Vorticity Boundary Conditions Research Articles

Boundary Conditions for Momentum and Vorticity at an Interface between Two Fluids

Boundary Conditions for Momentum and Vorticity at an Interface between Two Fluids

CFD informed design of bench-scale experiments to characterize air entrainment into fuel beds induced by columnar vortices

Recent experiments show that strong vortices, similar to fire whirls, can form far from a fire front in the region of smoldering fuel. These buoyancy-induced columnar vortices, visualized by entrained smolder smoke, were observed lofting hot embers into the air and in some cases lead to spot ignitions at the base of the vortex. Gaining insight on how the flow field of a buoyancy-induced columnar vortex could impact surrounding smoldering fuel is the focus of this study. Specifically, the potential air entrainment into a fuel substrate beneath the vortex. The flow field of such columnar vortices has been shown to drive air flow downward under certain conditions and, in the context of combustion, drive air deeper than typical entrainment, inducing spot ignitions and increasing burning and smolder rates. NIST's Fire Dynamics Simulator is utilized to successfully model buoyancy-induced columnar vortices. Then it is utilized to study the behavior of vortices as temperature and vorticity boundary conditions are changed. The flow field and fresh air entrainment potential are analyzed. The simulation results inform the experiment design and preliminary experimental results are presented. Understanding these high-risk phenomena will lead to better risk mitigation and more resilient Wildland-Urban interface communities.

Availability of the Boundary Condition for Vorticity

By applying a boundary condition for vorticity [1] in addition to that for velocity, a velocity distribution on a flat plate set in a parallel homogeneous flow has been numerically obtained through a one-way calculation from surface to infinity, without the “matching” procedure between an analysis from surface to infinity and that from infinity to surface. The numerical results obtained were in excellent agreement with those by Howarth [2]. The usage of the boundary condition for vorticity has raised the accuracy of velocity distribution near a plate’s surface and made it possible to realize the one-way calculation from surface to infinity.

On port-Hamiltonian formulations of 3-dimensional compressible Newtonian fluids

In this manuscript, a general formulation of 3-dimensional compressible fluids based on the port-Hamiltonian framework is presented, both for isentropic and non-isentropic assumptions, describing the energy flux between the mechanical, chemical, and thermal domains, with an explicit characterization of the first and the second law of thermodynamics. For isentropic fluids, the conversion of kinetic energy into heat by viscous friction is considered as energy dissipation associated with the rotation and compression of the fluid. A dissipative port-Hamiltonian formulation is derived for this class of fluids, including vorticity boundary conditions in the port variables. For non-isentropic fluids, we consider a fluid mixture with multiple chemical reactions. To describe the energy fluxes, we propose a pseudo port-Hamiltonian formulation, which includes the rate of irreversible entropy creation by heat flux, chemical reaction, diffusion of matter, and viscous friction.

Receptivity to Freestream Periodic Vorticity Disturbance on a Flat Plate with an Elliptic Leading Edge

The receptivity of the Blasius boundary layer over a semi-infinite flat plate with an elliptic leading edge and of aspect ratio five was investigated using a direct numerical solution of two-dimensional Navier-Stokes equations. The result of the computation where the slip condition is applied to the fluctuating component of velocity at the wall surface is compared with that of an ordinary computation using a nonslip condition. Another numerical experiment is performed where no vorticity fluctuation is supplied from a freestream while prerecorded values of vorticities at the wall in response to the passage of convecting fluctuations are used as the wall vorticity boundary condition. It is shown that vorticity fluctuations in the boundary layer can be classified according to their wavelengths. Waves with longer wavelengths originate from the freestream, whereas waves with shorter wavelengths close to T-S waves originate from the surface of the plate. In another numerical experiment, the slip boundary condition against the fluctuation component of vorticity is applied to the limited area of the wall surface. The aim of the study is to determine the part of the elliptic leading edge or flat plate that induces vorticity fluctuations, thereby resulting in the creation of T-S waves. The numerical results show that the contribution of vorticity fluctuations originating from the juncture is the most crucial, whereas the vorticities supplied in the elliptic leading-edge surface negatively affect the amplitude of vorticity fluctuations inside the boundary layer. And, the stagnation section did not show positive contribution.

Parallel implementation of a VIScous Vorticity Equation (VISVE) method in 3-D laminar flow

This paper presents a newly developed parallel implementation of solving the 3–D vorticity equation to fully simulate the incompressible laminar flow in the Eulerian frame. This method is designed to solve 3–D problems with irregular wall boundaries in small and compact computational domains in general shapes efficiently. The curl form of vorticity equation is discretized using the Finite Volume Method (FVM) by applying Stokes' theorem, which automatically guarantees the divergence–free condition of vorticity field at all times. The vorticity preserving velocity field is recovered by an explicit scheme without solving any linear system, and this velocity field is reprojected onto a divergence–free space by solving only one scalar velocity–potential Poisson's equation. The vorticity boundary condition is satisfied by employing a vorticity creation scheme, that can handle arbitrary wall boundary shapes. Numerical results of the flow past a 3–D NACA0012 hydrofoil with one periodic direction, the flow past a sphere, and the flow past a 3–D rectangular wing are presented to validate the scheme.

A New Boundary Condition for the Hall-Magnetohydrodynamics Equation with the Ion-Slip Effect

In this paper, we prove a regularity criteria and the local well-posedness of strong solutions to the magnetohydrodynamics with the Hall and ion-slip effects in a smooth bounded domain. Moreover, for the magnetohydrodynamics with the Hall effect and without ion-slip effect, we prove the global stability of strong large solutions. These results are obtained under an integrable property that depends on a new vorticity boundary condition which is motivated by the generalized Navier-slip boundary condition involving the vorticity.

A fourth-order accurate finite difference method to evaluate the true transient behaviour of natural convection flow in enclosures

PurposeModelling accurately the transient behaviour of natural convection flow in enclosures been a challenging task because of a variety of numerical errors which have limited achieving the higher order temporal accuracy. A fourth-order accurate finite difference method in both space and time is proposed to overcome these numerical errors and accurately model the transient behaviour of natural convection flow in enclosures using vorticity–streamfunction formulation.Design/methodology/approachFourth-order wide stencil formula with appropriate one-sided difference extrapolation technique near the boundary is used for spatial discretisation, and classical fourth-order Runge–Kutta scheme is applied for transient term discretisation. The proposed method is applied on two transient case studies, i.e. convection–diffusion of a Gaussian Pulse and Taylor Vortex flow having analytical solution.FindingsError magnitude comparison and rate of convergence analysis of the proposed method with these analytical solutions establish fourth-order accuracy and prove the ability of the proposed method to truly capture the transient behaviour of incompressible flow. Also, to test the transient natural convection flow behaviour, the algorithm is tested on differentially heated square cavity at high Rayleigh number in the range of 103-108, followed by studying the transient periodic behaviour in a differentially heated vertical cavity of aspect ratio 8:1. An excellent comparison is obtained with standard benchmark results.Research limitations/implicationsThe developed method is applied on 2D enclosures; however, the present methodology can be extended to 3D enclosures using velocity–vorticity formulations which shall be explored in future.Originality/valueThe proposed methodology to achieve fourth-order accurate transient simulation of natural convection flows is novel, to the best of the authors’ knowledge. Stable fourth-order vorticity boundary conditions are derived for boundary and external boundary regions. The selected case studies for comparison demonstrate not only the fourth-order accuracy but also the considerable reduction in error magnitude by increasing the temporal accuracy. Also, this study provides novel benchmark results at five different locations within the differentially heated vertical cavity of aspect ratio 8:1 for future comparison studies.

Associated Weber–Orr Transform, Biot–Savart Law and Explicit Form of the Solution of 2D Stokes System in Exterior of the Disc

In this article we derive the explicit solution of 2-D Stokes system in exterior of the disc with no-slip condition on inner boundary and given velocity $\mathbf{v}_\infty$ at infinity. It turned out it is the first application of the associated Weber-Orr transform to mathematical physics in comparison to classical Weber-Orr transform which is used in many researches. From no-slip condition for velocity field we will obtain Robin-type boundary condition for vorticity. Then the initial-boundary value problem for vorticity will be solved with help of the associated Weber-Orr transform. Also the explicit formula of Biot-Savart Law in polar coordinates will be given.

An Iteration-Free Approach to Solving the Navier–Stokes Equations by Implicit Finite Difference Schemes in the Vorticity-Stream Function Formulation

The paper introduces a new algorithm for solving the finite difference equations at the upper time level of an implicit scheme that approximates the Navier–Stokes system in the vorticity-stream function formulation. The algorithm requires no iterations and computes the corresponding discrete solution exactly. It is based on the method of difference potentials and allows one to efficiently address the well-known difficulties typical for this type of formulations—two boundary conditions for the stream function and no boundary conditions for vorticity. Abstract for the translation The paper is translated from the Russian by S. Tsynkov (tsynkov@math.ncsu.edu). The original [12] was published more than 20 years ago (the actual Russian citation is [14]). It has not been previously translated into English and went largely unnoticed by the numerical analysis and scientific computing research community. Yet it presents an important contribution to the discipline as it offers a full answer to the question that has long been outstanding. The incompressible Navier–Stokes equations in the vorticity-stream function formulation require two boundary conditions for the stream function and no boundary conditions for vorticity. A standard approach to addressing numerically the apparent overdetermination in one variable and underdetermination in the other was through the use of iterations. Instead, work [12] shows the unambiguous way of discretizing the Navier–Stokes system implicitly and solving the resulting finite difference equations on the upper time level exactly. It is equivalent to deriving the correct non-local boundary condition for vorticity. Note from the translator My Ph.D. advisor Prof. Ryaben’kii and my postdoc mentor Prof. Abarbanel were friends. They interacted closely and participated in joint projects at ICASE (NASA Langley Research Center) in the late 1990s and early 2000s. Prof. Ryaben’kii visited Tel Aviv several times, and Prof. Abarbanel traveled to Moscow in 2013 to celebrate Prof. Ryaben’kii’s 90th birthday. This paper, which was written by Prof. Ryaben’kii with his Ph.D. student at the time, V. Torgashov, provides a well-deserved tribute to a friend from a friend.

Inclusion of no-slip boundary conditions in the MEEVC scheme

This work presents three methods for enforcing tangential velocity boundary conditions for the MEEVC scheme, which was shown to be mass, enstrophy, energy and vorticity conserving scheme in the case of inviscid flow [1]. While the normal velocity component can be strongly imposed in a div-conforming formulation for the velocity field, inclusion of the tangential velocity needs to be set through an appropriate choice of vorticity boundary conditions. Three methods to impose the tangential velocity boundary condition will be discussed: The kinematic Dirichlet formulation, the kinematic Neumann formulation and the dynamic Neumann formulation. The conservation properties of each of the resulting schemes are analyzed and numerical results are shown for the Taylor–Green vortex and for the dipole collision test cases. These confirm that kinematic Neumann vorticity boundary conditions perform best.

Orthogonal Collocation on Finite Element Method for Lid-Driven Cavity Flow

Among the candidates for the Electrical Energy Storage (EES), Redox Flow Batteries (RFBs) are drawing attention towards a feasible solution. They are advantageous in flexibility and long-term stability of capacity and are therefore expected to play a critical role in the renewable energy and microgrid system. Meanwhile, the performance of the RFBs greatly depends on the motion of the electrolyte. And the problem becomes more complicated when we consider the edge effect in 2-dimensional modeling. Up to date, there is no analytical solution for the fluid dynamics of electrolyte flow in RFBs. To maximize the performance and enable optimum real-time battery management, fast and robust algorithm for the flow model and numerical simulation in multi-dimension is essential. In this presentation, we introduce Orthogonal Collocation on Finite Element (OCFE) approach in the successive domain for the classical 2-D lid-driven cavity flow problem. The Navier-Stokes equation is mathematically reformulated as Partial Differential Equations (PDE) of stream function and vorticity. To get the vorticity boundary condition, imaginary points, which are located at the outside of the cavity boundary, are introduced. The result shows good agreements with precedential values while using minimum computational time and memory.

On well-posedness of a velocity-vorticity formulation of the stationary Navier-Stokes equations with no-slip boundary conditions

We study well-posedness of a velocity-vorticity formulation of the Navier-Stokes equations, supplemented with no-slip velocity boundary conditions, a corresponding zero-normal condition for vorticity on the boundary, along with a natural vorticity boundary condition depending on a pressure functional. In the stationary case we prove existence and uniqueness of a suitable weak solution to the system under a small data condition. The topic of the paper is driven by recent developments of vorticity based numerical methods for the Navier-Stokes equations.

MHD Stokes flow and heat transfer in a lid‐driven square cavity under horizontal magnetic field

This study considers the steady flow of a viscous, incompressible and electrically conducting fluid in a lid‐driven square cavity under the effect of a uniform horizontally applied magnetic field. The governing equations are obtained from the Navier‐Stokes equations including buoyancy and Lorentz force terms and the energy equation including Joule heating and viscous dissipation terms. These equations are solved iteratively in terms of velocity components, stream function, vorticity, temperature, and pressure by using radial basis function approximation. Particular solution, which is approximated by radial basis functions to satisfy both differential equation and boundary conditions, becomes the solution of the differential equation itself. Vorticity boundary conditions are obtained from stream function equation using finite difference scheme. Normal derivative of pressure is taken as zero on the boundary. The numerical results are obtained for several values of Hartmann number and Grashof number for the Stokes approximation (Re << 1). The results show that when the viscous dissipation is present, the flow and isolines concentrate through the cold wall forming boundary layers as Grashof number increases. An increase in the magnetic field intensity retards the effect of buoyancy force in the square cavity, whereas the movement of the upper lid causes buoyancy force to be dominant. The solution is obtained in a considerably low computational expense through the use of radial basis function approximations for the MHD equations. Copyright © 2017 John Wiley & Sons, Ltd.

On the Development of a Nonprimitive Navier–Stokes Formulation Subject to Rigorous Implementation of a New Vorticity Integral Condition

In this paper, a new integral vorticity boundary condition has been developed and implemented to compute solution of nonprimitive Navier---Stokes equation. Global integral vorticity condition which is of primitive character can be considered to be of entirely different kind compared to other vorticity conditions that are used for computation in literature. The procedure realized as explicit boundary vorticity conditions imitates the original integral equation. The main purpose of this paper is to design an algorithm which is easy to implement and versatile. This algorithm based on the new vorticity integral condition captures accurate vorticity distribution on the boundary of computational flow field and can be used for both wall bounded flows as well as flows in open domain. The approach has been arrived at without utilizing any ghost grid point outside of the computational domain. Convergence analysis of this alternative vorticity integral condition in combination with semi-discrete centered difference approximation of linear Stokes equation has been carried out. We have also computed correct pressure field near the wall, for both attached and separated boundary layer flows, by using streamfunction and vorticity field variables. The competency of the proposed boundary methodology vis-a-vis other popular vorticity boundary conditions has been amply appraised by its use in a model problem that embodies the essential features of the incompressibility and viscosity. Subsequently the proposed methodology has been further validated by computing analytical solution of steady Stokes equation. Finally, it has been applied to three benchmark problems governed by the incompressible Navier---Stokes equations, viz. lid driven cavity, backward facing step and flow past a circular cylinder. The results obtained are in excellent agreement with computational and experimental results available in literature, thereby establishing efficiency and accuracy of the proposed algorithm. We were able to accurately predict both vorticity and pressure fields.

Natural vorticity boundary conditions on solid walls

We derive a new kind of boundary conditions for the vorticity equation with solid wall boundaries for fluid flow problems . The formulation uses a Dirichlet condition for the normal component of vorticity and Neumann type conditions for the tangential components . In a Galerkin (integral) formulation the tangential condition is natural, i.e., it is enforced by a right-hand side functional and does not impose a boundary constraint on trial and test spaces. The functional involves the pressure variable, and we discuss several velocity–vorticity formulations where the proposed condition is appropriate. Several numerical experiments are given that illustrate the validity of the new approach. • We devise vorticity boundary conditions on solid walls, which possess physical and geometrical information relevant to near-boundary vortex dynamics. • The conditions are local and enforced in the Galerkin formulation through a right-hand side functional depending on the pressure. • Two numerical schemes are suggested that benefit from the new conditions and solve for velocity, vorticity, and pressure in a decoupled time-stepping fashion.

Implementation of Neumann boundary condition with influence matrix method for viscous annular flow using pseudospectral collocation

The flowfield of an annular Couette flow is predicted numerically from an unsteady initial condition using the Chebyshev–Fourier collocation method. The numerical solution is obtained from the vorticity–velocity formulation of the unsteady Navier–Stokes equations. In this formulation the velocity boundary conditions are overspecified while vorticity boundary conditions are unspecified. This difficulty is resolved by a matrix influence method to convert the overspecified velocity boundary conditions to sufficiently specified Dirichlet boundary conditions for both velocity and vorticity. These boundary conditions are implemented by considering three methods: the traditional row replacement method, Fornberg’s fictitious points method, and Driscoll and Hale’s rectangular collocation method. The solution is advanced in time using the third-order Adams–Bashforth semi-implicit backward differentiation scheme. The accuracy of the numerical solutions are assessed in two ways. In one case, the accuracy of the converged steady state solution is compared to the analytical solution. In the second case, the residual of the continuity equation for the vorticity–velocity method is assessed in comparison to the vorticity streamfunction formulation that inherently satisfies the continuity equation. All methods demonstrate high accuracy for the continuity equation residual at the domain interior; however, the row replacement and fictitious points methods exhibit poor accuracy at the boundaries during the transient. The rectangular projection method exhibits excellent accuracy throughout the domain and boundaries at all times using the resampling points. On the other hand, for evaluating secondary quantities such as the wall shear stress, the rectangular projection method demonstrates several orders of magnitude less accuracy when evaluated using the resampling points, but high accuracy when evaluated directly at the original collocation points.

An operator-splitting scheme for the stream function–vorticity formulation of the unsteady Navier–Stokes equations

The stream function–vorticity formulation of the (Navier–)Stokes equations yields a coupled system of a parabolic equation for the vorticity and an elliptic equation for the stream function. The essential coupling between them occurs through the boundary conditions which in case of a Dirichlet boundary involve only the stream function. Therefore, the boundary condition for the vorticity must be derived from them and thus the vorticity equation must be coupled to the stream function equation via its boundary condition. In this paper we propose an unconditionally stable splitting scheme for the unsteady Stokes equations in a stream function–vorticity formulation, that decouples the vorticity and stream function computations at each time step. The spatial discretization is based on a finite volume discretization on (generally) unstructured Delaunay grids and corresponding Voronoi finite volume cells. A generalization of the well-known Thom vorticity boundary condition is derived for such grids and the corresponding discrete problem is decoupled by a two-step splitting scheme which results in a decoupled discrete parabolic problem for the vorticity and an elliptic problem for the stream function. Furthermore, the scheme is extended to the unsteady Navier–Stokes equations. Finally, the stability and accuracy of the resulting schemes are demonstrated on numerical examples.

DRBEM solution of natural convective heat transfer with a non-Darcy model in a porous medium

This study presents the dual reciprocity boundary element (DRBEM) solution of Brinkman–Forchheimer-extended Darcy model in a porous medium containing an incompressible, viscous fluid. The governing dimensionless equations are solved in terms of stream function, vorticity and temperature. The problem geometry is a unit square cavity with either partially heated top and bottom walls or hot steps at the middle of these walls. DRBEM provides one to obtain the expected behavior of the flow in considerably small computational cost due to the discretization of only the boundary, and to compute the space derivatives in convective terms as well as unknown vorticity boundary conditions using coordinate matrix constructed by radial basis functions. The Backward-Euler time integration scheme is utilized for the time derivatives. The decrease in Darcy number suppresses heat transfer while heat transfer increases for larger values of porosity, and the natural convection is pronounced with the increase in Rayleigh number.

FEM solution of natural convection flow in square enclosures under magnetic field

PurposeThe purpose of the paper is to obtain finite element method (FEM) solution of steady, laminar, natural convection flow in inclined enclosures in the presence of an oblique magnetic field. The momentum equations include the magnetic effect, and the induced magnetic field due to the motion of the electrically conducting fluid is neglected. Quadratic triangular elements are used to ensure accurate approximation for second order derivatives of stream function appearing in the vorticity equation.Design/methodology/approachGoverning equations in terms of stream function and vorticity are solved by FEM using quadratic triangular elements. Vorticity boundary conditions are obtained through Taylor series expansion of stream function equation by using more interior stream function values to improve the accuracy. Isothermally heated or cooled and/or adiabatic conditions for the temperature are imposed. Results are obtained for Rayleigh number values and Hartmann number values up to 1000000 and 100, respectively.FindingsIt is observed that streamlines form a thin boundary layer close to the heated walls as Ha increases. The same effect is seen in the vorticity contours, and isotherms are not affected much. As Ra increases streamlines are deformed moving from the heated walls through cooled walls. Vorticity starts to develop boundary layers close to heated and adjacent walls. Isotherms are pushed towards the sinusoidally heated wall whereas in the case of linearly heated left and bottom walls they expand towards cooled part of the cavity as Ra increases.Originality/valueThe application of FEM with quadratic elements for solving natural convection flow problem under the effect of a magnetic field is new in the sense that the results are obtained for large values of Rayleigh and Hartmann numbers.