Simulación de flujos compresibles utilizando un esquema explícito de Taylor-Galerkin modificado

Como ya fue demostrado por diferentes autores, el esquema de Taylor-Galerkin (TG) en el contexto del Método de Elementos Finitos (FEM: Finite Element Method) resulta particularmente adecuado para la solución de flujos compresibles en régimen supersónico que involucran ondas de choque. Sin embargo, el esquema TG presenta inestabilidades en flujos con números de Mach subsónicos. Aún en casos en los cuales el Mach freestream es supersónico, siempre existen regiones del flujo, cerca de las paredes de los obstáculos, en donde la velocidad es baja o próxima a cero y el Mach local es subsónico. Por esto, algunos investigadores desarrollaron el algoritmo “Characteristic Based Split” (CBS) buscando obtener un único esquema que presente un comportamiento adecuado, tanto en regímenes subsónicos como supersónicos. En las últimas dos décadas algunos trabajos establecieron ciertas ventajas en la convergencia del algoritmo CBS comparado con el algoritmo TG. Sin embargo, estas ventajas de convergencia son a costo de tiempo de simulación debido a la operación de “Split” típica del algoritmo CBS. En este trabajo se propone un esquema híbrido denominado Taylor-Galerkin Modificado (MTG: Modified Taylor-Galerkin) que presenta las ventajas de ambos esquemas: mejor convergencia sin costo adicional en términos de tiempo de cómputo. Se aplicaron los esquemas TG, CBS y MTG utilizando elementos finitos hexaédricos en conjunto con una técnica de captura de choque para la solución de flujos aerodinámicos compresibles supersónicos, viscosos y no viscosos. Además, con la intención de obtener un algoritmo eficiente, las matrices elementales son integradas analíticamente. Ésto se realiza con dos enfoques diferentes. En el primer enfoque, la matriz Jacobiana inversa y el determinante de la matriz Jacobiana a nivel de elemento se evalúan con una forma de integración reducida, utilizando el punto situado en el centro del elemento para las matrices de masa, convectivas, difusivas y matrices de estabilización; todas estas matrices están integradas analíticamente usando el Jacobiano en el centro del elemento. En el segundo enfoque, las matrices de masa y convectivas se calculan mediante un esquema de integración completa donde se integra considerando la dependencia del Jacobiano con la posición, mientras que sólo las matrices difusivas y las matrices de estabilización se calculan con integración reducida, utilizando el punto situado en el centro del elemento para calcular la matriz Jacobiana inversa y el determinante de la matriz Jacobiana a nivel de elemento. Por último, este trabajo incorpora el modelo de turbulencia de Spalart-Allmaras, pero utilizando una versión conservativa de la ecuación de transporte de la variable turbulenta. Se prueban los algoritmos para determinar las mejoras de convergencia tanto en casos laminares como turbulentos para diferentes números de Mach (flujos, subsónicos, transónicos y supersónicos).

A Divergence-Free Immersed Boundary Method and Its Finite Element Applications

ABSTRACTIn order to maintain the no-slip condition and the divergence-free property simultaneously, an iterative scheme of immersed boundary method in the finite element framework is presented. In this method, the Characteristic-based Split scheme is employed to solve the momentum equations and the formulation for the pressure and the extra body force is derived according to the no-slip condition. The extra body force is divided into two divisions, one is in relation to the pressure and the other is irrelevant. Two corresponding independent iterations are set to solve the two sections. The novelty of this method lies in that the correction of the velocity increment is included in the calculation of the extra body force which is relevant to the pressure and the update of the force is incorporated into the iteration of the pressure. Hence, the divergence-free properties and no-slip conditions are ensured concurrently. In addition, the current method is validated with well-known benchmarks.

Study on Mass Transports in Evolution of Separation Bubbles Using LCSs and Lobe Dynamics

The lobe dynamics andmass transport between separation bubble and main flow in flow over airfoil are studied in detail, using Lagrangian coherent structures (LCSs), in order to understand the nature of evolution of the separation bubble. For this problem, the transient flow over NACA0012 airfoil with low Reynolds number is simulated numerically by characteristic based split (CBS) scheme, in combination with dual time stepping. Then, LCSs and lobe dynamics are introduced and developed to investigate themass transport between separation bubble and main flow, from viewpoint of nonlinear dynamics. The results show that stable manifolds and unstable manifolds could be tangled with each other as time evolution, and the lobes are formed periodically to induce mass transport between main flow and separation bubble, with dynamic behaviors. Moreover, the evolution of the separation bubble depends essentially on the mass transport which is induced by lobes, ensuing energy and momentum transfers. As the results, it can be drawn that the dynamics of flow separation could be studied using LCSs and lobe dynamics, and could be controlled feasibly if an appropriate control is applied to the upstream boundary layer with high momentum.

AC-CBS-Based Partitioned Semi-Implicit Coupling Algorithm for Fluid-Structure Interaction Using Stabilized Second-Order Pressure Scheme

We analyze in this paper the pressure splitting scheme of a partitioned semi-implicit coupling algorithm for fluid-structure interaction (FSI) simulation. The semi-implicit coupling algorithm is developed on the ground of the artificial compressibility characteristic-based split (AC-CBS) scheme that serves not only for the fluid subsystem but also for the global FSI system. As the dual-time stepping procedure recommended for quasi-incompressible flows is incorporated into the implicit coupling stage, the fluctuating pressure may be unusually susceptible to the AC coefficient. Moreover, it is not trivial to devise an optimal AC formulation for pressure estimation. Instead, we consider a stabilized second-order pressure splitting scheme in the AC-CBS-based partitioned semi-implicit coupling algorithm. Computer simulation of a benchmark FSI experiment demonstrates that good agreement is exposed between the available and present data.

Stabilized MLPG-VF-based method with CBS scheme for laminar flow at high Reynolds and Rayleigh numbers

In the present work, the meshless local Petrov–Galerkin vorticity-stream function (MLPG-VF) method is extended to solve two-dimensional laminar fluid flow and heat transfer equations for high Reynolds and Rayleigh numbers. The characteristic-based split (CBS) scheme which uses unity test function is employed for discretization, and the moving least square (MLS) method is used for interpolation of the field variables. Four test cases are considered to evaluate the present algorithm, namely lid-driven cavity flow with Reynolds numbers up to and including [Formula: see text], flow over a backward-facing step at Reynolds number of [Formula: see text], natural convection in a square cavity for Rayleigh numbers up to and including [Formula: see text], and natural convection in a concentric square outer cylinder and circular inner cylinder annulus for Rayleigh numbers up to and including [Formula: see text]. In each case, the result obtained using the proposed algorithm is either compared with the results from the literatures or with those obtained using conventional numerical techniques. The present algorithm shows stable results at lower or equal computational cost compared to the other upwinding schemes usually employed in the MLPG method. Close agreements between the compared results as well as higher accuracy of the proposed method show the ability of this stabilized algorithm.