Turbulence Modeling

Nature of turbulence

Turbulence is an unsteady, three-dimensional, chaotic flow in which velocity and pressure fluctuate randomly around a mean value. Its main features:

• Irregularity: random fluctuations that cannot be predicted in detail; treated statistically.
• Diffusivity: rapid mixing of momentum, heat and mass, much faster than molecular diffusion.
• Rotationality and 3D vortex stretching: it is made of eddies of many sizes.
• Dissipation: viscous action turns kinetic energy into heat, so turbulence needs a continuous energy supply.
• High Reynolds number: it appears when $Re$ exceeds a critical value.
• A property of the flow, not of the fluid.

Energy cascade (Richardson, Kolmogorov)

 mean flow
    | energy input
    v
 [ large eddies  L ]   (size of the geometry)
    |  vortex stretching, breakup
    v
 [ medium eddies   ]   inertial range
    |  no dissipation here
    v
 [ small eddies eta ]  viscous dissipation
    |
    v
   heat

• Large eddies, of size comparable to the flow geometry ($\ell_0$), extract energy from the mean flow.
• They are unstable and break up into smaller eddies, passing energy down the scales (the inertial range) with little loss.
• At the smallest scales the Kolmogorov scales viscosity dissipates the energy to heat:

$$\eta=\left(\frac{\nu^3}{\varepsilon}\right)^{1/4},\qquad \tau_\eta=\left(\frac\nu\varepsilon\right)^{1/2},\qquad u_\eta=(\nu\varepsilon)^{1/4}$$

where $\varepsilon$ is the dissipation rate per unit mass. The ratio of largest to smallest scale grows as $\ell_0/\eta\sim Re^{3/4}$.

Engineering implications

• Increases drag, pressure losses and wall friction and heat transfer coefficients (useful in heat exchangers, harmful in pipelines).
• Improves mixing (combustors, reactors) and delays flow separation (golf ball dimples).
• Causes noise and vibration.
• For CFD: direct simulation (DNS) needs about $Re^{9/4}$ grid points, which is impractical, so modelling (RANS, LES) is used.

Reynolds decomposition and averaging

Each flow variable is split into a mean and a fluctuation:

$$u=\bar u+u',\qquad p=\bar p+p',\qquad \bar u=\frac1T\int_0^Tu\,dt,\qquad \overline{u'}=0$$

Rules: $\overline{\bar u}=\bar u$, $\overline{u'}=0$, but $\overline{u'v'}\neq0$ in general. For unsteady mean flow, ensemble averaging is used.

RANS equations

Substitute into the incompressible continuity and Navier-Stokes equations and average:

Continuity:

$$\frac{\partial\bar u_i}{\partial x_i}=0$$

Momentum:

$$\rho\left(\frac{\partial\bar u_i}{\partial t}+\bar u_j\frac{\partial\bar u_i}{\partial x_j}\right)=-\frac{\partial\bar p}{\partial x_i}+\frac{\partial}{\partial x_j}\left(\mu\frac{\partial\bar u_i}{\partial x_j}-\rho\overline{u'_iu'_j}\right)$$

The new term $-\rho\overline{u'_iu'_j}$ is the Reynolds stress tensor: 6 independent components (symmetric) which represent the extra momentum transport by turbulence. The term comes from the averaging of the non-linear convective term.

Closure problem

The averaged equations have 4 equations (continuity + 3 momentum) but 10 unknowns (4 mean variables + 6 Reynolds stresses). Writing equations for the stresses brings in triple correlations $\overline{u'u'u'}$, and so on: every level introduces more unknowns than equations. The set cannot be closed without turbulence models that relate the Reynolds stresses to mean quantities.

Boussinesq hypothesis

Reynolds stresses are assumed to act like viscous stresses, proportional to the mean strain rate, through an eddy viscosity $\mu_t$:

$$-\rho\overline{u'_iu'_j}=\mu_t\left(\frac{\partial\bar u_i}{\partial x_j}+\frac{\partial\bar u_j}{\partial x_i}\right)-\frac23\rho k\delta_{ij},\qquad k=\tfrac12\overline{u'_iu'_i}$$

The problem then becomes one of finding $\mu_t$: zero-equation (mixing length), one-equation (Spalart-Allmaras) and two-equation (k-$\varepsilon$, k-$\omega$) models. Alternatively, Reynolds stress models (RSM) solve transport equations for each stress, without the isotropy assumption.

(a) Statistics

Mean:

$$\bar u=\frac{10.2+9.8+10.5+9.6+10.1+10.4+9.7+10.3}{8}=\frac{80.6}{8}=10.075\ \text{m/s}$$

Fluctuations $u'=u-\bar u$: 0.125, -0.275, 0.425, -0.475, 0.025, 0.325, -0.375, 0.225.

Sum of squares $=0.0156+0.0756+0.1806+0.2256+0.0006+0.1056+0.1406+0.0506=0.795$

$$u'_{rms}=\sqrt{\frac{0.795}{8}}=0.315\ \text{m/s}$$ $$I=\frac{u'_{rms}}{\bar u}=\frac{0.315}{10.075}=0.0313=3.1\%$$

(If the fluctuations were isotropic, $k\approx\tfrac32u'^2_{rms}=0.149$ m²/s².)

(b) Inlet turbulence values

$$k=\tfrac32(UI)^2=1.5\,(12\times0.05)^2=1.5\times0.36=0.54\ \text{m}^2/\text{s}^2$$

Length scale: $\ell=0.07D=0.07\times0.2=0.014$ m.

Dissipation rate:

$$\varepsilon=C_\mu^{3/4}\frac{k^{3/2}}{\ell}=\frac{0.09^{0.75}\times0.54^{1.5}}{0.014}=\frac{0.1643\times0.3968}{0.014}=4.66\ \text{m}^2/\text{s}^3$$

Specific dissipation rate:

$$\omega=\frac{\varepsilon}{C_\mu k}=\frac{4.66}{0.09\times0.54}=95.8\ \text{s}^{-1}$$

(Equivalent: $\omega=\sqrt k/(C_\mu^{1/4}\ell)=0.735/(0.5477\times0.014)=95.8$.)

Answer: $\bar u=10.08$ m/s, $u'_{rms}=0.315$ m/s, $I=3.1\%$; inlet $k=0.54$ m²/s², $\varepsilon=4.66$ m²/s³, $\omega=95.8$ s$^{-1}$.

Both are two-equation eddy viscosity models: two transport equations give a velocity scale and a length (or time) scale, from which the eddy viscosity $\mu_t$ is found (Boussinesq hypothesis).

Standard k-$\varepsilon$ (Launder-Spalding)

Solves transport equations for turbulent kinetic energy $k$ and its dissipation rate $\varepsilon$:

$$\mu_t=\rho C_\mu\frac{k^2}{\varepsilon},\quad C_\mu=0.09$$ $$\frac{D(\rho k)}{Dt}=\nabla\cdot\left[\left(\mu+\frac{\mu_t}{\sigma_k}\right)\nabla k\right]+P_k-\rho\varepsilon$$ $$\frac{D(\rho\varepsilon)}{Dt}=\nabla\cdot\left[\left(\mu+\frac{\mu_t}{\sigma_\varepsilon}\right)\nabla\varepsilon\right]+C_{1\varepsilon}\frac\varepsilon kP_k-C_{2\varepsilon}\rho\frac{\varepsilon^2}{k}$$

with $C_{1\varepsilon}=1.44,\ C_{2\varepsilon}=1.92,\ \sigma_k=1.0,\ \sigma_\varepsilon=1.3$. Valid for fully turbulent flow, so it uses wall functions near walls.

k-$\omega$ SST (Menter)

A hybrid: the Wilcox k-$\omega$ model near walls, which is switched by blending functions $F_1,F_2$ to a transformed k-$\varepsilon$ model in the free stream. Uses $\omega=\varepsilon/(C_\mu k)$ (specific dissipation rate) and limits the shear stress in adverse pressure gradients through

$$\mu_t=\frac{\rho a_1k}{\max(a_1\omega,\ SF_2)}$$

(the "shear stress transport" limiter, $a_1=0.31$).

Comparison

Neither model captures anisotropy or strong swirl well; for those, RSM or LES is preferred.