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

Which of these represent ensemble averaging if ‘N’ represents the

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

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