The dynamics of a laminar flow in a symmetric channel with a sudden expansion

2001 ◽  
Vol 436 ◽  
pp. 283-320 ◽  
Author(s):  
T. HAWA ◽  
Z. RUSAK
Keyword(s):  
Reynolds Number ◽  
Laminar Flow ◽  
Numerical Simulations ◽  
Direct Numerical Simulations ◽  
Sudden Expansion ◽  
Asymptotically Stable ◽  
Two Dimensional ◽  
Stable Mode ◽  
Symmetric States ◽  
Asymmetric Equilibrium

Bifurcation analysis, linear stability study, and direct numerical simulations of the dynamics of a two-dimensional, incompressible, and laminar flow in a symmetric long channel with a sudden expansion with right angles and with an expansion ratio D/d (d is the width of the channel inlet section and D is the width of the outlet section) are presented. The bifurcation analysis of the steady flow equations concentrates on the flow states around a critical Reynolds number Rec(D/d) where asymmetric states appear in addition to the basic symmetric states when Re [ges ] Rec(D/d). The bifurcation of asymmetric states at Rec has a pitchfork nature and the asymmetric perturbation grows like √Re − Rec(D/d). The stability analysis is based on the linearized equations of motion for the evolution of infinitesimal two-dimensional disturbances imposed on the steady symmetric as well as asymmetric states. A neutrally stable asymmetric mode of disturbance exists at Rec(D/d) for both the symmetric and the asymmetric equilibrium states. Using asymptotic methods, it is demonstrated that when Re < Rec(D/d) the symmetric states have an asymptotically stable mode of disturbance. However, when Re > Rec(D/d), the symmetric states are unstable to this mode of asymmetric disturbance. It is also shown that when Re > Rec(D/d) the asymmetric states have an asymptotically stable mode of disturbance. The direct numerical simulations are guided by the theoretical approach. In order to improve the numerical simulations, a matching with the asymptotic solution of Moffatt (1964) in the regions around the expansion corners is also included. The dynamics of both small- and large-amplitude disturbances in the flow is described and the transition from symmetric to asymmetric states is demonstrated. The simulations clarify the relationship between the linear stability results and the time-asymptotic behaviour of the flow. The current analyses provide a theoretical foundation for previous experimental and numerical results and shed more light on the transition from symmetric to asymmetric states of a viscous flow in an expanding channel. It is an evolution from a symmetric state, which loses its stability when the Reynolds number of the incoming flow is above Rec(D/d), to a stable asymmetric equilibrium state. The loss of stability is a result of the interaction between the effects of viscous dissipation, the downstream convection of perturbations by the base symmetric flow, and the upstream convection induced by two-dimensional asymmetric disturbances.

Bifurcation Characteristics of Flows in Rectangular Sudden Expansion Channels

2005 ◽  
Author(s):  
Francine Battaglia ◽  
George Papadopoulos
Keyword(s):  
Reynolds Number ◽  
Laminar Flow ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Expansion Ratio ◽  
Sudden Expansion ◽  
Two Dimensional ◽  
Effective Expansion ◽  
Three Dimensional Simulations

The effect of three-dimensionality on low Reynolds number flows past a symmetric sudden expansion in a channel was investigated. The geometric expansion ratio of in the current study was 2:1 and the aspect ratio was 6:1. Both experimental velocity measurements and two- and three-dimensional simulations for the flow along the centerplane of the rectangular duct are presented for Reynolds numbers in the range of 150 to 600. Comparison of the two-dimensional simulations with the experiments revealed that the simulations fail to capture completely the total expansion effect on the flow, which couples both geometric and hydrodynamic effects. To properly do so requires the definition of an effective expansion ratio, which is the ratio of the downstream and upstream hydraulic diameters and is therefore a function of both the expansion and aspect ratios. When the two-dimensional geometry was consistent with the effective expansion ratio, the new results agreed well with the three-dimensional simulations and the experiments. Furthermore, in the range of Reynolds numbers investigated, the laminar flow through the expansion underwent a symmetry-breaking bifurcation. The critical Reynolds number evaluated from the experiments and the simulations was compared to other values reported in the literature. Overall, side-wall proximity was found to enhance flow stability, helping to sustain laminar flow symmetry to higher Reynolds numbers in comparison to nominally two-dimensional double-expansion geometries. Lastly, and most importantly, when the logarithm of the critical Reynolds number from all these studies was plotted against the reciprocal of the effective expansion ratio, a linear trend emerged that uniquely captured the bifurcation dynamics of all symmetric double-sided planar expansions.

Direct numerical simulations of bubbly flows Part 2. Moderate Reynolds number arrays

1999 ◽  
Vol 385 ◽  
pp. 325-358 ◽  
Author(s):  
ASGHAR ESMAEELI ◽  
GRÉTAR TRYGGVASON
Keyword(s):  
Reynolds Number ◽  
Numerical Simulations ◽  
Volume Fraction ◽  
Three Dimensional ◽  
Direct Numerical Simulations ◽  
Dimensional System ◽  
Two Dimensional ◽  
Reynolds Stresses ◽  
Bubble Distribution ◽  
Moderate Reynolds Number

Direct numerical simulations of the motion of two- and three-dimensional finite Reynolds number buoyant bubbles in a periodic domain are presented. The full Navier–Stokes equations are solved by a finite difference/front tracking method that allows a fully deformable interface between the bubbles and the ambient fluid and the inclusion of surface tension. The rise Reynolds numbers are around 20–30 for the lowest volume fraction, but decrease as the volume fraction is increased. The rise of a regular array of bubbles, where the relative positions of the bubbles are fixed, is compared with the evolution of a freely evolving array. Generally, the freely evolving array rises slower than the regular one, in contrast to what has been found earlier for low Reynolds number arrays. The structure of the bubble distribution is examined and it is found that while the three-dimensional bubbles show a tendency to line up horizontally, the two-dimensional bubbles are nearly randomly distributed. The effect of the number of bubbles in each period is examined for the two-dimensional system and it is found that although the rise Reynolds number is nearly independent of the number of bubbles, the velocity fluctuations in the liquid (the Reynolds stresses) increase with the size of the system. While some aspects of the fully three-dimensional flows, such as the reduction in the rise velocity, are predicted by results for two-dimensional bubbles, the structure of the bubble distribution is not. The magnitude of the Reynolds stresses is also greatly over-predicted by the two-dimensional results.

Direct numerical simulations of bubbly flows. Part 1. Low Reynolds number arrays

1998 ◽  
Vol 377 ◽  
pp. 313-345 ◽  
Author(s):  
ASGHAR ESMAEELI ◽  
GRÉTAR TRYGGVASON
Keyword(s):  
Reynolds Number ◽  
Numerical Simulations ◽  
Stokes Flow ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Direct Numerical Simulations ◽  
Two Dimensional ◽  
Rise Velocity ◽  
Regular Array ◽  
Average Rise

Direct numerical simulations of the motion of two- and three-dimensional buoyant bubbles in periodic domains are presented. The full Navier–Stokes equations are solved by a finite difference/front tracking method that allows a fully deformable interface between the bubbles and the ambient fluid and the inclusion of surface tension. The governing parameters are selected such that the average rise Reynolds number is O(1) and deformations of the bubbles are small. The rise velocity of a regular array of three-dimensional bubbles at different volume fractions agrees relatively well with the prediction of Sangani (1988) for Stokes flow. A regular array of two- and three-dimensional bubbles, however, is an unstable configuration and the breakup, and the subsequent bubble–bubble interactions take place by ‘drafting, kissing, and tumbling’. A comparison between a finite Reynolds number two-dimensional simulation with sixteen bubbles and a Stokes flow simulation shows that the finite Reynolds number array breaks up much faster. It is found that a freely evolving array of two-dimensional bubbles rises faster than a regular array and simulations with different numbers of two-dimensional bubbles (1–49) show that the rise velocity increases slowly with the size of the system. Computations of four and eight three-dimensional bubbles per period also show a slight increase in the average rise velocity compared to a regular array. The difference between two- and three-dimensional bubbles is discussed.

scholarly journals Lyapunov dimension of elastic turbulence

2017 ◽  
Vol 822 ◽  
Author(s):  
Emmanuel Lance Christopher VI M. Plan ◽  
Anupam Gupta ◽  
Dario Vincenzi ◽  
John D. Gibbon
Keyword(s):  
Reynolds Number ◽  
Constitutive Model ◽  
Numerical Simulations ◽  
Mathematical Analysis ◽  
Critical Value ◽  
Weissenberg Number ◽  
Direct Numerical Simulations ◽  
Chaotic Behaviour ◽  
Two Dimensional ◽  
Lyapunov Dimension

Low-Reynolds-number polymer solutions exhibit a chaotic behaviour known as ‘elastic turbulence’ when the Weissenberg number exceeds a critical value. The two-dimensional Oldroyd-B model is the simplest constitutive model that reproduces this phenomenon. To make a practical estimate of the resolution scale of the dynamics, one requires the assumption that an attractor of the Oldroyd-B model exists; numerical simulations show that the quantities on which this assumption is based are bounded. We estimate the Lyapunov dimension of this assumed attractor as a function of the Weissenberg number by combining a mathematical analysis of the model with direct numerical simulations.

scholarly journals Control of laminar breakdown in a supersonic boundary layer employing streaks

2021 ◽  
Vol 932 ◽  
Author(s):  
Simon Kneer ◽  
Zhengfei Guo ◽  
Markus J. Kloker
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Laminar Flow ◽  
Point Source ◽  
Numerical Simulations ◽  
Supersonic Boundary Layer ◽  
Direct Numerical Simulations ◽  
Roughness Elements ◽  
Controlling Mechanism ◽  
Laminar Flow Control

In this study direct numerical simulations are employed to investigate the effects of various parameters on the laminar-flow-control capabilities of narrowly spaced streaks in a supersonic boundary layer at Mach $2.0$ . Previous work by Sharma et al. (J. Fluid Mech., vol. 873, 2019, pp. 1072–1089) has found these streak modes, excited by a spanwise blowing-and-suction strip, to be highly effective at delaying pure oblique-type breakdown. In the present work it is shown that spectrum-enriching subharmonic modes, relevant with increasing running-length Reynolds number, do not destroy the controlling mechanism, and also a complex breakdown scenario, triggered by a multi-frequency point source, is found to be effectively controlled. Moreover, the control-streak excitation by roughness elements is compared in detail with the blowing-and-suction method, revealing relevant differing features.

Bifurcation Characteristics of Flows in Rectangular Sudden Expansion Channels

2005 ◽  
Vol 128 (4) ◽  
pp. 671-679 ◽  
Author(s):  
Francine Battaglia ◽  
George Papadopoulos
Keyword(s):  
Reynolds Number ◽  
Laminar Flow ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Expansion Ratio ◽  
Sudden Expansion ◽  
Two Dimensional ◽  
Effective Expansion ◽  
Three Dimensional Simulations

The effect of three dimensionality on low Reynolds number flows past a symmetric sudden expansion in a channel was investigated. The geometric expansion ratio in the current study was 2:1 and the aspect ratio was 6:1. Both experimental velocity measurements and two- and three-dimensional simulations for the flow along the centerplane of the rectangular duct are presented for Reynolds numbers in the range of 150 to 600. Comparison of the two-dimensional simulations with the experiments revealed that the simulations failed to capture completely the total expansion effect on the flow, which is influenced by both geometric and hydrodynamic effects. To properly do so requires the definition of an effective expansion ratio, which is the ratio of the downstream and upstream hydraulic diameters and is therefore a function of both the expansion and aspect ratios. When two-dimensional simulations were performed using the effective expansion ratio, the new results agreed well with the three-dimensional simulations and the experiments. Furthermore, in the range of Reynolds numbers investigated, the laminar flow through the expansion underwent a symmetry-breaking bifurcation. The critical Reynolds number evaluated from the experiments and the simulations were compared to other values reported in the literature. Overall, side-wall proximity was found to enhance flow stability, thus sustaining laminar flow symmetry to higher Reynolds numbers. Last, and most important, when the logarithm of the critical Reynolds number was plotted against the reciprocal of the effective expansion ratio, a linear trend emerged that uniquely captured the bifurcation dynamics of all symmetric double-sided planar expansions.

Bifurcation analysis of two-dimensional laminar flow through sudden expansion channel

2022 ◽  
Vol 13 (1) ◽  
pp. 0-0
Keyword(s):  
Laminar Flow ◽  
Expansion Ratio ◽  
Sudden Expansion ◽  
Two Dimensional ◽  
Governing Equations ◽  
Bifurcation Phenomena ◽  
Different Types ◽  
Fluent Software ◽  
Flow Through ◽  
Volume Method

In this work, bifurcation characteristics of unsteady, viscous, Newtonian laminar flow in two-dimensional sudden expansion and sudden contraction-expansion channels have been studied for different values of expansion ratio. The governing equations have been solved using finite volume method and FLUENT software has been employed to visualize the simulation results. Three different mesh studies have been performed to calculate critical Reynolds number (Recr) for different types of bifurcation phenomena. It is found that Recr decreases with the increase in expansion ratio (ER).

scholarly journals Unravelling turbulence near walls

2009 ◽  
Vol 630 ◽  
pp. 1-4 ◽  
Author(s):  
IVAN MARUSIC
Keyword(s):  
Laminar Flow ◽  
Turbulent Flow ◽  
Numerical Simulations ◽  
Turbulent Flows ◽  
Direct Numerical Simulations ◽  
Aircraft Wing ◽  
Physical Mechanisms ◽  
Drag And Lift

Turbulent flows near walls have been the focus of intense study since their first description by Ludwig Prandtl over 100 years ago. They are critical in determining the drag and lift of an aircraft wing for example. Key challenges are to understand the physical mechanisms causing the transition from smooth, laminar flow to turbulent flow and how the turbulence is then maintained. Recent direct numerical simulations have contributed significantly towards this understanding.

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