Detection of Energetic Low Dimensional Subspaces in Spatio-Temporal Space in Turbulent Pipe Flow

Abstract Low dimensional subspaces are extracted out of highly complex turbulent pipe flow at $$Re_{\tau }=181$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>R</mml:mi> <mml:msub> <mml:mi>e</mml:mi> <mml:mi>τ</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>181</mml:mn> </mml:mrow> </mml:math> using a Characteristic Dynamic Mode Decomposition (CDMD). Having lower degrees of freedom, the subspaces provide a more clear basis to detect events which meet our understanding of large-scale coherent structures. To this end, a temporal sequence of state vectors from direct numerical simulations are rotated in space-time such that persistent dynamical modes on a hyper-surface are found travelling along its normal in space-time, which serves as the new time-like coordinate. The main flow features are captured with a minimal number of modes on a moving frame of reference whose velocity matches that of the most energetic scale. Reconstruction of the candidate modes in physical space gives the low rank model of the flow. The structures living in this subspace have long lifetimes, posses wide range of length-scales and travel at group velocities close to that of the moving frame of reference. The modes within this subspace are highly aligned, but are separated from the remaining modes by larger angles. We are able to capture the essential features of the flow like the spectral energy distribution and Reynolds stresses with a subspace consisting of about 10 modes. The remaining modes are collected in two further subspaces, which distinguish themselves by their axial length scale and degree of isotropy.