Bradley Plohr - Academia.edu (original) (raw)

Papers by Bradley Plohr

We consider a model for immiscible three-phase (e.g., water, oil, and gas) flow in a porous mediu... more We consider a model for immiscible three-phase (e.g., water, oil, and gas) flow in a porous medium. We allow the relative permeability of the gas phase to exhibit hysteresis, in that it varies irreversibly along two extreme paths (the imbibition and drainage curves) that bound a region foliated by reversible paths (scanning curves). By numerically solving one-dimensional flow problems involving simultaneous and alternating injection of water and gas into a rock core, we demonstrate the effects of hysteresis.

arXiv (Cornell University), Aug 14, 2013

In large eddy simulations, the Reynolds averages of nonlinear terms are not directly computable i... more In large eddy simulations, the Reynolds averages of nonlinear terms are not directly computable in terms of the resolved variables and require a closure hypothesis or model, known as a subgrid scale term. Inspired by the renormalization group (RNG), we introduce an expansion for the unclosed terms, carried out explicitly to all orders. In leading order, this expansion defines subgrid scale unclosed terms, which we relate to the dynamic subgrid scale closure models. The expansion, which generalizes the Leonard stress for closure analysis, suggests a systematic higher order determination of the model coefficients. The RNG point of view sheds light on the nonuniqueness of the infinite Reynolds number limit. For the mixing of N species, we see an N + 1 parameter family of infinite Reynolds number solutions labeled by dimensionless parameters of the limiting Euler equations, in a manner intrinsic to the RNG itself. Large eddy simulations, with their Leonard stress and dynamic subgrid models, break this nonuniqueness and predict unique model coefficients on the basis of theory. In this sense large eddy simulations go beyond the RNG methodology, which does not in general predict model coefficients.

Contemporary mathematics, 1989

Spe Journal, Jun 1, 2001

In immiscible three-phase flow, the lead oil bank can split into two, a Buckley-Leverett shock wa... more In immiscible three-phase flow, the lead oil bank can split into two, a Buckley-Leverett shock wave followed by a new type of shock wave. Such a nonclassical "transitional" shock wave is common in three-phase flow. Its sensitivity to diffusion implies that capillary pressure must be modeled correctly to calculate the flow. In particular, transitional shock waves arise in water-alternating-gas (WAG) flow. They can be calculated by semi-analytic methods, which are helpful in the design of effective WAG recovery strategies.

Acta Mathematica Scientia, 2012

ABSTRACT Practical simulations of turbulent processes are generally cutoff, with a grid resolutio... more ABSTRACT Practical simulations of turbulent processes are generally cutoff, with a grid resolution that stops within the inertial range, meaning that multiple active regions and length scales occur below the grid level and are not resolved. This is the regime of large eddy simulations (LES), in which the larger but not the smaller of the turbulent length scales are resolved. Solutions of the fluid Navier-Stokes equations, when considered in the inertial regime, are conventionally regarded as solutions of the Euler equations. In other words, the viscous and diffusive transport terms in the Navier-Stokes equations can be neglected in the inertial regime and in LES simulations, while the Euler equation becomes fundamental.For such simulations, significant new solution details emerge as the grid is refined. It follows that conventional notions of grid convergence are at risk of failure, and that a new, and weaker notion of convergence may be appropriate. It is generally understood that the LES or inertial regime is inherently fluctuating and its description must be statistical in nature. Here we develop such a point of view systematically, based on Young measures, which are measures depending on or indexed by space time points. In the Young measure (ξ)x,t, the random variable ξ of the measure is a solution state variable, i.e., a solution dependent variable, representing momentum, density, energy and species concentrations, while the space time coordinates, x, t, serve to index the measure.Theoretical evidence suggests that convergence via Young measures is sufficiently weak to encompass the LES/inertial regime; numerical and theoretical evidence suggests that this notion may be required for passive scalar concentration and thermal degrees of freedom. Our objective in this research is twofold: turbulent simulations without recourse to adjustable parameters (calibration) and extension to more complex physics, without use of additional models or parameters, in both cases with validation through comparison to experimental data.

arXiv (Cornell University), Nov 18, 2022

We introduce the vanishing adsorption criterion for contact discontinuities and apply it to the G... more We introduce the vanishing adsorption criterion for contact discontinuities and apply it to the Glimm-Isaacson model of chemical flooding of a petroleum reservoir. We prove that this criterion, which derives from a physical effect, justifies the admissibility criteria adopted previously by Keyfitz-Kranzer, Isaacson-Temple, and de Souza-Marchesin for models such that the fractional flow function depends monotonically on chemical concentration. We also demonstrate that the adsorption criterion selects the undercompressive contact discontinuities required to solve the general Riemann problem in an example model with non-monotone dependence.

: A method for accelerating thin metallic plates to hypervelocities has been proposed by G. McCal... more : A method for accelerating thin metallic plates to hypervelocities has been proposed by G. McCall. In this method a shock in a propellant generates a strong expansion wave that smoothly accelerates the plate. We have studied the hydrodynamics of this process in one dimension, both analytically and computationally. The metal was modeled as a stiffened gas, and the corresponding Riemann problem was solved. The asymptotic behavior of the solution was determined analytically. The one-dimensional random choice method, modified so that material boundaries are tracked and the spatial mesh is refined locally, was used to compute the flow, comparison with the for within the accelerating plate were accurately resolved, so that possible structural demate to the plate could be evaluated. Keywords: Computation; Numerical methods and procedures; Shock waves; Thermodynamic models; flow; Tungsten plates.

Matemática Contemporânea

We study the stability and asymptotic behavior of transitional shock waves as solutions of a para... more We study the stability and asymptotic behavior of transitional shock waves as solutions of a parabolic system of conservation laws. In contrast tp classical shock waves, transitional shock waves are semitive to the precise form of the parabolic term, not only in their internal structure but also in terms of the end states that they connect. In our numerical investigation, these waves exbibit robust stability. Moreover, their response to perturbation differs from that of classical waves; in particular, the asymptotic state of a perturbed transitional wave depends on the location of the perturbation relative to the shock wave. We develop a linear scattering model that predicts behavior agreeing quantitatively with our numerical results. 192 K. ZUMBRUN, B. J. PLOHR, D. MARCHESIN SCATTERING O F TRANSITIONAL WAVES of Riemann problems for non-strictly-hyperbolk systems, for example in the @d F 1 (0) is a multiple of the identity matrix.

IV ic re*"....n 'P'P :011KfIC Ct 1mfYfmttort 11'd commenits fe~ar*-1 O's, cufaei qs'-ste v' .

Annals of Mathematical Sciences and Applications, 2016

In large eddy simulations, the Reynolds averages of nonlinear terms are not directly computable i... more In large eddy simulations, the Reynolds averages of nonlinear terms are not directly computable in terms of the resolved variables and require a closure hypothesis or model, known as a subgrid scale term. Inspired by the renormalization group (RNG), we introduce an expansion for the unclosed terms, carried out explicitly to all orders. In leading order, this expansion defines subgrid scale unclosed terms, which we relate to the dynamic subgrid scale closure models. The expansion, which generalizes the Leonard stress for closure analysis, suggests a systematic higher order determination of the model coefficients. The RNG point of view sheds light on the nonuniqueness of the infinite Reynolds number limit. For the mixing of N species, we see an N + 1 parameter family of infinite Reynolds number solutions labeled by dimensionless parameters of the limiting Euler equations, in a manner intrinsic to the RNG itself. Large eddy simulations, with their Leonard stress and dynamic subgrid models, break this nonuniqueness and predict unique model coefficients on the basis of theory. In this sense large eddy simulations go beyond the RNG methodology, which does not in general predict model coefficients.

We investigate solutions of Riemann problems for systems of two conservation laws in one spatial ... more We investigate solutions of Riemann problems for systems of two conservation laws in one spatial dimension. Our approach is to organize Riemann solutions into strata of successively higher codimension. The codimension-zero stratum consists of Riemann solutions that are structurally stable: the number and types of waves in a solution are preserved under small perturbations of the flux function and initial data. Codimension-one Riemann solutions, which constitute most of the boundary of the codimension-zero stratum, violate structural stability in a minimal way. At the codimension-one stratum, either the qualitative structure of Riemann solutions changes or solutions fail to be parameterized smoothly by the flux function and the initial data. In this paper, we give an overview of the phenomena associated with codimension-one Riemann solutions. We list the different kinds of codimension-one solutions, and we classify them according to their geometric properties, their roles in solving Riemann problems, and their relationships to wave curves.

We consider a model for immiscible three-phase (e.g., water, oil, and gas) flow in a porous mediu... more We consider a model for immiscible three-phase (e.g., water, oil, and gas) flow in a porous medium. We allow the relative permeability of the gas phase to exhibit hysteresis, in that it varies irreversibly along two extreme paths (the imbibition and drainage curves) that bound a region foliated by reversible paths (scanning curves). By numerically solving one-dimensional flow problems involving simultaneous and alternating injection of water and gas into a rock core, we demonstrate the effects of hysteresis.

arXiv (Cornell University), Aug 14, 2013

In large eddy simulations, the Reynolds averages of nonlinear terms are not directly computable i... more In large eddy simulations, the Reynolds averages of nonlinear terms are not directly computable in terms of the resolved variables and require a closure hypothesis or model, known as a subgrid scale term. Inspired by the renormalization group (RNG), we introduce an expansion for the unclosed terms, carried out explicitly to all orders. In leading order, this expansion defines subgrid scale unclosed terms, which we relate to the dynamic subgrid scale closure models. The expansion, which generalizes the Leonard stress for closure analysis, suggests a systematic higher order determination of the model coefficients. The RNG point of view sheds light on the nonuniqueness of the infinite Reynolds number limit. For the mixing of N species, we see an N + 1 parameter family of infinite Reynolds number solutions labeled by dimensionless parameters of the limiting Euler equations, in a manner intrinsic to the RNG itself. Large eddy simulations, with their Leonard stress and dynamic subgrid models, break this nonuniqueness and predict unique model coefficients on the basis of theory. In this sense large eddy simulations go beyond the RNG methodology, which does not in general predict model coefficients.

Contemporary mathematics, 1989

Spe Journal, Jun 1, 2001

In immiscible three-phase flow, the lead oil bank can split into two, a Buckley-Leverett shock wa... more In immiscible three-phase flow, the lead oil bank can split into two, a Buckley-Leverett shock wave followed by a new type of shock wave. Such a nonclassical "transitional" shock wave is common in three-phase flow. Its sensitivity to diffusion implies that capillary pressure must be modeled correctly to calculate the flow. In particular, transitional shock waves arise in water-alternating-gas (WAG) flow. They can be calculated by semi-analytic methods, which are helpful in the design of effective WAG recovery strategies.

Acta Mathematica Scientia, 2012

ABSTRACT Practical simulations of turbulent processes are generally cutoff, with a grid resolutio... more ABSTRACT Practical simulations of turbulent processes are generally cutoff, with a grid resolution that stops within the inertial range, meaning that multiple active regions and length scales occur below the grid level and are not resolved. This is the regime of large eddy simulations (LES), in which the larger but not the smaller of the turbulent length scales are resolved. Solutions of the fluid Navier-Stokes equations, when considered in the inertial regime, are conventionally regarded as solutions of the Euler equations. In other words, the viscous and diffusive transport terms in the Navier-Stokes equations can be neglected in the inertial regime and in LES simulations, while the Euler equation becomes fundamental.For such simulations, significant new solution details emerge as the grid is refined. It follows that conventional notions of grid convergence are at risk of failure, and that a new, and weaker notion of convergence may be appropriate. It is generally understood that the LES or inertial regime is inherently fluctuating and its description must be statistical in nature. Here we develop such a point of view systematically, based on Young measures, which are measures depending on or indexed by space time points. In the Young measure (ξ)x,t, the random variable ξ of the measure is a solution state variable, i.e., a solution dependent variable, representing momentum, density, energy and species concentrations, while the space time coordinates, x, t, serve to index the measure.Theoretical evidence suggests that convergence via Young measures is sufficiently weak to encompass the LES/inertial regime; numerical and theoretical evidence suggests that this notion may be required for passive scalar concentration and thermal degrees of freedom. Our objective in this research is twofold: turbulent simulations without recourse to adjustable parameters (calibration) and extension to more complex physics, without use of additional models or parameters, in both cases with validation through comparison to experimental data.

arXiv (Cornell University), Nov 18, 2022

We introduce the vanishing adsorption criterion for contact discontinuities and apply it to the G... more We introduce the vanishing adsorption criterion for contact discontinuities and apply it to the Glimm-Isaacson model of chemical flooding of a petroleum reservoir. We prove that this criterion, which derives from a physical effect, justifies the admissibility criteria adopted previously by Keyfitz-Kranzer, Isaacson-Temple, and de Souza-Marchesin for models such that the fractional flow function depends monotonically on chemical concentration. We also demonstrate that the adsorption criterion selects the undercompressive contact discontinuities required to solve the general Riemann problem in an example model with non-monotone dependence.

: A method for accelerating thin metallic plates to hypervelocities has been proposed by G. McCal... more : A method for accelerating thin metallic plates to hypervelocities has been proposed by G. McCall. In this method a shock in a propellant generates a strong expansion wave that smoothly accelerates the plate. We have studied the hydrodynamics of this process in one dimension, both analytically and computationally. The metal was modeled as a stiffened gas, and the corresponding Riemann problem was solved. The asymptotic behavior of the solution was determined analytically. The one-dimensional random choice method, modified so that material boundaries are tracked and the spatial mesh is refined locally, was used to compute the flow, comparison with the for within the accelerating plate were accurately resolved, so that possible structural demate to the plate could be evaluated. Keywords: Computation; Numerical methods and procedures; Shock waves; Thermodynamic models; flow; Tungsten plates.

Matemática Contemporânea

We study the stability and asymptotic behavior of transitional shock waves as solutions of a para... more We study the stability and asymptotic behavior of transitional shock waves as solutions of a parabolic system of conservation laws. In contrast tp classical shock waves, transitional shock waves are semitive to the precise form of the parabolic term, not only in their internal structure but also in terms of the end states that they connect. In our numerical investigation, these waves exbibit robust stability. Moreover, their response to perturbation differs from that of classical waves; in particular, the asymptotic state of a perturbed transitional wave depends on the location of the perturbation relative to the shock wave. We develop a linear scattering model that predicts behavior agreeing quantitatively with our numerical results. 192 K. ZUMBRUN, B. J. PLOHR, D. MARCHESIN SCATTERING O F TRANSITIONAL WAVES of Riemann problems for non-strictly-hyperbolk systems, for example in the @d F 1 (0) is a multiple of the identity matrix.

IV ic re*"....n 'P'P :011KfIC Ct 1mfYfmttort 11'd commenits fe~ar*-1 O's, cufaei qs'-ste v' .

Annals of Mathematical Sciences and Applications, 2016

In large eddy simulations, the Reynolds averages of nonlinear terms are not directly computable i... more In large eddy simulations, the Reynolds averages of nonlinear terms are not directly computable in terms of the resolved variables and require a closure hypothesis or model, known as a subgrid scale term. Inspired by the renormalization group (RNG), we introduce an expansion for the unclosed terms, carried out explicitly to all orders. In leading order, this expansion defines subgrid scale unclosed terms, which we relate to the dynamic subgrid scale closure models. The expansion, which generalizes the Leonard stress for closure analysis, suggests a systematic higher order determination of the model coefficients. The RNG point of view sheds light on the nonuniqueness of the infinite Reynolds number limit. For the mixing of N species, we see an N + 1 parameter family of infinite Reynolds number solutions labeled by dimensionless parameters of the limiting Euler equations, in a manner intrinsic to the RNG itself. Large eddy simulations, with their Leonard stress and dynamic subgrid models, break this nonuniqueness and predict unique model coefficients on the basis of theory. In this sense large eddy simulations go beyond the RNG methodology, which does not in general predict model coefficients.

We investigate solutions of Riemann problems for systems of two conservation laws in one spatial ... more We investigate solutions of Riemann problems for systems of two conservation laws in one spatial dimension. Our approach is to organize Riemann solutions into strata of successively higher codimension. The codimension-zero stratum consists of Riemann solutions that are structurally stable: the number and types of waves in a solution are preserved under small perturbations of the flux function and initial data. Codimension-one Riemann solutions, which constitute most of the boundary of the codimension-zero stratum, violate structural stability in a minimal way. At the codimension-one stratum, either the qualitative structure of Riemann solutions changes or solutions fail to be parameterized smoothly by the flux function and the initial data. In this paper, we give an overview of the phenomena associated with codimension-one Riemann solutions. We list the different kinds of codimension-one solutions, and we classify them according to their geometric properties, their roles in solving Riemann problems, and their relationships to wave curves.