Vorticity Function Research Articles

Natural convection combined with surface radiation in a rotating cavity with an element of variable volumetric heat generation

Thermogravitational convection and surface radiation in a rotating chamber with an element of variable volumetric thermal generation has been studied. The analyzed chamber has two thermally-insulated horizontal borders, two isothermal vertical walls of the low temperature and variable heat-generated element on the bottom border. The cavity rotates with a constant angular velocity. Mathematical model has been formulated in non-dimensional stream function and vorticity functions. The important equations have been worked out by the finite difference technique on uniform grid. The impacts of the Taylor number, surface emissivity, and heat source volumetric heat generation oscillation frequency on heat and mass transfer have been investigated. It has been found that the surface emissivity and the volumetric heat generation oscillation frequency can improve the heat transfer in a rotating cavity. A rise of the surface emissivity illustrates the total energy transport enhancement and a diminution of the average heater temperature, while a rise of the volumetric heat generation oscillation frequency characterizes a reduction of the average heater temperature.

Thermal radiation and natural convection in a large-scale enclosure heated from below: Building application

A computational research of radiative-convective energy transport in large-scale enclosure with a heat-generating heater under normal room conditions has been conducted. The heater (underfloor heating system) is located at the bottom of the room. Employing the Boussinesq assumption, the control equations have been solved contemporaneously to receive both the velocity fields and temperature patterns. To generate the systems of linear equations using vorticity and stream function, the finite difference technique has been employed. The developed convective-radiative model has been validated through a comparison with several problems. The influence of heater size and location, internal surfaces emissivity from 0 to 1, Ostrogradsky number for a wide range from 0 to 5 on Nusselt numbers and both stream function and temperature distributions has been investigated. The results demonstrate that the influence of the thermal radiation on total heat transfer increases with surface emissivity of walls and heater surfaces.

Stratosphere–Troposphere Exchange and O3 Variability in the Lower Stratosphere and Upper Troposphere over the Irene SHADOZ Site, South Africa

This study aims to investigate the Stratosphere-Troposphere Exchange (STE) events and ozone changes over Irene (25.5° S, 28.1° E). Twelve years of ozonesondes data (2000–2007, 2012–2015) from Irene station operating in the framework of the Southern Hemisphere Additional Ozonesodes (SHADOZ) was used to study the troposphere (0–16 km) and stratosphere (17–28 km) ozone (O3) vertical profiles. Ozone profiles were grouped into three categories (2000–2003, 2004–2007 and 2012–2015) and average composites were calculated for each category. Fifteen O3 enhancement events were identified over the study period. These events were observed in all seasons (one event in summer, four events in autumn, five events in winter and five events in spring); however, they predominantly occur in winter and spring. The STE events presented here are observed to be influenced by the Southern Hemisphere polar vortex. To strengthen the investigation into STE events, advected potential vorticity maps were used, which were assimilated using Modélisation Isentrope du transport Méso–échelle de l’Ozone Stratosphérique par Advection (MIMOSA) model for the 350 K (~12–13 km) isentropic level. These maps indicated transport of high latitude air masses responsible for the reduction of the O3 mole fractions at the lower stratosphere over Irene which coincides with the enhancement of ozone in the upper troposphere. In general, the stratosphere is dominated by higher Modern Retrospective Analysis for Research Application (MERRA-2) potential vorticity (PV) values compared to the troposphere. However, during the STE events, higher PV values from the stratosphere were observed to intrude the troposphere. Ozone decline was observed from 12 km to 24 km with the highest decline occurring from 14 km to 18 km. An average decrease of 6.0% and 9.1% was calculated from 12 to 24 km in 2004–2007 and 2012–2015 respectively, when compared with 2000–2003 average composite. The observed decline occurred in the upper troposphere and lower stratosphere with winter and spring showing more decline compared with summer and autumn.

Numerical Investigation of Magneto-Natural Heat Transfer in Nanofluid Filled Inclined Enclosure

This paper examines the natural convection in an inclined enclosure that is filled with a water-copper nanofluid and influenced by a magnetic field applied normal to the plane of the cavity. The horizontal walls are assumed to be insulated while the side vertical walls of the cavity are heated differentially. The governing equations are re-formulated in functions of stream function and vorticity. The resulting boundary value problem is solved numerically using an ADI method (alternating direction implicit). A variety of plots showing the velocity and temperature profiles and the influence of Hartmann number as well as Rayleigh number on the streamlines and isotherms are shown.

Shape and size of large-scale vortices: A generic fluid pattern in geophysical fluid dynamics

Planetary rotation organizes fluid motions into coherent, long-lived swirls, known as large-scale vortices (LSVs), which play an important role in the dynamics and long-term evolution of geophysical and astrophysical fluids. Here, using direct numerical simulations, we show that LSVs in rapidly rotating mixed convective and stably stratified fluids, which approximates the two-layer, turbulent-stratified dynamics of many geophysical and astrophysical fluids, have a generic shape and that their size can be predicted. We show that LSVs emerge in the convection zone from upscale energy transfers and can penetrate into the stratified layer. At the convective-stratified interface, the LSV cores have a positive buoyancy anomaly. Due to the thermal wind constraint, this buoyancy anomaly leads to winds in the stratified layer that decay over a characteristic vertical length scale. Thus LSVs take the shape of a depth-invariant cylinder with a finite-size radius in the turbulent layer and of a penetrating half dome in the stratified layer. Importantly, we demonstrate that when LSVs penetrate all the way through the stratified layer and reach a boundary that is no-slip, they saturate by boundary friction. We provide a prediction for the penetration depth and maximum radius of LSVs as a function of the LSV vorticity, the stratified layer depth, and the stratification. Our results, which apply for cyclonic LSVs, suggest that LSVs in slowly rotating stars and Earth's liquid core are confined to the convective layer, while in Earth's atmosphere and oceans they can penetrate far into the stratified layer.

Inviscid, Incompressible, Axisymmetric, Annular Jet Impingement

Abstract An analytical Bessel-sine series solution is presented for inviscid, incompressible, axisymmetric, annular impinging jets. The inflow velocity profile is relatable to an aircraft engine thrust reverser, and has a velocity deficit as well as nonzero vorticity function at the inner radius. As a consequence, inviscid recirculation appears at the impingement corner, the strength of which is made determinate by assuming a local patch of constant vorticity function. Results are first compared against prior round jet solutions, with rigorous convergence and accuracy assessment, before extension to annular impinging jets. Finally, while the Bessel-sine series are consistent with the governing equations and boundary conditions, computational issues arise with steep near wall gradients of a composite parameter that is inherent to the methodology, necessitating extremely fine mesh resolution. A hybrid of finite differences and Bessel series is proposed to address these numerical issues.

Solar Rossby waves observed in GONG++ ring-diagram flow maps

Context. Solar Rossby waves have only recently been unambiguously identified in Helioseimsic and Magnetic Imager (HMI) and Michelson Doppler Imager maps of flows near the solar surface. So far this has not been done with the Global Oscillation Network Group (GONG) ground-based observations, which have different noise properties. Aims. We use 17 years of GONG++ data to identify and characterize solar Rossby waves using ring-diagram helioseismology. We compare directly with HMI ring-diagram analysis. Methods. Maps of the radial vorticity were obtained for flows within the top 2 Mm of the surface for 17 years of GONG++ data. The data were corrected for systematic effects including the annual periodicity related to the B0 angle. We then computed the Fourier components of the radial vorticity of the flows in the co-rotating frame. We performed the same analysis on the HMI data that overlap in time. Results. We find that the solar Rossby waves have measurable amplitudes in the GONG++ sectoral power spectra for azimuthal orders between m = 3 and m = 15. The measured mode characteristics (frequencies, lifetimes, and amplitudes) from GONG++ are consistent with the HMI measurements in the overlap period from 2010 to 2018 for m ≤ 9. For higher-m modes the amplitudes and frequencies agree within two sigmas. The signal-to-noise ratio of modes in GONG++ power spectra is comparable to those of HMI for 8 ≤ m ≤ 11, but is lower by a factor of two for other modes. Conclusions. The GONG++ data provide a long and uniform data set that can be used to study solar global-scale Rossby waves from 2001.

Hamiltonian description of vortex systems

In the framework of 2D ideal Hydrodynamics a vortex system is defined as a smooth vorticity function having few positive local maxima and negative local minima separated by curves of zero vorticity. Invariants of such structures are discussed following from the vorticity conservation law and invertibility of Lagrangian motion. Hamiltonian formalism for vortex systems is developed by introducing new functional variables diagonalizing the original non-canonical Poisson bracket.

Vorticity and Stream Function Formulations for the 2D Navier–Stokes Equations in a Bounded Domain

The main purpose of this work is to provide a Hilbertian functional framework for the analysis of the planar incompressible Navier–Stokes (NS) equations either in vorticity or in stream function formulation. The fluid is assumed to occupy a bounded possibly multiply connected domain. The velocity field satisfies either homogeneous (no-slip boundary conditions) or prescribed Dirichlet boundary conditions. We prove that the analysis of the 2D Navier–Stokes equations can be carried out in terms of the so-called nonprimitive variables only (vorticity field and stream function) without resorting to the classical NS theory (stated in primitive variables, i.e. velocity and pressure fields). Both approaches (in primitive and nonprimitive variables) are shown to be equivalent for weak (Leray) and strong (Kato) solutions. Explicit, Bernoulli-like formulas are derived and allow recovering the pressure field from the vorticity fields or the stream function. In the last section, the functional framework described earlier leads to a simplified rephrasing of the vorticity dynamics, as introduced by Maekawa (Adv Differ Equ 18(1–2):101–146, 2013). At this level of regularity, the vorticity equation splits into a coupling between a parabolic and an elliptic equation corresponding respectively to the non-harmonic and harmonic parts of the vorticity equation. By exploiting this structure it is possible to prove new existence and uniqueness results, as well as the exponential decay of the palinstrophy (that is, loosely speaking, the $$H^1$$ norm of the vorticity) for large time.

Existence and uniqueness results for a second order differential equation for the ocean flow in arctic gyres

In this paper, existence and uniqueness of solutions for the nonlinear and linear models of the arctic gyres over $$84^{\circ }$$ north latitude are studied by using the stereographic projection, which represents the streamline and no jet at the outside boundary. By using the fixed point technique, we prove the existence and uniqueness of the local solution of the nonlinear model. Next, we present the existence and uniqueness of the solution in the semi-infinite interval under the suitable asymptotic conditions. In the case of linear vorticity function, we give the explicit solutions by adopt the idea for linear ODEs.

Exploring the latitude and depth dependence of solar Rossby waves using ring-diagram analysis

Context. Global-scale equatorial Rossby waves have recently been unambiguously identified on the Sun. Like solar acoustic modes, Rossby waves are probes of the solar interior. Aims. We study the latitude and depth dependence of the Rossby wave eigenfunctions. Methods. By applying helioseismic ring-diagram analysis and granulation tracking to observations by HMI aboard SDO, we computed maps of the radial vorticity of flows in the upper solar convection zone (down to depths of more than 16 Mm). The horizontal sampling of the ring-diagram maps is approximately 90 Mm (∼7.5°) and the temporal sampling is roughly 27 hr. We used a Fourier transform in longitude to separate the different azimuthal orders m in the range 3 ≤ m ≤ 15. At each m we obtained the phase and amplitude of the Rossby waves as functions of depth using the helioseismic data. At each m we also measured the latitude dependence of the eigenfunctions by calculating the covariance between the equator and other latitudes. Results. We conducted a study of the horizontal and radial dependences of the radial vorticity eigenfunctions. The horizontal eigenfunctions are complex. As observed previously, the real part peaks at the equator and switches sign near ±30°, thus the eigenfunctions show significant non-sectoral contributions. The imaginary part is smaller than the real part. The phase of the radial eigenfunctions varies by only ±5° over the top 15 Mm. The amplitude of the radial eigenfunctions decreases by about 10% from the surface down to 8 Mm (the region in which ring-diagram analysis is most reliable, as seen by comparing with the rotation rate measured by global-mode seismology). Conclusions. The radial dependence of the radial vorticity eigenfunctions deduced from ring-diagram analysis is consistent with a power law down to 8 Mm and is unreliable at larger depths. However, the observations provide only weak constraints on the power-law exponents. For the real part, the latitude dependence of the eigenfunctions is consistent with previous work (using granulation tracking). The imaginary part is smaller than the real part but significantly nonzero.

4D Flow Vorticity Visualization Predicts Regions of Quantitative Flow Inconsistency for Optimal Blood Flow Measurement.

To evaluate whether automated vorticity mapping four-dimensional (4D) flow MRI can identify regions of quantitative flow inconsistency. In this retrospective study, 35 consecutive patients who underwent MR angiography with 4D flow MRI at 3.0 T from December 2017 to October 2018 were analyzed using a λ 2-based technique for vorticity visualization and quantification. The patients were aged 58.6 years ± 14.4 (standard deviation), 12 were women, 18 had ascending aortic aneurysms (maximal diameter > 4.0 cm), and 10 had bicuspid aortic valves. Flow measurements were made in the ascending aorta (aAo), mid-descending aorta, main pulmonary artery, and superior vena cava. Statistical tests included t tests and F tests with a type I error threshold (α) of .05. The 35 patients were visually classified as having no (n = 9), mild (n = 8), moderate (n = 11), or severe vorticity (n = 7). Across all patients, standard deviation of cardiac output in the aAo (0.58 L/min ± 0.45) was significantly (P < .001) higher than in the pulmonary arteries (0.15 L/min ± 0.10) and descending aorta and superior vena cava (0.14 L/min ± 0.12). The standard deviation of cardiac output observed in the aAo was significantly greater (P < .01) in patients with moderate or severe vorticity (0.73 L/min ± 0.55) than in those with none or mild vorticity (0.44 L/min ± 0.26). Cardiac output and blood flow are essential MRI measurements in the evaluation of structural heart disease. Vorticity visualization may be used to help guide optimal location for flow quantification.© RSNA, 2020See also the commentary by Markl in this issue.

A spectral element Crank\u2013Nicolson model to the 2D unsteady conduction\u2013convection problems about vorticity and stream functions

In this study, a time semi-discretized Crank–Nicolson (CN) scheme of the two-dimensional (2D) unsteady conduction–convection problems for vorticity and stream functions is first built together with showing the existence and stability along with error estimates to the semi-discretized CN solutions. Afterwards, a fully discretized spectral element CN (SECN) model of the 2D unsteady conduction–convection problems as regards the vorticity and stream functions is set up together with showing the proof of the existence and stability along with error estimates of the SECN solution. Lastly, a set of numerical experiments are offered for checking the correctness of the theoretical conclusions.

Magnetohydrodynamic natural convection and entropy generation analyses inside a nanofluid-filled incinerator-shaped porous cavity with wavy heater block

The aim of the current study is natural convection analysis conjugated with entropy generation analysis in an incinerator shaped permeable enclosure loaded with Al2O3–H2O nanofluid subjected to the magnetic field with a rectangular wavy heater block positioned on the bottom of the cavity wall. The bottom and top horizontal walls are adiabatic; the inclined and vertical walls are thought to be cooled. Firstly, the governing expressions and standard k–e turbulence model are rewritten from dimensional form to non-dimensional form using dimensionless parameters such as vorticity and stream function. In the next step, the equation of entropy generation is written in dimensionless form. Then, the system of non-dimensional governing equations is solved by the finite volume method (FVM) conjugated with a non-dimensionalization scheme using ANSYS Fluent. Fine grids (wall y+ < 2) with inflated layers have been used for the higher Rayleigh number. The effects of the Rayleigh number in the laminar region (Ra = 103, 104, and 105) and turbulent region (Ra = 108, 0.5 × 109, and 109), Darcy number (Da = 0.01 and 100), Hartmann number (Ha = 0 and 40), and the nanoparticles ( $$ \phi = 2{{\% }} $$ ) on the entropy generation number and natural convection are investigated. The validation results were in good agreement with those of the literature. The results demonstrate that for the laminar region, the Nusselt number and entropy generation number increase as the Rayleigh number and the Darcy number grow, whereas both of them abate as Hartmann number increases. In the turbulent region, the average Nusselt number decreases by ascending the Darcy number. Also, for turbulent region (Ra = 109), convection flow strength decreases 6.28% when Hartmann number increases from 0 to 40, whereas the entropy generation number increases 31.5% at Da = 0.01.

Investigation of entropy generation in a square inclined cavity using control volume finite element method with aided quadratic Lagrange interpolation functions

One of the most concerned subjective in Mechanical science is natural convection study in a cavity. Furthermore, the investigation of entropy generation can be useful for better designing of the thermal systems. In the present study, natural convection flow and entropy generation are numerically investigated in the presence of a magnetic field in a square inclined cavity which known as the effect of magneto-hydrodynamic (MHD). Firstly, governing equations, including mass, momentum, and energy balance equations are applied to the problem. Then, governing equations are rewritten in dimensionless form using non-dimensional parameters, vorticity, and stream function. Furthermore, the entropy generation equation is written as a non-dimensional equation. Then, the contribution of the heat transfer entropy generation into the total entropy generation rate is determined using the Bejan number. A new measure for evaluation of the thermal performance of the cavity is presented that is the so-called ecological coefficient of performance (ECOP) based on the second law of thermodynamics. Flow and heat transfer characteristics are investigated for different values of the Rayleigh number, inclination angle, and Hartmann number. The new correlations of entropy generation as a function of Rayleigh number are obtained using the two-dimensional version of quadratic Lagrange interpolation functions (QLIFs). The obtained values for the average Nusselt number and the entropy generation number are compared with those of the literature and excellent agreement is observed. The results show that the entropy generation number rises with increasing of Hartmann number and whereas it has a maximum value for a specified inclination angle. Also, ECOP increases with increasing of Hartmann number for low values of Rayleigh number. The results discover that the optimum values of the inclination angle are 20°, 35° and 48°for Hartmann number 0, 25 and 75 at Ra = 105, respectively. Also, with increasing Ha from 25 to 75, the contribution of the Ngen due to magnetic field decreases from 23% to 9% for β = 0° and it decreases from 24% to 6% for β = 45°.

Simulations of axisymmetric, inviscid swirling flows in circular pipes with various geometries

The numerical simulations of the dynamics of high Reynolds number ($$Re>100{,}000$$) swirling flows in pipes with varying geometries of engineering applications continues to be a challenging computational problem, specifically when vortex-breakdown zones or wall-separation regions naturally evolve in the flows. To tackle this challenge, the present paper describes a simulation scheme of the evolution of inviscid, axisymmetric and incompressible swirling flows in expanding or contracting pipes. The integration in time of the circulation together with azimuthal vorticity uses an explicit, first-order accurate finite-difference scheme with a second-order accurate upwind difference formulation in the axial and radial directions. The Poisson solver for advancing in time the spatial distribution of the stream function as a function of azimuthal vorticity uses a second-order accurate over-relaxation difference scheme. No additional numerical steps are needed for computing the natural evolution of flows including the dynamics to states with slow-speed recirculation zones along the pipe centerline or attached to pipe wall. This numerical method shows convergence of the computed results with mesh refinement for various swirl levels and pipe geometry variations. The computed results of time-asymptotic states also present agreement with available theoretical predictions of steady vortex flows in diverging or contracting pipes. In addition, comparison with available experimental data demonstrates that the present algorithm accurately predicts instability processes and long-term mean-flow dynamics of vortex flows in pipes at high Re. The inviscid-flow simulation results support the theoretical predictions and clarify the nature of high-Re flow evolution.

Numerical investigation of bifurcation characteristics under perturbations in vortex induced vibration of cylinder with two degrees of freedom

In this paper, the bifurcation characteristics and perturbation response in vortex induced vibration of cylinder with two degrees of freedom is investigated, aiming to study the law and mechanism of the influence of perturbation on the bifurcation of vortex-induced vibration, and to give some engineering suggestions. The main content of this paper is to observe the variation of the amplitude, frequency, lift and drag coefficients of the vortex-induced vibration response of the cylinder after suddenly perturbing the cylinder displacement, to investigate the influence of displacement perturbations on bifurcation characteristics. Then through vorticity map analysis, the mechanism is further analyzed from the point of view of flow field. The results show that when the displacement perturbation is applied, the motion state of the cylinder will change, which will cause the change of the shedding mode and the wake mode of the vortex, resulting in the instability of the flow field. Then under the coupling action of disturbed flow field and cylinder motion, the flow field and the vibration state of the cylinder finally stabilize to the corresponding state of the upper branch or the lower branch, depending on the direction and magnitude of the perturbation. It can be concluded that in the velocity range near the branch jump points (upper and lower branches), the cylinder's vortex induced vibration may have two relatively stable states and can be transformed to each other under appropriate excitation of perturbations. Finally, some suggestions are given, based on the above results.

A Transformation for Imposing the Rigid Bodies on the Solution of the Vorticity-Stream Function Formulation of Incompressible Navier–Stokes Equations

A new penalization method is proposed for implementing the rigid bodies on the solution of the vorticity-stream function formulation of the incompressible Navier–Stokes equations. The method is based upon an active transformation of dependent variables. The transformation may be interpreted as time dilation. In this interpretation, the rigid body is considered as a region where the time is dilated infinitely, that is, time is stopped. The transformation is introduced in the vorticity and stream function equations to achieve a set of modified equations. The, in the modified equations, the time dilation of the solid region is approached to infinity. The mathematical and physical properties of the modified equations are investigated and implementation of the no-slip and no-penetration conditions are justified. Moreover, a suitable numerical method is presented for the solution of the modified equations. In the proposed numerical method, time integration is performed via the Crank–Nicolson method, and the semi-discrete equations are spatially discretized via second-order finite differencing on a uniform Cartesian grid. The method is applied to the fluid flow around a square obstacle placed in a channel, a sudden flow perpendicular to a thin flat plate, and the flow around a circular cylinder. The accuracy of the numerical solutions is evaluated.

Aerodynamic Characteristics of Mira Fastback Model in Experiment and CFD

The lasting high fuel cost has recently inspired the resurgence in drag reduction research for vehicles, which calls for a thorough understanding of the vehicle wake. The acknowledged MIRA fastback vehicle model is characterized by similar real vehicle geometry, thus it is especially suitable for the above purpose. In spite of a considerable number of previous investigations, our knowledge of flow around this model remains incomplete. This paper aims to visit turbulent flow structure behind this model. An investigation has been conducted to measure the near wake flow structure of the MIRA 1/8 scale mode, using both Particle Image Velocimetry (PIV) experimental method and Computational Fluid Dynamics (CFD) simulation method. In order to capture the flow structure accurately, PIV measurement was performed in different sections along three orthogonal directions, and the CFD method acquired additional simulation results to catch the better flow status. Through the maps of the time-averaged vorticity, the instantaneous vorticity and the handled velocity vector from the PIV and CFD methods, we found out the formation mechanism of the transient flow of fastback model and summed up the schematic of flow structure. This study not only analyzed the vortex shedding characteristics of turbulent near wake, but more importantly provided insight into the complex three-dimensional features of the flow structure in the wakes of MIRA fastback model.

Emergence of Self-Excited Oscillations in Flows of Inviscid Fluids in a Channel

The mechanism of self-excited oscillations arising in an ideal incompressible fluid flowing through a rectangular channel is studied numerically. The problem is formulated in the form of the Euler equations for ideal fluid dynamics in terms of vorticity and stream function with Yudovich’s boundary conditions. The vorticity intensity at the inlet of the channel is used as a bifurcation parameter. Grid approximations are employed to search for steady-state regimes and to analyze their stability, while the nonstationary problem is solved by applying the vortex-in-cell method. It is shown that a steady flow through the channel is established when the vorticity intensity at the inlet is low. As the vorticity intensity at the inlet grows, the steady-state regime becomes unstable in an oscillatory manner and self-excited oscillations emerge in its neighborhood. The evolution of the self-excited oscillations with an increasing bifurcation parameter is studied. In the case of high supercriticality, a chaotic flow regime is observed in the channel.