#### Date of Award

2016

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mechanical and Aerospace Engineering

#### Committee Chair

Sarma Rani

#### Committee Member

Kader Frendi

#### Committee Member

Shankar Mahalingam

#### Committee Member

Babak Shotorban

#### Committee Member

Sivaguru Ravindran

#### Subject(s)

Stochastic processes., Turbulence., Fluid mechanics.

#### Abstract

This dissertation presents an analytical and computational investigation of the relative motion of monodisperse, non-settling, inertial particle pairs in stationary, homogeneous isotropic turbulence. The research objectives of this dissertation were to: (1) develop a stochastic theory describing the relative velocities and separations of highly inertial particle pairs in isotropic turbulence; (2) analyze and validate the developed theory through comparisons with prior theories, and with data from direct numerical simulations; and (3) investigate the effects of stochastic and deterministic forcing schemes on the relative motion of particle pairs in direct numerical simulations (DNS) of isotropic turbulence. The PDF kinetic equation describing the relative motion of inertial particle pairs in a turbulent flow requires the closure of the phase-space diffusion current. A novel analytical closure for the diffusion current is presented that is applicable to high-inertia particle pairs with Stokes numbers $St_r \gg 1$, where $St_r$ is a Stokes number based on the time-scale $\tau_r$ of eddies whose size scales with pair separation $r$. In the asymptotic limit of $St_r \gg 1$, the pair PDF kinetic equation reduces to an equation of the Fokker-Planck form. Closure of the diffusivity tensor in the Fokker-Planck equation is achieved by converting the Lagrangian correlations of fluid relative velocities ``seen" by pairs into Eulerian fluid velocity correlations at pair separations that remain essentially constant during timescales of $O(\tau_r)$. A detailed quantitative analysis of the stochastic theory was performed by solving the Langevin equations that are statistically equivalent to the closed Fokker-Planck equation. The pair relative-motion statistics obtained from the Langevin simulations (LS) for $Re_\lambda = 76,131$ and $St_\eta = 10,20,40,80$ are compared with the results obtained in prior theoretical analyses, as well as with the data from DNS. Excellent comparison between LS and DNS results was found for the radial distribution functions, while a reasonable agreement was seen for the relative velocity statistics. Finally, the effects of deterministic and stochastic forcing schemes on the relative motion of heavy inertial particles in DNS of isotropic turbulence were studied. The effects of forcing time scale $T_f$, a key parameter in stochastic forcing, on the relative motion statistics of particle pairs were assessed by considering five $T_f$'s, ranging from $4T_E$ to $T_E/4$, where $T_E$ is the Eulerian integral time scale obtained from the DNS with deterministic forcing. Six DNS runs (one deterministic and five stochatic) are performed for each of the three grid resolutions $128^3$, $256^3$, $512^3$. Data from the runs with deterministic forcing and stochastic forcing (with five time scales) were compared, and their effects on particle statistics quantified.

#### Recommended Citation

Dhariwal, Rohit, "Stochastic theory and direct numerical simulations of the relative motion of high-stokes number particles in isotropic turbulence" (2016). *Dissertations*. 111.

https://louis.uah.edu/uah-dissertations/111