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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.

Numerical simulation of tidal flow in Danang Bay Based on non-hydrostatic shallow water equations
(2016)

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.

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.

Compressible flows are prevalent in natural and technological processes, particularly in the energy transition to renewable energy systems. Consequently, extensive research has focused on understanding the stability of tangential--velocity discontinuity in compressible media. Despite recent advancements that address industrial challenges more realistically, many studies have ignored viscous stress tensors' impact, leading to inaccuracies in predicting interface stability. This omission becomes critical, especially in high Reynolds or low Mach number flows, where viscous forces dissipate kinetic energy across interfaces, affect total energy dissipation, and dampen flow instabilities. Our work is thus motivated to analyze the viscosity force effect by including the viscous stress tensor terms in the motion equations. Our results show that by considering the effect of viscous forces, the tangential-velocity discontinuity interface is constantly destabilized for the entire range of the Mach number.

While energy-intensive industries like the steel industry plan to switch to renewable energy sources, other industries, such as the cement industry, have to rely on carbon capture storage and utilization technologies to reduce the inevitable carbon dioxide (CO2) emissions of their production processes. In this context, we investigate the problem of finding optimal pipeline diameters from a discrete set of diameters for a tree-shaped network transporting captured CO2 from multiple sources to a single sink.
The general problem of optimizing arc capacities in potential-based fluid networks is a challenging mixed-integer nonlinear program. Additionally, the behaviour of CO2 is highly sensitive and nonlinear regarding temperature and pressure changes. We propose an iterative algorithm splitting the problem into two parts: a) the pipe-sizing problem under a fixed supply scenario and temperature distribution and b) the thermophysical modelling including mixing effects, the Joule-Thomson effect, and heat exchange with the surrounding environment. We show the effectiveness of our approach by applying our algorithm to a real-world network planning problem for a CO2 network in Western Germany.

For industries like the cement industry, switching to a carbon-neutral production process is impossible. They must rely on carbon capture, utilization and storage (CCUS) technologies to reduce their production processes’ inevitable carbon dioxide (CO2) emissions. For transporting continuously large amounts of CO2, utilizing a pipeline network is the most effective solution; however, building such a network is expensive. Therefore minimizing the cost of the pipelines to be built is extremely important to make the operation financially feasible. In this context, we investigate the problem of finding optimal pipeline diameters from a discrete set of diameters for a tree-shaped network transporting captured CO2 from multiple sources to a single sink. The general problem of optimizing arc capacities in potential-based fluid networks is already a challenging mixed-integer nonlinear optimization problem. The problem becomes even more complex when adding the highly sensitive nonlinear behavior of CO2 regarding temperature and pressure changes. We propose an iterative algorithm splitting the problem into two parts: a) the pipe-sizing problem under a fixed supply scenario and temperature distribution and b) the thermophysical modeling, including mixing effects, the Joule-Thomson effect, and heat exchange with the surrounding environment. We demonstrate the effectiveness of our approach by applying our algorithm to a real-world network planning problem for a CO2 network in Western Germany. Further, we show the robustness of the algorithm by solving a large artificially created set of network instances.

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.

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.

Linear stability of a simple shear layer between two parallel streams in a shallow water flow
(2024)

The stability of shear layers in fluid flows is a crucial factor in forming vortices and jets and plays a fundamental role in the development of turbulence. Such shear layer instabilities are ubiquitous in natural phenomena, such as atmospheric and oceanic flows, contributing to the formation of weather systems and predicting tsunamis. This study specifically focuses on the stability of a shear layer sandwiched between two semi-infinite layers within a two-dimensional flow. The velocity profile of the shear layer is assumed to be linearly dependent on the vertical coordinate, while the velocity of the other layers remains uniform with differing strengths. The effect of viscosity and surface tension is ignored to simplify the analysis. The shallow water equations are used to analyze the interface stability of the shear layer, and the resulting dispersion relation between wave frequency and other wave characteristics is obtained. This relation incorporates Whittaker functions and their first derivatives and is used to derive appropriate limits corresponding to various physical conditions. Our study thus contributes to a deeper understanding of the stability of shear layers and their implications for natural phenomena.

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