Nesting an incompressible-flow code within a compressible-flow code: a two-dimensional study
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97 MATHEMATICS AND COMPUTING↗
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Optimization techniques in self-similar compressible flow
We investigate the one-dimensional (1D) inviscid compressible flow equations for an ideal gas through the lens of optimization techniques. It is the case that, to our knowledge, optimization analysis applied to the so-called “linear velocity” solutions of the Euler compressible flow equations has not been previously conducted. Through both gradient-based and variational techniques, new variants of well-studied flow scenarios, i.e., self-similar, 1D, linear velocity solution class to idealized inviscid compressible flow equations, are determined, as encoded in both the kinematic and thermodynamic properties of this self-similar solution class. With the kinematics of the said solutions being driven by a self-similar “scale radius” and the thermodynamics being driven separately through the appearance of an arbitrary function, a myriad of new solution classes is possible. Acting as a guide to more realistic physical circumstances as well as discovery, it is the hope that the presented cases serve as the framework for future investigations into the intersection of self-similarity and optimization techniques. Fields of study that may find this work to be of interest include aerodynamic design, flow control, inertial confinement fusion, physics-informed neural networks, and other related areas of interest.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Verification assessment of piston boundary conditions for Lagrangian simulation of compressible flow similarity solutions
This work is concerned with the use of similarity solutions of the compressible flow equations as benchmarks or verification test problems for finite-volume compressible flow simulation software. In practice, this effort can be complicated by the infinite spatial/temporal extent of many candidate solutions or “test problems.” Methods can be devised with the intention of ameliorating this inconsistency with the finite nature of computational simulation; the exact strategy will depend on the code and problem archetypes under investigation. For example, self-similar shock wave propagation can be represented in Lagrangian compressible flow simulations as rigid boundary-driven flow, even if no such “piston” is present in the counterpart mathematical similarity solution. The purpose of this work is to investigate in detail the methodology of representing self-similar shock wave propagation as a piston-driven flow in the context of various test problems featuring simple closed-form solutions of infinite spatial/temporal extent. The closed-form solutions allow for the derivation of similarly closed-form piston boundary conditions (BCs) for use in Lagrangian compressible flow solvers. Finally, the consequences of utilizing these BCs (as opposed to directly initializing the self-similar solution in a computational spatial grid) are investigated in terms of common code verification analysis metrics (e.g., shock strength/position errors and global convergence rates).
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Developing A Fully Implicit Compressible Flow Valve Model in RELAP-7 and Its Application in Short Term BWR Station Black-Out Analyses
This paper presents a set of dynamical models for SRV simulation and their implementation in the RELAP-7 code, which is a new reactor system analysis code being developed at INL. RELAP-7 is based on the MOOSE (the Multi-Physics Object-Oriented Simulation Environment) framework. The compressible flow SRV model covers three different sets of models: fully closed case, subsonic case, and choking case. The RELAP-7 is a fully implicit code and the preconditioned Jacobian-free Newton-Krylov (JFNK) method is used to solve the discretized nonlinear problem. The JFNK method generally needs a good preconditioning scheme to be effective and efficient. There are different ways to precondition the JFNK method. In RELAP-7, two main options are used to obtain the preconditioning matrix: (1) FDP (Finite Difference Preconditioning) option which uses the default finite differencing method provided in the PETSc package. PETSc is the numerical solver software developed by Argonne National Laboratory and used by MOOSE as the numerical solution engine; (2) SMP (Single Matrix Preconditioning) option which requires code developers to provide approximate analytical Jacobian inputs for the important diagonal and off-diagonal terms. Due to the strong discontinuity in the residual vector space when the compressible flow valve model switches from one set of model to another, FDP option cannot guarantee convergence. Instead, accurate analytical Jacobain terms are provided for the compressible flow valve model. With the SMP option selected, the compressible flow valve model can be solved efficiently under different cases. The developed compressible flow valve model has been used for SRV component simulation in several simplified BWR SBO (Station Black-Out) simulations with RELAP-7. This paper only presents cases for short-term SBO in which all the safety systems fail to start and function properly when SBO occurs. Consequently, only the SRVs automatically open and close periodically to control the primary system pressure and discharge the high temperature and high pressure steam into the suppression pool. Although simplified, this case is quite similar to what happened in Fukushima Daiichi Unit 1. Unit 1 had no RCIC system, while the isolation condenser system was believed to be non-functioning, or only available for a very short period of time during the accident. Therefore, the Unit 1 reactor core reached a fuel failure temperature only a few hours into the accident. Two short-term SBO cases will be compared: one with single SRV discharge line which represents all the SRV lines and another case with multi-group SRVs which represent each group of SRV line with different setting points.
22 GENERAL STUDIES OF NUCLEAR REACTORS↗
Three-dimensional nonlinear ideal MHD equilibria with field-aligned incompressible and compressible flows
The equilibrium properties of three-dimensional ideal magnetohydrodynamics (MHD) are investigated. Incompressible and compressible flows are considered. The governing equations are taken in a steady state such that the magnetic field is parallel to the plasma flow. Equations of stationary equilibrium for both of incompressible and compressible MHD flows are derived and described in a mathematical mode. For incompressible MHD flows, Alfvénic and non-Alfvénic flows with constant and variable magnetofluid density are investigated. For Alfvénic incompressible flows, the general three-dimensional solutions are determined with the aid of two potential functions of the velocity field. For non-Alfvénic incompressible flows, the stationary equilibrium equations are reduced to two differential constraints on the potential functions, flow velocity, magnetofluid density, and the static pressure. Some examples which may be of some relevance to axisymmetric confinement systems are presented. For compressible MHD flows, equations of the stationary equilibrium are derived with the aid of a single potential function of the velocity field. The existence of three-dimensional solutions for these MHD flows is investigated. Several classes of three-dimensional exact solutions for several cases of nonlinear equilibrium equations are presented.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
A high-order vertex-based central ENO finite-volume scheme for three-dimensional compressible flows
High-order discretization methods offer the potential to reduce the computational cost associated with modeling compressible flows. However, it is difficult to obtain accurate high-order discretizations of conservation laws that do not produce spurious oscillations near discontinuities, especially on multi-dimensional unstructured meshes. A novel, high-order, central essentially non-oscillatory (CENO) finite-volume method that does not have these difficulties is proposed for tetrahedral meshes. The proposed unstructured method is vertex-based, which differs from existing cell-based CENO formulations, and uses a hybrid reconstruction procedure that switches between two different solution representations. It applies a high-order k-exact reconstruction in smooth regions and a limited linear reconstruction when discontinuities are encountered. Both reconstructions use a single, central stencil for all variables, making the application of CENO to arbitrary unstructured meshes relatively straightforward. The new approach was applied to the conservation equations governing compressible flows and assessed in terms of accuracy and computational cost. For all problems considered, which included various function reconstructions and idealized flows, CENO demonstrated excellent reliability and robustness. Up to fifth-order accuracy was achieved in smooth regions and essentially non-oscillatory solutions were obtained near discontinuities. The high-order schemes were also more computationally efficient for high-accuracy solutions, i.e., they took less wall time than the lower-order schemes to achieve a desired level of error. In one particular case, it took a factor of 24 less wall-time to obtain a given level of error with the fourth-order CENO scheme than to obtain the same error with the second-order scheme.
97 MATHEMATICS AND COMPUTING↗
Symmetries of the Euler compressible flow equations for general equation of state
The Euler compressible flow equations exhibit different Lie symmetries depending on the equation of state (EOS) of the medium in which the flow occurs. This means that, in general, different types of similarity solution will be available in different flow media. We present a comprehensive classification of all EOS’s to which the Euler equations apply, based on the Lie symmetries admitted by the corresponding flow equations, restricting to the case of 1-D planar, cylindrical, or spherical geometry. The results are conveniently summarized in tables. This analysis also clarifies past work by Axford and Ovsiannikov on symmetry classification.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
An Efficient B-Spline Lagrangian/Eulerian Method for Compressible Flow, Shock Waves, and Fracturing Solids
This study presents a new method for modeling the interaction between compressible flow, shock waves, and deformable structures, emphasizing destructive dynamics. Extending advances in time-splitting compressible flow and the Material Point Methods (MPM), we develop a hybrid Eulerian and Lagrangian/Eulerian scheme for monolithic flow-structure interactions. We adopt the second-order WENO scheme to advance the continuity equation. To stably resolve deforming boundaries with sub-cell particles, we propose a blending treatment of reflective and passable boundary conditions inspired by the theory of porous media. The strongly coupled velocity-pressure system is discretized with a new mixed-order finite element formulation employing B-spline shape functions. Shock wave propagation, temperature/density-induced buoyancy effects, and topology changes in solids are unitedly captured.
97 MATHEMATICS AND COMPUTING↗
A set of parallel, implicit methods for a reconstructed discontinuous Galerkin method for compressible flows on 3D hybrid grids
A set of implicit methods are proposed for a third-order hierarchical WENO reconstructed discontinuous Galerkin method for compressible flows on 3D hybrid grids. An attractive feature in these methods are the application of the Jacobian matrix based on the P1 element approximation, resulting in a huge reduction of memory requirement compared with DG (P2). Also, three approaches -- analytical derivation, divided differencing, and automatic differentiation (AD) are presented to construct the Jacobian matrix respectively, where the AD approach shows the best robustness. A variety of compressible flow problems are computed to demonstrate the fast convergence property of the implemented flow solver. Furthermore, an SPMD (single program, multiple data) programming paradigm based on MPI is proposed to achieve parallelism. The numerical results on complex geometries indicate that this low-storage implicit method can provide a viable and attractive DG solution for complicated flows of practical importance.
97 MATHEMATICS AND COMPUTING↗
Entropy Stable Discretization of Compressible Flows in Thermochemical Nonequilibrium
Entropy stable numerical methods for compressible flow have been demonstrated to exhibit better robustness than purely linearly stable methods and need less overall artificial dissipation for long simulations in subsonic and transonic flows. In this work we seek to extend these benefits to multicomponent, multitemperature flows in thermochemical nonequilibrium such as combustion and hypersonic flight. We first derive entropy functions that symmetrize the governing equations and allow stability proofs for such systems. The impact of diffusion model selection on provable entropy stability is considered in detail, including both rigorous models of irreversible thermodynamics and simplified models of greater practical interest. Based on the proven entropy functions we develop affordable, entropy conservative two-point flux functions for solution in conservation form. We derive entropy conservative fluxes for calorically and thermally perfect mixtures, with heat capacities described by either polynomials of the temperature or formulas from statistical thermodynamics.
42 ENGINEERING↗
A Modified Johnson–Cook Constitutive Model for the Compressive Flow Behaviors of the SnSbCu Alloy at High Strain Rates
The SnSbCu alloy is widely used as the material for the main bearing in low-speed marine engines, and an accurate constitutive model is the foundation for studying the frictional behaviors of bearings. In this work, the compressive flow behaviors of the SnSbCu alloy were considered under the different strain rates (1000-5000 s{sup −1}) and temperatures (20-110 °C) by quasi-static and split-Hopkinson pressure bar dynamic compression tests. First, the original Johnson–Cook model was used to describe the constitutive relation of the SnSbCu alloy at high strain rates, and the results predicted by the original model showed relatively large errors compared with the experimental results since the coupled effect of temperature and strain rate was omitted. Then, a modified Johnson–Cook constitutive model was developed to describe the compressive flow behaviors of the SnSbCu alloy, and the results predicted by this modified model agreed well with the experimental data. Moreover, a finite element analysis was also conducted to verify the accuracy of the modified Johnson–Cook model.
36 MATERIALS SCIENCE↗
A cut-cell finite volume – finite element coupling approach for fluid–structure interaction in compressible flow
We present a loosely coupled approach for the solution of fluid–structure interaction problems between a compressible flow and a deformable structure. The method is based on staggered Dirichlet–Neumann partitioning. The interface motion in the Eulerian frame is accounted for by a conservative cut-cell Immersed Boundary method. The present approach enables sub-cell resolution by considering individual cut-elements within a single fluid cell, which guarantees an accurate representation of the time-varying solid interface. The cut-cell procedure inevitably leads to non-matching interfaces, demanding for a special treatment. A Mortar method is chosen in order to obtain a conservative and consistent load transfer. We validate our method by investigating two-dimensional test cases comprising a shock-loaded rigid cylinder and a deformable panel. Moreover, the aeroelastic instability of a thin plate structure is studied with a focus on the prediction of flutter onset. Finally, we propose a three-dimensional fluid–structure interaction test case of a flexible inflated thin shell interacting with a shock wave involving large and complex structural deformations.
42 ENGINEERING↗
Nemchinov-Dyson Solutions of the Two-Dimensional Axisymmetric Inviscid Compressible Flow Equations [Review]
We investigate the two-dimensional (2D) inviscid compressible flow equations in axisymmetric coordinates, constrained by an ideal gas equation of state (EOS). Beginning with the assumption that the 2D velocity field is space-time separable and linearly variable in each corresponding spatial coordinate, we proceed to derive an in nite family of elliptic or hyperbolic, uniformly expanding or contracting \gas cloud" solutions. Construction of specific example solutions belonging to this family is dependent on the solution of a system of nonlinear, coupled, second-order ordinary differential equations, and the prescription of an additional physical process of interest (e.g., uniform temperature or uniform entropy flow). The physical and computational implications of these solutions as pertaining to quantitative code verification or model qualification studies are discussed in some detail.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Faxén form of time-domain force on a sphere in unsteady spatially varying viscous compressible flows
An explicit expression for the time-dependent force on a stationary, finite-sized spherical particle located in an unsteady inhomogeneous ambient flow is presented. The force expression accounts for both viscous and compressible effects. Towards this end, a time-harmonic plane travelling wave of a given frequency propagating in a viscous compressible flow over a sphere is considered. Linearized compressible Navier–Stokes equations are solved to obtain an analytical expression for the force exerted on the particle in the frequency domain. The force obtained in the Laplace space due to a travelling wave of a given frequency and wavenumber is then generalized to any arbitrary incoming flow. This is achieved by relating the radial and tangential velocity components in the Laplace space to the surface-averaged radial velocity and volume-averaged velocity vectors respectively in the time space. Moreover an expression relating the surface-averaged radial velocity and volume-averaged velocity vector has been provided. The total force is written as a summation of the undisturbed and disturbed force (quasi-steady, inviscid-unsteady and viscous-unsteady) contributions. The force contributions thus obtained are expressed as comprising of two parts – that arising due to spatial variation in the ambient flow and the other arising due to temporal variation. The current formulation is applicable to inhomogeneous ambient flows, however in the limit of negligible Reynolds and Mach numbers. The results are applicable even for particles of sizes larger than the acoustic wavelength. The accuracy of the explicit time-domain force expression is first tested by computing the force on an 80 mm diameter particle due to a weak planar expansion fan. Extension of this formulation when nonlinear effects become important is also proposed and tested by considering strong expansion fans. The results thus obtained are compared against corresponding axisymmetric numerical simulations.
Mechanics↗
An interface capturing scheme for modeling atomization in compressible flows
The study of atomization in supersonic flow is critical to ensuring reliable ignition of scramjet combustors under startup conditions. Numerical methods incorporating surface tension effects have largely focused on the incompressible regime as most atomization applications occur at low Mach numbers. Simulating surface tension effects in compressible flow requires robust numerical methods that can handle discontinuities caused by both shocks and material interfaces with high density ratios. In this work, a shock and interface capturing scheme is developed that uses the Harten–Lax–van Leer–Contact (HLLC) Riemann solver while a Tangent of Hyperbola for INterface Capturing (THINC) interface reconstruction scheme retains the fluid immiscibility condition in the volume fraction and phasic densities in the context of the five equation model. The approach includes the effects of compressibility, surface tension, and molecular viscosity. One and two-dimensional benchmark problems demonstrate the desirable interface sharpening and conservation properties of the approach. Simulations of secondary atomization of a cylindrical water column after its interaction with a shockwave show good qualitative agreement with experimentally observed behavior. Three-dimensional examples of primary atomization of a liquid jet in a Mach 2 crossflow demonstrate the robustness of the method.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
A higher‐order unsplit 2D direct Eulerian finite volume method for two‐material compressible flows based on the MOOD paradigms
SUMMARY A higher‐order unsplit multi‐dimensional discretization of the diffuse interface model for two‐material compressible flows proposed by R. Saurel, F. Petitpas and R. A. Berry in 2009 is developed. The proposed higher‐order method is based on the concepts of the Multidimensional Optimal Order Detection (MOOD) method introduced in three recent papers for single‐material flows. The first‐order unsplit multi‐dimensional Finite Volume discretization presented by SPB serves as foundation for the development of the higher‐order unlimited schemes. Specific detection criteria along with a novel decrementing algorithm for the MOOD method are designed in order to deal with the complexity of multi‐material flows. Numerically, we compare errors and computational times on several 1D problems (stringent shock tube and cavitation problems) computed on 2D meshes with the second‐ and fourth‐order MOOD methods using a classical MUSCL method as reference. Several simulations of a 2D shocked R22 bubble in the air are also presented on Cartesian and unstructured meshes with the second‐ and fourth‐order MOOD methods, and qualitative comparisons confirm the conclusions obtained with 1D problems. These numerical results demonstrate the robustness of the MOOD approach and the interest of using more than second‐order methods even for locally singular solutions of complex physics models. Copyright © 2014 John Wiley & Sons, Ltd.
Diot, S.↗
A high-order discontinuous Galerkin method with unstructured space–time meshes for two-dimensional compressible flows on domains with large deformations
We introduce a high-order accurate space-time discontinuous Galerkin method for solving two-dimensional compressible flow problems on fully unstructured space-time meshes. The discretization is based on a nodal formulation, with appropriate numerical fluxes for the first and the second-order terms, respectively. The scheme is implicit, and we solve the resulting non-linear systems using a parallel Newton-Krylov solver. The meshes are produced by a mesh moving technique with element connectivity updates, and the corresponding space-time elements are produced directly based on these local operations. To obtain globally conforming tetrahedral meshes, we first derive the required conditions on a prism boundary mesh to allow for a valid local triangulation. Next, we present an efficient algorithm for finding a global mesh that satisfies these conditions. We additionally show how to add and remove mesh nodes, again using local constructs for the space-time mesh. Our approach is demonstrated on a number of test problems, showing the high-order accuracy for model problems, and the ability to solve flow problems on domains with complex large deformations.
97 MATHEMATICS AND COMPUTING↗