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We study the correlation function and mean linear response function of thevelocity Fourier mode of statistically steady-state, homogeneous and isotropicturbulence in the Eulerian and Lagrangian coordinates through direct numericalsimulation (DNS). As the Lagrangian velocity, we here adopt Kraichnan'sLagrangian history framework where Lagrangian particles are labelled withcurrent positions and their velocity are measured at some time before. ThisLagrangian velocity is numerically calculated with a method known as passivevector method. Our first goal is to study relation between the correlationfunction and the mean linear response function in the Eulerian and Lagrangiancoordinates. Such a relation is known to be important in analysing the closedset of equations for the two functions, which are obtained bydirect-interaction-approximation type closures. We demonstrate numerically thatthe fluctuation-dissipation theorem (proportionality between the two functions)does not hold. The relation is further investigated with general analyticalexpressions of the mean linear response function under stochastic settings,which are known as the fluctuation-response relations in non-equilibriumstatistical mechanics. Our second goal is to identify characteristic timesassociated with the two functions and to compare the times between the Eulerianand Lagrangian coordinates. Our DNS result supports the common view that theEulerian characteristic times have the sweeping-time scaling ($\propto k^{-1}$,where $k$ is the wavenumber) for both functions and the Lagrangiancharacteristic times in the inertial range have the Kolmogorov-time scaling($\propto k^{-2/3}$) for both functions.

Correlation function and linear response function of homogeneous isotropic turbulence in the Eulerian and Lagrangian coordinates