scholarly journals Clustering of rapidly settling, low-inertia particle pairs in isotropic turbulence. Part 2. Comparison of theory and DNS

2019 ◽  
Vol 871 ◽  
pp. 477-488 ◽  
Author(s):  
Sarma L. Rani ◽  
Rohit Dhariwal ◽  
Donald L. Koch
Keyword(s):  
Reasonable Agreement ◽  
Isotropic Turbulence ◽  
Diffusion Flux ◽  
Stochastic Theory ◽  
Polar Angle ◽  
Harmonic Coefficient ◽  
Gravitational Acceleration ◽  
Particle Clustering ◽  
Particle Pairs ◽  
And Diffusion

Part 1 (Rani et al. J. Fluid Mech., vol. 871, 2019, pp. 450–476) of this study presented a stochastic theory for the clustering of monodisperse, rapidly settling, low-Stokes-number particle pairs in homogeneous isotropic turbulence. The theory involved the development of closure approximations for the drift and diffusion fluxes in the probability density function (p.d.f.) equation for the pair relative positions $\boldsymbol{r}$. In this part 2 paper, the theory is quantitatively analysed by comparing its predictions of particle clustering with data from direct numerical simulations (DNS) of isotropic turbulence containing particles settling under gravity. The simulations were performed at a Taylor micro-scale Reynolds number $Re_{\unicode[STIX]{x1D706}}=77.76$ for three Froude numbers $Fr=\infty ,0.052,0.006$, where $Fr$ is the ratio of the Kolmogorov scale of acceleration and the magnitude of gravitational acceleration. Thus, $Fr=\infty$ corresponds to zero gravity, and $Fr=0.006$ to the highest magnitude of gravity among the three DNS cases. For each $Fr$, particles of Stokes numbers in the range $0.01\leqslant St_{\unicode[STIX]{x1D702}}\leqslant 0.2$ were tracked in the DNS, and particle clustering quantified both as a function of separation and the spherical polar angle. We compared the DNS and theory values for the exponent $\unicode[STIX]{x1D6FD}$ characterizing the power-law dependence of clustering on separation. The $\unicode[STIX]{x1D6FD}$ from the $Fr=0.006$ DNS case are in reasonable agreement with the theoretical predictions obtained using the second drift closure (referred to as DF2). To quantify the anisotropy in clustering, we calculated the leading–order coefficient in the spherical harmonics expansion of the p.d.f. of pair relative positions. The coefficients predicted by the theory (DF2) again show reasonable agreement with those calculated from the DNS clustering data for $Fr=0.006$. However, we note that in spite of the high magnitude of gravity, the clustering is only marginally anisotropic both in DNS and theory. The theory predicts that the spherical harmonic coefficient scales with $\unicode[STIX]{x1D6FD}(=\unicode[STIX]{x1D6FD}_{2}St_{\unicode[STIX]{x1D702}}^{2})$, where $\unicode[STIX]{x1D6FD}_{2}$ is the ratio of the drift and diffusion flux coefficients. Since the drift flux, and thereby $\unicode[STIX]{x1D6FD}_{2}$, is seen to decrease with gravity for $St_{\unicode[STIX]{x1D702}}<1$, the anisotropy is also correspondingly diminished.

scholarly journals Clustering of rapidly settling, low-inertia particle pairs in isotropic turbulence. Part 1. Drift and diffusion flux closures

2019 ◽  
Vol 871 ◽  
pp. 450-476 ◽  
Author(s):  
Sarma L. Rani ◽  
Vijay K. Gupta ◽  
Donald L. Koch
Keyword(s):  
Velocity Gradient ◽  
Dissipation Rate ◽  
Isotropic Turbulence ◽  
Diffusion Flux ◽  
Fluid Velocity ◽  
Polar Angle ◽  
Turbulent Dissipation ◽  
Drift Flux ◽  
Particle Pairs ◽  
And Diffusion

In this two-part study, we present the development and analysis of a stochastic theory for characterizing the relative positions of monodisperse, low-inertia particle pairs that are settling rapidly in homogeneous isotropic turbulence. In the limits of small Stokes number and Froude number such that $Fr\ll St_{\unicode[STIX]{x1D702}}\ll 1$, closures are developed for the drift and diffusion fluxes in the probability density function (p.d.f.) equation for the pair relative positions. The theory focuses on the relative motion of particle pairs in the dissipation regime of turbulence, i.e. for pair separations smaller than the Kolmogorov length scale. In this regime, the theory approximates the fluid velocity field in a reference frame following the primary particle as locally linear. In this part 1 paper, we present the derivation of closure approximations for the drift and diffusion fluxes in the p.d.f. equation for pair relative positions $\boldsymbol{r}$. The drift flux contains the time integral of the third and fourth moments of the ‘seen’ fluid velocity gradients along the trajectories of primary particles. These moments may be analytically resolved by making approximations regarding the ‘seen’ velocity gradient. Accordingly, two closure forms are derived specifically for the drift flux. The first invokes the assumption that the fluid velocity gradient along particle trajectories has a Gaussian distribution. In the second drift closure, we account for the correlation time scales of dissipation rate and enstrophy by decomposing the velocity gradient into the strain-rate and rotation-rate tensors scaled by the turbulent dissipation rate and enstrophy, respectively. An analytical solution to the p.d.f. $\langle P\rangle (r,\unicode[STIX]{x1D703})$ is then derived, where $\unicode[STIX]{x1D703}$ is the spherical polar angle. It is seen that the p.d.f. has a power-law dependence on separation $r$ of the form $\langle P\rangle (r,\unicode[STIX]{x1D703})\sim r^{\unicode[STIX]{x1D6FD}}$ with $\unicode[STIX]{x1D6FD}\sim St_{\unicode[STIX]{x1D702}}^{2}$ and $\unicode[STIX]{x1D6FD}<0$, analogous to that for the radial distribution function of non-settling pairs. An explicit expression is derived for $\unicode[STIX]{x1D6FD}$ in terms of the drift and diffusion closures. The $\langle P\rangle (r,\unicode[STIX]{x1D703})$ solution also shows that, for a given $r$, the clustering of $St_{\unicode[STIX]{x1D702}}\ll 1$ particles is only weakly anisotropic, which is in conformity with prior observations from direct numerical simulations of isotropic turbulence containing settling particles.

Stochastic theory and direct numerical simulations of the relative motion of high-inertia particle pairs in isotropic turbulence

2017 ◽  
Vol 813 ◽  
pp. 205-249 ◽  
Author(s):  
Rohit Dhariwal ◽  
Sarma L. Rani ◽  
Donald L. Koch
Keyword(s):  
Numerical Simulations ◽  
Relative Motion ◽  
Relative Velocity ◽  
Isotropic Turbulence ◽  
Stochastic Theory ◽  
Stokes Number ◽  
Time Correlation ◽  
Direct Numerical Simulations ◽  
Time Integral ◽  
Particle Pairs

The relative velocities and positions of monodisperse high-inertia particle pairs in isotropic turbulence are studied using direct numerical simulations (DNS), as well as Langevin simulations (LS) based on a probability density function (PDF) kinetic model for pair relative motion. In a prior study (Rani et al., J. Fluid Mech., vol. 756, 2014, pp. 870–902), the authors developed a stochastic theory that involved deriving closures in the limit of high Stokes number for the diffusivity tensor in the PDF equation for monodisperse particle pairs. The diffusivity contained the time integral of the Eulerian two-time correlation of fluid relative velocities seen by pairs that are nearly stationary. The two-time correlation was analytically resolved through the approximation that the temporal change in the fluid relative velocities seen by a pair occurs principally due to the advection of smaller eddies past the pair by large-scale eddies. Accordingly, two diffusivity expressions were obtained based on whether the pair centre of mass remained fixed during flow time scales, or moved in response to integral-scale eddies. In the current study, a quantitative analysis of the (Rani et al. 2014) stochastic theory is performed through a comparison of the pair statistics obtained using LS with those from DNS. LS consist of evolving the Langevin equations for pair separation and relative velocity, which is statistically equivalent to solving the classical Fokker–Planck form of the pair PDF equation. Langevin simulations of particle-pair dispersion were performed using three closure forms of the diffusivity – i.e. the one containing the time integral of the Eulerian two-time correlation of the seen fluid relative velocities and the two analytical diffusivity expressions. In the first closure form, the two-time correlation was computed using DNS of forced isotropic turbulence laden with stationary particles. The two analytical closure forms have the advantage that they can be evaluated using a model for the turbulence energy spectrum that closely matched the DNS spectrum. The three diffusivities are analysed to quantify the effects of the approximations made in deriving them. Pair relative-motion statistics obtained from the three sets of Langevin simulations are compared with the results from the DNS of (moving) particle-laden forced isotropic turbulence for $St_{\unicode[STIX]{x1D702}}=10,20,40,80$ and $Re_{\unicode[STIX]{x1D706}}=76,131$. Here, $St_{\unicode[STIX]{x1D702}}$ is the particle Stokes number based on the Kolmogorov time scale and $Re_{\unicode[STIX]{x1D706}}$ is the Taylor micro-scale Reynolds number. Statistics such as the radial distribution function (RDF), the variance and kurtosis of particle-pair relative velocities and the particle collision kernel were computed using both Langevin and DNS runs, and compared. The RDFs from the stochastic runs were in good agreement with those from the DNS. Also computed were the PDFs $\unicode[STIX]{x1D6FA}(U|r)$ and $\unicode[STIX]{x1D6FA}(U_{r}|r)$ of relative velocity $U$ and of the radial component of relative velocity $U_{r}$ respectively, both PDFs conditioned on separation $r$. The first closure form, involving the Eulerian two-time correlation of fluid relative velocities, showed the best agreement with the DNS results for the PDFs.

scholarly journals Spatial–temporal variation of nitrogen and diffusion flux across the water–sediment interface at the hydro-fluctuation belt of Danjiangkou reservoir in China

2020 ◽  
Vol 20 (4) ◽  
pp. 1241-1252
Author(s):  
Han Wang ◽  
Yuping Han ◽  
Lide Pan
Keyword(s):  
Diffusion Flux ◽  
Sample Collection ◽  
Sediment Pore Water ◽  
Overlying Water ◽  
Danjiangkou Reservoir ◽  
Water And Sediment ◽  
Water Quality Deterioration ◽  
Nitrogen Contents ◽  
And Diffusion ◽  
Sediment Interface

Abstract Based on overlying water and sediment sample collection from 15 sites during July, September, November 2018 and January 2019 in the hydro-fluctuation belt of Danjiangkou reservoir China, the variation of nitrogen (N) was studied. And the concentrations of NH4+-N, NO3−-N and NO2−-N in the sediment, pore water and overlying water were determined to evaluate the diffusion flux across the water–sediment interface. The results showed that the lowest sediment N concentration was 36.54 mg/L in July, and the highest one was 145.93 mg/L in November. Spatially, the sediment N concentrations were higher in tidal soil and loam than in sandy soil. According to the diffusion fluxes of NH4+, NO3− and NO2−, sediments at all sites tend to release N to the overlying water except in the sampling month of November, when the sediment acts as a sink of NO3−. The highest release rates of NH4+-N and NO3−-N were 17.66 mg m−2·d−1 and 80.15 mg m−2·d−1, respectively, which are much higher than the release rate of NO2−-N (0.29 mg m−2·d−1). The findings indicate that hydro-fluctuation belt sediment contributes a lot to the nitrogen contents in the overlying water, and internal pollution is a main reason for the water quality deterioration and even eutrophication.

Heat and Diffusion Fluxes in a Multicomponent Ionized Gas in a Magnetic Field

1969 ◽  
Vol 24 (6) ◽  
pp. 967-976
Author(s):  
R. S. Devoto
Keyword(s):  
Magnetic Field ◽  
Heat Flux ◽  
Thermal Diffusion ◽  
Diffusion Flux ◽  
Momentum Equation ◽  
Ionized Gas ◽  
Burnett Method ◽  
And Diffusion ◽  
Monatomic Gas

Ordinary and thermal diffusion as well as heat flux in a dilute, ionized, multicomponent monoatomic gas in a magnetic field are considered with the Chapman-Enskog-Burnett method. It is shown how, with certain modifications, the usual expressions for the properties of an un-ionized monatomic gas may be applied to this case. The expression for the diffusion flux is compared with the momentum equation suggested by Schliiter.

scholarly journals Lagrangian statistics of particle pairs in homogeneous isotropic turbulence

2005 ◽  
Vol 17 (11) ◽  
pp. 115101 ◽  
Author(s):  
L. Biferale ◽  
G. Boffetta ◽  
A. Celani ◽  
B. J. Devenish ◽  
A. Lanotte ◽  
...  
Keyword(s):  
Isotropic Turbulence ◽  
Homogeneous Isotropic Turbulence ◽  
Lagrangian Statistics ◽  
Particle Pairs

Concentration fluctuations in water–methanol mixtures as studied by ultrasonic absorption

1976 ◽  
Vol 54 (3) ◽  
pp. 355-366 ◽  
Author(s):  
William D. T. Dale ◽  
Peter A. Flavelle ◽  
Peeter Kruus
Keyword(s):  
Experimental Data ◽  
Intermolecular Interactions ◽  
Vapour Pressure ◽  
Reasonable Agreement ◽  
Concentration Fluctuations ◽  
Ultrasonic Absorption ◽  
Diffusion Data ◽  
And Diffusion ◽  
Excess Absorption

Measurements of the ultrasonic absorption and velocity, and viscosity have been carried out in the system methanol–water at 0, 15, and 25 °C at frequencies from 10 to 50 MHz. The data show the existence of two maxima in the excess absorption as a function of composition. The existence of the two maxima are predicted by the Romanov–Solovyev theory relating the absorption to concentration fluctuations. The magnitude of the excess absorption is calculated at 25 and 0 °C from density, vapour pressure, and diffusion data. New experimental data on density and vapour pressure are presented for this system at 0 °C. The calculated excess absorption is in reasonable agreement with the measured. This indicates that any specific localized intermolecular interactions present in this system could be responsible for at best a smaller portion of the excess ultrasonic absorption, as longer range concentration fluctuations account for essentially all of the excess. Ultrasonic data on the systems methanol–octanol and n-propanol–water are also presented in the course of examining the generality of these phenomena.

Experimental investigation and thermodynamic calculation in the Ag–Bi–Ni and Cu–Bi–Ni systems

2009 ◽  
Vol 24 (8) ◽  
pp. 2644-2653 ◽  
Author(s):  
Feng Gao ◽  
Cuiping Wang ◽  
X.J. Liu ◽  
Yoshikazu Takaku ◽  
I. Ohnuma ◽  
...  
Keyword(s):  
Experimental Data ◽  
Phase Equilibria ◽  
Intermetallic Compounds ◽  
Electron Probe ◽  
Reasonable Agreement ◽  
Diffusion Couples ◽  
Scanning Calorimetry ◽  
Temperatures Of Phase Transformations ◽  
Ternary Intermetallic Compound ◽  
And Diffusion

The phase equilibria at 300, 400, 500, and 600 °C in the Ag–Bi–Ni system and 300, 400, and 500 °C in the Cu–Bi–Ni system were experimentally determined by metallography and electron probe microanalysis on equilibrated alloys and diffusion couples. Differential scanning calorimetry was used to measure the temperatures of phase transformations. All the experimental results show that the solubilities of the ternary elements of the binary intermetallic compounds in the Ag–Bi–Ni system are limited. However, the binary intermetallic compounds have some solubilities of the ternary elements in the Cu–Bi–Ni system. No ternary intermetallic compound was found in the Ag–Bi–Ni and Cu–Bi–Ni systems. On the basis of the determined results, the phase equilibria in the Ag–Bi–Ni and Cu–Bi–Ni systems were thermodynamically assessed, and reasonable agreement between the calculated results and experimental data was obtained.

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