Stream Vorticity Research Articles

Numerical study of the impacts of stochastic forcing on the vortex in fluid flow

This paper focuses on a numerical study about the stochastic Navier–Stokes equations. Unlike previous studies, this paper focuses on studying these equations from the perspective of vortices. The vorticity–stream function method was proposed to deal with incompressible fluid flow. And a Crank–Nicolson Fourier pseudo-spectral method was put forward to solve the formulation of stream function equation. In addition, we have conducted some numerical experiments to observe the effects of random forcing on vortices in the fluid flow by utilizing stochastic solution.

Numerical simulation of natural convection in a rectangular enclosure filled with porous medium saturated with magnetic nanofluid using Buongiorno's two‐component model

AbstractMotivated by studying emerging nanofluid‐based magnetic fuel cells and hybrid direct absorber solar collectors, a numerical study is presented for buoyancy‐driven flow in a vertical enclosure containing a porous medium saturated with magnetized nanofluid flow under a transverse static magnetic field. The enclosure features adiabatic side walls and vertical heat slits, ensuring consistent cold temperatures on the lower and upper bounded walls. The side walls of the regime are insulated, and the hot slits are centrally located on these walls. The finite difference method (FDM) is applied to employ the transformed dimensionless vorticity–stream function (VSF) formulation of the transport equations. The impact of pertinent parameters on isotherm, streamline, iso‐concentration, and average Nusselt and Sherwood numbers are visualized with contour plots and graphs. Increasing Darcy number values tend to amplify the isotherm magnitudes. Higher Hartmann (magnetic) number values, on the other hand, lead to a reduction in local Nusselt number profiles but do not significantly modify the local Sherwood number. The porous medium permeability, as simulated via the Darcy number, has a strong impact on streamlines, thermal contours, and iso‐concentrations. A reduction in Darcy's number suppresses local Nusselt and Sherwood numbers, whereas an elevation in Rayleigh's number enhances them. Increasing the Buongiorno nanoscale Brownian motion parameter enhances local Nusselt and Sherwood numbers at both cold walls of the enclosure. The simulations provide a deeper insight into enclosure flows involving electrically conducting nanofluids in advanced processing systems for nanomaterials and hybrid fuel cells utilizing electromagnetic and liquid fuel technologies.

Blowup analysis for a quasi-exact 1D model of 3D Euler and Navier–Stokes

We study the singularity formation of a quasi-exact 1D model proposed by Hou and Li (2008 Commun. Pure Appl. Math. 61 661–97). This model is based on an approximation of the axisymmetric Navier–Stokes equations in the r direction. The solution of the 1D model can be used to construct an exact solution of the original 3D Euler and Navier–Stokes equations if the initial angular velocity, angular vorticity, and angular stream function are linear in r. This model shares many intrinsic properties similar to those of the 3D Euler and Navier–Stokes equations. It captures the competition between advection and vortex stretching as in the 1D De Gregorio (De Gregorio 1990 J. Stat. Phys. 59 1251–63; De Gregorio 1996 Math. Methods Appl. Sci. 19 1233–55) model. We show that the inviscid model with weakened advection and smooth initial data or the original 1D model with Hölder continuous data develops a self-similar blowup. We also show that the viscous model with weakened advection and smooth initial data develops a finite time blowup. To obtain sharp estimates for the nonlocal terms, we perform an exact computation for the low-frequency Fourier modes and extract damping in leading order estimates for the high-frequency modes using singularly weighted norms in the energy estimates. The analysis for the viscous case is more subtle since the viscous terms produce some instability if we just use singular weights. We establish the blowup analysis for the viscous model by carefully designing an energy norm that combines a singularly weighted energy norm and a sum of high-order Sobolev norms.

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.

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.

Heat transport in magnetohydrodynamic physiological blood flow through a constricted artery

AbstractIn this paper, we study how the magnetohydrodynamic (MHD) pulsatile flow of blood and heat transfer works through a constricted artery with a flexible wall. The human circulatory network consists of veins and arteries that sometimes contain constrictions, allowing the impact of the applied magnetic field on flow fields to be observed. The walls of the flowing medium are considered to be a function of time. The flowing blood is hypothesized as shear‐thinning fluid, emulating Yeleswarapu's viscosity replica. Additionally, we consider the energy equation to understand the impact of a magnetic field on heat transfer rates for such flows. The vorticity transport equation along with the stream function equation is obtained using the vorticity–stream function technique. Numerical solutions of the governing nonlinear MHD equations and energy equation in addition to physically pertinent flow conditions were achieved by adapting a finite difference scheme. Considerable attention has been paid to ensure an accurate comparison between the current and previous results. The two sets of numbers appear to match closely. For an even deeper understanding of the flow and heat transport process, the effects of height of stenosis and diverse physiological parameters on time‐averaged wall shear stress (TAWSS), rate of heat transport, and so on are explored in depth through their graphical depiction. In the vicinity of the constriction, it is observed that the separation becomes longer with increasing constriction height. Higher magnetic force strength leads to a reduction in separation length. Newtonian fluids transfer heat more rapidly in their narrowing regions and downstream than fluids with non‐Newtonian behavior.

The Mixed Finite Element Reduced-Dimension Technique with Unchanged Basis Functions for Hydrodynamic Equation

The mixed finite element (MFE) method is one of the most valid numerical approaches to solve hydrodynamic equations because it can be very suited to solving problems with complex computing domains. Regrettably, the MFE method for the hydrodynamic equations would include lots of unknowns. Especially, when it is applied to settling the practical engineering problems, it could contain hundreds of thousands and even tens of millions of unknowns. Thus, it would bring about many difficulties for actual applications, such as consuming a long CPU running time and accumulating many round-off errors, so as to be very difficult to obtain the desired numerical solutions. Therefore, we herein take the two-dimensional (2D) unsteady Navier–Stokes equation in hydrodynamics as an example. Using the proper orthogonal decomposition to lower the dimension of unknown Crank–Nicolson MFE (CNMFE) solution coefficient vectors for the 2D unsteady Navier–Stokes equation about vorticity–stream functions, we construct a reduced-dimension recursive CNMFE (RDRCNMFE) method with unchanged basis functions. In the circumstances, the RDRCNMFE method can keep the basis functions unchanged in an MFE subspace and has the same precision as the classical CNMFE method. We employ the matrix method to analyse the existence and stability along with errors to the RDRCNMFE solutions, leading to a very simple theory analysis. We use the numerical simulations for the backwards-facing step flow to verify the effectiveness of the RDRCNMFE method. The RDRCNMFE method with unchanged basis functions only reduces the dimension of the solution coefficient vectors of the CNMFE, which is completely different from previous order reduction methods which greatly affects the accuracy by reducing the dimension of the MFE subspace.

Reduced‐order extrapolation technique for coefficient vectors of numerical solutions about fluid mechanics equation

AbstractFor convenience, this study mainly focuses on the reduced order of coefficient vectors for the classical finite spectral element Crank–Nicolson (CFSECN) solutions about the two‐dimensional (2D) nonstationary Stokes problem, resulting that the reduced order model can possess the same basis functions together with same precision as the CFSECN one. To this end, the CFSECN format of the 2D nonstationary Stokes problem about vorticity function and stream function and, the existence and stability as well as error estimations for the CFSECN solutions are first proposed. Afterwards, a reduced order extrapolation finite spectral element Crank–Nicolson (ROEFSECN) model is created by means of proper orthogonal decomposition method, and the existence together with stability as well as errors of the ROEFSECN solutions is analyzed by the matrix tool, resulting that the theory analysis turns into very simple. Finally, two numerical examples are provided to confirm the correctness for the theory consequences. Therefore, the study is absolutely distinguished from the existed ones.

Double diffusive convection of CuO + MWCNT hybrid nanofluid in an inclined open cavity with internal heat generation/absorption

An intention of the current study is to analyse the effect of double-diffusive convection in a hybrid nanofluid filled in an inclined open cavity with heat generation/absorption. The cavity walls are insulated whereas the centrally placed baffle is maintained at a high temperature. The CuO + MWCNT nanoparticles suspended in the water are used as a working hybrid nanofluid. The angle of inclination γ = 0°–90° is applied for the open cavity. The non-dimensional governing equation in the form of vorticity stream function transport is solved numerically by using the finite difference method. The behaviour of fluid flow, heat transfer, and mass transfer are presented as a contour. At various parametric conditions (q = −4 to 4, Br = −4 to 4, Le = 1–6), the average Sherwood and Nusselt numbers are calculated and explained graphically. The results report that an inclination angle suppresses the heat and mass transfer rate in an open enclosure.

Disturbance growth on a NACA0008 wing subjected to free stream turbulence

The stability of an incompressible boundary layer flow over a wing in the presence of free stream turbulence (FST) has been investigated by means of direct numerical simulations and compared with the linearised boundary layer equations. Four different FST conditions have been considered, which are characterised by their turbulence intensity levels and length scales. In all cases the perturbed flow develops into elongated disturbances of high and low streamwise velocity inside the boundary layer, where their spacing has been found to be strongly dependent on the scales of the incoming free stream vorticity. The breakdown of these streaks into turbulent spots from local secondary instabilities is also observed, presenting the same development as the ones reported in flat plate experiments. The disturbance growth, characterised by its root mean squares value, is found to depend not only on the turbulence level, but also on the FST length scales. Particularly, higher disturbance growth is observed for our cases with larger length scales. This behaviour is attributed to the preferred wavenumbers that can exhibit maximum transient growth. We study this boundary layer preference by projection of the flow fields at the leading edge onto optimal disturbances. Our results demonstrate that optimal disturbance growth is the main cause of growth of disturbances on the wing boundary layer.

The Dimensionality Reduction of Crank–Nicolson Mixed Finite Element Solution Coefficient Vectors for the Unsteady Stokes Equation

By means of a proper orthogonal decomposition (POD) to cut down the dimensionality of unknown finite element (FE) solution coefficient vectors in the Crank–Nicolson (CN) mixed FE (CNMFE) method for two-dimensional (2D) unsteady Stokes equations in regard to vorticity stream functions, a reduced dimension recursive-CNMFE (RDR-CNMFE) method is constructed. In this case, the RDR-CNMFE method has the same FE basis functions and accuracy as the CNMFE method. The existence, stability, and errors of RDR-CNMFE solutions are analyzed by matrix analyzing, resulting in very simple theory analysis. Some numerical tries are used to check on the validity of the RDR-CNMFE method. The RDR-CNMFE method has second-order time accuracy and few unknowns so as to be able to shorten CPU runtime and retard the error cumulation in simulation calculating process, and improve real-time calculating accuracy.

Magneto-convective flow through a porous enclosure with Hall current and thermal radiation effects: numerical study

This paper reports the numerical study of magnetohydrodynamic radiative-convective flow in a square cavity containing a porous medium with Hall currents. This study is relent to hydromagnetic fuel cell design and thermofluidic dynamics of complex magnetic liquid fabrication in enclosures. The governing equations of this fluid system are solved by a finite-difference vorticity stream function approach executed in MATLAB software. A detailed parametric investigation of the impact of Rayleigh number (thermal buoyancy parameter), Hartman number (magnetic body force parameter), Darcy number (permeability parameter), Hall parameter and radiation parameter on the streamline, temperature contours, local Nusselt number along the hot wall and mid-section velocity profiles is computed. Validation with previous special cases in the literature is included. Hall current and radiative effects are found to significantly modify thermofluidic characteristics. From the numerical results, it is found that the magnetic field suppresses the natural convection only for small buoyancy ratios. But, for larger buoyancy ratio, the magnetic field is effective in suppressing the thermal convective flow.

Optimization of cooling of three radiant heat source inside an inclined porous medium using particle swarm optimization

In this study, an optimization of the maximum cooling of three radiant heat sources inside a closed square porous environment is performed. Three radiant-heat sources are cooled by airflow in a porous medium with an inclination angle. It is applied in the cooling of electronic equipment inside laptop computers. The cooling phenomenon is a function of five independent variables, such as the length of heat source, radiation parameter, Darcy number, the angle of inclination of a porous medium, and Rayleigh number. Differential equations are solved using the finite-difference approach. And those are solved by the Gauss-Seidel approach. Simulation is a numerical solution of the vorticity–stream equations. Coding is done in MATLAB software. And it is obtained the maximum objective function using the algorithm of particle swarm optimization. Cooling simulations are performed by drawing flow lines, constant temperature lines, dimensionless velocity component profiles, local Nusselt numbers, and mean Nusselt numbers. The main goal is to find the highest cooling rate in different amounts of five independent variables. The flow is in the laminar flow regime, and inclination angles are between 0 to 360°. The present study is the first study on the numerical solution of the cooling of three radiant-heat sources with an inclination angle by the porous media-filled square environment. And it is developed from the cooling of hot radiation walls problems in an inclined enclosure with constant temperature local radiant heat sources. This investigation showed the flow lines and their direction are influenced by the inclination angle of the porous medium strongly, and the maximum heat transfer is achieved at an angle of inclination of zero degrees. This investigation showed at the Darcy number, 0.001, and the radiation parameter, 1. The maximum and minimum Nusselt number is obtained for the inclination angle 60 and 180°, respectively.

Numerical analysis of MHD nanofluid flow and heat transfer in a circular porous medium containing a Cassini oval under the influence of the Lorentz and buoyancy forces

AbstractThis paper reports our study on the flow characteristics and heat transfer performance of magnetohydrodynamics (MHD) nanofluid in an innovative porous, circle‐shaped enclosure incorporating a Cassini oval cavity using the Darcy law. The MHD nanofluid considered in this study is Al2O3–H2O. A two‐dimensional (2D) mathematical model is developed based on a homogeneous model. The formulation of the vorticity stream function is then used to obtain coupled equations. Finally, the coupled partial differential equations are solved numerically using the finite element method. Model predictions are then compared against results from the previously published study to verify the accuracy and validity of the developed model, and a good agreement is achieved. Figures demonstrate the effects of nanoparticle volume fraction, inclined angle, Lorentz, and buoyancy forces on the MHD nanofluid flow. The results indicate that the convection mechanism becomes weaker with an increase in solid nanoparticle volume fraction. A significant increase in the Rayleigh number will lead to a stronger and more cohesive core vortex. In addition, when magnetic force is applied horizontally, favorable Nuave occurs. Based on the numerical results, a correlation to predict the average Nusselt number within the enclosure is developed as a function of Hartmann number (Ha), Rayleigh number (Ra), and inclined angle (γ).

Vorticity–Stream Function Formulation for Turbulent Oscillatory Boundary Layer over the Sea Bottom

Abstract The oscillatory flow within the oscillatory boundary layer at the bottom of propagating surface waves is determined by numerically solving the vorticity–stream function problem when the fl...

Physics-constrained intraventricular vector flow mapping by color Doppler

Color Doppler by transthoracic echocardiography creates two-dimensional fan-shaped maps of blood velocities in the cardiac cavities. It is a one-component velocimetric technique since it only returns the velocity components parallel to the ultrasound beams. Intraventricular vector flow mapping (iVFM) is a method to recover the blood velocity vectors from the Doppler scalar fields in an echocardiographic three-chamber view. We improved our iVFM numerical scheme by imposing physical constraints. The iVFM consisted in minimizing regularized Doppler residuals subject to the condition that two fluid-dynamics constraints were satisfied, namely planar mass conservation, and free-slip boundary conditions. The optimization problem was solved by using the Lagrange multiplier method. A finite-difference discretization of the optimization problem, written in the polar coordinate system centered on the cardiac ultrasound probe, led to a sparse linear system. The single regularization parameter was determined automatically for non-supervision considerations. The physics-constrained method was validated using realistic intracardiac flow data from a patient-specific computational fluid dynamics (CFD) model. The numerical evaluations showed that the iVFM-derived velocity vectors were in very good agreement with the CFD-based original velocities, with relative errors ranged between 0.3% and 12%. We calculated two macroscopic measures of flow in the cardiac region of interest, the mean vorticity and mean stream function, and observed an excellent concordance between physics-constrained iVFM and CFD. The capability of physics-constrained iVFM was finally tested with in vivo color Doppler data acquired in patients routinely examined in the echocardiographic laboratory. The vortex that forms during the rapid filling was deciphered. The physics-constrained iVFM algorithm is ready for pilot clinical studies and is expected to have a significant clinical impact on the assessment of diastolic function.

Numerical simulation of thermal management during natural convection in a porous triangular cavity containing air and hot obstacles

A numerical study is presented of laminar viscous magnetohydrodynamic natural convection flow in a triangular-shaped porous enclosure filled with electrically conducting air and containing two hot obstacles. The mathematical model is formulated in terms of dimensional partial differential equations. The pressure gradient term is eliminated by using the vorticity–stream (\(\omega - \psi\)) function approach. The emerging dimensionless governing equations are employed by the regular finite difference scheme along with thermofluidic boundary conditions. The efficiency of the obtained computed results for isotherms and streamlines is validated via comparison with previously published work. The impact of physical parameters on streamlines and temperature contours for an extensive range of Rayleigh number (Ra = 103–105), Hartmann magnetohydrodynamic number (Ha = 5–30), Darcy parameter (Da = 0.0001–0.1) for fixed Prandtl number (Pr = 0.71) is considered. Numerical results are also presented for local and average Nusselt numbers along the hot base wall. Interesting features of the thermofluid behaviour are revealed. At lower Rayleigh number, the isotherms are generally parallel to the inclined wall and only distorted substantially near the obstacles at the left vertical adiabatic wall; however, with increasing Rayleigh number, this distortion is magnified in the core zone and simultaneously warmer zones expand towards the inclined cold wall. The simulations are relevant to magnetic materials processing and hybrid magnetic fuel cells.

Vortex dynamics on a Möbius strip

We consider the dynamics of a two-dimensional incompressible perfect fluid on a Möbius strip embedded in $\mathbb {R}^{3}$ . The vorticity–stream function formulation of the Euler equations is derived from an exterior-calculus form of the momentum equation. The non-orientability of the Möbius strip and the distinction between forms and pseudo-forms this introduces lead to unusual properties: a boundary condition is provided by the conservation of circulation along the single boundary of the strip, and there is no integral conservation for the vorticity density or for any odd function thereof. A finite-difference numerical implementation is used to illustrate the Möbius-strip realisation of familiar phenomena: translation of vortices along boundaries, shear instability and decaying turbulence.

Cu and Cu-SWCNT Nanoparticles' Suspension in Pulsatile Casson Fluid Flow via Darcy-Forchheimer Porous Channel with Compliant Walls: A Prospective Model for Blood Flow in Stenosed Arteries.

The use of experimental relations to approximate the efficient thermophysical properties of a nanofluid (NF) with Cu nanoparticles (NPs) and hybrid nanofluid (HNF) with Cu-SWCNT NPs and subsequently model the two-dimensional pulsatile Casson fluid flow under the impact of the magnetic field and thermal radiation is a novelty of the current study. Heat and mass transfer analysis of the pulsatile flow of non-Newtonian Casson HNF via a Darcy–Forchheimer porous channel with compliant walls is presented. Such a problem offers a prospective model to study the blood flow via stenosed arteries. A finite-difference flow solver is used to numerically solve the system obtained using the vorticity stream function formulation on the time-dependent governing equations. The behavior of Cu-based NF and Cu-SWCNT-based HNF on the wall shear stress (WSS), velocity, temperature, and concentration profiles are analyzed graphically. The influence of the Casson parameter, radiation parameter, Hartmann number, Darcy number, Soret number, Reynolds number, Strouhal number, and Peclet number on the flow profiles are analyzed. Furthermore, the influence of the flow parameters on the non-dimensional numbers such as the skin friction coefficient, Nusselt number, and Sherwood number is also discussed. These quantities escalate as the Reynolds number is enhanced and reduce by escalating the porosity parameter. The Peclet number shows a high impact on the microorganism’s density in a blood NF. The HNF has been shown to have superior thermal properties to the traditional one. These results could help in devising hydraulic treatments for blood flow in highly stenosed arteries, biomechanical system design, and industrial plants in which flow pulsation is essential.

Numerical investigation of thermally developed MHD flow with pulsation in a channel with multiple constrictions

This article concerns heat transfer analysis in pulsating flow in a channel with walls having multiple symmetric constrictions. The flow is influenced by Lorentz force and thermal radiation. The unsteady governing equations, simplified for low conducting fluids, are solved by the finite difference method using the stream–vorticity function formulation. The effects of the emerging parameters, including the magnetic field parameter (Hartman number), Reynolds number, Prandtl number, and radiation parameter on various flow profiles, are studied. The profiles of dimensionless axial velocity, temperature, wall shear stress (WSS), skin friction coefficient, and local Nusselt number are discussed graphically. The profiles are examined at various prominent axial locations and time instants of the pulse cycle. The WSS has a direct relation with the Hartmann and Strouhal numbers. The WSS generated at the first constriction is higher than that at the second constriction. The WSS increases with an increase in the Strouhal number in the accelerating phase and decreases in the decelerating phase on both the constrictions. The temperature decreases with an increase in the Hartman and Prandtl numbers at the constricted portion of the channel. The radiation parameter directly affects the temperature and inversely affects the Nusselt number at the constricted part of the channel. However, in general, the flow profiles exhibit irregular patterns downstream of the constriction. The Nusselt profiles are higher at the first encountered constriction bump than that at the next bumps.