The Elder Problem

Abstract

This paper presents an autobiographical and biographical historical account of the genesis, evolution and resolution of the Elder Problem. It begins with John W. Elder and his autobiographical story leading to his groundbreaking work on natural convection at Cambridge in the 1960's. His seminal work published in the Journal of Fluid Mechanics in 1967 became the basis for the modern benchmark of variable density flow simulators that we know today as "The Elder Problem". There have been well known and major challenges with the Elder Problem model benchmark-notably the multiple solutions that were ultimately uncovered using different numerical models. Most recently, it has been shown that the multiple solutions are indeed physically realistic bifurcation solutions to the Elder Problem and not numerically spurious artefacts. The quandary of the Elder Problem has now been solved-a major scientific breakthrough for fluid mechanics and for numerical modelling. This paper-records, reflections, reminiscences, stories and anecdotes-is an historical autobiographical and biographical memoir. It is the personal story of the Elder Problem told by some of the key scientists who established and solved the Elder Problem. 2017 marks the 50 year anniversary of the classical work by John W. Elder published in Journal of Fluid Mechanics in 1967. This set the stage for this scientific story over some five decades. This paper is a celebration and commemoration of the life and times of John W. Elder, the problem named in his honour, and some of the key scientists who worked on, and ultimately solved, it.

The Elder Problem

“…Understanding and reporting on the genesis, evolution, quandary and solution to the Elder Problem has been an interesting and important exercise. Our full historical autobiographical and biographical memoir (Elder et al ) documents our personal account. We have enjoyed working with many of the key protagonists, including Diersch, Voss, Holzbecher, Frolkovič, and Johannsen to put the various bits and pieces of the jigsaw puzzle together.…”

“…The year 2017 marks the 50‐year anniversary of John W. Elder's pioneering experiments on natural convection published in 1967 in the Journal of Fluid Mechanics (JFM) (Elder , ). A historical paper was recently published narrating the personal autobiographical and biographical stories related to Elder's original work; the development of the popular groundwater modeling benchmark known today as the Elder Problem; the quandary created by the benchmark; and the eventual solution to it (Elder et al ). In this Historical Note, we wish to highlight the salient features of the story of the Elder Problem and describe its significance for the groundwater community.…”

“…John W. Elder with the Hele‐Shaw cell apparatus he used for his experiments at Cambridge in 1967 (photo taken on January 5, 2017; from Elder et al . (Figure A1 of Elder et al [] paper; published by MDPI, 2017.) …”

Numerical Simulation of Density-Driven Flow and Heat Transport Processes in Porous Media Using the Network Method

“…Elder's experiment is somewhat similar to Bénard´s, but chooses a distribution where the system is only partially heated from below (half of the lower boundary). An excellent autobiographical and biographical historical account of the genesis, evolution and resolution can be found in Elder et al [31]. Using the streamfunction formulation and normalized temperature variables for a 2-D domain, Elder presented the first numerical simulation of the laboratory set-up in the same publication.…”

Solution of the 3D density-driven groundwater flow problem with uncertain porosity and permeability

“…Thus, simulation and analysis of this density-driven flow play an important role in predicting how pollution can migrate through an aquifer [28,43].In contrast to the transport of pollution by molecular diffusion, convection due to density-driven flow is an unstable process that can undergo quite complicated patterns of distribution. The Elder problem is a simplified but comprehensive model that describes the intrusion of salt water from a top boundary into an aquifer [23,24, 25,79]. The evolution of the concentration profile is typically referred to as fingering.…”