \frac{|b|}{a\Delta x} \right) + (\Delta y - b) \left( 1 - \frac{|\Delta y - b|}{a\Delta x} \right) + (\Delta y - b - a\Delta x) \left( 1 + \frac{|\Delta y - b - a\Delta x|}{a\Delta x} \right) \right] \qquad (39)$$

The unknown $b$ is obtained by solving Eq. (39).

It is possible to express $V^I$ and $V_{tot}$ as functions of $f_1$ and $f_2$ in Fig. 1(b). The volume of the flame surface interacting with the wall is expressed following:

$$V^I = \frac{1}{2} \int_{0}^{\Delta x} (|f_1| - |f_2| + |h_2| - |h_1|)dx \qquad (40)$$
$f_1$ and $f_2$ are the boundary functions of the flame brush and $h_1$ and $h_2$ are the equations of these boundaries expressed in the opposite reference frame with origin $(\Delta x, \delta_I)$:

$$ \left\{ \begin{array}{ll} f_1(x) = ax + b_p, & b_p = b + \frac{\delta_T}{2|\cos\theta|} \\ f_2(x) = ax + b_m, & b_m = b - \frac{\delta_T}{2|\cos\theta|} \\ h_1(x) = ax - b_p - a\Delta x + \delta_I \\ h_2(x) = ax - b_m - a\Delta x + \delta_I \end{array} \right. \qquad (41) $$

which yields the following analytical solution:

$$ V^I = \frac{1}{4a\Delta x} [f_1(\Delta x) |f_1(\Delta x)| - f_2(\Delta x) |f_2(\Delta x)| - b_p |b_p| + b_m |b_m| + (\delta_I - f_1(\Delta x)) |\delta_I - f_1(\Delta x)| - (\delta_I - f_2(\Delta x)) |\delta_I - f_2(\Delta x)| - (\delta_I - b_p) |\delta_I - b_p| + (\delta_I - b_m) |\delta_I - b_m|] \qquad (42) $$

Similarly, the total volume of the flame brush is written following:

$$ V^{tot} = \frac{1}{2} \int_{0}^{\Delta x} (|f_1| - |f_2| + |g_2| - |g_1|)dx \qquad (43) $$

with

$$ \left\{ \begin{array}{l} g_1(x) = ax - b_p - a\Delta x + \Delta y \\ g_2(x) = ax - b_m - a\Delta x + \Delta y \end{array} \right. \qquad (44) $$

which leads to:

$$ V^{tot} = \frac{1}{4a\Delta x} [f_1(\Delta x) |f_1(\Delta x)| - f_2(\Delta x) |f_2(\Delta x)| - b_p |b_p| + b_m |b_m| + (\Delta y - f_1(\Delta x)) |\Delta y - f_1(\Delta x)| - (\Delta y - f_2(\Delta x)) |\Delta y - f_2(\Delta x)| - (\Delta y - b_p) |\Delta y - b_p| + (\Delta y - b_m) |\Delta y - b_m|] \qquad (45) $$

## References

1. J. Heywood, Internal Combustion Engine Fundamentals, Mc-Graw-Hill Eds, 1988.

2. P. Leduc, B. Dubar, A. Ranini, G. Monnier, Downsizing of gasoline engine: an efficient way to reduce co_2 emissions, Oil & Gas Sci. and Tech. Rev. IFP58 (1) (2003) 157–127.

3. B. Boust, J. Sotton, S. Labuda, M. Bellenoue, A thermal formulation for single-wall quenching of transient laminar flames, Combust. Flame 149 (2007) 286–294.

4. S. Labuda, M. Karrer, J. Sotton, M. Bellenoue, Experimental study of single-wall flame quenching at high pressures, Combust. Sci. Tech. 183 (2011) 409–426.

5. Z. Sroka, Some aspects of thermal load and operating indexes after downsizing for internal combustion engines, J. Therm. Anal. Calorim. 10973 (2011) 1–8.

6. F. Foucher, S. Burnel, C. Mounaïm-Roussel, Evaluation of burning rates in the vicinity of the piston in a spark-ignition engine, Proc. Combust. Inst. 29 (2002) 751–757.

7. F. Foucher, S. Burnel, C. Mounaïm-Rousselle, M. Boukhakfa, B. Renou, M. Trinité, Flame wall interaction: effect of stretch, Exp. Thermal Fluid Sci. 27 (2003) 431–437.

8. T. Alshaalan, C. Rutland, Turbulence, scalar transport, and reactions rates flame-wall interaction, in: 27th Symp. (Int.) on Combust., The Combustion Institute, 1998, pp. 793–799.

9. T. Alshaalan, C. Rutland, Wall heat flux in turbulent premixed reacting flow, Combust. Sci. Tech. 174 (2002) 793–799.

10. G. Bruneaux, T. Poinsot, J. Ferziger, Premixed flame-wall interaction in a turbulent channel flow: budget for the flame surface density evolution equation and modelling, J. Fluid Mech. 349 (1997) 191–219.
11. A. Gruber, R. Sankaran, E. Hawkes, J. Chen, Turbulent flame-wall interaction: a direct numerical simulation study, J. Fluid Mech. 658 (2010) 5–32.

12. C. Angelberger, T. Poinsot, B. Delhaye, Improving near-wall combustion and wall heat transfer modelling in si engine computations, SAE Paper 972881.

13. Z. Han, R. Reitz, A temperature wall function formulation for variable-density turbulent flows with application to engine convective heat transfer modeling, Int. J. Heat Mass Transfer 40 (3) (1997) 613–625.

14. W. Kays, M. Crawford, Convective Heat and Mass Transfer, 3rd Edition, McGraw Hill, 1994.

15. R. Dahms, M. Drake, R. Grover, A. Solomon, T. Fansler, Detailed simulations of ignition and combustion processes in a spray-guided gasoline engine using the sparkcimm/g-equation modeling framework, SAE paper 2012-01-0132.

16. S. Yang, R. Reitz, Improved combustion submodels for modelling gasoline engines with the level set g equation and detailed chemical kinetics, in: Proc. of the IMechE, Part D: J. of Automobile Eng. 223, 2009, pp. 703–726.

17. O. Cabrit, F. Nicoud, Direct simulations for wall modeling of multicomponent reacting compressible turbulent flows, Phys. Fluids 21 (2009) 055108.

18. C. Choi, K. Huh, Development of a coherent flamelet model for a spark-ignited turbulent premixed flame in a closed vessel, Combust. Flame 114 (1998) 336–348.

19. K. Bray, P. Libby, J. Moss, Flamelet crossing frequencies and mean reaction rates in premixed turbulent combustion, Combust. Sci. Tech. 41 (1984) 143–172.

20. N. Darabiha, V. Giovangigli, A. Trouvé, S. Candel, E. Esposito, Coherent flame description of turbulent premixed ducted flames, in: R. Borghi, S. Murthy (Eds.), Lecture Notes in Engineering, Vol. 40, Springer, 1987, pp. 591–637.

21. J. Duclos, D. Veynante, T. Poinsot, A comparison of flamelets models for premixed turbulent combustion, Combust. Flame 95 (1993) 101–117.

22. F. Marble, J. Broadwell, The coherent flamelet model for turbulent chemical reactions, Tech. Rep. Project Squid, TRW-9-PU (1977).

23. S. Richard, O. Colin, O. Vermorel, A. Benkenida, C. Angelberger, D. Veynante, Towards large eddy simulation of combustion in spark ignition engines, Proc. Combust. Inst. 31 (2007) 3059–3066.

24. B. Leveugle, Y. D'Angelo, J. Réveillon, Dns of flame / wall interaction in a strongly non isothermal case, in: Proc. European Combustion Meeting, Cardiff, UK, 2011.

25. O. Colin, A. Benkenida, C. Angelberger, 3d modeling of mixing, ignition and combustion phenomena in highly stratified gasoline engines, Oil & Gas Sci. and Tech. Rev. IFP58 (1) (2003) 47–62.

26. O. Colin, A. Benkenida, The 3-zones extended coherent flame model (ecfm3z) for computing premixed/diffusion combustion, Oil & Gas Sci. and Tech. Rev. IFP59 (6) (2004) 593–609.

27. C. Ferguson, J. Keck, On laminar flame quenching and its application to spark ignition engines, Combust. Flame 28 (1977) 197–205.

28. T. Poinsot, D. Haworth, G. Bruneaux, Direct simulation and modeling of flame-wall interaction for premixed turbulent combustion, Combust. Flame 95 (1993) 118–132.

29. C. Westbrook, A. Adamczyk, G. Lavoie, A numerical study of laminar flame wall quenching, Combust. Flame 40 (1981) 81–99.

30. A. Lipatnikov, J. Chomiak, Turbulent flame speed and thickness: phenomenology, evaluation and application in multi-dimensional simulations, Prog. Energy Comb. Sci. 28 (2002)
1-74.

31. V. Zimont, Gas premixed combustion at high turbulence. turbulent flame closure combus-
tion model, Exp. Thermal Fluid Sci. 21 (2000) 179–186.

32. S. Richard, Simulation aux grandes échelles de la combustion dans les moteurs à allumage commandé, Ph.D. thesis, Ecole Centrale Paris, France (2005).

33. T. Poinsot, D. Veynante, Theoretical and Numerical Combustion, 3rd Edition, R.T. Edwards, 2011.

34. M. Chauvy, B. Delhom, J. Reveillon, F.-X. Demoulin, Flame/wall interactions: Laminar study of unburnt hc formation, Flow Turb. Combust. 349 (2010) 369–396.

35. L. Vervisch, B. Labegorre, J. Reveillon, Hydrogen-sulphur oxy-flame analysis and single-step flame tabulated chemistry 83 (2004) 605–614.

36. W. Bush, F. Fendell, S. Fink, Effect of boundary thermal constraint on planar premixed-flame/wall interaction, Combust. Sci. Tech. 24 (1-2) (1980) 53–70.

37. W. Hocks, N. Peters, G. Adomeit, Flame quenching in front of a cold wall under two-step-kinetics, Combust. Flame 41 (1981) 157–170.

38. O. Ezekoye, R. Greif, R. Sawyer, Increased surface temperature effects on wall heat transfer during unsteady flame quenching, in: 24th Symp. (Int.) on Combust., The Combustion Institute, 1992, pp. 1465–1472.
<figure><img src="image_3.png" /><figcaption>Figure 2: Instantaneous 2D field of the fuel reaction rate obtained from the numerical simulation.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Figure 3: Numerical result: mean longitudinal evolutions of the Peclet number $\overline{Pe}$, the angle $\overline{\theta}$ and the average fuel consumption rate $\overline{\omega}_F$.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 4: Validation of the $\alpha$-model: (a) the effect of the fluctuation of $\theta$ in the modeling of the quenching distance is taken into account using a gaussian pdf following Eq. (26); (b) $\bar{q}_I$: contribution of the modeled flux during the interaction, $\bar{q}_{LnKC}$: contribution of the flux provided by the LnKC model, $\bar{q}_w$: total wall heat flux. Symbols: DNS result; solid lines: modeled values.</figcaption></figure>