## Computational Fluid Dynamics

Youngs PLIC-VOF technique is an accurate tool for interface representation and intrinsic volume conservation for multiphase simulations. Direction-splitting of the algorithm ensures stable performance upto Courant numbers of unity. However, baseline PLIC algorithms face an open problem of conservative volume advection in shearing velocity fields. Rather than modifying the advection equation of the baseline (Youngs PLIC-VOF) algorithm to achieve volume conservation for a limited range of solenoidal velocity fields, a coupling of the advection algorithm with a redistribution-based volume correction routine is proposed here. The advantage of such a framework is that it decouples the prerequisite for mass conservation from the direction-splitting of the advection equation. This happens because the advection is now made material-preserving upto machine accuracy for any velocity field, solenoidal or not. The volume conservation property of the redistribution algorithm proposed here is established through stringent shear tests at a Courant number of unity, dam break simulations and Rayleigh-Taylor instability. It is observed that the only mechanism through which such a formulation would violate volume conservation is by the addition of overall material source/sink terms to the advection equation.

A high-resolution, Navier–Stokes solver is developed for direct numerical simulation (DNS) of free shear flow. All terms in Navier–Stokes equations are discretized using higher order methods. Diffusion term is discretized using fourth order central difference scheme while second order Adams–Bashforth is used for time derivative. Advecting velocity is approximated using fourth order Lagrangian interpolation. For the approximation of advected velocity, a blended fifth order-upwind scheme is proposed. Developed high resolution solver is used for DNS of round jet in transitional and turbulent regimes. A novel open outlet boundary condition (OOBC) is proposed which has the ability to dynamically adjust according to prevailing local condition at the outlet thereby minimizing reflections from outlet. Ability of blended fifth order upwind scheme and fifth order WENO is assessed in terms of algorithmic efficiency as well as fidelity of simulations. It is demonstrated that the proposed blended fifth order upwind scheme outperforms the WENO scheme in terms of algorithmic efficiency. Assessment of fidelity of simulations reveals that WENO displays a tendency to over-predict momentum advection in transitional as well as fully turbulent regime of the round jet. In contrast, the proposed advection scheme is not faced with such limitation.

Generation of steep waves $(H/\lambda > 0.03)$ in a NSE-based NWT is a challenging task and is seldom attempted in near-shallow water $(kh<1.0)$. Failure towards attainment of the target steepness is characterized by an under-prediction of the wave height $(H)$ along the length of the NWT. The issue of height damping has received limited attention in literature. In this context, a NSE based NWT has been developed using a PLIC-VOF formulation and a mass source-based wave generator. Wave damping at the NWT boundaries has been achieved using sponge layers. A total of nine wave designs have been simulated in three categories each of steepness (low, moderate and large) and relative depth (near-shallow, intermediate and deep). {\textcolor{buliya}{Criteria for selecting spatio-temporal resolution within the tank have been determined through parametric analysis. For $H/\lambda>0.03$, it is seen that wave height reduction in excess of $5\%$ occurs in near-shallow and deep water. It is found that height reduction in deep water is largely attributable to free-surface damping as predicted by Airy theory. However, numerical height damping in near-shallow water is observed to be much stronger, especially near the source. A deeper analysis reveals that mass source-based wave generation induces strong and persistent vortical activity in near-shallow water which in turn leads to dissipation of source-injected momentum. Appropriate modifications in source design, that are aimed at reducing wave-vorticity interactions in the near-field, are proposed for steep wave generation in $kh<1$ and $kh>2.5$. Substantial improvement in the prediction of far-field wave height is hence reported.

Direct numerical simulation (DNS) of transitional and turbulent round jets is reported in a comparative framework. Such comparison is central towards revealing the roles that molecular viscosity and vorticity intensification play in the evolution of jets. The initial and intermediate evolution is differentiated based on assessment of the starting jet, roll-up frequency, dynamics of vortex rings and emergence of the secondary instability. Long term behavior is differentiated based on assessment of preferred mode frequency, time averaged vortical structures, half jet-width and volume flow rate obtained from the time averaged velocity field. The present study demonstrates that viscous damping of cross-stream vorticity plays a key role in establishing helical instability as the dominant mode in long term evolution of the transitional jet. On the contrary, varicose mode is dominant in the turbulent jet, despite preferred mode frequency being the same in both cases. Lastly a novel attempt is made towards comparing individual terms constituting turbulence budget between both regimes. Through such comparison, relative dominance of various transport mechanisms governing the evolution of turbulence kinetic energy $(\mathcal{K})$ is revealed. It is observed that terms accounting for forward cascade of $\mathcal{K}$ from inertial to smallest scales are comparatively larger for the turbulent jet whilst those accounting for backscatter of $\mathcal{K}$ are comparatively larger for the transitional jet. It is also established that turbulence dissipation is evidently the same for both jets. Thus, the property of turbulence dissipation being independent of Reynolds number for turbulent jets can also be extrapolated to transitional jets.

Development of mass-source function based numerical wave tank (NWT) algorithms in the Navier-Stokes (NSE) framework is impeded by multiple design issues such as: (a) optimization of a number of variables characterizing the source region, (b) wave-vorticity interactions and (c) a mandatory requirement of modeling the domain on both sides of the wavemaker. In this paper, we circumvent these hurdles by proposing a volume-preserving inflow-boundary based Navier-Stokes wave tank. Wave generation and propagation is modeled in a two-phase PLIC-VOF set-up. Near-exact volume preservation is achieved (at arbitrarily large steepness) using kinematic stretching that is aimed towards balancing the streamwise momentum between points lying above and below the still water level. Numerical damping of steep waves is prevented by using blended third-order and limiter schemes for momentum advection. In addition, a mesh stair-stepping strategy has been adopted for modeling non-Cartesian immersed boundaries on a staggered variable arrangement. The proposed NWT model is rigorously benchmarked against various wave-propagation scenarios. These include the simulation of: (a) monochromatic waves of various steepnesses, (b) monochromatic waves superimposed with free harmonics, (c) irregular waves in deep water and (d) wave transformation occurring over a submerged trapezoidal bar. Excellent agreement with analytical, numerical and experimental data is reported with both validation as well as verification of the proposed NWT model being established

State of the art in numerical wave tank (NWT) development demands that emerging algorithms be capable of utilizing most of the computing power available from today’s multi-core CPU architectures. Based on this motivation, we attempt MPI-based parallelization of our (in-house) Navier–Stokes equation (NSE)- based NWT algorithm. Parallelization strategy adopted in this paper involves domain decomposition along the direction of wave propagation. The parallelized NWT code is tested on single, multi-core, shared-memory nodes for both regular wave generation and wave-rigid structure interaction (WSI) scenarios. It is demonstrated that the resultant wave topology is independent of number of threads considered in the parallel computation *(**N**p**)*. It is further observed that proposed parallelization strategy results in appreciable reduction in computation time *(**CT **)*. However, maximum speedup *(ψ**max **) *is observed to be limited by the number of physical cores *(*N*) *available on the computing node. Nonetheless, the results demonstrate that, for the same grid size, proposed NWT code is significantly faster than ANSYS® FLUENT for WSI simulations.

Compressive Interface Capturing Scheme for Arbitrary Meshes (CICSAM) is a numerical scheme which is extensively used in commercial computational fluid dynamics solvers. However, it generates significant numerical diffusion at higher Courant number. In order to improve its performance a modification is proposed in the existing algorithm to improve its interface capturing capability at high Courant number. The ULTIMATE QUICKEST scheme for interpolation of face value of volume fraction in the existing CICSAM algorithm is re- placed by Fromm scheme in the modified CISCAM algorithm. The Fromm scheme imbibes better dispersion properties than conventional ULTIMATE QUICKEST. The modified algorithm is tested extensively for known two and three dimensional velocity fields. Also the scheme is coupled with Navier–Stokes’ solver and its performance is tested for collapse of water-column and bubble-rise with high surface tension coefficient. The modified scheme results in significant reduction of numerical diffusion and improves the interface capturing abilities.

An interface capturing scheme called modified switching technique for advection and capturing of surfaces (MSTACS) has been proposed. The proposed interface capturing scheme utilizes basic framework of switching technique for advection and capturing of surfaces (STACS) for solution of the volume fraction advection equation. MSTACS has been compared against three interface capturing schemes with the help of various two and three dimensional test cases. It is proved that MSTACS is able to capture sharp interfaces with minimum numerical diffusion over a wide range of Courant numbers. Further, the interface capturing capability of MSTACS is demonstrated by coupling it with the Navier-Stokes equations (NSE) to simulate the three dimensional flow problems such as Rayleigh-Taylor instability and dam break with an obstacle.

A Compressive volume-of-fluid (VOF) schemes exhibits numerical diffusion which inhibit them in obtaining a numerically sharp and wrinkle free description of fluid interface which is vital for understanding the complex interfacial dynamics. Therefore, the present study introduces a novel compressive VOF scheme capable of capturing sharp abrupt interfaces at stringent Courant conditions while demonstrating excellent computational efficiency. Moreover, the proposed method is also able to effectively conserve the fluid mass even when subjected to an involved flow field. The performance of the proposed method is evaluated against several canonical pure advection test problems and the results are compared with four established compressive VOF schemes. The quantitative and qualitative analysis of the numerical results assert the ascendancy of the newly introduced scheme. Furthermore, the method exhibits excellent agreement with the literature when applied to the high density ratio problems which involve complex interface topologies dominated by viscous and surface tension forces.

The liquid jet when perturbed sinusoidally will lead to instability under certain conditions. Understanding of the causes and consequences of such a behaviour is still obscure. Hence, numerical investigations are reported in the present study for two phase spatially oscillating planar jet in a quiescent air. Simulations are performed by solving the Navier-Stokes equations and using volume of uid method (VOF) to track the air-water interface. It is demonstrated that an increase in amplitude of oscillation is caused due to the formation of low pressure region created by the vortical structures in air near the leading edge of the jet when detected. This two way coupling between air and water is analysed with the help of enstrophy, divergence of lamb vector and vortex forces. It is found through parametric study that surface tension and viscosity stabilize the perturbations in an oscillating planar jet. On the other hand, increase in Froude number (Fr) initially leads to an augmentation of perturbation amplitude and later causes its damping when surface tension forces become dominant. The numerical analysis for different inlet velocity profiles establishes that the jet is more stable subjected to parabolic inlet velocity profile as compared to uniform profile due to lower relative velocity at the interface. Present work also reveals the role of capillary instability in addition to Kelvin Helmholtz and Rayleigh Taylor instabilities that induces primary breakup in the jet.

A Lagrangian-Eulerian advection scheme (LEAS) with Moment-of-Fluid (MoF) interface reconstruction is developed for interfacial flows. In MoF method, in addition to the volume fraction field, material centroid is used for interface reconstruction. Physically material centroid indicates the material location inside a mixed cell. Therefore, MoF method reconstructs linear interfaces exactly and is second order accurate for curved interfaces. Despite being second order accurate for static interface reconstruction, MoF recosntruction is inaccurate when centroid is not properly advected. An accurate centroid advection method based on barycenter of centroid is discussed here. A comparison of this centroid advection with the pure Lagrangian advection based on Runge-kutta integrator is demonstrated for two different advection tests. The superiority of MoF is established by comparing the volume error norm with other interface reconstructions. Significant improvement in accuracy is observed when LEAS material advection is used in conjuction with accurate centroid advection for MoF reconstruction

The approach followed here does not deal with edge flux and corner flux computation as well as corner flux correction typical of unsplit Eulerian advection. The method discussed is a Lagrangian-Eulerian advection scheme (LEAS) which is CFL condition independent, in contrast to most of the unsplit Eulerian advection algorithms. Also the scheme is equally applicable on general polygonal grids. The method works on the assumption that fluid content remain constant inside a stream tube generated between Eulerian cell and Lagrangian precell. The Lagrangian precell is obtained by back tracking of cell vertices in time. Volume fraction remapping is performed by a series of polygon intersection between Lagrangian precell and vertex connected neighbor cell fluid polygons. The fluid polygons are obtained from interface reconstruction. The volume fraction inconsistencies are removed by conservative volume fraction repair algorithm. Since interface reconstruction is also an integral part of the overall volume tracking algorithm, details of the same is presented for the sake of completeness. Overall volume tracking error is solely due to approximate interface reconstruction while the scheme is derived naturally from the fact that volume inside a stream tube remain constant. LEAS performs better compared to the existing unsplit algorithms for the same interface reconstruction

Moment-of-fluid (MOF) method is an extended volume-of-fluid method which incorporates material centroid in addition to material volume fraction for interface reconstruction. MOF is an exact method for linear interfaces and second order accurate for curved interfaces. The interface reconstruction in MOF is based on the best possible approximation of material centroid. Exact matching of the reconstructed centroid with the reference centroid produces the exact material configuration. This is however possible only for linear interfaces when the material volume and centroid are consistent. Consistent moments can be obtained by adaptive mesh refinement (AMR-MOF) in the interfacial cells based on a refinement criterion. In the case of AMR-MOF, centroid error is used as the refinement criterion. Since the material centroid is approximated to the reference centroid in interface reconstruction, centroid error is always higher than the prescribed tolerance. Therefore, intermediate interface reconstructions in AMR-MOF are insignificant and increases the computational time as the refinement always proceeds to the maximum level. In the present article, a refined moment-of-fluid (RMOF) method for evolving interfaces is proposed. In RMOF, each mixed cell is refined to prescribed level of refinement as opposed to AMR-MOF where refinement is dependent on a refinement criterion. Since the number of interface reconstructions performed at each level of refinement in AMR-MOF is much higher than that of fixed level of refinement in RMOF, the computation time is significantly reduced in RMOF. The reduction in computational time for the proposed RMOF method in comparison to standard MOF and AMR-MOF is demonstrated in this work through series of time reversed advection tests.

Inconsistent implementation of surface tension in an interfacial solver for incompressible two-phase flow results in spurious currents near the interface. The balanced force algorithm en-forces consistent implementation of surface tension force and pressure gradient at the cell faces thereby generating conservative face centre velocity field, even though the cell center velocity may not be divergence free and spurious current free. The implication of this inconsistency is directly reflected in interface smoothness while using a Lagrangian-Eulerian interface evolution in contrary to Eulerian advection where divergence free face velocities are used. This paper describes a consistent balanced force algorithm in collocated grid for consistent implementation of surface tension force. The interface evolution is tracked by re moment-of-uid (RMOF) method using Lagrangian-Eulerian advection scheme (LEAS). In consistent balanced force algorithm, additional Poisson equation is solved to obtain cell face corrections for conservative cell center velocity computation. The effectiveness of the proposed method is demonstrated for several test cases.