# The Elder Problem

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## Abstract

**:**

## 1. Prologue

## 2. Model Benchmarks

## 3. Our Story

## 4. Part 1: The Origin of This Paper—Craig Simmons’ Story

## 5. Part 2: John Elder’s Story—An Autobiographical Reflection

#### 5.1. Origin of the Model System

#### 5.1.1. 1953/1954 The Dawn: Geothermal and All, Department of Scientific and Industrial Research (DSIR), Wellington

#### The Wairakei Geothermal Project

#### 5.1.2. 1951–1955 at University in New Zealand

[And so during November-December 1955, by ship with 1200 others: I sailed away across the Pacific; through the Panama Canal; across the Atlantic to the land of chimney pots (first impression on board at the wharf).]

#### 5.1.3. 1956–1958 Cavendish Laboratory, Cambridge University

^{2}= another constant.

Some other process must be at work. The heat is transferred by convection: by the flow of the fluid moving through the porous/permeable medium because of unbalanced pressures arising from density differences produced from temperature differences.

#### 5.1.4. 1959–1961 Geophysics Division, DSIR Wellington

^{2}) [contrary to the suggestion of a high regional heat flux].

#### 5.1.5. 1961–1962 Where Next?

#### 5.1.6. 1962–1964 Here Again: Back in Cambridge

#### 5.1.7. 1964–1965 Institute of Geophysics and Planetary Physics (IGPP)

#### 5.1.8. 1966–1970 All Together Now: DAMTP

#### The Maths Lab

#### 5.1.9. And Not Least We Have 1966–1967

**1967**, 27, 609–623 [2]) was focused on the comparison of the experimental observations together with simple numerical methods for both a steady state solution and the corresponding (with the same boundary conditions) numerical time-dependent solution—as much as anything to give confidence in the use of the numerical methods.

#### 5.2. The Model Behaviour

#### 5.3. Looking Ahead

- [Set the salinity = 0 on all boundaries except for t ≥ 0, on the top boundary z = 1, from x = −l/2 to x = l/2 set the salinity = 1.
- The container is 2l wide and 1 high.
- The saline source occupies half of the top boundary.]

## 6. Part 3: The Elder Problem Benchmark—Uptake of Elder’s 1967 JFM Paper into Modern Fluid Mechanics and Modelling in the 1980s and 1990s

#### 6.1. Key Questions

- (1)
- How did it come into existence as the formally titled “Problem” we know today?
- (2)
- How did it obtain this nomenclature?
- (3)
- Who were the earliest scientists who studied it as a potential model code benchmark and what motivated their work?

Prof. Hans Diersch worked on one side of The Wall, in East Berlin at the Institute of Mechanics of the Academy of Sciences of German Democratic Republic (GDR); Ekkehard Holzbecher worked on the other side of The Wall in West Berlin at the Technical University of Berlin.

#### 6.2. Hans-Jörg Diersch’s Story

#### 6.3. Ekkehard Holzbecher’s Story

#### 6.4. Cliff Voss’ Story

## 7. Part 4: Quandary and Resolution—Late 1990s and Early 2000s

#### 7.1. The Quandary

#### 7.2. The Resolution

#### 7.3. Peter Frolkovič’s Story

^{3}f (distributed density driven flow) to simulate variable density groundwater flow in 3D using parallel computations.

^{3}f a question arose if more than one stable steady solution of Elder Problem exists.

^{3}f it was possible to find three stable steady solutions. [The same three solutions have also been found by other authors using different models. The three stable steady solutions for the Elder Problem are shown in Figure 5.] Such results were for the first time reported by Schwarz in 1997 and published later in his PhD thesis Dichteabhängige Strömungen in homogenen und heterogenen porösen Medien that was submitted in 1999 at ETH Zürich. According to Schwarz in his words “the Elder’s experiment and the corresponding simulation was an ever-present notion during my doctoral thesis”. Schwarz left the academic community after finishing his PhD thesis and has worked at UBS Bank since 1999.

^{3}f, but he found time to continue on the results from the previous project. Together with his colleague Hennie De Schepper he was interested on the development of upwind methods for convection dominated transport coupled with variable density flow. The Elder Problem was a perfect example to show advantages of such methods. To publish related numerical simulations it was obvious to try also to explain the seemingly incompatible results published by various researchers for this problem.

#### 7.4. Klaus Johannsen’s Story

#### 7.5. Craig Simmons’ Story

_{1}. From Ra = 76 onwards, S

_{2}comes into existence via a fold bifurcation. Similarly, S

_{3}comes into existence at Ra = 172”. They also noted: “An interesting detail is that near the bifurcation points, Sh for S

_{2}and S

_{3}is lower than for S

_{1}. Sufficiently far away from the bifurcation points, solutions with more plumes have a higher Sh. The bifurcation points are comparable with Johannsen (2003) who reports that S

_{2}and S

_{3}come into existence at Ra = 67 and Ra = 151, respectively. The slight differences are most likely caused by the different numerical methods”.

## 8. Epilogue

## 9. Additional Remarks

_{c}= 4π

^{2}for the HRL Problem. He had even met Ernest Ralph Lapwood (known as Ralph) during his first time in Cambridge. Elder’s work was preoccupied with finding what happened to the convective system at the highest possible Rayleigh number he could get in the laboratory or on the computer—to compare with the natural systems. The earliest work by others which stimulated Elder most was that in Iceland by T. Einarsson (1942), G. Bodvarsson (various papers 1948–1961) and Italy (Larderello) by F. Penta 1954, R. Burgassi 1961, R. Burgassi (et al.) 1961.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix

**Figure A1.**John W. Elder with the actual Hele-Shaw cell apparatus he used for his experiments at Cambridge in 1967 (Photo taken 5 January 2017).

^{3}f (distributed density driven flows) and r

^{3}t (radionuclides, reaction, retardation and transport) that have been used to produce results published by several research groups in Europe. The method of consistent velocity approximation for variable density flow published in 1996 and 1998 (inspired by works of Voss and Souza (1987) and Leijnse (1992)) is available as “Frolkovic-Knabner algorithm” in the commercial software FEFLOW.

## References and Notes

- This reference list only contains the key works on the Elder Problem by the protagonists of this paper. Other examples and related works listed in the main text may be found on literature search engines.
- Elder, J.W. Steady free convection in a porous medium heated from below. J. Fluid Mech.
**1967**, 27, 29–48. [Google Scholar] [CrossRef] - Elder, J.W. Transient convection in a porous medium. J. Fluid Mech.
**1967**, 27, 609–623. [Google Scholar] [CrossRef]. (For other works by Elder please see also: J. Fluid Mech.**1965**, 23, 77–98, 99–111,**1966**, 24, 823–843;**1968**, 32, 69–96.). - Frolkovič, P.; De Schepper, H. Numerical modelling of convection dominated transport coupled with density driven flow in porous media. Adv. Water Resour.
**2001**, 24, 63–72. [Google Scholar] [CrossRef] - Diersch, H.-J.G.; Kolditz, O. Variable-density flow and transport in porous media: Approaches and challenges. Adv. Water Resour.
**2002**, 25, 899–944. [Google Scholar] [CrossRef] - Johannsen, K. The Elder problem—Bifurcations and steady state solutions. In Proceedings of the XIVth International Conference on Computational Methods in Water Resources (CMWR XIV), Delft, The Netherlands, 23–28 June 2002; Volume 47, pp. 485–492.
- Johannsen, K. On the Validity of the Boussinesq approximation for the Elder Problem. Comput. Geosci.
**2003**, 7, 169–182. [Google Scholar] [CrossRef] - Van Reeuwijk, M.; Mathias, S.A.; Simmons, C.T.; Ward, J.D. Insights from a pseudospectral approach to the Elder problem. Water Resour. Res.
**2009**, 45. [Google Scholar] [CrossRef] - Diersch, H.J. Primitive Variable Finite Element Solutions of Free Convective Flows in Porous Media. J. Appl. Math. Mech.
**1981**, 61, 325–337. [Google Scholar] - Voss, C.I.; Souza, W.R. Variable Density Flow and Solute Transport Simulation of Regional Aquifers Containing a Narrow Freshwater-Saltwater Transition Zone. Water Resour. Res.
**1987**, 23, 1851–1866. [Google Scholar] [CrossRef] - Organisation for Economic Co-operation and Development (OECD). The International Hydrocoin Project. Groundwater Hydrology Modelling Strategies for Performance Assessment of Nuclear Waste Disposal: Summary Report. 1992. Available online: http://www.iaea.org/inis/collection/NCLCollectionStore/_Public/24/002/24002761.pdf (accessed on 16 March 2017).
- Holzbecher, E. Numerische Modellierung von Dichteströmungen im porösen Medium. Ph.D. Thesis, Inst. Für Wasserbau und Wasserwirtschaft, Berlin, Germany, 1991. [Google Scholar]
- Holzbecher, E. Modeling Density-Driven Flow in Porous Media; This book includes his work on the HYDROCOIN Project & PhD 1991; Springer: Berlin, Germany, 1998. [Google Scholar]
- Holzbecher, E. FEMLAB Performance on 2D porous media variable density benchmarks. In Proceedings of the FEMLAB Conference, Frankfurt, Germany, 2005; pp. 203–208.

**Figure 1.**The equations of the 1967 model (

**top**) and flow plane diagram (

**bottom**). Diagram of the flow plane (bottom) showing the heated part of the base; and (for example) a 16 × 4 mesh. In a finite-difference numerical-representation the flow and temperature are determined/calculated only at the mesh joins (17 × 5 here). [Modified from Journal of Fluid Mechanics 1967, 27, 29–48 [1]; Journal of Fluid Mechanics 1967, 27, 609–623 [2]. Reproduced with permission from Elder, J.W., Steady free convection in a porous medium heated from below, Transient convection in a porous medium; published by Cambridge University Press, 1967.]

**Figure 2.**A = 400: stream-function and temperature distribution for the short heater problem at various dimensionless times (t = 0.005, 0.01, 0.02, 0.05, 0.075, 0.1). Numerical model, Journal of Fluid Mechanics 1967, 27, 609–623 [2]. The curves show 0.2 and 0.6 of the maximum value. [Figure 4, p. 615, Journal of Fluid Mechanics 1967, 27, 609–623 [2]. Reproduced with permission from Elder, J.W., Transient convection in a porous medium; published by Cambridge University Press, 1967.]

**Figure 3.**Visualization of the streamlines of a flow in a Hele-Shaw cell. [Figure 5, Plate 2 (after Plate 1, facing page 624) Journal of Fluid Mechanics 1967, 27, 609–623 [2]. Copyright Permission from Elder, J.W., Transient convection in a porous medium; published by Cambridge University Press, 1967.]

**Figure 4.**Numerical model, Voss and Souza (1987) [9], A = 400: compare Figure 2. SUTRA solid lines. Elder (1967) [2] dashed lines. Here, salinity model results from Voss and Souza (1987) [9] are shown upside down to correspond with the original thermal model. Salinity distributions are given for different times in nominal years: 1 year = 0.005 dimensionless time. [Modified from Figure 9a of Voss and Souza (1987) paper [9]. Reproduced with permission from Voss, C.I. and Souza, W.R., Variable Density Flow and Solute Transport Simulation of Regional Aquifers Containing a Narrow Freshwater-Saltwater Transition Zone; published by American Geophysical Union, 1987.]

**Figure 5.**The temperature distribution for the 3 steady-state solutions S1, S2 and S3 at Ra = 400 obtained by using different initial conditions. (The salinity results shown upside down as in the thermal version.) Bifurcations and possible steady state modes: 0 < Ra < 76, S1 only; 76 < Ra < 172, S1 and S2; 172 < Ra, S1, S2 and S3 (only the S1 mode was found in JFM 1967 [2]) [For details see: Frolkovič and De Schepper (2001) [3], Johannsen (2003) [6], van Reeuwijk et al. (2009) [7]]. [Modified from Figure 3 of van Reeuwijk et al. (2009) paper [7]. Reproduced with permission from Van Reeuwijk, M., et al., Insights from a pseudospectral approach to the Elder problem; published by American Geophysical Union, 2009.]

**Table 1.**Plume development stages in the numerical model of the Hele-Shaw cell (Journal of Fluid Mechanics 1967, 27, 609–623) [2].

Time t | ||
---|---|---|

0.005 | 2 independent small plumes | 2 eddy pairs |

0.01 | 2 independent plumes | 2 eddy pairs |

0.02 | 5 interactive plumes | 4 eddy pairs |

the growing hot layer in the middle of the heater has become unstable | ||

0.05 | 3 interactive plumes | 2 eddy pairs |

the independent flows at the ends of the heater | ||

are now dominated by the flow generated by the whole heater | ||

0.075 | 1 central plume + 2 peripheral plumes | 1 eddy pair |

0.10 | 1 central plume | 1 eddy pair |

near the steady state | ||

We have 3 overlapping stages: | ||

the independent flows near the heater ends; | ||

the interaction of these flows with that above the whole heater; | ||

the development of a single upwelling plume. |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Elder, J.W.; Simmons, C.T.; Diersch, H.-J.; Frolkovič, P.; Holzbecher, E.; Johannsen, K.
The Elder Problem. *Fluids* **2017**, *2*, 11.
https://doi.org/10.3390/fluids2010011

**AMA Style**

Elder JW, Simmons CT, Diersch H-J, Frolkovič P, Holzbecher E, Johannsen K.
The Elder Problem. *Fluids*. 2017; 2(1):11.
https://doi.org/10.3390/fluids2010011

**Chicago/Turabian Style**

Elder, John W., Craig T. Simmons, Hans-Jörg Diersch, Peter Frolkovič, Ekkehard Holzbecher, and Klaus Johannsen.
2017. "The Elder Problem" *Fluids* 2, no. 1: 11.
https://doi.org/10.3390/fluids2010011