Crystals stirred up: 1. Direct numerical simulations of crystal settling in nondilute magmatic suspensions

Abstract

[1] This is the first paper in a two-part series examining the fluid dynamics of crystal settling and flotation in the lunar magma ocean. A key challenge in constraining solidification processes is determining the ability of individual crystals to decouple from vigorous thermal convection and settle out or float. The goal of this paper is to develop a computational methodology capable of capturing the complex solid-fluid interactions that determine settling and flotation. In the second paper, we use this computational approach to explore the conditions under which plagioclase feldspar would be able to buoyantly float and form the earliest crust on the Moon. The direct numerical method described in this paper relies on a fictitious domain approach and captures solid-body motion in 2D and 3D with little overhead beyond single fluid calculations. The two main innovations of our numerical implementation of a fictitious domain approach are an analytical quadrature scheme, which increases accuracy and reduces computational expense, and the derivation of a multibody collision scheme. Advantages of this approach over previous simulations of crystal-bearing magmatic suspensions include the following: (1) we fully resolve the two-way interaction between fluid and solid phases, implying that crystals are not only passively advected in an ambient flow field but are also actively driving flow, and (2) we resolve the flow around each individual crystal without assuming specific settling speeds or drag coefficients. We present several benchmark problems and convergence tests to validate our approach.

The origin of eucrites, diogenites, and olivine diogenites: Magma ocean crystallization and shallow magma chamber processes on Vesta

“…If the vestan magma ocean retained at least a 10–20% crystal fraction at all stages, settling on a Myr time scale would have been essentially impossible due to crystal–crystal interactions (Suckale et al. ).…”

Crystals stirred up: 2. Numerical insights into the formation of the earliest crust on the Moon

“…An approach of this type is commonly referred to as “direct” because it resolves the multiphase flow dynamics of the suspension without relying on approximate drag formulas or settling speeds and without limiting the range and nature of hydrodynamic interactions between crystals and between crystals and magma. Direct numerical simulations are thus particularly advantageous for comparing crystal settling and flotation over a range of different crystal fractions because, contrary to many previous models (see Suckale et al [2012] for a more detailed discussion), they resolve the effect of increasing crystal fraction on both the motion of the crystals and on the flow field in the interstitial liquid. For further details regarding our numerical methodology and for extensive benchmarks of our approach, please refer to the companion paper [ Suckale et al , 2012].…”

“…Because the boundary layer in which settling occurs is defined by the flow speed becoming comparable to the settling speed, we set the influx speed to the free settling speed of the heaviest mineral phase in suspension. Once the inflow is specified, the outflow is determined self‐consistently from the flow field inside the domain in order to capture the effect of the presence of crystals on the down‐wind flow field [ Suckale et al , 2012]. The boundary conditions along the sidewalls also allow for mass‐preserving inflow and outflow.…”

“…The crystal fraction in suspension in the lunar magma ocean depends on the competition between two timescales: the solidification time, which we define as the time it takes for the magma ocean to solidify by one additional volume percent, and the residence time of the crystals, which we define as the time it takes 99% of the crystals originally in suspension to settle out assuming that no additional nucleation occurs during said time period [ Martin and Nokes , 1988]. To estimate these two timescales throughout the solidification history of a magma ocean, we couple a computation of the thermal and petrological evolution of the magma ocean at the planetary scale (following Meyer et al [2010] and Elkins‐Tanton et al [2011]) with direct numerical simulations of the fluid dynamics at the scale of individual crystals [ Suckale et al , 2012]. While the first suite of computations yields an estimate for the solidification time, the second suite of computations constrains the characteristic crystal settling time at varying crystal fractions.…”

M2Di: Concise and efficient MATLAB 2‐D Stokes solvers using the Finite Difference Method

“…The dynamics of systems consisting of multiple material phases of distinct rheologies and densities can be studied using direct numerical simulations. Magmatic suspensions and crystal mushes are often considered as laminar fluids [ Glazner , ] and processes such as phase separation, mixing, and segregation can, to some extent, be investigated using Stokes flow models [ Suckale et al ., ; Yamato et al ., ]. Numerical models, however, require high resolution in order to render the hydrodynamic interactions between crystals and the net effect of the crystalline load on the overall flow.…”