A spectral approach to a constrained optimization problem for the Helmholtz equation in unbounded domains

We study some convergence issues for a recent approach to the problem of transparent boundary conditions for the Helmholtz equation in unbounded domains. The approach is based on the minimization on an integral functional which arises from an integral formulation of the radiation condition at infinity. In this Letter, we implement a Fourier-Chebyschev collocation method and show that this approach reduce the computational cost significantly. As a consequence, we give numerical evidence of some convergence estimates available in literature and we study the robustness of the algorithm at low and mid-high frequencies

Poloidal-toroidal decomposition in a finite cylinder. II. Discretization, regularization and validation

The Navier-Stokes equations in a finite cylinder are written in terms of poloidal and toroidal potentials in order to impose incompressibility. Regularity of the solutions is ensured in several ways: First, the potentials are represented using a spectral basis which is analytic at the cylindrical axis. Second, the non-physical discontinuous boundary conditions at the cylindrical corners are smoothed using a polynomial approximation to a steep exponential profile. Third, the nonlinear term is evaluated in such a way as to eliminate singularities. The resulting pseudo-spectral code is tested using exact polynomial solutions and the spectral convergence of the coefficients is demonstrated. Our solutions are shown to agree with exact polynomial solutions and with previous axisymmetric calculations of vortex breakdown and of nonaxisymmetric calculations of onset of helical spirals. Parallelization by azimuthal wavenumber is shown to be highly effective

A High-Order Radial Basis Function (RBF) Leray Projection Method for the Solution of the Incompressible Unsteady Stokes Equations

A new projection method based on radial basis functions (RBFs) is presented for discretizing the incompressible unsteady Stokes equations in irregular geometries. The novelty of the method comes from the application of a new technique for computing the Leray-Helmholtz projection of a vector field using generalized interpolation with divergence-free and curl-free RBFs. Unlike traditional projection methods, this new method enables matching both tangential and normal components of divergence-free vector fields on the domain boundary. This allows incompressibility of the velocity field to be enforced without any time-splitting or pressure boundary conditions. Spatial derivatives are approximated using collocation with global RBFs so that the method only requires samples of the field at (possibly scattered) nodes over the domain. Numerical results are presented demonstrating high-order convergence in both space (between 5th and 6th order) and time (up to 4th order) for some model problems in two dimensional irregular geometries.Comment: 34 pages, 8 figure

A fast and well-conditioned spectral method for singular integral equations

We develop a spectral method for solving univariate singular integral equations over unions of intervals by utilizing Chebyshev and ultraspherical polynomials to reformulate the equations as almost-banded infinite-dimensional systems. This is accomplished by utilizing low rank approximations for sparse representations of the bivariate kernels. The resulting system can be solved in ${\cal O}(m^2n)$ operations using an adaptive QR factorization, where $m$ is the bandwidth and $n$ is the optimal number of unknowns needed to resolve the true solution. The complexity is reduced to ${\cal O}(m n)$ operations by pre-caching the QR factorization when the same operator is used for multiple right-hand sides. Stability is proved by showing that the resulting linear operator can be diagonally preconditioned to be a compact perturbation of the identity. Applications considered include the Faraday cage, and acoustic scattering for the Helmholtz and gravity Helmholtz equations, including spectrally accurate numerical evaluation of the far- and near-field solution. The Julia software package SingularIntegralEquations.jl implements our method with a convenient, user-friendly interface

Numerical investigation of Von Karman swirling bioconvective nanofluid transport from a rotating disk in a porous medium with Stefan blowing and anisotropic slip effects

In recent years, significant progress has been made in modern micro- and nanotechnologies related to applications in micro/nano-electronic devices. These technologies are increasingly utilizing sophisticated fluent media to enhance performance. Among the new trends is the simultaneous adoption of nanofluids and biological micro-organisms. Motivated by bio-nanofluid rotating disk oxygenators in medical engineering, in the current work, a mathematical model is developed for steady convective Von Karman swirling flow from an impermeable power-law radially stretched disk rotating in a Darcy porous medium saturated with nanofluid doped with gyrotactic micro-organisms. Anisotropic slip at the wall and blowing effects due to concentration are incorporated. The nano-bio transport model is formulated using non-linear partial differential equations (NPDEs), which are transformed to a set of similarity ordinary differential equations (SODEs) by appropriate transformations. The transformed boundary value problem is solved by a Chebyshev collocation method. The impact of key parameters on dimensionless velocity components, concentration, temperature and motile microorganism density distributions are computed and visualized graphically. Validation with previous studies is included. It is found that that the effects of suction provide a better enhancement of the heat, mass and microorganisms transfer in comparison to blowing. Moreover, physical quantities decrease with higher slip parameters irrespective of the existence of blowing. Temperature is suppressed with increasing thermal slip whereas nanoparticle concentration is suppressed with increasing wall mass slip. Micro-organism density number increases with the greater microorganism slip. Radial skin friction is boosted with positive values of the power law stretching parameter whereas it is decreased with negative values. The converse response is computed for circumferential skin friction, nanoparticle mass transfer rate and motile micro-organism density number gradient. Results from this study are relevant to novel bioreactors, membrane oxygenators, food processing and bio-chromatography

A 3D pseudospectral method for cylindrical coordinates. Application to the simulations of rotating cavity flows

International audienceThe present work proposes a collocation spectral method for solving the three-dimensional Navier-Stokes equations using cylindrical coordinates. The whole diameter -R < r < R is discretized with an even number of radial Gauss-Lobatto collocation points and an angular shift is introduced in the Fourier transform that avoid pole and parity conditions usually required. The method keeps the spectral convergence that reduces the number of grid points with respect to lower-order numerical methods. The grid-points distribution densifies the mesh only near the boundaries that makes the algorithm well-suited to simulate rotating cavity flows where thin layers develop along the walls. Comparisons with reliable experimental and numerical results of the literature show good quantitative agreements for flows driven by rotating discs in tall cylinders and thin inter-disc cavities. Associated to a spectral vanishing viscosity [E. Séverac, E. Serre, A spectral vanishing viscosity for the LES of turbulent flows within rotating cavities, J. Comp. Phys. 226 (2007) 1234-1255], the method provides very promising LES results of turbulent cavity flows

Large Eddy Simulation of Non-Isothermal Turbulent Rotor-Stator Flows

Non-isothermal turbulent flows in an enclosed rotorstator cavity are here investigated using large eddy simulation (LES). Besides their fundamental importance as three-dimensional prototype flows, such flows arise in many industrial applications and especially in turbomachineries. The LES is performed using a Spectral Vanishing Viscosity technique, which is shown leading to stable discretizations without sacrificing the formal accuracy of the spectral approximation. The LES results have been favorably compared to velocity measurements in the isothermal case. The Boussinesq approximation is then used to take into account the centrifugal-buoyancy effects. The thermal effects have been examined for Re equal to 1 million in a rotor-stator cavity of aspect ratio G=(b-a)/h=5 and curvature parameter Rm=(b-a)/(b+a)=1.8 (a, b the inner and outer radii of the rotor and h the interdisk spacing) and for Rayleigh numbers up to Ra=108. These LES results provide accurate, instantaneous quantities which are of interest in understanding the physics of turbulent flows and heat transfers in an interdisk cavity. The averaged results show small effects of density variation on the mean and turbulent fields

Compressed absorbing boundary conditions via matrix probing

Absorbing layers are sometimes required to be impractically thick in order to offer an accurate approximation of an absorbing boundary condition for the Helmholtz equation in a heterogeneous medium. It is always possible to reduce an absorbing layer to an operator at the boundary by layer-stripping elimination of the exterior unknowns, but the linear algebra involved is costly. We propose to bypass the elimination procedure, and directly fit the surface-to-surface operator in compressed form from a few exterior Helmholtz solves with random Dirichlet data. The result is a concise description of the absorbing boundary condition, with a complexity that grows slowly (often, logarithmically) in the frequency parameter.Comment: 29 pages with 25 figure

Turbulence characteristics of the B\"{o}dewadt layer in a large enclosed rotor-stator system

A three-dimensional (3D) direct numerical simulation is combined with a laboratory study to describe the turbulent flow in an enclosed annular rotor-stator cavity characterized by a large aspect ratio G=(b-a)/h=18.32 and a small radius ratio a/b=0.152, where a and b are the inner and outer radii of the rotating disk and h is the interdisk spacing. The rotation rate Omega under consideration is equivalent to the rotational Reynolds number Re=Omegab2/nu=9.5 x 104, where nu is the kinematic viscosity of the fluid. This corresponds to a value at which an experiment carried out at the laboratory has shown that the stator boundary layer is turbulent, whereas the rotor boundary layer is still laminar. Comparisons of the 3D computed solution with velocity measurements have given good agreement for the mean and turbulent fields. The results enhance evidence of weak turbulence at this Reynolds number, by comparing the turbulence properties with available data in the literature. An approximately self-similar boundary layer behavior is observed along the stator side. The reduction of the structural parameter a1 under the typical value 0.15 and the variation in the wall-normal direction of the different characteristic angles show that this boundary layer is three-dimensional. A quadrant analysis of conditionally averaged velocities is performed to identify the contributions of different events (ejections and sweeps) on the Reynolds shear stress producing vortical structures. The asymmetries observed in the conditionally averaged quadrant analysis are dominated by Reynolds stress-producing events in this B\"{o}dewadt layer. Moreover, case 1 vortices (with a positive wall induced velocity) are found to be the major source of generation of special strong events, in agreement with the conclusions of Lygren and Andersson.Comment: 16 page