Abstract
For applications regarding transition prediction, wing design and control of boundary layers, the fundamental understanding of disturbance growth in the flat plate boundary layer is an important issue. In the present work we investigate the stability of boundary layer in Poiseuille flow. We normalize pressure and time by inertial and viscous effects. The disturbances are taken to be periodic in the spanwise direction and time. We present a set of linear governing equations for the parabolic evolution of wavelike disturbances. Then, we derive the so-called modified Orr-Sommerfeld equation that can be applied in the layer. Contrary to what one might think of, we find that Squire’s theorem is not applicable for the boundary layer. We find also that normalization by inertial or viscous effects leads to the same order of stability or instability. For the 2-D disturbances flow, we find the same critical Reynolds number for our two normalizations. This value coincides with the one we know for neutral stability of the known Orr-Sommerfeld equation. We notice also that for all over values of k in the case , correspond the same values of at whatever the normalization. We therefore conclude that in the boundary layer with 2-D disturbances, we have the same neutral stability curve whatever the normalization. We find also that for a flow with high hydrodynamic Reynolds number, the neutral disturbances in the boundary layer are two dimensional. At last, we find that transition from stability to instability or the opposite can occur according to the Reynolds number and the wave number.
Highlights
Boundary-layer theory is crucial in understanding why certain phenomena occur
We find that normalization by inertial or viscous effects leads to the same order of stability or instability
We have investigated the stability of boundary layer in Poiseuille flow
Summary
Boundary-layer theory is crucial in understanding why certain phenomena occur. It is well known that the instability of boundary layer is sensitive to the mean velocity profile, so that a small distortion to the basic flow may have a detrimental effect on its stability. Ho and Denn (1977) studied low Reynolds number stability for plane Poiseuille flow by using a numerical scheme based on the shooting method. Samad, and Chaudhury (2000) investigated the linear stability of plane Poiseuille flow at small Reynolds number of a conducting Oldroyd fluid in the presence of magnetic field. They found that viscoelastic parameters have destabilizing effect and magnetic field has a stabilizing effect in the field of flow but no instabilities are found. In the third section we present the general formulation, highlighting the fundamental equations that model the flat-plate boundary layer flow according to the normalization by inertial and viscous effects.
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