Kinematic simulation of homogeneous turbulence by unsteady random Fourier modes

The velocity field of homogeneous isotropic turbulence is simulated by a large number (38–1200) of random Fourier modes varying in space and time over a large number (> 100) of realizations. They are chosen so that the flow field has certain properties, namely (i) it satisfies continuity, (ii) the two-point Eulerian spatial spectra have a known form (e. g. the Kolmogorov inertial subrange), (iii) the time dependence is modelled by dividing the turbulence into large- and small-scales eddies, and by assuming that the large eddies advect the small eddies which also decorrelate as they are advected, (iv) the amplitudes of the large- and small-scale Fourier modes are each statistically independent and each Gaussian. The structure of the velocity field is found to be similar to that computed by direct numerical simulation with the same spectrum, although this simulation underestimates the lengths of tubes of intense vorticity. Some new results and concepts have been obtained using this kinematic simulation: (a) for the inertial subrange (which cannot yet be simulated by other means) the simulation confirms the form of the Eulerian frequency spectrum ^ E₁₁ = C^ E^2{3}U^2{3}₀^-5{3}, where ε, U 0, ω are the rate of energy dissipation per unit mass, large-scale r. m. s. velocity, and frequency. For isotropic Gaussian large-scale turbulence at very high Reynolds number, C E ≈ 0. 78, which is close to the computed value of 0. 82; (b) for an observer moving with the large eddies the ‘Eulerian—Lagrangian’ spectrum is ϕ EL 11 = C EL εω −2, where C EL ≈ 0. 73; (c) for an observer moving with a fluid particle the Lagrangian spectrum ϕ L 11 = C L εω −2, where C L ≈ 0. 8, a value consistent with the atmospheric turbulence measurements by Hanna (1981) and approximately equal to C EL ; (d) the mean-square relative displacement of a pair of particles 〈Δ 2 〉 tends to the Richardson (1926) and Obukhov (1941) form 〈Δ 2 〉 = G Δ ε t 3, provided that the subrange extends over four decades in energy, and a suitable origin is chosen for the time t. The constant G Δ is computed and is equal to 0. 1 (which is close to Tatarski's 1960 estimate of 0. 06) ; (e) difference statistics (i. e. displacement from the initial trajectory) of single particles are also calculated. The exact result that Y 2 = G Y ε t 3 with G Y = 2π C L is approximately confirmed (although it requires an even larger inertial subrange than that for 〈Δ 2 〉). It is found that the ratio Rscr G = 2〈 Y 2 〉/〈Δ 2 〉≈ 100, whereas in previous estimates Rscr G ≈ 1, because for much of the time pairs of particles move together around vortical regions and only separate for the proportion of the time (of O (f c) ) they spend in straining regions where streamlines diverge. It is estimated that Rscr G ≈ O (f c −3). Thus relative diffusion is both a ‘structural’ (or ‘topological’) process as well as an intermittent inverse cascade process determined by increasing eddy scales as the particles separate; (f) statistics of large-scale turbulence are also computed, including the Lagrangian timescale, the pressure spectra and correlations, and these agree with predictions of Batchelor (1951), Hinzc (1975) and George et al. (1984).