• Journal of Korean Society of Coastal and Ocean Engineers
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• v.31 no.2
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• pp.41-49
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• 2019

We approximately obtain heights of cnoidal waves overtopping on a porous breakwater using both the one-layer Boussinesq equations (Vu et al., 2018) and the two-layer Boussinesq equations (Huynh et al., 2017). For cnoidal waves overtopping on a porous breakwater, we find through numerical experiments that the heights of cnoidal waves overtopping on a low-crested breakwater (obtained by the Navier-Stokes equations) are smaller than the heights of waves passing through a high-crested breakwater (obtained by the one-layer Boussinesq equations) and larger than the heights of waves passing through a submerged breakwater (obtained by the two-layer Boussinesq equations). As the cnoidal wave nonlinearity becomes smaller or the porous breakwater width becomes narrower, the heights of transmitting waves obtained by the one-layer and two-layer Boussinesq equations become closer to the height of overtopping waves obtained by the Navier-Stokes equations.

• Journal of the Korean Mathematical Society
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• v.46 no.5
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• pp.1041-1069
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• 2009

We study the periodic homogenization of the non-stationary Stokes equations. The fundamental homogenization theorem and corrector theorem are proved under a very general assumption on the viscosity coefficients and data. The proofs are based on a weak formulation suitable for an application of classical Tartar's method of oscillating test functions. Such a weak formulation is derived by adapting an argument in Teman's book [Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland, Amsterdam, 1984].

• Journal of the Chungcheong Mathematical Society
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• v.35 no.4
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• pp.307-313
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• 2022

In this paper, we will consider the periodic stationary Stokes equations on Rⁿ. For the cube of the period, we set Ω = ∏ⁿ_i=1(0, L_i). And we will study the stability of the solutions on various functional spaces, for the Stokes equations on Rⁿ.

• Communications of the Korean Mathematical Society
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• v.35 no.1
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• pp.339-345
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• 2020

This note deals with the question of the regularity of (Leray) weak solutions of the Navier-Stokes equations in terms of the pressure. This criterion improves on the existing results.

• Journal of KIISE:Computer Systems and Theory
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• v.32 no.9
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• pp.445-456
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• 2005

We propose a fast and accurate fluid solver of the Wavier-Stokes equations for the physics-based fluid simulations. Our method utilizes the solution of the Stokes equation as an initial guess for the velocity of the nonlinear term in the Wavier-Stokes equations. By guessing the initial velocity close to the exact solution of the given nonlinear differential equations, we can develop remarkably accurate and stable fluid solver. Our solver is based on the implicit scheme of finite difference methods, that makes it work well for large time steps. Since we employ the ADI method, our solver is also fast and has a uniform computation time. The experimental results show that our solver is excellent for fluids with high Reynolds numbers such as smoke and clouds.

• Transactions of the Korean Society of Mechanical Engineers B
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• v.25 no.11
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• pp.1581-1593
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• 2001

An efficient variable-reordering method for finite element meshes is used and the effect of variable-reordering is investigated. For the element renumbering of unstructured meshes, Cuthill-McKee ordering is adopted. The newsy reordered global matrix has a much narrower bandwidth than the original one, making the ILU preconditioner perform bolter. The effect of variable reordering on the convergence behaviour of saddle point type matrix it studied, which results from P2/P1 element discretization of the Navier-Stokes equations. We also propose and test 'level(0) preconditioner'and 'level(2) ILU preconditioner', which are another versions of the existing 'level(1) ILU preconditioner', for the global matrix generated by P2/P1 finite element method of incompressible Navier-Stokes equations. We show that 'level(2) ILU preconditioner'performs much better than the others only with a little extra computations.

• Journal of the Chungcheong Mathematical Society
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• v.19 no.3
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• pp.283-289
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• 2006

Roh[1] derived 2D g-Navier-Stokes equations from 3D Navier-Stokes equations. In this paper, we will see the space $L^2({\Omega},\;g)$, which is the weighted space of $L^2({\Omega})$, as natural generalized space of $L^2({\Omega})$ which is mathematical setting for Navier-Stokes equations. Our future purpose is to use the space $L^2({\Omega},\;g)$ as mathematical setting for the g-Navier-Stokes equations. In addition, we will see Helmoltz-Leray projection on $L^2_{per}({\Omega},\;g)$) and compare with the one on $L^2_{per}({\Omega})$.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.29 no.6
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• pp.326-334
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• 2017

We obtain height of waves overtopping on a porous breakwater using both the one-layer and two-layer Boussinesq equations. The one-layer Boussinesq equations of Lee et al. (2014) are used and the two-layer Boussinesq equations are derived following Cruz et al. (1997). For solitary waves overtopping on a porous breakwater, we find through numerical experiments that the height of waves overtopping on a low-crested breakwater (obtained by the Navier-Stokes equations) are smaller than the height of waves passing through a high-crest breakwater (obtained by the one-layer Boussinesq equations) and larger than the height of waves passing through a submerged breakwater (obtained by the two-layer Boussinesq equations). As the wave nonlinearity becomes smaller or the porous breakwater width becomes narrower, the heights of transmitting waves obtained by the one-layer and two-layer Boussinesq equations become closer to the height of overtopping waves obtained by the Navier-Stokes equations.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.21 no.4
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• pp.203-214
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• 2017

In this paper we propose and analyze two a posteriori error estimators for the stabilized $P_1/P_1$ finite element discretization of the Stokes equations. These error estimators are computed by solving local Poisson or Stokes problems on elements of the underlying triangulation. We establish their asymptotic exactness with respect to the velocity error under certain conditions on the triangulation and the regularity of the exact solution.

• 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.