SPH基本问题研究及其在高速水下物体流场模拟中的应用
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摘要
光滑粒子流体动力学(SPH)中,物质域由具有质量和体积的点(粒子)离散;跟踪每个粒子的运动,可得到物质构型的变化。因此计算介质大变形时,可避免基于网格算法常遇到的网格畸变。SPH常采用基于密度的求解过程,能方便处理连续介质力学中高速、超高压情况。
高速水下物体,会导致超空穴。基于Euler描述研究空穴流的方法,主要包括势流理论和“流体输运方程结合两类空化模型”的方法。势流方法,不考虑空穴内汽体;这是因为汽体质量太小,其惯性对水流动力学条件影响很小。基于势流中边界元或面源法需要数值迭代得到空穴界面。第二种方法,考虑汽体与水流关系,因此能揭示更多空穴流图景,也需要数值迭代得到空穴界面;该类方法常基于压力的求解过程,处理速度尺度为1000 m·s~(-1)的高亚声速物体驱动的显著压缩流动,没有基于密度求解过程的算法方便。
空穴表面是物质界面。界面在SPH中是这些界面上的粒子集合。因此SPH可自然跟踪物质界面。因此,若SPH能计算高速水下物体驱动的流动,则它可能会自然得到空穴发展。其基于密度求解的过程也能方便处理显著压缩流动。因此探索这样的工作是有必要的。本文面向于自编程实现SPH计算100 m·s~(-1)低亚声速和1000m·s~(-1)高亚声速的水下物体驱动的流动(简称高速驱动流动),并相应研究SPH几个基本问题——稳定性、固壁条件施加、后处理及空穴表面提取。
本文的高速驱动流动计算模型中,固壁多,固壁折角多,运动与静止固壁位置关系复杂;这要求对复杂形状固壁,固壁条件施加算法统一、简便,否则程序通用性大大降低。而SPH对函数的近似在边界处被截断,即边界缺陷需要在施加边界条件时得到改善。但能改善该缺陷的镜像虚粒子法(Ghost Particle Method)与仿粒子法(Dummy Particle Method),对不同形状固壁,施加算法没有统一性,对复杂形状固壁,算法也不简便。本文提出简便性与通用性改善虚粒子方法(CUI-GPM),对折角和弯曲固壁,算法统一;并用于流动和热传导计算。高速驱动流动中,空穴导致流体与物体发生很大分离,应用CUI-GPM时,无法准确计算空穴回射流。于是提出DPM与GPM结合方法:DPM施加无滑移条件,物体速度直接赋予虚粒子;采用GPM思路,虚粒子的密度更新和流体相同;由此计算得到较符合实际的回射流。
为提取SPH所得空穴界面以与空穴的理论形状对比,研究了SPH的后处理。SPH传统后处理方法,无法直接得到连续云图、等值线、流线,以及微积分运算和切片等,限制了其结果显示。本文考虑将粒子集网格化形成三角单元集,将粒子作为节点,利用FEM后处理技术实现SPH后处理与成熟后处理技术的链接。Delaunay三角化提供粒子集的网格化技术,但它在非凸区域粒子集上会得到不包含质量的空白单元。本文提出所谓“单元称重”法去除空白单元,保留下的单元作为后处理的有限单元。基于上述思路,可方便提取自由表面和超空穴界面。
算法稳定性研究及其结果,为高速驱动流动计算提供参考依据。本文引入SPH矩阵格式研究作为粒子集的SPH稳定性。研究表明,张力不稳定的原因是采用空间坐标来标记粒子位置[72];也得到了Swegle关于张力不稳定性的充分条件。张力稳定必要条件,要求光滑长度因子(光滑长度比粒子间距)取在光滑函数一阶导数极值点;该取值同时也使光滑函数Fourier变换分量达到极大,同时也是光滑函数对光滑长度的极大值点。
稳定性分析得到了满足张力稳定必要条件的数值声速。稳定必要条件指出应选择与流动相契合的物理模型,否则计算出现虚假压缩率。因此在高速驱动流动计算中,为水选择了合适的物态方程。对稳定必要条件的分析得知,误差频率应远小于时间积分频率,并据此得到了CFL条件的Courant数,为时间步长取值提供了依据。根据该必要条件,还得到流动压缩率和“误差频率与时间积分频率比值”的关系,并计算得到Monaghan[10]与Morris[95]提出的微可压流动的计算保持稳定时压缩率的变化范围,为考察高速驱动流动的计算稳定与合理,提供了参考。
基于以上研究结果并应用于高速驱动流动的计算,本文得到了稳定的计算过程,流动压缩率保持在合理范围。SPH所得空穴、与独立膨胀原理和实验所得空穴形状一致,表明SPH计算高速驱动流动的适用性。计算显示,物体在水中启动时空穴尾部有明显回射流;而从发射筒射入水时,空穴尾部回射流较小,尾部界面近似垂直在物体表面。而物体后体的存在,使空穴相对变小。高亚声速与低亚声速下的空穴形态发展表明,显著压缩流动使空穴在长度方向不对称,并使空穴尺寸比流动不可压时更大,这符合既有的理论结果。
本文实现了SPH计算高速水下物体流场的工作。相应提出的固壁条件施加算法,为利用SPH自编程计算“含较复杂形状的固壁”的问题提供了途径。相应提出的后处理方法,为SPH形成通用后处理器提供了可供实用的途径。算法稳定性研究,给出了时间步长、光滑长度取值及物理模型选择等方面的理论依据。
Smoothed Particle Hydrodynamics (SPH) discretizes the configuration by a set of particles with the properties of mass and volume. By tracing the motion of each particle, the variation history of the configuration could be derived. The mesh distortion, which occurs usually in the methods based on girds when they are applied to compute the large deformation in continuum mechanics, is thus avoided in SPH because of its meshless property. It also masters the high speed and hyperpressure problems in continuum mechanis well with a density-based algorithm.
Supercavities are induced for the high speed underwater moving bodies. Two classes of methods based on the Euler formulation, i.e. potential theory and transportation equations with two kinds of cavitation models, are the two main ways for studying the supercavity flows. The potential theory neglects the vapor in the supercavity, because the mass in the cavity is so tiny that its inertial applies little effect on the dynamics of the water flow. Massive iterations are performed to seek the cavity surface in this method. The relationship between the vapor and water flows is included in the second type methods and it thus reveals much more characteristics in the cavity flows. Much more iterations are performed to derive the supercavity in the second type methods. The second type ones, which are mainly realized with the pressure-based algorithm, are relatively inconvenient, comparing with the ones with density-based algorithm, to process the flows with considerable compression.
The supercavity surface, in fact, is the interface between mediums. Mediums’interface, in SPH, is regarded as a set of particles. Theoretically, SPH could conveniently and naturally provides the cavity’s development, if it could realize the flows driven by the high speed underwater bodies. Because it is performed with a density-based algorithm, it thus treats the considerable compression well in the flows. The work in this thesis, which is performed to numerically study the low-subsonic and high-subsonic flows driven respectively by underwater high speed bodies with speeds of 100 m·s~(-1) and 1000 m·s~(-1) (for simplicity, we’ll refrence it as high speed driven flows), is realized by the code developed by us with SPH. Along with the work, a basic problem, i.e. the stabitliy of SPH, is also studied, and two important techniques are proposed to implement the wall boundary conditions and to process the SPH data respectively. Especially, the supercavity surface could be easily extracted by the post-processing techniques proposed here.
Complicate relative position between static and dynamic walls which have complex geometries exist in the computational models of the high speed driven flows. Too many walls with too many angles and relative motion between walls challenge the universality and simplicity of the implementation algorithm for the wall boudanry conditions. If the algorithm is restrict only to simple shape walls or changes for different shape walls, the universality of the complete SPH code is thus reduced rapidly. And the Boundary Deficience of SPH (BDS), i.e. the SPH approximation being truncated near the wall, requires a remedy when the boundary conditions are being implemented. However, the methods, i.e. the Ghost Particle Method (GPM) and Dummy Particle Method (DPM), which could remedy BDS, are neither inconvenient to be implemented on complex geometrical walls and neither universal for different shape walls. A Convenience and Universality Improved Ghost Particle Method (CUI-GPM) is thus proposed to universally treat the walls with angles or curvatures. Its applications in the fluids and heat conduction exhibit its feasibilities and reasonabilities. Regretfully, CUI-GPM encounters difficulties in the computations of high speed driven flows. Because the supercavity separates the fluid from the body, which arouse the errors in the interpolation of CUI-GPM, and re-entry jet at the tail of the supercavity could not be mastered well in SPH. Another boundary implementation algorithm, which combines the DPM and GPM, is thus advised. DPM realizes the non-slippery condition by directly assigning the body speed to the ghost particles, and GPM refreshes the density on ghost particles as that on fluid particles. With the latter route, the re-entry jets derived by SPH accords the practical ones.
For extracting the supercavity surface and comparing it with the theoretical shape, the post-processing of SPH is studied. Traditional post-processing technique could not supply continuous color contours, continuous contour lines and streamlines, and slices of the datas, etc. A relatively new post-process technology is thus presented, which supplies a route to tansforme the SPH data into the FEM data, and the developed post-processing technology for FEM are employed for SPH. The Delaunay triangulation algorithm transforms the SPH particle sets into the meshes with a set of triangular elements. However, for the particle set on the non-convex domain, the Delaunay triangulation results some empty elements which contains no mass, these empty elements should be deleted from the original element set. Thus, a so called“Element Mass Weighting”Method (EMWM) is presented to delete those empty elements. And the rest elements with their nodes (particles) and the datas on them are employed as the FEM data to process the SPH data. The EMWM with Delaunay triangulation is very convenient to extract the supercavity’s surface.
The stability analysis with its results supplies the basis for the computation of the high speed driven flows. The matrix form of SPH is imported to study the stability of the SPH particle set. The intrinsic reason of the Tensile Instability (TI) is found to be caused by the Euler coordinates of particles, and the sufficient condition of TI, which was ever given by Swegle, is also derived here. The Tensile Stability Prerequisite (TSP) requires the smoothing length factor (the ratio of smoothing length to the particle interval) equals the extreme point of the first order derivative of the smoothing function. The value of the smoothing factor is also found to be equal to the smoothing function’s extreme point with respect to the smoothing length, which also makes the Fourier tansformation of the smoothing function attain its extreme value.
The TSP yields the value of the numerical sound speed. The TSP also requires that a correct and suitable physical model of the medium should be chosen for the flow otherwise the spurious compression occurs in the computation. Consequently, a proper state equation of the water is chosen in the computation of high speed driven flow. The implication of the TSP, i.e. the frequence of the error should be far smaller than that of the time integration, is also revealed and which yields the Courant number of CFL condition, which supplies the reasonable refrence to choose the time step size. It could consequently yield the flow compression that satisfies the stability prerequisite, which yields the empirical values of the flow compression from the studies of Monaghan[10] and Morris[95] for observing and checking the stability and reasonability of a computation case with weakly compression flows. This supplies the theoretical route to observe and check the reasonability of the computation case of the high speed driven flow.
Based on the results derived above and with their application in the computation of the high speed driven flow, stable and reasonable computational cases are perfomed, during which the compression varies in a reasonable limit. The supercavity derived by SPH accords well with the ones given by the Principle of Independent Expansion of the Cross Sections of the Supercavity and experiments, which shows the feasibility of applying SPH to calculate the high speed driven flows. The numerical results also indicate that, the re-entry jet occurs obviously at the tails of the supercavities when the bodies start in the water, and that the re-entry jet is not obvious when the bodies start in a lauch canister and the cavity tail is almost perpendicular to the bodies’s surface. It can also be found that the length of the caity is reduced for the body with a longer afterbody. The asymmetry along the axis of the supercavity is found to be induced by the compression in the flow which also futher enlarges the size of the supercavity.
The numerical studies of the flows driven by high speed underwater bodies are realized by SPH. The implementation algorithms of the wall boundary, which was proposed as a technique to realize the work above, relatively enable SPH to be conveniently developed for complex shapes walls in engineering. And the post-processing algorithms advised above supplies a practical route to realize a universal post-processor of SPH. And the stability analysis of SPH provides the information to construct and perform a reasonable and stable computation case.
高速水下物体,会导致超空穴。基于Euler描述研究空穴流的方法,主要包括势流理论和“流体输运方程结合两类空化模型”的方法。势流方法,不考虑空穴内汽体;这是因为汽体质量太小,其惯性对水流动力学条件影响很小。基于势流中边界元或面源法需要数值迭代得到空穴界面。第二种方法,考虑汽体与水流关系,因此能揭示更多空穴流图景,也需要数值迭代得到空穴界面;该类方法常基于压力的求解过程,处理速度尺度为1000 m·s~(-1)的高亚声速物体驱动的显著压缩流动,没有基于密度求解过程的算法方便。
空穴表面是物质界面。界面在SPH中是这些界面上的粒子集合。因此SPH可自然跟踪物质界面。因此,若SPH能计算高速水下物体驱动的流动,则它可能会自然得到空穴发展。其基于密度求解的过程也能方便处理显著压缩流动。因此探索这样的工作是有必要的。本文面向于自编程实现SPH计算100 m·s~(-1)低亚声速和1000m·s~(-1)高亚声速的水下物体驱动的流动(简称高速驱动流动),并相应研究SPH几个基本问题——稳定性、固壁条件施加、后处理及空穴表面提取。
本文的高速驱动流动计算模型中,固壁多,固壁折角多,运动与静止固壁位置关系复杂;这要求对复杂形状固壁,固壁条件施加算法统一、简便,否则程序通用性大大降低。而SPH对函数的近似在边界处被截断,即边界缺陷需要在施加边界条件时得到改善。但能改善该缺陷的镜像虚粒子法(Ghost Particle Method)与仿粒子法(Dummy Particle Method),对不同形状固壁,施加算法没有统一性,对复杂形状固壁,算法也不简便。本文提出简便性与通用性改善虚粒子方法(CUI-GPM),对折角和弯曲固壁,算法统一;并用于流动和热传导计算。高速驱动流动中,空穴导致流体与物体发生很大分离,应用CUI-GPM时,无法准确计算空穴回射流。于是提出DPM与GPM结合方法:DPM施加无滑移条件,物体速度直接赋予虚粒子;采用GPM思路,虚粒子的密度更新和流体相同;由此计算得到较符合实际的回射流。
为提取SPH所得空穴界面以与空穴的理论形状对比,研究了SPH的后处理。SPH传统后处理方法,无法直接得到连续云图、等值线、流线,以及微积分运算和切片等,限制了其结果显示。本文考虑将粒子集网格化形成三角单元集,将粒子作为节点,利用FEM后处理技术实现SPH后处理与成熟后处理技术的链接。Delaunay三角化提供粒子集的网格化技术,但它在非凸区域粒子集上会得到不包含质量的空白单元。本文提出所谓“单元称重”法去除空白单元,保留下的单元作为后处理的有限单元。基于上述思路,可方便提取自由表面和超空穴界面。
算法稳定性研究及其结果,为高速驱动流动计算提供参考依据。本文引入SPH矩阵格式研究作为粒子集的SPH稳定性。研究表明,张力不稳定的原因是采用空间坐标来标记粒子位置[72];也得到了Swegle关于张力不稳定性的充分条件。张力稳定必要条件,要求光滑长度因子(光滑长度比粒子间距)取在光滑函数一阶导数极值点;该取值同时也使光滑函数Fourier变换分量达到极大,同时也是光滑函数对光滑长度的极大值点。
稳定性分析得到了满足张力稳定必要条件的数值声速。稳定必要条件指出应选择与流动相契合的物理模型,否则计算出现虚假压缩率。因此在高速驱动流动计算中,为水选择了合适的物态方程。对稳定必要条件的分析得知,误差频率应远小于时间积分频率,并据此得到了CFL条件的Courant数,为时间步长取值提供了依据。根据该必要条件,还得到流动压缩率和“误差频率与时间积分频率比值”的关系,并计算得到Monaghan[10]与Morris[95]提出的微可压流动的计算保持稳定时压缩率的变化范围,为考察高速驱动流动的计算稳定与合理,提供了参考。
基于以上研究结果并应用于高速驱动流动的计算,本文得到了稳定的计算过程,流动压缩率保持在合理范围。SPH所得空穴、与独立膨胀原理和实验所得空穴形状一致,表明SPH计算高速驱动流动的适用性。计算显示,物体在水中启动时空穴尾部有明显回射流;而从发射筒射入水时,空穴尾部回射流较小,尾部界面近似垂直在物体表面。而物体后体的存在,使空穴相对变小。高亚声速与低亚声速下的空穴形态发展表明,显著压缩流动使空穴在长度方向不对称,并使空穴尺寸比流动不可压时更大,这符合既有的理论结果。
本文实现了SPH计算高速水下物体流场的工作。相应提出的固壁条件施加算法,为利用SPH自编程计算“含较复杂形状的固壁”的问题提供了途径。相应提出的后处理方法,为SPH形成通用后处理器提供了可供实用的途径。算法稳定性研究,给出了时间步长、光滑长度取值及物理模型选择等方面的理论依据。
Smoothed Particle Hydrodynamics (SPH) discretizes the configuration by a set of particles with the properties of mass and volume. By tracing the motion of each particle, the variation history of the configuration could be derived. The mesh distortion, which occurs usually in the methods based on girds when they are applied to compute the large deformation in continuum mechanics, is thus avoided in SPH because of its meshless property. It also masters the high speed and hyperpressure problems in continuum mechanis well with a density-based algorithm.
Supercavities are induced for the high speed underwater moving bodies. Two classes of methods based on the Euler formulation, i.e. potential theory and transportation equations with two kinds of cavitation models, are the two main ways for studying the supercavity flows. The potential theory neglects the vapor in the supercavity, because the mass in the cavity is so tiny that its inertial applies little effect on the dynamics of the water flow. Massive iterations are performed to seek the cavity surface in this method. The relationship between the vapor and water flows is included in the second type methods and it thus reveals much more characteristics in the cavity flows. Much more iterations are performed to derive the supercavity in the second type methods. The second type ones, which are mainly realized with the pressure-based algorithm, are relatively inconvenient, comparing with the ones with density-based algorithm, to process the flows with considerable compression.
The supercavity surface, in fact, is the interface between mediums. Mediums’interface, in SPH, is regarded as a set of particles. Theoretically, SPH could conveniently and naturally provides the cavity’s development, if it could realize the flows driven by the high speed underwater bodies. Because it is performed with a density-based algorithm, it thus treats the considerable compression well in the flows. The work in this thesis, which is performed to numerically study the low-subsonic and high-subsonic flows driven respectively by underwater high speed bodies with speeds of 100 m·s~(-1) and 1000 m·s~(-1) (for simplicity, we’ll refrence it as high speed driven flows), is realized by the code developed by us with SPH. Along with the work, a basic problem, i.e. the stabitliy of SPH, is also studied, and two important techniques are proposed to implement the wall boundary conditions and to process the SPH data respectively. Especially, the supercavity surface could be easily extracted by the post-processing techniques proposed here.
Complicate relative position between static and dynamic walls which have complex geometries exist in the computational models of the high speed driven flows. Too many walls with too many angles and relative motion between walls challenge the universality and simplicity of the implementation algorithm for the wall boudanry conditions. If the algorithm is restrict only to simple shape walls or changes for different shape walls, the universality of the complete SPH code is thus reduced rapidly. And the Boundary Deficience of SPH (BDS), i.e. the SPH approximation being truncated near the wall, requires a remedy when the boundary conditions are being implemented. However, the methods, i.e. the Ghost Particle Method (GPM) and Dummy Particle Method (DPM), which could remedy BDS, are neither inconvenient to be implemented on complex geometrical walls and neither universal for different shape walls. A Convenience and Universality Improved Ghost Particle Method (CUI-GPM) is thus proposed to universally treat the walls with angles or curvatures. Its applications in the fluids and heat conduction exhibit its feasibilities and reasonabilities. Regretfully, CUI-GPM encounters difficulties in the computations of high speed driven flows. Because the supercavity separates the fluid from the body, which arouse the errors in the interpolation of CUI-GPM, and re-entry jet at the tail of the supercavity could not be mastered well in SPH. Another boundary implementation algorithm, which combines the DPM and GPM, is thus advised. DPM realizes the non-slippery condition by directly assigning the body speed to the ghost particles, and GPM refreshes the density on ghost particles as that on fluid particles. With the latter route, the re-entry jets derived by SPH accords the practical ones.
For extracting the supercavity surface and comparing it with the theoretical shape, the post-processing of SPH is studied. Traditional post-processing technique could not supply continuous color contours, continuous contour lines and streamlines, and slices of the datas, etc. A relatively new post-process technology is thus presented, which supplies a route to tansforme the SPH data into the FEM data, and the developed post-processing technology for FEM are employed for SPH. The Delaunay triangulation algorithm transforms the SPH particle sets into the meshes with a set of triangular elements. However, for the particle set on the non-convex domain, the Delaunay triangulation results some empty elements which contains no mass, these empty elements should be deleted from the original element set. Thus, a so called“Element Mass Weighting”Method (EMWM) is presented to delete those empty elements. And the rest elements with their nodes (particles) and the datas on them are employed as the FEM data to process the SPH data. The EMWM with Delaunay triangulation is very convenient to extract the supercavity’s surface.
The stability analysis with its results supplies the basis for the computation of the high speed driven flows. The matrix form of SPH is imported to study the stability of the SPH particle set. The intrinsic reason of the Tensile Instability (TI) is found to be caused by the Euler coordinates of particles, and the sufficient condition of TI, which was ever given by Swegle, is also derived here. The Tensile Stability Prerequisite (TSP) requires the smoothing length factor (the ratio of smoothing length to the particle interval) equals the extreme point of the first order derivative of the smoothing function. The value of the smoothing factor is also found to be equal to the smoothing function’s extreme point with respect to the smoothing length, which also makes the Fourier tansformation of the smoothing function attain its extreme value.
The TSP yields the value of the numerical sound speed. The TSP also requires that a correct and suitable physical model of the medium should be chosen for the flow otherwise the spurious compression occurs in the computation. Consequently, a proper state equation of the water is chosen in the computation of high speed driven flow. The implication of the TSP, i.e. the frequence of the error should be far smaller than that of the time integration, is also revealed and which yields the Courant number of CFL condition, which supplies the reasonable refrence to choose the time step size. It could consequently yield the flow compression that satisfies the stability prerequisite, which yields the empirical values of the flow compression from the studies of Monaghan[10] and Morris[95] for observing and checking the stability and reasonability of a computation case with weakly compression flows. This supplies the theoretical route to observe and check the reasonability of the computation case of the high speed driven flow.
Based on the results derived above and with their application in the computation of the high speed driven flow, stable and reasonable computational cases are perfomed, during which the compression varies in a reasonable limit. The supercavity derived by SPH accords well with the ones given by the Principle of Independent Expansion of the Cross Sections of the Supercavity and experiments, which shows the feasibility of applying SPH to calculate the high speed driven flows. The numerical results also indicate that, the re-entry jet occurs obviously at the tails of the supercavities when the bodies start in the water, and that the re-entry jet is not obvious when the bodies start in a lauch canister and the cavity tail is almost perpendicular to the bodies’s surface. It can also be found that the length of the caity is reduced for the body with a longer afterbody. The asymmetry along the axis of the supercavity is found to be induced by the compression in the flow which also futher enlarges the size of the supercavity.
The numerical studies of the flows driven by high speed underwater bodies are realized by SPH. The implementation algorithms of the wall boundary, which was proposed as a technique to realize the work above, relatively enable SPH to be conveniently developed for complex shapes walls in engineering. And the post-processing algorithms advised above supplies a practical route to realize a universal post-processor of SPH. And the stability analysis of SPH provides the information to construct and perform a reasonable and stable computation case.
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