This study numerically analyzes the characteristics of shock wave propagation and attenuation in different mediums of explosion near the air-water interface (free surface). This study discussed flow physics like shock wave propagation, reflection, transmission, and cavitation qualitatively and quantitatively. The numerical simulation is carried out with air, water, and TNT (Tri Nitro Toluene), which were modeled using the ideal gas, Mie-Gruneisen (shock), and Jones-Wilkins-Lee (JWL) equation of state, respectively. The Coupled Eulerian-Lagrangian model is employed. In an explosion above the air-water interface, the shock wave propagates and reaches the free surface. Due to the acoustic impedance of water, the incident shock wave reflects, and part of the shock wave is transmitted into the water. The acoustic impedance of water is much higher than that of air, so this free surface acts like a solid wall. On the other hand, in an explosion below the interface, the incident shock wave reaches the free surface and the shock wave reflects as an expansion wave, resulting in cavitation. In an explosion at the free surface, shock wave propagates in both air and water; the propagation and attenuation of shock wave were studied. Hence, the free surface near the medium of explosion plays a significant role in the characteristics of shock propagation and attenuation effects.
This work addresses the statistical mechanics of vortex gases and the thermodynamics of the initial formation (genesis) of tornado-like vortices. In particular, it discusses how the thermodynamics of discrete vortices in a quasi-two-dimensional (thin three-dimensional) boundary layer drives the creation of ``hairpins'' and ``cusp'' vortices, their two-dimensional rearrangement, and stretching of vorticity in the vertical direction that originates inside a turbulent boundary layer that leads to a tornado-like flow. Turbulence of that process is analyzed with the help of the vortex filament model and its connection to the fluid flow equations via the Hasimoto and Madelung transforms of the cubic nonlinear Schrödinger and Gross--Pitaevskii equations. Finally, a formula for the local non-equilibrium entropy density flux associated with the development of quasi-two-dimensional turbulence in the boundary layer is provided.
This study examines the mathematical foundations of the Euler and Navier-Stokes equations of fluid dynamics, identifying some inconsistencies in the mathematical definitions of flow velocity and the material derivative. We show that the flow velocity of a fluid parcel, which in the Lagrangian description is traditionally modeled as a bivariate function of the presumed independent variables of initial parcel position and time, is more accurately defined as a parametric function of time, with the initial parcel position treated as a time-dependent parameter. This finding leads to the result that the standard form of the material derivative in the Lagrangian description is mathematically inconsistent. We also show that if the fluid flow is non-unidirectional, then the map from parcel position to flow velocity becomes a one-to-many map, leading to the conclusion that the flow velocity is not a valid mathematical function of position in both the Lagrangian and Eulerian descriptions under such conditions. Therefore, if flow velocity is not a valid mathematical function of position, we conclude that the inability to integrate the Euler and Navier-Stokes differential equations in the spatial domain implies the nonexistence of a mathematical solution of these equations under these conditions. Additionally, through mathematical and theoretical analysis, supported by experimental and numerical simulations, we uncover challenges in the material consistency of the definition of the material derivative in the Eulerian description. This inconsistency leads to a decoupling between the Lagrangian and Eulerian descriptions, especially under complex non-unidirectional flow conditions and multi-directional flows with intersecting pathlines. We also show that the Eulerian description is a quasi-continuum mechanics model that, when applied to certain fluids, especially gases and low-viscosity liquids where intermolecular forces are weak or intermediate, limits the ability to accurately model the bi-directional transmission of deformation and force continuously between neighboring parcels. While the Euler and Navier-Stokes equations remain largely valid and effective for modeling unidirectional flows in viscous fluids, our findings suggest the need to refocus on developing fluid dynamics solutions rooted in the Lagrangian model to more accurately capture complex flow behaviors and improve applicability across fields such as atmospheric sciences, oceanography, and plasma physics. These insights aim to advance our understanding of the limits of existing fluid dynamics models by addressing foundational inconsistencies, the understanding of which can contribute to refining these mathematical models.
