scholarly journals Incompressible Navier-Stokes-Fourier limit from the Landau equation

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Mohamad Rachid
Keyword(s):  
Landau Equation ◽  
Navier Stokes ◽  
Fourier Limit

STATIONARY SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH MEMORY AND STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

2005 ◽  
Vol 07 (05) ◽  
pp. 553-582 ◽  
Author(s):  
YURI BAKHTIN ◽  
JONATHAN C. MATTINGLY
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Stationary Solution ◽  
Landau Equation ◽  
Stationary Solutions ◽  
Navier Stokes ◽  
Partial Differential ◽  
Navier Stokes Equation ◽  
With Memory

We explore Itô stochastic differential equations where the drift term possibly depends on the infinite past. Assuming the existence of a Lyapunov function, we prove the existence of a stationary solution assuming only minimal continuity of the coefficients. Uniqueness of the stationary solution is proven if the dependence on the past decays sufficiently fast. The results of this paper are then applied to stochastically forced dissipative partial differential equations such as the stochastic Navier–Stokes equation and stochastic Ginsburg–Landau equation.

Wave-Packet Models for Jet Dynamics and Sound Radiation

2019 ◽  
Vol 71 (2) ◽  
Author(s):  
André V. G. Cavalieri ◽  
Peter Jordan ◽  
Lutz Lesshafft
Keyword(s):  
Sound Source ◽  
Stochastic Systems ◽  
Landau Equation ◽  
Pressure Fluctuations ◽  
Turbulent Jets ◽  
Navier Stokes ◽  
Acoustic Analogy ◽  
Ginzburg Landau ◽  
Flow Fluctuations ◽  
Radiation Properties

Organized structures in turbulent jets can be modeled as wavepackets. These are characterized by spatial amplification and decay, both of which are related to stability mechanisms, and they are coherent over several jet diameters, thereby constituting a noncompact acoustic source that produces a distinctive directivity in the acoustic field. In this review, we use simplified model problems to discuss the salient features of turbulent-jet wavepackets and their modeling frameworks. Two classes of model are considered. The first, that we refer to as kinematic, is based on Lighthill's acoustic analogy, and allows an evaluation of the radiation properties of sound-source functions postulated following observation of jets. The second, referred to as dynamic, is based on the linearized, inhomogeneous Ginzburg–Landau equation, which we use as a surrogate for the linearized, inhomogeneous Navier–Stokes system. Both models are elaborated in the framework of resolvent analysis, which allows the dynamics to be viewed in terms of an input–ouput system, the input being either sound-source or nonlinear forcing term, and the output, correspondingly, either farfield acoustic pressure fluctuations or nearfield flow fluctuations. Emphasis is placed on the extension of resolvent analysis to stochastic systems, which allows for the treatment of wavepacket jitter, a feature known to be relevant for subsonic jet noise. Despite the simplicity of the models, they are found to qualitatively reproduce many of the features of turbulent jets observed in experiment and simulation. Sample scripts are provided and allow calculation of most of the presented results.

Computing Optimal Forcing Using Laplace Preconditioning

2017 ◽  
Vol 22 (5) ◽  
pp. 1508-1532 ◽  
Author(s):  
M. Brynjell-Rahkola ◽  
L. S. Tuckerman ◽  
P. Schlatter ◽  
D. S. Henningson
Keyword(s):  
Large Scale ◽  
Stokes Equations ◽  
Landau Equation ◽  
Normal Operator ◽  
Navier Stokes ◽  
System Response ◽  
Dominant Eigenvalue ◽  
Navier Stokes Equations ◽  
Driven Cavity ◽  
Forcing Function

AbstractFor problems governed by a non-normal operator, the leading eigenvalue of the operator is of limited interest and a more relevant measure of the stability is obtained by considering the harmonic forcing causing the largest system response. Various methods for determining this so-called optimal forcing exist, but they all suffer from great computational expense and are hence not practical for large-scale problems. In the present paper a new method is presented, which is applicable to problems of arbitrary size. The method does not rely on timestepping, but on the solution of linear systems, in which the inverse Laplacian acts as a preconditioner. By formulating the search for the optimal forcing as an eigenvalue problem based on the resolvent operator, repeated system solves amount to power iterations, in which the dominant eigenvalue is seen to correspond to the energy amplification in a system for a given frequency, and the eigenfunction to the corresponding forcing function. Implementation of the method requires only minor modifications of an existing timestepping code, and is applicable to any partial differential equation containing the Laplacian, such as the Navier-Stokes equations. We discuss the method, first, in the context of the linear Ginzburg-Landau equation and then, the two-dimensional lid-driven cavity flow governed by the Navier-Stokes equations. Most importantly, we demonstrate that for the lid-driven cavity, the optimal forcing can be computed using a factor of up to 500 times fewer operator evaluations than the standard method based on exponential timestepping.

scholarly journals Koopman mode expansions between simple invariant solutions

2019 ◽  
Vol 879 ◽  
pp. 1-27 ◽  
Author(s):  
Jacob Page ◽  
Rich R. Kerswell
Keyword(s):  
State Space ◽  
Stokes Equations ◽  
Time Window ◽  
Landau Equation ◽  
Dynamic Mode Decomposition ◽  
Navier Stokes ◽  
Dynamic Mode ◽  
Invariant Solutions ◽  
Mode Decomposition ◽  
Heteroclinic Connection

A Koopman decomposition is a powerful method of analysis for fluid flows leading to an apparently linear description of nonlinear dynamics in which the flow is expressed as a superposition of fixed spatial structures with exponential time dependence. Attempting a Koopman decomposition is simple in practice due to a connection with dynamic mode decomposition (DMD). However, there are non-trivial requirements for the Koopman decomposition and DMD to overlap, which mean it is often difficult to establish whether the latter is truly approximating the former. Here, we focus on nonlinear systems containing multiple simple invariant solutions where it is unclear how to construct a consistent Koopman decomposition, or how DMD might be applied to locate these solutions. First, we derive a Koopman decomposition for a heteroclinic connection in a Stuart–Landau equation revealing two possible expansions. The expansions are centred about the two fixed points of the equation and extend beyond their linear subspaces before breaking down at a cross-over point in state space. Well-designed DMD can extract the two expansions provided that the time window does not contain this cross-over point. We then apply DMD to the Navier–Stokes equations near to a heteroclinic connection in low Reynolds number ($Re=O(100)$) plane Couette flow where there are multiple simple invariant solutions beyond the constant shear basic state. This reveals as many different Koopman decompositions as simple invariant solutions present and once more indicates the existence of cross-over points between the expansions in state space. Again, DMD can extract these expansions only if it does not include a cross-over point. Our results suggest that in a dynamical system possessing multiple simple invariant solutions, there are generically places in phase space – plausibly hypersurfaces delineating the boundary of a local Koopman expansion – across which the dynamics cannot be represented by a convergent Koopman expansion.

Dynamical system approach to instability of flow past a circular cylinder

2010 ◽  
Vol 656 ◽  
pp. 82-115 ◽  
Author(s):  
TAPAN K. SENGUPTA ◽  
NEELU SINGH ◽  
V. K. SUMAN
Keyword(s):  
Dynamical System ◽  
Differential Equations ◽  
Numerical Simulations ◽  
Landau Equation ◽  
Navier Stokes ◽  
Modal Interactions ◽  
Navier Stokes Equation ◽  
Flow Past ◽  
Instability Modes ◽  
Compound Matrix Method

The main aim of this paper is to relate instability modes with modes obtained from proper orthogonal decomposition (POD) in the study of global spatio-temporal nonlinear instabilities for flow past a cylinder. This is a new development in studying nonlinear instabilities rather than spatial and/or temporal linearized analysis. We highlight the importance of multi-modal interactions among instability modes using dynamical system and bifurcation theory approaches. These have been made possible because of accurate numerical simulations. In validating computations with unexplained past experimental results, we noted that (i) the primary instability depends upon background disturbances and (ii) the equilibrium amplitude obtained after the nonlinear saturation of primary growth of disturbances does not exhibit parabolic variation with Reynolds number, as predicted by the classical Stuart–Landau equation. These are due to the receptivity of the flow to background disturbances for post-critical Reynolds numbers (Re) and multi-modal interactions, those produce variation in equilibrium amplitude for the disturbances that can be identified as multiple Hopf bifurcations. Here, we concentrate on Re = 60, which is close to the observed second bifurcation. It is also shown that the classical Stuart–Landau equation is not adequate, as it does not incorporate multi-modal interactions. To circumvent this, we have used the eigenfunction expansion approach due to Eckhaus and the resultant differential equations for the complex amplitudes of disturbance field have been called here the Landau–Stuart–Eckhaus (LSE) equations. This approach has not been attempted before and here it is made possible by POD of time-accurate numerical simulations. Here, various modes have been classified either as a regular mode or as anomalous modes of the first or the second kind. Here, the word anomalous connotes non-compliance with the Stuart–Landau equation, although the modes originate from the solution of the Navier–Stokes equation. One of the consequences of multi-modal interactions in the LSE equations is that the amplitudes of the instability modes are governed by stiff differential equations. This is not present in the traditional Stuart–Landau equation, as it retains only the nonlinear self-interaction. The stiffness problem of the LSE equations has been resolved using the compound matrix method.

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