Long-term behavior of cooling fluid in a rectangular container.
Long-term behavior of cooling fluid in a rectangular container.
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DOI：
10.1103/physreve.69.056315
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发表时间：
2004-05
期刊：
影响因子：
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通讯作者：
Wenxian Lin;S. Armfield
中科院分区：
文献类型：
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作者：
Wenxian Lin;S. Armfield
在这项研究中，通过尺度分析和直接数值模拟，研究了由于壁温固定而导致的不稳定自然对流在无限长度的矩形容器中冷却初始静态等温牛顿流体的长期行为。考虑两种具体情况。情况 1 假设流体的冷却是由垂直侧壁上施加的固定温度引起的，而顶部和底部边界是绝热的。情况 2 假设冷却是由垂直侧壁和底部边界上施加的固定温度引起的，而顶部边界是绝热的。表示容器中流体冷却的长期行为的适当参数是时间 t 时容器整个体积上单位长度的瞬态平均流体温度 T(a)(t)（即瞬态区域平均流体温度，如随后的数值模拟中所使用）和冷却边界上的平均努塞尔数。进行的标度分析表明，对于这两种情况，theta(a)(tau) 的标度为 e(-C(ARa)(-1/4) tau)，其中 theta(a)(tau) 是 T(a)(t) 的无量纲形式，tau 是无量纲时间，A 是容器的长宽比，Ra 是瑞利数，C 是比例常数。我们对这两种情况进行了一系列直接数值模拟，其中 A、Ra 和 Pr（Pr 是普朗特数）的选定值在 1/3< 或 =A< 或 =3、6 x 10(6) < 或 =Ra< 或 =6 x 10(10) 和 1< 或 =Pr< 或 =1000 范围内，以验证所建立的比例关系。发现这些数值结果与标度关系非常吻合。数值结果也用于量化比例关系，发现当 Ra、A 和 Pr 在上述范围内时，情况 1 和 2 的 C 分别为 0.645 和 0.705。
In this study, the long-term behavior of cooling an initially quiescent isothermal Newtonian fluid in a rectangular container with an infinite length by unsteady natural convection due to a fixed wall temperature has been investigated by scaling analysis and direct numerical simulation. Two specific cases are considered. Case 1 assumes that the cooling of the fluid is caused by the imposed fixed temperature on the vertical sidewall while the top and bottom boundaries are adiabatic. Case 2 assumes that the cooling is caused by the imposed fixed temperature on both the vertical sidewall and the bottom boundary while the top boundary is adiabatic. The appropriate parameters to represent the long-term behavior of the fluid cooling in the container are the transient average fluid temperature T(a)(t) over the whole volume of the container per unit length (i.e., the transient area average fluid temperature, as used in the subsequent numerical simulations) at time t and the average Nusselt number on the cooling boundary. A scaling analysis has been carried out which shows that for both cases theta(a)(tau) scales as e(-C(ARa)(-1/4) tau), where theta(a)(tau) is the dimensionless form of T(a)(t), tau is the dimensionless time, A is the aspect ratio of the container, Ra is the Rayleigh number, and C is a proportionality constant. A series of direct numerical simulations with the selected values of A, Ra, and Pr (Pr is the Prandtl number) in the ranges of 1/3< or =A< or =3, 6 x 10(6) < or =Ra< or =6 x 10(10), and 1< or =Pr< or =1000 have been carried out for both cases to validate the developed scaling relations. It is found that these numerical results agree well with the scaling relations. The numerical results have also been used to quantify the scaling relations and it is found that C=0.645 and 0.705 respectively for Cases 1 and 2 with Ra, A and Pr in the above-mentioned ranges.