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.

Effect of Tee and Valve on the Flow Measurement Accuracy for Turbulent Pipe Flow With Flow Conditioner

2011 ◽  
Author(s):  
Boualem Laribi ◽  
Nahla Bouricha
Keyword(s):  
Experimental Study ◽  
Reynolds Number ◽  
Pipe Flow ◽  
Flow Measurement ◽  
Measurement Accuracy ◽  
Stokes Equations ◽  
Turbulent Pipe Flow ◽  
Navier Stokes ◽  
Turbulent Model ◽  
Navier Stokes Equations

This work describes the effect of a Tee and a Valve on the flow measurement accuracy and the performances of the E´toile flow straightener described by the standard ISO 5167 to produce the fully developed pipe flow with these disturbances. Simulation is carried out for an air flow in 100mm pipe diameter with a Reynolds number between 104 and 106. The code used for this work is Fluent V6.3, where the Navier-Stokes equations are solved by the finite volumes method with K-ε model like turbulent model. The results show that for the disturbance valve 50% closed, the length of establishment seems to be reached at 25D downstream the E´toile where the flow gyration angle is reduced practically to zero value. But for the Tee disturbance the results show that the flow needs more than 25D to reach the profiles requested by the standards. An experimental study is essential to validate these results for choosing a standard disturbance which will be examined with conditioners quoted in standard 5167 and thereafter the development of a new flow conditioner.

Stability of Hagen-Poiseuille flow with superimposed rigid rotation

1976 ◽  
Vol 73 (1) ◽  
pp. 153-164 ◽  
Author(s):  
P.-A. Mackrodt
Keyword(s):  
Poiseuille Flow ◽  
Pipe Flow ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Neutral Curve ◽  
Rigid Rotation ◽  
Transition To Turbulence ◽  
Navier Stokes ◽  
Navier Stokes Equations

The linear stability of Hagen-Poiseuille flow (Poiseuille pipe flow) with superimposed rigid rotation against small three-dimensional disturbances is examined at finite and infinite axial Reynolds numbers. The neutral curve, which is obtained by numerical solution of the system of perturbation equations (derived from the Navier-Stokes equations), has been confirmed for finite axial Reynolds numbers by a few simple experiments. The results suggest that, at high axial Reynolds numbers, the amount of rotation required for destabilization could be small enough to have escaped notice in experiments on the transition to turbulence in (nominally) non-rotating pipe flow.

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 New variational principles for locating periodic orbits of differential equations

2011 ◽  
Vol 369 (1944) ◽  
pp. 2211-2218 ◽  
Author(s):  
Bruce M. Boghosian ◽  
Luis M. Fazendeiro ◽  
Jonas Lätt ◽  
Hui Tang ◽  
Peter V. Coveney
Keyword(s):  
Periodic Orbits ◽  
Variational Principles ◽  
Stokes Equations ◽  
Iterative Algorithms ◽  
Navier Stokes ◽  
Unstable Periodic Orbits ◽  
Navier Stokes Equations ◽  
New Methods ◽  
Discrete Representations

We present new methods for the determination of periodic orbits of general dynamical systems. Iterative algorithms for finding solutions by these methods, for both the exact continuum case, and for approximate discrete representations suitable for numerical implementation, are discussed. Finally, we describe our approach to the computation of unstable periodic orbits of the driven Navier–Stokes equations, simulated using the lattice Boltzmann equation.

Swirling Laminar Pipe Flow of Suspensions

1973 ◽  
Vol 40 (2) ◽  
pp. 331-336 ◽  
Author(s):  
S. K. Tung ◽  
S. L. Soo
Keyword(s):  
Pipe Flow ◽  
Stokes Equations ◽  
Fluid Phase ◽  
Navier Stokes ◽  
Swirl Ratio ◽  
Particulate Phase ◽  
Flow Properties ◽  
Navier Stokes Equations ◽  
Flow Reynolds Number ◽  
Laminar Pipe Flow

Vortex pipe flow of suspensions with laminar motion in the fluid phase is treated. The pipe consists of two smoothly joined sections, one stationary and the other rotating with a constant angular velocity. The flow properties of the fluid phase are determined by solving the complete Navier-Stokes equations numerically. The governing parameters are the flow Reynolds number and swirl ratio. Subsequent numerical solution to the momentum equations governing the particulate phase provides for both particle velocity and concentration distributions.

Laminar Pipe Flow With Mass Transfer Cooling

1965 ◽  
Vol 87 (2) ◽  
pp. 252-258 ◽  
Author(s):  
Y. Peng ◽  
S. W. Yuan
Keyword(s):  
Temperature Distribution ◽  
Pipe Flow ◽  
Energy Equation ◽  
Stokes Equations ◽  
Porous Wall ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Coolant Injection ◽  
Laminar Pipe Flow ◽  
The Heat Transfer Coefficient

The effect of foreign coolant injection at the wall on the temperature distribution of a laminar flow of a fluid with variable transport and thermodynamic properties in a porous-wall pipe has been investigated. The velocity components, mass concentration, and temperature distribution were obtained by the solution of the Navier-Stokes equations, the diffusion equation, and the energy equation. A perturbation method was used to solve the first equations for small flows through the porous wall, and the eigenvalues in the latter two equations were calculated with the aid of the CDC 1604 computer. The results from this investigation depict the significant differences in both velocity distribution and temperature distribution between the present case of hydrogen coolant and the case of air coolant [1]. The results also show that the heat transfer coefficient at the wall in the present case is considerably smaller than the case of air-coolant injection.

The evolution of piston-driven pipe flow: thermal transpiration

2006 ◽  
Vol 463 (2079) ◽  
pp. 675-696
Author(s):  
A.M.J Davis
Keyword(s):  
Pipe Flow ◽  
State Transition ◽  
Stokes Equations ◽  
Solvability Condition ◽  
Navier Stokes ◽  
Entry And Exit ◽  
Flow Conditions ◽  
Additional Result ◽  
Navier Stokes Equations ◽  
Thermal Transpiration

The steady-state transition from and to the uniform entry and exit flow profiles is well described, at large aspect ratio, in terms of the stream function by the pipe eigenfunctions. But these latter are unsuited to oscillatory motion or the time evolution of the symmetric piston-driven pipe flow, for which an appropriate solution has a combination of a Fourier series along the finite pipe and a Fourier–Bessel series in the transverse direction. A non-uniqueness requires the identification of a solvability condition and care is needed in demonstrating its satisfaction. An additional result is that the solution must be constructed to satisfy the normal flow conditions identically. Application is made to thermal transpiration, recently explained by the revised Navier–Stokes equations and boundary conditions.

Time-dependent helical waves in rotating pipe flow

1990 ◽  
Vol 221 ◽  
pp. 289-310 ◽  
Author(s):  
Michael J. Landman
Keyword(s):  
Pipe Flow ◽  
Stokes Equations ◽  
Equations Of Motion ◽  
Reynolds Numbers ◽  
Linear Instability ◽  
Two Dimensions ◽  
Navier Stokes ◽  
Helical Symmetry ◽  
Navier Stokes Equations ◽  
Turbulent Transition

The Navier-Stokes equations for flow in a rotating circular pipe are solved numerically, subject to imposing helical symmetry on the velocity field v = v(r, θ + αz,t). The helical symmetry is exploited by writing the equations of motion in helical variables, reducing the problem to two dimensions. A limited study of the pipe flow is made in the parameter space of the wavenumber α, and the axial and azimuthal Reynolds numbers. The steadily rotating waves previously studied by Toplosky & Akylas (1988), which arise from the linear instability of the basic steady flow, are found to undergo a series of bifurcations, through periodic to aperiodic time dependence. The relevance of these results to the mechanism of laminar-turbulent transition in a stationary pipe is discussed.

INVARIANT MANIFOLDS FOR STOCHASTIC MODELS IN FLUID DYNAMICS

2011 ◽  
Vol 11 (02n03) ◽  
pp. 439-459
Author(s):  
SALAH-ELDIN A. MOHAMMED
Keyword(s):  
Fluid Dynamics ◽  
Invariant Manifolds ◽  
Stokes Equations ◽  
Stationary Solutions ◽  
Navier Stokes ◽  
Joint Work ◽  
Navier Stokes Equations ◽  
Infinite Dimensional ◽  
Stable Manifold Theorem ◽  
Local Stable Manifold

This paper is a survey of recent results on the dynamics of Stochastic Burgers equation (SBE) and two-dimensional Stochastic Navier–Stokes Equations (SNSE) driven by affine linear noise. Both classes of stochastic partial differential equations are commonly used in modeling fluid dynamics phenomena. For both the SBE and the SNSE, we establish the local stable manifold theorem for hyperbolic stationary solutions, the local invariant manifold theorem and the global invariant flag theorem for ergodic stationary solutions. The analysis is based on infinite-dimensional multiplicative ergodic theory techniques developed by D. Ruelle [22] (cf. [20, 21]). The results in this paper are based on joint work of the author with T. S. Zhang and H. Zhao ([17–19]).

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