Pressure-velocity Coupling Research Articles

Implicit–explicit Crank–Nicolson scheme for Oseen’s equation at high Reynolds number

In this paper, we continue the work on implicit–explicit (IMEX) time discretizations for the incompressible Oseen’s equations that we started in Burman et al. [Implicit–explicit time discretization for Oseen’s equation at high Reynolds number with application to fractional step methods, SIAM J. Numer. Anal. 61 (2023) 2859–2886]. The pressure velocity coupling and the viscous terms are treated implicitly, while the convection term is treated explicitly using extrapolation. Herein, we focus on the IMEX Crank–Nicolson method for time discretization. For the discretization in space we consider finite element methods with stabilization on the gradient jumps. The stabilizing terms ensure inf–sup stability for equal order interpolation and robustness at high Reynolds number. Under suitable Courant conditions, we prove the stability of the IMEX Crank–Nicolson scheme in this regime. The stabilization allows us to prove error estimates of order [Formula: see text]. Here [Formula: see text] is the mesh parameter, [Formula: see text] the polynomial order and [Formula: see text] the time step. Finally, we discuss some fractional step methods that are implied by the IMEX scheme. Numerical examples are reported comparing the different methods when applied to the Navier–Stokes’ equations.

Numerical Simulation of Droplet Coalescence Using Meshless Radial Basis Function and Domain Decomposition Method

The present investigation of the dynamic two-binary droplet interactions has gained attention since its use to expand and improve several numerical methods. Generally, its interactions are classified into coalescence, bouncing, reflective, and stretching separation. This study simulated droplet coalescence using the meshless radial basis function (RBF) method. These methods are used to solve the Navier-Stokes equations combined with the Cahn-Hilliard equations to track the interface between two fluids. This work uses the fractional step method to calculate the pressure-velocity coupling in the Navier-Stokes equations. The numerical results were compared with the available data in the literature to validate the proposed method. Based on the validation, the proposed method conforms well with the literature. To identify further coalescence characteristics, the model considered different values in viscosity (2, 4, and 8 cP), collision velocity (1.5 m/s and 3 m/s), and surface tension (0.014, 0.028, and 0.056 N/m) parameters. The increasing viscosity was linearly proportional to the collision time, whereas increased surface tension and collision velocity shortened the collision time.

Evaluation of thermal destratification performance and optimization of jetting parameters in a spray-bar thermodynamic vent system

Future operations in space exploration require the spray-bar thermodynamic vent system (TVS) to control self-pressurization in the cryogenic propellant tank. The submerged buoyant jetting, which is followed by thermal destratification due to mixing, leads to a decrease in tank pressure. In this study, a three-dimensional (3D) numerical model is developed to study the mixing and cooling characteristics induced by submerged buoyant jets in a fully filled, thermally stratified cryogenic propellant tank. The finite volume method is employed to perform transient simulations, solving the governing equations and turbulence models. The SIMPLEC algorithm is applied for pressure–velocity coupling, and the pressure terms are discretized spatially using the PRESTO! technique. Subsequently, dynamic response studies for various parameters are carried out. The results reveal that as the jetting cryogenic energy increases from 5.5 W to 88 W, both the average and maximum temperatures decrease by 0.8 K and 0.4 K, respectively. Adjusting the jetting angle to an upward 45° is beneficial in reducing the maximum temperature by approximately 0.2 K. Moreover, the Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) is integrated with the Entropy Weight Method (EWM) to derive a comprehensive evaluation index that includes the average temperature, maximum temperature, temperature standard deviation, and jetting cryogenic energy. Additionally, by correlating dimensionless jetting parameters with the evaluation index, it is identified that an inlet Reynolds number of 158 and a Grashof number of 19,161 correspond to the optimal thermal destratification performance. This study offers significant insights for the design and improvement of the TVS.

Numerical Study on the Impact of the Richardson Number on Convection in an Enclosure, with Moving Sidewalls for Three Aspect Ratios: A = 1, 2 and 0.5

Using mathematical models, this research investigation seeks to explore how varying aspect ratios and the Richardson number impact the mixing effectiveness, fluid flow velocity, and heat transfer characteristics in a rectangle-shaped space where cooling occurs along the sides and warm with a steady temperature exists throughout the 80% of the bottom section's center. Discretizing the governing equations through the finite volume approach, we solved them iteratively using the Tri-diagonal Matrix Algorithm, where pressure-velocity coupling was successfully implemented via the SIMPLER procedure. Innovative approaches for convection and diffusion terms involved employing a power-law difference scheme alongside a central differential scheme. In various simulations with specified parameters (Re = 100, Pr = 0.71), we visualized results in forms like streamlines, isothermal lines, and temperature/velocity profiles, along with local and average Nusselt numbers. We observe five unique structure formations during flow behavior by analyzing how Richardson numbers impact heat transmission within a specific geometric configuration (enclosure size A = 1). A change in aspect ratio leads to improved control over fluid movement while minimizing structural distortion. A subtle rise in heat transmission emerged as the average Nusselt number and Richardson number intersection strengthened force convection's domination while experiencing significant escalation under conditions conductive to natural convection. Decreasing the aspect ratio increased heat transfer efficiency when the Richardson number rose.

Phase-field formulated meshless simulation of axisymmetric Rayleigh-Taylor instability problem

A formulation of the immiscible Newtonian two-liquid system with different densities and influenced by gravity is based on the Phase-Field Method (PFM) approach. The solution of the related governing coupled Navier-Stokes (NS) and Cahn-Hillard (CH) equations is structured by the meshless Diffuse Approximate Method (DAM) and Pressure Implicit with Splitting of Operators (PISO). The variable density is involved in all the terms. The related moving boundary problem is handled through single-domain, irregular, fixed node arrangement in Cartesian and axisymmetric coordinates. The meshless DAM uses weighted least squares approximation on overlapping subdomains, polynomial shape functions of second-order and Gaussian weights. This solution procedure has improved stability compared to Chorin's pressure-velocity coupling, previously used in meshless solutions of related problems. The Rayleigh-Taylor instability problem simulations are performed for an Atwood number of 0.76. The DAM parameters (shape parameter of the Gaussian weight function and number of nodes in a local subdomain) are the same as in the authors’ previous studies on single-phase flows. The simulations did not need any upwinding in the range of the simulations. The results compare well with the mesh-based finite volume method studies performed with the open-source code Gerris, Open-source Field Operation and Manipulation (OpenFOAM®) code and previously existing results.

Enhancing the heat transfer in CuO-MWCNT oil hybrid nanofluid flow in a pipe

In this study, three-dimensional steady-state laminar flow simulations were conducted in a horizontal pipe using CuO and multi-walled carbon nanotubes (MWCNT) nanoparticles with engine oil as the base fluid. Various nanoparticle volume fractions were examined under a constant heat flux boundary condition applied to the pipe wall. The main goal was to assess and compare the effects of different nanoparticle volume concentrations, including CuO and MWCNT in ratios of 1:1 and 1:2, on convective heat transfer. A second-order discretisation method was employed for solving the equations, and the SIMPLE algorithm was used for pressure–velocity coupling in the CFD code. The study focused on the impact of nanoparticle volume fraction on the convective heat transfer coefficient and the Nusselt number at a Reynolds number of 750. The findings indicate that increasing the nanoparticle volume fraction enhances both the convective heat transfer coefficient and the Nusselt number, with MWCNT having a more pronounced effect compared to CuO. Specifically, adding 2% CuO increases the heat transfer coefficient by 65%, while a mixture of 1% CuO and 1% MWCNT boosts it by 75%. The thermal boundary layer thickness also grows with higher nanoparticle concentrations, with 1% CuO and 3% CuO increasing the thickness by 1.5% and 3.6%, respectively. A formula for the thermal boundary layer thickness in CuO-oil nanofluids is provided based on volume fraction, and a scale analysis of the average heat transfer coefficient confirms that the simulation results are consistent with this analysis.

CFD study of heat transfer in power‐law fluids over multiple corrugated circular cylinders in a heat exchanger

AbstractThe heat transfer in power‐law fluids across three corrugated circular cylinders placed in a triangular pitch arrangement is studied computationally in a confined channel. Continuity, momentum, and energy balance equations were solved using ANSYS FLUENT (Version 18.0). The flow is assumed to be steady, incompressible, two‐dimensional, and laminar. A square domain of side 300Dh is selected after a detailed domain study. An optimized grid with 98,187 cells is used in the study. The convergence criteria of 10−7 for the continuity, x‐momentum, and y‐momentum balances and 10−12 for the energy equation were used. Constant density and non‐Newtonian power‐law viscosity modules were used. The diffusive term is discretized using a central difference scheme. Convective terms are discretized using the Second‐Order Upwind scheme. Pressure–velocity coupling between continuity and momentum equations was implemented using the semi‐implicit method for pressure‐linked equation scheme. Streamlines show wake development behind the cylinders, which is very dominant at large ReN and n. Isotherm contours are cramped at higher values of ReN and PrN, implying higher heat transfer. Global parameters, like, Cd and Nu, are computed for the wide ranges of controlling dimensionless parameters, such as power‐law index (0.3 ≤ n ≤ 1.5), Reynolds (0.1 ≤ ReN ≤ 40), and Prandtl (0.72 ≤ PrN ≤ 500) numbers. The NuLocal plot attains a pitch near the corrugation of the surface due to abrupt changes in velocity and temperature gradients. Nu increases with ReN and/or PrN and decreases with n under ot herwise identical situations. Nu is correlated with pertinent parameters, namely, ReN, PrN, and n.

Effects of cooler shape and position on solidification of phase change material in a cavity

BackgroundFor balancing the imbalance between the energy supply and demand, phase-change materials (PCMs) provide an efficient means in terms of thermal energy storage and release. The performance of the energy storage is primarily dependent on the melting as well as the solidification process of the storage medium. Faster charging or discharging of the thermal energy is a primary concern for any thermal energy storage unit. On this background, the present study explores the novel approach for enhancing the solidification process of PCM considering the effects of cooler shape (namely semi-circular, triangular, and rectangular) and their position (namely top, side, and bottom) in a molten PCM-filled enclosure. The middle portion of the cooler wall is curved; whereas the remaining cooler wall is straight maintaining the same cooler wall length. MethodsTo analyze the solidification process, the involved transport equations are solved numerically following a finite volume-based computational approach using Ansys Fluent solver in conjunction with the appropriate boundary conditions. The computational model is generated for all the geometry comprising different shapes, as well as positions of the cooler wall. The third-order upwind scheme (QUICK) technique is utilized to discretize the momentum and energy equations. This scheme is well capable to accurately capture the gradients in the temperature and flow domains. Furthermore, the semi-implicit pressure-linked equation (SIMPLE) technique is utilised to address the pressure-velocity coupling. The resolved data are then saved as selective variables (U, V, and θ), which undergo post-processing to produce a local thermo-fluid flow field and extract average data. Significant findingsThe shape, as well as the position of a cooler, dictates the solidification process in an energy storage system. Thermal energy storage with a triangular-shaped cold wall positioned at the top could be opted as an appropriate design approach of an efficient energy storage system compared to a semi-circular or rectangular-shaped cooler model. The shortest solidification time of PCM occurs when the cooler wall is positioned at the top. The top position of the cooler having a triangular shape with higher Grashof number (Gr) values leads to a faster solidification process. Some ideas for possible future research areas in this field are provided after a comprehensive examination.

Integrated wellbore-reservoir modeling based on 3D Navier–Stokes equations with a coupled CFD solver

The occurrence of fluid flow near a wellhead is the major concern of the petroleum industry, as pressure drop, loss of formation, and other variables of interest are mostly affected in this region. The fluid flows from the hydrocarbon reservoir to the wellbore can be characterized as laminar to turbulent; thus, it is important to model this phenomenon with the integrated wellbore-reservoir model. Using 3D Navier–Stokes equations, an integrated wellbore-reservoir model is created in this study, and it incorporates the formation damage zone. For the porous-porous and porous-fluid interfaces, the General Grid Interface (GGI) approach is applied in conjunction with the conservative mass flux interface model. Model equations are solved using a velocity-pressure coupling solver that is pressure-based. For reliable and quick results, the system of equations is solved using an algebraic multigrid approach. The pressure diffusivity equation’s analytical solution under steady-state flow circumstances is used to validate the model. The integrated wellbore-reservoir model is applied to different reservoir scenarios, for example, different production rates, formation zones, and reservoir formation conditions. The results indicate that the present Computational Fluid Dynamics (CFD) model can be extended to simulate the real field scale model. integrated wellbore-reservoir modeling based on 3D Navier–Stokes equations with efficient computational techniques can lead the field of petroleum industries to advance current knowledge.

Application of machine learning to model the pressure poisson equation for fluid flow on generic geometries

This study addresses the importance of enhancing traditional fluid-flow solvers by introducing a Machine Learning procedure to model pressure fields computed by standard fluid-flow solvers. The conventional approach involves enforcing pressure–velocity coupling through a Poisson equation, combining the Navier–Stokes and continuity equations. The solution to this Poisson equation constitutes a substantial percentage of the overall computational cost in fluid flow simulations, therefore improving its efficiency can yield significant gains in computational speed. The study aims to create a versatile method applicable to any geometry, ultimately providing a more efficient alternative to the conventional pressure solver. Machine Learning models were trained with flow fields generated by a Computational Fluid Dynamics solver applied to the confined flow over multiple geometries, namely wall-bounded cylinders with circular, rectangular, triangular, and plate cross-sections. To achieve applicability to any geometry, a method was developed to estimate pressure fields in fixed-shape blocks sampled from the flow domain and subsequently assemble them to reconstruct the entire physical domain. The model relies on multilayer perceptron neural networks combined with Principal Component Analysis transformations. The developed Machine Learning models achieved acceptable accuracy with errors of around 3%. Furthermore, the model demonstrated enhanced computational efficiency, outperforming the classical PISO algorithm by up to 30 times.

Numerical modelling of continuous casting of round billets with turbulence in the melt

The present work aims to solve the continuous casting benchmark problem in axisymmetry with turbulence in the melt by the meshless method. The physical model of the liquid-solid system that involves mass, momentum and energy conservation is formulated in the mixture continuum approximation. The melt is assumed Newtonian and incompressible. A k-epsilon Reynolds averaged Navier-Stokes turbulence model is implemented with Abe-Kondoh-Nagano closures. The mushy region is modelled as a Darcy porous media with the Kozeny-Carman permeability model. The solution of partial differential equations is implemented locally using collocation with radial basis functions and explicit time-stepping. The velocity pressure coupling is performed using the fractional step method. The validation of the model is assessed by comparing its outcomes with the results of the finite volume method. A sensitivity study of the varying casting speed and temperature on the velocity, temperature, and solid fraction fields is shown.

Laminar convective heat transfer combining buoyancy and thermal radiation of Carreau fluid within a concentric and eccentric annulus

The present study concerns a steady-state laminar combined thermal radiation and buoyancy-driven conjugate magnetohydrodynamic (MHD) natural convective flow and heat transfer inside a fully porous medium eccentric circular annulus bounded by two infinite horizontal cylinders occupied by a Carreau fluid permeated by a uniform transverse magnetic field (B0 ) and inclined by an angle (α). The convective and thermal radiation are approximated by the Boussinesq and Roseland models, respectively. The numerical integration was performed using the implicit finite volume method. The convective terms in the momentum and energy equations were Discretized using the second-order upwind method. On the other hand, the diffusive terms were Discretized using the central difference scheme. The pressure-velocity coupling was solved using the SIMPLE approach. The findings suggest a clear trend of enhancement in the heat transfer rate when the aspect ratio and annulus eccentricity increase. This pattern is more pronounced in the negative vertical eccentricities. It is worth noting that the eccentric annulus, which has an annulus aspect ratio of 1.6 and a vertical eccentricity of −0.8, offers the best performance regarding the heat transfer mechanism inside the annulus gap. The heat transfer mechanism turns out to be strongly affected by thermal radiation, Hartmann number, and magnetic field inclination angle. The increased thermal radiation parameter and the decreased flow behavior index improve the heat transfer mechanism between the Carreau fluid and the inner cylinder in the annular gap. Furthermore, the average Nusselt number increases considerably as the induced magnetic field inclines.

Phase-field formulated meshless simulation of Rayleigh-Taylor instability problem

The interface between two immiscible Newtonian liquids with different densities and the same viscosity, influenced by gravity, is based on the Phase-Field Method (PFM) formulation. The solution of the related governing coupled Navier-Stokes (NS) and Cahn-Hillard (CH) equations is structured by the meshless Diffuse Approximate Method (DAM) and Pressure Implicit with Splitting of Operators (PISO). The variable density is involved in the inertial and buoyancy terms (non-Boussinesq formulation). The related moving boundary problem is handled through single-domain, irregular, fixed node arrangement in two-dimensional Cartesian coordinates. The meshless DAM uses weighted least squares approximation on overlapping subdomains, polynomial shape functions of second-order and Gaussian weights. Implicit time discretisation is performed for the NS and CH equations in the momentum predictor and Phase-Field (PF) variable corrector steps of PISO, while the momentum corrector steps solve the NS equation explicitly. This solution procedure has improved stability compared to Chorin’s pressure-velocity coupling, previously used in meshless solutions of related problems. The Rayleigh-Taylor instability problem simulations are performed for an Atwood number of 0.76. The DAM parameters (shape parameter of the Gaussian weight function and number of nodes in a local subdomain) are the same as in the author’s previous studies on single-phase flows. The simulations did not need any upwinding in the range of the simulations. The results compare well with the mesh-based finite volume method studies performed with the open-source code Gerris.

DNS of Marangoni effects on a suspension of droplets in microgravity using the unstructured conservative level-set method

The multi-marker unstructured conservative level-set (UCLS) method for two-phase flow with variable surface tension is used for the Direct Numerical Simulation of thermocapillary-driven motion of a bi-dispersed suspension of droplets in microgravity. The so-called Marangoni stresses induced by surface tension gradients on the interface lead to a coupling of the momentum transport equation and the thermal energy transport equation. The finite-volume method discretizes the transport equations on three-dimensional collocated unstructured meshes. The UCLS method performs interface capturing with mass conservation of the fluid phases, whereas the multiple marker approach circumvents the numerical coalescence of fluid particles. The fractional-step projection method solves the pressure-velocity coupling. Unstructured flux-limiters schemes discretize the convective term in transport equations. Numerical and physical findings are reported.

Unstructured Conservative Level-Set (UCLS) method for reactive mass transfer in bubble swarms at high density ratio

This research presents a parallel Unstructured Conservative Level-Set (UCLS) method for reactive mass transfer in bubbles with a high-density ratio. This method uses the multiple-marker UCLS approach to circumvent the numerical coalescence of bubbles, combined with the finite-volume method to discretize transport equations on 3D collocated unstructured meshes. The fractional-step projection method solves the pressure-velocity coupling. The central difference scheme discretizes the diffusive term, whereas convective term within the momentum transport equation, level-set advection equations, and mass transfer equation are discretized by unstructured flux-limiters schemes. Such a combination of numerical techniques preserves the numerical stability in bubbly flows with a high Reynolds number and high-density ratio. Numerical and physical findings on the effect of physical properties ratios on the reactive mass transfer are reported, including their effect on the Reynolds number, interfacial area and Sherwood number.

A Compressible Formulation of the One-Fluid Model for Two-Phase Flows

In this paper, we introduce a compressible formulation for dealing with 2D/3D compressible interfacial flows. It integrates a monolithic solver to achieve robust velocity–pressure coupling, ensuring precision and stability across diverse fluid flow conditions, including incompressible and compressible single-phase and two-phase flows. Validation of the model is conducted through various test scenarios, including Sod’s shock tube problem, isothermal viscous two-phase flows without capillary effects, and the impact of drops on viscous liquid films. The results highlight the ability of the scheme to handle compressible flow situations with capillary effects, which are important in computational fluid dynamics (CFD).

On wave-current interaction with a horizontal cylinder near free surface by a fourth-order-accurate finite volume compact solver

We present a numerical solver for dealing with the problem of wave-current interaction with a horizontal cylinder near free surface. The solver solves the filtered Navier-Stokes equations which are discretized using a fourth-order-accurate compact finite volume scheme on a collocated grid. The turbulence effect is modelled by a subgrid-scale model using a large eddy simulation, and the free surface and the cylinder are modelled by a volume of fluid method and a ghost cell immersed boundary method. The pressure-velocity coupling is ensured via a momentum interpolation method. We obtain the hydrodynamic force and its spectra characteristics, dynamic pressure distribution, streamline topology and vorticity patterns with different immersed depths under uniform currents, regular waves and wave-current flows. Validation is performed by comparing the calculated hydrodynamic force with the measured data within one wave period. The results show that the position of vortex shedding and dissipation is different for the cases under different immersed depth in regular waves and wave-current flows. The dynamic pressure around the cylinder as the force reaches its peak value changes with immersed depth. The value and phase of the force are sensitive to immersed depth.

Investigation of numerical phase transition of nano-enhanced SiC/paraffin wax PCM in solar-assisted water desalination system

Adding nanoparticles in phase change material (PCM) is a new trend for enhancing their thermal energy-storing ability as well as thermal conductivity. From this point of view, a numerical study is carried out to examine the addition of silicon carbide nanoparticles in paraffin wax PCM, used as heat energy storing material in a semi-cylindrical solar water desalination system. The PCM temperature is maintained at 52 °C and the semi-cylindrical tubular section has a wall temperature of 85 °C. The top plane section of the tubular solar collector is made up of graphite material. A semi-circular cross-section is selected for numerical analysis. A finite volume solver is used for solving thermal and fluid flow governing equations. A pressure staggering option algorithm is applied for pressure–velocity coupling. The temporal parameters like temperature, velocity contours, melting fraction, enthalpy, and entropy of nano silicon embedded paraffin wax PCM are widely discussed. The results clearly show that the phase transition of solid PCM to fluid PCM is greatly influenced by nanoparticle addition and enhances the rate of heat transfer. Initially for the first 60 min the melting fraction and temperature of PCM remain uniform as the time step increases above 60 min the behavior of PCM changes abruptly which clearly indicates the random distribution of nanoparticles within the PCM. A critical time and temperature limit exists for nanoparticles-based PCM beyond which, the thermal efficiency of the solar water desalination system gets influenced.

Temporal numerical analysis of beeswax PCM melting in a cube geometry subjected to a constant wall temperature condition

The ability of phase change materials (PCM) to store thermal energy has gained wide application area, like battery thermal management, solar water desalination and many other. The melting process of beeswax phase change material within the cube geometry with constant wall temperature (65 °C) boundary condition has been investigated using solidification and melting model. The fluid flow and heat transfer governing equations are solved using second order finite volume scheme. A PRESTO algorithm is applied for pressure-velocity coupling. The convergence criteria of 10−10 have been selected for energy equation, while 10−8 is selected for both momentum and continuity equations. The results like percentage variation along length-height and height-width plane for transient liquid fraction and temperature has been plotted, along with velocity streamlines within the cube geometry. From the obtained results it is concluded that the melting fraction and temperature of beeswax PCM is different in different planes and the major factors which affect the complete melting process is wall temperature, and the geometry. A difference of more than 0.1 °C in temperature has been recorded between mid-length-height and height-width plane while a difference of more than 2% in liquid fraction of PCM is observed. Even the uniformity of temperature and liquid fraction is notably influenced and vary along length, height, and width of cube geometry. Thus, it is concluded that melting process of PCM may affect the ability to store and release the heat energy which further affect the performance parameters of applied physical system.

Numerical Study of a Thermal Plume Developing in a Neutral Environment and in Interaction: Application to Fires Problems

This study focuses on the mathematical and numerical modelling of how different properties, such as channel length and hot source height, affect the flow of thermal plumes and their interaction with the surrounding environment. Studying the behaviour of a directed thermal plume flow can provide a good visualisation of the control parameters for various applications related to fires and pollutants. To determine the control parameters affecting this flow, it is imperative to initially analyse the flow through a natural convection consisting of a disc-shaped heat source at the inlet of a vertical cylinder using mathematical and numerical methods. The flow in this study is characterised by natural convection, turbulent, steady, two-dimensional and incompressible. Numerical results were obtained via computational fluid dynamics analysis, which is based on mathematical flow modelling using the system of Navier–Stokes equations and the finite volume method. The SIMPLE algorithm was used for the velocity–pressure coupling. After introducing the concept of Reynolds averaging and analysis, different RANS turbulence models are presented, particularly two-equation models, using the Ansys Fluent software. Comparison of the different profiles revealed the ideal flow control parameters. An improvement in thermal properties was observed when the height of the heat source was = 1.5 cm and the vertical length of the channel was D = 20cm.