Sommario
Si studia la forma di instabilità del moto alla Couette per vortici con simmetria a spirale. In una opportuna metrica “a spirale” le equazioni risultano analoghe a quelle che descrivono i vortici a simmetria assiale. La soluzione approssimata delle equazioni è ottenuta per discretizzazione mediante differenze finite, usando opportuni accorgimenti per ridurre notevolmente le dimensioni delle matrici su cui si eseguono i calcoli. Viene fatto il calcolo del numero critico di Taylor per le equazioni linearizzate in funzione dei parametri significativi e si stabiliscono confronti con risultati noti. Alcuni grafici mostrano i risultati ottenuti.
Summary
The instability of Couette flow caused by vortices with spiral symmetry is studied; the equations of motion in a suitable “spiral” metric turn out to be analogous with those describing vortices with axial symmetry. The approximate solution is obtained by discretization of the differential equations. Special numerical techniques reduce drastically the dimensions of the matrices involved in the calculations. The Taylor number for transition is computed; comparisons with known results are established. The results are shown in a number of graphs.
This is a preview of subscription content, log in via an institution to check access.
Abbreviations
- v=(v r,v 0,v z):
-
speed of the perturbation
- v0 :
-
speed of the steady motion
- r, ϑ, z :
-
cylindrical coordinates
- t :
-
time
- P :
-
pressure change due to the perburbation
- ϱ :
-
density
- ν :
-
kinematic viscosity
- Ω 1,Ω 2 :
-
angular speed of the two cylinders
- R 1,R 2 :
-
radii of the coaxial rotating cylinders
- η :
-
R 1 R 2 −1
- c :
-
R 1 −R 2
- λ :
-
Ω 2 Ω 1 −1
- T :
-
R 1 Ω 21 c 3 v −2 (Taylor number)
- V :
-
(v0)0
- ξ :
-
(r −R 1)c −1
- x :
-
zc −1
- τ :
-
vtc −2
- m :
-
pitch of the spiral
- q :
-
πmc −1
- ɛ :
-
cR 1 −1
- ζ :
-
(z −mϑ)c −1.
- μ :
-
m 2 r −2
References
G. I. Taylor,On the stability of a viscous liquid contained between two rotating cylinders, Phil. Trans. Roy. Soc. A. 223, 289–343, London, 1923.
G. Capriz, G. Ghelardoni andG. Lombardi,Numerical study of the stability problem for Couette flow, Physics of Fluids, 9, 1934–36, 1966.
G. Capriz,The numerical approach to hydrodynamic problems, In. J. L. Lions ed., Numerical Analysis of Partial Differential Equations, Cremonese, 109–159, Roma, 1968.
E. R. Bond et a.,FORMAC: An Experimental FO Rmula M Anipulation Compiler, Share 0016.
S. Chandrasekhar,Hydrodynamic and Hydromagnetic Stability, Oxford Univ. Press, 1961.
G. Ghelardoni, Questioni connesse con l'impostazione analitica e la soluzione numerica di un problema di idrodinamica, Calcolo, 2, suppl. no. 1, 51–66, 1965.
G. Gherlardoni andG. Lombardi,The stability problem for Couette flow: a finite difference approach, Meccanica, Vol. 6, no. 1, 43–52, 1971.
G. Ghelardoni and G. Lombardi, Soluzione numerica di un problema di stabilità idrodinamica, Calcolo, 2, suppl. no. 1, 67–80, 1965.
T. Muir,A treatise on the theory of determinants, Dover Publications, New York, 1960.
W. Velte, Stabilität und Verzweigung stationärer Lösungen der Navier-Stokesschen Gleichungen beim Taylor-problem, Arch. Rat. Mech. Anal., 22, 1–14, 1966.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Andreussi, F., Menchi, O. On the spiral form of instability of couette flow of a viscous fluid. Meccanica 6, 212–217 (1971). https://doi.org/10.1007/BF02128915
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02128915