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28

The progress in turbulence has come in fits and spurts, and it is very active in the last few years, due to the influence of AdS/CFT. I think it will be solved soon, but this opinion was shared by many in previous generations, and may be much too optimistic. Navier Stokes equations The basic equations of motion for turbulent flows have been known since the ...


11

There is a simple general argument for why you get small-scale motion from large-scale motion in any nonlinear nonintegrable continuous mechanical system, whether it is fluids, or electromagnetic waves interacting with charged plasmas, or surface waves on water, or anything nonlinear at all. This argument must break down for those special cases where the ...


9

General Mumbo-Jumbo about Statistics When you have any Hamiltonian mechanical system, with degrees of freedom $q_i$, conjugate variables $p_i$, and Hamiltonian $H(q_i,p_i)$ there is a conserved phase space volume, which is just the area in q,p space, defined by the volume element $$\prod_i dp_i dq_i$$ The conservation of phase space volume is Liouville's ...


8

From the Wikipedia article for Reynolds number: In fluid mechanics, the Reynolds number (Re) is a dimensionless number that gives a measure of the ratio of inertial forces to viscous forces and consequently quantifies the relative importance of these two types of forces for given flow conditions. In addition to measuring the ratio of inertial to ...


7

The statement that "fluids are fractal" is not quite correct (or at the very least is not precise). Instead what really happens is that energy in fluids transitions to higher and higher frequencies via a recursive formula which looks slightly fractal-like (called the "Selection rule"). This is one of the most famous results in fluid mechanics and is due to ...


6

The standard explanation is that there is a constant flux of energy from large eddies to smaller eddies. The time scale for an eddy of scale $r$ to turn over is $\tau \sim \frac{r}{v(r)}\sim \frac{1}{kv}$, the energy density for scale $r$ is $\sim v(r)^2$, so you get an energy flux rate $\epsilon \sim v^2/\tau \sim v^3k$ which is assumed constant. You then ...


6

Let's make some assumptions. First, assume the fish is rigid. Second, let's assume he's not flapping. Third, I guess let's assume it's a male fish since I said "he." We'll also assume this is 2D because we're looking for an approximation. I would approximate the fish as an airfoil. NACA airfoils are a pretty good choice because they are analytically defined ...


5

There are three ways you can proceed in: 1. Homogeneous Flow Model Herein, you would assume single averaged flow quantities and then solve the Navier-Stokes equations as if it were arising from the flow of an averaged liquid. What I mean is that if you had water and steam flowing together, you would take the average density, viscosity and so on. ...


5

Turbulence is indeed an unsolved problem both in physics and mathematics. Whether it is the "greatest" might be argued but for lack of good metrics probably for a long time. Why it is an unsolved problem from a mathematical point of view read Terry Tao (Fields medal) here : ...


5

The high speed expression, proportional to $v^2$ is the ram pressure, which is wholly a momentum transfer effect and has nothing to do with viscosity - in contrast with the low flow speed Stokes law you cite above. To understand the ram pressure, which arises particularly for supersonic objects, witness the object is just shoving fluid out of its way, and ...


5

Strictly speaking, turbulence doesn't exist in two dimensions. The energy cascade required for turbulence to develop (transfer energy from large scales to small scales) is due to the (incompressible for illustration) vorticity equation: $\frac{D\vec{\omega}}{Dt} = \left(\vec{\omega}\cdot\nabla\right)\vec{v} + \nu\nabla^2\vec{\omega}$ specifically the ...


5

The onset of turbulence in fluids is determined by the Reynolds number $$ \mathrm{Re} = \frac{vL}{\nu}, $$ where $L$ is the characteristic length scale, $v$ the characteristic velocity, and $\nu$ the viscosity. The onset of turbulence in fluids occurs for $\mathrm{Re}$ greater than about 1000 or more, depending on geometry. If we want to see the equivalent ...


5

From Wikipedia: A spoiler is an automotive aerodynamic device whose intended design function is to 'spoil' unfavorable air movement across a body of a vehicle in motion, usually described as turbulence or drag. A rear spoiler is designed to change the flow of air over the rear of the vehicle. This change in air flow increases downforce on the rear of ...


4

The key is the Reynolds number, $$ Re=\frac{\rho LV}{\mu}=\frac{LV}{\nu}\tag{1} $$ where $L$ and $V$ are characteristic lengths and velocities of the particular problem and $\mu$ & $\nu$ are the dynamic & kinematic viscosities, respectively. If you multiply (1) by $\rho LV/\rho LV$, you get $$ Re=\frac{\rho L^2V^2}{\mu LV} $$ The numerator is the ...


4

No, because the boundary breaks the symmetry. Turbulence will also occur if a fast moving object (such as a boat or aircraft) moves through stationary fluid. For the observer travelling with the fluid in your example, the boundary is a surface that's moving rapidly with respect to the fluid, and that's what's causing the turbulence. Invariance with respect ...


4

I don't think that such a computation of a theoretical limit of accuracy is possible. There are several sources of uncertainty in weather models: initial and boundary data, parameterizations, numerical instability, rounding and approximation errors of the numerical scheme employed to solve the Navier-Stokes equations for the atmosphere. The term ...


4

This is a reasonable question. At the scale of a waterspout, the inertial forces of fast-moving air should be large compared to the viscous forces (i.e., very large Reynolds number). Yet the inflow along the surface of the water is laminar, where we would ordinarily expect boundary-layer vorticity (i.e., turbulence). A detailed description of the expected ...


4

The size of the Kolmogorv scale is not universal, it is dependent on the flow phenomena you are looking at. I don't know the details for compressible flows, so I will give you some hints on incompressible flows. From the quotes poem, you can anticipate that everything that is dissipated at the smallest scales, has to be present at larger scale first. ...


4

Reynold's number is defined to be: $$ \text{Re} = \frac{ v D }{ \nu } $$ where $v$ is the characteristic velocity for the flow, $D$ is a characteristic size and $\nu$ is the kinematic viscosity. Now, why should we care? Why is Reynold's number important? Well, the first thing to realize is that the Reynolds number is a dimensionless number. This means ...


4

The truck will have in its wake some unknown mass of air almost moving with a speed $v$ comparable to the truck's speed $\bf V$. The pressure behind the truck will be lower than the pressure at the sidewalk because air pressure follows the Bernoulli equation, $$ P_\mathbf{P} = P_\text{road} + \frac{1}{2}\rho v^2, $$ where $\rho \approx 1~$kg/m$^3$ is the ...


4

The fly is carried away within the turbulent motion of the air the moving car generates. Therefore, it stays close to the car (for a short while) and returns without actually having to fly at 80 mph. -> Answer to your second question: No! A google search for "turbulence around car/obstacle/plane" gives colourful pictures of the wind field around moving ...


3

I think the simplest answer to this question would be that the stream of water has a number of forces acting on it (gravity, air drag) from many directions. Some torque is bound to be produced as the stream falls through the air. If you throw a ball or any small object from a height, it rotates, no matter how you drop it. Same logic applies here. As far as ...


3

Firstly, some general answer on nondimensional groups such as the Reynolds number e.g.: they do not generally characterise the flow as a whole, but a feature that you choose in the flow. If the flow is not an academic problem, you will have several such features, which have different lengths, velocities... In the case of multiphase flow, this is obvious ...


3

IANAFD but I'll stick my neck out and say this: resolving the Clay problem one way or another won't cause people doing CFD to lose any more sleep than they already do. First of all, Jean Leray proved the existence of weak solutions to Navier Stokes in $R^3$ way back in the 1930s, and that is pretty much what matters for the task of getting numerical ...


3

The reference that you show, makes it overly complicated to see the steps that are taken. In fluid dynamics, it is worth to familiarize yourself with index notation. You can reduced the three sets of equations for $u$, $v$ and $w$ to a single equation for $u_i$, where $i\in{1,2,3}$. Furthermore, you sum over the repeated index (often $j$). In ...


3

There are many examples of 2D turbulence. Many experiments involving 2D turbulence use flow in a thin film of soap. 2D turbulence is pretty easy to model too, and I've even seen online web-based examples: http://www.ibiblio.org/e-notes/webgl/gpu/fluid.htm One super interesting thing about 2D turbulence is the "Inverse energy cascade." In traditional 3D ...


3

Yes I have always interpreted that Feynman was saying that we had the equations but little idea about the properties of solutions. Mathematically this is concentrated in the Clay problem because the very first and most important question to a system of equations is whether there are unique and preferably continuous solutions. This problem has been sofar ...


3

I would say the most difficult part of numerical simulations of hydrodynamics is writing your own code. This would involve: understanding & implementing algorithms (e.g., Riemann solvers, Runge-Kutta integrators, etc) running & comparing test cases (e.g., Kelvin-Helmholtz instabilities) parallelization (?) lots and lots of debugging! If this is ...


3

For Newtonian fluids (such as water and air), the viscous stress tensor, $T_{ij}$, is proportional to the rate of deformation tensor, $D_{ij}$: $$D_{ij} = \frac{1}{2}\left(\frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i}\right)$$ $$T_{ij} = \lambda\Delta\delta_{ij} + 2\mu D_{ij}$$ where $\Delta \equiv D_{11} + D_{22} + D_{33}$. The ...


3

Basically, the scale of a certain parameter is the order of magnitude of that parameter. Being able to determine the scales of a parameters in a complex system (like turbulence problems) is very useful. For turbulence, the size of the largest eddies is given by the characteristic length scale you are working with, $L$, and the smallest eddy size is given ...



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