# Subgrid Reaction-Diffusion Closure for Large Eddy Simulations Using the Linear-Eddy Model

## Abstract

Turbulent combustion models approximate the interaction between turbulence, molecular transport and chemical reactions. Among the many available turbulent combustion models, the present focus is the linear-eddy model (LEM) used as a subgrid combustion model for large eddy simulations. In particular this paper introduces a new LEM closure with the reaction-rate approach to close the filtered chemical source terms in the governing equations for species mass fractions and enthalpy. The new approach is tested using a non-premixed syngas flame and a bluff-body stabilized premixed flame problem. Simulation results are compared to data from a direct numerical simulation and experiments. This comparison shows that mean and rms quantities compare well with experiments and are in the range of previous simulation studies. These results are obtained with a pressure-based and unstructured computational-fluid-dynamics solver, an approach that is preferred in industry.

## Keywords

Turbulent combustion model Large eddy simulations Linear-eddy model Chemical source term closure Bluff-body stabilized flame Reacting temporal jet Splicing## 1 Introduction

Turbulent combustion happens in combustion devices such as internal combustion engines and gas-turbine engines [1, 2]. It involves an interaction between highly non-linear chemical reactions, turbulence and molecular mixing. Turbulence is characterized by a broad range of scales in spacetime, ranging from the smallest Kolmogorov scales to the largest flow structures characterized by the scales of the flow geometry. Computational fluid dynamics (CFD) provides useful tools to study turbulent combustion. One such CFD tool is direct numerical simulations (DNS), which aims at fully resolving the flow physics. However, DNS is currently computationally affordable only for relatively simple problems. In contrast, CFD approaches for engineering problems approximate the above physics with turbulent combustion models. There are numerous turbulent combustion models available [2, 3] and several studies have compared them [4, 5, 6].

From a practical point of view it is desirable to have a turbulent combustion model that is generally valid, which means that it is predictive for both premixed and non-premixed combustion modes and independent of chemical time scales (fast/flamelet-type chemistry vs. non-fast chemistry). One such type of model is the transported probability-density-function (TPDF) method [43, 44, 45]. Another one is the linear-eddy model (LEM) of Kerstein [7] when used as a subgrid combustion model for large eddy simulations (LES) [8, 9], in which case it is called LES-LEM.

LEM was initially formulated as a scalar mixing model for non-reacting flows [7] and later extended to predict reactive flows [10]. LEM relies on a one-dimensional line on which all spatial and temporal scales are resolved, hence reducing computational cost relative to three-dimensional DNS while retaining high resolution. Modeling of turbulence is done via stochastic mapping events. Diffusion and reaction are represented with reactive-diffusive partial differential equations.

LES-LEM has been employed in a large variety of combustion problems [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. McMurtry et al. [24] showed simulation results of turbulent scalar mixing. Non-premixed turbulent combustion simulations were done by Calhoon [25] and Menon and Calhoon [26], and premixed turbulent combustion by Smith [27] and Chakravarthy and Menon [28]. Spray combustion simulations were performed by Pannala and Menon [29], and by incorporating radiative heat loss and soot modeling by Zimberg et al. [30]. Predictions of a premixed turbulent methane/air flame were presented by Sankaran and Menon [31]. Sen and Menon employed artificial neural networks to speed up chemistry calculations [32, 33]. LES-LEM with a low-Mach-number numerical scheme was used to simulate the Sandia flame D by Ochoa et al. [34]. Choi and Menon applied LES-LEM for simulations of a cavity-stabilized combustor [35]. A few examples of simulation on reciprocating engines include work done in KIVA for simulating a direct injection spark ignition engine using LES-LEM [36], and an URANS-LEM method in order to investigate pressure histories in an automotive HCCI engine [37]. Martinez et al. [38, 39] studied turbulent combustion of hydrogen-enriched fuels. Lovett et al. [40] studied the flame structure of bluff-body stabilized flames. Srinivasan et al. [41] used LES-LEM for investigating combustion instabilities in a continuous variable resonance combustor (CVRC) and also performed spray combustion simulations [42].

Turbulent combustion models can be used in at least two different ways [3]. One is the so-called primitive-variable method in which the models provide closure for the filtered or averaged thermochemical quantities, i.e. the species mass fractions and an energy variable such as enthalpy or temperature. This method is perhaps the most popular one and is typically used with flamelets [2], transported-PDF methods [43, 44, 45], and in most previous LES-LEM studies as well. The other method is the reaction-rate approach in which the turbulent combustion model provides closure for the chemical source terms in the conservation equations of the thermochemical quantities, i.e. models of this type solve transport equations for all species and a form of the energy equation. Thus, this approach is computationally more expensive than the primitive-variable method. However, the approach can be regarded as more broadly applicable as it simplifies the inclusion of more physics into the modeling [3].

An additional advantage of using the reaction-rate approach is that it can potentially ease the use of more than one turbulent combustion model during a simulation. For example, consider a combustion problem involving a vigorously burning flame in some region in spacetime, a flame undergoing extinction-reignition in another region, and ignition in yet another region. This is a multi-physics combustion problem. Knowing the fact that turbulent combustion models can be more cost-effective (e.g. in terms of accuracy and computational cost) for a particular type of physics but not necessarily for other physics, it becomes desirable to use one particularly suitable model for one region in spacetime and other suitable models in other regions. The potential for such an approach to lead to more cost-effective combustion simulations is beginning to be recognized [46, 47]. This is the case not only in the combustion community but, for instance, in the multi-phase-flow community as well [48]. Within this hybrid framework, the primitive-variable method runs into the difficulty that turbulent combustion models handle the turbulent transport in different ways, as discussed shortly for the case of LEM. In comparison, the reaction-rate approach does not run into this problem as it solves for the conservation equations of the thermochemical quantities. Furthermore, the reaction-rate approach isolates modeling errors due to the turbulent combustion model in the chemical source terms, simplifying as well the comparison of errors from different models.

In the LEM framework, the reaction-rate approach could also help in reducing potential artifacts produced by the splicing algorithm. This algorithm, as clarified later, is used to transport the thermochemical quantities residing on the individual LEM domains between LES cells. It can result in a spatial distribution of scalars that may differ from the distribution that would be obtained by solving the filtered or averaged conservation equations. Thus, by solving such equations concurrently with the splicing, it is possible to make adjustment to the spatial distribution of a thermochemical quantity if such an inconsistency is considered unacceptably large.

The primitive-variable approach is standard for previous LES-LEM implementation so it is valuable then to explore how well a new LEM closure with the reaction-rate approach performs. Therefore, the present work and a parallel one [49] use LEM following the reaction-rate approach. The present work differs from this previous one in that it uses a significantly different code framework, and it retains the splicing algorithm instead of getting rid of it. Importantly, the latter feature allows the present approach to retain the time-history of the subgrid solution rather than disregard it, at the cost of making the code more complex. In principle this facilitates a more accurate modeling of highly unsteady phenomena, e.g. extinction and re-ignition. This potential capability, however, remains to be shown and is not addressed here. The present work makes use of unstructured meshes and a pressure-based approach, both of which are usual industry practice. The current paper is a major revision and expansion of a previous conference paper [50]. Common material is reprinted with permission from AIAA.

The present paper is arranged as follows. Section 2 introduces the new proposed methodology based on LEM. Thereafter, Section 3 compares predictions with this methodology with those from DNS and traditional LES-LEM for a syngas flame problem [51]. Section 4 compares simulation results with experiments for a bluff-body stabilized flame combustor called the Volvo combustor from now on [52, 53, 54]. These two test problems are selected since they have been used in the development of various turbulent combustion models and serve the purpose of determining whether the present approach works well. These tests, however, suffer from two serious shortcomings discussed later on. Due to these shortcomings, Section 5 presents a sensitivity study for high-Reynolds-number flames. Finally, simulation results are used to further explain how the present model works, and then the paper finishes with conclusions in Section 7.

## 2 Methodology

### 2.1 LES equations

The current work uses the OpenFOAM library version 2.3.1 [55, 56] and considers the following filtered conservation equations for global mass, momentum, sensible enthalpy, and species mass for gases that are variable density, viscous, heat-conducting and multiple-component and move at low-Mach-number speeds:

*ρ*denotes the density,

*p*the pressure,

*u*

_{i}is the velocity component in spatial direction

*i*,

*h*is the sensible enthalpy, and

*Y*

_{α}is the mass fraction of the species

*α*. The bar denotes conventional filtering and the tilde density weighted Favre filtering. \(\bar \tau _{ji}\), \(\bar q_{j}\), and \(\bar j_{\alpha ,j}\) are respectively the filtered viscous stress tensor, heat flux, and Fickian molecular flux of species

*α*. Likewise, \(\tau _{ji}^{sgs}\), \(q_{j}^{sgs}\), and \(j_{\alpha ,j}^{sgs}\) are the subgrid viscous stress tensor, heat flux, and molecular flux of species

*α*, all of which need closure. These conservation equations are complemented with the equation of state for ideal gases

*W*

_{α}is the molecular weight of species

*α*and

*R*

_{m}is the universal gas constant. The averaged caloric equation of state is given as,

*h*

_{α}(

*T*) given by the NASA polynomials [57].

The terms \(\tau _{ij}^{sgs}\), \(q_{j}^{sgs}\), and \(j_{\alpha ,j}^{sgs}\) are the subgrid stress tensor, heat flux, and mass flux of species *k*. They are closed with

*μ*

^{sgs}is computed with OpenFOAM’s implementation of the Smagorinsky model [58]

*C*

_{k}being a model constant (

*C*

_{k}= 0.02), Δ the cube root of a given finite volume, and

*k*

^{sgs}the SGS kinetic energy obtained from

*a*=

*C*

_{e}/Δ, \(b=2 \widetilde {S}_{kk} / 3\), \(c=2 C_{k} {\Delta } (\widetilde {S}_{ij} - \delta _{ij} \widetilde {S}_{kk} /3) \widetilde {S}_{ij}\), and

*C*

_{e}= 1.048. For incompressible flows, the constants

*C*

_{k}and

*C*

_{e}are related to the constant

*C*

_{s}of the typical implementation of the Smagorinsky model [59] through \(C_{s}=C_{k}^{3/4}C_{e}^{-1/4}\). For the Volvo combustor simulations,

*μ*

^{sgs}is computed with OpenFOAM’s implementation of Yoshizawa & Horiuti’s one-equation eddy model [60].

Closure of the averaged source terms \(\bar S_{h}\) and \(\bar S_{\alpha }\) is done as follows.

### 2.2 Closure of the chemical source terms with the no-model approach

With the no-model (also called the perfectly stirred reactor model) approach, the averaged chemical source term is given by:

*α*. It means that the chemical source term is simply evaluated using directly the filtered values of the resolved fields.

### 2.3 LEM subgrid modeling inside a LES cell

The LES-LEM model is composed of the following elements: the LES equations discussed before; the LEM modeling inside a LES cell; the exchange of information between the LEM and the LES; the LEM modeling “between” LES cells; and, for the current formulation the closure of \(\bar S_{h}\) and \(\bar S_{\alpha }\).

The hallmark feature of the LEM is a 1D domain resolving all scales [9] having an inflow on one end and an outflow at the other end. In LES-LEM, there is one 1D domain per computational LES cell, as shown in Fig. 1. All the cells of the LEM domain within a LES cell are uniformly initialized with the thermochemical conditions in that LES cell. The initial LEM domain length is chosen to be equal to the local LES mesh spacing Δ. The solution on the LEM domain time advances the conservation equations in their 1D form as:

*M*

_{LEM}is the total mass in a LES cell, the summation (index i) is done over all cells of the LEM domain, the

*x*coordinate is parallel to the 1D array, and

*S*

_{T}is the heat-source term in the temperature equation. Constant pressure is assumed on the LEM which allows updating of the LEM density resulting in change of the LEM domain length. Here unity Prandtl and Lewis numbers are used to compute

*q*

_{x}and

*j*

_{α,x}.

*M*(

*x*) is a mapping operation known as the triplet map. The effect of the triplet map on a flow property profile defined in [

*x*

_{0},

*x*

_{0}+

*l*] is to replace the profile with three compressed images of the original with the middle image flipped. This is how the rotational and compressive motions observed in the turbulent flows are represented in the LEM [62].

The triplet maps are parametrized by a location *x*_{0}, a length *l*, and an eddy rate *λ* per unit domain length. They are implemented in a stochastic way by sampling *x*_{0} from a uniform distribution and *l* from

*l*

_{p}and

*l*

_{max}are the smallest and the largest unresolved length-scales characterizing the turbulence, and are specified by the user. Here

*l*

_{p}is taken as the Kolmogorov length-scale

*η*, and

*l*

_{max}equals the local mesh spacing of the LES solver Δ. Implied in Eq. 17 are Kolmogorov’s scalings in the inertial subrange (e.g. the time-scale of an eddy

*τ*scales as

*τ*∼

*l*

^{2/3}). An estimate for the Kolmogorov length-scale is given by

*N*

_{η}being a model constant, and

*R*

*e*

_{Δ}the subgrid Reynolds number:

*ν*is the kinematic viscosity and

*u*

_{sgs}is the characteristic subgrid velocity fluctuation obtained from the LES subgrid turbulent kinetic energy \(\bar k_{sgs}\) as \(u_{sgs}=\sqrt {2\bar k_{sgs}/3}\).

The eddy rate *λ* per unit domain length with unit (length x time)^{-1} is obtained through an analogy between eddy sampling statistics and inertial range scaling laws and is computed via

*C*

_{λ}being a model constant. The average time interval between triplet maps is:

*l*

_{LEM}is the LEM domain length and the actual eddy occurrences are sampled from a Poisson process. The sampling rejected the eddies that extend beyond the LEM domain boundary. The values of

*C*

_{λ}= 1 and

*N*

_{η}= 1.1 are used throughout unless said otherwise. These values were chosen to ensure that triplet maps are occurring during the simulation.

### 2.4 LES-LEM advection modeling

Another element of the present LES-LEM formulation is an algorithm that transfers the subgrid-scale quantities, i.e. those defined in the 1D arrays, between 1D arrays in the adjacent LES computational cells. The present work uses the splicing algorithm since it has proven to be robust for practical combustion implementations. The splicing algorithm cuts and pastes the end portions of the 1D arrays of the adjacent LES cells to emulate a Lagrangian type of mass transfer. Decisions of how much to cut and paste are based on values of the mass fluxes across the faces of LES computational cells. For example, if the mass flux at a given face of cell A is large and directed from cell A to cell B, a large portion of the 1D array in A will be cut and then pasted onto one of the two endpoints of the array in B. There is some arbitrariness, however, in how this is exactly done. If this mass flux were small, a small portion would be cut and pasted. Details about the splicing algorithm are given in [50, 64].

### 2.5 Closure of the chemical source terms with LEM

Figure 2 shows the coupling between the LES and the LEM subgrid combustion model. The LES provides information about the cell spacing Δ, the subgrid turbulent kinetic energy \(\bar k_{sgs}\), the time step of the simulation Δ*t* and the pressure to the LEM. With this information, the LEM subgrid-scale simulation advances Eqs. 13–20 for the LES time interval Δ*t*. The chemical source terms \(\bar S_{h}\) and \(\bar S_{\alpha }\) are obtained by taking the median or mean value along various LEM elements for the current LES time step. Mean values are calculated (e.g. for species mass fractions) with

*x*

_{i}denotes the non-uniform length of an individual LEM cell. The median source term, on the other hand, is calculated by first sorting the source term values of all the LEM elements in ascending order, and then picking up the middle value. Apparently, the chosen value can come from a different cell for each species. Comparison of predictions using the mean and the median is discussed later in Section 4.4. Although the use of the mean is intuitive, the median is considered here as well because it is a more robust measure since it prevents outliers. These source terms are then used in the LES conservation equations for the species mass fractions and sensible enthalpy. The temperature on the LES side is then calculated from the caloric equation of state, Eq. 6.

In the present implementation, the PISO algorithm [63] is used. It has an outer loop which starts with predicting a velocity (using Eq. 2), which is followed by an inner iterative loop to correct the velocity field using the outcome of a pressure Poisson equation obtained by fulfilling the continuity equation. In the inner iterative loop, the LES density is obtained from the equation of state Eq. 5. After splicing is done with the predicted LES mass flux, the LEM lines are restored back to the state before the predicted splicing was done, in order to do the splicing with the correct LES mass flux as described in [64].

The last step is to correct the temperatures and the species mass fractions in the LEM domains to take into account the fact that the convection due to the solution of Eqs. 1-4 differs from that of the splicing algorithm. The need for this step is discussed further in Section 6. There is not a unique way to conduct this correction, and the following algorithm is a result of trial and error and has been found to be robust, as demonstrated later.

The first step of the correction computes the difference *T*^{′} between the LES temperature \(\tilde T\) and the Favre-averaged LEM temperature \(\tilde T_{LEM}\), the latter of which is obtained via:

*T*

_{i}denotes the LEM cell temperature, and the summation (index i) is over all cells of the LEM array. In a second step,

*T*

^{′}is added to each

*T*

_{i}. This updates \(\tilde T_{LEM}\). If the new updated difference

*T*

^{′}is within a specified tolerance the correction is stopped, otherwise the updated difference

*T*

^{′}is added again to each

*T*

_{i}and the aforementioned steps are repeated.

At this point, a distinctive conceptual feature of the present closure must be highlighted. It is common in LES to make a sharp distinction between the closure of diffusive processes (micromixing) and reaction. This is evident by noticing that in Eqs. 1–4 separate closure is needed for, on one hand, \(\bar q_{j}\) and \(\bar j_{\alpha ,j}\), and, on the other hand, \(\bar S_{h}\) and \(\bar S_{\alpha }\). However, a more realistic picture is that diffusive and reacting processes are closely coupled at the subgrid level (meaning reaction and diffusive terms can be of the same order), for example, in some premixed flames. Now, with LEM \(\bar S_{h}\) and \(\bar S_{\alpha }\) are computed as explained above using a 1D structure that represents both reaction and diffusion. Thus, it is perhaps more accurate to say that the present closure, although given for \(\bar S_{h}\) and \(\bar S_{\alpha }\), is not a closure for subgrid reaction, but for subgrid reaction-diffusion processes, i.e. with proper resolution the LEM is capable of fully resolving flame structures in 1D. This is why the approach is termed a subgrid reaction-diffusion closure with LEM.

### 2.6 Numerical method

The conservation equations Eqs. 1–4 are solved with an adaptation of reactingFoam which is the standard solver of the OpenFOAM library for chemically reacting flow. This solver is transient, pressure-based and can handle unstructured meshes [64]. A low-Mach-number assumption is used as done in reactingLMFoam [65, 66] by splitting the pressure into a fluid-mechanical-induced pressure and a thermodynamic pressure, the latter of which is spatially constant and which is used to evaluate the equation of state. As a result, acoustic waves are eliminated from the solution. The OpenFOAM library has various temporal and spatial discretization options available. In the current work, the time discretization is done with a second-order backward-differencing scheme that uses the current and previous two time step values. OpenFOAM’s limited linear differencing scheme is used for the convective and diffusive fluxes.

## 3 Medium-Reynolds-Number Syngas Flame

### 3.1 Configuration

*L*

_{x}×

*L*

_{y}×

*L*

_{z}studied with DNS by Hawkes et al. [51]. Fig. 4 shows a schematic of the jet. The fuel stream is an initially planar jet moving in the

*x*-direction, with the oxidizer moving in the opposite direction on either side of the fuel stream. Turbulence is generated by shear at the interface between the two jets. The fuel is syngas, comprising 50

*%*CO, 10

*%*H

_{2}, and 40

*%*N

_{2}by volume. The oxidizer is composed of 25

*%*O

_{2}and 75

*%*N

_{2}by volume. Both fuel and oxidizer are initially at 500 K and react according to a skeletal mechanism for C

_{1}-kinetics [51]. The pressure is atmospheric. The characteristic jet velocity is

*U*, and at

*t*= 0 both streams have a velocity of

*U*/2 in their respective directions. A characteristic length-scale of the problem is the jet height

*H*, and a characteristic jet time scale is

*t*

_{j}=

*H*/

*U*. Thus, a Reynolds number can be defined as

*R*

*e*= (

*U*

*H*)/

*ν*

_{f}, where

*ν*

_{f}is the kinematic viscosity of the pure fuel. The Damköhler number is defined as

*D*

*a*=

*χ*

_{q}

*t*

_{j}, with

*χ*

_{q}being the quenching scalar dissipation rate for the flamelet solution. The size of the computational domain is given in terms of the characteristic length as

*L*

_{x}= 12

*H*,

*L*

_{y}= 14

*H*, and

*L*

_{z}= 8

*H*. The (global) Mach number is

*M*=

*U*/(

*c*

_{1}+

*c*

_{2}), where

*c*

_{1}= 476 ms

^{− 1}, and

*c*

_{2}= 445 ms

^{− 1}are the sound speeds in the fuel and oxidizer streams respectively. Periodic boundary conditions are specified in the streamwise (

*x*) and the spanwise (

*z*) directions. The mesh spacing is uniform and equal in all directions.

For the simulations presented in this work, *H* = 0.96 mm, and *U* = 193 ms^{− 1}, which correspond to case M of Hawkes et al. [51]. The jet Reynolds number is 4478, and the jet characteristic time is *t*_{j} ∼ 5 *μ* s. The LES is set up closely following the setup used in the DNS study. However, whereas the DNS used a uniform mesh size of 768 × 896 × 512, the LES uses a mesh 8 times coarser, 96 × 112 × 64. The DNS mesh spacing corresponds roughly to the Kolmogorov length-scale.

In the DNS, combustion is started by superimposing a laminar flamelet solution and by using broadband turbulent velocity fluctuations. In the present LES, all fields are initialized by mapping the DNS initial condition onto the LES mesh. One more difference with the DNS is treatment of the boundary conditions at the ends of the vertical extent of the domain. Here Neumann boundary conditions are used with all quantities except pressure, which is set to an atmospheric value.

For the syngas jet, comparisons are presented here for two different simulations: a simulation with no turbulent combustion model (the no-model approach), and with the LES-LEM model. The LES-LEM simulations used the median source terms. Using the mean source terms produced almost identical results. The mesh used to simulate this jet is formed by uniform cubical control volumes. For the advective terms in the transport equations, a second-order center-difference scheme is used.

### 3.2 Comparison with DNS

*t*

_{j}. This is likely due to the present use of a simplified molecular transport modeling (e.g., unity Lewis numbers for all species) since a preliminary no-model-approach simulation (not shown) not using this simplified modeling captures well the profiles at 10

*t*

_{j}. The impact of more realistic molecular diffusion models will be investigated in the future. Nonetheless, overall the simulation results agree well with the DNS, which was the main purpose of evaluating the new implementation here. In addition, Figs. 5 and 6 show that for this case LES-LEM does not show a clear benefit over the no-model approach. One possible explanation is that the Reynolds number associated with this problem is not large enough for the detailed turbulent combustion models to fully kick in. This is further discussed in Section 6.

### 3.3 Comparison with traditional LES-LEM

## 4 Volvo Combustor

The Volvo combustor is a standard test case for the evaluation of turbulent premixed combustion models which has been investigated extensively in the past. Here the case is used to compare the new LES-LEM implementation with experiments and other modeling approaches and to evaluate the impact of certain model parameters within LES-LEM on the simulation results.

### 4.1 Configuration

The experimental setup of the Volvo test problem [52, 53, 54] consists of a rectilinear channel, with an inlet section where air and (gaseous) propane mix before entering a honeycomb, and a downstream section where the premixed reactants burn in a flame that gets stabilized downstream of a wedge [52]. The cross section of the wedge is an equilateral triangle with side of 40 mm. The end of the downstream section is connected to a round exhaust duct with a cross sectional area about 3.4 times larger than that of the channel [67]. Unfortunately, the length of this exhaust section has not been documented in the open literature. Atmospheric conditions are used.

*h*

_{1}= 40 mm,

*h*

_{2}=

*h*

_{1}

*s*

*i*

*n*60,

*x*

_{min}= -820 mm,

*x*

_{max}= 680 mm,

*y*

_{min}= -60 mm,

*y*

_{max}= 60 mm, and a spanwise length of 40 mm. The honeycomb is ignored as done in previous CFD studies. The pressure gradient normal to the inlet plane is set to zero. No turbulent fluctuations are imposed at the inlet because the region of interest is downstream of the wedge which is responsible for generation of strong turbulence. All walls are modeled as no-slip and adiabatic, unless stated otherwise. At the outlet, velocity, temperature and species are set to a zero-normal-gradient condition or a fixed value, depending on the direction of the flow. Pressure is dealt with a similar way at the outlet. Cyclic boundary conditions are used in the z-direction, unless indicated otherwise. An unstructured hexahedral 3D mesh with 77400 cells is used with the simulation time step of 10

^{− 6}s.

Summary of the simulation settings for case 1

Simulation | Denomination | | Correction | sgs fluxes in | Source terms | Combustion in cold |
---|---|---|---|---|---|---|

No. | splicing | calculation | regions | |||

1 | No-model | NA | NA | NA | NA | NA |

2 | LES-LEM baseline | 1 | on | on | Median | on |

3 | LES-LEM | 10 | on | on | Median | on |

6 | LES-LEM correction off | 1 | off | on | Median | on |

7 | LES-LEM sgs splicing off | 1 | on | off | Median | on |

10 | LES-LEM combustion off | 1 | on | on | Median | off |

cold regions | ||||||

11 | LES-LEM mean source terms | 1 | on | on | Mean | on |

Single step chemical reaction and five species mechanism for propane combustion for case 1 (rate constants are written as *A**T*^{b} x exp(− *E*/*R**T*); units are cm, mol, s, K, cal)

Reaction No. | Reactions | A | b | E (cal) |
---|---|---|---|---|

1 | C | 1.50E + 10 | 0 | 20000 |

### 4.2 Comparison with experiments

*y*= 0 plane. This agrees with experiments (cf. Fig. 9 in Sjunnesson et al. [54]) and many other simulation studies. However, the transition towards an asymmetric (sinusoidal) flame shape further downstream agrees with some simulation studies (Fig. 4b in Moller et al. [69], Fig. 8a in Fureby [70] and Fig. 3 in Comer et al. [71]) and disagrees with others where the flame is predominantly symmetric all the way towards the outlet (Fig. 4 in Giacomazzi et al. [72], Cocks et al. [73], and perhaps also Baudoin et al. [74]). This latter predominantly symmetric shape appears to be the physically correct one for the present boundary conditions because many of the most recent simulation studies show this shape [75, 76, 77, 78, 79]. The cause for the transition from symmetric to asymmetric flame shape seen in the present simulations may be the use of a low-Mach-number formulation instead of a fully-compressible one [77]. More importantly, since many studies diverge on whether the flame further downstream is symmetric or not, the fact that this portion of the flow (in a non-linear dynamics sense) is close to a bifurcation cannot be ruled out, in which case its accurate prediction is very difficult. Furthermore, it must be stressed that, from the open literature, it is unclear how the flame looks like close to the outlet in the experiments.

*C*

_{λ}= 1 (simulation 2 cf. Tab. 1) and

*C*

_{λ}= 10 (simulation 3 cf. Tab. 1). These quantities are averaged in the spanwise direction and over time, the latter of which is done for a time interval of 0.02 s. (Through trial and error this interval is found to be adequate to evaluate the mean quantities.) Overall, Figs. 11 and 12 show that the sensitivity to the type of model and LES-LEM model constant is small, although the best agreement with the experiments is given by the LES-LEM simulation with

*C*

_{λ}= 10. In Fig. 11, perhaps the largest disagreement between the experimental data and predictions is seen in the temperature profiles near the outlet. This disagreement could be due to the lack of near-wall mesh refinement in the present simulations, or due to the fact that the wall thermal boundary condition from the experiments is unknown. Nonetheless, the overall agreement between the simulations and experiments shown in Figs. 11 and 12 is within that seen in some simulation studies [70, 72, 73, 74, 80] but slightly less accurate as those seen in other studies [13, 81], the latter of which includes a previous LES-LEM study, which is compared to the current study in the Section 4.3.

*x*/

*h*= 9.4, where the LES-LEM model with

*C*

_{λ}= 10 gives the best agreement with experiments.

### 4.3 Comparison with traditional LES-LEM

### 4.4 Sensitivity to model parameters

In terms of the effect of using the median or mean to compute the source terms, the effect of this choice was found to be negligible. This has been observed in another study as well [61].

Figures 15 and 16 also compare predictions between the baseline LES-LEM case (simulation 2 cf. Table 1) and a case where the source term calculations in Eqs. 14–15 is only done at locations where the resolved temperature is larger than 285 K (simulation 10 cf. Table 1). This simple change to the model formulation reduces the computational cost by a factor of 4.7 and provides predictions comparable to the baseline case, as can be seen in Figs. 15 and 16.

## 5 High-Reynolds-Number syngas Flame

The apparent insensitivity to the choice of turbulent combustion model seen above motivated the simulation of two cases not considered in the DNS study. Two simulations with an increased Reynolds number to demonstrate the influence of increased subgrid turbulence-chemistry interactions are shown here. Case M10 is case M of Hawkes et al. [51], which has been discussed in Section 3 already, but with all length-scales of the mesh multiplied by a factor of 10. This was done by scaling length of the mesh points by 10 in the 3 Cartesian directions, resulting in increasing size of the computational domain by 10 in all the 3 directions. The same is done for case M10U2 but, in addition, the initial velocity field is multiplied by a factor of 2. In cases M10 and M10U2 the Reynolds number is 44780 and 179120 respectively. The simulations of these cases use the same number of cells (in the 3 directions) as indicated above for the M case. So by increasing the length of each cell, the cell-based Reynolds number was increased. This also increased the LEM domain lengths.

*t*= 30

*t*

_{j}. This observation is consistent with a larger turbulent mixing between these layers, which in the case of LEM is represented with triplet maps and the splicing as clarified in Section 6. Furthermore, this merging seen in LEM but not in the no-model simulations is similar to that seen in LEM and not in eddy-breakup-model (EBU) simulations of reacting wakes [82, 83]. In these reacting wakes this merging is observed in the experiments.

Moving to case M10U2, Fig. 17 shows that LEM produces a larger extinction region in spacetime (represented by the yellow region) than the no-model approach. This comparison is similar to the observation of stronger extinction, in the form of lift-off distance, seen in LEM simulations of a stagnation-point-combustor in comparison with EBU simulations [14]. Thus, these effects of LEM are consistent with those seen in other studies.

In view of the lack of DNS data for cases M10 and M10U2, however, it is not possible to judge whether the no-model approach or LEM provide more accurate predictions. Nonetheless, for the present purposes, the key message from Fig. 17 is that, in contrast to the previous results, the effect of the turbulent combustion model in cases M10 and M10U2 is profound since it affects the qualitative evolution of the flow. More investigation with LES-LEM focusing on high-Reynolds-number flames will follow in the future.

## 6 Discussion

*N*

_{eddies}, being implemented per LES time step. On the other hand, values of

*N*

_{eddies}of near zero indicate that LEM behaves closely to the no-model approach. Note that

*N*

_{eddies}is related to the

*μ*

^{sgs}through Eqs. 10, 19 and 20 since a higher

*λ*implies higher values of

*N*

_{eddies}. The second way the above interaction is captured in LEM is through the movement of LEM elements between adjacent LES cells (not “inside” a LES cell) due to the splicing. From this explanation, the low values of

*μ*

^{sgs}and spatially averaged (in xz plane)

*N*

_{eddies}shown in Figs. 18a and 18b for case M suggest the LEM behaves in a no-model-approach way. In other words, the elements of LEM that represent the subgrid-scale interaction between turbulence and other physics are not active. In contrast, notice in Fig. 18 that for the cases M10 and M10U2 both

*μ*

^{sgs}and

*N*

_{eddies}increase by an order of magnitude. (Note the change of the color scale.) In these cases the elements of LEM to model turbulence-chemistry interactions are active.

Having discussed how predictions from the present LES-LEM model compare with the experiments and a benchmark model (the no-model approach), attention is now drawn towards the design of such a LES-LEM model. For this purpose, Fig. 10 compares the effect of the correction step, cf. Figure 2. Notice in Fig. 10 that by not using this correction there was an artificial extinction of the flame. Thus, the correction step or some alternative is required. This is not surprising because the convection of species by the solution of Eq. 4 differs from that produced by the splicing in at least two ways. First, the solution of Eq. 4 numerically smears the species in a way that the splicing is not prone to because the splicing corresponds to diffusion-free advection in a Lagrangian way. Second, the splicing produces an artifact that the solution of Eq. 4 is not prone to. Such an artifact can be demonstrated with the following test problem of passive scalar advection.

*Y*

_{α}denotes the mass fraction of the scalar

*α*, and the summation (index i) is over the elements of the LEM array. First, it is evident by comparing Fig. 19b and c that the splicing distorts the shape of the blob in a way that should be considered an artifact. However, also notice that the numerical dissipation produced by the splicing is much less than that from the solution of Eqs. 1–4 (as usually done). This is a result of the fact that the splicing is a Lagrangian algorithm. Finally, a comparison of Figs. 19b and d shows that the present correction step does indeed correct for the splicing artifact just discussed. Thus, in problems such as this blob-convection test the present correction can be useful to prevent artifacts from the splicing. However, if the mesh is fine enough the splicing artifacts are small and the effect of the correction is negligible [61].

## 7 Summary and Conclusions

The present paper introduces a new LEM closure for LES with the reaction-rate approach. Using LEM in this way, instead of closing the filtered thermochemical quantities as traditionally done, offers some conceptual advantages. For example, this new closure facilitates the assessment of inconsistency due to the turbulence-transport model. Moreover, for the purpose of reducing computational cost, it simplifies the use of more than one turbulent combustion model within one simulation, which may well include the use of no-model in very refined regions.

Predictions with this new closure agree with DNS results for a syngas flame problem with an accuracy comparable to previous traditional LES-LEM simulations. Furthermore, present predictions for the Volvo combustor are within those of previous simulations, but not as accurate as those from simulations using meshes ten or more times finer than those used here.

In spite of the above promising results, LES-LEM does not show more accurate predictions over the no-model approach for low-Reynolds-number reactive turbulence. The level of turbulence, given by the Reynolds number for instance, in the DNS syngas flame and the Volvo tests are fairly low. This statement is supported by the fact that, at least in the syngas flames, while the LEM model is barely active in the DNS syngas flame (case M), it is very active in the higher-Reynolds-number syngas flames (cases M10 and M10U2). Unfortunately, there are no DNS nor experiments for these flames. Furthermore, the heat release rate in the DNS syngas flame and the Volvo tests is also low, in which case the effects of combustion (e.g., flow expansion) on the fluid dynamics is mild. This is a well known problem with the Volvo combustor and similar academic combustors [73].

In practice, however, much higher Reynolds numbers than those of the medium-Reynolds-number syngas flame and Volvo test case can occur, in particular in gas-turbine-engine combustors [84]. In principle LES-LEM is better suited for these problems, but this capability remains to be demonstrated. Thus, future work on its development should consider test problems closer to actual applications, problems such as the high-Reynolds-number, premixed jet burner of Dunn et al. [85], or the very-high-Reynolds-number, gas-turbine-like LM6000 [86, 87] or GE-E3 [88] combustors, and conduct careful comparisons with, say, the no-model approach at various levels of mesh refinement.

Finally, unlike traditional LES-LEM implementations, the present work demonstrates the use of LES-LEM in a pressure-based and unstructured CFD solver, an approach that is preferred in industry.

## Notes

### Acknowledgements

The authors thank the Swedish Research Council Vetenskapsrådet and the Combustion Engine Research Center (CERC) for funding this project at Chalmers. Special thanks to Alan Kerstein and Venkateswaran Sankaran for helpful discussions. Thanks to Prof. Evatt Hawkes for kindly providing the DNS data used in the present study. Also thanks to the financial support from the Small-Business-Innovation-Research / Small-Business-Technology-Transfer (SBIR/STTR) project titled ‘Physical submodel development for turbulence combustion closure’, contract number FA9550-15-C-0044.

### Funding Information

This study was funded by the Swedish Research Council Vetenskapsrådet, the Combustion Engine Research Center (CERC) at Chalmers University of Technology and the Small-Business-Innovation-Research / Small-Business-Technology-Transfer (SBIR/STTR) project titled ‘Physical submodel development for turbulence combustion closure’(contract number FA9550-15-C-0044).

### Compliance with Ethical Standards

### **Conflict of interests**

The authors declare that they have no conflict of interest.

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