Characteristic-based Split Scheme Research Articles

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.

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.

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.

MHD convective process due to rotation of cylinders and movement of a wavy wall of two-sided wavy enclosures with radiation

PurposeThis paper aims to study the mixed convective process due to various dynamics, namely, inner rotating cylinders and upper-wavy wall movement for the first time.Design/methodology/approachThe Galerkin finite element method together with the characteristic-based split scheme is applied to solve the governing system.FindingsThe main outcomes revealed that the direction of the rotation of the cylinders, radius and locations of the rotating shapes are beneficial controlling elements for the enhancement of heat transfer. Also, for all the considered cases, values of the Bejan number indicate that the fluid friction irreversibility is dominance compared to the heat transfer irreversibility. Further, average values of the heat transfer entropy, fluid friction entropy and total entropy are minimized in the case of fixed cylinders regardless of the cylinder radius.Originality/valueThe authors are interested in the mixed convection case due to regular boundaries and hence this simulation purposes a first attempt to examine the mixed convective flow due to irregular wavy boundaries. This study considered various dynamics, namely, inner rotating cylinders and wavy-lid driven wall which makes it more attractive to the readers. Various cases based on radius of the cylinder and direction of the rotations together with several locations of the rotating shapes are taken into account which makes the current simulation is comprehensive. Various studies presented in this field are made by commercial software and these treatments need special conditions (having limitation) but the current solution methodology is based on a finite element method home-code. Various important impacts, are, also, examined, namely, inclined geometry, inclined magnetic field, thermal radiation and heat generation/absorption. The entropy of the current complex system is analyzed based on the second law of thermodynamics.

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.

Nanofluid Flows within Porous Enclosures Using Non-Linear Boussinesq Approximation

In this paper, the Galerkin finite element method (FEM) together with the characteristic-based split (CBS) scheme are applied to study the case of the non-linear Boussinesq approximation within sinusoidal heating inclined enclosures filled with a non-Darcy porous media and nanofluids. The enclosure has an inclination angle and its side-walls have varying sinusoidal temperature distributions. The working fluid is a nanofluid that is consisting of water as a based nanofluid and Al2O3 as nanoparticles. The porous medium is modeled using the Brinkman Forchheimer extended Darcy model. The obtained results are analyzed over wide ranges of the non-linear Boussinesq parameter 0 ≤ ζ ≤ 1, the phase deviation 0° ≤ Φ ≤ 180°, the inclination angle 0° ≤ γ ≤ 90°, the nanoparticles volume fraction 0% ≤ φ ≤ 4%, the amplitude ratio 0 ≤ a ≤1 and the Rayleigh number 104 ≤ Ra ≤ 106. The results revealed that the average Nusselt number is enhanced by 0.73%, 26.46% and 35.42% at , 105 and 106, respectively, when the non-linear Boussinesq parameter is varied from 0 to 1. In addition, rate of heat transfer in the case of a non-uniformly heating is higher than that of a uniformly heating. Non-linear Boussinesq parameter rises the flow speed and heat transfer in an enclosure. Phase deviation makes clear changes on the isotherms and heat transfer rate on the right wall of an enclosure. An inclination angle varies the flow speed and it has a slight effect on heat transfer in an enclosure.

Effects of flow-induced vibration on forced convection heat transfer from two tandem circular cylinders in laminar flow

Flow-induced vibrations (FIVs) and forced convection heat transfers of two tandem circular cylinders in laminar flow with Re=150 and Pr=0.7 are investigated using fluid-structure interaction (FSI) simulation. The cylinder is heated, elastically supported and only allowed to vibrate in the transverse direction, with the mass ratio, damping ratio, centre-to-centre distance fixed respectively at m*=2, ζ=0, L/D=4 and the reduced velocity varied in Ur=1~20. The two-dimensional incompressible Navier-Stokes equations and energy equation are coupled with the vibrating equation of the spring-mass-damper model to describe FIV of two tandem cylinders involving forced convection heat transfer process. An FSI solver is proposed by combing the modified characteristic-based split scheme, dual-time stepping method, segment spring analogy method and loosely coupled partitioned approach, the accuracy and stability of which are validated using two benchmark models including the forced convection heat transfer from a fixed circular cylinder and FIV of an elastically supported circular cylinder. Using FSI simulation, characteristics of FIVs and their effects on forced convection heat transfers of two tandem circular cylinders are analyzed in detail. The obtained numerical results are insightful to the understanding of the heat transfer features of a group of cylindrical structures under FIVs, such as the tube bundles in heat exchangers and power plants, fuel rod bundles in nuclear reactors and riser system in offshore oil and gas platforms.

A Divergence-Free Immersed Boundary Method and Its Finite Element Applications

ABSTRACTIn order to maintain the no-slip condition and the divergence-free property simultaneously, an iterative scheme of immersed boundary method in the finite element framework is presented. In this method, the Characteristic-based Split scheme is employed to solve the momentum equations and the formulation for the pressure and the extra body force is derived according to the no-slip condition. The extra body force is divided into two divisions, one is in relation to the pressure and the other is irrelevant. Two corresponding independent iterations are set to solve the two sections. The novelty of this method lies in that the correction of the velocity increment is included in the calculation of the extra body force which is relevant to the pressure and the update of the force is incorporated into the iteration of the pressure. Hence, the divergence-free properties and no-slip conditions are ensured concurrently. In addition, the current method is validated with well-known benchmarks.

FEM-CBS algorithm for convective transport of nanofluids in inclined enclosures filled with anisotropic non-Darcy porous media using LTNEM

PurposeThe Galerkin finite element method (FEM) based on the characteristic-based split (CBS) scheme is applied to simulate the nanofluid flow and thermal fields inside an inclined geometry filled by a heat-generating hydrodynamically and thermally anisotropic non-Darcy porous medium using the local thermal non-equilibrium model (LTNEM). Property of the hydrodynamic anisotropy is taken in both the Forchheimer coefficient and permeability and these tools are considered as functions of inclination of the principal axes. Also, the thermal conductivity for the porous phase is assumed to be anisotropic.Design/methodology/approachThe Galerkin FEM based on the CBS scheme is applied to solve the partial differential equations governing the flow and thermal fields.FindingsIt is noted that the net rate of the heat transfer between the nanofluid and solid phases are influenced by variations of the anisotropic properties. Also, the system is reached to the thermal equilibrium state at H > 100. Further, the maximum nanofluid temperature is reduced by 12.27% when the nanoparticles volume fraction is varied from 0% to 4%.Originality/valueThis paper aims to study the nanofluid flow and heat transfer characteristics inside an inclined enclosure filled with a heat-generating, hydrodynamically and thermally anisotropic porous medium using the CBS scheme. The LTNEM is considered between the nanofluid and porous phases while the local thermal equilibrium model (LTEM) between the base fluid (water) and the nanoparticles (alumina) is taken into account. The Galerkin FEM is introduced to discretize the governing system of equations. Also, examine influences of the anisotropic properties (permeability, Forchheimer terms and thermal conductivity of the porous medium), inclination angle and nanoparticles volume fraction on the net rate of the heat transfer between the nanofluid and porous phases and on the local thermal non-equilibrium state is one of the concerns of this paper.

An iterative divergence-free immersed boundary method in the finite element framework for moving bodies

A novel iterative direct-forcing immersed boundary (IB) method, which meets the divergence-free and no-slip conditions simultaneously, is proposed in the finite element framework for incompressible flow problems. The coupled velocity and the pressure of the fluid field are solved by the Characteristic-based Split scheme. The formulation for the extra body force is derived according to the no-slip boundary condition. The novelty of the proposed IB method lies in that the calculation of the pressure, the iterations of the velocity, the pressure and the extra body force are accomplished in a loop at the same time. Therefore, the divergence-free and no-slip conditions are ensured fully at the same time. The accuracy of the current IB method is verified by several cases, including static and moving boundaries. Meanwhile, the contributing factors of the spurious force oscillations for moving boundaries are studied. A brand-new viewpoint is proposed, which is different with the common standpoint. The major source of spurious force oscillations pointed out in this paper is that the change of interpolation coefficients when boundaries sweep within an element, instead of the change of interpolation points when boundaries move across adjacent elements, as discussed in existing references. The oscillations which are caused by the inherent properties of the immersed boundary method cannot be avoided, while effective methods can be employed to suppress them, including refining the meshes and applying a proper discrete delta function.

A strongly-coupled cell-based smoothed finite element solver for unsteady viscoelastic fluid–structure interaction

This paper describes a cell-based smoothed finite element method (CS-FEM) for computing unsteady viscoelastic fluid–structure interaction (VFSI) within the arbitrary Lagrangian–Eulerian framework. The incompressible Navier–Stokes equations incorporating the Oldroyd-B constitutive relation are decoupled via the smoothed characteristic-based split scheme that allows equal low-order interpolations for the triple primitive variables. The elastodynamics equation of a geometrically nonlinear solid is solved by the modified Newton–Raphson method in conjunction with the generalized-α method. Following an efficient moving mesh strategy, the block-Gauss–Seidel procedure is utilized to tightly couple three interacting fields. In particular, CS-FEM is adopted to spatially discretize the global VFSI system where all gradient related terms are readily smoothed. The cell-based smoothing concept is also introduced to evaluate viscoelastic fluid forces acting on the deformable body. Two transient VFSI examples are analyzed to demonstrate the enlarged applicability of CS-FEM. The main characteristics of viscoelastic flow-induced oscillations are successfully captured.

A cell-based smoothed CBS finite element formulation for computing the Oldroyd-B fluid flow

In this paper, a cell-based smoothed finite element method (CS-FEM) incorporating characteristic-based split (CBS) scheme is developed to compute viscoelastic fluid flow. The system of the Navier–Stokes and Oldroyd-B constitutive equations is decoupled in tandem with equal low-order approximation of the triple primitive variables. The spatial discretization is based upon CS-FEM which readily smooths all gradient-related entries of the governing equations over bilinear four-node quadrilateral elements. Moreover, the cell-based smoothing concept is applied to evaluation of viscoelastic fluid force. Four benchmark problems are replicated to demonstrate the applicability and robustness of the smoothed CBS finite element formulation. A good agreement is exposed between the present and previous results.

A modified immersed smoothed FEM with local field reconstruction for fluid–structure interactions

A modified immersed smoothed finite element method (mIS-FEM) is proposed for fluid–structure interactions (FSI) using linear triangular elements in this paper. A sharp interface treatment is proposed and implemented to modify the original IS-FEM. The proposed treatment reconstructs the fluid velocity field near interface using auxiliary triangles, leading to better resolutions without much extra efforts. In the proposed method, the fluid solver is the FEM with a semi-implicit characteristic-based split (CBS) scheme using linear triangles. For the solid solver, the S-FEM based on explicit total Lagrangian formulation is used to solve finite solid deformation. Several numerical examples are employed to verify proposed mIS-FEM, and to examine the improvements. The results demonstrate that mIS-FEM is a reliable and robust numeric method for interactions between fluid and solids undergo large deformation.

The cell-based smoothed finite element method for viscoelastic fluid flows using fractional-step schemes

This paper generalizes the cell-based smoothed finite element method (CS-FEM) to viscoelastic fluid flows under the fractional-step umbrella. The characteristic-based split (CBS) scheme is employed to decouple the Navier–Stokes equations and the Oldroyd-B constitutive model. Moreover, the discrete elastic-viscous-split-stress (DEVSS) procedure and stabilized pressure gradient projection (SPGP) are introduced into a second-order CBS framework. The projections of velocity and pressure gradients therein combine to form the DEVSS-G/CBS(B)-SPGP algorithm that further stabilizes the discretized model. Both proposed schemes are suitable for use in CS-FEM because they accept equal low-order interpolations for primitive variables. The spatial discretization is based upon bilinear four-node quadrilateral elements where all gradient-related items are smoothed without any methodological difficulties. In particular, the cell-based smoothing concept enables accurate evaluation of viscoelastic fluid force. The developed methods are validated against previously published data for several benchmark problems. A satisfactory agreement exists between the present and earlier results.

Forced convection heat transfer from a circular cylinder with a flexible fin

Forced convection heat transfer from a circular cylinder with a flexible fin in laminar flow with Re = 200 and Pr = 0.7 is investigated numerically. The two-dimensional incompressible Navier-Stokes equations and energy equation are coupled with the Euler-Bernoulli beam equation to describe the flow-induced vibration (FIV) of the flexible fin considering the convection heat transfer process. The modified characteristic-based split scheme, Galerkin finite element method, semi-torsional spring analogy method and loosely coupled partitioned approach are employed irrespectively for the flow and convection heat transfer, fin vibration, mesh movement and fluid–structure interaction. The accuracy and stability of the proposed numerical method are validated using three benchmark models including the forced convection heat transfer from a stationary cylinder, forced convection heat transfer from a transversely oscillating cylinder and FIV of a flexible plate behind a square cylinder. Finally, forced convection heat transfer characteristics from a circular cylinder with a flexible fin with fin length l = 0.5D–1.5D (D is the cylinder diameter) and elastic modulus E = 104 - 5 × 105 are analyzed in detail. The numerical results show that, when the vortex shedding frequency approaches the natural frequency of the flexible fin, the FIV frequency is locked on the natural frequency and the fin exhibits large-amplitude vibration. As a result, the ‘dead water’ region behind the cylinder is reduced and the convection heat transfer is improved. In the combinations of parameters considered, a maximum of 11.07% enhancement in heat transfer is obtained by the flexible fin.

The use of artificial compressibility to improve partitioned semi-implicit FSI coupling within the classical Chorin–Témam projection framework

Over the last decade the classical Chorin–Témam projection method has been utilized to address fluid-structure interaction in a semi-implicit manner. In previous studies the fluid projection step is fully coupled with the structural motion due to the divergence-free constraint. A set of simultaneous equations thus have to be iteratively solved. To overcome this difficulty, a simple and accurate partitioned semi-implicit coupling method is proposed based on the artificial compressibility (AC) in this paper. The iterated AC parameter decouples the pressure, end-of-step velocity and structural motion within the characteristic-based split scheme. The present approach is completely matrix-free and has unlimited access to the finite elements. Its performance is demonstrated for an oscillating bluff body subjected to uniform flows.