• not a fluid property

analogy: molucular stress <–> turbulent stress

TKE

mean kinetic energy per unit mass associated with eddies (u=U-<U>) 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

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

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

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

• \( C_\mu \) model constant

kinetic energy disspated by viscosity per unit mass unit time

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