USSR missiles problem

When the development of the jet engines started Russians engineers thought tried to adjust an extra security defensive system with missiles attached in the back area of the plane. If the plane was under attack the missiles where supposed to be fired at the enemies following. The results were surprisingly not so good though, since as tested the missiles almost instantly changed direction (rotating at turning against the plane itself  and destroying it !!!

it looks like the blades of the missiles change its direction since the wind conforming vector has the same direction with the missiles velocity vector …

 

mesh refinement

MESH REFINEMENT:

– description

– development

– test cases

– results

mesh_refinement_presentation

Lax-Wendroff scheme

The shock wave propagation equation and numerical approach. The governing equation is:

with:   and 

where are the (non-dimensional) density, pressure and velocity.

FINITE VOLUME METHODS

   The functions of interest (the ones expanded with Taylor formula) give a big remainder that is not smaller than the first terms. Moreover it is necessary to conserve the integral of such functions along the computational domain. These demands lead to a design of FV methods that are basically arise from Finite Difference methods. The physical domain is considered to be divided into cells. Between time and time , the increment of a physical (extensive) quantity, let us say the mass in cell , denoted by:

is given by the sum of flux $latex flux_{jk}$ exchanged with each neighbouring cell :

  (1)

Conservation of total mass is ensured by the equality:

(2)

In the case of linear conservation laws, this may be written, for example, as:

in which the elementary flux is assumed to satisfy a consistency equality and to approximate the normal flux of through the common interface of the two cells:

The Lax-Wendroff theorem for steady linear case can be summarized as:

If the scheme is conservtive according to the flux definition (1),(2), and if the discrete solution converges to a limit, then the limit is a weak solution ( i.e a distribution satisfying the equations).

Accuaracy is influenced by the specific method of discretization and cell type. In cell-center schemes primal cells give the partition. In vertex-centered schemes a dual partition is used for flux-derivation.

Lax-Wendroff scheme was intruduced as the first stable three-point scheme applicable for compressible flow problems. It is commonly used at hyperbolic partial differential equations. It is second-order accurate in both space and time. This method is an example of explicit time integration where the function that defines governing equation is evaluated at the current time. The first step in the Lax–Wendroff method calculates values for the quantity at half time steps, and half grid points, . In the second step values at  are calculated using the data for  and .

1) LAX-WENDROFF scheme:

2) MACCORMACK scheme:

Shock Wave propagation with pressure ratio = 2.5 and γ =1.4

 

Fluids and Physics – Physics and Fluids

This article from MIT might explain why the two fields of study are more alike than different.

When fluid dynamics mimic quantum mechanics

MIT researchers expand the range of quantum behaviors that can be replicated in fluidic systems, offering a new perspective on wave-particle duality.

In the early days of quantum physics, in an attempt to explain the wavelike behavior of quantum particles, the French physicist Louis de Broglie proposed what he called a “pilot wave” theory. According to de Broglie, moving particles — such as electrons, or the photons in a beam of light — are borne along on waves of some type, like driftwood on a tide.

Several years ago a fluid system was shown to reproduce the classic “double slit” experiment from physics. Now researchers from MIT have reproduced another phenomena from physics – the statistical behavior of electrons when confined in a circular region by ions. The experiment involves bouncing a droplet on the vibrating surface of a fluid and monitoring its path as it moves around the surface. The video at the link explains everything.

This might be startling or a manifestation of particle-wave duality.

Read this article above or watch the experiment here:

kind of refinement tri-elements :P

 

 

 

 

 

from http://www.sciencedirect.com/science/article/pii/S0306454909003752

MATURE HURRICANE – A CARNOT ENGINE

IT’s a bit trivial this days to talk about hurricanes , so i decided to search a bit further , considering what exactly is a hurricane and how scientists like meteorologists , engineers and mathematicians approach this problem .

According to  this paper  :

* there are tropical cyclones , which should not considered hurricanes . (~60% of them only are called so …)

* mature hurricanes  may be idealized as an axisymmetric vortex in hydrostatic and rotational balance. The cyclonic azimuthal flow reaches
its maximum intensity near the surface and decreases slowly upward, becoming anticyclonic near the top of the storm, roughly 15 kmabove the surface.
* hurricanes act like a CARNOT-ENGINE ! Bernulli’s equation and 1st law of thermodynamics applied for Carnot model .

Let z be the azimuthal  and r the radious of the hurricane  then starting a point (a)  and looking the photo below :

Air begins spiraling in toward the storm center at point a , acquiring entropy from the ocean surface at fixed temperature T. It then ascends adiabatically from point c, flowing out near the storm top to some large radius, denoted symbolically by point o. The excess entropy lost by export or by electromagnetic radiation  to space between o and o’ at a much lower temperature To. The cycle is closed by integrating along an absolute vortex line between o’ and a. The curves c-o and o’-a also represent surfaces of constant absolute angular momentum about the storm’s axis.

all this and much more from NY university maths department here : emanuel91