How long do particles spend in vortical regions in turbulent flows?

Akshay Bhatnagar, Anupam Gupta, Dhrubaditya Mitra, Rahul Pandit, and Prasad Perlekar

Phys. Rev. E 94, 053119 – Published 23 November 2016

Abstract

We obtain the probability distribution functions (PDFs) of the time that a Lagrangian tracer or a heavy inertial particle spends in vortical or strain-dominated regions of a turbulent flow, by carrying out direct numerical simulations of such particles advected by statistically steady, homogeneous, and isotropic turbulence in the forced, three-dimensional, incompressible Navier-Stokes equation. We use the two invariants, $Q$ and $R$ , of the velocity-gradient tensor to distinguish between vortical and strain-dominated regions of the flow and partition the $Q - R$ plane into four different regions depending on the topology of the flow; out of these four regions two correspond to vorticity-dominated regions of the flow and two correspond to strain-dominated ones. We obtain $Q$ and $R$ along the trajectories of tracers and heavy inertial particles and find out the time $t_{pers}$ for which they remain in one of the four regions of the $Q - R$ plane. We find that the PDFs of $t_{pers}$ display exponentially decaying tails for all four regions for tracers and heavy inertial particles. From these PDFs we extract characteristic time scales, which help us to quantify the time that such particles spend in vortical or strain-dominated regions of the flow.

Received 8 September 2016

DOI:https://doi.org/10.1103/PhysRevE.94.053119

Physics Subject Headings (PhySH)

Particle-laden flows Turbulence

Fluid Dynamics

Authors & Affiliations

Akshay Bhatnagar^1,2,*, Anupam Gupta^3,†, Dhrubaditya Mitra^2,‡, Rahul Pandit^1,§, and Prasad Perlekar^4,∥

¹Centre for Condensed Matter Theory, Department of Physics, Indian Institute of Science, Bangalore 560012, India
²Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, 10691 Stockholm, Sweden
³Laboratoire de Génie Chimique, Universite de Toulouse, INPT-UPS, 31030 Toulouse, France
⁴TIFR Centre for Interdisciplinary Sciences, 21 Brundavan Colony, Narsingi, Hyderabad 500075, India

^*akshayphy@gmail.com
^†anupam1509@gmail.com
^‡dhruba.mitra@gmail.com
^§rahul@physics.iisc.ernet.in
^∥perlekar@tifrh.res.in

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 94, Iss. 5 — November 2016

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
The flow is topologically different for values of $Q$ and $R$ that lie in the four regions shown in the $Q - R$ plane (after Ref. [20]); the black curve is the zero-discriminant line $Δ = 0$ . Regions A and B are vorticity-dominated regions; in region A vortices are stretched and in region B they are compressed. By contrast, regions C and D correspond to strain-dominated or extensional regions; in region C fluid elements experience biaxial strain, whereas, in region D, they feel axial strain. The red dashed curve shows an illustrative path, in the $Q - R$ plane, as a tracer moves through the fluid in our DNS.
Reuse & Permissions
Figure 2
Plots of $R$ (red dashed line) and the discriminant $Δ$ of the characteristic equation for the velocity-gradient tensor (solid blue line), calculated along the trajectory of a tracer as a function of the dimensionless time $t / T_{eddy}$ . The intersection of any one of these curves with the black horizontal line indicates the migration of a particle from one region of the $Q - R$ plane to another. $R$ and $Δ$ are nondimensionalized by $Λ^{3}$ and $Λ^{6}$ , respectively, where $Λ = u_{η} / η$ .
Reuse & Permissions
Figure 3
Contour plots of the joint PDFs of $Q$ and $R$ , on log scales, calculated along the trajectories of particles with different Stokes numbers, from the top left corner, (a) tracers, (b) $St = 0.1$ , (c) $St = 0.5$ , (d) $St = 1.0$ , (e) $St = 1.4$ , and (f) $St = 2.0$ . $Q$ and $R$ are nondimensionalized by $Λ^{2}$ and $Λ^{3}$ , respectively, where $Λ = u_{η} / η$ . The $Δ = 0$ curve is shown by the solid black line; $Δ > 0$ corresponds to the vorticity-dominated region and $Δ < 0$ corresponds to the strain-dominated one.
Reuse & Permissions
Figure 4
Semilog plots of the persistence-time PDFs $P_{ϕ} (t_{pers})$ of the times $t_{pers}$ for the four regimes in the $Q - R$ plane, for different Stokes numbers; the inset shows $P_{ϕ} (t_{pers})$ for small $t_{pers}$ .
Reuse & Permissions
Figure 5
Semilog plots of the cumulative persistence time PDFs (obtained by the rank-order method) for the four regimes in the $Q - R$ plane, for different values of the Stokes number.
Reuse & Permissions
Figure 6
Semilog plots of the cumulative persistence-time PDFs (obtained by the rank-order method) for vortical ( $Δ > 0$ , left panel) and strain-dominated ( $Δ < 0$ , right panel) regions, for different values of the Stokes number $St$ (the plot for tracers is labeled by $St = 0$ ). From the slopes of the tails of these PDFs we extract the times $T_{vortical}$ and $T_{strain}$ , for particle residence in vortical or strain-dominated regions of the flow, respectively.
Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics