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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

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 ...


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

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 ...


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

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 : ...


4

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 ...


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 assume you mean the critical exponents of the velocity correlation functions? First of all, I don't think they have been derived using fluid/gravity (this is a very difficult problem), at best the problem was mapped onto a different problem. There was a series of papers by Oz and others about incompressible (and compressible) Navier-Stokes some years ...


3

I think this is just straightforward from linear wave problems. For example, a single wave with fix wavelength $\lambda$ has the form $E(x) \sim exp(-ik_0x)$ in position space can be very simple in spectral space as $E(k) \sim \delta(k-k_0)$. For nonlinear wave (turbulence and so on) problems, many information can also be more clearly in spectral space, ...


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

The short answer is that the Navier-Stokes equation, which describes all aspects of fluid motion, cannot be solved for turbulent flow, unless certain simplifications are made. There are a number of reasons for this, some of which are described on this page. As computer power increases eventually we should be able to solve the equation directly. This is what ...


3

I'm not competent to review the literature for you, but one of the Clay Millenium prizes concerns the Navier-Stokes equations, which is part of what Feynman is talking about, so to the extent that no-one has claimed that particular prize, No. One measure of how well we can deal with turbulent flow in practice can be found in how much better we can predict ...


3

Mister S.Emeis from Institute for Meteorology and Climate Research, Karlsruhe Institute of Technology, gives an answer: Link here Here is the abstract of his explanation: The occurrence of wake clouds at Horns Rev wind farm is explained as mixing fog. Mixing fog forms when two nearly saturated air masses with different temperature are mixed. Due to the ...


3

I am not sure how useful this "back of the envelope" calculation of reliability of Numerical Weather Prediction is going to be. Several of the assumptions in the question are not correct, and there are other factors to consider. Here are some correcting points: The Weather is 3 dimensional and resides on the surface of the planet up to a height of at ...


3

Turbulence is not one of the great unsolved problems in physics. Physics tells us exactly how turbulence emerges as a direct consequence of local mass and momentum conservation. We can create multiparticle computer models such as lattice gas automata that generate turbulence at large length and time scales. We can write down the equations that govern ...


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

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

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 ...


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

The problem with the phase space flow in Hamiltonian mechanics is that the flow itself is non-dynamical, that is, the flow is immediately defined for a given Hamiltonian, so there is no independent equation governing its evolution. Thus, Liouville equation is simply a transport of a scalar variable in a given flow. So, dimensional analysis of the flow ...


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 ...


2

I believe you're exactly right: it's the complexity of hills, buildings, trees, asphalt, water, etc that make surface winds complicated. As you go higher in the atmosphere, these surface effects disappear and the winds become much more steady. You can see this in the winds aloft forecasts issued by the FAA for use in aviation: ...


2

Let me here just derive the equation (6.11) that follows the sentence, you mention. The Navier-Stokes equation (6.6a) reads $$\partial_t v_i + v_j\partial_j v_i = -\partial_i p + f_i + \nu~\partial_j\partial_j v_i.$$ The incompressibility condition (6.6b) reads $$\partial_jv_j=0. $$ Hence we have in the unprimed and the primed point that $$\partial_t ...



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