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Abstract A novel technique based on the Full Orthogonalization Arnoldi (FOA) is proposed to perform Dynamic Mode Decomposition (DMD) for a sequence of snapshots. A modification to FOA is presented for situations where the matrix A is unknown, but the set of vectors { A i − 1 v 1 } i = 1 N are known. The modified FOA is the kernel for the proposed projected DMD algorithm termed, FOA based DMD. The proposed algorithm to compute DMD modes and eigenvalues i) does not require Singular Value Decomposition (SVD) for snapshot matrices X with κ 2 (X) ≪ 1 / ϵ m , where κ 2 (X) is the 2-norm condition number of the snapshot matrix and ϵ m is the relative round-off error or machine epsilon, ii) has an optional rank truncation step motivated by round off error analysis for snapshot matrices X with κ 2 (X) ≈ 1 / ϵ m , iii) requires only one snapshot at a time, thus making it a 'streaming' method even with the optional rank truncation step, iv) consumes less memory and requires less floating point operations to obtain the projected matrix than existing projected DMD methods and v) lends itself to easy parallelism as the main computational kernel involves only vector additions, dot products and matrix vector products. The new technique is therefore well-suited for DMD of large datasets on parallel computing platforms. We show both theoretically and using numerical examples that for FOA based DMD without rank truncation, the finite precision error in the computed projection of the linear mapping is O (ϵ m κ 2 (X)). The proposed method is also compared to existing projected DMD methods for computational cost, memory consumption and relative round off error. Error indicators are presented that are useful to decide when to stop acquiring new snapshots. The proposed method is applied to several examples of numerical simulations of fluid flow. Highlights • Novel algorithm – 'FOA based DMD' based on Arnoldi proposed for large datasets. • Algorithm is parallel and streaming. • Less memory and floating point operations than existing DMD methods. • Error indicators proposed to decide when to stop acquiring new snapshots. • Finite precision error analysis of FOA based DMD and other DMD methods performed. [ABSTRACT FROM AUTHOR]

Abstract We present a method that dynamically and efficiently performs connectivity calculations between many (O (10 5)) moving, unstructured overset meshes in parallel. In order to connect overset meshes, elements exterior to the solution domain must be removed from the simulation. In regions with many overlapping meshes, elements must be selectively removed to reduce redundancy while maintaining a solution over the entire domain. Around masked regions interpolation partner pairing is required between meshes to provide boundary conditions. For general unstructured meshes, these steps involve challenging computational geometry calculations which must be efficient and automatic. For many moving meshes each step must be massively parallelized and scalable to large numbers of computational cores. To establish communication patterns a parallelized master/slave algorithm is used which minimizes global communication and storage. To remove elements a parallel 'Forest Fire' flood-fill algorithm is used to set a masking variable. For interpolation partner pairing, and other necessary searches, k-dimensional tree data structures (k-d trees) are extensively used. Often in a calculation, the connectivity between overset meshes remains largely the same between time steps. The temporal coherence of the various objects in the connectivity calculation is directly used to only update necessary information with time, resulting in substantial cost savings. Details of the different algorithms are presented. Resulting connectivity and timings are shown for complex geometries. Parallel scaling is demonstrated for 100,000 spherical particles within a channel up to 492,000 processors. Highlights • Dynamic, parallel method for connectivity between unstructured overset grids. • Approaches for mesh cutting, interpolation partner pairing and parallel communication. • Scaling shown for 100,000 particles in a channel up to 492,000 processors. [ABSTRACT FROM AUTHOR]

A polar coordinate system introduces a singularity at the pole, r = 0 , where terms with a factor 1 / r can be ill-defined. While there are several approaches to eliminate this pole singularity in finite difference methods, finite volume methods largely bypass this issue by not storing or computing data at the pole. However, all methods face a very restrictive time step when using an explicit time advancement scheme in the azimuthal direction, where cell sizes are of the order O ( Δ r ( r Δ θ ) ) . We use a conservative finite volume approach of merging cells on a structured O-mesh to remove this time step limit imposed by the CFL condition. The cell-merging procedure is implemented as a corrector step and incurs no changes to the underlying data structure for a structured grid. This short note describes the procedure and presents the validation and application of the algorithm to various problems. The algorithm is shown to be inexpensive and scalable. In addition, the cell-merging procedure is easily coupled with a line implicit scheme in the radial direction. [ABSTRACT FROM AUTHOR]

Abstract: Purely dissipative eddy-viscosity subgrid models have proven very successful in large-eddy simulations (LES) at moderate resolution. Simulations at coarse resolutions where the underlying assumption of small-scale universality is not valid, warrant more advanced models. However, non-eddy viscosity models are often unstable due to the lack of sufficient dissipation. This paper proposes a simple modeling approach which incorporates the dissipative nature of existing eddy viscosity models into more physically appealing non-eddy viscosity SGS models. The key idea is to impose the SGS dissipation of the eddy viscosity model as a constraint on the non-eddy viscosity model when determining the coefficients in the non-eddy viscosity model. We propose a new subgrid scale model (RSEM), which is based on estimation of the unresolved velocity field. RSEM is developed in physical space and does not require the use of finer grids to estimate the subgrid velocity field. The model coefficient is determined such that total SGS dissipation matches that from a target SGS model in the mean or least-squares sense. The dynamic Smagorinsky model is used to provide the target dissipation. Results are shown for LES of decaying isotropic turbulence and turbulent channel flow. For isotropic turbulence, RSEM displays some level of backward dissipation, while yielding as good results as the dynamic Smagorinsky model. For channel flow, the results from RSEM are better than those from the dynamic Smagorinsky model for both statistics and instantaneous flow structures. [Copyright &y& Elsevier]

Abstract: This paper develops a dynamic error analysis procedure for the numerical errors arising from spatial discretization in large-eddy simulation. The analysis is based on EDQNM closure theory, and is applied to the LES of decaying isotropic turbulence. First, the effects of finite-differencing truncation error, aliasing error and the dynamic Smagorinsky model are independently considered. The time-evolution of kinetic energy and spectra predicted by the analysis are compared to actual LES using the Navier–Stokes equations, and good agreement is obtained. The analysis is then extended to simultaneously consider all sources of error in a second-order discretely energy conserving, central-difference LES solver. Good agreement between the analysis and actual LES is obtained. The analysis is used to compare the contribution of the subgrid model to that of numerical errors, and it is shown that the contribution of the subgrid scale model is much higher than the numerical errors. The proposed one-dimensional EDQNM-LES model shows potential as a more general tool for the analysis of numerical error, and SGS model in simulations of turbulent flow. [Copyright &y& Elsevier]

Abstract: A non-dissipative, robust, implicit algorithm is proposed for direct numerical and large-eddy simulation of compressible turbulent flows. The algorithm addresses the problems caused by low Mach numbers and under-resolved high Reynolds numbers. It colocates variables in space to allow easy extension to unstructured grids, and discretely conserves mass, momentum and total energy. The Navier–Stokes equations are non-dimensionalized using an incompressible scaling for pressure, and the energy equation is used to obtain an expression for the velocity divergence. A pressure-correction approach is used to solve the resulting equations, such that the discrete divergence is constrained by the energy equation. As a result, the discrete equations analytically reduce to the incompressible equations at very low Mach number, i.e., the algorithm overcomes the acoustic time-scale limit without preconditioning or solution of an implicit system of equations. The algorithm discretely conserves kinetic energy in the incompressible inviscid limit, and is robust for inviscid compressible turbulence on the convective time-scale. These properties make it well-suited for DNS/LES of compressible turbulent flows. Results are shown for acoustic propagation, the incompressible Taylor problem, periodic shock tube problem, and isotropic turbulence. [Copyright &y& Elsevier]

• Fully GPU accelerated unstructured overset method to simulate flows over moving bodies. • Novel overset assembly method inspired by real-time ray tracing algorithms. • Accurate and cost effective artificial compressibility pressure regularization shown. • Primal-dual reconstruction introduced for accurate calculations on general meshes. • Speedups of 15-50x found compared to benchmark overset solver for high cell loadings. Hardware acceleration consists of offloading computational work to devices such as graphics processing units (GPUs) to produce overall speed-up. Algorithms and numerical methods must be constructed to suit the available hardware in order to effectively produce speed-up. In this work a numerical method is presented which can effectively use hardware acceleration to simulate incompressible turbulent fluid flow. The method is an unstructured overset method where unstructured meshes are attached to individual bodies and connected throughout the flow domain to produce a single domain solution through an overset assembly process. The unstructured overset method shown in Horne and Mahesh [1] and Horne and Mahesh [2] was found capable of scaling to O(105) computational cores for O(105) moving bodies in turbulent flow fields while producing accurate flow results. This highly scalable method is modified and extended to effectively utilize on-node hardware acceleration. Overset assembly algorithms which use hardware acceleration are presented based on successful accelerated algorithms in real-time ray tracing and computational geometry. Timing results for core overset assembly operations are presented showing a maximum O(100x) speedup when using hardware acceleration. A novel method for turbulent fluid flow is presented which utilizes over-decomposition of the flow domain to produce task-parallelism allowing asynchronous calculation of the different steps of the method while also providing overlap between data transfer and computation. A mixed precision solver is utilized which provides a balance between optimal performance and numerical accuracy. A cost effective and accurate artificial compressibility pressure regularization is used which has minimal memory complexity and minimizes computational cost while maintaining accuracy. A primal-dual Laplacian operator is introduced which produces accurate results on skewed meshes. Results for canonical flow cases with overset meshes are shown illustrating the method's accuracy and numerical properties. Substantial speed-up is demonstrated for the numerical method reaching upwards of 50 times as fast as the non-accelerated method for high cell loadings. [ABSTRACT FROM AUTHOR]

The paper performs simulation of a rectangular plate excited by turbulent channel flow at friction Reynolds numbers of 180 and 400. The fluid–structure interaction is assumed to be one-way coupled, i.e, the fluid affects the solid and not vice versa. We solve the incompressible Navier–Stokes equations using finite volume direct numerical simulation in the fluid domain. In the solid domain, we solve the dynamic linear elasticity equations using a time-domain finite element method. The obtained plate averaged displacement spectra collapse in the low frequency region in outer scaling. However, the high frequency spectral levels do not collapse in inner units. This spectral behavior is reasoned using theoretical arguments. The resonant vibration is stronger at the third natural frequency than at the first natural frequency. We explain this behavior by comparing the fluid and solid length scales. We further study the sources of plate excitation using a novel formulation. This formulation expresses the average displacement spectrum of the plate as an integrated contribution from the fluid sources within the channel. Analysis of the sources reveals that at the plate natural frequencies, the contribution of the fluid sources to the plate excitation peaks in the buffer layer. The corresponding wall-normal width is found to be ≈ 0. 75 δ. The integrated contribution of the overlap and outer regions together to the plate response is comparable to that from the buffer region for R e τ = 180 and exceeds the buffer region contribution for R e τ = 400. We analyze the decorrelated features of the sources using spectral Proper Orthogonal Decomposition (POD) of the net displacement source. We enforce the orthogonality of the modes in an inner product with a symmetric positive definite kernel. The dominant spectral POD mode contributes to the entire plate excitation. The contribution of the remaining modes from the different wall-normal regions undergo destructive interference resulting in zero net contribution. The envelope of the dominant mode further shows that the intensity of the sources peaks in the buffer region and the wall-normal width of the sources extend well into the outer region of the channel. [ABSTRACT FROM AUTHOR]

An unstructured overset method capable of performing direct numerical simulation (DNS) and large eddy simulation (LES) of many (O (10 5)) moving bodies, utilizing many computational cores (O (10 5)), in turbulent, incompressible fluid flow is presented. Unstructured meshes are attached to bodies and placed within a fixed background domain. Body meshes are allowed to arbitrarily overlap and move throughout the domain. Within each mesh a high resolution, unstructured, non-dissipative finite volume method is used to solve for the flow field. Boundary conditions for each mesh are provided by interpolation from flow solutions on overlapping meshes. When many unstructured meshes of different resolution overlap, care is required in the connection between the different flow solutions. An interpolant is created which seeks to preserve volume conservation of flow quantities between meshes regardless of overlapping mesh differences. An implicit fractional step method is used for time advancement, requiring the calculation of a predicted fluid velocity and corrector pressure field. For the predictor step, the resulting interpolation is directly introduced into the implicit equations for the predicted flow field. For the corrected pressure field, the continuity between meshes is weakly enforced using a penalty formulation. The pressure formulation is symmetric, positive-definite and non-singular resulting in a formulation which is readily solvable using traditional iterative matrix inversion techniques. An Arbitrary Euler-Lagrangian (ALE) method coupled to a 6 degrees of freedom rigid body equation system (6-DOF) is used for body motion. For rotation, a quaternion representation is used to solve Euler's equations of rigid body motion. A linear spring damper model, which uses geometry information readily available from the overset assembly process, is used for collisions. Validation of the method for canonical flow fields is presented including assessment of order of accuracy and kinetic energy conservation properties. Particle-resolved direct numerical simulation (PR-DNS) of single particles in various flow fields are presented for validation. PR-DNS results of 50,000 spherical particles freely moving within turbulent channel flow are shown as a demonstration of the method at full scale. LES results of a marine propeller under crashback conditions are shown to demonstrate the ability to simulate highly unsteady turbulent flows over complex, moving geometries. • Unstructured overset method shown for LES/DNS of many (O (10 5)) moving bodies, using many cores (O (10 5)), in turbulent flow. • Novel overset supercell reconstruction introduced which yields superior kinetic energy properties to other reconstructions. • Novel penalty formulation shown that produces efficient linear systems to enforce pressure continuity between overset meshes. • LES of propeller crashback illustrates method's ability to simulate turbulent flow over complex moving geometry. • PR-DNS of 50,000 freely moving particles in a turbulent channel demonstrates the method at full scale. [ABSTRACT FROM AUTHOR]

Abstract: A spatially non-dissipative, implicit numerical method to simulate turbulent reacting flows over a range of Mach numbers, is described. The compressible Navier–Stokes equations are rescaled so that the zero Mach number equations are discretely recovered in the limit of zero Mach number. The dependent variables are co-located in space, and thermodynamic variables are staggered from velocity in time. The algorithm discretely conserves kinetic energy in the incompressible, inviscid, non-reacting limit. The chemical source terms are implicit in time to allow for stiff chemical mechanisms. The algorithm is readily extended to complex chemical mechanisms. Numerical examples using both simple and complex chemical mechanisms are presented. [Copyright &y& Elsevier]

• Robust variational level set methodology is proposed to capture equilibrium interface shapes over arbitrary rough surfaces. • Distance regularization is used instead of reinitialization. Mass conservation and no-penetration constraints are incorporated. • A harmonic mean based surface tension distribution is used that leads to accurate numerical results. • Parallel implementation validated with canonical level set test cases and analytical solutions of Gibbs free energy. • Applied to problems for the prediction of equilibrium interface shapes over grooves, posts and arbitrary surface roughness. A robust numerical methodology to predict equilibrium interfaces over arbitrary solid surfaces is developed. The kernel of the proposed method is the distance regularized level set equations (DRLSE) with techniques to incorporate the no-penetration and volume-conservation constraints. In this framework, we avoid reinitialization that is typically used in traditional level set methods. This allows for a more efficient algorithm since only one advection equation is solved, and avoids numerical error associated with the re-distancing step. A novel surface tension distribution, based on harmonic mean, is prescribed such that the zero level set has the correct liquid-solid surface tension value. This leads to a more accurate prediction of the triple contact point location. The method uses second-order central difference schemes which facilitates easy parallel implementation, and is validated by comparing to traditional level set methods for canonical problems. The application of the method in the context of Gibbs free energy minimization, to obtain liquid-air interfaces is validated against existing analytical solutions. The capability of the methodology to predict equilibrium shapes over both structured and realistic rough surfaces is demonstrated. [ABSTRACT FROM AUTHOR]