Characteristic-based Split Research Articles

Physics-informed neural network combined with characteristic-based split for solving forward and inverse problems involving Navier–Stokes equations

This paper presents a novel approach for solving the shallow-water transport equation using a physics-informed neural network (PINN) combined with characteristic-based split (CBS). Simulation of the tide in East Coast of China is performed to verify the applicability of the present method in a practical problem. Furthermore, we propose a boundary condition that accounts for the second-order partial derivative term, which is more appropriate for solving the diffusion equation with open domains than the commonly used assumption of zero boundary values. Our numerical results demonstrate that this boundary condition leads to improved convergence of the network. In addition, we introduce a parameter estimation method that requires information from the field at only two different times, yet yields accurate parameter estimates. We observe that excessive participation of variables in gradient backpropagation can lead to neural networks getting trapped in local optima. We use PINN combined with CBS method to solve 3-D incompressible flow. As the number of variables involved in gradient backpropagation increases, the accuracy of the solution decreases, which can partially support our viewpoint. The source codes for the numerical examples in this work are available at https://github.com/double110/PINN-cbs-.git.

Physics-informed neural network combined with characteristic-based split for solving Navier–Stokes equations

This paper presents a novel approach for solving the shallow-water transport equation using a physics-informed neural network (PINN) combined with characteristic-based split (CBS). Our method separates the variables, so it is no need to consider the weight coefficients between different loss functions. As not all partial derivatives are involved in the gradient backpropagation, our method can save half of the memory occupation without losing accuracy, and resulting in a significant reduction in computation time compared to traditional PINN. We solving the progress of the dispersion of hot water under known flow fields. Furthermore, we propose a boundary condition that accounts for the second-order partial derivative term, which is more appropriate for solving the diffusion equation with open domains than the commonly used assumption of zero boundary values. Our numerical results demonstrate that this boundary condition leads to improved convergence of the network. In addition, we introduce a parameter estimation method to estimate the diffusion coefficient of hot water flow, which requires information from the field at only two different times. We observe that excessive participation of variables in gradient backpropagation can lead to neural networks getting trapped in local optima. We use PINN combined with CBS method to solve 3-D incompressible flow. As the number of variables involved in gradient backpropagation increases, the accuracy of the solution decreases, which can partially support our viewpoint. The source codes for the numerical examples in this work are available at https://github.com/double110/PINN-cbs-.git.

A nodal position finite element model for fluid-structure interaction analysis of floating bodies with mooring

A numerical model based on the Nodal Position Finite Element Method (NPFEM) is proposed in this work for fluid-structure interaction (FSI) analysis of moored bodies subject to multiphase free surface flows. The semi-implicit CBS (Characteristic-Based Split) method is utilized here for discretization of the flow fundamental equations in the context of the Finite Element Method, where linear tetrahedral elements are adopted. Turbulence is analyzed using Large Eddy Simulation (LES) with the dynamic sub-grid scale model and the Level Set method is utilized for simulation of multiphase free surface flows. The structural system is modeled using a three-dimensional rigid body approach for the floating object, while the mooring is discretized using a geometrically nonlinear cable formulation based on the NPFEM. A partitioned coupling scheme is adopted for fluid-structure interaction analysis taking into account the fluid-structure and cable-structure couplings, where the flow equations are kinematically described employing an Arbitrary Lagrangean-Eulerian (ALE) formulation with a numerical scheme for mesh motion. The algorithms constituting the numerical model are verified first considering classical applications on fluid dynamics and FSI, in addition to applications involving the dynamic analysis of cables. Finally, problems involving floating bodies with and without mooring are simulated to demonstrate the applicability and accuracy of the proposed numerical model.

Modeling of particle-laden flows with n-sided polygonal smoothed finite element method and discrete phase model

In this work, an n-sided polygonal smoothed finite element method (nSFEM) using mean value shape function for concave polygonal element coupled with the discrete phase model (DPM) is developed to solve particle-laden flows. The fluid phase is solved using nSFEM stabilized by the well-developed characteristic-based split (CBS) algorithm. The concave polygonal element is successfully constructed via the intrinsic characteristic of mean value shape function and a new smoothing domain (SD) construction with more physical meaning; thus, addressing the severely distorted local mesh, as well as retaining good geometric adaptability. In the meantime, the modeling of the coupling between the fluid and the particle is achieved by reusing the mean value interpolation to obtain the fluid drag force acting on the particle. The accurate capture of fluid velocities at particle positions within arbitrary polygonal elements via mean value interpolation ensures fluid drag with high-fidelity. Several classical numerical examples, including particle-laden flow around a circular cylinder, are presented to demonstrate the high accuracy and good robustness, and the capability of the proposed method for predicting particle motion in the analysis of complex particle-laden flows.

Adaptative finite element simulation of unsteady turbulent compressible flows on unstructured meshes

The simulation of high-speed turbulent compressible flows using Large Eddy Simulation (LES) combined with the Characteristic-Based Split scheme (CBS) and a mesh adaptation strategy is presented in this work. A compressible dynamic Smagorinsky model is employed for the compressible LES model and the CBS scheme is employed in a Finite Element Method (FEM) context for space and time discretization using unstructured meshes. An accurate mesh adaptation scheme is frequently applied along the simulation including mesh refinement, mesh coarsening, edge or face swapping and a vertex smoothing technique in order to control the interpolation error and the mesh size. A feature-based error estimation is employed considering multi-scale and multi-metric intersection in order to capture turbulence effects. Two and three-dimensional applications are evaluated in order to investigate the capabilities of the present approach for evaluating external turbulent and compressible aerodynamics problems. The turbulent transonic flow around a circular cylinder in 2D and 3D domains and the turbulent supersonic flow past a blunt cylinder-flare model are investigated. Aerodynamic coefficients, frequency analysis and pressure distributions are compared against experimental data and other numerical simulations.

NUMERICAL ANALYSIS OF CIRCULATION AND DISPERSION OF POLLUTANTS IN VENTILATED BUILDINGS UNDER NON-ISOTHERMAL CONDITIONS

Air quality, thermal comfort and internal circulation in urban areas are closely linked to the promotion of natural ventilation in buildings. In this sense, the main objectives of the present work are to study the effect of internal circulation in buildings with different openings and the phenomenon of pollutant dispersion in naturally ventilated buildings under non-isothermal conditions using a numerical model for incompressible flows with heat and mass transport. For the flow simulation, a semi-implicit Characteristic-Based Split (CBS) scheme is used in the context of the Finite Element Method (FEM), where linear tetrahedral elements are used for spatial discretization. Turbulence is treated using the Large Eddy Simulation (LES) methodology and thermal effects are considered in the momentum balance equation through buoyancy forces, which are calculated taking into account the Boussinesq approximation. Classical examples are analyzed to verify the numerical model proposed here, as well as the forms of natural ventilation and, finally, a numerical investigation is carried out considering the dispersion of pollutants and the thermal effects simultaneously. In addition to ensuring air exchange while maintaining a healthy environment, ventilation helps to disperse pollutants and promotes thermal comfort, both inside the building and in the street canyon, as indicated by results obtained.

MAGNETIC CONVECTION-RADIATION INTERACTION IN WAVY POROUS TRIANGULAR CONTAINERS USING HYBRID NANOFLUIDS: ENTROPY ANALYSIS

The free convection in triangular enclosures can be found in various practical applications such as microelectronic devices, design of building roofs and attics, geophysical applications, solar collector electric machinery, ventilation in rooms, and cooling of electronic devices. Therefore, this study aims to examine the free convection in sinusoidal, wavy porous triangle-shaped containers full of Al<sub>2</sub>O<sub>3</sub>-CuO/water hybrid nanosuspension under the impacts of internal heat generation, inclined Lorentz force, and thermal radiation. The two heating mechanisms are classified into two distinct situations: Case 1 (C1) is a triangle-shaped domain that has a heated portion on the left edge and the lower edge is cold; Case 2 (C2) is a triangle-shaped domain with a cold part on the left edge and a lower heated wavy edge. The dimensionless forms of governing system with the considered conditions are transformed numerically using the finite elements approach via the characteristic-based split (CBS) algorithm. The gained outcomes are portrayed graphically via streamlines, isotherms, entropy features, and the Bejan issue. The heat transfer rate and fluid flow in view of internal heated and wavy walls play a significant role. The influence of the magnetic force highly controls the flow structures inside the cavity; fluid velocity moves in below the heated wavy surface because of the magnetic force performed in the incline orientation. The growth in the undulation number weakens the highest absolute value of the streamlines, or the convective case strength in the cavity. The vertical temperature increases the nanoparticle volume fraction and (St)<sub>avg</sub> into the cavity at α = 30° to 60°.

N-Side Cell-Based Smoothed Finite Element Method for Incompressible Flow with Heat Transfer Problems

In this paper, an n-sided cell-based smoothed finite element method (nCS-FEM) is presented to solve the convective heat transfer in incompressible viscous fluid flow. The Characteristic-based-split (CBS) scheme is used to stabilize the spatial and pressure oscillations of CS-FEM that occur during the process of solving incompressible Navier-Stokes equations. Meanwhile, the characteristic-Galerkin method in CBS is adopted as the convective stabilization in the heat transfer equation. In order to verify the proposed method, typical numerical examples are used to test the computational accuracy of CS-FEM. Meanwhile, the investigation of the robustness of highly distorted element is conducted in a natural convection test case. The calculation results of numerical examples show that CS-FEM is qualified to solve the steady and unsteady convective heat transfer problems for incompressible laminar flow with high reliability. The flow across tube banks with complex geometry in the heat sinks is also simulated to prove the capabilities of the proposed method. Well agreed comparison with the results of CFD software, Fluent, proves that the present method can effectively simulate realistic and complex flow with heat transfer problems.

Study for the cross-flow coupling mechanism of tandem cylinder groups at low Reynolds number

A numerical analysis of flow-induced vibrations for tandem cylinder groups which include two and three tandem cylinders with a spacing ratio of L/D = 5.5 (where L is the distance between the circular cylinders and D is the circular diameter) in planar shear flow is carried out. The cylinders can only vibrate freely in the cross-flow direction. A finite element method called four-step semi-implicit characteristic-based split (4-SICBS) is used to solve the Navier-Stokes equation. On the other hand, in order to maximize the oscillation response of the dynamic system, the damping ratio of it is ξ = 0. And the mass ratio of structure is set to mr = 2.0. The four main parameters, such as the number of cylinders (Num = 2, 3), Reynolds number (Re = 100, 160), the shear ratio (k = 0.0, 0.05, 0.1) as well as the reduced velocity (Ur = 3–30) are mainly studied the dynamic behaviors, frequency and spectrum characteristics, phase characteristics as well as energy characteristics of tandem cylinder groups. The results show that vibration responses of the upstream cylinders in tandem cylinder groups are similar to that of a single cylinder, while the parameters play a key role in flow-induced vibrations of the midstream and downstream cylinders. As Re = 100, the downstream cylinder of two tandem cylinders and the midstream cylinder of three tandem cylinders are in the state of increasing and then decreasing, and the downstream cylinder of three tandem cylinders has a larger vibration response with the change of k. As k = 0.1, the amplitude drops to zero with the increase of Ur. In addition, the situation becomes more complicated with the increase of Re. For the downstream cylinder of three tandem cylinders at Re = 160, there are two peaks of the cross-flow amplitude. The oscillation remains strong at larger Ur for the case of k = 0.1. Finally, the interactions between cylinders are revealed, together with the vortex dynamic mechanism underlying the oscillation characteristics of tandem cylinders exposed to planar shear flow.

A coupled cell-based smoothed finite element method and discrete phase model for incompressible laminar flow with dilute solid particles

In this paper, the cell-based smoothed finite element method (CS-FEM) empowered by the discrete phase model (DPM) is developed to solve dilute solid particles movements induced by incompressible laminar flow. In the present method, the fluid phase is solved by CS-FEM in the Eulerian framework, while particles are treated as discrete phases traced using Newton's second law in the Lagrangian framework. Meanwhile, the fluidic drag force on particles is considered to realize the one-way coupling of fluid to particles. For the fluid phase, the semi-implicit characteristic-based split (CBS) method is employed to suppress the spatial and pressure oscillations arising from the numerical solution of the Navier-Stokes equations discretized by the CS-FEM. To accurately capture the fluid velocity at an arbitrary particle position inside quadrilateral elements, the mean value coordinates interpolation is introduced. Furthermore, the motion equations for particles are solved by the fourth-order Runge-Kutta method to ensure high accuracy on particle trajectories. Several numerical examples in this paper demonstrate that the proposed method can effectively predict the effect of fluid flow on particle trajectories and position distributions in the analysis of practical and complex flow problems.

An immersed edge-based smoothed finite element method with the stabilized pressure gradient projection for fluid–structure interaction

In this paper, an effective and stable immersed edge-based smoothed finite element method (IES-FEM) is presented for fluid–structure interaction (FSI) problems. Under the framework of immersed algorithm, the system equation can be decomposed into three components, i.e, equations of nonlinear structure, incompressible viscous fluid, and FSI force. The characteristic-based split (CBS) scheme is first used to alleviate the spatial oscillation in the Navier-Stokes (N-S) equation. After that, the second-order pressure accuracy is formulated in the original CBS via stabilized pressure gradient projection (SPGP). Then, the problem domains are discretized using the simplest three-node triangular element and the smoothing domains are established. In order to enhance the accuracy of lower-order interpolation, the edge-based smoothing operation is further performed on all gradient-related terms for structure and fluid, namely, the coupled ES/ES-FEM. Moreover, an accurate form of FSI force evaluation is developed by introducing the whole function that considers the pressure and viscous force on the interface. Numerical examples demonstrate that the proposed scheme has higher computational precision (even for distorted mesh), faster convergence rate, good robustness, and lower computed costs.

Magnetic mixed convection confined two-sided wavy enclosures having two-sided lid-driven walls containing obstacles with thermal radiation

The present research provides numerical handling of a laminar flow mixed-radiation interaction convection heat transfer within two-sided wavy enclosures containing adiabatic obstacles. The convection process is due to the movements of the enclosure side walls together with the difference in temperature and density. Two designs of the flow domain are recognized, namely, a wavy enclosure containing a square obstacle (D1) or including a cylindrical obstacle (D2). Both the flow domain and the magnetic field are inclined with different angles and impacts of the thermal radiation and heat absorption(generation) are conducted. The Galerkin Finite Element Method associated with the Characteristic-Based Split approach is utilized to solve the governing system. The prime outcomes detected that the growth of the radiation and heat generation parameters enhances the convective process. Also, the increase in the magnetic parameter causes a dominance of the nanofluid friction irreversibility compared to the heat transfer one.

Analysis/application of stabilization by the over-integration technique in CBS-SEM for incompressible flow

When the characteristic-based split (CBS) scheme is combined with spectral element method (SEM), high-order nonlinear terms in CBS scheme produce non-negligible aliasing error, which causes severe stability problems. In this paper, a stabilized framework based on CBS-SEM for incompressible flow simulation is established with stabilization by over-integration. The introduced over-integration addresses the aliasing-driven instability by approximating nonlinear terms with higher-order interpolations. A numerical experiment is presented to assess the performance of the stabilized framework in terms of the stability range, solution quality, and computational efficiency. The main conclusions are as follows: (1) Over-integration effectively extends the range of stability for CBS-SEM, where the increase in low-order interpolations is more significant. (2) For extremely under-resolved problems, two types of instabilities, including instability caused by the limitation of CFL condition and instability triggered by aliasing error, are observed in the numerical experiment. The stabilized framework is verified to be effective in handling both types of instability. (3) The order of over-integration only slightly affects the stability, but it markedly improves the capture of complex flow structure for high-order interpolations. In addition, a numerical investigation on incompressible flow past an airfoil is performed to further demonstrate the effectiveness of the framework.

FEM treatments for MHD radiative convective flow of MWCNTs C2H6O2 nanofluids between inclined hexagonal/hexagonal or hexagonal/cylinder

This paper aims to analyze the second law of the thermodynamic for the non-Newtonian power-law nanofluids flow between inclined polygonal/polygonal or polygonal/cylinder shapes. Multi-wall carbon nanotubes are incorporated in the ethylene glycol as a base fluid to formulate the worked suspension. Electromagnetic forces together with the thermal radiation are taken into account. Unusually, the magnetic field is assumed to be horizontal while the flow domain is inclined and a semi implicit treatment is presented for the magnetic terms. The solution methodology is depending on the Galerkin finite element method (FEM) and the Characteristic-Based Split (CBS) scheme. Various designs for the considered models are considered based on the areas and shapes of the inner polygonal/cylinder. The main outcomes revealed that area of the flow and power-law index N control in the convective and heat transfer process. Also, the fluid friction irreversibility takes its maximum values at high values of the power-law index. Furthermore, ratios of the increase in values of the entropy tools ( S T + , S F + , S tot + ) due to the variations of γ take their maximum at γ = 30 and low values of Ha .

Magnetic radiative buoyancy-driven convection of MWCNTs -C2H6O2 power-law nanofluids in inclined enclosures with wavy walls

The current numerical investigation focuses on the power-law nanofluids behaviors in slant wavy enclosures in the presence of thermal radiation. The entropy of the system together with the flow and thermal features are examined for several patterns of the side wavy walls, namely, Out-Out, In-In and In-out. The components of the suspension are multi-wall carbon nanotubes and ethylene glycol. The magnetic field is considered in the current study to be inclined and its components are expressed in terms of the enclosure inclination angle. The solution scheme is based on the Characteristic-Based Split (CBS) scheme and the resulting system is discretized using the finite element method (FEM). It is noted that the flow structures, temperature, and entropy features have strong responses for changing the patterns of the side walls. Also, for the non-inclined domain, an augmentation in the non-Newtonian fluid flow up to 44.2% is given as the radiation coefficient is altered from 0 to 5.

Efficient semi-implicit coupling fluid-structure interaction analysis via model-order reduction of dynamic grids

In this paper, a fluid-structure interaction (FSI) using the model-order reduction (MOR) of dynamic grids is proposed. Within the interface problem between the fluid and the structure, the grid deformation is based on the spring analogy, and it is further extended in order for the efficient computation. In this procedure, the projection-based MOR technique is employed. The relevant MOR technique is realized by using the proper orthogonal decomposition and the discrete empirical interpolation method (POD/DEIM). The resulting FSI framework is based on a semi-implicit coupling approach. In order to achieve this, a characteristic-based split (CBS) scheme is used for an arbitrary Lagrangian–Eulerian (ALE) formulation of the two-dimensional Navier-Stokes equation. Also, a structural analysis is based on the co-rotational (CR) formulation in order to simulate the geometrically nonlinear behavior. Both the fluid and structural analyses are combined with the semi-implicit coupling approach, and the relevant algorithm is embedded in the ALE/CBS scheme. Then, the present FSI analysis is verified using three examples, and the computational efficiency is evaluated. The resulting numerical data demonstrated the validity of the present analysis, and it is found that a significant reduction in the computational cost is possibly achieved with the proposed approach.

Exploration of the entropy due to the nano–encapsulated phase change materials (NEPCMs) within oblique prismatic containers

The finite element treatments (FEM) together with the characteristic-based split (CBS) scheme are applied to examine the entropy generation and convective flow due to the presence of the nano–encapsulated phase change materials (NEPCMs) confined an inclined prismatic containers. The enclosure is filled with elements of a porous medium (glass balls) and a sine formula is considered for the latent heat of the phase change. A combination of the shell and core heat capacity is assumed to formulate the overall heat capacity of the mixture and the computations are carried out for two different designs termed as D1 and D2. Features of the flow, solidification/melting zones, entropy generation, local and average Bejan numbers are illustrated. The main outcomes uncovered that using the second design D2 gives a reduction in the maximum values of the stream functions up to 36%. Also, the ribbon solidification/melting zones are occurred near the cold wall at the low values of while they are extending and moving away from the right inclined wall towards the heated area as is altered.

Finite element treatment for nanofluids flow within enclosures filled with hydrodynamically and thermally anisotropic porous media: Impacts of sinusoidal heating

AbstractThe local thermal non‐equilibrium model (LTNEM) in case of a sinusoidal heating is applied to investigate the unsteady nanofluid flow and thermal fields inside an inclined enclosure filled with a thermally and hydrodynamically anisotropic porous medium. The Galerkin finite element method (GFEM) together with the Characteristic‐based Split (CBS) Scheme is used to solve the dimensionless governing equations. Impacts of the controlling parameters are studied in two cases, namely, non‐Darcy Regime in which values of the Darcy number and the Rayleigh number are, respectively, and the Darcy regime where . Parametric studies are performed for wide ranges of the permeability ratio , the Nield number , the inclination angle , the nanoparticles volume fraction , the inclination of the principal axes , the amplitude ratio and the phase deviation . The main outcomes revealed that the sinusoidal heating results in multicellular flow components inside the geometry. Also, the increase in the Nield number causes that the system converts from the LTNE state to local thermal equilibrium (LTE) state. For the non‐Darcy regime, the convective process and the rate of the heat transfer are augmented as the inclination of the principal axes is increased.

Stabilization of a smoothed finite element semi-implicit coupling scheme for viscoelastic fluid–structure interaction

We propose in this work a stabilization approach for semi-implicit coupling of viscoelastic fluid–structure interaction (VFSI) using the cell-based smoothed finite element method (CS-FEM). The viscoelastic fluid and nonlinear solid equations are spatially discretized by the CS-FEM and then are semi-implicitly coupled via a partitioned solution strategy. The current semi-implicit coupling framework depends on a second-order characteristic-based split (CBS(B)) scheme that solves the Navier–Stokes equations together with the Oldroyd-B constitutive model in the fractional-step manner. To enhance the stability of the semi-implicit coupling algorithm, the discrete elastic-viscous split stress-gradient (DEVSS-G) procedure is introduced into the explicit stage while the stabilized pressure gradient projection (SPGP) is earmarked for the implicit stage. Moreover, the iterated end-of-step velocity begins with the intermediate velocity during the subiterations. The DEVSS-G/CBS(B)-SPGP technique is readily applied to the CBS-based partitioned semi-implicit coupling algorithm for VFSI. Visible improvements in stabilization and efficiency are revealed in a benchmark test. • A semi-implicit coupling scheme is proposed for viscoelastic fluid–structure interaction. • DEVSS-G/CBS(B)-SPGP stabilization is presented for the semi-implicit scheme. • VFSI system is entirely formulated by cell-based smoothed finite element method. • The developed technique boosts the stability and efficiency considerably.

Impacts of the Variable Properties of a Porous Medium on the Entropy Analysis Within Odd-Shaped Enclosures Filled by Hybrid Nanofluids

The entropy generation from convective hybrid nanofluid flow within the odd-shaped geometries using the finite element method is examined is examined. The flow domain is filled by variable porosity and permeability porous media. Also, the thermal conductivity is assumed to be heterogeneous, and the Brinkman extended non-Darcy model is applied to simulate the porous medium. The geometry is considered partially/fully filled by the porous medium and four cases are assumed, namely, only the horizontal channel is a porous (case 1), only the vertical channel is a porous (case 2), the entire domain is a porous (case 3) and the entire domain is a non-porous (case 4). Different designs of the odd-geometry are taken into account based on the aspect ratio. The characteristic-based split (CBS) based on semi-implicit (SI) scheme is used to treat the governing equations. Simulations are carried out for various values of the maximum Darcy parameter $${\text{Da}}_{\hbox{max} }$$ , alumina-copper volume fraction $$\phi_{Al} , \phi_{Cu}$$ and different conditions of the porous domain. It is noted that the case of the heterogeneous thermal conductivity maximizes the irreversibility from heat transfer and total entropy generation. Also, a weakness in values of the average Bejan number is obtained as the Rayleigh number is grown, regardless the porous conditions.