학술논문
Development of Parallel Lattice Boltzmann Method on the 3D Lid-Driven Semi-Elliptical Cavity Flow and Mass-Conserved Boundary Condition
Document Type
Dissertation/ Thesis
Source
Subject
Language
English
Abstract
이 논문은 유체 유동 문제를 시뮬레이션하기 위해 격자 Boltzmann 방법을 구현 한 나의 경험에 대한 포괄적 인 설명이다. 3 차원 뚜껑 구동 반 타원형 유동을 연구하는 데 적용됩니다. 그리고 질량 보전 된 경계 조건은 격자 볼츠만 시뮬레이션에서 제안된다. 시뮬레이션을 가속화하고이 수치 모델의 재사용 가능한 구현을 생성하기 위해 병렬 프로그램 기반 MPI & CUDA 기술이 개발되었습니다. 그 정확성은 몇 가지 벤치 마크 문제로부터 검증되고, 성과는 측정되고 분석됩니다.격자 Boltzmann 시뮬레이션에서, Guo 등이 제안한 속도와 압력 경계 조건에 대해 널리 사용되는 비평 형 외삽 방법. 특정 환경에서 일정한 질량 누출을 일으킬 수 있습니다.특히 외력이 유체 영역에 존재할 때. 이 경계 조건에서, 경계 노드에서의 비평 형 분포함수는 인접 유체 노드에서의 분포 함수의 비평 형 부분으로부터 직접적으로 근사되어 1차 외삽을 이룬다. 경계 노드에서의 거시적 인 속도와 밀도는 같은 외삽 법에 기초하여 얻어진다. 그러면 분포 함수의 상응하는 평형 부분이 명시 적으로 계산 될 수있다. 벽이 없는 슬립 경계에 대해이 방법의 자세한 구현을 분석하고 질량 누설은 인접 유체 노드에서 직접 외삽해서는 안되는 밀도에 대한 외삽 방법의 결과라는 것을 알 수 있습니다. 경계 노드에서 분포 함수의 평형 부분에 대한 밀도 항을 수정하여 질량 보존을 보장 할 수 있는 질량 보전 규칙을 제안한다. 대량 벤치마킹 문제는 대량 보존 경계 조건을 테스트하기 위해 시뮬레이션됩니다.질량 보존 경계법의 제안을 제외하고, 다중 이완 시간 격자 볼츠만 방법을 사용하여, 종횡 대 종횡비 K가 다른 반 타원형 공동에서 뚜껑 구동 흐름을 시뮬레이트한다 0.5 ~2.0이다. 결과의 그리드 독립성이 확립되었습니다. 개발 된 프로그램은 400 및 1000 Reynolds 수의 정사각형 및 입방 형 공동에서 뚜껑 구동 흐름을 시뮬레이션하여 유효성을 검사합니다. 참고로, 2 차원 반 타원형 공동의 뚜껑 구동 흐름이 먼저 연구됩니다. 와류 구조에 대한 레이놀즈 수 Re와 종횡비 K의 효과가 조사되었다. 고정 된 종횡비의 경우, 2 차 와류가 큰 Re로 캐비티의 왼쪽 상단 모서리에 나타나고 Reynolds 수가 증가함에 따라 그 크기가 천천히 확장됩니다. 레이놀즈 수가 충분히 크면 새로운 3 차 와류가 캐비티에서 발생합니다. 그러나 그 위치는 공동의 종횡비에 달려 있습니다. 종횡비가 작으면이 3 차 와류가 캐비티의 하단 왼쪽 모서리에 나타나고 높은 레이놀즈 수의 경우 캐비티의 왼쪽 상단 모서리에서 발생합니다. 이 3 차 와류의 크기와 위치는 미드 플레인의 속도 프로파일에 큰 영향을 미친다. 와류 구조에 중요한 영향을 미치는 것 외에도 레이놀즈 수는 정상 상태에서 진동 상태로의 유체 흐름의 변화에 매우 중요합니다. 이는 호프 분기를 말합니다. 진폭 계수를 정의함으로써 임계 레이놀즈 수는 캐비티 종횡비의 증가에 따라 감소한다는 것을 알 수있다. 동일한 캐비티 종횡비에 대해, 반 타원형 캐비티의 임계 레이놀즈 수는 장방형 캐비티의 레이놀즈 수보다 크다. 낮은 레이놀즈 수에서, 2 차원 모델은 유동 구조를 합리적으로 예측할 수 있지만, 반타원형 캐비티의 측벽 근처에 경계층이 미치는 영향으로 인해 높은 레이놀즈 수에서 3차원 결과와 큰 불일치가 나타난다 . 3 차원 반 타원형 공동에서 뚜껑 구동 흐름의 시뮬레이션을 위해 수평 중심선을 따라 수직 중심선 및 z- 성분을 따르는 x- 성분의 속도프로파일이 얻어지고 2 차원 경우의 결과와 비교된다. 첫 번째 호프 분기점 레이놀즈 수는 구해지고 캐비티 종횡비와 음의 상관 관계가 있습니다. 그리고 그것들은 위의 2 차원반 타원형 캐비티의 임계 값보다 훨씬 낮습니다.
This thesis is a comprehensive account of my experiences implementing the lattice Boltzmann method to simulate fluid flow problems. It’s applied to study the three-dimensional lid-driven semi-elliptical cavity flow. And a mass-conserved boundary condition is proposed in the lattice Boltzmann simulations. To accelerate the simulation and produce reusable implementations of this numerical model, parallel programs based MPI and CUDA techniques are developed. Its accuracy is validated from some benchmark problems, and the performances are measured and analyzed.In lattice Boltzmann simulations, the widely used non-equilibrium extrapolation method for velocity and pressure boundary conditions proposed by Guo et al. can produce constant mass leakage in certain circumstances, especially when external force field exists in the fluid domain. In this boundary condition, the non-equilibrium distribution function at boundary node is approximated directly from the non-equilibrium part of distribution function at the adjacent fluid node with a first-order extrapolation. And the macroscopic velocity and density at boundary node are obtained based on the same extrapolation method. Then the corresponding equilibrium part of distribution function can be calculated explicitly. For no-slip wall boundary, we analyze the detailed implementation of this method, and find that the mass leakage results from the extrapolation method for density, which shouldn’t be extrapolated directly from the adjacent fluid node. Then we propose a mass-conserved rule to revise the density term for the equilibrium part of distribution function at boundary node, which can guarantee the mass conservation. Several benchmark problems are simulated to test our mass-conserved boundary condition.Except for the proposal of mass-conserved boundary method, the multiple-relaxation-time lattice Boltzmann method is adopted to simulate the lid-driven flow in semi-elliptical cavities with different vertical-to-horizontal aspect ratio K, which is in the range of 0.5~2.0. Grid-independence of the results is established. The developed program is also validated by simulating the lid-driven flow in square and cubic cavities at 400 and 1000 Reynolds number. As a reference, the lid-driven flow in two-dimensional semi-elliptical cavities is firstly studied. The effects of Reynolds number Re and aspect ratio K on the vortex structure are investigated. For a fixed aspect ratio, a secondary vortex appears in the upper-left corner of the cavity at a large Re and its size expands slowly as Reynolds number increases. When the Reynolds number is large enough, a new tertiary vortex will arise in the cavity. But its position depends on the aspect ratio of cavity. When aspect ratio is small, this tertiary vortex appears at the bottom-left corner of cavity, and it will arise at the upper-left corner of cavity for high Reynolds number. The size and position of this tertiary vortex have great effects on the velocity profiles of midplanes. In addition to the significant effect on vortex structure, the Reynolds number is critically important for the transition of fluid flow from steady state to oscillatory state, which refers to the Hopf bifurcation. By defining an amplitude coefficient, we find that the critical Reynolds number decreases with the increase of the cavity aspect ratio. For same cavity aspect ratio, the critical Reynolds number of semi-elliptical cavity is larger than that of rectangular cavity.At low Reynolds numbers, the two-dimensional models may predict the flow structures reasonably well, but significant discrepancy with three-dimensional results appear at high Reynolds numbers, which results from the effect of boundary layers near the side walls of the semi-elliptical cavity. For the simulation of lid-driven flow in three-dimensional semi-elliptical cavities, the velocity profiles of x-component along the vertical centerline and z-component along the horizontal centerline are obtained, and compared with those results of two-dimensional cases. The first Hopf bifurcation Reynolds numbers are obtained, which correlate negatively with the cavity aspect ratio. And they are much lower than the critical values from two-dimensional semi-elliptical cavity above.
This thesis is a comprehensive account of my experiences implementing the lattice Boltzmann method to simulate fluid flow problems. It’s applied to study the three-dimensional lid-driven semi-elliptical cavity flow. And a mass-conserved boundary condition is proposed in the lattice Boltzmann simulations. To accelerate the simulation and produce reusable implementations of this numerical model, parallel programs based MPI and CUDA techniques are developed. Its accuracy is validated from some benchmark problems, and the performances are measured and analyzed.In lattice Boltzmann simulations, the widely used non-equilibrium extrapolation method for velocity and pressure boundary conditions proposed by Guo et al. can produce constant mass leakage in certain circumstances, especially when external force field exists in the fluid domain. In this boundary condition, the non-equilibrium distribution function at boundary node is approximated directly from the non-equilibrium part of distribution function at the adjacent fluid node with a first-order extrapolation. And the macroscopic velocity and density at boundary node are obtained based on the same extrapolation method. Then the corresponding equilibrium part of distribution function can be calculated explicitly. For no-slip wall boundary, we analyze the detailed implementation of this method, and find that the mass leakage results from the extrapolation method for density, which shouldn’t be extrapolated directly from the adjacent fluid node. Then we propose a mass-conserved rule to revise the density term for the equilibrium part of distribution function at boundary node, which can guarantee the mass conservation. Several benchmark problems are simulated to test our mass-conserved boundary condition.Except for the proposal of mass-conserved boundary method, the multiple-relaxation-time lattice Boltzmann method is adopted to simulate the lid-driven flow in semi-elliptical cavities with different vertical-to-horizontal aspect ratio K, which is in the range of 0.5~2.0. Grid-independence of the results is established. The developed program is also validated by simulating the lid-driven flow in square and cubic cavities at 400 and 1000 Reynolds number. As a reference, the lid-driven flow in two-dimensional semi-elliptical cavities is firstly studied. The effects of Reynolds number Re and aspect ratio K on the vortex structure are investigated. For a fixed aspect ratio, a secondary vortex appears in the upper-left corner of the cavity at a large Re and its size expands slowly as Reynolds number increases. When the Reynolds number is large enough, a new tertiary vortex will arise in the cavity. But its position depends on the aspect ratio of cavity. When aspect ratio is small, this tertiary vortex appears at the bottom-left corner of cavity, and it will arise at the upper-left corner of cavity for high Reynolds number. The size and position of this tertiary vortex have great effects on the velocity profiles of midplanes. In addition to the significant effect on vortex structure, the Reynolds number is critically important for the transition of fluid flow from steady state to oscillatory state, which refers to the Hopf bifurcation. By defining an amplitude coefficient, we find that the critical Reynolds number decreases with the increase of the cavity aspect ratio. For same cavity aspect ratio, the critical Reynolds number of semi-elliptical cavity is larger than that of rectangular cavity.At low Reynolds numbers, the two-dimensional models may predict the flow structures reasonably well, but significant discrepancy with three-dimensional results appear at high Reynolds numbers, which results from the effect of boundary layers near the side walls of the semi-elliptical cavity. For the simulation of lid-driven flow in three-dimensional semi-elliptical cavities, the velocity profiles of x-component along the vertical centerline and z-component along the horizontal centerline are obtained, and compared with those results of two-dimensional cases. The first Hopf bifurcation Reynolds numbers are obtained, which correlate negatively with the cavity aspect ratio. And they are much lower than the critical values from two-dimensional semi-elliptical cavity above.