scholarly journals Inviscid limit for the damped generalized incompressible Navier-Stokes equations on $ \mathbb{T}^2 $

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yang Liu ◽  
Chunyou Sun
Keyword(s):  
Stokes Equations ◽  
Inviscid Limit ◽  
Navier Stokes ◽  
Dissipative Operator ◽  
Navier Stokes Equations ◽  
Long Time ◽  
Time Averages ◽  
Weaker Conditions ◽  
Anomalous Dissipation

<p style='text-indent:20px;'>In this paper, for the damped generalized incompressible Navier-Stokes equations on <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{T}^{2} $\end{document}</tex-math></inline-formula> as the index <inline-formula><tex-math id="M3">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> of the general dissipative operator <inline-formula><tex-math id="M4">\begin{document}$ (-\Delta)^{\alpha} $\end{document}</tex-math></inline-formula> belongs to <inline-formula><tex-math id="M5">\begin{document}$ (0,\frac{1}{2}] $\end{document}</tex-math></inline-formula>, we prove the absence of anomalous dissipation of the long time averages of entropy. We also give a note to show that, by using the <inline-formula><tex-math id="M6">\begin{document}$ L^{\infty} $\end{document}</tex-math></inline-formula> bounds given in Caffarelli et al. [<xref ref-type="bibr" rid="b4">4</xref>], the absence of anomalous dissipation of the long time averages of energy for the forced SQG equations established in Constantin et al. [<xref ref-type="bibr" rid="b12">12</xref>] still holds under a slightly weaker conditions <inline-formula><tex-math id="M7">\begin{document}$ \theta_{0}\in L^{1}(\mathbb{R}^{2})\cap L^{2}(\mathbb{R}^{2}) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ f \in L^{1}(\mathbb{R}^{2})\cap L^{p}(\mathbb{R}^{2}) $\end{document}</tex-math></inline-formula> with some <inline-formula><tex-math id="M9">\begin{document}$ p&gt;2 $\end{document}</tex-math></inline-formula>.</p>

Transonic nozzle flow of dense gases

1993 ◽  
Vol 247 ◽  
pp. 661-688 ◽  
Author(s):  
A. Kluwick
Keyword(s):  
Stokes Equations ◽  
Equations Of State ◽  
Weak Shock ◽  
Inviscid Limit ◽  
Navier Stokes ◽  
Perturbation Equation ◽  
Navier Stokes Equations ◽  
Sonic Point ◽  
Specific Heats ◽  
Dense Gases

The paper deals with the flow properties of dense gases in the throat area of slender nozzles. Starting from the Navier–Stokes equations supplemented with realistic equations of state for gases which have relatively large specific heats a novel form of the viscous transonic small-perturbation equation is derived. Evaluation of the inviscid limit of this equation shows that three sonic points rather than a single sonic point may occur during isentropic expansion of such media, in contrast to the case of perfect gases. As a consequence, a shock-free transition from subsonic to supersonic speeds cannot, in general, be achieved by means of a conventional converging–diverging nozzle. Nozzles leading to shock-free flow fields must have an unusual shape consisting of two throats and an intervening antithroat. Additional new results include the computation of the internal thermoviscous structure of weak shock waves and a phenomenon referred to as impending shock splitting. Finally, the relevance of these results to the description of external transonic flows is discussed briefly.

Three-dimensional impulsive vortex formation from slender orifices

2011 ◽  
Vol 666 ◽  
pp. 506-520 ◽  
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.

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

2013 ◽  
Vol 143 (5) ◽  
pp. 905-927 ◽  
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.

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