Fraktal geometri ve hidrolik pürüzlülük
The Fractal geometry and the hydraulic roughness
- Tez No: 19288
- Danışmanlar: PROF.DR. CAHİT ÖZGÜR
- Tez Türü: Yüksek Lisans
- Konular: Makine Mühendisliği, Mechanical Engineering
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1991
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 167
Özet
ÖZET“Fraktal Geometri ve Hidrolik Pürüzlülük”isimli bu YUksek Lisans Tezi çalışmasının ilk bölümlerinde kaosun bilimdeki yeri üzerinde durularak fraktal geometriye geçis yapılmıştır. Bilinen boyut kavramlarının doğadaki şekilleri tanımlamadaki yetersizlikleri Üzerine hem düzgün şekilleri, hem de kaosu ve doğanın geometrisini veren şekilleri kapsayan genelleştirilmiş bir boyut ifadesi kurularak konunun matematiğine girilmiştir. Bundan sonra değişik fraktal örnekler Üzerinde durularak, fraktal şekillerin özellikleri tanıtılmış ve topolojik boyuttan fraktal boyuta geçis için önemli bir asama olarak nitelenen Hausdorff Boyutu üzerinde durulmuştur. Çünkü Hausdorff boyutu, pek çok bakımdan fraktal boyut ile aynı karakterleri göstermekte olup teorisi ise kümeler konusuna dayanmaktadır. Bu boyuta göre ölçü ve boyut kavramları her bir küme elemanı için yapılarak daha sonra ise bu elemanlardan tüm kümeye erişilmektedir. Bundan sonra ise ilk baslıklarda basit örneklerle verilen fraktal şekillerin, kompleks düzlemdeki iterasyonlar la fraktal fonksiyonlarla elde edilmesi açıklanmıştır. Fraktal geometri konusu yeterince tanıtıldıktan sonra ise, fraktal geometrinin doğadaki varlıkların geometrisini temsil ettiği savından yola çıkılarak doğal pürüzlü yüzeylerin de fraktal bir yapıya sahip oldukları fikri üzerinde durulmuş ve bu yüzeyler ile aynı istatistik parametrelere sahip olan fraktal yüzeylerin elde edilişi anlatılmıştır. Bu yüzeylerin elde edilmesinde faydalanılan fraktal fonksiyon olarak İse Weierstrass Fonksiyonu tanıtılmıştır. Ayrıca mekanik olarak islenmiş yüzeylerin dahi belli bir frekanstan sonra yine doğal pürüzlü karakterlerini koruduğu ifade edilmiştir. Bundan sonra ise konunun hidrolikteki önemi üzerinde durularak sürekli yük kayıp katsayısının değişiminde yüzey pürüzlülüğünün rolü detaylı olarak açıklandıktan sonra, bu etkileşimin Moody diyagramından da görüldüğü gibi suni ve doğal pürüzlü yüzeyler için içerdiği farklılık araştırılmıştır. Bunun için ise doğal yüzeylerin fraktal, suni yüzeylerin ise fraktal olmayan bir yapıya sahip olmalarının etkisi üzerinde durularak yapılan deneylerle doğal yüzeylerin davranışı bir kez daha incelenmiştir. Ayrıca kullanılan borular da standart, ölçü »enerji kayıpları ve içerdikleri suni kum pürüzlülük değerleri bakımından detaylı olarak tanıtılarak gerekli Moody diyagramları çizilmiştir. 2111
Özet (Çeviri)
SUMMARY THE FRACTAL GEOMETRY AND THE HYDRAULIC ROUGHMESSI Today so »any kind of mathematical picture Implies a smooth relationship between an object's form and the forces acting on it. In the examples of the planets and pendulum, it also implies that the physics is deterministic, meaning that we can predict the future of these systems from their past. Two recent developments have deeply affected the relationship between geometry and phsics, however. The first comes from the recognition that nature is full of something called deterministic chaos. The reason for the interest in the thesis subject named“Fractal Geometry and Hydraulic Roghness”is that they are connected with chaos. In mathematics, chaos has a specialised meaning. The easiest way to understand chaos is by some examples. Suppose a particle Is moving In a confined region of space according to a definite deterministic law. Following the path traced out by the particle, we are likely to observe that it settles down to one of three possible behaviours- the geometrical description of which is called an attractor. The particle may be attracted to a final resting position. In this case, the attractor is Just a point. Or the particle may settle down in periodic cycle. Here the attractor is an ellipse and the future motion can be predicted with astonishingly high accuracy as far ahead as we want. The last possibility is that the particle may continue to move wildly and erratically while, nevertheless, remaining in some bounded region of space. Once a particle is attracted to a strange attractor there is no escaping. Almost anywhere we start inside the attractor, the point moves, on the average, in the same, Just as no matter how we start of a pendulum, it always eventually comes to rest at the same point. Although the motion is specified by the precise laws, for all practical purposes, the particle behaves as if it were moving randomly. The interesting point here is that strange attractors are very frequently fractals. There are many apparently simple physical systems in the Universe that obey deterministic laws but nevertheless behave unpredictably. The second development came from efferts to find mathematical descriptions for some of the most irregular and complicated phenomena seen around us: the shapes of mountains and clouds, how XIV'/ i' galaxies are distributed in the Universe, and nearer home, the way prices in the finsncJa] marVi=t? fluctuate. One way of obtaining such a discription is to seek a model. The word fractal has been coined from the latin“fractus”which describes a broken stone- broken up and irregular. Fractals are geometrical shapes that, contrary to those of Euclid, are not regular at all. First, they are irregular allover. Secondly, they have the same degree of irregularity on all scales. A fractal object looks the same examined from far away or nearby - it is self similar. As we approach it, however we find that small pieces of the whole, which seemed from a distance to be formless blobs, become well defined objects whose shape is roughly that of previously examined whole. Nature provides many examples of fractals. The rules governing growth ensure that become translated into large scale ones. A striking mathematical model of the way fractals work is the Sierpinski gasket. Take a black triangle and divide it into four smaller triangles and erase the central fourth triangle so that it leaves a white triangle. Each new black triangle will have sides that are half as long as the initial triangle. Repeat the exercise with each new triangle and we obtain the same structure on an ever decreasing scale with a detail that is twice as fine as that in the preceding stage.“When parts of the object are exactly like the whole, the object is said to be linearly self similar. However, the most importants fractals deviate from linear self similarity. Some of these are fractals that describe general randomness, while others are fractals that can describe chaotic and nonlinear systems. For complicated geometrical objects the ordinary notion of dimension may vary with scale. As an example, take a ball with a diameter of 10 cm made of a thread of 1 mm. From far away, the ball appears as a point. From a distance of 10 cm, the ball of thread is three dimensional. At 10 mm, its a mess of one dimensional threads. At 1 mm, each thread becomes a column and the whole becomes a three dimensional object again. At 0.1 mm, each column disolves into fibres, and the ball again becomes one dimensional and so on, with the dimension crossing over repeatedly from one value to another. When the ball is represented by a finite number of atom like pin points, it becomes zero dimensional again. For fractals, the counterparts of the familiar dimensions CO, 1,2. 3) are known as fractal dimensions. Usually, their values are not whole numbers. XV..The techniques discussed in the preceding section generate smooth curves and surfaces. But many objects, such as mountains and clouds have irregular or Iragment features. Such objects can be modeled using the fractal geometry methods developed by Mandelbrot. Basically, fractal objects are described as geometric entities that cannot be represented with Euclidian geometry methods. This means that a fractal curve cannot be described as one dimensional, and a fractal surface is not two dimensional. Fractal shapes have a fractional dimension. Smooth curves are one dimensional objects whose length can be precisely defined betwen two points. A fractal curve, on the other hand contains inf initevariety of detail at each point along the curve, we cannot say exactly what is length is. In fact as we continue to zoom in more and more detail, the length of the fractal curve grows longer.and longer. An outline of a mountain, for example, shows more variation to closer we get to it. As we near the mountain, the detail in the individual ledges and boulders becomes apparent. Moving even closer, we see the outlines of rocks, then stones, and then grains of sand. At each step, the outline reveals more twists and turns, and the overall length of curve tends to infinity as we move closer and closer. Similar types of curves describe coastlines and the edge of clouds. Such curves, represented in a two dimensional coordinate system, can be described -mathematically with fractional dimensions between 1 and 2. Whena fractal curves is described in three dimensional space, it has a dimension between 1 and 3. A fractal curve is generated by repeatedly applying a specified transformation function to points within a region of space. The amount of detail included in the final display of the curve depends on the number of iterations performed and the resolution of the display system. If P =(x iV ) is a selected initial, point. each iteration of transformation function F generates the next level of detail with the calculations. P =FCP ). P =FCP ) » P =FCP 5 i o 2 1 S 2 The transformation function can be specified in various ways to generate either regular or random variations along the curve at each iteration. One meyhod for determining the dimension of fractal curve such as that in basic fractal curves in Chapter 3 is the calculation log N logC-> XVJ.where N is the number of subdivisions at each step and c is the scaling factor, AI + hrvTrrh fractal cbjects by definition contain infinite detail. we generate a fractal curve with finite number of iterations. Therefore the curves we display with repeated patterns actually have finite length. Our presentation approaches a true fractal curve as the number of transformations is increased to produce more and more detail. Many fractal curves are generated with functions in the complex plane. That is, each two dimensional point Cx.y) is represented as the complex expression z=x+iy, where x and y are real numbers and i is used to represent the square root of -1. A complex function fCz) is then used to map points repeatedly from one position to another. De[ending on the initial point selected, this iteration could cause points to diverge to infinity, or the points could remain on the some curve. For example the function fCz5=z transforms points according to their relation to the unit circle. Any point z whose magnitute |z| is greater than 1 is transformed through a sequence of points that tend to infinity. A point with |z| >1. i cos2no> x 00 i yCx)= A £ n 1 1 The frequencies üs form a discrete Weierstrass spectrum ranging from 6>n to infinity in geometric progression. The paramet e r 6i determines the density of the spectrum. The Weierstrass function has interesting mathematical properties which are important to the present study. In this expressin of the Weierstrass function the phases of all the frequncy modes match x=0 and as x increases the phase differences between the modes appear. This function can be made stochastic by introducing a random phase for each mode of the Weierstrass spectrum. However, the apearance of randomness in the profile generated by the function can also be achieved by evaluating the function starting at x=x,x£0 instead of x=0. The choice of x determines the relative phase differences between the frequency modes. The power spectrum SCoj) follows as 2A2 SCu) = C 5-2D) n 1 The dimension D of the profile determines the slope of the power spectrum in logSCü>}~logcı> plot. It is observed that the power spectrum for a large number of different types of surfaces have similar characteristic and follow 1 SC6>) - k to xvixiFor most experimental cases the value of k varies between 1.7 and £.3. üuriace processing uattens the power spectrum at lower frequencies thus creating a corner in it. In order to characterize processed surfaces the Weierstrass function is split into two parts, one characterizing the surface at frequencies below the corner frequency and the other for higher frequencies where the surface behaves as anatural one. It is concluded that the fractal dimension of 1.0 and a scaling constant in the Weierstrass function are universal for all natural surfaces is length of the sample. This physicallly explanation for this phenomenon is that every surface processing technique, such as machining, has a finite frequency of influence beyond which the surface remains unaffected and behaves as a natural surface. This provides a very useful method. of generating artificial profiles which can simulate real profiles. Since the characterization parameters A. D and u> depend on the machining process, standardization of these values for different machining processes would enable this method to generate profiles whitout the use of extensive and costly exper imentation. It is known that the friction loss in the turbulent pipe flow is a function of the Reynolds Number of flow and the relative roughness k/'D of the -pipe wall. This nondimensional functional relationship has been verified by Nikuradse's experimental data on turbulent flow in smooth and various artificially roughened pipes. Numerous empirical formulas”were proposed along this line of reasoning. In the smooth pipes the coefficient of friction X depends on only the Reynolds number. On the other hand, for turbulent flow in completely rough pipes X is independent of the Reynolds number because the turbulence in completely rough pipes is largely due to the pipe wall roughnesses The varification of turbulent pipe flow theory presented in preceding articles has been based on Nikuradse's experimental data on turbulent flow in smooth and various artificially roughened pipes. However, Nikuradse's experimental curves cannot be used directly to evaluate the coefficient of friction X for commercial pipes because the actual wall roughness of commercial pipes does not have a pattern geometrically similar to the uniform sand roughness used by Nikuradse In most commercial pipes both the roughness distribution and the geometrical shape of roughness are different from those of artificial sand roughness. Obviously, it is impossible to describe the irregular surface roughnesses found in commercial pipes in termd of the height of uneveness k or by a single nondimensional parameter k/D. Nevertheless, it has been found convenient to set up an arbtrary measuring scale corresponding to Nikuradse's sand roughnesses forall commercial pipes. The roughness factors for commercial pipes are reduced to a common scale cf equivalent sand roughness factor k/D. Since fluid turbulance in completely rough pipes depends only the pipe wall roughnesses, the equivalent roughness factor k/D for any commercial pipe may be determined experimentally by first computing X for a given turbulent flow in a completely rough region, and then substituting the value of X into the following equation 1 IV2 = 1.74 + 2.01og VX 10 k to solve for an equivalent value of k/D. In the transition region where the value of X depends on both the pipe wall roughness k and the fluid viscosity (i.e., the Reynolds number of flow), the results obtained from commercial pipes would be expected to differ from those of sand- r oughened pipes. This difference can readily be be seen from the figures in which test results from both Nikuradse's artificially roughened pipes and commercial pipes are plotted. The plotting for all commercial pipes was worked out by C.F. Colebrook, and the experimental data for commercial pipes in the transition region are seen to deviate from Nikuradse's data on turbulent flow in sand- roughened pipes. The coordinates are chosen to be IvVX -2,01og CEV2k) and ReVX/CD^k), so that the equation for smooth pipe appears as a straight line with a slope of 2.0, and the rough-pipr equation plots as a horizontal line. The experimental points for commercial pipes are seen to.follow closely a single curve which is asymptotic to both the smooth-pipe and the rough-pipe equations. Colebrook developed an empirical equation for the curve representing the transition region for commercial pipes, rV2k ?* 1+18.7 ReVX - 2.01og =1.74. -2.01og VX lû 2k which may now be used to predict the coefficient of friction X in commercial pipes. However, the complex form of equation is not entirely suitable for engineering applications; L.F.Moody has plotted the equation to appear in the form of a series of X versus Re curves for various values of D/k. Moodys diagram is essentially the same as Nikuradse's, except for the transition regions. The experimental results in this study about the commercial pipes included the actual, ;wall roughness »have shown geometrically similar curves ( to the Moody's diagram except the magnltute of IVk. And then using the Colebrook* s formula, the artificially sand roughness of the commercial pipes has been IX.calculated as mm. The difference between the Colebrook's and Nikuradse's curve appears due to the diiierent behaviour ol the iluld on the actual wailand artificial roughened surface for turbulent flow. While the curve begins to leave the Blasius line for the transition region, the viscos effects show a decreeasingand the turbulent effects show an increasing and the otherwise the thickness of boundary layer C6) and also laminar region tends to decrease. If the thickness of the boundary layer equals to the roughness k, the fluid elements which shows the turbulent character contact to the roghnesses. On the artificial roghness, the profile contact to the turbulent flow shows different character (mean slope, mean curvature) as geometrically for the different boundary layer thickness. It tends to occur the different turbulent effects for the thin and thick boundary layers for k>6>0.. For the the thick boundary layers the turbulent flow contact to the bottom side of the sand roughness and the laminar affect still goes on. But for the thin boundary layers the turbulent flow contact to all of the body of the sand roughness and it also apears turbulent affect. The decreasing of the laminar affects and the increasing of the turbulent affects does not occur balanced on the transition region. But for the actual wall rougness, the profile has a fractal character and -the geometrical shape does not change increasing or dicreasing the boundary layer thickness. Because, the fractal profiles are self similar and the profile region which contacts to the turbulent flow shows the same geometrical character for different boundary layer thickness. Thus, The transition of the curve from the Blasius line to the X.=constant line is balanced. rscxi
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