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

2016 ◽  
Vol 93 (2) ◽  
Author(s):  
Ati S. Sharma ◽  
Rashad Moarref ◽  
Beverley J. McKeon ◽  
Jae Sung Park ◽  
Michael D. Graham ◽  
...  
Keyword(s):  
Coherent States ◽  
Stokes Equations ◽  
Wall Turbulence ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Low Dimensional

The engine behind (wall) turbulence: perspectives on scale interactions

2017 ◽  
Vol 817 ◽  
Author(s):  
B. J. McKeon
Keyword(s):  
Coherent States ◽  
Systems Approach ◽  
Stokes Equations ◽  
Stokes Operator ◽  
Wall Turbulence ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Scale Interactions ◽  
High Reynolds ◽  
Data Informed

Known structures and self-sustaining mechanisms of wall turbulence are reviewed and explored in the context of the scale interactions implied by the nonlinear advective term in the Navier–Stokes equations. The viewpoint is shaped by the systems approach provided by the resolvent framework for wall turbulence proposed by McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382), in which the nonlinearity is interpreted as providing the forcing to the linear Navier–Stokes operator (the resolvent). Elements of the structure of wall turbulence that can be uncovered as the treatment of the nonlinearity ranges from data-informed approximation to analysis of exact solutions of the Navier–Stokes equations (so-called exact coherent states) are discussed. The article concludes with an outline of the feasibility of extending this kind of approach to high-Reynolds-number wall turbulence in canonical flows and beyond.

scholarly journals Local improvements to reduced-order models using sensitivity analysis of the proper orthogonal decomposition

2009 ◽  
Vol 629 ◽  
pp. 41-72 ◽  
Author(s):  
ALEXANDER HAY ◽  
JEFFREY T. BORGGAARD ◽  
DOMINIQUE PELLETIER
Keyword(s):  
Sensitivity Analysis ◽  
Proper Orthogonal Decomposition ◽  
Stokes Equations ◽  
Orthogonal Decomposition ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Reduced Order ◽  
Selection Step ◽  
Low Dimensional ◽  
Proper Orthogonal

The proper orthogonal decomposition (POD) is the prevailing method for basis generation in the model reduction of fluids. A serious limitation of this method, however, is that it is empirical. In other words, this basis accurately represents the flow data used to generate it, but may not be accurate when applied ‘off-design’. Thus, the reduced-order model may lose accuracy for flow parameters (e.g. Reynolds number, initial or boundary conditions and forcing parameters) different from those used to generate the POD basis and generally does. This paper investigates the use of sensitivity analysis in the basis selection step to partially address this limitation. We examine two strategies that use the sensitivity of the POD modes with respect to the problem parameters. Numerical experiments performed on the flow past a square cylinder over a range of Reynolds numbers demonstrate the effectiveness of these strategies. The newly derived bases allow for a more accurate representation of the flows when exploring the parameter space. Expanding the POD basis built at one state with its sensitivity leads to low-dimensional dynamical systems having attractors that approximate fairly well the attractor of the full-order Navier–Stokes equations for large parameter changes.

scholarly journals A framework for studying the effect of compliant surfaces on wall turbulence

2015 ◽  
Vol 768 ◽  
pp. 415-441 ◽  
Author(s):  
M. Luhar ◽  
A. S. Sharma ◽  
B. J. McKeon
Keyword(s):  
Rational Design ◽  
Large Scale ◽  
Stokes Equations ◽  
Positive Influence ◽  
Wall Turbulence ◽  
Navier Stokes ◽  
Spectral Space ◽  
Linear Superposition ◽  
Navier Stokes Equations ◽  
Compliant Surfaces

This paper extends the resolvent formulation proposed by McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382) to consider turbulence–compliant wall interactions. Under this formulation, the turbulent velocity field is expressed as a linear superposition of propagating modes, identified via a gain-based decomposition of the Navier–Stokes equations. Compliant surfaces, modelled as a complex wall admittance linking pressure and velocity, affect the gain and structure of these modes. With minimal computation, this framework accurately predicts the emergence of the quasi-two-dimensional propagating waves observed in recent direct numerical simulations. Further, the analysis also enables the rational design of compliant surfaces, with properties optimized to suppress flow structures energetic in wall turbulence. It is shown that walls with unphysical negative damping are required to interact favourably with modes resembling the energetic near-wall cycle, which could explain why previous studies have met with limited success. Positive-damping walls are effective for modes resembling the so-called very-large-scale motions, indicating that compliant surfaces may be better suited for application at higher Reynolds number. Unfortunately, walls that suppress structures energetic in natural turbulence are also predicted to have detrimental effects elsewhere in spectral space. Consistent with previous experiments and simulations, slow-moving spanwise-constant structures are particularly susceptible to further amplification. Mitigating these adverse effects will be central to the development of compliant coatings that have a net positive influence on the flow.

Stochastic low-dimensional modelling of a random laminar wake past a circular cylinder

2008 ◽  
Vol 606 ◽  
pp. 339-367 ◽  
Author(s):  
DANIELE VENTURI ◽  
XIAOLIANG WAN ◽  
GEORGE EM KARNIADAKIS
Keyword(s):  
Circular Cylinder ◽  
Turbulent Flows ◽  
Stokes Equations ◽  
Stochastic Simulations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Spatial Modes ◽  
Laminar Wake ◽  
Low Dimensional ◽  
Proper Orthogonal

We present a new compact expansion of a random flow field into stochastic spatial modes, hence extending the proper orthogonal decomposition (POD) to noisy (non-coherent) flows. As a prototype problem, we consider unsteady laminar flow past a circular cylinder subject to random inflow characterized as a stationary Gaussian process. We first obtain random snapshots from full stochastic simulations (based on polynomial chaos representations), and subsequently extract a small number of deterministic modes and corresponding stochastic modes by solving a temporal eigenvalue problem. Finally, we determine optimal sets of random projections for the stochastic Navier–Stokes equations, and construct reduced-order stochastic Galerkin models. We show that the number of stochastic modes required in the reconstruction does not directly depend on the dimensionality of the flow system. The framework we propose is general and it may also be useful in analysing turbulent flows, e.g. in quantifying the statistics of energy exchange between coherent modes.

scholarly journals Alternative physics to understand wall turbulence: Navier–Stokes equations with modified linear dynamics

2020 ◽  
Vol 1522 ◽  
pp. 012003
Author(s):  
Adrián Lozano-Durán ◽  
Marios-Andreas Nikolaidis ◽  
Navid C. Constantinou ◽  
Michael Karp
Keyword(s):  
Stokes Equations ◽  
Wall Turbulence ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Linear Dynamics

scholarly journals Relative periodic orbits form the backbone of turbulent pipe flow

2017 ◽  
Vol 833 ◽  
pp. 274-301 ◽  
Author(s):  
N. B. Budanur ◽  
K. Y. Short ◽  
M. Farazmand ◽  
A. P. Willis ◽  
P. Cvitanović
Keyword(s):  
Periodic Orbits ◽  
Pipe Flow ◽  
Chaotic Dynamics ◽  
Stokes Equations ◽  
Turbulent Pipe Flow ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Infinite Dimensional ◽  
Unstable Manifolds ◽  
Low Dimensional

The chaotic dynamics of low-dimensional systems, such as Lorenz or Rössler flows, is guided by the infinity of periodic orbits embedded in their strange attractors. Whether this is also the case for the infinite-dimensional dynamics of Navier–Stokes equations has long been speculated, and is a topic of ongoing study. Periodic and relative periodic solutions have been shown to be involved in transitions to turbulence. Their relevance to turbulent dynamics – specifically, whether periodic orbits play the same role in high-dimensional nonlinear systems like the Navier–Stokes equations as they do in lower-dimensional systems – is the focus of the present investigation. We perform here a detailed study of pipe flow relative periodic orbits with energies and mean dissipations close to turbulent values. We outline several approaches to reduction of the translational symmetry of the system. We study pipe flow in a minimal computational cell at $Re=2500$, and report a library of invariant solutions found with the aid of the method of slices. Detailed study of the unstable manifolds of a sample of these solutions is consistent with the picture that relative periodic orbits are embedded in the chaotic saddle and that they guide the turbulent dynamics.

Centrifugal effects in rotating convection: nonlinear dynamics

2009 ◽  
Vol 628 ◽  
pp. 269-297 ◽  
Author(s):  
J. M. LOPEZ ◽  
F. MARQUES
Keyword(s):  
Nonlinear Dynamics ◽  
Linear Stability ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Limited Attention ◽  
Navier Stokes Equations ◽  
Rotating Convection ◽  
Simplifying Assumptions ◽  
Basic States ◽  
Low Dimensional

Rotating convection in cylindrical containers is a canonical problem in fluid dynamics, in which a variety of simplifying assumptions have been used in order to allow for low-dimensional models or linear stability analysis from trivial basic states. An aspect of the problem that has received only limited attention is the influence of the centrifugal force, because it makes it difficult or even impossible to implement the aforementioned approaches. In this study, the mutual interplay between the three forces of the problem, Coriolis, gravitational and centrifugal buoyancy, is examined via direct numerical simulation of the Navier–Stokes equations in a parameter regime where the three forces are of comparable strengths in a cylindrical container with the radius equal to the depth so that wall effects are also of order one. Two steady axisymmetric basic states exist in this regime, and the nonlinear dynamics of the solutions bifurcating from them is explored in detail. A variety of bifurcated solutions and several codimension-two bifurcation points acting as organizing centres for the dynamics have been found. A main result is that the flow has simple dynamics for either weak heating or large centrifugal buoyancy. Reducing the strength of centrifugal buoyancy leads to subcritical bifurcations, and as a result linear stability is of limited utility, and direct numerical simulations or laboratory experiments are the only way to establish the connections between the different solutions and their organizing centres, which result from the competition between the three forces. Centrifugal effects primarily lead to the axisymmetrization of the flow and a reduction in the heat flux.

scholarly journals Transition delay using control theory

2011 ◽  
Vol 369 (1940) ◽  
pp. 1365-1381 ◽  
Author(s):  
S. Bagheri ◽  
D. S. Henningson
Keyword(s):  
Control Theory ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Control Approach ◽  
Disciplinary Field ◽  
Flow Configurations ◽  
Transition Delay ◽  
Low Dimensional

This review gives an account of recent research efforts to use feedback control for the delay of laminar–turbulent transition in wall-bounded shear flows. The emphasis is on reducing the growth of small-amplitude disturbances in the boundary layer using numerical simulations and a linear control approach. Starting with the application of classical control theory to two-dimensional perturbations developing in spatially invariant flows, flow control based on control theory has progressed towards more realistic three-dimensional, spatially inhomogeneous flow configurations with localized sensing/actuation. The development of low-dimensional models of the Navier–Stokes equations has played a key role in this progress. Moreover, shortcomings and future challenges, as well as recent experimental advances in this multi-disciplinary field, are discussed.

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