Spatio-temporal growth of disturbances in a boundary layer and energy based receptivity analysis

Abstract

In fluid dynamical systems, it is not known a priori whether disturbances grow either in space or in time or as spatio-temporal structures. However, for boundary layers, 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 receptivity problem of a zero pressure gradient boundary layer excited by a localized harmonic source is investigated under the general spatio-temporal framework, using the Bromwich contour integral method. While this approach has been shown to be equivalent to the spatial study, for unstable systems excited by a single frequency source ͓T. K. Sengupta, M. Ballav, and S. Nijhawan, Phys. Fluids 6, 1213 ͑1994͔͒, here we additionally show, how the boundary layer behaves when it is excited ͑i͒ at a single frequency that corresponds to a stable condition ͑given by spatial normal-mode analysis͒ and ͑ii͒ by wideband frequencies, that shows the possibility of flow transition due to a spatio-temporally growing forerunner or wave front. An energy based receptivity analysis tool is also developed as an alternative to traditional instability theory. Using this, we reinterpret the concept of critical layer that was originally postulated to explain the mathematical singularity of inviscid disturbance field in traditional instability theory of normal modes.

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].…”

“…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.…”

“…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]. 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.…”