• ATUL SHARMA

• Level set method for computational multi-fluid dynamics: A review on developments, applications and analysis

Functions and conservation as well as subsidiary equations in Level Set Method (LSM) are presented. After the mathematical formulation, improvements in the numerical methodology for LSM are reviewed here for advection schemes, reinitialization methods, hybrid methods, adaptive-grid LSM, dual-resolution LSM, sharp-interface LSM, conservative LSM, parallel computing and extension from two to multi fluid/phase as well as to various types of two-phase flow. In the second part of this article, LSM method based Computational Multi-Fluid Dynamics (CMFD) applications and analysis are reviewed for four different types of multi-phase flow: separated and parallel internal flow, drop/bubble dynamics during jet break-up, drop impact dynamics on a solid or liquid surface and boiling. In the last twenty years, LSM has established itself as a method which is easy to program and is accurate as well as computationally-efficient.

• Break-up of a non-Newtonian jet injected downwards in a Newtonian liquid

The present work on downward injection of non-Newtonian jet is an extension of our recent work (Lakdawala et al, Int. J. Multiphase Flow. 59: 206–220, 2014) on upward injection of Newtonian jet. The non-Newtonian rheology of the jet is described by a Carreau type generalized Newtonian fluid (GNF) model, which is a phenomenological constitutive equation that accounts for both rate-thinning and rate-thickening. Level set method based numerical study is done for Newtonian as well as various types of shear thinning and thickening jet fluid. Effect of average injection velocity ($V_{av,i}$) is studied at a constant Reynolds number Re = 14.15, Weber number W e = 1, Froude number F r = 0.25, density ratio $\chi$ = 0.001 and viscosity ratio $\eta$ = 0.01. CFD analysis of the temporal variation of interface and jet length ($L_{j}$) is done to propose different types of jet breakup regimes. At smaller, intermediate and larger values of $V_{av,i}$, the regimes found are periodic uniform drop (P-UD), quasi-periodic non-uniform drop (QP-NUD) and no breakup (NB) regimes for a shear thinning jet; and periodic along with Satellite Drop (P+S), jetting (J) and no breakup (NB) regimes for a shear thickening jet, respectively. This is presented as a drop-formation regime map. Shear thickening (thinning) is shown to produce long (short) jet length. Diameter of the primary drop increases and its frequency of release decreases, due to increase in stability of the jet for shear thickening as compared to thinning fluid.

• Thrust generation and wake structure for flow across a pitching airfoil at low Reynolds number

In this work, we present detailed particle image velocimetry (PIV) based investigation of wake structure of a pitching airfoil. PIV measurements have been carried out for NACA0015 airfoil at Re = 2900 with reduced frequency range of 1.82–10.92 and pitching angle of 5°. Two different wake structures (reverse Kármán shedding and deflected vortex shedding) are observed over this parameter range. The vorticity decreases substantially over a distance of two chord-lengths. The velocity profile indicates a jet-like flow downstream of the airfoil. It is shown that the jet-like flow downstream of the airfoil is however not a sufficient condition for the generation of thrust. The vortex strength is found to be invariant of the pitching frequency. Certain differences from the reported results are noted, which may be because of difference in the airfoil shape. These results can help improve understanding of the flow behavior as the low Reynolds number range is not well studied.

• Effect of wavelength of fish-like undulation of a hydrofoil in a free-stream flow

Fish-like undulating body was proposed as an efficient propulsion system, and various mechanisms of thrust generation in this type of propulsion are found in the literature—separately for undulating and pitching fishes/foil. The present work proposes a unified study for undulating and pitching foil, by varying wavelength l (from 0.8 to 8.0) of a wave traveling backwards over the NACA0012 hydrofoil in a free-stream flow; the larger wavelength is shown to lead to the transition from the undulating motion to pitching motion. The effect ofwavelength of undulation is studied numerically at a Reynolds number Re=4000, maximum amplitude of undulation Amax 0:1 and non-dimensional frequency of undulation St=0:4, using level-set immersedboundary-method based in-house 2D code. The Navier–Stokes equation governing the fluid flow is solved using a fully implicit finite-volume method, while level-set equation governing the movement of the hydrofoil is solved using an explicit finite-difference method. It is presented here that the thrust generation mechanism for the low wavelength case undulating (l=0.8) foil is different from the mechanism for the high wavelength pitching foil. With increasing wavelength, mean thrust coefficient of the undulating foil increases and asymptotes to value for the pure pitching foil. Furthermore, the ratio of maximum thrust coefficient to maximum lateral force coefficient is found to be larger for the smaller wavelength undulating foil as compared with the larger wavelength pitching foil.

• Cut-cell-based Direct Simulation Monte Carlo method on a Cartesian grid for rarefied gas flow around complex geometries

The present work proposes a cut-cell-based Direct Simulation Monte Carlo (DSMC) solver, for computing rarefied flows around complex geometries on Cartesian grids, wherein analytical expression for the surface of the immersed boundary (IB) is considered to evaluate cut-cell volume as well as to implement the particle–boundary interactions. Consequently the proposed DSMC solver models an accurate collision rate in the cut cells and ensures an analytically expressed IB-based implementation of the boundary conditions at thesurface of the immersed geometry, as in the IB methods for the continuum flows. Performance of the present Cartesian cut-cell-based DSMC solver is tested on a variety of rarefied gas flows around three complex geometries (cylinder, NACA 0012 airfoil and double-wedge airfoil) for various flow speeds (ranging from Ma = 2 to 10) and degrees of rarefication (varying from Kn = 0:25 to around 0.0032). Results of our computations on Cartesian grids show a very good agreement with the corresponding DSMC results in literature computed on body-fitted grids. Furthermore, the present results show a good agreement with the corresponding experimental data in the literature. Straightforward and analytically expressed IB-based implementation in the proposed DSMC solver can make it a natural choice for its coupling with an immersed boundary method (IBM)- based continuum solver for a novel coupled IBM–DSMC method for continuum–rarefied gas flows.

• Adaptive interface-Mesh un-Refinement (AiMuR) based sharp-interface level-set-method for two-phase flow

Adaptive interface-Mesh un-Refinement (AiMuR) based Sharp-Interface Level-Set-Method (SILSM) is proposed for both uniform and non-uniform Cartesian-Grid. The AiMuR involves interface location based dynamic un-refinement (with merging of the four control volumes) of the Cartesian grid away from theinterface. The un-refinement is proposed for the interface solver only. A detailed numerical methodology is presented for the AiMuR and ghost-fluid method based SI-LSM. Advantage of the novel as compared to the traditional SI-LSM is demonstrated with a detailed qualitative as well as quantitative performance study, involving the SI-LSMs on both coarse grid and fine grid, for three sufficiently different two-phase flow problems:dam break, breakup of a liquid jet and drop coalescence. A superior performance of AiMuR based SI-LSM is demonstrated - the AiMuR on a coarser non-uniform grid (NUcAiMuR) is almost as accurate as the traditional SILSM on a uniform fine grid (Uƒ) and takes a computational time almost same as that by the traditional SI-LSM on a uniform coarse grid (Uc). The AMuR is different from the existing Adaptive Mesh Refinement (AMR) asthe former involves only mesh un-refinement while the later involves both refinement and un-refinement of the mesh. Moreover, the proposed computational development is significant since the present adaptive un-refinement strategy is much simpler to implement as compared to that for the commonly used adaptive refinement strategies. The proposed numerical development can be extended to various other multi-physics, multi-disciplinaryand multi-scale problems involving interfaces.