## Stability of Boundary Layers

None of the velocity profiles which arise in a parallel channel have an inflexion point and would be classified as stable based on the Rayleigh equation. If the effect of viscosity were to be viewed as purely stabilizing then flow in a parallel channel would be unconditionally stable. This is contrary to real-life experience and also contrary to what is predicted by OSE. Hence the origin of instability may be traced to the viscosity of the fluid itself. It can be shown that the wave speed of disturbances introduced in the flow (cr) is always less than the maximum fluid velocity. As a result, at some point in the channel the fluid velocity equals cr, i.e., u- cr = 0. This is called the critical layer in the flow where disturbances growth rapidly. The OSE has a singularity at this point and must be treated carefully in this region.

Flow over a flat plate produces velocity profiles which are free of inflexion points. From an inviscid view point they are stable. However, as described above, they undergo viscous instability initiated at the critical layer within the boundary-layer. The critical Reynolds number of approaching uniform flow is infinity and continually decreases along the plate with no minimum being ever attained. It may be loosely correlated with the increase in the boundary-layer thickness over the plate. Instability sets in when the local Reynolds number exceeds the critical Reynolds number at that location. The point of intersection occurs at a Reynolds number of about 60000. This is followed by a transition region where some disturbances preferentially grow over others. Finally, the flow becomes fully turbulent with practically a continuous range of frequencies being present in the velocity fluctuations.

Transition in the boundary-layer is poorly understood and is the subject matter of current research. For many practical purposes, it may be treated as a combination of laminar and turbulent flows. For example, the drag on the portion of the plate exposed to transitional flow can be calculated as the average value of drag based on purely laminar and purely turbulent conditions.

Flow over a flat plate produces velocity profiles which are free of inflexion points. From an inviscid view point they are stable. However, as described above, they undergo viscous instability initiated at the critical layer within the boundary-layer. The critical Reynolds number of approaching uniform flow is infinity and continually decreases along the plate with no minimum being ever attained. It may be loosely correlated with the increase in the boundary-layer thickness over the plate. Instability sets in when the local Reynolds number exceeds the critical Reynolds number at that location. The point of intersection occurs at a Reynolds number of about 60000. This is followed by a transition region where some disturbances preferentially grow over others. Finally, the flow becomes fully turbulent with practically a continuous range of frequencies being present in the velocity fluctuations.

Transition in the boundary-layer is poorly understood and is the subject matter of current research. For many practical purposes, it may be treated as a combination of laminar and turbulent flows. For example, the drag on the portion of the plate exposed to transitional flow can be calculated as the average value of drag based on purely laminar and purely turbulent conditions.