• 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.

• 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.

• Transactions of the Korean Society of Mechanical Engineers B
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• v.21 no.3
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• pp.392-404
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• 1997

The flow induced by the oscillatory motion of a solid body is important in a number of practical problems. As the solid boundary oscillates harmonically, there is steady streaming motion invoked by the Reynolds stresses, which could cause extensive migration of the fluid during a period of fluid motion. We here analyzed the flow in a square cavity with an oscillating top wall for the parameters which make the time derivatives and the convective terms equally important in the entire cavity flow. The full Navier-Stokes equations are solved by the second-order time accurate Momentum Coupling Method which is devised by the authors. The particular numerical scheme does not need subiteration at each time step which is usually a required process to calculate the incompressible Navier-Stokes equations. The effect of two parameters, the Reynolds number and the frequency parameter, on the oscillatory flow has been investigated.

• 한국전산유체공학회:학술대회논문집
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• 2003.10a
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• pp.33-34
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• 2003

• 한국전산유체공학회:학술대회논문집
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• 2000.05a
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• pp.66-72
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• 2000

Aerodynamic sensitivity analysis is performed for the Navier-Stokes equations coupled with two-equation turbulence models using a discrete adjoint method and a direct differentiation method respectively. Like the mean flow equations, the turbulence model equations are also hand-differentiated to accurately calculate the sensitivity derivatives of flow quantities with respect to design variables in turbulent viscous flows. Both the direct differentiation code and the adjoint variable code adopt the same time integration scheme with the flow solver to efficiently solve the differentiated equations. The sensitivity codes are then compared with the flow solver in terms of solution accuracy, computing time and computer memory requirements. The sensitivity derivatives obtained from the sensitivity codes with different turbulence models are compared with each other. Using two-equation turbulence models, it is observed that a usual assumption of constant turbulent eddy viscosity in adjoint methods may lead to seriously inaccurate results in highly turbulent flows.

• Journal of computational fluids engineering
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• v.7 no.4
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• pp.19-27
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• 2002

An efficient implicit multigrid method is presented for the Navier-Stokes and k-ω turbulence equations. Freezing and limiting strategies are applied to improve the robustness and convergence of the multigrid method. The eddy viscosity and strongly nonlinear production terms of turbulence are frozen in the coarser grids by passing down the values without update of them. The turbulence equations together with the Navier-Stokes equations, however, are consecutively solved on the coarser grids in a loosely coupled fashion. A simple limit for k is also introduced to circumvent slow-down of convergence. Numerical results for the unseparated and separated transonic airfoil flows show that all computations converge well without any robustness problem and the computing time is reduced to a factor of about 3 by the present multigrid method.

• Proceedings of the KSME Conference
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• 2002.08a
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• pp.35-38
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• 2002

The convergence acceleration methods for the compressible Wavier-Stokes equations are studied ,which are multigrid method and implicit preconditioned multistage time stepping method. In this paper, the performance of implicit preconditioning methods are studied for the full-coarsening multigrid methods on the high Reynolds number compressible flow computations. The effect of numerical flux on the convergence are investigated for the inviscid and viscous calculations.

• Journal of the Society of Naval Architects of Korea
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• v.48 no.6
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• pp.581-591
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• 2011

This paper describes the air compressibility effect in the CFD simulation of water impact load prediction. In order to consider the air compressibility effect, two sets of governing equations are employed, namely the incompressible Navier-stokes equations and compressible Navier-Stokes equations that describe general compressible gas flow. In order to describe violent motion of free surface, volume-of-fluid method is utilized. The role of air compressibility is presented by the comparative study of water impact load obtained from two different air models, i.e. the compressible and incompressible air. For both cases, water is considered as incompressible media. Compressible air model shows oscillatory behavior of pressure on the solid surface that may attribute to the air-cushion effect. Incompressible air model showed no such oscillatory behavior in the pressure history. This study also showed that the CFD simulation can capture the formation of air pockets enclosed by water and solid surface, which may be the location where the air compressibility effect is dominant.

• Journal of computational fluids engineering
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• v.10 no.1
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• pp.67-72
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• 2005

This research is based on the iterative space-marching method for incompressible and compressible Navier-Stokes equations[1-4]. A principle of parallel computational schemes construction for steady and unsteady problems is suggested. It is analytically proven that convergence of these schemes is unconditional for incompressible case. When the parallel scheme is used the total volume of computations is the sum of a large number of independent and equal parts. Estimation of the speed-up K shows that K > 1000 in ideal case. First results of using the parallel schemes are presented.

• Journal of computational fluids engineering
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• v.8 no.4
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• pp.1-5
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• 2003

A Thin-layer Wavier-Stokes equations are applied for the hypersonic flow over blunt bodies with applications to laminar as well as turbulent flows. The equations are expressed in the forms of flux-vector splitting and explicit algorithm. The upwind schemes of Steger-Warming and Van Leer are investigated to predict accurately the heating loads along the surface of the body. A mixed scheme has been presented for the differencing the convective terms and the mixed scheme is found to be less dissipative producing accurate solutions.