Spatiotemporal Growing Wave Fronts in Spatially Stable Boundary Layers

Abstract

In fluid dynamical systems, it is not known a priori whether disturbances grow either in space or in time or as spatiotemporal structures. For a zero pressure gradient boundary layer (also known as the Blasius boundary layer), it is customary to treat it as a spatial problem, and some limited comparison between prediction and laboratory experiments exist. In the present work, the two-dimensional receptivity problem of a Blasius boundary layer excited by a localized harmonic source is investigated under the general spatiotemporal framework, by using the Bromwich contour integral method. While this approach is seen to be equivalent to the spatial study for unstable systems, here we show for the first time how spatially stable systems show spatiotemporally growing wave fronts.

Mixed convection flow past a vertical plate: Stability analysis and its direct simulation

“…Thus, the spatio-temporal growing wavefront is seen to occur due to interaction of multiple modes for fluid flows-with or without heat transfer. The spatio-temporal wave-front seen here is the same that was described in [1,2] for the Blasius profile. It is not a transient response-as established in Sengupta et al [1,2] by solving the problem using both linear receptivity approach and solution of full NavierStokes equation.…”

“…The spatio-temporal wave-front seen here is the same that was described in [1,2] for the Blasius profile. It is not a transient response-as established in Sengupta et al [1,2] by solving the problem using both linear receptivity approach and solution of full NavierStokes equation. In the linear receptivity study, governing OrrSommerfeld equation was solved by Bromwich contour integral method described in [2,20].…”

“…7(a). This type of spatio-temporal growing wave-front has been reported also in Sengupta et al [1] for a pure hydrodynamic stability study, where the phenomenon was established to be related to interactions of multiple decaying modes for spatially stable systems. Presented results in Fig.…”

“…The establishment of the presence of spatio-temporal growing wave-front is worthwhile, as it can explain discrepancies between experimental observation and normal-mode linear stability theory results. We noted in [1,2] and once again here that the leading wave-packet displays variation at lower wave numbers and frequencies as compared to the asymptotic waves. This has an important significance.…”

“…In comparison, the mixed convection opposing flows past vertical plate are more unstable (as shown in Section 3) with significantly higher instability indicated by the growth rate that is two orders of magnitude higher. Also, near critical Reynolds number, many stable eigenvalues with similar attenuation rates cause modal interactions of the type described in [1,2]. These interactions give rise to additional spatio-temporal excitation-usually at lower frequency.…”