Question

Which of these represent ensemble averaging if ‘N’ represents the number of identical quantities?

(a) \(\frac{1}{N}\int_N \phi dN\)

(b) \(lim_{N→∞}\frac{⁡1}{N}\int_N \phi dN\)

(c) \(\frac{1}{N}\sum_{i=1}^N\phi_i \)

(d) \(lim_{N→∞}\frac{⁡1}{N} \sum_{i=1}^N\phi_i \)

The question was posed to me in a national level competition.

My doubt stems from Turbulence Modelling in chapter Turbulence Modelling of Computational Fluid Dynamics

Answer 1

The correct choice is (d) \(lim_{N→∞}\frac{⁡1}{N} \sum_{i=1}^N\phi_i \)

Explanation: Ensemble averaging is based on identical flow variables. It sums up the identical variables and then takes the average. The equation is \(lim_{N→∞}\frac{⁡1}{N} \sum_{i=1}^N\phi_i \).