Non-orthogonal Grids Research Articles

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.

Lattice Boltzmann simulations on transient radiative transfer problems in irregular geometries with constant or graded refractive index

This work is concerned with the transient radiative transport in inhomogeneous two- or three-dimensional semitransparent graded-index media with irregular geometry. The lattice Boltzmann method is applied to the graded-index media mapped with non-orthogonal grids to solve the radiative transfer while the finite volume method expresses the angular processing. The solution of the transient radiation from the methodology developed is firstly validated with good accuracy by considering benchmark solutions available in the literature for comparisons for various forms of geometry including a cube, elliptic cylinder as well as irregular shapes. Afterwards, the solver is used to analyze the radiative transfer in graded-index media including regular and irregular geometries, and various governing parameters such as laser and optical properties, scattering laws, incident angle of the incoming beam, refractive index, and temperature distributions. Suitable calculations of the transient radiometric heat flux and incident energy help to evaluate the time-resolved reflectance and transmittance. The results from this study indicate that the present methodology and boundary treatments are simple and suitable to deal with multidimensional radiative transfer in irregular domains.

An affine transform center line extraction and discontinuity deviation correction method for non-orthogonal grid laser

The extraction of the laser center line is one of the key steps of laser scanning, which is related to the accuracy of the coordinate calculation. Non - orthogonal grid laser scanning has the advantages of a large measuring range and small translation error. For the extraction of the center line of the laser cross region of non-orthogonal grid, the discontinuity and deviation of the results obtained by the existing methods will cause the error of center line coordinate extraction. To solve these problems, according to the properties of image affine, an improved method of laser center extraction for non-orthogonal grid is proposed in this paper. Firstly, the image is divided into sub-images by affine transformation according to slopes, and the orthogonal grid laser image is obtained. Secondly, in order to solve the problem of centerline discontinuity, the orthogonal grid laser image was rotated 0° and 90°, and the Steger method was used to process images. Inverse affine transformation is performed on the results. Finally, the sub-images are clipped and splicing according to slopes, and the two rotation results are union to get the complete laser center line. By comparing various methods and 3D reconstruction experiments, the accuracy of the proposed method is verified in the center line extraction of grid laser cross-region.

Evaluation of Different Discretization Schemes for Non-Orthogonal Grids and Highly Skewed Flow

Evaluation of Different Discretization Schemes for Non-Orthogonal Grids and Highly Skewed Flow

Cattaneo–Christov heat flux impacts on MHD radiative natural convection of Al2O3-Cu-H2O hybrid nanofluid in wavy porous containers using LTNE

This paper aims to examine impacts of Cattaneo–Christov heat flux on the magnetohydrodynamic convective transport within irregular containers in the presence of the thermal radiation. Both of the magnetic field and flow domain are slant with the inclination angles Ω and γ, respectively. The worked fluid is consisting of water (H2O) and Al2O3-Cu hybrid nanoparticles. The enclosures are filled with a porous medium, and the local thermal nonequilibrium (LTNE) model between the hybrid nanofluids and the porous elements are considered. Influences of various types of the obstacles are examined, namely, horizontal cold elliptic, vertical elliptic and cross section ellipsis. The solution methodology is depending on the finite volume method with nonorthogonal grids. The major outcomes revealed that the location (0.75, 0.5) is better for the rate of the flow and temperature gradients. The higher values of H* causes that the solid phase temperature has a similar behavior of the fluid phase temperature indicating to the thermal equilibrium state. Also, the fluid-phase average Nusselt number is maximizing by increasing Cattaneo–Christov heat flux factor.

Entropy Optimization and Slant MHD Mixed Convection Hybrid Nanofluid Flow Within an Oblique Irregular Lid-Driven Enclosure Contains Baffles: Local Thermal Non-equilibrium Model

This paper explores the heat transfer by mixed convection within double lid-driven enclosures filled with Cu–Al_2O_3 hybrid nanofluids. The domain contains T-shaped baffles and is filled with local thermal non-equilibrium (LTNE) porous medium. The wavy boundaries are partially cooled, while the inner baffles have constant heat flux conditions. Mathematical formulations for the system irreversibility in cases of LTNE, constant heat flux conditions and double diffusive are presented and analyzed. The solution methodology is depending on the finite volume scheme in case of non-orthogonal grids. The major results revealed that the alteration of the undulation number lambda from 1 to 5 gives an augmentation in values of theta _text {f,max} up to 4.57%. Also, the increase in the lengths of the baffles causes a reduction in the flow, while theta _text {f,max} is rising. Furthermore, the alteration in Ri from 0.5 to 10 gives an augmentation in theta _text {f,max} up to 12.97%.

A generic balanced-force algorithm for finite volume method on polyhedral unstructured grids with non-orthogonality

A generic balanced-force algorithm is proposed to solve incompressible multiphase flows with complex interfaces and large density ratios on the polyhedral unstructured grids of large non-orthogonality, skewness and non-uniformity. This algorithm combines the finite volume method based on the volume of fluid (VOF) approach with the fractional step method in a collocated framework, which retains a complete balance between the gravity, surface tension with exact curvature, and the resulting pressure gradients. In the original scheme, the gravity and surface tension terms are represented as gradients of the volume fraction and then discretized in the identical fashion as the pressure gradient, which results in a balanced formulation on the structured grids. However, this scheme cannot be readily applied to unstructured grids, despite that it has been misused for more than one decade. As demonstrated in the paper, it is insufficient to achieve force balance merely by the identical discretization of scalar gradients due to the non-orthogonality of unstructured grids, which will generate severe spurious currents that intensify with larger non-orthogonality angle and higher fluid density ratio. Therefore, we present a novel balanced-force algorithm to treat the non-orthogonal term in a different manner, which significantly suppresses, or even eliminates, the spurious velocity for two-phase flows with large density ratio on arbitrary mesh topology. Various numerical examples have demonstrated that the present algorithm and computational framework offer a promising platform to provide high-fidelity and robust predictions for practical multiphase flow simulations involving highly complex geometry.

Modeling the remediation of aquifers from the lenses of petroleum products

An improved algorithm for modeling the dynamics of petroleum product lenses based on the planned filtration equations is presented. The program code «Oilfem 2», which implements the algorithm, is supplemented with a generator of non-orthogonal grids, the ability to create groundwater models or import them, the dependence between the pumping rate of the lens of petroleum products and its thickness is taken into account. The improved program code was applied on a test task to eliminate oil pollution, which is a round lens of petroleum products. The results of this work showed a good correspondence between the two solutions according to two criteria: the change in the flow rate of pumping oil products and the change in the volume of free oil products over time.

Flexural Properties of Textile-Reinforced Concrete with Nonorthogonal Grids

Textile-reinforced concrete (TRC) is a cement-based composite material that uses textile as a reinforcement material. The weft-direction fiber bundles in the traditional orthogonally arranged warp-weft textile hardly bear force, and its bonding strength with the weft fiber bundle is not ideal. Under the action of force, a small included angle between the stressed fiber bundle and the stressed direction can effectively increase the anchoring effect of their fibers in the matrix, resulting in higher bonding and reinforcement efficiency. To improve the utilization rate of fibers and the bonding strength between the textile and the concrete matrix, an arrangement along the diagonal of the grids was proposed in this paper. The flexural properties of basalt TRC plates with orthogonal grids (OG-BTRC) and plates with nonorthogonal grids (NOG-BTRC) with different grid angles and grid sizes with different laying methods, namely, a side layout (SL) and diagonal layout (DL), were studied through four-point bending tests. A comparative analysis was carried out with an ABAQUS simulation and the test results. The results showed that with a decrease in the grid angle, the BTRC specimens gradually showed a failure mode of multiple cracks, and most of the cracks appeared in the pure bending area; as the grid angle decreased, the BTRC specimens exhibited excellent flexural bearing capacity, good ductility, and high toughness. The total number of cracks on the specimen increased when it failed, while the spacing of the cracks decreased, and the fracture morphology appeared as fine and uniform features. The toughness of the specimen with a small grid angle using the DL laying method was greater than that using the SL laying method. The software simulation value matched the test data well, which proved that the test result was reliable.

A high-fidelity solver based on hybrid numerical methods on unstructured grids for incompressible multiphase flows

A high-fidelity solver is developed on the unstructured grids to simulate incompressible multiphase flows with free surface based on volume of fluid (VOF) method. In this framework, the velocity is defined by both volume integrated average (VIA) and point value (PV) as computational variables which are updated simultaneously by volume-average/point-value multi-moment (VPM) method at each time step. Compared with our previous work, a node-based finite element (FE) method is used instead of cell-centered finite volume (FV) method for the discretization of pressure Poisson equation, which provides a new pattern that can naturally treat with the velocity and pressure coupling. It does not require Rhie–Chow interpolation or the use of a Petrov–Galerkin finite element formulation with tunable parameters to circumvent the well-known instability conditions. Moreover, we propose a novel balanced-force formulation with FE discretization to accommodate the surface tension force, which effectively eliminates the large numerical errors associated with the non-orthogonality of unstructured grids. In VOF method, the volume fraction is defined as VIA to identify the moving interface which is updated by solving advection equation through the FV formulation. The resulting model that combines hybrid discretizations, rigorously preserves the mass and momentum conservation, substantially improves the numerical accuracy and robustness of pressure field, and significantly suppresses the spurious velocity to nearly machine error even for multiphase flows with large density ratios. Various numerical examples have been presented to demonstrate the excellent capabilities of present solver which can provide high-fidelity solutions for practical applications in presence of largely deformed interface and highly complex geometry.

Numerical 3D Model of Suspended Sediment Transport Downstream Al-Amarah Barrage, Iraq

This research is an analytical study for simulation both sediment transport and flow within the Tigris river reach located downstream of the Al-Amarah barrage within the Maysan province. This study adopted a three-dimensional program (SSIIM) which use the Navier-Stokes equations for calculating the flow, and the convection-diffusion equations for calculating the sediment transport by the finite volume method as approximated method. A structured non-orthogonal three-dimensional grid is employed to perform the simulation. The obtained results are subsequently compared to the field measurements. The determination coefficient (R2) for this comparison is 0.96 for flow velocity distribution and 0.94 for sediment concentration distribution. The results also showed through the simulation of the water flow, the state of the secondary flow and its effect on both the main flow and the erosion of the river bed in the studied cross sections. According to the high convergence of the results of this model with the field measurements, this model is a powerful tool for simulating flow and sediment concentrations in river systems and channels.

Non-linear Boltzmann equation on hybrid-unstructured non-conforming multi-domains

Adaptivity is crucial for addressing practical challenges of the next decades. In this regard, recent surveys [117] highlight that it continues to be a major bottleneck in computational fluid dynamics workflow. We introduce mixed non-conforming discontinuous Galerkin discretization for the full Boltzmann equation in 2D/3D. These schemes have been designed for efficiency — motivated in part by spectral and isogeometric weighted collocation methods — and retain an optimal O(p+1) convergence for a p order approximation for non-linear kinetic systems on non-orthogonal grids. In this setting, it is possible to analyze highly complex problems of industrial strength i.e., structured, unstructured, mixed, irregular, multi-domain (multi-block) adaptive geometries at massively parallel scale (ten thousand cores or beyond). Mixed domains permit flexible mesh generation, whereas local nature of discontinuous Galerkin permits construction of adaptive numerical schemes that scale well. To address flows in mixing regime (low/high rarefaction), we couple the scheme with an asymptotic preserving implicit-explicit time discretization. These schemes are iteration free and applicable for a wide range of rarefied flows from free-molecular to continuum. To ensure stability in presence of shocks, we describe a method of constructing limiters on non-conforming grids. Finally, we show that the computational overhead for solving kinetic equations on non-conforming structured/unstructured domains is negligible relative to conforming domains. So, there is no reason to not prefer non-conforming unstructured domains.

Non-orthogonal computational grids for studying dislocation motion in phase field approaches

In this work, new non-orthogonal computational grids are implemented into a phase field model called Phase Field Dislocation Dynamics (PFDD). We demonstrate that the new non-orthogonal grid can accommodate multiple slip planes in either the face centered cubic (FCC) or body centered cubic (BCC) crystallographic systems. We show that they avoid numerical errors induced when modeling glide on inclined slip planes in an orthogonal grid. The Gibbs effect that arises in the orthogonal or rotated orthogonal grids is substantially diminished when a non-orthogonal grid is employed. A few test cases demonstrate the effectiveness of using non-orthogonal grids in solving systems with multiple non-planar slip systems.

Mixed convective transport in inclined porous open arc-shaped enclosures saturated by nanofluids using a second-order Boussinesq approximation

Numerical simulations are performed for the mixed convective transport in an inclined open arc-shaped channel filled by isotropic porous media using nanofluids. The Darcy regime is considered and the non-linear Boussinesq approximation (NBA) is applied for the mixture density. The vorticity-stream function scheme based on the finite volume method in case of the non-orthogonal grids is conducted to solve the dimensionless governing system of equations. The local thermal equilibrium model between the mixture and the porous medium is taken into account and water is considered as a base fluid while the Al2O3 is assumed as nanoparticles. The computations are carried out for various values of the curvature parameter ε ranging from 0.1 to 0.5, the Darcy number Da ranging from 10−2 to 10−5, the Rayleigh number ranging from 104 to 107, the Reynolds number Re ranging from 10 to 5×102, the inclination angle γ ranging from 0 to π2, the nonlinear Boussinesq parameter ξ ranging from 0 to 3 and the nanoparticles volume fraction is varied from 0 % to 4 %. The results reveal that an increase in the nonlinear Boussinesq parameter enhances the average Nusselt number regardless of the values of the inclination angle.

Entropy Generation and MHD Convection within an Inclined Trapezoidal Heated by Triangular Fin and Filled by a Variable Porous Media

Analyses of the entropy of a thermal system that consists of an inclined trapezoidal geometry heated by a triangular fin are performed. The domain is filled by variable porosity and permeability porous materials and the working mixture is Al2O3-Cu hybrid nanofluids. The porosity is varied exponentially with the smallest distance to the nearest wall and the permeability is depending on the particle diameter. Because of using the two energy equations model (LTNEM), sources of the entropy are entropy due to the transfer of heat of the fluid phase, entropy due to the fluid friction and entropy due to the porous phase transfer of heat. A computational domain with new coordinates (ξ,η) is created and Finite Volume Method (FVM) in case of the non-orthogonal grids is used to solve the resulting system. Various simulations for different values of the inclination angle, Hartmann number and alumina-copper concentration are carried out and the outcomes are presented in terms of streamlines, temperature, fluid friction entropy and Bejan number. It is remarkable that the increase in the inclination angle causes a diminishing of the heat transfer rate. Additionally, the irreversibility due to the temperature gradients is dominant near the heated fins, regardless of the values of the Hartmann number.

Projection-based monotone embedded discrete fracture method for flow and transport in porous media

The extension of the monotone embedded discrete fracture method (monotone EDFM) (Nikitin and Yanbarisov, 2020) is considered in this work. The new method is based on the two coupled discretization schemes: projection-based EDFM (pEDFM) (Tene et al., 2017) and nonlinear compact multi-point flux approximation scheme (Lipnikov et al., 2012). The main novelty of the extended method is the satisfaction of the discrete maximum principle and applicability to the anisotropic porous media with conductive and blocking fractures. The resulting projection-based monotone EDFM combines effectiveness and simplicity of pEDFM approach with accuracy and physical relevance of the nonlinear FV schemes on non-orthogonal grids and anisotropic media.

Dynamic effects of fault modeling on stair-step and corner-point grids

Fault modeling has become an integral element of reservoir simulation for structurally complex reservoirs. Modeling of faults in general has major implications for the simulation grid. In turn, the grid quality control is very important in order to attain accurate simulation results. We investigate the dynamic effects of using stair-step grid (SSG) and corner-point grid (CPG) approaches for fault modeling from the perspective of dynamic reservoir performance forecasting. We have performed a number of grid convergence and grid-type sensitivity studies for a variety of simple, yet intuitive faulted flow simulation problems with gradually increasing complexity. We have also explored the added value of the multipoint flux approximation (MPFA) method over the conventional two-point flux approximation (TPFA) to increase the accuracy of reservoir simulation results obtained on CPGs. Effects of fault seal modeling on grid-resolution convergence and grid-type sensitivity have also been briefly examined. For simple geometries, both SSG and CPG can be used for fault modeling with similar accuracy in conjunction with the pillar-grid approach. This is evidenced by the fact that simulation results from SSG and CPG converge to identical solutions. SSG and CPG yield different results for more complex geometries. Simulation results approach to a converged solution for relatively fine SSGs. However, a SSG only provides an approximation to the fault geometry and reservoir volumes when the grid is coarse. On the other hand, non-orthogonality errors are increasingly evident in relatively more complex faulted models on CPGs and such errors cannot be addressed by grid refinement. It has been observed that MPFA partially addresses the discretization errors on non-orthogonal grids but only from the total flux accuracy perspective. However, transport related errors are still evident. Grid convergence behaviors and grid effects are quite similar with or without fault seal modeling (i.e., dedicated fault-zone modeling by use of scaled-up seal factors) for simple geometries. However, in more complex test cases, we have observed that it is more difficult to achieve converged results in conjunction with fault seal modeling due to increased heterogeneity of the underlying problem.

Multi-dimensional dissipation strategy within advection upstream splitting methods in hypersonic flows

The work develops the multi-dimensional dissipation strategy within advection upstream splitting methods (AUSM) in hypersonic flows based on the affordable shock-stable item by Chen et al. (J. Comput. Phys. 373 (2018) 662-672). This strategy is related to the sear velocity difference and the pressure-based sensing function. It strengthens the robustness of AUSM-family schemes against shock anomalies and meantime preserves shear layer. A series of numerical cases illuminate its potential capability for hypersonic flows, particularly in the extreme situations such as the large aspect ratio of cells, skewed non-orthogonal grid, and unstructured grid.

Fully implicit multiple graphics processing units’ schemes for hypersonic flows with lower upper symmetric Gauss-Seidel preconditioner on unstructured non-orthogonal grids

The paper discusses the simulation of the hypersonic flow problems on unstructured grids using fully implicit scheme and the analysis of the orthogonal corrections on the quality of heat fluxes on the surfaces of the bodies subject to hypersonic flow. The governing equations are the Navier-Stokes equations for viscous calorically perfect gas. Second order finite volume discretization is applied to the problem on arbitrary unstructured grids with implicit temporal treatment. The problem is solved using the Newton-Raphson method. Such methods require the solution of large linear systems with iterative solvers and require a preconditioning operator to converge. In this paper a well known Lower-Upper Symmetric Gauss Seidel (LU-SGS) preconditioner is applied. The new method is based on the tricky reordering for the factored Jacobi matrix that allows one to execute block triangular matrix solvers on GPUs in parallel without the loss of algebraic properties of the original non-factored operator. Good convergence properties are demonstrated for large scale problems of external aerodynamics with implicit Courant number around 1000 - 10000 for flows with Mach number around 13-25. The influence of the non-orthogonal corretions on the fluxes for highly scewed anisotropic grids is demonstrated at Mach numbers around 20. It is shown that the non-orthogonality corrections are of the second infinitesimal order compared to the effects of the shock fix influence on the heat fluxes on the body.

A new model for simulation of collection and conveyance sections of Qanat

In this paper, a new numerical model has been developed for simulation of Qanat-aquifer system. This model employs quasi-3D mixed-form of Richards’ equation and 1D fully-hydrodynamic form of Saint-Venant equations to simulate subsurface and overland flow, respectively. In order to handle non-orthogonal grids, subsurface flow module benefits from coordinate transformation technique. Using the above-mentioned governing equations, the presented model is able to simulate water flow inside both collection and conveyance sections of the gallery as well as dynamics of groundwater and vadose zone from impermeable bed rock to the soil-air interface. Since measured data corresponding to the hydraulics of Qanats is scarce, the overland and subsurface modules have been validated with analytical, numerical and experimental benchmarks in the literature. Subsequently, the model was employed to simulate ten different hypothetical aquifer-Qanat systems with different properties including the depth of groundwater aquifer, roughness of the gallery and saturated hydraulic conductivity of the gallery-aquifer boundary and the influence of each the parameters was monitored on the outflow rate at the appearance point of each Qanat. Furthermore, the advance of water inside two initially dry galleries were simulated at different time levels up to steady state. Eventually, the streamlines have been shown at the steady state for two Qanat-aquifer systems. Although, the presented study sheds light on some aspects of Qanat-aquifer hydraulics, the validation of the presented model with in-lab or on-field data remains ongoing for the future researches.