• Title, Summary, Keyword: Navier-Stokes equations

### DERIVATION OF THE g-NAVIER-STOKES EQUATIONS

• Roh, Jaiok
• Journal of the Chungcheong Mathematical Society
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• v.19 no.3
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• pp.213-218
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• 2006
• The 2D g-Navier-Stokes equations are a certain modified Navier-Stokes equations and have the following form, $$\frac{{\partial}u}{{\partial}t}-{\nu}{\Delta}u+(u{\cdot}{\nabla})u+{\nabla}p=f$$, in ${\Omega}$ with the continuity equation ${\nabla}{\cdot}(gu)=0$, in ${\Omega}$, where g is a suitable smooth real valued function. In this paper, we will derive 2D g-Navier-Stokes equations from 3D Navier-Stokes equations. In addition, we will see the relationship between two equations.

### Temperature Preconditioning for Improving Convergence Characteristics in Calculating Low Mach Number Flows, II: Navier-Stokes Equations (저속 유동 계산의 수렴성 개선을 위한 온도예조건화 II: 나비어스톡스 방정식)

• Lee, Sang-Hyeon
• Journal of the Korean Society for Aeronautical & Space Sciences
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• v.35 no.12
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• pp.1075-1081
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• 2007
• The temperature preconditioning is applied to the Navier-Stokes equations. Also, a new concept of diffusion Mach numbers is introduced to modify the reference Mach number for the Navier-Stokes equations. Flows over a circular cylinder were calculated at different Reynolds numbers. It is shown that the temperature preconditioning improves the convergence characteristics of Navier-Stokes equations. Also, it is shown that the modified reference Mach number alleviates the convergence problems at locally low speed regions.

### GEOMETRY OF L2(Ω, g)

• Roh, Jaiok
• 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 $L^2_{per}({\Omega})$.

### THE GLOBAL ATTRACTOR OF THE 2D G-NAVIER-STOKES EQUATIONS ON SOME UNBOUNDED DOMAINS

• Kwean, Hyuk-Jin;Roh, Jai-Ok
• Communications of the Korean Mathematical Society
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• v.20 no.4
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• pp.731-749
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• 2005
• In this paper, we study the two dimensional g-Navier­Stokes equations on some unbounded domain ${\Omega}\;{\subset}\;R^2$. We prove the existence of the global attractor for the two dimensional g-Navier­Stokes equations under suitable conditions. Also, we estimate the dimension of the global attractor. For this purpose, we exploit the concept of asymptotic compactness used by Rosa for the usual Navier-Stokes equations.

### LARGE TIME BEHAVIOR FOR THE DEGENERATE COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DAMPING

• Kwon, Young-Sam
• Bulletin of the Korean Mathematical Society
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• v.56 no.3
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• pp.555-561
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• 2019
• In this paper we consider degenerate compressible Navier-Stokes equations giving rise to a variety of mathematical problems in many areas. We study the long time behavior for the degenerate compressible Navier-Stokes equations with damping.

### INCOMPRESSIBLE NAVIER-STOKES EQUATIONS IN HETEROGENEOUS MEDIA

• Pak, Hee Chul
• Journal of the Chungcheong Mathematical Society
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• v.19 no.4
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• pp.335-347
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• 2006
• The homogenization of non-stationary Navier-Stokes equations on anisotropic heterogeneous media is investigated. The effective coefficients of the homogenized equations are found. It is pointed out that the resulting homogenized limit systems are of the same form of non-stationary Navier-Stokes equations with suitable coefficients. Also, steady Stokes equations as cell problems are identified. A compactness theorem is proved in order to deal with time dependent homogenization problems.

### COMPARISON OF COUPLING METHODS FOR NAVIER-STOKES EQUATIONS AND TURBULENCE MODEL EQUATIONS (Navier-Stokes 방정식과 난류모델 방정식의 연계방법 비교)

• Lee, Seung-Soo;Ryu, Se-Hyun
• 한국전산유체공학회:학술대회논문집
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• pp.111-116
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• 2005
• Two coupling methods for the Navier-Stokes equations and a two-equation turbulence model equations are compared. They are the strongly coupled method and the loosely coupled method. The strongly coupled method solves the Navier-Stokes equations and the two-equation turbulence model equations simultaneously, while the loosely coupled method solves the Navier-Stokes equation with the turbulence viscosity fixed and subsequently solves the turbulence model equations with all the flow quantities fixed. In this paper, performances of two coupling methods are compared for two and three-dimensional problems.

### CONVERGENCE OF THE NEWTON'S METHOD FOR AN OPTIMAL CONTROL PROBLEMS FOR NAVIER-STOKES EQUATIONS

• Choi, Young-Mi;Kim, Sang-Dong;Lee, Hyung-Chun
• Bulletin of the Korean Mathematical Society
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• v.48 no.5
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• pp.1079-1092
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• 2011
• We consider the Newton's method for an direct solver of the optimal control problems of the Navier-Stokes equations. We show that the finite element solutions of the optimal control problem for Stoke equations may be chosen as the initial guess for the quadratic convergence of Newton's algorithm applied to the optimal control problem for the Navier-Stokes equations provided there are sufficiently small mesh size h and the moderate Reynold's number.

### ANALYSIS AND COMPUTATIONS OF OPTIMAL AND FEEDBACK CONTROL PROBLEMS FOR NAVIER-STOKES EQUATIONS

• Lee, Hyung-Chun
• Journal of the Korean Mathematical Society
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• v.34 no.4
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• pp.841-857
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• 1997
• We present analysis and some computational methods for boundary optimal and feedback control problems for Navier-Stokes equations. We use one example to illustrate our methodology and ideas which are applicable to general control problems for Navier-Stokes equations. First, we discuss the existence of optimal solutions and derive an optimality system of equations from which an optimal solution may be computed. Then we present a gradient type iterative method. Finally, we present some numerical results.

### Numerical Simulation of Overtopping of Cnoidal Waves on a Porous Breakwater Using the Boussinesq Equations: Comparison with Solutions of the Navier-Stokes Equations (Boussinesq 식을 사용하여 Cnoid 파의 투수방파제 월파 해석: Navier-Stokes 식 결과와 비교)

• Huynh, Thanh Thu;Lee, Changhoon;Ahn, Suk Jin
• 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.
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