--- title: "Turbulence intensity and inlet k, epsilon" subject: "Applied Computational Fluid Dynamics" chapter: 5 key: "semester_7/applied_computational_fluid_dynamics:turbulence-intensity-inlet-numerical" source: https://www.bctengineer.com/q/semester_7/applied_computational_fluid_dynamics/turbulence-intensity-inlet-numerical site: BCT Engineering --- # Turbulence intensity and inlet k, epsilon Applied Computational Fluid Dynamics · Chapter 5: Turbulence Modeling - Asked in: year not given, 5 marks - Video: [OpenFOAM Tutorial ( Chap 5 - Section 2): Turbulence Field: k, epsilon, nut Boundary Conditions](https://www.youtube.com/watch?v=gnoJDPDHESA) — Abecator CFD | Engineering Simulation Academy - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/applied_computational_fluid_dynamics/turbulence-intensity-inlet-numerical - Whole chapter: https://www.bctengineer.com/md/semester_7/applied_computational_fluid_dynamics/chapter-5 ## Question Eight instantaneous velocity readings (m/s) taken at a point in a turbulent duct flow are 10.2, 9.8, 10.5, 9.6, 10.1, 10.4, 9.7 and 10.3. (a) Find the mean velocity, the rms fluctuation and the turbulence intensity. (b) For a CFD inlet of a 0.2 m diameter pipe with mean velocity 12 m/s and turbulence intensity 5%, estimate the turbulent kinetic energy $k$, the turbulence length scale (0.07 D), the dissipation rate $\varepsilon$ and the specific dissipation rate $\omega$ (take $C_\mu=0.09$). ## Short answer Mean velocity = 10.08 m/s, rms fluctuation = 0.315 m/s, turbulence intensity = 3.1%. Inlet k = 1.5(UI)² = 0.54 m²/s², length scale ℓ=0.014 m, ε = C_μ^{3/4} k^{3/2}/ℓ = 4.66 m²/s³, ω = ε/(C_μ k) = 95.8 s⁻¹. ## Answer #### (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$^{-1}$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For the eight readings, what are the mean and rms fluctuation? A. 10.075 m/s and 0.315 m/s B. 10.075 m/s and 0.795 m/s C. 10.0 m/s and 0.315 m/s D. 9.8 m/s and 0.1 m/s Answer: A. Mean = 80.6/8 = 10.075; rms = sqrt(0.795/8) = 0.315 m/s. (Key point: Mean and rms) 2. What is the measured turbulence intensity? A. 31% B. 0.315% C. 3.1% D. 5% Answer: C. I = u'_rms/mean = 0.315/10.075 = 0.0313, i.e. 3.1%. (Key point: Turbulence intensity) 3. For U = 12 m/s and I = 5%, which formula and value give k? A. k = (UI)^2 = 0.36 B. k = 1.5 (UI)^2 = 0.54 m^2/s^2 C. k = 1.5 U I = 0.9 D. k = 0.5 (UI)^2 = 0.18 Answer: B. k = 1.5 (12 x 0.05)^2 = 1.5 x 0.36 = 0.54 m^2/s^2. (Key point: Inlet k) 4. How are epsilon and omega estimated from k and the length scale l = 0.014 m? A. epsilon = Cmu^(3/4) k^(3/2)/l = 4.66; omega = epsilon/(Cmu k) = 95.8 B. epsilon = k/l; omega = k/epsilon C. epsilon = Cmu k^2/l; omega = 4.66 D. epsilon = Cmu^(3/4) k^(3/2)/l = 4.66; omega = epsilon/k = 8.6 Answer: A. epsilon = 0.1643 x 0.3968/0.014 = 4.66, and omega = epsilon/(Cmu k) = 4.66/(0.09 x 0.54) = 95.8 1/s. (Key point: Epsilon and omega) ## Related in Applied Computational Fluid Dynamics - [Nature of turbulence and energy cascade](https://www.bctengineer.com/question.md?key=semester_7%2Fapplied_computational_fluid_dynamics%3Aturbulence-energy-cascade-nature) — Applied Computational Fluid Dynamics, chapter 5 - [Integral continuity for a compressing duct](https://www.bctengineer.com/question.md?key=semester_7%2Fapplied_computational_fluid_dynamics%3Aintegral-continuity-duct-numerical) — Applied Computational Fluid Dynamics, chapter 2 - [Stokes flow and settling velocity](https://www.bctengineer.com/question.md?key=semester_7%2Fapplied_computational_fluid_dynamics%3Astokes-flow-settling-particle) — Applied Computational Fluid Dynamics, chapter 2 - [Reynolds number and flow regime in pipes](https://www.bctengineer.com/question.md?key=semester_7%2Fapplied_computational_fluid_dynamics%3Areynolds-number-flow-regime) — Applied Computational Fluid Dynamics, chapter 1 - [Pressure drop in a pipe network](https://www.bctengineer.com/question.md?key=semester_7%2Fapplied_computational_fluid_dynamics%3Apipe-network-pressure-drop-numerical) — Applied Computational Fluid Dynamics, chapter 6 - [k-epsilon and k-omega SST models](https://www.bctengineer.com/question.md?key=semester_7%2Fapplied_computational_fluid_dynamics%3Ak-epsilon-k-omega-sst-models) — Applied Computational Fluid Dynamics, chapter 5 ## Related in other subjects - [Velocity for fully turbulent flow in rough pipe](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Afully-turbulent-velocity-1200mm-pipe) — Hydraulics, chapter 1 - [Pipe flow regimes and velocity profiles](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Apipe-flow-regimes-profiles) — Hydro Power, chapter 1 - [Laminar/turbulent flow and friction factor](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Alaminar-turbulent-friction-factor) — Hydro Power, chapter 1 - [Parabolic velocity profile in laminar pipe flow](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Alaminar-parabolic-velocity-profile) — Hydraulics, chapter 1