concepts in Turbulent Flow

Table of Contents

• 1. Concepts/Glossary
  • 1.1. Turbulent eddy viscosity ,μ_t
  • 1.2. Turbulent kinetic energy (TKE)
  • 1.3. Kolmogorov length scale, η
  • 1.4. turbulent length scale, ℓ
  • 1.5. turbulent kinetic energy dissipation rate, ε
  • 1.6. Turbulence intensity, I

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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