Stream Function-vorticity Equations Research Articles

Intracardiac Flow Analysis of the Right Ventricle in Pediatric Patients With Repaired Tetralogy of Fallot Using a Novel Color Doppler Velocity Reconstruction

Repaired tetralogy of Fallot (RTOF) patients will develop right ventricular (RV) dysfunction from chronic pulmonary regurgitation (PR). Cardiac magnetic resonance sequences such as four-dimensional flow can demonstrate altered vorticity and flow energy loss (FEL); however, they are not as available as conventional echocardiography (echo). The study determined whether a novel, vendor-independent Doppler velocity reconstruction (DoVeR) could measure RV intracardiac flow in conventional echo of RTOF patients. The primary hypothesis was that DoVeR could detect increased vorticity and diastolic FEL in RTOF patients. Repaired tetralogy of Fallot patients with echo were retrospectively paired with age-/size-matched controls. Doppler velocity reconstruction employed the stream function-vorticity equation to approximate intracardiac flow fields from color Doppler. A velocity field of the right ventricle was reconstructed from the apical 4-chamber view. Vortex strength (VS, area integral of vorticity) and FEL were derived from DoVeR. Cardiac magnetic resonance and exercise stress parameters (performed within 1year of echo) were collected for analysis. Twenty RTOF patients and age-matched controls were included in the study. Mean regurgitant fraction was 40.5%±7.6%, and indexed RV end-diastolic volume was 158±36mL/m2. Repaired tetralogy of Fallot patients had higher total, mean diastolic, and peak diastolic VS (P=.0013, P=.0012, P=.0032, respectively) and higher total, mean diastolic, and peak diastolic body surface area-indexed FEL (P=.0016, P=.0022, P<.001, respectively). Peak diastolic indexed FEL and peak diastolic VS had weak-to-moderate negative correlation with RV ejection fraction (r=-0.52 [P=.019] and r=-0.49 [P=.030], respectively) and left ventricular ejection fraction (r=-0.47 [P=.034] and r=-0.64 [P=.002], respectively). Mean diastolic indexed FEL and VS had moderate-to-strong negative correlation with percent predicted maximal oxygen consumption (r=-0.69 [P=.012] and r=-0.75 [P=.006], respectively). DoVeR can detect alterations to intracardiac flow in RTOF patients from conventional color Doppler imaging. Echo-based measures of diastolic VS and FEL correlated with ventricular function. DoVeR has the potential to provide serial evaluation of abnormal flow dynamics in RTOF patients.

Colour-Doppler echocardiography flow field velocity reconstruction using a streamfunction-vorticity formulation.

We introduce a new method (Doppler Velocity Reconstruction or DoVeR), for reconstructing two-component velocity fields from colour Doppler scans. DoVeR employs the streamfunction-vorticity equation, which satisfies mass conservation while accurately approximating the flow rate of rotation. We validated DoVeR using artificial colour Doppler images generated from computational fluid dynamics models of left ventricle (LV) flow. We compare DoVeR against the conventional intraventricular vector flow mapping (iVFM1D) and reformulated iVFM (iVFM2D). LV model error analysis showed that DoVeR is more robust to noise and probe placement, with noise RMS errors (nRMSE) between 3.81% and 6.67%, while the iVFM methods delivered 4.16-24.17% for iVFM1D and 4.06-400.21% for iVFM2D. We test the DoVeR and iVFM methods using in vivo mouse LV ultrasound scans. DoVeR yielded more haemodynamically accurate reconstructions, suggesting that it can provide a more reliable approach for robust quantification of cardiac flow.

MHD natural convection in a cavity in the presence of cross partial magnetic fields and Al[formula omitted]O[formula omitted]-water nanofluid

In this study, a numerical investigation on nanofluid flow and heat transfer is presented in a cavity under the effects of cross partial magnetic fields and a partial heater. The governing stream function–vorticity equations are solved by thin plate spline radial basis functions (Rbfs) in space derivatives and backward Euler method in time derivatives. The vorticity transport equation involves either only buoyancy force or both buoyancy and cross partial magnetic field force. The results are depicted as streamlines, isotherms and vorticity contours as well as mean Nusselt number. The convective heat transfer increases with the rise in the length of the partial heater, in the concentration of nanoparticles and in Rayleigh number while it decreases with the augmentation of the Lorentz force.

Multidomain BEM for laminar flow in complex fractal geometry

This paper demonstrates the highly efficient 2D multidomain Boundary Element Method (BEM) for solving stream functionvorticity equations on fractal geometry containing a thousand corners and a hundred recirculation zones. Considering the sharp corners, two problems are solved. The problem of undefined unit normal in corner is overridden using mixed boundary element discretisation. The second problem is on determining the boundary vorticity values used as a boundary condition in a vorticity transport equation for a no-slip boundary. The implicit computation of boundary vorticities from a stream function governing equation as an integral constraint is applied. The numerical example is an intricate fractal geometry of the Koch snowflake solved in a large variation of Reynolds number values ranging from 10−6 to 100. The flow pattern self-similarity is obtained, equivalent to fractal geometry similarity.

Suppression mechanism of two-degree-of-freedom vortex-induced vibration by Lorentz forces in the uniform flow

In this paper, the control of 2DOF VIV (two-degree-of-freedom vortex-induced vibration) by Lorentz force has been investigated numerically based on the derivation of stream function-vorticity equations together with the initial and boundary conditions in exponential-polar coordinates attached on a moving cylinder, hydrodynamics forces and the cylinder motion equation. From the derivations of force components, the lift/drag induced by the inertial force only depend on the motion along the corresponding direction, while the lift/drag induced by flow field is affected by the cylinder motion along the two directions. Based on the calculation results, the displacement variation of 2DOF VIV along the transverse direction is similar with that of 1DOF VIV (one-degree-of-freedom vortex-induced vibration). However, the secondary vortex is strengthened with the effect of the pressure side and weakened with the effect of the suction side. With the application of symmetrical Lorentz force, the effects of the pressure/suction side and vortex shedding are weakened, which lead to the suppression of 2DOF VIV. Moreover, the cylinder vibration is fully suppressed and the drag is negative due to the net thrust generated if Lorentz force is large enough, which means the final position of cylinder is at the upstream of the initial position. Furthermore, the fluid-structure interactions from the quiescent cylinder to the steady vibration and then vibration control by Lorentz force are investigated. The shear layers and secondary vortexes grow with the increase of cylinder amplitude and decay with the decrease of cylinder amplitude due to the suppression of cylinder vibration controlled by Lorentz force in the whole process.

Numerical investigations of two-degree-of-freedom vortex-induced vibration in shear flow

Exponential-polar coordinates attached to a moving cylinder are used to deduce the stream function-vorticity equations for two-degree-of-freedom vortex-induced vibration, the initial and boundary conditions, and the distribution of the hydrodynamic force, which consists of the vortex-induced force, inertial force, and viscous damping force. The fluid-structure interactions occurring from the motionless cylinder to the steady vibration are investigated numerically, and the variations of the flow field, pressure, lift/drag, and cylinder displacement are discussed. Both the dominant vortex and the cylinder shift, whose effects are opposite, affect the shear layer along the transverse direction and the secondary vortex along the streamwise direction. However, the effect of the cylinder shift is larger than that of the dominant vortices. Therefore, the former dominates the total effects of the flow field. Moreover, the symmetry of the flow field is broken with the increasing shear rate. With the effect of the background vortex, the upper vortices are strengthened, and the lower vortices are weakened; thus, the shear layer and the secondary vortices induced by the upper shedding vortices are strengthened, while the shear layer and the secondary vortices induced by the lower shedding vortices are weakened. Therefore, the amplitudes of the displacement and drag/lift dominated by the upper vortex are larger than those of the displacement and drag/lift dominated by the lower vortex.

Investigations on the Effects of Vortex-Induced Vibration with Different Distributions of Lorentz Forces

The control of vortex-induced vibration (VIV) in shear flow with different distributions of Lorentz force is numerically investigated based on the stream function–vorticity equations in the exponential-polar coordinates exerted on moving cylinder for Re = 150. The cylinder motion equation coupled with the fluid, including the mathematical expressions of the lift force coefficient C l , is derived. The initial and boundary conditions as well as the hydrodynamic forces on the surface of cylinder are also formulated. The Lorentz force applied to suppress the VIV has no relationship with the flow field, and involves two categories, i.e., the field Lorentz force and the wall Lorentz force. With the application of symmetrical Lorentz forces, the symmetric field Lorentz force can amplify the drag, suppress the flow separation, decrease the lift fluctuation, and then suppress the VIV while the wall Lorentz force decreases the drag only. With the application of asymmetrical Lorentz forces, besides the above-mentioned effects, the field Lorentz force can increase additional lift induced by shear flow, whereas the wall Lorentz force can counteract the additional lift, which is dominated on the total effect.

The mechanism investigation of two-degree-of-freedom vortex-induced vibration with electro-magnetic forces

The electro-magnetic forces generated by electromagnetic field take control of the flow in the electrolyte solution. In this paper, the mechanism of two-degree-of-freedom vortex-induced vibration controlled by electro-magnetic forces is investigated numerically. With the coordinate at the moving cylinder, the stream function-vorticity equations, the initial and boundary conditions and distribution of hydrodynamic force are deduced in the exponential-polar coordinate. The equation of vorticity transport is solved by the alternative-direction implicit algorithm. The equation of stream function is integrated by means of a fast Fourier transform algorithm. The cylinder motion is calculated by the Runge-Kutta method. The flow field, pressure, lift/drag and cylinder displacement are interacted along the transverse and streamwise direction, where the instantaneous variations are discussed. The derivation shows that the vibration displacement in one direction, whose effects on the flow field influence the vortex-induced forces in both directions, affects the inertial force only in the corresponding direction and is independent of that in the other direction. The numerical calculations show that the vortex-induced vibration is affected by two factors, i.e., the vortex shedding and the cylinder shift. Both of the two factors have influences on the shear layers in the transverse direction and the secondary vortex in the streamwise direction, which further leads to the variations of lift/drag and the cylinder motion. Along the transverse direction, the strength of shear layer on one side is increased by the vortex shedding while the strength of shear layer on the other side is increased by the cylinder shift. Along the streamwise direction, the pressure of cylinder tail is varied with the effect of shedding vortex on the secondary vortex while the effect of cylinder shift on the secondary vortex is also opposite to that of shedding vortex. Notably, the effect of cylinder shift prevails over the effect of shedding vortex so that the former is dominated in the total effects. The flow separation and vortex shedding are suppressed as the fluid of boundary layer is accelerated under the action of electro-magnetic forces. Meanwhile, the vibration displacements decrease gradually along both the transverse and streamwise directions, which also suppresses the effects of pressure/suction sides. Therefore, the vibration is suppressed and the cylinder turns steady rapidly. In addition, the thrust generated by the wall electro-magnetic force counteracts the drag generated by the fluid electro-magnetic force, which means that the final position is at the upstream of the initial position. The experimental results show that the vortexes on the cylinder are suppressed fully and the flow field is steady under the action of electro-magnetic force, which agrees well with the numerical results.

A Numerical Study of the Flow Past an Impermeable Sphere Embedded in a Porous Medium

The flow past an impermeable sphere embedded in a fluid-saturated porous medium was investigated numerically. The flow is considered viscous, laminar, axisymmetric, steady and incompressible. The generalized model for fluid flow through an isotropic, rigid and homogeneous porous medium was used. The stream function–vorticity equations were solved numerically in spherical coordinates system by a defect correction multigrid method. The computations were focused on the influence of the Darcy number (Da) and Forchheimer constant on the flow field for two boundary conditions on the surface of the sphere: slip and no-slip. The conditions when the macroscopic inertial terms of the generalized model can be neglected are presented. For high-porosity porous media, the macroscopic inertial terms of the generalized model can be neglected if Da \(Re< 1\) (Re is the sphere Reynolds number).

Effect of Undulations on the Natural Convection Heat Transfer and Entropy Generation Inside a Porous Right-Angled Triangular Enclosure

In the present study, natural convection in a two-dimensional porous right-angled triangular enclosure with one wavy wall is studied numerically. Three cases with one, two, and three undulations on the left wall are studied in this analysis. The stream function-vorticity equations are solved using finite-difference technique and a structured nonorthogonal body-fitted mesh is used for computations. The effect of Rayleigh number (Ra = 103–106), Darcy number (Da = 10−4–10−2) and undulations on the heat transfer, fluid flow, and entropy generation is investigated. It is found that average Nusselt number increases with Darcy number and number of undulations present on the left wall at fixed Darcy number.

An in-depth study on vortex-induced vibration of a circular cylinder with shear flow

To investigate the in-depth mechanism of vortex-induced vibration of a circular cylinder with shear flow, in this paper, with the use of exponential-polar coordinate attached on the moving cylinder, the stream function-vorticity equations of vortex-induced vibration, the initial/boundary conditions and distribution of hydrodynamic force together with cylinder motion equation in shear flow are deduced, the hydrodynamic force consists of inertial force, the vortex-induced force and viscous damping force. Similarly, the cylinder motion equation with virtual mass is induced where the virtual mass consists of the cylinder mass, the potential added mass and the apparent added mass induced by viscosity. Our numerical results revealed that there are three factors affecting fluid-structure interactions from the fixed cylinder to its steady vibration: The first is the vortex shedding where one side shear layer of cylinder strengthens with the effect of the dominated vortex. The second is the vibration of cylinder which pushes the fluid on the pressure side and pumps that on the suction side. The third is the vortexes strengthen in one side and weaken in the other side together with the shift of front stagnation point with the effect of background vortex which is generated by shear flow. The character of vortex-induced vibration in shear flow are affected by the above three factors.

Numerical study of the suppression mechanism of vortex-induced vibration by symmetric Lorentz forces

In this paper, the electro-magnetic control of vortex-induced vibration (VIV) of a circular cylinder is investigated numerically based on the stream function–vorticity equations in the exponential–polar coordinates attached on the moving cylinder for Re=150. The effects of the instantaneous wake geometries and the corresponding cylinder motion on the hydrodynamic forces for one entire period of vortex shedding are discussed using a drag–lift phase diagram. The drag–lift diagram is composed of the upper and lower closed curves due to the contributions of the vortex shedding but is magnified, translated and turned under the action of the cylinder motion. The Lorentz force for controlling the vibration cylinder is classified into the field Lorentz force and the wall Lorentz force. The symmetric field Lorentz force will symmetrize the flow passing over the cylinder and decreases the lift oscillation, which, in turn, suppresses the VIV, whereas the wall Lorentz force has no effect on the lift. The cylinder vibration increases as the work performed by the lift dominates the energy transfer. Otherwise, the cylinder vibration decreases. If the net transferred energy per motion is equal to zero, the cylinder will vibrate steadily or be fixed.

Analysis of natural convection heat transfer and entropy generation inside porous right-angled triangular enclosure

Natural convection in a two-dimensional porous right-angled triangular enclosure having undulation on the left wall is analyzed numerically. The bottom wall is having a sinusoidal temperature and other two walls are maintained at isothermal cold temperature. The stream function-vorticity equations are solved using finite-difference technique. The computations are done on a structured non-orthogonal body fitted mesh. Standard QUICK scheme is used for convective term discretization and diffusive term is discretized using central difference scheme. In this study, the effect of Rayleigh number (Ra=103−106), Darcy number (Da=10−4−10−2) and undulations on the left wall is investigated on the heat transfer and fluid flow. It is found that for small Ra, the heat transfer is dominated by conduction across the fluid layers but with the increase of Ra, the process began to be dominated by convection and the undulation on the left wall play an important role in increasing heat transfer. It is also found that heat transfer is also a strong function of Darcy number. As Darcy number increases, heat transfer increases in an appreciable manner.

ADI method based on [formula omitted]-continuous two-node integrated-RBF elements for viscous flows

We propose a C2-continuous alternating direction implicit (ADI) method for the solution of the streamfunction–vorticity equations governing steady 2D incompressible viscous fluid flows. Discretisation is simply achieved with Cartesian grids. Local two-node integrated radial basis function elements (IRBFEs) [D.-A. An-Vo, N. Mai-Duy, T. Tran-Cong, A C2-continuous control-volume technique based on Cartesian grids and two-node integrated-RBF elements for second-order elliptic problems, CMES: Comput. Model. Eng. Sci. 72 (2011) 299–334] are used for the discretisation of the diffusion terms, and then the convection terms are incorporated into system matrices by treating nodal derivatives as unknowns. ADI procedure is applied for the time integration. Following ADI factorisation, the two-dimensional problem becomes a sequence of one-dimensional problems. The solution strategy consists of multiple use of a one-dimensional sparse matrix algorithm that helps saving the computational cost. High levels of accuracy and efficiency of the present methods are demonstrated with solutions of several benchmark problems defined on rectangular and non-rectangular domains.

Numerical study of the effects of lid oscillation on the periodic flow pattern and convection heat transfer in a triangular cavity

This study is concerned with a periodic flow pattern with mixed convection in a triangular cavity caused by the effects of lid oscillation and buoyancy. The dimensionless stream function–vorticity formulation is adopted, and a curvilinear grid method for solving the stream function–vorticity equations in irregular geometries is used. Attention is in particular focused on the flow behaviour under the interaction between the frequency of the oscillation of the lid velocity and the frequency of the natural periodic flow. Meanwhile, numerical predictions of the thermal characteristics represented by local and average Nusselt numbers on the walls as well as the transient flow patterns are also provided. Results show that the frequency of oscillation of lid velocity and the natural periodic flow frequency both appear to be major frequencies in the frequency spectrum of the cavity flow, for cases with a dimensionless lid oscillation frequency less than 0.5. When the dimensionless frequency of the oscillation of the lid velocity is equal to or higher than 0.5, the flow and thermal fields are completely locked-on to the lid oscillation, and the natural periodic flow frequency is no longer visible in the spectrum.

High order compact computation and nonuniform grids for streamfunction vorticity equations

Fourth order compact difference schemes for steady streamfunction vorticity formulation of 2D incompressible Navier–Stokes equations on nonuniform grids are derived using Maple software package. Especially we deduce fourth order compact difference scheme of the first partial derivative terms. In order to resolve boundary layers, grid transformation techniques are used, which maps a nonuniform grid onto a uniform one for use with the fourth order compact difference scheme. A Krylov subspace iterative method with an ILUT preconditioning technique is employed to solve the resulting linear system. The proposed high accuracy computation method is applied to two model problems. Computational results are compared with results computed by other schemes in the literature.

A Cartesian Grid Method for Solving the Two-Dimensional Streamfunction-Vorticity Equations in Irregular Regions

We describe a method for solving the two-dimensional Navier–Stokes equations in irregular physical domains. Our method is based on an underlying uniform Cartesian grid and second-order finite-difference/finite-volume discretizations of the streamfunction-vorticity equations. Geometry representing stationary solid obstacles in the flow domain is embedded in the Cartesian grid and special discretizations near the embedded boundary ensure the accuracy of the solution in the cut cells. Along the embedded boundary, we determine a distribution of vorticity sources needed to impose the no-slip flow conditions. This distribution appears as a right-hand-side term in the discretized fluid equations, and so we can use fast solvers to solve the linear systems that arise. To handle the advective terms, we use the high-resolution algorithms in CLAWPACK. We show that our Stokes solver is second-order accurate for steady state solutions and that our full Navier–Stokes solver is between first- and second-order accurate and reproduces results from well-studied benchmark problems in viscous fluid flow. Finally, we demonstrate the robustness of our code on flow in a complex domain.

A Cartesian Grid Method for Solving the Two-Dimensional Streamfunction-Vorticity Equations in Irregular Regions

We describe a method for solving the two-dimensional Navier Stokes equations in irregular physical domains. Our method is based on an underlying uniform Cartesian grid and second-order finite-diffe...

Transition of Natural Convective Flows in a Horizontal Cylindrical Annulus: Pr=0.2

Transition of flows in natural convection in a horizontal cylindrical annulus is investigated for the fluid with Pr=0.2. The unsteady streamfunction-vorticity equation is solved with finite difference method. As Rayleigh number is increased, the steady crescent-shaped eddy flow bifurcates to a time-periodic flow with like-rotating eddies. After the first Hopf bifurcation, however, a reverse transition from oscillatory to a steady flow occurs by the flow pattern variation. Hysteresis phenomenon occurs between the solution branches of up-scan and down-scan stages, and dual solutions with one steady and one oscillatory flow are found. Overall Nusselt of the flows at the flows at the down-scan stage is greater than that at the up-scan stage.

Chaotic Thermal Convection of a Intermediate Prandtl-Number Fluid in a Horizontal Annulus: Pr=0.2

Natural convection of a fluid with intermediate Prand시 number of Pr=0.2 in a horizontal annulus is considered, and the bifurcation phenomena and chaotic flows are numerically investigated. The unsteady two-dimensional streamfunction-vorticity equation is solved with finite difference method. The steady downward flow with two counter-rotating eddies bifurcates to a simple periodic flow with a fundamental frequency. And afterwards, second Hopf bifurcation occurs, and a quasi-periodic flow with two incommensurable frequencies appears. However, a new time-periodic flow is established after experiencing quasi-periodic states. As Rayleigh number is increased further, the chaotic flow regime is reached after a sequence of successive Hopf bifurcation to quasi-periodic and chaotic flow regimes. A scenario similar to the Ruelle-Takens-Newhouse scenario of the onset of chaos is observed.