Second order structure functions for higher powers of turbulent velocity

Area of Science:

• Fluid Dynamics
• Turbulence Research
• Statistical Mechanics

Background:

• The temporal second-order structure functions of turbulent fluid velocity fluctuations are crucial for understanding turbulence.
• Previous studies primarily focused on single Eulerian points (N=1), with limited exploration of spatial averaging effects (N>1).
• The Kolmogorov framework predicts smoothing of fluctuations with spatial averaging, potentially losing essential flow information.

Purpose of the Study:

• To experimentally investigate the impact of spatial averaging on temporal second-order structure functions of turbulent velocity fluctuations.
• To analyze the evolution of scaling exponents with increasing spatial averaging (N) in both 3D and 2D turbulence.
• To compare the observed behaviors with theoretical predictions and explore implications for turbulence theory and experimental design.

Main Methods:

• Conducted experiments in a 3D turbulent tank with rotating jets.
• Performed 2D turbulence experiments using a gravity-assisted soap film setup.
• Calculated temporal second-order structure functions for integer powers of velocity fluctuations, averaged over N Eulerian points.

Main Results:

• In 3D turbulence, scaling exponents evolve exponentially towards an asymptotic value, consistent with prior findings.
• In 2D turbulence, scaling exponents show polynomial dependence on power 'm', with logarithmic evolution and a distinct asymptotic behavior.
• The convergence rates differ significantly between 3D (exponential) and 2D (logarithmic) turbulence, suggesting different inter-scale coupling mechanisms.

Conclusions:

• Spatial averaging of turbulent velocity fluctuations does not simply smooth out information but reveals distinct scaling properties.
• The observed differences in convergence forms between 3D and 2D turbulence highlight fundamental distinctions in their dynamics.
• Results have implications for determining resolution requirements in 2D turbulence studies and understanding atmospheric boundary layer turbulence.