scholarly journals Atmospheric Predictability: Revisiting the Inherent Finite-Time Barrier

2019 ◽  
Vol 76 (12) ◽  
pp. 3883-3892 ◽  
Author(s):  
Tsz Yan Leung ◽  
Martin Leutbecher ◽  
Sebastian Reich ◽  
Theodore G. Shepherd
Keyword(s):  
Finite Time ◽  
Lipschitz Continuity ◽  
Initial Conditions ◽  
Analytic Solutions ◽  
Navier Stokes ◽  
Model Resolution ◽  
Spectral Slope ◽  
Potential Conflict ◽  
Apparent Contradiction ◽  
Atmospheric Flows

Abstract The accepted idea that there exists an inherent finite-time barrier in deterministically predicting atmospheric flows originates from Edward N. Lorenz’s 1969 work based on two-dimensional (2D) turbulence. Yet, known analytic results on the 2D Navier–Stokes (N-S) equations suggest that one can skillfully predict the 2D N-S system indefinitely far ahead should the initial-condition error become sufficiently small, thereby presenting a potential conflict with Lorenz’s theory. Aided by numerical simulations, the present work reexamines Lorenz’s model and reviews both sides of the argument, paying particular attention to the roles played by the slope of the kinetic energy spectrum. It is found that when this slope is shallower than −3, the Lipschitz continuity of analytic solutions (with respect to initial conditions) breaks down as the model resolution increases, unless the viscous range of the real system is resolved—which remains practically impossible. This breakdown leads to the inherent finite-time limit. If, on the other hand, the spectral slope is steeper than −3, then the breakdown does not occur. In this way, the apparent contradiction between the analytic results and Lorenz’s theory is reconciled.

Highly accelerated, free-surface flows

1993 ◽  
Vol 248 ◽  
pp. 449-475 ◽  
Author(s):  
Michael S. Longuet-Higgins
Keyword(s):  
Free Surface ◽  
Finite Time ◽  
Initial Conditions ◽  
Analytic Solutions ◽  
Free Surface Flows ◽  
Parametric Form ◽  
Surface Flows ◽  
Pass Through ◽  
Accelerated Flow ◽  
Special Case

Accelerations exceeding 20g in surface waves have been observed both in experiments and in numerically computed flows with a free surface. The present paper describes a family of analytic solutions which display such behaviour. They are expressible in parametric form as z = F sinh ω + iG cosh ω + γω + iH, where F, G and H are functions of the time t only, and γ is linear in t. ω is a complex parameter which is real at the free surface. The functions F(t) and G(t) satisfy two nonlinear, coupled ODEs, which can be solved numerically. Typically the solutions pass through an ‘inertial shock’, or singularity in the time, where the displacements vary as t2/3, the velocities as t1/3 and the accelerations as t-4/3. In this class of solution the free surface develops a cusp as t → ∞. In a special case, F and G vary as t4/7 and the cusp is reached in finite time. Gravity is neglected, but plays a part in setting up the initial conditions for the highly accelerated flow.In future papers it will be shown that more general solutions exist in which the acceleration is momentarily large but bounded.

scholarly journals Effect of an oscillating time-dependent pressure gradient on Dean flow: transient solution

2020 ◽  
Vol 9 (1) ◽  
Author(s):  
Basant K. Jha ◽  
Dauda Gambo
Keyword(s):  
Pressure Gradient ◽  
Initial Conditions ◽  
Fluid Velocity ◽  
Outer Cylinder ◽  
Time Dependent ◽  
Navier Stokes ◽  
Dean Flow ◽  
Continuity Equations ◽  
Riemann Sum ◽  
Flow Formation

Abstract Background Navier-Stokes and continuity equations are utilized to simulate fully developed laminar Dean flow with an oscillating time-dependent pressure gradient. These equations are solved analytically with the appropriate boundary and initial conditions in terms of Laplace domain and inverted to time domain using a numerical inversion technique known as Riemann-Sum Approximation (RSA). The flow is assumed to be triggered by the applied circumferential pressure gradient (azimuthal pressure gradient) and the oscillating time-dependent pressure gradient. The influence of the various flow parameters on the flow formation are depicted graphically. Comparisons with previously established result has been made as a limit case when the frequency of the oscillation is taken as 0 (ω = 0). Results It was revealed that maintaining the frequency of oscillation, the velocity and skin frictions can be made increasing functions of time. An increasing frequency of the oscillating time-dependent pressure gradient and relatively a small amount of time is desirable for a decreasing velocity and skin frictions. The fluid vorticity decreases with further distance towards the outer cylinder as time passes. Conclusion Findings confirm that increasing the frequency of oscillation weakens the fluid velocity and the drag on both walls of the cylinders.

Computational Characterization of Passive Fluid Mixing in Microfluidics

Volume 3 ◽  
2004 ◽  
Author(s):  
Erik D. Svensson
Keyword(s):  
Stokes Equations ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Fluid Mixing ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Fluid Systems ◽  
Sensitive Dependence ◽  
Microfluidic Mixers

In this work we computationally characterize fluid mixing in a number of passive microfluidic mixers. Generally, in order to systematically study and characterize mixing in realistic fluid systems we (1) compute the fluid flow in the systems by solving the stationary three-dimensional Navier-Stokes equations or Stokes equations with a finite element method, and (2) compute various measures indicating the degree of mixing based on concepts from dynamical systems theory, i.e., the sensitive dependence on initial conditions and mixing variance.

scholarly journals PARALLEL COMPUTATIONS IN STUDIES OF DEPENDENCE OF GAS DYNAMIC PARAMETERS OF UPWARD SWIRLING FLOW OF GAS ON BLOWING VELOCITY

2016 ◽  
pp. 92-97
Author(s):  
R. E. Volkov ◽  
A. G. Obukhov
Keyword(s):  
Stokes Equations ◽  
Swirling Flow ◽  
Initial Conditions ◽  
Exact Analytical Solution ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Heat Conducting ◽  
Computational Domain ◽  
Navier Stokes Equations ◽  
Gas Dynamic

The rectangular parallelepiped explicit difference schemes for the numerical solution of the complete built system of Navier-Stokes equations. These solutions describe the three-dimensional flow of a compressible viscous heat-conducting gas in a rising swirling flows, provided the forces of gravity and Coriolis. This assumes constancy of the coefficient of viscosity and thermal conductivity. The initial conditions are the features that are the exact analytical solution of the complete Navier-Stokes equations. Propose specific boundary conditions under which the upward flow of gas is modeled by blowing through the square hole in the upper surface of the computational domain. A variant of parallelization algorithm for calculating gas dynamic and energy characteristics. The results of calculations of gasdynamic parameters dependency on the speed of the vertical blowing by the time the flow of a steady state flow.

Two-dimensional flow of a viscous fluid in a channel with porous walls

1991 ◽  
Vol 227 ◽  
pp. 1-33 ◽  
Author(s):  
Stephen M. Cox
Keyword(s):  
Finite Time ◽  
Stokes Equations ◽  
Pitchfork Bifurcation ◽  
Time Dependent ◽  
High Rate ◽  
Navier Stokes ◽  
Recent Theory ◽  
Navier Stokes Equations ◽  
Two Dimensional Flow ◽  
Steady Solutions

We consider the flow of a viscous incompressible fluid in a parallel-walled channel, driven by steady uniform suction through the porous channel walls. A similarity transformation reduces the Navier-Stokes equations to a single partial differential equation (PDE) for the stream function, with two-point boundary conditions. We discuss the bifurcations of the steady solutions first, and show how a pitchfork bifurcation is unfolded when a symmetry of the problem is broken.Then we describe time-dependent solutions of the governing PDE, which we calculate numerically. We analyse these unsteady solutions when there is a high rate of suction through one wall, and the other wall is impermeable: there is a limit cycle composed of an explosive phase of inviscid growth, and a slow viscous decay. The inviscid phase ‘almost’ has a finite-time singularity. We discuss whether solutions of the governing PDE, which are exact solutions of the Navier-Stokes equations, may develop mathematical singularities in a finite time.When the rates of suction at the two walls are equal so that the problem is symmetrical, there is an abrupt transition to chaos, a ‘homoclinic explosion’, in the time-dependent solutions as the Reynolds number is increased. We unfold this transition by perturbing the symmetry, and compare direct numerical integrations of the governing PDE with a recent theory for ‘Lorenz-like’ dynamical systems. The chaos is found to be very sensitive to symmetry breaking.

Stochastic storage networks: stationarity and the feedforward case

1997 ◽  
Vol 34 (02) ◽  
pp. 498-507 ◽  
Author(s):  
Offer Kella
Keyword(s):  
Total Variation ◽  
Finite Time ◽  
Initial Conditions ◽  
Stability Conditions ◽  
Input Process ◽  
Stationary Increments ◽  
Stationary Version ◽  
Storage Network ◽  
The Difference ◽  
Storage Networks

We show that for a certain storage network the backward content process is increasing, and when the net input process has stationary increments then, under natural stability conditions, the content process has a stationary version under which the cumulative lost capacities have stationary increments. Moreover, for the feedforward case, we show that under some minimal conditions, two content processes with net input processes which differ only by initial conditions can be coupled in finite time and that the difference of two content processes vanishes in the limit if the difference of the net input processes monotonically approaches a constant. As a consequence, it is shown that for the natural stability conditions, when the net input process has stationary increments, the distribution of the content process converges in total variation to a proper limit, independent of initial conditions.

scholarly journals Physiologic blood flow is turbulent

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Khalid M. Saqr ◽  
Simon Tupin ◽  
Sherif Rashad ◽  
Toshiki Endo ◽  
Kuniyasu Niizuma ◽  
...  
Keyword(s):  
Blood Flow ◽  
Stokes Equation ◽  
Initial Conditions ◽  
Hydrodynamic Instability ◽  
Transition To Turbulence ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Frequency Spectra ◽  
Hydrodynamic Stability Theory

Abstract Contemporary paradigm of peripheral and intracranial vascular hemodynamics considers physiologic blood flow to be laminar. Transition to turbulence is considered as a driving factor for numerous diseases such as atherosclerosis, stenosis and aneurysm. Recently, turbulent flow patterns were detected in intracranial aneurysm at Reynolds number below 400 both in vitro and in silico. Blood flow is multiharmonic with considerable frequency spectra and its transition to turbulence cannot be characterized by the current transition theory of monoharmonic pulsatile flow. Thus, we decided to explore the origins of such long-standing assumption of physiologic blood flow laminarity. Here, we hypothesize that the inherited dynamics of blood flow in main arteries dictate the existence of turbulence in physiologic conditions. To illustrate our hypothesis, we have used methods and tools from chaos theory, hydrodynamic stability theory and fluid dynamics to explore the existence of turbulence in physiologic blood flow. Our investigation shows that blood flow, both as described by the Navier–Stokes equation and in vivo, exhibits three major characteristics of turbulence. Womersley’s exact solution of the Navier–Stokes equation has been used with the flow waveforms from HaeMod database, to offer reproducible evidence for our findings, as well as evidence from Doppler ultrasound measurements from healthy volunteers who are some of the authors. We evidently show that physiologic blood flow is: (1) sensitive to initial conditions, (2) in global hydrodynamic instability and (3) undergoes kinetic energy cascade of non-Kolmogorov type. We propose a novel modification of the theory of vascular hemodynamics that calls for rethinking the hemodynamic–biologic links that govern physiologic and pathologic processes.

Melting and Resolidification of a Substrate Caused by Molten Microdroplet Impact

2001 ◽  
Vol 123 (6) ◽  
pp. 1110-1122 ◽  
Author(s):  
D. Attinger ◽  
D. Poulikakos
Keyword(s):  
Heat Transfer ◽  
Initial Conditions ◽  
Navier Stokes ◽  
Element Formulation ◽  
Material Transport ◽  
Melting Depth ◽  
Flat Substrate ◽  
Substrate Melting ◽  
Lagrangian Finite Element ◽  
Good Agreement

This paper describes the main features and results of a numerical investigation of molten microdroplet impact and solidification on a colder flat substrate of the same material that melts due to the energy input from the impacting molten material. The numerical model is based on the axisymmetric Lagrangian Finite-Element formulation of the Navier–Stokes, energy and material transport equations. The model accounts for a host of complex thermofluidic phenomena, exemplified by surface tension effects and heat transfer with solidification in a severely deforming domain. The dependence of the molten volume on time is determined and discussed. The influence of the thermal and hydrodynamic initial conditions on the amount of substrate melting is discussed for a range of superheat, Biot number, and impact velocity. Multidimensional and convective heat transfer effects, as well as material mixing between the droplet and the substrate are found and quantified and the underlying physics is discussed. Good agreement in the main features of the maximum melting depth boundary between the present numerical results and published experiments of other investigators for larger (mm-size) droplets was obtained, and a complex mechanism was identified, showing the influence of the droplet fluid dynamics on the substrate melting and re-solidification.

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