Finite element error analysis of surface Stokes equations in stream function formulation

19 - Viscous and Non-Newtonian Flows

Model equations describing large scale buoyant convection in an enclosure are formulated with the vorticity and stream function as dependent variahles. The mathematical model, based on earlier work of the authors, is unique in two respects. First, it neglects viscous and thermal conductivity effects. Second the fluid is taken to be thermally expandable: large density variations are allowed while acoustic waves are filtered out. A volumetric heat source of specified spatial and temporal variation drives the flow in a two-dimensional rectangular enclosure. An algorithm for solution of the equations in this vorlicily, stream-function formulation is presented. Results of computations using this algorithm are presented. Comparison of these results with those obtained earlier by the authors using a finite difference code to integrate the primitive equations show excellent agreement. A method for periodically smoothing the computational results during a calculation, using Lanczos smoothing, is also presented. Computations with smoothing at different time intervals are presented and discussed.

Extending FEniCS to work in higher dimensions using tensor product finite elements

A spectral multigrid method for the stokes problem in streamfunction formulation

Free convection in oblique enclosures filled with a porous medium

Application of orthogonal collocation to simulation and control of first order hyperbolic systems

The problem of creeping flow past a swarm of porous circular cylinders with Happel and Kuwabara boundary conditions is investigated. The Brinkman equation for the flow inside the porous cylinder and the Stokes equation outside the porous cylinder in their stream function formulations are used. The force experienced by each porous circular cylinder in a cell is evaluated. Explicit expressions of stream functions are obtained for both the inside and outside flow fields. The earlier results reported by Happel and Kuwabara for flow past a solid cylinder in Happel’s and Kuwabara’s cell model, have been deduced. Analytical expressions for the velocity components, pressure, vorticity and stresstensor are also obtained

This paper concerns the Stokes flow of an incompressible viscous fluid past a swarm of porous nanocylindrical particles enclosing a solid cylindrical core with Kuwabara boundary condition. An aggregate of porous nanocylindrical particles is considered as a hydro‐dynamically equivalent to a solid cylindrical core with concentric porous cylindrical shell. The Brinkman equation inside the porous cylindrical shell and the Stokes equation outside the porous cylindrical shell in their stream function formulations are used. Explicit expressions for the stream functions in both regions have been investigated. The drag force acting at each nanoporous cylindrical particle in a cell is evaluated. Also, we solved the same problem by using Happel boundary condition on the hypothetical cell. In certain limiting cases, drag force converges to pre‐existing analytical results, such as the drag on a porous circular cylinder and the drag on a solid cylinder in Kuwabara′s cell or Happel′s cell. Representative results are then discussed and compared for both cases and presented in graphical form by using Mathematica software.

Density-driven flow and solute transport problems. A 2-D numerical model based on the network simulation method

The second variation of the stream function energy of water waves with vorticity

Introduction. Computer Program Packages. Typical Problems. Classification of Equations. Discrete Methods. Finite Differences and Computational Molecules. Finite Difference Operators. Method of Weighted Residuals. Finite Elements. Method of Lines. Errors. Stability and Convergence. Irregular Boundaries. Choice of Discrete Network. Dimensionless Forms. Parabolic Equations: Introduction. Properties of a Simple Explicit Method. Fourier Stability Method. Implicit Methods. Additional Stability Considerations. Matrix Stability Analysis. Extension of Matrix Stability Analysis. Consistency, Stability, And Convergence. Pure Initial Value Problems. Variable Coefficients. Examples of Equations with Variable Coefficients. General Concepts of Error Reduction. Methods of Lines (MOL) for Parabolic Equations. Weighted Residuals and the Method of Lines. Bubnov-Galerkin Scheme for Parabolic Equations. Finite Elements and Parabolic Equations. Hermite Basis. Finite Elements and Parabolic Equations. General Basis Fucntion. Finite Elements and Parabolic Equations. Special Basis Functions. Explicity Finite Difference Methods for Nonlinear Problems. Further Applications on One Dimentions. Asymmetric Approximations. Elliptic Equations: Introduction. Simple Finite Difference Schemes. Direct Methods. Iterative Methods. Linear Elliptic Equations. Some Poit Iterative Methods. Convergence of Point Iterative Methods. Rates of Convergence. Accelerations. Conjugate Gradient Method. Extensions of SOR. Auliative Examples of Over-Relaxation. Other Point Iterative Methods. Block Iterative Methods. Alternating Direction Methods. Summary of ADI Results. Triangular Elements. Boundary Element Method (BEM). Spectral Methods. Some Nonlinear Examples. Hyperbolic Equations: Introduction. The Quasilinear System. Introductory Examples. Method of Characteristics. Constant States and Simple Waves. Typical Application of Characteristics. Finite Differences for First-Order Equations. Lax-Wendroff Methods and Other Algorithms Dissipation and Dispersion. Explicity Finite Difference Methods. Attenuation. Implicit Methods for Second-Order Equations. Time Quasilinear Examples. Simultaneous First-Order Equations. Explicit Methods. An Implicity Method for First-Order Equations. Hybrid Methods for First-Order Equations. Finite Elements and the Wave Equation. Spectral Methods and Periodic Systems. Gas Dyunamics in One Space Variable. Eulerian Difference Equations. Lagrangian Difference Equations. Hopscotch Methods for Conservation Laws. Explicity-Implicity Schemes for Conservation Laws. Special Topics: Introduction. Singularities. Shocks. Eigenvale Problems. Parabolic Equations in Segeral Space Variables. Additional Comments on Elliptic Equations. Hyperbolic Equations in Higher Dimensions. Mixed Systems. Higher Order Equations in Elasticity and Vibrations. Computational Fluid Mechanics. Stream Function. Vorticity Method for Fluid Mechanics. Primitive Variable Methods for Fluid Mechanics. Vector Potential Methods for Fluid Mechanics. Introduction to Monte Carlo Mehtods. Fast Fourier Transform and Applications. Method of Fractional Steps. Applications of Group Theory in Computation. Computational Ocean Acoustics. Enclosure Methods. Chapter References. Author Index. Subject Index.

Recent works showed that pressure-robust modifications of mixed finite element methods for the Stokes equations outperform their standard versions in many cases. This is achieved by divergence-free reconstruction operators and results in pressure independent velocity error estimates which are robust with respect to small viscosities. In this paper we develop a posteriori error control which reflects this robustness. The main difficulty lies in the volume contribution of the standard residual-based approach that includes the $L^2$-norm of the right-hand side. However, the velocity is only steered by the divergence-free part of this source term. An efficient error estimator must approximate this divergence-free part in a proper manner, otherwise it can be dominated by the pressure error. To overcome this difficulty a novel approach is suggested that uses arguments from the stream function and vorticity formulation of the Navier--Stokes equations. The novel error estimators only take the $\mathrm{curl}$ of the right-hand side into account and so lead to provably reliable, efficient and pressure-independent upper bounds in case of a pressure-robust method in particular in pressure-dominant situations. This is also confirmed by some numerical examples with the novel pressure-robust modifications of the Taylor--Hood and mini finite element methods.

We are interested to the natural convection flow resulting from the combined buoyancy effects of thermal and mass diffusion in an inclined cavity. The first part of the study was set up to predict the fresh water production by resolving a set of energy balance equation including the top glass cover, the brackish water and the absorber layer. The influence of parameters such as inlet water temperature, water flow rate and climate conditions was investigated. The second part deals with the problem in a Cartesian coordinate system involves the use of a control-volume based finite element method (CVFEM) and solves the full vorticity transport equation together with the stream function, concentration and energy equations. The predicted stream function patterns, isoconcentration and isotherms are presented for different Rayleigh numbers. The results were compared with the available data and the agreement is satisfactory

Drag force on a circular cylinder midway between two parallel plates at [formula omitted] Part 2: moving uniformly (numerical and experimental)

Drag force on a circular cylinder midway between two parallel plates at very low Reynolds numbers—Part 1: Poiseuille flow (numerical)