A Penalty-projection based Efficient and Accurate Stochastic Collocation Method for Magnetohydrodynamic Flows

We propose, analyze, and test a penalty projection-based efficient and accurate algorithm for the Uncertainty Quantification (UQ) of the time-dependent Magnetohydrodynamic (MHD) flow problems in convection-dominated regimes. The algorithm uses the Els\"asser variables formulation and discrete Hodge decomposition to decouple the stochastic MHD system into four sub-problems (at each time-step for each realization) which are much easier to solve than solving the coupled saddle point problems. Each of the sub-problems is designed in a sophisticated way so that at each time-step the system matrix remains the same for all the realizations but with different right-hand-side vectors which allows saving a huge amount of computer memory and computational time. Moreover, the scheme is equipped with ensemble eddy-viscosity and grad-div stabilization terms. The stability of the algorithm is proven rigorously. We prove that the proposed scheme converges to an equivalent non-projection-based coupled MHD scheme for large grad-div stabilization parameter values. We examine how Stochastic Collocation Methods (SCMs) can be combined with the proposed penalty projection UQ algorithm. Finally, a series of numerical experiments are given which verify the predicted convergence rates, show the algorithm's performance on benchmark channel flow over a rectangular step, and a regularized lid-driven cavity problem with high random Reynolds number and magnetic Reynolds number.Comment: 28 pages, 13 figure

Adaptive stochastic trajectory modelling in the chaotic advection regime

Motivated by the goal of improving and augmenting stochastic Lagrangian models of particle dispersion in turbulent flows, techniques from the theory of stochastic processes are applied to a model transport problem. The aim is to find an efficient and accurate method to calculate the total tracer transport between a source and a receptor when the flow between the two locations is weak, rendering direct stochastic Lagrangian simulation prohibitively expensive. Importance sampling methods that combine information from stochastic forward and back trajectory calculations are proposed. The unifying feature of the new methods is that they are developed using the observation that a perfect strategy should distribute trajectories in proportion to the product of the forward and adjoint solutions of the transport problem, a quantity here termed the ‘density of trajectories’ D(x,t). Two such methods are applied to a ‘hard’ model problem, in which the prescribed kinematic flow is in the large-Péclet-number chaotic advection regime, and the transport problem requires simulation of a complex distribution of well-separated trajectories. The first, Milstein’s measure transformation method, involves adding an artificial velocity to the trajectory equation and simultaneously correcting for the weighting given to each particle under the new flow. It is found that, although a ‘perfect’ artificial velocity v∗ exists, which is shown to distribute the trajectories according to D, small errors in numerical estimates of v∗ cumulatively lead to difficulties with the method. A second method is Grassberger’s ‘go-with-the-winners’ branching process, where trajectories found unlikely to contribute to the net transport (losers) are periodically removed, while those expected to contribute significantly (winners) are split. The main challenge of implementation, which is finding an algorithm to select the winners and losers, is solved by a choice that explicitly forces the distribution towards a numerical estimate of D generated from a previous back trajectory calculation. The result is a robust and easily implemented algorithm with typical variance up to three orders of magnitude lower than the direct approach

Magnetic diffusivity tensor and dynamo effects in rotating and shearing turbulence

The turbulent magnetic diffusivity tensor is determined in the presence of rotation or shear. The question is addressed whether dynamo action from the shear-current effect can explain large-scale magnetic field generation found in simulations with shear. For this purpose a set of evolution equations for the response to imposed test fields is solved with turbulent and mean motions calculated from the momentum and continuity equations. The corresponding results for the electromotive force are used to calculate turbulent transport coefficients. The diagonal components of the turbulent magnetic diffusivity tensor are found to be very close together, but their values increase slightly with increasing shear and decrease with increasing rotation rate. In the presence of shear, the sign of the two off-diagonal components of the turbulent magnetic diffusion tensor is the same and opposite to the sign of the shear. This implies that dynamo action from the shear--current effect is impossible, except perhaps for high magnetic Reynolds numbers. However, even though there is no alpha effect on the average, the components of the alpha tensor display Gaussian fluctuations around zero. These fluctuations are strong enough to drive an incoherent alpha--shear dynamo. The incoherent shear--current effect, on the other hand, is found to be subdominant.Comment: 12 pages, 13 figures, improved version, accepted by Ap

Multiscale Modelling Of Platelet Aggregation

During clotting under flow, platelets bind and activate on collagen and release autocrinic factors such ADP and thromboxane, while tissue factor (TF) on the damaged wall leads to localized thrombin generation. Toward patient-specific simulation of thrombosis, a multiscale approach was developed to account for: platelet signaling (neural network trained by pairwise agonist scanning, PAS-NN), platelet positions (lattice kinetic Monte Carlo, LKMC), wall-generated thrombin and platelet-released ADP/thromboxane convection-diffusion (PDE), and flow over a growing clot (lattice Boltzmann). LKMC included shear-driven platelet aggregate restructuring. The PDEs for thrombin, ADP, and thromboxane were solved by finite element method using cell activation-driven adaptive triangular meshing. At all times, intracellular calcium was known for each platelet by PAS-NN in response to its unique exposure to local collagen, ADP, thromboxane, and thrombin. The model accurately predicted clot morphology and growth with time on collagen/TF surface as compared to microfluidic blood perfusion experiments. The model also predicted the complete occlusion of the blood channel under pressure relief settings. Prior to occlusion, intrathrombus concentrations reached 50 nM thrombin, ~1 μM thromboxane, and ~10 μM ADP, while the wall shear rate on the rough clot peaked at ~1000-2000 sec-1. Additionally, clotting on TF/collagen was accurately simulated for modulators of platelet cyclooxygenase-1, P2Y1, and IP-receptor. The model was then extended to a rectangular channel with symmetric Gaussian obstacles representative of a coronary artery with severe stenosis. The upgraded stenosis model was able to predict platelet deposition dynamics at the post-stenotic segment corresponding to development of artery thrombosis prior to severe myocardial infarction. The presence of stenosis conditions alters the hemodynamics of normal hemostasis, showing a different thrombus growth mechanism. The model was able to recreate the platelet aggregation process under the complex recirculating flow features and make reasonable prediction on the clot morphology with flow separation. The model also detected recirculating transport dynamics for diffusible species in response to vortex features, posing interesting questions on the interplay between biological signaling and prevailing hemodynamics. In future work, the model will be extended to clot growth with a patient cardio-vasculature under pulsatile flow conditions