Point particles randomly distributed within a vertically periodic cylinder of period L* and radius R* in an unbounded fluid.
Application of quasi-periodicity in numerical simulations. The velocity of the particle at height z is calculated by summing all the contributions from the periodically extended region (z − L, z + L) of length 2L centred on that particle.
Numerical simulation of a particle plume with period L = 40, particle number density n = 12 at t = 25. The plume initially had a statistically uniform radius R = 1. (a) Projection of particle positions onto the x − z plane, (b) projection onto the y − z plane, (c) radial position, , against z. To a good approximation the disturbances are axisymmetric.
Spatiotemporal evolution of a particle plume with period L = 200 and particle number density n = 25. The shading represents the value of R at that particular time and height.
Evolution of σ in 12 particle plumes with periodic length L = 100 and number density n = 12 (grey solid lines). The average behaviour (from these 12 plus a further 80 simulations) is also shown (black dashed line).
(a) Variation of σavg with t for various values of n (heavy lines). The statistical uncertainty is estimated by (thin lines), where . (b) Scaling σavg by n −1/2 demonstrates a good collapse onto a single curve for t ≲ 25. The duration of the collapse increases with n. An analytic prediction (Eq. (29)) of the initial growth is also shown (straight line).
Comparison of values of σavg at t = 200 in the present quasi-periodic simulations with the earlier experimental and numerical data of PNGS. Their data have been non-dimensionalised with respect to the mean plume radius and show a better collapse when plotted against n here than against ϕ in the original Figure 9 of PNGS.
Evolution of RMS Fourier amplitudes of the radial variation (Eq. (7)) for (a) n = 6 and (b) n = 25 with L = 100. One error bar is plotted in each figure to give an indication of the typical statistical uncertainty.
(a) Evolution of the autocorrelation with time for average of 152 simulations with n = 25 and L = 100. The typical statistical uncertainty is indicated for t = 80. (b) Autocorrelations evaluated at t = 200 for various n. Location of the first minimum is marked with a cross.
Variation of the typical length scale with n for both the previous results of PNGS and the new quasi-periodic results.
(a) Growth of the RMS average radius with t, ensemble averaged over multiple simulations with L = 40 (thick lines). An estimate of the statistical uncertainty is shown for n = 6 (thin lines). (b) Log-log plot of against t (thick lines) demonstrating t 2/3 behaviour (thin lines) at long times. (c) D n v n for the fitted 1 + D n t 2/3 growth of . The line of best fit has gradient −0.648 ≈ −2/3.
(a) Growth of with t, ensemble averaged over multiple simulations with L = 40 (thick lines). An estimate of the statistical uncertainty is shown for n = 6 (thin lines). (b) Comparison of the growth in the maximum radius () with growth in the average radius (); their ratio is plotted against time. The long-time constant behaviour of n = 6 is also shown.
Evidence of nonlinear wave-breaking near z = 13 as visualised by the radial location of particles at four successive times for a simulation with n = 25 and L = 40.
(a) Variation of σavg with t for various values of n (heavy lines) showing long-term growth. An estimate of the statistical uncertainty is shown for the case n = 6 (thin lines). (b) Variation of σavg with t when scaled by . Horizontal lines have been fitted to the region t > 400 for each value of n.
Response I(λ), as given by Eq. (16), to density variations in the axial direction. Also shown is the response measured from numerical simulations via the RMS average of coefficients in the DFT of , where the average is taken over 1000 simulations with n = 25 and L = 100. The small difference is due to density variations with radial or azimuthal structure.
Behaviour of radial bulges on the surface of a cylinder of continuous fluid with constant density contrast. (a) Phase speed c p and group speed c g of small-amplitude axisymmetric sinusoidal disturbances of wavelength λ. These wavespeeds are calculated in the frame of no net flux along the cylinder, in which the surface of the cylinder has a velocity of −0.125. (In the frame moving with the surface, c p would always be positive.) (b) Nonlinear wave-breaking of large-amplitude disturbances with wavelength λ = 5. Here the continuum behaviour is approximated by a plume of particles with n = 200.
Estimated time scales T shear and T prop limiting the growth of radial bulges due to density fluctuations being sheared out in the axial direction and due to radial bulges propagating relative to the density fluctuations, respectively. The shaded region corresponds to times less than the minimum of the two time scales.
Response to density modes with low radial and azimuthal wavenumbers (m, p).
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