• Bulletin of the Korean Mathematical Society
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• v.61 no.3
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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.

• Korean Journal of Mathematics
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• v.16 no.1
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• pp.57-84
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• 2008

In this paper, a boundary control problem for a flow governed by the time-dependent two dimensional Navier-Stokes equations is considered. We derive a mathematical formulation and a relevant process for an appropriate control along the part of the boundary to minimize the drag due to the flow. After showing the existence of an optimal solution, the first order optimality conditions are derived. The strict differentiability of the state solution in regard to the control parameter shall be exposed rigorously, and the necessary conditions along with the system for the optimal solution shall be deduced in conjunction with the evaluation of the first order Gateaux derivative to the performance functional.

• Journal of computational fluids engineering
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• v.1 no.1
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• pp.112-122
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• 1996

A finite element scheme using the concept of PISO method has been developed to solve the Navier-Stokes viscous flows in all speed range. This scheme includes development of new pressure equation that retains both the hyperbolic term related with the density variation and the elliptic term reflecting the incompressibility constraint. The present method is applied to the incompressible two-dimensional driven cavity flow problems(Re=100, 400 and 1,000). For compressible flows, the Carter plate problem(M=3 and Re=1,000) is computed. Finally, we have simulated the shock-boundary layer interaction(M=2 and Re=2.96×10/sup 5/), a more difficult problem, and compared its results with the experiment to demonstrate the shock capturing capability of the present solution algorithm.

• Journal of the Korean Mathematical Society
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• v.43 no.1
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• pp.45-63
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• 2006

This paper is concerned with a boundary control problem for the vorticity minimization, in which the flow is governed by the stationary two dimensional Stokes equations. We wish to find a mathematical formulation and a relevant process for an appropriate control along the part of the boundary to minimize the vorticity due to the flow. After showing the existence and uniqueness of an optimal solution, we derive the optimality conditions. The differentiability of the state solution in regard to the control parameter shall be conjunct with the necessary conditions for the optimal solution. For the minimizer, an algorithm based on the conjugate gradient method shall be proposed.

• Korean Journal of Mathematics
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• v.23 no.2
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• pp.293-312
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• 2015

We deal with a boundary control problem for the vorticity minimization, in which the ow is governed by the time-dependent two dimensional incompressible Navier-Stokes equations. We derive a mathematical formulation and a process for an appropriate control along the portion of the boundary to minimize the vorticity motion due to the ow in the fluid domain. After showing the existence of an optimal solution, we derive the optimality system for which optimal solutions may be determined. The differentiability of the state solution in regard to the control parameter shall be conjunct with the necessary conditions for the optimal solutions.

• 한국전산유체공학회:학술대회논문집
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• 1999.05a
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• pp.177-185
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• 1999

The airfoil design optimization procedures based on the Navier-Stokes equations were developed, This procedure enables more realistic and practical transonic airfoil designs. The modified Hicks-Henne functions were used to generate the shape of airfoils. Five Hick-Henne functions were used to design upper surface of airfoil only. To enhance the ability of Hick-Henne function to generate various airfoil shape with limited number of functions, the positions of control points were adjusted through optimization procedure. The design procedure was applied to the single-point design for the drag minimization problem with lift and area constraints. The result shows the capability of the procedure to generate much realistic airfoils with very small drag-creep in the low transonic regime. This is mainly due to the viscosity effect of Navier-Stokes flow analysis. However, in the higher transonic range tile drag-creep appears. The multi-point design is shown to be an effective way to avoid the drag-creep and improve off-design performance which is very similar in the Euler design.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.23 no.2
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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.

• 한국전산유체공학회:학술대회논문집
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• 1999.05a
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• pp.113-123
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• 1999

In this study a pure vector potential method (PVPM) for a three dimensional, unsteady, incompressible flow is proposed. A simplified version for a two dimensional problem is described in detail, and a method to prescribe appropriate boundary conditions is also presented. The resulting numerical algorithm is applied to the cavity flow driven by an impulsively started wall and also to the Stokes' first problem. Some important unsteady/steady features are captured for these two flows, and quantitative agreements of flow variables with available reference database are good.

• Journal of the Society of Naval Architects of Korea
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• v.44 no.2 s.152
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• pp.130-138
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• 2007

A coupled variational equation for fluid-structure interaction (FSI) problems is derived from a steady state Navier-Stokes equation for incompressible Newtonian fluid and an equilibrium equation for geometrically nonlinear structures. For a fully coupled FSI formulation, between fluid and structures, a traction continuity condition is considered at interfaces where a no-slip condition is imposed. Under total Lagrange formulation in the structural domain, finite rotations are well described by using the second Piola-Kirchhoff stress and Green-Lagrange strain tensors. An adjoint shape design sensitivity analysis (DSA) method based on material derivative approach is applied to the FSI problem to develop a shape design optimization method. Demonstrating some numerical examples, the accuracy and efficiency of the developed DSA method is verified in comparison with finite difference sensitivity. Also, for the FSI problems, a shape design optimization is performed to obtain a maximal stiffness structure satisfying an allowable volume constraint.

• Journal of computational fluids engineering
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• v.17 no.2
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• pp.35-41
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• 2012

This paper presents incompressible Navier-Stokes solution algorithm for 2D Free-surface flow problems on the Cartesian mesh, which was implemented to run on Graphics Processing Units(GPU). The INS solver utilizes the variable arrangement on the Cartesian mesh, Finite Volume discretization along Constrained Interpolation Profile-Conservative Semi-Lagrangian(CIP-CSL). Solution procedure of incompressible Navier-Stokes equations for free-surface flow takes considerable amount of computation time and memory space even in modern multi-core computing architecture based on Central Processing Units(CPUs). By the recent development of computer architecture technology, Graphics Processing Unit(GPU)'s scientific computing performance outperforms that of CPU's. This paper focus on the utilization of GPU's high performance computing capability, and presents an efficient solution algorithm for free surface flow simulation. The performance of the GPU implementations with double precision accuracy is compared to that of the CPU code using an representative free-surface flow problem, namely. dam-break problem.