Statistical Modeling of Energy Amplification of Inflow Perturbations in Boundary Layer Flows

Linear growth of disturbances plays a crucial role in the laminar-to-turbulent transition in boundary layers as well as other wall-bounded flows. Though the linearized Navier-Stokes operator may be stable in such flows, the non-normality of the operator can lead to significant transient growth of disturbances, which can ultimately lead to transition. The importance of this transition mechanism is usually quantified by the maximum energy growth undergone by any disturbance to the boundary layer. Clearly, this is an upper bound for the (linear) transient growth experienced by real disturbances, and a statistical framework that studies the average, rather than extreme, disturbance behavior has recently been proposed. The framework is based on the assumption that disturbances evolve linearly, and it gives an exact relation between the two-point correlation tensor of the incipient disturbances and the turbulent intensity of downstream ones. The primary aims of this work are to implement this framework for a transitional Blasius boundary layer and to conduct a direct numerical simulation (DNS) of the same flow in order to test the predictions of the statistical framework. Applying the framework to spatial transient growth requires the use of a spatial marching method, and we use the one-way Navier-Stokes equations (OWNS). We also develop a reduced-basis technique to approximate formulae in the statistical framework that would otherwise be intractable for this flow. To model the statistics of the free-stream turbulence, the inflow disturbances are prescribed using the von K\'arm\'an energy spectrum. Finally, we present a comparison of the kinetic energy growth obtained from the DNS with that predicted by the statistical framework.