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Title: Implementation of a simple model for linear and nonlinear mixing at unstable fluid interfaces in hydrodynamics codes

Abstract

A simple model was recently described for predicting the time evolution of the width of the mixing layer at an unstable fluid interface [J. D. Ramshaw, Phys. Rev. E 58, 5834 (1998); ibid. 61, 5339 (2000)]. The ordinary differential equations of this model have been heuristically generalized into partial differential equations suitable for implementation in multicomponent hydrodynamics codes. The central ingredient in this generalization is a nun-diffusional expression for the species mass fluxes. These fluxes describe the relative motion of the species, and thereby determine the local mixing rate and spatial distribution of mixed fluid as a function of time. The generalized model has been implemented in a two-dimensional hydrodynamics code. The model equations and implementation procedure are summarized, and comparisons with experimental mixing data are presented.

Authors:
Publication Date:
Research Org.:
Lawrence Livermore National Lab., CA (US)
Sponsoring Org.:
US Department of Energy (US)
OSTI Identifier:
15006143
Report Number(s):
UCRL-JC-139800
TRN: US200405%%282
DOE Contract Number:
W-7405-ENG-48
Resource Type:
Conference
Resource Relation:
Conference: 2000 Nuclear Explosives Code Development Conference, Oakland, CA (US), 10/23/2000--10/27/2000; Other Information: PBD: 1 Oct 2000
Country of Publication:
United States
Language:
English
Subject:
45 MILITARY TECHNOLOGY, WEAPONRY, AND NATIONAL DEFENSE; 73 NUCLEAR PHYSICS AND RADIATION PHYSICS; 70 PLASMA PHYSICS AND FUSION TECHNOLOGY; DIFFERENTIAL EQUATIONS; HYDRODYNAMICS; IMPLEMENTATION; NUCLEAR EXPLOSIVES; PARTIAL DIFFERENTIAL EQUATIONS; SPATIAL DISTRIBUTION

Citation Formats

Ramshaw, J D. Implementation of a simple model for linear and nonlinear mixing at unstable fluid interfaces in hydrodynamics codes. United States: N. p., 2000. Web.
Ramshaw, J D. Implementation of a simple model for linear and nonlinear mixing at unstable fluid interfaces in hydrodynamics codes. United States.
Ramshaw, J D. Sun . "Implementation of a simple model for linear and nonlinear mixing at unstable fluid interfaces in hydrodynamics codes". United States. doi:. https://www.osti.gov/servlets/purl/15006143.
@article{osti_15006143,
title = {Implementation of a simple model for linear and nonlinear mixing at unstable fluid interfaces in hydrodynamics codes},
author = {Ramshaw, J D},
abstractNote = {A simple model was recently described for predicting the time evolution of the width of the mixing layer at an unstable fluid interface [J. D. Ramshaw, Phys. Rev. E 58, 5834 (1998); ibid. 61, 5339 (2000)]. The ordinary differential equations of this model have been heuristically generalized into partial differential equations suitable for implementation in multicomponent hydrodynamics codes. The central ingredient in this generalization is a nun-diffusional expression for the species mass fluxes. These fluxes describe the relative motion of the species, and thereby determine the local mixing rate and spatial distribution of mixed fluid as a function of time. The generalized model has been implemented in a two-dimensional hydrodynamics code. The model equations and implementation procedure are summarized, and comparisons with experimental mixing data are presented.},
doi = {},
journal = {},
number = ,
volume = ,
place = {United States},
year = {Sun Oct 01 00:00:00 EDT 2000},
month = {Sun Oct 01 00:00:00 EDT 2000}
}

Conference:
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  • A simple model is described for predicting the time evolution of the half-width h of a planar mixing layer between two immiscible incompressible fluids driven by an arbitrary time-dependent variable acceleration history a(l)a (t): The model is based on a heuristic expression for the kinetic energy per unit area of the mixing layer. This expression is based on that for the kinetic energy of a linearly perturbed interface, but with a dynamically renormalized wavelength which becomes proportional to h in the nonlinear regime. An equation of motion for h is then derived by means of Lagrange's equations. This model reproducesmore » the known linear growth rates of the Rayleigh-Taylor (RT) and Richtmyer-Meshkov (RM) instabilities, as well as the quadratic RT and power-law RM growth laws in the nonlinear regime. The time exponent in the RM power law depends on the rate of kinetic energy dissipation. In the case of zero dissipation, this exponent reduces to 2/3 in agreement with elementary scaling arguments. A conservative numerical scheme is proposed to solve the model equations, and is used to perform calculations that agree well with published mixing data from linear electric motor experiments. Considerations involved in implementing the model in hydrodynamics codes are briefly discussed.« less
  • A simple model was recently described for predicting linear and nonlinear mixing at an unstable planar fluid interface subjected to an arbitrary time-dependent variable acceleration history [J. D. Ramshaw, Phys. Rev. E {bold 58}, 5834 (1998)]. Here we present an analogous model for describing the mixing of two adjacent spherical fluid shells of different density resulting from an arbitrary time-dependent mean interface radius R(t). As in the planar case, the model is based on a heuristic expression for the kinetic energy of the system. This expression is based on that for the kinetic energy of a linearly perturbed interface, butmore » with a dynamically renormalized effective wavelength which becomes proportional to the half-width a(t) of the mixing layer in the nonlinear regime. An equation of motion for s=R{sup 2}a is then derived from Lagrange{close_quote}s equations. This evolution equation properly reduces to Plesset{close_quote}s equation for small perturbations, and to the previous planar model in the limit of very large R. The conservation properties of the model are established, and a suitable numerical scheme which preserves these properties is proposed. {copyright} {ital 1999} {ital The American Physical Society}« less
  • A simple model is described for predicting the time evolution of the half-width h of a mixing layer between two initially separated immiscible fluids of different density subjected to an arbitrary time-dependent variable acceleration history a(t). The model is based on a heuristic expression for the kinetic energy per unit area of the mixing layer. This expression is based on that for the kinetic energy of a linearly perturbed interface, but with a dynamically renormalized wavelength which becomes proportional to h in the nonlinear regime. An equation of motion for h is then derived from Lagrange{close_quote}s equations. This model reproducesmore » the known linear growth rates of the Rayleigh-Taylor (RT) and Richtmyer-Meshkov (RM) instabilities, as well as the nonlinear RT growth law h={alpha}Aat{sup 2} for constant a (where A is the Atwood number) and the nonlinear RM growth law h{approximately}t{sup {theta}} for impulsive a, where {alpha} and {theta} depend on the rate of kinetic energy dissipation. In the case of zero dissipation, {theta}=2/3 in agreement with elementary scaling arguments. A conservative numerical scheme is proposed to solve the model equations, and is used to perform calculations that agree well with published experimental mixing data for four different acceleration histories. {copyright} {ital 1998} {ital The American Physical Society}« less
  • A simple model was recently described for predicting linear and nonlinear mixing at an unstable planar interface between two fluids of different density subjected to an arbitrary time-dependent variable acceleration history [J. D. Ramshaw, Phys. Rev. E 58, 5834 (1998)]. Here we generalize this model to include the Kelvin-Helmholtz (KH) instability resulting from a tangential velocity discontinuity {delta}u, as well as the effects of a uniform anisotropic compression or expansion of the mixing layer as a whole. The model consists of a second-order nonlinear ordinary differential equation of motion for the half-width h of the mixing layer. This equation ismore » derived by combining the wavelength renormalization hypothesis used in the earlier model with a suitable expression for the rate of change of the kinetic energy of the mixing layer. The resulting generalized model contains no additional free parameters, and reduces to the previous model in the absence of tangential velocities and compression. It also reduces in the linear regime to the correct linearized stability equation for an accelerated shear layer with compression [J. D. Ramshaw, Phys. Rev. E 61, 1486 (2000)]. For a pure incompressible KH instability in the nonlinear regime, the model predicts that h={eta}|{delta}u|t, where {eta}=[{alpha}(2-{theta})/(sq root)({theta}(1-{theta}))]{radical}({rho}{sub 1}{rho}{sub 2})/({rho}{sub 1}+{rho}{sub 2}), and {alpha} and {theta} are parameters appearing in the nonlinear Rayleigh-Taylor and Richtmyer-Meshkov growth laws. For equal densities and the same parameter values previously used to match variable-acceleration experimental data, we find {eta}=0.10, in close agreement with experimental data for free shear layers. (c) 2000 The American Physical Society.« less
  • We report applications of several high-speed photographic techniques to diagnose fluid instability and the onset of turbulence in an ongoing experimental study of the evolution of shock-accelerated, heavy-gas cylinders. Results are at Reynolds numbers well above that associated with the turbulent and mixing transitions. Recent developments in diagnostics enable high-resolution, planar (2D) measurements of velocity fields (using particle image velocimetry, or PIV) and scalar concentration (using planar laser-induced fluorescence, or PLIF). The purpose of this work is to understand the basic science of complex, shock-driven flows and to provide high-quality data for code validation and development. The combination of thesemore » high-speed optical methods, PIV and PLIF, is setting a new standard in validating large codes for fluid simulations. The PIV velocity measurements provide quantitative evidence of transition to turbulence. In the PIV technique, a frame transfer camera with a 1 ms separation is used to image flows illuminated by two 10 ns laser pulses. Individual particles in a seeded flow are tracked from frame to frame to produce a velocity field. Dynamic PLIF measurements of the concentration field are high-resolution, quantitative dynamic data that reveal finely detailed structure at several instances after shock passage. These structures include those associated with the incipient secondary instability and late-time transition. Multiple instances of the flow are captured using a single frame Apogee camera and laser pulses with 140 {mu}s spacing. We describe tradeoffs of diagnostic instrumentation to provide PLIF images.« less