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A family of compressible exact periodic solutions is reported for the first time in an open cavity flow setup. These are found using a novel framework which permits the computation of such solutions in an arbitrary complex geometry. The periodic orbits arise from a synchronised concatenation of convective and acoustic events which strongly depend on the Mach number. This flow-acoustic interaction furnishes the periodic solutions with a remarkable stability and it is found to completely dominate the system's dynamics and the sound directivity. The periodic orbits, which could be called `exact Rossiter modes', collapse with a family of equilibrium solutions at a subcritical Hopf bifurcation, occurring in the quasi-incompressible regime. This shows compressibility has a destabilising effect in cavity flows, which we analyse in detail. By establishing a connection with previous 2D and 3D stability studies of cavity flows, we are able to isolate the effect of purely compressible two-dimensional flow phenomena across Mach number. A linear stability analysis of the equilibria provides insight into the compressible flow mechanisms responsible for the instability. A close look at the adjoint modes suggests that an eigenvalue merge occurs at a Mach number between 0.35 and 0.4, which boosts the receptivity of the leading mode and determines the onset of the unstable character of the system. The effect of the choice of base flow over the transition dynamics is also discussed, where in the present case, the frequencies associated to the leading eigenmodes show a strong connection with the frequencies of the periodic orbits at the same Mach numbers.

A novel adjoint-based framework oriented to optimal flow control in compressible direct numerical simulations is presented. Also, a new formulation of the adjoint characteristic boundary conditions is introduced, which enhances the stability of the adjoint simulations. The flow configuration chosen as a case study consists of a two dimensional open cavity flow with aspect ratio $L/H=3$ and Reynolds number $Re=5000$. This flow configuration is of particular interest, as the turbulent and chaotic nature of separated flows pushes the adjoint approach to its limit. The target of the flow actuation, defined as cost, is the reduction of the pressure fluctuations at the sensor location. To exploit the advantages of the adjoint method, a large number of control parameters is used. The control consists of an actuating sub-domain where a two-dimensional body force is applied at every point within the sub-volume. This results in a total of $2.256 \cdot 10^6$ control parameters. The final actuation achieved a successful reduction of the cost of $79.6\%$, by altering the directivity of the sound radiated by the trailing edge of the cavity and breaking up the large shear layer instabilities into smaller structures.

An analytical framework for studying the logarithmic region of turbulent channels is formulated. We build on recent findings (Moarref et al., J. Fluid Mech., 734, 2013) that the velocity fluctuations in the logarithmic region can be decomposed into a weighted sum of geometrically self-similar resolvent modes. The resolvent modes and the weights represent the linear amplification mechanisms and the scaling influence of the nonlinear interactions in the Navier-Stokes equations (NSE), respectively (McKeon & Sharma, J. Fluid Mech., 658, 2010). Originating from the NSE, this framework provides an analytical support for Townsend's attached-eddy model. Our main result is that self-similarity enables order reduction in modeling the logarithmic region by establishing a quantitative link between the self-similar structures and the velocity spectra. Specifically, the energy intensities, the Reynolds stresses, and the energy budget are expressed in terms of the resolvent modes with speeds corresponding to the top of the logarithmic region. The weights of the triad modes -the modes that directly interact via the quadratic nonlinearity in the NSE- are coupled via the interaction coefficients that depend solely on the resolvent modes (McKeon et al., Phys. Fluids, 25, 2013). We use the hierarchies of self-similar modes in the logarithmic region to extend the notion of triad modes to triad hierarchies. It is shown that the interaction coefficients for the triad modes that belong to a triad hierarchy follow an exponential function. The combination of these findings can be used to better understand the dynamics and interaction of flow structures in the logarithmic region. The compatibility of the proposed model with theoretical and experimental results is further discussed., Submitted to J. Fluid Mech