General variational model reduction applied to incompressible viscous flows

2008 ◽  
Vol 617 ◽  
pp. 31-50 ◽  
Author(s):  
THORSTEN BOGNER
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Reynolds Numbers ◽  
Spatial Dimension ◽  
Reduced Model ◽  
Navier Stokes ◽  
Variational Model ◽  
Navier Stokes Equation ◽  
Nonlinear Dynamical ◽  
Density Matrix Renormalization ◽  
Spatial Dimensions

In this paper, a method is introduced that allows calculation of an approximate proper orthogonal decomposition (POD) without the need to perform a simulation of the full dynamical system. Our approach is based on an application of the density matrix renormalization group (DMRG) to nonlinear dynamical systems, but has no explicit restriction on the spatial dimension of the model system. The method is not restricted to fluid dynamics. The applicability is exemplified on the incompressible Navier–Stokes equation in two spatial dimensions. Merging of two equal-signed vortices with periodic boundary conditions is considered for low Reynolds numbers Re≤800 using a spectral method. We compare the accuracy of a reduced model, obtained by our method, with that of a reduced model obtained by standard POD. To this end, error functionals for the reductions are evaluated. It is observed that the proposed method is able to find a reduced system that yields comparable or even superior accuracy with respect to standard POD method results.

Suppression of vortex shedding behind a circular cylinder by another control cylinder at low Reynolds numbers

2007 ◽  
Vol 573 ◽  
pp. 171-190 ◽  
Author(s):  
A. DIPANKAR ◽  
T. K. SENGUPTA ◽  
S. B. TALLA
Keyword(s):  
Vortex Shedding ◽  
Physical Mechanism ◽  
Grid Method ◽  
Reynolds Numbers ◽  
Accurate Solution ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Low Reynolds Numbers ◽  
Low Reynolds ◽  
Control Cylinder

Vortex shedding behind a cylinder can be controlled by placing another small cylinder behind it, at low Reynolds numbers. This has been demonstrated experimentally by Strykowski & Sreenivasan (J. Fluid Mech. vol. 218, 1990, p. 74). These authors also provided preliminary numerical results, modelling the control cylinder by the innovative application of boundary conditions on some selective nodes. There are no other computational and theoretical studies that have explored the physical mechanism. In the present work, using an over-set grid method, we report and verify numerically the experimental results for flow past a pair of cylinders. Apart from providing an accurate solution of the Navier–Stokes equation, we also employ an energy-based receptivity analysis method to discuss some aspects of the physical mechanism behind vortex shedding and its control. These results are compared with the flow picture developed using a dynamical system approach based on the proper orthogonal decomposition (POD) technique.

Steady Laminar Flow through Twisted Pipes: Fluid Flow in Square Tubes

1981 ◽  
Vol 103 (4) ◽  
pp. 785-790 ◽  
Author(s):  
J. H. Masliyah ◽  
K. Nandakumar
Keyword(s):  
Laminar Flow ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Dissipation Term ◽  
Axial Velocity Profile ◽  
Secondary Motion ◽  
Qualitative Nature ◽  
Flow Through ◽  
Steady Laminar

The Navier-Stokes equation in a rotating frame of reference is solved numerically to obtain the flow field for a steady, fully developed laminar flow of a Newtonian fluid in a twisted tube having a square cross-section. The macroscopic force and energy balance equations and the viscous dissipation term are presented in terms of variables in a rotating reference frame. The computed values of friction factor are presented for dimensionless twist ratios, (i.e., length of tube over a rotation of π radians normalized with respect to half the width of tube) of 20, 10, 5 and 2.5 and for Reynolds numbers up to 2000. The qualitative nature of the axial velocity profile was observed to be unaffected by the swirling motion. The secondary motion was found to be most important near the wall.

scholarly journals Numerical Simulations of the Impact and Spreading of a Particulate Drop on a Solid Substrate

2012 ◽  
Vol 2012 ◽  
pp. 1-10
Author(s):  
Hyun Jun Jeong ◽  
Wook Ryol Hwang ◽  
Chongyoup Kim
Keyword(s):  
Numerical Simulations ◽  
Solid Surface ◽  
Reynolds Numbers ◽  
Suspended Particles ◽  
Solid Substrate ◽  
Oscillatory Behavior ◽  
Navier Stokes ◽  
Droplet Spreading ◽  
Navier Stokes Equation ◽  
The Impact

We present two-dimensional numerical simulations of the impact and spreading of a droplet containing a number of small particles on a flat solid surface, just after hitting the solid surface, to understand particle effects on spreading dynamics of a particle-laden droplet for the application to the industrial inkjet printing process. The Navier-Stokes equation is solved by a finite-element-based computational scheme that employs the level-set method for the accurate interface description between the drop fluid and air and a fictitious domain method for suspended particles to account for full hydrodynamic interaction. Focusing on the particle effect on droplet spreading and recoil behaviors, we report that suspended particles suppress the droplet oscillation and deformation, by investigating the drop deformations for various Reynolds numbers. This suppressed oscillatory behavior of the particulate droplet has been interpreted with the enhanced energy dissipation due to the presence of particles.

The Onset of Turbulence: Insights Into the Navier-Stokes Equations

2017 ◽  
Author(s):  
Carl E. Rathmann
Keyword(s):  
Stokes Equations ◽  
Direct Consequence ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Fluid Behavior ◽  
Onset Of Turbulence ◽  
And Mathematics ◽  
High Reynolds

For well over 150 years now, theoreticians and practitioners have been developing and teaching students easily visualized models of fluid behavior that distinguish between the laminar and turbulent fluid regimes. Because of an emphasis on applications, perhaps insufficient attention has been paid to actually understanding the mechanisms by which fluids transition between these regimes. Summarized in this paper is the product of four decades of research into the sources of these mechanisms, at least one of which is a direct consequence of the non-linear terms of the Navier-Stokes equation. A scheme utilizing chaotic dynamic effects that become dominant only for sufficiently high Reynolds numbers is explored. This paper is designed to be of interest to faculty in the engineering, chemistry, physics, biology and mathematics disciplines as well as to practitioners in these and related applications.

Cylinder rolling on a wall at low Reynolds numbers

2011 ◽  
Vol 685 ◽  
pp. 461-494 ◽  
Author(s):  
Alain Merlen ◽  
Christophe Frankiewicz
Keyword(s):  
Three Dimensional ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Flow Around A Cylinder ◽  
Low Reynolds Numbers ◽  
Initial Location ◽  
Upstream Pressure ◽  
Small Reynolds Numbers ◽  
Onset Of Cavitation

AbstractThe flow around a cylinder rolling or sliding on a wall was investigated analytically and numerically for small Reynolds numbers, where the flow is known to be two-dimensional and steady. Both prograde and retrograde rotation were analytically solved, in the Stokes regime, giving the values of forces and torque and a complete description of the flow. However, solving Navier–Stokes equation, a rotation of the cylinder near the wall necessarily induces a cavitation bubble in the nip if the fluid is a liquid, or compressible effects, if it is a gas. Therefore, an infinite lift force is generated, disconnecting the cylinder from the wall. The flow inside this interstice was then solved under the lubrication assumptions and fully described for a completely flooded interstice. Numerical results extend the analysis to higher Reynolds number. Finally, the effect of the upstream pressure on the onset of cavitation is studied, giving the initial location of the phenomenon and the relation between the upstream pressure and the flow rate in the interstice. It is shown that the flow in the interstice must become three-dimensional when cavitation takes place.

Phase Field Modeling of Nonequilibrium Patterns on the Surface of a Liquid Film Under Lateral Oscillations at the Substrate

2014 ◽  
Vol 24 (09) ◽  
pp. 1450110 ◽  
Author(s):  
Rodica Borcia ◽  
Michael Bestehorn
Keyword(s):  
Liquid Film ◽  
Phase Field ◽  
Field Model ◽  
Phase Field Model ◽  
Solid Substrate ◽  
Navier Stokes ◽  
Bottom Plate ◽  
Navier Stokes Equation ◽  
Harmonic Vibrations ◽  
Spatial Dimensions

We use a phase field model which couples the generalized Navier–Stokes equation (including the Korteweg stress tensor) with the continuity equation for studying nonlinear pattern formation on the surface of a liquid film under (linear and circular) lateral harmonic vibrations at the solid substrate. First, we prove the thermodynamic consistency of our phase field model. Next, we present computer simulations in three spatial dimensions. We illustrate nonequilibrium patterns at the instability onset, confirming in this way the results recently reported in Phys. Rev. E 88, 023025 (2013). The lateral profiles of the deflected surface are compared with those reported in J. Fluid Mech. 686, 409 (2011) for Faraday instability excited by vertical harmonic vibrations at the bottom plate.

A two-dimensional vortex condensate at high Reynolds number

2013 ◽  
Vol 715 ◽  
pp. 359-388 ◽  
Author(s):  
Basile Gallet ◽  
William R. Young
Keyword(s):  
Stokes Equation ◽  
Numerical Solutions ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Finite Limit ◽  
Two Dimensional ◽  
Navier Stokes Equation ◽  
Quasilinear Theory ◽  
Zero Integral ◽  
High Reynolds

AbstractWe investigate solutions of the two-dimensional Navier–Stokes equation in a $\lrm{\pi} \ensuremath{\times} \lrm{\pi} $ square box with stress-free boundary conditions. The flow is steadily forced by the addition of a source $\sin nx\sin ny$ to the vorticity equation; attention is restricted to even $n$ so that the forcing has zero integral. Numerical solutions with $n= 2$ and $6$ show that at high Reynolds numbers the solution is a domain-scale vortex condensate with a strong projection on the gravest mode, $\sin x\sin y$. The sign of the vortex condensate is selected by a symmetry-breaking instability. We show that the amplitude of the vortex condensate has a finite limit as $\nu \ensuremath{\rightarrow} 0$. Using a quasilinear approximation we make an analytic prediction of the amplitude of the condensate and show that the amplitude is determined by viscous selection of a particular solution from a family of solutions to the forced two-dimensional Euler equation. This theory indicates that the condensate amplitude will depend sensitively on the form of the dissipation, even in the undamped limit. This prediction is verified by considering the addition of a drag term to the Navier–Stokes equation and comparing the quasilinear theory with numerical solution.

Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder

1957 ◽  
Vol 2 (3) ◽  
pp. 237-262 ◽  
Author(s):  
Ian Proudman ◽  
J. R. A. Pearson
Keyword(s):  
Circular Cylinder ◽  
Boundary Conditions ◽  
Reynolds Numbers ◽  
Polar Coordinates ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Flow Past ◽  
Small Reynolds Numbers ◽  
Cylindrical Polar Coordinates ◽  
Flow Past A Sphere

This paper is concerned with the problem of obtaining higher approximations to the flow past a sphere and a circular cylinder than those represented by the well-known solutions of Stokes and Oseen. Since the perturbation theory arising from the consideration of small non-zero Reynolds numbers is a singular one, the problem is largely that of devising suitable techniques for taking this singularity into account when expanding the solution for small Reynolds numbers.The technique adopted is as follows. Separate, locally valid (in general), expansions of the stream function are developed for the regions close to, and far from, the obstacle. Reasons are presented for believing that these ‘Stokes’ and ‘Oseen’ expansions are, respectively, of the forms $\Sigma \;f_n(R) \psi_n(r, \theta)$ and $\Sigma \; F_n(R) \Psi_n(R_r, \theta)$ where (r, θ) are spherical or cylindrical polar coordinates made dimensionless with the radius of the obstacle, R is the Reynolds number, and $f_{(n+1)}|f_n$ and $F_{n+1}|F_n$ vanish with R. Substitution of these expansions in the Navier-Stokes equation then yields a set of differential equations for the coefficients ψn and Ψn, but only one set of physical boundary conditions is applicable to each expansion (the no-slip conditions for the Stokes expansion, and the uniform-stream condition for the Oseen expansion) so that unique solutions cannot be derived immediately. However, the fact that the two expansions are (in principle) both derived from the same exact solution leads to a ‘matching’ procedure which yields further boundary conditions for each expansion. It is thus possible to determine alternately successive terms in each expansion.The leading terms of the expansions are shown to be closely related to the original solutions of Stokes and Oseen, and detailed results for some further terms are obtained.

SOLVING BURGERS EQUATION BY A MESHLESS METHOD

2005 ◽  
Vol 19 (28n29) ◽  
pp. 1651-1654 ◽  
Author(s):  
B. J. SHI ◽  
D. W. SHU ◽  
S. J. SONG ◽  
Y. M. ZHANG
Keyword(s):  
Shock Wave ◽  
Burgers Equation ◽  
Applied Mathematics ◽  
Reynolds Numbers ◽  
Least Square ◽  
Navier Stokes ◽  
Large Reynolds Number ◽  
Navier Stokes Equation ◽  
Point Collocation ◽  
Shock Wave Formation

Burgers equation is a fundamental partial differential equation of second order to describe the integrated process of convection-diffusion in physics. It occurs in various areas of applied mathematics and physics, such as modeling of turbulence, boundary layer behavior, shock wave formation, and mass transport. The convective and diffusive terms in Navier-Stokes equation are included in Burgers equation while the pressure term is neglected. A least-square point collocation meshless formula is proposed to discretize the Burgers equation. To verify the present meshless approach, the distributions of velocity for Burgers equation under different Reynolds numbers are investigated. Numerical results show that the proposed approach presents a good simulation of shock wave for Burgers equation with large Reynolds number.

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