Second-order Discretization Scheme Research Articles

Numerical Study of the Effect of Primary Nozzle Geometry on Supersonic Gas-Solid Jet of Bypass Injected Dry Powder Fire Extinguishing Device

A two-way coupled model between polydisperse particle phases with compressible gases and a density-based coupling implicit solution method, combining the third-order MUSCL with QUICK spatial discretization scheme and the second-order temporal discretization scheme, are constructed based on the discrete-phase model (DPM) and the stochastic wander model (DRWM) in the Eulerian–Lagrangian framework in conjunction with a unitary particulate source (PSIC) approach and the SST k-ω turbulence model. The accuracy of the numerical prediction method is verified using previous supersonic nozzle gas-solid two-phase flow experiments. Numerical simulation of a two-phase jet of dry powder extinguishing agent gas with pilot-type supersonic nozzle was performed to analyze the influence of geometrical parameters, such as the length ratio rL and the area ratio rA of the main nozzle on the two-phase flow field, as well as on the jet performance indexes, such as the particle mean velocity vp,a, velocity inhomogeneity Φvp, particle dispersion Ψp, particle mean acceleration ap,a, etc. By analyzing the parameters, we indicate the requirements for the combination of jet performance metrics for different flame types such as penetrating, spreading, and dispersing.

Development of a Fully-Implicit Second-Order system Thermal-Hydraulics analysis code for reactor modeling and simulations

There has been continuous advancement in numerical techniques and software engineering in the field of system thermal-hydraulics over the past decade. The current state-of-the-art involves the use of fully-implicit high-resolution schemes in conjunction with the Jacobian-Free Newton-Krylov (JFNK) method for both single-phase and two-phase flows. However, the application of these advanced schemes in system thermal-hydraulics codes is not a common practice. Given the opportunity and the need to develop a new modern system thermal–hydraulic code from scratch, there is no doubt that these recent advancements should be leveraged. In this work, we introduce a new modern object-oriented system thermal-hydraulics code named RETA. RETA is characterized by several key features, including: second-order temporal and spatial discretization schemes for both incompressible and compressible flows, as well as single-phase and two-phase flows; fully-implicit solution schemes utilizing both the Newton and PJFNK methods; and an object-oriented design of major fluid components and physical models, which allows for easy extension. A series of verification and demonstration studies are conducted to verify the implementation, demonstrate the accuracy improvement, and evaluate the performance of the new code. It is concluded that results from the second-order scheme are generally more reliable. Without the concern of a code stability issue, the second-order scheme should be the preferred and default choice for a code user or an analyst.

A second-order fully-balanced structure-preserving variational discretization scheme for the Cahn–Hilliard–Navier–Stokes system

We propose and analyze a structure-preserving space-time variational discretization method for the Cahn–Hilliard–Navier–Stokes system. Uniqueness and stability for the discrete problem is established in the presence of concentration-dependent mobility and viscosity parameters by means of the relative energy estimates and order optimal convergence rates are established for all variables using balanced approximation spaces and relaxed regularity conditions on the solution. Numerical tests are presented to demonstrate the proposed method is fully practical and yields the predicted convergence rates. The discrete stability estimates developed in this paper may also be used to analyse other discretization schemes, which is briefly outlined in the discussion.

Effect of cube spacings on the three-dimensional flow structure over an array of wall-mounted cube

The turbulent flow over an array of cubes mounted on one of the walls of a channel has been investigated using direct numerical simulation for cube spacing that ranges between 2.0 and 4.0. The Reynolds number based on the cube size and the average streamwise velocity is chosen to be 4000. The Navier–Stokes equations have been discretized using second-order spatial and temporal discretization schemes. The present investigation focuses on the flow structures and comprehensive characterization of the separated zones surrounding the cubes, as well as the associated wall-shear stress. A vortex shedding has been observed for the cube spacings of 3.0 and 4.0 without any evidence of vortex shedding for the lowest pitch of 2.0. For the two cases having pitches of 3.0 and 4.0, the presence of the unsteady separation bubbles at the cube's top and side surfaces results in a decrease in wall-shear stress. The quadrant analysis for the region close to the top surface of the cube is performed with the help of the joint probability density function, which reveals dual peaks within the recirculation bubble at the top surface of the cube for higher cube spacings. By conducting an invariant analysis of the Reynolds stress tensor for different cube spacings, we have explored the characteristic of Reynolds stress anisotropy due to the total fluctuations. The production of negative turbulent kinetic energy (TKE) is observed in different regions within the flow domain, among which the horseshoe vortex region for each cube spacing reveals its dominant presence. The physical mechanism responsible for the production of the negative TKE has also been attempted by decomposing the production term into two parts, namely, normal and shear components.

Scalar auxiliary variable approache for the surface quasi-geostrophic equation

We present a numerical method for approximating the two-dimensional surface quasi-geostrophic equation. We first reformulate the equation into an equivalent system by using the scalar auxiliary variable approach. Then we propose first-order and second-order discretization schemes for solving the new surface quasi-geostrophic system in time. Furthermore, we prove that both of these schemes based on the scalar auxiliary variable approach satisfy an unconditionally energy stability property. Since the method gives two linear equations with constant coefficients in each time step. We have an efficient approach and accurate for solving these spacial fractional diffusion equations. Ample numerical experiments are carried out to validate the correctness of these schemes and their accuracy for inviscid problems and the problems with small viscosity.

Entropy generation analysis of a dual-height plate-fin heat sink with Nanofluid coolant applying two-phase mixture model

In this work, the two-phase mixture model is applied to solve the velocity field, the pressure field and obtain the entropy generation rates in a dual-height plate-fin heatsink. The governing equations were solved using the second-order upwind discretization scheme as well as SIMPLE scheme used in coupling the pressure and velocity fields. The Re, nanoparticle concentration (ψ), secondary fin height (hsf), and fin spacing (dsf) were considered within the ranges of 500-2000, 0%-1%, 5 mm-17.5 mm, and 0.25 mm-0.5 mm, respectively. The results demonstrated that S˙th almost decrease by 30% as hsf increases from 5 mm to 17.5 mm. The increase in hsf escalates S˙fr by 308% and 238% at Re numbers of 500 and 2000, respectively. In addition, the escalation of Re from 500 to 2000 increases S˙fr by 2125% and 1748% for hsf 5 mm and 17.5 mm, respectively. The fin spacing analysis revealed that the lowest S˙th and highest S˙fr belong to dsf=0.5 mm. The lower the dsf the higher the backward flow and mixing flow at the heatsink outlet, thereby delay in flow passage and intensification of S˙th and S˙fr.

Numerically Detection Fluid Characteristic Effects in Porous Media for Plastic Manufacturing Process Reconstruction

Research work aimed to prove the cooling effectiveness and efficiency of heating numerically. It was carried out by reconstructing the mold to reduce condensation while keeping the production cycle time. The simulation was done using CFD Fluent, Finite Volume Method (FVM) with a solution method of pressure-velocity, SIMPLE coupling, and second-order upwind discretization scheme. The model's boundary condition represents plastic packaging's production process. The simulation was done to optimize performance during production. Reconstruction and simulation were undertaken. The initial investigation was to produce data for cooling time versus the whole production time. The research result indicated that the average cooling time was 3/5 of the total production time. There were 15.5706 of 26.4206 sec and 15.641 of 26.491 sec, respectively. Further investigation was focused on investigating heat absorption and transfer through cells in the matrix. The porous effect showed temperature gradient reduction between cooled and cooler temperatures by simulation in color. The phenomenon predicted that the Ra contributed to the rate of temperature decrease revealed. There is also an apparent increase in temperature interfaces is evidence. The porous media application with The Rayleigh Number ≥ 105 retained the potential cooling effectivity and heating efficiency. By simulation, a series of discussions of the result of research, the potential may apply in the plastic mold to reconstruct the space and its methods using filling-porous. It is relevant because the heating ability in the phase change of fluid that creates particular characteristics has the potential to make effective cooling.

Effect of Magnetic Baffles and Magnetic Nanofluid on Thermo-Hydraulic Characteristics of Dimple Mini Channel for Thermal Energy Applications

The combined effect of a magnetic baffle and a dimple turbulator on the heat transfer and pressure drop is investigated computationally in a mini channel. Fe3O4 magnetic nanofluid is used as a working fluid. The Reynolds number (Re) is varied from 150 to 210 and the magnetic field intensities range from 1200 G to 2000 G. Finite-volume based commercial computational fluid dynamics (CFD) solver ANSYS-Fluent 18.1 was used for the numerical simulations. A laminar viscous model is used with pressure-velocity coupling along with second-order upwind discretization and QUICK scheme for discretizing the momentum and energy equations. The results show that there is an increase of 3.53%, 10.77%, and 25.39% in the Nusselt numbers when the magnetic fields of 1200 G, 1500 G and 2000 G, respectively, are applied at x = 15 mm, as compared to the flow without a magnetic field when the pitch = 10 mm. These values change to 1.51%, 6.14% and 18.47% for a pitch = 5 mm and 0.85%, 4.33%, and 15.25% for a pitch = 2.5 mm, when compared to the flow without a magnetic field in the respective geometries. When the two sources are placed at x = 7.5 mm and 15 mm, there is an increase of 4.52%, 13.93%, and 33.08% in the Nusselt numbers when magnetic fields of 1200 G, 1500 G, and 2000 G are applied when the pitch = 10 mm. The increment changed to 1.82%, 8.16%, and 22.31% for a pitch = 5 mm and 1.01%, 5.96%, and 21.38% for a pitch = 2.5 mm. This clearly shows that the two sources at the front have a higher increment in the Nusselt numbers compared to one source, due to higher turbulence. In addition, there is a decrease in the pressure drop of 10.82%, 16.778%, and 26.75% when magnetic fields of 1200 G, 1500 G, and 2000 G, respectively, are applied at x = 15 mm, as compared to flow without magnetic field when the pitch = 10 mm. These values change to 2.46%, 4.98%, and 8.54% for a pitch = 5 mm and 1.62%, 3.52%, and 4.78% for a pitch = 2.5 mm, when compared to flow without magnetic field in the respective geometries. When two sources are placed at x = 7.5 mm and 15 mm, there is an decrease of 19.02%, 31.3%, and 50.34% in the pressure drop when the magnetic fields of 1200 G, 1500 G and 2000 G are applied when the pitch = 10 mm. These values change to 4.18%, 9.52%, and 16.52% for a pitch = 5 mm and 3.08%, 6.88%, and 14.88% for a pitch = 2.5 mm. Hence, with the increase in the magnetic field, there is a decrease in pressure drop for both the cases and the pitches. This trend is valid only at lower magnetic field strength, because the decrease in the pressure drop dominates over the increase in pressure drop due to turbulence.

A fast second-order discretization scheme for the linearized Green-Naghdi system with absorbing boundary conditions

In this paper, we present a fully discrete second-order finite-difference scheme with fast evaluation of the convolution involved in the absorbing boundary conditions to solve the one-dimensional linearized Green-Naghdi system. The Padé expansion of the square-root function in the complex plane is used to implement the fast convolution. By introducing a constant damping parameter into the governing equations, the convergence analysis is developed when the damping term fulfills some conditions. In addition, the scheme is stable and leads to a highly reduced computational cost and low memory storage. A numerical example is provided to support the theoretical analysis and to illustrate the performance of the fast numerical scheme.

Fast fluid dynamics simulation of airflow around a single bluff body under different turbulence models and discretization schemes

Fast and accurate simulation of outdoor airflow distribution is important for studying urban microclimate. Choosing a reasonable turbulence model and discretization scheme is not only related to the computational accuracy but also to efficiency. However, conventional CFD methods are computationally intensive and slow for unsteady problems, and thus cannot meet the demand for fast simulation of urban microclimate. In this paper, three pressure-correction schemes (i.e., NIPC, SIPC, and NSPF) for solving the N–S equation item by item are implemented in OpenFOAM, and then the differences are compared when applying different turbulence models and discretization schemes to quickly simulate the airflow distribution around a single 1:1:2 bluff body. All pressure-correction schemes can accurately predict the main airflow characteristics around the bluff body. The three schemes are about 2.5–3.5 times faster than the PISO algorithm, and they take the shortest computational time when applying RKE, followed by SQKE and RNG, while the longest computational time is required when applying SKE and LBKE. NIPC and SIPC have similar computational speeds, while NSPF is about 10–16% faster than them. The pressure-correction scheme with the first-order upwind scheme is about 6–10% faster than the second-order discretization scheme. Considering both computational accuracy and efficiency, the combination of NSPF with RNG or SQKE turbulence model and first-order upwind scheme may be a reasonable choice to quickly simulate the urban airflow distributions.

On fully decoupled MSAV schemes for the Cahn–Hilliard–Navier–Stokes model of two-phase incompressible flows

We construct first- and second-order time discretization schemes for the Cahn–Hilliard–Navier–Stokes system based on the multiple scalar auxiliary variables (MSAV) approach for gradient systems and (rotational) pressure-correction for Navier–Stokes equations. These schemes are linear, fully decoupled, unconditionally energy stable, and only require solving a sequence of elliptic equations with constant coefficients at each time step. We carry out a rigorous error analysis for the first-order scheme, establishing optimal convergence rate for all relevant functions in different norms. We also provide numerical experiments to verify our theoretical results.

Fast Fluid Dynamics Simulation of the Airflow Around a Single Bluff Body

Fast and accurate simulation of the outdoor airflow distribution is important for studying urban microclimate. In this paper, two pressure-correction schemes (i.e., NIPC and NSPF) for solving the N-S equation item by item are implemented in OpenFOAM and their differences from the PISO algorithm in simulating the airflow around a single 1:1:2 bluff body are analyzed. The RNG k-ε turbulence model is chosen to study the airflow disturbance, while the second-order discretization scheme of Gauss limitedLinear is used to solve the advection term in the N-S equation. The results show that the NIPC can accurately predict the main airflow characteristics around the bluff body, while the NSPF cannot predict the recirculation region on its top. The two pressure-correction schemes underestimate the TKE distribution on the top and leeward sides of the bluff body when applying the RNG k-ε turbulence model, and the maximum relative error is about 30%. However, they are consistent with the results of the PISO algorithm under the same conditions. The two schemes are about 2.5-3.0 times faster than the PSIO algorithm when run on a CPU, and the NSPF is about 12% faster than the NIPC scheme.

Systematic Validation Study of an Unsteady Cavitating Flow over a Hydrofoil Using Conditional Averaging: LES and PIV

We present results of Large-eddy simulations (LES) modeling of steady sheet and unsteady cloud cavitation on a two-dimensional hydrofoil which are validated against Particle image velocimetry (PIV) data. The study is performed for the angle of attack of 9° and high Reynolds numbers ReC of the order of 106 providing a strong adverse pressure gradient along the surface. We employ the Schnerr–Sauer and Kunz cavitation models together with the adaptive mesh refinement in critical flow regions where intensive phase transitions occur. Comparison of the LES and visualization results confirms that the flow dynamics is adequately reproduced in the calculations. To correctly match averaged velocity distributions, we propose a new methodology based on conditional averaging of instantaneous velocity fields measured by PIV which only provides information on the liquid phase. This approach leads to an excellent overall agreement between the conditionally averaged fields of the mean velocity and turbulence intensity obtained experimentally and numerically. The benefits of second-order discretization schemes are highlighted as opposed to the lower-order TVD scheme.

Error estimates of local discontinuous Galerkin method with implicit-explicit Runge Kutta for two-phase miscible flow in porous media

Two-phase miscible flow in porous media is described by a coupled system of nonlinear partial differential equations. In this paper, we develop and analyze the fully discrete local discontinuous Galerkin method coupled with the implicit-explicit Runge Kutta (IMEX-RK) method for the problems. Second-order time discretization scheme in two adjacent time layer can be obtained with the IMEX-RK method. For the temporal-spatial discretization schemes, we give the optimal convergence analysis for both the pressure and concentration in L2-norm. We present several numerical examples for two-phase miscible flow problem to demonstrate the effectiveness and robustness of our method.

A simple multiscale method for mean field games

This paper proposes a multiscale method for solving the numerical solution of mean field games which accelerates the convergence and addresses the problem of determining the initial guess. Starting from an approximate solution at the coarsest level, the method constructs approximations on successively finer grids via alternating sweeping, which not only allows for the use of classical time marching numerical schemes, but also enables applications to both local and nonlocal problems. At each level, numerical relaxation is used to stabilize the iterative process. A second-order discretization scheme is derived for higher order convergence. Numerical examples are provided to demonstrate the efficiency of the proposed method in both local and nonlocal, 1-dimensional and 2-dimensional cases.

CFD analysis for airflow distribution of a conventional building plan for different wind directions

Abstract Computational fluid dynamics analysis of a building plan has been investigated with predominant wind velocity for different wind directions. The flow properties’ variation in the computational domain has been modeled by solving the Reynolds-Averaged Navier–Stokes (RANS) equations with the finite volume second-order discretization scheme. The turbulence of airflow distribution in and around the building has been modeled with the Shear Stress Transport (SST) k–ω turbulence model from the analysis of different turbulence models. Numerical results are analysed by evaluating and comparing the various flow properties at different building plan locations with different wind directions. The development of pressure coefficients, wind-driven driving force, and air change per hour are studied for different wind directions. From the analysis of numerical results, it is identified that better ventilation with sufficient airflow distribution has existed when the wind is coming from the west direction.

Two-grid scheme of expanded mixed finite element method for semilinear parabolic integro-differential equations

ABSTRACT In this paper, we present a two-grid scheme of expanded mixed finite element method combined with two second-order time discretization schemes for semilinear parabolic integro-differential equations and give a detailed convergence analysis. On the coarse grid space, we first use the Crank–Nicolson scheme to solve the original nonlinear problem at the first time step, then we utilize the Leap-Frog scheme at the rest time levels. Next, we make use of the known coarse mesh solution and Taylor expansion to infer the solution on the fine mesh space. Thus, we only need to solve the nonlinear problem once on the coarse grid space of the two-grid scheme. Finally, a numerical example is presented to verify the effectiveness of the proposed two-grid scheme.

Numerical Study of Bach-bladed Savonius Wind Turbine with Varying Blade Shape Factor

Savonius wind turbine with Bach-profile blades is considered in this study. Previous studies have shown that a rotor with the Bach-profile blade produces better performance than a standard Savonius turbine. This study focuses on the blade shape factor variations of the Bach-profile blade to give the best performance. Two-dimensional unsteady simulations are performed with moving mesh. The configuration being tested is the Savonius rotor with Bach-profile blades with an arc angle of 135◦ . The blade shape factor is varied 0.2, 0.3, 0.4 at a constant freestream velocity of 4 m/s, with a corresponding Reynolds number of 20,000. The k-ω Shear Stress Transport turbulence model was used, with secondorder discretization schemes for the pressure and momentum equations. The boundary conditions were set as velocity inlet for the inlet, outflow for the outlet, and walls for the blade surfaces. The top and bottom sides were set as symmetric. Results showed that the configuration with a shape factor of 0.4 gave the best performance among the others. This configuration gave a higher moment coefficient and power coefficient of about 6.8% and 7.3%, respectively. Results extracted from the simulation includes the flow structure, and the distribution of the pressure coefficients along the blade surface.

Computational fluid dynamics study of Savonius rotors using OpenFOAM

In this work, two-dimensional models of Savonius rotors are simulated using OpenFOAM® in order to predict the aerodynamic performance of small-scale vertical-axis wind turbines. The results are reported analyzing the aerodynamic performance and forces acting on the rotors. Power coefficient, [Formula: see text], is compared with experimental data for each operation point, and for three different geometries. Simulations with first- and second-order discretization schemes are carried out and compared, both quantitative and qualitative. Since usual grid dimensions result not to be suitable for simulations of Savonius rotors, an analysis of different domains is performed and compared. Finally, a set up for computational fluid dynamics simulation of two-dimensional Savonius rotors is proposed. The fluid–rotor interaction is analyzed and the vortex shedding is correlated with [Formula: see text] values and wake description.

Effect of sinus size and position on hemodynamics during pulsatile flow in a carotid artery bifurcation

Background and objectivesHemodynamics plays a crucial role in the progression of atherosclerosis and the treatment of arterial diseases. Stroke is one of the arterial diseases and a leading cause of death worldwide. Hemodynamics in the carotid artery plays a vital role in the stroke. The common carotid artery bifurcates into the internal carotid artery and the external carotid artery. Carotid sinus, a slightly dilated area, exists in the internal carotid artery just after the bifurcation and acts as a pressure receptor and regulator. The location and size of the sinus can vary in different people; the change in sinus size and location may affect the hemodynamics. It is necessary to study the shift in hemodynamics due to changes in sinus size and position on atherosclerosis. The change in flow behavior may suggest the probable sites of backflow and low wall shear stress, and therefore the sites prone to atherosclerosis. MethodsThe model of the carotid artery has been constructed using patient data. Transient computational fluid dynamics simulations have been performed using a finite volume method for the numerical solution in a three-dimensional computational domain using ANSYS Fluent 19.2. Pulsatile flow is specified at the inlet boundary. The coupled scheme is used for the pressure-velocity coupling. The second-order discretization scheme is used for pressure interpolation and second-order upwind scheme is used for the discretisation of momentum equation. The temporal term is discretized using the first-order implicit scheme. ResultsThe effect of sinus size and location on the overall flow behavior, wall shear stress, and secondary flow are presented. Results show that the outer wall of bifurcation has low wall shear stress and bigger recirculation as compared with that on the inner wall of bifurcation. Numerical results obtained for varying sinus size and position are shown in graphs and contours, including wall shear stress, secondary flow, and velocity streamlines. ConclusionNumerical results reveal that sinus away from bifurcation, and larger diameter sinus has more recirculation and low wall shear stress. Therefore, the person having sinus away from bifurcation and larger sinus diameter are more susceptible to plaque formation.