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Research Papers: Multiphase Flows

Global Linear Instability of Flow Through a Converging–Diverging Channel

[+] Author and Article Information
Mamta R. Jotkar

Tata Institute of Fundamental Research,
Centre for Interdisciplinary Sciences,
Hyderabad 500075, India
e-mail: mamtaj@tifrh.res.in

Gayathri Swaminathan

Airbus Group India Pvt Ltd,
Bangalore 560048, India
e-mail: gayathri.swaminathan@airbus.com

Kirti Chandra Sahu

Department of Chemical Engineering,
Indian Institute of Technology Hyderabad,
Yeddumailaram, Telangana 502205, India
e-mail: ksahu@iith.ac.in

Rama Govindarajan

Tata Institute of Fundamental Research,
Centre for Interdisciplinary Sciences,
Hyderabad 500075, India
e-mail: rama@tifrh.res.in

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received December 31, 2014; final manuscript received July 11, 2015; published online October 1, 2015. Assoc. Editor: Sharath S. Girimaji.

J. Fluids Eng 138(3), 031301 (Oct 01, 2015) (8 pages) Paper No: FE-14-1801; doi: 10.1115/1.4031429 History: Received December 31, 2014; Revised July 11, 2015

The global linear stability, where we assume no homogeneity in either of the spatial directions, of a two-dimensional laminar base flow through a spatially periodic converging–diverging channel is studied at low Reynolds numbers. A large wall-waviness amplitude is used to achieve instability at critical Reynolds numbers below ten. This is in contrast to earlier studies, which were at lower wall-waviness amplitude and had critical Reynolds numbers an order of magnitude higher. Moreover, our leading mode is a symmetry-breaking standing mode, unlike the traveling modes which are standard at higher Reynolds numbers. Eigenvalues in the spectrum lie on distinct branches, showing varied structure spanning the geometry. Our global stability study suggests that such modes can be tailored to give enhanced mixing in microchannels at low Reynolds numbers.

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References

McAlpine, A. , and Drazin, P. G. , 1998, “ On the Spatio-Temporal Development of Small Perturbations of Jeffery–Hamel Flows,” Fluid Dyn. Res., 22(3), pp. 123–138. [CrossRef]
Swaminathan, G. , Sameen, A. , Sahu, K. C. , and Govindarajan, R. , 2011, “ Global Instabilities in Diverging Channel Flows,” Theor. Computat. Fluid Dyn., 25, pp. 53–64. [CrossRef]
Vinod, N. , and Govindarajan, R. , 2012, “ Secondary Instabilities in Incompressible Axisymmetric Boundary Layers: Effect of Transverse Curvature,” ASME J. Fluid Eng., 134(2), p. 024503. [CrossRef]
Peixinho, J. , and Besnard, H. , 2013, “ Transition to Turbulence in Slowly Divergent Pipe Flow,” Phys. Fluids, 25(11), p. 111702. [CrossRef]
Deka, R. K. , and Paul, A. , 2013, “ Stability of Dean Flow Between Two Porous Concentric Cylinders With Radial Flow and a Constant Heat Flux at the Inner Cylinder,” ASME J. Fluid Eng., 135(4), p. 041203. [CrossRef]
Duryodhan, V. , Singh, S. , and Agarwal, A. , 2013, “ Liquid Flow Through a Diverging Microchannel,” Microfluid. Nanofluid., 14(1), pp. 53–67. [CrossRef]
Tripathi, S. , Prabhakar, A. , Kumar, N. , Singh, S. G. , and Agrawal, A. , 2013, “ Blood Plasma Separation in Elevated Dimension T-Shaped Microchannel,” Biomed. Microdevices, 15(3), pp. 415–425. [CrossRef] [PubMed]
Sobey, I. J. , 1980, “ On Flow Through Furrowed Channels—Part 1: Calculated Flow Patterns,” J. Fluid Mech., 96(1), pp. 1–26. [CrossRef]
Stephanoff, K. D. , Sobey, I. J. , and Bellhouse, B. J. , 1980, “ On Flow Through Furrowed Channels—Part 2: Observed Flow Patterns,” J. Fluid Mech., 96(1), pp. 27–32. [CrossRef]
Nishimura, T. , Ohori, Y. , and Kawamura, Y. , 1984, “ Flow Characteristics in a Channel With Symmetric Wavy Wall for Steady Flow,” J. Chem. Eng. Jpn., 17(5), pp. 466–471. [CrossRef]
Blancher, S. , Creff, R. , Batina, J. , and Andre, P. , 1994, “ Hydrodynamic Stability in Periodic Geometry,” Finite Elem. Anal. Des., 16(3–4), pp. 261–270. [CrossRef]
Cho, K. J. , Kim, M. , and Shin, H. D. , 1998, “ Linear Stability of Two-Dimensional Steady Flow in Wavy-Walled Channels,” Fluid Dyn. Res., 23(6), pp. 349–370. [CrossRef]
Rush, T. A. , Newell, T. A. , and Jacobi, A. M. , 1999, “ An Experimental Study of Flow and Heat Transfer in Sinusoidal Wavy Passages,” Int. J. Heat Mass Transfer, 42(9), pp. 1541–1553. [CrossRef]
Selvarajan, S. , Tulapurkara, E. G. , and Ram, V. V. , 1999, “ Stability Characteristics of Wavy Walled Channel Flows,” Phys. Fluids, 11(3), pp. 579–589. [CrossRef]
Stone, K. , and Vanka, S. , 1999, “ Numerical Study of Developing Flow and Heat Transfer in a Wavy Passage,” ASME J. Fluid Eng., 121(4), pp. 713–720. [CrossRef]
Floryan, J. M. , 2003, “ Vortex Instability in a Diverging-Converging Channel,” J. Fluid Mech., 482, pp. 17–50. [CrossRef]
Eagles, P. M. , 1966, “ The Stability of a Family of Jeffery–Hamel Solutions for Divergent Channel Flow,” J. Fluid Mech., 24(part 1), pp. 191–207. [CrossRef]
Sahu, K. C. , and Govindarajan, R. , 2005, “ Stability of Flow Through a Slowly Diverging Pipe,” J. Fluid Mech., 531, pp. 325–334. [CrossRef]
Blancher, S. , and Creff, R. , 2004, “ Analysis of Convective Hydrodynamics Instabilities in a Symmetric Wavy Channel,” Phys. Fluids, 16(10), pp. 3726–3737. [CrossRef]
Kim, S. K. , 2001, “ An Experimental Study of Flow in a Wavy Channel by Piv,” Sixth Asian Symposium on Visualization, Pusan, Korea, pp. 349–370.
Guzman, A. M. , and Amon, C. H. , 1994, “ Transition to Chaos in Converging–Diverging Channel Flows: Ruelle–Takens–Newhouse Scenario,” Phys. Fluids, 6(6), pp. 1994–2002. [CrossRef]
Cabal, A. , Szumbarski, J. , and Floryan, J. M. , 2002, “ Stability of Flow in a Wavy Channel,” J. Fluid Mech., 457, pp. 191–212. [CrossRef]
Floryan, J. M. , 2005, “ Two-Dimensional Instability of Flow in a Rough Channel,” Phys. Fluids, 17(4), p. 044101. [CrossRef]
Floryan, J. M. , 2007, “ Three-Dimensional Instabilities of Laminar Flow in a Rough Channel and the Concept of Hydraulically Smooth Wall,” Eur. J. Mech. B/Fluids, 26(3), pp. 305–329. [CrossRef]
Floryan, J. M. , and Floryan, C. , 2010, “ Traveling Wave Instability in a Diverging-Converging Channel,” Fluid Dyn. Res., 42(2), p. 025509. [CrossRef]
Floryan, J. , and Asai, M. , 2011, “ On the Transition Between Distributed and Isolated Surface Roughness and Its Effect on the Stability of the Channel Flow,” Phys. Fluids, 23(10), p. 104101. [CrossRef]
Szumbarski, J. , and Floryan, J. M. , 2006, “ Transient Disturbance Growth in a Corrugated Channel,” J. Fluid Mech., 568, pp. 243–272. [CrossRef]
Laval, J.-P. , and Marquillie, M. , 2011, “ Direct Numerical Simulations of Converging-Diverging Channel Flow,” Prog. Wall Turbul.: Understanding Model., ERCOFTAC Series, 14, pp. 203–209.
Akbari, M. , Sinton, D. , and Bahrami, M. , 2011, “ Viscous Flow in Variable Cross-Section Microchannels of Arbitary Shapes,” Int. J. Heat Mass Transfer, 54(17–18), pp. 3970–3978.
Jose, B. M. , and Cubaud, T. , 2012, “ Droplet Arrangement and Coalescence in Diverging/Converging Microchannels,” Microfluid. Nanofluid., 12(4), pp. 687–696. [CrossRef]
Sahu, K. S. , 2011, “ The Instability of Flow Through a Slowly Diverging Pipe With Viscous Heating,” ASME J. Fluid Eng., 133(7), p. 071201. [CrossRef]
Drazin, P. G. , and Reid, W. H. , 1981, Hydrodynamic Stability, Cambridge University, Cambridge, UK.
Theofilis, V. , Duck, P. W. , and Owen, J. , 2004, “ Viscous Linear Stability Analysis of Rectangular Duct and Cavity Flows,” J. Fluid Mech., 505, pp. 249–286. [CrossRef]
Venkatesh, T. N. , Sarasamma, V. R. , Rajalakshmy, S. , Sahu, K. C. , and Govindarajan, R. , 2005, “ Super-Linear Speedup of a Parallel Multigrid Navier–Stokes Solver on Flosolver,” Curr. Sci., 88(4), pp. 589–593.
Alleborn, N. , Nandakumar, K. , Raszillier, H. , and Durst, F. , 1997, “ Further Contributions on the Two-Dimensional Flow in a Sudden-Expansion Flow,” J. Fluid Mech., 330, pp. 169–188. [CrossRef]
Lanzerstorfer, D. , and Kuhlmann, H. C. , 2012, “ Global Stability of Multiple Solutions in Plane Sudden-Expansion Flow,” J. Fluid Mech., 702, pp. 378–402. [CrossRef]
Theofilis, V. , 2003, “ Advances in Global Linear Instability Analysis of Non-Parallel and Three-Dimensional Flows,” Progr. Aerosp. Sci., 39(4), pp. 249–315. [CrossRef]
Theofilis, V. , Sherwin, S. J. , and Abdessemed, N. , 2004, “ On Global Instabilities of Separated Bubble Flows and Their Control in External and Internal Aerodynamic Applications,” Report No. NATO RTO-AVT-111, p. 21.
Theofilis, V. , Federov, A. , Obrist, D. , and Dallmann, U. C. , 2003, “ The Extended Gortler–Hammerlin Model for Linear Instability of Three-Dimensional Incompressible Swept Attachment-Line Boundary Layer Flow,” J. Fluid Mech., 487, pp. 271–313. [CrossRef]
Ehrenstein, U. , and Gallaire, F. , 2005, “ On Two-Dimensional Temporal Modes in Spatially Evolving Open Flows: The Flat-Plate Boundary Layer,” J. Fluid Mech., 536, pp. 209–218. [CrossRef]
Akervik, E. , Ehrenstein, U. , Gallaire, F. , and Henningson, D. S. , 2008, “ Global Two-Dimensional Stability Measures of the Flat Plate Boundary-Layer Flow,” Eur. J. Mech. B/Fluids, 27(5), pp. 501–513. [CrossRef]
Chedevergne, F. , Casalis, G. , and Feraille, T. , 2006, “ Biglobal Linear Stability Analysis of the Flow Induced by Wall Injection,” Phys. Fluids, 18(1), p. 014103. [CrossRef]
Tatsumi, T. , and Yoshimura, T. , 1990, “ Stability of the Laminar Flow in a Rectangular Duct,” J. Fluid Mech., 212, pp. 437–449. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Schematic of a symmetric sinusoidal channel under study. Ratio of the amplitude of the wall waviness and to the minimum width of the channel, ϵ=2.3 and domain length is fixed at L = 10. Note that in a typical microchannel, several geometries like this one will be connected in series.

Grahic Jump Location
Fig. 2

Variation of the critical Reynolds number Recr with the wall-waviness amplitude ε in selected previous studies. Although different studies have used different aspect ratios, different domain length and different wall shapes, a trend is observed. Note that the result of Nishimura et al. [10] shown here does not correspond to the critical Reynolds number. The study of Blancher et al. [11] is on a different geometry. Please see text for further details.

Grahic Jump Location
Fig. 3

Contours of streamwise, U (top) and transverse, V (bottom) velocities of the base flow obtained from numerical simulations, for a symmetric case with ε=2.3 and L = 10 as shown in the parameter setting explained in Fig. 1, at Re = 5

Grahic Jump Location
Fig. 4

Contours of streamlines for a symmetric case with ε=2.3 and L = 10 and the corresponding laminar separation bubble formation for different Re as shown in the figure. The shaded region shows the separation bubble formed at the walls, containing recirculating flow. Notice that the bubbles at Re = 20 at the top and bottom walls differ slightly.

Grahic Jump Location
Fig. 5

Eigenspectra for flow through a converging–diverging channel with ε=2.3 and L = 10. Grid independence is shown using the stability grids (n × m, n and m being the resolution in x and y direction, respectively) as shown in the legend for Re = 20 (unstable case). The least stable mode is marked by the arrow.

Grahic Jump Location
Fig. 6

Eigenspectra for flow through a converging–diverging channel with ε=2.3 and L = 10. Grid independence is shown using the mean flow grids (n × m, n and m being the resolution in x and y direction, respectively) as shown in the legend for Re = 20 for a stability grid of size 51 × 51. This is an unstable case, where grid insensitivity is usually harder to obtain.

Grahic Jump Location
Fig. 7

Eigenspectra for flow through a converging–diverging channel with ε=1.3 and L = 10. Grid independence is shown using the stability grids, n × m, n and m being the resolution in x and y direction, respectively, as shown in the legend for Re = 50 (unstable case).

Grahic Jump Location
Fig. 8

The real part of the streamfunction ϕ (top), streamwise velocity u (middle), and the transverse velocity v (bottom) of the leading (unstable and stationary) mode. The imaginary parts of the components are equal to zero.

Grahic Jump Location
Fig. 9

Eigenspectrum for the flow in a converging–diverging channel with ε=2.3 and L = 10 for Re = 10 (unstable case) and resolution 51 × 51. Periodic boundary condition is imposed at the inlet and outlet.

Grahic Jump Location
Fig. 10

Perturbations (real/imaginary parts of the streamwise velocity component, u) marked by the arrows in Figs. 9(a) and 9(b). The real parts of u for the stationary modes ((c)–(e)). The real (top) and imaginary (bottom) parts of u for the traveling modes.

Grahic Jump Location
Fig. 11

The least stable branch of eigenspectra and the leading eigenmode for the different Re

Grahic Jump Location
Fig. 12

The real part of the streamwise velocity (u) of the leading eigenmode for (a) ε=2.3 (b)ε=1.3

Grahic Jump Location
Fig. 13

Converging–diverging channel with L = 10 for different ε, maximum amplitude of wall-waviness to minimum half-width of the channel. The growth rate, ωi, of the least stable eigenmode as a function of the Reynolds number is shown. The flow is neutrally stable at ωi=0, which defines the Recr.

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