• Journal of Korean Society of Coastal and Ocean Engineers
• /
• v.30 no.4
• /
• pp.143-152
• /
• 2018

Numerical convergence and stability of TVD schemes have been applied in the FUNWAVE-TVD model were compared. The fourth order accurate MUSCL-TVD scheme using minmod limiter suggested by Yamamoto and Daiguji (1993), the fourth order accurate MUSCL-TVD scheme using van-Leer limiter suggested by Erduran et al. (2005) and the second order accurate MUSCL-TVD scheme using van-Leer limiter in Zhou et al. (2001) were compared. Comparisons of the numerical scheme were conducted with experimental data of Vincent and Briggs irregular wave experiments. In comparison with the fourth order accurate scheme using van-Leer limiter, the fourth order accurate scheme using minmod limiter is less dissipative but required lower CFL condition for stable numerical solution. On the other hand, the scheme using van-Leer limiter required smaller resolution spatial grid due to numerical dissipation, but relatively higher CFL condition can be used compared to the scheme using minmod limiter. In the breaking wave experiments which were conducted using high resolution spatial grid to reduce numerical dissipation, the characteristic of the schemes can be clearly observed. Numerical instabilities and blow-up of the numerical solutions were found in the irregular wave breaking simulation with the scheme using minmod limiter. However, the simulation can be completed with the scheme using van-Leer limiter, but required low CFL condition. Good agreements with the observed data were also observed in the results using van-Leer limiter.

• Journal of the Korean Mathematical Society
• /
• v.54 no.1
• /
• pp.141-175
• /
• 2017

We present a high-resolution van Leer-type numerical scheme for the isentropic model of fluid flows in a nozzle with variable cross-section. Basically, the scheme is an improvement of the Godunov-type scheme. The scheme is shown to be well-balanced, as it can capture exactly equilibrium states. Numerical tests are conducted which include comparisons between the van Leer-type scheme and the Godunov-type scheme. It is shown that the van Leer-type scheme achieves a very good accuracy for initial data belong to both supersonic and supersonic regions, and the exact solution eventually possesses a resonant phenomenon.

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

• Journal of the Society of Naval Architects of Korea
• /
• v.58 no.3
• /
• pp.182-190
• /
• 2021

Computational simulations of turbulent flows around a model ship have been performed to investigate an influence of TVD schemes on the accuracy of advective terms associated with ship resistances. Several TVD schemes including upwind, second-order upwind, vanLeer, and QUICK as well as a nonTVD linear scheme were studied by examining temporal and spatial characteristics of accuracy transition in adjacent cells to the hull. Even though vanLeer scheme was the most accurate among TVD schemes in both structured and unstructured grid systems, the ratio of accuracy switch from 2^nd order to 1^st order in vanLeer scheme was considerable compared with the 2^nd order linear scheme. Also, the accuracy transition was observed to be overally scattered in the unstructured grid while the accuracy transition in the structured grid appeared relatively clustered. It concluded that TVD schemes had to be carefully used in computational simulations of turbulent flows around a model ship due to the loss of accuracy despite its attraction of numerical stability.

• Proceedings of the KSME Conference
• /
• 2001.06e
• /
• pp.311-316
• /
• 2001

The thin-layer Navier-Stokes equations are solved for the hypersonic flow over blunt cone configurations 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 in their ability to accurately predict the heating loads along the surface of the body. A comparison with the second order extensions of these schemes is made and a hybrid scheme incorporating a combination of central differencing and flux-vector-splitting is presented. This scheme is also investigated in its ability to accurately predict heat transfer distributions.

• Journal of computational fluids engineering
• /
• v.9 no.3
• /
• pp.39-48
• /
• 2004

In the present study, an efficient implementation technique of the van Leer's implicit operator is suggested in accordance with the Roe's explicit operator. By using an efficient treatment of the off-diagonal terms, which occupy most of the memory requirement for the linear system of equations, it is shown that the improved scheme only requires less than 30% of memory and is approximately 10-20% faster than the baseline scheme.

• Journal of the Korean Society for Aeronautical & Space Sciences
• /
• v.34 no.8
• /
• pp.47-54
• /
• 2006

In this paper, numerical algorithms applied in two-step eulerian scheme are investigated and implemented. Element quantities are advected with donor or van Leer algorithm. Nodal quantities are advected with Simplifed ALE [SALE] algorithm. Material interfaces are determined with Simple Line Interface Calculation [SLIC] algorithm. Practical aspects considered for code development are addressed in detail. The results of developed two-step Eulerian code are verified by comparing with those from pure lagrangian scheme and commercial code.

• Journal of computational fluids engineering
• /
• v.10 no.1
• /
• pp.99-106
• /
• 2005

Performance of first, second and third order accurate methods for calculation of in viscid fluxes in fluid flow governing equations are investigated here. For the purpose, an upwind method based on Roe's scheme is used to solve 2-dimensional Euler equations. To increase the accuracy of the method two different schemes are applied. The first one is a second and third order upwind-based algorithm with the MUSCL extrapolation Van Leer (1979), based on primitive variables. The other one is an upwind-based algorithm with the Chakravarthy extrapolation to the fluxes of mass, momentum and energy. The results show that the thickness of shock layer in the third order accuracy is less than its value in second order. Moreover, applying limiter eliminates the oscillations near the shock while increases the thickness of shock layer especially in MUSCL method using Van Albada limiter.

• Journal of the Korean Society for Aeronautical & Space Sciences
• /
• v.31 no.2
• /
• pp.10-16
• /
• 2003

The unsteady supersonic flow over tandem cavities has been analyzed by the integration of Navier-Stokes equations with the k-$\varepsilon$ turbulence model. The unsteady flow is characterized by the periodicity due to the mutual relation between the shear layer and the internal flow in cavities. The upwind TVD scheme based on the flux vector split with the van Leer limiters is used. The results show the principal frequency is very reasonable. The principal frequency of the rear cavity due to the front cavity has been analyzed by the combination of the several aspect ratios of cavities. In the case of the front cavity of low aspect ratio, the frequencies of tandem cavities are almost same, because two shear layers developed from each cavity are mixed and developed to one shear layer. However, in the case of the front cavity of high aspect ratio, the characteristis of frequency are very different, because the second shear layer is developed in the diffused first shear layer.

• Proceedings of the Korea Water Resources Association Conference
• /
• 2011.05a
• /
• pp.148-148
• /
• 2011

Riemann 문제는 천수방정식과 같은 쌍곡선형 방정식과 단일한 도약에 의해 불연속인 어떤 점의 좌 우에서 상수인 자료로 구성되는 초기치 문제로서 그 해법은 Godunov 방법과 같이 정확해에 의하면 정확 Riemann 해법, 근사 기법에 의하면 근사 Riemann 해법으로 불린다. 지금까지 이용되는 근사 Riemann 해법으로는 1981년에 P. L. Roe가 제안한 Roe의 선형화 기법과 1983년에 A. Harten, P. D. Lax, 그리고 B. van Leer가 제안한 HLL 기법의 수정 기법들이다. 최대 및 최소 파속만 고려하는 것으로 알려진 HLL 기법은 1988년에 B. Einfeldt의 제안에 의해 두 파속의 결정에서 Roe의 선형화 기법에 따른 고유치와 비교하는 것으로 수정되었다(HLLE 기법). 또한, 1994년에 E. F. Toro 등은 접촉파를 고려하기 위해 선형화된 지배방정식의 정확해로부터 중앙 파속을 고려하는 기법을 제안하였고, 이를 HLLC 기법으로 불렀다. 2002년에 T. Linde는 중앙 파속을 평가하기 위해 일반화된(수학적) 엔트로피 함수를 도입하였으며, van Leer는 이를 HLLL 기법으로 불렀다. 이 기법에서는 접촉파의 평가를 위해 보존변수에 대한 일반화된 엔트로피 함수로부터 중앙 파속이 유도되며, 이것과 특성 속도의 비교를 통해 최대 및 최소 파속이 결정된다. 따라서 이 기법에서는 모든 파속이 초기치로부터 결정되므로 HLLE 기법과 달리 Roe의 선형화 기법과 완전히 결별되고 HLLC 기법과 달리 정확해에 의존되지 않는 점에서 HLLL 기법은 모태인 HLL 기법의 온전한 계승으로 볼 수 있다. HLLL 기법은 여러 분야에 적용된 바 있으나, 수공학 분야에 적용된 사례는 알려진 바 없다. 이는 천수방정식에 대한 (물리적) 엔트로피 함수가 명확하지 않기 때문인 것으로 보인다. 이 연구에서는 보존변수로부터 정의되는 총 에너지를 일반화된 엔트로피 함수로 간주하여 모형을 구성하고, 정확해가 알려진 1차원 문제에 대해 적용성을 검토하였다. 정확해가 알려진 경우에 대해 모의한 결과, 1차 정도 수치해의 한계에도 불구하고, HLLL 기법의 결과는 대체로 정확해와 잘 일치하였으며 그 외의 HLL-형 기법의 그것에 비해 우수한 것으로 나타났다. 특히, 물이 빠져 바닥이 드러나는 상태에 대한 접촉 파속의 추정에서 Riemann 불변량을 이용하는 HLLC 기법에 비해 물이 빠지는 전선을 더 정확하게 포착하는 HLLL 기법의 결과는 매우 고무적이었다.