Vorticity Formulation Research Articles

Nonconforming finite element discretization for semilinear problems with trilinear nonlinearity

Abstract The Morley finite element method (FEM) is attractive for semilinear problems with the biharmonic operator as a leading term in the stream function vorticity formulation of two-dimensional Navier–Stokes problem and in the von Kármán equations. This paper establishes a best-approximation a priori error analysis and an a posteriori error analysis of discrete solutions close to an arbitrary regular solution on the continuous level to semilinear problems with a trilinear nonlinearity. The analysis avoids any smallness assumptions on the data, and so has to provide discrete stability by a perturbation analysis before the Newton–Kantorovich theorem can provide the existence of discrete solutions. An abstract framework for the stability analysis in terms of discrete operators from the medius analysis leads to new results on the nonconforming Crouzeix–Raviart FEM for second-order linear nonselfadjoint and indefinite elliptic problems with $L^\infty $ coefficients. The paper identifies six parameters and sufficient conditions for the local a priori and a posteriori error control of conforming and nonconforming discretizations of a class of semilinear elliptic problems first in an abstract framework and then in the two semilinear applications. This leads to new best-approximation error estimates and to a posteriori error estimates in terms of explicit residual-based error control for the conforming and Morley FEM.

Hybrid nanomaterial migration due to MHD within a tank

Transportation of hybrid ferrofluid has been illustrated in the current article with the imposition of Lorentz forces. A numerical approach with the help of vorticity formulation was involved. Outputs demonstrate the opportunity to manage the thermal behavior with the magnetic field. As Lorentz force is imposed, the strength of eddies reduces and velocity of nanopowders reduces. Reductive impact of Hartmann is ignorable in the lower permeability, but it becomes stronger with the rise in Da. As Rd enhances, isotherms become more disturbed and higher Nu is obtained. Unfavorable effect of permeability on the thickness of boundary layer makes Nu to increase, and a similar impact was demonstrated for buoyancy force.

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.

Boundary layer formulations in orthogonal curvilinear coordinates for flow over wind-generated surface waves

The development of the governing equations for fluid flow in a surface-following coordinate system is essential to investigate the fluid flow near an interface deformed by propagating waves. In this paper, the governing equations of fluid flow, including conservation of mass, momentum and energy balance, are derived in an orthogonal curvilinear coordinate system relevant to surface water waves. All equations are further decomposed to extract mean, wave-induced and turbulent components. The complete transformed equations include explicit extra geometric terms. For example, turbulent stress and production terms include the effects of coordinate curvature on the structure of fluid flow. Furthermore, the governing equations of motion were further simplified by considering the flow over periodic quasi-linear surface waves wherein the wavelength of the disturbance is large compared to the wave amplitude. The quasi-linear analysis is employed to express the boundary layer equations in the orthogonal wave-following curvilinear coordinates with the corresponding decomposed equations for the mean, wave and turbulent fields. Finally, the vorticity equations are also derived in the orthogonal curvilinear coordinates in order to express the corresponding velocity–vorticity formulations. The equations developed in this paper proved to be useful in the analysis and interpretation of experimental data of fluid flow over wind-generated surface waves. Experimental results are presented in a companion paper.

Thermal and fluid flow modeling of the molten pool behavior during TIG welding by stream vorticity method

The present paper deals with the numerical simulation of weld pool development in Tungsten Inert Gas (TIG) process. A mathematical model is developed in order to solve the Navier–Stokes equations expressed in the stream–vorticity formulation coupled with heat equation taking into account the liquid solid phase change. Using the stream–vorticity formulation in incompressible fluid flow, the same problem is solved with reducing the number of transport equations. Therefore, only one transport equation (vorticity) and one Poisson equation (stream) are considered in this model. The FORTRAN programming and the numerical simulation are then achieved using appropriate discretization that ensures the convergence of the numerical methods to solve a large and sparse linear algebraic systems. Furthermore, to solve the radiation phenomena during welding described by the Stefen law, another method is proposed. The obtained numerical results are discussed and validate with experimental.

Uniform Approximation of 2 Dimensional Navier--Stokes Equation by Stochastic Interacting Particle Systems

We consider an interacting particle system modeled as a system of $N$ stochastic differential equations driven by Brownian motions. We prove that the (mollified) empirical process converges, uniformly in time and space variables, to the solution of the two-dimensional Navier-Stokes equation written in vorticity form. The proofs follow a semigroup approach.

Convergence of transport noise to Ornstein–Uhlenbeck for 2D Euler equations under the enstrophy measure

We consider the vorticity form of the 2D Euler equations which is perturbed by a suitable transport type noise and has white noise initial condition. It is shown that stationary solutions of this equation converge to the unique stationary solution of the 2D Navier–Stokes equation driven by the space-time white noise.

Weak vorticity formulation of the incompressible 2D Euler equations in bounded domains

In this article we examine the interaction of incompressible 2D flows with material boundaries. Our focus is the dynamic behavior of the circulation of velocity around boundary components and the possible exchange between flow vorticity and boundary circulation in flows with vortex sheet initial data, in the case of bounded domains. Our point of departure is the observation that ideal flows with vortex sheet regularity have well-defined circulation around each connected component of the boundary. In addition, we show that the velocity can be uniquely reconstructed from the vorticity and boundary component circulations, which allows to recast 2D Euler evolution using vorticity and the circulations as dynamic variables. The weak form of this vortex dynamics formulation of the equations is called the weak vorticity formulation. Existence of a weak solution for the 2D Euler equations, in velocity form, is guaranteed by Delort’s Theorem, when the initial vorticity is a bounded measure satisfying a sign condition. The main result in this article is the equivalence between the weak velocity and weak vorticity formulations, without sign assumptions. Despite their being equivalent, the qualitative information concerning weak solutions is more apparent from the weak vorticity formulation than from the velocity formulation, and the remainder of the article is devoted to several consequences which can be derived from our main result. First, we consider weak solutions obtained by mollifying initial data and passing to the limit, with the portion of vorticity singular with respect to the Lebesgue measure assumed to be nonnegative. For these solutions we prove a set of inequalities which restrict the possible generation of vorticity by the boundary. Next, we prove that, if the weak solution conserves circulation around the boundary components, then it is a boundary coupled weak solution, a stronger version of the weak vorticity formulation. We prove existence of a weak solution which conserves circulation around the boundary components if the initial vorticity is integrable, i.e. if the singular part vanishes. Finally, we discuss the definition of the net mechanical force which the flow exerts on each material boundary component and its relation with conservation of circulation.

A hybrid [formula omitted]-[formula omitted] HOC approach for surface tension driven flows in level set framework

In this study, a higher order compact (HOC) finite difference scheme is proposed for the simulation of multiphase flows with surface tension effects by coupling the pure streamfunction formulation of Navier–Stokes (N–S) equations with the level set method. Unlike most of the existing methods for multiphase flows that solve the incompressible Navier–Stokes equations in the velocity–pressure form or the stream-function vorticity form, we recast the governing equations into the biharmonic form of the N–S equations which carries stream-function ψ as the only variable to be computed. The coupling of this formulation with level set (LS) enables the simulation of two phase flows by the core computation of only the single variable ψ free from discontinuities in the whole physical domain. In the process, we also propose a novel hybrid HOC algorithm for post-processing the pressure on either side of the interface including the jump across it. The robustness of this method is established through a series of benchmark problems such as dispersion of capillary waves, static bubble and oscillating bubble at equilibrium in a fluid at rest, and deforming circular drops in shear flows. We compare our numerical results with available benchmark results and excellent match is obtained in all the cases along with accurate representation of overall trends. Our approach is proved to be very efficient in terms of its easy implementation, relatively low computational time and high order of accuracy.

Computational hemodynamic investigation of a new bileaflet mechanical heart valve

In this paper, we perform a numerical analysis for simulating steady, two-dimensional, laminar blood flow through our proposed design, known as the Butterfly mechanical heart valve, where the leaflets are fully opened. Blood has been assumed to be Newtonian and non-Newtonian fluid using the Casson model for shear-thinning behavior. A non-uniform Cartesian grid generation technique is presented to generate a two-dimensional grid for the irregular geometry of the Butterfly valve. The governing Navier–Stokes equations of flow, written in a stream function–vorticity formulation, are solved by the finite difference method with hybrid differencing of the convective terms. The computed results show that the blood’s non-Newtonian nature significantly affects the flow field with the existence of recirculation and consequently stagnation causing thrombus formation, as well as an increase of the shear stress along the wall, which contributes to hemolytic blood damage. The results demonstrate that the model is capable of predicting the hemodynamic features most interesting to physiologists. It can be used to assess thromboembolic problems occurring with heart valves and in the design of cardiac prostheses.

Global solutions for random vorticity equations perturbed by gradient dependent noise, in two and three dimensions

The aim of this work is to prove an existence and uniqueness result of Kato–Fujita type for the Navier–Stokes equations, in vorticity form, in 2D and 3D, perturbed by a gradient-type multiplicative Gaussian noise (for sufficiently small initial vorticity). These equations are considered in order to model hydrodynamic turbulence. The approach was motivated by a recent result by Barbu and Rockner (J Differ Equ 263:5395–5411, 2017) that treats the stochastic 3D Navier–Stokes equations, in vorticity form, perturbed by linear multiplicative Gaussian noise. More precisely, the equation is transformed to a random nonlinear parabolic equation, as in Barbu and Rockner (2017), but the transformation is different and adapted to our gradient-type noise. Then, global unique existence results are proved for the transformed equation, while for the original stochastic Navier–Stokes equations, existence of a solution adapted to the Brownian filtration is obtained up to some stopping time.

Stream function - vorticity formulation of mixture mass flow

Employing a generalized quasi two-phase bulk mixture mass flow model derived from a general two-phase model (Pudasaini, 2012), here, we formulate a stream function - vorticity and vorticity-transport equation for a rapid flow of mixture of viscous fluid and solid particles down a channel. The original system of partial differential equations (PDEs) in velocity and pressure is converted into the reduced stream function - vorticity form as a close system of equations that is free of pressure which replaces the original system of three equations by a set of just two PDEs as an advantage of the new model. A novel pressure Poisson equation in terms of stream function, vorticity and the rate-dependent mixture viscosity is derived. For given mixture viscosity, our pressure Poisson equation can provide mixture pressure. The two equations are coupled through the Poisson equation to provide the full system in stream function, vorticity and mixture pressure to describe the dynamics of mixture flow. However, the pressure is decoupled and can also be computed separately using stream function and vorticity. We further reduce the new system to obtain exact expressions for stream function. One of the most important advancements here is the construction of a new pressure Poisson equation for shear mixture flows that includes yield strength of the mixture. We also discuss the importance of pressure Poisson equation induced by the flow field intensity, the yield strength, and free surface geometry. Our results show that the pressure Poisson equation is mainly characterized by the non-linear diffusion of the free surface. Furthermore, mixture pressures are derived analytically for thin and thick flows. Similarly, different flow scenarios such as pressure dominated flow; thick, low yield strength flow; and thin, high yield strength flows are analyzed. Several exact/analytical solutions are constructed for the pressure and flow depth distributions for incipient flow, shearing flow, free surface flow, propagating bore front and mass deposition. The novel models developed here and analytical results are consistent with the observed phenomena indicating their application potential in detailed description of the mixture flow dynamics more efficiently than the existing complex models.

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.

Similarity reductions and exact solutions for two-dimensional Euler–Boussinesq equations

In this paper, by introducing a stream function and new coordinates, we transform classical Euler–Boussinesq equations into a vorticity form. We further construct traveling wave solutions and similarity reduction for the vorticity form of Euler–Boussinesq equations. In fact, our similarity reduction provides a kind of linearization transformation of Euler–Boussinesq equations.

Meshless velocity – vorticity local boundary integral equation (LBIE) method for two dimensional incompressible Navier-Stokes equations

PurposeThis paper aims to describe the 2D meshless local boundary integral equation (LBIE) method for solving the Navier–Stokes equations.Design/methodology/approachThe velocity–vorticity formulation is selected to eliminate the pressure gradient of the equations. The local integral representations of flow kinematics and transport kinetics are derived. The integral equations are discretized using the local RBF interpolation of velocities and vorticities, while the unknown fluxes are kept as independent variables. The resulting volume integrals are computed using the general radial transformation algorithm.FindingsThe efficiency and accuracy of the method are illustrated with several examples chosen from reference problems in computational fluid dynamics.Originality/valueThe meshless LBIE method is applied to the 2D Navier–Stokes equations. No derivatives of interpolation functions are used in the formulation, rendering the present method a robust numerical scheme for the solution of fluid flow problems.

Poisson structure of the three-dimensional Euler equations in Fourier space

We derive a simple Poisson structure in the space of Fourier modes for the vorticity formulation of the Euler equations on a three-dimensional periodic domain. This allows us to analyse the structure of the Euler equations using a Hamiltonian framework. The Poisson structure is valid on the divergence free subspace only, and we show that using a projection operator it can be extended to be valid in the full space. We then restrict the simple Poisson structure to the divergence-free subspace on which the dynamics of the Euler equations take place, reducing the size of the system of ordinary differential equations by a third. The projected and the restricted Poisson structures are shown to have the helicity as a Casimir invariant. We conclude by showing that periodic shear flows in three dimensions are equilibria that correspond to singular points of the projected Poisson structure, and hence that the usual approach to study their nonlinear stability through the energy-Casimir method fails.

FINMHD: An Adaptive Finite-element Code for Magnetic Reconnection and Formation of Plasmoid Chains in Magnetohydrodynamics

Abstract Solving the problem of fast eruptive events in magnetically dominated astrophysical plasmas requires the use of particularly well adapted numerical tools. Indeed, the central mechanism based on magnetic reconnection is determined by a complex behavior with quasi-singular forming current layers enriched by their associated small-scale magnetic islands called plasmoids. A new code is thus presented for the solution of two-dimensional dissipative magnetohydrodynamics (MHD) equations in cartesian geometry specifically developed to this end. A current–vorticity formulation representative of an incompressible model is chosen in order to follow the formation of the current sheets and the ensuing magnetic reconnection process. A finite-element discretization using triangles with quadratic basis functions on an unstructured grid is employed, and implemented via a highly adaptive characteristic-Galerkin scheme. The adaptivity of the code is illustrated on simplified test equations and finally for magnetic reconnection associated with the nonlinear development of the tilt instability between two repelling current channels. Varying the Lundquist number S has allowed us to study the transition between the steady-state Sweet–Parker reconnection regime (for S ≲ 104) and the plasmoid-dominated reconnection regime (for S ≳ 105). The implications for the understanding of the mechanism explaining the fast conversion of free magnetic energy in astrophysical environments such as the solar corona are briefly discussed.

Stochastic modelling of nanofluids using the fast Boundary-Domain Integral Method

In this paper, we couple a numerical method aimed at simulation of flow and heat transfer of nanofluids with stochastic modelling of input and material parameters. In order to simulate nanofluids, an in-house numerical method was developed, based on the solution of 3D velocity–vorticity formulation of Navier–Stokes equations. A fast Boundary-Domain Integral Method has been employed to solve the governing equations and set up the deterministic flow and heat transfer solver. The developed algorithm is used to simulate natural convection of a nanofluid in a closed cavity. The uncertainty present in the input parameters is propagated to the output of interest via the Stochastic Collocation Method. The stochastic mean, variance, and higher-order moments of the output values are presented. The non-intrusive nature of the Stochastic Collocation Method facilitates the previously validated deterministic code to remain unchanged. The stochastic analysis reveals that the uncertainty of input parameters influences the output results most in the areas where high flow field gradients appear.

A pseudospectral approach applicable for time integration of linearized N‐S operator that removes pole singularity and physically spurious eigenmodes

SummaryThis paper addresses a modified singularity removal technique for the eigenvalue or optimal mode problems in pipe flow using a pseudospectral method. The current approach results in the linear stability operator to be devoid of any unstable physically spurious modes, and thus, it provides higher numerical stability during time‐based integration. The correctness of the numerical operator is established by calculating the known eigenvalues of pipe Poiseuille flow. Subsequently, the optimal modes are determined with Farrell's approach and compared with the existing literature. The usefulness of this approach is further demonstrated in the time‐based numerical integration of the linearized Navier‐Stokes operator for the adjoint method–based optimal mode determination. The numerical scheme is implemented with the radial velocity‐radial vorticity formulation. Even number of Chebyshev‐Lobatto grid points are distributed over the domain r∈[−1,1] omitting the centerline, which also efficiently provides higher resolution near the wall boundary. The boundary conditions are imposed with homogeneous wall boundary conditions, whereas the analytic nature of a proper set of base functions enforces correct centerline conditions. The resulting redundancy introduced in the process is eliminated with the proper usage of parity.

Effect of isoflux thermal boundary condition on mixed convective heat transfer from a sphere for liquid metals

Numerical simulation is performed for mixed convective heat transfer past a sphere with uniform heat flux (Isoflux) thermal boundary condition for liquid metals. The steady, incompressible, laminar, axisymmetric and Newtonian fluid is considered in a vertically upward direction. A spherical geometry higher order compact scheme is utilised to solve the governing equations in stream function vorticity formulation together with the energy equation. The fourth order convergence of the current scheme is demonstrated. The velocity, streamline, isotherm contour profiles together with vorticity, pressure, local Nusselt number and average Nusselt number on the surface of the sphere are presented for the Reynolds number in the range , Richardson number in the range and Prandtl number in case of 0.020, 0.0208, 0.065 (liquid Gallium, liquid metal alloy , liquid lithium). The critical Richardson number is identified beyond which the recirculation phenomena of the flow are completely suppressed. The rate of isoflux heat transfer is more when compare to the isothermal case for the same range of parameters.