A frame invariant and maximum principle enforcing second‐order extension for cell‐centered ALE schemes based on local convex hull preservation

Two difficulties are clearly identified for high‐order extensions of ALE schemes for Euler equations: strict respect of the maximum principle and preservation of the Galilean invariance. We deal with these two issues in this paper. Our approach is closely related to the concepts of a posteriori limiting and convex hull spanning. We introduce the notion of local convex hull preservation schemes, which embodies these two concepts. We lean on this notion to propose a fully Galilean invariant ALE scheme. Moreover, we provide a new limiter (called Apitali for A Posteriori ITerAtive LImiter) for the remap step, enforcing the local convex hull preservation property. Copyright © 2014 John Wiley & Sons, Ltd.     

Comparisons of some difference forms for compressible flow in cylindrical geometry on arbitrary Lagrangian and Eulerian framework

The study of cylindrically symmetric compressible fluid is interesting from both theoretical and numerical points of view. In this paper, the typical spherical symmetry properties of the numerical schemes are discussed, and an area weighted scheme is extended from a Lagrangian method to an arbitrary Lagrangian and Eulerian (ALE) method. Numerical results are presented to compare three discrete configurations, i.e., the control volume scheme, the area weighted scheme, and the plane scheme with the addition of a geometrical source. The fact that the singularity arises from the geometrical source term in the plane scheme is illustrated. A suggestion for choosing the discrete formulation is given when the strong shock wave problems are simulated.     

Performance analysis of a 3D unstructured mesh hydrodynamics code on multi‐core and many‐core architectures

Several next generation high performance computing platforms are or will be based on the so‐called many‐core architectures, which represent a significant departure from commodity multi‐core architectures. A key issue in transitioning large‐scale simulation codes from multi‐core to many‐core systems is closing the serial performance gap, that is, overcoming the large difference in single‐core performance between multi‐core and many‐core systems. In this paper, we discuss how this problem was addressed for a 3D unstructured mesh hydrodynamics code, describe how Amdahl's law can be used to estimate performance targets and guide optimization efforts, and present timing studies performed on multi‐core and many‐core platforms. Published 2014. This article is a U.S. Government work and is in the public domain in the USA.     

Interface reconstruction with least‐square fit and split Eulerian–Lagrangian advection

Two new volume‐of‐fluid (VOF) reconstruction algorithms, which are based on a least‐square fit technique, are presented. Their performance is tested for several standard shapes and is compared to a few other VOF/PLIC reconstruction techniques, showing in general a better convergence rate. The geometric nature of Lagrangian and Eulerian split advection algorithms is investigated in detail and a new mixed split Eulerian implicit–Lagrangian explicit (EI–LE) scheme is presented. This method conserves the mass to machine error, performs better than split Eulerian and Lagrangian algorithms, and it is only slightly worse than unsplit schemes. However, the combination of the interface reconstruction with the least‐square fit and its advection with the EI–LE scheme appears superior to other existing approaches. Copyright © 2003 John Wiley & Sons, Ltd.     

Mesh regularization for an ALE code based on the limitation of the Lagrangian mesh velocity

The Lagrangian approach is usually used for the simulation of flow with strong shock waves. Moreover, this approach is particularly well suited to treatment of material interfaces in the case of multimaterial flows.Unfortunately, this formulation leads to very large deformations in the mesh. The arbitrary Lagrangian‐Eulerian method overcomes this drawback by using a mesh regularization that is based on an analysis of cell geometry. The regularization step may be considered as a method used to correct the nonconvex and potentially tangled cells that constitute the mesh. In this paper, we present a new approach to mesh regularization. Instead of using a purely geometric criterion, we propose that the mesh evolution is computed on the basis of the flow vorticity. This approach is called the large Eddy limitation method, and it is aimed here to be used in finite volume direct arbitrary Lagrangian‐Eulerian methods. The large Eddy limitation method is general, which means that it is not restricted to applications in the finite volume framework dedicated to fluid flow simulation; for instance, it could also be naturally applied to the finite element framework.     

Quantitative comparison of Taylor flow simulations based on sharp‐interface and diffuse‐interface models

A pressure‐driven flow of elongated bullet‐shaped bubbles in a narrow channel is known as Taylor flow or bubble‐train flow. This process is of relevance in various applications of chemical engineering. In this paper, we describe a typical simplified experimental setting, with surface tension, density and viscosity as prescribed input parameters. We compare a sharp‐interface model based on a moving grid aligned with the bubble boundary (ALE coordinates) and a diffuse‐interface model where the bubble shape is implicitly given by a phase‐field function. Four independent implementations based on the two modeling approaches are introduced and described briefly. Besides the simulation of the bubble shapes, we compare some resulting quantities such as pressure difference and film widths within the implementations and to existing analytical and experimental results. The simulations were conducted in 2D and 3D (rotationally symmetric). Good accordance of the results indicate the applicability and the usability of all approaches. Differences between the models and their implementations are visible but in no contradiction to theoretical results. Copyright © 2013 John Wiley & Sons, Ltd.     

A meshless finite point method for three‐dimensional analysis of compressible flow problems involving moving boundaries and adaptivity

A finite point method for solving compressible flow problems involving moving boundaries and adaptivity is presented. The numerical methodology is based on an upwind‐biased discretization of the Euler equations, written in arbitrary Lagrangian–Eulerian form and integrated in time by means of a dual‐time steeping technique. In order to exploit the meshless potential of the method, a domain deformation approach based on the spring network analogy is implemented, and h‐adaptivity is also employed in the computations. Typical movable boundary problems in transonic flow regime are solved to assess the performance of the proposed technique. In addition, an application to a fluid–structure interaction problem involving static aeroelasticity illustrates the capability of the method to deal with practical engineering analyses. The computational cost and multi‐core performance of the proposed technique is also discussed through the examples provided. Copyright © 2013 John Wiley & Sons, Ltd.     

Residual‐based turbulence models for moving boundary flows: hierarchical application of variational multiscale method and three‐level scale separation

This paper presents residual‐based turbulence models for problems with moving boundaries and interfaces. The method is developed via a hierarchical application of variational multiscale ideas and the models are cast in an arbitrary Lagrangian–Eulerian (ALE) frame to accommodate the deformation of domain boundaries. An overlapping additive decomposition of velocity and pressure fields into coarse and fine scale components leads to coarse and fine scale mixed‐field problems. The problem governing fine scales is subjected to a further decomposition of the fine scale velocity into overlapping components termed as fine scales level I and level II. In turn, in the bottom‐up integration of scales, the model for level II fine scales serves to stabilize the problem governing level I fine scales, and model for level I fields yields the turbulence models. From the computational perspective, the coarse scales are represented in terms of the standard Lagrange shape functions, whereas level I and level II scales are represented via quadratic and fourth order polynomial bubbles, respectively. Because of the bubble functions approach employed in the consistently derived fine scale models, the resulting method is free of any embedded or tunable parameters. The proposed turbulence models share a common feature with the LES models in that the largest scales in the flow are numerically resolved, whereas the subgrid scales are modeled. The method is applied to flow around a plunging airfoil at Re = 40,000, and results are compared with experimental and numerical data published in the literature. Also presented are the results for the plunging airfoil at Re = 60,000 to show the robustness and range of applicability of the method. Copyright © 2013 John Wiley & Sons, Ltd.     

Three‐dimensional computation for flow‐induced vibrations in in‐line and cross‐flow directions of a circular cylinder

We present numerical results for in‐line and cross‐flow vibrations of a circular cylinder, which is immersed in a uniform flow and is elastically supported by damper‐spring systems to compute vibrations of a rigid cylinder. In the case of a circular cylinder with a low Scruton number, it is well‐known that two types of self‐excited vibrations appear in the in‐line direction in the range of low reduced velocities. On the other hand, a cross‐flow vibration of the circular cylinder can be excited in the range of high reduced velocities. Therefore, we compute the flow‐induced vibrations of the circular cylinder in the wide range of the reduced velocities at low and high Scruton numbers and discuss about excitation mechanisms in the in‐line and cross‐flow directions. Copyright © 2011 John Wiley & Sons, Ltd.     

Some considerations for high‐order ‘incremental remap’‐based transport schemes: edges,reconstructions, and area integration

The problem of two‐dimensional tracer advection on the sphere is extremely important in modeling of geophysical fluids and has been tackled using a variety of approaches. A class of popular approaches for tracer advection include ‘incremental remap’ or cell‐integrated semi‐Lagrangian‐type schemes. These schemes achieve high‐order accuracy without the need for multistage integration in time, are capable of large time steps, and tend to be more efficient than other high‐order transport schemes when applied to a large number of tracers over a single velocity field. In this paper, the simplified flux‐form implementation of the Conservative Semi‐LAgrangian Multi‐tracer scheme (CSLAM) is reformulated using quadratic curves to approximate the upstream flux volumes and Gaussian quadrature for integrating the edge flux. The high‐order treatment of edge fluxes is motivated because of poor accuracy of the CSLAM scheme in the presence of strong nonlinear shear, such as one might observe in the midlatitudes near an atmospheric jet. Without the quadratic treatment of upstream edges, we observe at most second‐order accuracy under convergence of grid resolution, which is returned to third‐order accuracy under the improved treatment. A shallow‐water barotropic instability also reveals clear evidence of grid imprinting without the quadratic correction. Consequently, these tests reveal a problem that might arise in tracer transport near nonlinearly sheared regions of the real atmosphere, particularly near cubed‐sphere panel edges. Although CSLAM is used as the foundation for this analysis, the conclusions of this paper are applicable to the general class of incremental remap schemes. Copyright © 2012 John Wiley & Sons, Ltd.     

Application of pressure‐ and density‐based methods for different flow speeds

In this paper, pressure‐based and density‐based methods are studied at different flow speeds. The methods are intended for steady flows, and the goal is to find as general an approach as possible to cover different Mach number regimes. The solution methods are based on a finite‐volume approach. Various forms of inviscid fluxes are applied and connected with either a pressure‐based or density‐based implicit solution. For this purpose, a new pressure‐correction method is developed that can be applied for incompressible and for compressible flows. Another option is a standard density‐based approximate factorization method. In both cases, a convergence is accelerated with a Full Approximation Scheme (FAS) multigrid approach. Sample problems in the range of Ma = 0…6 are simulated using different approaches, and their efficiency and accuracy are compared. On the basis of the quality of the solutions, recommendations are made. © 2015 The Authors. International Journal for Numerical Methods in Fluids published by John Wiley & Sons Ltd.     

High‐order ADER‐WENO ALE schemes on unstructured triangular meshes—application of several node solvers to hydrodynamics and magnetohydrodynamics

In this paper, we present a class of high‐order accurate cell‐centered arbitrary Lagrangian–Eulerian (ALE) one‐step ADER weighted essentially non‐oscillatory (WENO) finite volume schemes for the solution of nonlinear hyperbolic conservation laws on two‐dimensional unstructured triangular meshes. High order of accuracy in space is achieved by a WENO reconstruction algorithm, while a local space–time Galerkin predictor allows the schemes to be high order accurate also in time by using an element‐local weak formulation of the governing PDE on moving meshes. The mesh motion can be computed by choosing among three different node solvers, which are for the first time compared with each other in this article: the node velocity may be obtained either (i) as an arithmetic average among the states surrounding the node, as suggested by Cheng and Shu, or (ii) as a solution of multiple one‐dimensional half‐Riemann problems around a vertex, as suggested by Maire, or (iii) by solving approximately a multidimensional Riemann problem around each vertex of the mesh using the genuinely multidimensional Harten–Lax–van Leer Riemann solver recently proposed by Balsara et al. Once the vertex velocity and thus the new node location have been determined by the node solver, the local mesh motion is then constructed by straight edges connecting the vertex positions at the old time level tⁿ with the new ones at the next time level t^{n + 1}. If necessary, a rezoning step can be introduced here to overcome mesh tangling or highly deformed elements. The final ALE finite volume scheme is based directly on a space–time conservation formulation of the governing PDE system, which therefore makes an additional remapping stage unnecessary, as the ALE fluxes already properly take into account the rezoned geometry. In this sense, our scheme falls into the category of direct ALE methods. Furthermore, the geometric conservation law is satisfied by the scheme by construction. We apply the high‐order algorithm presented in this paper to the Euler equations of compressible gas dynamics as well as to the ideal classical and relativistic magnetohydrodynamic equations. We show numerical convergence results up to fifth order of accuracy in space and time together with some classical numerical test problems for each hyperbolic system under consideration. Copyright © 2014 John Wiley & Sons, Ltd.     

Improved production implicit continuous‐fluid Eulerian method for compressible flow problems in Uintah

The implicit continuous‐fluid Eulerian (ICE) method is a successful and widely used semi‐implicit finite‐volume method that applies to flows that range from supersonic to subsonic regimes. The classical ICE method has been expanded to problems in multiphase flow, which spans a wide area of science and engineering. The ICE method is utilized by the Center for the Simulation of Accidental Fires and Explosions code Uintah written at the University of Utah to simulate explosions, fires and other fluid and fluid‐structure interaction phenomena. The ICE method used in Uintah (referred to here as Production ICE) is described in many papers by Kashiwa at Los Alamos National Laboratory and Harman at University of Utah. However, Production ICE does not perform as well as many current methods for compressible flow problems governed by the Euler equations. We show, via examples, that changing the nonconservation form of the solver in Production ICE to a conservation form improves the numerical solutions. In addition, the use of slope limiters makes it possible to suppress the nonphysical oscillations generated by the ICE method in conservation form. This new form of ICE is referred to as IMPICE, the IMproved Production ICE method. The accuracy of IMPICE for one‐dimensional Euler equations is investigated by using a number of test cases. Copyright © 2011 John Wiley & Sons, Ltd.     

Efficient computation of operator‐type response sensitivities for uncertainty quantification and predictive modeling: illustrative application to a spent nuclear fuel dissolver model

This work honors the 75th birthday of Professor Ionel Michael Navon by presenting original results highlighting the computational efficiency of the adjoint sensitivity analysis methodology for function‐valued operator responses by means of an illustrative paradigm dissolver model. The dissolver model analyzed in this work has been selected because of its applicability to material separations and its potential role in diversion activities associated with proliferation and international safeguards. This dissolver model comprises eight active compartments in which the 16 time‐dependent nonlinear differential equations modeling the physical and chemical processes comprise 619 scalar and time‐dependent model parameters, related to the model's equation of state and inflow conditions. The most important response for the dissolver model is the time‐dependent nitric acid in the compartment furthest away from the inlet, where measurements are available at 307 time instances over the transient's duration of 10.5 h. The sensitivities to all model parameters of the acid concentrations at each of these instances in time are computed efficiently by applying the adjoint sensitivity analysis methodology for operator‐valued responses. The uncertainties in the model parameters are propagated using the above‐mentioned sensitivities to compute the uncertainties in the computed responses. A predictive modeling formalism is subsequently used to combine the computational results with the experimental information measured in the compartment furthest from the inlet and then predict optimal values and uncertainties throughout the dissolver. This predictive modeling methodology uses the maximum entropy principle to construct an optimal approximation of the unknown a priori distribution for the a priori known mean values and uncertainties characterizing the model parameters and the computed and experimentally measured model responses. This approximate a priori distribution is subsequently combined using Bayes' theorem with the “likelihood” provided by the multi‐physics computational models. Finally, the posterior distribution is evaluated using the saddle‐point method to obtain analytical expressions for the optimally predicted values for the parameters and responses of both multi‐physics models, along with corresponding reduced uncertainties. This work shows that even though the experimental data pertains solely to the compartment furthest from the inlet (where the data were measured), the predictive modeling procedure used herein actually improves the predictions and reduces the predicted uncertainties for the entire dissolver, including the compartment furthest from the measurements, because this predictive modeling methodology combines and transmits information simultaneously over the entire phase‐space, comprising all time steps and spatial locations. Copyright © 2016 John Wiley & Sons, Ltd.