Akışkanlar mekaniğinde tansörler
Tensors in fluid mechanics
- Tez No: 39628
- Danışmanlar: Y.DOÇ.DR. A. KORHAN BİNARK
- Tez Türü: Yüksek Lisans
- Konular: Makine Mühendisliği, Mechanical Engineering
- Anahtar Kelimeler: Akışkanlar mekaniği, Enerji, Tensör analizi, Fluid mechanics, Energy, Tensor analysis
- Yıl: 1994
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
Bu çalışmada birçok mühendislik probleminde adını duyduğumuz, çok boyutlu incelemelerde işlekçe kullanılabilen tansörter ve tansör notasyonu, bir mühendisin matematiğe bakması gereken gözle bakılarak ele almağa çalışılmıştır. Konunun özgünlüğü ve nadirliği nedeniyle temel kavramlara inilerek tarifler yapılmıştır. Tansörlerin diğer matematiksel uygulamalardan notasyon olarak farklılığı ve bunun sonucu olarak uygulamadaki kolaylığı, bilgisayara kolay adaptasyonu; boyut, değişken bağımlılığı ve geometri gibi konuların doğrudan probleme dahil edilebilmesi, bilhassa yurdumuzdaki uygulamalarda örneklerine pek fazla rastlanmaması gibi nedenler konuyu cazip kılmaktadır. Konuyu anlaşılır işlemek için örneklerden ve özel durumlardan sık sık faydalanılm ıştır. Bu çalışmada, yüzey üzerindeki akışların dinamik sınır tabakası problemlerine yönelmek üzere akışkanlara ait temel hareket denklemleri tansörel formda çıkarılmış, bir kanat profiline uygulamak üzere, örnek bir dönüşüm ve bunun ilgili parametreleri tarif ve hesap edilmiştir. vu
Özet (Çeviri)
This study is related to the tensor analysis and its applications, especially on Fiuid Mechanics. in the study, tensors and basic related concepts are given, equations of fluid motion in tensorial notation are presented, and by considering a problem, a coordinate transformation and its parameters are calculated. Tensor calculus, as a subject, is wide and complicated. Some applications of it are presented in available studies on Fluid Mechanics, Elasticity, Thin Layer Theory, ete. Ali these tensorial applications are based on the physical meaning and mathematical representation of tensor concept. Tensors are magnitudes that may contain more than öne dimension and component Therefore, tensorial equations may represent many scalar equations at önce. Describing the tensor as a mathematical magnitude including some componertts is not enough. it also has some physical meanings wfth the components shown by subscripts ör superscripts. Covariant components are shovvn by using subscripts and contravariant ones are shovvn by using superscripts. Contravariant ones are described parallel to the axes and covariant ones are perpendicular. Due to these descriptions, covariant and contravariant characteristics are physically same in Cartesian coordinates. On the contrary, in the case of most other kinds of coordinate, especially for curvilinear ones, distinguishing them from each other is of vrtal importance both mathematicaily and physicaliy. The space, which the problem is considered in must be selected as convenient as possible to the behavior of the solution. Therefore, most of characteristics in tensor calculus are based upon the coordinate transformations. The equations in tensorial notation may indude some dimensional magnitudes that do not have ail of the tensorial properties. Christoffel Symbol (r) vvhich is three indiced is such a magnitude. it is not a tensor but it plays some important roles in the equations due to the characteristics of transformation. To describe a transformation and its inverse properly, the condition J*0 must have been satisfied, vvhere J (Jacobian) represents the determinant of the transformation matrbcAlgebraic calculations are described for tensors, similar to vectors. Therefore, multiplication, summation and the other characteristic operations are described for tensor calculus. In addition, since tensors may be multidimensional, they also have some other peculiarities. Among these, summation convention and many kinds of symmetry are of great importance. Generally, with an equation, the range of a suffix is written on the right, separately. For instance, in a three-dimensional coordinate system, the range of index i is shown asI «1,2, 3. This representation is used for all suffixes in a tensorial equation. According to this notation, the stated suffixes are to be considered for all integer values of the range. Due to summation (Einstein) convention, a repeated index of a variable has a different meaning. In such a case, the variable is equal to a summation that is described as follows : A'B,, = ± A%i = A1^ + A2B2J + A3B3J (1) In tensor algebra, the metric tensor (g) as a special tensor of fundamental importance is calculated by considering the coordinate transformation. For the case of Cartesian coordinates, metric is the same with the Kronecker delta (S) which can be described similar to unit matrix in matrix algebra. In a tensorial equation, appropriateness of suffixes for all terms separated by summations or equilibrium must be satisfied property. For instance, a superscript must have been located on every term (single or production) in the same order. This is why, obeying this appropriateness indices are interchangeable. This property is used to combine two or more equations. However, it is necessary to prevent any confusion of index that may occur after such substitutions or during the calculations. When a certain problem is considered, first of all, the metrics and symbols of Christoffel are calculated with all components due to the geometry and transformation. Covariant or absolute derivatives of a tensor of any order are described by using these calculated values. Among these covariant derivatives Vka> = |£ + Tia1 J,k,l=1,2n, (2) *k v*»* = İ5r - rİ»ı i,k,l=1,2,...,n, (3) Ö3T aVAj v^î-^-r^î + riAîIJ,U=1,2n, (4) dy VA** vkA? = -f-riA? + r^*-r^J ıj,k,p,M,2,...,n (5) may be shown. Furthermore, some well-known operatöre (Div., Grad, Rot, Lap.) depend on these vaiues and derivatives by the formulas Grad^ = --j- öy DivV = VjV1, RotV = VjV1 - V,vJ I^V^^-r;^).(6) Added to these, according to Ricci's Theorem Vkg* - o\ T.k9*“n°)ia,k=1,2n, (7) vff* * ° / Vg« = 0/ Vh»i-Vk(g^*).g^Vk»aU,K=1,2,...,n, (8) Vk85 = oIJ,k=1,2,...,n (9) may be written. Thus, needed magnitudes for a problem can be calculated. As an application, equations of motion for a Nevvtonian fluid may be written in tensorial notation. Comparing wrth vectorial equations, especially for curvilinear coordinates, these basic equations of fluid motion are very complicated; however, in the case of curvilinear surface flow, usage of tensors is much more convenientIn the case of incompressible flow, negligible temperature differences and ideal gas as fluid, after some suitable assumptions, 3v' + ripr1 = 0 İJ-1,2,3 (10) as Continuity Equation, ?t aV,”aV, av- av' +^ + rir ?ay'ay* -ay'“-ay* ”ay- ^ay m kl -rplH* v<g + iy)+ V^ - x* = o IJ,k,l,m*1,2,3 (11) as Momentum Equation, as Energy Equation and p«pRT h*CpT e*CvT Cp-CV*R (13) as Constitutive Equations may be used. The metrics and Christoffel symbols in the equations of motion are calculated by the formulas &j = g«P. i., j, a,p,l,J=1,2,...,n (14) g" = ^nr, IJ=1,2n (15) XITL - Vfe + %£- %t) IJ,M=1,2n (16) 2 oy öy öy The value of metric is zero for perpendicular coordinates, and gcc = 8CC = 1 may be written for linear coordinates. These simplifications make the calculations of the flow around an airfoil easier. By using the tensor analysis it is also possible to examine a flow around a body in motion. Flow around a compressor blade of an aeroplane is such a complicated problem. It is very difficult to examine these kinds of motion in scalar notation. Also some of the time dependent problems are more suitable to examine in tensor notation. Albert Einstein did such a study and presented the results that are known as Relativity Theories. Xll
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