• Title/Summary/Keyword: stokes' approximation

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ASYMPTOTIC BEHAVIOR OF STRONG SOLUTIONS TO 2D g-NAVIER-STOKES EQUATIONS

  • Quyet, Dao Trong
    • Communications of the Korean Mathematical Society
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    • v.29 no.4
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    • pp.505-518
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    • 2014
  • Considered here is the first initial boundary value problem for the two-dimensional g-Navier-Stokes equations in bounded domains. We first study the long-time behavior of strong solutions to the problem in term of the existence of a global attractor and global stability of a unique stationary solution. Then we study the long-time finite dimensional approximation of the strong solutions.

Modification of the Cubic law for a Sinusoidal Aperture using Perturbation Approximation of the Steady-state Navier-Stokes Equations (섭동 이론을 이용한 정상류 Navier-Stokes 방정식의 주기함수 간극에 대한 삼승 법칙의 수정)

  • 이승도
    • Tunnel and Underground Space
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    • v.13 no.5
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    • pp.389-396
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    • 2003
  • It is shown that the cubic law can be modified regarding the steady-state Navier-Stokes equations by using perturbation approximation method for a sinusoidal aperture variation. In order to adopt the perturbation theory, the sinusoidal function needs to be non-dimensionalized for the amplitude and wavelength. Then, the steady-state Navier-Stokes equations can be solved by expanding the non-dimensionalized stream function with respect to the small value of the parameter (the ratio of the mean aperture to the wavelength), together with the continuity equation. From the approximate solution of the Navier-Stokes equations, the basic cubic law is successfully modified for the steady-state condition and a sinusoidal aperture variation. A finite difference method is adopted to calculate the pressure within a fracture model, and the results of numerical experiments show the accuracy and applicability of the modified cubic law. As a result, it is noted that the modified cubic law, suggested in this study, will be used for the analysis of fluid flow through aperture geometry of sinusoidal distributions.

Finite Element Analysis of Incompressible Transient Navier-Stokes Equation using Fractional-Step Methods (Fractional-Step법을 이용한 비압축성 비정상 Navier-Stokes 방정식의 유한 요소해석)

  • Kim, Hyung-Min;Lee, Shin-Pyo
    • Transactions of the Korean Society of Mechanical Engineers B
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    • v.27 no.4
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    • pp.458-465
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    • 2003
  • The main objective of the research is to develop a research code solving transient incompressible Navier-Stokes equation. In this research code, Adams-Bashforth method was applied to the convective terms of the navier stokes equation and the splitted equations were discretized spatially by finite element methods to solve the complex geometry problems easily. To reduce the divergence on the boundaries of pressure poisson equation due to the unsuitable pressure boundary conditions, multi step approximation pressure boundary conditions derived from the boundary linear momentum equations were used. Simulations of Lid Driven Flow and Flow over Cylinder were conducted to prove the accuracy by means of the comparison with results of the previous workers.

FE Approximation of the Vorticity-Stream function Equations for Incompressible 2-D flows (비압축성 2-D 유동에 대한 와도-흐름함수 방정식의 유한요소 근사)

  • Pak, Seong-Kwan;Kim, Do-Wan;Kweon, Young Cheol
    • Proceedings of the Computational Structural Engineering Institute Conference
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    • 2003.10a
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    • pp.437-443
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    • 2003
  • The object of this paper is the treatment of how to make the vorticity boundary condition instead of pressure in the primitive variable case. An improved algorithm for solving the vorticity-stream function equation is presented. The linear finite element approximation for the solution of Wavier-Stokes and Stokes flows is constructed. Not only regular domain but also complicate domain can be analyze d, using this formulation.

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A Fourier Series Approximation for Deep-water Waves

  • Shin, JangRyong
    • Journal of Ocean Engineering and Technology
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    • v.36 no.2
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    • pp.101-107
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    • 2022
  • Dean (1965) proposed the use of the root mean square error (RMSE) in the dynamic free surface boundary condition (DFSBC) and kinematic free-surface boundary condition (KFSBC) as an error evaluation criterion for wave theories. There are well known wave theories with RMSE more than 1%, such as Airy theory, Stokes theory, Dean's stream function theory, Fenton's theory, and trochodial theory for deep-water waves. However, none of them can be applied for deep-water breaking waves. The purpose of this study is to provide a closed-form solution for deep-water waves with RMSE less than 1% even for breaking waves. This study is based on a previous study (Shin, 2016), and all flow fields were simplified for deep-water waves. For a closed-form solution, all Fourier series coefficients and all related parameters are presented with Newton's polynomials, which were determined by curve fitting data (Shin, 2016). For verification, a wave in Miche's limit was calculated, and, the profiles, velocities, and the accelerations were compared with those of 5th-order Stokes theory. The results give greater velocities and acceleration than 5th-order Stokes theory, and the wavelength depends on the wave height. The results satisfy the Laplace equation, bottom boundary condition (BBC), and KFSBC, while Stokes theory satisfies only the Laplace equation and BBC. RMSE in DFSBC less than 7.25×10-2% was obtained. The series order of the proposed method is three, but the series order of 5th-order Stokes theory is five. Nevertheless, this study provides less RMSE than 5th-order Stokes theory. As a result, the method is suitable for offshore structural design.

STABLE LOW ORDER NONCONFORMING QUADRILATERAL FINITE ELEMENTS FOR THE STOKES PROBLEM

  • Kim, Young-Deok;Kim, Se-Ki
    • Journal of the Korean Mathematical Society
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    • v.39 no.3
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    • pp.363-376
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    • 2002
  • Stability result is obtained for the approximation of the stationary Stokes problem with nonconforming elements proposed by Douglas et al [1] for the velocity and discontinuous piecewise constants for the pressure on qudrilateral elements. Optimal order $H^1$and $L^2$error estimates are derived.

FINITE ELEMENT APPROXIMATIONS OF THE OPTIMAL CONTROL PROBLEMS FOR STOCHASTIC STOKES EQUATIONS

  • Choi, Youngmi;Kim, Soohyun;Lee, Hyung-Chun
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.3
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    • pp.847-862
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    • 2014
  • Finite element approximation solutions of the optimal control problems for stochastic Stokes equations with the forcing term perturbed by white noise are considered. Error estimates are established for the fully coupled optimality system using Brezzi-Rappaz-Raviart theory. Numerical examples are also presented to examine our theoretical results.

A NONCONFORMING PRIMAL MIXED FINITE ELEMENT METHOD FOR THE STOKES EQUATIONS

  • Cho, Sungmin;Park, Eun-Jae
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.6
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    • pp.1655-1668
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    • 2014
  • In this article, we propose and analyze a new nonconforming primal mixed finite element method for the stationary Stokes equations. The approximation is based on the pseudostress-velocity formulation. The incompressibility condition is used to eliminate the pressure variable in terms of trace-free pseudostress. The pressure is then computed from a simple post-processing technique. Unique solvability and optimal convergence are proved. Numerical examples are given to illustrate the performance of the method.

FINITE ELEMENT APPROXIMATION AND COMPUTATIONS OF BOUNDARY OPTIMAL CONTROL PROBLEMS FOR THE NAVIER-STOKES FLOWS THROUGH A CHANNEL WITH STEPS

  • Lee, Hyung-Chun;Lee, Yong-Hun
    • Journal of the Korean Mathematical Society
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    • v.36 no.1
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    • pp.173-192
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    • 1999
  • We study a boundary optimal control problem of the fluid flow governed by the Navier-Stokes equations. the control problem is formulated with the flow through a channel with steps. The first-order optimality condition of the optimal control is derived. Finite element approximations of the solutions of the optimality system are defined and optimal error estimates are derived. finally, we present some numerical results.

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SLOW VISCOUS FLOW PAST A CAVITY WITH INFINITE DEPTH

  • Kim, D.W;Kim, S.B;Chu, J.H
    • Journal of applied mathematics & informatics
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    • v.7 no.3
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    • pp.801-812
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    • 2000
  • Two-dimensional slow viscous flow on infinite half-plane past a perpendicular infinite cavity is considered on the basis of the Stokes approximation. Using complex representation of the two-dimensional Stokes flow, the problem is reduced to solving a set of Fredholm integral equations of the second kind. The streamlines and the pressure and vorticity distribution on the wall are numerically determined.