scholarly journals Attractor local dimensionality, nonlinear energy transfers and finite-time instabilities in unstable dynamical systems with applications to two-dimensional fluid flows

2013 ◽  
Vol 469 (2153) ◽  
pp. 20120550 ◽  
Author(s):  
Themistoklis P. Sapsis
Keyword(s):  
Finite Time ◽  
Stokes Equations ◽  
Fractal Dimensionality ◽  
Fluid Flows ◽  
Synergistic Activity ◽  
Energy Transfer Mechanism ◽  
Mean Flow ◽  
Low Dimensionality ◽  
The Mean ◽  
Nonlinear Energy

We examine the geometry of the finite-dimensional attractor associated with fluid flows described by Navier–Stokes equations and relate its nonlinear dimensionality to energy exchanges between dynamical components (modes) of the flow. Specifically, we use a stochastic framework based on the dynamically orthogonal equations to perform efficient order-reduction and describe the stochastic attractor in the reduced-order phase space in terms of the associated probability measure. We introduce the notion of local fractal dimensionality to describe the geometry of the attractor and we establish a connection with the number of positive finite-time Lyapunov exponents. Subsequently, we illustrate in specific fluid flows that the low dimensionality of the stochastic attractor is caused by the synergistic activity of linearly unstable and stable modes as well as the action of the quadratic terms. In particular, we illustrate the connection of the low-dimensionality of the attractor with the circulation of energy: (i) from the mean flow to the unstable modes (due to their linearly unstable character), (ii) from the unstable modes to the stable ones (due to a nonlinear energy transfer mechanism) and (iii) from the stable modes back to the mean (due to the linearly stable character of these modes).

Improved κ-ɛ Modelling of Impeller Flow Performance of a Mixed-Flow Pump under off-Design Operating States

1993 ◽  
Vol 207 (4) ◽  
pp. 219-229 ◽  
Author(s):  
S M Fraser ◽  
Y Zhang
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Careful Analysis ◽  
Navier Stokes ◽  
Mean Flow ◽  
Mixed Flow ◽  
Navier Stokes Equations ◽  
Flow Through ◽  
The Mean ◽  
Flow Pump

Three-dimensional turbulent flow through the impeller passage of a model mixed-flow pump has been simulated by solving the Navier-Stokes equations with an improved κ-ɛ model. The standard κ-ɛ model was found to be unsatisfactory for solving the off-design impeller flow and a converged solution could not be obtained at 49 per cent design flowrate. After careful analysis, it was decided to modify the standard κ-ɛ model by including the extra rates of strain due to the acceleration of impeller rotation and geometrical curvature and removing the mathematical ill-posedness between the mean flow turbulence modelling and the logarithmic wall function.

scholarly journals Parametric forcing approach to rough-wall turbulent channel flow

2012 ◽  
Vol 712 ◽  
pp. 169-202 ◽  
Author(s):  
A. Busse ◽  
N. D. Sandham
Keyword(s):  
Numerical Simulations ◽  
Channel Flow ◽  
Stokes Equations ◽  
Rough Surfaces ◽  
Turbulent Channel Flow ◽  
Rough Wall ◽  
Force Term ◽  
Shape Functions ◽  
Mean Flow ◽  
The Mean

AbstractThe effects of rough surfaces on turbulent channel flow are modelled by an extra force term in the Navier–Stokes equations. This force term contains two parameters, related to the density and the height of the roughness elements, and a shape function, which regulates the influence of the force term with respect to the distance from the channel wall. This permits a more flexible specification of a rough surface than a single parameter such as the equivalent sand grain roughness. The effects of the roughness force term on turbulent channel flow have been investigated for a large number of parameter combinations and several shape functions by direct numerical simulations. It is possible to cover the full spectrum of rough flows ranging from hydraulically smooth through transitionally rough to fully rough cases. By using different parameter combinations and shape functions, it is possible to match the effects of different types of rough surfaces. Mean flow and standard turbulence statistics have been used to compare the results to recent experimental and numerical studies and a good qualitative agreement has been found. Outer scaling is preserved for the streamwise velocity for both the mean profile as well as its mean square fluctuations in all but extremely rough cases. The structure of the turbulent flow shows a trend towards more isotropic turbulent states within the roughness layer. In extremely rough cases, spanwise structures emerge near the wall and the turbulent state resembles a mixing layer. A direct comparison with the study of Ashrafian, Andersson & Manhart (Intl J. Heat Fluid Flow, vol. 25, 2004, pp. 373–383) shows a good quantitative agreement of the mean flow and Reynolds stresses everywhere except in the immediate vicinity of the rough wall. The proposed roughness force term may be of benefit as a wall model for direct and large-eddy numerical simulations in cases where the exact details of the flow over a rough wall can be neglected.

Oscillatory forcing of flow through porous media. Part 1. Steady flow

2002 ◽  
Vol 465 ◽  
pp. 213-235 ◽  
Author(s):  
D. R. GRAHAM ◽  
J. J. L. HIGDON
Keyword(s):  
Porous Media ◽  
Flow Rate ◽  
Stokes Equations ◽  
Nonlinear Flow ◽  
Navier Stokes ◽  
Full Time ◽  
Inertial Effects ◽  
Mean Flow ◽  
Navier Stokes Equations ◽  
The Mean

Oscillatory forcing of a porous medium may have a dramatic effect on the mean flow rate produced by a steady applied pressure gradient. The oscillatory forcing may excite nonlinear inertial effects leading to either enhancement or retardation of the mean flow. Here, in Part 1, we consider the effects of non-zero inertial forces on steady flows in porous media, and investigate the changes in the flow character arising from changes in both the strength of the inertial terms and the geometry of the medium. The steady-state Navier–Stokes equations are solved via a Galerkin finite element method to determine the velocity fields for simple two-dimensional models of porous media. Two geometric models are considered based on constricted channels and periodic arrays of circular cylinders. For both geometries, we observe solution multiplicity yielding both symmetric and asymmetric flow patterns. For the cylinder arrays, we demonstrate that inertial effects lead to anisotropy in the effective permeability, with the direction of minimum resistance dependent on the solid volume fraction. We identify nonlinear flow phenomena which might be exploited by oscillatory forcing to yield a net increase in the mean flow rate. In Part 2, we take up the subject of unsteady flows governed by the full time-dependent Navier–Stokes equations.

A Numerical Study of Vortex Breakdown in Turbulent Swirling Flows

1999 ◽  
Vol 122 (1) ◽  
pp. 179-183 ◽  
Author(s):  
Robert E. Spall ◽  
Blake M. Ashby
Keyword(s):  
Reynolds Stress ◽  
Stokes Equations ◽  
Vortex Breakdown ◽  
Numerical Study ◽  
Reynolds Stress Model ◽  
Navier Stokes ◽  
Stress Model ◽  
Mean Flow ◽  
Navier Stokes Equations ◽  
The Mean

Solutions to the incompressible Reynolds-averaged Navier–Stokes equations have been obtained for turbulent vortex breakdown within a slightly diverging tube. Inlet boundary conditions were derived from available experimental data for the mean flow and turbulence kinetic energy. The performance of both two-equation and full differential Reynolds stress models was evaluated. Axisymmetric results revealed that the initiation of vortex breakdown was reasonably well predicted by the differential Reynolds stress model. However, the standard K-ε model failed to predict the occurrence of breakdown. The differential Reynolds stress model also predicted satisfactorily the mean azimuthal and axial velocity profiles downstream of the breakdown, whereas results using the K-ε model were unsatisfactory. [S0098-2202(00)01601-1]

scholarly journals Linear iterative method for closed-loop control of quasiperiodic flows

2019 ◽  
Vol 868 ◽  
pp. 26-65 ◽  
Author(s):  
Colin Leclercq ◽  
Fabrice Demourant ◽  
Charles Poussot-Vassal ◽  
Denis Sipp
Keyword(s):  
Turbulent Flows ◽  
Linear Models ◽  
Stokes Equations ◽  
Linear Time ◽  
Base Flow ◽  
Nonlinear Flow ◽  
Mean Flow ◽  
Loop Control ◽  
Linear Controller ◽  
The Mean

This work proposes a feedback-loop strategy to suppress intrinsic oscillations of resonating flows in the fully nonlinear regime. The frequency response of the flow is obtained from the resolvent operator about the mean flow, extending the framework initially introduced by McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382) to study receptivity mechanisms in turbulent flows. Using this linear time-invariant model of the nonlinear flow, modern control methods such as structured ${\mathcal{H}}_{\infty }$-synthesis can be used to design a controller. The approach is successful in damping self-sustained oscillations associated with specific eigenmodes of the mean-flow spectrum. Despite excellent performance, the linear controller is however unable to completely suppress flow oscillations, and the controlled flow is effectively attracted towards a new dynamical equilibrium. This new attractor is characterized by a different mean flow, which can in turn be used to design a second controller. The method can then be iterated on subsequent mean flows, until the coupled system eventually converges to the base flow. An intuitive parallel can be drawn with Newton’s iteration: at each step, a linearized model of the flow response to a perturbation of the input is sought, and a new linear controller is designed, aiming at further reducing the fluctuations. The method is illustrated on the well-known case of two-dimensional incompressible open-cavity flow at Reynolds number $Re=7500$, where the fully developed flow is initially quasiperiodic (2-torus state). The base flow is reached after five iterations. The present work demonstrates that nonlinear control problems may be solved without resorting to nonlinear reduced-order models. It also shows that physically relevant linear models can be systematically derived for nonlinear flows, without resorting to black-box identification from input–output data; the key ingredient being frequency-domain models based on the linearized Navier–Stokes equations about the mean flow. Applicability to amplifier flows and turbulent dynamics has, however, yet to be investigated.

A self-consistent formulation for the sensitivity analysis of finite-amplitude vortex shedding in the cylinder wake

2016 ◽  
Vol 800 ◽  
pp. 327-357 ◽  
Author(s):  
P. Meliga ◽  
E. Boujo ◽  
F. Gallaire
Keyword(s):  
Sensitivity Analysis ◽  
Limit Cycle ◽  
Stokes Equations ◽  
Finite Amplitude ◽  
Cylinder Wake ◽  
Mean Flow ◽  
Cycle Frequency ◽  
Flow Perturbation ◽  
Self Consistent ◽  
The Mean

We use the adjoint method to compute sensitivity maps for the limit-cycle frequency and amplitude of the Bénard–von Kármán vortex street in the wake of a circular cylinder. The sensitivity analysis is performed in the frame of the semi-linear self-consistent model recently introduced by Mantič et al. (Phys. Rev. Lett., vol. 113, 2014, 084501), which allows us to describe accurately the effect of the control on the mean flow, but also on the finite-amplitude fluctuation that couples back nonlinearly onto the mean flow via the formation of Reynolds stress. The sensitivity is computed with respect to arbitrary steady and synchronous time-harmonic body forces. For a small amplitude of the control, the theoretical variations of the limit-cycle frequency predict well those of the controlled flow, as obtained from either self-consistent modelling or direct numerical simulation of the Navier–Stokes equations. This is not the case if the variations are computed in the simpler mean flow approach overlooking the coupling between the mean and fluctuating components of the flow perturbation induced by the control. The variations of the limit-cycle amplitude (that falls out the scope of the mean flow approach) are also correctly predicted, meaning that the approach can serve as a relevant and systematic guideline to control strongly unstable flows exhibiting non-small, finite amplitudes of oscillation. As an illustration, we apply the method to control by means of a small secondary control cylinder and discuss the obtained results in the light of the seminal experiments of Strykowski & Sreenivasan (J. Fluid Mech., vol. 218, 1990, pp. 71–107).

Turbulent diffusion near a free surface

2000 ◽  
Vol 407 ◽  
pp. 145-166 ◽  
Author(s):  
LIAN SHEN ◽  
GEORGE S. TRIANTAFYLLOU ◽  
DICK K. P. YUE
Keyword(s):  
Boundary Layer ◽  
Free Surface ◽  
Turbulent Diffusion ◽  
Similarity Solution ◽  
Stokes Equations ◽  
Numerical Study ◽  
Turbulent Shear ◽  
Surface Boundary ◽  
Mean Flow ◽  
The Mean

We study numerically and analytically the turbulent diffusion characteristics in a low-Froude-number turbulent shear flow beneath a free surface. In the numerical study, the Navier–Stokes equations are solved directly subject to viscous boundary conditions at the free surface. From an ensemble of such simulations, we find that a boundary layer develops at the free surface characterized by a fast reduction in the value of the eddy viscosity. As the free surface is approached, the magnitude of the mean shear initially increases over the boundary (outer) layer, reaches a maximum and then drops to zero inside a much thinner inner layer. To understand and model this behaviour, we derive an analytical similarity solution for the mean flow. This solution predicts well the shape and the time-scaling behaviour of the mean flow obtained in the direct simulations. The theoretical solution is then used to derive scaling relations for the thickness of the inner and outer layers. Based on this similarity solution, we propose a free-surface function model for large-eddy simulations of free-surface turbulence. This new model correctly accounts for the variations of the Smagorinsky coefficient over the free-surface boundary layer and is validated in both a priori and a posteriori tests.

Flow Properties of Fluids Confined in Parallel-Plate Nanochannels

2012 ◽  
Vol 229-231 ◽  
pp. 22-25
Author(s):  
Zhong Qiang Zhang ◽  
Guang Gui Cheng ◽  
Jian Ning Ding ◽  
Zhi Yong Ling
Keyword(s):  
Pressure Gradient ◽  
Flow Velocity ◽  
Parallel Plate ◽  
Stokes Equations ◽  
Channel Width ◽  
Mean Flow ◽  
The Mean ◽  
Dynamics Simulations ◽  
Mean Flow Velocity ◽  
Pressure Driven

Molecular dynamics simulations are carried out to explore the fluid flows in parallel-plate nanochannels. A “channel moving” pressure-driven model is utilized to study the planar Poiseuille flows. Considering the slip boundary conditions, relationships among the pressure gradient, mean flow velocity and the channel width are investigated to couple the atomistic regime to continuum. The results show that the mean flow velocity almost linearly increases with the increase of the pressure gradient. The slope of the linear relationship between the pressure gradient and the mean flow velocity is nonlinearly decreased with increasing the channel width. The results indicate that the approximate accuracy is reduced with decreasing the channel width while the pressure-driven flows confined in nanochannels are approximately described by the Navier-Stokes equations.

Twin Inclined Circular Jets Evolution Through a Cool Crossflow

2007 ◽  
Author(s):  
Amina Radhouane ◽  
Nejla Mahjoub Sai¨d ◽  
Hatem Mhiri ◽  
George Lepalec ◽  
Philippe Bournot
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Mean Flow ◽  
Navier Stokes Equations ◽  
Vortical Structures ◽  
Circular Jets ◽  
Turbulent Closure ◽  
The Mean ◽  
Good Agreement

The aim of this paper is to examine experimentally as well as numerically the flowfield resulting from the interaction between a twin circular inclined hot jets emerging into a cooling crossflow. The resulting flowfield is quite complex due to the presence of different vortical structures including the kidney vortex, the horse-shoe vortex, etc... The evolution of the twin inclined jets through the crossflow could be depicted by tracking the mean-flow velocity field and its associated turbulence statistics by means of the PIV technique. This evolution can be influenced by many factors. Herein, we will deal with that resulted by the injection nozzles’ inclination and the jets’ spacing. Then, we performed a three dimensional sample of the studied configuration in order to simulate the evolution of the resulting flowfield. For that, the Navier Stokes equations were simulated with an RSM second order turbulent closure model. Then a non uniform meshing was applied. A good agreement was obtained between the experimental data and the numerical modeling. After validation we could represent in addition to the available results, the temperature distribution and the effects the variation of the injection inclination and that of the jets’ spacing bring on it (on its spatial evolution).

Numerical Flow Prediction in Inlet Pipe of Vertical Inline Pump

2017 ◽  
Vol 140 (5) ◽  
Author(s):  
Christopher Stephen ◽  
Shouqi Yuan ◽  
Ji Pei ◽  
Xing Cheng G
Keyword(s):  
Stokes Equations ◽  
Reverse Flow ◽  
Pressure Coefficient ◽  
Navier Stokes ◽  
Mean Flow ◽  
Inlet Pipe ◽  
Navier Stokes Equations ◽  
Computational Fluid Dynamics Cfd ◽  
The Mean ◽  
Good Agreement

For a pump, the inlet condition of flow determines the outlet conditions of fluid (i.e., energy). As a rule to minimize the losses at the entry of pump, the bends should be avoided as one of the methods. But for the case of vertical inline pump, it is unavoidable in order to save the space for installation. For the purpose of investigation in inlet pipe of vertical inline pump, the unsteady Reynolds-averaged Navier–Stokes equations are solved using the computational fluid dynamics (CFD) code. The results have been shown that there is a good agreement between the performance characteristics obtained from the simulation and experiments. The velocity coefficient from the simulation along the inlet pipe sections is well matched with the theoretical values and found to have variation near the exit of inlet pipe. The pressure and velocity coefficients studies depict the flow physics at each section along with the study of helicity at the exit of inlet pipe to determine the recirculation effects. It is observed that the vortices associated with the motion of the particles are moved toward the surfaces and are more intense than the mean flow. The trends of pressure coefficient at the exit of inlet pipe were addressed with reference to the various flow rates for eight set of radial lines. Hence, this work concludes that for inlet pipe, the generation of circulation was due to the stream path and the reverse flow from the impeller and was reconfirmed with the literature.

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