General Curvilinear Coordinates Research Articles

Computation of three-dimensional incompressible flows using high-order weighted essentially non-oscillatory finite-difference lattice Boltzmann method

In the present work, an accurate and robust solution methodology based on the high-order weighted essentially non-oscillatory (WENO) finite-difference lattice Boltzmann method (LBM) in the three-dimensional generalized curvilinear coordinates is presented and applied for simulating the three-dimensional incompressible flows over complicated configurations with curved boundaries. Here, the incompressible form of the lattice Boltzmann equation in three dimensions is considered and the discretization of the spatial derivative terms is performed with the fifth-order WENO finite-difference method and the temporal derivative term is discretized with the fourth-order Runge–Kutta scheme to ensure the accuracy and stability of the solution method for both the steady and unsteady problems. The three-dimensional lattice Boltzmann equation applied here is based on a nineteen discrete velocity model for transforming the microscopic properties to the macroscopic ones. To assess the accuracy and robustness of the present three-dimensional high-order finite-difference LBM solver, different incompressible flow benchmarks and practical test cases are studied that are the cavity flow, the Beltrami flow, the flow in the curved ducts of rectangular cross sections, and the flow over a sphere for different flow conditions. The decay of the homogeneous isotropic turbulence is also computed to examine the suitability of the present solution method to be applied as the direct numerical simulation of turbulent flows. It is demonstrated that the solution methodology presented based on the high-order WENO finite-difference LBM in the three-dimensional generalized curvilinear coordinate can be used for accurately and effectively computing the three-dimensional practical incompressible flow problems.

Exact solutions of steady Euler equations in two dimension by functional separation

Exact solutions of two dimensional Euler equations for incompressible fluid flows are found by the functional separation of variables. More specifically, two independent variables are considered in the Cartesian coordinates, polar coordinates and also general orthogonal curvilinear coordinates which include elliptic, parabolic and hyperbolic systems. Several new solutions are obtained and other classical solutions are rediscovered.

Strongly conservative discretization of governing equations in cylindrical coordinates

Cylindrical coordinates are often used in computational fluid dynamics (CFD), particularly when one considers axisymmetric flow. Although cylindrical coordinates have several advantages in accurately describing the rotational flow as well as the complex boundary, there are limits to how far they can be taken. These are (1) the governing equations are mostly in a non-conservative form in current standard models, which cannot preserve the conservation of numerical simulations, (2) the singularity along the axis at the coordinate origin (z-axis), leading to difficulty reproducing the flow across the z-axis, and (3) the time step is extremely shortened to satisfy the CFL condition near the z-axis because the numerical cell thereof is narrow in the azimuthal direction for a given angular resolution. A strongly conservative model is proposed in this paper to overcome these difficulties. First, the strongly conservative form of the governing equations is derived by considering changes in the direction of the base vector within a cell, and a concept of direction error is defined to reveal the inherent error of the standard model. Second, inspired by the derivation of the first, a strongly conservative method for evaluating the flux across each cell surface is proposed, preserving the conservation of discretization and enlarging the time step. Finally, the proposed model is tested by three classical problems, i.e., uniform flow, Sod shock tube problem, and expanding outflows. The results imply that the proposed model is strongly conservative for the cylindrical coordinate system, and it can overcome the limits of the standard model and eliminate the inherent error in the standard model. Furthermore, the strongly conservative model and method are extended to arbitrary coordinates for the numerical conservation in general curvilinear coordinates of the problem with complex boundaries.

Derivation of entropy production in a fluid flow in a general curvilinear coordinate system

The paper deals with the derivation of the entropy production in the fluid flow performed in a general curvilinear coordinate system. The derivation of the entropy production is based on the thermodynamics laws as well as on the balances of mass, momentum, and energy. A brief description of the differential geometry used in general curvilinear coordinates is presented here as well to define the used notation.The application of this approach is then shown in the evaluation of the entropy production along the suction side of the blade, where the calculation was performed using available experimental data.

Lattice Boltzmann model in general curvilinear coordinates applied to exactly solvable 2D flow problems

Numerical simulation results of basic exactly solvable fluid flows using the previously proposed by H. Chen Lattice Boltzmann Method (LBM) formulated on a general curvilinear coordinate system are presented. As was noted in the theoretical work of H. Chen, such curvilinear Lattice Boltzmann Method preserves a fundamental one-to-one exact advection feature in producing minimal numerical diffusion, as the Cartesian lattice Boltzmann model. As we numerically show, the new model converges to exact solutions of basic fluid flows with the increase of grid resolution in the presence of both natural curvilinear geometry and/or grid non-uniform contraction, both for near equilibrium and non-equilibrium LBM parameter choices.

A Numerical Code for a Wide Range of Compressible Flows on Hybrid Computational Architectures

The major points in the development of the parallel multiplatform multipurpose numerical code solving the full unsteady Navier–Stokes equations are presented. The developed code is primarily designed for running on multi-GPU computational devices but can also be used on traditional multicore CPUs and even on manycore processors such as Intel Xeon Phi. Physical models include calorically perfect inert gas, single- and multi-temperature approaches for chemically reactive flows and an Euler–Euler model for gas-particle suspensions. Main details of the implementation are described. Shock capturing TVD and WENO schemes in general curvilinear coordinates are used for spatial approximation. Explicit, semi-implicit and fully implicit schemes are employed for advancing solution in time. The code is written in C++ with CUDA API and opensource libraries, such as MPI, zlib and VTK. A few examples of numerical simulations are briefly described to provide general idea of the numerical code capabilities. They include a supersonic flow past a wedge, a jet exhausting from a square nozzle, a heavy gas bubble descending in a lighter medium and a heterogeneous detonation in gas-particle suspension.

Thermo-hydrodynamic Airflow Behavior Analysis in Solar Chimney Device

A numerical methodology has been developed to analyze the thermo-hydrodynamic aspect of airflow occurring in solar chimney power plants (SCPP) according to some dominant parameters. The general curvilinear coordinates finite volume method (GCCFVM), which is necessary in the case of turbulent flow through complex geometries, is used in this work. The governing equations describing the steady state turbulent fluid flow are solved numerically using this technique. It is shown that the chimney tower dimensions control directly the hydrodynamic field. However, the collector dimensions control directly the thermal field and indirectly the hydrodynamic field. It is demonstrated that the solar radiation influences strongly and positively the thermo-hydrodynamic field by increasing the mass flow rate. The mass flow decreases with the increase of the ambient temperature and then the system is more efficient with low ambient temperature. Indeed, the mass flow rate increases from 0.8 kg/s up to approximately 2 kg/s when the solar radiation varies between 200 W/m2 and 1000 W/m2 for fixed ambient temperature value of 30 °C. When ambient temperature increases from 10 °C up to 50 °C, the mass flow rate decreases slightly and in a linear manner from 1.7 kg/s to 1.5 kg/s for fixed solar radiation intensity value of 600 W/m2. Contrasting to other studies, conclusion based on simplified analytical models, ambient temperature affects adversely the performance of a SCPP in decreasing the mass flow rate. This conclusion should be taken into consideration when analyzing models dedicated to the prediction of solar chimney power plant performance.

An intuitive form of the energy functional for resistive plasma perturbations

The energy functional of ideal magnetohydrodynamics (MHD) is extended to the regime of finite resistivity and transformed into an intuitive form, which separates pressure-gradient, current-density drive and stabilising phenomena, analogously to the ideal version of the functional derived by Greene and Johnson (Plasma Phys., vol. 10, issue 8, 1968, pp. 729–745). The resulting resistive energy functional is not in the form of a solvable eigenvalue problem but is intended to analyse the energetic composition of eigenfunctions which were calculated in advance by solving the MHD equations. The new resistive energy functional provides information on the different energetic drives and stabilising phenomena of an eigenfunction. The new energy functional has been implemented in the CASTOR3D code in general 3D curvilinear coordinates and successfully validated across three different coordinate systems for a numerical test case.

2D Fourier finite element formulation for magnetostatics in curvilinear coordinates with a symmetry direction

We present a numerical method for the solution of three-dimensional linear magnetostatic problems by embedding their geometry into a symmetric domain and Fourier expansion in the symmetry coordinate, leading to a set of decoupled two-dimensional problems. Special cases include axial and translational symmetry. To keep the treatment universal, a framework for the Fourier decomposition of the vector potential formulation is developed in the covariant formalism with general curvilinear coordinates. Under the restriction of homogeneous material parameters along the symmetry direction, but not on fields and current densities, this leads to a decoupled set of two-dimensional equations in both, zeroth (non-oscillatory) and non-zero (oscillatory) harmonics. For the latter it is possible to eliminate one component of the vector potential resulting in a fully transverse vector potential orthogonal to the transverse magnetic field. In addition to the Poisson-like equation for the longitudinal component of the non-oscillatory problem, a general curl-curl Helmholtz equation describes the transverse problem covering both, non-oscillatory and oscillatory case. The resulting variational forms are treated by the usual nodal finite element method for the longitudinal problem and by a two-dimensional edge element method for the transverse problem. The numerical solution can be computed independently for each harmonic using fewer degrees of freedom than a three-dimensional simulation at comparable accuracy. These properties make the developed approach useful for the analysis of perturbation harmonics in magnetic plasma confinement devices in cylindrical coordinates or magnetic flux coordinates. The method is validated on analytical and numerical test cases and applied to a racetrack-shaped coil setup.

Characteristic boundary conditions for magnetohydrodynamic equations

In the present study, a characteristic-based boundary condition scheme is developed for the compressible magnetohydrodynamic (MHD) equations in the general curvilinear coordinate system, which is an extension of the characteristic boundary scheme for the Navier–Stokes equations. The eigenstructure and the complete set of characteristic waves are derived for the ideal MHD equations in general curvilinear coordinates (ξ,η,ζ). The characteristic boundary conditions are derived and implemented in a high-order MHD solver where the sixth-order compact scheme is used for the spatial discretization. The fifth-order Weighted Essentially Non-Oscillatory (WENO) scheme is also employed for the spatial discretization of problems with discontinuities. In our MHD solver, the fourth-order Runge–Kutta scheme is utilized for time integration. The characteristic boundary scheme is first verified for the non-magnetic (i.e., B=0) Sod shock tube problem. Then, various in-house test cases are designed to examine the derived MHD characteristic boundary scheme for three different types of boundaries: non-reflecting inlet and outlet, solid wall, and single characteristic wave injection. The numerical examples demonstrate the accuracy and robustness of the MHD characteristic boundary scheme.

Fully conservative overset mesh to overcome the blunt body metric singularity in finite difference methods

A fully conservative overset mesh method is proposed and applied to overcome the metric singularity at the symmetry line for blunt bodies, e.g., capsules and blunted cones, in general curvilinear coordinates. The overset mesh is placed automatically at the symmetry line by avoiding the collapse of the grid lines using a hexahedral structure in contrast to the prismatic structure of a body-orientated mesh. In addition, the grid points of the overset mesh coincide with those of the body-orientated mesh to avoid interpolation techniques to interchange the flow variables between the two meshes. This coincidence ensures the conservation of the flow variables and avoids uncertainties at the shock as the method is naturally conservative. The thin-layer Navier–Stokes equations for high Reynolds number flows are solved using an AUSM+ or an AUSMPW+ flux vector splitting in combination with a mesh adaption to capture the shock accurately. For verification purposes of the proposed method, a supersonic 2D-axisymmetric hemisphere cylinder is chosen and the results along the wall are verified. Furthermore, the conservative properties of the applied overset mesh method are shown and the results on the stagnation line are presented. In addition, a supersonic 3D calculation is investigated to show the applicability of the presented method for simulations with an angle of attack.

Strain-gradient theory for shear deformation free-form microshells: Governing equations of motion and general boundary conditions

A first-order shear deformation (FOSD) free-form microshell model described in general curvilinear coordinates was developed within the complete framework of Mindlin’s form II linear isotropic strain-gradient theory (SGT), considering both strain-gradient and micro-inertia effects. The high-order governing equations of motion and consistent boundary conditions were simultaneously obtained through a variational formulation based on Hamilton’s principle. The established microshell model contains five strain-gradient material constants and one length parameter resulting from micro-inertia effects in addition to two classical Lamé constants, thereby capturing microstructure-induced size-dependent phenomena in both static and dynamic analyses. The constructed model within the general SGT can be flexibly reduced to those based on the modified strain-gradient, modified couple stress, and simplified strain-gradient theories. Moreover, by enforcing geometric restrictions, the general formulation for free-form microshells can be specialized for doubly curved microshells. Finally, the forced vibration results of a simply supported closed cylindrical microshell confirmed the reliability and accuracy of the proposed model and strain-gradient and high-order inertia effects on the size-dependent dynamic characteristics of microshells. Altogether, the presented mathematical formulations can provide a foundation for the in-depth understanding of microscale shell structures.

A three-dimensional off-lattice Boltzmann method for the simulation of blood flow through a model irregular stenosis

In the present work, a three-dimensional characteristic-based off-lattice Boltzmann method is developed in general cylindrical curvilinear coordinates to handle body-fitted non-uniform meshes that typically arise in blood flow simulation of stenosed arteries. To handle the singularity point at r = 0, the azimuthal mapping approach together with a special periodic boundary condition is developed. The numerical solver is validated, using reference data from literature, for steady flow through a stenosed lumen and for pulsatile flow through an abdominal aortic aneurysm. Thereafter, the solver is applied to study pulsatile blood flow through a model irregular arterial stenosis with an aerial occlusion of 75%. The surface irregularity of the stenosis is modeled using a sine function while keeping the cosine-shaped occlusion. The degree of irregularity is controlled by the amplitude and frequency of the sine function. Flow characteristics such as wall shear stress (WSS), divergence of WSS, oscillatory shear index, relative residence time (RRT), the turbulence kinetic energy (TKE), and power spectral density are used to investigate the near-wall vascular remodeling caused by the resulting disturbed flow. The present study demonstrates that for a given areal occlusion, an increase in the amplitude and frequency of the surface irregularity increases the number of locations susceptible for perfusion of low-density lipoproteins and promotes flow disturbances in the stenotic and post-stenotic regions. An increase in the Reynolds number not only increases the peaks of the WSS but also brings the peaks together causing alternating stress in the stenosed region. Furthermore, a positive correlation is found between the Reynolds number and the TKE production. On the other hand, an increase in the Womersley number reduces the TKE production, curtails the disturbed flow, and reduces the RRT of the solutes, all of which further reduce the risk factors. Overall, the developed solver demonstrates the importance of shape of the stenosis on the blood flow dynamics for physiological inflow conditions.

Design & Optimization of Large Cylindrical Radomes with Subcell and Non-Orthogonal FDTD Meshes Combined with Genetic Algorithms

The word radome is a contraction of radar and dome. The function of radomes is to protect antennas from atmospheric agents. Radomes are closed structures that protect the antennas from environmental factors such as wind, rain, ice, sand, and ultraviolet rays, among others. The radomes are passive structures that introduce return losses, and whose proper design would relax the requirement of complex front-end elements such as amplifiers. The radome consists mostly in a thin dielectric curved shape cover and sometimes needs to be tuned using metal inserts to cancel the capacitive performance of the dielectric. Radomes are in the near field region of the antennas and a full wave analysis of the antenna with the radome is the best approach to analyze its performance. A major numerical problem is the full wave modeling of a large radome-antenna-array system, as optimization of the radome parameters minimize return losses. In the present work, the finite difference time domain (FDTD) combined with a genetic algorithm is used to find the optimal radome for a large radome-antenna-array system. FDTD uses general curvilinear coordinates and sub-cell features as a thin dielectric slab approach and a thin wire approach. Both approximations are generally required if a problem of practical electrical size is to be solved using a manageable number of cells and time steps in FDTD inside a repetitive optimization loop. These approaches are used in the full wave analysis of a large array of crossed dipoles covered with a thin and cylindrical dielectric radome. The radome dielectric has a thickness of ~λ/10 at its central operating frequency. To reduce return loss a thin helical wire is introduced in the radome, whose diameter is ~0.0017λ and the spacing between each turn is ~0.3λ. The genetic algorithm was implemented to find the best parameters to minimize return losses. The inclusion of a helical wire reduces return losses by ~10 dB, however some minor changes of radiation pattern could distort the performance of the whole radome-array-antenna system. A further analysis shows that desired specifications of the system are preserved.

Volumetric Lattice Boltzmann Models in General Curvilinear Coordinates: Theoretical Formulation

A theoretical formulation of lattice Boltzmann models on a general curvilinear coordinate system is presented. It is based on a volumetric representation so that mass and momentum are exactly conserved as in the conventional lattice Boltzmann on a Cartesian lattice. In contrast to some previously existing approaches for arbitrary meshes involving interpolation approximations among multiple neighboring cells, the current formulation preserves the fundamental one-to-one advection feature of a standard lattice Boltzmann method on a uniform Cartesian lattice. The new approach is built on the concept that a particle is moving along a curved path. A discrete space-time inertial force is derived so that the momentum conservation is exactly ensured for the underlying Euclidean space. We theoretically show that the new scheme recovers the Navier-Stokes equation in general curvilinear coordinates in the hydrodynamic limit, along with the correct mass continuity equation.

Nodal Integral Methods in General 2d Curvilinear Coordinates - Applied to Convection–Diffusion Equation in Domains Discretized Using Quadrilateral Elements

Nodal Integral Methods in General 2d Curvilinear Coordinates - Applied to Convection–Diffusion Equation in Domains Discretized Using Quadrilateral Elements

Modified couple stress theory for three-dimensional elasticity in curvilinear coordinate system: application to micro torus panels

This research considers size effects in the linear three-dimensional elasticity analysis of micro-tori. The fundamental relations (displacement form) are derived for isotropic toroidal shells in the framework of the modified couple stress theory in the curvilinear coordinate system to predict the mechanical responses. A numerical solution for the displacement field is obtained using the GDQ method. The numerical results are in a close agreement with those found by the finite element and the Galerkin method. Parametric studies are conducted to explore the effect of size-dependency, micro-tori geometry, meridional and circumferential angle, toroidal shell thickness, and different boundary conditions on the distribution of the displacement fields. Numerical results for displacement also show that natural frequencies of micro-toroidal shells, predicted by modified couple stress theory, are less than those predicted by the classical theory, due to the significant effect of length scale parameter (related to material microstructures) on the mechanical responses. The use of general curvilinear coordinates in toroidal structures enables us to also study the mechanical behavior of irregular geometries, cap-shaped panel, saddle-shaped panel, and sectorial-shaped panel.

High-Order Gas-Kinetic Scheme in Curvilinear Coordinates for the Euler and Navier-Stokes Solutions

The high-order gas-kinetic scheme (HGKS) has achieved success in simulating compressible flow in Cartesian mesh. To study the flow problem in general geometry, such as the flow over a wing-body configuration, the development of a three-dimensional HGKS in general curvilinear coordinates becomes necessary. In this paper, a two-stage fourth-order gas-kinetic scheme is developed for the Euler and Navier-Stokes solutions in the curvilinear coordinates. Based on the coordinate transformation, the kinetic equation is transformed first to the computational space, and the flux function in the gas-kinetic scheme is obtained there and is transformed back to the physical domain for the update of conservative flow variables inside each control volume. To achieve the expected order of accuracy, the dimension-by-dimension reconstruction based on the WENO scheme is adopted in the computational domain, where the reconstructed variables are the cell averaged Jacobian and the Jacobian-weighted conservative variables, and the conservative variables are obtained by ratio of the above reconstructed data at Gaussian quadrature points of each cell interface. In the two-stage fourth-order gas kinetic scheme (GKS), similar to the generalized Riemann solver (GRP), the initial spatial derivatives of conservative variables have to be used in the evaluation of the time dependent flux function in GKS, which are reconstructed as well through orthogonalization in physical space and chain rule. A variety of numerical examples from the order tests to the solutions with strong discontinuities are presented to validate the accuracy and robustness of the current scheme. The precise satisfaction of the geometrical conservation law in non-orthogonal mesh is also demonstrated through the numerical example.

A high‐order compact scheme for solving the 2D steady incompressible Navier‐Stokes equations in general curvilinear coordinates

SummaryIn this paper, a high‐order compact finite difference algorithm is established for the stream function‐velocity formulation of the two‐dimensional steady incompressible Navier‐Stokes equations in general curvilinear coordinates. Different from the previous work, not only the stream function and its first‐order partial derivatives but also the second‐order mixed partial derivative is treated as unknown variable in this work. Numerical examples, including a test problem with an analytical solution, three types of lid‐driven cavity flow problems with unusual shapes and steady flow past a circular cylinder as well as an elliptic cylinder with angle of attack, are solved numerically by the newly proposed scheme. For two types of the lid‐driven trapezoidal cavity flow, we provide the detailed data using the fine grid sizes, which can be considered the benchmark solutions. The results obtained prove that the present numerical method has the ability to solve the incompressible flow for complex geometry in engineering applications, especially by using a nonorthogonal coordinate transformation, with high accuracy.

Definition of the Spatial Propagator and Implications for Magnetic Field Properties

We present a theoretical framework to analyze the 3D coronal vector magnetic-field structure. We assume that the vector magnetic field exists and is a priori smooth. We introduce a generalized connectivity phase space associated with the vector magnetic field in which the basic elements are the field line and its linearized variation: the Spatial Propagator. We provide a direct formulation of these elements in terms of the vector magnetic field and its spatial derivatives, constructed with respect to general curvilinear coordinates and the equivalence class of general affine parameterizations. The Spatial Propagator describes the geometric organization of the local bundle of field lines, equivalent to the kinematic deformation of a propagated volume tied to the bundle. The Spatial Propagator’s geometric properties are characterized by dilation, anisotropic stretch, and rotation. Extreme singular values of the Spatial Propagator describe quasi-separatrix layers (QSLs), while true separatrix surfaces and separator lines are identified by the vanishing of one and two singular values, respectively. Finally, we show that, among other possible applications, the squashing factor [Q] is easily constructed from an analysis of particular sub-matrices of the Spatial Propagator.