concepts in Turbulent Flow
Table of Contents
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 ,μt
- 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
