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FIG. 6. Sample of tracer pathlines in DNS (above) and the steady (A = 0) KS (below). The width of each dimension of the box in the visualization is approximately four times the Kolmogorov microscale.

Figure 6 Sample of tracer pathlines in DNS (above) and the steady (A = 0) KS (below). The width of each dimension of the box in the visualization is approximately four times the Kolmogorov microscale.

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TABLE I. Relevant parameters and output statistics for the incompressible DNS.
TABLE I. Relevant parameters and output statistics for the incompressible DNS.
FIG. 1. A typical Eulerian energy spectrum obtained from the DNS, once approximately stationary turbulence was achieved.
FIG. 1. A typical Eulerian energy spectrum obtained from the DNS, once approximately stationary turbulence was achieved.
TABLE II. Relevant parameters for particle tracking methods. It should be noted that in Type I] KS (bounded domain), the tendency of the domain to “bleed” particles to the absorbing boundary condition implies a limited period during which the mean concentration of particles in the inner domain remains roughly constant. For Type I KS (un- bounded domain), there exists no such concern, and for single-particle Lagrangian statistics, long simulations are preferred to ensure all particles have “forgotten” their ballistic range behaviour. All single-particle Lagrangian statistics (in the no-gravity case) were simulated in Type I KS. In the sravity case, particles which rapidly traverse a periodic domain in the DNS will see a spuriously highly correlated fluid field,™* thus there exists no benefit in running a very long simulation for the corresponding KS case. Consequently for Lagrangian statistics in KS with gravity, Type II KS was
TABLE II. Relevant parameters for particle tracking methods. It should be noted that in Type I] KS (bounded domain), the tendency of the domain to “bleed” particles to the absorbing boundary condition implies a limited period during which the mean concentration of particles in the inner domain remains roughly constant. For Type I KS (un- bounded domain), there exists no such concern, and for single-particle Lagrangian statistics, long simulations are preferred to ensure all particles have “forgotten” their ballistic range behaviour. All single-particle Lagrangian statistics (in the no-gravity case) were simulated in Type I KS. In the sravity case, particles which rapidly traverse a periodic domain in the DNS will see a spuriously highly correlated fluid field,™* thus there exists no benefit in running a very long simulation for the corresponding KS case. Consequently for Lagrangian statistics in KS with gravity, Type II KS was
FIG. 2. Illustration of particle-laden Type II KS. The computational domain consists of an outer domain with an absorbing boundary condition, and a smaller inner subdomain in which the statistics of the particles are calculated. The ensemble of particles is depicted as dots. In contrast, Type I KS occurs on an arbitrarily large spatial domain.
FIG. 2. Illustration of particle-laden Type II KS. The computational domain consists of an outer domain with an absorbing boundary condition, and a smaller inner subdomain in which the statistics of the particles are calculated. The ensemble of particles is depicted as dots. In contrast, Type I KS occurs on an arbitrarily large spatial domain.
TABLE III. KS Type for various statistics of interest. employed. All correlation dimension values were calculated in Type II KS, again due to the need for a high mean concentration to obtain meaningful measurements of clustering. This is summarized in Table III.
TABLE III. KS Type for various statistics of interest. employed. All correlation dimension values were calculated in Type II KS, again due to the need for a high mean concentration to obtain meaningful measurements of clustering. This is summarized in Table III.
FIG. 3. Dispersion of tracers in Types A and B KS, with a range of unsteadiness parameters A =0, 0.5, 1, 5. DNS results are shown for comparison. In Fig. 5, the t-separation of one diagonal component of the flatness tensor of the Lagrangian velocity is shown for DNS and Type A KS. KS underpredicts the flatness of velocity histories in all cases, though A = 0 yields the most intermittent velocity histories. More insight is offered by
FIG. 3. Dispersion of tracers in Types A and B KS, with a range of unsteadiness parameters A =0, 0.5, 1, 5. DNS results are shown for comparison. In Fig. 5, the t-separation of one diagonal component of the flatness tensor of the Lagrangian velocity is shown for DNS and Type A KS. KS underpredicts the flatness of velocity histories in all cases, though A = 0 yields the most intermittent velocity histories. More insight is offered by
FIG. 4. Lagrangian autocorrelation of tracers in Type A KS, with 2 =0, 0.5, 1, 5, with comparison to DNS results. Type B results are similar.
FIG. 4. Lagrangian autocorrelation of tracers in Type A KS, with 2 =0, 0.5, 1, 5, with comparison to DNS results. Type B results are similar.
FIG. 5. F a (component of the flatness of the Lagrangian velocity) for tracers for DNS, and for Type A KS, with A=0, 0.5, 1, 5. For the DNS case, FF (0) ~ 24.
FIG. 5. F a (component of the flatness of the Lagrangian velocity) for tracers for DNS, and for Type A KS, with A=0, 0.5, 1, 5. For the DNS case, FF (0) ~ 24.
FIG. 7. Time history of D2 for tracers, for DNS and steady (A = 0) KS.
FIG. 7. Time history of D2 for tracers, for DNS and steady (A = 0) KS.
FIG. 8. Dispersion in DNS, for tracers and all inertial particle types, without gravity. Inset plot is zoomed-in view of long-range dispersion.
FIG. 8. Dispersion in DNS, for tracers and all inertial particle types, without gravity. Inset plot is zoomed-in view of long-range dispersion.
FIG. 9. Dispersion in steady (A =0) KS, for tracers and all inertial particle types, without gravity.
FIG. 9. Dispersion in steady (A =0) KS, for tracers and all inertial particle types, without gravity.
FIG. 10. Long-range dispersion (evaluated at 95 7,), for DNS and steady (A = 0) KS, for all inertial particles, without gravity.
FIG. 10. Long-range dispersion (evaluated at 95 7,), for DNS and steady (A = 0) KS, for all inertial particles, without gravity.
FIG. 11. Number of particles in inner subdomain vs time (for steady Type II KS), for St ~ 10 inertial particles. the best results are obtained for steady (A = U) Ks (See Fig. 15 Tor Lype A Ks) Indeed, as noted in Chen, Goto, and Vassilicos,*” strong unsteadiness causes a lack of persis- tence of the flow field, a feature upon which depends the organization of the ensemble of inertial particles. Correlation dimension as a function of St, for DNS and KS, is shown in Fig. 14. The profiles show good qualitative agreement with observations in the literature, for both DNS?!*4 and KS,'? with the minimum D> occurring for moderate Stokes numbers (St ~ 1), within either DNS or KS—again low D> indicates greater clustering. A lack of quantitative agreement with previous DNS results may be explained by the inherent difficulty in estimating the Stokes number (as pointed out by Monchaux, Bourgin, and Cartellier**), and qualitative differences in the spectra (perhaps arising from the choice of forcing method). The lack of quantitative agreement with previous KS results may be explained by fundamental differences in the implementation of KS; Ijzermans, Meneguz, and Reeks!* employed an unsteady field (corresponding to neither Type A nor Type B KS as described here), and enforced periodicity of the KS modes. All results hereafter are
FIG. 11. Number of particles in inner subdomain vs time (for steady Type II KS), for St ~ 10 inertial particles. the best results are obtained for steady (A = U) Ks (See Fig. 15 Tor Lype A Ks) Indeed, as noted in Chen, Goto, and Vassilicos,*” strong unsteadiness causes a lack of persis- tence of the flow field, a feature upon which depends the organization of the ensemble of inertial particles. Correlation dimension as a function of St, for DNS and KS, is shown in Fig. 14. The profiles show good qualitative agreement with observations in the literature, for both DNS?!*4 and KS,'? with the minimum D> occurring for moderate Stokes numbers (St ~ 1), within either DNS or KS—again low D> indicates greater clustering. A lack of quantitative agreement with previous DNS results may be explained by the inherent difficulty in estimating the Stokes number (as pointed out by Monchaux, Bourgin, and Cartellier**), and qualitative differences in the spectra (perhaps arising from the choice of forcing method). The lack of quantitative agreement with previous KS results may be explained by fundamental differences in the implementation of KS; Ijzermans, Meneguz, and Reeks!* employed an unsteady field (corresponding to neither Type A nor Type B KS as described here), and enforced periodicity of the KS modes. All results hereafter are
FIG. 12. Dispersion of St ~ 10 inertial particles in steady (A =0) KS without gravity, for Type I KS (unbounded domain) and Type II KS (bounded domain). St ~ 10 particles require about 4000 time steps to “forget” their initial conditions, hence the reduced time interval in this graph.
FIG. 12. Dispersion of St ~ 10 inertial particles in steady (A =0) KS without gravity, for Type I KS (unbounded domain) and Type II KS (bounded domain). St ~ 10 particles require about 4000 time steps to “forget” their initial conditions, hence the reduced time interval in this graph.
FIG. 13. Time history of correlation dimension for Types A KS, with A =0, 0.5, 1, 5, for St ~ 1 inertial particles, with DNS shown for comparison. Type B results are similar.
FIG. 13. Time history of correlation dimension for Types A KS, with A =0, 0.5, 1, 5, for St ~ 1 inertial particles, with DNS shown for comparison. Type B results are similar.
FIG. 14. Correlation dimension (D2) as a function of St, without gravity, for DNS and steady (A =0) KS
FIG. 14. Correlation dimension (D2) as a function of St, without gravity, for DNS and steady (A =0) KS
FIG. 15. Preferential concentration of St ~ 1 inertial particles, from a 0.04L-thick slice of particle-laden DNS.
FIG. 15. Preferential concentration of St ~ 1 inertial particles, from a 0.04L-thick slice of particle-laden DNS.
FIG. 16. Lateral (( Ye, )) and longitudinal ({ y, ma dispersion, in DNS and steady (A =0) KS, for St ~ | inertial particles, with gravity. The results for longitudinal and lateral (mean-drift-corrected) dispersion of St = | inertial parti- cles, in both DNS and KS, are shown in Fig. 16. There exists a clear reduction in lateral dispersion compared with longitudinal for steady (A = 0) KS, indicating that KS indeed qualitatively repro- duces the continuity effect. The corresponding DNS case is shown for comparison, though the
FIG. 16. Lateral (( Ye, )) and longitudinal ({ y, ma dispersion, in DNS and steady (A =0) KS, for St ~ | inertial particles, with gravity. The results for longitudinal and lateral (mean-drift-corrected) dispersion of St = | inertial parti- cles, in both DNS and KS, are shown in Fig. 16. There exists a clear reduction in lateral dispersion compared with longitudinal for steady (A = 0) KS, indicating that KS indeed qualitatively repro- duces the continuity effect. The corresponding DNS case is shown for comparison, though the
FIG. 17. Correlation dimension as a function of St, with gravity, for DNS and steady (A =0) KS.
FIG. 17. Correlation dimension as a function of St, with gravity, for DNS and steady (A =0) KS.
FIG. 18. RIP for St ~ 1 inertial particles, for DNS and steady (A =0) KS in gravity, alongside RE 1(7e3/(udTn)). KS appears to yield a more favourable comparison of D> over the range of St (Fig. 17), a compared with the case without gravity. As was noted in the no-gravity case, increasing particle inertia appears to reduce the sensitivity of particle behaviour to the spurious Lagrangian propertie: of a KS field, and drift appears to enhance this corrective effect. In the case of high mean drift the correlation of drift-normal fluid velocities conditioned on the (drifting) inertial particle location (RIP ) becomes dominated by the spatial component (by Taylor’s frozen turbulence hypothesis) which is given as the one-time, two- point Eulerian velocity autocorrelation in the direction of th mean drift (mathematically, RE(re3 /(UaT)), Where ug is the enforced drift speed (ug = 0.1, whicl exceeds Unms = V(2/3)k) and e3 is the unit vector in the downward direction—see Fig. 18). KS enforces spatial statistics of the DNS, hence the domination of Lagrangian autocorrelations by the spatial properties consequently i speaking, the drift causes the inertial and DNS, hence they exhibit similar d mproves the prediction of dispersion (see Fig. 19). Loosely particles to “see” similar turbulence structure between the K‘ ynamics.
FIG. 18. RIP for St ~ 1 inertial particles, for DNS and steady (A =0) KS in gravity, alongside RE 1(7e3/(udTn)). KS appears to yield a more favourable comparison of D> over the range of St (Fig. 17), a compared with the case without gravity. As was noted in the no-gravity case, increasing particle inertia appears to reduce the sensitivity of particle behaviour to the spurious Lagrangian propertie: of a KS field, and drift appears to enhance this corrective effect. In the case of high mean drift the correlation of drift-normal fluid velocities conditioned on the (drifting) inertial particle location (RIP ) becomes dominated by the spatial component (by Taylor’s frozen turbulence hypothesis) which is given as the one-time, two- point Eulerian velocity autocorrelation in the direction of th mean drift (mathematically, RE(re3 /(UaT)), Where ug is the enforced drift speed (ug = 0.1, whicl exceeds Unms = V(2/3)k) and e3 is the unit vector in the downward direction—see Fig. 18). KS enforces spatial statistics of the DNS, hence the domination of Lagrangian autocorrelations by the spatial properties consequently i speaking, the drift causes the inertial and DNS, hence they exhibit similar d mproves the prediction of dispersion (see Fig. 19). Loosely particles to “see” similar turbulence structure between the K‘ ynamics.
FIG. 19. Lateral dispersion for St ~ | inertial particles, for DNS and steady KS, with and without gravity.
FIG. 19. Lateral dispersion for St ~ | inertial particles, for DNS and steady KS, with and without gravity.
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