Turbulence intensity and inlet k, epsilon

(a) Statistics

Mean:

$$\bar u=\frac{10.2+9.8+10.5+9.6+10.1+10.4+9.7+10.3}{8}=\frac{80.6}{8}=10.075\ \text{m/s}$$

Fluctuations $u'=u-\bar u$: 0.125, -0.275, 0.425, -0.475, 0.025, 0.325, -0.375, 0.225.

Sum of squares $=0.0156+0.0756+0.1806+0.2256+0.0006+0.1056+0.1406+0.0506=0.795$

$$u'_{rms}=\sqrt{\frac{0.795}{8}}=0.315\ \text{m/s}$$ $$I=\frac{u'_{rms}}{\bar u}=\frac{0.315}{10.075}=0.0313=3.1\%$$

(If the fluctuations were isotropic, $k\approx\tfrac32u'^2_{rms}=0.149$ m²/s².)

(b) Inlet turbulence values

$$k=\tfrac32(UI)^2=1.5\,(12\times0.05)^2=1.5\times0.36=0.54\ \text{m}^2/\text{s}^2$$

Length scale: $\ell=0.07D=0.07\times0.2=0.014$ m.

Dissipation rate:

$$\varepsilon=C_\mu^{3/4}\frac{k^{3/2}}{\ell}=\frac{0.09^{0.75}\times0.54^{1.5}}{0.014}=\frac{0.1643\times0.3968}{0.014}=4.66\ \text{m}^2/\text{s}^3$$

Specific dissipation rate:

$$\omega=\frac{\varepsilon}{C_\mu k}=\frac{4.66}{0.09\times0.54}=95.8\ \text{s}^{-1}$$

(Equivalent: $\omega=\sqrt k/(C_\mu^{1/4}\ell)=0.735/(0.5477\times0.014)=95.8$.)

Answer: $\bar u=10.08$ m/s, $u'_{rms}=0.315$ m/s, $I=3.1\%$; inlet $k=0.54$ m²/s², $\varepsilon=4.66$ m²/s³, $\omega=95.8$ s$^{-1}$.