Question

Which of these represent the Gaussian filter function?

(a) \((\frac{\gamma}{\pi \Delta^2})^{\frac{3}{2}} exp(-\gamma \big|\vec x,\vec{x’}\big|)\)

(b) \((\frac{\gamma}{\pi \Delta^2})^{\frac{3}{2}} exp(-\gamma \frac{\big|\vec x,\vec{x’}\big|}{\Delta})\)

(c) \((\frac{\gamma}{\Delta^2})^{\frac{3}{2}} exp(-\gamma \frac{\big|\vec x,\vec{x’}\big|^2}{\Delta^2})\)

(d) \((\frac{\gamma}{\pi \Delta^2})^{\frac{3}{2}} exp(-\gamma \frac{\big|\vec x,\vec{x’}\big|^2}{\Delta^2})\)

This question was addressed to me in an online interview.

This question is from Large Eddy Simulation for Turbulent Models topic in portion Turbulence Modelling of Computational Fluid Dynamics

Answer 1

Right answer is (d) \((\frac{\gamma}{\pi \Delta^2})^{\frac{3}{2}} exp(-\gamma \frac{\big|\vec x,\vec{x’}\big|^2}{\Delta^2})\)

Easiest explanation: The filter function, in general, is a function of the spatial vector (\(\vec{x}\)), its derivative (\(\vec{x^{‘}}\)) and the cut-off width (Δ). The Gaussian filter has an additional parameter (γ). The typical value of γ is 6. The function is given as

\(G(\vec{x},\vec{x’},\Delta)=(\frac{\gamma}{\pi \Delta^2})^{\frac{3}{2}} exp(-\gamma \frac{\big|\vec x,\vec{x’}\big|^2}{\Delta^2})\)