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The Camassa–Holm equations and turbulence
 PHYSICA D 133 (1999) 49–65
, 1999
"... In this paper we will survey our results on the Camassa–Holm equations and their relation to turbulence as discussed in S. Chen, C. Foias, D.D. Holm, E. Olson, E.S. Titi, S. Wynne, The Camassa–Holm equations as a closure model for turbulent ..."
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Cited by 183 (24 self)
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In this paper we will survey our results on the Camassa–Holm equations and their relation to turbulence as discussed in S. Chen, C. Foias, D.D. Holm, E. Olson, E.S. Titi, S. Wynne, The Camassa–Holm equations as a closure model for turbulent
A connection between the CamassaHolm equations and turbulent flows in channels and pipes
 Physics of Fluids
, 1999
"... Abstract. In this paper we discuss recent progress in using the CamassaHolm equations to model turbulent flows. The CamassaHolm equations, given their special geometric and physical properties, appear particularly well suited for studying turbulent flows. We identify the steady solution of the Cam ..."
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Cited by 67 (19 self)
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Abstract. In this paper we discuss recent progress in using the CamassaHolm equations to model turbulent flows. The CamassaHolm equations, given their special geometric and physical properties, appear particularly well suited for studying turbulent flows. We identify the steady solution of the CamassaHolm equation with the mean flow of the Reynolds equation and compare the results with empirical data for turbulent flows in channels and pipes. The data suggests that the constant α version of the CamassaHolm equations, derived under the assumptions that the fluctuation statistics are isotropic and homogeneous, holds to order α distance from the boundaries. Near a boundary, these assumptions are no longer valid and the length scale α is seen to depend on the distance to the nearest wall. Thus, a turbulent flow is divided into two regions: the constant α region away from boundaries, and the near wall region. In the near wall region, Reynolds number scaling conditions imply that α decreases as Reynolds number increases. Away from boundaries, these scaling conditions imply α is independent of Reynolds number. Given the agreement with empirical and numerical data, our current work indicates that the CamassaHolm equations provide a promising theoretical framework from which to understand some turbulent flows.
Optimal and Robust Control and Estimation of Linear Paths to Transition
 J. Fluid Mech
, 1998
"... this paper is not valid. Iterative optimal control approaches over finite time intervals, which make use of full state information, may still be formulated (Abergel & Temam 1990) and performed (Moin & Bewley 1995) with impressive results. In order to make such schemes practical, one must und ..."
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Cited by 63 (10 self)
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this paper is not valid. Iterative optimal control approaches over finite time intervals, which make use of full state information, may still be formulated (Abergel & Temam 1990) and performed (Moin & Bewley 1995) with impressive results. In order to make such schemes practical, one must understand how to account for disturbances in a rigorous fashion and how to estimate accurately the necessary components of the state (for instance, the location and strength of the nearwall coherent structures) based on limited flow measurements. The present paper makes these concepts clear in a fluidmechanical sense, albeit for a linear problem, and thus provides a step in this development. Techniques to extend the robust control concept, introduced for problems in fluid mechanics in the present work, to nonlinear problems (such as turbulence) are discussed in Bewley, Moin & Temam (1997) and Bewley, Temam & Ziane (1998). 1.1. Outline of paper The structure of the remainder of the paper is: Section 2: the governing equations for the flow stability problem are put in a standard notation which makes subsequent application of control theory straightforward. Two specific cases are identified to be examined in detail: one supercritical and one subcritical. Section 3: the control approach and numerical method used are briefly summarized. Section 4: the methods used to analyse the openloop and closedloop systems are reviewed. Section 5: the uncontrolled (`openloop') systems are studied in detail. Section 6: the controlled (`closedloop') systems are studied in detail. Root loci, which demonstrate the movement of the closedloop system eigenvalues with respect to control parameters, are shown to illuminate some general trends, but fail to provide a quantitative measure of system performa...
2004 Scaling of the energy spectra of turbulent channels
 J. Fluid Mech
"... The spectra and correlations of the velocity fluctuations in turbulent channels, especially above the buffer layer, are analysed using new direct numerical simulations with friction Reynolds numbers up to Reτ =1900. It is found, and explained, that their scaling is anomalous in several respects, inc ..."
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Cited by 62 (6 self)
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The spectra and correlations of the velocity fluctuations in turbulent channels, especially above the buffer layer, are analysed using new direct numerical simulations with friction Reynolds numbers up to Reτ =1900. It is found, and explained, that their scaling is anomalous in several respects, including a squareroot behaviour of their width with respect to their length, and a velocity scaling of the largest modes with the centreline velocity Uc. It is shown that this implies a logarithmic correction to the k−1 energy spectrum, and that it leads to a scaling of the total fluctuation intensities away from the wall which agrees well with the mixed scaling of de Graaff & Eaton (2000) at intermediate Reynolds numbers, but which tends to a pure scaling with Uc at very large ones. 1.
Recurrent motions within plane Couette turbulence
 Journal of Fluid Mechanics
"... We describe accurate computations of threedimensional periodic and relative periodic motions within plane Couette turbulence at Re = 400. To ensure that the computed solutions are true solutions of the NavierStokes equations, careful attention is paid to time discretization errors and to spatial r ..."
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Cited by 53 (6 self)
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We describe accurate computations of threedimensional periodic and relative periodic motions within plane Couette turbulence at Re = 400. To ensure that the computed solutions are true solutions of the NavierStokes equations, careful attention is paid to time discretization errors and to spatial resolution. All the computed solutions are linearly unstable. While direct numerical simulation helps us understand the statistics of turbulent fluid flows, elucidation of the geometry of turbulent flows in phase space requires the computation of steady states, traveling waves, periodic motions, and close recurrences. The computed solutions are used as a basis to discuss the manner in which the geometry of turbulent dynamics in phase space can be understood. The method used for computing these solutions is described in detail.
Spanwise structure and scale growth in turbulent boundary layers
 2006 ANALYSIS OF DATA ON THE RELATION BETWEEN EDDIES 783
, 2003
"... Spanwise structure and growth mechanisms in a turbulent boundary layer are investigated experimentally. PIV measurements are obtained in the streamwise– spanwise (x–z)plane from the buffer layer to the top of the logarithmic region at Reθ =1015 and 7705. The dominant motions of the flow are shown t ..."
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Cited by 50 (1 self)
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Spanwise structure and growth mechanisms in a turbulent boundary layer are investigated experimentally. PIV measurements are obtained in the streamwise– spanwise (x–z)plane from the buffer layer to the top of the logarithmic region at Reθ =1015 and 7705. The dominant motions of the flow are shown to be largescale regions of momentum deficit elongated in the streamwise direction. Throughout the logarithmic layer, the regions are consistently bordered by vortices organized in the streamwise direction, offering strong support for a vortex packet model. Additionally, evidence is presented for the existence and organization of hairpin vortices in the region y+ < 60. Statistical evidence is also presented for two important aspects of the vortex packet paradigm: vortex organization in the streamwise direction, and the clear association of the hairpin signature with local minima in streamwise velocity. Several spanwise lengthscales are shown to vary linearly with distance from the wall, revealing selfsimilar growth of spanwise structure in an average sense. Inspection of the data, however, suggests that individual structures do not grow strictly selfsimilarly in time. It is proposed that additional scale growth occurs by the merging of vortex packets on an eddybyeddy basis via a vortex reconnection mechanism similar to that suggested by Wark & Nagib (1989). The proposed mechanism provides a link between vortexpairing concepts and the observed coalescence of streaky lowspeed regions in the inner layer. 1.
Symmetrypreserving discretization of turbulent flow
 J. Comp. Phys
, 2003
"... We propose to perform turbulent flow simulations in such manner that the difference operators do have the same symmetry properties as the underlying differential operators, i.e., the convective operator is represented by a skewsymmetric coefficient matrix and the diffusive operator is approximated ..."
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Cited by 47 (4 self)
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We propose to perform turbulent flow simulations in such manner that the difference operators do have the same symmetry properties as the underlying differential operators, i.e., the convective operator is represented by a skewsymmetric coefficient matrix and the diffusive operator is approximated by a symmetric, positivedefinite matrix. Mimicking crucial properties of differential operators forms in itself a motivation for discretizing them in a certain manner. We give it a concrete form by noting that a symmetrypreserving discretization of the Navier–Stokes equations is stable on any grid, and conserves the total mass, momentum and kinetic energy (for the latter the physical dissipation is to be turned off, of coarse). Being stable on any grid, the choice of the grid may be based on the required accuracy solely, and the main question becomes: how accurate is a symmetrypreserving discretization? Its accuracy is tested for a turbulent flow in a channel by comparing the results to those of physical experiments and previous numerical studies. The comparison is carried out for a Reynolds number of 5600, which is based on the channel width and the mean bulk velocity (based on the channel halfwidth and wall shear velocity the Reynolds number becomes 180). The comparison shows that with a fourthorder, symmetrypreserving method a 64 64 32 grid suffices to perform an accurate numerical simulation.
A spectral vanishing viscosity method for largeeddy simulations
 J. Comput. Phys
"... A new simulation approach for high Reynolds number turbulent flows is developed, combining concepts of monotonicity in nonlinear conservation laws with concepts of largeeddy simulation. The spectral vanishing viscosity (SVV), first introduced by E. Tadmor [SIAM J. Numer. Anal. 26, 30 (1989)], is in ..."
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Cited by 45 (3 self)
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A new simulation approach for high Reynolds number turbulent flows is developed, combining concepts of monotonicity in nonlinear conservation laws with concepts of largeeddy simulation. The spectral vanishing viscosity (SVV), first introduced by E. Tadmor [SIAM J. Numer. Anal. 26, 30 (1989)], is incorporated into the Navier– Stokes equations for controlling highwavenumber oscillations. Unlike hyperviscosity kernels, the SVV approach involves a secondorder operator which can be readily implemented in standard finite element codes. In the work presented here, discretization is performed using hierarchical spectral/hp methods accommodating effectively an ab initio intrinsic scale separation. The key result is that monotonicity is enforced via SVV leading to stable discretizations without sacrificing the formal accuracy, i.e., exponential convergence, in the proposed discretization. Several examples are presented to demonstrate the effectiveness of the new approach including a comparison with eddyviscosity spectral LES of turbulent channel flow. In its current implementation the SVV approach for controlling the small scales is decoupled from the large scales, but a procedure is proposed that will provide coupling similar to the classical LES formulation. c ○ 2000 Academic Press 1.
A systems theory approach to the feedback stabilization of infinitesimal and finiteamplitude disturbances in plane Poiseuille flow
, 1997
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