Exact Coherent Structures in Viscoelastic Turbulence

粘弹性，即弹性和粘性的存在，越来越多地被认识到是当今世界许多常见液体（例如血液、洗发水、油漆、DNA悬浮液等）的重要特征。然而，由于目前用于描述粘弹性流体的数学模型的复杂性，尽管粘弹性流体具有丰富的有趣特性，但仍然知之甚少（例如，70多年来人们已经知道，只有微量的弹性就足以使湍流施加在表面上的粘性阻力至少减半）。特别是，直到最近才意识到可能存在三种形式的湍流：（经典）牛顿湍流（NT），其存在于没有弹性的情况下，弹性湍流（ET），其存在于没有惯性的情况下，以及第三种，显然是中间形式的湍流，称为弹性惯性湍流（EIT），其需要惯性和弹性的平衡存在。这个建议是针对试图通过建立在最近发现的单向粘弹性流动的不稳定性，以确定EIT的动力学起源。我们已经发现，从这种不稳定性中出现的有限振幅状态与EIT中看到的相似，这表明它们是代理，可用于理解EIT的底层物理原理，并绘制出来以查看EIT何时存在于参数空间中。还需要解决的悬而未决的问题包括试图建立这些国家之间的联系和ET和NT。在这里建立联系将有助于解开是否真的有3种不同类型的湍流或同一湍流的更多不同限制。最终，所提出的工作将提高我们对在给定的一组参数下将实现何种类型的粘弹性流动（层流、湍流或介于两者之间的其他流动）的理解，这将有助于工程师设计工业过程或设计产品。

Viscoelasticity, which is the presence of both elasticity and viscosity, is increasingly realised to be an important feature of many common liquids in today's world (e.g. blood, shampoo, paints, DNA suspensions etc). However, due to the complexity of the mathematical models currently used to describe them, viscoelastic fluids remain poorly understand despite a wealth of interesting properties (e.g. it has been known for over 70 years that only a minute amount of elasticity is enough to at least halve the viscous drag on a surface exerted by a turbulent flow). In particular, it has only relatively recently been realised that there are possibly three forms of turbulence which can occur: (classical) Newtonian turbulence (NT) which exists in the absence of elasticity, Elastic turbulence (ET) which exists in the absence of inertia, and a third, apparently intermediate, form of turbulence called Elasto-inertial turbulence (EIT) which requires a balance of inertia and elasticity to exist. This proposal is directed at trying to identify the dynamical origins of EIT by building upon a recently discovered instability of unidirectional viscoelastic flows. Finite-amplitude states already found by us to emerge from this instability resemble what is seen in EIT suggesting that they are proxy which can be used to understand the underlying physics of EIT as well as mapped out to see when EIT exists in parameter space. Outstanding questions to be also addressed include trying to establish connections between these states and both ET and NT. Establishing connections here would help untangle whether there really are 3 distinct types of turbulence or more different limits of the same turbulence. Ultimately, the proposed work will improve our understanding of what type of viscoelastic flow (laminar, turbulent or something else in between) will be realised at a given set of parameters which will help engineers design industrial processes or design products.