液体晃动数值模拟及刚—液耦合动力学研究
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摘要
充液航天器在任务执行过程中,面临着大范围变轨机动、大角度姿态控制等指令,星载液体的晃动对控制精度、系统的稳定性都有不可忽视的影响。多年来,液体晃动的研究取得了丰富的理论和实验成果。鉴于理论研究进行了模型的简化而实验研究费用又十分高昂,本文采用有限元方法实现在多种形状贮箱内液体的大幅晃动数值仿真,并将充液航天器作为刚体与粘性液体组成的刚-液耦合系统进行数值研究。
首先利用ALE有限元方法建立带自由液面的粘性液体晃动的计算模型。采用等阶插值函数完成速度和压力变量的有限元空间离散,以Crank-Nicolson二阶精度方法作为时间离散格式,结合稳定化分步法来减小因时间离散精度提高对压力计算的数值影响。该计算方案提高了对流项和粘性项的离散精度,改善了速度计算结果,在计算步中有选择地加入压力迭代,保证液体不可压约束条件的满足,提高计算效率。数值结果表明,它适用于模拟长时间的液体大幅晃动,具有很高的稳定性和很小的数值阻尼。
其次,研究ALE描述下的动网格技术。结合上述数值方案,完成了弧形壁面贮箱内液体晃动计算。将ALE网格运动界面节点的速度定义为一个标量和形状向量的乘积,用它修正自由液面网格速度与流场速度关系式,以增加自由面网格节点运动的自由度。对内部网格用Laplace平滑技术计算运动速度,它将ALE有限元方法推广到计算非直壁运动边界的内流问题,如在轨贮箱的壁面形状多是弧形。数值算例表明使用该网格移动方法的计算结果与理论值吻合。
最后,利用Jourdain变分原理,建立了充粘性液体的刚-液耦合系统的数学模型,采用交替积分格式完成计算。该数学模型的特点是:以充液系统的整体响应为研究对象,计算中实时地记入液体晃动带来的质量分布、转动惯量、晃动力、力矩等对系统动力学影响。比较不同充液比下,椭圆形腔内液体晃动对受俯仰激励系统的动力学影响;在水平激励下,椭圆形腔充液刚体的响应在液体晃动频率附近最强烈;在该频率的水平激励下系统呈现非线性的耦合特征。
Liquid-filled spacecraft will implement many commands, such as large-scale or-bit maneuver, a wide-angle attitude control during the mission. And on-board liquidsloshing will bring distinct in?uence to the accuracy of the instructions and the stabilityof the whole system. The rich theoretical results have been achieved from the study onliquid sloshing during these years. However, the theoretical results almost are stemmedfrom the simplified models, and the experiments cost are expensive. This paper adoptsfinite element method in the numerical simulation of the ?uid sloshing in tanks withvarious shapes, and studies the spacecraft as a rigid-?uid interaction coupled system,where the liquid-filled spacecraft is a rigid tank and the liquid is viscous.
Firstly, the ALE finite element method is applied to model the free surface slosh-ing of viscous ?uid. Equal-order interpolation functions are used to discrete the finiteelement space. The Crank-Nicolson, a second order accuracy discrete method, is usedas the time discretion method. A stabilized fractional-step method is imposed to re-duce the in?uence of the pressure results driven by the improved discrete accuracy.
This calculation procedure enhances the discrete precision of convection and viscousterms, and improves the velocity results. The pressure iteration procedure is chose toensure the incompressible constraint of ?uid, and it can also improve the computationale?ciency. Numerical experiments indicate that this method is very stable and can beused to simulate the ?uid sloshing in a long time. Moreover, this method has smallernumerical damping than other methods.
Secondly, the moving mesh technology is studied under ALE description. Com-bined with the numerical method above, the sloshing in the tank with a variety ofcurved walls is completed. Defined the nodal velocity on the moving interface of theALE mesh as a product of a scalar and a shape vector, the kinematical boundary con-ditions on the free surface is modified to increase the freedom of the movements of thegrid on free interface. Then the interior nodes’velocities are smoothed by a Laplace technique. Numerical experiments indicate that this moving grid method does not af-fect the results of the sloshing, and can extend the ALE finite element method to theinterior ?ow computations with non-straight boundary. It is utility to simulation slosh-ing in the spacecraft tanks.
Finally, the model of rigid-liquid coupled system in which liquid is viscous isestablished by using the Jourdain principle, the stagger algorithem is used in the inte-gral calculation of interaction system. The features of this mathematical model is thatit regards the liquid-filled system’s response as the research object, the in?uences tothe system dynamics including the mass distribution, the inertia moment, the sloshingforce, sloshing torque and others, which are brought by ?uid sloshing, are recordedreal-time during the calculation. Numerical experiments indicate that the response ofthe liquid-filled system is strongest near the resonance frequency of the liquid itself.The characteristics of the nonlinear response exist under this frequency.
首先利用ALE有限元方法建立带自由液面的粘性液体晃动的计算模型。采用等阶插值函数完成速度和压力变量的有限元空间离散,以Crank-Nicolson二阶精度方法作为时间离散格式,结合稳定化分步法来减小因时间离散精度提高对压力计算的数值影响。该计算方案提高了对流项和粘性项的离散精度,改善了速度计算结果,在计算步中有选择地加入压力迭代,保证液体不可压约束条件的满足,提高计算效率。数值结果表明,它适用于模拟长时间的液体大幅晃动,具有很高的稳定性和很小的数值阻尼。
其次,研究ALE描述下的动网格技术。结合上述数值方案,完成了弧形壁面贮箱内液体晃动计算。将ALE网格运动界面节点的速度定义为一个标量和形状向量的乘积,用它修正自由液面网格速度与流场速度关系式,以增加自由面网格节点运动的自由度。对内部网格用Laplace平滑技术计算运动速度,它将ALE有限元方法推广到计算非直壁运动边界的内流问题,如在轨贮箱的壁面形状多是弧形。数值算例表明使用该网格移动方法的计算结果与理论值吻合。
最后,利用Jourdain变分原理,建立了充粘性液体的刚-液耦合系统的数学模型,采用交替积分格式完成计算。该数学模型的特点是:以充液系统的整体响应为研究对象,计算中实时地记入液体晃动带来的质量分布、转动惯量、晃动力、力矩等对系统动力学影响。比较不同充液比下,椭圆形腔内液体晃动对受俯仰激励系统的动力学影响;在水平激励下,椭圆形腔充液刚体的响应在液体晃动频率附近最强烈;在该频率的水平激励下系统呈现非线性的耦合特征。
Liquid-filled spacecraft will implement many commands, such as large-scale or-bit maneuver, a wide-angle attitude control during the mission. And on-board liquidsloshing will bring distinct in?uence to the accuracy of the instructions and the stabilityof the whole system. The rich theoretical results have been achieved from the study onliquid sloshing during these years. However, the theoretical results almost are stemmedfrom the simplified models, and the experiments cost are expensive. This paper adoptsfinite element method in the numerical simulation of the ?uid sloshing in tanks withvarious shapes, and studies the spacecraft as a rigid-?uid interaction coupled system,where the liquid-filled spacecraft is a rigid tank and the liquid is viscous.
Firstly, the ALE finite element method is applied to model the free surface slosh-ing of viscous ?uid. Equal-order interpolation functions are used to discrete the finiteelement space. The Crank-Nicolson, a second order accuracy discrete method, is usedas the time discretion method. A stabilized fractional-step method is imposed to re-duce the in?uence of the pressure results driven by the improved discrete accuracy.
This calculation procedure enhances the discrete precision of convection and viscousterms, and improves the velocity results. The pressure iteration procedure is chose toensure the incompressible constraint of ?uid, and it can also improve the computationale?ciency. Numerical experiments indicate that this method is very stable and can beused to simulate the ?uid sloshing in a long time. Moreover, this method has smallernumerical damping than other methods.
Secondly, the moving mesh technology is studied under ALE description. Com-bined with the numerical method above, the sloshing in the tank with a variety ofcurved walls is completed. Defined the nodal velocity on the moving interface of theALE mesh as a product of a scalar and a shape vector, the kinematical boundary con-ditions on the free surface is modified to increase the freedom of the movements of thegrid on free interface. Then the interior nodes’velocities are smoothed by a Laplace technique. Numerical experiments indicate that this moving grid method does not af-fect the results of the sloshing, and can extend the ALE finite element method to theinterior ?ow computations with non-straight boundary. It is utility to simulation slosh-ing in the spacecraft tanks.
Finally, the model of rigid-liquid coupled system in which liquid is viscous isestablished by using the Jourdain principle, the stagger algorithem is used in the inte-gral calculation of interaction system. The features of this mathematical model is thatit regards the liquid-filled system’s response as the research object, the in?uences tothe system dynamics including the mass distribution, the inertia moment, the sloshingforce, sloshing torque and others, which are brought by ?uid sloshing, are recordedreal-time during the calculation. Numerical experiments indicate that the response ofthe liquid-filled system is strongest near the resonance frequency of the liquid itself.The characteristics of the nonlinear response exist under this frequency.
引文
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