In many simulations of turbulent flows, the viscous forces $\nu\nabla^2 \bu$ are replaced by a hyper-viscous term $-\nu_p(-\nabla^2)^{p}\bu$ in order to suppress the effect of viscosity at the large scales. In this work we examine the effect of hyper-viscosity on decaying turbulence for values of $p$ ranging from $p=1$ (ordinary viscosity) up to $p=100$. Our study is based on direct numerical simulations of the Taylor-Green vortex for resolutions from ${512}^3$ to ${2048}^3$. Our results demonstrate that the evolution of the total energy $E$ and the energy dissipation $ϵ$ remain almost unaffected by the order of the hyper-viscosity used. However, as the order of the hyper-viscosity is increased, the energy spectrum develops a more pronounced bottleneck that contaminates the inertial range. At the largest values of $p$ examined, the spectrum at the bottleneck range has a positive power-law behavior $E(k)\propto k^{\alpha }$ with the power-law exponent $\alpha$ approaching the value obtained in flows at thermal equilibrium $\alpha =2$. This agrees with the prediction of Frisch et al. [Phys. Rev. Lett. 101, 144501 (2008)] who suggested that at high values of $p$, the flow should behave like the truncated Euler equations (TEE). Nonetheless, despite the thermalization of the spectrum, the flow retains a finite dissipation rate up to the examined order, which disagrees with the predictions of the TEE system implying suppression of energy dissipation. We reconcile the two apparently contradictory results, predicting the value of $p$ for which the hyper-viscous Navier-Stokes goes over to the TEE system and we discuss why thermalization appears at smaller values of $p$.

中文翻译：

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高粘流中的湍流级联，瓶颈和热光谱

在湍流的许多模拟流动时，粘性力$ \ NU \ nabla ^ 2 \ BU $由超粘性项替换$ - ^ {P} \ BU $ - \ nu_p（\ nabla ^ 2）中，为了抑制粘度的影响很大。在这项工作中，我们研究了高粘度对衰减湍流的影响，其值为$p$ 从 $p=\mathrm{1个}$ （普通粘度）至 $p=100$。我们的研究基于Taylor-Green涡旋的直接数值模拟，其分辨率为${512}^3$ 至 ${2048}^3$。我们的结果表明总能量的演化$Ë$ 和能量耗散 $ϵ$几乎不受使用的高粘度顺序的影响。但是，随着高粘度阶数的增加，能谱会形成更明显的瓶颈，污染惯性范围。在...的最大值$p$ 经检查，瓶颈范围内的频谱具有正幂律行为 $Ë（ķ）\propto {ķ}^{\alpha }$ 幂律指数 $\alpha$ 接近热平衡时在流量中获得的值 $\alpha =2$。这与Frisch等人的预测一致。[物理 牧师 101，144501（2008）]的建议，$p$，该流的行为应类似于截断的Euler方程（TEE）。尽管如此，尽管频谱发生了热化，该流仍保持有限的耗散率，直至检查的阶次，这与TEE系统的预测不符，暗示了能量耗散的抑制。我们调和这两个明显矛盾的结果，从而预测了$p$ 为此，超粘性的Navier-Stokes转到了TEE系统，我们讨论了为什么热化会在较小的温度下出现。 $p$。