scholarly journals Correspondence between Koopman mode decomposition, resolvent mode decomposition, and invariant solutions of the Navier-Stokes equations

2016 ◽  
Vol 1 (3) ◽  
Author(s):  
Ati S. Sharma ◽  
Igor Mezić ◽  
Beverley J. McKeon
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Invariant Solutions ◽  
Navier Stokes Equations ◽  
Mode Decomposition

Directivity and intermittency in the nearfield of a Mach 1.3 jet

2017 ◽  
Vol 16 (3) ◽  
pp. 135-164 ◽  
Author(s):  
S Unnikrishnan ◽  
Datta V Gaitonde ◽  
Lionel Agostini
Keyword(s):  
Acoustic Field ◽  
Stokes Equations ◽  
Base Flow ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Local Fluctuations ◽  
Mode Decomposition ◽  
Gradual Evolution ◽  
Large Eddy ◽  
High Frequency Content

Local fluctuations in a Mach 1.3 cold jet are tracked to understand the genesis of nearfield directivity and intermittency. A newly developed approach leveraging two synchronized large-eddy simulations is employed to solve the forced Navier–Stokes equations, linearized about the evolving unsteady base flow. The results are summarized by exposing the effect of two acoustically significant turbulent regions: the lip-line and core collapse location. The near-acoustic field displays the clear signature of the two regions. However, for both regions, the nearfield evolution of the perturbation field is characterized by generation of intermittent wavepackets, which propagate into the near-acoustic field and gradually acquire their expected broadband and narrowband characteristics at sideline and downstream angles respectively. The simulations elucidate how higher frequencies are obtained in the sideline directions as lower frequencies are filtered out of the forcing fluctuations. Likewise, shallow-angle acoustic signals arise through filtering of high frequency content in that direction. The directivity and intermittency are connected to the filtering of scales by jet turbulence with empirical mode decomposition. The observations highlight the gradual evolution of seemingly random core turbulence into well-defined intermittent wavepackets in the nearfield of the jet. The manner in which centerline fluctuations are segregated into upstream, sideline, and downstream components is examined through narrowband correlations. A similar analysis for the lipline contribution shows primarily upstream and downstream patterns because of the larger structures in the shear layer.

Similarity in non-rotating and rotating turbulent pipe flows

1999 ◽  
Vol 379 ◽  
pp. 1-22 ◽  
Author(s):  
MARTIN OBERLACK
Keyword(s):  
Reynolds Number ◽  
Scaling Laws ◽  
Stokes Equations ◽  
Scaling Law ◽  
Navier Stokes ◽  
Invariant Solutions ◽  
Mean Velocity ◽  
Navier Stokes Equations ◽  
Pipe Flows ◽  
The Mean

The Lie group approach developed by Oberlack (1997) is used to derive new scaling laws for high-Reynolds-number turbulent pipe flows. The scaling laws, or, in the methodology of Lie groups, the invariant solutions, are based on the mean and fluctuation momentum equations. For their derivation no assumptions other than similarity of the Navier–Stokes equations have been introduced where the Reynolds decomposition into the mean and fluctuation quantities has been implemented. The set of solutions for the axial mean velocity includes a logarithmic scaling law, which is distinct from the usual law of the wall, and an algebraic scaling law. Furthermore, an algebraic scaling law for the azimuthal mean velocity is obtained. In all scaling laws the origin of the independent coordinate is located on the pipe axis, which is in contrast to the usual wall-based scaling laws. The present scaling laws show good agreement with both experimental and DNS data. As observed in experiments, it is shown that the axial mean velocity normalized with the mean bulk velocity um has a fixed point where the mean velocity equals the bulk velocity independent of the Reynolds number. An approximate location for the fixed point on the pipe radius is also given. All invariant solutions are consistent with all higher-order correlation equations. A large-Reynolds-number asymptotic expansion of the Navier–Stokes equations on the curved wall has been utilized to show that the near-wall scaling laws for at surfaces also apply to the near-wall regions of the turbulent pipe flow.

scholarly journals Koopman mode expansions between simple invariant solutions

2019 ◽  
Vol 879 ◽  
pp. 1-27 ◽  
Author(s):  
Jacob Page ◽  
Rich R. Kerswell
Keyword(s):  
State Space ◽  
Stokes Equations ◽  
Time Window ◽  
Landau Equation ◽  
Dynamic Mode Decomposition ◽  
Navier Stokes ◽  
Dynamic Mode ◽  
Invariant Solutions ◽  
Mode Decomposition ◽  
Heteroclinic Connection

A Koopman decomposition is a powerful method of analysis for fluid flows leading to an apparently linear description of nonlinear dynamics in which the flow is expressed as a superposition of fixed spatial structures with exponential time dependence. Attempting a Koopman decomposition is simple in practice due to a connection with dynamic mode decomposition (DMD). However, there are non-trivial requirements for the Koopman decomposition and DMD to overlap, which mean it is often difficult to establish whether the latter is truly approximating the former. Here, we focus on nonlinear systems containing multiple simple invariant solutions where it is unclear how to construct a consistent Koopman decomposition, or how DMD might be applied to locate these solutions. First, we derive a Koopman decomposition for a heteroclinic connection in a Stuart–Landau equation revealing two possible expansions. The expansions are centred about the two fixed points of the equation and extend beyond their linear subspaces before breaking down at a cross-over point in state space. Well-designed DMD can extract the two expansions provided that the time window does not contain this cross-over point. We then apply DMD to the Navier–Stokes equations near to a heteroclinic connection in low Reynolds number ($Re=O(100)$) plane Couette flow where there are multiple simple invariant solutions beyond the constant shear basic state. This reveals as many different Koopman decompositions as simple invariant solutions present and once more indicates the existence of cross-over points between the expansions in state space. Again, DMD can extract these expansions only if it does not include a cross-over point. Our results suggest that in a dynamical system possessing multiple simple invariant solutions, there are generically places in phase space – plausibly hypersurfaces delineating the boundary of a local Koopman expansion – across which the dynamics cannot be represented by a convergent Koopman expansion.

Symmetry Reductions and Exact Solutions of the (2+1)-Dimensional Navier-Stokes Equations

2010 ◽  
Vol 65 (6-7) ◽  
pp. 504-510 ◽  
Author(s):  
Xiaorui Hua ◽  
Zhongzhou Dongb ◽  
Fei Huangc ◽  
Yong Chena
Keyword(s):  
Vector Fields ◽  
Stokes Equations ◽  
Stationary Wave ◽  
Navier Stokes ◽  
Invariant Solutions ◽  
Navier Stokes Equations ◽  
One Dimensional ◽  
Group Invariant ◽  
Dimensional Group ◽  
Group Invariant Solutions

By means of the classical symmetry method, we investigate the (2+1)-dimensional Navier-Stokes equations. The symmetry group of Navier-Stokes equations is studied and its corresponding group invariant solutions are constructed. Ignoring the discussion of the infinite-dimensional subalgebra, we construct an optimal system of one-dimensional group invariant solutions. Furthermore, using the associated vector fields of the obtained symmetry, we give out the reductions by one-dimensional and two-dimensional subalgebras, and some explicit solutions of Navier-Stokes equations are obtained. For three interesting solutions, the figures are given out to show their properties: the solution of stationary wave of fluid (real part) appears as a balance between fluid advection (nonlinear term) and friction parameterized as a horizontal harmonic diffusion of momentum.

Semi-invariant solutions of the Navier-Stokes equations

2018 ◽  
Vol 41 (8) ◽  
pp. 2853-2893
Author(s):  
V. Rosenhaus ◽  
Ravi Shankar ◽  
Cody Squellati
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Invariant Solutions ◽  
Navier Stokes Equations

scholarly journals Computing heteroclinic orbits using adjoint-based methods

2018 ◽  
Vol 858 ◽  
Author(s):  
M. Farano ◽  
S. Cherubini ◽  
J.-C. Robinet ◽  
P. De Palma ◽  
T. M. Schneider
Keyword(s):  
Stokes Equations ◽  
Travelling Waves ◽  
Heteroclinic Orbits ◽  
Navier Stokes ◽  
Invariant Solutions ◽  
Shooting Methods ◽  
Equilibrium Solutions ◽  
Connecting Orbits ◽  
Navier Stokes Equations ◽  
Exact Invariant

Transitional turbulence in shear flows is supported by a network of unstable exact invariant solutions of the Navier–Stokes equations. The network is interconnected by heteroclinic connections along which the turbulent trajectories evolve between invariant solutions. While many invariant solutions in the form of equilibria, travelling waves and periodic orbits have been identified, computing heteroclinic connections remains a challenge. We propose a variational method for computing orbits dynamically connecting small neighbourhoods around equilibrium solutions. Using local information on the dynamics linearized around these equilibria, we demonstrate that we can choose neighbourhoods such that the connecting orbits shadow heteroclinic connections. The proposed method allows one to approximate heteroclinic connections originating from states with multi-dimensional unstable manifold and thereby provides access to heteroclinic connections that cannot easily be identified using alternative shooting methods. For plane Couette flow, we demonstrate the method by recomputing three known connections and identifying six additional previously unknown orbits.

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