Approximation of the Eigenvalues of a Nonselfadjoint Operator Arising in the Study of the Stability of Stationary Solutions of the Navier–Stokes Equations

1976 ◽  
Vol 13 (2) ◽  
pp. 185-197 ◽  
Author(s):  
John E. Osborn
Keyword(s):  
Stokes Equations ◽  
Stationary Solutions ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Nonselfadjoint Operator ◽  
The Stability

The exponential behavior and stabilizability of the stochastic 3D Navier–Stokes equations with damping

2019 ◽  
Vol 31 (07) ◽  
pp. 1950023 ◽  
Author(s):  
Hui Liu ◽  
Lin Lin ◽  
Chengfeng Sun ◽  
Qingkun Xiao
Keyword(s):  
Weak Solutions ◽  
Multiplicative Noise ◽  
Stokes Equations ◽  
Stationary Solutions ◽  
Navier Stokes ◽  
Exponential Behavior ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
The Mean ◽  
The Stability

The stochastic 3D Navier–Stokes equation with damping driven by a multiplicative noise is considered in this paper. The stability of weak solutions to the stochastic 3D Navier–Stokes equations with damping is proved for any [Formula: see text] with any [Formula: see text] and [Formula: see text] as [Formula: see text]. The weak solutions converge exponentially in the mean square and almost surely exponentially to the stationary solutions are proved for any [Formula: see text] with any [Formula: see text] and [Formula: see text] as [Formula: see text]. The stabilization of these equations is obtained for any [Formula: see text] with any [Formula: see text] and [Formula: see text] as [Formula: see text].

scholarly journals CONVERGENCE RATE FOR THE COMPRESSIBLE NAVIER–STOKES EQUATIONS WITH EXTERNAL FORCE

2006 ◽  
Vol 03 (03) ◽  
pp. 561-574 ◽  
Author(s):  
SEIJI UKAI ◽  
TONG YANG ◽  
HUIJIANG ZHAO
Keyword(s):  
Convergence Rate ◽  
Stokes Equations ◽  
Stationary Solutions ◽  
Navier Stokes ◽  
Optimal Convergence Rate ◽  
Potential Force ◽  
Stationary Potential ◽  
Navier Stokes Equations ◽  
The Stability ◽  
External Forces

For the compressible Navier–Stokes equations with a stationary potential force, the stability of the stationary solutions was studied by Matsumura and Nishida. The convergence rate to the stationary solutions in time was later studied by Deckelnick which was improved by Shibata and Tanaka for more general external forces. This paper deals with the case for the stationary potential force under some smallness condition, to establish an almost optimal convergence rate in L2(ℝN)-norm for N ≥ 3.

Advanced Numerical Methods for the Prediction of Tonal Noise in Turbomachinery—Part I: Implicit Runge–Kutta Schemes

2013 ◽  
Vol 136 (2) ◽  
Author(s):  
Graham Ashcroft ◽  
Christian Frey ◽  
Kathrin Heitkamp ◽  
Christian Weckmüller
Keyword(s):  
Stokes Equations ◽  
Temporal Integration ◽  
Aerodynamic Noise ◽  
Navier Stokes ◽  
Noise Generation ◽  
Academic Problems ◽  
Tonal Noise ◽  
Navier Stokes Equations ◽  
Runge Kutta ◽  
The Stability

This is the first part of a series of two papers on unsteady computational fluid dynamics (CFD) methods for the numerical simulation of aerodynamic noise generation and propagation. In this part, the stability, accuracy, and efficiency of implicit Runge–Kutta schemes for the temporal integration of the compressible Navier–Stokes equations are investigated in the context of a CFD code for turbomachinery applications. Using two model academic problems, the properties of two explicit first stage, singly diagonally implicit Runge–Kutta (ESDIRK) schemes of second- and third-order accuracy are quantified and compared with more conventional second-order multistep methods. Finally, to assess the ESDIRK schemes in the context of an industrially relevant configuration, the schemes are applied to predict the tonal noise generation and transmission in a modern high bypass ratio fan stage and comparisons with the corresponding experimental data are provided.

Investigation of the stability of boundary layers by a finite-difference model of the Navier—Stokes equations

1976 ◽  
Vol 78 (2) ◽  
pp. 355-383 ◽  
Author(s):  
H. Fasel
Keyword(s):  
Finite Difference ◽  
Stability Theory ◽  
Stokes Equations ◽  
Time Dependent ◽  
Linear Stability Theory ◽  
Navier Stokes ◽  
Experimental Measurements ◽  
Navier Stokes Equations ◽  
The Stability ◽  
Small Periodic

The stability of incompressible boundary-layer flows on a semi-infinite flat plate and the growth of disturbances in such flows are investigated by numerical integration of the complete Navier–;Stokes equations for laminar two-dimensional flows. Forced time-dependent disturbances are introduced into the flow field and the reaction of the flow to such disturbances is studied by directly solving the Navier–Stokes equations using a finite-difference method. An implicit finitedifference scheme was developed for the calculation of the extremely unsteady flow fields which arose from the forced time-dependent disturbances. The problem of the numerical stability of the method called for special attention in order to avoid possible distortions of the results caused by the interaction of unstable numerical oscillations with physically meaningful perturbations. A demonstration of the suitability of the numerical method for the investigation of stability and the initial growth of disturbances is presented for small periodic perturbations. For this particular case the numerical results can be compared with linear stability theory and experimental measurements. In this paper a number of numerical calculations for small periodic disturbances are discussed in detail. The results are generally in fairly close agreement with linear stability theory or experimental measurements.

scholarly journals The Asymptotic Stability of the Generalized 3D Navier-Stokes Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Wen-Juan Wang ◽  
Yan Jia
Keyword(s):  
Weak Solution ◽  
Asymptotic Stability ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Perturbed System ◽  
Regular Class ◽  
Stability Issue ◽  
The Stability

We study the stability issue of the generalized 3D Navier-Stokes equations. It is shown that if the weak solutionuof the Navier-Stokes equations lies in the regular class∇u∈Lp(0,∞;Bq,∞0(ℝ3)),(2α/p)+(3/q)=2α,2<q<∞,0<α<1, then every weak solutionv(x,t)of the perturbed system converges asymptotically tou(x,t)asvt-utL2→0,t→∞.

scholarly journals Disconnected stationary solutions for 3D Kolmogorov flow problem: preliminary results

2021 ◽  
Vol 2090 (1) ◽  
pp. 012046
Author(s):  
Nikolay M. Evstigneev
Keyword(s):  
Flow Problem ◽  
Stokes Equations ◽  
Continuation Method ◽  
Fourier Method ◽  
Stationary Solutions ◽  
Navier Stokes ◽  
System Of Nonlinear Equations ◽  
Arc Length ◽  
Navier Stokes Equations ◽  
Numerical Bifurcation

Abstract The extension of the classical A.N. Kolmogorov’s flow problem for the stationary 3D Navier-Stokes equations on a stretched torus for velocity vector function is considered. A spectral Fourier method with the Leray projection is used to solve the problem numerically. The resulting system of nonlinear equations is used to perform numerical bifurcation analysis. The problem is analyzed by constructing solution curves in the parameter-phase space using previously developed deflated pseudo arc-length continuation method. Disconnected solutions from the main solution branch are found. These results are preliminary and shall be generalized elsewhere.

scholarly journals On the Importance of the Influence of both Velocity and Aspect Ratios on the Occurrence of Bifurcation Phenomena within a Two-Sided Lid-Driven Enclosure

2015 ◽  
Vol 55 ◽  
pp. 160-172
Author(s):  
Fayçal Hammami ◽  
Nader Ben Cheikh ◽  
Brahim Ben Beya
Keyword(s):  
Stokes Equations ◽  
Numerical Study ◽  
Numerical Results ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Left Wall ◽  
Driven Cavity ◽  
Aspect Ratios ◽  
The Stability ◽  
The Right

This paper deals with the numerical study of bifurcations in a two-sided lid driven cavity flow. The flow is generated by moving the upper wall to the right while moving the left wall downwards. Numerical simulations are performed by solving the unsteady two dimensional Navier-Stokes equations using the finite volume method and multigrid acceleration. In this problem, the ratio of the height to the width of the cavity are ranged from H/L = 0.25 to 1.5. The code for this cavity is presented using rectangular cavity with the grids 144 × 36, 144 × 72, 144 × 104, 144 × 136, 144 × 176 and 144 × 216. Numerous comparisons with the results available in the literature are given. Very good agreements are found between current numerical results and published numerical results. Various velocity ratios ranged in 0.01≤ α ≤ 0.99 at a fixed aspect ratios (A = 0.5, 0.75, 1.25 and 1.5) were considered. It is observed that the transition to the unsteady regime follows the classical scheme of a Hopf bifurcation. The stability analysis depending on the aspect ratio, velocity ratios α and the Reynolds number when transition phenomenon occurs is considered in this paper.

scholarly journals Spectral Direction Splitting Schemes for the Incompressible Navier-Stokes Equations

2011 ◽  
Vol 1 (3) ◽  
pp. 215-234 ◽  
Author(s):  
Lizhen Chen ◽  
Jie Shen ◽  
Chuanju Xu
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Splitting Schemes ◽  
Time Step ◽  
Order Of Accuracy ◽  
Navier Stokes Equations ◽  
Pressure Stabilization ◽  
Legendre Spectral Method ◽  
Direction Splitting ◽  
The Stability

AbstractWe propose and analyze spectral direction splitting schemes for the incompressible Navier-Stokes equations. The schemes combine a Legendre-spectral method for the spatial discretization and a pressure-stabilization/direction splitting scheme for the temporal discretization, leading to a sequence of one-dimensional elliptic equations at each time step while preserving the same order of accuracy as the usual pressure-stabilization schemes. We prove that these schemes are unconditionally stable, and present numerical results which demonstrate the stability, accuracy, and efficiency of the proposed methods.

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