Simple model for linear and nonlinear mixing at unstable fluid interfaces with variable acceleration
Abstract
A simple model is described for predicting the time evolution of the halfwidth h of a planar mixing layer between two immiscible incompressible fluids driven by an arbitrary timedependent 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 reproduces the known linear growth rates of the RayleighTaylor (RT) and RichtmyerMeshkov (RM) instabilities, as well as the quadratic RT and powerlaw 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 aremore »
 Authors:
 Publication Date:
 Research Org.:
 Lawrence Livermore National Laboratory, Livermore, CA
 Sponsoring Org.:
 USDOE Office of Defense Programs (DP)
 OSTI Identifier:
 3402
 Report Number(s):
 UCRLJC132872
DP0101031; ON: DE00003402
 DOE Contract Number:
 W7405Eng48
 Resource Type:
 Conference
 Resource Relation:
 Conference: 1998 Nuclear Explosives Development Conference, Las Vegas, NV, October 2530, 1998
 Country of Publication:
 United States
 Language:
 English
 Subject:
 66 PHYSICS; Mixing; Instability; RayleighTaylor Instability; Helmholtz Instability; Fluids; Interfaces; Acceleration
Citation Formats
Ramshaw, J D, and Rathkopf, J. Simple model for linear and nonlinear mixing at unstable fluid interfaces with variable acceleration. United States: N. p., 1998.
Web.
Ramshaw, J D, & Rathkopf, J. Simple model for linear and nonlinear mixing at unstable fluid interfaces with variable acceleration. United States.
Ramshaw, J D, and Rathkopf, J. 1998.
"Simple model for linear and nonlinear mixing at unstable fluid interfaces with variable acceleration". United States.
doi:. https://www.osti.gov/servlets/purl/3402.
@article{osti_3402,
title = {Simple model for linear and nonlinear mixing at unstable fluid interfaces with variable acceleration},
author = {Ramshaw, J D and Rathkopf, J},
abstractNote = {A simple model is described for predicting the time evolution of the halfwidth h of a planar mixing layer between two immiscible incompressible fluids driven by an arbitrary timedependent 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 reproduces the known linear growth rates of the RayleighTaylor (RT) and RichtmyerMeshkov (RM) instabilities, as well as the quadratic RT and powerlaw 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.},
doi = {},
journal = {},
number = ,
volume = ,
place = {United States},
year = 1998,
month =
}

A simple model is described for predicting the time evolution of the halfwidth h of a mixing layer between two initially separated immiscible fluids of different density subjected to an arbitrary timedependent 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 »

Implementation of a simple model for linear and nonlinear mixing at unstable fluid interfaces in hydrodynamics codes
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 nundiffusional 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.more » 
Simple model for linear and nonlinear mixing at unstable fluid interfaces in spherical geometry
A simple model was recently described for predicting linear and nonlinear mixing at an unstable planar fluid interface subjected to an arbitrary timedependent 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 timedependent 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 » 
Simple model for mixing at accelerated fluid interfaces with shear and compression
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 timedependent variable acceleration history [J. D. Ramshaw, Phys. Rev. E 58, 5834 (1998)]. Here we generalize this model to include the KelvinHelmholtz (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 secondorder nonlinear ordinary differential equation of motion for the halfwidth h of the mixing layer. This equation ismore » 
Planar velocity and scalar concentration measurements in shockaccelerated,unstable fluid interfaces.
We report applications of several highspeed photographic techniques to diagnose fluid instability and the onset of turbulence in an ongoing experimental study of the evolution of shockaccelerated, heavygas cylinders. Results are at Reynolds numbers well above that associated with the turbulent and mixing transitions. Recent developments in diagnostics enable highresolution, planar (2D) measurements of velocity fields (using particle image velocimetry, or PIV) and scalar concentration (using planar laserinduced fluorescence, or PLIF). The purpose of this work is to understand the basic science of complex, shockdriven flows and to provide highquality data for code validation and development. The combination of thesemore »