Dynamical System Theory of Flow Instability Using the Impulse and the Frequency Response Approaches

Abstract

We study the causal relation in a fluid dynamical system, for the impulse and frequency response approaches as instability theories and corresponding experiments. The zero-pressure-gradient (ZPG) boundary layer is analyzed to find complementary aspects of these approaches. The drawbacks of instability study are in formulating it as a homogeneous system. Another difficulty for the instability is in classifying it for either temporal or spatial growth. When viscous effects were included in the spatial theory, it predicted wave solution (known as Tollmien–Schlichting (TS) waves), which left many scientists unconvinced. Experimental verification remained difficult as instability does not require explicit excitation, and dependence on background noise makes experiment non-repeatable. The classic experiment of Schubauer and Skramstad for the boundary layer (J Aero Sci 14(2), 69–78, [24]) excited a monochromatic source inside to obtain spatially growing TS waves—considered as the frequency response of the boundary layer. In contrast, Gaster and Grant (Proc R Soc A 347(1649), 253–269, [13]) tried to create TS waves by a localized impulse excitation and ended up creating a wave-packet by the impulse response of the dynamical system. Here, we focus mainly on the impulse response of the ZPG boundary layer using Bromwich contour integral method (BCIM) developed by the authors for spatio-temporal growth of disturbance field in creating spatio-temporal wave-front (STWF). The main achievement of BCIM is in identifying the cause for the creation of STWF by both the approaches.