More efficient time integration for Fourier pseudo-spectral DNS of incompressible turbulence

Time integration of Fourier pseudo-spectral DNS is usually performed using the classical fourth-order accurate Runge--Kutta method, or other methods of second or third order, with a fixed step size. We investigate the use of higher-order Runge-Kutta pairs and automatic step size control based on local error estimation. We find that the fifth-order accurate Runge--Kutta pair of Bogacki \& Shampine gives much greater accuracy at a significantly reduced computational cost. Specifically, we demonstrate speedups of 2x-10x for the same accuracy. Numerical tests (including the Taylor-Green vortex, Rayleigh-Taylor instability, and homogeneous isotropic turbulence) confirm the reliability and efficiency of the method. We also show that adaptive time stepping provides a significant computational advantage for some problems (like the development of a Rayleigh-Taylor instability) without compromising accuracy

A hybrid discrete exterior calculus and finite difference method for anelastic convection in spherical shells

The present work develops, verifies, and benchmarks a hybrid discrete exterior calculus and finite difference (DEC-FD) method for density-stratified thermal convection in spherical shells. Discrete exterior calculus (DEC) is notable for its coordinate independence and structure preservation properties. The hybrid DEC-FD method for Boussinesq convection has been developed by Mantravadi et al. (Mantravadi, B., Jagad, P., & Samtaney, R. (2023). A hybrid discrete exterior calculus and finite difference method for Boussinesq convection in spherical shells. Journal of Computational Physics, 491, 112397). Motivated by astrophysics problems, we extend this method assuming anelastic convection, which retains density stratification; this has been widely used for decades to understand thermal convection in stars and giant planets. In the present work, the governing equations are splitted into surface and radial components and discrete anelastic equations are derived by replacing spherical surface operators with DEC and radial operators with FD operators. The novel feature of this work is the discretization of anelastic equations with the DEC-FD method and the assessment of a hybrid solver for density-stratified thermal convection in spherical shells. The discretized anelastic equations are verified using the method of manufactured solution (MMS). We performed a series of three-dimensional convection simulations in a spherical shell geometry and examined the effect of density ratio on convective flow structures and energy dynamics. The present observations are in agreement with the benchmark models.Comment: 32 pages, 13 figure

Entropy stable wall boundary conditions for the three-dimensional compressible Navier-Stokes equations

Non-linear entropy stability and a summation-by-parts framework are used to derive entropy stable wall boundary conditions for the three-dimensional compressible Navier--Stokes equations. A semi-discrete entropy estimate for the entire domain is achieved when the new boundary conditions are coupled with an entropy stable discrete interior operator. The data at the boundary are weakly imposed using a penalty flux approach and a simultaneous-approximation-term penalty technique. Although discontinuous spectral collocation operators on unstructured grids are used herein for the purpose of demonstrating their robustness and efficacy, the new boundary conditions are compatible with any diagonal norm summation-by-parts spatial operator, including finite element, finite difference, finite volume, discontinuous Galerkin, and flux reconstruction/correction procedure via reconstruction schemes. The proposed boundary treatment is tested for three-dimensional subsonic and supersonic flows. The numerical computations corroborate the non-linear stability (entropy stability) and accuracy of the boundary conditions.Comment: 43 page

Entropy Stable Wall Boundary Conditions for the Compressible Navier-Stokes Equations

Non-linear entropy stability and a summation-by-parts framework are used to derive entropy stable wall boundary conditions for the compressible Navier-Stokes equations. A semi-discrete entropy estimate for the entire domain is achieved when the new boundary conditions are coupled with an entropy stable discrete interior operator. The data at the boundary are weakly imposed using a penalty flux approach and a simultaneous-approximation-term penalty technique. Although discontinuous spectral collocation operators are used herein for the purpose of demonstrating their robustness and efficacy, the new boundary conditions are compatible with any diagonal norm summation-by-parts spatial operator, including finite element, finite volume, finite difference, discontinuous Galerkin, and flux reconstruction schemes. The proposed boundary treatment is tested for three-dimensional subsonic and supersonic flows. The numerical computations corroborate the non-linear stability (entropy stability) and accuracy of the boundary conditions

Characterization of pressure fluctuations within a controlled-diffusion blade boundary layer using the equilibrium wall-modelled LES

In this study, the generation of airfoil trailing edge broadband noise that arises from the interaction of turbulent boundary layer with the airfoil trailing edge is investigated. The primary objectives of this work are: (i) to apply a wall-modelled large-eddy simulation (WMLES) approach to predict the fow of air passing a controlled-difusion blade, and (ii) to study the blade broadband noise that is generated from the interaction of a turbulent boundary layer with a lifting surface trailing edge. This study is carried out for two values of the Mach number, Ma∞ = 0.3 and 0.5, two values of the chord Reynolds number, Re = 8.30 × 10^5 and 2.29 × 10^6, and two angles of attack, AoA= 4 and 5. To examine the influence of the grid resolution on aerodynamic and aeroacoustic quantities, we compare our results with experimental data available in the literature. We also compare our results with two in-house numerical solutions generated from two wall-resolved LES (WRLES) calculations, one of which has a DNS-like resolution. We show that WMLES accurately predicts the mean pressure coefficient distribution, velocity statistics (including the mean velocity), and the traces of Reynolds tensor components. Furthermore, we observe that the instantaneous flow structures computed by the WMLES resemble those found in the reference WMLES database, except near the leading edge region. Some of the differences observed in these structures are associated with tripping and the transition to a turbulence mechanism near the leading edge, which are significantly affected by the grid resolution. The aeroacoustic noise calculations indicate that the power spectral density profiles obtained using the WMLES compare well with the experimental data

Preparing the path for the efficient simulation of turbulent compressible industrial flows with robust collocated RK-DG solvers

We present an analysis of the performance of some standard and optimized explicitly Runge­ Kutta schemes that are equipped with CFL-based and error-based time step adaptivity when they are coupled with the relaxation procedure to achieve fully-discrete entropy stability for complex compressible flow simulations. We investigate the performance of the temporal integration algorithms by simulating the flow past the NASA juncture flow model using the in-house KAUST SSDC hp-adaptive collocated entropy stable discontinuous Galerkin solver. In addition, we present a preliminary analysis of the performance of the SSDC framework on the Amazon web service cloud computing. The results indicate that SSDC scales well on the most recent and exotic computing architectures available on the Amazon cloud platform. Our findings might help select a more robust and efficient temporal integration algorithm and guide the choice of the EC2 AWS instances that give the best price and wall-clock-time performance to simulate industrially relevant turbulent flow problems