• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.19 no.2
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• pp.103-121
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• 2015

In this paper, we briefly review and describe a projection algorithm for numerically computing the two-dimensional time-dependent incompressible Navier-Stokes equation. The projection method, which was originally introduced by Alexandre Chorin [A.J. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comput., 22 (1968), pp. 745-762], is an effective numerical method for solving time-dependent incompressible fluid flow problems. The key advantage of the projection method is that we do not compute the momentum and the continuity equations at the same time, which is computationally difficult and costly. In the projection method, we compute an intermediate velocity vector field that is then projected onto divergence-free fields to recover the divergence-free velocity. Numerical solutions for flows inside a driven cavity are presented. We also provide the source code for the programs so that interested readers can modify the programs and adapt them for their own purposes.

• 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.

• Transactions of the Korean Society of Mechanical Engineers
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• v.16 no.10
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• pp.1915-1927
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• 1992

A three-dimensional Navier-Stokes code has been developed for analysis of viscous flows through turbomachinery blade rows or other internal passages. The Navier-Stokes equations are written in a cartesian coordinate system, then mapped to a general body-fitted coordinate system. Streamwise viscous terms are neglected and turbulent effects are modeled using the baldwin-Lomax model. Equations are discretized using finite difference method on the stacked C-type grids and solved using LU-ADI decomposition scheme. calculations are made for a two-dimensional cascade in a transonic wind-tunnel to see the infuence of the endwalls. The flow pattern of the three-dimensional flow near the endwall is found to be different from that of the two-dimensional flow due to the existence of the endwalls.

• Journal of the Korean Mathematical Society
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• v.50 no.5
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• pp.1129-1163
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• 2013

Based on finite element discretization, two linearization approaches to the defect-correction method for the steady incompressible Navier-Stokes equations are discussed and investigated. By applying $m$ times of Newton and Picard iterations to solve an artificial viscosity stabilized nonlinear Navier-Stokes problem, respectively, and then correcting the solution by solving a linear problem, two linearized defect-correction algorithms are proposed and analyzed. Error estimates with respect to the mesh size $h$, the kinematic viscosity ${\nu}$, the stability factor ${\alpha}$ and the number of nonlinear iterations $m$ for the discrete solution are derived for the linearized one-step defect-correction algorithms. Efficient stopping criteria for the nonlinear iterations are derived. The influence of the linearizations on the accuracy of the approximate solutions are also investigated. Finally, numerical experiments on a problem with known analytical solution, the lid-driven cavity flow, and the flow over a backward-facing step are performed to verify the theoretical results and demonstrate the effectiveness of the proposed defect-correction algorithms.

• Journal of the Korean Society for Aeronautical & Space Sciences
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• v.35 no.6
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• pp.474-482
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• 2007

An analysis is made for flow and rocket motion during a supersonic separation stage of an air-launching rocket(ALR) from the mother plane. Three-dimensional compressible Navier-Stokes equations are numerically solved to analyze the steady/unsteady flow fields around the rocket which is being separated from the mother plane configuration(F-4E Phantom). Simulation results clearly demonstrate the effect of shock-expansion wave interaction around both of the rocket and the mother plane. To predict the behavior of the ALR by the change of the center-of-gravity, three cases of numerical analysis are performed. As a result, a design-guideline of supersonic air-launching rockets for safe separation is proposed.

• Journal of Navigation and Port Research
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• v.28 no.8
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• pp.753-757
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• 2004

The free-surface flow is simulated to make clear the viscous interaction of stem waves and the sub-breaking phenomena around a high speed vehicle. The Navier-Stokes equation is solved by a finite difference method where the body-fitted coordinate system, the wall function and the triple-grid system are invoked They are applied to study precisely on the stem flow of S-103 as to which extensive experimental data are available. Computations are extended to the submerged revolutional body. The numerical result shows that the gradient of M/Us is greatly influenced by the submerged depth And the stem wave is influenced by the separation due to the bow wave.

• 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.

• 한국전산유체공학회:학술대회논문집
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• 2009.11a
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• pp.27-33
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• 2009

In this paper, the flow past a rotary oscillating circular cylinder is simulated. The high-order and high-resolution numerical schemes with the characteristic boundary conditions are used for the compressible Navier-Stokes equation. The frequencies of rotating oscillation are $0.19\;{\leq}\;S_f\;{\leq}\;0.25$ for the maximum angular $\theta_{max}=10^{\circ}$ and $17^{\circ}$. The flow conditions are Mach number of 0.3 and Reynolds number of 1000. At Lock-on and Non-lock-on region which are defined by the relation between the vortex shedding frequency and the oscillating frequency, the drag and lift coefficient are analyzed.

• 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.

• 유체기계공업학회:학술대회논문집
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• 2006.08a
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• pp.423-426
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• 2006

An analysis is made of flow and rocket motion during a supersonic separation stage of air-launching rocket(ALR) from the mother plane. Three-dimensional compressible Navier-Stokes equations is numerically solved to analyze the steady/unsteady flow field around the rocket which is being separated from the mother plane configuration(F-4E Phantom). The simulation results clearly demonstrate the effect of shock-expansion wave interaction between the rocket and the mother plane. To predict the behavior of the ALR according to the change of the C.G., three cases of numerical analysis are performed. As a result, a design-guideline of supersonic air-launching rocket for the safe separation is proposed.