scholarly journals Convergence rates for the numerical approximation of the 2D stochastic Navier–Stokes equations

2021 ◽  
Vol 147 (3) ◽  
pp. 553-578
Author(s):  
Dominic Breit ◽  
Alan Dodgson
Keyword(s):  
Convergence Rates ◽  
Stokes Equations ◽  
Linear Convergence ◽  
Careful Analysis ◽  
Two Dimensions ◽  
Convergence In Probability ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Cylindrical Wiener Process ◽  
Numer Anal

AbstractWe study stochastic Navier–Stokes equations in two dimensions with respect to periodic boundary conditions. The equations are perturbed by a nonlinear multiplicative stochastic forcing with linear growth (in the velocity) driven by a cylindrical Wiener process. We establish convergence rates for a finite-element based space-time approximation with respect to convergence in probability (where the error is measured in the $$L^\infty _tL^2_x\cap L^2_tW^{1,2}_x$$ L t ∞ L x 2 ∩ L t 2 W x 1 , 2 -norm). Our main result provides linear convergence in space and convergence of order (almost) 1/2 in time. This improves earlier results from Carelli and Prohl (SIAM J Numer Anal 50(5):2467–2496, 2012) where the convergence rate in time is only (almost) 1/4. Our approach is based on a careful analysis of the pressure function using a stochastic pressure decomposition.

Perturbation Procedures for Nonlinear Viscous Flows

1983 ◽  
Vol 50 (2) ◽  
pp. 265-269
Author(s):  
D. Nixon
Keyword(s):  
Perturbation Theory ◽  
Shock Waves ◽  
Transonic Flow ◽  
Stokes Equations ◽  
Viscous Flows ◽  
Two Dimensions ◽  
Experimental Results ◽  
Navier Stokes ◽  
Navier Stokes Equations

The perturbation theory for transonic flow is further developed for solutions of the Navier-Stokes equations in two dimensions or for experimental results. The strained coordinate technique is used to treat changes in location of any shock waves or large gradients.

A Numerical Study of the Hydrodynamics of Reversing Flows Around a Cylinder

1994 ◽  
Vol 116 (4) ◽  
pp. 202-208 ◽  
Author(s):  
K. Nakajima ◽  
Y. Kallinderis ◽  
I. Sibetheros ◽  
R. W. Miksad ◽  
K. Lambrakos
Keyword(s):  
Experimental Data ◽  
Finite Element Method ◽  
Stokes Equations ◽  
Numerical Study ◽  
Offshore Structures ◽  
Two Dimensions ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Random Behavior ◽  
Marching Scheme

A numerical study of the nonlinear and random behavior of flow-induced forces on offshore structures and experimental verification of the results are presented. The numerical study is based on a finite-element method for the unsteady incompressible Navier-Stokes equations in two dimensions. The momentum equations combined with a pressure correction equation are solved employing fourth-order artificial dissipation with a nonstaggered grid, instead of the more commonly used staggered meshes. The solution is advanced in time with a combined explicit and implicit marching scheme. Emphasis is placed on study of reversing flows around a cylinder. Comparisons with experimental data evaluate accuracy and robustness of the method.

Improved κ-ɛ Modelling of Impeller Flow Performance of a Mixed-Flow Pump under off-Design Operating States

1993 ◽  
Vol 207 (4) ◽  
pp. 219-229 ◽  
Author(s):  
S M Fraser ◽  
Y Zhang
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Careful Analysis ◽  
Navier Stokes ◽  
Mean Flow ◽  
Mixed Flow ◽  
Navier Stokes Equations ◽  
Flow Through ◽  
The Mean ◽  
Flow Pump

Three-dimensional turbulent flow through the impeller passage of a model mixed-flow pump has been simulated by solving the Navier-Stokes equations with an improved κ-ɛ model. The standard κ-ɛ model was found to be unsatisfactory for solving the off-design impeller flow and a converged solution could not be obtained at 49 per cent design flowrate. After careful analysis, it was decided to modify the standard κ-ɛ model by including the extra rates of strain due to the acceleration of impeller rotation and geometrical curvature and removing the mathematical ill-posedness between the mean flow turbulence modelling and the logarithmic wall function.

Large deviation principles of 2D stochastic Navier–Stokes equations with Lévy noises

2021 ◽  
pp. 1-49
Author(s):  
Huaqiao Wang
Keyword(s):  
Weak Convergence ◽  
Stokes Equations ◽  
Large Deviation ◽  
Careful Analysis ◽  
Energy Estimates ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Convergence Method ◽  
Weak Convergence Method ◽  
Weak Convergence Approach

Taking the consideration of two-dimensional stochastic Navier–Stokes equations with multiplicative Lévy noises, where the noises intensities are related to the viscosity, a large deviation principle is established by using the weak convergence method skillfully, when the viscosity converges to 0. Due to the appearance of the jumps, it is difficult to close the energy estimates and obtain the desired convergence. Hence, one cannot simply use the weak convergence approach. To overcome the difficulty, one introduces special norms for new arguments and more careful analysis.

Time-dependent helical waves in rotating pipe flow

1990 ◽  
Vol 221 ◽  
pp. 289-310 ◽  
Author(s):  
Michael J. Landman
Keyword(s):  
Pipe Flow ◽  
Stokes Equations ◽  
Equations Of Motion ◽  
Reynolds Numbers ◽  
Linear Instability ◽  
Two Dimensions ◽  
Navier Stokes ◽  
Helical Symmetry ◽  
Navier Stokes Equations ◽  
Turbulent Transition

The Navier-Stokes equations for flow in a rotating circular pipe are solved numerically, subject to imposing helical symmetry on the velocity field v = v(r, θ + αz,t). The helical symmetry is exploited by writing the equations of motion in helical variables, reducing the problem to two dimensions. A limited study of the pipe flow is made in the parameter space of the wavenumber α, and the axial and azimuthal Reynolds numbers. The steadily rotating waves previously studied by Toplosky & Akylas (1988), which arise from the linear instability of the basic steady flow, are found to undergo a series of bifurcations, through periodic to aperiodic time dependence. The relevance of these results to the mechanism of laminar-turbulent transition in a stationary pipe is discussed.

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