102 research outputs found

    Dynamic p-enrichment schemes for multicomponent reactive flows

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    We present a family of p-enrichment schemes. These schemes may be separated into two basic classes: the first, called \emph{fixed tolerance schemes}, rely on setting global scalar tolerances on the local regularity of the solution, and the second, called \emph{dioristic schemes}, rely on time-evolving bounds on the local variation in the solution. Each class of pp-enrichment scheme is further divided into two basic types. The first type (the Type I schemes) enrich along lines of maximal variation, striving to enhance stable solutions in "areas of highest interest." The second type (the Type II schemes) enrich along lines of maximal regularity in order to maximize the stability of the enrichment process. Each of these schemes are tested over a pair of model problems arising in coastal hydrology. The first is a contaminant transport model, which addresses a declinature problem for a contaminant plume with respect to a bay inlet setting. The second is a multicomponent chemically reactive flow model of estuary eutrophication arising in the Gulf of Mexico.Comment: 29 pages, 7 figures, 3 table

    Accelerated Temporal Schemes for High-Order Unstructured Methods

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    The ability to discretize and solve time-dependent Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs) remains of great importance to a variety of physical and engineering applications. Recent progress in supercomputing or high-performance computing has opened new opportunities for numerical simulation of the partial differential equations (PDEs) that appear in many transient physical phenomena, including the equations governing fluid flow. In addition, accurate and stable space-time discretization of the partial differential equations governing the dynamic behavior of complex physical phenomena, such as fluid flow, is still an outstanding challenge. Even though significant attention has been paid to high and low-order spatial schemes over the last several years, temporal schemes still rely on relatively inefficient approaches. Furthermore, academia and industry mostly rely on implicit time marching methods. These implicit schemes require significant memory once combined with high-order spatial discretizations. However, since the advent of high-performance general-purpose computing on GPUs (GPGPU), renewed interest has been focused on explicit methods. These explicit schemes are particularly appealing due to their low memory consumption and simplicity of implementation. This study proposes low and high-order optimal Runge-Kutta schemes for FR/DG high-order spatial discretizations with multi-dimensional element types. These optimal stability polynomials improve the stability of the numerical solution and speed up the simulation for high-order element types once compared to classical Runge-Kutta methods. We then develop third-order accurate Paired Explicit Runge-Kutta (P-ERK) schemes for locally stiff systems of equations. These third-order P-ERK schemes allow Runge-Kutta schemes with different numbers of active stages to be assigned based on local stiffness criteria, while seamlessly pairing at their interface. We then generate families of schemes optimized for the high-order flux reconstruction spatial discretization. Finally, We propose optimal explicit schemes for Ansys Fluent finite volume density-based solver, and we investigate the effect of updating and freezing reconstruction gradient in intermediate Runge-Kutta schemes. Moreover, we explore the impact of optimal schemes combined with the updated gradients in scale-resolving simulations with Fluent's finite volume solver. We then show that even though freezing the reconstruction gradients in intermediate Runge-Kutta stages can reduce computational cost per time step, it significantly increases the error and hampers stability by limiting the time-step size

    Direct Numerical Simulation of Interfacial Flows: Implicit Sharp-Interface Method (I-SIM)

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    In recent work (Nourgaliev, Liou, Theofanous, JCP in press) we demonstrated that numerical simulations of interfacial flows in the presence of strong shear must be cast in dynamically sharp terms (sharp interface treatment or SIM), and that moreover they must meet stringent resolution requirements (i.e., resolving the critical layer). The present work is an outgrowth of that work aiming to overcome consequent limitations on the temporal treatment, which become still more severe in the presence of phase change. The key is to avoid operator splitting between interface motion, fluid convection, viscous/heat diffusion and reactions; instead treating all these non-linear operators fully-coupled within a Newton iteration scheme. To this end, the SIM’s cut-cell meshing is combined with the high-orderaccurate implicit Runge-Kutta and the “recovery” Discontinuous Galerkin methods along with a Jacobian-free, Krylov subspace iteration algorithm and its physics-based preconditioning. In particular, the interfacial geometry (i.e., marker’s positions and volumes of cut cells) is a part of the Newton-Krylov solution vector, so that the interface dynamics and fluid motions are fully-(non-linearly)-coupled. We show that our method is: (a) robust (L-stable) and efficient, allowing to step over stability time steps at will while maintaining high-(up to the 5th)-order temporal accuracy; (b) fully conservative, even near multimaterial contacts, without any adverse consequences (pressure/velocity oscillations); and (c) highorder-accurate in spatial discretization (demonstrated here up to the 12th-order for smoothin-the-bulk-fluid flows), capturing interfacial jumps sharply, within one cell. Performance is illustrated with a variety of test problems, including low-Mach-number “manufactured” solutions, shock dynamics/tracking with slow dynamic time scales, and multi-fluid, highspeed shock-tube problems. We briefly discuss preconditioning, and we introduce two physics-based preconditioners – “Block-Diagonal” and “Internal energy-Pressure-Velocity Partially Decoupled”, demonstrating the ability to efficiently solve all-speed flows with strong effects from viscous dissipation and heat conduction

    A discontinuous Galerkin method for the solution of compressible flows

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    This thesis presents a methodology for the numerical solution of one-dimensional (1D) and two-dimensional (2D) compressible flows via a discontinuous Galerkin (DG) formulation. The 1D Euler equations are used to assess the performance and stability of the discretisation. The explicit time restriction is derived and it is established that the optimal polynomial degree, p, in terms of efficiency and accuracy of the simulation is p = 5. Since the method is characterised by minimal diffusion, it is particularly well suited for the simulation of the pressure wave generated by train entering a tunnel. A novel treatment of the area-averaged Euler equations is proposed to eliminate oscillations generated by the projection of a moving area on a fixed mesh and the computational results are validated against experimental data. Attention is then focussed on the development of a 2D DG method implemented using the high-order library Nektar++. An Euler and a laminar Navier–Stokes solvers are presented and benchmark tests are used to assess their accuracy and performance. An artificial diffusion term is implemented to stabilise the solution of the Euler equations in transonic flow with discontinuities. To speed up the convergence of the explicit method, a new automatic polynomial adaptive procedure (p-adaption) and a new zonal solver are proposed. The p-adaptive procedure uses a discontinuity sensor, originally developed as an artificial diffusion sensor, to assign appropriate polynomial degrees to each element of the domain. The zonal solver uses a modification of a method for matching viscous subdomains to set the interface conditions between viscous and inviscid subdomains that ensures stability of the flow computation. Both the p-adaption and the zonal solver maintain the high-order accuracy of the DG method while reducing the computational cost of the simulation
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