基于Logvinovich原理的通气超空泡理论及其数值研究
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摘要
超空化是空化发展的极端状态，即空泡把整个内体包裹起来。一方面，由于内体的沾湿面积十分有限，即仅仅头部和尾部与水接触，巨大的减阻潜力为发展水下高速航行体提供了依据；另一方面，在流体力学领域超空泡涉及气（汽）液相变、湍流、回射流、气液压缩性等诸多流动难题,是一种有待认识的物理现象。正因如此，超空泡吸引着来自世界范围内的学者和工程技术人员的广泛关注。尽管到现在为止关于超空泡流的研究已经取得了很大发展，但是还有许多机理性和技术性的问题制约着对这个物理现象的理解和在工程上的应用，例如非定常通气超空泡的泄气机理及稳定性、水洞通气超空泡的堵塞效应、无界流域通气超空泡及其航行体的机动运动和超空泡的形态控制等。
正是基于这样的考虑，本文从试验、理论和数值模拟三个方面对通气超空泡流展开了研究。具体研究内容如下：
（1）基于前支撑模型，对通气超空泡流展开了试验研究。研究了水洞堵塞比对通气超空泡形态及空化数的影响；比较分析了在相同空化数下的水洞通气超空泡和无界流域通气超空泡的形态差异；研究了通气量对超空泡形态、尾部闭合机制及空泡流动的影响和所产生的延迟效应；研究了重力对通气超空泡形态及空化数的影响。
（2）提出了一个非定常通气超空泡的泄气模型，基于由Logvinovich原理推导的空泡截面方程，采用数值—解析的方法，获得了一个非定常通气超空泡的泄气率计算公式，并得到了实验的证实；基于势流理论，建立了水洞通气超空泡的尾部双涡管闭合模型，并推导了在水平均匀来流下重力和攻角对通气超空泡影响的数学模型；在此基础上，建立了一个非定常通气超空泡模型，即在给定通气率的情况下，将本文中建立的空泡影响模型、尾部闭合模型、泄气模型联立空泡截面模型构成了一个可用来描述非定常通气超空泡的封闭的方程组；针对水洞超空泡流的堵塞现象，基于势流理论，提出了一个最小空化数的数学模型。
（3）基于建立的非定常通气超空泡模型，发展了能数值模拟非定常通气超空泡的空化数耦合系数修正算法，对水洞通气超空泡流动特征进行了数值分析。建立了一个水洞通气超空泡的最小空化数模型，采用数值—解析的方法，获得了最小空化数的计算公式，并通过实验证实了该公式的有效性；数值分析了水洞壁面和重力对超空泡形态、阻力和空化数的影响以及最小空化数与堵塞空化数之间的关系。
（4）基于势流理论，分别推导了在无界流域中的垂向平面内和水平面内，攻角、重力和惯性力对通气超空泡曲线运动的影响模型；基于建立的非定常通气超空泡模型，采用空化数耦合系数修正算法，数值模拟了在平面内做任意曲线运动的通气超空泡，并分析了惯性力对超空泡形态的影响以及由空泡控制量的改变而产生的空泡变形和脱落。
（5）基于建立的非定常通气超空泡模型，采用空化数耦合系数修正算法，数值分析了各种空泡控制量扰动对通气超空泡稳定性的影响，包括速度扰动、空化器偏转角扰动、通气率扰动和通气射流扰动；数值研究了有攻角的非定常通气超空泡的稳定性，并基于数值结果，采用FFT（快速傅立叶变换）方法，分析了空泡脉动特性，最后采用数值—解析的方法得到了有攻角的非定常通气超空泡的稳定性判断准则；数值模拟了在速度扰动下的不同攻角的通气超空泡的脱落，并分析了空泡脱落频率和空泡内气体动力学参数及攻角之间的关系。
（6）基于建立的非定常通气超空泡模型，提出了一个适用于做任意运动的通气超空泡的形态控制算法；也提出了一个通气和运动速度联合控制算法，可以实现在垂向平面内做变深运动的通气超空泡的形态控制，并基于这个控制算法，对保持定常形态做垂向运动的通气超空泡进行了控制仿真。
（7）基于刚体动力学理论，对超空泡航行体进行了动力学建模，并结合建立的非定常通气超空泡模型，发展了一个能模拟在纵向平面内做曲线运动的超空泡航行体的数值算法；基于这个算法对做下潜运动的超空泡航行体进行了数值模拟，并计算了滑行力。通过与经验模型的计算结果相比较，在一定程度上证实了文中建立的非定常通气超空泡模型及其航行体动力学模型的有效性。
Supercavitation that envelops its vehicle fully is the extreme state of cavitation.On the one hand, it provides the evidence for developing the high-speed underwatervehicle to get great potential of drag reduction because of very limited wetted area.In this case only the head and tail of the vehicle contact water. A betterunderstanding of it requires a deeper research of the academic circles as a physicalphenomenon in the field of fluid mechanics on the other hand, which invloves somechallenges such as phase change between gas and liquid, turbulence, re-entrant jets,gas and liquid compressibility and so on. As much supercavitating flows haveattracted widespread attention from researchers and engineers around the world.Although research on supercavitating flows has made great advances so far, somemechanism and technical questions still restrict its understanding and application,such as the gas-leakage mechanism, the stability of the unsteady ventilatedsupercavity, the blockage effect of the ventialted supercavity in water tunnel, themaneuvering motion and shape control of the ventilated supercavity and its mobilevehicle in unbounded flows, etc.
From this view, this thesis is devoted to ventilated supercavitating flows fromexperiment, theory and numerical simulation. The concrete research contents are asfollows:
(1) Research on the ventilated supercavity is done by experiment based on theforward facing model. The effects of blockage ratio on the dimensions of theventilated supercavity and its cavitation number in water tunnel are discussed; Thedimension differences of the ventilated supercavity in between water tunnel andunbounded flows are analyzed comparatively at the same cavitation number; Someexperiments are carried out to research the effects of ventilation on supercavitydimensions, its closure mechanism and the flows and its delayed effects; Gravityeffects on the dimensions of the ventilated supercavity and its cavitation number arealso discussed.
(2) The gas-leakage rate of the unsteady ventilated supercavity is modeled, andis formulated based on the equation of cavity section from Logvinovich’s principleusing the numerical-analytical method. The obtained formula is validated byexperiment; The closure model of the two vortex tubes of the ventilated supercavity is derived in water tunnel based on the potential theory, and effect models of gravityand angle of attack (AOA) on it are established respectively in uniform flows; Onthis basis, the mathematical model of the unsteady ventilated supercavity isestablished, which consists of the established effect models, the closure model of itstail, and its gas-leakage model combined with the cavity section model for the givenventilation rate; For the choking flows in water tunnel, the mathematical model ofthe minimum cavitation number is also proposed based on the potential theory.
(3) The cavitation number embedded coefficient algorithm is developed tosimulate the ventilated supercavity in water tunnel and analyze its characteristicsbased on the established unsteady ventilated supercavity model; The model of theminimum cavitation number of the ventilated supercavity in water tunnel ispresented, and its formula is obtained using the numerical-analytical method andvalidated by experiments; Ventilated supercavity is simulated to analyze the effectsof walls and gravity on its shape, drag and cavitation number in water tunnel, andthen the relation between the minimum cavitation number and the blockagecavitation number is discussed.
(4) Effect models of AOA, gravity and inertial force on the ventilatedsupercavity in the curvilinear motion of the vertical and horizontal plane areestablished respectively based on the potential theory; The ventilated supercavity issimulated in the arbitrary curvilinear motion of the plane using the cavitationnumber embedded coefficient algorithm based on the established unsteadysupercavity model, and the effects of inertial force on the shape and its deformationand shedding caused by the change of control variables of supercavity are alsoanalyzed.
(5) The effects of the disturbances of various control variables of ventilatedsupercavity from velocity, deflection angle of cavitator, ventilation rate and jets onthe supercavity stability are analyzed numerically using the cavitation numberembedded coefficient algorithm based on the established unsteady supercavitymodel; The stability of the unsteady ventilated supercavity with AOA is researchednumerically, and its pulsation characteristics are analyzed based on the numericalresults using FFT (Fast Fourier Transform Algorithm). Then the stability criterion ofthe supercavity with AOA is obtained using the numerical-analytical method; Theshedding of the ventilated supercavity with velocity disturbance is simulated tomake an analysis of the relation between its stability and the dynamics parameter ofthe gas in supercavity, AOA.
(6) The shape control algorithm is developed based on the established unsteadyventilated supercavity model for the ventilated supercavity in the arbitrary motion;Another control algorithm combining the ventilation and the motion velocity is alsopresented to control supercavity dimensions in the vertical plane, and thesupercavity control with constant shape in the vertical motion is simulated.
(7) Supercavitating vehicle is modeled, and based on the model in combinationwith the established supercavity model, the algorithm is developed to simulate thevehicle in the curvilinear motion of its longitudinal plane; Uing this algorithm, thevehicle making a diving movement is simulated and its planing forces are alsocomputed. The model is validated by comparing the results with the empericalmodel’s results to some extent.
正是基于这样的考虑，本文从试验、理论和数值模拟三个方面对通气超空泡流展开了研究。具体研究内容如下：
（1）基于前支撑模型，对通气超空泡流展开了试验研究。研究了水洞堵塞比对通气超空泡形态及空化数的影响；比较分析了在相同空化数下的水洞通气超空泡和无界流域通气超空泡的形态差异；研究了通气量对超空泡形态、尾部闭合机制及空泡流动的影响和所产生的延迟效应；研究了重力对通气超空泡形态及空化数的影响。
（2）提出了一个非定常通气超空泡的泄气模型，基于由Logvinovich原理推导的空泡截面方程，采用数值—解析的方法，获得了一个非定常通气超空泡的泄气率计算公式，并得到了实验的证实；基于势流理论，建立了水洞通气超空泡的尾部双涡管闭合模型，并推导了在水平均匀来流下重力和攻角对通气超空泡影响的数学模型；在此基础上，建立了一个非定常通气超空泡模型，即在给定通气率的情况下，将本文中建立的空泡影响模型、尾部闭合模型、泄气模型联立空泡截面模型构成了一个可用来描述非定常通气超空泡的封闭的方程组；针对水洞超空泡流的堵塞现象，基于势流理论，提出了一个最小空化数的数学模型。
（3）基于建立的非定常通气超空泡模型，发展了能数值模拟非定常通气超空泡的空化数耦合系数修正算法，对水洞通气超空泡流动特征进行了数值分析。建立了一个水洞通气超空泡的最小空化数模型，采用数值—解析的方法，获得了最小空化数的计算公式，并通过实验证实了该公式的有效性；数值分析了水洞壁面和重力对超空泡形态、阻力和空化数的影响以及最小空化数与堵塞空化数之间的关系。
（4）基于势流理论，分别推导了在无界流域中的垂向平面内和水平面内，攻角、重力和惯性力对通气超空泡曲线运动的影响模型；基于建立的非定常通气超空泡模型，采用空化数耦合系数修正算法，数值模拟了在平面内做任意曲线运动的通气超空泡，并分析了惯性力对超空泡形态的影响以及由空泡控制量的改变而产生的空泡变形和脱落。
（5）基于建立的非定常通气超空泡模型，采用空化数耦合系数修正算法，数值分析了各种空泡控制量扰动对通气超空泡稳定性的影响，包括速度扰动、空化器偏转角扰动、通气率扰动和通气射流扰动；数值研究了有攻角的非定常通气超空泡的稳定性，并基于数值结果，采用FFT（快速傅立叶变换）方法，分析了空泡脉动特性，最后采用数值—解析的方法得到了有攻角的非定常通气超空泡的稳定性判断准则；数值模拟了在速度扰动下的不同攻角的通气超空泡的脱落，并分析了空泡脱落频率和空泡内气体动力学参数及攻角之间的关系。
（6）基于建立的非定常通气超空泡模型，提出了一个适用于做任意运动的通气超空泡的形态控制算法；也提出了一个通气和运动速度联合控制算法，可以实现在垂向平面内做变深运动的通气超空泡的形态控制，并基于这个控制算法，对保持定常形态做垂向运动的通气超空泡进行了控制仿真。
（7）基于刚体动力学理论，对超空泡航行体进行了动力学建模，并结合建立的非定常通气超空泡模型，发展了一个能模拟在纵向平面内做曲线运动的超空泡航行体的数值算法；基于这个算法对做下潜运动的超空泡航行体进行了数值模拟，并计算了滑行力。通过与经验模型的计算结果相比较，在一定程度上证实了文中建立的非定常通气超空泡模型及其航行体动力学模型的有效性。
Supercavitation that envelops its vehicle fully is the extreme state of cavitation.On the one hand, it provides the evidence for developing the high-speed underwatervehicle to get great potential of drag reduction because of very limited wetted area.In this case only the head and tail of the vehicle contact water. A betterunderstanding of it requires a deeper research of the academic circles as a physicalphenomenon in the field of fluid mechanics on the other hand, which invloves somechallenges such as phase change between gas and liquid, turbulence, re-entrant jets,gas and liquid compressibility and so on. As much supercavitating flows haveattracted widespread attention from researchers and engineers around the world.Although research on supercavitating flows has made great advances so far, somemechanism and technical questions still restrict its understanding and application,such as the gas-leakage mechanism, the stability of the unsteady ventilatedsupercavity, the blockage effect of the ventialted supercavity in water tunnel, themaneuvering motion and shape control of the ventilated supercavity and its mobilevehicle in unbounded flows, etc.
From this view, this thesis is devoted to ventilated supercavitating flows fromexperiment, theory and numerical simulation. The concrete research contents are asfollows:
(1) Research on the ventilated supercavity is done by experiment based on theforward facing model. The effects of blockage ratio on the dimensions of theventilated supercavity and its cavitation number in water tunnel are discussed; Thedimension differences of the ventilated supercavity in between water tunnel andunbounded flows are analyzed comparatively at the same cavitation number; Someexperiments are carried out to research the effects of ventilation on supercavitydimensions, its closure mechanism and the flows and its delayed effects; Gravityeffects on the dimensions of the ventilated supercavity and its cavitation number arealso discussed.
(2) The gas-leakage rate of the unsteady ventilated supercavity is modeled, andis formulated based on the equation of cavity section from Logvinovich’s principleusing the numerical-analytical method. The obtained formula is validated byexperiment; The closure model of the two vortex tubes of the ventilated supercavity is derived in water tunnel based on the potential theory, and effect models of gravityand angle of attack (AOA) on it are established respectively in uniform flows; Onthis basis, the mathematical model of the unsteady ventilated supercavity isestablished, which consists of the established effect models, the closure model of itstail, and its gas-leakage model combined with the cavity section model for the givenventilation rate; For the choking flows in water tunnel, the mathematical model ofthe minimum cavitation number is also proposed based on the potential theory.
(3) The cavitation number embedded coefficient algorithm is developed tosimulate the ventilated supercavity in water tunnel and analyze its characteristicsbased on the established unsteady ventilated supercavity model; The model of theminimum cavitation number of the ventilated supercavity in water tunnel ispresented, and its formula is obtained using the numerical-analytical method andvalidated by experiments; Ventilated supercavity is simulated to analyze the effectsof walls and gravity on its shape, drag and cavitation number in water tunnel, andthen the relation between the minimum cavitation number and the blockagecavitation number is discussed.
(4) Effect models of AOA, gravity and inertial force on the ventilatedsupercavity in the curvilinear motion of the vertical and horizontal plane areestablished respectively based on the potential theory; The ventilated supercavity issimulated in the arbitrary curvilinear motion of the plane using the cavitationnumber embedded coefficient algorithm based on the established unsteadysupercavity model, and the effects of inertial force on the shape and its deformationand shedding caused by the change of control variables of supercavity are alsoanalyzed.
(5) The effects of the disturbances of various control variables of ventilatedsupercavity from velocity, deflection angle of cavitator, ventilation rate and jets onthe supercavity stability are analyzed numerically using the cavitation numberembedded coefficient algorithm based on the established unsteady supercavitymodel; The stability of the unsteady ventilated supercavity with AOA is researchednumerically, and its pulsation characteristics are analyzed based on the numericalresults using FFT (Fast Fourier Transform Algorithm). Then the stability criterion ofthe supercavity with AOA is obtained using the numerical-analytical method; Theshedding of the ventilated supercavity with velocity disturbance is simulated tomake an analysis of the relation between its stability and the dynamics parameter ofthe gas in supercavity, AOA.
(6) The shape control algorithm is developed based on the established unsteadyventilated supercavity model for the ventilated supercavity in the arbitrary motion;Another control algorithm combining the ventilation and the motion velocity is alsopresented to control supercavity dimensions in the vertical plane, and thesupercavity control with constant shape in the vertical motion is simulated.
(7) Supercavitating vehicle is modeled, and based on the model in combinationwith the established supercavity model, the algorithm is developed to simulate thevehicle in the curvilinear motion of its longitudinal plane; Uing this algorithm, thevehicle making a diving movement is simulated and its planing forces are alsocomputed. The model is validated by comparing the results with the empericalmodel’s results to some extent.
引文
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