Open Boundary Conditions for the Streamfunction Vorticity Formulation of Unsteady Laminar Convection (original) (raw)
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The paper deals with the Boundary Element Method (BEM) for modelling 2D unsteady laminar flow using stream function-vorticity formulation of the Navier-Stokes equations. The numerical algorithm for solving a general parabolic diffusion-convection equation is based on linear mixed elements and a multidomain model also known as subdomain technique. Robustness, accuracy and economy of the developed numerical algorithm is shown on a standard case of steady backward facing step flow and a periodic flow past a circular cylinder test case. q
International Journal for Numerical Methods in Fluids, 1994
The streamfunction-vorticity equations for incompressible two-dimensional flows are uncoupled and solved in sequence by the finite element method. The vorticity at no-slip boundaries is evaluated in the framework of the streamfunction equation. The resulting scheme achieves convergence, even for very high values of the Reynolds number, without the traditional need for upwinding. The stability and accuracy of the approach are demonstrated by the solution of two well-known benchmark problems: flow in a lid-driven cavity at Re ⩽ 10,000 and flow over a backward-facing step at Re = 800.
Open Journal of Fluid Dynamics
A numerical study is presented on the problem of 2D natural convection in a differentially heated cavity. The equations governing this unsteady flow phenomenon were solved using the vorticity-stream function formulation of the Navier-Stokes equations and heat. The results obtained are compared with the results of the literature and make it possible to validate this approach. In this work, we studied the heat transfer in a cavity and we determined the variation of the local Nusselt number which allows obtaining the rate of thermal transfer by convection in an enclosure. We analyzed thermal fields for different Rayleigh numbers by selecting two points to visualize temperature fluctuations over time. Thus, the creation of the ascending and descending movements of the fluid inside the cavity was analyzed. We have also established temperature histograms for the graphical presentation of the temperature distribution. The modeling of the two-dimensional problem was established using a "Fortran 90" calculation code. The results also show the different vorticity contour maps in laminar flow regime. We have presented our results of numerical simulations using a visualization tool. The Rayleigh number varies in the range of 10³ to 10⁶ for a Prandtl number equal to 0.72 corresponding to air.
Finite element stream function-vorticity solutions of the incompressible Navier-Stokes equations
International Journal for Numerical Methods in Fluids, 1987
The incompressible, two-dimensional Navier-Stokes equations are solved by the finite element method (FEM) using a novel stream function/vorticity formulation. The no-slip solid walls boundary condition is applied by taking advantage of the simple implementation of natural boundary conditions in the FEM, eliminating the need for an iterative evaluation of wall vorticity formulae. In addition, with the proper choice of elements, a stable scheme is constructed allowing convergence to be achieved for all Reynolds numbers, from creeping to inviscid flow, without the traditional need for upwinding and its associated false diffusion. Solutions are presented for a variety of geometries.
Cmes-computer Modeling in Engineering & Sciences, 2009
This paper reports a new discretisation technique for the streamfunction-vorticity-temperature ($\psi - \omega - T$) formulation governing natural convection defined in 2D enclosured domains. The proposed technique combines strengths of three schemes, i.e. smooth discretisations (Galerkin formulation), powerful high-order approximations (one-dimensional integrated radial-basis-function networks) and pressure-free low-order system ($\psi-\omega-T$ formulation). In addition, a new effective way of deriving computational boundary conditions for the vorticity is proposed. Two benchmark test problems, namely free convection in a square slot and a concentric annulus, are considered, where a convergent solution for the former is achieved up to the Rayleigh number of 10/8.
Unsteady forced convection laminar boundary layer flow over a moving longitudinal cylinder
Acta Mechanica, 1992
The unsteady nonsimilar forced convection flow over a longitudinal cylinder, which is moving in the same or in the opposite direction to the free stream, has been investigated. The unsteadiness is due to the free stream velocity, cylinder velocity, surface temperature of the cylinder and the mass transfer, and nonsimilarity is due to the transverse curvature. The partial differential equations, governing the flow, have been solved numerically, using an implicit finite-difference scheme in combination with a quasilinearization technique. The results show that both, skin friction and heat-transfer, are appreciably affected by the free stream velocity distributions and by the cylinder velocity. Also, skin friction as well as heat-transfer are found to increase as the transverse curvature or the suction increases, but the effect of injection is just the opposite. The heat-transfer is significantly affected by the viscous dissipation and variation of surface temperature with time. It is observed that results of this problem are crucially dependent on the parameter c~, which is the ratio of the velocity of the cylinder to the velocity of the free stream. In particular, it is found that solutions for the upstream moving cylinder exist only for a certain range of this parameter (c0, and they are nonunique in a small range of c~ too.
A theoretical study of laminar mixed convection from a horizontal cylinder in a cross stream
International Journal of Heat and Mass Transfer, 1983
The problem oflaminar mixed convection from a horizontal isothermal cylinder is considered. The free stream direction is assumed to be horizontal and perpendicular to the cylinder axis. The study is based on the solution of the full Navier-Stokes and energy equations for 2-dim. flow of a Boussinesq fluid. The free stream is assumed to start impulsively from rest and the velocity and thermal boundary layers are developed in time until reaching steady conditions. The investigation covered the ranges of Reynolds number 1 < Re < 40 and Grashof numbers up to Gr = 5 Re2 while keeping Prandtl number at a constant value of 0.7. Comparison of results with previous experimental correlations shows a good agreement. The streamline and isotherm patterns are plotted and different aspects of the phenomenon are discussed.
Ocean Engineering, 2019
A conservative vorticity method for solving the 2-D incompressible unsteady viscous flow in the eulerian frame is proposed. The numerical method implements the vorticity-stream function formulation of the Navier-Stokes (N-S) equations, solved by the cell-centered Finite Volume Method (FVM). A convection limiter, coupling the first order upwind scheme and the Quadratic Upstream Interpolation for Convective Kinematics (QUICK) scheme, is introduced to maintain the accuracy and keep the monotonicity of vorticity interpolations. The vorticity induced stream functions are recovered by a closed-form expression, which also accurately predicts the far-field boundary conditions. The convective flux is evaluated by the difference of stream functions on the neighboring two nodes, which spontaneously eliminates the net mass flux out of a control cell. The conservative FVM and QUICK scheme and the definite evaluations of stream functions and convective flux guarantee the conservation of circulation and mass. The impulsively started flow past a circular cylinder at Reynolds numbers ranging from 100 to 9500 and the inline oscillatory flow around a circular cylinder at Reynolds numbers 100 and 200 are numerically simulated using this method. The current results under the inertial and non-inertial translational frame are well validated by comparisons with other numerical and experimental benchmark tests.
Numerical simulation of natural and mixed convection flows by Galerkin-characteristic method
International Journal for Numerical Methods in Fluids, 2007
A numerical investigation is performed to study the solution of natural and mixed convection flows by Galerkin-characteristic method. The method is based on combining the modified method of characteristics with a Galerkin finite element discretization in primitive variables. It can be interpreted as a fractional step technique where convective part and Stokes/Boussinesq part are treated separately. The main feature of the proposed method is that, due to the Lagrangian treatment of convection, the Courant-Friedrichs-Lewy (CFL) restriction is relaxed and the time truncation errors are reduced in the Stokes/Boussinesq part. Numerical simulations are carried out for a natural convection in squared cavity and for a mixed convection flow past a circular cylinder. The computed results are compared with those obtained using other Eulerian-based Galerkin finite element solvers, which are used for solving many convective flow models. The Galerkin-characteristic method has been found to be feasible and satisfactory.