Velocity Discontinuity Research Articles

Modulation Instability of Surface Waves in the Model with the Uniform Wind Profile

The modulation instability of surface capillary-gravity water waves is analysed in a shear flow model with a tangential discontinuity of velocity. It is assumed that air blows along the surface of the water with a uniform profile in the vertical direction. Such a model, despite its simplicity, plays an important role in hydrodynamics as the reference model for investigating basic physical phenomena of wave–current interactions and acquiring insights into a series of complex phenomena. In certain cases where the wavelength of interfacial perturbations is much bigger than the width of the shear flow profile, the model with the tangential discontinuity in the velocity is adequate for describing physical phenomena at least within limited spatial and temporal frameworks. A detailed analysis of the air-flow conditions under which modulation instability sets in is presented. It is also shown that the interfacial waves are subject to dissipative or radiative instability when negative-energy waves appear at the interface.

A stabilizer-free pressure-robust finite element method for the Stokes equations

In this paper, we introduce a new finite element method for solving the Stokes equations in the primary velocity-pressure formulation using H(div) finite elements to approximate velocity. Like other finite element methods with velocity discretized by H(div) conforming elements, our method has the advantages of an exact divergence-free velocity field and pressure-robustness. However, most of H(div) conforming finite element methods for the Stokes equations require stabilizers to enforce the weak continuity of velocity in tangential direction. Some stabilizers need to tune penalty parameter and some of them do not. Our method is stabilizer free although discontinuous velocity fields are used. Optimal-order error estimates are established for the corresponding numerical approximation in various norms. Extensive numerical investigations are conducted to test accuracy and robustness of the method and to confirm the theory. The numerical examples cover low- and high-order approximations up to the degree four, and 2D and 3D cases.

Joint inversion for 1-D crustal seismicS- andP-wave velocity structures with interfaces and its application to the Wabash Valley Seismic Zone

SUMMARYInterfaces are important part of Earth’s layering structure. Here, we developed a new model parametrization and iterative linearized inversion method that determines 1-D crustal velocity structure using surface wave dispersion, teleseismic P-wave receiver functions and Ps and PmP traveltimes. Unlike previous joint inversion methods, the new model parametrization includes interface depths and layer Vp/Vs ratios so that smoothness constraint can be conveniently applied to velocities of individual layers without affecting the velocity discontinuity across the interfaces. It also allows adding interface-related observation such as traveltimes of Ps and PmP in the joint inversion to eliminate the trade-off between interface depth and Vp/Vs ratio and therefore to reduce the uncertainties of results. Numerical tests show that the method is computationally efficient and the inversion results are robust and independent of the initial model. Application of the method to a dense linear array across the Wabash Valley Seismic Zone (WVSZ) produced a high-resolution crustal image in this seismically active region. The results show a 51–55-km-thick crust with a mid-crustal interface at 14–17 km. The crustal Vp/Vs ratio varies from 1.69 to 1.90. There are three pillow-like, ∼100 km apart high-velocity bodies sitting at the base of the crust and directly above each of them are a low-velocity anomaly in the middle crust and a high-velocity anomaly in the upper crust. They are interpreted to be produced by mantle magmatic intrusions and remelting during rifting events in the end of the Precambrian. The current diffuse seismicity in the WVSZ might be rooted in this ancient distributed rifting structure.

Lithospheric structures beneath the western Mongolian Plateau: Insight from S wave receiver function

• S wave receiver functions reveals lithospheric structures in western Mongolia. • Two velocity discontinuities (i.e., LAB and MLD) are notable in the upper mantle . • Lower part of the lithosphere beneath the Hangay Dome has been delaminated. The Mongolian Plateau is located in the western segment of the Central Asian Orogenic Belt. In the Cenozoic, the western plateau was affected by the India-Asian collision; the Hangay Dome of a low-relief plateau has been formed. However, the mechanism of the uplift of the plateau and particularly the Hangay Dome is debated. In this study, we image the lithosphere-asthenosphere boundary (LAB) with S wave receiver functions. The LAB is much shallower beneath the western Mongolian Plateau (~80–90 km) than beneath the North China Craton (~150 km) and Siberian Craton (~180 km). We also found another possible deep LAB at 200–250 km depths beneath the Hangay Dome, indicating a lithospheric delamination beneath western Mongolia. Thus we conclude that the consequential asthenospheric upwelling caused the tectonic uplift of the Hangay Dome.

A lattice Boltzmann modeling of the bubble velocity discontinuity (BVD) in shear-thinning viscoelastic fluids

The bubble velocity discontinuity (BVD), when single bubble rising in shear-thinning viscoelastic fluids, is studied numerically. Our three-dimensional numerical scheme employs a phase-field lattice Boltzmann method together with a lattice Boltzmann advection-diffusion scheme, the former to model the macroscopic hydrodynamic equations for multiphase fluids, and the latter to describe the polymer dynamics modeled by the exponential Phan–Thien–Tanner (ePTT) constitutive model. An adaptive mesh refinement technique is implemented to reduce computational cost. The multiphase solver is validated by simulating the buoyant rise of single bubble in a Newtonian fluid. The critical bubble size for the existence of the BVD and the velocity-increasing factor of the BVD are accurately predicted, and the results are consistent with the available experiments. Bubbles of different sizes are characterized as subcritical (smaller than critical size) and supercritical (larger than critical size) according to their transient rising velocity behaviors, and the polymeric stress evolution affecting the local flow pattern and bubble deformation is discussed. Pseudo-supercritical bubbles are observed with transition behaviors in bubble velocity, and their sizes are smaller than the critical value. The formation of bubble cusp and the existence of negative wake are observed for both the pseudo-supercritical and the supercritical bubbles. For the supercritical bubble, the trailing edge cusp and the negative wake arise much earlier. The link between the BVD, the bubble cusp, and the negative wake is discussed, and the mechanism of the BVD is explained.

Vortex sheet turbulence as solvable string theory

We study steady vortex sheet solutions of the Navier–Stokes in the limit of vanishing viscosity at fixed energy flow. We refer to this as the turbulent limit. These steady flows correspond to a minimum of the Euler Hamiltonian as a functional of the tangent discontinuity of the local velocity parametrized as [Formula: see text]. This observation means that the steady flow represents the low-temperature limit of the Gibbs distribution for vortex sheet dynamics with the normal displacement [Formula: see text] of the vortex sheet as a Hamiltonian coordinate and [Formula: see text] as a conjugate momentum. An infinite number of Euler conservation laws lead to a degenerate vacuum of this system, which explains the complexity of turbulence statistics and provides the relevant degrees of freedom (random surfaces). The simplest example of a steady solution of the Navier–Stokes equation in the turbulent limit is a spherical vortex sheet whose flow outside is equivalent to a potential flow past a sphere, while the velocity is constant inside the sphere. Potential flow past other bodies provide other steady solutions. The new ingredient we add is a calculable gap in tangent velocity, leading to anomalous dissipation. This family of steady solutions provides an example of the Euler instanton advocated in our recent work, which is supposed to be responsible for the dissipation of the Navier–Stokes equation in the turbulent limit. We further conclude that one can obtain turbulent statistics from the Gibbs statistics of vortex sheets by adding Lagrange multipliers for the conserved volume inside closed surfaces, the rate of energy pumping, and energy dissipation. The effective temperature in our Gibbs distribution goes to zero as [Formula: see text] with Reynolds number [Formula: see text] in the turbulent limit. The Gibbs statistics in this limit reduces to the solvable string theory in two dimensions (so-called [Formula: see text] critical matrix model). This opens the way for nonperturbative calculations in the Vortex Sheet Turbulence, some of which we report here.

Regional characteristics of the lithospheric structure in the southeast of North China revealed by explosion seismology

Based on the blasting seismic detection data obtained in the southeast of North China in recent years, this paper comprehensively analyzes and studies the crust-mantle lithospheric structure and seismological characteristics of different tectonic regions, such as offshore basins, west Shandong uplift, Tanlu fault zone and Jiangsu-Shandong orogenic belt. The low-velocity Pg waves in Dongying depression and Northern Jiangsu basin reveal the unstable basement structure with extremely thick sediments. The travel time of Pg wave is characterized by relatively low propagation velocity and small crustal thickness of offshore continental margin; the first break time and high apparent velocity of Pg wave in west Shandong uplift indicate that the sedimentary basement is relatively thin. The Pm wave shows the characteristic of dominant wave in the first-order velocity discontinuity of the crust-mantle interface, which reflects the high crustal velocity and stable structure in west Shandong uplift. The Pm and Pl wave are obviously complicated, which can reflect the crust-mantle lithospheric structure of the transitional zone between Tanlu fault zone and Jiangsu-Shandong orogenic belt. The small time difference between Pn and PL waves can be regarded as the highly destructive seismological manifestation of Tanlu fault zone on the crust-lithosphere scale. Based on many geophysical phenomena such as electrical structure, density structure and terrestrial heat flow, it is believed that the lithospheric destruction degree of Tanlu fault zone and Jiangsu-Shandong orogenic belt was high during the destruction of the North China Craton.

Stability of interface between liquids with high viscosity contrast in unevenly rotating cavity

The effect of modulation of the cavity rotation rate on the interface of liquids of different densities and high viscosity contrast is studied experimentally. The cavity is a short horizontal cylinder rotating around its axis. The end walls of the cavity form a narrow gap. Under uniform rotation the interface has an axisymmetric shape. With the modulation growth, the axisymmetric boundary loses stability. Instability manifests itself in the appearance of a regular quasistationary relief at the interphase. The relief excitation is associated with the Kelvin-Helmholtz instability. Tangential velocity discontinuity under modulation arises due to various interaction of liquids with the cavity end walls because of viscosity contrast. The viscosity contrast, on the one hand, is responsible for tangential velocity discontinuity; on the other, it has a significant effect on the threshold of Kelvin-Helmholtz instability. It results in decrease of stability threshold and an increase of relief wavelength.

Introduction of Plastic Block Method in the Upper Bound Limit Analysis of Soil Stability Problems

In this paper, a new method, called the Plastic Block method, was introduced to improve the current rigid block solutions in framework of rigorous limit analysis method. It is a combination of rigid block limit analysis and finite-element limit analysis methods used in the upper bound limit analysis solutions of stability problems in soil mechanics. The method is similar to the first one in terms of speed of solution process and consideration of a collapse mechanism as a priority, and also is similar to the second one in terms of capability of considering plastic deformability of the blocks in calculation of internal dissipation of energy (besides that of velocity discontinuities between them). Comparison of the results obtained by the proposed method, using a few number of blocks (in case of the classical bearing capacity problem of shallow foundations selected to illustrate capability of the method), showed the better (smaller) upper bound not only in comparison with the well-known rigid block solutions, but also in comparison with some cases of finite-element limit analysis solutions obtained by the models including a large number of elements.

Seismic features of the Permian Basin Region from receiver function analysis

We used receiver function (RF) analysis to derive new seismic models of Moho depth, crustal Vp/Vs ratio, depth of sedimentary velocity discontinuity, and corresponding sedimentary Vp/Vs ratio in the Permian Basin region from the seismic waveform data of 260 moment magnitude (Mw) ≥ 6 teleseismic events. We obtained the optimal models by searching one- and two-layer models and comparing similarity between the synthetic and observed RFs. We find the first-order velocity discontinuity in sedimentary basins is shallower than the basin basement. The Delaware Basin (DB) exhibits a bowl-shaped sedimentary velocity discontinuity, with depths ranging from 1.7 km to 5.8 km, whereas the Central Basin Platform (CBP) has a shallow sedimentary velocity discontinuity, with 0.2 km to 2.7 km depths. Moho depth ranges from 36 km to 46 km in the Permian Basin region. The median Vp/Vs ratios of the crust are 1.75, 1.79, and 1.78 in the DB, CBP, and the Diablo Platform (DP), respectively. Our interpretation of the different Vp/Vs ratio in the three subregions is that the crust in the DB includes more low-Vp/Vs-ratio felsic rocks, and the crust in the CBP and DP contains more high-Vp/Vs-ratio gabbroic and basaltic rocks from the Proterozoic and Phanerozoic mafic intrusion.

Double droplet splashing on a thin liquid film with a pseudopotential lattice Boltzmann method

This paper studies the interaction of two droplets splashing on a stationary film. A source term is included in the large-density-ratio pseudopotential lattice Boltzmann method to achieve tuneable surface tension. This model offers excellent numerical accuracy and stability for droplet impacts on liquid films. The influence of the Reynolds number, Weber number, film thickness, and horizontal/vertical distance between the droplets on the crown geometry evolution is investigated. The energy loss during the impact process and the velocity discontinuity in the liquid film are the two key factors affecting the stability and evolution process of the crown. A smaller Reynolds number or thicker liquid film enhances the energy loss and decreases the velocity discontinuity, leading to more stable side and central jets. An increase in the horizontal distance between the droplets reduces the velocity discontinuity, causing the central jet height to decrease. An increase in the Weber number does not affect the energy loss or velocity discontinuity, but the lower surface tension leads to a dramatic deformation in both the central and side jets. A vertical distance between the two droplets causes an asymmetrical evolution of the crown geometry, and postpones the breakup time of the central jet .

A multiclass Lighthill-Whitham-Richards traffic model with a discontinuous velocity function

The well-known Lighthill-Whitham-Richards (LWR) kinematic model of traffic flow models the evolution of the local density of cars by a nonlinear scalar conservation law. The transition between free and congested flow regimes can be described by a flux or velocity function that has a discontinuity at a determined density. A numerical scheme to handle the resulting LWR model with discontinuous velocity was proposed in [J.D. Towers, A splitting algorithm for LWR traffic models with flux discontinuities in the unknown, J. Comput. Phys., 421 (2020), article 109722]. A similar scheme is constructed by decomposing the discontinuous velocity function into a Lipschitz continuous function plus a Heaviside function and designing a corresponding splitting scheme. The part of the scheme related to the discontinuous flux is handled by a semi-implicit step that does, however, not involve the solution of systems of linear or nonlinear equations. It is proved that the whole scheme converges to a weak solution in the scalar case. The scheme can in a straightforward manner be extended to the multiclass LWR (MCLWR) model, which is defined by a hyperbolic system of \begin{document}$ N $\end{document} conservation laws for \begin{document}$ N $\end{document} driver classes that are distinguished by their preferential velocities. It is shown that the multiclass scheme satisfies an invariant region principle, that is, all densities are nonnegative and their sum does not exceed a maximum value. In the scalar and multiclass cases no flux regularization or Riemann solver is involved, and the CFL condition is not more restrictive than for an explicit scheme for the continuous part of the flux. Numerical tests for the scalar and multiclass cases are presented.

A G3 continuous tool path smoothing method for 5-axis CNC machining

The chain of short linear tool path segments is the most common form for the five-axis machining by Computer Numerical Control (CNC) machines. The connection points of the linear path segments lead to the discontinuities in position, velocity, acceleration, and jerk, resulting in fluctuation of machine motion which leaves undesired marks on the finished surface. This paper presents a geometric smoothing algorithm by enforcing the continuity of first, second, and third geometric derivatives of the splined path segments at their meeting points for the five-axis machining of curved surfaces. The linear tool path segments are first smoothed by Bezier splines while constraining the path errors within a set tolerance. The tool path segments, that cannot be smoothed within the path error limit, are smoothly connected at the connection points. The tooltip position and tool axis orientation displacements are synchronized to avoid discontinuous displacements along the tool axis. The proposed algorithms are demonstrated with simulations and experiments in following a five-axis, curved tool path on a CNC machine tool. It is shown that the proposed G3 continuous tool path algorithm gives smoother tool-tip and tool axis orientation, reduces the tracking errors, contour errors and amplitudes of jounce at the connections of segments along the tool path, leading to the smoother surface for the precision five-axis machining of sculptured surfaces.

Lateral Structural Variation of the Lithosphere‐Asthenosphere System in the Northeastern to Eastern Iranian Plateau and Its Tectonic Implications

AbstractThe Iranian plateau is a natural laboratory for deciphering the lithospheric deformation and deep dynamics in response to the Neo‐Tethyan subduction and subsequent Arabia‐Eurasia continental collision. Here we used S‐wave receiver function data from a dense seismic array to construct the structural image of the lithosphere‐asthenosphere system across the northeastern to eastern Iranian plateau. The lithosphere‐asthenosphere boundary (LAB) is consistently imaged, being relatively flat at 80–90 km depth in eastern Iran, but deepening northward to ∼120 km depth beneath the Kopeh Dagh‐Turan platform boundary with an overall “root‐like” shape in northeastern Iran. A shallower (<45 km) and a deeper Moho (∼50–55 km) are also observed beneath eastern and northeastern Iran, respectively, suggesting similar crustal deformation to mantle lithosphere but different patterns laterally. Moreover, a strong positive velocity discontinuity (PVD) is detected at ∼170 km depth beneath eastern Iran, possibly representing the base of an asthenospheric low‐velocity layer (LVL). Waveform modeling reveals large velocity contrasts of ∼4%–6% over a depth range of less than 30 and 20 km, respectively, across the LAB and the asthenospheric PVD. Combining our new results with geological and petrological constraints, we deduced that the asthenospheric LVL contains a small amount of melt, with sharp boundaries separating from the nearly melt‐free lithosphere above (LAB) and deeper mantle below (PVD). The observed structural variations of the lithosphere‐asthenosphere system across the study region may result from the complex lithospheric deformation in response to the Arabia‐Eurasia collision and effects of multiple tectono‐thermal events associated with the earlier Neo‐Tethyan subduction.

Instability of a tangential discontinuity surface in a three-dimensional compressible medium

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.

Upper-Bound Analysis for Equal Channel Angular Pressing (ECAP) with an Intersecting Channel Angle of 120°

Abstract The exact calculations of the stress and strain distributions based on the controlling equations for a forming process with large deformation are often difficult. To circumvent such difficulties, some analytical methods such upper-bound analysis and slip-line field theory have been established by making a number of simplifying assumptions regarding the material properties and deformation modes. In this work an analytical model based on the upper-bound theory was successfully developed to predict material flow pattern and maximum process loads for an Equal Channel Angular Pressing (ECAP) die with circular cross-section and an intersecting channel angle of 120°. Based on the model, the power dissipated on all frictional and velocity discontinuity surfaces were determined and optimized in order predict the maximum process force as function of the channel geometry and the material plastic behavior. To validate the developed model, the ECAP die were produced and used to determine experimental load-displacement curves of AA6061-T6 specimens with different lengths. A good correlation between theoretical and experimental results was observed. In addition, the constant friction factor demonstrated to have a strong effect on the relative extrusion pressure.

DAM BREAK WAVE, TIDAL BORE, IN-RIVER TSUNAMI SURGE: WHAT THE HELL?

Flood waves resulting from dam breaks have been responsible for numerous losses of life through centuries. Both the 26 December 2004 tsunami and 11 March 2011 Tohoku catastrophes were human tragedies of international significance. An important point is the physical analogy between dam break waves travelling downstream, tidal bores progressing deep inland, in-river tsunami propagating upstream, as well rejection surges in hydropower canals. The leading edge is a hydrodynamic shock, with a marked discontinuity in free-surface elevation and velocity and pressure fields, and a tri-phase flow with three distinct flowing phases, i.e. liquid (water), solid (sediment) and gas (air). Seminal features of bores and surges include a net mass flux, the breaking in shallow waters, and the intense turbulence at the front associated with massive sedimentary processes and air entrainment in the breaking roller. In this keynote talk, physical experiments, numerical CFD modelling and field observations are presented and compared. Current knowledge gaps are discussed. Ultimately it is argued that the 'solitary wave' analogy is not directly relevant to model the unsteady turbulent mixing of in-river tsunami surges, tidal bores and dam break waves.Recorded Presentation from the vICCE (YouTube Link): https://youtu.be/SQaPoSj2lP4

Control Volume Isogeometric Analysis for groundwater flow modeling in heterogeneous porous media

Many important groundwater transport applications require solving the Darcy flow in heterogeneous porous media. Flow simulations, especially in large, highly heterogeneous aquifers, require extensive computational resources, a multiresolution (multiscale) approach to resolve the different heterogeneity scales and an accurate calculation of the velocity field. Common methods, such as finite volumes and elements, assume a discontinuous conductivity field introducing velocity discontinuities along the cell or element interfaces due to using classic discrete operators or Lagrangian basis functions. Over the last decade, the development of isogeometric analysis (IGA) eliminates many of the aforementioned limitations bridging the gap between CAD and numerical analysis. Since classic IGA uses the Galerkin and collocation approach, in this paper, we present a third concept in the form of Control Volume IsoGeometric Analysis (CV-IGA) enabling local and global mass conservation as well as multiresolution description of all heterogeneity scales. Due to the approximation properties of spline basis functions, the velocity field and its derivatives are continuous and are obtained by an optimal convergence rate. The CV-IGA methodology is verified with 2-D numerical and stochastic benchmark flow simulations, including comparisons with classic methods and two other IGA formulations as well as the convergence analysis of the head and velocity fields for different orders of Fup and B-spline basis functions.

Modelling transverse solute mixing across a vegetation generated shear layer

ABSTRACT Transverse solute mixing across a vegetation generated horizontal shear layer was quantified using laser induced fluorometry techniques for artificial and real vegetation. A two-dimensional finite difference model (FDM) was developed to describe transverse concentration profiles for flows containing transverse variations in velocity and transverse dispersion, from a steady solute input. The FDM was employed inversely, to optimize the parameters describing the transverse distribution of the transverse dispersion coefficient for vegetation generated shear layers. When laboratory data are available, continuous function descriptions produce slightly improved FDM modelled solute concentration profiles compared with simplified step discontinuity velocity and dispersion inputs. When laboratory data are not available, estimates of step or continuous transverse distributions from other work enable concentration profiles to be predicted with a similar goodness of fit. This paper presents a validated, simple, robust finite difference model to describe the mixing of solutes in a channel containing marginal vegetation.

Hybrid Large Eddy Simulation for low-order Discontinuous Galerkin methods using an explicit filter

In this paper we present a simple, easily implemented and effective approach for explicitly-filtered Large Eddy Simulation with a Discontinuous Galerkin (DG) discretisation for velocity. DG formulations are often desirable due to their stability and increased accuracy, however this can come at greater computational expense due to the additional degrees of freedom in the velocity field. Additionally, data output can also be an issue, due to the increased storage requirements. Here we present a hybrid approach, based upon the construction of an approximation of the velocity shear tensor using information from a projected Continuous Galerkin (CG) version of the discontinuous velocity field. The resulting turbulence algorithm is implemented within Fluidity, an open-source computational fluid dynamics solver. The model is then validated with a well known test case, and shown to agree favourably with published results. Comparisons are also made between the CG/DG hybrid LES with DG-only LES, which demonstrate the superior computational performance of the hybrid model.