Using a data base generated by a numerical simulation, the three-dimensional coherent structures of a transitional, spatially evolving boundary layer are determined and their spatio-temporal behaviour is investigated in detail. The coherent structures are calculated by the proper orthogonal decomposition method (POD), which leads to an expansion of the flow field variables into Karhunen-Loéve eigenfunctions. It is shown that the dynamical coherent structures of the flat-plate boundary layer can be described by pairs of eigenfunctions that contain complete information on the spatial evolution of the structures. It is further demonstrated that first-order coherent structures determined by POD correspond to structures that are observed in experiments. In the region of the boundary layer where the spike signals of transition occur, higher-order coherent structures also play an essential role. By considering these higher-order structures as well as their dynamical behaviour in time, a compact description of the flow phenomena in the boundary layer can be obtained. The description of the events occurring at the spike stages of the transitional boundary layer shows, from a coherent structures point of view, striking similarities to the bursting event of fully turbulent boundary layers.

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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.

We revisit the issue of finding proper boundary conditions for the field equations describing incompressible flow problems, for quantities like pressure or vorticity, which often do not have immediately obvious “physical” boundary conditions. Most of the issues are discussed for the example of a primitive-variables formulation of the incompressible Navier-Stokes equations in the form of momentum equations plus the pressure Poisson equation. However, analogous problems also exist in other formulations, some of which are briefly reviewed as well. This review article cites 95 references.

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▪ Abstract  This review describes some of the important developments in the numerical investigation of transition to turbulence in wall-bounded and free shear flows during the past decade. The evolution of numerical techniques and models as well as the advances in our theoretical understanding of the physics of laminar-turbulent transition that were achieved using these tools are described. For wall-bounded flows, particular emphasis is placed on investigations studying various scenarios of “bypass transition” in flows that are asymptotically stable. A brief review of investigations into receptivity and control of transitional flows is included.

In this paper we present results from a numerical investigation of turbulent channel flow in the presence of a compliant wall. The compliant wall is modelled as a homogeneous spring-supported plate. The simulation code is validated both by comparison with an alternative code and by reproducing results of linear stability theory. Our results demonstrate that with the wall compliance we used in the simulation there is little change in the very long-time behaviour of the turbulent skin friction drag and little modification to the near-wall turbulent coherent structures. The values of pertinent statistical quantities of the turbulence near the compliant walls converge to those near a rigid wall and the statistical effect of the wall compliance on the turbulent channel flow is small.

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