Linear Inviscid Analysis

  • Peter J. Schmid
  • Dan S. Henningson
Part of the Applied Mathematical Sciences book series (AMS, volume 142)

Abstract

We begin this section by deriving the stability equations for infinitesimal disturbances when effects due to viscosity are negligible. Stability calculations of this sort were among the first in the field of hydrodynamic stability theory. We will assume parallel flow. Let U i = U(y)δ1i be the base flow, i.e., a flow in the x-direction that varies with y (see Figure 2.1). If this flow is substituted into the disturbance equations (1.6) and the nonlinear and viscous terms are omitted, the resulting equations can be written as
$$ \frac{{\partial u}} {{\partial t}} + U\frac{{\partial u}} {{\partial x}} + vU' = - \frac{{\partial p}} {{\partial x}} $$
(2.1)
$$ \frac{{\partial v}} {{\partial t}} + U\frac{{\partial v}} {{\partial x}} = - \frac{{\partial p}} {{\partial y}} $$
(2.2)
$$ \frac{{\partial w}} {{\partial t}} + U\frac{{\partial w}} {{\partial x}} = - \frac{{\partial p}} {{\partial z}} $$
(2.3)
and the continuity equation is
$$ \frac{{\partial u}} {{\partial x}} + \frac{{\partial v}} {{\partial y}} + \frac{{\partial w}} {{\partial z}} = 0. $$
(2.4)

Keywords

Vortex Coherence Vorticity Advection cosB 

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Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Peter J. Schmid
    • 1
  • Dan S. Henningson
    • 2
  1. 1.Applied Mathematics DepartmentUniversity of WashingtonSeattleUSA
  2. 2.Department of MechanicsRoyal Institute of Technology (KTH)StockholmSweden

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