湍流是一个很宽泛的概念，我们平时计算的问题常常带有复杂的流场结构，但是却不能称为湍流。那么什么样的流动才算湍流呢？二维甚至一维是否有湍流呢？
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湍流尚没有严格的数学定义，但是有一些特性。
2020 Computational Methods for Fluid Dynamics (4th edition) 的第347页有较好的阐述：
Most flows encountered in engineering practice are turbulent (Pope 2000, and
Jovanovi´c 2004) and therefore require different treatment compared to laminar flows
studied so far.Turbulent flows are characterized by the following properties:
• Turbulent flows are highly unsteady. A plot of the velocity as a function of time at
most points in the flow would appear random to an observer unfamiliar with these
flows. The word ‘chaotic’ could be used but it has been given another definition
in recent years.
• They are three-dimensional. The time-averaged velocity may be a function of only
two coordinates, but the instantaneous field fluctuates rapidly in all three spatial
dimensions.
• They contain a great deal of vorticity. Indeed, vortex stretching is one of the
principal mechanisms by which the intensity of turbulence is increased.
• Turbulence increases the rate at which conserved quantities are stirred. Stirring is a
process in which parcels of fluid with differing concentrations of at least one of the
conserved properties are brought into contact. The actual mixing is accomplished
by diffusion. Nonetheless, this process is often called turbulent diffusion.
• By means of the processes just mentioned, turbulence brings fluids of differing
momentum content into contact. The reduction of the velocity gradients due to the
action of viscosity reduces the kinetic energy of the flow; in other words, mixing is
a dissipative process. The lost kinetic energy is irreversibly converted into internal
energy of the fluid.
• It has been shown in recent years that turbulent flows contain coherent structures—
repeatable and essentially deterministic events that are responsible for a large
part of the mixing. However, the random component of turbulent flows causes
these events to differ from each other in size, strength, and time interval between
occurrences, making study of them very difficult.
• Turbulent flows fluctuate on a broad range of length and time scales. This property
makes direct numerical simulation of turbulent flows very difficult. (See below.)
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有垂直于流管轴线方向的分速度产生，放弃了神圣时间线
害怕，什么是神圣时间线呐
流动具有随机性或脉动性。
关键是普通的数值模拟过程中，如果雷诺数足够大，网格足够细，也可以看到小的涡结构。比如二维Rieamnn问题，这些算脉动嘛？
一维怎么模拟湍流啊？一维流动方向仅有x啊
个人认为湍流还是有序的，从仿真角度，无序的话应当是chaos，混沌等概念
来回来的无序运动是不是算湍流？
有一些策略是通过使用波函数对速度给定初值，如果速度本身的能谱符合湍流的能谱，就认为他是一维的湍流。但是似乎这个东西只能用来做一些测试，直接算是算不出来这种湍流的
湍流一定是chaos，chaos不一定是湍流。
湍流是自然界普遍存在的不规则运动，属于随机过程，主要特征是不规则性，其不规则性可表现在流动变量（速度、压强等）的时间序列呈现不规则的振荡状态，也能表现在流动变量在空间上的不规则分布。随机过程具有统计意义，故而湍流有平均值、脉动值以及能谱等统计量。------《湍流理论与模拟》张兆顺等 2017
哇！大佬好！那么请问数值模拟过程中是否也会产生这种随机性呢
应该算是吧，有随机性，有一些统计特征应该就是了
烟雾算是湍流吗？烟雾和水最不同的一点就是，烟雾里有非常非常多的漩涡，这些漩涡会流动，拉伸，挤压并且相互影响，导致难以预测。模拟烟雾时也经常加各种噪声
对，我也纠结这个问题，因为数值模拟的话，你格式确定，基本上结果就是确定性的，最多添加一些扰动，那么这个时候算是湍流嘛？反过来讲，是不是如果我扰动加的合适，就是湍流了，和是不是DNS无关？
简单粗暴地说，我觉得层流之外都是湍流hhhh----开个玩笑。
具体还是要看应用场景吧，不同需求有不同的界定范围
对于简单的模型，我直接
根据雷诺数判断