Turbulent Combustion and Reactive Flows — Coupling of Chemistry, Transport, and Turbulence

Scope: rigorous treatment of chemically reactive turbulent flows, coupling fluid mechanics, transport, and thermodynamics. Covers governing equations, turbulence–chemistry interaction, statistical models (PDF, flamelet), and entropy/exergy aspects.

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1. Governing Conservation Equations for Reactive Flows

Turbulent combustion involves the simultaneous conservation of mass, momentum, species, and energy. The instantaneous equations are:

1.1 Continuity

$\frac{∂ρ}{∂t} + ∇·(ρ\mathbf{v}) = 0.$

1.2 Momentum (Navier–Stokes)

$ρ\frac{D\mathbf{v}}{Dt} = -∇p + ∇·\boldsymbol{τ} + ρ\mathbf{g}.$

1.3 Species Conservation (for species i)

$\frac{∂(ρY_i)}{∂t} + ∇·(ρY_i\mathbf{v}) = -∇·\mathbf{J}_i + \dot{ω}_i,$ where:

• $Y_i$: mass fraction of species i,
• $\mathbf{J}_i = -ρD_i∇Y_i$: diffusive flux,
• $\dot{ω}_i$: chemical production rate (kg/m³·s).

1.4 Energy (Total or Enthalpy Form)

$ρ\frac{Dh}{Dt} = \frac{Dp}{Dt} + ∇·(k∇T) + ∑_i h_i ∇·\mathbf{J}_i + \dot{q}_{chem},$ with $\dot{q}_{chem} = -∑_i h_i\dot{ω}_i$ (heat release by reactions).

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2. Chemical Kinetics and Source Terms

2.1 Reaction Rate

For a generic reaction: $∑_i ν'_i A_i ⇌ ∑_i ν''_i A_i,$ reaction rate (mol/m³·s): $\dot{ω}_r = k_f ∏_i [C_i]^{ν'_i} - k_b ∏_i [C_i]^{ν''_i}.$

Temperature dependence (Arrhenius law): $k_f = A T^n e^{−E_a/(RT)}.$

2.2 Species Source Term

$\dot{ω}_i = M_i ∑_r (ν''_{i,r} - ν'_{i,r})\dot{ω}_r.$

2.3 Heat Release Rate

$\dot{q}_{chem} = -∑_i h_i\dot{ω}_i = ∑_r ΔH_r\dot{ω}_r.$

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3. Turbulence–Chemistry Interaction: Averaging and Closure

For turbulent reacting flows, decompose using Reynolds or Favre (density-weighted) averaging: $\tilde{φ} = \frac{\overline{ρφ}}{\bar{ρ}}, \quad φ'' = φ - \tilde{φ}.$

Averaging species equation gives: $\frac{∂(\bar{ρ}\tilde{Y}_i)}{∂t} + ∇·(\bar{ρ}\tilde{Y}_i\tilde{\mathbf{v}}) = -∇·(\bar{ρ} \widetilde{Y_i''\mathbf{v}''}) + \bar{ρ}\tilde{\dot{ω}}_i.$

Unknown correlations $\widetilde{Y_i''\mathbf{v}''}$ and $\tilde{\dot{ω}}_i$ require modeling — the core challenge of turbulence–chemistry closure.

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4. Time-Scale Ratios and Regime Classification

Two fundamental time scales:

• Turbulent mixing time: $t_t = k/ε$
• Chemical time: $t_c = 1/\dot{ω}_{max}$

4.1 Damköhler Number

$Da = \frac{t_t}{t_c} = \frac{\text{mixing time}}{\text{reaction time}}.$

• $Da ≫ 1$: fast chemistry (mixing-limited regime)
• $Da ≪ 1$: slow chemistry (kinetics-limited regime)

4.2 Karlovitz Number (Ka)

$Ka = \frac{t_η}{t_c} = \frac{ε^{1/2}}{ν^{1/2}\dot{ω}_c}.$

• $Ka < 1$: flamelet regime (chemistry faster than smallest eddies)
• $Ka > 1$: distributed reaction regime (mixing dominates).

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5. Regime Diagram (Peters’ Classification)

Regime	Damköhler	Karlovitz	Description
Flamelet	Da≫1, Ka<1	Chemistry fast, thin flame sheet
Thin reaction zones	Da≈1, Ka≈1	Partial disruption by turbulence
Distributed reaction	Da≪1, Ka≫1	Chemistry slow, volumetric reaction
Well-stirred reactor	Da≪1	Perfectly mixed turbulence

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6. PDF Formulation and Closure

The probability density function (PDF) approach represents the distribution of scalar variables (species, temperature) within turbulent flow.

Define joint PDF $P(ξ, t)$ where $ξ = (Y_1, Y_2, T, ...)$. The mean value of any function $f(ξ)$: $\tilde{f} = ∫ f(ξ) P(ξ) dξ.$

The PDF transport equation: $\frac{∂(ρP)}{∂t} + ∇·(ρ\tilde{\mathbf{v}}P) = -\frac{∂}{∂ξ_i}(\langle \dot{ξ}_i P\rangle) + D_{mix},$ where $D_{mix}$ models micro-mixing between scalar states.

6.1 β-PDF for Mixture Fraction

For nonpremixed combustion, scalar fluctuations (mixture fraction Z) are well approximated by a β-distribution: $P(Z) = \frac{Z^{α−1}(1−Z)^{β−1}}{B(α,β)}.$

Favre mean: $\tilde{Z} = α/(α+β)$, variance: $\tilde{Z''^2} = αβ/[(α+β)^2(α+β+1)].$

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7. Flamelet Model for Nonpremixed Combustion

Assume thin flame sheets embedded within turbulent field. Chemistry is fast relative to turbulent fluctuations, so local flame structure ≈ laminar flame.

7.1 Mixture Fraction Coordinate

Define scalar $Z$ (mass fraction of fuel): $\frac{∂(ρZ)}{∂t} + ∇·(ρZ\mathbf{v}) = ∇·(ρD∇Z).$

7.2 Flamelet Equation

$ρχ/2\frac{∂^2φ}{∂Z^2} = \dot{ω}_φ(T(Z), Y_i(Z)),$ where χ = scalar dissipation rate: $χ = 2D |∇Z|^2.$

Flame structure depends only on $Z$ and $χ$; turbulent effects enter via PDF averaging.

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8. Premixed Turbulent Combustion

For premixed flames, fuel and oxidizer are mixed before reaction.

Flame speed $S_L$ enhanced by turbulence: $S_T = S_L(1 + u'/S_L)^{n}.$ Empirically, $n ≈ 0.5–1.0$, depending on regime.

Flame front modeled via level-set or G-equation: $\frac{∂G}{∂t} + (\mathbf{v}·∇)G = S_L|∇G|.$

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9. Eddy Dissipation Concept (EDC)

Proposed by Magnussen (1981) — reaction rate controlled by fine-scale turbulence: $\dot{ω}_i = ρ\frac{ε}{k}\frac{Y_i - Y_{i,eq}}{τ^*},$ where $τ^* ∼ (k/ε)^{1/2}$ is eddy turnover time.

Effective rate = min(chemical rate, turbulent rate). Suitable for CFD implementations with k–ε turbulence models.

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10. Conditional Moment Closure (CMC)

Conditional averaging conditioned on scalar Z: $\langle Y_i | Z \rangle, \quad \langle T | Z \rangle.$

Transport equation for conditional mean: $ρ\frac{D\langle Y_i | Z \rangle}{Dt} = ρD∇^2\langle Y_i | Z \rangle + \langle \dot{ω}_i | Z \rangle + M(Z),$ where $M(Z)$ is the micro-mixing term.

CMC bridges between detailed chemistry and turbulence statistics.

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11. Heat Release, Density Fluctuations, and Flow Coupling

Heat release modifies density via equation of state: $ρ = \frac{pM}{RT}.$

Buoyancy and expansion cause strong coupling between combustion and flow field — leading to flame stretch, wrinkling, and acoustic instabilities.

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12. Entropy Generation and Exergy Destruction in Combustion

Total local entropy production: $σ_s = \frac{\dot{q}_{chem}}{T} + \frac{k(∇T)^2}{T^2} + \frac{μ(∇v)^2}{T}.$

Exergy destruction density: $\dot{e}_D = T_0 σ_s.$

Irreversibilities arise from:

• Finite-rate chemistry (chemical irreversibility)
• Thermal gradients (heat conduction)
• Viscous dissipation (momentum diffusion)

These losses define the thermodynamic efficiency limits of combustion systems.

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13. Turbulent Combustion Modeling in CFD

Model	Principle	Regime of Validity
Eddy Break-Up (EBU)	Reaction rate ∝ turbulence dissipation	Fast-chemistry (mixing-controlled)
Flamelet	Pre-tabulated laminar flame library	Nonpremixed, Da≫1
PDF	Statistical closure for chemistry and mixing	All regimes, detailed kinetics
EDC	Finite-rate model using k–ε turbulence	Moderate-to-high Re flames
CMC	Conditional averaging on mixture fraction	General, detailed but expensive

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14. Example: Jet Flame Scaling

For turbulent nonpremixed jet flames: $x_f ∝ D Re^{1/2} Sc^{1/2}, \quad L_f ∝ Re^{1/2}.$

Flame length scales with turbulent mixing rate; collapse occurs when chemistry time ≈ mixing time.

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15. Summary Equations

Concept	Equation
Species conservation	$∂(ρY_i)/∂t + ∇·(ρY_i\mathbf{v}) = -∇·\mathbf{J}_i + \dot{ω}_i$
Reaction rate	$\dot{ω}_r = k_f ∏[C_i]^{ν'_i} - k_b ∏[C_i]^{ν''_i}$
Damköhler number	$Da = t_t/t_c$
Karlovitz number	$Ka = t_η/t_c$
Scalar dissipation rate	$χ = 2D
EDC rate	$\dot{ω}_i = ρ(ε/k)(Y_i - Y_{i,eq})/τ^*$
Flamelet equation	$ρχ/2 ∂^2φ/∂Z^2 = \dot{ω}_φ$
Entropy generation	$σ_s = \dot{q}_{chem}/T + k(∇T)^2/T^2 + μ(∇v)^2/T$

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16. Cross-Links

• 04_Turbulence_and_Mixing.md — turbulent transport and diffusion fundamentals.
• Thermodynamics/08_Chemical_Thermodynamics.md — equilibrium, Gibbs energy, and reaction spontaneity.
• Thermodynamics/10_NonEquilibrium_Thermodynamics.md — entropy production in reactive systems.
• Heat_Transfer/Combustion_Applications.md — radiative and convective heat transfer in flames.