Numerical Solutions of 2-D Steady Incompressible Flow in a Driven Skewed Cavity

The benchmark test case for non-orthogonal grid mesh, the "driven skewed cavity flow", first introduced by Demirdzic et al. (1992, IJNMF, 15, 329) for skew angles of alpha=30 and alpha=45, is reintroduced with a more variety of skew angles. The benchmark problem has non-orthogonal, skewed grid mesh with skew angle (alpha). The governing 2-D steady incompressible Navier-Stokes equations in general curvilinear coordinates are solved for the solution of driven skewed cavity flow with non-orthogonal grid mesh using a numerical method which is efficient and stable even at extreme skew angles. Highly accurate numerical solutions of the driven skewed cavity flow, solved using a fine grid (512x512) mesh, are presented for Reynolds number of 100 and 1000 for skew angles ranging between 15<alpha<165

Workshops for Learning in Computational Fluid Mechanics

The must-do of teaching Computational Fluid Dynamics (CFD) is the practical component. The main aim of the present paper is to propose test cases used in research. When these benchmarks are conveniently scaled-down and parameterized, students improve their understanding of the strong and weak points of the numerical models and gain an insight into the fluid dynamics processes learnt in the classroom. The goal is to prepare students who are new in this area to become self-sufficient in engineering practice.Comunicación del proyecto PID1415_030 presentada en la Second International Conference on Technological Ecosystem for Enhancing Multiculturality (TEEM'14

Stationary solutions of incompressible viscous flow in a wall-driven semi-circular cavity

Stationary numerical solutions of incompressible viscous flow inside a wall-driven semicircular cavity are presented. After a conformal mapping of the geometry, using a body-fitted mesh, the Navier-Stokes equations are solved numerically. The stationary solutions of the flow in a wall-driven semi-circular cavity are computed up to Re = 24000. The present results are in good agreement with the published results found in the literature. Our results show that as the Reynolds number increases, the sizes of the secondary and tertiary vortices increase, whereas the size of the primary vortex decreases. At large Reynolds numbers, the vorticity at the primary vortex centre increases almost linearly stating that Batchelor’s mean-square law is not valid for wall-driven semi-circular cavity flow. Detailed results are presented and also tabulated for future references and benchmark purposes

A Novel Approach for Solving Navier-Stokes Equations on Complex Geometries

Wind turbines in a wind farm undergo significant interference through wake and terrain interaction. Numerical modeling of a complex terrain necessitates the use of curvilinear body fitted coordinates. This paper proposes a novel mixed basis formulation of the governing conservation equations for general curvilinear non-orthogonal grids with the physical covariant velocity as the primary solution variable. The result is an algorithm which has many advantages of orthogonal equations. The conservation equations written in this form retains the diagonal dominance of the pressure equation. The newly formed conservation equations are solved using the SIMPLER algorithm and are shown to converge well for non-orthogonal grids. Standard K - e model is used for turbulence closure

Numerical simulation and energy flux vector visualization of radiative-convection heat transfer in a porous triangular enclosure

A detailed theoretical examination laminar natural convection heat flow in a triangular porous cavity with significant radiative heat transfer and porosity variation is presented. Twodimensional laminar incompressible flow is considered with the left slant and right walls are low and high temperature respectively, and the remaining (top) wall prescribed as adiabatic. The Darcy-Brinkman isotropic model is utilized, and the coupled governing equations are solved by a numerical method utilizing finite differences. Visualization of isotherms and streamlines is achieved with the method of Energy Flux Vectors (EFVs). The impacts of the different model parameters (Rayleigh number Ra, Darcy number-Da, porosity-E and radiation parameter-Rd) on the thermo fluid characteristics are studied in detail. The computations show that convective heat transfer is enhanced with greater Darcy parameter (permeability) which also leads to intensification in the density of energy flux vector patterns. The flow is accelerated with increasing buoyancy effect (Rayleigh number) and temperatures are also increased with greater radiative flux. Average Nusselt number is decreased with higher porosity. The simulations are relevant to hybrid porous media solar collectors

Numerical simulation and energy flux vector visualization of radiative-convection heat transfer in a porous triangular enclosure

A detailed theoretical examination laminar natural convection heat flow in a triangular porous cavity with significant radiative heat transfer and porosity variation is presented. Twodimensional laminar incompressible flow is considered with the left slant and right walls are low and high temperature respectively, and the remaining (top) wall prescribed as adiabatic. The Darcy-Brinkman isotropic model is utilized, and the coupled governing equations are solved by a numerical method utilizing finite differences. Visualization of isotherms and streamlines is achieved with the method of Energy Flux Vectors (EFVs). The impacts of the different model parameters (Rayleigh number Ra, Darcy number-Da, porosity-E and radiation parameter-Rd) on the thermo fluid characteristics are studied in detail. The computations show that convective heat transfer is enhanced with greater Darcy parameter (permeability) which also leads to intensification in the density of energy flux vector patterns. The flow is accelerated with increasing buoyancy effect (Rayleigh number) and temperatures are also increased with greater radiative flux. Average Nusselt number is decreased with higher porosity. The simulations are relevant to hybrid porous media solar collectors

Numerical simulation of thermal radiation influence on natural convection in a trapezoidal enclosure : heat flow visualization through energy flux vectors

A theoretical and numerical study of natural convection intwo-dimensional laminar incompressible flow in a trapezoidal enclosurein the presence of thermal radiation is conducted, motivated by energy systems applications. Heat flow visualization via the method of energy flux vectors (EFVs) is also included. The trapezoidal cavity has an inclined top wall which in addition to the bottom wall is maintained at constant temperature, whereas the remaining (vertical side) walls are adiabatic. The governing partial differential conservation equations are transformed using a vorticity-stream function formulation and non-dimensional variables and the resulting nonlinear boundary value problem is solved using a finite difference method with incremental time steps. EFVs provide abundant details of the heat flow at the core of the enclosure. The larger energy flux vectors indicate high temperature gradient zones and the sparse EFVs correspond to low temperature gradient zone. Heat flow distribution in the trapezoidal enclosure can be clearly elaborated via energy flux vectors and provides a deeper insight into thermal characteristics. A comprehensive parametric study is performed to evaluate the impact of Rayleigh number (buoyancy parameter) and radiation parameter on transport phenomena. The computations indicate that local Nusselt number and velocity are increasing functions of the Rayleigh number and radiation parameter. Significant changes in streamlines, temperature contours and energy streamlines for high Rayleigh number are observed. The energy flux vectors show that a large eddy is formed within the enclosure which migrates towards the cold wall. Greater thermal buoyancy force accelerates the primary flow whereas it decelerates the secondary flow. The simulations are relevant to solar collector systems, enclosure fire dynamics, electronic cooling and fuel cell systems. Furthermore, the computations furnish a good benchmark for more general computational fluid dynamics (CFD) analysis with commercial software e.g. ANSYS FLUENT

A High Order Numerical Scheme for Incompress ible Navier-Stokes Equations

To solve the incompressible Navier-Stokes equations in a generalized coordinate system, a high order solver is presented. An exact projection method/fractional-step scheme is used in this study. Convective terms of the Navier-Stokes (N-S) equations are solved using fifth-order WENO spatial operators, and for the diffusion terms, a sixth-order compact central difference scheme is employed. The third-order Runge-Kutta (R-K) explicit time-integrating scheme with total variation diminishing (TVD) is adopted for the unsteady flow computations. The advantage of using a WENO scheme is that it can resolve applications using less number of grid points. Benchmark cases such as, driven cavity flow, Taylor-Green (TG) vortex, double shear layer, backward-facing step, and skewed cavity are used to investigate the accuracy of the scheme for two dimensional flow

A Comparative Study of the SIMPLE and Fractional Step Time Integration Methods for Transient Incompressible Flows

Time integration methods are necessary for the solution of transient flow problems. In recent years, interest in transient flow problems has increased, leading to a need for better understanding of the costs and benefits of various time integration schemes. The present work investigates two common time integration schemes, namely the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) and the Fractional Step (FS) method. Three two-dimensional, transient, incompressible flow problems are solved using a cell centered, finite volume code. The three test cases are laminar flow in a lid-driven skewed cavity, laminar flow over a square cylinder, and turbulent flow over a square cylinder. Turbulence is modeled using wall functions and the k - ε turbulence model with the modifications suggested by Kato and Launder. Solution efficiency as measured by the effort carried out by the flow equation solver and CPU time is examined. Accuracy of the results, generated using the SIMPLE and FS time integration schemes, is analyzed through a comparison of the results with existing experimental and/or numerical solutions. Both the SIMPLE and FS algorithms are shown to be capable of solving benchmark flow problems with reasonable accuracy. The two schemes differ slightly in their prediction of flow evolution over time, especially when simulating very slowly changing flows. As the time step size decreases, the SIMPLE algorithm computational cost (CPU time) per time step remains approximately constant, while the FS method experiences a reduction in cost per time step. Also, the SIMPLE algorithm is numerically stable for time steps approaching infinity, while the FS scheme suffers from numerical instability if the time step size is too large. As a result, the SIMPLE algorithm is recommended to be used for transient simulations with large time steps or steady state problems while the FS scheme is better suited for small time step solutions, although both time-stepping schemes are found to be most efficient when their time steps are at their maximum stable value

A new mixed basis Navier-Stokes formulation for simulating wind turbines on complex terrain

Wind farms are subjected to significant aerodynamic interference due to the unsteady wakes of the individual turbines as well as the complex terrains on which they are erected. Numerical modeling of complex geometries necessitates the use of curvilinear body-fitted coordinates. In the present research, conservation equations efficient for convection-dominated flows over complex terrain, are developed. The Navier-Stokes equations written in the novel mixed basis form allows discretization of the convective flux to be compactly represented, while also preserving the diagonal dominance in the discrete pressure equation. The resulting generalized conservation equations closely resemble the orthogonal equations. Hence they lend themselves suitable to algorithms for orthogonal systems with the addition of a source term. Additionally, the efficiency of the mixed basis formulation is illustrated by simplifying the equations to various geometries of practical applications like tubular, rotated, extruded, and orthogonal. By developing a single solver using the mixed formulation that retains the majority of the terms to be invariant and implementing geometry-based simplifications in the source term, a general, efficient solution procedure can be obtained for all geometries ranging from the complex body-fitted coordinates to the Cartesian coordinates. A momentum source method is used to model the wind turbines. The newly formed conservation equations are solved on a structured grid using the SIMPLER algorithm. Three different RANS closure models, including the standard, RNG, and realizable K − ε, are implemented. Results validating the ability of the numerical procedure to simulate flow over complex terrains and wind turbines are presented. Applications providing insights into the performance and loading of the turbines as well as the turbine-wake and turbine-terrain interactions are analyzed