Long Time Stability of High Order MultiStep Numerical Schemes for Two-Dimensional Incompressible Navier--Stokes Equations

A novel high-order finite volume method for the resolution of the Navier-Stokes equations is presented. The approach combines a third order finite volume method in an unstructured two-dimensional grid, with a spectral approximation in the third dimension. The method is suitable for the resolution of complex two-dimensional geometries that require the third dimension to capture three-dimensional non-linear unsteady effects, such as those for instance present in linear cascades with separated bubbles. Its main advantage is the reduction in the computational cost, for a given accuracy, with respect standard finite volume methods due to the inexpensive high-order discretization that may be obtained in the third direction using fast Fourier transforms. The method has been applied to the resolution of transitional bubbles in flat plates with adverse pressure gradients and realistic two-dimensional airfoils.

Download Full-text

Three-dimensional impulsive vortex formation from slender orifices

Journal of Fluid Mechanics ◽

10.1017/s0022112010004994 ◽

2011 ◽

Vol 666 ◽

pp. 506-520 ◽

Cited By ~ 20

Author(s):

F. DOMENICHINI

Keyword(s):

Stokes Equations ◽

Intermediate Phase ◽

Vortex Formation ◽

Three Dimensional ◽

Navier Stokes ◽

Two Dimensional ◽

Navier Stokes Equations ◽

Vortex Ring Formation ◽

Long Time ◽

Definition Of

The vortex formation behind an orifice is a widely investigated phenomenon, which has been recently studied in several problems of biological relevance. In the case of a circular opening, several works in the literature have shown the existence of a limiting process for vortex ring formation that leads to the concept of critical formation time. In the different geometric arrangement of a planar flow, which corresponds to an opening with straight edges, it has been recently outlined that such a concept does not apply. This discrepancy opens the question about the presence of limiting conditions when apertures with irregular shape are considered. In this paper, the three-dimensional vortex formation due to the impulsively started flow through slender openings is studied with the numerical solution of the Navier–Stokes equations, at values of the Reynolds number that allow the comparison with previous two-dimensional findings. The analysis of the three-dimensional results reveals the two-dimensional nature of the early vortex formation phase. During an intermediate phase, the flow evolution appears to be driven by the local curvature of the orifice edge, and the time scale of the phenomena exhibits a surprisingly good agreement with those found in axisymmetric problems with the same curvature. The long-time evolution shows the complete development of the three-dimensional vorticity dynamics, which does not allow the definition of further unifying concepts.

Download Full-text

Metastability and rapid convergence to quasi-stationary bar states for the two-dimensional Navier–Stokes equations

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210511001478 ◽

2013 ◽

Vol 143 (5) ◽

pp. 905-927 ◽

Cited By ~ 31

Author(s):

Margaret Beck ◽

C. Eugene Wayne

Keyword(s):

Decay Rate ◽

Stokes Equations ◽

Navier Stokes ◽

Two Dimensional ◽

Navier Stokes Equation ◽

Navier Stokes Equations ◽

Optimal Decay ◽

Optimal Decay Rate ◽

Long Time ◽

High Reynolds

Quasi-stationary, or metastable, states play an important role in two-dimensional turbulent fluid flows, where they often emerge on timescales much shorter than the viscous timescale, and then dominate the dynamics for very long time intervals. In this paper we propose a dynamical systems explanation of the metastability of an explicit family of solutions, referred to as bar states, of the two-dimensional incompressible Navier–Stokes equation on the torus. These states are physically relevant because they are associated with certain maximum entropy solutions of the Euler equations, and they have been observed as one type of metastable state in numerical studies of two-dimensional turbulence. For small viscosity (high Reynolds number), these states are quasi-stationary in the sense that they decay on the slow, viscous timescale. Linearization about these states leads to a time-dependent operator. We show that if we approximate this operator by dropping a higher-order, non-local term, it produces a decay rate much faster than the viscous decay rate. We also provide numerical evidence that the same result holds for the full linear operator, and that our theoretical results give the optimal decay rate in this setting.

Download Full-text

Well-Posedness and Long-Time Behavior of Solutions for Two-Dimensional Navier-Stokes Equations with Infinite Delay and General Hereditary Memory

10.20944/preprints201909.0008.v2 ◽

2021 ◽

Author(s):

Yasi Zheng ◽

Wenjun Liu ◽

Yadong Liu

Keyword(s):

Stokes Equations ◽

Infinite Delay ◽

Navier Stokes ◽

Time Behavior ◽

Two Dimensional ◽

Navier Stokes Equations ◽

Bounded Absorbing Set ◽

Long Time ◽

The One ◽

Class Of Functions

We address the dynamics of two-dimensional Navier-Stokes models with infinite delay and hereditary memory, whose kernels are a much larger class of functions than the one considered in the literature, on a bounded domain. We prove the existence and uniqueness of weak solutions by means of Faedo-Galerkin method. Moreover, we establish the existence of global attractor for the system with the existence of a bounded absorbing set and asymptotic compact property.

Download Full-text