Low-dimensional representations of exact coherent states of the Navier-Stokes equations from the resolvent model of wall turbulence.

We report that many exact invariant solutions of the Navier-Stokes equations for both pipe and channel flows are well represented by just a few modes of the model of McKeon and Sharma [J. Fluid Mech. 658, 336 (2010)]. This model provides modes that act as a basis to decompose the velocity field, ordered by their amplitude of response to forcing arising from the interaction between scales. The model was originally derived from the Navier-Stokes equations to represent turbulent flows and has been used to explain coherent structure and to predict turbulent statistics. This establishes a surprising new link between the two distinct approaches to understanding turbulence.

[1]  Fabian Waleffe,et al.  THREE-DIMENSIONAL COHERENT STATES IN PLANE SHEAR FLOWS , 1998 .

[2]  B. Eckhardt,et al.  Dynamical systems and the transition to turbulence in linearly stable shear flows , 2008, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[3]  S. Sherwin,et al.  Streamwise vortices in shear flows: harbingers of transition and the skeleton of coherent structures , 2010, Journal of Fluid Mechanics.

[4]  D. Viswanath,et al.  The critical layer in pipe flow at high Reynolds number , 2008, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[5]  J. Tropp,et al.  Model-based scaling of the streamwise energy density in high-Reynolds-number turbulent channels , 2013, Journal of Fluid Mechanics.

[6]  P. Schmid,et al.  Dynamic mode decomposition of numerical and experimental data , 2008, Journal of Fluid Mechanics.

[7]  F. Waleffe.  On a self-sustaining process in shear flows , 1997 .

[8]  M. Farge.  Wavelet Transforms and their Applications to Turbulence , 1992 .

[9]  B. J. McKeon,et al.  A critical-layer framework for turbulent pipe flow , 2010, Journal of Fluid Mechanics.

[10]  R. Moarref,et al.  A low-order decomposition of turbulent channel flow via resolvent analysis and convex optimization , 2014, 1401.6417.

[11]  R. Kerswell,et al.  Recent progress in understanding the transition to turbulence in a pipe , 2005 .

[12]  P. Hall,et al.  The high-Reynolds-number asymptotic development of nonlinear equilibrium states in plane Couette flow , 2014, Journal of Fluid Mechanics.

[13]  D. Dennis,et al.  Distinct organizational States of fully developed turbulent pipe flow. , 2014, Physical review letters.

[14]  Y. Duguet,et al.  Highly symmetric travelling waves in pipe flow , 2008, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[15]  D. Viswanath.  Recurrent motions within plane Couette turbulence , 2006, Journal of Fluid Mechanics.

[16]  J. Gibson,et al.  Visualizing the geometry of state space in plane Couette flow , 2007, Journal of Fluid Mechanics.

[17]  B. J. McKeon,et al.  Experimental manipulation of wall turbulence: a systems approach , 2013 .

[18]  F. Waleffe,et al.  Lower branch coherent states in shear flows: transition and control. , 2007, Physical review letters.

[19]  Rich R. Kerswell,et al.  Recurrence of Travelling Waves in Transitional Pipe Flow , 2007 .

[20]  P. Cvitanović,et al.  Geometry of the turbulence in wall-bounded shear flows: periodic orbits , 2010 .

[21]  P. Cvitanović,et al.  Symmetry reduction in high dimensions, illustrated in a turbulent pipe. , 2015, Physical review. E.

[22]  H. Wedin,et al.  Exact coherent structures in pipe flow: travelling wave solutions , 2003, Journal of Fluid Mechanics.

[23]  Genta Kawahara,et al.  Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst , 2001, Journal of Fluid Mechanics.

[24]  P. Holmes,et al.  The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows , 1993 .

[25]  B. McKeon,et al.  On the origin of frequency sparsity in direct numerical simulations of turbulent pipe flow , 2014 .

[26]  B. J. McKeon,et al.  On coherent structure in wall turbulence , 2013, Journal of Fluid Mechanics.

[27]  B. Eckhardt,et al.  Traveling waves in pipe flow. , 2003, Physical review letters.