Bérengère Podvin

A low-dimensional approach for the minimal flow unit

The proper orthogonal decomposition (POD) is applied to the minimal flow unit (MFU) of a turbulent channel flow. Our purpose is to establish a numerical validation of low-dimensional models based on the POD. The simplest (two-mode) model possible is built for the simplified flow in the minimal unit. The dynamical behaviour predicted by the model is compared with that actually occurring in the direct numerical simulation of the flow. The various modelling assumptions which underlie the construction of low-dimensional models are examined and confronted with numerical evidence. The relationship between intermittency in the MFU and intermittent low-dimensional parameters is investigated closely. The agreement observed is quite satisfactory, especially given the crudeness of the truncation considered. To further demonstrate the adequacy of the model, we develop a dynamical filtering procedure to recover information from realistic (partial) measurements. The success obtained illustrates the versatility of the low-dimensional paradigm

Reconstructing the flow in the wall region from wall sensors

In this paper we examine how the flow in the wall region can be predicted from realistic (partial) measurements. The underlying motivation of this study is the improvement of control schemes for near-wall flows. We propose a method based on the proper orthogonal decomposition which provides estimated amplitudes for the coherent structures (i.e., the large scales) of the flow from wall measurements. The method is tested for the direct numerical simulation of a minimal flow unit. The large scales obtained by reconstruction from wall data are compared to those of the velocity field in the wall layer. The dominant structures—the streamwise streaks—are well recovered, the cross-stream motions less so since they are associated with higher-order structures unaccounted for in our truncation. We defined “rescaled” eigenfunctions to try to improve the representation of the cross-stream components of the flow. Aliasing effects due to realistic (large) sensor spacings were examined. We find that the spanwise spacing is the limiting factor for the estimation, so that a realistic grid will affect the reconstruction by at least 20% compared to full wall information.

Proper orthogonal decomposition investigation of turbulent Rayleigh-Bénard convection in a rectangular cavity

We consider the large-eddy simulation (LES) of turbulent Rayleigh-Benard convection for air in a parallepipedic cavity of ratio (1:5:1) over the range Ra = 6 × 108 up to Ra = 1010 previously studied in Sergent and Le Quere (Proceedings of the 13th European Turbulence Conference, 2011). Using proper orthogonal decomposition (POD) analysis, we confirm the existence of a large-scale circulation (LSC) consisting of quasi-stationary cross-stream rolls (y-rolls) which are aligned with the small direction of the box. Strong changes in the LSC are observed to take place over a few hundred convective time units, defined as κ/(Lx2Ra1/2), where κ is the fluid diffusivity, Lx is the height of the box and Ra is the Rayleigh number. We also show the existence of a secondary flow, which consists of horizontal rolls (z-rolls) surrounding the core of the cavity and orthogonal to the cross-stream rolls. The amplitude of these longitudinal rolls oscillates on a time scale of 50 convective units. The longitudinal rolls are a...

Lagrangian and Eulerian view of the bursting period

Low-dimensional models for the turbulent wall layer display an intermittent phenomenon with an ejection phase and a sweep phase that strongly resembles the bursting phenomenon observed in experimental flows. The probability distribution of inter-burst times has the observed shape [E. Stone and P. J. Holmes, Physica D 37, 20 (1989); SIAM J. Appl. Math. 50, 726 (1990); Phys. Lett. A 5, 29 (1991); P. J. Holmes and E. Stone, in Studies in Turbulence, edited by T. B. Gatski, S. Sarkar, and C. G. Speziale (Springer, Heidelberg, 1992)]. However, the time scales both for bursts and interburst durations are unrealistically long, a fact that was not appreciated until recently. We believe that the long time scales are due to the model’s inclusion of only a single coherent structure, when in fact a succession of quasi-independent structures are being swept past the sensor in an experiment. A simple statistical model of this situation restores the magnitude of the observed bursting period, although there is a great de...

Dynamical systems theory and extra rates of strain in turbulent flows

Peter Bradshaw has shed considerable light on turbulent flows with extra rates of strain. Here we apply a different approach to two of the flows that Bradshaw studied. We begin with a brief introduction reminding the reader of our application of dynamical systems theory to the flat-plate boundary layer. There we constructed a low-dimensional model that simulated the bursting in the boundary layer, in qualitative agreement with experiment. The bursting frequency responded to modeled polymer additives and flow accelerations by decreasing, resulting in a decreased skin friction, in agreement with experiment. Now we consider the application of this same technique to incompressible turbulent shear flows with streamline curvature in the direction of the mean flow, and to such flows with a transverse strain rate. We show qualitatively how the additional physical effects in these two flows will result in a changed rate of bursting, which will result in a change in the skin friction, likewise in agreement with experiment.

A few thoughts on proper orthogonal decomposition in turbulence

Proper orthogonal decomposition was originally introduced in turbulence to identify large-scale patterns in turbulent flows. Over the years, several extensions have been formulated in order to strengthen its model-predictive abilities, with limited success in the case of fully developed turbulence. We argue that physics-based insight obtained from the proper orthogonal decomposition structures and other turbulence analysis techniques could lead to significant developments in that respect. Numerical results from channel flow simulations are used to illustrate our conjectures.