We prove that the plane Couette and Poiseuille flows are nonlinearly stable if the Reynolds number is less than Re-Orr (2 pi/(lambda sin theta))/sin theta when a perturbation is a tilted perturbation in the direction x' which forms an angle theta is an element of(0, pi/2] with the direction i of the basic motion and does not depend on x'. Re-Orr is the critical Orr-Reynolds number for spanwise perturbations which is computed for wave number 2 pi/(lambda sin theta), with lambda being any positive wavelength. By taking the minimum with respect to lambda, we obtain the critical energy Reynolds number for a fixed inclination angle and any wavelength: for plane Couette flow, it is Re-Orr = 44.3/sin theta, and for plane Poiseuille flow, it is Re-Orr = 87.6/sin theta (in particular, for theta = pi/2 we have the classical values Re-Orr = 44.3 for plane Couette flow and Re-Orr = 87.6 for plane Poiseuille flow). Here the nondimensional interval between the planes bounding the channel is [-1, 1]. In particular, these results improve those obtained by Joseph, who found for streamwise perturbations a critical nonlinear value of 20.65 in the plane Couette case, and those obtained by Joseph and Carmi who found the value 49.55 for plane Poiseuille flow for streamwise perturbations. If we fix some wavelengths from the experimental data and the numerical simulations, the critical Reynolds numbers that we obtain are in a very good agreement both with the the experiments and the numerical simulation. These results partially solve the Couette-Sommerfeld paradox

Nonlinear stability results for plane Couette and Poiseuille flows

Falsaperla paolo;Giacobbe Andrea
;
Mulone giuseppe
2019

Abstract

We prove that the plane Couette and Poiseuille flows are nonlinearly stable if the Reynolds number is less than Re-Orr (2 pi/(lambda sin theta))/sin theta when a perturbation is a tilted perturbation in the direction x' which forms an angle theta is an element of(0, pi/2] with the direction i of the basic motion and does not depend on x'. Re-Orr is the critical Orr-Reynolds number for spanwise perturbations which is computed for wave number 2 pi/(lambda sin theta), with lambda being any positive wavelength. By taking the minimum with respect to lambda, we obtain the critical energy Reynolds number for a fixed inclination angle and any wavelength: for plane Couette flow, it is Re-Orr = 44.3/sin theta, and for plane Poiseuille flow, it is Re-Orr = 87.6/sin theta (in particular, for theta = pi/2 we have the classical values Re-Orr = 44.3 for plane Couette flow and Re-Orr = 87.6 for plane Poiseuille flow). Here the nondimensional interval between the planes bounding the channel is [-1, 1]. In particular, these results improve those obtained by Joseph, who found for streamwise perturbations a critical nonlinear value of 20.65 in the plane Couette case, and those obtained by Joseph and Carmi who found the value 49.55 for plane Poiseuille flow for streamwise perturbations. If we fix some wavelengths from the experimental data and the numerical simulations, the critical Reynolds numbers that we obtain are in a very good agreement both with the the experiments and the numerical simulation. These results partially solve the Couette-Sommerfeld paradox
File in questo prodotto:
File Dimensione Formato  
Falsaperla, Giacobbe, Mulone_2019_Nonlinear stability results for plane Couette and Poiseuille flows.pdf

accesso aperto

Descrizione: Articolo
Tipologia: Versione Editoriale (PDF)
Dimensione 376.46 kB
Formato Adobe PDF
376.46 kB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11769/365600
Citazioni
  • ???jsp.display-item.citation.pmc??? 0
  • Scopus 10
  • ???jsp.display-item.citation.isi??? 10
social impact