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Description: It is well known that the interface between two regions of an incompressible ideal fluid flow moving in a relative motion is necessarily destabilized, regardless of the velocity difference's strength. This phenomenon is the so-called Kelvin-Helmholtz instability (KHI). However, a large number of works demonstrated a surprising result that the instability is suppressed for shallow water flows; the interface is stabilized if the Froude number, defined by the velocity difference's ratio to the gravity wave's speed, is sufficiently large. In a limited way, these authors have been used the shallow-water equations without the higher-order effect of the dispersive terms. Thus, this investigation aims to examine these higher-order dispersive effects to analyze the interface stability problem of tangential-velocity discontinuity in shallow-water flows. In particular, we use the Green-Naghdi equations to introduce the dispersive terms related to the depth and the depth-averaged horizontal velocities of the fluid. We show that the interface stability depends on the Froude number (i.e., the velocity difference's strength) and the water depth. A critical value of the Froude number to stabilize the interface is smaller than the case of no dispersive terms, and the flow in a deeper region is more stable than in a shallower one. We also consider the distribution of kinetic and potential energy to clarify a feature characteristic of a large class of instabilities in shallow water flow. The instability of flows is caused by the decrease in the kinetic energy during the perturbation of waves. This phenomenon is known as negative energy modes and plays a vital role in applying the model to industrial equipment. A conclusion is that the equipartition of energies occurs if and only if the velocity difference is zero and the water depth is shallow enough to ignore the dispersive terms.

Description: We examine a frictional effect on the linear stability of an interface of discontinuity in tangential velocity. The fluid is moving with uniform velocity U in a region but is at rest in the other, and the bottom surface is assumed to exert drag force, quadratic in velocity, on the thin fluid layer. In the absence of the drag, the instability of the Kelvin-Helmholtz type is suppressed for U〉√8 c, with c being the propagating speed of the gravity wave. We find by asymptotic analyses for both small and large values of the drag strength that the drag, regardless of its strength, makes the flow unstable for the whole range of the Froude number U/c.

Description: We examine an effect of side walls on the linear stability of an interface of tangential-velocity discontinuity in shallow-water flow. The flow is pure horizontal in the plane xy, and the fluid is bounded in a finite width 2d in the y− direction. In region 0 〈 y 〈 d, the fluid is moving with uniform velocity U but is at rest for −d 〈 y 〈 0. Without side walls, the flow is unstable for a velocity difference U〈√8c U 〈 √8 c, with c being the velocity of gravity waves. In this work, we show that if the velocity difference U is smaller than 2c, the interface is always destabilized, also known as the flow is unstable. The unstable region of an infinite width model is shrunken by the effects of side walls in the case of narrow width, while there is no range for the Froude number for stabilization in the case of large width. These results play an important role in predicting the wave propagations and have a wide application in the fields of industry. As a result of the interaction of waves and the mean flow boundary, the flow is unstable, which is caused by a decrease in the kinetic energy of disturbance.

Description: The stability of flows in porous media plays a vital role in transiting energy supply from natural gas to hydrogen, especially for estimating the usability of existing underground gas storage infrastructures. Thus, this research aims to analyze the interface stability of the tangential-velocity discontinuity between two compressible gases by using Darcy's model to include the porosity effect. The results shown in this research will be a basis for considering whether underground gas storages in porous material can be used to store hydrogen. We show the relation between the Mach number M, the viscosity \mu, and the porosity \epsilon on the stability of the interface. This interface stability affects gases' withdrawal and injection processes, thus will help us to determine the velocity which with gas can be extracted and injected into the storage effectively. By imposing solid walls along the flow direction, the critical values of these parameters regarding the stability of the interface are smaller than when considering no walls. The consideration of bounded flows approaches the problem more realistically. In particular, this analysis plays a vital role when considering two-dimensional gas flows in storages and pipes.

Description: The stability of a flow in porous media relates to the velocity rate of injecting and withdrawing natural gases inside porous storage. We thus aim to analyze the stability of flows in porous media to accelerate the energy transition process. This research examines a flow model of a tangential--velocity discontinuity with porosity and viscosity changes in a three-dimensional (3D) compressible medium because of a co-existence of different gases in a storage. The fluids are assumed to move in a relative motion where the plane y=0 is a tangential-velocity discontinuity surface. We obtain that the critical value of the Mach number to stabilize a tangential discontinuity surface of flows via porous media is smaller than the one of flows in a plane. The critical value of the Mach number M to stabilize a discontinuity surface of the 3D flow is different by a factor |cosθ| compared to the two-dimensional (2D) flow. Here, θ is the angle between velocity and wavenumber vectors. Our results also show that the flow model with viscosity and porosity effects is stable faster than those without these terms. Our analysis is done for both infinite and finite flows. The effect of solid walls along the flow direction could suppress the instability, i.e., the tangential-discontinuity surface is stabilized faster

Description: Compressible flows appear in many natural and technological processes, for instance, the flow of natural gases in a pipe system. Thus, a detailed study of the stability of tangential velocity discontinuity in compressible media is relevant and necessary. The first early investigation in two-dimensional (2D) media was given more than 70 years ago. In this article, we continue investigating the stability in three-dimensional (3D) media. The idealized statement of this problem in an infinite spatial space was studied by Syrovatskii in 1954. However, the omission of the absolute sign of cos θ with θ being the angle between vectors of velocity and wave number in a certain inequality produced the inaccurate conclusion that the flow is always unstable for entire values of the Mach number M. First, we revisit this case to arrive at the correct conclusion, namely that the discontinuity surface is stabilized for a large Mach number with a given value of the angle θ. Next, we introduce a real finite spatial system such that it is bounded by solid walls along the flow direction. We show that the discontinuity surface is stable if and only if the dispersion relation equation has only real roots, with a large value of the Mach number; otherwise, the surface is always unstable. In particular, we show that a smaller critical value of the Mach number is required to make the flow in a narrow channel stable.

Description: It is well known as the Kelvin-Helmholtz instability (KHI) that an interface of tangential velocity discontinuity is necessarily unstable, regardless of the velocity difference's strength. However, the KHI is suppressed for shallow water flows if the Froude number, defined by the ratio of the velocity difference to the gravity wave's speed, is sufficiently large. In this investigation, we examine the effect of the depth difference of two fluid layers on the KHI. The depth difference enhances instability. Given the Froude number in the instability range, the growth rate sensitively depends on the depth ratio and increases monotonically with the depth ratio difference from unity. The critical value of the Froude number for stabilization varies with the depth ratio and attains the minimum value √8 for equal depth. This behavior is verified by asymptotic analysis.

Description: The present study investigates the stability of a tangential-velocity discontinuity in porous media during the withdrawing and injecting processes of natural gases from and into an underground gas storage. The focus is placed on analyzing the impact of inertia forces on the interface stability using the Forchheimer equations. Other publications have relied primarily on Darcy's law to describe flow stability in porous media. However, Darcy's law only adequately describes flows in which viscous forces dominate over inertia forces. As the flow rate increases, the significance of inertia forces becomes more pronounced, and Darcy's law becomes insufficient for considering such flows. Our findings indicate that even a slight consideration of the inertia effect leads to permanent destabilization of the discontinuity interface, regardless of the fluid viscosity or the Mach number. In contrast, when the inertia effect is neglected, the interface is stabilized across the entire Mach number range if the fluid viscosity is strong enough.

Description: This paper presents a numerical simulation of the tidal flow in Danang Bay (Vietnam) based on the non-hydrostatic shallow water equations. First, to test the simulation capability of the non-hydrostatic model, we have made a test simulation comparing it with the experiment by Beji and Battjes 1993. Simulation results for this case are compared with both the experimental data and calculations obtained from the traditional hydrostatic model. It is shown that the non-hydrostatic model is better than the hydrostatic model when the seabed topography variation is complex. The usefulness of the non-hydrostatic model is father shown by successfully simulating the tidal flow of Danang Bay.

Description: Due to the coexistence of different gases in underground storage, this work explores the interface stability's impact on energy storage, specifically during the injection and withdrawal of gases such as hydrogen and natural gas. A new approach of combing simulation and time series analysis is used to accurately predict instability modes in energy systems. Our simulation is based on the 2D Euler equations, solved using a second-order finite volume method with a staggered grid. The solution is validated by comparing them to experimental data and analytical solutions, accurately predicting the instability's behavior. We use time series analysis and state-of-the-art regime-switching methods to identify critical features of the interface dynamics, providing crucial insights into system optimization and design.