3. Receptivity and Instability

In this chapter, linear stability and receptivity analysis of the zero-pressure gradient (ZPG) boundary layer, under the parallel flow assumption, is discussed. This assumption implies that the equilibrium flow quantities do not grow in the streamwise direction and requires solving the Orr-Sommerfeld equation (OSE) to study evolution of disturbance field in a linearized analysis. The concept of the spatio-temporal wave-front (STWF) originates from the receptivity analysis with the OSE solved for the response field. First, the simplified description of equilibrium flow in terms of a similarity solution for ZPG boundary layer is presented. Following which the OSE is derived for boundary layers, making use of the parallel flow approximation (Drazin and Reid, Hydrodynamic stability, Cambridge University Press, UK, 1981, [19], Sengupta, Instabilities of flows and transition to turbulence, CRC Press, Taylor & Francis Group, Florida, USA, 2012, [53]). This equation have been solved for the ZPG boundary layer using analytical approaches in Heisenberg (Annalen der Physik Leipzig, 379:577–627, 1924, [28]), Schlichting (Nach Gesell d Wiss z G\(\ddot{\mathrm{o}}\)tt., MPK 42:181–208, 1933, [48]), Tollmien (NACA TM 609, 1931, [71]). We instead introduce the compound matrix method, a robust method for stiff differential equation useful for the OSE. Finally, the receptivity analysis of the ZPG boundary layer flow is provided, with results taken from Sengupta et al. (Phys Rev Lett, 96(22):224504, 2006, [61]), Sengupta et al. (Phys Fluids, 18:094101, 2006, [62]). The unique feature of the materials in this chapter is the topic of instability of mixed convection flows for which two theorems are enunciated for an inviscid linear mechanism, based on materials extensively taken from Sengupta et al., Physics of Fluids, 25, 094102 (2013).