Dynamics of three-dimensional coherent structures in a flat-plate boundary layer

Abstract

An investigation is presented that analyses the energy flows that are connected to the dynamical behaviour of coherent structures in a transitional flat-plate boundary layer. Based on a mathematical description of the three-dimensional coherent structures of this flow as provided by the Karhunen–Loève procedure, energy equations for the coherent structures are derived by Galerkin projection of the Navier–Stokes equations in vorticity transport formulation onto the corresponding basis of eigenfunctions. In a first step, the time-averaged energy balance – showing the energy flows that support the different coherent structures and thus maintain the fluctuations of the velocity field – is considered. In a second step, the instantaneous power budget is investigated for the particularly interesting case of a coherent structure providing a prime contribution to the characteristic spike events of the transitional boundary layer. As this structure shows a strong variation in energy, the question about which mechanisms cause these variations is addressed. Our results show that the occurrence of a spike must be attributed to an autonomous event and cannot be interpreted as just an epiphenomenon of the passage of a Λ-vortex.

On the need for a nonlinear subscale turbulence term in POD models as exemplified for a high-Reynolds-number flow over an Ahmed body

“…Their ROM describes the coherent structures in the turbulent boundary layer, particularly sweeps or ejections. Other examples are the vortex shedding flow behind a circular cylinder at low Reynolds number (Deane et al 1991;Noack et al 2003), transitional and turbulent boundary layers (Aubry et al 1988;Rempfer & Fasel 1994b), a turbulent jet and the mixing layer (Rajaee et al 1994;Ukeiley et al 2001) and the lid-driven cavity flow (Cazemier et al 1998).…”

“…In the proposed POD models, we employ the modal eddy viscosity refinement by Rempfer & Fasel (1994b) and the nonlinear eddy viscosity scaling based on the FTT framework proposed by Noack et al (2011) to stabilize the long-term solution behaviour. The POD model utilizes a dataset of time-resolved flow fields of the flow around the bluff body.…”

Low-dimensional modelling of high-Reynolds-number shear flows incorporating constraints from the Navier–Stokes equation

“…Inclusion of a large number of POD basis functions creates very large POD-Galerkin ROMs that are still very computationally expensive to solve. Addition of turbulence models is equally undesirable because the empirical terms modify the dynamics of the Navier-Stokes equations (Rempfer & Fasel 1994;Aubry et al 1988;Ukeiley et al 2001;Sirisup & Karniadakis 2004;Iliescu & Wang 2012;Bailon-Cuba et al 2012).…”

On long-term boundedness of Galerkin models

“…Noack, Morzyński & Tadmor 2011). Examples of low-order models include boundary layers (Rempfer & Fasel 1994), cylinder wakes (Deane et al 1991;Noack et al 2003), mixing layers (Noack, Papas & Monkewitz 2005;Wei & Rowley 2009), lid-driven cavities (Cazemier, Verstappen & Veldman 1998;Balajewicz, Dowell & Noack 2013) and supersonic diffuser flows (Willcox & Megretski 2005). However, these models tend to be fragile: small changes of system parameters may give rise to unphysical divergent solutions, at least for a subset of initial conditions.…”