On the adequacy of the ten-dimensional model for the wall layer

Abstract

In this paper, we provide a numerical validation of the ten-dimensional Proper Orthogonal Decomposition-based model constructed by Aubry et al. [J. Fluid Mech. 192, 115 (1988)] for the wall region of the turbulent boundary layer. Under certain conditions, this model was shown to display intermittent features highly reminiscent of the experimental observations of the bursting process in the wall layer, which makes it a potential key player in understanding and possibly controlling the dynamics of wall-bounded flows. In the same spirit as in our previous study [Podvin and Lumley, J. Fluid Mech. 362, 121 (1998)], we carried out a numerical simulation of a channel flow with relatively small horizontal dimensions which matched those in the 10-D model. The closure hypotheses used to build up the model were confronted with numerical results. Time histories of the modes in the model were compared to those of the simulation. Emphasis was put on identifying long-term characteristics such as a “mean” intermittency period. Our model, quite similar to Aubry’s, was found to display the same heteroclinic cycles under conditions consistent with the numerical experiment. The intermittency period in the model was found to agree well with that found in the simulation. However, the well-ordered character of 10-D bursts is significantly different from the simulation. To try and understand this discrepancy, we simulated a model with streamwise modes (32-D) and found evidence of increasing complexity in the bursts displayed.

Low-dimensional models for turbulent plane Couette flow in a minimal flow unit

“…As in § 2.3 we denote the 'true' modal amplitude obtained by projection of the DNS data onto the (n, n x , n z ) mode byâ (n) n x ,n z . We follow Podvin & Lumley (1998) and Podvin (2001) and calculate, for each of these true modal amplitudes, the effect of the neglected modes on the POD mode under consideration, given by…”

“…That paper and the present one take up earlier work begun by Aubry et al (1988) and Sanghi & Aubry (1993), cf. Berkooz, Holmes & Lumley (1991), Berkooz et al (1994), Podvin & Lumley (1998), Podvin (2001), and Sirovich (1987), Zhou & Sirovich (1992), Sirovich & Zhou (1994a, b), Webber, Handler & Sirovich (1997), on near-wall turbulence and channel flows, in which the proper orthogonal or Karhumen-Loève decomposition (POD), first suggested for use in turbulence by Lumley (1971), is used to construct optimal bases in the…”

Two-level discretizations of nonlinear closure models for proper orthogonal decomposition

“…, / r }) has always played a central role in POD reduced-order model strategies. Closure modeling has developed in two main directions (that often overlap): (i) improving the numerical stability [24][25][26]30,73,74]; and (ii) improving the physical accuracy [9,14,[16][17][18][19][20][21]75]. This paper is concerned with the latter.…”

“…The POD has its roots in statistics [1,2] and was initially used in the study of coherent structures in turbulent flows [3,4]. Over the past three decades, POD has been steadily developed in two main directions: (i) understanding the fundamental mechanisms in fluid flows, e.g., [5][6][7][8][9][10][11][12][13]; and (ii) developing reduced-order models (ROMs), i.e., models that predict the evolution of the dominant structures in the underlying complex system at a very low computational cost [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]. In this report, we are concerned with the latter.…”

Reduced-order description of fluid flow with moving boundaries by proper orthogonal decomposition

“…Aubry et al (1988) built a five-mode, reduced-order model for the wall region of a fully turbulent channel. Their effort was extended by Podvin (2001), who provided numerical validation of a ten-dimensional model for the wall layer. The ability of the ten-dimensional model to produce intermittent features, which are reminiscent of bursting process in a wall layer, was also demonstrated.…”