• Journal of the Korean Mathematical Society
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• v.40 no.6
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• pp.1061-1074
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• 2003

We first show the analyticity of Stokes operator in Besov spaces $B_{p,q}$$^{a}$ ( $R_{＋}$$^{n}$). Then, we estimate the asymptotic behavior of the Stokes solutions. We also show the Hodge decomposition.n.

• 한국전산유체공학회:학술대회논문집
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• 1998.11a
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• pp.96-101
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• 1998

Various discretiation methods of Laplacian operator in the Pressure-Poisson equation are investigated for the solution of incompressible Navier-Stokes equations using the non-staggered grid. Laplacian operators previously proposed by other researchers are applied to a Driven-Cavity problem. The computational results are compared with those of Ghia. The results show the characteristics of the discrete Laplacian operators.

• Journal of the Korean Mathematical Society
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• v.34 no.1
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• pp.87-118
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• 1997

This paper is concerned with a mixed Lagrangian formulation of the wiscous, stationary, incompressible Navier-Stokes equations $$ (1.1) -\nu\Delta u + (u \cdot \nabla)u + \nabla_p = f in \Omega $$ and $$ (1.2) \nubla \cdot u = 0 in \Omega $$ along with inhomogeneous Dirichlet boundary conditions on a portion of the boundary $$ (1.3) u = ^{0 on \Gamma_0 _{g on \Gamma_g, $$ where $\Omega$ is a bounded open domain in $R^d, d = 2 or 3$, or with a boundary $\Gamma = \partial\Omega$, which is composed of two disjoint parts $\Gamma_0$ and $\Gamma_g$.

• Proceedings of the Korea Committee for Ocean Resources and Engineering Conference
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• 2004.11a
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• pp.179-182
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• 2004

This paper deals with a new mathematical formulation of nonlinear wave profile based on Banach fixed point theorem. As application of the formulation and its solution procedure, some numerical solutions was presented in this paper and nonlinear equation was derived. Also we introduce a new operator for iteration and getting solution. A numerical study was accomplished with Stokes' first-order solution and iteration scheme, and then we can know the nonlinear characteristic of Stokes' high-order solution. That is, using only Stokes' first-oder(linear) velocity potential and an initial guess of wave profile, it is possible to realize the corresponding high-oder Stokian wave profile with tile new numerical scheme which is the method of iteration. We proved the mathematical convergence of tile proposed scheme. The nonlinear strategy of iterations has very fast convergence rate, that is, only about 6-10 iterations arc required to obtain a numerically converged solution.

• Transactions of the Korean Society of Mechanical Engineers
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• v.19 no.8
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• pp.1950-1963
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• 1995

A modification of the approximate-factorization method is made to accelerate the convergency rate and to take sufficiently large Courant number without loss of accuracy. And a stable implicit finite-difference scheme for solving the incompressible Navier-Stokes equations employed above modified method is developed. In the present implicit scheme, the volume fluxes with contravariant velocity components and the pressure formulation in curvilinear coordinates is adopted. In order to satisfy the continuity condition completely and to remove spurious errors for the pressure, the Navier-Stokes equations are solved by a modified SMAC scheme using a staggered gird. The upstream-difference scheme such as the QUICK scheme is also employed to the right hand side. The implicit scheme is unconditionally stable and satisfies a diagonally dominant condition for scalar diagonal linear systems of implicit operator on the left hand side. Numerical results for some test calculations of the two-dimensional flow in a square cavity and over a backward-facing step are obtained using both usual approximate-factorization method and the modified one, and compared with each other. It is shown that the present scheme allows a sufficiently large Courant number of O(10$^{2}$) and reduces the computing time.

• Bulletin of the Korean Mathematical Society
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• v.58 no.5
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• pp.1109-1127
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• 2021

We study the long-term dynamics for a family of incompressible three-dimensional Leray-α-like models that employ the spectral fractional Laplacian operators. This family of equations interpolates between incompressible hyperviscous Navier-Stokes equations and the Leray-α model when varying two nonnegative parameters 𝜃₁ and 𝜃₂. We prove the existence of a finite-dimensional global attractor for the continuous semigroup associated to these models. We also show that an operator which projects the weak solution of Leray-α-like models into a finite-dimensional space is determining if it annihilates the difference of two "nearby" weak solutions asymptotically, and if it satisfies an approximation inequality.

• 한국전산유체공학회:학술대회논문집
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• 1996.05a
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• pp.131-136
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• 1996

The numerical method for the flow field of a gas atomization process is presented. For the analysis of the compressible supersonic jet flow of a gas. an axisymmetric Navier-Stokes equations are solved using a LU-factored upwind method. The MUSCL type TVD scheme is used for the discretization of inviscid flux, whereas Steger-Warming splitting and LU factorization is applied to the implicit operator. For the validation of the present method, we computed the flow field around the simple gas atomizer proposed by Issac. The numerical results has shown excellent agreement with the experimental data.

• Bulletin of the Korean Mathematical Society
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• v.57 no.3
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• pp.661-676
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• 2020

To understand the interaction of a fluid and a rigid body, we use the concept of B-evolution. Then in a similar way to the usual Navier-Stokes system, we obtain a Helmholtz type decomposition. Using B-evolution theory and the decomposition, we work on the semigroup to analyze the linear part of the system.

• International Journal of Contents
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• v.4 no.2
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• pp.7-12
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• 2008

In this paper, we present a real-time physical simulation model of water surfaces with a novel method to represent the water mass flow in full three dimensions. In a physical simulation model, the state of the water surfaces is represented by a set of physical values, including height, velocity, and the gradient. The evolution of the velocity field in previous works is handled by a velocity solver based on the Navier-Stokes equations, which occurs as a result of the unevenness of the velocity propagation. In this paper, we integrate the principle of the mass conservation in a fluid of equilateral density to upgrade the height field from the unevenness, which in mathematical terms can be represented by the divergence operator. Thus the model generates waves induced by horizontal velocity, offering a simulation that puts forces added in all direction into account when calculating the values for height and velocity for the next frame. Other effects such as reflection off the boundaries, and interactions with floating objects are involved in our method. The implementation of our method demonstrates to run with fast speed scalable to real-time rates even for large simulation domains. Therefore, our model is appropriate for a real-time and large scale water surface simulation into which the animator wishes to visualize the global fluid flow as a main emphasis.

• 한국전산유체공학회:학술대회논문집
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• 2004.03a
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• pp.161-165
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• 2004

In the vortex particle method based on the vorticity-velocity formulation for solving the Wavier-Stokes equations, the unsteady, incompressible, viscous laminar flow over a NACA 0012 foil is simulated. By applying an operator-splitting method, the 'convection' and 'diffusion' equations are solved sequentially at each time step. The convection equation is solved using the vortex particle method, and the diffusion equation using the particle strength exchange(PSE) scheme which is modified to avoid a spurious vorticity flux. The scheme is improved for variety body shape using one image layer scheme. For a validation of the present method, we illustrate the early development of the viscous flow about an impulsively started NACA 0012 foil for Reynolds number 550.