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Fundamental Issues and Canonical Flows

Mixing Analysis in a Lid-Driven Cavity Flow at Finite Reynolds Numbers

[+] Author and Article Information
Pradeep Rao1

 Department of Mechanical Engineering, Texas A&M University, College Station, TX 77845prao@vt.edu

Andrew Duggleby

 Department of Mechanical Engineering, Texas A&M University, College Station, TX 77845

Mark A. Stremler

 Department of Engineering Science and Mechanics, Virginia Tech, Blacksburg, VA 24061

1

Corresponding author. Present address: Department of Engineering Science and Mechanics, Virginia Tech, Blacksburg, VA 24061.

J. Fluids Eng 134(4), 041203 (Apr 20, 2012) (8 pages) doi:10.1115/1.4006361 History: Received March 08, 2011; Revised March 08, 2012; Published April 20, 2012; Online April 20, 2012

The influence of inertial effects on chaotic advection and mixing is investigated for a two-dimensional, time-dependent lid-driven cavity flow. Previous work shows that this flow exhibits exponential stretching and folding of material lines due to the presence of figure-eight stirring patterns in the creeping flow regime. The high sensitivity to initial conditions and the exponential growth of errors in chaotic flows necessitate an accurate solution of the flow in order to calculate metrics based on Lagrangian particle tracking. The streamfunction-vorticity formulation of the Navier-Stokes equations is solved using a Fourier-Chebyshev spectral method, providing the necessary exponential convergence and machine-precision accuracy. Poincaré sections and mixing measures are used to analyze chaotic advection and quantify the mixing efficiency. The calculated mixing characteristics are almost identical for Re ≤ 1. For the time range investigated, the best mixing in this system is observed for Re = 10. Interestingly, increasing the Reynolds number to the range 10 < Re ≤ 100 results in an observed decrease in mixing efficacy.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

General flow domain, and representative streamlines in the Stokes’ flow limit with U2/U1≃0.8413 for (a) the first half of the flow period and (b) the second half of the flow period. Filled circles show the stagnation points used to define the flow protocol, and open circles show the points that exchange positions along the dotted streamlines when taking the parameter values h = 1, Umax≃1.0, and τ≃15.261

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Figure 2

(a) Flow period n* at which the bulk flow becomes periodic in time, (b) spectral convergence of the stream function, and (c) order of convergence. Since the convergence is geometric for a spectral method instead of algebraic, the order of convergence (rate) varies with degrees of freedom, and often machine precision is reached.

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Figure 3

Instantaneous streamlines at time t = n*τ, i.e. at the end of an advection cycle, once the flow has become periodic for Re = (a) 10, (b) 30, (c) 50 and (d) 100

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Figure 4

Poincaré sections for Re = (a) 0.01, (b) 1, (c) 10, (d) 30, (e) 50, and (f) 100

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Figure 5

Comparison of dispersion of passively advected particles for Re = 1, Re = 10, Re = 30, Re = 50 and Re = 100 (top to bottom) at 1, 4, and 10 advection cycles (left to right)

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Figure 6

Quantifying the time-dependent distribution of passively advected particles using (a) the spreading index ɛ1 , (b) the homogeneity index ɛ4 , and (c) the variance of particle density σ, each as a function of the flow period p = t

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