concepts in Turbulent Flow

1. Concepts/Glossary

1 Concepts/Glossary

eddy: any spatial flow pattern that persists for a short time.
Eddy: (oxford dictionary) a circular movement of air,water
homogeneous: the statistical characteristics of turbulence is independent of location of space.
homogeneous turbulence: a turbulent flow, on the average, is uniform in all directions ( Wilcox)
integral length scale: length scale of largest eddy, L
Isotropic: the statistical characteristics of turbulence is independent of the direction of space.
stationary process (turbulence): the statistics of a random variable, U(mean property), are independent of time (stationary / stationary process)
Turbulence intensity: I = u'_rms/ U , where u'_rms : root-mean-square of the fluctuating velocity
Production of TKE (production): \(\mathcal{P}\), rate of production of turbulent kinetic energy (Eq. (5.133), pope)

\[ \mathcal{P}= - <u_i u_j> \frac{ \partial <U_i> }{\partial x_j } \]

1.1 Turbulent eddy viscosity ,μ_t

not a fluid property

analogy: molucular stress <-> turbulent stress

1.2 Turbulent kinetic energy (TKE)

TKE: mean kinetic energy per unit mass associated with eddies in turbulent flow.

\[ k = \frac{1}{2} \left(\, \overline{(u')^2} + \overline{(v')^2} + \overline{(w')^2} \,\right) \]

\[ k=0.5<u_i u_i>= 0.5 < \mathbf{u} \cdot \mathbf{u}> \] (4.24, pope)

physically, TKE is the mean kinetic energy per unit mass in the fluctuating velocity field (u').
half the trace of the Reynolds stress tensor

1.3 Kolmogorov length scale, η

\[ \eta = (\frac{\nu^3}{\epsilon})^{1/4} \]

1.4 turbulent length scale, ℓ

integral length scale: L, the size of the largest eddies, which are constrained by the physical boundaries of the flow
(Kolmogorove length scale): η, the size of smallest eddies which is determined by viscosity
(no term): the size of large eddy

Figure 1: Different length scales and ranges in turbulence energy cascade (Fig. 6.1 Pope)

k-ε model \[ \ell = C_\mu^{3/4} \, \frac{k^\frac{3}{2}}{\epsilon} \]

\( C_\mu \) model constant

1.5 turbulent kinetic energy dissipation rate, ε

kinetic energy disspated by viscosity per unit mass unit time

1.6 Turbulence intensity, I

Ideally, you will have a good estimate of the turbulence intensity at the inlet boundary from external, measured data. For example, if you are simulating a wind-tunnel experiment, the turbulence intensity in the free stream is usually available from the tunnel characteristics. In modern low-turbulence wind tunnels, the free-stream turbulence intensity may be as low as 0.05%.

\[ I = u'_rms/ U \]

u'_rms : root-mean-square of the fluctuating velocity

< 1% low > 10% high

Author: kaiming

Created: 2019-02-27 Wed 18:18

Emacs 24.5.1 (Org mode 8.2.10)

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