Beauty is a relative concept, and its canons sometimes change very quickly. Our review contains photos nine custom models from around the world, destroying stereotypes and proving that you can be beautiful, desirable and in demand, no matter what.
Girl without a leg - Victoria Modesta
The Latvian singer and model has undergone 15 ineffective surgeries on her leg, which was injured during childbirth. The leg had to be amputated soon after. But her self-confidence did not diminish. Rather, on the contrary. She inspires many with her example.
Two meters of beauty - Erica Irvine
35-year-old model from California, Erica Irvine, due to her height - 2 meters 5 centimeters - is considered one of the tallest and most popular models in the world.
Alien on the podium - Masha Telna
Ukrainian model with surprisingly big eyes. She is often called "alien" and "beautiful elf".
Sexy Lily McMenamy
The rising star of the modeling business, to put it mildly, too non-standard model Lily McMenamy conquered the catwalks of world brands and won a myriad of men's hearts.
Provocative model - Melanie Gaydos
Melanie Gaidos is a German-born art model based in New York City. Melanie's unusual appearance attracts photographers and artists. The girl often participates in the most provocative photo shoots, not embarrassed to be naked. A bald head, the absence of eyebrows and eyelashes, a cleft lip - all this creates a vivid image that makes a strong impression on others. Melanie became an Internet star after starring in the video of the German band Rammstein for the song Mein Herz Brennt (My heart is on fire) in 2012.
Appetizing Denise Bido
The 28-year-old Puerto Rican model is considered one of the most sought after XXL models. Bido is proud that she has become a clear example of the fact that you can always be successful and beautiful, for all women heavier than 55 kilograms
Androgynous Casey Legler
This French model was the first to be contracted as a man. She advertises menswear. Legler's height is 188 centimeters. A short haircut, sharp cheekbones and a strong-willed chin make Casey look very much like a young man.
Ì Ministry of General and Vocational Education of the Russian Federation
Ï Yerm State Technical University
Department of Mathematical Modeling of Systems and Processes
Ï .Ã.Ô ðèê
TURBULENENCE: MODELS AND APPROACHES
April 1998
UC 532.517.4
Turbulence: models and approaches. Lecture course. Part I /
Ï .Ã.Ô ðèê; Ï åðì. ãîñ. òåõí. un-t. P Erm', 1998. 108 p.
Ï The first part of the course of lectures includes an introduction and three of the seven sections of the course "Turbulence: Models and Approaches". The first section contains basic information from fluid mechanics, which is necessary for its further presentation. The second is devoted to questions related to the stochastic behavior of low-mode systems of the hydrodynamic type. The third section derives equations for the statistical moments of velocity pulsations and gives a brief overview of the models used to close them.
For students and graduate students of physical and mathematical specialties. And l.64. Bibliography 12 titles
Reviewers: |
Department of Physics Perm |
state technical university, |
|
Doctor of Physical and Mathematical Sciences, Professor D.V. Lyubimov |
© Perm State Technical University,
INTRODUCED AND E ............................................... ................................................. ....................... |
||
ÎÑÍ Î ÂÛ ...................................................................................................................... |
||
Equations of fluid motion .................................................................. ................................................. ......... |
||
Stability of currents .................................................. ................................................. .................... |
||
Free convection of an incompressible fluid............................................................... ................................. |
||
Convective stability .................................................................. ................................................. ........... |
||
Small mode model of convection (Lorenz system) .............................................................. ............................... |
||
CHAOS IN D INAMIC SYSTEMS .............................................. ................... |
||
Conservative and dissipative systems.................................................................... ...................................... |
||
Bifurcations .............................................................. ................................................. .................................... |
||
How to describe transition and chaos? ................................................. ................................................. ...................... |
||
Fourier spectra ............................................................... ................................................. ................................. |
||
Strange Attractor .................................................. ................................................. ....................... |
||
Fractals................................................... ................................................. ......................................... |
||
Subharmonic cascade .................................................................. ................................................. .............. |
||
Some examples ............................................................... ................................................. ......................... |
||
SEMIEM P E RICAL MODELS .............................................. ............................... |
||
Developed turbulence .............................................................. ................................................. ................ |
||
Equations for statistical moments............................................................... ......................................... |
||
Turbulent viscosity .................................................................. ................................................. ................. |
||
Mixing path length .......................................................... ................................................. ...................... |
||
Turbulent Viscosity Transfer Models .............................................................. ......................................... |
||
Two-parameter models .................................................................. ................................................. ..... |
||
INTRODUCED AND E
Turbulence remains one of the most difficult objects of study in fluid and gas mechanics. For almost a century of history of its study, dozens of different approaches have been proposed, almost always reflecting the most actively developing promising areas of mathematics and physics of the corresponding period of time. Statistical physics and probability theory, dimensional theory, Fourier analysis and direct numerical methods, dynamic systems theory, fractal theory and wavelet analysis - this is not a complete list of areas of science that gave basic ideas to turbulence researchers.
The theory of turbulence is far from complete. New approaches to key study continue to appear. A growing number of models are being proposed to better understand its individual properties. To give an idea of the main ideas driving this process, to demonstrate the possibilities of various approaches and to show the problems that they have not resolved, to present modern models that have not yet been included in textbooks and have not become textbooks - this is the goal of the proposed course of lectures.
The course is intended for students of the specialty "applied mathematics" who are oriented towards work in research institutions and departments, especially those related to solving problems of fluid and gas mechanics. At the same time, the course also discusses general approaches to modeling complex dynamic systems, which can be useful for specialists involved in modeling a wide variety of (and not only mechanical) systems and phenomena. The course is designed for students who have received broad basic training in basic mathematical disciplines, including methods of mathematical physics, functional analysis and probability theory, as well as who have attended special courses in mechanics (continuum mechanics, theory of constitutive relations).
The course of lectures consists of two parts. The first part includes three chapters, including mainly information that can be found in various textbooks and monographs, but collected together and presented in the light of the problems discussed in this course. The second part contains results that, with a rare exception, have not yet been included in books and can only be found in original papers.
The first chapter contains basic information on the dynamics of incompressible fluids, including the derivation of equations of motion for ideal and viscous fluids and examples of problems that have exact solutions. The foundations of the theo-
stability theory, which is of the utmost importance in understanding the problems of transition from laminar to turbulent flows. Two problems are discussed in detail: the stability of plane Poiseuille flows (the Orr-Sommerfeld problem) and the Rayleigh problem of the convective stability of a horizontal layer of an incompressible fluid heated from below. The last problem is preceded by the derivation of the equations of free convection in the Boussinesq approximation and a discussion of the necessary conditions for the stability of an inhomogeneously heated fluid in the field of gravity. Particular attention is paid to the question of the dimensionless representation of the equations of motion, the laws of similarity and dimensionless parameters and their role in describing the processes of transition to chaotic behavior. The chapter ends with the derivation of the low-mode model of convection (the Lorentz model). This derivation has a methodological purpose - to show and discuss the problem of designing nonlinear equations of motion on a finite-dimensional basis and the transition from partial differential equations to ordinary differential equations. At the same time, a detailed derivation of the model is useful, since the resulting system of equations is widely used in the next chapter, where its properties are discussed in detail.
Significant progress in understanding the nature and properties of turbulence has occurred in recent decades due to advances in dynamical systems theory that have enabled them to understand how chaotic behavior occurs in deterministic systems. The second chapter is devoted to these results, in which basic information from the theory of dynamical systems is given and some applications are discussed. The concept of phase space is introduced and examples of phase portraits of some simple dynamical systems are given. The features of the evolution of conservative and dissipative systems are discussed. For dissipative systems, the concept of an attractor is introduced, and the properties of attractors of stochastic systems are discussed. And brief information from the theory of fractals is presented, the concept of generalized dimension is given, and algorithms for determining the dimension of attractors of stochastic systems are described. The foundations of the theory of bifurcations are given, and some methods of studying the transition to chaos and characterizing dynamical systems in the case of periodic and chaotic behavior (Poincare sections, Lyapunov exponents, Kolmogorov entropy, Fourier spectra) are considered. The main scenarios of the transition from order to chaos are described and discussed: the Landau scenario, the Ryuelle and Takkens scenario, and the subharmonic cascade. The chapter concludes with examples of hydrodynamic systems that demonstrate their chaotic behavior. A detailed analysis of the behavior of the Lorentz model, the equations of which were derived in the first chapter, has been carried out. The simplest model of the generation of the Earth's magnetic field (the Rikitaka dynamo) is also considered, which reproduces the effect of random reversals of the magnetic field direction. The results are also shown and discussed.
experimental observation of chaotization of a convective flow in a closed cavity.
In the third chapter, acquaintance with the methods of describing developed turbulence begins, namely, with the historically first and most developed approach to the description of turbulent flows. This is the Reynolds approach and the numerous semi-empirical models of turbulence that have grown out of it. The chapter begins with the definition of the statistical moments of random fields, characterizing the turbulent flow. Further, the derivation of the Reynolds equation for mean fields is given, and issues related to the appearance of Reynolds stresses in the equations of the tensor are discussed. It is shown how the chain of Friedmann-Keller equations is obtained and the closure problem is formulated. The discussion of ways to solve this problem begins with a description of the Boussinesq hypothesis for the stress tensor, a definition of the concept of turbulent viscosity, and a description and discussion of the Prandtl mixing path model. In the following sections, more complex models are considered: turbulent viscosity transfer models and two parameter models of the k − ε model type. A comparatively modest place is given to semi-empirical models in the proposed course of lectures for two reasons. First, it is this approach that is most fully covered in the literature and can be freely studied from textbooks. Secondly, the main goal of this course is to get acquainted with the methods of studying the properties of small scale turbulence (homogeneous isotropic turbulence), which just remains beyond the field of view of semiempirical models. Therefore, the description of these approaches is necessary only for a general acquaintance with the ideology of the method, enabling him to refer to it in the future and make the necessary comparisons.
1 ÎÑÍ Î ÂÛ
1.1 Equations of fluid motion
Hydrodynamics is a branch of continuum mechanics that describes the movement of liquids and gases within the framework of a continuum model. The latter means that we are considering the scale l >> λ , ãäå λ is the mean free path of molecules.
A physically infinitesimal volume is considered, and the characteristics of the medium are introduced: velocity v and two thermodynamic quantities: pressure P and density ρ.
1.1.1 Continuity equation
The laws of motion are derived from the laws of conservation. First, the law of conservation of matter is used. A certain volume V is fixed in space, bounded by the surface S, whose mass is equal to
m = ò ρ dV .
È the change in the mass of this volume is
∂ m = ∂ ò ρ dV ,
∂ t ∂ t V
and the fluid flow flowing out of the volume
ò ρ vn dS .
If we take the direction of motion of the considered volume as a positive direction, then the mass conservation condition can be written as
∂ t ò ρdV = − ò ρv n dS .
The right side of the equality is transformed according to the Ostrogradsky–Gauss theorem
ò ρ vn dS = ò div(ρ v) dV . |
|||
¶ρ |
|||
òêé |
Div(ρ v ) ú ù dV = 0 |
||
ë¶t |
|||
and since the equality must be valid for any volume, the integrand must satisfy the equation
which is called continuity equation (continuity). For an incompressible fluid, the density is a constant value (ρ = const ) and equation (1.1) is simplified:
1.1.2 And the ideal fluid
We will first derive the equations for the velocity for an ideal fluid. And a perfect fluid is a fluid without viscosity and thermal conductivity.
The law of conservation of momentum for a moving liquid volume is
(ò ρ vdv) = å Fi |
||
where on the right side is the sum of all forces acting on the selected volume. Restricting ourselves to the consideration of gravity and pressure forces, we write
ò ρ vdV = ò ρ gdV + ò (− P) dS . |
|||
Considering that ò d dt ρ dV º 0 (the integral is taken over the liquid particle, then
is for a given amount of liquid, and not for a given volume), you can rewrite the equation in the form
òρ |
(v )= ò (ρ g |
- С P ) dV |
||
and, again based on an arbitrary choice of particle volume, go to the differential form
Ñ P |
||||||||
Entering |
derivative equation |
It's substantive |
||||||
derivative that describes the change in the velocity of a fluid particle. The consideration of the motion of individual fluid particles is called Lagrange's approach to describing the motion of a fluid. In most cases, Euler's approach is preferable, which consists in describing the characteristics of the fluid at a given point. To get the equation of motion in the Euler form, you need to get the connection between the substantial and local derivatives. We write the speed increment
dt + |
dx + |
dy + |
||||||||
and get from it the connection of the substantial (total) derivative with respect to time with the partial derivative of the speed with respect to time (change in speed at a given point)
dt ¶ t ¶ x dt ¶ y dt ¶ z dt dt |
x ¶ x |
y ¶ y |
z¶z |
|||||||||||||||||||||||
And using the resulting relation, we arrive at Euler equation received by him back in 1755:
(v с )v |
Ñ P + |
|||||||
The hydrostatic approximation is obtained under the condition that there is no movement, that is, the velocity and the time derivative are equal to zero:
v = 0 . |
|||||||
Thus, |
|||||||
С p + |
|||||||
or p = ρ g . Given that gravity is directed vertically downwards and assuming that the coordinate z is directed vertically, ò.å. g = − ge z , we obtain
We now write the momentum flux in tensor notation. Note that in what follows we will sometimes denote the time derivative as ¶ t .
¶ t (ρ v i ) = ρ ¶ t v i + ¶ t ρ v i
We rewrite the continuity equation in the form
∂t ρ − ∂ (ρ v k ) = 0 , ∂ x k
and the Euler equation (1.5) in the form
∂v |
= −v |
∂v i |
∂P |
||||||
k ∂ x л |
ρ ∂x i |
||||||||
Let us substitute the last two formulas into the expression for the change in momentum:
(ρ v |
) = - ρv |
¶v i |
¶ (ρ v k ) |
(ρ v |
|||||||||||||||||||||||
k ¶ x k |
¶x k |
||||||||||||||||||||||||||
¶x i |
i ¶ x k |
¶x i |
|||||||||||||||||||||||||
= - δ |
(ρ v |
) = - |
(δP + ρv |
||||||||||||||||||||||||
ik ¶ x k |
¶x k |
¶x k |
|||||||||||||||||||||||||
and introduce the momentum flux density tensor, which describes the transfer of the i-th component of the momentum through the area perpendicular to k - îé îñ
A bitchy viper, prone to a bohemian life, eccentric and capricious... But can anyone go against the firm decisions of the head, which is adorned with an impudent eroquoise of the ultra-lightened "Blond" color ...? Once again, insomnia against the background of internal hysteria, and so, in work and adventures, a summer smelling of sweets and raspberries passes, and the body, trained from childhood by ballet, is again drowning in size XS ...
Anorexia seems romantic only from the pages of cheap modern novels. In fact, it is so dreary to try to stuff at least yogurt into yourself in a day! The back aches so habitually from a new dose of dances, in which there is a real drive, real life, real energy, real salvation, real beauty, real passion, a real desire for magic and the REAL itself ... True, sitting at the computer for a long time, sometimes even passing after midnight, with a weighty whip, boldly leaves the memory with tension and pain in the spine ... Such habitual insomnia, in the sensation of which the most daring ideas are born ... Susceptibility and very big problems with the nerves and heart, like a visiting card of a vulnerable, but strong character human... But many emotions will remain inside. So it is necessary. Green eyes with a devilish twinkle... All because HE loves me...
Thoughts, a mug of cold green tea, old jeans, a loose T-shirt that makes it much more comfortable to feel like a queen, pink slippers and thoughts, thoughts, thoughts... How confidently a story about different people with dissimilar destinies develops, which it would be easier to even call characters ... But all of them are united by one thing - the soul and unpredictably changed appearance of their creator. It is so difficult and pleasant to work on your images, so hysterically writing out each of their features... A bunch of things and objects occupying all conceivable space, but this is all solely for the sake of a higher goal... In reality, only one mirror is important, observing the evolution of the soul. And the sharp clicks of the camera lens, which will leave forever in that very cherished world these images, born from the union of impressionability and fantasy, during a photo session that reveals the secrets of the soul ...!
And strange creatures ride public transport. Those that are called men inadvertently cast glances at the outline of that very beautiful chest, slightly short of the first size. Female creatures are interested in androgynous appearance, which rather even has something more from an incredibly beautiful boy who strives to seem like a girl ... Young girls curiously eat into a stylish range of clothes in the Unisex style with pretty eyes and so cutely slip through resolute, but gentle lips look at boys over fifteen ... Pensioners squeamishly turn up their old noses, but deep down they don’t give a damn about this indistinct city, only the thought of how SHE is there is important? We haven't seen each other again for so long... So that again, after saying a lot of tenderness to each other, shamelessly bite into each other's lips...
In the ears of electronic music... Life is a rhythm. After a couple of hours, you can again fall into the dance. In the meantime, there is an opportunity to imagine how lucky you will be to arrange a real rock - roll on tour with your group. Everything will happen soon... The main thing is to remember every second how fucking lilac smells in early June!!! Because he loves me...
Guys, we put our soul into the site. Thanks for that
for discovering this beauty. Thanks for the inspiration and goosebumps.
Join us at Facebook And In contact with
Just a few years ago, a girl with imperfect facial proportions could not be imagined as a model. But today everything is different: fashion photographers are increasingly choosing models with a memorable appearance.
website reminds: do not worry about the imperfections of your face, because they make our appearance unique. These girls do just that, and now they are one of the most sought-after models in the world.
Daphne Groeneveld
Daphne is the face of Dior Addict perfume, she also shoots for brands Calvin Klein, Dior, Miu Miu, Gucci and Prada. In 2011, the girl was awarded the title of "Best Dutch Model" at the Marie Claire Fashion Awards.
Julia Zimmer
The face of this model of German origin combines delicate and coarse features, which gives her a special charm. Today, Julia collaborates with Vivienne Westwood and Prada and, apparently, we will hear more about her in a couple of years.
Liza Ostanina
Lisa is a young model from Izhevsk. When a representative of the world-famous modeling agency wrote to her and invited her to work, she did not believe it, because she never considered herself pretty. But it all ended with the fact that already at the age of 15, Lisa left to work in Japan and continues to conquer the fashion capitals of the world.
Issa Lish
The model is half Mexican, half Japanese. She became famous for her angular, androgynous and even slightly alien features. Issa studied at the Faculty of Art History and was going to be a sculptor, but life turned out differently. By the way, this model has a rather non-trivial and psychedelic instagram.
Laura O'Grady
As a child, Laura was teased for her large protruding ears, but she says it never occurred to her to get rid of this feature with surgery. The girl has already managed to make a brilliant modeling career.
Sierra Sky
The appearance of the girl is often compared with the appearance of the heroines from the paintings of the Renaissance: wavy long hair, an oval face, barely noticeable eyebrows, sensual lips and a thin nose. An interesting personality is hidden behind her unusual appearance - Sierra is fond of literature, writes prose and poetry.
Magdalena Frakowiak
One of the most famous models in Poland. The main highlight of her appearance was high cheekbones and an unusual oval face. Magdalena collaborated with Karl Lagerfeld, and French Vogue included her in the top 30 models of the 2000s.
Damaris Goddry
The Belgian with Latin American roots is equally good at both feminine looks and unisex looks. Vogue Italia and Garage Magazine are already working with the girl, and the tabloids write that she has a great future ahead of her.
Lindsay Wixon
One of the most successful and promising models in the world is Lindsay Wixon. A very small mouth with full lips, a dimple in her chin and a gap between her teeth make her appearance childishly cute - you certainly cannot confuse her with anyone.
Alek Wek
When Alek was little, she and her parents had to flee to Europe from Sudan. In London, the girl was noticed by a representative of a modeling agency. Now Alec is a successful model. It is impossible to look away from her - and all because of her perky smile and proud posture.
Kelly Mittendorf
It was the extravagant appearance that became Kelly's pass to the big world of fashion. She is not shy about appearing ugly and loves bold looks, which is why fashion photographers love her. The images that Kelly embodies are mesmerizing.
Ikeline Stange
At her first casting, the girl showed up in the most inappropriate form for any model: she was wearing green tights, glasses without glasses, a casually worn jacket and long dreadlocks. Under all this tinsel, the agents discerned the appearance of an aristocrat: a chiseled nose, sharp cheekbones and slanting eyes.
