Towards a mass and volume conserving interface reinitialization scheme for a diffuse interface methodology (for shock-particle interaction). Ju Zhang, Thomas L.
Towards a mass and volume conserving interface reinitialization scheme for a diffuse interface methodology (for shock-particle interaction) Ju Zhang, Thomas L. Jackson, Prashanth Sridharan, and S. Balachandar
Citation: AIP Conference Proceedings 1793, 150005 (2017); View online: https://doi.org/10.1063/1.4971734 View Table of Contents: http://aip.scitation.org/toc/apc/1793/1 Published by the American Institute of Physics
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Towards a Mass and Volume Conserving Interface Reinitialization Scheme for A Diffuse Interface Methodology (for shock-particle interaction) Ju Zhang1,a) , Thomas L. Jackson2 , Prashanth Sridharan2 and S. Balachandar2 1
Department of Mechanical and Aerospace Engineering, Florida Institute of Technology, Melbourne, FL 32901 USA 2 Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL 32611 USA a)
Corresponding author: jzhang@fit.edu
Abstract. Recent work using a diffuse interface numerical method for multiphase problems is found to have relatively poor conservation properties of the particles. A novel constrained interface reinitialization scheme is proposed and demonstrated to be effective in conserving particle mass and particle volume when appropriate.
INTRODUCTION We are developing a computational framework to provide improved description of multiphase blast wave dynamics, where solid particles can deform and can even undergo phase transitions; see [1, 2] for details. The framework includes a reactive compressible flow solver with sophisticated material interface tracking capability and realistic equations of state (EOS) for multiphase flow modeling. The numerical framework is based on the diffuse interface method of Shukla et al. [3]. When phase transitions are ignored, we show that mass conservation of the particle during longtime integration is poor. To preserve particle mass and particle volume, a novel constrained interface reinitialization method is proposed. We show that this new method is found to be effective and can precisely preserve particle mass and volume, independent of grid resolution.
THEORETICAL BACKGROUND A robust numerical framework is needed that can handle different microstructures, strong shocks, multi-material interfaces, material deformation, general equations of state (EOS), and chemistry. A two-dimensional Euler solver, for both planar and axisymmetric geometries, that can handle all of these issues is described in detail in [1]. A numerical method was recently developed by Shukla et al. for handling such problems [3], albeit in the absence of chemistry. A key ingredient of [3] is the use of a material marker φ, advected according to ∂φ + u · ∇φ = 0. ∂t
(1)
In the physical system φ jumps from 0 to 1 across a material interface, but in the numerical scheme it changes smoothly over a small number of mesh points in a narrow region we designate S. Before moving to the next time step a correction is calculated on S by means of the equation ∂φ = n · ∇(h |∇φ| − φ(1 − φ)), ∂ˆτ
(2)
where τˆ is a pseudo-time, n is the interface normal, and h is a grid-dependent parameter. This equation is integrated to steady state. The nonlinear term leads to convective steepening of the interface domain, whereas the diffusion term
Shock Compression of Condensed Matter - 2015 AIP Conf. Proc. 1793, 150005-1–150005-4; doi: 10.1063/1.4971734 Published by AIP Publishing. 978-0-7354-1457-0/$30.00
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smears it out; these effects balance, as in Burgers equation. The density ρ is treated in a similar fashion using ∂ρ = H(φ)n · (∇(h n · ∇ρ) − (1 − 2φ)∇ρ), ∂ˆτ
(3)
where H(φ) numerically describes the set S. This scheme was fully demonstrated in [3].
MOTIVATION A numerical issue associated with the interface reinitialization scheme Eqns. (2-3) in multi-dimensional simulations of multiphase flows is the “shrinking” of the particle. This shrinking is especially pronounced when the particle is resolved with only moderate grid resolution; such coarse resolutions are necessary for large scale simulations when thousands of particles are included. The particle shrinking effect can be demonstrated by a coarse resolution (d/20; i.e., 20 points across the particle diameter) simulation of shock NM-Al interaction shown in Figure 1. The Al particle is numerically overly compressed by the interface reinitialization scheme, and shrinks as it is advected downstream until it disappears at t/τ = 21. The corresponding time history of the mass of the particle is shown in Figure 2. Clearly, the mass of the particle is poorly conserved even with a resolution of d/80, although we note that particle mass is better preserved with grid refinement. The reinitialization scheme Eqns. (2-3) preserves mass only in the asymptotic sense as resolution increases.
FIGURE 1. φ contours of shock-inert-particle interaction without constrained interface reinitialization at t/τ = 8.6, 18, 20 and 21 from left to right, top to bottom. Shock pressure ps = 1 GPa; grid resolution is d/20. τ = d/U s is the shock particle interaction time, with d the particle diameter and U s the shock speed.
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FIGURE 2. Time history of the mass of the Al particle at different resolutions without constrained interface reinitialization. Resolutions are d/20 (dash-dot), d/40 (dash) and d/80 (solid).
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We next propose a solution to the problem. To this end, we modify the interface reinitialization scheme by adding a new term in the RHS of Eqn. (2), ∂φ = n · ∇(h |∇φ| − φ(1 − φ)) + H(φ)λφ . ∂ˆτ
(4)
Here, λφ is a Lagrange multiplier calibrated to conserve particle volume. In effect, the H(φ)λφ term changes φ over the set S in pseudo-time τˆ , and thus increases the volume of the particle and counter-balances the shrinking effect of reinitialization. Similarly, a Lagrange multiplier λρ can also be added to the density correction Eqn. (3) and calibrated to conserve particle mass, namely ∂(ρ1 φ1 ) = H(φ1 )n · (∇(h n · ∇(ρ1 φ1 )) − (1 − 2φ1 )∇(ρ1 φ1 )) + H(φ1 )λρ , ∂ˆτ
(5)
where ρ1 is the particle density. Thus, ρ1 is adjusted within the set S to precisely preserve particle mass, independent of grid resolution. We still solve equation (3) for the mixture density ρ, and then the medium density ρ2 is computed according to φ2 ρ2 = ρ − φ1 ρ1 .
RESULTS Shock Al-NM Interface Interaction Problem The simulation previously shown in Figure 1 is repeated with the new scheme, and results are shown in Figure 3. Note that with the new constrained reinitialization scheme the particle no longer disappears at large times, and that mass and volume are precisely preserved at all grid resolutions (results not shown). Future work on the modelling and simulation of particle deformation will be presented elsewhere; [4].
FIGURE 3. φ contours of shock-Al-NM interaction with constrained reinitialization λφ , at t/τ = 24 to be compared with Figure 1d. Resolution is d/20.
Bubble Collapse Problem The single bubble collapse test problem in [5] is simulated. Since for this problem only mass should be preserved, we set λφ = 0. The simulation is carried out in 1-D spherical coordinates with mass conservation correction using λρ in Eqns. (5). Figure 4 compares our results with those of [5]. Note from the figure that the scheme with mass correction captures not only the evolution of the radius of the bubble, but also precisely preserves the mass of the air within the bubble. Without showing the results, we note that the solution (density, pressure, radial velocity, energy) with mass conservation correction is essentially identical to that by the original scheme without mass conservation correction, the only noticeable difference being in density in the air near the interface. Further investigation where an analytical solution exists will be conducted in the future; [6].
ACKNOWLEDGMENTS This work was supported in part by the U.S. Department of Energy, National Nuclear Security Administration, Advanced Simulation and Computing Program, as a Cooperative Agreement under the Predictive Science Academic Alliance Program, under Contract No. DE-NA0002378. TLJ was also supported in part by the Defense Threat Reduction Agency, Basic Research Award No. HDTRA1-14-1-0031 to University of Florida; SB was also supported in part by the Defense Threat Reduction Agency, Basic Research Award No. HDTRA1-14-1-0028 to University of Florida;
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FIGURE 4. (a) Comparison of the evolutions of the radius of the air bubble by [5] (black), the 1-D simulation in spherical coordinates without (red) and with (blue) mass correction. (b) Time history of the mass of bubble without (red) and with (blue) mass correction.
JZ was also supported in part by the Defense Threat Reduction Agency, Basic Research Award under sub-award No. UFOER00010237 to FIT with primary award to University of Florida under award No. HDTRA1-14-1-0031; Dr. Suhithi Peiris, program manager.
REFERENCES [1] [2] [3] [4] [5] [6]
Zhang, J., Jackson, T.L., Buckmaster, J., Freund, J.B. (2012). Numerical modeling of shock-to-detonation transition in energetic materials. Combustion and Flame, 159, pp. 1769-1778. Sridharan, P., Jackson, T.L., Zhang, J. and Balachandar, S. (2015). Shock interaction with one-dimensional array of particles in air. J. Applied Physics, 117, 075902. Shukla, R., Pantano, C., Freund, J.B. (2010). An interface capturing method for the simulation of multi-phase compressible flows. J. Comput. Physics, 229, pp. 7411-7439. Sridharan, P., Jackson, T.L., Zhang, J. and Balachandar, S. (2015). Shock interaction with deformable particles using a constrained interface reinitialization scheme. (in preparation). Tiwari A., Freund, J.B., Pantano, C. (2013). A diffuse interface model with immiscibility preservation. J. Comput. Physics, 252, pp. 290-309. Zhang, J., Jackson, T.L., Sridharan, P. and Balachandar, S. (2015). Towards a mass, volume, and shape conserving interface reinitialization scheme for a diffuse interface methodology (for shock-particle interaction). (in progress).
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