Turbulence Mixing

Phenomenon
AliasesTurbulence, Reynolds Decomposition

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\to\infty}\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.\rho(\partial u_i/\partial t + u_j \partial u_i/\partial x_j) = -\partial p/\partial x_i + \mu \partial^2u_i/\partial x_j^2.

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

The additional term −ρui′uj′‾-\rho\overline{u'_i u'_j}is the Reynolds stress tensor: τijR=−ρui′uj′‾.\tau^R_{ij} = -\rho\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'_iand average: ∂k∂t+uˉj∂k/∂xj=P−ε+Tk,\frac{\partial k}{\partial t} + \bar{u}_j\partial k/\partial x_j = P - \varepsilon + 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} \partial\bar{u}_i/\partial x_j: production,
  • ε=ν(∂ui′/∂xj)(∂ui′/∂xj)‾\varepsilon = \nu \overline{(\partial u'_i/\partial x_j)(\partial u'_i/\partial 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 ε\varepsilonand kinematic viscosity ν\nu.

5.1 Kolmogorov Scales

η=(ν3/ε)1/4,uη=(νε)1/4,τη=(ν/ε)1/2.\eta = (\nu^3/\varepsilon)^{1/4}, \quad u_\eta = (\nu\varepsilon)^{1/4}, \quad \tau_\eta = (\nu/\varepsilon)^{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 \varepsilon^{2/3} k^{−5/3}, where CK≈1.5C_K \approx 1.5is 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} \to \text{transfer} \to \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 ε\varepsilonis 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^\infty 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}{2\pi} \int_{−\infty}^{\infty} 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,\tau_t = \rho(l_m)^2\left|\frac{d\bar{u}}{dy}\right|\frac{d\bar{u}}{dy}, where lml_mis the mixing length (analogous to mean free path of eddies).

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

Relating to Reynolds stress: −u′v′‾=νt∂uˉ/∂y,νt=(lm)2∣∂uˉ/∂y∣.-\overline{u'v'} = \nu_t \partial\bar{u}/\partial y, \quad \nu_t = (l_m)^2 |\partial\bar{u}/\partial 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_\tau}{\kappa y}, \quad u_\tau = \sqrt{}{\tau_w/\rho}. Integrate to get: uˉuτ=1κln⁡(yuτ/ν)+B.\frac{\bar{u}}{u_\tau} = \frac{1}{\kappa}\ln(yu_\tau/\nu) + B. Constants κ=0.41,B=5.0\kappa = 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 kkand its dissipation are introduced.

10.1 k–ε Model

∂k∂t+uˉj∂k/∂xj=P−ε+∂/∂xj[(ν+νt/σk)∂k/∂xj],\frac{\partial k}{\partial t} + \bar{u}_j\partial k/\partial x_j = P - \varepsilon + \partial/\partial x_j[(\nu + \nu_t/\sigma_k)\partial k/\partial x_j], ∂ε∂t+uˉj∂ε/∂xj=Cε1εkP−Cε2ε2k+∂/∂xj[(ν+νt/σε)∂ε/∂xj].\frac{\partial\varepsilon}{\partial t} + \bar{u}_j\partial\varepsilon/\partial x_j = C_{\varepsilon1}\frac{\varepsilon}{k}P - C_{\varepsilon2}\frac{\varepsilon^2}{k} + \partial/\partial x_j[(\nu + \nu_t/\sigma_\varepsilon)\partial\varepsilon/\partial x_j].

Turbulent viscosity: νt=Cμk2ε.\nu_t = C_\mu \frac{k^2}{\varepsilon}. Empirical constants: Cμ=0.09,Cε1=1.44,Cε2=1.92,σk=1.0,σε=1.3.C_\mu=0.09, C_{\varepsilon1}=1.44, C_{\varepsilon2}=1.92, \sigma_k=1.0, \sigma_\varepsilon=1.3.

10.2 k–ω Model

Uses specific dissipation rate ω=ε/k\omega = \varepsilon/k: νt=k/ω,\nu_t = k/\omega, 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'} = -\alpha_t \partial\bar{T}/\partial y, \quad \overline{v'c'} = -D_t \partial\bar{c}/\partial y.

Define turbulent Prandtl and Schmidt numbers: Prt=νt/αt,Sct=νt/Dt.Pr_t = \nu_t/\alpha_t, \quad Sc_t = \nu_t/D_t. Empirically, Prt≈0.85,Sct≈0.7−1.0.Pr_t \approx 0.85, Sc_t \approx 0.7-1.0.

Effective diffusivities: αeff=α+αt,Deff=D+Dt.\alpha_{eff} = \alpha + \alpha_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) \sim x^{−1/2}, \quad \delta(x) \sim x.

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

Momentum flux conservation: ∫ρu2dy=const.\int \rho u^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) = \int \hat{u}(k)e^{ikx} dk. Energy spectrum E(k)E(k)satisfies: ∂E/∂t=T(k)−2νk2E(k),\partial E/\partial t = T(k) - 2\nu k^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_\eta.

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

14. Scalar Mixing and Turbulent Diffusion

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

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

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

15. Entropy Production and Energy Dissipation

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

16. Summary Equations

ConceptExpression
Reynolds stressτijR=−ρui′uj′‾\tau^R_{ij} = -\rho\overline{u'_i u'_j}
TKE equation∂k/∂t+Uj∂k/∂xj=P−ε+Tk\partial k/\partial t + U_j\partial k/\partial x_j = P - \varepsilon + T_k
Kolmogorov scalesη=(ν3/ε)1/4,uη=(νε)1/4,τη=(ν/ε)1/2\eta=(\nu^3/\varepsilon)^{1/4}, u_\eta=(\nu\varepsilon)^{1/4}, \tau_\eta=(\nu/\varepsilon)^{1/2}
Spectrum lawE(k)=CKε2/3k−5/3E(k)=C_K \varepsilon^{2/3} k^{−5/3}
Eddy viscosity$\nu_t=(l_m)^2
k–ε modelνt=Cμk2/ε\nu_t=C_\mu k^2/\varepsilon
Log-lawU+=1/κln⁡y++BU^+=1/\kappa\ln y^+ + B
Turbulent PrandtlPrt=νt/αt≈0.85Pr_t=\nu_t/\alpha_t\approx0.85
Turbulent SchmidtSct=νt/Dt≈1Sc_t=\nu_t/D_t\approx1
  • 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.