Internally heated convection

热驱动的湍流在自然界和技术中无处不在。它们的发生是由于在对流电池边界设置的特定热条件和/或由于电池内部的热源。在这个项目中，我们将研究后一种驱动形式的湍流热对流，即内部加热驱动，无论是在经典的恒定驱动情况下，还是在空间和时间调制的驱动情况下，这更接近于内部加热湍流的许多应用。我们的结合理论和数值研究将基于二维和三维直接数值模拟，这将在广泛的控制参数范围内进行：普朗特数长达50年，瑞利-罗伯茨数长达60年，热调制频率长达40年。首先，对于恒热驱动，我们将通过直接数值模拟来验证和进一步发展内部加热对流中动量输运（雷诺数）和体温的标度理论。在对广泛参数范围内的数值数据进行深入分析的基础上，我们将扩展标度理论来预测通过流体层的任何水平表面（包括顶部和底部）的热传输，即相应的努塞尔数，在内部加热对流中，努塞尔数取决于与底板的距离。我们还将揭示全局流响应参数与局部流组织之间的联系。特别地，我们将把边界层理论扩展到考虑的热对流类型，并从扩展的边界层方程中推导出速度和温度的垂直平均剖面。内热流体层上部以浮力为主的部分和下部以渗透为主的部分的流动结构和剖面有很大的不同。最后，研究了时间和空间时间调制内部加热湍流对流的结果。特别是，我们想要了解热驱动源的空间和/或时间调制对全球流动组织、热量和动量输运以及系统总体温度的影响。我们期望根据调制频率来识别不同的状态，并从理论上解释这些状态和它们之间的转换。

Thermally driven turbulent flows are omnipresent in nature and technology. They occur due to specific thermal conditions set at the boundaries of a convection cell and/or due to internal sources of heat inside the cell. In this project, we will study turbulent thermal convection with this latter form of driving - namely, driving by internal heating, both for the classical case with constant driving and for spatially and temporally modulated driving, which is closer to many applications of internally heated turbulence. Our combined theoretical and numerical study will be based on two- and three-dimensional direct numerical simulations, which will be conducted in a broad range of control parameters: up to five decades in Prandtl number, up to six decades in Rayleigh--Roberts number, and up to four decades in the thermal modulation frequency. First, for constant thermal driving, with the direct numerical simulations, we will verify and further develop the scaling theory for the momentum transport (Reynolds number) and bulk temperature in internally heated convection. Based on a deep analysis of the numerical data in a broad parameter range, we will then extend the scaling theory to also predict the heat transport through any horizontal surface (including top and bottom) of the fluid layer, i.e. the corresponding Nusselt numbers, which in internally heated convection depend on the distance from the bottom plate. We will also reveal the connection between the global flow response parameters with the local flow organization. In particular, we will extend the boundary layer theory to the considered type of thermal convection and derive the vertical mean profiles of the velocity and of the temperature from the extended boundary layer equations. The structure of the flow and the profiles are quite different for the upper, buoyancy-dominated part of the internally heated fluid layer, and the lower, penetrative part. Finally, the consequences of temporally and spatiotemporally modulating the internally heated turbulent convection will be studied. In particular, we want to understand the effect of the spatial and/or temporal modulation of the thermal driving source on the global flow organization, the heat and momentum transport, and the bulk temperature of the system. We expect to identify different regimes, depending on the modulation frequency, and want to theoretically explain these regimes and the transitions between them.