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Optimal Disturbance Growth in Watertable Flow
Author(s) -
Olsson P. J.,
Henningson D. S.
Publication year - 1995
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm1995942183
Subject(s) - wavenumber , hagen–poiseuille equation , reynolds number , exponential decay , physics , vorticity , shear flow , exponential growth , mathematics , exponential function , free surface , mechanics , eigenvalues and eigenvectors , flow (mathematics) , mathematical analysis , vortex , turbulence , optics , quantum mechanics
The linear stability of a liquid flow down an inclined plane is investigated. The equations governing the evolution of the disturbance are written in vector form where the dependent variables are the normal velocity and the normal vorticity. Similar to other shear flows, it is shown that there can be transient growth in the energy of a disturbance followed by an exponential decay although all eigenvalues predict decay only. Parameter studies reveal that the maximum amplification occur for waves with no streamwise dependence and with a spanwise wavenumber of (1). The mechanism involved in this growth is analyzed. A free surface parameter ( S ) can be identified that is related to the extent gravity and surface tension influence the free surface. A scaling of the equations is studied which revealed that the maximum transient growth scales with the Reynolds number as Re 2 if k 2 S Re 2 is kept constant, where k is the absolute value of the wavenumber vector. For small values of S exponential growth of free‐surface modes also exists. In general, however, we have found that for moderate times the transient growth dominates over the exponential growth and that its characteristics are similar to the transient growth found in other shear flows, e.g., plane Poiseuille flow.