Nonlinear effects in two-layer large-amplitude geostrophic dynamics. Part 1. The strong-beta case

2000 ◽  
Vol 412 ◽  
pp. 125-160 ◽  
Author(s):  
RICHARD H. KARSTEN ◽  
GORDON E. SWATERS
Keyword(s):  
Large Amplitude ◽  
Nonlinear Effects ◽  
Breaking Waves ◽  
Short Wave ◽  
Finite Amplitude ◽  
Length Scale ◽  
Leading Order ◽  
Length Scales ◽  
Geostrophic Balance ◽  
Suitable Framework

Baroclinic large-amplitude geostrophic (LAG) models, which assume a leading-order geostrophic balance but allow for large-amplitude isopycnal deflections, provide a suitable framework to model the large-amplitude motions exhibited in frontal regions. The qualitative dynamical characterization of LAG models depends critically on the underlying length scale. If the length scale is sufficiently large, the effect of differential rotation, i.e. the β-effect, enters the dynamics at leading order. For smaller length scales, the β-effect, while non-negligible, does not enter the dynamics at leading order. These two dynamical limits are referred to as strong-β and weak-β models, respectively.A comprehensive description of the nonlinear dynamics associated with the strong- β models is given. In addition to establishing two new nonlinear stability theorems, we extend previous linear stability analyses to account for the finite-amplitude development of perturbed fronts. We determine whether the linear solutions are subject to nonlinear secondary instabilities and, in particular, a new long-wave–short-wave (LWSW) resonance, which is a possible source of rapid unstable growth at long length scales, is identified. The theoretical analyses are tested against numerical simulations. The simulations confirm the importance of the LWSW resonance in the development of the flow. Simulations show that instabilities associated with vanishing potential- vorticity gradients can develop into stable meanders, eddies or breaking waves. By examining models with different layer depths, we reveal how the dynamics associated with strong-β models qualitatively changes as the strength of the dynamic coupling between the barotropic and baroclinic motions varies.

Nonlinear effects in two-layer large-amplitude geostrophic dynamics. Part 2. The weak-beta case

2000 ◽  
Vol 412 ◽  
pp. 161-196 ◽  
Author(s):  
RICHARD H. KARSTEN ◽  
GORDON E. SWATERS
Keyword(s):  
Numerical Simulations ◽  
Nonlinear Stability ◽  
Large Amplitude ◽  
Baroclinic Instability ◽  
Finite Amplitude ◽  
Linear Instability ◽  
Steady Flows ◽  
Leading Order ◽  
Unstable Flow ◽  
Amplitude Analysis

This paper is a continuation of our study on nonlinear processes in large-amplitude geostrophic (LAG) dynamics. Here, we examine the so-called weak-β models. These models arise when the intrinsic length scale is large enough so that the dynamics is geostrophic to leading order but not so large that the β-effect enters into the dynamics at leading order (but remains, nevertheless, dynamically non-negligible). In contrast to our previous analysis of strong-β LAG models in Part 1, we show that the weak-β models allow for vigorous linear baroclinic instability.For two-layer weak-β LAG models in which the mean depths of both layers are approximately equal, the linear instability problem can exhibit an ultraviolet catastrophe. We argue that it is not possible to establish conditions for the nonlinear stability in the sense of Liapunov for a steady flow. We also show that the finite-amplitude evolution of a marginally unstable flow possesses explosively unstable modes, i.e. modes for which the amplitude becomes unbounded in finite time. Numerical simulations suggest that the development of large-amplitude meanders, squirts and eddies is correlated with the presence of these explosively unstable modes.For two-layer weak-β LAG models in which one of the two layers is substantially thinner than the other, the linear stability problem does not exhibit an ultraviolet catastrophe and it is possible to establish conditions for the nonlinear stability in the sense of Liapunov for steady flows. A finite-amplitude analysis for a marginally unstable flow suggests that nonlinearities act to stabilize eastward and enhance the instability of westward flows. Numerical simulations are presented to illustrate these processes.

scholarly journals Subject-specific multiscale modeling of aortic valve biomechanics

2021 ◽  
Author(s):  
G. Rossini ◽  
A. Caimi ◽  
A. Redaelli ◽  
E. Votta
Keyword(s):  
Aortic Valve ◽  
Boundary Conditions ◽  
Narrow Range ◽  
Radial Strain ◽  
Length Scale ◽  
Length Scales ◽  
Anatomical Features ◽  
Wide Range ◽  
Subject Specific ◽  
Cell Strains

AbstractA Finite Element workflow for the multiscale analysis of the aortic valve biomechanics was developed and applied to three physiological anatomies with the aim of describing the aortic valve interstitial cells biomechanical milieu in physiological conditions, capturing the effect of subject-specific and leaflet-specific anatomical features from the organ down to the cell scale. A mixed approach was used to transfer organ-scale information down to the cell-scale. Displacement data from the organ model were used to impose kinematic boundary conditions to the tissue model, while stress data from the latter were used to impose loading boundary conditions to the cell level. Peak of radial leaflet strains was correlated with leaflet extent variability at the organ scale, while circumferential leaflet strains varied over a narrow range of values regardless of leaflet extent. The dependency of leaflet biomechanics on the leaflet-specific anatomy observed at the organ length-scale is reflected, and to some extent emphasized, into the results obtained at the lower length-scales. At the tissue length-scale, the peak diastolic circumferential and radial stresses computed in the fibrosa correlated with the leaflet surface area. At the cell length-scale, the difference between the strains in two main directions, and between the respective relationships with the specific leaflet anatomy, was even more evident; cell strains in the radial direction varied over a relatively wide range ($$0.36-0.87$$ 0.36 - 0.87 ) with a strong correlation with the organ length-scale radial strain ($$R^{2}= 0.95$$ R 2 = 0.95 ); conversely, circumferential cell strains spanned a very narrow range ($$0.75-0.88$$ 0.75 - 0.88 ) showing no correlation with the circumferential strain at the organ level ($$R^{2}= 0.02$$ R 2 = 0.02 ). Within the proposed simulation framework, being able to account for the actual anatomical features of the aortic valve leaflets allowed to gain insight into their effect on the structural mechanics of the leaflets at all length-scales, down to the cell scale.

Nonlinear Sloshing in Fixed and Vertically Excited Containers

2002 ◽  
Author(s):  
Jannette B. Frandsen ◽  
Alistair G. L. Borthwick
Keyword(s):  
Perturbation Theory ◽  
Free Surface ◽  
Small Perturbation ◽  
Numerical Solutions ◽  
Vertical Direction ◽  
Nonlinear Effects ◽  
Breaking Waves ◽  
Surface Flow ◽  
Nonlinear Behaviour ◽  
Computational Domain

Nonlinear effects of standing wave motions in fixed and vertically excited tanks are numerically investigated. The present fully nonlinear model analyses two-dimensional waves in stable and unstable regions of the free-surface flow. Numerical solutions of the governing nonlinear potential flow equations are obtained using a finite-difference time-stepping scheme on adaptively mapped grids. A σ-transformation in the vertical direction that stretches directly between the free-surface and bed boundary is applied to map the moving free surface physical domain onto a fixed computational domain. A horizontal linear mapping is also applied, so that the resulting computational domain is rectangular, and consists of unit square cells. The small-amplitude free-surface predictions in the fixed and vertically excited tanks compare well with 2nd order small perturbation theory. For stable steep waves in the vertically excited tank, the free-surface exhibits nonlinear behaviour. Parametric resonance is evident in the instability zones, as the amplitudes grow exponentially, even for small forcing amplitudes. For steep initial amplitudes the predictions differ considerably from the small perturbation theory solution, demonstrating the importance of nonlinear effects. The present numerical model provides a simple way of simulating steep non-breaking waves. It is computationally quick and accurate, and there is no need for free surface smoothing because of the σ-transformation.

Solitary Filaments of Coupled Electromagnetic Wave and Finite Amplitude Ion Fluctuations

1976 ◽  
Vol 31 (12) ◽  
pp. 1517-1519 ◽  
Author(s):  
P. K. Shukla ◽  
M. Y. Yu ◽  
S. G. Tagare
Keyword(s):  
Electromagnetic Wave ◽  
Electromagnetic Radiation ◽  
Plasma Density ◽  
Large Amplitude ◽  
Small Amplitude ◽  
Finite Amplitude ◽  
Nonlinear Coupling ◽  
Incoming Radiation

Abstract We show analytically that the nonlinear coupling of a large amplitude electromagnetic wave with finite amplitude ion fluctuations leads to filamentation. The latter consists of striations of the electromagnetic radiation trapped in depressions of the plasma density. The filamentation is found to be either standing or moving normal to the direction of the incoming radiation. Criteria for the existence of localized filaments are obtained. Small amplitude results are discussed.

scholarly journals High-Fidelity Simulations of a Linear HPT Vane Cascade Subject to Varying Inlet Turbulence

2017 ◽  
Author(s):  
Richard Pichler ◽  
Richard D. Sandberg ◽  
Gregory Laskowski ◽  
Vittorio Michelassi
Keyword(s):  
Heat Flux ◽  
Turbulence Intensity ◽  
Side Surface ◽  
Suction Side ◽  
Transition Process ◽  
Length Scale ◽  
Length Scales ◽  
Inflow Turbulence ◽  
High Turbulence ◽  
Turbulence Length

The effect of inflow turbulence intensity and turbulence length scales have been studied for a linear high-pressure turbine vane cascade at Reis = 590,000 and Mis = 0.93, using highly resolved compressible large-eddy simulations employing the WALE turbulence model. The turbulence intensity was varied between 6% and 20% while values of the turbulence length scales were prescribed between 5% and 20% of axial chord. The analysis focused on characterizing the inlet turbulence and quantifying the effect of the inlet turbulence variations on the vane boundary layers, in particular on the heat flux to the blade. The transition location on the suction side of the vane was found to be highly sensitive to both turbulence intensity and length scale, with the case with turbulence intensity 20% and 20% length scale showing by far the earliest onset of transition and much higher levels of heat flux over the entire vane. It was also found that the transition process was highly intermittent and local, with spanwise parts of the suction side surface of the vane remaining laminar all the way to the trailing edge even for high turbulence intensity cases.

A Parametric Numerical Study of the Effects of Freestream Turbulence Intensity and Length Scale on Anti-Vortex Film Cooling Design at High Blowing Ratio

2013 ◽  
Author(s):  
Timothy W. Repko ◽  
Andrew C. Nix ◽  
James D. Heidmann
Keyword(s):  
Turbulence Intensity ◽  
Film Cooling ◽  
Numerical Study ◽  
Length Scale ◽  
Freestream Turbulence ◽  
Cooling Effectiveness ◽  
Length Scales ◽  
Film Cooling Effectiveness ◽  
Blowing Ratio ◽  
Cooling Design

An advanced, high-effectiveness film-cooling design, the anti-vortex hole (AVH) has been investigated by several research groups and shown to mitigate or counter the vorticity generated by conventional holes and increase film effectiveness at high blowing ratios and low freestream turbulence levels. [1, 2] The effects of increased turbulence on the AVH geometry were previously investigated and presented by researchers at West Virginia University (WVU), in collaboration with NASA, in a preliminary CFD study [3] on the film effectiveness and net heat flux reduction (NHFR) at high blowing ratio and elevated freestream turbulence levels for the adjacent AVH. The current paper presents the results of an extended numerical parametric study, which attempts to separate the effects of turbulence intensity and length-scale on film cooling effectiveness of the AVH. In the extended study, higher freestream turbulence intensity and larger scale cases were investigated with turbulence intensities of 5, 10 and 20% and length scales based on cooling hole diameter of Λx/dm = 1, 3 and 6. Increasing turbulence intensity was shown to increase the centerline, span-averaged and area-averaged adiabatic film cooling effectiveness. Larger turbulent length scales were shown to have little to no effect on the centerline, span-averaged and area-averaged adiabatic film-cooling effectiveness at lower turbulence levels, but slightly increased effect at the highest turbulence levels investigated.

scholarly journals Fracture Mechanics of Human Blood Clots: Measurements of Toughness and Critical Length scales

2021 ◽  
Author(s):  
Shiyu Liu ◽  
Guangyu Bao ◽  
Zhenwei Ma ◽  
Christian Kastrup ◽  
Jianyu Li
Keyword(s):  
Fracture Mechanics ◽  
Whole Blood ◽  
Significant Contribution ◽  
Critical Length ◽  
Length Scale ◽  
Length Scales ◽  
Platelet Poor Plasma ◽  
Lap Shear ◽  
Blood Clots ◽  
Nonlinear Elastic

Blood coagulates to plug vascular damage and stop bleeding, and thus the function of blood clots in hemostasis depends on their resistance against rupture (toughness). Despite the significance, fracture mechanics of blood clots remains largely unexplored, particularly the measurements of toughness and critical length scales governing clot fracture. Here, we study the fracture behavior of human whole blood clots and platelet-poor plasma clots. The fracture energy of whole blood clots and platelet-poor plasma clots determined using modified lap-shear method is 5.90 +- 1.18 J/m2 and 0.96 +- 0.90 J/m2, respectively. We find that the measured toughness is independent of the specimen geometry and loading conditions. These results reveal a significant contribution of blood cells to the clot fracture, as well as the dissipative length scale and nonlinear elastic length scale governing clot fracture.

scholarly journals Large amplitude, short wave peristalsis and its implications for transport

2015 ◽  
Author(s):  
Lindsay D Waldrop ◽  
Laura A. Miller
Keyword(s):  
Fluid Flow ◽  
Large Amplitude ◽  
Compression Wave ◽  
Wave Speed ◽  
Flow Direction ◽  
Short Wave ◽  
Flow Speed ◽  
Biological Pumps ◽  
Flow Speeds ◽  
Flow Compression

Valveless, tubular pumps are widespread in the animal kingdom, but the mechanism by which these pumps generate fluid flow are often in dispute. Where the pumping mechanism of many organs was once described as peristalsis, other mechanisms, such as dynamic suction pumping, have been suggested as possible alternative mechanisms. Peristalsis is often evaluated using criteria established in a technical definition for mechanical pumps, but this definition is based on a small-amplitude, long-wave approximation which biological pumps often violate. In this study, we use a direct numerical simulation of large-amplitude, short-wave peristalsis to investigate the relationships between fluid flow, compression frequency, compression wave speed, and tube occlusion. We also explore how the flows produced differ from the criteria outlined in the technical definition of peristalsis. We find that many of the technical criteria are violated by our model: fluid flow speeds produced by peristalsis are greater than the speeds of the compression wave; fluid flow is pulsatile; and flow speed have a non-linear relationship with compression frequency when compression wave speed is held constant. We suggest that the technical definition is inappropriate for evaluating peristalsis as a pumping mechanism for biological pumps because they too frequently violate the assumptions inherent in these criteria. Instead, we recommend that a simpler, more inclusive definition be used for assessing peristalsis as a pumping mechanism based on the presence of non-stationary compression sites that propagate uni-directionally along a tube without the need for a structurally fixed flow direction.

Finite-amplitude internal wavepacket dispersion and breaking

2001 ◽  
Vol 429 ◽  
pp. 343-380 ◽  
Author(s):  
BRUCE R. SUTHERLAND
Keyword(s):  
Numerical Simulations ◽  
Internal Waves ◽  
Large Amplitude ◽  
Finite Amplitude ◽  
Critical Amplitude ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
The Stability ◽  
The Waves ◽  
Horizontal Wavelength

The evolution and stability of two-dimensional, large-amplitude, non-hydrostatic internal wavepackets are examined analytically and by numerical simulations. The weakly nonlinear dispersion relation for horizontally periodic, vertically compact internal waves is derived and the results are applied to assess the stability of weakly nonlinear wavepackets to vertical modulations. In terms of Θ, the angle that lines of constant phase make with the vertical, the wavepackets are predicted to be unstable if [mid ]Θ[mid ] < Θc, where Θc = cos−1 (2/3)1/2 ≃ 35.3° is the angle corresponding to internal waves with the fastest vertical group velocity. Fully nonlinear numerical simulations of finite-amplitude wavepackets confirm this prediction: the amplitude of wavepackets with [mid ]Θ[mid ] > Θc decreases over time; the amplitude of wavepackets with [mid ]Θ[mid ] < Θc increases initially, but then decreases as the wavepacket subdivides into a wave train, following the well-known Fermi–Pasta–Ulam recurrence phenomenon.If the initial wavepacket is of sufficiently large amplitude, it becomes unstable in the sense that eventually it convectively overturns. Two new analytic conditions for the stability of quasi-plane large-amplitude internal waves are proposed. These are qualitatively and quantitatively different from the parametric instability of plane periodic internal waves. The ‘breaking condition’ requires not only that the wave is statically unstable but that the convective instability growth rate is greater than the frequency of the waves. The critical amplitude for breaking to occur is found to be ACV = cot Θ (1 + cos2 Θ)/2π, where ACV is the ratio of the maximum vertical displacement of the wave to its horizontal wavelength. A second instability condition proposes that a statically stable wavepacket may evolve so that it becomes convectively unstable due to resonant interactions between the waves and the wave-induced mean flow. This hypothesis is based on the assumption that the resonant long wave–short wave interaction, which Grimshaw (1977) has shown amplifies the waves linearly in time, continues to amplify the waves in the fully nonlinear regime. Using linear theory estimates, the critical amplitude for instability is ASA = sin 2Θ/(8π2)1/2. The results of numerical simulations of horizontally periodic, vertically compact wavepackets show excellent agreement with this latter stability condition. However, for wavepackets with horizontal extent comparable with the horizontal wavelength, the wavepacket is found to be stable at larger amplitudes than predicted if Θ [lsim ] 45°. It is proposed that these results may explain why internal waves generated by turbulence in laboratory experiments are often observed to be excited within a narrow frequency band corresponding to Θ less than approximately 45°.

Finite-amplitude convection in the presence of an unsteady shear flow

1995 ◽  
Vol 287 ◽  
pp. 225-249 ◽  
Author(s):  
Philip Hall
Keyword(s):  
Shear Flow ◽  
Critical Size ◽  
Low Frequency ◽  
Nonlinear Effects ◽  
Finite Amplitude ◽  
Present Calculation ◽  
Wide Range ◽  
Prandtl Numbers ◽  
Boussinesq Fluid

The effect of an unsteady shear flow on the planform of convection in a Boussinesq fluid heated from below is investigated. In the absence of the shear flow it is well-known, if non-Boussinesq effects can be neglected, that convection begins in the form of a supercritical bifurcation to rolls. Subcritical convection in the form of say hexagons can be induced by non-Boussinesq behaviour which destroys the symmetry of the basic state. Here it is found that the symmetry breaking effects associated with an unsteady shear flow are not sufficient to cause subcritical convection so the problem reduces to the determination of how the orientations of roll cells are modified by an unsteady shear flow. Recently Kelly & Hu (1993) showed that such a flow has a significant stabilizing effect on the linear stability problem and that, for a wide range of Prandtl numbers, the effect is most pronounced in the low-frequency limit. In the present calculation it is shown that the stabilizing effects found by Kelly & Hu (1993) do survive for most frequencies when nonlinear effects and imperfections are taken into account. However a critical size of the frequency is identified below which the Kelly & Hu (1993) conclusions no longer carry through into the nonlinear regime. For frequencies of size comparable with this critical size it is shown that the convection pattern changes in time. The cell pattern is found to be extremely complicated and straight rolls exist only for part of a period.

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