• 대한수학회보
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• 제50권5호
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• pp.1737-1751
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• 2013

An efficient and stable point collocation scheme based on a meshfree method is studied for the stationary incompressible Navier-Stokes equations. We describe the diffuse derivatives associated with the moving least square method. Using these diffuse derivatives, we propose a point collocation method to fit in solving the Navier-Stokes equations which improves the stability of the direct point collocation scheme. The convergence of the numerical solution is investigated from numerical examples. The driven cavity ow and the backward facing step ow are implemented for the reliability of the scheme. Also, the viscous ow on complicated geometry is successfully calculated such as the ow past a circular cylinder in duct.

• 대한수학회논문집
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• 제34권3호
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• pp.819-839
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• 2019

We study the stabilization of 2D g-Navier-Stokes equations in bounded domains with no-slip boundary conditions. First, we stabilize an unstable stationary solution by using finite-dimensional feedback controls, where the designed feedback control scheme is based on the finite number of determining parameters such as determining Fourier modes or volume elements. Second, we stabilize the long-time behavior of solutions to 2D g-Navier-Stokes equations under action of fast oscillating-in-time external forces by showing that in this case there exists a unique time-periodic solution and every solution tends to this periodic solution as time goes to infinity.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제23권2호
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• pp.115-138
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• 2019

The dual singular function method [DSFM] is a solver for corner sigulaity problem. We already construct DSFM in previous reserch to solve the Stokes equations including one singulairity at each reentrant corner, but we find out a crucial incorrection in the proof of well-posedness and regularity of dual singular function. The goal of this paper is to prove accuracy and well-posdness of DSFM for Stokes equations including two singulairities at each corner. We also introduce new applicable algorithms to slove multi-singulrarity problems in a complicated domain.

• 한국전산유체공학회지
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• 제16권4호
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• pp.72-83
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• 2011

A high-order discontinuous Galerkin method for the two-dimensional compressible Navier-Stokes equations was developed on unstructured triangular meshes. For this purpose, the BR2 methd(the second Bassi and Rebay discretization) was adopted for space discretization and an implicit Euler backward method was used for time integration. Numerical tests were conducted to estimate the convergence order of the numerical solutions of the Poiseuille flow for which analytic solutions are available for comparison. Also, the flows around a flat plate, a 2-D circular cylinder, and an NACA0012 airfoil were numerically simulated. The numerical results showed that the present implicit discontinuous Galerkin method is an efficient method to obtain very accurate numerical solutions of the compressible Navier-Stokes equations on unstructured meshes.

• 유체기계공업학회:학술대회논문집
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• 유체기계공업학회 2000년도 유체기계 연구개발 발표회 논문집
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• pp.169-175
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• 2000

A design method for transonic turbine blades is developed based on Navier-Stokes equations. The present computing process is done on the four separate steps, 1.e., determination of the blade profile, generation of the computational grids, cascade flow simulation and analysis of the computed results in the sense of the aerodynamic performance. The blade shapes are designed using the cubic polynomials under the control of the design parameters. Numerical methods for the flow equations are based on Van-Leer's FVS with an upwind TVD scheme on the finite volume. Applications are made to the VKI transonic rotor blades. Computed results are analyzed with respect to the aerodynamic performance and are compared with the experimental data.

• 대한수학회보
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• 제61권3호
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• pp.637-669
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• 2024

The goal of this paper is to study decay properties of weak solutions to Cauchy problem of the non-stationary fractional Navier-Stokes equations. By using the Fourier splitting method, we give the time L²-decay rate of weak solutions, which reveals that L²-decay is generally determined by its linear generalized Stokes flow. In second part, we establish various decay results and the uniqueness of the two dimensional fractional Navier-Stokes flows. In the end of this article, as an appendix, the existence of global weak solutions is given by making use of Galerkin' method, weak and strong compact convergence theorems.

• 대한수학회지
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• 제36권1호
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• pp.159-171
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• 1999

We formulate and study a finite element method for a linearized steady state, compressible, viscous Navier-Stokes equations in 2D, based on the discontinuous Galerkin method. Dislike the standard discontinuous galerkin method, we do not assume that the triangle sides be bounded away from the characteristic direction. the unique stability follows from the inf-sup condition established on the finite dimensional spaces for the (incompressible) Stokes problem. An error analysis having a jump discontinuity for pressure is shown.

• 대한수학회지
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• 제45권2호
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• pp.537-558
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• 2008

Consider a weak solution u of the Navier-Stokes equations in the class $L^2((0,T);X_1(\mathbb{R}^d)^d)$. We establish a new approach to treat the regularity problem for the Navier-Stokes equation in term of the multiplier space $X_1(\mathbb{R}^d)$.

• 대한수학회지
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• 제33권4호
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• pp.1129-1137
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• 1996

• 한국항해학회지
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• 제9권2호
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• pp.43-63
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• 1985

As a circular cylinder has a comparatively simple shape and becomes a basic problem for flows around other various shapes of bodies, the problem of two-dimensional viscous flow around the circular cylinder has been investigated, both theoretically and experimentally. But not a few problems are left unsolved. It is well known that the calculations are successfully made with the approximations of Stokes or Oseen for very low Reynolds numbers, but as Reynolds number is increased, Oseen's approximations as well as Stokes's ones become more and more remote from the exact solution of the Navier-Stokes equations. Therefore, in this paper, the authors transform the Navier-Stokes equations into the finite difference equations in the steady two-dimensional viscous flow at Reynolds number up to 45, and then solve the solution of the Navier-Stokes equations numerically. Also, the authors examine the accuracy of the solution by means of flow visualization with aluminum powder. The main results are as follows; (1) The critical Reynolds number at which twin vortices begin to form in the rear of the circular cylinder is found to be 6 in the experiment and 4 in the numerical solution. (2) As Reynolds number is increased, it is proved that the ratio of the length of the twin vortices to the diameter is grown almost linearly, both experimentally and numerically. (3) Separation angle is also increased according to reynolds number. But it is found that it would converge into 101.3 degrees, both experimentally and numerically.