Turbulent Combustion

Phenomenon

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.

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

∂ρ∂t+∇⋅(ρv)=0.\frac{\partial\rho}{\partial t} + \nabla\cdot(\rho\mathbf{v}) = 0.

1.2 Momentum (Navier–Stokes)

ρDvDt=−∇p+∇⋅τ+ρg.\rho\frac{D\mathbf{v}}{Dt} = -\nabla p + \nabla\cdot\boldsymbol{\tau} + \rho\mathbf{g}.

1.3 Species Conservation (for species i)

∂(ρYi)∂t+∇⋅(ρYiv)=−∇⋅Ji+ω˙i,\frac{\partial(\rho Y_i)}{\partial t} + \nabla\cdot(\rho Y_i\mathbf{v}) = -\nabla\cdot\mathbf{J}_i + \dot{\omega}_i, where:

  • YiY_i: mass fraction of species i,
  • Ji=−ρDi∇Yi\mathbf{J}_i = -\rho D_i\nabla Y_i: diffusive flux,
  • ω˙i\dot{\omega}_i: chemical production rate (kg/m³·s).

1.4 Energy (Total or Enthalpy Form)

ρDhDt=DpDt+∇⋅(k∇T)+∑ihi∇⋅Ji+q˙chem,\rho\frac{Dh}{Dt} = \frac{Dp}{Dt} + \nabla\cdot(k\nabla T) + \sum_i h_i \nabla\cdot\mathbf{J}_i + \dot{q}_{chem}, with q˙chem=−∑ihiω˙i\dot{q}_{chem} = -\sum_i h_i\dot{\omega}_i(heat release by reactions).

2. Chemical Kinetics and Source Terms

2.1 Reaction Rate

For a generic reaction: ∑iνi′Ai⇌∑iνi′′Ai,\sum_i \nu'_i A_i ⇌ \sum_i \nu''_i A_i, reaction rate (mol/m³·s): ω˙r=kf∏i[Ci]νi′−kb∏i[Ci]νi′′.\dot{\omega}_r = k_f \prod_i [C_i]^{\nu'_i} - k_b \prod_i [C_i]^{\nu''_i}.

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

2.2 Species Source Term

ω˙i=Mi∑r(νi,r′′−νi,r′)ω˙r.\dot{\omega}_i = M_i \sum_r (\nu''_{i,r} - \nu'_{i,r})\dot{\omega}_r.

2.3 Heat Release Rate

q˙chem=−∑ihiω˙i=∑rΔHrω˙r.\dot{q}_{chem} = -\sum_i h_i\dot{\omega}_i = \sum_r \Delta H_r\dot{\omega}_r.

3. Turbulence–Chemistry Interaction: Averaging and Closure

For turbulent reacting flows, decompose using Reynolds or Favre (density-weighted) averaging: φ~=ρφ‾ρˉ,φ′′=φ−φ~.\tilde{\varphi} = \frac{\overline{\rho\varphi}}{\bar{\rho}}, \quad \varphi'' = \varphi - \tilde{\varphi}.

Averaging species equation gives: ∂(ρˉY~i)∂t+∇⋅(ρˉY~iv~)=−∇⋅(ρˉYi′′v′′~)+ρˉω˙~i.\frac{\partial(\bar{\rho}\tilde{Y}_i)}{\partial t} + \nabla\cdot(\bar{\rho}\tilde{Y}_i\tilde{\mathbf{v}}) = -\nabla\cdot(\bar{\rho} \widetilde{Y_i''\mathbf{v}''}) + \bar{\rho}\tilde{\dot{\omega}}_i.

Unknown correlations Yi′′v′′~\widetilde{Y_i''\mathbf{v}''}and ω˙~i\tilde{\dot{\omega}}_irequire modeling — the core challenge of turbulence–chemistry closure.

4. Time-Scale Ratios and Regime Classification

Two fundamental time scales:

  • Turbulent mixing time: tt=k/εt_t = k/\varepsilon
  • Chemical time: tc=1/ω˙maxt_c = 1/\dot{\omega}_{max}

4.1 Damköhler Number

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

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

4.2 Karlovitz Number (Ka)

Ka=tηtc=ε1/2ν1/2ω˙c.Ka = \frac{t_\eta}{t_c} = \frac{\varepsilon^{1/2}}{\nu^{1/2}\dot{\omega}_c}.

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

5. Regime Diagram (Peters’ Classification)

RegimeDamköhlerKarlovitzDescription
FlameletDa≫1, Ka<1Chemistry fast, thin flame sheet
Thin reaction zonesDa≈1, Ka≈1Partial disruption by turbulence
Distributed reactionDa≪1, Ka≫1Chemistry slow, volumetric reaction
Well-stirred reactorDa≪1Perfectly mixed turbulence

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)P(\xi, t)where ξ=(Y1,Y2,T,...)\xi = (Y_1, Y_2, T, ...). The mean value of any function f(ξ)f(\xi): f~=∫f(ξ)P(ξ)dξ.\tilde{f} = \int f(\xi) P(\xi) d\xi.

The PDF transport equation: ∂(ρP)∂t+∇⋅(ρv~P)=−∂∂ξi(⟨ξ˙iP⟩)+Dmix,\frac{\partial(\rho P)}{\partial t} + \nabla\cdot(\rho\tilde{\mathbf{v}}P) = -\frac{\partial}{\partial\xi_i}(\langle \dot{\xi}_i P\rangle) + D_{mix}, where DmixD_{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)=Zα−1(1−Z)β−1B(α,β).P(Z) = \frac{Z^{\alpha−1}(1−Z)^{\beta−1}}{B(\alpha,\beta)}.

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

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 ZZ(mass fraction of fuel): ∂(ρZ)∂t+∇⋅(ρZv)=∇⋅(ρD∇Z).\frac{\partial(\rho Z)}{\partial t} + \nabla\cdot(\rho Z\mathbf{v}) = \nabla\cdot(\rho D\nabla Z).

7.2 Flamelet Equation

ρχ/2∂2φ∂Z2=ω˙φ(T(Z),Yi(Z)),\rho\chi/2\frac{\partial^2\varphi}{\partial Z^2} = \dot{\omega}_\varphi(T(Z), Y_i(Z)), where χ = scalar dissipation rate: χ=2D∣∇Z∣2.\chi = 2D |\nabla Z|^2.

Flame structure depends only on ZZand χ\chi; turbulent effects enter via PDF averaging.

8. Premixed Turbulent Combustion

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

Flame speed SLS_Lenhanced by turbulence: ST=SL(1+u′/SL)n.S_T = S_L(1 + u'/S_L)^{n}. Empirically, n≈0.5−1.0n \approx 0.5-1.0, depending on regime.

Flame front modeled via level-set or G-equation: ∂G∂t+(v⋅∇)G=SL∣∇G∣.\frac{\partial G}{\partial t} + (\mathbf{v}\cdot\nabla)G = S_L|\nabla G|.

9. Eddy Dissipation Concept (EDC)

Proposed by Magnussen (1981) — reaction rate controlled by fine-scale turbulence: ω˙i=ρεkYi−Yi,eqτ∗,\dot{\omega}_i = \rho\frac{\varepsilon}{k}\frac{Y_i - Y_{i,eq}}{\tau^*}, where τ∗∼(k/ε)1/2\tau^* \sim (k/\varepsilon)^{1/2}is eddy turnover time.

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

10. Conditional Moment Closure (CMC)

Conditional averaging conditioned on scalar Z: ⟨Yi∣Z⟩,⟨T∣Z⟩.\langle Y_i | Z \rangle, \quad \langle T | Z \rangle.

Transport equation for conditional mean: ρD⟨Yi∣Z⟩Dt=ρD∇2⟨Yi∣Z⟩+⟨ω˙i∣Z⟩+M(Z),\rho\frac{D\langle Y_i | Z \rangle}{Dt} = \rho D\nabla^2\langle Y_i | Z \rangle + \langle \dot{\omega}_i | Z \rangle + M(Z), where M(Z)M(Z)is the micro-mixing term.

CMC bridges between detailed chemistry and turbulence statistics.

11. Heat Release, Density Fluctuations, and Flow Coupling

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

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

12. Entropy Generation and Exergy Destruction in Combustion

Total local entropy production: σs=q˙chemT+k(∇T)2T2+μ(∇v)2T.\sigma_s = \frac{\dot{q}_{chem}}{T} + \frac{k(\nabla T)^2}{T^2} + \frac{\mu(\nabla v)^2}{T}.

Exergy destruction density: e˙D=T0σs.\dot{e}_D = T_0 \sigma_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.

13. Turbulent Combustion Modeling in CFD

ModelPrincipleRegime of Validity
Eddy Break-Up (EBU)Reaction rate ∝ turbulence dissipationFast-chemistry (mixing-controlled)
FlameletPre-tabulated laminar flame libraryNonpremixed, Da≫1
PDFStatistical closure for chemistry and mixingAll regimes, detailed kinetics
EDCFinite-rate model using k–ε turbulenceModerate-to-high Re flames
CMCConditional averaging on mixture fractionGeneral, detailed but expensive

14. Example: Jet Flame Scaling

For turbulent nonpremixed jet flames: xf∝DRe1/2Sc1/2,Lf∝Re1/2.x_f \propto D Re^{1/2} Sc^{1/2}, \quad L_f \propto Re^{1/2}.

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

15. Summary Equations

ConceptEquation
Species conservation∂(ρYi)/∂t+∇⋅(ρYiv)=−∇⋅Ji+ω˙i\partial(\rho Y_i)/\partial t + \nabla\cdot(\rho Y_i\mathbf{v}) = -\nabla\cdot\mathbf{J}_i + \dot{\omega}_i
Reaction rateω˙r=kf∏[Ci]νi′−kb∏[Ci]νi′′\dot{\omega}_r = k_f \prod[C_i]^{\nu'_i} - k_b \prod[C_i]^{\nu''_i}
Damköhler numberDa=tt/tcDa = t_t/t_c
Karlovitz numberKa=tη/tcKa = t_\eta/t_c
Scalar dissipation rate$\chi = 2D
EDC rateω˙i=ρ(ε/k)(Yi−Yi,eq)/τ∗\dot{\omega}_i = \rho(\varepsilon/k)(Y_i - Y_{i,eq})/\tau^*
Flamelet equationρχ/2∂2φ/∂Z2=ω˙φ\rho\chi/2 \partial^2\varphi/\partial Z^2 = \dot{\omega}_\varphi
Entropy generationσs=q˙chem/T+k(∇T)2/T2+μ(∇v)2/T\sigma_s = \dot{q}_{chem}/T + k(\nabla T)^2/T^2 + \mu(\nabla v)^2/T
  • 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.