Vortex Problem Research Articles

Morse Theory and Relative Equilibria in the Planar n-Vortex Problem

Morse theoretical ideas are applied to the study of relative equilibria in the planar $n$-vortex problem. For the case of positive circulations, we prove that the Morse index of a critical point of the Hamiltonian restricted to a level surface of the angular impulse is equal to the number of pairs of real eigenvalues of the corresponding relative equilibrium periodic solution. The Morse inequalities are then used to prove the instability of some families of relative equilibria in the four-vortex problem with two pairs of equal vorticities. We also show that, for positive circulations, relative equilibria cannot accumulate on the collision set.

Development of an Ideal Magnetohydrodynamics Flowsolver for High Speed Flow Control

This paper presents the baseline development of an ideal magnetohydrodynamics (MHD) solver towards enhancing the knowledge base on the numerical and flow physics complexities associated with MHD flows. The ideal MHD governing equations consisting of the coupled fluid flow equations and the Maxwell’s equations of electrodynamics are implemented in the three dimensional finite volume flowsolver, CERANS. Upwind flux functions such as AUSM-PW+, KFVS and the local Lax-Friedrichs schemes were used for solving the discretized form of governing equations. The solenoidal constraint which requires that the magnetic field to be divergence free all through the flow field evolution is ensured using the artificial compressibility analogy method or the Powell’s source term method. The code had been validated for standard MHD test cases involving complex flowfields such as the MHD shock tube, blast, vortex, cloud-shock interaction and cylinder shock interaction problems. The flow control effect of MHD interaction had been demonstrated for supersonic flow past a wedge and the results are compared with analytical results obtained by solving the MHD Rankine Hugoniot relations. Further, MHD flow control for high speed flows had been demonstrated for the hypersonic blunt body problem. Through rigorous testing and validation, it is observed that the CERANS-MHD code is able to mimic the complex flows due to MHD interactions and the comparison of results are found to be in good agreement with similar literature.

Hybrid sixth order spatial discretization scheme for non-uniform Cartesian grids

A class of high accuracy compact schemes used to solve wave problems and the Navier Stokes equation (NSE) on a non-uniform Cartesian grid is presented here. Global spectral analysis (GSA) performed with the help of model one-dimensional (1D) convection equation reveals that the scheme has excellent dispersion relation preserving properties and scale resolution. The developed analysis tool is used to provide resolution and numerical properties for grids with randomly varying spacing between the nodes, for the first time.The results of the benchmark problem of two-dimensional (2D) convection equation are used to validate with exact solution. Two-dimensional NSE is also solved for (i) square lid-driven cavity (LDC) at different Reynolds numbers and (ii) Taylor-Green vortex problem, as evidences of effectiveness and accuracy of the new scheme. These establish the robustness of the non-uniform high order compact scheme developed here for simulations of fluid flow and wave phenomena.

A high‐order flux reconstruction adaptive mesh refinement method for magnetohydrodynamics on unstructured grids

SummaryWe report our recent development of the high‐order flux reconstruction adaptive mesh refinement (AMR) method for magnetohydrodynamics (MHD). The resulted framework features a shock‐capturing duo of AMR and artificial resistivity (AR), which can robustly capture shocks and rotational and contact discontinuities with a fraction of the cell counts that are usually required. In our previous paper, we have presented a shock‐capturing framework on hydrodynamic problems with artificial diffusivity and AMR. Our AMR approach features a tree‐free, direct‐addressing approach in retrieving data across multiple levels of refinement. In this article, we report an extension to MHD systems that retains the flexibility of using unstructured grids. The challenges due to complex shock structures and divergence‐free constraint of magnetic field are more difficult to deal with than those of hydrodynamic systems. The accuracy of our solver hinges on 2 properties to achieve high‐order accuracy on MHD systems: removing the divergence error thoroughly and resolving discontinuities accurately. A hyperbolic divergence cleaning method with multiple subiterations is used for the first task. This method drives away the divergence error and preserves conservative forms of the governing equations. The subiteration can be accelerated by absorbing a pseudo time step into the wave speed coefficient, therefore enjoys a relaxed CFL condition. The AMR method rallies multiple levels of refined cells around various shock discontinuities, and it coordinates with the AR method to obtain sharp shock profiles. The physically consistent AR method localizes discontinuities and damps the spurious oscillation arising in the curl of the magnetic field. The effectiveness of the AMR and AR combination is demonstrated to be much more powerful than simply adding AR on finer and finer mesh, since the AMR steeply reduces the required amount of AR and confines the added artificial diffusivity and resistivity to a narrower and narrower region. We are able to verify the designed high‐order accuracy in space by using smooth flow test problems on unstructured grids. The efficiency and robustness of this framework are fully demonstrated through a number of two‐dimensional nonsmooth ideal MHD tests. We also successfully demonstrate that the AMR method can help significantly save computational cost for the Orszag‐Tang vortex problem.

A high-order semi-explicit discontinuous Galerkin solver for 3D incompressible flow with application to DNS and LES of turbulent channel flow

We present an efficient discontinuous Galerkin scheme for simulation of the incompressible Navier–Stokes equations including laminar and turbulent flow. We consider a semi-explicit high-order velocity-correction method for time integration as well as nodal equal-order discretizations for velocity and pressure. The non-linear convective term is treated explicitly while a linear system is solved for the pressure Poisson equation and the viscous term. The key feature of our solver is a consistent penalty term reducing the local divergence error in order to overcome recently reported instabilities in spatially under-resolved high-Reynolds-number flows as well as small time steps. This penalty method is similar to the grad–div stabilization widely used in continuous finite elements. We further review and compare our method to several other techniques recently proposed in literature to stabilize the method for such flow configurations. The solver is specifically designed for large-scale computations through matrix-free linear solvers including efficient preconditioning strategies and tensor-product elements, which have allowed us to scale this code up to 34.4 billion degrees of freedom and 147,456 CPU cores. We validate our code and demonstrate optimal convergence rates with laminar flows present in a vortex problem and flow past a cylinder and show applicability of our solver to direct numerical simulation as well as implicit large-eddy simulation of turbulent channel flow at Reτ=180 as well as 590.

Robust second-order scheme for multi-phase flow computations

A robust high-order scheme for the multi-phase flow computations featuring jumps and discontinuities due to shock waves and phase interfaces is presented. The scheme is based on high-order weighted-essentially non-oscillatory (WENO) finite volume schemes and high-order limiters to ensure the maximum principle or positivity of the various field variables including the density, pressure, and order parameters identifying each phase. The two-phase flow model considered besides the Euler equations of gas dynamics consists of advection of two parameters of the stiffened-gas equation of states, characterizing each phase. The design of the high-order limiter is guided by the findings of Zhang and Shu (2011) [36], and is based on limiting the quadrature values of the density, pressure and order parameters reconstructed using a high-order WENO scheme. The proof of positivity-preserving and accuracy is given, and the convergence and the robustness of the scheme are illustrated using the smooth isentropic vortex problem with very small density and pressure. The effectiveness and robustness of the scheme in computing the challenging problem of shock wave interaction with a cluster of tightly packed air or helium bubbles placed in a body of liquid water is also demonstrated. The superior performance of the high-order schemes over the first-order Lax-Friedrichs scheme for computations of shock-bubble interaction is also shown. The scheme is implemented in two-dimensional space on parallel computers using message passing interface (MPI). The proposed scheme with limiter features approximately 50% higher number of inter-processor message communications compared to the corresponding scheme without limiter, but with only 10% higher total CPU time. The scheme is provably second-order accurate in regions requiring positivity enforcement and higher order in the rest of domain.

Vortex characteristics in Fourier and non-Fourier heat conduction based on heat flux rotation

Vortex is one of the most important characteristics for flow field, but for heat conduction, vortex problems have not been studied. In this paper, heat vortex is mainly investigated from the viewpoint of heat flux rotation. The heat flux rotation is calculated and discussed for Fourier’s law and typical non-Fourier heat conduction models, including the Cattaneo-Vernotte (CV) model, Jeffrey model, Guyer-Krumhansl (GK) model and dual-phase-lagging (DPL) model. It is found that the change rule of heat flux rotation is only determined by the heat conduction law, and the change rule can also reflect certain relaxation process in non-Fourier heat conduction. These conclusions can provide a perspective for proving non-Fourier heat conduction, which is more rigorous than heat wave phenomena or finite propagation velocity. The heat flux rotation can also shows particular characteristics of non-Fourier heat conduction, which is quite different from mechanical phenomena. Different from the heat flux rotation, the entropy flux rotation for non-Fourier heat conduction does not change in an established rule.

A density-based method with semi-discrete central-upwind schemes for ideal magnetohydrodynamics

In the present work, the central-upwind schemes proposed by Kurganov et al. (SIAM J Sci Comput 23:707–740, 2001) for hydrodynamics are extended and combined with the divergence cleaning method of Dedner (J Comput Phys 175:645–673, 2002) in order to approximate the equations of the ideal magnetohydrodynamics in a finite volume discretization framework with Gaussian integration. To improve the quality of the solution, the Van Leer interpolation scheme is used. The accuracy and the robustness of the obtained solver are demonstrated through numerical simulations of benchmark problems such as the Brio–Wu shock tube problem, the Orszag–Tang vortex problem and the 2D cloud–shock interaction problem.

Global continua of periodic solutions of singular first-order Hamiltonian systems of N-vortex type

The paper deals with singular first order Hamiltonian systems of the form \[ \Gamma_k\dot{z}_k(t)=J\nabla_{z_k} H\big(z(t)\big),\quad z_k(t) \in \Omega \subset \mathbb{R}^2,\ k=1,\dots,N, \] where $J\in\mathbb{R}^{2\times2}$ defines the standard symplectic structure in $\mathbb{R}^2$, and the Hamiltonian $H$ is of $N$-vortex type: \[ H(z_1,\dots,z_N) = -\frac1{2\pi} \sum_{j\neq k=1}^N \Gamma_j \Gamma_k \log|z_j-z_k| - F(z). \] This is defined on the configuration space $\{(z_1,\ldots,z_N)\in \Omega^{2N}:z_j\neq z_k\text{ for }j\neq k\}$ of $N$ different points in the domain $\Omega\subset\mathbb{R}^2$. The function $F:\Omega^N\to\mathbb{R}$ may have additional singularities near the boundary of $\Omega^N$. We prove the existence of a global continuum of periodic solutions $z(t)=(z_1(t),\dots,z_N(t))\in\Omega^N$ that emanates, after introducing a suitable singular limit scaling, from a relative equilibrium $Z(t)\in\mathbb{R}^{2N}$ of the $N$-vortex problem in the whole plane (where $F=0$). Examples for $Z$ include Thomson's vortex configurations, or equilateral triangle solutions. The domain $\Omega$ need not be simply connected. A special feature is that the associated action integral is not defined on an open subset of the space of $2\pi$-periodic $H^{1/2}$ functions, the natural form domain for first order Hamiltonian systems. This is a consequence of the singular character of the Hamiltonian. Our main tool in the proof is a degree for $S^1$-equivariant gradient maps that we adapt to this class of potential operators.

Numbers of Relative Equilibria in the Planar Four-Vortex Problem: Some Special Cases

Three planar four-vortex problems are considered in this paper. In the $$(3+1)$$ -vortex problem, we study the relative equilibria of the four point vortices when one vortex has zero vorticity and the other three with nonzero vorticities form an equilateral triangle. In the $$(1+3)$$ -vortex problem, we study the limiting cases of the relative equilibria when one of the four point vortices has fixed nonzero vorticity and other vorticities approach zero. The third problem is the case of vanishing total vorticity. All problems involve two real vorticity parameters. We consider all meaningful pairs of parameters and find there can only be 4, 8, 9 or 10 relative equilibria in the $$(3+1)$$ -vortex problem, and 8, 10, 12 or 14 relative equilibria in the $$(1+3)$$ -vortex problem. For the case of zero total vorticity, there are 0, 1 or 2 collinear relative equilibria and 2, 3 or 4 strictly planar relative equilibria. We completely classify parameters according to the different numbers of relative equilibria. For all cases, we reduce them to the problems of counting common zeros in an open region of $${{{\mathbb {R}}}}^{2}$$ for polynomial systems with two equations, two variables, and two parameters. We propose a method to count zeros for such type of systems for all parameters in an open region of $${\mathbb {R}}^{2}$$ through symbolic computations. Therefore, all of our results are proved rigorously.

IMPROVED PERFORMANCES IN SUBSONIC FLOWS OF AN SPH SCHEME WITH GRADIENTS ESTIMATED USING AN INTEGRAL APPROACH

ABSTRACT In this paper, we present results from a series of hydrodynamical tests aimed at validating the performance of a smoothed particle hydrodynamics (SPH) formulation in which gradients are derived from an integral approach. We specifically investigate the code behavior with subsonic flows, where it is well known that zeroth-order inconsistencies present in standard SPH make it particularly problematic to correctly model the fluid dynamics. In particular, we consider the Gresho–Chan vortex problem, the growth of Kelvin–Helmholtz instabilities, the statistics of driven subsonic turbulence and the cold Keplerian disk problem. We compare simulation results for the different tests with those obtained, for the same initial conditions, using standard SPH. We also compare the results with the corresponding ones obtained previously with other numerical methods, such as codes based on a moving-mesh scheme or Godunov-type Lagrangian meshless methods. We quantify code performances by introducing error norms and spectral properties of the particle distribution, in a way similar to what was done in other works. We find that the new SPH formulation exhibits strongly reduced gradient errors and outperforms standard SPH in all of the tests considered. In fact, in terms of accuracy, we find good agreement between the simulation results of the new scheme and those produced using other recently proposed numerical schemes. These findings suggest that the proposed method can be successfully applied for many astrophysical problems in which the presence of subsonic flows previously limited the use of SPH, with the new scheme now being competitive in these regimes with other numerical methods.

On the eddy-resolving capability of high-order discontinuous Galerkin approaches to implicit LES / under-resolved DNS of Euler turbulence

We present estimates of spectral resolution power for under-resolved turbulent Euler flows obtained with high-order discontinuous Galerkin (DG) methods. The ‘1% rule’ based on linear dispersion–diffusion analysis introduced by Moura et al. (2015) [10] is here adapted for 3D energy spectra and validated through the inviscid Taylor–Green vortex problem. The 1% rule estimates the wavenumber beyond which numerical diffusion induces an artificial dissipation range on measured energy spectra. As the original rule relies on standard upwinding, different Riemann solvers are tested. Very good agreement is found for solvers which treat the different physical waves in a consistent manner. Relatively good agreement is still found for simpler solvers. The latter however displayed spurious features attributed to the inconsistent treatment of different physical waves. It is argued that, in the limit of vanishing viscosity, such features might have a significant impact on robustness and solution quality. The estimates proposed are regarded as useful guidelines for no-model DG-based simulations of free turbulence at very high Reynolds numbers.

A pseudo-compressible variational multiscale solver for turbulent incompressible flows

In this work, we design an explicit time-stepping solver for the simulation of the incompressible turbulent flow through the combination of VMS methods and artificial compressibility. We evaluate the effect of the artificial compressibility on the accuracy of the explicit formulation for under-resolved LES simulations. A set of benchmarks have been solved, e.g., the 3D Taylor---Green vortex problem in turbulent regimes. The resulting method is proven to be an effective alternative to implicit methods in some application ranges (in terms of problem size and computational resources), providing comparable results with very low memory requirements. As an example, with the explicit approach, we are able to solve accurately the Taylor-Green vortex benchmark in a fine mesh with $$512^3$$5123 cells on a 12 cores 64 GB ram machine.

Quantum calculus of classical vortex images, integrable models and quantum states

From two circle theorem described in terms of q-periodic functions, in the limit q→1 we have derived the strip theorem and the stream function for N vortex problem. For regular N-vortex polygon we find compact expression for the velocity of uniform rotation and show that it represents a nonlinear oscillator. We describe q-dispersive extensions of the linear and nonlinear Schrodinger equations, as well as the q-semiclassical expansions in terms of Bernoulli and Euler polynomials. Different kind of q-analytic functions are introduced, including the pq-analytic and the golden analytic functions.

A truly incompressible smoothed particle hydrodynamics based on artificial compressibility method

In the present study, a truly incompressible smoothed particle hydrodynamics based on the artificial compressibility method for simulating steady and unsteady incompressible flows is proposed and assessed. The incompressible Navier–Stokes equations in the primitive variables formulation using the artificial compressibility method proposed by Chorin in the Eulerian reference frame are written in a Lagrangian reference frame to provide an appropriate incompressible SPH algorithm. The proposed SPH formulation implemented here is based on an implicit dual-time stepping scheme to be capable of time-accurate analysis of unsteady flows. The advantage of the Artificial Compressibility-based Incompressible SPH (ACISPH) method proposed over the Weakly Compressible SPH (WCSPH) and the pseudo-compressibility SPH methods is that the ACISPH formulation is a truly incompressible SPH algorithm and it does not involve any approximate enforcement of the incompressibility condition that usually causes to use a large magnitude speed of sound implying a small time step in the computations. The approximate enforcement of the incompressibility condition used in the WCSPH method also causes spurious oscillations in the density and pressure fields which is not the case for the ACISPH formulation proposed. Unlike the projection SPH algorithm and the moving-particle semi-implicit (MPS) method formulation, the ACISPH formulation presented herein does not involve the iterative solution of a Poisson equation for the pressure field. The accuracy and robustness of the proposed incompressible SPH (ACISPH) are demonstrated by solving different incompressible flow problems. A sensitivity study is also conducted to evaluate the effects of particle resolution and the value of artificial compressibility parameter on the accuracy and convergence rate of the solution. To validate the results, different test cases are considered herein that are incompressible flows in a 2-D simple channel, a 2-D locally expanded channel, a 2-D cavity, the unsteady Couette flow with pressure gradient, the Taylor vortex problem and a dam break problem with dry bed. Results obtained for these cases are in good agreement with the available analytical and numerical results. The study shows the artificial compressibility-based ISPH (ACISPH) method proposed is accurate and robust for simulating steady and unsteady incompressible flow problems.

Low-Dissipation Low-Dispersion Second-Order Scheme for Unstructured Finite Volume Flow Solvers

A new low-dissipation low-dispersion second-order scheme suitable for unstructured finite volume flow solvers is presented that is designed for vortical flows and for scale-resolving simulations of turbulence. The idea is that, by optimizing its dispersion properties, a standard second-order method can be improved significantly for such flows. The key is to include gradient information for computing face values of the fluxes and to use this additional degree of freedom to improve the dispersion properties of the scheme. The scheme is motivated by a theoretical consideration of the dispersion properties for a one-dimensional scalar transport equation problem. Then, the new scheme is applied using the DLR TAU-code for compressible flows and the DLR THETA-code for incompressible flows for simulations of the moving vortex problem and the Taylor–Green vortex flow. The improved accuracy for small-scale transportation and the easy implementation make this scheme a promising candidate for efficient scale-resolving simulations of turbulent flows.

SpectralPlasmaSolver: a Spectral Code for Multiscale Simulations of Collisionless, Magnetized Plasmas

We present the design and implementation of a spectral code, called SpectralPlasmaSolver (SPS), for the solution of the multi-dimensional Vlasov-Maxwell equations. The method is based on a Hermite-Fourier decomposition of the particle distribution function. The code is written in Fortran and uses the PETSc library for solving the non-linear equations and preconditioning and the FFTW library for the convolutions. SPS is parallelized for shared- memory machines using OpenMP. As a verification example, we discuss simulations of the two-dimensional Orszag-Tang vortex problem and successfully compare them against a fully kinetic Particle-In-Cell simulation. An assessment of the performance of the code is presented, showing a significant improvement in the code running-time achieved by preconditioning, while strong scaling tests show a factor of 10 speed-up using 16 threads.

Dynamics of restricted three and four vortices problem on the plane

The dynamics of a test particle (a particle with zero vorticity) advected by the velocity field of N point-vortices with vorticities Γj, j = 1, …N, is considered. Making an analogy with similar studies in celestial mechanics, we call such a study a “restricted N-vortex problem” or (N + 1)-vortex problem. In particular, we study and characterize the global planar dynamics of some restricted 3 and 4-vortex problems, as a function of the vorticities Γj of the vortices.

Existence, Stability, and Symmetry of Relative Equilibria with a Dominant Vortex

We analyze existence, stability, and symmetry of point vortex relative equilibria with one dominant vortex and $N$ vortices with infinitesimal circulation. The dimension of the problem can be reduced by taking an infinitesimal circulation limit, resulting in the so-called $(1+N)$-vortex problem. In this work, we first generalize the reduction to allow for circulations of varying signs and weights. We then prove that symmetric configurations require equality of two circulation parameters in the $(1+3)$-vortex problem and show that there are stable asymmetric relative equilibria. In a number of examples, we use rigorous methods from algebraic geometry to count all relative equilibria.

Assessment of Five SGS Models for Low-R e m MHD Turbulence

Assessment of three regularization-based and two eddy-viscosity-based subgrid-scale (SGS) turbulence models for large eddy simulations (LES) are carried out in the context of magnetohydrodynamic (MHD) decaying homogeneous turbulence (DHT) with a Taylor scale Reynolds number (Reλ) of 120 and a MHD transition-to-turbulence Taylor-Green vortex (TGV) problems with a Reynolds number of 3000, through direct comparisons to direct numerical simulations (DNS). Simulations are conducted using the low-magnetic Reynolds number approximation (Rem<<1). LES predictions using the regularization-based Leray- α,LANS- α, and Clark- α SGS models, along with the eddy viscosity-based non-dynamic Smagorinsky and the dynamic Smagorinsky models are compared to in-house DNS for DHT and previous results for TGV. With regard to the regularization models, this work represents their first application to MHD turbulence. Analyses of turbulent kinetic energy decay rates, energy spectra, and vorticity fields made between the varying magnetic field cases demonstrated that the regularization models performed poorly compared to the eddy-viscosity models for all MHD cases, but the comparisons improved with increase in magnitude of magnetic field, due to a decrease in the population of SGS eddies within the flow field.