Circular Couette Flow between Porous Cylinders

2005 ◽  
Vol 219 (8) ◽  
pp. 743-747 ◽  
Author(s):  
V. D. Djordjevic
Keyword(s):  
Couette Flow ◽  
Effective Viscosity ◽  
Stokes Equations ◽  
Outer Cylinder ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Circular Couette Flow ◽  
Special Case ◽  
Porous Boundaries ◽  
Porous Cylinders

Circular Couette flow between porous cylinders is treated in this paper. Exact solutions of Navier-Stokes equations for the flow in the gap between cylinders and of Darcy-Brinkman-Lapwood equations for the flow in porous rings are found analytically by matching velocities and stresses on the porous boundaries, without making any previous assumptions concerning the slip velocities. In the special case in which the inner cylinder is fully porous and stationary, and the outer cylinder is fully rigid and rotating, the torque exerted by the rotation of the outer cylinder on the inner one is found, and it is shown how the effective viscosity of the liquid can be determined in possible experiments.

Pulsatile Waterhammer Subject to Laminar Friction

1972 ◽  
Vol 94 (2) ◽  
pp. 467-472 ◽  
Author(s):  
D. A. P. Jayasinghe ◽  
H. J. Leutheusser
Keyword(s):  
Stokes Equations ◽  
Fluid Motion ◽  
Present Theory ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
One Dimensional ◽  
Arbitrary Velocity ◽  
Velocity Input ◽  
Special Case ◽  
Signalling Problem

This paper deals with elastic waves which may be generated in a fluid by the sudden movement of a flow boundary. In particular, an analysis of the classical piston, or signalling problem is presented for the special case of arbitrary velocity input into a stationary fluid contained in a circular, semi-infinite waveguide. The decay of the pulse, as well as the resulting flow development in the inlet region of the pipe are analyzed by means of an asymptotic expansion of the suitably nondimensionalized Navier-Stokes equations for a compressible, nonheat-conducting Newtonian fluid. The results differ significantly from those of the more conventional one-dimensional approach based on the so-called telegrapher’s equation of mathematical physics. The present theory realistically predicts the growth of a boundary layer both in time and position and, hence, it appears to represent the transient fluid motion in a manner which is physically more appealing.

scholarly journals A doubly localized equilibrium solution of plane Couette flow

2014 ◽  
Vol 750 ◽  
Author(s):  
E. Brand ◽  
J. F. Gibson
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Equilibrium Solution ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Edge States ◽  
Plane Couette Flow ◽  
Turbulent Spot ◽  
Navier Stokes Equations ◽  
Exponential Decay Rate

AbstractWe present an equilibrium solution of plane Couette flow that is exponentially localized in both the spanwise and streamwise directions. The solution is similar in size and structure to previously computed turbulent spots and localized, chaotically wandering edge states of plane Couette flow. A linear analysis of dominant terms in the Navier–Stokes equations shows how the exponential decay rate and the wall-normal overhang profile of the streamwise tails are governed by the Reynolds number and the dominant spanwise wavenumber. Perturbations of the solution along its leading eigenfunctions cause rapid disruption of the interior roll-streak structure and formation of a turbulent spot, whose growth or decay depends on the Reynolds number and the choice of perturbation.

scholarly journals Doubly localized states in plane Couette flow

2014 ◽  
Vol 758 ◽  
pp. 1-4 ◽  
Author(s):  
Bruno Eckhardt
Keyword(s):  
Couette Flow ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Linear Instability ◽  
Localized States ◽  
Transition To Turbulence ◽  
Navier Stokes ◽  
Plane Couette Flow ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations

AbstractMuch of our understanding of the transition to turbulence in flows without a linear instability came with the discovery and characterization of fully three-dimensional solutions to the Navier–Stokes equation. The first examples in plane Couette flow were periodic in both spanwise and streamwise directions, and could explain the transitions in small domains only. The presence of localized turbulent spots in larger domains, the spatiotemporal decoherence on larger scales and the ability to trigger turbulence with pointwise perturbations require solutions that are localized in both directions, like the one presented by Brand & Gibson (J. Fluid Mech., vol. 750, 2014, R3). They describe a steady solution of the Navier–Stokes equations and characterize in unprecedented detail, including an analytic computation of its localization properties. The study opens up new ways to describe localized turbulent patches.

A Contribution to the Numerical Solution of Developing Laminar Flow in the Entrance Region of Concentric Annuli With Rotating Inner Walls

1974 ◽  
Vol 96 (4) ◽  
pp. 333-340 ◽  
Author(s):  
J. E. R. Coney ◽  
M. A. I. El-Shaarawi
Keyword(s):  
Boundary Layer ◽  
Laminar Flow ◽  
Stokes Equations ◽  
Complete Solution ◽  
Numerical Differentiation ◽  
Velocity Profiles ◽  
Entrance Region ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Special Case

The boundary layer simplification of the Navier-Stokes equations for hydrodynamically developing laminar flow with constant physical properties in the entrance region of concentric annuli with rotating inner walls have been numerically solved using a simple linearized finite-difference scheme. Additional results to those existing in the literature by Martin and Payne [1–2] will be presented here. An advantage of the analysis used in this paper is that it does not solve for the stream function and vorticity, but predicts the development of tangential, axial and radial velocity profiles directly, thus avoiding numerical differentiation. Results for the development of these velocity profiles, pressure drop and friction factor are presented for five annuli radii ratios (0.3, 0.5, 0.674, 0.727 and 0.90) at various values of the parameter Re2/Ta. The paper may be considered as a direct comparison between the boundary layer solution and the complete solution of the Navier-Stokes equations [1–2] for that special case.

Implementation of a Cavitation Algorithm into a Procedure for the Prediction of Non-Newtonian Flow in Hydrodynamic Journal Bearings

1987 ◽  
Vol 201 (6) ◽  
pp. 449-456
Author(s):  
P D Williams ◽  
G R Symmons
Keyword(s):  
Effective Viscosity ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Journal Bearings ◽  
Navier Stokes ◽  
Newtonian Flow ◽  
Film Rupture ◽  
Navier Stokes Equations ◽  
Finite Breadth ◽  
Newtonian Liquids

A procedure for solving the Navier–Stokes equations for the steady, three-dimensional, cavitated flow of non-Newtonian liquids within finite-breadth journal bearings is described. The method uses a finite difference approach, together with a technique known as SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) which has now become well established in the field of computational fluid dynamics. The concept of ‘effective viscosity’ to describe the non-linear dependence of shear stress on shear rate is used to predict the performance of bearings having a single axial inlet groove situated at the position of maximum clearance between the shaft and housing. The implementation of a cavitation algorithm into the equation set allows the loci of film rupture and reformation in the vicinity of the supply groove and elsewhere to be traced, these having a particularly important influence on the predicted lubricant flowrate. Results are obtained for a range of non-linearity factors and lead to the conclusion that all the important indicators of bearing performance can be determined using the technique described.

scholarly journals Analytic and Numerical Solutions of Couette Flow Problem: A Comparative Study

2017 ◽  
Vol 12 (1) ◽  
pp. 105-113
Author(s):  
Dhak Bahadur Thapa ◽  
Kedar Nath Uprety
Keyword(s):  
Differential Equation ◽  
Couette Flow ◽  
Numerical Solutions ◽  
Flow Problem ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Parabolic Partial Differential Equation ◽  
Navier Stokes Equations ◽  
One Dimensional ◽  
Nicolson Scheme

In this work, an incompressible viscous Couette flow is derived by simplifying the Navier-Stokes equations and the resulting one dimensional linear parabolic partial differential equation is solved numerically employing a second order finit difference Crank-Nicolson scheme. The numerical solution and the exact solution are presented graphically.Journal of the Institute of Engineering, 2016, 12(1): 105-113

Numerical experiments on time-dependent rotational Couette flow

1973 ◽  
Vol 59 (1) ◽  
pp. 77-95 ◽  
Author(s):  
D. C. S. Liu ◽  
C. F. Chen
Keyword(s):  
Couette Flow ◽  
Numerical Experiments ◽  
Stokes Equations ◽  
Axial Direction ◽  
Navier Stokes ◽  
Simultaneous Occurrence ◽  
Navier Stokes Equations ◽  
Finite Difference Approximations ◽  
Critical Wavelength ◽  
Explicit Finite Difference

The flow induced by impulsively starting the inner cylinder in a Couette flow apparatus is investigated by using a nonlinear analysis. Explicit finite-difference approximations are used to solve the Navier–Stokes equations for axisymmetric flows. Random small perturbations are distributed initially and periodic boundary conditions are applied in the axial direction over a length which, in general, is chosen to be the critical wavelength observed experimentally. Simultaneous occurrence of Taylor vortices is obtained at supercritical Reynolds numbers. The development of streamlines, perturbation velocity components and the kinetic energy of the perturbations is examined in detail. Many salient features of the physical flow are observed in the numerical experiments.

scholarly journals A streamwise constant model of turbulence in plane Couette flow

2010 ◽  
Vol 665 ◽  
pp. 99-119 ◽  
Author(s):  
D. F. GAYME ◽  
B. J. McKEON ◽  
A. PAPACHRISTODOULOU ◽  
B. BAMIEH ◽  
J. C. DOYLE
Keyword(s):  
Couette Flow ◽  
Small Amplitude ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Turbulent Shear ◽  
Theoretic Approach ◽  
Plane Couette Flow ◽  
Navier Stokes Equations ◽  
Turbulent Plane ◽  
3C Model

Streamwise and quasi-streamwise elongated structures have been shown to play a significant role in turbulent shear flows. We model the mean behaviour of fully turbulent plane Couette flow using a streamwise constant projection of the Navier–Stokes equations. This results in a two-dimensional three-velocity-component (2D/3C) model. We first use a steady-state version of the model to demonstrate that its nonlinear coupling provides the mathematical mechanism that shapes the turbulent velocity profile. Simulations of the 2D/3C model under small-amplitude Gaussian forcing of the cross-stream components are compared to direct numerical simulation (DNS) data. The results indicate that a streamwise constant projection of the Navier–Stokes equations captures salient features of fully turbulent plane Couette flow at low Reynolds numbers. A systems-theoretic approach is used to demonstrate the presence of large input–output amplification through the forced 2D/3C model. It is this amplification coupled with the appropriate nonlinearity that enables the 2D/3C model to generate turbulent behaviour under the small-amplitude forcing employed in this study.

scholarly journals On the self-sustained nature of large-scale motions in turbulent Couette flow

2015 ◽  
Vol 782 ◽  
pp. 515-540 ◽  
Author(s):  
Subhandu Rawat ◽  
Carlo Cossu ◽  
Yongyun Hwang ◽  
François Rincon
Keyword(s):  
Couette Flow ◽  
Turbulent Flows ◽  
Large Scale ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Buffer Layers ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Steady Solutions ◽  
Smagorinsky Constant

Large-scale motions in wall-bounded turbulent flows are frequently interpreted as resulting from an aggregation process of smaller-scale structures. Here, we explore the alternative possibility that such large-scale motions are themselves self-sustained and do not draw their energy from smaller-scale turbulent motions activated in buffer layers. To this end, it is first shown that large-scale motions in turbulent Couette flow at $Re=2150$ self-sustain, even when active processes at smaller scales are artificially quenched by increasing the Smagorinsky constant $C_{s}$ in large-eddy simulations (LES). These results are in agreement with earlier results on pressure-driven turbulent channel flows. We further investigate the nature of the large-scale coherent motions by computing upper- and lower-branch nonlinear steady solutions of the filtered (LES) equations with a Newton–Krylov solver, and find that they are connected by a saddle–node bifurcation at large values of $C_{s}$. Upper-branch solutions for the filtered large-scale motions are computed for Reynolds numbers up to $Re=2187$ using specific paths in the $Re{-}C_{s}$ parameter plane and compared to large-scale coherent motions. Continuation to $C_{s}=0$ reveals that these large-scale steady solutions of the filtered equations are connected to the Nagata–Clever–Busse–Waleffe branch of steady solutions of the Navier–Stokes equations. In contrast, we find it impossible to connect the latter to buffer-layer motions through a continuation to higher Reynolds numbers in minimal flow units.

ON THE NON-NEWTONIAN INCOMPRESSIBLE FLUIDS

1993 ◽  
Vol 03 (01) ◽  
pp. 35-63 ◽  
Author(s):  
JOSEF MÁLEK ◽  
JINDŘICH NEČAS ◽  
MICHAEL RŮŽIČKA
Keyword(s):  
Weak Solution ◽  
Stress Tensor ◽  
Growth Condition ◽  
Stokes Equations ◽  
Periodic Problem ◽  
Incompressible Fluids ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Nonlinear Stress ◽  
Special Case

The Navier-Stokes equations can be included as a special case into the class of non-Newtonian incompressible fluids with the nonlinear stress tensor τ=τ(e), the components of which satisfy the p-growth condition. Measure-valued solutions already exist for p>2n/(n+2). For the space periodic problem, the existence of the weak solution is then obtained for p>3n/(n+2). These solutions are regular and unique for p≥1+2n/(n+2).

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