Viscosity and Turbulence

Viscosity

Def Viscosity
Viscosity prevents the formation of discontinuities in real fluids. In general, (kinematic) viscosity is the diffusion of momentum. Commonly, the viscosity is related to temperature. Let $\mu$ be dynamic viscosity and $\rho$ be density, we have
$\nu =\frac{\mu }{\rho }(m/s^2)$

Def Burgers’ Equation
For a given field $u(x,t)$ and diffusion coefficient $\nu$ (i.e. viscosity in fluid mechanics), the Burger’s equation is defined as:
$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=\nu \frac{{\partial }^{2}u}{\partial x^2}$

Def Newtonian Fluid
A newtonian fluid is a type of fluid such that stress is proportional to strain.

Prop Viscosity as Stickiness
The equation of motion of treating viscosity as stickiness between layers is given by:
$\ddot{x}=\nu \frac{{\partial }^{2}u}{\partial z^2}$

Def Navier-Stokes Equations
The Navier-Stokes equations model essentially all fluid flows are defined by:

& \frac{D\vec{u}}{Dt}=-\frac{1}{\rho}\nabla P + \vec g + \nu \nabla^2\vec{u} \\ &\frac{D \rho}{Dt}+\rho \nabla \cdot \vec{u}=0 \end{aligned}$$ Specially, if $\nabla P = \vec g=0$ then the first equation reduces to the Burger's equation. ## Cascade to Turbulence **Def** <i><u>Reynold’s Number</u></i> Th Reynold’s number is a non-dimensional number that quantifies the relative strength of the viscous terms compared to the inertial, assuming a velocity scale of $u$ and a length scale of $L$: $$ \text{Re}=\frac{uL}{\nu} $$ When Reynold’s number is considerably larger, the fluid behaves as an inviscid fluid except the boundaries. When it is order one, we have a fluid system which is influenced by the viscosity everywhere. **Def** <i><u>Turbulence</u></i> Turbulence is defined as continual cascade of instabilities at various scales. Turbulence is chaotic. **Def** <i><u>Reynold’s Decomposition</u></i> Let $\bar u = \frac{1}{T} \int_0^Tu \mathrm{d}t$ be the average velocity over time interval $T$, which is independent on $t$. Then define the following decomposition as Reynold’s decomposition: $$ u^\prime = u-\bar u $$ **Prop** $\frac{1}{T} \int_0^Tu^\prime \mathrm{d}t=0$ **Def** <i><u>Turbulent Viscosity</u></i> The turbulent or eddy viscosity $\nu_T \gg \nu$ is used to model the turbulence, which is known as large eddy simulation. $$ \frac{D\bar{u}}{Dt}= \nabla \cdot ( (\nu_T+\nu) \nabla\bar{u} ) $$ where $\bar{u}$ is the mean velocity.