General Curvilinear Coordinate System Research Articles

More considerations about the symmetry of the stress tensor of fluids.

Regarding a recent dispute about the symmetry of the stress tensor of fluids, more considerations are presented. The usual proofs of this symmetry are reviewed, and contradictions between this symmetry and the mechanism of gas viscosity are analyzed for simple gas flows. It is emphasized that these proofs depend on the theorem of angular momentum that assumes that internal forces between any two fluid particles are always along the line connecting them. Without this assumption, from Newton's laws of motion one can only obtain the theorem of angular momentum with an additional term. It is proved within classical continuum mechanics that this additional term represents the total moment of internal forces, and its volume density is similar to the body couple introduced in generalized continuum mechanics. When discrete structure of matter is considered, this term corresponds to the total moment of the forces exerted on the nuclei by the internal electrons. This moment of internal forces may lead to nonsymmetry of the stress tensor and is, in general, nonzero as long as shear stress exists. A nonsymmetrical stress tensor suggested in the literature is discussed in terms of its effects in eliminating the contradictions and simplifying the Navier-Stokes equation. The derivation of this stress tensor for ideal gases based on the kinetic theory of gas molecules is presented, and its form in a general orthogonal curvilinear coordinate system is given. Finally, a possible experimental verification of this nonsymmetrical stress tensor is discussed.

Energy-conserving finite difference scheme based on velocity interpolation applicable to unsteady flows using collocated grids

The collocation method uses the Rhie–Chow scheme to find the cell interface velocity by pressure-weighted interpolation. The accuracy of this interpolation method in unsteady flows has not been fully clarified. This study constructs a finite difference scheme for incompressible fluids using a collocated grid in a general curvilinear coordinate system. The velocity at the cell interface is determined by weighted interpolation based on the pressure difference to prevent pressure oscillations. The Poisson equation for the pressure correction value is solved with the cross-derivative term omitted to improve calculation efficiency. In addition, simultaneous relaxation of velocity and pressure is applied to improve convergence. Even without the cross-derivative term, calculations can be stably performed, and convergent solutions are obtained. In unsteady inviscid flow, the conservation of kinetic energy is excellent even in a non-orthogonal grid, and the calculation result has second-order accuracy to time. In viscous analysis at a high Reynolds number, the error decreases compared with that of the Rhie–Chow interpolation method. The present numerical scheme improves calculation accuracy in unsteady flows. The possibility of applying this computational method to high Reynolds number flows is demonstrated through several analyses.

Numerical study of shock wave boundary layer interaction flows using a novel turbulence model

This study proposes a novel partial average fluctuation velocity (PAFV) turbulence based on the PAFV and the average strain rate tensor, which is used to calculate the shock wave boundary layer interaction flows, including cascade flow and transonic compressor flow. In order to illustrate the adequacy, the calculation results of several other turbulence models were also compared with PAFV, including k-ω, shear stress transport, SA-neg (negative Spalart-Allmaras), and stress-BSL (ω-based Reynolds stress model). The suitability of the PAFV turbulence model was initially tested by calculating the transonic flow in the L030-4 cascade. Subsequently, numerical computations were performed for the 3D (three-dimensional) National Aeronautics and Space Administration Rotor 67 transonic compressor rotor. In the numerical simulation, the flow control equations were transformed in a general curvilinear coordinate system for precision enhancement, and the spatial discretization of control equations was executed using the finite difference method. The convection term was processed using the Steger–Warming flux vector splitting method, the diffusion term was discretized using a second-order central difference scheme, and the time derivative term was discretized using the third-order Runge–Kutta method. A parallel computing strategy with multi-block partitioning was employed to speed up the calculations and facilitate faster convergence alongside a local time step method. All numerical computation procedures were written in Python, revealing the following: (1) The PAFV turbulence model aptly simulates the transonic flow within the cascade channel. The shock wave pattern in the cascade channel aligns well with the experimental schlieren images. (2) During the Rotor 67 transonic compressor rotor calculation, the flow–pressure ratio, flow efficiency, and inlet and outlet total temperatures and pressures align well with the experimental data. Furthermore, the resolution of the flow field structure related to the interaction between the tip leakage vortex and channel shock wave is high. (3) The turbulence model uses PAFV as the turbulence velocity scale, thereby suppressing the fluctuation velocity and turbulence viscosity near shock wave areas, preventing excessive turbulence viscosity. Moreover, PAFV inherently exhibits transport properties and anisotropic characteristics, making it well-suited for handling transonic compressor flows.

Asymptotic Study of Flows Induced by Oscillations of Cylindrical Bodies

Hydrodynamic flows induced by translational oscillations of cylindrical bodies of various cross-sectional shapes are studied. The motion of fluid around oscillating bodies is described using the system of Navier–Stokes equations written in a generalized curvilinear coordinate system. Transition to a given body shape is implemented using a conformal mapping. The problem is solved using the method of successive asymptotic expansions under the assumption that the oscillation amplitudes are small. In each asymptotic approximation, the subproblems are solved numerically using the finite-difference method. Based on the results of the work, estimates of the hydrodynamic effect are obtained, the applicability of the high-frequency asymptotic approximation is estimated, and secondary stationary flows near cylinders are studied, in particular, the occurrence of directed stationary flows near an oscillating asymmetric body is considered with reference to the Joukowski airfoil.

FLEW: A DNS Solver for Compressible Flows in Generalized Curvilinear Coordinates

We present FLEW, an in-house high-fidelity solver for direct numerical simulation (DNS) of turbulent compressible flows over arbitrary shaped geometries. FLEW solves the Navier–Stokes equations written in a generalized curvilinear coordinate system, in which the surface coordinates are non-orthogonal, whereas the third axis is normal to the surface. Spatial discretization relies on high-order finite-difference schemes. The convective terms are discretized using an hybrid approach, combining the near-zero numerical dissipation provided by central approximations with the robustness of weighted essentially non-oscillatory (WENO) schemes, required to capture shock waves. Central schemes are stabilized using a skew-symmetric-like splitting of convective derivatives, endowing the solver with the energy-preserving property in the inviscid limit. The maximum order of accuracy is eighth for central schemes (also used for viscous terms discretization) and seventh for WENO. The code is oriented to modern high-performance computing (HPC) platforms thanks to message passing interface (MPI) parallelization and the ability to run on graphics processing unit (GPU) architectures. Reliability, accuracy and robustness of the code are assessed in the low-subsonic, transonic and supersonic regimes. We present the results of several benchmarks, namely the inviscid Taylor–Green vortex, turbulent curved channel flow, transonic laminar flow over a NACA 0012 airfoil and turbulent supersonic ramp flow. The results for all configurations proved to be in excellent agreement with previous studies.

Collocated Finite Difference Method in General Coordinates for Simulation of Flow Over Arbitrary Geometry

A finite difference method to be used in simulations of turbulent flows such as Direct Numerical Simulation (DNS) and Large-Eddy Simulation (LES) has been developed and verified for basic benchmark flows. The method is constructed using the general curvilinear coordinate system with the collocated grid arrangement of the variables so that it can be easily applied to various engineering flows including hydraulic applications involving complex geometry and moving boundary. The computer code is verified by test calculation of steady laminar internal flow in a curved cavity and an external flow above a curved hill. Furthermore the same code is used to conduct a DNS of fully-developed turbulent flow in a two-dimensional open channel and a LES of flow over curved hill. All results indicate that the method is sound with sufficient accuracy for the intended applications.

Comparative analysis of dynamic stability of cylindrical and conical shells under periodic axial compression

A comparative analysis of the dynamic stability of cylindrical and conical shells with the same geometric and mechanical characteristics under periodic uniformly distributed axial compression was presented. The study of the stability of steady periodic vibrations of thin elastic shells was based on the joint use of the method of curvilinear grids, the projection method and the parameter continuation method combined with the Newton–Kantorovich method. Geometrically nonlinear relations of the thin elastic shells theory are formulated on the basis of the vector approximation of the displacements function in the general curvilinear coordinate system in tensor form and satisfy the Kirchhoff-Love hypothesis. The discretization of the differential equations of the steady forced vibrations in the direction of the generating shells using the method of curvilinear grids was carried out. The components of the elements displacement vectors of the shells middle surface in the circular directionare approximated by trigonometric series. Reduction of the number of generalized coordinates of the discrete dynamic model of shells steady forced vibrations was performed by the Bubnov-Galerkin basis reduction method. A transition from vector ordinary differential equations to a nonlinear system of algebraic equations was made. The construction of a mathematical model of the dynamic stability of steady forced nonlinear vibrations of thin elastic shells was performed according to Floquet's theory using the projection method. The criterion for the loss of stability was the equality to zero of the determinant of the matrix of linearized equations of steady forced nonlinear vibrations of shells according to the Lyapunov theorem. A comparative analysis of frequencies and modes of natural vibrations of cylindrical and conical shells with the same geometric and mechanical characteristics and boundary conditions was performed. Nonlinear steady vibrations of the shells due to periodic axial compression were studied. The critical values of the dynamic load and the corresponding forms of loss of shell stability in the range of lower frequencies of their natural vibrations were obtained.

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.

Numerical modeling of debris flows using basic equations in generalized curvilinear coordinate system and its application to debris flows in Kinryu River Basin in Saga City, Japan

Each year, structures installed in rivers and valleys (e.g., bank protections, bridges, and sabo dams) are damaged by debris flows; therefore, it is necessary to predict the flow characteristics of debris flows accurately to design river and valley structures that are robust against such debris flows. In recent years, several two-dimensional models have been developed to simulate debris flow. However, the basic equations in the numerical simulation models are in the Cartesian coordinate system, and the shape of each numerical analysis grid is square or rectangular. It is impossible to use numerical grids that match the horizontal shape of the river or valley structures, such as bank protections and sabo dams, when employing square or rectangular grids. This makes it difficult to calculate the fluid forces acting on river or valley structures. In this study, the basic equations of debris flow are provided in the generalized curvilinear coordinate system and a numerical simulation model of debris flow designed using an orthogonal curved grid along the horizontal shape of rivers and valleys is developed. Numerical simulations of a periodical meandering valley are conducted using the basic equations in the generalized curvilinear and Cartesian coordinate systems; the difference between the two results is discussed. Further, numerical simulations of the debris flow that occurred in the Kinryu River Basin in Saga City, Japan, in 2019 are performed using the basic equations in the generalized curvilinear and Cartesian coordinate systems, and the difference between the two simulation results is discussed. The numerical simulation results of debris flow in a periodical meandering channel suggest a slow propagation of debris flow when equations in the Cartesian coordinate system are used; this is attributed to the rough boundaries of debris flow when a square grid is used, which causes numerical energy dissipation that does not exist physically. The inundation areas and land surface deformation of debris flow in Kinryu River Basin were reproduced well by the developed numerical analysis model; the basic equations were written in the generalized curvilinear coordinate system. The development and decrescence of debris flows were reproduced using the developed model; the first debris flow in Kinryu River Basin was stopped in the upstream mild slope area, while the second flowed into the residential area. Reproducing the flow depth, velocity, and discharge could be improved by the model using basic equations in the generalized curvilinear coordinate system, as compared with those reproduced using basic equations in the Cartesian coordinate system. When the Cartesian coordinate system was used, the volume of the debris flow was evaluated to be small and the travel distance of the debris flow was evaluated to be short; thus, the debris flow does not arrive at the residential area and the risk of debris flows is underestimated.

A Wave-Targeted Essentially Non-Oscillatory 3D Shock-Capturing Scheme for Breaking Wave Simulation

A new three-dimensional high-order shock-capturing model for the numerical simulation of breaking waves is proposed. The proposed model is based on an integral contravariant form of the Navier–Stokes equations in a time-dependent generalized curvilinear coordinate system. Such an integral contravariant form of the equations of motion is numerically integrated by a new conservative numerical scheme that is based on three elements of originality: the time evolution of the state of the system is carried out using a predictor–corrector method in which exclusively the conserved variables are used; the point values of the conserved variables on the cell face of the computational grid are obtained using an original high-order reconstruction procedure called a wave-targeted essentially non-oscillatory scheme; the time evolution of the discontinuity on the cell faces is calculated using an exact Riemann solver. The proposed model is validated by numerically reproducing several experimental tests of breaking waves on computational grids that are significantly coarser than those used in the literature to validate the existing 3D shock-capturing models. The results obtained with the proposed model are also compared with those obtained with a previously published model, which is based on second-order total variation diminishing reconstructions and an approximate Riemann solver usually adopted in the existing 3D shock-capturing models. Through the above comparison, the main drawbacks of the existing 3D shock-capturing models and the ability of the proposed model to simulate breaking waves and wave-induced currents are shown. The proposed 3D model is able to correctly simulate the wave height increase in the shoaling zone and to effectively predict the location of the wave breaking point, the maximum wave height, and the wave height decay in the surf zone. The validated model is applied to the simulation of the interaction between breaking waves and an emerged breakwater. The numerical results show that the proposed model is able to simulate both the large-scale circulation patterns downstream of the barrier and the onset of quasi-periodic vortex structures close to the edge of the barrier.

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.

A General Curvilinear Magnetic Field‐Line‐Following Coordinate System for Ionosphere‐Plasmasphere Modeling

AbstractWe propose a simple way to define a field‐line‐following, general curvilinear coordinate system for a general magnetic field. This definition of field‐line‐following coordinate system reduces to a usual definition of dipole coordinate system when the magnetic field is approximated by an axisymmetric dipole. In this way, it can facilitate the numerical implementation by enabling validation of various metric terms computed numerically against those defined analytically in the case of the dipole field. Steps involved in grid generation are also sketched. Highly accurate results are obtained using the high‐order ordinary differential equation (ODE) solver to solve the general magnetic field line equations. The accuracy and consistency of the numerical implementation are validated against analytical results in the case of a dipole field. Numerical results show that this field‐line‐following coordinate system for the general magnetic field, like the coordinates for the dipole field, is also an Euler potential or Clebsch type coordinate system.

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.

Near-wall flow structures and related surface quantities in wall-bounded turbulence

By applying the Taylor-series expansion solution of the Navier–Stokes equations, an analysis is given to elucidate the relationships between near-wall flow structures and the fundamental surface quantities (skin friction and surface pressure). The derived results are used to understand the physical features of near-wall flow structures around a typical strong wall-normal velocity event (SWNVE) in a turbulent channel flow based on the direct numerical simulation data at Reτ=180. The simulation is carefully done using a multiple-relaxation-time lattice Boltzmann method combined with an improved on-wall bounce-back implementation. It is found that both the skin friction divergence and the Laplacian of surface pressure have good correspondence with sweep and ejection motions induced by the quasi-streamwise vortex above the viscous sublayer. Interestingly, the surface pressure variation induced by a quasi-streamwise vortex tends to attenuate the wall-normal velocity magnitude in both the sweep and ejection sides through the Laplacian of surface pressure. Similar physical effects of surface-pressure-related terms are also observed for the near-wall Reynolds stress. The concentrated enstrophy and dissipation are associated with the SWNVE and high skin friction magnitude. It is found that the SWNVE is dynamically important in generating the boundary enstrophy flux, greatly enhancing the intermittency of turbulence inside the viscous sublayer. In addition, by applying the methods of differential geometry, the near-wall Taylor-series expansions are generalized for a stationary curved surface in a general curvilinear coordinate system. The generalized results could be useful in evaluating the curvature effect in the near-wall region for complex flows.

Numerical Simulation of Unsteady Subsonic Viscous Gas Flows Based on High-Order Composite Compact Schemes

A family of high-order multioperator compact schemes for computation of viscous gas flows on curvilinear grids is considered. Depending on the number of operators used, the order of a scheme approximating the convective terms of the original equations can vary from the 6th to 22nd. The viscous terms and the metric coefficients of a generalized curvilinear coordinate system are approximated with the same order. An example of three schemes, including a five-operator one, which is the most computationally intensive in the described family, is used to examine their structure. The change in the approximation and dissipation properties of the schemes caused by an increase in their order is investigated. The schemes are used for comparative computations of subsonic gas flows based on the Euler and Navier–Stokes equations. Numerical results based on the 22nd-order scheme are presented for a viscous subsonic airfoil flow in a wide range of angles of attack and for an impermeable parachute canopy.

Numerical modeling for compressible two-phase flows and application to near-field underwater explosions

In this study, an accurate shock- and interface-capturing method using curvilinear body-fitted structured grids is introduced to simulate compressible multiphase flows with shockwaves. A five-equation model—proficient in capturing unsteady shocks in compressible multiphase flows without nonphysical spurious oscillations—was enhanced by extending it to a multidimensional general curvilinear coordinate system. A new eigensystem was constructed for a monotonic upstream-centered scheme for conservation laws/Godunov-type finite volume scheme that can capture strong shocks in compressible multiphase flows. The developed method was validated using a typical test involving a free-field underwater explosion that produces a strong shockwave in addition to bubble interfaces. Numerical results were compared and found to agree well with the empirical equation and previously published results. Further, the interaction of a shockwave with a wedge was computed to evaluate the shockwave propagation characteristics and appropriate treatment of solid wall boundary conditions. The numerical results show good agreement with the experimental data. Finally, the impact and propagation characteristics of shockwaves in two more complex cases, involving one and two explosions near a rigid cylinder, were numerically analyzed, and the results indicate high impacts of primary blast waves on the structure.

The Regularity Criteria and the A Priori Estimate on the 3D Incompressible Navier-Stokes Equations in Orthogonal Curvilinear Coordinate Systems

The paper considers the regularity problem on three-dimensional incompressible Navier-Stokes equations in general orthogonal curvilinear coordinate systems. We establish one regularity criteria of the weak solutions involving only in a vorticity component ω 3 and one a priori estimate on the solution that H 3 u 3 L ∞ 0 , T ; L p ℝ 3 is bounded for 1 ≤ p ≤ ∞ to three-dimensional incompressible Navier-Stokes equations in orthogonal curvilinear coordinate systems. These extent greatly the corresponding results on axisymmetric cylindrical flow.

A novel sharp interface immersed boundary framework for viscous flow simulations at arbitrary Mach number involving complex and moving boundaries

This work presents a robust and efficient sharp interface immersed boundary (IBM) framework, which is applicable for all-speed flow regimes and is capable of handling arbitrarily complex bodies (stationary or moving). The work deploys an in-house, parallel, multi-block structured finite volume flow solver, which employs a 3D unsteady Favre averaged Navier Stokes equations in a generalized curvilinear coordinate system; while we employ a combination of HCIB (Hybrid Cartesian Immersed boundary) method and GC(Ghost-cell) for solution reconstruction near immersed boundary interface. A significant difficulty for these sharp interface approaches is of handling sharp features/edges of complex geometries. In this study, we observe that apart from the need for robust node classification strategy and higher order boundary formulations, the direction in which the reconstruction procedures are performed plays an important role in handling sharp edges. Taking this into account we present a versatile interface tracking procedure based on ray tracing algorithm and a novel three step solution reconstruction procedure that computes pseudo-normals in the regions where the normal is not well-defined and reconstructs the flow field along those directions. We demonstrate that this procedure enables solver to efficiently handle and accurately represent sharp-edged regions. A fifth-order weighted essentially non-oscillatory (WENO) scheme is used for capturing shock-induced discontinuities and complex fluid-solid interactions with high resolution. The developed IBM framework is applied to a wide range of flow phenomena encompassing all-speed regimes (M=0.001 to M = 2.0). A total of seven benchmark cases (three stationary and four moving bodies) are presented involving various geometries (cylinder, airfoil, wedge) and the predictions are found to be in excellent agreement with the published results.

A tribological model for geometrically structured anisotropic surfaces in a covariant form

AbstractThis contribution proposes a tribological model within a three‐dimensional contact formulation considering structural anisotropy of the contact interface. A simple elastoplastic constitutive law is adopted for the description of the behavior on the anisotropic contact interface. Starting with the establishment of the thermodynamic framework of the contact problem, the dissipative, irreversible process is described. By applying the principle of maximum dissipation, the evolution equations and the expressions of the tangential contact forces for the cases of sticking and sliding are obtained and, subsequently, formulated in algorithmic form, in order to enable their implementation into finite element codes. The anisotropic behavior is incorporated through the definition of a tensor of anisotropy. The form of this tensor is defined in a general curvilinear coordinate system. The cases of both constant and nonconstant anisotropic tensor are studied. The analytical solution of a numerically computed problem, serves the validation of the proposed model.