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 co ntinuity, (ii) the two-point Eulerian spatial spectra have a known for m (e.g. the Kolmogorov inertial subrange), (iii) the time dependence i s modelled by dividing the turbulence into large- and small-scales edd ies, and by assuming that the large eddies advect the small eddies whi ch also decorrelate as they are advected, (iv) the amplitudes of the l arge- 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 tube s of intense vorticity. Some new results and concepts have been obtain ed using this kinematic simulation: (a) for the inertial subrange (whi ch cannot yet be simulated by other means) the simulation confirms the form of the Eulerian frequency spectrum phi(11)E = C(E)epsilon(2/3)U0 (2/3)omega(-5/3), where epsilon, U0, omega are the rate of energy diss ipation per unit mass, large-scale r.m.s. velocity, and frequency. For isotropic Gaussian large-scale turbulence at very high Reynolds numbe r, C(E) almost-equal-to 0.78, which is close to the computed value of 0.82; (b) for an observer moving with the large eddies the 'Eulerian-L agrangian' spectrum is phi(11)EL = C(EL)epsilon-omega(-2), where C(EL) almost-equal-to 0.73; (c) for an observer moving with a fluid particl e the Lagrangian spectrum phi(11)L = C(L)epsilon-omega(-2), where C(L) almost-equal-to 0.8, a value consistent with the atmospheric turbulen ce measurements by Hanna (1981) and approximately equal to C(EL); (d) the mean-square relative displacement of a pair of particles <DELTA(2) > tends to the Richardson (1926) and Obukhov (1941) form <DELTA(2)> = G(DELTA)epsilon-t3, provided that the subrange extends over four decad es in energy, and a suitable origin is chosen for the time t. The cons tant G(DELTA) is computed and is equal to 0.1 (which is close to Tatar ski's 1960 estimate of 0.06); (e) difference statistics (i.e. displace ment from the initial trajectory) of single particles are also calcula ted. The exact result that <Y2> = G(Y)epsilon-t3 with G(Y) = 2-pi-C(L) is approximately confirmed (although it requires an even larger inert ial subrange than that for <DELTA(2)>). It is found that the ratio R(G ) = 2<Y2>/<DELTA(2)> almost-equal-to 100, whereas in previous estimate s R(G) almost-equal-to 1, because for much of the time pairs of partic les move together around vortical regions and only separate for the pr oportion of the time (of O(f(c))) they spend in straining regions wher e streamlines diverge. It is estimated that R(G) almost-equal-to O(f(c )-3). Thus relative diffusion is both a 'structural' (or 'topological' ) process as well as an intermittent inverse cascade process determine d 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), Hinze (1975) and George et al. (1984 ).