In this work, we aim to shed light to the following research question: can we find a nonlinear tensorial subgrid-scale (SGS) heat flux model with good physical and numerical properties, such that we can obtain satisfactory predictions for buoyancy-driven turbulent flows? This is motivated by our findings showing that the classical (linear) eddy-diffusivity assumption, $$\varvec{q}^{eddy} \propto \nabla \overline{T}$$ q eddy ∝ ∇ T ¯ , fails to provide a reasonable approximation for the actual SGS heat flux, $$\varvec{q}= \overline{\varvec{u}T} - \overline{\varvec{u}} \overline{T}$$ q = u T ¯ - u ¯ T ¯ : namely, a priori analysis for air-filled Rayleigh-Bénard convection (RBC) clearly shows a strong misalignment. In the quest for more accurate models, we firstly study and confirm the suitability of the eddy-viscosity assumption for RBC carrying out a posteriori tests for different models at very low Prandtl numbers (liquid sodium, $$Pr=0.005$$ P r = 0.005 ) where no heat flux SGS activity is expected. Then, different (nonlinear) tensor-diffusivity SGS heat flux models are studied a priori using DNS data of air-filled ( $$Pr=0.7$$ P r = 0.7 ) RBC at Rayleigh numbers up to $$10^{11}$$ 10 11 . Apart from having good alignment trends with the actual SGS heat flux, we also restrict ourselves to models that are numerically stable per se and have the proper cubic near-wall behavior. This analysis leads to a new family of SGS heat flux models based on the symmetric positive semi-definite tensor $$\mathsf {G}\mathsf {G}^{T}$$ G G T where $$\mathsf {G}\equiv \nabla \overline{\varvec{u}}$$ G ≡ ∇ u ¯ , i.e. $$\varvec{q}\propto \mathsf {G}\mathsf {G}^{T}\nabla \overline{T}$$ q ∝ G G T ∇ T ¯ , and the invariants of the $$\mathsf {G}\mathsf {G}^{T}$$ G G T tensor.

中文翻译：

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浮力驱动湍流大涡模拟的合适张量-扩散率模型

在这项工作中，我们的目标是阐明以下研究问题：我们能否找到具有良好物理和数值特性的非线性张量亚网格尺度（SGS）热通量模型，以便我们可以获得对浮力驱动湍流的满意预测? 这是由于我们的研究结果表明经典（线性）涡扩散假设，$$\varvec{q}^{eddy}\propto\nabla\overline{T}$$ q eddy ∝ ∇ T¯ ，未能提供实际 SGS 热通量的合理近似值，$$\varvec{q}= \overline{\varvec{u}T} - \overline{\varvec{u}} \overline{T}$$ q = u T ¯ - u ¯ T¯ ：即，对充气瑞利-贝纳对流 (RBC) 的先验分析清楚地显示出强烈的未对准。为了寻求更准确的模型，我们首先研究并确认了 RBC 涡粘性假设的适用性，在非常低的 Prandtl 数（液态钠，$$Pr=0.005$$P r = 0.005）下对不同模型进行后验检验，其中没有热通量 SGS 活动是期待。然后，不同（非线性）张量-扩散系数 SGS 热通量模型先验地使用充满空气的 DNS 数据（$$Pr=0.7$$P r = 0.7 ）RBC 在瑞利数高达 $$10^{11}$ 10 美元 11 . 除了与实际 SGS 热通量具有良好的对齐趋势外，我们还将自己限制在数值稳定且具有适当立方近壁行为的模型上。这种分析导致了基于对称正半定张量 $$\mathsf {G}\mathsf {G}^{T}$$ GGT 的新 SGS 热通量模型系列，其中 $$\mathsf {G}\equiv \nabla \overline{\varvec{u}}$$ G ≡ ∇ u ¯ , i. e. $$\varvec{q}\propto \mathsf {G}\mathsf {G}^{T}\nabla \overline{T}$$ q ∝ GGT ∇ T ¯ ，以及 $$\mathsf {G }\mathsf {G}^{T}$$ GGT 张量。