# Tag Info

32

Gravity can, of course, become turbulent if it is coupled to a turbulent fluid. The interesting question is thus, as John Rennie points out, whether a vacuum solution can be "turbulent". As far as I'm aware this is not known. If turbulence does occur in vacuum gravity, it is remarkably hard to stir up. Even in very extreme situations like colliding black ...

16

Thanks to holography, we now know that solutions to the Einstein equation in certain $d+1$ dimensional spaces are equivalent (dual) to solutions of the Navier-Stokes equation in $d$ dimensions. This is the fluid-gravity correspondence. As a result, turbulence can be studied using the Einstein equations, see, for example, http://arxiv.org/abs/1307.7267.

14

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 : http://terrytao.wordpress.com/2007/03/18/why-global-regularity-for-navier-stokes-is-...

10

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 ...

7

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 ...

7

The Reynolds number, with $\rho$ the density, $u$ the velocity magnitude, $\mu$ the viscosity and $L$ some characteristic length scale (e.g. channel height or pipe diameter) is given by $$\text{Re}=\frac{\rho~u~L}{\mu}.$$ This is a dimensionless relation of the ratio of inertial forces ($\rho u u$) to viscous forces ($\mu\frac{u}{L}$). It therefore signifies ...

7

Update Recently, there was a talk titled Turbulent gravity in asymptotically AdS spacetimes which may be of interest. In these papers, spacetimes which are anti-de Sitter asymptotically with reflecting boundary conditions are considered, and the notion of turbulence in this case is that small perturbations about these spacetimes exhibit 'turbulent behavior.'...

6

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 ...

6

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 "...

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

The size of the Kolmogorov 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. ...

6

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 ...

6

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 ...

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 ...

6

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 ...

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 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

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

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 ...

5

The fluid-gravity correspondence that Thomas referred to in his answer is a very concrete set-up where we can import intuition from fluid dynamics to suggest how we might get turbulence in vacuum GR (with negative cosmological constant). I thought it deserved more explanation. First, fluid dynamics is a universal description, applicable in any system (e.g. ...

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

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 ...

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

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 ...

4

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 ...

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 ...

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