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 principles 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, <a href="/goto?url=http://www.witpress.com" target="_blank">www.witpress.com, ISSN 1755-8336 (on-line) </a>doi:10.2495/1-84564-001-2/1b © 2006 WIT Press 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 compareonelengthwithanother. Ifwecompareonelengthwithanother, whichisapartofthesame 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], magnetohydrodynamics [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, <a href="/goto?url=http://www.witpress.com" target="_blank">www.witpress.com, ISSN 1755-8336 (on-line) </a> © 2006 WIT Press 22 Flow Phenomena in Nature Table 1: Basic quantities, their dimensions and units, given in the SI system. <ul style="display: flex;"><li style="flex:1">Basic unit </li><li style="flex:1">Dimension </li><li style="flex:1">SI unit </li></ul>Length Mass Temperature Time LMϑτm (metre) kg (kilogram) K (kelvin) s (second) Species amount Electric current Luminous intensity NISmol (mole) A (ampere) cd (candela) Table 2: Some often used physical quantities with their dimensions. <ul style="display: flex;"><li style="flex:1">Physical quantity </li><li style="flex:1">Symbol </li><li style="flex:1">Dimension </li></ul>−1 Velocity of sound Acceleration Angular velocity Diffusion coefficient Density as aLτ −2 L−τ1 <ul style="display: flex;"><li style="flex:1">ω</li><li style="flex:1">τ</li></ul>DρL2τ−1 ML−3 Dynamic viscosity Energy Frequency ηML−1τ−1 J = ML2τ−2 τ−1 E f F C hν<ul style="display: flex;"><li style="flex:1">Force </li><li style="flex:1">N = MLτ−2 </li></ul>−2 −1 Heat capacity Heat transfer coefficient Kinematic viscosity Power Pressure Momentum Resistance force Surface tension Shear stress Thermal conductivity Thermal diffusivity Velocity L2τ ϑMτ−−31ϑ −1 L2τ P p I WW = ML2τ−3 Pa = ML−1τ−2 MLτ−1 N = MLτ−2 σMτ−2 τjk kαML−1τ−2 MLτ−3ϑ−1 L2τ−1 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, <a href="/goto?url=http://www.witpress.com" target="_blank">www.witpress.com, ISSN 1755-8336 (on-line) </a> © 2006 WIT Press 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 mꢂ ꢀ ꢁ pj =(Xi)aij . (2) i=1 In this equation Xi denotes one of the basic units powered by the exponent ai j, 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 <ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul>[U] = p1 = L τ−1 ,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 <ul style="display: flex;"><li style="flex:1">f (ꢀ1, ꢀ2, . . . , ꢀd) = 0, </li><li style="flex:1">(4) </li></ul>where d is the number of the dimensionless products. These dimensionless quantities are products of the physical quantities, which means nꢂ ꢃ ꢄ bj ꢀ = pj <ul style="display: flex;"><li style="flex:1">,</li><li style="flex:1">(5) </li></ul>(6) j=1 with the dimension nꢂ[ꢀ] = 1 = [pj]bj . j=1 Inserting eqn (2) into eqn (6) leads to <ul style="display: flex;"><li style="flex:1">n</li><li style="flex:1">m</li></ul>ꢂ ꢂ (Xi)a b .(7) <ul style="display: flex;"><li style="flex:1">ij </li><li style="flex:1">j</li></ul>[ꢀ] = 1 = 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 nꢅai jbj = 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. Theequivalencebetweeneqns(1)and(4)isknownintheliteratureasthe‘Buckinghamtheorem’ or ‘pi theorem’. The reader is referred to the work by Görtler [9] or Spurk [11] for a more detailed discussionofthistopic.Thistheoremalsogivesarelationbetweenthenumberofphysicalvariables involved, n, and the number of independent dimensionless groups, d, according to d = n − r, where r is the rank of the dimension matrix. (9) WIT Transactions on State of the Art in Science and Engineering, Vol 3, <a href="/goto?url=http://www.witpress.com" target="_blank">www.witpress.com, ISSN 1755-8336 (on-line) </a> © 2006 WIT Press 24 Flow Phenomena in Nature As an example to illustrate the use of dimensional analysis, we consider the resistance of a sphere that is exposed to a fluid flowing at a uniform velocity. Let us assume that the diameter of the sphere is D and the velocity of the fluid is U. Furthermore, let us assume that the flow is incompressible and the fluid properties can be considered to be constant. We now determine the resistance W of the sphere. Investigating the problem, one finds that the following quantities have to be a part of the solution: the approach velocity U, the viscosity η, the density of the fluid ρ and the diameter of the sphere D. Therefore, W = f (U, D, η, ρ). (10) This means that we can construct the dimension matrix as shown in Table 3, where the dimensions of all quantities are expressed in basic units.The rank of the dimension matrix r = 3, which implies that the number of dimensionless products is d = n − r = 5 − 3 = 2. In order to determine the exponents bj, one has to solve eqn (8). However, sometimes one can find the dimensionless groups much quicker by multiplying (or dividing) the physical quantities in such a way that the elements of the dimension matrix become zero. Working on Table 3 in this way, we obtain the modified dimension matrix as shown in Table 4. We conclude from Table 4 that η cannot be included in the problem, because it is the only quantity left with the dimension of time. If we drop η, then ρ cannot be included in the problem, because it is the only quantity left with the dimension of mass. Finally, if we drop η, then D cannot be included in the problem, because it is the only quantity left with the dimension of length. Therefore, it follows that the functional relationship can only be of the form <ul style="display: flex;"><li style="flex:1">ꢆ</li><li style="flex:1">ꢇ</li></ul>WρU2D2 UDρ = f .(11) ηEquation (11) shows that the non-dimensional resistance of two different spheres is similar, if the dimensional groups of this equation are the same. It has to be highlighted here that it has been Table 3: Dimension matrix for the problem of a sphere in uniform flow. W b1 U b2 <ul style="display: flex;"><li style="flex:1">η</li><li style="flex:1">ρ</li></ul>D<ul style="display: flex;"><li style="flex:1">b5 </li><li style="flex:1">b3 </li><li style="flex:1">b4 </li></ul>LMτ11 −2 10 −1 −1 1 −1 −3 10 100Table 4: Modified dimension matrix for the problem of a sphere in uniform flow. <ul style="display: flex;"><li style="flex:1">W/(ρU2D2) </li><li style="flex:1">UDρ/η </li></ul><ul style="display: flex;"><li style="flex:1">η</li><li style="flex:1">ρ</li></ul>DLMτ000 000 −1 1 −1 −3 10 100WIT Transactions on State of the Art in Science and Engineering, Vol 3, <a href="/goto?url=http://www.witpress.com" target="_blank">www.witpress.com, ISSN 1755-8336 (on-line) </a> © 2006 WIT Press Laws of Similarity in Fluid Mechanics 25 possible to obtain a functional form of the solution without considering any specific equation. This is a very important fact: even if we do not know the equations that describe a particular process, dimensional analysis provides us with the functional dependence of the solution. 3 Dimensional groups and their physical significance By applying dimensional analysis to the problem discussed above, we already obtained the important dimensionless group <ul style="display: flex;"><li style="flex:1">UDρ </li><li style="flex:1">UD </li></ul>Re = =.(12) <ul style="display: flex;"><li style="flex:1">η</li><li style="flex:1">ν</li></ul>This dimensionless group is called the Reynolds number, in honour of the researcher O. Reynolds. The tradition of naming dimensionless groups in the honour of famous researchers is based on a suggestion made by Gröber [14]. In order to obtain all dimensionless groups that are important for fluid mechanics processes one can use the underlying conservation equations <ul style="display: flex;"><li style="flex:1">∂ρ </li><li style="flex:1">∂</li></ul>+(ρuk) = 0 Mass, (13) ∂t ∂xk <ul style="display: flex;"><li style="flex:1">∂uj </li><li style="flex:1">∂uj </li></ul>∂τjk ∂p ρ+ ρuk <ul style="display: flex;"><li style="flex:1">= − </li><li style="flex:1">+</li></ul>+ ρfj Momentum, (14) (15) ∂t <ul style="display: flex;"><li style="flex:1">∂xk </li><li style="flex:1">∂xj </li><li style="flex:1">∂xk </li></ul>∂uj <ul style="display: flex;"><li style="flex:1">∂T </li><li style="flex:1">∂p </li></ul>∂qk <ul style="display: flex;"><li style="flex:1">ρcpuk </li><li style="flex:1">= uk </li></ul>++ τij ∂xi ,i, j, k = 1, 2, 3 Energy, ∂t <ul style="display: flex;"><li style="flex:1">∂xk </li><li style="flex:1">∂xk </li></ul>where uj denotes the velocity components associated with the coordinates xj, and ρ, p and cp are the fluid properties (density, pressure and specific heat at constant pressure, respectively). τij are the shear stresses and qk is the specific heat flux. The shear stress tensor in eqns (14) and (15) has to be related to the working fluid under investigation (e.g. Newtonian fluid, viscoelastic fluid). The reader is referred to the work by Spurk [15] for a detailed discussion of these aspects. In addition, a relation between the heat flux vector and the temperature field has to be prescribed. Equations (13)–(15) have to be solved with appropriate boundary conditions, which are normallytheno-slipconditionsatsolidsurfacesandboundaryconditionsfortheflowattheinlet/outlet of the domain. In addition, boundary conditions have to be specified for the energy equation. Here, normally the surface temperature or the heat flux distribution at the surface (or a combination of both) is given in combination with suitable boundary conditions at the inlet/outlet of the flow domain. If we non-dimensionalize eqns (13)–(15), we obtain several dimensionless groups. When these are set constant, full similarity is obtained for differently scaled processes. In order to elucidate the method for obtaining these dimensionless groups, we provide a simple example here for illustration. Let us consider the flow through a parallel plate channel, which is oriented vertically upwards. The configuration is depicted in Fig. 1. We consider a laminar, boundary layer type flow of a Newtonian fluid. The fluid properties are assumed to be constant (except the density) and we use the Boussinesq approximation [16] to relate the density difference to the temperature difference. Using the simplifications already mentioned, the conservation equations can be simplified to

Laws of Similarity in Fluid Mechanics 21

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