Stream Function Formulation Research Articles

A triangular spectral method for the Stokes eigenvalue problem by the stream function formulation

An efficient spectral method is developed in this paper for the two‐dimensional Stokes eigenvalues on arbitrary triangle. By using the spectral theory of compact operator and approximate property of orthogonal polynomial, we give the error estimate of the approximate eigenvalues and eigenfunctions. In addition, we also present some numerical results to show the validity of our algorithm and the correctness of the theoretical results.

Heat transfer of Fe3O4–water nanofluid in a permeable medium with thermal radiation in existence of constant heat flux

Influence of non-uniform magnetic field on Fe3O4–water ferrofluid flow in a porous cavity is simulated. Shape factor effect on nanofluid properties is taken into consideration. Vorticity stream function formulation is reported. The solutions of final equations are obtained using CVFEM. Figures are depicted for different values of radiation parameter (Rd), Darcy number (Da), Fe3O4–water volume fraction (ϕ), Rayleigh (Ra) and Hartmann (Ha) numbers. Results show that augmenting in Hartmann number leads to decrease in velocity of nanofluid and heat transfer rate. Utilizing Platelet shaped nanoparticles results the highest Nusselt number.

Formula omitted]-discontinuous Galerkin methods for a wind-driven ocean circulation model: Two-grid algorithm

This paper presents a nonconforming finite element method for a streamfunction formulation of the stationary quasi-geostrophic equations, which describe the large scale wind-driven ocean circulation. The streamfunction formulation is a fourth order nonlinear PDE and the nonconforming method is based on C0-elements instead of C1-elements. Existence and uniqueness of the approximation are proved and optimal error estimates in several norms of interest are demonstrated under a small data assumption. Two-grid algorithms based on Picard and Newton type linearizations are then presented to efficiently resolve nonlinearities and computational results are given to demonstrate the efficiency of the algorithm. The Mediterranean sea example is tested with real world coastline data, which illustrates the effectiveness of the two-grid approach in the wind-driven ocean circulation simulation.

Mixed Convection in a Ventilated Cavity Filled with a Triangular Porous Layer

Mixed convection in a square cavity having a triangular porous layer and a local heater has been investigated numerically. The governing partial differential equations with corresponding boundary conditions have been solved by the finite difference method using the dimensionless stream function, vorticity and temperature formulation. The effects of the Richardson number ( $${ Ri} = 0.01 - 10$$ ), Darcy number ( $${ Da} = 10^{-7} -10^{-1}$$ ), heater length ( $${\delta = H/L = 0.2-0.4}$$ ) and different locations of the porous layer on the streamlines and isotherms as well as the average and local Nusselt numbers at the heater have been analyzed. It has been found that all these key parameters essentially affect the flow and heat transfer patterns.

Smooth Travelling-Wave Solutions to the Inviscid Surface Quasi-Geostrophic Equation

We construct families of smooth travelling-wave solutions to the inviscid surface quasi-geostrophic equation (SQG). These solutions can be viewed as the equivalents for this equation of the vortex anti-vortex pairs in the context of the incompressible Euler equation. Our argument relies on the stream function formulation and eventually amounts to solving a fractional nonlinear elliptic equation by variational methods.

Numerical investigation of MHD nanofluid free convective heat transfer in a porous tilted enclosure

PurposeThe effect of a magnetic field on nanofluid natural convection in a porous annulus is simulated. Control volume-based finite element method (CVFEM) is applied to find the influence of tilted angle and Darcy, Rayleigh and Hartmann numbers on nanofluid hydrothermal behavior. Vorticity stream function formulation is taken into account. Also, Brownian motion effect on nanofluid thermal conductivity is considered. Results reveal that Hartmann number and tilted angle make changes in nanofluid flow style. Nusselt number enhances with augment of Darcy number and buoyancy forces but reduces with rise of tilted angle and Hartmann number.Design/methodology/approachThe influence of adding CuO nanoparticles in water on the velocity and temperature distribution in an inclined half-annulus was studied considering constant heat flux. CVFEM is applied to the simulation procedure.FindingsInfluences of CuO volume fraction, inclination angle and Rayleigh number on hydrothermal manners are presented.Originality/valueResults indicate that inclination angle makes changes in flow style. The temperature gradient enhances with rise of buoyancy forces, whereas it reduces with augment of inclination angle.

Characterization of columnar inertial modes in rapidly rotating spheres and spheroids.

We consider fluid-filled spheres and spheroidal containers of eccentricity ϵ in rapid rotation, as a proxy for the interior dynamics of stars and planets. The fluid motion is assumed to be quasi-geostrophic (QG): horizontal motions are invariant parallel to the rotation axis z, a characteristic which is handled by use of a stream function formulation which additionally enforces mass conservation and non-penetration at the boundary. By linearizing about a quiescent background state, we investigate a variety of methods to study the QG inviscid inertial wave modes which are compared with fully three-dimensional (3D) calculations. We consider the recently proposed weak formulation of the inviscid system valid in spheroids of arbitrary eccentricity, to which we present novel closed-form polynomial solutions. Our modal solutions accurately represent, in both spatial structure and frequency, the most z-invariant of the inertial wave modes in a spheroid, and constitute a simple basis set for the analysis of rotationally dominated fluids. We further show that these new solutions are more accurate than those of the classical axial-vorticity equation, which is independent of ϵ and thus fails to properly encode the container geometry. We also consider the effects of viscosity for the cases of both no-slip and stress-free boundary conditions for a spherical container. Calculations performed under the columnar approximation are compared with 3D solutions and excellent agreement has been found despite fundamental differences in the two formulations.

Magnetic source influence on nanofluid flow in porous medium considering shape factor effect

In this research, Fe3O4–water nanofluid flow in a permeable medium under the influence of external magnetic source is simulated. Effect of nanoparticle's shape on thermal conductivity is taken into consideration. CVFEM is employed to solve the final equations which are obtained with vorticity stream function formulation. Figures are depicted for various values of Darcy number (Da), radiation parameter (Rd), Fe3O4–water volume fraction (ϕ), Rayleigh (Ra) and Hartmann (Ha) numbers. Results demonstrate that Greatest Nusselt number obtains for Platelet shaped Fe3O4 nanoparticle. Increasing in Hartmann number results in decreasing in velocity of nanofluid and Nusselt number.

CVFEM for influence of external magnetic source on Fe3O4-H2O nanofluid behavior in a permeable cavity considering shape effect

This research is made to show the impact of external magnetic source on Fe3O4 – water nanofluid treatment in a permeable cavity. Shape factor effect on nanofluid properties are taken into account. Final equations are derived by means of vorticity stream function formulation. Control volume based finite element method is employed for solving final formulae. Figures are depicted for various values of radiation parameter, Darcy number, Fe3O4 –water volume fraction, Rayleigh number and Hartmann number. Results reveal that selecting Platelet shaped nanoparticles results the highest heat transfer rate. Nanofluid velocity and heat transfer rate decrease with augment of Hartmann number.

Nanofluid heat transfer in a permeable enclosure in presence of variable magnetic field by means of CVFEM

This research is about modeling the influence of Lorentz forces on Fe3O4–water nanofluid heat transfer enhancement in a permeable cavity. Shape factor influence on thermal conductivity of nanofluid is taken into consideration. Vorticity stream function formulation is utilized. The solutions of final equations are obtained by Control Volume based Finite Element Method. Graphs are shown for different values of radiation parameter (Rd), Darcy number (Da), nanofluid volume fraction (ϕ), Rayleigh number (Ra) and Hartmann number (Ha). Results indicate that heat transfer rate enhances with augment of Hartmann number. Selecting Platelet shaped nanoparticles leads to greatest heat transfer rate.

Numerical modeling for Fe3O4-water nanofluid flow in porous medium considering MFD viscosity

The goal of this research is to demonstrate the influence of variable magnetic field on ferrofluid flow in a porous cavity. Shape factor impact on nanofluid properties is taken into consideration. Final equations are derived via vorticity stream function formulation. Then they are solved via CVFEM. Graphs are shown for different values of radiation parameter, Darcy number, Fe3O4-water volume fraction, Rayleigh and Hartmann numbers. Results prove that Nusselt number reduces with increase of Lorentz forces. In order to reach highest Nusselt number, Platelet shaped nanoparticles should be used.

Creeping flow dynamics over superhydrophobic ball: Slip effects and drag reduction

Slip and drag over superhydrophobic ball in the creeping flow regime (Reynolds number <0.1) are investigated due to its relevance in physiological systems. Experiments involve comparing the settling velocities of superhydrophobic (SH) balls (3mm–11.96mm) fabricated using etching method with that of smooth balls, in highly viscous Newtonian mediums: golden syrup and honey. Due to the high Capillary number (Ca>1) of the system, air is trapped in the surface asperities without formation of film (plastron). The flow over SH ball is modeled using the stream function formulation, that incorporates wall-slip and relates it to the enhanced settling velocities and drag reduction. Slip length decreases (178–24μm) with the increase in Reynolds number, establishing the dependence of slippage on flow apart from surface morphology. The drag reduction is comparatively low 8% due to the absence of plastron. A component based analysis reveal that form drag contributes towards one-third of the total drag while skin drag and normal stress drag contribute towards the remaining two-third. The skin drag takes over the normal stress drag with the increase in Reynolds number, eventually replacing it. The experimental method and modeling constitute a generic technique for evaluating the potential of SH surface to reduce drag.

Magnetic field influence on CuO–H2O nanofluid convective flow in a permeable cavity considering various shapes for nanoparticles

In this paper, CuO–H2O nanofluid forced convection in a lid driven porous cavity is investigated under the impact of magnetic field. Shape effect of nanoparticles and Brownian motion impact on nanofluid properties are taken into consideration. Vorticity stream function formulation is utilized. The solutions of final equations are obtained by CVFEM. Graphs are shown for different values of Darcy number (Da), CuO–H2O volume fraction (ϕ), Reynolds (Ra) and Hartmann (Ha) numbers. Outputs indicate that selecting Platelet shaped nanoparticles results the highest heat transfer rate. Nusselt number augments with rise of Darcy and Reynolds number while it decreases with augment of Lorentz forces.

Numerical modeling of magnetohydrodynamic CuO—Water transportation inside a porous cavity considering shape factor effect

Impact of shape factor on nanofluid forced convection is examined considering the impact of Lorentz forces. Brownian motion effect on nanofluid properties is taken into consideration. Vorticity stream function formulation is utilized. The solutions of final equations are obtained by CVFEM. Graphs are shown for various values of Darcy number(Da), CuO – water volume fraction(ϕ), Reynolds (Ra) and Hartmann(Ha) numbers. Results prove that temperature gradient augments with rise of Darcy and Reynolds number while it decreases with increase of Lorentz forces. Selecting Platelet shaped nanoparticles results the greatest heat transfer rate.

Multi-layer PCM solidification in a finned triplex tube considering natural convection

Efficient latent heat storage systems represent an important opportunity to improve the thermal performance and operational capability of industrial systems (e.g. solar thermal). In this paper, numerical study is performed to evaluate the heat transfer and solidification features, phase change period and temperature distribution of double-layer Phase Changing Materials (PCM) in a finned triplex tube. A two-dimensional finite volume numerical technique is used to solve the governing equations considering conduction and convection heat transfer mechanisms at a fixed Rayleigh number of 106. After comparing the results for single and double layer configurations, possible arrangements of two different PCMs are discussed in terms of different thicknesses of each PCM, fin sizes and different heat transfer fluid temperatures. Results are illustrated as the form of temperature, liquid fraction, stream function and velocity magnitude contours and their changes over the freezing time. Variation of liquid fraction values, average and minimum temperatures of layers in a double-layer PCM are reported for better insight into the heat transfer features of the latent heat thermal energy storage system to enable uniform discharging designs and balance the phase changing rate within the whole annulus.

Aerodynamics of planar counterflowing jets

The planar laminar flow resulting from the impingement of two gaseous jets of different density issuing into an open space from aligned steadily fed slot nozzles of semi-width $H$ separated by a distance $2L$ is investigated by numerical and analytical methods. Specific consideration is given to the high Reynolds and low Mach number conditions typically present in counterflow-flame experiments, for which the flow is nearly inviscid and incompressible. It is shown that introduction of a density-weighted vorticity–streamfunction formulation effectively reduces the problem to one involving two jets of equal density, thereby removing the vortex-sheet character of the interface separating the two jet streams. Besides the geometric parameter $L/H$, the solution depends only on the shape of the velocity profiles in the feed streams and on the jet momentum-flux ratio. While conformal mapping can be used to determine the potential solution corresponding to uniform velocity profiles, numerical integration is required in general to compute rotational flows, including those arising with Poiseuille velocity profiles, with simplified solutions found in the limits $L/H\ll 1$ and $L/H\gg 1$. The results are used to quantify the near-stagnation-point region, of interest in counterflow-flame studies, including the local value of the strain rate as well as the curvature of the separating interface and the variations of the strain rate away from the stagnation point.

Forced convection of nanofluid in presence of constant magnetic field considering shape effects of nanoparticles

Nanofluid forced convective heat transfer in a porous semi-annulus is studied in presence of uniform magnetic field. Various shapes of nanoparticles are considered. Brownian motion impact on viscosity of nanofluid is taken into account. Governing equations are presented in vorticity stream function formulation. Control volume based finite element method (CVFEM) is utilized to obtain the results. At first, the best shape of nanoparticles is selected and then influences of nanofluid volume fraction, Darcy, Reynolds and Hartmann numbers are presented. Results showed that velocity of nanofluid enhances with increase of Darcy and Reynolds numbers. Platelet shape has the highest rate of heat transfer. Nusselt number enhances with increase of nanofluid volume fraction, Darcy, and Reynolds number while it reduces with increase of Lorentz forces.

The Interface Between Fresh and Salt Groundwater in Horizontal Aquifers: The Dupuit\u2013Forchheimer Approximation Revisited

We analyze the motion of a sharp interface between fresh and salt groundwater in horizontal, confined aquifers of infinite extend. The analysis is based on earlier results of De Josselin de Jong (Proc Euromech 143:75–82, 1981). Parameterizing the height of the interface along the horizontal base of the aquifer and assuming the validity of the Dupuit–Forchheimer approximation in both the fresh and saltwater, he derived an approximate interface motion equation. This equation is a nonlinear doubly degenerate diffusion equation in terms of the height of the interface. In that paper, he also developed a stream function-based formulation for the dynamics of a two-fluid interface. By replacing the two fluids by one hypothetical fluid, with a distribution of vortices along the interface, the exact discharge field throughout the flow domain can be determined. Starting point for our analysis is the stream function formulation. We derive an exact integro-differential equation for the movement of the interface. We show that the pointwise differential terms are identical to the approximate Dupuit–Forchheimer interface motion equation as derived by De Josselin de Jong. We analyze (mathematical) properties of the additional integral term in the exact interface motion formulation to validate the approximate Dupuit–Forchheimer interface motion equation. We also consider the case of flat interfaces, and we study the behavior of the toe of the interface. In particular, we give a criterion for finite or infinite speed of propagation.

Viscous and inviscid simulations of the start-up vortex

Inviscid, unsteady simulations of the roll up of the start-up vortex issuing from a semi-infinite plate are compared with previous simulations of the viscous flow. The inviscid equations were solved by a lumped-vortex method, the two-dimensional, incompressible Navier–Stokes equations in the vorticity–streamfunction formulation modelled the viscous problem. The purpose is to verify whether the irregular behaviour found by the inviscid solution well approximates the unstable evolution of the viscous spiral vortex in the limit of infinitely large time (or equivalently Reynolds number).

Equivalent formulations for steady periodic water waves of fixed mean-depth with discontinuous vorticity

In this work we prove the equivalence between three different weak formulations of the steady periodic water wave problem where the vorticity is discontinuous. In particular, we prove that generalised versions of the standard Euler and stream function formulation of the governing equations are equivalent to a weak version of the recently introduced modified-height formulation. The weak solutions of these formulations are considered in Holder spaces.