Turbulence Mixing

Phenomenon

Turbulence and Mixing — Statistical, Spectral, and Modeling Foundations

Scope: first-principles derivation of turbulence equations, energetics, and modeling concepts. Includes RANS formulation, turbulent kinetic energy balance, Kolmogorov similarity, eddy viscosity models, and turbulent transport of momentum, heat, and mass.

1. Nature of Turbulence

Turbulence is characterized by irregular, three-dimensional, and chaotic motion resulting in enhanced momentum, heat, and mass transport.

Key features:

  • Irregularity: stochastic velocity and pressure fluctuations.
  • Diffusivity: increased transport due to eddy mixing.
  • Dissipation: conversion of kinetic energy into internal energy by viscosity.
  • Continuum: despite randomness, governed by Navier–Stokes equations.
  • High Reynolds number: inertia ≫ viscosity (Re > ~4000 for pipes).

2. Reynolds Decomposition and Averaging

Decompose instantaneous quantities into mean and fluctuating parts: ui=uˉi+ui′,p=pˉ+p′.u_i = \bar{u}_i + u'_i, \quad p = \bar{p} + p'.

Mean (ensemble or time average): uˉi=lim⁡T→∞1T∫0Tui(t) dt.\bar{u}_i = \lim_{T→∞}\frac{1}{T}\int_0^T u_i(t)\,dt.

Fluctuations satisfy ui′‾=0\overline{u'_i} = 0.

3. Reynolds-Averaged Navier–Stokes (RANS) Equations

Start with incompressible Navier–Stokes: ρ(∂ui/∂t+uj∂ui/∂xj)=−∂p/∂xi+μ∂2ui/∂xj2.ρ(∂u_i/∂t + u_j ∂u_i/∂x_j) = -∂p/∂x_i + μ ∂^2u_i/∂x_j^2.

Substitute decomposition and average: ρ(∂uˉi/∂t+uˉj∂uˉi/∂xj)=−∂pˉ/∂xi+μ∂2uˉi/∂xj2−ρ∂(ui′uj′‾)/∂xj.ρ(∂\bar{u}_i/∂t + \bar{u}_j ∂\bar{u}_i/∂x_j) = -∂\bar{p}/∂x_i + μ ∂^2\bar{u}_i/∂x_j^2 - ρ∂(\overline{u'_i u'_j})/∂x_j.

The additional term −ρui′uj′‾-ρ\overline{u'_i u'_j} is the Reynolds stress tensor: τijR=−ρui′uj′‾.τ^R_{ij} = -ρ\overline{u'_i u'_j}. It represents the momentum transfer due to turbulent eddies.

4. Turbulent Kinetic Energy (TKE) Equation

Multiply fluctuating velocity equation by ui′u'_i and average: ∂k∂t+uˉj∂k/∂xj=P−ε+Tk,\frac{∂k}{∂t} + \bar{u}_j∂k/∂x_j = P - ε + T_k, where:

  • k=12ui′ui′‾k = \tfrac{1}{2}\overline{u'_i u'_i}: turbulent kinetic energy,
  • P=−ui′uj′‾∂uˉi/∂xjP = -\overline{u'_i u'_j} ∂\bar{u}_i/∂x_j: production,
  • ε=ν(∂ui′/∂xj)(∂ui′/∂xj)‾ε = ν \overline{(∂u'_i/∂x_j)(∂u'_i/∂x_j)}: dissipation,
  • TkT_k: turbulent transport (redistribution).

Energy cascade interpretation: energy is injected at large scales (production), transferred through intermediate scales (inertial subrange), and dissipated at smallest scales (Kolmogorov scale).

5. Kolmogorov Similarity Hypothesis (1941)

Assume local isotropy and universality at small scales — the only relevant quantities are dissipation rate εε and kinematic viscosity νν.

5.1 Kolmogorov Scales

η=(ν3/ε)1/4,uη=(νε)1/4,τη=(ν/ε)1/2.η = (ν^3/ε)^{1/4}, \quad u_η = (νε)^{1/4}, \quad τ_η = (ν/ε)^{1/2}.

These represent smallest eddies where viscous dissipation occurs.

5.2 Inertial Subrange and Energy Spectrum

Energy spectrum E(k)E(k) satisfies: E(k)=CKε2/3k−5/3,E(k) = C_K ε^{2/3} k^{−5/3}, where CK≈1.5C_K ≈ 1.5 is the Kolmogorov constant.

This -5/3 power law is observed universally in high-Re turbulence.

6. Energy Cascade

Turbulent kinetic energy is transferred from large to small scales: production→transfer→dissipation.\text{production} → \text{transfer} → \text{dissipation}.

  • Large eddies (size L): energy-containing range, governed by mean shear.
  • Intermediate eddies: inertial subrange, self-similar.
  • Small eddies: dissipative range.

Energy flux through scales εε is constant in stationary turbulence.

7. Statistical Description of Turbulence

Define velocity autocorrelation function: Ruu(r)=u′(x)u′(x+r)‾.R_{uu}(r) = \overline{u'(x)u'(x+r)}. Integral scale (large-eddy size): L=∫0∞Ruu(r)/Ruu(0) dr.L = \int_0^∞ R_{uu}(r)/R_{uu}(0)\,dr.

The energy spectrum is its Fourier transform: E(k)=12π∫−∞∞Ruu(r)e−ikr dr.E(k) = \frac{1}{2π} \int_{−∞}^{∞} R_{uu}(r)e^{−ikr}\,dr.

8. Eddy Viscosity and Mixing-Length Model

Prandtl proposed the mixing-length hypothesis: τt=ρ(lm)2∣duˉdy∣duˉdy,τ_t = ρ(l_m)^2\left|\frac{d\bar{u}}{dy}\right|\frac{d\bar{u}}{dy}, where lml_m is the mixing length (analogous to mean free path of eddies).

In wall-bounded flows: lm=κy,κ≈0.41.l_m = κ y, \quad κ ≈ 0.41.

Relating to Reynolds stress: −u′v′‾=νt∂uˉ/∂y,νt=(lm)2∣∂uˉ/∂y∣.-\overline{u'v'} = ν_t ∂\bar{u}/∂y, \quad ν_t = (l_m)^2 |∂\bar{u}/∂y|.

9. Turbulent Boundary Layer and Log-Law of the Wall

From momentum balance and mixing-length assumption: duˉdy=uτκy,uτ=√τw/ρ.\frac{d\bar{u}}{dy} = \frac{u_τ}{κ y}, \quad u_τ = √{τ_w/ρ}. Integrate to get: uˉuτ=1κln⁡(yuτ/ν)+B.\frac{\bar{u}}{u_τ} = \frac{1}{κ}\ln(yu_τ/ν) + B. Constants κ=0.41,B=5.0κ = 0.41, B = 5.0 (smooth wall). This empirical relation forms the foundation of wall models.

10. Two-Equation Turbulence Models

To generalize mixing-length models, transport equations for kk and its dissipation are introduced.

10.1 k–ε Model

∂k∂t+uˉj∂k/∂xj=P−ε+∂/∂xj[(ν+νt/σk)∂k/∂xj],\frac{∂k}{∂t} + \bar{u}_j∂k/∂x_j = P - ε + ∂/∂x_j[(ν + ν_t/σ_k)∂k/∂x_j], ∂ε∂t+uˉj∂ε/∂xj=Cε1εkP−Cε2ε2k+∂/∂xj[(ν+νt/σε)∂ε/∂xj].\frac{∂ε}{∂t} + \bar{u}_j∂ε/∂x_j = C_{ε1}\frac{ε}{k}P - C_{ε2}\frac{ε^2}{k} + ∂/∂x_j[(ν + ν_t/σ_ε)∂ε/∂x_j].

Turbulent viscosity: νt=Cμk2ε.ν_t = C_μ \frac{k^2}{ε}. Empirical constants: Cμ=0.09,Cε1=1.44,Cε2=1.92,σk=1.0,σε=1.3.C_μ=0.09, C_{ε1}=1.44, C_{ε2}=1.92, σ_k=1.0, σ_ε=1.3.

10.2 k–ω Model

Uses specific dissipation rate ω=ε/kω = ε/k: νt=k/ω,ν_t = k/ω, with improved near-wall resolution.

11. Turbulent Transport of Heat and Species

Analogous to momentum transport: v′T′‾=−αt∂Tˉ/∂y,v′c′‾=−Dt∂cˉ/∂y.\overline{v'T'} = -α_t ∂\bar{T}/∂y, \quad \overline{v'c'} = -D_t ∂\bar{c}/∂y.

Define turbulent Prandtl and Schmidt numbers: Prt=νt/αt,Sct=νt/Dt.Pr_t = ν_t/α_t, \quad Sc_t = ν_t/D_t. Empirically, Prt≈0.85,Sct≈0.7–1.0.Pr_t ≈ 0.85, Sc_t ≈ 0.7–1.0.

Effective diffusivities: αeff=α+αt,Deff=D+Dt.α_{eff} = α + α_t, \quad D_{eff} = D + D_t.

12. Free Shear Flows — Jets, Wakes, and Mixing Layers

Turbulent mixing layers grow linearly downstream due to eddy entrainment.

For plane jet: Uc(x)∼x−1/2,δ(x)∼x.U_c(x) ∼ x^{−1/2}, \quad δ(x) ∼ x.

Self-similar profile: u/Uc=f(η),η=y/δ(x).u/U_c = f(η), \quad η = y/δ(x).

Momentum flux conservation: ∫ρu2dy=const.∫ ρu^2 dy = const.

These flows exemplify entrainment and turbulent diffusion processes.

13. Spectral Energy Transfer and Dissipation

Fourier transform of velocity field: u(x)=∫u^(k)eikxdk.u(x) = ∫ \hat{u}(k)e^{ikx} dk. Energy spectrum E(k)E(k) satisfies: ∂E/∂t=T(k)−2νk2E(k),∂E/∂t = T(k) - 2νk^2E(k), where T(k)T(k) represents nonlinear energy transfer between wavenumbers.

In steady turbulence, T(k)T(k) balances dissipation for k>kηk>k_η.

Kolmogorov inertial range: E(k)∝k−5/3.E(k) ∝ k^{−5/3}.

14. Scalar Mixing and Turbulent Diffusion

For a passive scalar (e.g., dye, temperature): ∂c/∂t+ui∂c/∂xi=D∇2c.∂c/∂t + u_i ∂c/∂x_i = D ∇^2 c. In turbulent flow: ∂cˉ/∂t+uˉi∂cˉ/∂xi=∂/∂xi[(D+Dt)∂cˉ/∂xi].∂\bar{c}/∂t + \bar{u}_i ∂\bar{c}/∂x_i = ∂/∂x_i[(D + D_t)∂\bar{c}/∂x_i].

Empirical relation for turbulent diffusivity: Dt∼u′l′,D_t ∼ u' l', where u′u' and l′l' are characteristic velocity and length of large eddies.

Turbulent mixing time: tm∼L/u′.t_m ∼ L/u'.

15. Entropy Production and Energy Dissipation

At small scales, dissipation rate: ε=ν(∂ui′/∂xj)2‾.ε = ν \overline{(∂u'_i/∂x_j)^2}. Energy lost as heat contributes to local entropy generation: σs=εT.σ_s = \frac{ε}{T}. This ties turbulence energetics directly to thermodynamic irreversibility.

16. Summary Equations

ConceptExpression
Reynolds stressτijR=−ρui′uj′‾τ^R_{ij} = -ρ\overline{u'_i u'_j}
TKE equation∂k/∂t+Uj∂k/∂xj=P−ε+Tk∂k/∂t + U_j∂k/∂x_j = P - ε + T_k
Kolmogorov scalesη=(ν3/ε)1/4,uη=(νε)1/4,τη=(ν/ε)1/2η=(ν^3/ε)^{1/4}, u_η=(νε)^{1/4}, τ_η=(ν/ε)^{1/2}
Spectrum lawE(k)=CKε2/3k−5/3E(k)=C_K ε^{2/3} k^{−5/3}
Eddy viscosity$ν_t=(l_m)^2
k–ε modelνt=Cμk2/εν_t=C_μ k^2/ε
Log-lawU+=1/κln⁡y++BU^+=1/κ\ln y^+ + B
Turbulent PrandtlPrt=νt/αt≈0.85Pr_t=ν_t/α_t≈0.85
Turbulent SchmidtSct=νt/Dt≈1Sc_t=ν_t/D_t≈1
  • transport-equations.md — convection–diffusion framework for scalar mixing.
  • 02_Boundary_Layers_and_Separation.md — turbulent wall-bounded flow.
  • Thermodynamics/10_NonEquilibrium_Thermodynamics.md — entropy generation by turbulent dissipation.
  • 05_Turbulent_Combustion_and_Reactive_Flows.md — coupling of turbulence and chemical reactions.