Laws of Similarity in Fluid Mechanics 21

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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], 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

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.

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 W UDρ = f . (11) ρU2D2 η

Equation (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.

WU ηρD b1 b2 b3 b4 b5

L1 1−1 −31 M10110 τ −2 −1 −100

Table 4: Modiﬁed dimension matrix for the problem of a sphere in uniform ﬂow.

W/(ρU2D2) UDρ/η η ρ D

L0 0−1 −31 M0 0110 τ 00−100

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 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 UDρ UD Re = = . (12) η ν 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

∂ρ ∂ + (ρuk) = 0 Mass, (13) ∂t ∂xk

∂uj ∂uj ∂p ∂τjk ρ + ρuk =− + + ρfj Momentum, (14) ∂t ∂xk ∂xj ∂xk

∂T ∂p ∂qk ∂uj ρcpuk = uk + + τij , i, j, k = 1, 2, 3 Energy, (15) ∂t ∂xk ∂xk ∂xi 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). τ ij 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 normally the no-slip conditions at solid surfaces and boundary conditions for the flow at the inlet/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

∂u ∂v + = 0, (16) ∂x ∂y

Figure 1: A vertical two-dimensional channel with laminar internal ﬂow.

∂u ∂u 1 ∂p ∂2u u + v =− + gβ(T − T ) + ν , (17) ∂x ∂y ρ ∂x w ∂y2

∂T ∂T ∂2T u + v = α , (18) ∂x ∂y ∂y2 with the boundary conditions

y = L: u = v = 0, T = Tw, ∂u ∂T y = 0: = 0, = 0, (19) ∂y ∂y

x = 0: u = u0, T = T0.

In eqns (16)–(19) u and v are the velocity components in the x and y directions, T and p denote the temperature and the pressure, β is the volumetric coefﬁcient of thermal expansion [16], ν is the kinematic viscosity, and α is the heat diffusivity. Furthermore, we assume that the problem under consideration is steady and that the constant velocity and temperature at the entrance of the duct are given by u0 and T0. Let us now non-dimensionalize the equations given above. This can be achieved by introducing the following quantities:

x y u v p T − T x˜ = , y˜ = , u˜ = , v˜ = , p˜ = , = w (20) L L u0 u0 p T0 − Tw into eqns (16)–(19). This results in ∂u˜ ∂v˜ + = 0, (21) ∂x˜ ∂y˜

∂u˜ ∂u˜ p ∂p˜ gβ(T − T )L ν ∂2u˜ u˜ +˜v =− + 0 w + , (22) ˜ ˜ 2 ˜ 2 ˜2 ∂x ∂y ρu0 ∂x u0 u0L ∂y

∂ ∂ α ∂2 u˜ +˜ = v 2 , (23) ∂x˜ ∂y˜ u0L ∂y˜

y˜ = 1: u˜ =˜v = 0, = 0,

∂u˜ ∂ y˜ = 0: = 0, = 0, (24) ∂y˜ ∂y˜

x˜ = 0: u˜ = 1, = 1.

From eqns (20)–(24) we can see that the following dimensionless groups have been generated during the scaling process: u L 0 = Re, (25) ν

p = 2 Eu, (26) ρu0 − − 3 2 gβ(T0 Tw)L = gβ(T0 Tw)L ν = Gr 2 2 2 , (27) u0 ν u0L Re u L u L ν 0 = 0 = Re Pr. (28) α ν α As already mentioned, the dimensionless group given by eqn (25) is the Reynolds number. From the above derivations it is clear that the Reynolds number compares the inertial forces to the viscous forces; it determines the behaviour and characteristics of viscous flow patterns; it is one of the key parameters to indicate whether a flow is laminar or turbulent. In pipe flow, for example, the flow is laminar as long as the Reynolds number (based on the pipe diameter and the mean throughflow velocity) is well below a critical value of about 2000–3000. For Reynolds numbers higher than 3000, the flow turns turbulent in technical piping systems. Very low values of the Reynolds number indicate that the flow is dictated by viscous forces and therefore the conservation equations might be simplified by ignoring the inertial terms on the left hand side of the Navier–Stokes equations. For more details on this subject, the reader is referred to Schlichting [17]. Equation (26) defines the Euler number. It is the ratio of the pressure force to the inertial force. In eqn (27), we have introduced the Grashof number. This quantity is important for all processes in which buoyancy has an effect. If the flow is only driven by density differences, this number has a similar importance as the Reynolds number. Depending on the value of the Grashof number, the flow might be considered to be laminar or turbulent. If the problem contains a superimposed flow velocity (as in the present example), the group Gr/Re2 is a measure of the relative importance of free convection effects in a forced convection problem [16]. The Prandtl number, appearing in eqn (28), contains only fluid properties. It can be shown [16] that the Prandtl number compares the hydrodynamic with the thermal boundary layer thickness. This means that a flow at a high Prandtl number has a thinner thermal boundary layer than a flow at a smaller Prandtl number, provided all other dimensionless quantities are unchanged. Similar to the derivation given above, several other dimensionless groups can be obtained from the conservation equations for different applications (two-phase flows, mass transfer, compressible flows, etc.). A very good summary of many dimensionless groups can be found in [18]. Table 5 gives a selection of important dimensionless groups together with their definition and area of use.

Table 5: Important dimensionless groups (dimensionless numbers) and their area of application.

Number name Symbol Deﬁnition Area of application

Archimedes Ar gρL3/(ν2ρ) Motion of liquids due to density differences Biot Bi hL/k Heat/mass transfer problems Capillary Ca ηU/σ Atomization, two-phase flows 2 Eckert Ec U /(2cpT) Compressible flow Fourier Fo αt/L2 Unsteady heat conduction processes Froude Fr U2/(Lg) Waves and surface behaviour Grashof Gr gβTL3/ν2 Free convection Knudsen Kn lm/L Gas dynamics Lewis Le Sc/Pr Combined heat and mass transfer processes Mach Ma U/a Compressible flow Nusselt Nu hL/k Heat transfer Ohnesorge Oh η/(ρσL)1/2 = We1/2/Re Atomization and sprays Peclet Pe Re Pr Heat transfer Prandtl Pr ν/α Heat transfer in free and forced convection flows Schmidt Sc ν/D Mass transfer Rayleigh Ra Gr Pr Free convection Reynolds Re UL/ν Viscous flow problems Strouhal Sr fL/U Unsteady flows Stanton St h/(Uρcp) = Nu/(Re Pr ) Heat transfer Sherwood Sh hL¯ /D Mass transfer Weber We ρU2L/σ Droplet dynamics, atomization, sprays

4 Examples

In Section 3, we have seen that it is a straightforward matter to find the specific dimensionless groups relevant for a given problem whenever it is possible to explicitly write down the governing equations and boundary conditions. This always reduces the effort to find a solution for the equations. But even without solving the equations, much can be learned and great insight can be gained. Very often, however, the governing equations may be too complex or may even be unknown, and it is impossible, or at least impractical, to write them down explicitly. In these cases, dimensional analysis can still provide exact results that greatly help to understand the problem. The method also represents a valuable tool to design meaningful experiments and to properly scale the test results to the original. In this section, we give simple examples of problems that occur in nature and apply dimensional analysis, using as much physical insight as we have, but without writing down the governing equations.

4.1 The dynamics of planetary rings

Saturn’s ring system is one of the most fascinating natural phenomena. The ring consists of particles that, following the Keplerian law, travel in circular orbits around the planet with angular

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 29 and circumferential velocities GM = (29) r3 and u = r, (30) respectively. Here, G is the gravitational constant, M is the mass of the central planet and r is the radial position of the particle. Particles that are closer to the planet travel at a greater speed, so that two particles located on neighbouring radial positions will eventually collide. This is the source of a fluctuation velocity of the particles, which is superimposed on the mean circumferential flow velocity. As the particles are inelastic, a fraction of the particle kinetic energy is lost in dissipation: in a collision between two particles, their normal velocity components are related through the coefficient of restitution ε. When ε equals one, collisions are perfectly elastic and energy is conserved. Values of ε less than one involve a loss of kinetic energy. We would like to mention that if the particles were ideally elastic, the particle flow would not be confined to a thin ring system. Rather, the particles would consume all available space around the planet. Only a small amount of inelasticity is necessary to confine the particle flow to a thin ring. The dynamics of the ring system is thus governed by the steady balance between collisional production of the velocity fluctuations as driven by the mean Keplerian shear at a rate du 1 =− (31) dr 2 and its dissipation in collisions. As the analysis of planetary ring dynamics has successfully employed methods from the classical kinetic theory of gases, the mean fluctuation velocity is also called the ‘granular temperature’ and is denoted by θ, having the dimension of velocity squared. We restrict the analysis to a dilute flow of identical smooth spherical particles with diameter D and density ρs. The flow density is denoted by ρ. We now want to determine the magnitude of the granular temperature θ as a function of D, , ρ, ρs and ε, and find

θ = f (D, , ρ, ρs, ε). (32)

Dimensional analysis yields at once √ θ = f (µ, ε), (33) D where the solid volume fraction µ is deﬁned as ρ µ = . (34) ρs √ Since θ increases when µ decreases, we try to expand (D )/ θ in terms of µ to get

2 D µ √ = f1(0, ε) + µf (0, ε) + f (0, ε) +··· . (35) θ 1 2 1

With f1(0, ε) = 0 and neglecting powers of µ higher than the ﬁrst, we get √ µ θ = f (ε). (36) D

The precise functional form of f (ε) can only be determined√ by solving the governing equations of the flow [19]. The dimensionless quantity S = (D/µ θ) appears in every granular shear flow and can be interpreted as a non-dimensional shear rate. We may now define a ring thickness t1 by 2 ∞ t1 = ρ dz, (37) ρs 0 where z denotes the axial coordinate normal to the ring plane. Obviously, t1 is the thickness of√ the ring if the particles were pressed together to form a densely packed solid disc. Using ξ = z / θ as the non-dimensional axial distance, we get √ t µ θ ∞ 1 = 2 µ dξ. (38) D D 0 In astronomy, it is more customary to use a quantity called ‘optical depth’ φ, which is related to t1 by 3 t 3 ∞ φ = 1 = µ dξ. (39) 2 D S 0 Another, second measure of the ring’s thickness may be defined by 2 ∞ t2 = ρ dz, (40) ρ 0 hence t 2 ∞ 2 = µ dξ. (41) D µ S 0 The thickness t2 is a rough measure of the apparent visual thickness of the ring, as it is an exact measure of the ring’s axial extent if ρ were constant. Of course, the value of S cannot be found from dimensional analysis alone. A detailed analytical treatment of the problem [19] provides values of S for a range of values of ε. Roughly, we find that for ε ranging between√ 0 and 0.63 µ ranges between 0.003 and 0.009, D ranges from a few centimetres up to 10 m, θ ranges from 0.002 to 0.005 m/s, φ ranges from 1 to 4, and t2 ranges from 20 m up to 2000 m.

4.2 The run-off from a watershed

The watershed of a river is the territory that is drained by the river. If rain falls on a watershed, the river rises. The discharge of the river will continue to rise for a number of days after the rainfall, as some time is needed for water to drain from the watershed into the river. Obviously, it is of practical importance to determine the amount of discharge V˙ of the river as a function of the time t that has elapsed after the rainfall. Since the run-off V˙ depends on the topography, the vegetation, the nature and saturation of the soil, etc., we will restrict the analysis to geologically and geometrically similar watersheds. In this case, the discharge V˙ depends on the time t, the amount of rainfall H, which is usually expressed as ‘metres of rain’ [i.e. the total rainfall of the watershed (in L3) divided by the area of the watershed (in L2)], the area A of the watershed, the acceleration due to gravity g, the density of water ρ and its kinematic viscosity ν. This leads to an expression of the form f (V˙ , t, ν, H, A, g, ρ) = 0. (42) As ρ is the only quantity that contains the dimension of mass, it is impossible to form a dimensionless product containing ρ. Consequently, ρ cannot be included in the problem. Then, we

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 31 have n = 6 physical variables. The rank of the dimension matrix is r = 2, and we anticipate ﬁnding d = n − r = 2 dimensionless products. These are − − − = VA˙ 5/4 g 1/2, = tA 1/4 g1/2, (43) 1 2 − − − = v A 3/4 g 1/2 and = HA 1/2. (44) 3 4 so that we may express eqn (42) in the equivalent dimensionless form V˙ tg1/2 v H = f , , . (45) A5/4 g1/2 A1/4 A3/4 g1/2 A1/2

It is reasonable to assume that the discharge V˙ is proportional to the amount of rainfall H,so ˙ ˙ that Vand H appear only in the combination V/H. Instead of 1, we introduce ∗ −1 ˙ −3/4 −1/2 −1 = 1 = VA g H , (46) 1 4 which leads to the relation V˙ tg1/2 ν = f , . (47) A3/4 g1/2H A1/4 A3/4 g1/2 This example was given by Langhaar [20] together with experimental data. For different watersheds, the data of non-dimensional discharge as a function of non-dimensional time falls on a = −3/4 −1/2 single curve, suggesting that the parameter 3 νA g has little inﬂuence on this problem. The discharge increases with time until it reaches a maximum for tA−1/4g1/2 ≈ 3, then it decreases. If one considers the complexity of the problem, it is astonishing to see how much insight and information can be gained without writing down or even solving the governing equations, but using purely dimensional reasoning.

4.3 The velocity of ﬂight of birds

Dimensional analysis is routinely used in fluid mechanics and thermodynamics. In biology, however, the method has not received as much attention, even though it can provide quite some insight and explain many of the phenomena that we are used to seeing in daily life. Let us consider, for example, the flight of birds. From observation we know that, in general, large birds fly faster than small birds. For simplicity, we assume that a bird’s wings are shaped like the wings of an airplane. Then, the parameters that most influence the velocity of flight U are the mass of the bird m, the acceleration due to gravity g, the area of the bird’s wings A (i.e. the projection of the wings in the vertical direction) and the density of air ρ. Another important parameter is the angle of attack γ , measured as the angle between the horizontal and the chord of the wing. For simplicity, we restrict the analysis to geometrically similar birds and neglect the effects of viscosity and unsteadiness of the flow. Therefore, we seek a functional relation of the form

U = f (m, g, A, ρ, γ ). (48)

From dimensional analysis, we ﬁnd mg = f (γ ). (49) ρAU 2

This expression states that the weight of the bird mg and the aerodynamic lifting force ρAU 2 must be in balance and are a function of the angle of attack. In other words, the angle of attack must be adjusted in such a way that the aerodynamically produced lift carries the bird’s weight. For geometrically similar birds A ∝ L2 and L ∝ m1/3 so that we have A ∝ m2/3. (50) Then, we find from eqn (49), for constant angle of attack, m 1/2 U ∝ ∝ m1/6. (51) m2/3 This implies, that the characteristic velocity of flight of birds increases with the weight and the size of the bird. The increase in velocity is, however, slower than the increase in weight. Consider, for example, two birds with mass m1 and m2. Their velocities scale as U m 1/6 2 = 2 . (52) U1 m1 If bird 2 weighs twice as much as bird 1, it will fly only about 12% faster; if bird 2 flies at twice the speed of bird 1, it will weigh 64 times as much as bird 1. Another interesting result is found when we look at the frequency ω at which a bird flaps its wings. During flapping, the wings move downward (or upward) with a velocity V ≈ ω L, where L is the length of the wing measured from the tip to the body. Therefore, we find V ωL = . (53) U U From aerodynamics we know that in an ideal flow the ratio V/U is equal to the ratio of the drag force FD to the lifting force FL, therefore, V F c = D = D = tan (γ ) (54) U FL cL where cD and cL are the drag and the lift coefficients, respectively, defined by F F c = D and c = L . (55) D ρAU 2 L ρAU 2

Both cD and cL are functions of γ , much like eqn (49). It is reasonable to assume that birds adjust the angle of attack γ of their wings in such a way that drag becomes a minimum while lift becomes a maximum, so that most of the bird’s energy is used for forward flight and only little energy is consumed to overcome resistance. Thus, cD/cL = constant, and this constant will be the same for all geometrically similar birds. It follows that V/U = constant, and with L ∝ m1/3 and eqn (51) we find from eqn (53) − ω ∝ m 1/6. (56) This confirms the observation that small birds flap their wings faster than large birds [21]. We would like to mention that a more refined description of a bird’s flight must take into account the unsteady movement of the wings, the viscosity of air and, of course, the fact that neither are all birds geometrically similar nor do they all move their wings in a similar fashion, and so the aerodynamics are more complicated. The reader is referred to [22, 23] for more detailed investigations of these problems. Many other more interesting examples of the application of dimensional analysis in biology are given in [13] and [21].

4.4 The resistance of a sphere in low Reynolds number ﬂow

So far, we have used the [LMτ] system as the basic system for dimensional analysis. This is, however, not necessary. As the number of dimensionless products d = n − r, we might try to increase the number of basic units, anticipating that the rank r of the dimension matrix will likewise increase. If this was the case, the number of dimensionless products d would decrease, giving a simpler and sharper result. In order to explore the benefits and limitations of this approach, we consider the problem of determining the resistance of a small animal, characterized by a typical length D, which is immersed in a very viscous fluid flowing at a uniform free stream velocity. The same assumptions that led to eqn (10) hold, so that

W = f (U, D, η, ρ). (57)

Instead of using the [LMτ] system, we now increase the number of basic units by using the [LMFτ] system, treating the force [F] as a primary unit. Then, the dimension matrix is given in Table 6. Now, n = 5 and r = 4, and the only dimensionless product that can be constructed is W = = constant. (58) ηUD The apparent discrepancy between eqns (11) and (58) can be resolved by noting that, based on the [LMFτ] system, Newton’s second law

F = ma (59) is no longer dimensionally homogeneous. In order to render Newton’s law dimensionally homogeneous when using the [LMFτ] system, we must introduce a dimensional constant C, such that

F = Cma, (60) with − − [C] = L 1M 1F1τ 2. (61) In fact, all physical equations require the introduction of dimensional quantities, either in the form of dimensional constants or in the form of physical properties, in order to render them dimensionally homogeneous. Therefore, the proper form of eqn (57) would have been

W = f (U, D, η, ρ, C). (62)

In this case, n = 6 and r = 4, and we again ﬁnd two dimensionless groups similar to eqn (11). The omission of C in the list of variables in eqn (57) implies that we have considered

Table 6: Dimension matrix.

WU ηρD

L0 1 −2 −31 M00010 F10100 τ 0 −1100

Newton’s second law [eqn (59)] to be of no importance in our problem. However, inertial forces that arise from acceleration of the flow can only be neglected as compared to viscous forces if the Reynolds number of the flow is very small. If the Reynolds number cannot be considered to be small, we must include the constant C in our list of variables. In all cases, however, where the Reynolds number is small, dimensional analysis based on the [LMFτ] system without considering the constant C will produce a sharper result. From theory, we find that for a sphere the constant in eqn (58) has the value constant = 3π so that

W = 3πηUD. (63)

Rearranging the terms, we get the well-known result for small values of the Reynolds number 8W 24 c = = . (64) w ρU2D2π Re For intermediate values of the Reynolds number, we have

cw = f (Re) (65) and for large Reynolds numbers experiments show that

f (Re →∞) = 0.4. (66)

It is worthwhile to mention that the number of basic units may also be increased by using distinctive units for length in each direction, say Lx, Ly and Lz. By effectively increasing the number of primary units we can increase the rank of the dimension matrix and thus get fewer non-dimensional variables. Huntley [24] has given many examples of this method, and, more recently, Rudolph [25] has shown that this approach can fruitfully be applied to the analysis and reconstruction of images. It must be noted, however, that these methods are not based on mathematical or physical grounds. The method rather implicitly assumes that the dynamics of a physical system in one space direction is entirely independent of the dynamics of the system in another direction. Therefore, if there is good reason to believe that in a particular problem the dynamics in one direction are independent of the dynamics in the other directions, the use of multiple length scales can be of great beneﬁt; in all other cases, their use will produce false results.

5 Conclusions

In conclusion, we would like to emphasize that dimensional analysis cannot substitute physical understanding. Dimensional analysis can be applied to any problem and in any discipline. It always reduces the number of variables, simplifying the search for a solution, giving a sharper result and providing deeper insight. The results are always exact. In conjunction with physical understanding and abstraction, dimensional analysis is a very powerful tool.

References

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