Hinton, Edward

Hogg, Andrew J

Huppert, Herbert E

English

The consequences of lava flowing downhill around and over topography and interacting with human-made constructions is modelled by considering the flow of a Newtonian fluid. Small obstacles can be overtopped by the flow, but topography of sufficient height will deflect the flow around it and form dry regions in the wake. Both numerical solutions and the results of laboratory experiments are discussed. We provide numerous pictures of flow patterns and evaluate the force they exert. The experimental results, focusing on flows past circular cylinders, are in good agreement with our numerical evaluations. Flows over depressions, which act to concentrate the flow, are also discussed

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                          Huppert, H. E., Hinton, E., & Hogg, A. J. (2020). Defending againstlava flows: theory, experiments and field confirmation. Paperpresented at 3rd Conference of the Arabian Journal of Geoosciences,Sousse, Tunisia.Peer reviewed versionLink to publication record in Explore Bristol ResearchPDF-documentUniversity of Bristol - Explore Bristol ResearchGeneral rightsThis document is made available in accordance with publisher policies. Please cite only thepublished version using the reference above. Full terms of use are available:http://www.bristol.ac.uk/red/research-policy/pure/user-guides/ebr-terms/  Defending against lava flows: theory, experiments and 1 field confirmation  2 Herbert E. Huppert1, Edward M. Hinton2, Andrew J. Hogg3 3 1 Institute of Theoretical Geophysics, King’s College, Cambridge, CB2 1ST, UK 4 2 School of Mathematics and Statistics, University of Melbourne, Victoria, Australia 5 3 Schoool of Mathematics, University of Bristol, Bristol BS8 1TW, UK 6 heh1@cam.ac.uk  7 Abstract. The consequences of lava flowing downhill around and over topogra-8 phy and interacting with human-made constructions is modelled by considering 9 the flow of a Newtonian fluid. Small obstacles can be overtopped by the flow, but 10 topography of sufficient height will deflect the flow around it and form dry regions 11 in the wake. Both numerical solutions and the results of laboratory experiments 12 are discussed. We provide numerous pictures of flow patterns and evaluate the 13 force they exert. The experimental results, focusing on flows past circular cylin-14 ders, are in good agreement with our numerical evaluations. Flows over depres-15 sions, which act to concentrate the flow, are also discussed.  16 Keywords: lava flows, topographic forcings, gravity currents, flow-structure 17  interactions 18 1 Introduction  19 Volcanic eruptions and the subsequent flow of lava lead to deaths of both humans and 20 animals, as well as resulting in destruction of many properties and dire financial prob-21 lems. Approximately 2000 people have been killed by lava flows in the last 20 years. On 22 average, tens of millions of cubic meters of lava are erupted each year onto the Earth’s 23 surface, either into the atmosphere or under the oceans, travelling along the ground at 24 speeds of up to 100km/hr. Can we predict how lava flows are diverted by natural topog-25 raphy and by buildings? Where and in what orientation should constructions be placed 26 to maximise the ‘dry spots’, free from lava? What is the anticipated force on a defending 27 wall and to what height and length need it be built to play a useful role? These are some 28 of the questions addressed in this review-like paper, which summarises material spelt 29 out in greater detail in Hinton et al. [1,2,3].   30  31 2  2 The model  32 Consider a time independent two-dimensional flow of flux 𝑄 per unit width of thin, vis-33 cous, Newtonian liquid, of kinematic viscosity ν, to model a lava flow down an inclined 34 plane at angle β to the horizontal. The thickness of the flow is then given by [4] 35  𝐻∞ = (3ν 𝑄 𝑔⁄ sin 𝛽)13. (2.1) 36 To this (vertical) lengthscale can be added horizontal and vertical lengthscales 𝐿 and 𝐷 37 dependent on the topography or building on the slope encountered by the lava flow. 38 Introducing downslope and cross-slope dimensionless variables 𝑥 and 𝑦, and a dimen-39 sionless axis perpendicular to the slope 𝑧 by  40  (𝑥, 𝑦) = (𝑋, 𝑌) 𝐿⁄ ,       𝑧 = 𝑍 𝐻∞⁄ , (2.2) 41 we find that the dimensionless depth ℎ(𝑥, 𝑦) of the lava satistfies [1] 42  (𝜕ℎ3)/𝜕𝑥 = 𝛻[ℎ3𝛻~(ℱ(ℎ + ℳ𝑚)], (2.3) 43 where 𝑚(𝑥, 𝑦) is a dimensionless expression for the underlying topography, along with 44 the governing non-dimensional parameters  45  ℱ = 𝐻∞/𝐿 tan 𝛽  =  (3𝜈𝑄/𝑔 sin 𝛽)1/3/(𝐿 tan 𝛽).      (2.4) 46 and           ℳ = 𝐷 𝐿 tan 𝛽⁄ .                    (2.5) 47 3 Flow patterns  48 3.1 One-dimensional mounds  49 Consider, to start and to illustrate some of the fundamental aspects of the flows, a one-50 dimensional situation (independent of the cross-flows co-ordinate, 𝑦), with the mound 51 given by 𝑚(𝑥). (2.3) can then be integrated once, using the boundary condition ℎ → 1 52 as 𝑥 → −∞, to obtain  53  ℎ3(1 − ℳ𝑑𝑚𝑑𝑥) =  1 +  ℱℎ3  𝑑ℎ𝑑𝑥. (3.1) 54 Because it is one-dimensional, all the flow must go over the mound. The most important 55 consequence, determined from numerical solution of (3.1) for a variety of 𝑚(𝑥), ℱ and  56 ℳ is that for small ℳ, ℳ< ℳ𝑐, where ℳ𝑐 is a critical value, dependent on the details 57 of 𝑚(𝑥) and the value of ℱ, the flow progresses uniformly over the mound, with a down-58 ward sloping upper surface everywhere. However, for  ℳ > ℳ𝑐 a pond develops up-59 stream of the obstacle, the surface of which is horizontal. The value ℳ𝑐 is the smallest 60 value of ℳ so that 1 −  ℳ𝑚′(𝑥) is somewhere negative. As an example, for 𝑚 =61 exp(−𝑥2), ℳ𝑐 = (𝑒/2)1/2 ≈1.16.  62  63 3  3.2 Two-dimensional mounds  64 For mounds described by 𝑚(𝑥, 𝑦), the flow can: go over the mound; around the mound; 65 not reach the top of the mound (if higher than some critical value); not completely cover 66 the ground, that is, develop ‘dry patches’ - relatively safe places to be during a lava flow. 67 Figure 1-4 display numerically determined flow fields for a variety of ℱ, ℳ and 𝑚(𝑥, 𝑦). 68 An interesting series of examples is provided by an elliptical mound given by  69  𝑚(𝑥, 𝑦) = exp{−[𝑥2 + (𝑦 𝑏⁄ )2]}, (3.2) 70 which tends to a long barrier as 𝑏 → ∞. Figure 5 shows the expected flow thickness for 71 two values of 𝑏. What is the force exerted on such a topographic feature, envisaged as a 72 defending wall to an oncoming lava flow? In the limit 𝑏 → ∞, for a barrier just suffi-73 ciently high to stop the oncoming flow (which climbs up the barrier) the maximum force 74 ~𝜌𝑔(𝐿 tan 𝛽)2, which for the illustrative values 𝐿 = 50𝑚 and tan 𝛽 = 0.25, leads to a 75 maximum force of the order 107𝑁𝑚−1. 76  77  78 Figure 1: Contour plots of the thickness of flow over topography specified by 𝑚 =79 𝑒𝑥𝑝(−𝑟2) for ℱ = 0.1. a) ℳ = 0.5 and b) ℳ = 1.5. Red crosses mark the points of 80 maximum thickness. Note the dry zone in b) 81  82  83 Figure 2: Contour plots of the thickness of flow past a circular cylinder under the 84 condition of no normal flow at the boundry, for ℱ = 20, 1 and 0.025.  85  86 4  87 Figure 3: Contour plots of the thickness of flow around a square-on square obstacle for 88 ℱ = 10 and 0.25. Note that the flow remains attached to the square in both cases and 89 there is no dry region for these values of ℱ.  90  91  92 Figure 4: Contour plot of the thickness of flow around a square rotated 45° to the 93 oncoming flow for ℱ = 0.25.  94  95  96 Figure 5: Contour plots of the thickness of flow over an elliptical mound with ℱ = 0.05 97 and ℳ = 1.4 for a) 𝑏 = 0.2 and b) 𝑏 = 4.  98  99 Figure 6: Calculated and experimental results for the maximum and minimum flow 100 thickness as a function of ℱfor flow past a cylinder. A zero flow thickness indicates the 101 existence of a dry zone downstream of the cylinder. 102 5     103 3.3 Experimental verification  104 We carried out a series of experiments on a slope of width 30cm, length 120cm inclined 105 at angles between 3.5 and 23 degrees on which we affixed (tall) cylinders of radius be-106 tween 2.4 and 4.8cm [2]. The upstream flow thickness varied between 0.5 and 1.5cm, 107 leading to values of ℱ between 0.5 and 7.1 (the cylinders were all tall and so ℳ is not a 108 relevant parameter.) Figure 6 displays a compendium of the results for the maximum 109 and minimum flow thickness, with good agreement between theoretical predictions, ob-110 tained by numerically solving (2.3), and the experimental results.  111  112 3.4 Depressions 113 Real topography includes not only mounds and hills, but also depressions; and both 114 together. An initial analysis of some effects due solely to depressions is contained in [3] 115 and the flow thickness for two cases make up figure 7. For smallish depressions the flow 116 thickness is but slightly perturbed. For deeper depressions large ponds of fluid accumu-117 late and have a significant effect on the flow downstream.  118  119  120 Figure 7: Flow thickness over a circular Gaussian depression for ℱ = 0.1 and a) ℳ =121 −0.8 and b) ℳ = −1.6. 122  123 Depressions are significantly different from hills because a sufficiently high hill, not 124 touched by the flow at its higher points, does not come into contact with the flow; and 125 hence the higher parts of the hill don’t influence the flow. No matter how deep the de-126 pression it will influence the flow and there will be some flow (though maybe small) 127 right to the bottom. In principle this resembles the influence of Moffatt eddies, slow 128 motions in a sharp corner, well away from the forcing flow [5]. 129  130 Of considerable interest and novelty are flows over topography containing both hills 131 and depressions. We plan to publish on this topic in the future.  132  133 3.5 Field observations 134 Here is not the best place to compare our model results with real data taken in the 135 field. However, numerous opportunities present themselves as outlined on Hawaii [6], 136 Santorini [7] and elsewhere. This, too, will be reported elsewhere (Hinton et al. 2022).  137 6   138 4 Conclusions   139 Lava flows are frequent on the Earth and can cause much damage. Defending people 140 and property in such situations is a very worthwhile endeavor. Our work has begun to 141 lay down some of the foundations and principles that might be employed.  Many further 142 questions remain, including what shape of cross-sectional area 𝐴 (of a building) maxim-143 ises the area of the dry zone. How sensitive is the result to the input parameters? How 144 will the concepts we have developed be used in any way usefully during forthcoming 145 volcanic eruptions?  146 References 147 1. Hinton, E.M., Hogg, A.J. and Huppert, H.E. Interaction of viscous free-surface flows with 148 topography. Journal of Fluid Mechanics, 876, 912-938 (2019). 149 2. Hinton, E.M., Hogg, A.J. and Huppert, H.E. Viscous free-surface flows past cylinders. Phys-150 ical Review Fluids, 5, 1-30 (2020). 151 3. Hinton, E.M., Hogg, A.J. and Huppert, H.E. Free-surface viscous flows over a depression. 152 22nd Australasian Fluid Mechanics Conference AFMC 2020 (2020). 153 4. Lister, J.R. Viscous flows down an inclined plane from point and line sources. Journal of 154 Fluid Mechanics, 242, 631-653. (1992) 155 5. Moffatt, H.K. Viscous and resistive eddies near a sharp corner. Journal of Fluid Mechanics, 156 18, 1-18. (1964) 157 6. Neal, A.C. et al. The 2018 rift eruption and summit collapse of Kilauea Volcano. Science 158 363, 367-374 (2019).  159 7. Druitt, T.H., Edwards, L., Mellors, R.M., Pyle, D.M., Sparks, R.S.J., Lanphere M., Davies 160 M. and Barreiro B. Santorini volcano. Geological Society Memoir 19 (1999). 161  162 

Defending against lava flows:theory, experiments and field confirmation

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Defending against lava flows:theory, experiments and field confirmation

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