三维气体流动与纵波的理论表述:兼议气动声学

A Theoretical Formulation of Three-Dimensional Gas Flow and Longitudinal Wave,with Comments on Aeroacoustics

流动和在其中传播的纵波是可压缩流体的两种运动形式,分别由气体动力学和气动声学的理论所刻画。经典流动理论旨在求得方程的近似解,基本上只能处理无黏、定常、附着流,其大部分功能已被直接计算-测量N-S方程的离散解所取代。声比拟理论主导的气动声学则在声学远场预测上成就斐然,但对噪声源的辨识一直不尽如人意。为把经典的流动理论转变成对N-S方程离散解实施物理诊断的理论,并对捕捉声源问题寻找思路,本文把原始N-S方程组转化为一个单变量方程集,包括一个矢量速度方程和一个独立热力学变量的方程,二者都是非线性的二阶运流波动方程,分别主控流动和纵波。方程的算子显示主变量演化的定性特征,方程的结构显示主变量与其他变量的耦合机制,其中能无中生有地产生主变量的耦合机制就是它的源。矢量速度方程是经典气体动力学对定常无黏流得到的标量速度方程的全面推广,它和声速方程以及熵方程的结合等价于原始N-S方程组,但其应用价值尚待今后开展研究。本文着重讨论纵波和气动声学问题,达到两个认识:热力学量的方程以扰动密度方程(5.2节)为代表,可作为纵波低阶理论的基本方程,具有确定的、由运流非线性导致的运动学源;矢量速度方程的散度或扰动密度方程的物质导数导致同一个胀量运流波动方程,它既主控流动的纵分量又主控纵波的产生与传播,能辨识与剪切过程和熵过程的耦合所形成的动力学源。期望通过与计算-实验的密切结合,这个单变量方程组的理论框架有助于推动实现上述目的,并能引导构型设计和流动控制的实践。

Flows and longitudinal waves propagating thereon are two motion forms of compressible fluids,which are described by gas dynamics and aeroacoustics,respectively.Since basically classical flow theory can only deal with inviscid,steady,and attached flows,most of its functions has been replaced by direct computing discrete solutions of the N-S equations.Aeroacoustics dominated by acoustic analogy theory has made great achievements in acoustic far-field prediction,but the identification of noise sources has been unsatisfactory.In order to upgrade the flow theory to have the same generality as the N-S equations,and to help capturing sources of noise,in this article the original N-S equation set is cast to a set of univariate equations,including a vector velocity equation and a scalar equation for independent thermodynamic variables.Both equations are nonlinear and of second-order advective-hyperbolic type,which control flows and longitudinal waves respectively.The operators of each equation reveal the qualitative characteristics of the mainvariable evolution,and the structures of the equation reveal the coupling mechanism between the main variable and others,among which those that can generate the main variable from none are its sources.The vector velocity equation is a comprehensive generalization of the scalar velocity equation of classical gas dynamics for steady inviscid flow.Its combination with the sound-speed equation and the entropy equation is equivalent to the original N-S equations,but its application remains to be further explored.This article focuses on longitudinal waves and aeroacoustics,and amounts to two major findings:The thermodynamic-variable equation(represented by the density disturbance equation(Sec.5.2)can serve as the prototype for low-order theory of aeroacoustics,which has definite kinematic source caused by the advection nonlinearity;The divergence of the vector velocity equation or the material derivative of the density disturbance equation leads to the same advective wave equation for the dilatation,which dominates both the longitudinal part of the flow and the generation and propagation of longitudinal waves,and can identify the dynamic sources from the coupling of compressing process with shearing and entropy processes.It is expected that,through a close alliance with computational and experimental studies,this theoretical framework for single-variable equations can contribute to deeper physical diagnostics of numerical-experimental solutions to the N-S equations,and guiding the practice of configuration design and flow control.