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Special Section Articles

Nonlinear Rayleigh–Taylor Instability of a Cylindrical Interface in Explosion Flows

[+] Author and Article Information
Subramanian Annamalai

Department of Mechanical
and Aerospace Engineering,
University of Florida,
Gainesville, FL 32611
e-mail: subbu.ase@ufl.edu

Manoj K. Parmar

Department of Mechanical
and Aerospace Engineering,
University of Florida,
Gainesville, FL 32611
e-mail: mparmar@ufl.edu

Yue Ling

Department of Mechanical
and Aerospace Engineering,
University of Florida,
Gainesville, FL 32611
e-mail: yueling@ufl.edu

S. Balachandar

Department of Mechanical
and Aerospace Engineering,
University of Florida,
Gainesville, FL 32611
e-mail: bala1s@ufl.edu

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received May 14, 2013; final manuscript received November 5, 2013; published online April 28, 2014. Assoc. Editor: Ye Zhou.

J. Fluids Eng 136(6), 060910 (Apr 28, 2014) (15 pages) Paper No: FE-13-1320; doi: 10.1115/1.4026021 History: Received May 14, 2013; Revised November 05, 2013

The nonlinear growth of instabilities of an outward propagating, but decelerating, cylindrical interface separated by fluids of different densities is investigated. Single mode perturbations are introduced around the contact-surface, and their evolution is studied by conducting inviscid 2D and 3D numerical simulations. In the past, a significant amount of work has been carried out to model the development of the perturbations in a planar context where the contact surface is stationary or in a spherical context where a point-source blast wave is initiated at the origin. However, for the finite-source cylindrical blast-wave problem under consideration, there is a need for a framework which includes additional complexities such as compressibility, transition from linear to nonlinear stages of instability, finite thickness of the contact interface (CI), and time-dependent deceleration of the contact surface. Several theoretical potential flow models are presented. The model which is able to capture the above mentioned effects (causing deviation from the classical Rayleigh–Taylor Instability (RTI)) is identified as it compares reasonably well with the DNS results. Only for higher wavenumbers, the early development of secondary instabilities (Kelvin–Helmholtz) complicates the model prediction, especially in the estimation of the high-density fluid moving into low-density ambient.

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Figures

Grahic Jump Location
Fig. 1

(a) Schematic of a cylindrical shock tube. The shaded region represents the region encompassing the high-pressure gas. Here, PS denotes primary shock, CI is the contact interface, SS represents secondary shock and EFH denotes the head of expansion fan. (b) r-t diagram of a cylindrical shock tube for a pressure ratio of 22.

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Fig. 2

Density contour for m=24 at τ=0 (left) and at τ=1.38 (right). Only a portion of the sector is shown. Here, PS denotes primary shock, CI represents the contact interface and SS is the secondary shock.

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Fig. 3

Undisturbed CI acceleration (g) as a function of time, local minima (bubble) and local maxima (spike) densities across the undisturbed CI

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Fig. 4

(a) Schematic showing the bubble and spike amplitudes and (b) development of the initial Gaussian profile perturbation with time showing the evolution of the imposed mode (m=24) and the generation of harmonics (2m=48, 4m=96) at subsequent times. Also seen is the numerical definition of bubble and spike amplitudes.

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Fig. 5

(a) Ratio of bubble amplitude to wavelength and (b) growth of bubble amplitude for axial wave numbers (n) of 6,12,24, and 48

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Fig. 6

Comparison of DNS bubble amplitude with various buoyancy-drag models for a axial wave number (n) of (a) 6 and (b) 12

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Fig. 7

(a) Comparison of DNS spike amplitude with various buoyancy-drag models and (b) The ratio of bubble to spike amplitude (using model III) for a axial wave number (n) of 6

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Fig. 8

Comparison of (a) DNS bubble amplitude and (b) DNS spike amplitude for axial wave number (n) of 48 with various buoyancy-drag models

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Fig. 9

(a) Figure showing substantial mixing for high wave number (n=48) while (b) clear bubble and spike structures are visible for low wave number (n=6) at late times (τ=8.26)

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Fig. 10

Figure showing the total bubble velocity (u), bubble velocity purely due to instabilities (uinst), base flow velocity (ubase), and divergence of base flow velocity (D) at the instantaneous CI for the axial wave number (n) of 6

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Fig. 11

(a) Ratio of bubble amplitude to wavelength and (b) growth of bubble amplitude for circumferential wave numbers (m) of 6, 12, 24, and 48

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Fig. 12

Comparison of DNS bubble amplitude with various buoyancy-drag models for circumferential wave number (m) of (a) 6 and (b) 12

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Fig. 13

Comparison of (a) DNS bubble amplitude and (b) DNS spike amplitude for circumferential wave number (m) of 48 with various buoyancy-drag models

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Fig. 14

Distinguishable bubble and spike structures seen for circumferential wave number (m=48) at τ=6.9 (left) while no such clear bubble-spike pattern can be observed for axial wave number (n=48) at τ=6.9 (right)

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Fig. 15

(a) Ratio of bubble amplitude to wavelength and (b) Growth of bubble amplitude for mixed wave numbers of m=6,n=6 and m=24,n=6

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Fig. 16

(a) Comparison of DNS bubble amplitude for the (a) mixed wave number of m=6,n=6 and (b) m=24,n=6 with various buoyancy-drag models

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Fig. 17

(a) Comparison of the mixed wave number m=6,n=6 DNS bubble amplitude with the corresponding 2D wave numbers in the isolated circumferential case (m=6) and isolated axial case (n=6) and (b) comparison of the mixed wave number m=24,n=6 DNS bubble amplitude with the corresponding 2D counterparts in the isolated circumferential case (m=24) and isolated axial case (n=6)

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Fig. 18

Iso-density contour for m=6,n=6 case at (a) τ=2.75, (b) τ=6.89, and (c) τ=9.1

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Fig. 19

Iso-density contour for m=24,n=6 case at (a) τ=2.75, (b) τ=6.89, and (c) τ=9.1

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