Laws of Similarity in Fluid Mechanics 21

Laws of similarity in fluid mechanics B. Weigand1 & V. Simon2 1Institut für Thermodynamik der Luft- und Raumfahrt (ITLR), Universität Stuttgart, Germany. 2Isringhausen GmbH & Co KG, Lemgo, Germany. Abstract All processes, in nature as well as in technical systems, can be described by fundamental equations—the conservation equations. These equations can be derived using conservation princi- ples and have to be solved for the situation under consideration. This can be done without explicitly investigating the dimensions of the quantities involved. However, an important consideration in all equations used in fluid mechanics and thermodynamics is dimensional homogeneity. One can use the idea of dimensional consistency in order to group variables together into dimensionless parameters which are less numerous than the original variables. This method is known as dimensional analysis. This paper starts with a discussion on dimensions and about the pi theorem of Buckingham. This theorem relates the number of quantities with dimensions to the number of dimensionless groups needed to describe a situation. After establishing this basic relationship between quantities with dimensions and dimensionless groups, the conservation equations for processes in fluid mechanics (Cauchy and Navier–Stokes equations, continuity equation, energy equation) are explained. By non-dimensionalizing these equations, certain dimensionless groups appear (e.g. Reynolds number, Froude number, Grashof number, Weber number, Prandtl number). The physical significance and importance of these groups are explained and the simplifications of the underlying equations for large or small dimensionless parameters are described. Finally, some examples for selected processes in nature and engineering are given to illustrate the method. 1 Introduction If we compare a small leaf with a large one, or a child with its parents, we have the feeling that a ‘similarity’ of some sort exists. On the other hand, the leg of a mouse looks quite different from the leg of an elephant and it is hard to imagine that they might be similar in some respect. So the question arises what ‘similarity’means and how it can be expressed mathematically. This question is of great importance for all biological as well as technical systems, and it might be worthwhile to investigate this question here in the present book. WIT Transactions on State of the Art in Science and Engineering, Vol 3, © 2006 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) doi:10.2495/1-84564-001-2/1b Laws of Similarity in Fluid Mechanics 21 In daily life, we are used to thinking in terms of dimensions and to measuring them using a clock or a metre stick. For example, we think of a mouse as an animal that is about 7 cm in length with legs that are relatively small. One might ask, what we mean by ‘relatively small’, and we might answer that the legs of a mouse are relatively small compared with the size of our own legs. This clearly shows that it is meaningless to state that a certain length is small or large. One must always compare one length with another. If we compare one length with another, which is a part of the same problem (e.g. the height or length of the animal), then we obtain a dimensionless quantity. With this dimensionless quantity we can scale the problem, leading to geometric similarity. Geometric similarity was already known to the ancient Greek mathematicians in connection with geometrical problems, as has been noted by Galileo in 1638 [1]. If additional quantities like time or mass are present in the problem, dimensional analysis can be used to obtain a set of dimensionless groups that completely determine the problem. If all these dimensionless groups are set to be constant, one obtains full similarity between two processes, even though the ‘real’ dimensional quantities like length and weight are different. In 1914, Buckingham [2] stated a basic theorem (Buckingham or pi theorem) which showed that every equation that is written in terms of dimensional variables can be converted into an equation that is governed only by dimensionless quantities, and, most important, that the number of dimensionless quantities is lower than the number of dimensional quantities. In other words, dimensional analysis is a means to reduce the number of independent variables and therefore reduces the effort to find a solution to a given problem. Lord Rayleigh [3] developed a systematic method for manipulating the exponents of the various dimensional entities to yield dimensionless groups. Nowadays, similarity and dimensional methods are used in numerous engineering problems (e.g. in hydrodynamics [4], heat transfer [5], magnetohydro- dynamics [6] and meteorology [7] among others). General discussions of the method and many interesting applications can be found in [8–12]. Similarity and dimensional analyses have also been used in biology and for a more detailed discussion the reader is referred to the paper by Stahl [13]. In the present paper, we first explain the basics of dimensional analysis. The method is explained by solving a simple example to show its ease of use. Then, we introduce some well-known dimensionless groups. These dimensionless groups describe the magnitude of certain physical processes that are important in fluid flow. Finally, some examples of similarity in nature and engineering are given, demonstrating the benefits of the method. 2 Dimensional analysis Dimensional analysis is a powerful method, which can be used in all physical applications. The method is based on the fact that all physical quantities have units, which can be split into ‘basic’ or ‘primary’ units and ‘secondary’ units. The basic units are characterized by the fact that they are independent of each other. A basic unit cannot be constructed from a combination of other basic units. It is customary to use the International System of dimensions, the SI system, with the units given in Table 1 as basic units. However, it is possible to use other basic units than those of the SI system. Beside the basic units given in Table 1, there are derived or secondary units. Velocity, for example, is the unit length L divided by time τ. This can be denoted by [Lτ −1], whereas density is the unit mass M divided by length cubed L3, which can be written as [ML−3]. Table 2 shows a list of important secondary quantities in fluid mechanics. In general, one can say that for any problem there is always the task to express a physical quantity p1 as a function of other physical quantities p2, p3, ..., pn, where n is the number of WIT Transactions on State of the Art in Science and Engineering, Vol 3, © 2006 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) 22 Flow Phenomena in Nature Table 1: Basic quantities, their dimensions and units, given in the SI system. Basic unit Dimension SI unit Length L m (metre) Mass M kg (kilogram) Temperature ϑ K (kelvin) Time τ s (second) Species amount N mol (mole) Electric current I A (ampere) Luminous intensity S cd (candela) Table 2: Some often used physical quantities with their dimensions. Physical quantity Symbol Dimension −1 Velocity of sound as Lτ Acceleration a Lτ −2 Angular velocity ωτ−1 Diffusion coefficient D L2τ −1 Density ρ ML−3 Dynamic viscosity η ML−1τ −1 Energy E J = ML2τ −2 Frequency f τ −1 Force F N = MLτ −2 Heat capacity C L2τ −2ϑ−1 Heat transfer coefficient h Mτ −3ϑ−1 Kinematic viscosity ν L2τ −1 Power P W = ML2τ −3 Pressure p Pa = ML−1τ −2 Momentum I MLτ −1 Resistance force W N = MLτ −2 Surface tension σ Mτ −2 −1 −2 Shear stress τ jk ML τ Thermal conductivity k MLτ −3ϑ−1 Thermal diffusivity α L2τ −1 Velocity U, V, W Lτ −1 physical variables needed to describe the problem under consideration ([9, 11, 12]). This means that a relation of the form f ( p1, p2, ..., pn) = 0 (1) WIT Transactions on State of the Art in Science and Engineering, Vol 3, © 2006 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Laws of Similarity in Fluid Mechanics 23 will always exist. Each of the physical variables pj has dimensions that can be constructed from the basic units given above. The dimension of the variable pj, denoted by pj , is then given by m aij pj = (Xi) . (2) i=1 In this equation Xi denotes one of the basic units powered by the exponent aij, m is the number of basic units involved and the index j can vary from 1 to n. For example, the dimension of the physical quantity ‘velocity’ is −1 [U] = p1 = L τ , X1 = L, X2 = τ, a11 = 1, a21 =−1. (3) Since every physical process must be independent of the arbitrary units used, the relation given by eqn (1) can be reduced to a relationship between dimensionless quantities j, so that f ( 1, 2, ..., d) = 0, (4) where d is the number of the dimensionless products. These dimensionless quantities are products of the physical quantities, which means n bj = pj , (5) j=1 with the dimension n bj [ ] = 1 = [pj] . (6) j=1 Inserting eqn (2) into eqn (6) leads to n m aijbj [ ] = 1 = (Xi) . (7) j=1 i=1 In eqn (7) the coefficients aij are known and the exponents bj have to be determined such that the equation is satisfied. This is always possible if the system of equations n aijbj = 0 (8) j=1 has non-trivial solutions for the n unknown coefficients bj. The coefficient matrix aij is also known as the dimension matrix. The equivalence between eqns (1) and (4) is known in the literature as the ‘Buckingham theorem’ or ‘pi theorem’. The reader is referred to the work by Görtler [9] or Spurk [11] for a more detailed discussion of this topic.This theorem also gives a relation between the number of physical variables involved, n, and the number of independent dimensionless groups, d, according to d = n − r, (9) where r is the rank of the dimension matrix.

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