Vortex Problem Research Articles

Discretely Nonlinearly Stable Weight-Adjusted Flux Reconstruction High-Order Method for Compressible Flows on Curvilinear Grids

To achieve genuine predictive capability, an algorithm must consistently deliver accurate results over prolonged temporal integration periods, avoiding the unwarranted growth of aliasing errors that compromise the discrete solution. Provable nonlinear stability bounds the discrete approximation and ensures that the discretization does not diverge. Nonlinear stability is accomplished by satisfying a secondary conservation law, namely for compressible flows; the second law of thermodynamics. For high-order methods, discrete nonlinear stability and entropy stability, have been successfully implemented for discontinuous Galerkin (DG) and residual distribution schemes, where the stability proofs depend on properties of L2-norms. In this paper, nonlinearly stable flux reconstruction (NSFR) schemes are developed for three-dimensional compressible flow in curvilinear coordinates. NSFR is derived by merging the energy stable flux reconstruction (ESFR) framework with entropy stable DG schemes. NSFR is demonstrated to use larger time-steps than DG due to the ESFR correction functions, at the cost of larger error levels at design order convergence for equivalent degrees of freedom, while preserving discrete nonlinear stability. NSFR differs from ESFR schemes in the literature since it incorporates the FR correction functions on the volume terms through the use of a modified mass matrix. We also prove that discrete kinetic energy stability cannot be preserved to machine precision for quadrature rules where the surface quadrature is not a subset of the volume quadrature. This result stems from the inverse mapping from the kinetic energy variables to the conservative variables not existing for the kinetic energy projected variables. This paper also presents the NSFR modified mass matrix in a weight-adjusted form. This form reduces the computational cost in curvilinear coordinates because the dense matrix inversion is approximated by a pre-computed projection operator and the inverse of a diagonal matrix on-the-fly and exploits the tensor product basis functions to utilize sum-factorization. The nonlinear stability properties of the scheme are verified on a nonsymmetric curvilinear grid for the inviscid Taylor-Green vortex problem and the correct orders of convergence were obtained on a curvilinear mesh for a manufactured solution. Lastly, we perform a computational cost comparison between conservative DG, overintegrated DG, and our proposed entropy conserving NSFR scheme, and find that our proposed entropy conserving NSFR scheme is computationally competitive with the conservative DG scheme.

Dynamics of a restricted (6 + 1)-vortex problem

This paper investigates the dynamics of a restricted (6 + 1)-vortex problem. The existence of nested equilateral triangle configurations depends on the position of two triangles, which can be divided into two situations: staggered arrangement and symmetrical arrangement. Firstly, we provide the existence conditions for the configuration and further discuss the stability of the configuration numerically. Secondly, we have established the global dynamics of the restricted point-vortex problem with configurations under different arrangements, including the dynamic behavior of equilibrium points, periodic orbits, homoclinic orbits, and heteroclinic orbits.

Dynamics of the Restricted $$(N+1)$$-Vortex Problem with a Regular Polygon Distribution

Dynamics of the Restricted $$(N+1)$$-Vortex Problem with a Regular Polygon Distribution

A new compact scheme-based Lax–Wendroff method for high fidelity simulations

The classical Lax–Wendroff (LW) method employs explicit second order central difference schemes to solve partial differential equations. Recently, the authors in Sengupta et al. (2023) have identified numerical parameter ranges in performing large eddy simulation (LES) using this classical LW method. In a bid to improve these numerical parameter ranges for high fidelity computations, a new non-uniform grid based compact scheme for LW method (NUCLW) is developed here. This new method uses sixth order non-uniform compact scheme (NUC6) for the first and higher order derivatives. Reported global spectral analysis confirms substantial improvement in the resolution and accuracy of NUCLW scheme over the classical LW scheme. A significant advantage of the present scheme is its ability to compute solutions for non-uniform, structured grids, and that does not suffer degradation in accuracy. The potential of the NUCLW scheme for high accuracy computations is demonstrated by solving 2D incompressible Navier–Stokes equations for the benchmark unsteady flow inside a square lid driven cavity problem for a subcritical Reynolds number of 3200 and a post-critical Reynolds number of 10,000 and simulating the instability of the Taylor–Green vortex problem. These demonstrate the ability of the NUCLW scheme to solve the Navier–Stokes equations for high fidelity simulations such as LES/DNS with enhanced ranges of numerical parameters with respect to enhanced accuracy and faster computations.

Convergence analysis of high-order IMEX-BDF schemes for the incompressible Navier–Stokes equations

In this paper, we consider developing high-order temporal integration schemes for the unsteady incompressible Navier–Stokes equations in bounded two-dimensional domain subjected to the periodic boundary conditions. Utilizing the k-step (k=3,4,5) backward differentiation formula (BDF) coupled with the implicit–explicit (IMEX) treatment of the nonlinear convective term in an anti-symmetry form, a class of IMEX-BDFk schemes up to fifth-order in time are constructed and analyzed. By imposing a zero-mean constrain on the finite-dimensional space for the pressure, the proposed numerical schemes are proven to be uniquely solvable. Based on the recent theoretical framework consisting of a class of discrete orthogonal convolution kernels, rigorous L2 norm error estimates for both the velocity and the pressure are established by using a novel divergence free projection system. The proposed schemes are then implemented in two benchmark experiments, including a Taylor–Green vortex problem and a double shear layer flow at various high Reynolds numbers. Numerical results demonstrate the expected solution accuracy and the computational effectiveness in simulating the realistic flow dynamics.

A Hybrid Computer Vision and Machine Learning Approach for Robust Vortex Core Detection in Fluid Mechanics Applications

Abstract Vortex core detection remains an unsolved problem in the field of experimental and computational fluid dynamics. Available methods such as the Q, delta, and swirling strength criterion are based on a decomposed velocity gradient tensor but detect spurious vortices (false positives and false negatives), making these methods less robust. To overcome this, we propose a new hybrid machine learning approach in which we use a convolutional neural network to detect vortex regions within surface streamline plots and an additional deep neural network to detect vortex cores within identified vortex regions. Furthermore, we propose an automatic labeling approach based on K-means clustering to preprocess our input images. We show results for two classical test cases in fluid mechanics: the Taylor–Green vortex problem and two rotating blades. We show that our hybrid approach is up to 2.6 times faster than a pure deep neural network-based approach and furthermore show that our automatic K-means clustering labeling approach achieves within 0.45% mean square error of the more labour-intensive, manual labeling approach. At the same time, by using a sufficient number of samples, we show that we are able to reduce false positives and negatives entirely and thus show that our hybrid machine learning approach is a viable alternative to currently used vortex detection tools in fluid mechanics applications.

Gas-kinetic scheme for partially ionized plasma in hydrodynamic regime

Most plasmas are only partially ionized. To better understand the dynamics of these plasmas, the behaviors of a mixture of neutral species and plasma in ideal magnetohydrodynamic states are investigated. The current approach is about the construction of coupled kinetic models for the neutral gas, electron, and proton, and the development of the corresponding gas-kinetic scheme (GKS) for the solution in the continuum flow regime. The scheme is validated in the 1D Riemann problem for an enlarged system with the interaction from the Euler waves of the neutral gas and magnetohydrodynamic ones of the plasma. Additionally, the Orszag–Tang vortex problem across different ionized states is tested to examine the influence of neutrals on the MHD wave evolution. These tests demonstrate that the proposed scheme can capture the fundamental features of ideal partially ionized plasma, and a transition in the wave structure from the ideal MHD solution of the fully ionized plasma to the Euler solution of the neutral gas is obtained.

An immersed boundary velocity correction method combined with virtual body-fitted grid for simulation of incompressible flows

In this work, a virtual body-fitted grid is introduced into the velocity correction-based immersed boundary method (IBM) to simulate incompressible flows. The impact of the immersed boundary is indirectly transmitted to the flow field via a virtual body-fitted grid. In this method, the fractional step technique consisting of the predictor and the corrector is adopted. The prediction step is executed on the Eulerian mesh, and the correction step is done on the virtual grid to fulfill the no-slip boundary condition. After the correction step, the corrected velocity field on the virtual grid is then assigned to that on the Eulerian mesh to update the flow field. Being able to adjust the grid spacing flexibly, the virtual body-fitted grid alleviates the shortcomings of the conventional IBM that uses the smooth Dirac delta function to associate Lagrangian points with their surrounding Eulerian points. As a result, the present method is easy to apply to non-uniform Cartesian grids, which is inapplicable to the conventional IBM with the smooth Dirac delta function. Numerical experiments concerning flow past a circular cylinder and a NACA0012 airfoil demonstrate the advantages of the present method, i.e., fewer Lagrangian points are required to avoid the streamline penetration of boundary and the range of “diffuse interface” can be narrowed by reducing the normal grid spacing of the virtual body-fitted grid to improve numerical results on a coarse mesh. In addition, an accuracy assessment on the decaying vortex problem reveals that the present IBM has a second-order accuracy.

Advancing Finite Difference Solutions for Two‐Dimensional Incompressible Navier–Stokes Equations Using Artificial Compressibility Method and Sparse Matrix Computation

In this article, a numerical scheme for solving two‐dimensional (2D) time‐dependent incompressible Navier–Stokes equations is presented. The artificial compressibility technique is used to incorporate a time derivative of pressure term to the continuity equation. It is employed for pressure–velocity coupling. The scheme consists of backward difference approximation for time derivatives and central difference approximation for spatial derivatives, implemented on a collocated grid. The discretization of the differential equations yields a system of algebraic equations with a block coefficient matrix. To solve this system efficiently, matrix inversion with sparse matrix computation is employed. The proposed numerical scheme is applied to solve three flow problems (lid‐driven cavity flow, rectangular channel flow, and Taylor–Green vortex problem) to validate the accuracy and applicability of the scheme. The results affirm the scheme’s capability to provide precise approximations for solutions to the Navier–Stokes equations. With slight modifications, the scheme can be applied to solve various flow problems with high accuracy, less memory usage, and reduced computational time.

Approximate deconvolution discretisation

A new strategy is presented for the construction of high-order spatial discretisations extracted from a lower-order basic discretisation. The key consideration is that any spatial discretisation of a derivative of a solution can be expressed as the exact differentiation of a corresponding ‘filtered’ solution. Hence, each numerical discretisation method may be directly linked to a unique spatial filter, expressing the truncation error of the basic method. By approximately deconvolving the implied filter of the basic numerical discretisation an augmented high-order method can be obtained. In fact, adopting a deconvolution of the implied filter of suitable higher order enables the formulation of a new spatial discretisation method of correspondingly higher order. This construction is illustrated for finite difference (FD) discretisation schemes, solving partial differential equations in fluid mechanics. Knowing the implied filter of the basic discretisation, one can derive a corresponding higher order method by approximately eliminating the implied spatial filter to a certain desired order. We use deconvolution to compensate for the implied filter. This corresponds to a ‘sharpening’ of numerical solution features before the application of the basic FD method. The combination will be referred to as Approximate Deconvolution Discretisation (ADD). The accuracy of the deconvolved FD scheme depends on the order of approximation of the deconvolution filter. We present the ‘sharpening’ of several well-known FD operators for first- and second-order derivatives and quantify the achieved accuracy in terms of the modified wavenumber spectrum. Examples include high-order extensions up to new schemes with spectral accuracy. The practicality of the deconvolved FD schemes is illustrated in various ways: (i) by investigation of exactly solvable advection and diffusion problems, (ii) by tracking the evolution of the numerical solution to the Taylor-Green vortex problem and (iii) by showing that ADD yields spectral accuracy for the Burgers equation and for double-jet flow of an incompressible fluid.

Inverse distance weighting interpolation-based immersed boundary velocity correction method for incompressible flows

In this work, the inverse distance weighting (IDW) interpolation is introduced into the implicit velocity correction-based immersed boundary method (IBM) for simulation of incompressible flows. In the original implicit velocity correction-based IBM, the solid body must be immersed in a uniform mesh region due to the use of the smooth Dirac delta function, which is utilized to associate Lagrangian points with their surrounding Eulerian points and only works with uniform meshes. The IDW method has the advantage that the interpolation range can be set flexibly. The introduction of the IDW interpolation can extend the application of the IBM to non-uniform meshes while reducing the number of Lagrangian points. The numerical test by the decaying vortex problem proves that the IDW interpolation does not significantly affect the overall accuracy of the IBM. In addition, numerical experiments for the flows around a circular cylinder and a NACA0012 airfoil demonstrate the advantages of the proposed method, including allowing fewer Lagrangian points while ensuring no streamline penetration to the solid body, as well as its adaptability to non-uniform meshes which can improve the computational efficiency due to the use of fewer mesh points. Finally, the simulation of the flow past a stationary sphere illustrates that the proposed method can effectively simulate the three-dimensional flow.

Point vortex dynamics on Kähler twistor spaces

In this paper, we provide tools to study the dynamics of point vortex dynamics on CPn and the flag manifold F1,2(C3). These are the only Kähler twistor spaces arising from four-manifolds. We give an explicit expression for Green’s function on CPn, which enables us to determine the Hamiltonian H and the equations of motions for the point vortex problem on CPn. Moreover, we determine the momentum map μ:F1,2(C3)→su*(3) on the flag manifold.

Jump penalty stabilization techniques for under-resolved turbulence in discontinuous Galerkin schemes

Jump penalty stabilization techniques for under-resolved turbulence have been recently proposed for continuous and discontinuous high order Galerkin schemes [1–3]. The stabilization relies on the gradient or solution discontinuity at element interfaces to incorporate localised numerical diffusion in the numerical scheme. This diffusion acts as an implicit subgrid model and stabilizes under-resolved turbulent simulations.This paper investigates the effect of jump penalty stabilization methods (penalising gradient or solution) for stabilization and improvement of high-order discontinuous Galerkin schemes in turbulent regime. We analyze these schemes using an eigensolution analysis, a 1D non-linear Burgers equation (mimicking a turbulent cascade) and 3D turbulent Navier-Stokes simulations (Taylor-Green Vortex and Kelvin-Helmholtz Instability problems).We show that the two jump penalty stabilization techniques can stabilize under-resolved simulations thanks to the improved dispersion-dissipation characteristics (when compared to non-penalized schemes) and provide accurate results for turbulent flows. The numerical results indicate that the proposed jump penalty methods stabilize under-resolved simulations and improve the simulations, when compared to the original unpenalized scheme and to classic explicit subgrid models (Smagorinsky and Vreman).

Compact moving particle semi-implicit method for incompressible free-surface flow

Recently, consistent meshfree particle methods have been intensively studied. It has been pointed out that numerical inaccuracy or instability could easily occur with incomplete or biased neighbor support. This study proposes a new meshfree particle method called the compact moving particle semi-implicit (CMPS) method to decrease the condition number. In the proposed CMPS, the first-order and second-order derivatives are discretized separately, enhancing the numerical stability significantly. By adopting a small dilation parameter of the compact support, the CMPS can remarkably improve accuracy and reduce computational costs. Formulations for zeroth-order, first-order, and second-order derivatives are derived, and various boundary conditions, e.g., Dirichlet and Neumann, are discussed. In order to better deal with complex free-surface flows using the CMPS, some new numerical techniques, i.e., optimized regularization and reconstructed particle shifting schemes, are also developed. Furthermore, the surface fitting method is extended to address the surface tension. A convergence study is conducted in complex geometry to verify the stability, accuracy, and efficiency of the CMPS. Then, second-order accuracy is confirmed using the Taylor–Green vortex problem. After that, numerical examples concerning various free-surface flows, including square patch, hydrostatic pressure, dam break, droplet oscillation, and droplet coalescence, are calculated to demonstrate the potential of the CMPS.

Physical/numerical duality of explicit/implicit subgrid-scale modelling

The development of implicit approaches has prompted debate on the actual usefulness of any explicit subgrid-scale modelling in large-eddy simulation. This question is addressed here by considering two generic turbulent flows: (i) the Taylor-Green vortex problem; (ii) the pipe flow. For both flow configurations, implicit modelling is found to overtake the very popular Smagorinsky model. To understand this robust observation, an analysis in the Fourier space is presented for the Taylor-Green vortex problem. The concept of spectral eddy viscosity, widely used in the pioneer work of Marcel Lesieur in two-point closure and subgrid-scale modelling, is revisited in a general framework based on explicit/implicit subgrid-scale modelling. In particular, the essentially anisotropic nature of implicit modelling is exhibited, as a favourable feature in terms of consistency with the computational mesh. Smagorinsky's model, considered as a generic explicit subgrid-scale model in the framework of Boussinesq's hypothesis, is found to be highly sensitive to numerical errors. Removing the latter is easy but makes computationally inefficient this type of explicit modelling. Comparisons between a priori and a posteriori spectral eddy viscosities show that neither Smagorinsky's model nor implicit modelling can mimic the expected spectral behaviour. Smagorinsky's model is observed to be weakly scale-selective with a poor ability to actually filter the solution. The feature of scale-selectivity is well replicated by implicit modelling which exhibits excellent capabilities for filtering. However, its lack of influence at the largest scales is against the expected behaviour for the spectral eddy viscosity at low wavenumber through the establishment of a non-zero plateau value. This lack of consistency of implicit LES could be overcome thanks to an extra explicit modelling but the attempt to mix Smagorinsky's model and implicit LES is not successful in this study. The potential of implicit large-eddy simulation is also exhibited for the accurate computation of near-wall turbulence inside a pipe flow despite the use of a regular Cartesian mesh with an immersed boundary method. Interestingly, the resulting coarse wall-normal resolution in the near-wall direction does not prevent the reliable prediction of statistical profiles up to the capture of subgrid-scale details. It is suggested that the regularisation associated with implicit modelling is a necessary condition to reach numerical accuracy. However, to faithfully represent the large-scale dynamics, present results confirm that non-local triad interactions must be taken into account as widely discussed in the inspiring textbook Lesieur [Turbulence in fluids. 4th ed. Springer; 2008] of Marcel Lesieur.

Assessing Saiph, a task-based DSL for high-performance computational fluid dynamics

Scientific applications face the challenge of efficiently exploiting increasingly complex parallel and distributed systems. Developing hand-tuned codes is a time-consuming, tedious and hardly reusable task. Reaching high performance appears detrimental to productivity and portability and unreasonable to expect from scientists. Domain-Specific Languages (DSLs) are collaborative environments aiming to overcome such difficulties by decoupling the problem description from the algorithmic implementation. However, developing a competitive tool in High-Performance Computing (HPC) is challenging: DSLs for HPC environments have two additional critical requirements, performance and scalability. Moreover, documented and successful cases are few, making it difficult to popularise DSLs as problem-solving environments for scientific HPC code development. In this context, Saiph is a task-based DSL easing the simulation of physical phenomena from Computational Fluid Dynamics (CFD), developed to meet HPC productivity and performance requirements. This work reports the tuning and evaluation of Saiph using the Taylor–Green Vortex (TGV) problem as a case study. We assess Saiph’s productivity, numerical methods, and high-performance strategies to illustrate its use and demonstrate its competitiveness, viability and benefits for CFD software developments in HPC environments. Hence, we contribute to the popularisation of HPC DSLs as suitable problem-solving environments able to unify modern computational and scientific knowledge.

A parallel unstructured multi-color SOR solver for 3D Navier–Stokes equations on graphics processing units

In this paper, we develop a GPU parallelization technique to accelerate the pressure Poisson solver. We apply it in three-dimensional Navier–Stokes simulations with arbitrary geometries. The numerical method is based on a projection method, and the discretization is obtained by combining a second-order unstructured finite-volume method and a σ transformation. The pressure solution is responsible for this formulation’s most time-consuming computation tasks. This is because the Poisson equation must be approximated. This work proposes a GPU-based Multi-Color Successive Over-Relaxation (MC-SOR) method to solve the resulting sparse linear system. The proposed parallel finite-volume code is verified and validated by solving the Poisson problem. We employ analytical solutions and well-known benchmark problems with complex flows. These include the Taylor–Green vortex problem and the problem of turbulent flow around a circular cylinder.

Motion of a superfluid vortex according to holographic quantum dissipation

Vortices are topological defects associated with superfluids and superconductors, which, when mobile, dissipate energy destroying the dissipationless nature of the superfluid. The nature of this ``quantum dissipation'' is rooted in the quantum physical nature of the problem, which has been the subject of an extensive literature. However, this has mostly been focused on the measures applicable in weakly interacting systems wherein they are tractable via conventional methods. Recently, it became possible to address such dynamical quantum thermalization problems in very strongly interacting systems using the holographic duality discovered in string theory, mapping the quantum problem on a gravitational problem in one higher dimension, having as benefit offering a more general view on how dissipation emerges from such intricate quantum physical circumstances. We study here the elementary problem of a single vortex in two space dimensions, set in motion by a sudden quench in the background superflow formed in a finite-density Reissner-Nordstrom holographic superfluid. This reveals a number of surprising outcomes addressing questions of principle. By fitting the trajectories unambiguously to the Hall-Vinen-Iordanskii phenomenological equation of motion we find that these are characterized by a large inertial mass at low temperature that, however, diminishes upon raising temperature. For a weak drive the drag is found to increase when temperature is lowered, which reveals a simple shear drag associated with the viscous metallic vortex cores, supplemented by a conventional normal fluid component at higher temperatures. For a strong drive we discover a unique dynamical phenomenon: the core of the vortex deforms accompanied by a large increase of the drag force.

High-Order WENO-based Semi-Implicit Projection Method for Incompressible Turbulent Flows: Development, Accuracy, and Reynolds Number Effects

A novel high-order projection method is presented in this study for the simulation of incompressible turbulent flows, utilising weighted essentially non-oscillatory (WENO) schemes. Spatial discretization involves using WENO schemes for nonlinear convection and standard central differences (CD) for the viscous term. The approach combines various orders of WENO and central difference schemes (WENO3/CD2, WENO5/CD4, and WENO7/CD6) to achieve third, fifth, and seventh-order spatial accuracy for velocity fields, respectively. Validating this method involved a thorough examination using the 2D Taylor Green vortex problem in both space and time, along with simulations of 3D Taylor-Green vortex problems at increasing Reynolds numbers (Re) of 1, 600, 16, 000, and 160, 000. The outcomes encompassed analyses of kinetic energy, enstrophy, energy spectra, and vortex fields, indicating the method's effectiveness in accurately simulating high Reynolds number turbulent flows without additional sub-grid scale models.

A superconsistent collocation method for high Reynolds number flows

We use a novel implicit second-order projection method together with a superconsistent collocation scheme (Funaro, 1993; Fatone et al., 2005; De l’Isle and Owens, 2021) for the solution of the primitive variable formulation of the Navier–Stokes equations at very high Reynolds numbers. In particular, we apply a superconsistent collocation scheme to the convection–diffusion equation arising from one step of the projection method and this represents the first time that superconsistent collocation methods have been employed for the solution of a convection–diffusion equation having unsteady convection velocity. In order to evidence the second-order (in space and time) convergence of our scheme for both the velocity and pressure fields we choose to solve the two-dimensional unsteady Taylor–Green vortex problem (Taylor and Green, 1937). The numerical results presented for the solution of the two-dimensional square lid-driven cavity problem are in excellent agreement over the whole range of Reynolds numbers considered (5000≤Re≤1000) with some others in the literature (Ghia et al., 1982; Wang and Liu, 2019; Bruneau and Saad, 2006; Auteri et al., 2002; Pan and Glowinski, 2000).