Unsteady turbulent buoyant plumes
Unsteady turbulent buoyant plumes
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不稳定的湍流浮力羽流
作者：
M. Woodhouse;J. Phillips;A. Hogg
我们对源条件随时间变化时湍流浮力羽流的非定常演化进行建模。积分模型是通过对描述羽流中质量、轴向动量和浮力演化的控制方程进行径向积分推导出来的。羽流中轴向速度和密度亏损的非均匀径向分布由积分方程中的形状因子明确地捕捉；通常假定的平顶分布导致形状因子等于1。当由羽流中平均轴向速度的径向分布确定的动量形状因子不等于1时，所得的非定常羽流模型是双曲线型的。当源条件保持恒定值时，该模型的解被证明保留了已确立的定常羽流解的形式。通过对这些定常解进行线性稳定性分析，我们表明在控制方程中包含不等于1的动量形状因子会导致一个适定的积分模型。因此，我们的模型不会出现先前提出的湍流羽流非定常积分模型中出现的数学病态。还确定了形状因子值的一个稳定性阈值，导致在其值的一个范围内，对定常解的小扰动幅度随距源的距离而衰减。方程系统的双曲线特性允许在描述羽流特性的场中在非定常演化过程中形成不连续性，并且我们计算数值解以说明源条件突然改变后羽流的瞬态发展。羽流对新源条件的调整是通过流体脉冲在羽流中的传播发生的。这个脉冲的动力学由一个相似解描述，并且通过构建这个新的相似解，我们确定了在源调整后瞬态脉冲的演化在性质上不同的三种情况。
We model the unsteady evolution of turbulent buoyant plumes following temporal changes to the source conditions. The integral model is derived from radial integration of the governing equations expressing the evolution of mass, axial momentum and buoyancy in the plume. The non-uniform radial profiles of the axial velocity and density deficit in the plume are explicitly captured by shape factors in the integral equations; the commonly assumed top-hat profiles lead to shape factors equal to unity. The resultant model for unsteady plumes is hyperbolic when the momentum shape factor, determined from the radial profile of the mean axial velocity in the plume, differs from unity. The solutions of the model when source conditions are maintained at constant values are shown to retain the form of the well-established steady plume solutions. We demonstrate through a linear stability analysis of these steady solutions that the inclusion of a momentum shape factor in the governing equations that differs from unity leads to a well-posed integral model. Therefore, our model does not exhibit the mathematical pathologies that appear in previously proposed unsteady integral models of turbulent plumes. A stability threshold for the value of the shape factor is also identified, resulting in a range of its values where the amplitudes of small perturbations to the steady solutions decay with distance from the source. The hyperbolic character of the system of equations allows the formation of discontinuities in the fields describing the plume properties during the unsteady evolution, and we compute numerical solutions to illustrate the transient development of a plume following an abrupt change in the source conditions. The adjustment of the plume to the new source conditions occurs through the propagation of a pulse of fluid through the plume. The dynamics of this pulse is described by a similarity solution and, through the construction of this new similarity solution, we identify three regimes in which the evolution of the transient pulse following adjustment of the source qualitatively differs.