Scales of turbulence

10.2. Scales of turbulence#

Kolmogorov scales: large, inertial, viscous scales

Power-law for inertial scales…

Turbulence involves different characteristic scales1, with their own characteristic length, time and relevant physical processes. While large scales usually depends on teh geometry of the flow, smaller scales have universal features for different flows: within this universal range, inertial subrange and dissipation range can be identified.

Large scales.

Inertial subrange. Assumption: turbulence in equilibrium. Spectral energy density \(E(\kappa)\)2 is written as a function of dissipation \(\varepsilon\) (equal to production, in equilibrium), and a length scale \(\ell\) or a wave number \(\kappa\). From dimensional analysis

\[[ E(\kappa) ] = \text{L}^3 \text{T}^{-2} \quad , \quad [ \varepsilon ] = \text{L}^2 \text{T}^{-3} \quad , \quad [ \kappa ] = \text{L}^{-1} \ ,\]

it follows

\[E \propto \varepsilon^{\frac{2}{3}} \kappa^{-\frac{5}{3}}\]

Kolmogorov scales. Two relevant physical quantities: \(\nu\) kinematic viscosity, average rate of dissipation per unit-mass \(\varepsilon\) or \(\mathscr{D}\), with physical dimensions

\[[\nu] = \frac{\text{L}^2}{\text{T}} \quad , \quad [ \mathscr{D} ] = \frac{\text{L}^2}{\text{T}^3} \ .\]

With these two relevant physical quantities, it’s possible to build characteristic length, time, velocity and Reynolds number (built with the characteristic dimensions, no surprise it goes to 1)

\[ \ell_{\eta} := \left( \frac{\nu^3}{\varepsilon} \right)^{\frac{1}{4}} \quad , \quad \tau_{\eta} := \left( \frac{\nu}{\varepsilon} \right)^{\frac{1}{2}} \quad , \quad U_{\eta} := \frac{\ell_{\eta}}{\tau_{\eta}} = \left( \nu \varepsilon \right)^{\frac{1}{4}} \quad , \quad Re_{\eta} := \frac{U_{\eta} \ell_{\mu}}{\nu} = 1 \]

todo evaluate the ratio of characteristic dimension of the viscous range with the chacteristic dimension of the problem in terms of Reynolds” number. Which assumptions to be made to get a Reynolds number of the large scales? Especially about \(\varepsilon\).

Under the assumption of turbulence in equilibrium, dimensional analysis gives

\[\varepsilon = \frac{U^3}{L} \ ,\]

and thus

\[ \frac{\ell_{\eta}}{L} = \text{Re}^{-\frac{3}{4}} \quad , \quad \frac{\tau_{\eta}}{T} = \text{Re}^{-\frac{1}{2}} \quad , \quad \frac{U_{\eta}}{U} = \text{Re}^{-frac{1}{4}} \ . \]

These relation gives an estimate of the spatial resolution — i.e. the dimension of the mathematical problem — required by a numerical methods to fully resolve all the structures in the flow. In a 3-dimensional domain of side \(L\), grid spacing should be of order \(\Delta x \sim \ell_{\eta} \sim L \, \text{Re}^{-\frac{3}{4}}\), and thus the number of grid cells approximately \(N \sim \left\frac{L}{\ell_{\tau}}\right)^3 = \text{Re}^{\frac{9}{4}}\).

1: Roughly speaking, the characteristic dimension, velocity,… of the vortices. But what’s a vortex?
2: Spectral energy density is defined as the energy density per unit wave-length, \(E = \int_{\kappa = 0}^{+\infty} E(\kappa) \, d \kappa\), and so it has physical dimension \([ E(\kappa) ] = [ E ][ L ] = L^3 T^{-2}\).