# Turbulent Premixed Flame Modeling Using the Algebraic Flame Surface Wrinkling Model: A Comparative Study between OpenFOAM and Ansys Fluent

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

#### 2.1. Premixed Combustion Model

#### 2.2. Energy Equation Treatment

#### 2.3. Thermophysical and Transport Properties

^{o}refers to standard state pressure conditions or $P=1\phantom{\rule{4pt}{0ex}}\mathrm{atm}$. In Equations (21)–(23), R is the universal gas constant and H and S refer to the standard state enthalpy and entropy, respectively, both defined at a reference temperature of ${T}_{ref}=298$ K.

## 3. Solver Implementation in OpenFOAM

**XiFoam.C**to

**myXiFoam.C**. In this step, we also modified the file

**myXiFoam.C**. The modification consisted of two steps. In the first step, we added the following include directive to

**myXiFoam.C**(Listing 1),

**myXiFoam.C**consisted of changing the name of the include directive

**bEqn.H**to

**mybEqn.H**(Listing 2),

**bEqn.H**to

**mybEqn.H**, and then, we proceeded to modify the file

**mybEqn.H**. This modification consisted of changing the way the turbulent flame speed flux is computed, as follows (Listing 3),

**rho_unb**computed with Cantera (this value is defined later in Step 3). In this step, we also added the following include directive to access the AFSW model implemented in the file

**AFSW.H**(Listing 4),

**AFSW.H**and

**myCreateFields.H**. In the

**AFSW.H**header file, we defined the function that implements the computation of the turbulent flame speed ${S}_{t}$ in the AFSW model. Notice the correspondence between Code Listing 5 and Equation (7), which we rewrite here for clarity.

**myCreateFields.H**, we declared all the variables required by the AFSW model implementation. Namely, $R{e}_{t}$ (

**Reynolds_number**in the source code), ${l}_{t}$ (

**L_x**in the source code), ${u}^{\prime}$ (

**up**in the source code), $L{e}_{eff}$ (

**Le**in the source code), ${\rho}_{u}$ (

**rho_unb**in the source code), ${\mu}_{u}$ (

**mu_unb**in the source code), and ${S}_{L0}$ (

**Su**in the source code). Notice that $R{e}_{t}$, ${l}_{t}$, and ${u}^{\prime}$ were computed using Equations (9)–(11), whereas $L{e}_{eff}$, ${\rho}_{u}$, ${\mu}_{u}$, and ${S}_{L0}$ are constant values (computed using Cantera).

**Make**. In this sub-directory, we only needed to modify the source code file named

**file**. The modification consisted of changing the name of the input file to

**myXiFoam.C**and the name of the output executable and its default location to

**$(FOAM_USER_APPBIN)/myXiFoam**.

**wmake**command. This command compiles the new solver, and it creates the executable

**myXiFoam**. It is worth stressing that all the variables declared in the header file

**myCreateFields.H**are hardwired. This means that in order to run a new simulation with a different species concentration and boundary conditions, we must update this file and recompile the solver. In the solver repository [34], the interested reader can find the header file

**myCreateFields.H**with the physical properties for both cases listed in Table 2.

**phiSt**computed as per Listing 3. The variable

**phiSt**represents the scalar contribution ${\rho}_{u}{S}_{t}$ in the reaction rate source term. Notice that

**phiSt**is a vector quantity, but by taking the divergence of it, we can convert it to a scalar quantity. Then, the reaction rate source term is computed using an implicit source, where the contribution of the term $|\nabla \tilde{b}|$ is implicitly computed in the source term.

## 4. Experimental Setup and Data

## 5. Numerical Background

#### 5.1. Computational Domain and Mesh

#### 5.2. Boundary and Initial Conditions

#### 5.3. Numerical Setup and Discretization Schemes

## 6. Results and Discussion

## 7. Conclusions and Perspectives

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

$\overline{}$ | Reynolds average. |

$\tilde{}$ | Favre average. |

$\alpha $ | Thermal diffusivity (molecular), measured in ${\mathrm{m}}^{2}/\mathrm{s}$. |

$\mathrm{\Delta}{H}_{c}^{o}$ | Lower heating value of combustion, measured in $\mathrm{J}/\mathrm{kg}$. |

$\u03f5$ | Turbulence dissipation rate, measured in ${\mathrm{m}}^{2}/{\mathrm{s}}^{3}$. |

$\lambda $ | Thermal conductivity, measured in $\mathrm{W}/\mathrm{m}\xb7\mathrm{K}$. |

$\mu $ | Dynamic viscosity, measured in $\mathrm{kg}/\mathrm{m}\xb7\mathrm{s}$. |

∇ | Gradient operator. |

$\nu $ | Kinematic viscosity, ${\mathrm{m}}^{2}/\mathrm{s}$. |

$\overline{\dot{\omega}}$ | Reaction rate source term, measured in ${\mathrm{kg}/\mathrm{m}}^{3}\xb7\mathrm{s}$. |

${\overline{\dot{\omega}}}_{b}$ | $\overline{\dot{\omega}}$ in the regress variable b equation, measured in $\mathrm{kg}/{\mathrm{m}}^{3}\xb7\mathrm{s}$. |

${\overline{\dot{\omega}}}_{c}$ | $\overline{\dot{\omega}}$ in the progress variable c equation, measured in $\mathrm{kg}/{\mathrm{m}}^{3}\xb7\mathrm{s}$. |

$\varphi $ | Equivalence ratio, non-dimensional. |

$\rho $ | Density, measured in $\mathrm{kg}/{\mathrm{m}}^{3}$. |

$\tau $ | Heat release factor, non-dimensional. |

a | JANAF table coefficient, non-dimensional. |

${A}_{s}$ | Sutherland model coefficient, measured in $\mathrm{kg}/\mathrm{m}\xb7\mathrm{s}\xb7\sqrt{\mathrm{K}}$. |

$AFR$ | Air-to-fuel ratio, non-dimensional. |

${C}_{p}$ | Specific heat, measured in $\mathrm{J}/\mathrm{kg}\xb7\mathrm{K}$. |

${C}_{p}^{o}$ | Standard state specific heat, measured in $\mathrm{J}/\mathrm{kg}\xb7\mathrm{K}$. |

$C{H}_{4}$ | Methane species. |

D | Binary mass diffusivity, measured in ${\mathrm{m}}^{2}/\mathrm{s}$. |

h | Enthalpy, measured in $\mathrm{J}/\mathrm{kg}$. |

${H}^{o}$ | Standard state enthalpy, measured in $\mathrm{J}/\mathrm{kg}$. |

${H}_{2}$ | Hydrogen species. |

K | Specific kinetic energy, measured in ${\mathrm{m}}^{2}/{\mathrm{s}}^{2}$. |

${l}_{t}$ | Turbulence length scale, measured in $\mathrm{m}$. |

m | Mass, measured in $\mathrm{kg}$. |

n | Mole number, measured in $\mathrm{mol}$. |

$NOx$ | Nitrogen Oxide. |

${N}_{2}$ | Nitrogen species. |

${O}_{2}$ | Oxygen species. |

R | Universal gas constant, measured in $\mathrm{J}/\mathrm{kg}\xb7\mathrm{K}$. |

${S}^{o}$ | Standard state entropy, measured in $\mathrm{J}/\mathrm{kg}\xb7\mathrm{K}$. |

${S}_{t}$ | Turbulent flame speed, measured in $\mathrm{m}/\mathrm{s}$. |

${S}_{L0}$ | Unstrained adiabatic laminar flame speed, measured in $\mathrm{m}/\mathrm{s}$. |

T | Temperature, measured in $\mathrm{K}$. |

${T}_{s}$ | Sutherland model coefficient, measured in $\mathrm{K}$. |

${T}_{ref}$ | Standard state temperature, measured in $\mathrm{K}$. |

${u}^{\prime}$ | Turbulent fluctuating velocity, measured in $\mathrm{m}/\mathrm{s}$. |

X | Molar ratio, non-dimensional. |

${Y}_{f}$ | Fuel mixture mass fraction, non-dimensional. |

b | Regress variable, non-dimensional. |

c | Progress variable, non-dimensional. |

k | Turbulent kinetic energy, measured in ${\mathrm{m}}^{2}/{\mathrm{s}}^{2}$. |

$Le$ | Lewis number, non-dimensional. |

P | Absolute pressure, measured in $\mathrm{Pa}$. |

$Pr$ | Prandtl number, non-dimensional. |

$Re$ | Reynolds number, non-dimensional. |

$Sc$ | Schmidt number, non-dimensional. |

t | Time, measured in $\mathrm{s}$. |

u | Velocity, measured in $\mathrm{m}/\mathrm{s}$. |

x | Spatial coordinates, measured in $\mathrm{m}$. |

Sub-indices | |

${}_{ad}$ | Adiabatic conditions. |

${}_{air}$ | Mix of ${O}_{2}$ and ${N}_{2}$ species. |

${}_{a}$ | Absolute. |

${}_{b}$ | Burnt mixture property. |

${}_{chem}$ | Chemical. |

${}_{C{H}_{4}}$ | Methane species. |

${}_{eff}$ | Effective. |

${}_{f}$ | Fuel. |

${}_{fuel}$ | $100\%\phantom{\rule{0.166667em}{0ex}}C{H}_{4}$ or $60\%\phantom{\rule{0.166667em}{0ex}}C{H}_{4}\phantom{\rule{0.166667em}{0ex}}+40\%\phantom{\rule{0.166667em}{0ex}}{H}_{2}$. |

${}_{{H}_{2}}$ | Hydrogen species. |

${}_{i}$ | Index of the spatial dimension, where $i=1,2,3$. |

${}_{mhrr}$ | Property temperature at which the maximum heat is released. |

${}_{oxidizer}$ | ${O}_{2}$ species. |

${}_{st}$ | Stoichiometric condition. |

${}_{s}$ | Sensible. |

${}_{t}$ | Turbulent property. |

${}_{u}$ | Unburnt mixture property. |

Acronyms | |

AFSW | Algebraic Flame Surface Wrinkling. |

CFD | Computational Fluid Dynamics. |

JANAF | Joint Army, Navy, and Air Force. |

Probability Density Function. | |

PSI | Paul Scherrer Institute |

RANS | Reynolds-Averaged Navier–Stokes. |

RHS | Right-Hand Side. |

SIMPLE | Semi-Implicit Method for Pressure Linked Equations. |

$k-\u03f5$ | Standard $k-\u03f5$ turbulence model. |

UDF | User-Defined Function. |

URANS | Unsteady Reynolds-Averaged Navier–Stokes. |

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**Figure 2.**Schematic view of the PSI generic, high-pressure Bunsen burner. Image taken from reference [10].

**Figure 5.**Inlet velocity profile computed using a precursor simulation. The inlet extreme is located at 12.5 mm from the pipe axis line.

**Figure 6.**Comparison of flame shapes (brush). Top row: experimental normalized OH* chemiluminescence [10]. Middle row: numerical flame shapes visualization using $\overline{c}$ (Ansys Fluent results). Bottom row: numerical flame shapes using $\overline{c}$ (OpenFOAM results). The left column of all rows corresponds to the case 100%$C{H}_{4}$. The right column of all rows corresponds to the case 60%$C{H}_{4}$+ 40%${H}_{2}$.

**Figure 7.**Plot of the $\overline{c}$ distribution over the axisymmetric axis. In the figure, the horizontal green line represents $\overline{c}=0.5$ (a value of $\overline{c}=0.5$ corresponds to the mid-isosurface between the unburned state and the burned state). The intersection of this line with the axial $\overline{c}$ distributions (the numerical results) was used to compute the flame height plotted in Figure 8.

**Figure 8.**Comparison of the numerical and experimental flame heights estimated at $\overline{c}=0.5$ (${h}_{\overline{c}=0.5}$). The flame height was calculated as shown in Figure 7.

**Figure 9.**Comparison of the numerical and experimental flame speeds at $\overline{c}=0.5$ (${S}_{t,\overline{c}=0.5}$). The flame speed was calculated using Kobayashi’s method [8].

**Table 1.**Premixed mixture properties. In the table, the sub-index u refers to unburnt conditions. The sub-index $ad$ refers to adiabatic conditions. The sub-index $mhrr$ refers to the temperature at which the maximum heat is released. The sub-indices $C{H}_{4}$ and ${H}_{2}$ refer to the species. The variable $\varphi $ is the equivalence ratio and is computed using Equation (26).

${\mathit{X}}_{{\mathbf{CH}}_{4}}$ | ${\mathit{X}}_{{\mathit{H}}_{2}}$ | $\mathit{\varphi}$ | ${\mathit{T}}_{\mathit{u}}$ | P | ${\mathit{\rho}}_{\mathit{u}}$ | ${\mathit{\mu}}_{\mathit{u}}$ | ${\mathit{D}}_{{\mathit{CH}}_{4}}$ | ${\mathit{D}}_{{\mathit{H}}_{2}}$ | $\mathit{\alpha}$ | ${\mathit{S}}_{\mathit{L}0}$ | ${\mathit{T}}_{\mathit{mhrr}}$ | ${\mathit{T}}_{\mathit{ad}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|

$(-)$ | $(-)$ | $(-)$ | $\left(\mathbf{K}\right)$ | $\left(\mathbf{atm}\right)$ | $\left(\mathbf{kg}/{\mathbf{m}}^{3}\right)$ | $\left({\mathbf{m}}^{2}/\mathbf{s}\right)$ | $\left({\mathbf{m}}^{2}/\mathbf{s}\right)$ | $\left({\mathbf{m}}^{2}/\mathbf{s}\right)$ | $\left({\mathbf{m}}^{2}/\mathbf{s}\right)$ | $\left(\mathbf{m}/\mathbf{s}\right)$ | $\left(\mathbf{K}\right)$ | $\left(\mathbf{K}\right)$ |

1.0 | 0.0 | 0.5 | 673 | 5 | 2.52 | 3.25 × ${10}^{-5}$ | 7.93 × ${10}^{-5}$ | 2.60 × ${10}^{-4}$ | 7.56 × ${10}^{-5}$ | 0.232 | 1606 | 1777 |

0.6 | 0.4 | 0.5 | 673 | 5 | 2.46 | 3.25 × ${10}^{-5}$ | 7.79 × ${10}^{-5}$ | 2.55 × ${10}^{-4}$ | 8.05 × ${10}^{-5}$ | 0.334 | 1589 | 1803 |

Species Volumetric Ratio | ||
---|---|---|

$C{H}_{4}$ | ${H}_{2}$ | |

Fuel mixture—Case 1 | 100% | 0% |

Fuel mixture—Case 2 | 60% | 40% |

T | P | $\mathit{\varphi}$ | ${\mathit{u}}_{\mathit{inlet}}$ | ${\mathit{u}}^{\prime}$ | ${\mathit{L}}_{\mathit{t}}$ |
---|---|---|---|---|---|

(K) | (atm) | (−) | $\left(\mathbf{m}/\mathbf{s}\right)$ | $\left(\mathbf{m}/\mathbf{s}\right)$ | $\left(\mathbf{mm}\right)$ |

673 | 5 | 0.5 | 40 | 6.16 | 2.2 |

**Table 4.**Boundary conditions and initialization type used in OpenFOAM. In the table, the abbreviations stand for: FV = fixedValue, C = Calculated, ZG = zeroGradient, NS = noSlip, UDP = User-Defined velocity profile (plotted in Figure 5), KWF = kqRWallFunction, EWF = epsilonWallFunction, NKWF = nutkWallFunction.

Boundary Condition | Initial Conditions | |||
---|---|---|---|---|

Variable | Inlet | Outlet | Wall | Initialization Type |

k | FV | ZG | KWF | uniform |

$\u03f5$ | FV | ZG | EWF | uniform |

${\nu}_{t}$ | C | ZG | NKWF | uniform |

U | UDP | ZG | NS | uniform |

p | ZG | FV | ZG | uniform |

T | FV | ZG | ZG | uniform |

b | FV | ZG | ZG | uniform |

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**MDPI and ACS Style**

Kutkan, H.; Guerrero, J. Turbulent Premixed Flame Modeling Using the Algebraic Flame Surface Wrinkling Model: A Comparative Study between OpenFOAM and Ansys Fluent. *Fluids* **2021**, *6*, 462.
https://doi.org/10.3390/fluids6120462

**AMA Style**

Kutkan H, Guerrero J. Turbulent Premixed Flame Modeling Using the Algebraic Flame Surface Wrinkling Model: A Comparative Study between OpenFOAM and Ansys Fluent. *Fluids*. 2021; 6(12):462.
https://doi.org/10.3390/fluids6120462

**Chicago/Turabian Style**

Kutkan, Halit, and Joel Guerrero. 2021. "Turbulent Premixed Flame Modeling Using the Algebraic Flame Surface Wrinkling Model: A Comparative Study between OpenFOAM and Ansys Fluent" *Fluids* 6, no. 12: 462.
https://doi.org/10.3390/fluids6120462