Stream Function-vorticity Formulation Research Articles

Fourth-Order Conservative Non-splitting Semi-Lagrangian Hermite WENO Schemes for Kinetic and Fluid Simulations

AbstractWe present fourth-order conservative non-splitting semi-Lagrangian (SL) Hermite essentially non-oscillatory (HWENO) schemes for linear transport equations with applications for nonlinear problems including the Vlasov–Poisson system, the guiding center Vlasov model, and the incompressible Euler equations in the vorticity-stream function formulation. The proposed SL HWENO schemes combine a weak formulation of the characteristic Galerkin method with two newly constructed HWENO reconstruction methods. The new HWENO reconstructions are meticulously designed to strike a delicate balance between curbing numerical oscillation and introducing excessive dissipation. Mass conservation naturally holds due to the weak formulation of the semi-Lagrangian discontinuous Galerkin method and the design of the HWENO reconstructions. We apply a positivity-preserving limiter to maintain the positivity of numerical solutions when needed. Abundant benchmark tests are performed to verify the effectiveness of the proposed SL HWENO schemes.

Novel enhanced thermal capabilities of Cu-Al2O3/water hybrid nanofluid in an oblique U-shape cavity: On the molecular behaviour through vorticity analysis

This paper presents the numerical investigation of natural convection heat transfer of copper-alumina/pure water hybrid nanofluid in an oblique U-shaped enclosure. The dimensionless governing equation is formed in the stream function-vorticity formulation using dimensionless variables. The Galerkin weighted residual finite element method with the damped Newton-Raphson iteration method is applied to the problem. The heat transfer performance for the system is investigated by varying the Rayleigh numbers, nanoparticle volume fractions and their ratios, and the oblique angles of the enclosure. The fluid flow, heat transfer, and vorticity analysis are conducted extensively for the Rayleigh number up to 106. It was found that the oblique enclosure dwindled the overall heat transfer performance. Correspondingly, the maximum vorticity is at the lowest with zero obliquenesses. The results are used to gain a deeper understanding of the fluid flow and thermal behaviour of the system to achieve more sustainable and greener energy management.

Combustion of a nanoparticles-laden chemical in a vented cavity

Mixed convective characteristics of the combustion of a nanoparticles-laden fuel (n-butanol nanofluid) in a vented cavity are investigated. The nanofluid and the oxidizer enter the cavity through the inlets on the left and right vertical walls, respectively. However, the resulting product produced from the oxidation process of the fuel exits the cavity through the outlet at the bottom wall. Heat generated from the oxidation process causes natural convection within the cavity. The conjugate effect of natural and forced convection finally gives rise to mixed convection phenomena. In this regard, a mathematical model for mixed convection flow in a vented cavity is formulated with no-slip and isothermal boundary conditions. Having transformed the model into a dimensionless form, the stream function-vorticity formulation is used. The resulting equations are then solved numerically using the finite difference method. Numerical results are illustrated with the streamlines, isotherms, and isolines of fuel and oxidizer concentrations. The maximum values of the stream function (ψmax) and the temperature (θmax) are found to increase with an increase in the Frank–Kamenetskii number (Λ), volume fraction of nanoparticles (φ), and stoichiometric ratio (χ). On the contrary, they decrease with the increase in the Reynolds number (Re). When the Grashof number (Gr) is increased, ψmax increases and θmax decreases. The remaining concentrations of fuel, (CF)min, and oxidizer, (CO)min, are higher for an increase in Gr, whereas the opposite is recognized for increasing Λ. With the increase in Gr and Λ, the steady-state flow in the cavity tends to be oscillating and then chaotic.

General vorticity‐streamfunction formulation for incompressible binary flow with arbitrary density ratio

AbstractThe classical vorticity‐streamfunction formulation (VSF) can avoid the difficulty in the calculation of pressure gradient term of the Navier Stokes equation via eliminating pressure gradient term from the theoretical basis. Within this context we propose a general VSF, together with redefined vorticity and streamfunction, so as to realize numerically stable and reliable simulations of binary fluids with an arbitrary density contrast. By incorporating the interface‐tracking phase‐field model based on the conservative Allen‐Cahn equation [Phys. Rev. E 94, 023311 (2016)], the binary flow simulation framework is established. Numerical tests are conducted using the Lattice Boltzmann method (LBM), which is usually regarded as an easy‐to‐use tool for solving the Navier–Stokes equation but generally suffers from the drawback of not being capable of enforcing incompressibility. The LBM herein functions as a numerical tool for solving the vorticity transport equation, the streamfunction equation, and the conservative Allen‐Cahn equation. Three two‐dimensional benchmark cases, i.e., the Capillary wave, the Rayleigh–Taylor instability, and the droplet splashing on a thin liquid film, are discussed in detail to verify the present methodology. Results show good agreements with both analytical predictions and literature data, as well as good numerical stability in terms of high density ratio and high Reynolds number. Overall, the general VSF inherits the intrinsic superiority of the classical VSF in enforcing incompressibility, and offers a useful and reliable alternative for binary flow modeling.

Magnetohydrodynamic and Ferrohydrodynamic Fluid Flow Using the Finite Volume Method

Many problems in fluid mechanics describe the change in the flow under the effect of electromagnetic forces. The present study explores the behaviour of an electric conducting, Newtonian fluid flow applying the magnetohydrodynamics (MHD) and ferrohydrodynamics (FHD) principles. The physical problems for such flows are formulated by the Navier–Stokes equations with the conservation of mass and energy equations, which constitute a coupled non-linear system of partial differential equations subject to analogous boundary conditions. The numerical solution of such physical problems is not a trivial task due to the electromagnetic forces which may cause severe disturbances in the flow field. In the present study, a numerical algorithm based on a finite volume method is developed for the solution of such problems. The basic characteristics of the method are, the set of equations is solved using a simultaneous direct approach, the discretization is achieved using the finite volume method, and the solution is attained solving an implicit non-linear system of algebraic equations with intense source terms created by the non-uniform magnetic field. For the validation of the overall algorithm, comparisons are made with previously published results concerning MHD and FHD flows. The advantages of the proposed methodology are that it is direct and the governing equations are not manipulated like other methods such as the stream function vorticity formulation. Moreover, it is relatively easily extended for the study of three-dimensional problems. This study examines the Hartmann flow and the fluid flow with FHD principles, that formulate MHD and FHD flows, respectively. The major component of the Hartmann flow is the Hartmann number, which increases in value the stronger the Lorentz forces are, thus the fluid decelerates. In the case of FHD fluid flow, the major finding is the creation of vortices close to the external magnetic field source, and the stronger the magnetic field of the source, the larger the vortices are.

Effect of discrete heating on the key parameters and entropy generation in a corrugated enclosure filled with hybrid nanofluid

This research work aims to study the effect of discrete heating on natural convective heat transfer, fluid flow, and entropy generation within a corrugated enclosure filled with water-based molybdenum disulfide (MoS2)–silicon dioxide (SiO2) hybrid nanofluid. In the study, various heights of the discrete heaters in terms of aspect ratios ( AR = 0.2 , 0.4 , 0.6 , 0.8 ) are considered for various Rayleigh numbers ( Ra = 10 3 − 10 6 ) with varying nanoparticle volume fractions ( ϕ = 0 % − 4 % ). Stream function-vorticity formulations of two-dimensional (2D) Navier-Stokes equations and energy equation are used as the governing equations, and a higher-order compact (HOC) finite difference scheme is employed to discretize those equations. The results obtained from the current developed code are first validated with the existing numerical and experimental data. Finally, the present computed results are studied, both qualitatively and quantitatively, in terms of streamlines, isotherms, entropy generations, average Nusselt numbers, and Bejan numbers. At Ra = 10 6 and the aspect ratio of heater length AR = 0.2 , a different and interesting flow and heat transfer phenomenon is observed. The results show that an increase in the aspect ratio of heater length from 0.2 to 0.8 decreases the average Nusselt number and total entropy generation by 57.49% and 13.76%, respectively, for Ra = 10 6 and ϕ = 4 % . In addition, the effect of nanoparticle concentration on total entropy generation increases up to 73.40% when the aspect ratio of heater length increases to 0.8.

Numerical simulation of flow and heat transfer in a pipe with nail-shaped hurdles

We investigate fluid flow and heat transfer in a channel obstructed by nail-shaped hurdles under forced convection using the Finite Difference Method. Due to the absence of an analytical solution, numerical techniques are employed. The channel is geometrically transformed into a rectangular configuration using curvilinear coordinates. Elliptic grid generation is used to discretize the domain. The Navier–Stokes equations are reformulated through the vorticity stream function formulation, and the Finite Difference Method incorporates a coordinate transformation approach. Effects of Reynolds numbers between (20≤Re≤600), Prandtl numbers (0.71,17.1), and nail spans (0.1≤h≤0.7) on velocity, temperature profiles is studied. Local and Average Nusselt numbers on Nail-shaped hurdle are also calculated. The observed maximum Nusselt numbers are 13.49,27.83, and 11.86 at Ra=600,Pr=7.1, and h=0.1, respectively Results show that higher Reynolds and Prandtl numbers increase Nusselt numbers, enhancing heat transfer. Narrower nail spans improve heat transfer efficiency.

Non-orthogonal multiple-relaxation-time lattice Boltzmann method for vorticity-streamfunction formulation

In this work, we propose a non-orthogonal multiple-relaxation-time (MRT) lattice Boltzmann method (LBM) for the two-dimensional (2D) incompressible Navier-Stokes equations in vorticity-stream function formulation. In this method, the vorticity and the streamfunction equations are calculated by two separate LB equations, which are constructed by using the non-orthogonal transformation matrix. Most importantly, to eliminate the residual term that may appear in the recovered vorticity equation, the convection term is handled as a source term in the LB equation rather than being directly recovered by the evolution equation. In addition, due to the convection term in our method is computed with a local scheme, it retains the parallel feature of the standard LBM. We then test the accuracy and the stability of the present LBM by considering various benchmark examples. The results show that the present method has a second-order convergence rate in space. Our numerical results indicate that the present LBM is a good candidate for solving streamfunction-vorticity formulation.

Abstract 12854: Early and Late Features of Intracardiac Flow in Repaired Tetralogy of Fallot (rTOF) Patients Using Doppler Velocity Reconstruction (DoVeR)

Background: Intracardiac flow analysis has implicated flow energy loss (FEL), kinetic energy (KE) and vortex strength (VS) as markers of ventricular dysfunction in repaired Tetralogy of Fallot (TOF); no studies have investigated early vs. late intracardiac flow conditions after repair. We utilized novel echocardiography (echo) based Doppler Velocity Reconstruction (DoVeR) to determine FEL, KE and VS in pre-operative TOF vs early post-repair (<1 month) and late post-repair (>10 years). Methods: This was a single-center retrospective cross-sectional study on TOF patients with pre- and post-operative surveillance transthoracic echo. Apical 4 chamber color Doppler views were collected. The streamfunction-vorticity formulation was used to reconstruct left ventricular (LV) blood velocity fields by DoVeR. Velocity flow fields were evaluated for peak FEL, KE and VS. The FEL was also normalized against KE to account for physiologic variations. Data were compared via one-way ANOVA. Results: Echo studies from 14 pre-operative (1.7±0.5 months), 14 early-post repair (3.6±2.3 months) and 15 late-post repair (16.5±6.6 years) TOF patients were included. There was significant variation in FEL, KE, VS, and normalized FEL across timepoints (all p<0.05) (Figure B). In general, when compared to pre-operative TOF, the early post-repair cohort had lower FEL/KE/VS corresponding to reduction in ventricular volume load; however late post-repair cohort had higher FEL/KE/VS, suggestive of effects from long-term interventricular interactions. Conclusion: The LV intracardiac flow in TOF changes early after initial repair and late after repair. DoVeR could potentially be used to serially evaluate TOF patients.

Numerical Simulation and Computational Fluid Dynamics Analysis of Two-Dimensional Lid-Driven Cavity Flow Within the Weapon Bay of an Autonomous Fighter Drone

The efficient deployment of weapons in military operations is critical for mission success, and the flow of air within the weapon bay of an autonomous fighter drone plays a vital role in achieving this objective. In this paper, we present a comprehensive numerical simulation and computational fluid dynamics (CFD) analysis of the three-dimensional lid-driven cavity flow within the weapon bay of an autonomous fighter drone. To address this challenging problem, we employ CFD analysis and a multigrid approach to solve the Navier-Stokes equations for the aerodynamic problem. Our simulations include high Reynolds numbers of up to 10,000, which demonstrates the potential of CFD analysis in optimizing the design of autonomous fighter drones for military operations. We evaluate the effectiveness of the linked strongly implicit multigrid technique in estimating high-Re fine-mesh flow solutions using the vorticity-stream function formulation of the two-dimensional incompressible Navier-Stokes equations. The model issue is the driven flow in a square cavity, and we consider meshes up to 1024 x 1024 points and combinations with Reynolds numbers as high as 1000. To further improve the accuracy of our simulations, we employ one-dimensional grid clustering coordinate transformations instead of uniform mesh refinement, as the flow field exhibits one or more secondary vortices. Our findings demonstrate that CFD analysis can provide valuable insights into the complex flow dynamics within the weapon bay of autonomous fighter drones, which can lead to the optimization of their design for enhanced mission capabilities. Overall, our study highlights the significance of numerical simulations and CFD analysis in the design and optimization of autonomous fighter drones for military applications. Our results can serve as a basis for future research in the field of UAV aerodynamics and contribute to the development of more efficient and effective military operations.

Application extension of the meshless local Petrov-Galerkin method: Non-Newtonian fluid flow implementations

A computer code is developed to enhance the meshless local Petrov-Galerkin method capability to solve non-Newtonian fluid flow problems. In particular, the mesh-free method here, is to be empowered to simulate and solve the two-dimensional laminar incompressible power-law cases for the first time. The newly developed computer code expresses the appropriate governing equations in terms of the vorticity-stream function formulation and sustains a weighting function of unity. The local quadrature domain integrated by parts over the appropriate control volumes provides the local weak form of the considered governing equations. The extended scheme implements the moving least square interpolation technique to approximate the obtained field variables. The ability of the developed code to handle the Ostwald-de Waele (also known as the power-law) fluid flow occurrences were assessed through comparing the obtained results of the extended method to those of the conventional mesh-based methods for some benchmarking cases available in the paper. Based on this comparison, the extended code demonstrates a good capability to address the power-law fluid flow problems for different indices for one of the most popular non-Newtonian fluid flow models.

Fluid Cohomology

The vorticity-streamfunction formulation for incompressible inviscid fluids is the basis for many fluid simulation methods in computer graphics, including vortex methods, streamfunction solvers, spectral methods, and Monte Carlo methods. We point out that current setups in the vorticity-streamfunction formulation are insufficient at simulating fluids on general non-simply-connected domains. This issue is critical in practice, as obstacles, periodic boundaries, and nonzero genus can all make the fluid domain multiply connected. These scenarios introduce nontrivial cohomology components to the flow in the form of harmonic fields. The dynamics of these harmonic fields have been previously overlooked. In this paper, we derive the missing equations of motion for the fluid cohomology components. We elucidate the physical laws associated with the new equations, and show their importance in reproducing physically correct behaviors of fluid flows on domains with general topology.

Unified a priori analysis of four second-order FEM for fourth-order quadratic semilinear problems

A unified framework for fourth-order semilinear problems with trilinear nonlinearity and general sources allows for quasi-best approximation with lowest-order finite element methods. This paper establishes the stability and a priori error control in the piecewise energy and weaker Sobolev norms under minimal hypotheses. Applications include the stream function vorticity formulation of the incompressible 2D Navier-Stokes equations and the von Kármán equations with Morley, discontinuous Galerkin, C^{0} interior penalty, and weakly over-penalized symmetric interior penalty schemes. The proposed new discretizations consider quasi-optimal smoothers for the source term and smoother-type modifications inside the nonlinear terms.

Maximum-principle-preserving high-order discontinuous Galerkin methods for incompressible Euler equations on overlapping meshes

In this paper, we construct a new local discontinuous Galerkin (LDG) algorithm to solve the incompressible Euler equation in two space dimensions on overlapping meshes. This method solves the vorticity, velocity field and potential function on different meshes. Different from the traditional LDG method, the overlapping meshes used in this paper make the velocity to be continuous along the interfaces of the primitive meshes. Therefore, the upwind fluxes can be applied. We derive two sufficient conditions to obtain the maximum principle of vorticity. The first one is the divergence-free numerical approximation of the velocity field. This condition further grants that the scheme of the vorticity equation keeps constant solutions. The second one is to preserve the positivity of the numerical vorticity. We select suitable time step sizes to construct positive numerical cell averages of the vorticity provided the vorticity in the previous time step is positive. Then a slope limiter can be applied to enforce the positivity of the numerical approximation of the vorticity. Thanks to the above two conditions, we can arbitrarily add constants to the vorticity function and construct high-order MPP LDG methods on overlapping meshes for the two-dimensional incompressible Euler equation in the vorticity stream function formulation. Numerical tests will be given to demonstrate the good performance of the proposed method.

Stefan problem coupled with natural convection: An application to dissolution process

This paper investigates the process of dissolution of a solute in a solvent placed in an horizontal concentric cylinder. The theoretical investigation solves a Stefan problem with phase transition due to natural convective flow. To realize the objective, the governing equation for the concentration distribution, stream function–vorticity form of the Navier–Stokes equation for the flow field, and a Stefan condition for calculating the timescale evolution of the front are coupled together with different parameters. These non-linear equations are solved using a stable and second-order accurate boundary-fitted alternating direction implicit scheme with first-order upwind difference approximation for convective terms. The numerical scheme is validated initially by applying it to solve a natural convection problem with no phase transition, for which benchmark solutions are available. The validated scheme is then applied to the chosen problem followed by a refinement study to obtain a reliable solution. The obtained results are used to analyze the effect of physical parameters such as the Stefan number (Ste), geometric aspect ratio of solute to fluid, the Rayleigh number (Ra) and the Schmidt (Sc) number on dissolution rates as well as the flow patterns. It is observed that the solute dissolution, without the temperature influence, mainly depends on the annulus gap width (L) and the convection rate. Additionally, it is also observed that, for the Rayleigh numbers greater than 105, the unit circular-shaped solute initially dissolves uniformly from the outer surface, but as the time progresses, due to the influence of laminar boundary layer flow around the solute, it changes into an egg-shape.

Oscillatory Marangoni flow in a deep layer of a Carreau fluid

We study the spreading dynamics of an insoluble and non-diffusive surfactant on the free surface of a deep layer of a Carreau fluid. When a non-uniform distribution of surfactant is imposed on the free surface of an initially motionless fluid, variations in surface tension are induced, causing the fluid motion from regions of lower surface tension to those of higher tension. Such movement redistributes the surfactant concentration until a uniform condition is reached. The two-dimensional momentum and convection–diffusion equations are employed to determine the hydrodynamics in the deep fluid layer and the evolution of the surfactant on the fluid surface. The modified vorticity-stream function formulation is used to solve numerically the hydrodynamic field, where the non-Newtonian dependent variables of the problem are decomposed into a Newtonian part and a non-Newtonian contribution. One of the most critical variables in practical applications regarding this phenomenon concerns the time required for the surfactant to reach a uniform distribution over the interface. The results show that the dimensionless parameters that control the decay of the variations in surfactant concentration in time are the Reynolds number Re, the fluid behavior index n, and ε, reflecting the influence of the inertia of the fluid. The results show that the temporal decay of the surfactant concentration on the fluid surface increases significantly for high values of the Reynolds number, it is more attenuated in pseudoplastic fluids than in Newtonian fluids, and the surfactant concentration decay exhibits asymmetric oscillations when inertial effects increase.

Numerical simulation of buoyant plumes using a fixed point iterative method

A numerical simulation of unsteady plumes for driven thermal high convection dominated flows is presented in this work. The problem is of practical interest for example in the cooling of electronic devices and heat transfer from pipes in heat exchange systems. This kind of thermal flows may be modeled using the unsteady Boussinesq approximation in the Stream function-vorticity formulation. Results are obtained with a simple numerical method previously reported for isothermal/natural and mixed convection flows. The numerical method is based on a fixed point iterative process to solve the non-linear elliptic system that results after an appropriate discretization in time. The iterative process leads us to the solution of uncoupled, well-conditioned and symmetric linear elliptic problems for which very efficient solvers are known to exist regardless of the space discretization.

A full-Eulerian approach for simulation of a system of fluid–rigid–elastic structure interaction based on the vorticity-stream function formulation

A monolithic mathematical framework for understanding the fluid–rigid–elastic structure interaction problem is proposed. A numerical method in a secondary formulation of the Navier–Stokes equations accompanying a technique for imposing the rigid boundaries is applied. The one-fluid formulation of the incompressible Navier–Stokes equation, containing the terms governing the elastic structure, is transformed into the vorticity-stream function formulation. The rigid structure is imposed in the flow field based on the velocity–vorticity kinematic relation and harmonic function theorem. The vorticity, level-set function, and left Cauchy–Green deformation tensor are updated utilizing three transport equations to investigate the evolution of the velocity field, elastic structure(s) configuration, and elastic stress tensor. The method is implemented to solve three challenging problems, and the results show its capabilities in proper imposing the rigid structures in the flow field and also the simultaneous modeling the rigid and elastic structure interactions with incompressible fluid flow.

INFLUENCE OF THE ASPECT RATIO AND LOCATION OF AN ISOTHERMAL BLOCK ON THE BUOYANCY-INDUCED HEAT TRANSFER IN AN ENCLOSURE

The objective of the current study's numerical analysis is to find the effect of the aspect ratio and placement of an isothermal block on fluid flow and heat transfer within an air-filled square cavity and having a Prandtl number of 0.71. The flow is considered to be taking place by natural convection and Rayleigh's numbers range between 10<sup>3</sup> and 10<sup>6</sup>. Between 1.00 and 2.00, the block's aspect ratio is evaluated. In a cavity with a centrally positioned hot block, the impact of Rayleigh's number and block aspect ratio on heat and fluid transport is evaluated. The effect of the location of the block is examined at three different locations as the top wall, center, and bottom wall of the cavity, respectively. A finite-difference-based stream function vorticity formulation is considered for the computation. The findings indicate that Rayleigh's number is a positive function of the heat-transfer rate and fluid velocity which shows that increase in Ra enhances buoyancy. The increase in the aspect ratio of the internal block enhances the area of hydrodynamic blockage, which modifies the rate of convection and flow velocity of the fluid within the cavity. The results also show that shifting the block's location has a significant impact on the buoyancy force generated inside the enclosure. A higher total heat transmission rate is produced by the bottom and center-placed blocks.

Thermal drift in an inclined viscous fluid flow

This paper considers viscous fluid flow in a slot between two parallel plates which start inclining with respect to the horizontal line. The lower plate was heated and had non-homogeneous temperature distribution while the upper plate was cooled and with homogeneous temperature distribution. The spatially periodic temperature distribution was gradually applied at the lower plate, after which the plates were slowly inclined in the positive-counterclockwise direction, and the fields of vorticity, stream function, and temperature are presented for different values of the angle of inclination. We used the vorticity-stream function formulation of Navier-Stokes equations, Fourier-Galerkin, and Chebyshev collocation method for numerical simulation of 2-D viscous fluid flow. We carried out numerical simulation using our in-house MATLAB code for subcritical uniform Rayleigh number, Rauni, and periodic Rayleigh number, Rap, on the lower plate. An accurate numerical scheme was developed to capture the full time-dependent behavior here. The interest lied in how the intensities of the vortexes and convection rolls changed as the inclination angle was increased with respect to time. Convection rolls rotating in the clockwise direction expanded and the rolls rotating in the counterclockwise direction shrank and their centers moved closer to the lower wall. Thermal drift appeared between them when the inclination angle started increasing.