On the Origins of Lagrangian Hydrodynamic Methods
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73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
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Numerical relativity in spherical coordinates: A new dynamical spacetime and general relativistic MHD evolution framework for the Einstein Toolkit
We present SphericalNR, a new framework for the publicly available Einstein Toolkit that numerically solves the Einstein field equations coupled to the equations of general relativistic magnetohydrodynamic (GRMHD) in a 3+1 split of spacetime in spherical coordinates without symmetry assumptions. The spacetime evolution is performed using reference-metric versions of either the Baumgarte-Shapiro-Shibata-Nakamura equations or the fully covariant and conformal Z4 system with constraint damping. We have developed a reference-metric version of the Valencia formulation of GRMHD with a vector potential method, guaranteeing the absence of magnetic monopoles during the evolution. In our framework, every dynamical field (both spacetime and matter) is evolved using its components in an orthonormal basis with respect to the spherical reference metric. Furthermore, all geometric information about the spherical coordinate system is encoded in source terms appearing in the evolution equations. This allows for the straightforward extension of Cartesian high-resolution shock-capturing finite volume codes to use spherical coordinates with our framework. To this end, we have adapted GRHydro, a Cartesian finite volume GRMHD code already available in the Einstein Toolkit, to use spherical coordinates. We present the full evolution equations of the framework, as well as details of its implementation in the Einstein Toolkit. Finally, we validate SphericalNR by demonstrating it passes a variety of challenging code tests in static and dynamical spacetimes.
79 ASTRONOMY AND ASTROPHYSICS↗
Simulating magnetized neutron stars with discontinuous Galerkin methods
Discontinuous Galerkin methods are popular because they can achieve high order where the solution is smooth, because they can capture shocks while needing only nearest-neighbor communication, and because they are relatively easy to formulate on complex meshes. We perform a detailed comparison of various limiting strategies presented in the literature applied to the equations of general relativistic magnetohydrodynamics. We compare the standard minmod /ΛΠ N limiter, the hierarchical limiter of Krivodonova, the simple WENO limiter, the HWENO limiter, and a discontinuous Galerkin-finite-difference hybrid method. The ultimate goal is to understand what limiting strategies are able to robustly simulate magnetized Tolman-Oppenheimer-Volkoff stars without any fine-tuning of parameters. Among the limiters explored in the paper, the only limiting strategy we can endorse is a discontinuous Galerkin-finite-difference hybrid method.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
A hybrid finite volume method and smoothed particle hydrodynamics approach for efficient and accurate blast simulations
Modeling strong shock waves in fluids remains a persistent challenge in computational physics. Essential to research efforts in industry and defense, numerous methods have been devised to improve the accuracy and efficiency of shock simulations. A novel, hybrid Finite Volume Method (FVM)-Smoothed Particle Hydrodynamics (SPH) approach is capable of further improving efficiency and retaining accuracy by exploiting the favorable characteristics of each respective method. This hybrid approach is presented for shock capturing in compressible fluids. The Python framework Pyro2 is employed to simulate a coarse FVM mesh, while the Python framework PySPH is utilized to model the fluid in regions with high gradients through SPH particles. The performance of the hybrid FVM-SPH scheme, compared to the individual FVM and SPH methods, is assessed in 1 kt and 10 kt blast simulations. Our results indicate that the hybrid approach offers higher computational efficiency than SPH while preserving its accuracy and characteristics. The hybrid approach had a relative speedup of 11.3x and 22.3x over the FVM and SPH approaches for the 1 kt simulation and a relative speedup of 14.7x and 20.9x over the FVM and SPH approaches for the 10 kt simulation. The hybrid SPH algorithm enables future compressible fluid simulations with more extensive capabilities than grid-based methods alone, presenting potential applications in modeling fluid-structure interactions and solid deformation and fracturing in blast simulations.
Myers, Conner↗
A study of methods which predict supersonic flow fields from body geometry, distance, and Mach number
A study of seven methods for predicting flow-field pressure signatures from the parameters Mach number, body geometry, and field-path distance has been made. The methods included the method of characteristics, which served as a standard of comparison; a shock-capturing method; three Whitham theory methods; a modified characteristics method; and a bicharacteristics method. Results from each method were also compared with recently obtained wind-tunnel data for a cone-cylinder model at Mach numbers of 2.96 and 4.63 with ratios of radial distance to cone length of 2 and 5. The comparisons at a Mach number of 2.96 showed that signatures from all the methods correlated well with wind-tunnel data and with the signatures predicted by the method of characteristics. At a Mach number of 4.63, however, the agreement between the signatures obtained in the wind tunnel and those predicted by theory varied from good to poor, as did the agreement between the signatures obtained by the method of characteristics and the other six methods. It should be noted that these results and comparisons indicate pressure prediction capabilities only for the near-field flow about bodies of revolution.
Mack, R. J.↗
On the shock-on-shock interaction problem
The unsteady, inviscid flowfield that results when a supersonic vehicle strikes a planar oblique shock wave, though difficult to simulate experimentally, is quite easy to model and compute numerically. The complicated flowfield, which contains multiple shock wave interactions, is determined using a second-order, shock-capturing, finite-difference approach which solves the time-dependent Euler equations under a self-similar transformation. A series of numerical results for a simple two-dimensional wedge is presented which describes the entire disturbed region, including the wave structure, and shows good agreement with the available two- and three-dimensional experimental data.-
Kutler, P.↗
Supercritical wing sections 2, volume 108
A mathematical theory for the design and analysis of supercritical wing sections was previously presented. Examples and computer programs showing how this method works were included. The work on transonics is presented in a more definitive form. For design, a better model of the trailing edge is introduced which should eliminate a loss of fifteen or twenty percent in lift experienced with previous heavily aft loaded models, which is attributed to boundary layer separation. How drag creep can be reduced at off-design conditions is indicated. A rotated finite difference scheme is presented that enables the application of Murman's method of analysis in more or less arbitrary curvilinear coordinate systems. This allows the use of supersonic as well as subsonic free stream Mach numbers and to capture shock waves as far back on an airfoil as desired. Moreover, it leads to an effective three dimensional program for the computation of transonic flow past an oblique wing. In the case of two dimensional flow, the method is extended to take into account the displacement thickness computed by a semi-empirical turbulent boundary layer correction.
Bauer, F.↗
Numerical solutions for inviscid supersonic corner flows
Analytical solutions for inviscid supersonic corner flows are virtually nonexistent due to the complexity of the interference geometry. In view of this, numerical solutions for compressive-compressive and expansive-compressive corner flows are obtained. The governing equations are written in strong conservation-law form and are solved iteratively in nonorthogonal conical coordinates by use of a second-order, shock-capturing, finite-difference technique. The computed wave structure and surface pressure distributions are compared with high Reynolds number (Re greater than 2,500,000 ft) experimental data. The results clearly show that the wave structure in the corner is dominated by the inviscid field.
Shankar, V. S. V.↗
Numerical solutions for supersonic corner flow
Analytical solutions for inviscid supersonic corner flows are virtually nonexistent due to the complexity of the interference geometry. In view of this, numerical solutions for swept-compressive and swept-expansive corner flows are obtained. The governing equations are written in strong conservation-law form and are solved iteratively in nonorthogonal conical coordinates by use of a second-order, shock-capturing, finite-difference technique. The computed wave structure and surface pressure distributions are compared with high Reynolds number experimental data and show very good agreement. The results clearly show that supersonic corner flow at reasonably high Reynolds numbers including the effect of sweep is dominated by the inviscid field.
Shankar, V.↗
Diffraction of a shock wave by a compression corner. I - Regular reflection
The unsteady, two-dimensional flowfield resulting from the interaction of a moving planar shock wave with a compression corner is determined using a second-order, discontinuity-fitting, finite-difference approach. The time-dependent Euler equations are transformed to normalize the distance between the body and peripheral shock and to include the existing self-similar property of the flow. The resulting set of partial differential equations in conservation-law form is then solved in a time-dependent fashion using MacCormack's scheme. The vortical singularity, which lies on the body surface, and the single reflected shock are both treated as discontinuities in the numerical procedure. The results of the numerical simulation compare quite favorably with existing experimental interferograms and yield better flowfield resolution than previous first-order, shock-capturing, numerical solutions.
Kutler, P.↗
Diffraction of a shock wave by a compression corner; regular and single Mach reflection
The two dimensional, time dependent Euler equations which govern the flow field resulting from the injection of a planar shock with a compression corner are solved with initial conditions that result in either regular reflection or single Mach reflection of the incident planar shock. The Euler equations which are hyperbolic are transformed to include the self similarity of the problem. A normalization procedure is employed to align the reflected shock and the Mach stem as computational boundaries to implement the shock fitting procedure. A special floating fitting scheme is developed in conjunction with the method of characteristics to fit the slip surface. The reflected shock, the Mach stem, and the slip surface are all treated as harp discontinuities, thus, resulting in a more accurate description of the inviscid flow field. The resulting numerical solutions are compared with available experimental data and existing first-order, shock-capturing numerical solutions.
Vijayashankar, V. S.↗
Diffraction of a shock wave by a compression corner. II - Single Mach reflection
The two-dimensional time-dependent Euler equations which govern the flow field resulting from the interaction of a planar shock with a compression corner are solved for initial conditions which result in single Mach reflection of the incident planar shock. The Euler equations are first transformed to include the self-similarity of the flow field. A second transformation is employed to normalize the distances between the ramp and the reflected shock and between the wall and the Mach stem. The resulting equations in strong conservation-law form are solved using a second-order discontinuity-fitting finite-difference approach. The results are compared with experimental interferograms and existing first-order shock-capturing numerical solutions.
Shankar, V.↗
Comparison of numerical and experimental 'conical' flow fields in supersonic corners with compression and/or expansion
The flow field produced by the intersection of two plane solid surfaces in a supersonic stream is a complex interference flow. These flows can be fully compressive, fully expansive, or of mixed compression-expansion nature. This paper presents a comparison of the experimentally obtained flow-field structure in an axial corner with that predicted numerically by using a shock-capturing finite-difference method. The effect of sweep and surface deflection are evaluated, and the general influence of each is presented for the three classes of corner flow. The results show that the numerical method is a valuable aid in understanding the flow structure for simple configurations. In addition, confidence in the numerical method is gained for use in solving more general three-dimensional configurations where the flow is nonconical and several wave interaction may be presented.
Anderson, D. A.↗
Transonic flow over the NACA 64A006 with an oscillating flap-calculations based on the Euler equations
Exploratory calculations of transonic flows over the airfoil with a quarter-chord oscillating flap were made using a program which obtains approximate solutions to the Euler equations with an explicit, shock-capturing, finite-difference scheme. The calculations, essentially inviscid and for the airfoil at zero angle-of-attack in a free-stream, are at Mach numbers and reduced frequencies which were tested in experiments by Tijdeman. The oscillatory lifts from analogus calculations by various investigators generally agree with one another better than they agree with Tijdeman's data. Inclusion in the calculations of an approximate modelling of boundary conditions expected at slotted wind tunnel walls tends to shift some of the results closer to the experimental values.
Magnus, R. J.↗
Numerical solution of the Navier-Stokes equations for arbitrary blunt bodies in supersonic flows
A time-dependent, two-dimensional Navier-Stokes code employing the body-fitted coordinate technique has been developed for supersonic flows past blunt bodies of arbitrary shape. The computer program is based on the finite-difference approximation of the compressible Navier-Stokes equations transformed to nonorthogonal curvilinear coordinates with the contravariant components of the velocity vector as dependent variables. The bow shock ahead of the body is obtained as part of the solution, by 'shock capturing'. Numerical solutions of the complete equations are presented in detail for free-stream Mach number 4.6, Reynolds number 10,000, and an isothermal wall temperature of 556 K for a circular cylinder with the free-stream outer boundaries forming a hyperbola in the front and a circular arc in the back.
Warsi, Z. U. A.↗
Calculation of two-dimensional inlet flow fields in a supersonic free stream by an implicit marching code with nonorthogonal mesh generation: User's manual
An implicit, shock-capturing finite-difference code which is used to calculate two-dimensional inlet flow fields in a supersonic free stream is explained. The Euler equations are subjected to general nonorthogonal transformation and a body-fitted coordinate system is employed. The mathematical formulation of the problem is given along with the numerical algorithm. Initial and boundary conditions, numerical stability, program limitations, and accuracy is discussed. An overall program logic as well as instructions for program use and operation are also furnished.
Biringen, S. H.↗
A supersonic, three-dimensional code for flow over blunt bodies: User's manual
A computer code is described which may be used to calculate the steady, supersonic, three-dimensional, inviscid flow over blunt bodies. The theoretical and numerical formulation of the problem is given (shock-capturing, downstream marching), including exposition of the boundary and initial conditions. The overall flow logic of the program, its usage, accuracy, and limitations are discussed.
Chaussee, D. S.↗
Computational models for the viscous/inviscid analysis of jet aircraft exhaust plumes
Computational models which analyze viscous/inviscid flow processes in jet aircraft exhaust plumes are discussed. These models are component parts of an NASA-LaRC method for the prediction of nozzle afterbody drag. Inviscid/shock processes are analyzed by the SCIPAC code which is a compact version of a generalized shock capturing, inviscid plume code (SCIPPY). The SCIPAC code analyzes underexpanded jet exhaust gas mixtures with a self-contained thermodynamic package for hydrocarbon exhaust products and air. A detailed and automated treatment of the embedded subsonic zones behind Mach discs is provided in this analysis. Mixing processes along the plume interface are analyzed by two upgraded versions of an overlaid, turbulent mixing code (BOAT) developed previously for calculating nearfield jet entrainment. The BOATAC program is a frozen chemistry version of BOAT containing the aircraft thermodynamic package as SCIPAC; BOATAB is an afterburning version with a self-contained aircraft (hydrocarbon/air) finite-rate chemistry package. The coupling of viscous and inviscid flow processes is achieved by an overlaid procedure with interactive effects accounted for by a displacement thickness type correction to the inviscid plume interface.
Dash, S. M.↗