Boussinesq近似在定常湍流中的适用性
Applicability of Boussinesq approximation in a turbulent fluid with constant properties
摘要：
描述浮力流体的运动方程通常使用J. Boussinesq在一个世纪前提出的一组近似来简化。为了恢复，它们包括假设恒定的流体性质，不可压缩性和热量守恒在热传输。假设满足第一个要求（恒定的流体性质），我们推导出一组4个标准，用于评估其他两个要求在湍流瑞利-贝纳德对流的有效性。第一个标准$αΔ\ll 1 $简单地由热边界层中的不可压缩性条件产生（$α$和$Δ$是热膨胀系数和驱动流动的温差）。其他3个标准与密度分层成正比或成二次方，或等效地与跨单元$Δ_{h}$的绝热梯度产生的温差成正比或成二次方。空气，水和低温氦的数值计算表明，只要满足第一个标准，大多数实验室实验是免费的，从这种Boussinesq违反。在超高瑞利数（$Ra>10^{16}$）实验中，在氦，分层标准之一，缩放与$αΔ_{h}$，可能会被违反。该标准保证压力波动对密度变化和通过压缩/膨胀循环的传热方程的影响可以忽略不计。外推到更高的$Ra$表明，强违反Boussinesq近似可能发生在大气对流。
英文摘要：
The equations of motion describing buoyant fluids are often simplified using a set of approximations proposed by J. Boussinesq one century ago. To resume, they consist in assuming constant fluid properties, incompressibility and conservation of calories during heat transport. Assuming fulfilment of the first requirement (constant fluid properties), we derive a set of 4 criteria for assessing the validity of the two other requirements in turbulent Rayleigh-Bénard convection. The first criterion $αΔ\ll 1 $ simply results from the incompressibility condition in the thermal boundary layer ($α$ and $Δ$ are the thermal expansion coefficient and the temperature difference driving the flow). The 3 other criteria are proportional or quadratic with the density stratification or, equivalently with the temperature difference resulting from the adiabatic gradient across the cell $Δ_{h}$. Numerical evaluations with air, water and cryogenic helium show that most laboratory experiments are free from such Boussinesq violation as long as the first criterion is fulfilled. In ultra high Rayleigh numbers ($Ra>10^{16}$) experiments in He, one of the stratification criteria, scaling with $αΔ_{h}$, could be violated. This criterion garanties that pressure fluctuations have a negligible influence both on the density variation and on the heat transfer equation through compression/expansion cycles. Extrapolation to higher $Ra$ suggests that strong violation of Boussinesq approximation could occur in atmospheric convection.
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