Rehm Ronald G, Baum Howard R
National Bureau of Standards, Washington, D. C. 20234.
J Res Natl Bur Stand (1977). 1978 May-Jun;83(3):297-308. doi: 10.6028/jres.083.019.
In this paper a set of approximate equations is derived which is applicable to very nonadiabatic, nondissipative, buoyant flows of a perfect gas. The flows are assumed to be generated by a heat source in which the heat is added slowly. The study is motivated by the occurrence of such flows in fires. There, the time scale associated with the fire growth and resultant fluid motion is usually long compared with the transit time of an acoustic signal (based on the temperature derived from the heat added) across the spatial extent of the fire. The approximate equations are characterized by a spatially uniform mean pressure appearing in both the energy equation and the equation of state with the spatially nonuniform portion of the pressure only appearing in the momentum equation. Therefore, the pressure remains almost constant in space while significant density and temperature variations, such as might occur in a fire, are allowed. The approximate equations are shown to reduce to the Boussinesq equations when the heat addition is mild. These equations are also shown in general to admit internal-wave motions while "filtering out" high-frequency, acoustic waves. In addition, they are shown to be expressible in conservation form, the pressure satisfying an elliptic equation whose homogeneous terms are derivable from the wave equation by letting the sound speed become infinite. An equation for the mean pressure is also obtained. For the special case of a room heated at a uniform rate with a small leak to the outside, an approximate solution for the mean pressure is determined explicitly.
本文推导了一组近似方程，该方程适用于理想气体的非常非绝热、无耗散的浮力流。假定这些流动是由一个缓慢添加热量的热源产生的。此项研究的动机源于火灾中出现的此类流动。在火灾中，与火灾发展及由此产生的流体运动相关的时间尺度通常比声信号（基于添加热量所产生的温度）在火灾空间范围内的传播时间长得多。这些近似方程的特点是，能量方程和状态方程中出现空间均匀的平均压力，而压力的空间非均匀部分仅出现在动量方程中。因此，压力在空间中几乎保持恒定，同时允许出现显著的密度和温度变化，比如火灾中可能出现的情况。结果表明，当热量添加较小时，这些近似方程可简化为布辛涅斯克方程。一般情况下，这些方程还允许内部波运动，同时“滤除”高频声波。此外，它们可以表示为守恒形式，压力满足一个椭圆方程，其齐次项通过令声速变为无穷大由波动方程导出。还得到了平均压力的一个方程。对于以均匀速率加热且有小泄漏到外部的房间这一特殊情况，明确确定了平均压力的近似解。
