scholarly journals The Jeffery–Hamel similarity solution and its relation to flow in a diverging channel

2011 ◽  
Vol 687 ◽  
pp. 404-430 ◽  
Author(s):  
P. E. Haines ◽  
R. E. Hewitt ◽  
A. L. Hazel
Keyword(s):  
Reynolds Number ◽  
Symmetry Breaking ◽  
Similarity Solution ◽  
Pitchfork Bifurcation ◽  
Numerical Results ◽  
Finite Domain ◽  
Infinite Domain ◽  
Neutral Curves ◽  
Diverging Channel ◽  
Radial Outflow

AbstractWe explore the relevance of the idealized Jeffery–Hamel similarity solution to the practical problem of flow in a diverging channel of finite (but large) streamwise extent. Numerical results are presented for the two-dimensional flow in a wedge of separation angle $2\ensuremath{\alpha} $, bounded by circular arcs at the inlet/outlet and for a net radial outflow of fluid. In particular, we show that in a finite domain there is a sequence of nested neutral curves in the $(\mathit{Re}, \ensuremath{\alpha} )$ plane, each corresponding to a midplane symmetry-breaking (pitchfork) bifurcation, where $\mathit{Re}$ is a Reynolds number based on the radial mass flux. For small wedge angles we demonstrate that the first pitchfork bifurcation in the finite domain occurs at a critical Reynolds number that is in agreement with the only pitchfork bifurcation in the infinite-domain similarity solution, but that the criticality of the bifurcation differs (in general). We explain this apparent contradiction by demonstrating that, for $\ensuremath{\alpha} \ll 1$, superposition of two (infinite-domain) eigenmodes can be used to construct a leading-order finite-domain eigenmode. These constructed modes accurately predict the multiple symmetry-breaking bifurcations of the finite-domain flow without recourse to computation of the full field equations. Our computational results also indicate that temporally stable, isolated, steady solutions may exist. These states are finite-domain analogues of the steady waves recently presented by Kerswell, Tutty, & Drazin (J. Fluid Mech., vol. 501, 2004, pp. 231–250) for an infinite domain. Moreover, we demonstrate that there is non-uniqueness of stable solutions in certain parameter regimes. Our numerical results tie together, in a consistent framework, the disparate results in the existing literature.

Flow past a sphere up to a Reynolds number of 300

1999 ◽  
Vol 378 ◽  
pp. 19-70 ◽  
Author(s):  
T. A. JOHNSON ◽  
V. C. PATEL
Keyword(s):  
Reynolds Number ◽  
Symmetry Breaking ◽  
Unsteady Flow ◽  
Axisymmetric Flow ◽  
Reynolds Numbers ◽  
Numerical Results ◽  
Near Wake ◽  
Hairpin Vortices ◽  
Breaking Mechanism ◽  
Visualization Experiments

The flow of an incompressible viscous fluid past a sphere is investigated numerically and experimentally over flow regimes including steady and unsteady laminar flow at Reynolds numbers of up to 300. Flow-visualization experiments are used to validate the numerical results and to provide additional insight into the behaviour of the flow. Near-wake visualizations are presented for both steady and unsteady flows. Calculations for Reynolds numbers of up to 200 show steady axisymmetric flow and compare well with previous experimental and numerical observations. For Reynolds numbers of 210 to 270, a steady non-axisymmetric regime is found, also in agreement with previous work. To advance the basic understanding of this transition, a symmetry breaking mechanism is proposed based on a detailed analysis of the calculated flow field.Unsteady flow is calculated at Reynolds numbers greater than 270. The results at a Reynolds number of 300 show a highly organized periodic flow dominated by vortex shedding. An analysis of the calculated vortical structure of the wake reveals a sequence of shed hairpin vortices in combination with a sequence of previously unidentified induced hairpin vortices. The numerical results compare favourably with experimental flow visualizations which, interestingly, fail to reveal the induced vortices. Based on the deduced symmetry-breaking mechanism, an analysis of the unsteady kinematics, and the experimental results, a mechanism driving the transition to unsteady flow is proposed.

Stability of thermocapillary flows in non-cylindrical liquid bridges

2002 ◽  
Vol 458 ◽  
pp. 35-73 ◽  
Author(s):  
CH. NIENHÜSER ◽  
H. C. KUHLMANN
Keyword(s):  
Reynolds Number ◽  
Prandtl Number ◽  
Critical Reynolds Number ◽  
Volume Fraction ◽  
Thermocapillary Flow ◽  
Surface Shape ◽  
Liquid Bridges ◽  
Gravity Level ◽  
Neutral Curves ◽  
Prandtl Numbers

The thermocapillary flow in liquid bridges is investigated numerically. In the limit of large mean surface tension the free-surface shape is independent of the flow and temperature fields and depends only on the volume of liquid and the hydrostatic pressure difference. When gravity acts parallel to the axis of the liquid bridge the shape is axisymmetric. A differential heating of the bounding circular disks then causes a steady two-dimensional thermocapillary flow which is calculated by a finite-difference method on body-fitted coordinates. The linear-stability problem for the basic flow is solved using azimuthal normal modes computed with the same discretization method. The dependence of the critical Reynolds number on the volume fraction, gravity level, Prandtl number, and aspect ratio is explained by analysing the energy budgets of the neutral modes. For small Prandtl numbers (Pr = 0.02) the critical Reynolds number exhibits a smooth minimum near volume fractions which approximately correspond to the volume of a cylindrical bridge. When the Prandtl number is large (Pr = 4) the intersection of two neutral curves results in a sharp peak of the critical Reynolds number. Since the instabilities for low and high Prandtl numbers are markedly different, the influence of gravity leads to a distinctly different behaviour. While the hydrostatic shape of the bridge is the most important effect of gravity on the critical point for low-Prandtl-number flows, buoyancy is the dominating factor for the stability of the flow in a gravity field when the Prandtl number is high.

A new Reliable Numerical Algorithm Based on the First Kind of Bessel Functions to Solve Prandtl–Blasius Laminar Viscous Flow over a Semi-Infinite Flat Plate

2012 ◽  
Vol 67 (12) ◽  
pp. 665-673 ◽  
Author(s):  
Kourosh Parand ◽  
Mehran Nikarya ◽  
Jamal Amani Rad ◽  
Fatemeh Baharifard
Keyword(s):  
Viscous Flow ◽  
Flat Plate ◽  
Numerical Algorithm ◽  
Bessel Functions ◽  
Nonlinear Ordinary Differential Equation ◽  
Finite Domain ◽  
Algebraic Equations ◽  
Infinite Domain ◽  
Third Order ◽  
Domain Truncation

In this paper, a new numerical algorithm is introduced to solve the Blasius equation, which is a third-order nonlinear ordinary differential equation arising in the problem of two-dimensional steady state laminar viscous flow over a semi-infinite flat plate. The proposed approach is based on the first kind of Bessel functions collocation method. The first kind of Bessel function is an infinite series, defined on ℝ and is convergent for any x ∊ℝ. In this work, we solve the problem on semi-infinite domain without any domain truncation, variable transformation basis functions or transformation of the domain of the problem to a finite domain. This method reduces the solution of a nonlinear problem to the solution of a system of nonlinear algebraic equations. To illustrate the reliability of this method, we compare the numerical results of the present method with some well-known results in order to show the applicability and efficiency of our method.

Evolution of clusters of sedimenting low-Reynolds-number particles with Oseen interactions

2008 ◽  
Vol 603 ◽  
pp. 63-100 ◽  
Author(s):  
G. SUBRAMANIAN ◽  
DONALD L. KOCH
Keyword(s):  
Reynolds Number ◽  
Velocity Field ◽  
Aspect Ratio ◽  
Evolution Equation ◽  
Similarity Solution ◽  
Initial Conditions ◽  
Radial Distance ◽  
Two Dimensions ◽  
Number Density ◽  
Cluster Radius

A theoretical framework is developed to describe, in the limit of small but finite Re, the evolution of dilute clusters of sedimenting particles. Here, Re =aU/ν is the particle Reynolds number, where a is the radius of the spherical particle, U its settling velocity, and ν the kinematic viscosity of the suspending fluid. The theory assumes the disturbance velocity field at sufficiently large distances from a sedimenting particle, even at small Re, to possess the familiar source--sink character; that is, the momentum defect brought in via a narrow wake behind the particle is convected radially outwards in the remaining directions. It is then argued that for spherical clusters with sufficiently many particles, specifically with N much greater than O(R0U/ν), the initial evolution is strongly influenced by wake-mediated interactions; here, N is the total number of particles, and R0 is the initial cluster radius. As a result, the cluster first evolves into a nearly planar configuration with an asymptotically small aspect ratio of O(R0U/N ν), the plane of the cluster being perpendicular to the direction of gravity; subsequent expansion occurs with an unchanged aspect ratio. For relatively sparse clusters with N smaller than O(R0U/ν), the probability of wake interactions remains negligible, and the cluster expands while retaining its spherical shape. The long-time expansion in the former case, and that for all times in the latter case, is driven by disturbance velocity fields produced by the particles outside their wakes. The resulting interactions between particles are therefore mutually repulsive with forces that obey an inverse-square law. The analysis presented describes cluster evolution in this regime. A continuum representation is adopted with the clusters being characterized by a number density field (n(r, t)), and a corresponding induced velocity field (u (r, t)) arising on account of interactions. For both planar axisymmetric clusters and spherical clusters with radial symmetry, the evolution equation admits a similarity solution; either cluster expands self-similarly for long times. The number density profiles at different times are functions of a similarity variable η = (r/t1/3), r being the radial distance away from the cluster centre, and t the time. The radius of the expanding cluster is found to be of the form Rcl (t) = A (ν a)1/3N1/3t1/3, where the constant of proportionality, A, is determined from an analytical solution of the evolution equation; one finds A = 1.743 and 1.651 for planar and spherical clusters, respectively. The number density profile in a planar axisymmetric cluster is also obtained numerically as a solution of the initial value problem for a canonical (Gaussian) initial condition. The numerical results compare well with theoretical predictions, and demonstrate the asymptotic stability of the similarity solution in two dimensions for long times, at least for axisymmetric initial conditions.

Stability of Horizontal Boundary Layers

2006 ◽  
Author(s):  
Jean-Yves Chemin ◽  
Benoit Desjardins ◽  
Isabelle Gallagher ◽  
Emmanuel Grenier
Keyword(s):  
Reynolds Number ◽  
Linear Stability ◽  
Mathematical Proof ◽  
Vertical Component ◽  
Ekman Layer ◽  
Numerical Results ◽  
Energy Estimates ◽  
Navier Stokes ◽  
The 1960S ◽  
The Stability

Let us now detail the stability properties of an Ekman layer introduced in Part I, page 11. First we will recall how to compute the critical Reynolds number. Then we will describe briefly what happens at larger Reynolds numbers. The first step in the study of the stability of the Ekman layer is to consider the linear stability of a pure Ekman spiral of the form where U∞ is the velocity away from the layer and ζ is the rescaled vertical component ζ = x3/√εν. The corresponding Reynolds number is Let us consider the Navier–Stokes–Coriolis equations, linearized around uE The problem is now to study the (linear) stability of the 0 solution of the system (LNSCε). If u=0 is stable we say that uE is linearly stable, if not we say that it is linearly unstable. Numerical results show that u=0 is stable if and only if Re<Rec where Rec can be evaluated numerically. Up to now there is no mathematical proof of this fact, and it is only possible to prove that 0 is linearly stable for Re<Re1 and unstable for Re>Re2 with Re1<Rec<Re2, Re1 being obtained by energy estimates and Re2 by a perturbative analysis of the case Re=∞. We would like to emphasize that the numerical results are very reliable and can be considered as definitive results, since as we will see below, the stability analysis can be reduced to the study of a system of ordinary differential equations posed on the half-space, with boundary conditions on both ends, a system which can be studied arbitrarily precisely, even on desktop computers (first computations were done in the 1960s by Lilly).

Turbulent Flow Between Two Disks Contrarotating at Different Speeds

1996 ◽  
Vol 118 (2) ◽  
pp. 408-413 ◽  
Author(s):  
M. Kilic ◽  
X. Gan ◽  
J. M. Owen
Keyword(s):  
Experimental Study ◽  
Reynolds Number ◽  
Unsteady Flow ◽  
Turbulent Flow ◽  
Boundary Layers ◽  
Axial Flow ◽  
Radial Component ◽  
Velocity Measurements ◽  
Type Flow ◽  
Radial Outflow

This paper describes a combined computational and experimental study of the turbulent flow between two contrarotating disks for −1 ≤ Γ ≤ 0 and Reφ ≈ 1.2 × 106, where Γ is the ratio of the speed of the slower disk to that of the faster one and Reφ is the rotational Reynolds number. The computations were conducted using an axisymmetric elliptic multigrid solver and a low-Reynolds-number k–ε turbulence model. Velocity measurements were made using LDA at nondimensional radius ratios of 0.6 ≤ x ≤ 0.85. For Γ = 0, the rotor–stator case, Batchelor-type flow occurs: There is radial outflow and inflow in boundary layers on the rotor and stator, respectively, between which is an inviscid rotating core of fluid where the radial component of velocity is zero and there is an axial flow from stator to rotor. For Γ = −1, antisymmetric contrarotating disks, Stewartson-type flow occurs with radial outflow in boundary layers on both disks and inflow in the viscid nonrotating core. At intermediate values of Γ, two cells separated by a streamline that stagnates on the slower disk are formed: Batchelor-type flow and Stewartson-type flow occur radially outward and inward, respectively, of the stagnation streamline. Agreement between the computed and measured velocities is mainly very good, and no evidence was found of nonaxisymmetric or unsteady flow.

Fluid Flow and Lateral Fluid Dispersion in Bounded Granular Beds

2002 ◽  
Author(s):  
Azita Soleymani ◽  
Eveliina Takasuo ◽  
Piroz Zamankhan ◽  
William Polashenski
Keyword(s):  
Reynolds Number ◽  
Porous Media ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Numerical Study ◽  
Fluid Velocity ◽  
Three Dimensional ◽  
Numerical Results ◽  
Transition To Turbulence ◽  
Wide Range

Results are presented from a numerical study examining the flow of a viscous, incompressible fluid through random packing of nonoverlapping spheres at moderate Reynolds numbers (based on pore permeability and interstitial fluid velocity), spanning a wide range of flow conditions for porous media. By using a laminar model including inertial terms and assuming rough walls, numerical solutions of the Navier-Stokes equations in three-dimensional porous packed beds resulted in dimensionless pressure drops in excellent agreement with those reported in a previous study (Fand et al., 1987). This observation suggests that no transition to turbulence could occur in the range of Reynolds number studied. For flows in the Forchheimer regime, numerical results are presented of the lateral dispersivity of solute continuously injected into a three-dimensional bounded granular bed at moderate Peclet numbers. Lateral fluid dispersion coefficients are calculated by comparing the concentration profiles obtained from numerical and analytical methods. Comparing the present numerical results with data available in the literature, no evidence has been found to support the speculations by others for a transition from laminar to turbulent regimes in porous media at a critical Reynolds number.

Numerical Computation for Flow-Induced Oscillations of a Circular Cylinder

2010 ◽  
Author(s):  
Norio Kondo
Keyword(s):  
Reynolds Number ◽  
Circular Cylinder ◽  
High Reynolds Number ◽  
Numerical Results ◽  
High Mass ◽  
Mass Ratio ◽  
Reynolds Number Flow ◽  
Wide Range ◽  
Number Flow ◽  
High Reynolds

This paper presents numerical results for flow-induced oscillations of an elastically supported circular cylinder, which is immersed in a high Reynolds number flow. The flow-induced oscillations of the circular cylinder at subcritical Reynolds numbers have been investigated by many researchers, and the interested phenomena with respect to the oscillations have been found in a wide range of the Scruton number. For the flow-induced oscillation of the circular cylinder with high mass ratio, it is well-known that there is the peak value of amplitudes at near the critical reduced velocity. Therefore, we computer flow-induced oscillations of a circular cylinder with a mass ratio of 8, which is placed in a high Reynolds number flow, by three-dimensional simulation, and the numerical results are compared with the results of flow-induced oscillations of the circular cylinder immersed in a subcritical Reynolds number flow.

scholarly journals Turbulent Flow Computations in Rotating Cavities Using Low-Reynolds-Number Models

1996 ◽  
Author(s):  
Hector Iacovides ◽  
Kostas S. Nikas ◽  
Marcel A. F. Te Braak
Keyword(s):  
Reynolds Number ◽  
Dissipation Rate ◽  
Correction Term ◽  
Low Reynolds Number ◽  
Moment Closure ◽  
Rotating Disc ◽  
Moment Coefficient ◽  
Second Moment Closure ◽  
Low Reynolds ◽  
Radial Outflow

This study is concerned with the use of low-Reynolds-number models of turbulence transport in the computation of flows through rotating cavities. The models tested are the Launder and Sharma low-Re k-ε (L-S) and a low-Re differential second-moment closure (DSM), first used by Iacovides and Toumpanakis, both with and without the Yap correction term to the dissipation rate equation. The cases examined include rotor-stator systems without throughflow, rotor-stator systems with radial outflow, contra-rotating disc systems without throughflow and also with radial outflow, co rotating discs with radial outflow and also rotor-stator systems with radial inflow. Earlier studies have shown that, when no throughflow or when radial outflow is involved, the L-S tends to over-estimate the size of the regions over which the boundary layers remain laminar, while the zonal k-ε/l-eqn model is unable to predict partially laminarized flows. A modification to the ε equation proposed here, which in regions of low turbulence reduces the dissipation rate when the fluid is in solid body rotation, provides a simple empirical way to significantly improve the L-S predictions of partially laminarized flows through rotating cavities, to acceptable levels. The DSM model used, in some cases led to some further predictive improvements and, for rotor-stator systems without throughflow, to a significant improvement in the predicted value of the moment coefficient. The Yap length scale correction term, while in most cases it has either a beneficial or a neutral effect on the flow predictions, in cases involving radial inflow it leads to poorer predictions. Models that do not rely on wall distance thus appear more likely to have a wider range of applicability.

PERIOD-DOUBLING SCENARIO WITHOUT FLIP BIFURCATIONS IN A ONE-DIMENSIONAL MAP

2005 ◽  
Vol 15 (04) ◽  
pp. 1267-1284 ◽  
Author(s):  
V. AVRUTIN ◽  
M. SCHANZ
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Symmetry Breaking ◽  
Period Doubling ◽  
Pitchfork Bifurcation ◽  
Period Doubling Bifurcation ◽  
Coexisting Attractors ◽  
One Dimensional ◽  
Poincaré Return Map ◽  
Smooth Dynamical System

In this work a one-dimensional piecewise-smooth dynamical system, representing a Poincaré return map for dynamical systems of the Lorenz type, is investigated. The system shows a bifurcation scenario similar to the classical period-doubling one, but which is influenced by so-called border collision phenomena and denoted as border collision period-doubling bifurcation scenario. This scenario is formed by a sequence of pairs of bifurcations, whereby each pair consists of a border collision bifurcation and a pitchfork bifurcation. The mechanism leading to this scenario and its characteristic properties, like symmetry-breaking and symmetry-recovering as well as emergence of coexisting attractors, are investigated.

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