Hydrostatic Limit Research Articles

On the kinetics of internal gravity waves beyond the hydrostatic regime

We present a new derivation of the kinetic equation for weak, non-hydrostatic internal gravity wave turbulence. The equation is equivalent to the one obtained by Caillol & Zeitlin (Dyn. Atmos. Oceans, vol. 32, issue 2, 2000, pp. 81–112), but it takes a canonical form. We show that it conserves the energy without involving the resonance condition in frequency, and look for the isotropic part of the steady, scale-invariant solutions. We provide a parametrization of the resonant manifold of non-hydrostatic internal gravity wave triadic interactions. This allows us to simplify the collision integral, and to evaluate the transfer coefficients of all triadic interactions. In the hydrostatic limit, our equation is equivalent to the Hamiltonian description of Lvov & Tabak (Phys. Rev. Lett., vol. 87, issue 16, 2001, 168501).

On the hydrostatic limit of stably stratified fluids with isopycnal diffusivity

This article is concerned with rigorously justifying the hydrostatic limit for continuously stratified incompressible fluids under the influence of gravity. The main distinction of this work compared to previous studies is the absence of any (regularizing) viscosity contribution added to the fluid-dynamics equations; only thickness diffusivity effects are considered. Motivated by applications to oceanography, the diffusivity effects in this work arise from an additional advection term, the specific form of which was proposed by Gent and McWilliams in the 1990s to model the effective contributions of geostrophic eddy correlations in non-eddy-resolving systems. The results of this paper heavily rely on the assumption of stable stratification. We establish the well-posedness of the hydrostatic equations and the original (non-hydrostatic) equations for stably stratified fluids, along with their convergence in the limit of vanishing shallow-water parameter. These results are obtained in high but finite Sobolev regularity and carefully account for the various parameters involved. A key element of our analysis is the reformulation of the systems using isopycnal coordinates, enabling us to provide meticulous energy estimates that are not readily apparent in the original Eulerian coordinate system.

Rigorous justification of the hydrostatic approximation limit of viscous compressible flows

This paper considers the asymptotic limit of small aspect ratio between vertical and horizontal spatial scales for viscous isothermal compressible flows. In particular, it is observed that fast vertical acoustic waves arise and induce an averaging mechanism of the density in the vertical variable, which at the limit leads to the hydrostatic approximation of compressible flows, i.e., the compressible primitive equations of atmospheric dynamics. We justify the hydrostatic approximation for general as well as “well-prepared” initial data. The initial data is called well-prepared when it is close to the hydrostatic balance in a strong topology. Moreover, the convergence rate is calculated in the well-prepared initial data case in terms of the aspect ratio, as the latter goes to zero.

On the hydrostatic limit for thin film flow with applications to thermosyphons

Heat pipes frequently encounter challenges in effectively returning condensed liquid from the condenser to the evaporator through capillary pumping, which limits their overall efficacy. When capillary pumping becomes irrelevant due to factors like the absence of wick, or overfilling, the heat pipe is more properly termed as thermosyphon. In a close-to-horizontal orientation, the driving force for liquid return in a thermosyphon relies on the difference in liquid pool depth between the evaporator and condenser. An excessively deep liquid pool in the condenser can hinder radial heat transfer, necessitating a design that favours intermediate-depth liquid pools. This study utilizes a theoretical approach based on the lubrication approximation to the Navier–Stokes equations to determine the fill ratio that maximizes thermosyphon performance. We explore the relationship between this fill ratio and factors such as the axial temperature difference along the thermosyphon. Our analysis thereby highlights the hydrostatic-driven flow limit in the thermosyphon as an analogue to the capillary limit in heat pipes. Regarding the hydrostatic limiting curve as an alternative to the capillary limiting curve offers valuable insights for enhancing thermosyphon performance and provides means of comparing the performance of one vs. the other passive heat transfer device.

Mathematical Insights into Hydrostatic Modeling of Stratified Fluids

Abstract We review recent mathematical results concerning the analysis of hydrostatic equations in the context of stably stratified fluids. Beginning with the simpler and better understood setting of homogeneous fluids, we emphasize the additional mathematical challenges posed by non-homogeneous framework. We present both positive and negative results, including well-posedness and proof of the hydrostatic limit with a suitable regularization, alongside ill-posedness in the fully inviscid setting and the breakdown of the hydrostatic limit in specific scenarios.

The hydrostatic limit of the Beris-Edwards system in dimension two

The hydrostatic limit of the Beris-Edwards system in dimension two

Testing of metal waterstops for waterproofing tack welds

The existing information on the study and application of tack weld metal waterstops in Russian construction practice is limited. The authors of the present paper investigate the operation and effectiveness of metal waterstops designed for waterproofing tack welds. An experimental study of metal waterstops made of zinc-coated sheets and installed in a factory-fabricated capacitive monolithic reinforced concrete structure was conducted. In order to provide different exposure conditions for a waterstop, a vessel structure was manufactured with two split-level tack welds on its perimeter. As a result of filling the vessel with water and monitoring the waterstops through special cavities, it was determined that the waterstops mounted with epoxy glue did not provide sufficient sealing, unlike those mounted with a silicone sealant. In addition, some technological advantages and disadvantages of metal waterstops were experimentally revealed. The disadvantages included a relatively high labor intensity of their installation, in particular, joining the elements along the length. The main technological advantage consisted in the inherent rigidity of the product. Further research should investigate the chemical behavior of the waterstop metal in the concrete body. To this end, the steel behavior under increased water pressure and the influence of the waterstop geometry and the chemical activity of the "structural concrete – waterstop metal" complex on the value of the hydrostatic pressure resistance limit should be elucidated.

Stability Properties of the Constant Coefficients Semi-Implicit Time Schemes Solving Nonfiltering Approximation of the Fully Compressible Equations

Abstract A set of control parameters is introduced in the fully elastic nonhydrostatic Euler equations formulated in the mass-based vertical coordinate of Laprise. Contrary to the classical approach, the hydrostatic limit is represented by a subspace of control parameters, instead of a single point. By finding a suitable path from the fully compressible equations to the hydrostatic subspace, we are able to construct a blended system with acoustic modes slowed down and gravity modes nearly unaffected. Numerical stability of the discretized system is thus improved, and the solution remains essentially the fully compressible one. Alternatively, control parameters can be used to redefine the linear model of the constant coefficients semi-implicit time scheme, increasing the numerical stability of the fully compressible system. With a careful choice of the control parameters in both, the linear model used in the semi-implicit temporal scheme, and in the full model, the blended system does not deteriorate the compressible solution while its semi-implicit temporal discretization is more stable. We illustrate the potential of the method in several simple examples and in real case studies using the ALADIN system.

A New Hydrodynamic Spherical Accretion Exact Solution and Its Quasi-spherical Perturbations

We present an exact γ = 5/3 spherical accretion solution that modifies the Bondi boundary condition of as r → ∞ to ρ → 0 as r → ∞. This change allows for simple power-law solutions on the density and infall velocity fields, ranging from a cold empty freefall condition where pressure tends to zero, to a hot hydrostatic equilibrium limit with no infall velocity. As in the case of the Bondi solution, a maximum accretion rate appears. As in the γ = 5/3 case of the Bondi solution, no sonic radius appears, this time however, because the flow is always characterized by a constant Mach number. This number equals 1 for the case of the maximum accretion rate, diverges toward the cold empty state, and becomes subsonic toward the hydrostatic equilibrium limit. It can be shown that in the limit r → 0, the Bondi solution tends to the new solution presented, extending the validity of the Bondi accretion value to cases where the accretion density profile does not remain at a fixed constant value out to infinity. We then explore small deviations from sphericity and the presence of angular momentum through an analytic perturbative analysis. Such perturbed solutions yield a rich phenomenology through density and velocity fields in terms of Legendre polynomials, which we begin to explore for simple angular velocity boundary conditions having zeros on the plane and pole. The new solution presented provides complementary physical insight into accretion problems in general.

Hydrodynamic and hydrostatic limit for a generalized contact process with mixed boundary conditions

Hydrodynamic and hydrostatic limit for a generalized contact process with mixed boundary conditions

Non-equilibrium Stationary Properties of the Boundary Driven Zero-Range Process with Long Jumps

We consider the zero-range process with long jumps and in contact with infinitely many reservoirs in its non-equilibrium stationary state. We derive the hydrostatic limit and the Fick's law, which are a consequence of a relationship between the exclusion process and the zero-range process. We also obtain the large deviation principle for the empirical density, i.e. we compute the non-equilibrium free energy.

The primitive equations of the polluted atmosphere as a weak and strong limit of the 3D Navier-Stokes equations in downwind-matching coordinates

<p style='text-indent:20px;'>A widely used approach to mathematically describe the atmosphere is to consider it as a geophysical fluid in a shallow domain and thus to model it using classical fluid dynamical equations combined with the explicit inclusion of an <inline-formula><tex-math id="M1">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> parameter representing the small aspect ratio of the physical domain. In our previous paper [<xref ref-type="bibr" rid="b14">14</xref>] we proved a weak convergence theorem for the polluted atmosphere described by the Navier-Stokes equations extended by an advection-diffusion equation. We obtained a justification of the generalised hydrostatic limit model including the pollution effect described for the case of classical, east-north-upwards oriented local Cartesian coordinates. Here we give a two-fold improvement of this statement. Firstly, we consider a meteorologically more meaningful coordinate system, incorporate the analytical consequences of this coordinate change into the governing equations, and verify that the weak convergence still holds for this altered system. Secondly, still considering this new, so-called downwind-matching coordinate system, we prove an analogous strong convergence result, which we make complete by providing a closely related existence theorem as well.</p>

Hydrostatic limit for the symmetric exclusion process with long jumps: supper-diffusive case

Hydrostatic behavior for the one dimensional exclusion process with long jumps in contact with infinite reservoirs at different densities are derived. The jump rate is described by a transition probability p which is proportional to | · |−(γ+1) for 1 < γ < 2 (supper-diffusive case). The reservoirs add or remove particles with rate proportional to κ > 0.

Hydrodynamics for SSEP with Non-reversible Slow Boundary Dynamics: Part I, the Critical Regime and Beyond

The purpose of this article is to provide a simple proof of the hydrodynamic and hydrostatic behavior of the SSEP in contact with slowed reservoirs which inject and remove particles in a finite size windows at the extremities of the bulk. More precisely, the reservoirs inject/remove particles at/from any point of a window of size K placed at each extremity of the bulk and particles are injected/removed to the first open/occupied position in that window. The hydrodynamic limit is given by the heat equation with non-linear Robin boundary conditions or Neumann boundary conditions, the latter being in the case when the reservoirs are too slow. The proof goes through the entropy method of Guo, Papanicolaou and Varadhan. We also derive the hydrostatic limit for this model, whose proof is based on the method developed by Landim and Tsunoda. We observe that we do not make use of correlation estimates in none of our results.

Evaporation driven detachment of a liquid bridge from a syringe needle in repose

In this paper, a study of the stability of an evaporating semi-unbounded axisymmetric liquid bridge that forms between a syringe needle tip and a horizontal interface by using both theory and experiments is presented. Here, the evaporation produces slow quasistatic motion such that it allows one to use hydrostatics to analyze interface profiles via solutions to the Young–Laplace equation. The two main parameters, in the hydrostatic limit, are the familiar Bond number and a slenderness parameter that often appears in the literature that studies liquid bridge stability. The axisymmetric Young–Laplace equation yields a semi-analytical solution for capillary pressure at zero Bond number using boundary conditions appropriate for this study. At finite Bond numbers, computation of interface profiles is used to estimate the maximum slenderness. Experiments using water for Bond numbers 0.01 < Bo < 0.1 show good agreement for the maximum slenderness when comparing those results with predictions based on solutions to the Young–Laplace equation.

Magnetic Network on Demand: Pressure Tunes Square Lattice Coordination Polymers Based on {[Cu(pyrazine)2]2+}n.

We report the pressure-induced structural and magnetic changes in [CuCl(pyz)2](BF4) (pyz = pyrazine) and [CuBr(pyz)2](BF4), two members of a family of three-dimensional coordination polymers based on square mesh {[Cu(pyz)2]2+}n layers. High-pressure X-ray diffraction and density functional theory calculations have been used to investigate the structure–magnetic property relationship. Although structurally robust and almost undeformed within a large pressure range, the {[Cu(pyz)2]2+}n network can be electronically modified by adjusting the interaction of the apical linkers interconnecting the layers, which has strong implications for the magnetic properties. It is then demonstrated that the degree of covalent character of the apical interaction explains the difference in magnetic exchange between the two species. We have also investigated the mechanical deformation of the network induced by nonhydrostatic compression that affects the structure depending on the crystal orientation. The obtained results suggest the existence of “Jahn–Teller frustration” triggered at the highest hydrostatic pressure limit.

Wave Emission From Bottom Vibrations in Subsurface Open-channel Shear Flow

The linearized water-wave radiation problem for a 2D oscillating bottom source in an inviscid shear flow with a free surface is investigated analytically. The fluid depth is constant. The velocity of the basic flow varies linearly with depth (uniform vorticity), with zero surface velocity. The far-field surface waves radiated out from the 2D source are calculated, based on Euler’s equation of motion with the application of radiation conditions. There are always two waves, one emitted in the upstream direction and the other in the downstream direction. The energy fluxes of these two waves are calculated. The hydrostatic limit of zero wave number is related to the theory of undular bores.

Indentation densification of fused silica assessed by raman spectroscopy and constitutive finite element analysis

AbstractInelastic deformation of anomalous glasses manifests in shear flow and densification of the glass network; the deformation behavior during indentation testing is linked strongly to both processes. In this paper, the indentation densification field of fused silica is investigated using depth‐resolved Raman spectroscopy and finite element simulations. Through affecting the size of the indent, the normal load and the Raman laser spot size determine the spatial sampling resolution, leading to a certain degree of structural averaging. For appropriate combinations of normal load (indent size) and laser spot diameter, a maximum densification of 18.4% was found at the indent center. The indentation behavior was modeled by extended Drucker‐Prager‐Cap (DPC) plasticity, assuming a sigmoidal hardening behavior of fused silica with a densification saturation of 21%. This procedure significantly improved the reproduction of the experimental densification field, yielding a maximum densification of 18.2% directly below the indenter tip. The degree of densification was found to be strongly linked to the hydrostatic pressure limit below the indenter in accordance to Johnson's expanding cavity model (J. Mech. Phys. Solids, 18 (1970) 115). Based on the good overlap between FEA and Raman, an alternative way to extract the empirical correlation factor m, which scales structural densification to Raman spectroscopic observations, is obtained. This approach does not require the use of intensive hydrostatic compaction experiments.

A DLM immersed boundary method based wave-structure interaction solver for high density ratio multiphase flows

In this paper we present a robust immersed boundary (IB) method for high density ratio multiphase flows that is capable of modeling complex wave-structure interaction (WSI) problems arising in marine and coastal engineering applications. The IB/WSI methodology is enabled by combining the distributed Lagrange multiplier (DLM) method of Sharma and Patankar (2005) [13] with a robust level set method based multiphase flow solver. The fluid solver integrates the conservative form of the variable-coefficient incompressible Navier-Stokes equations using a hybrid preconditioner and ensures consistent transport of mass and momentum at a discrete level. The consistent transport scheme preserves the numerical stability of the method in the presence of large density ratios found in problems involving air, water, and an immersed structure. The air-water interface is captured by the level set method on an Eulerian grid, whereas the free-surface piercing immersed structure is represented on a Lagrangian mesh. The Lagrangian structure is free to move on the background Cartesian grid without conforming to the grid lines. The fluid-structure interaction (FSI) coupling is mediated via Peskin's regularized delta functions in an implicit manner, which obviates the need to integrate the hydrodynamic stress tensor on the complex surface of the immersed structure. The IB/WSI numerical scheme is implemented within an adaptive mesh refinement (AMR) framework, in which the Lagrangian structure and the air-water interface are embedded on the finest mesh level to capture the thin boundary layers and the vortical structures arising from WSI. We use a well balanced gravitational force discretization that eliminates spurious velocity currents in the hydrostatic limit due to density variation in the three phases (air, water and solid). We also show that using a non-conservative and an inconsistent fluid solver can lead to catastrophic failure of the numerical scheme for large density ratio variations that are prevalent in WSI applications. An effective wave generation and absorption technique for a numerical wave tank is presented and used to simulate a benchmark case of water wave distortion due to a submerged structure. The numerical scheme is tested on several benchmark WSI problems from numerical and experimental literature in both two and three dimensions to demonstrate the applicability of the IB/WSI method to practical marine and coastal engineering problems.

A robust incompressible Navier-Stokes solver for high density ratio multiphase flows

This paper presents a robust, adaptive numerical scheme for simulating high density ratio and high shear multiphase flows on locally refined staggered Cartesian grids that adapt to the evolving interfaces and track regions of high vorticity. The algorithm combines the interface capturing level set method with a variable-coefficient incompressible Navier-Stokes solver that is demonstrated to stably resolve material contrast ratios of up to six orders of magnitude. The discretization approach ensures second-order pointwise accuracy for both velocity and pressure with several physical boundary treatments, including velocity and traction boundary conditions. The paper includes several test cases that demonstrate the order of accuracy and algorithmic scalability of the flow solver. To ensure the stability of the numerical scheme in the presence of high density and viscosity ratios, we employ a consistent treatment of mass and momentum transport in the conservative form of discrete equations. This consistency is achieved by solving an additional mass balance equation, which we approximate via a strong stability preserving Runga-Kutta time integrator and by employing the same mass flux (obtained from the mass equation) in the discrete momentum equation. The scheme uses higher-order total variation diminishing (TVD) and convection-boundedness criterion (CBC) satisfying limiter to avoid numerical fluctuations in the transported density field. The high-order bounded convective transport is done on a dimension-by-dimension basis, which makes the scheme simple to implement. We also demonstrate through several test cases that the lack of consistent mass and momentum transport in non-conservative formulations, which are commonly used in practice, or the use of non-CBC satisfying limiters can yield very large numerical error and very poor accuracy for convection-dominant high density ratio flows. Our numerical scheme also uses well-balanced surface tension and gravity force discretizations. In the hydrostatic limit, we show that the well-balanced formulation mitigates spurious flow currents and achieves discrete force-balance between the pressure gradient and surface tension or gravity.