Mutsuto Kawahara

Mixed finite element method for analysis of viscoelastic fluid flow

Abstract In the present paper, numerical analysis of incompressible viscoelastic fluid flow is discussed using mixed finite element Galerkin method. Because Maxwellian viscoelasticity is assumed as the constitutive equation, stress components could not be eliminated from the governing equation system. Because of this, mixed finite element method is utilized to discretize the basic equations. For the solution procedures to solve discretized equation system, Newton-Raphson method for steady flow and perturbation method for unsteady flow is employed. As the numerical examples, comparison was made on the finite element computational results between by direct method and by mixed method. Effects of the viscoelasticity is analyzed for the flows at Reynolds numbers 30, 50 and 70.

Three‐dimensional linear stability analysis of incompressible viscous flows using the finite element method

The linear stability of incompressible flows is investigated on the basis of the finite element method. The two-dimensional base flows computed numerically over a range of Reynolds numbers are perturbed with three-dimensional disturbances. The three-dimensionality in the flow associated with the secondary instability is identified precisely. First, by using linear stability theory and normal mode analysis, the partial differential equations governing the evolution of perturbation are derived from the linearized Navier–Stokes equation with slight compressibility. In terms of the mixed finite element discretization, in which six-node quadratic Lagrange triangular elements with quadratic interpolation for velocities (P2) and three-node linear Lagrange triangular elements for pressure (P1) are employed, a non-singular generalized eigenproblem is formulated from these equations, whose solution gives the dispersion relation between complex growth rate and wave number. Then, the stabilities of two cases, i.e. the lid-driven cavity flow and flow past a circular cylinder, are examined. These studies determine accurately stability curves to identify the critical Reynolds number and the critical wavelength of the neutral mode by means of the Krylov subspace method and discuss the mechanism of instability. For the cavity flow, the estimated critical results are Rec=920.277±0.010 for the Reynolds number and kc=7.40±0.02 for the wave number. These results are in good agreement with the observation of Aidun et al. and are more accurate than those by the finite difference method. This instability in the cavity is associated with absolute instability [Huerre and Monkewitz, Annu. Rev. Fluid Mech., 22, 473–537 (1990)]. The Taylor–Goertler-like vortices in the cavity are verified by means of the reconstruction of three-dimensional flows. As for the flow past a circular cylinder, the primary instability result shows that the flow has only two-dimensional characteristics at the onset of the von Karman vortex street, when Re<49. The estimated critical values of primary instability are Rec=46.389±0.010 and Stc=0.126 for the Strouhal number. These values are very close to the observation data [Williamson, J. Fluid Mech., 206, 579–627 (1989)] and other stability results [Morzynski and Thiele, Z. Agnew. Math. Mech., 71, T424–T428 (1991); Jackson, J. Fluid Mech., 182, 23–45 (1987)]. This onset of vortex shedding is associated with the symmetry-breaking bifurcation at the Hopf point. Copyright

Shape Identification for Fluid-Structure Interaction Problem Using Improved Bubble Element

Abstract Numerical solutions for identification of the shape of a circular cylinder are addressed in this paper. The Sakawa-Shindo method is used to minimize the algorithm. A unified computational approach for simulation of flow and shape identification is presented. As a numerical approach for spatial discretization, mixed interpolation by the bubble and linear elements is used for the velocity and pressure fields, respectively.

Structural oscillation control using tuned liquid damper

Abstract This paper presents a numerical study for structural control using a tuned liquid damper (TLD), which consists of solid tank filled with liquid. In order to determine the complete behavior, it is necessary to investigate the phenomenon coupling with structure and fluid. In the fluid analysis, the Navier–Stokes equation is used in the form of the arbitrary Lagrangian–Eulerian (ALE) formulation. For the discretization of the incompressible Navier–Stokes equation, the improved-balancing-tensor-diffusivity method and the fractional-step method are used. For computational stability, smoothing on the free surface is carried out. In the structural analysis, the Newmark’s β method is used for the time discretization. It is shown that the method presented in this paper is suitable for the analysis and the tuned liquid damper is efficient in reducing the oscillation.

LINEAR STABILITY OF INCOMPRESSIBLE FLUID FLOW IN A CAVITY USING FINITE ELEMENT METHOD

Numerical methods have been applied to theoretical studies of instability and transition to turbulence. In this study an analysis of the linear stability of incompressible flow is undertaken. By means of the finite element method the two-dimensional base flow is computed numerically over a range of Reynolds numbers and is perturbed with three-dimensional disturbances. The partial differential equations governing the evolution of perturbation are obtained from the non-linear Navier–Stokes equations with a slight compressibility by using linear stability and normal mode analysis. In terms of the finite element discretization a non-singular generalized eigenproblem is formulated from these equations whose solution gives the dispersion relation between complex growth rate and wave number. This study presents stability curves to identify the critical Reynolds number and critical wavelength of the neutral mode and discusses the mechanism of instability. The stability of lid-driven cavity flow is examined. Taylor–Goertler-like vortices in the cavity are obtained by means of reconstruction of three-dimensional flows.

Stabilized Bubble Function Method for Shallow Water Long Wave Equation

A relationship between the stabilized bubble function method and the stabilized finite element method is shown in this paper. The Petrov–Galerkin formulation with bubble function, i.e. a stabilized bubble function method, is proposed for the shallow water long wave equation. The Petrov–Galerkin formulation with the bubble function formulation possesses better stability than the Bubnov–Galerkin formulation with the bubble function.

Optimal Control in Navier-Stokes Equations

This paper presents a formulation for optimal control of a forced convection flow. The state equation that governs the forced convection flow can be expressed as the incompressible Navier-Stokes equations and energy equation. The optimal control can be formulated as finding a control force to minimize a performance function that is defined to evaluate a control object. The stabilized finite element method is used for the spatial discretization, while the Crank-Nicolson scheme is used for the temporal discretization. The Sakawa-Shindo method, which is an iterative procedure, is applied for minimizing the performance function.

Shape Optimization of Body Located in Incompressible Viscous Flow Based on Optimal Control Theory

This paper presents a numerical method of shape optimization of a body located in an incompressible viscous flow described by the Stokes and Oseen equations. The purpose of this study is to find the optimal shape that minimizes the fluid forces subjected to the body. The formulation of the shape optimization is based on the optimal control theory. The first thing that should be carried out in the optimal control theory is to define a performance function, which expresses the optimal shape. In this study, the fluid forces minimization problem is treated, i.e. fluid forces are directly used in the performance function. The performance function must be minimized subject to the basic equation. The optimal shape, which minimizes the fluid force, is pursued in this paper. This problem can be transformed into the minimization problem without constraint conditions by the Lagrange multiplier. As a numerical example, drag force minimization problems of a body located in low Reynolds number flows are carried out.

2-D Fluid-Structure Interaction Problems by an Arbitrary Lagrangian-Eulerian Finite Element Method

Abstract This paper presents a finite element analysis of a fluid-structure interaction problem by the Arbitrary Lagrangian-Eulerian (ALE) method. For the analysis of the Navier-Stokes equation, a fractional step method is used based the linear interpolation functions for both velocity and pressure. For a numerical example, the present method is applied to the flow analysis around an oscillating rectangular cylinder. The total calculation domain is divided into several sub-domains to improve the computational efficiency. In the inner domain, the mesh moves based on the ALE description. The outmost domain is the Eulerian domain. The intermediate domain is located as a transition.

Shallow Water Flow Analysis with Moving Boundary Technique Using Least-squares Bubble Function

This paper presents a computational simulation method for a river problem. For the actual flow problem, it is necessary to compute flow velocity, water elevation and water region at the same time. For the basic formulation, the unsteady shallow water equations are used. As the numerical approach, implicit FEM is proposed by bubble function. To control numerical stability and accuracy, LSBF (Least-Squares Bubble Function) is used to solve the finite element equations. Also, the fixed boundary technique is combined to deal with wet and dry areas in the moving finite element mesh. Some numerical tests are shown to check this method.