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In this study, an Unsteady Reynolds-Averaged Navier-Stokes (URANS) model is demonstrated its suitability for studying the flow and performance of open marine propellers and waterjet pumps. First, the accuracy of the URANS model is validated by studying turbulent flow past counter-rotating propellers (CRPs). Specifically, experimental data from Miller (1976) is employed for comparison against the URANS results. Subsequently, URANS is used to study the flow and performance of an Office of Naval Research (ONR) axial flow waterjet pump (AxWJ-2). Due to the large number of degrees of freedom for both simulations, parallel computations over 80 cores are performed. For the CRP study, torque and thrust coefficients are assessed against a range of advance ratios, ensuring a Reynolds number of less than 600,000. For the waterjet, torque and head coefficients are computed for a range of flow rates at a Reynolds number of 1.25 million. For both studies, two levels of mesh resolution are utilized. The finer meshes of both studies contained roughly four times the total number of cells employed in their respective coarser counterparts. These refinements lead to minor improvements, suggesting good grid resolutions with the coarser grids. Across all advance ratios for the CRP set, the URANS torque and thrust coefficients show good agreement with experimental results, remaining within 10% difference. The torque and head coefficients for the waterjet displayed even better agreement, with the greatest error across all flow conditions remaining under 3%. Moreover, URANS studies revealed that the stator is responsible for 20% of the waterjet’s power production. 

We present two major improvements over the Compressible High-ORder Unstructured Spectral difference (CHORUS) code published in Wang et al., “A compressible high-order unstructured spectral difference code for stratified convection in rotating spherical shells,” J. Comput. Phys. 290, 90–111 (2015). The new code is named CHORUS++ in this paper. Subsequently, we perform a series of efficient simulations for rotationally constrained convection (RCC) in spherical shells. The first improvement lies in the integration of the high-order spectral difference method with a boundary-conforming transfinite mapping on cubed-sphere grids, thus ensuring exact geometric representations of spherical surfaces on arbitrary sparse grids. The second improvement is on the adoption of higher-order elements (sixth-order) in CHORUS++ vs third-order elements for the original CHORUS code. CHORUS++ enables high-fidelity RCC simulations using sixth-order elements on very coarse grids. To test the accuracy and efficiency of using elements of different orders, CHORUS++ is applied to a laminar solar benchmark, which is characterized by columnar banana-shaped convective cells. By fixing the total number of solution degrees of freedom, the computational cost per time step remains unchanged. Nevertheless, using higher-order elements in CHORUS++ resolves components of the radial energy flux much better than using third-order elements. To obtain converged predictions, using sixth-order elements is 8.7 times faster than using third-order elements. This significant speedup allows global-scale fully compressible RCC simulations to reach equilibration of the energy fluxes on a small cluster of just 40 cores. In contrast, CHORUS simulations were performed by Wang et al. on supercomputers using approximately 10 000 cores. Using sixth-order elements in CHORUS++, we further carry out global-scale solar convection simulations with decreased rotational velocities. Interconnected networks of downflow lanes emerge and surround broader and weaker regions of upflow fields. A strong inward kinetic energy flux compensated by an enhanced outward enthalpy flux appears. These observations are all consistent with those published in the literature. Furthermore, CHORUS++ can be extended to magnetohydrodynamic simulations with potential applications to the hydromagnetic dynamo processes in the interiors of stars and planets. 

The Hall Magnetohydrodynamic (MHD) equations are an extension of the standard MHD equations that include the “Hall” term from the general Ohm’s law. The Hall term decouples ion and electron motion physically on the ion inertial length scales. Implementing the Hall MHD equations in a numerical solver allows more physical simulations for plasma dynamics on length scales less than the ion inertial scale length but greater than the electron inertial length. The present effort is an important step towards producing physically correct results to important problems, such as the Geospace Environmental Modeling (GEM) Magnetic Reconnection problem. The solver that is being modified is currently capable of solving the resistive MHD equations on unstructured grids using the spectral difference scheme which is an arbitrarily high-order method that is relatively simple to parallelize. The GEM Magnetic Reconnection problem is used to evaluate whether the Hall MHD equations have been correctly implemented in the solver using the spectral difference method with divergence cleaning (SDDC) algorithm by comparing against the reconnection rates reported in the literature. 

Abstract This paper reports a recent development of the high-order spectral difference method with divergence cleaning (SDDC) for accurate simulations of both ideal and resistive magnetohydrodynamics (MHD) on curved unstructured grids consisting of high-order isoparametric quadrilateral elements. The divergence cleaning approach is based on the improved generalized Lagrange multiplier, which is thermodynamically consistent. The SDDC method can achieve an arbitrarily high order of accuracy in spatial discretization, as demonstrated in the test problems with smooth solutions. The high-order SDDC method combined with the artificial dissipation method can sharply capture shock interfaces with the oscillation-free property and resolve small-scale vortex structures and density fluctuations on relatively sparse grids. The robustness of the codes is demonstrated through long time simulations of ideal MHD problems with progressively interacting shock structures, resistive MHD problems with high Lundquist numbers, and viscous resistive MHD problems on complex curved domains.