# Direct Numerical Simulation of 3D Sparse Multiscale Grid Generated Turbulence and Kinematic Simulation of Inertial Particle Pair Diffusion

Direct Numerical Simulation of 3D Sparse Multiscale Grid Generated Turbulence and Kinematic Simulation of Inertial Particle Pair Diffusion. PhD thesis, King Fahd University of Petroleum and Minerals.

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Fundamental problems in turbulence were investigated using numerical methods. For the first time, a new type of turbulence generating grid, the Sparse 3D Multiscale Grid Turbulence Generator [Malik, N. A. Patent US 9,599,269 B2, (2017); Malik N. A. Patent EP 2,965,805 B1, (2017)], or 3DS for short, is investigated using Direct Numerical Simulations (DNS). In the 3DS each scale in a (typically) fractal grid is held in its own co-frame and is separated from the next co-frame by some distance in an overall co-planar arrangement. This expands the parameter space in the 3DS as compared to 2D flat fractal grids (2DF) and classical regular grids (RG). The objective is that the 3DS system could provide better control of turbulence than previous grids. Turbulence characteristics downstream of all three grids placed in a conduit with periodic walls and inflow and outflow in the streamwise direction are compared for Kolmogorov scale Reynolds number $Re_{\eta} = 300$. The blockage ratio (solidity) of the RG and 2DF grids was $\sigma =32\%$, while in the 3DS it was varied: $\sigma_{3DS}^{Max} = 15\%$, 24\%, and 32\%. Variations in the ordering of the co-frames in the 3DS for the case: $\sigma_{3DS}^{Max} = 32\%$ was also investigated. However, the separation distances between the three co-frames were kept constant throughout this work. It was found that: (1) the 3DS generates much higher peak turbulence intensities, $I = u'/U_{mean}$, than the RG and 2DF for the same blockage ratio. (2) 3DS with $\sigma_{3DS}^{Max} = 24\%$ is closest to the performance of the 2DF. (3) The downstream turbulence can be divided in to three regions: (a) near-field region close to the grid where the turbulence is generated, (b) the intermediate region where the turbulence decays rapidly, (c) the far-field region downstream where the turbulence intensity is sustained and even increases along the channel centerline. (4) There exists a central core region around the centerline which is approximately homogeneous and isotropic in the far-field. (5) The main mechanism that sustains the turbulence intensity in the far-field is re-entrainment of turbulent fluid from outside of the central core region in to it; if the rate of entrainment is high enough then the turbulence intensity may increase downstream. (6) The order of arrangement of co-frames in the 3DS has comparatively less effect on the turbulence intensities than changes in blockage ratio, the case with decreasing scales (the natural case) produces the highest intensities. (7) Fully developed Kolmogorov spectrum was observed in the 3DS systems with blockage ratio 32\%, but the best spectrum was in the natural case. (8) The distribution of vorticity intensity $I_{\omega} (x)=\left\| \omega' \right\| L/ U_{mean}$ shows much less variation suggesting that vorticity generation plays a relatively weak role in 3DS systems. (9) Passive scalar transport was investigated for different Schmidt numbers $0.25 \le Sc = \nu / D \le 2$; the overall trends in scalar fluctuations decay were similar in all 3DS systems. In a second part of this study, a new theory for turbulent inertial particle pair diffusion in the inertial subrange was postulated and investigated numerically using Kinematic Simulations (KS) [N. A. Malik, PLoS ONE, 10(10): e0189917 (2017)]. The new theory is an extension of the non-local fluid particle pair diffusion theory; here we postulate that for all Stokes numbers, in the limit of heavy point particles released close together inside the inertial subrange, with one-way coupling, pair separation will be ballistic because the particle kinetic energy dominates over the small scale turbulence energy, and the pair diffusion coefficient will scale like $K_p(l) \sim l$ where $l$ is the pair separation variable. As the particle pair separation increases, and if the subrange is large enough, the pair diffusion transits slowly towards the fluid pair diffusion, $K_p (l) \to K_f (l) \sim l^{\gamma}$, which has been shown to follow non-Richardson scaling laws because of non-local effects [Malik, N. A. PLoS ONE (2018)] with $\gamma > 4/3$ for Kolmogorov turbulence. By considering the balance of momentum and energies, the transition length scale $l_c$ at which the ballistic regime starts to transit towards the non-ballistic regimes scales like, $l_c / \eta \sim St^{0.5}$ if $St<1$, and scales like $l_c / \eta \sim St^1$ if $St>1$, where $\eta$ is the Kolmogorov scale; a crossover between these two scalings occurs at $St = 1$. All of the predictions of the new theory have been confirmed numerically using KS.