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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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.

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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

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Fig. 12

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

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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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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

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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. 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.




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