arXiv subjects

Giuseppe Mulone

Publications and source records attributed to Giuseppe Mulone.

7 recordsLinked to original sources

arXiv2024

Energy Method and Stability of Shear Flows: an Elementary Tutorial

This paper provides a pedagogical introduction to the classical nonlinear stability analysis of the plane Poiseuille and Couette flows. The whole procedure is kept as simple as possible by presenting all the logical steps involved in the application of the energy method and leading to the Euler-Lagrange equations. Then, the eigenvalue problems needed for the evaluation of the nonlinear energy threshold of the Reynolds number for stability are formulated for transverse modes and for longitudinal modes. Such formulations involve the streamfunction and, in the case of longitudinal modes, also the streamwise component of velocity. An accurate numerical solution of the eigenvalue problems, based on Galerkin's method of weighted residuals with the test functions expressed in terms of Chebyshev polynomials, is discussed in details. The numerical codes developed for the software Mathematica 14 (\copyright{} Wolfram Research, Inc.) are also presented. A critical analysis of the obtained results is finally proposed.

physics.flu-dyn↗

arXiv2023

Monotone energy stability for Poiseuille flow in a porous medium

We study the monotone energy stability of ``Poiseuille flow" in a plane-parallel channel with a saturated porous medium modeled by the Brinkman equation, on the basis of an analogy with a magneto-hydrodynamic problem (Hartmann flow) (cf. \cite{Hill.Straughan.2010}, \cite{Nield.2003}). We prove that the least stabilizing perturbations, in the energy norm, are the two-dimensional spanwise perturbations. This result implies a Squire theorem for monotone nonlinear energy stability. Moreover, for Reynolds numbers less than the critical Reynolds number $R_E $ there can be no transient energy growth.

math-ph↗

arXiv2023

Nonlinear monotone energy stability of plane shear flows: Joseph or Orr critical thresholds?

Critical Reynolds numbers for the monotone exponential energy stability of Couette and Poiseuille plane flows were obtained by Orr 1907 \cite{Orr1907} in a famous paper, and by Joseph 1966 \cite{Joseph1966}, Joseph and Carmi 1969 \cite{JosephCarmi1969} and Busse 1972 \cite{Busse1972}. All these authors obtained their results applying variational methods to compute the maximum of a functional ratio derived from the Reynolds-Orr energy identity. Orr and Joseph obtained different results, for instance in the Couette case Orr computed the critical Reynolds value of 44.3 (on spanwise perturbations) and Joseph 20.65 (on streamwise perturbations). Recently in Falsaperla et al. 2022 \cite{FalsaperlaMulonePerrone2022}, the authors conjectured that the search of the maximum should be restricted to a subspace of the space of kinematically admissible perturbations. With this conjecture, the critical nonlinear energy Reynolds number was found among spanwise perturbations (a Squire theorem for nonlinear systems). With a direct proof and an appropriate and original decomposition of the dissipation terms in the Reynolds-Orr identity we show the validity of this conjecture in the space of three dimensional perturbations.

physics.flu-dyn↗

arXiv2023

Squire's theorem for nonlinear monotone energy stability

The Squire's theorem holds for parallel shear flows governed by the linearized Navier-Stokes equations. Squire writes ``For the study of the stability of flow between parallel walls it is sufficient to confine attention to disturbances of two-dimensional type", Squire \cite[p. 627]{Squire1933}. Instead, for nonlinear Navier-Stokes system it is supposed that the theorem does not hold in general (see Drazin and Reid \cite[pp. 429--430]{DrazinReid2004}). Here we prove that the Squire theorem holds also for nonlinear monotone energy stability of parallel shear flows that include Couette and Poiseuille flows between parallel walls.

math-ph↗

arXiv2023

Monotone energy stability of magnetohydrodynamics Couette and Hartmann flows

We study the monotone nonlinear energy stability of \textit{magnetohydrodynamics plane shear flows, Couette and Hartmann flows}. We prove that the least stabilizing perturbations, in the energy norm, are the two-dimensional spanwise perturbations and give some criti\-cal Reynolds numbers Re$_E$ for some selected Prandtl and Hartmann numbers. This result solves a conjecture given in a recent paper by Falsaperla et al. \cite{FMP.2022} and implies a Squire theorem for nonlinear energy: the less stabilizing perturbations in the \textit{energy norm} are the two-dimensional spanwise perturbations. Moreover, for Reynolds numbers less than Re$_E $ there can be no transient energy growth.

math-ph↗

arXiv2021

Energy stability of plane Couette and Poiseuille flows and Couette paradox

We study the nonlinear stability of plane Couette and Poiseuille flows with the Lyapunov second method by using the classical L2-energy. We prove that the streamwise perturbations are L2-energy stable for any Reynolds number. This contradicts the results of Joseph [10], Joseph and Carmi [12] and Busse [4], and allows us to prove that the critical nonlinear Reynolds numbers are obtained along two-dimensional perturbations, the spanwise perturbations, as Orr [16] had supposed. This conclusion combined with recent results by Falsaperla et al. [8] on the stability with respect to tilted rolls, provides a possible solution to the Couette-Sommerfeld paradox.

physics.flu-dyn↗

arXiv2018

Nonlinear stability results for plane Couette and Poiseuille flows

In this article we prove, choosing an appropriately weighted $L_2$-energy equivalent to the classical energy, that the plane Couette and Poiseuille flows are nonlinearly stable with respect to streamwise perturbations for any Reynolds number. In this case the coefficient of time-decay of the energy is $π^2/(2 {\rm Re})$, and it is a bound from above of the time-decay of streamwise perturbations of linearized equations. We also prove that the plane Couette and Poiseuille flows are nonlinearly stable if the Reynolds number is less then ${\rm Re}_{Orr}/\sin φ$ when the perturbation is a tilted perturbation, i.e. 2D perturbations with wave vector which forms an angle $φ\in [0, π/2]$ with the direction ${\bf i}$ of the motion. ${\rm Re}_{Orr}$ is the Orr (1907) critical Reynolds number for spanwise perturbations which, for the Couette flow is ${\rm Re}_{Orr}=177.22$ and for the Poiseuille flow is ${\rm Re}_{Orr}=175.31$. In particular these results improve those obtained by Joseph (1966), who found for streamwise perturbations a critical nonlinear value of $82.6$ in the Couette case, and those obtained by Joseph and Carmi (1969), who found the value $99.1$ for plane Poiseuille flow for streamwise perturbations. The results we obtain here are, for any angle, in a good agreement with the experiments (see Prigent et al. 2003) and the numerical simulations (see Barckley and Tuckerman 2005, 2007).

physics.flu-dyn↗