1 Equations, General Concepts and Nondimensional Numbers

1

Equations, General Concepts andNondimensional Numbers

Prior to expanding on the subject of convection in rotating fluids and related myriad manifestations,some propaedeutical concepts and accompanying fundamental mathematics must be provided to helpthe reader in the understanding of the descriptions and elaborations given later.

Along these lines, the goal of this introductory chapter is to stake out some common ground byproviding a survey of overarching principles, characteristic nondimensional parameters and governingequations .

Such a theoretical framework, in its broadest sense, attempts to classify and characterize all forcespotentially involved in the class of phenomena considered in the present book.

As the chapter progresses, in particular, balance equations are first introduced assuming an inertialframe of reference, hence providing the reader with fundamental information about the nature andproperties of forces of nonrotational origin (Sections 1.1–1.5); then such equations are reformulated ina rotating coordinate system (Section 1.6) in which the so-called centrifugal and Coriolis forces emergenaturally as ‘noninertial’ effects.

While such a practical approach justifies the use of continuum mechanics and of macrophysicaldifferential equations for the modelling of the underlying processes, it is insufficient, however, for theunderstanding/introduction of a microscopic phenomenological theory. Such development requires somemicrophysical reasoning. The cross-link between macro- and micro-scales is, in general, a challengingproblem. Due to page limits, here we limit ourselves to presenting the Navier–Stokes and energyequations directly in their macroscopic (continuum) form, the reader being referred to other texts (e.g.Lappa, 2010) for a complete elaboration of the approach leading from a microscopic phenomenologicalmodel to the continuum formalism.

1.1 The Navier-Stokes and Energy Equations

The Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes (Navier,1822; Stokes, 1845), describe the motion of a variety of fluid substances, including gases, liquidsand even solids of geological sizes and time-scales. Thereby, they can be used to model flows of

Rotating Thermal Flows in Natural and Industrial Processes, First Edition. Marcello Lappa.© 2012 John Wiley & Sons, Ltd. Published 2012 by John Wiley & Sons, Ltd.

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ATERIAL

2 Rotating Thermal Flows in Natural and Industrial Processes

technological interest (too many to mention; e.g. fluid motion inside a crucible used for crystal growthor for the processing of metal alloys), but also weather, ocean currents and even motions of cosmolog-ical interest.

In their macroscopic (continuum) form these equations establish that the overall mass must be con-served and that changes in momentum can be simply expressed as the sum of dissipative viscous forces,changes in pressure, gravity, surface tension (in the presence of a free surface) and other forces (electric,magnetic, etc.) acting on the fluid.

1.1.1 The Continuity Equation

The mass balance equation (generally referred to in the literature as the continuity equation) reads:∂ρ

∂t+ ∇ · (

ρV) = 0 (1.1a)

that, in terms of the substantial derivative D /Dt = ∂/∂t + V · ∇ (also known as ‘material’ or ‘total’derivative), can be rewritten as

Dρ

Dt+ ρ∇ · V = 0 (1.1b)

where ρ and V are, respectively, the fluid density and velocity.

1.1.2 The Momentum Equation

The momentum equation reads:∂

∂t

(ρV

) + ∇ · �mt

= Fb (1.2)

where Fb is the generic body force acting on the fluid and �mt

the flux of momentum, which can bewritten as

�mt

= ρV V − τ (1.3)

where τ is known as the stress tensor . Such a tensor can be regarded as a stochastic measure of theexchange of microscopic momentum induced at molecular level by particle random motion (it providesclear evidence of the fact that viscous forces originate in molecular interactions; we shall come backto this concept later).

Substituting Equation 1.3 into Equation 1.2, it follows that:∂

∂t

(ρV

) + ∇ ·(ρV V − τ

)= Fb (1.4a)

ρDV

Dt− ∇ · τ = Fb (1.4b)

1.1.3 The Total Energy Equation

Introducing the total energy as:

E = 1

2ρV 2 + ρuint (1.5)

the total energy balance equation can be cast in condensed form as:∂

∂t

[ρ

(1

2V 2 + uint

)]+ ∇ ·

[ρ

(1

2V 2 + uint

)V + J u − V · τ

]= Fb · V (1.6a)

Equations, General Concepts and Nondimensional Numbers 3

or in terms of the substantial derivative:

ρD

Dt

(1

2V 2 + uint

)+ ∇ ·

[J u − V · τ

]= Fb · V (1.6b)

where

�e =(

1

2ρV 2 + ρuint

)V + J u − V · τ (1.7)

J u being the diffusive flux of internal energy (it measures transport at the microscopic level of molecularkinetic energy due to molecular random motion).

1.1.4 The Budget of Internal Energy

A specific balance equation for the single internal energy can be obtained from subtracting the kineticenergy balance equation from the total energy balance equation (Equation 1.6).

Obviously, a balance equation for the pure kinetic energy can be introduced by taking the productof the momentum balance equation with V :

ρD

Dt

(V 2

2

)−

(∇ · τ

)· V = Fb · V (1.8a)

this equation, using the well-known vector identity ∇ ·(V · τ

)=

(∇ · τ

)· V + τ : ∇V can be rewritten

as

ρD

Dt

(V 2

2

)− ∇ ·

(V · τ

)= Fb · V − τ : ∇V (1.8b)

from which, among other things, it is evident that the diffusive flux of kinetic energy can be simplyexpressed as the scalar product between V and the stress tensor. Subtracting, as explained before,Equation 1.8b from Equation 1.6b, one obtains:

ρDuint

Dt+ ∇ · J u = τ : ∇V (1.9a)

or∂ρuint

∂t+ ∇ · [

ρuintV + J u

] = τ : ∇V (1.9b)

that is the aforementioned balance equation for the internal energy.

1.1.5 Closure Models

In general, the ‘closure’ of the thermofluid–dynamic balance equations given in the preceding sections,i.e. the determination of a precise mathematical formalism relating the diffusive fluxes (stress tensorand the diffusive flux of internal energy) to the macroscopic variables involved in the process, is notas straightforward as many would assume.

For a particular but fundamental category of fluids, known as ‘newtonian fluids’ the treatment of thisproblem, however, becomes relatively simple.

For the case considered in the present book (nonpolar fluids and absence of torque forces), the stresstensor can be taken symmetric, i.e. τ ij = τ ji (conversely, a typical example of fluids for which thestress tensor is not symmetric is given by ‘micropolar fluids’, which represent fluids consisting of rigid,randomly oriented particles suspended in a viscous medium).


If the considered fluid is in quiescent conditions (i.e. there is no macroscopic motion) the stresstensor is diagonal and simply reads

τ = −pI (1.10a)

where I is the unity tensor and p is the pressure.In the presence of bulk convection the above expression must be enriched with the contributions

induced by macroscopic fluid motion.In the most general case such a contribution should be related to the gradient of velocity ∇V via a

tensor having tensorial order 4 (from a mathematical point of view a relationship between two tensorshaving order two has to be established using a four-dimension tensor). According to some simpleconsiderations based on the isotropy of fluids (i.e. their property of not being dependent upon a specificdirection) and other arguments provided over the years by various authors (Isaac Newton’s landmarkobservations in liquids; later, the so-called Chapman–Enskog expansion elaborated for gases by Grad(1963) and Rosenau (1989)), the four-dimensional tensor relating the stress tensor to the gradientsof mass velocity simply reduces to a proportionality (scalar) constant that does not depend on suchgradients:

τ = −pI + 2μ(∇V

)s

o(1.10b)

where the constant of proportionality μ is known as the dynamic viscosity (it may be regarded asa macroscopic measure of the intermolecular attraction forces) and the tensor

(∇V)s

o(known as the

viscous stress tensor or the dissipative part of the stress tensor) comes from the following decompositionof ∇V :

∇V = 1

3

(∇ · V)I + (∇V

)s

o+ (∇V

)a(1.11)

where (∇V)s

o= (∇V

)s − 1

3

(∇ · V)I ,

(∇V)s = ∇V + ∇V T

2(1.12a)

(∇V)a = ∇V − ∇V T

2(1.12b)

The three contributions in Equation 1.11 are known to be responsible for volume changes, deformationand rotation, respectively, of a generic (infinitesimal) parcel of fluid (moving under the effect of thevelocity field V ; the reader being referred to Section 1.2 for additional details about the meaning of(∇V

)aand its kinship with the concept of vorticity).

Moreover, in general, the diffusive flux of internal energy can be written as (Fourier law):

J u = −λ∇T (1.13)

where λ is the thermal conductivity and T the fluid temperature.Using such closure relationships, and taking into account the following vector and tensor identities:

∇ ·(pI

)= ∇p (1.14)

∇ · (pV

) = p∇ · V + V · ∇p (1.15)

pI : ∇V = p∇ · V (1.16)(∇V)s

o: ∇V = (∇V

)s

o:

[1

3

(∇ · V)I + (∇V

)s

o+ (∇V

)a

]= (∇V

)s

o:(∇V

)s

o(1.17)

the balance equations can be therefore rewritten in compact form as follows:


Momentum:

∂

∂t

(ρV

) + ∇ · (ρV V

) + ∇p = ∇ · [2μ

(∇V)s

o

] + Fb (1.18a)

ρDV

Dt+ ∇p = ∇ · [

2μ(∇V

)s

o

] + Fb (1.18b)

Kinetic energy:

∂

∂t

(ρ

V 2

2

)+ ∇ ·

(ρ

V 2

2V

)+ V · ∇p = ∇ · [

2μV · (∇V)s

o

] − 2μ(∇V

)s

o:(∇V

)s

o+ Fb · V (1.19a)

ρD

Dt

(V 2

2

)+ V · ∇p = ∇ · [

2μV · (∇V)s

o

] − 2μ(∇V

)s

o:(∇V

)s

o+ Fb · V (1.19b)

Internal energy:

∂ρuint

∂t+ ∇ · [ρuintV

] = ∇ · (λ∇T

) − p∇ · V + 2μ(∇V

)s

o:(∇V

)s

o(1.20a)

ρDuint

Dt= ∇ · (

λ∇T) − p∇ · V + 2μ

(∇V)s

o:(∇V

)s

o(1.20b)

Total (Internal+Kinetic) energy

∂

∂t

[ρ

(1

2V 2 + uint

)]+ ∇ ·

[ρ

(1

2V 2 + uint

)V

]= ∇ · (

λ∇T − pV + 2μV · (∇V)s

o

) + Fb · V

(1.21a)

ρD

Dt

(1

2V 2 + uint

)= ∇ · (

λ∇T − pV + 2μV · (∇V)s

o

) + Fb · V (1.21b)

1.2 Some Considerations about the Dynamics of Vorticity

1.2.1 Vorticity and Circulation

Apart from the classical fluid-dynamic variables such as mass, momentum, (kinetic, internal or total)energy, whose balance equations have been shortly presented in the preceding section, ‘vorticity’ shouldbe regarded as an additional useful mathematical concept for a better characterization of certain typesof flow. Generally used in synergetic combination with the other classical concepts, this quantity hasbeen found to play a fundamental role in the physics of vortex-dominated flows, its dynamics being theprimary tool to understand the time evolution of dissipative vortical structures .

In the following we provide some related fundamental notions, together with a short illustration of therelated interdependencies with other variables, as well as a derivation of the related balance equation.

Along these lines, it is worth starting the discussion with the observation that, in general, vorticitycan be related to the amount of ‘circulation’ or ‘rotation’ (or more strictly, the local angular rate ofrotation) in a fluid (it is intimately linked to the moment of momentum of a generic small fluid particleabout its own centre of mass). The average vorticity in a small region of fluid flow, in fact, can be


Figure 1.1 Vorticity as a measure of the rate of rotational spin in a fluid.

defined as the circulation � around the boundary of the small region, divided by the area A of thesmall region.

ζ = �

A(1.22a)

where the fluid circulation � is defined as the line integral of the velocity V around the closed curve

in Figure 1.1.

� =∮

V · td (1.22b)

t being the unit vector tangent to .In practice, the vorticity at a point in a fluid can be regarded as the limit of Equation 1.22a as the

area of the small region of fluid approaches zero at the point:

ζ = d�

dA(1.22c)

In addition to the previous modelling, using the Stokes theorem (purely geometrical in nature), whichequates the circulation � around to the flux of the curl of V through any surface area bounded by :

� =∮

V · td =∫A

(∇ ∧ V) · n dS (1.23)

where n is the unit vector perpendicular to the surface A bounded by the closed curve (it is implicitlyassumed that is smooth enough, i.e. it is locally lipschitzian; this implies that the existence of theunit vector perpendicular to the surface is guaranteed), it becomes evident from a mathematical pointof view that the vorticity at a point can be defined as the curl of the velocity:

ζ = ∇ ∧ V (1.24)

Therefore, it is a vector quantity, whose direction is along the axis of rotation of the fluid.Notably, ζ has the same components as the anti-symmetric part of the tensor ∇V , that is in line with

the explanation given in Section 1.1.5 about the physical meaning of(∇V

)a.

Related concepts are the vortex line, which is a line that is at any point tangential to the local vorticity;and a vortex tube which is the surface in the fluid formed by all vortex lines passing through a given(reducible) closed curve in the fluid. The ‘strength’ of a vortex tube is the integral of the vorticityacross a cross-section of the tube, and is the same everywhere along the tube (because vorticity haszero divergence).

In general, it is possible to associate a vector vorticity with each point in the fluid; thus the whole fluidspace may be thought of as being threaded by vortex lines which are everywhere tangental to the localvorticity vector. These vortex lines represent the local axis of spin of the fluid particle at each point.


The related scalar quantity:

Z = (∇V)a

:(∇V

)a = ζ 2

2(1.25)

is generally referred to in the literature as the ‘density of enstrophy’. It plays a significant role in sometheories and models for the characterization of turbulence (as will be discussed in Chapter 6) and insome problems related to the uniqueness of solutions of the Navier–Stokes equations (Lappa, 2010).By simple mathematical manipulations it can also appear in global budgets of kinetic energy.

1.2.2 Vorticity in Two Dimensions

Apart from the point of view provided by mathematics (illustrated in Section 1.2.1 and related mathe-matical developments), there is another interesting way to introduce the notion of vorticity and to obtaininsights into its properties, which is more adherent to the ‘way of thinking’ of experimentalists.

Due to intrinsic properties of the curl operator, for two-dimensional flows, vorticity reduces to avector perpendicular to the plane.

Experimentalists have shown that, for such conditions, as an alternative to the classical definition,the strength of this vector at a generic point at any instant may be defined as the sum of the angularvelocities of any pair of mutually perpendicular, infinitesimal fluid lines (contained in the plane of the2D flow) passing through such a point (see Figure 1.2).

Equivalently (under a more physics-related perspective), Shapiro (1969) defined the vorticity of ageneric fluid particle as exactly twice the angular velocity of the solid particle at the instant of itsbirth originating (‘by magic’, e.g. by suddenly freezing) from the considered fluid particle (resorting tothis definition, one may therefore regard ζ /2 as the average angular velocity of the considered fluidelement; it is in this precise sense that the vorticity acts as a measure of the local rotation, or spin, offluid elements, as mentioned before).

At this stage it is also worth highlighting how, as a natural consequence of such arguments, it becomesobvious that for a fluid having locally a ‘rigid rotation’ around an axis, i.e. moving like a solid rotatingcylinder, vorticity will be simply twice the system angular velocity .

Another remarkable consequence of this observation is that for a fluid contained in a cylindricaltank rotating around its symmetry axis and being in relative motion with respect to the tank walls, thevorticity of any generic fluid particle will be given by the sum of two contributions, one related to theoverall rotation of the container, as discussed above (rigid-rotation contribution), and the other (relative

Figure 1.2 Vorticity as the sum of the angular velocity of two short fluid line elements that happen, at that instant, to bemutually perpendicular (Shapiro, 1969).


contribution) due to the motion displayed by the particle with respect to the rotating frame of reference(a coordinate system rotating at the same angular velocity of the container). These two componentsof vorticity are known as the solid-body vorticity (2) and the relative vorticity (ζ ), their sum beinggenerally referred to the absolute vorticity (ζ + 2).

1.2.3 Vorticity Over a Spherical Surface

Several useful generalizations of the concepts provided in the preceding Section 1.2.2 can be made. Asa relevant example, it is worth providing some fundamental information about the related concept ofsolid-body vorticity over the surface of a sphere, which, among other things, will also prove to be veryuseful in the context of the topics treated in the present book (in particular, this notion has extensivebackground applications to planetary atmosphere dynamics).

By simple geometrical arguments, the component of vorticity perpendicular to a spherical surfacedue to solid rotation can be written as:

f = 2 sin(ϕ) (1.26)

where ϕ is the latitude shown in Figure 1.3.

Ω

N. Pole

S. Pole

iy iz

ixzy

x

r

Equator

j

J

Figure 1.3 Typical global and local coordinate systems used for a planet. The orthogonal unit vectors ix, iy and iz pointin the direction of increasing longitude ϑ , latitude ϕ and altitude z. Locally, however, the mean motion can be consideredplanar and a rectangular reference system (x,y,z) can conveniently be introduced with coordinates measuring distancealong ix, iy and iz, respectively, i.e. x increasing eastward, y northward and z vertically upward.


The parameter f above is generally referred to as the Coriolis parameter (or frequency). As evidentin Equation 1.26, it accounts for the local intensity of the contribution brought to local fluid vorticityby planetary rotation and depends on the latitude through the sine function.

A useful simplification, however, can be introduced considering an observation made originally byRossby (1939). Rossby’s point (which can be justified formally by a Taylor series expansion) is thatthe ‘sphericity’ of Earth can be accommodated in a relatively simple way if the local planetary vorticityis properly interpreted and allowed a simplified variation with the latitude. In practice, with such amodel (generally referred to as the ‘β-plane approximation’) the Coriolis parameter, f , is set to varylinearly in space as

f = fREF + βy (1.27)

where f REF is the value of f at a given latitude (a reference value) and β = ∂f /∂y is the rate at whichthe Coriolis parameter increases northward (the local y-axis being assumed to be directed from theequator towards the north pole; the reader being referred to Figure 1.3 and its caption for additionaldetails):

β = ∂f

∂y= 1

REarth

∂f

∂ϕ= 2 cos(ϕ)

REarth(1.28)

REarth being the average radius of Earth.The component of the absolute fluid vorticity perpendicular to the Earth’s surface, therefore, will

simply read

ζ + fREF + βy (1.29)

The name ‘beta’ for this approximation derives from the convention to denote the linear coefficient ofvariation by the Greek letter β. The associated reference to a ‘plane’, however, must not be confusedwith the idea of a tangent plane touching the surface of the sphere at the considered latitude; the β-planemodel, in fact, does describe the dynamics on a hypothetical tangent plane. Rather, such an approach isemployed to take into proper account the latitudinal gradient of the planetary vorticity, while retaininga relatively simple form of the dynamical equations (indeed, it can be shown in a relatively easy waythat the linear variation of f does not contribute nonlinear terms to the balance equations).

The reader will easily realize that the practical essence of such a simplification is that it only retainsthe effect of the Earth’s curvature on the meridional (along a meridian) variations of the Coriolisparameter, while discarding all other curvature effects.

This approximation is generally valid in midlatitudes. Interestingly, however, it also works as an exactreference model for laboratory experiments in which the gradient of planetary vorticity is simulatedusing cylindrical containers with ‘inclined’ planar (‘sloping’) end walls (for which the vertical distancebetween the top and bottom boundaries varies linearly with the distance from the rotation axis).

A more restrictive simplification for a planetary atmosphere is the so-called f -plane approach inwhich the latitudinal variation of f is ignored, and a constant value of f appropriate for a particularlatitude is considered throughout the domain. This is typically used in latitudes where f is large, andfor scales that do not feel the curvature of the Earth. The closest thing on Earth to an f -plane is theArctic Ocean (β = 0 at the Pole). Continuing the analogy with laboratory experiments, moreover, thismodel would correspond to a classical rotating tank with purely horizontal top and bottom boundaries(for which f would reduce to twice the angular velocity, i.e. f = 2).


1.2.4 The Curl of the Momentum Equation

In general, for any flow (2D or 3D) a specific balance equation for vorticity can be derived by simplytaking the curl of the momentum equations (Equation 1.18) and taking into account the followingidentities:

∇ · ζ = ∇ · (∇ ∧ V) = 0 (1.30)

V · ∇V = ∇(

V 2

2

)+ ζ ∧ V (1.31)

∇ ∧ ∇(

V 2

2

)= 0 (1.32)

∇ ∧(V ∧ ζ

)= V

(∇ · ζ

)− ζ

(∇ · V) + ζ · ∇V − V · ∇ζ (1.33)

∇ ∧(

1

ρ∇p

)= 1

ρ∇ ∧ ∇p − 1

ρ2∇ρ ∧ ∇p = − 1

ρ2∇ρ ∧ ∇p (1.34)

This leads toDζ

Dt= ∂ζ

∂t+ V · ∇ζ = ζ · ∇V − ζ

(∇ · V) + 1

ρ2∇ρ ∧ ∇p

+ ∇ ∧[

∇ · 2μ(∇V

)s

o

ρ

]+ ∇ ∧

(1

ρFb

)(1.35)

The first term on the right member of this equation ζ · ∇V , is known to be responsible for possiblestretching of vortex filaments along their axial direction; this leads to contraction of the cross-sectionalarea of filaments and, as a consequence of the conservation of angular momentum , to an increase invorticity (this term is absent in the case of 2D flows). The second term ζ

(∇ · V)

describes possiblestretching of vorticity due to flow compressibility. The third term is generally known as the baroclinicterm (it accounts for changes in vorticity due to interaction of density and pressure gradients actinginside the fluid). The fourth term shows that vorticity can be produced or damped by the action ofviscous stresses. The last term accounts for possible production of vorticity due to other body forces.

1.3 Incompressible Formulation

A cardinal simplification traditionally used (this monograph is not an exception to this common rule)in the context of studies dealing with thermal convection in both natural and industrial processes isto consider the density constant (ρ = const = ρo). Resorting to such approximation, all the governingequations derived in the preceding subsection can be rewritten in a simpler form (in general, theapproximation of constant density is considered together with that of constant transport coefficients, μ

and λ, which leads to additional useful simplifications).Indeed, the continuity equations can be simplified as:

∇ · V = 0 (1.36)

as a consequence, in Equation 1.18a

∇ · [2μ

(∇V)s

o

] = μ∇ · [∇V + ∇V T] = μ

[∇2V + ∇ (∇ · V)] = μ∇2V (1.37)


and the momentum equation reads

ρo

∂V

∂t+ ρo∇ · [

V V] + ∇p = μ∇2V + Fb (1.38)

The internal energy equation becomes

ρo

∂uint

∂t+ ρo∇ · [

uintV] = λ∇2T + 2μ

(∇V)s

o:(∇V

)s

o(1.39)

where the last term 2μ(∇V

)s

o:(∇V

)s

orepresents the production of internal energy due to viscous

stresses (also referred to in the literature as density of viscous heating or kinetic energy degradation:the rate at which the work done against viscous forces is irreversibly converted into internal energy).

In general, the order of magnitude of this term is negligible with respect to the other terms and forthis reason it can be ignored in laboratory experiments (as shown by Gebhart (1962), the effect ofviscous dissipation in natural convection becomes significant only when the induced kinetic energy isappreciable compared to the amount of heat transferred; this occurs when either the equivalent bodyforce is large or when the convection region is extensive).

It is also worth noting that using thermodynamic relationships, the internal energy can be written asa function of the temperature T . In fact:

duint = cvdT (1.40a)

where cv is the specific heat at constant volume:

cv =(

∂uint

∂T

)v=const

(1.40b)

v being the specific volume v = 1/ρ.Taking into account that, in particular, for liquids cv

∼= cp where cp is the specific heat at constantpressure and introducing the thermal diffusivity α defined as α = λ/ρcp , the energy equation can becast in compact form as:

∂T

∂t+ ∇ · [

V T] = α∇2T (1.41)

Equations 1.36, 1.38 and 1.41 represent a set of three coupled equations whose solution is sufficientfor the determination of the problem unknowns, i.e. V , p and T .

As an alternative, the momentum equation can be replaced by the vorticity equation, which in theincompressible case reduces to:

Dζ

Dt= ∂ζ

∂t+ V · ∇ζ = ζ · ∇V − ζ

(∇ · V) + ∇ ∧

[∇ · 2μ

(∇V)s

o

ρo

]+ ∇ ∧

(1

ρo

F b

)(1.42)

which, taking into account Equation 1.37, can be rewritten as:

Dζ

Dt= ∂ζ

∂t+ V · ∇ζ = ζ · ∇V − ζ

(∇ · V) + 1

ρo

∇ ∧ [μ∇2V

] + 1

ρo

∇ ∧ (Fb

)(1.43)

Considering also the following vector identities (and using the fact that both V and ζ = ∇ ∧ V arediv-free):

∇ ∧ (∇ ∧ V) = ∇ (∇ · V

) − ∇2V → ∇2V = −∇ ∧ (∇ ∧ V) = −∇ ∧ ζ (1.44)

∇ ∧(∇ ∧ ζ

)= ∇

(∇ · ζ

)− ∇2ζ → −∇ ∧

(∇ ∧ ζ

)= ∇2ζ (1.45)


Equation 1.42 can be finally written as

ρo

∂ζ

∂t+ ρoV · ∇ζ = ρoζ · ∇V + μ∇2ζ + ∇ ∧ Fb (1.46)

Such equations can be made dimensionless by choosing characteristic scales for length, time, velocityand so on. The resulting grouping of physical properties and characteristic scales form dimensionlessnumbers which represent ratios of various forces or quantities. Theoreticians often communicate throughthis set of dimensionless parameters.

Here the attention is limited to the typical (most general) choice of characteristic reference quantitiesfor thermal convection (already used in Lappa (2010), hereafter simply referred to as ‘conventionalscalings’).

Lengths are scaled by a reference distance (L) and the velocity by the energy diffusion velocityVα = α/L; the scales for time and pressure are, respectively, L2/α and ρoα

2/L2. The temperature,measured with respect to a reference value T o, is scaled by a reference temperature gradient �T .

This approach leads to (in the following, for the sake of simplicity the same symbols used for theequations in dimensional form are also used for the nondimensional formulation):

∇ · V = 0 (1.47)∂V

∂t+ ∇ · [

V V] + ∇p = Pr ∇2V + Fb (1.48a)

∂T

∂t+ ∇ · [

V T] = ∇2T (1.49a)

where Pr is the Prandtl number (Pr = ν/α and ν is the constant kinematic viscosity ν = μ/ρ). This firstnondimensional parameter measures the relative importance of transport at a molecular level of momen-tum (via ν) and kinetic energy (via α), respectively. It is often regarded as a clear ‘signature’ of the fluidconsidered (this is the reason why researchers often identify considered fluids with the related valuesof the Pr instead of providing details (nomenclature, composition, etc.) about the chemical structure).

According to Equation 1.47, Equations 1.48a and 1.49a can be also written as

∂V

∂t+ V · ∇V + ∇p = Pr ∇2V + Fb (1.48b)

∂T

∂t+ V · ∇T = ∇2T (1.49b)

The nondimensional form of the vorticity equation, similarly reads:

∂ζ

∂t+ V · ∇ζ = ζ · ∇V + Pr ∇2ζ + ∇ ∧ Fb (1.50)

At this stage, we cannot proceed further without providing specific information on the nature of thedriving forces responsible of the genesis of the considered phenomena, as well as on the relatedfundamental models introduced over the years by the investigators.

In this context it is worth noting that physicists have often looked to applied mathematicians andengineers of various sorts for turning such effects into precise mathematical relationships. This strategyhas been largely beneficial to advancements of understanding. Remarkably, it has been fed by a fruitfulinteraction between theoreticians on one side and experimenters on the other side. In particular, in sucha process, theoreticians have brought forward their own peculiar way of thinking about flows and theireffects, such as the pervasive use of scaling analysis and dimensionless numbers. Direct experimentalanalysis has permitted us to assess the validity of such a way of thinking, feeding back, in turn, vital


information for further refinement and/or theoretical elaboration. The next sections provide some simpleand fundamental information along these lines.

1.4 Buoyancy Convection

Everybody knows (it is a fundamental law of Nature) that the presence of a planet creates a gravitationalfield that acts to attract objects with a force inversely proportional to the square of the distance betweenthe centre of the object and the centre of the planet. A remarkable impact of this body force on fluidsis the creation of flows due to density differences (so-called buoyancy-induced convection).

Consider, for instance, what happens on Earth when a container of water is heated from below. Asthe water on the bottom is heated by conduction through the container, it becomes less dense than theunheated, cooler water. Because of gravity, the cooler, more dense water sinks to the bottom of thecontainer and the heated water rises to the top due to buoyancy; thereby, a circulation pattern is producedthat mixes the hot water with the colder water. This is an example of buoyancy-driven (or gravity-driven)convection. The convection causes the water to be heated more quickly and uniformly than if it wereheated by conduction (thermal diffusion) alone. This is the same density-driven convection process towhich we refer when we state matter-of-factly that ‘hot air rises’.

From a mathematical point of view the buoyancy force can be simply obtained by multiplying thedensity of the considered fluid by gravity acceleration g. This means the body force term in themomentum Equation 1.38 will simply read:

Fb = ρg (1.51)

1.4.1 The Boussinesq Model

Following the usual approximation for incompressible fluids (introduced by Boussinesq, 1903), thephysical properties of the considered fluid can be assumed constant (as stated in Section 1.3), except ,however, for the density ρ in the above term (Equation 1.51), which, without introducing a significantdeparture from real life, can be assumed to be a linear function of temperature, i.e.:

ρ = ρ(T ) = ρ(TREF ) +∞∑

k=1

1

k!

dkρ

dT k

∣∣∣∣TREF

(T − TREF)k

withdρ

dT

∣∣∣∣TREF

= −βT �= 0 for k = 1 anddkρ

dT k

∣∣∣∣TREF

∼= 0 for k ≥ 2 (1.52)

→ ρ ∼= ρ0 [1 − βT (T − TREF)] (1.53)

where βT is simply known as the thermal expansion coefficient and T REF is a reference value fortemperature.

In his attempts to explain the motion of light in the aether, the above-mentioned Boussinesq (1903)opened a wide perspective of mechanics and thermodynamics. With a theory of heat convection influids and of propagation of heat in deforming or vibrating solids he showed that density fluctuationsare of minor importance in the conservation of mass. The motion of a fluid initiated by heat resultsmostly in an excess of buoyancy and is not due to internal waves excited by density variations.

In practice, this approximation states that density differences are sufficiently small to be neglected,except where they appear in terms multiplied by g , the acceleration due to gravity. The essence ofthis model lies in the fact that the difference in inertia is negligible, but gravity is sufficiently strong tomake the specific weight appreciably different between two fluid particles with different temperatures.


As a consequence, the continuity equation may be reduced to the vanishing of the divergence of thevelocity field (that is typical of incompressible flows, as shown in Section 1.3), and variations of thedensity can be ignored in the momentum equation too, except insofar as they give rise to a gravitationalforce.

The derivation of conditions for the validity of the Boussinesq approximation is not as straightforwardas many would assume. In the literature, a variety of sets of conditions have been assumed which, ifsatisfied, allow application of this approximation (see, e.g. Mihaljan, 1962; Mahrt, 1986).

Basically, as illustrated by Gray and Giorgini (1976) for the case of gases, such an assumption leadsto reliable results if both the product (βT �T ) and the ratio �T /T bulk are well below a value of 0.1,taken as the limit for the applicability of this model.

Although used before him (Oberbeck, 1879), Boussinesq’s theoretical approach established a cardinalsimplification that is extremely accurate for many flows, and makes the mathematics and physics simpler(see the discussions in Chandrasekhar, 1961).

1.4.2 The Grashof and Rayleigh Numbers

In the framework of such an approximation, the momentum equation in dimensional form can bewritten as:

∂V

∂t+ ∇ · [

V V] + 1

ρ0∇p = ν∇2V − [βT (T − TREF)] g (1.54)

The identification of the significant parameters in the momentum equation requires this equation tobe posed in nondimensional form through the choice of relevant reference quantities. Following theapproach defined in Section 1.3 (the conventional scalings), the nondimensional form reduces to:

∂V

∂t= −∇p − ∇ · [

V V] + Pr ∇2V − Pr RaT ig (1.55)

where

Ra = gβT �T L3

να(1.56)

is the Rayleigh number and ig the unit vector along the direction of gravity. This nondimensionalparameter measures the magnitude of the buoyancy velocity V g (it is traditionally employed as areference quantity in this kind of problem) to the thermal diffusive velocity, where V g reads

Vg = gβT �T L2

ν(1.57)

Moreover,

Gr = Ra/ Pr = gβT �T L3

ν2(1.58)

is the Grashof number that represents the ratio of buoyant to molecular viscous transport (obviously,only two of the Gr, Ra and Pr numbers are independent).

1.5 Surface-Tension-Driven Flows

Instability of flows of gravitational origin and their transition to turbulence are widespread phenomena inthe natural environment at several scales, and are at the root of typical problems in thermal engineering,materials and environmental sciences, meteorology, oceanography, geophysics and astrophysics.


The possible origin of natural flows, however, is not limited to the action of the gravitational force.Other volume or ‘surface’ forces may be involved in the process related to the generation of fluidmotion and ensuing evolutionary progress.

In particular, in the presence of a free interface (e.g. a surface separating two immiscible liquids,or a liquid and a gas) surface-tension-driven convection may arise as a consequence of temperature orconcentration gradients.

This phenomenon is usually referred to in the literature as ‘Marangoni convection’ (named afterthe Italian physicist Carlo Giuseppe Matteo Marangoni (1871), whose important discovery is nowexemplified by the famous example that he gave to explain his discovery: ‘If for any reasons, differencesof surface tension exist along a free liquid surface, the liquid will move towards the region of highersurface tension’).

This law is now nicknamed the ‘Marangoni effect’.

1.5.1 Stress Balance

For an exhaustive and consistent analysis of the conditions for which a system of two immiscible fluidphases with their ‘interphase layer’ can be modelled at the microscopic level as two volume phasesseparated by an ‘interface’, the reader is referred to Napolitano (1979).

In such a theoretical study thermodynamic and dynamic theories of the surface phase were developedfor increasing levels of sophistication according to the nature and relevance of the interactions betweenthe considered volume phases and their interphase layer. Here we limit ourselves to considering thecanonical case in which the interface separating a liquid and a gas can be modelled as a mathematicalboundary with no mass and zero thickness, assumed to be undeformable in time and with a fixedlocation in space.

The surface tension σ = σ (T ) for many cases of practical interest exhibits a linear dependence ontemperature, i.e.:

σ = σ(T ) = σ(TREF) +∞∑

k=1

1

k!

dkσ

dT k

∣∣∣∣TREF

(T − TREF)k

withdσ

dT

∣∣∣∣TREF

= −σT �= 0 for k = 1 anddkσ

dT k

∣∣∣∣TREF

∼= 0 for k ≥ 2 (1.59)

→ σ ∼= σ0 [1 − σT (T − TREF)] (1.60)

where σ 0 is the surface tension for T = T REF (T REF is a reference value), σ T = –dσ /dT > 0 (σ is adecreasing function of T ).

If a nonisothermal free surface is involved in the considered category of phenomena thensurface-tension forces FT

σ = ∇Sσ (∇S derivative tangential to the interface) arise that must bebalanced by viscous stresses in the liquid (throughout the present book the dynamic viscosity of thegas surrounding the free liquid surface will be assumed to be negligible with respect to the viscosityof the considered liquid); from a mathematical point of view this condition can be written as:

τd

· n = −σT

(I − nn

)· ∇T (1.61a)

where τd

= 2μ(∇V

)s

ois the dissipative part of the stress tensor (see Equation 1.10b), n is the unit

vector perpendicular to the liquid/gas interface (directed from liquid to gas) and I is the unity matrix.


For a planar surface, the balance above simply yields:

μ∂V S

∂n= −σT ∇ST (1.61b)

where n is the direction perpendicular to the free interface and V s the surface velocity vector.

1.5.2 The Reynolds and Marangoni Numbers

In nondimensional form (using the conventional scalings defined in Section 1.3), Equation 1.61b reads:

∂V S

∂n= −Ma∇ST (1.62)

where

Ma = σT �T L

μα(1.63)

(μ being the dynamic viscosity) is the so-called Marangoni number . This condition enforces a flowby tangential variation of the surface tension. The motion (thermocapillary convection) immediatelyresults whenever a temperature gradient exists along the considered interface, no matter how small.The surface moves from the region with a low surface tension (relatively hot) to that with a highsurface tension (relatively cold). The viscosity transfers this motion to the underlying fluid, i.e. theflow penetrates into the bulk through viscous coupling to the motion at the interface.

A related parameter, the Reynolds number (Re):

Re = σT �T L

ρν2(1.64)

measures the magnitude of the tangential stress σ T �T /L to the viscous stress ρν2/L2; the Prandtlnumber (Pr already introduced in Section 1.3) measures the rate of momentum diffusion ν to that ofheat diffusion α, and the Marangoni number is Ma = RePr (only two of these three numbers areindependent).

Other features of interest are the fundamental scales of velocity and temperature that determine thestrength of the flow.

It has been noted by Rybicki and Floryan (1987) that the appropriate scaling of the dimensionalvelocity is the Marangoni velocity:

V TMa = σT �T

μ(1.65)

Then the Marangoni number can also be seen as the measure of the relative importance of the Marangoniand the thermal diffusion velocities.

Moreover, for small Reynolds and Prandtl numbers, the dimensional temperature field scales likeRe × Pr × �T .

An additional relevant parameter with which researchers have often to deal is represented by the Biotnumber. It is defined as:

Bi = hL

λ(1.66)

where h is the so-called convective heat transfer coefficient at the free surface.This nondimensional number is used to take into account possible thermal coupling of liquid with the

ambient. On the free surface, it is generally assumed, in fact, that the heat transfer between the liquid


and the surrounding gas can be conveniently approximated by the following nondimensional relation:

∂T

∂n= −Bi (T − Ta) (1.67)

where T a is the gas (ambient) temperature: T < T a means the liquid is heated from the surrounding gasand, vice versa, T > T a means there exists a flux of heat from the liquid to the ambient; an adiabaticsurface can be seen as a special case of Equation 1.67 with Bi = 0.

Marangoni convection has been the subject of increasing interest in recent years with regard to manydifferent geometrical configurations and heating conditions (see Figure 1.4). Indeed, the mechanics ofresponse of these fluid-dynamical systems depends on the type of heating applied to the interface.

In the geometrically simplest case of a liquid contained in a rectangular cavity open from above,the heating can be applied either through the bottom (or from above) or through the sidewalls. Theresponse of the system is markedly different in each of these cases.

The direction of the imposed �T plays a crucial role: if the externally imposed �T yields imposedtemperature gradients that are primarily perpendicular to the interface, the basic state is static with adiffusive temperature distribution and motion (Marangoni–Benard convection) ensues with the onset ofinstability when �T exceeds some threshold; if the externally imposed �T yields imposed temperaturegradients that are primarily parallel to the interface, as anticipated, in these cases motion occurs for anyvalue of �T .

z

z

Lx

Ly

L r ΔT

j

y

x

z

b

a

d

Lz = d

(a)

(c)

(b)

Figure 1.4 Fundamental geometrical models for the study of buoyancy and Marangoni convection: (a) rectangular layeror slot; (b) annular pool and (c) liquid column.


1.5.3 The Microgravity Environment

Because in many circumstances gravity’s influence on fluids is strong and masks or overshadowsimportant factors, a number of experiments have been carried out in recent years on orbiting platforms(so-called ‘microgravity’ conditions). The peculiar behaviour of physical systems in space, and ulti-mately the interest in this ‘new’ environment, has come from the virtual disappearance of the gravityforces and related effects mentioned above, and the appearance of phenomena unobservable on Earth,especially those driven by surface forces (that become largely predominant when terrestrial gravityis removed).

In practice, gravity cannot simply be switched off, but its effects can be compensated with the helpof an appropriate acceleration force. This acceleration force must have exactly the same absolute valueas the gravity force and it must point in the opposite direction of the local gravity vector. The resultingequilibrium of forces is called in normal language: ‘weightlessness’.

As an example, the propulsionless flight of a space vehicle or a space station around the Earthis a special form of free-fall trajectory. The attraction force of the Earth’s gravity is permanentlycompensated by the centrifugal force resulting from the curved shape of the orbit (see, e.g. Lappa,2004).

In general, however, an exact equilibrium state is difficult to obtain and a very small gravity forcealways remains. This is the reason why specialists speak of ‘microgravity’ rather than ‘weightlessness’.

Over recent years, both through the results of such space experiments and through relatedground-based research (normally, the effect of microgravity environment is judged on the basisof comparison of experiments under identical conditions and in an identical set-up under groundconditions and in microgravity), a significant amount has been learned about gravitational andnongravitational contributions in a variety of natural phenomena and technological activities.

Even so it should be stressed that at the present stage the results obtained in microgravity are mostlyof a fundamental nature (quantifying theoretical models of gravity influences on fluid phenomena, orleading to better insights into the significance of forces and interactions which, during experiments onEarth are masked by gravity-induced flows), indeed, it has been demonstrated that such effects can berelevant in a number of phenomena of scientific and technological interest.

The most intensively studied type of convection in space is fluid flow induced bysurface-tension-driven forces; as outlined previously, in fact, microgravity gives the possibilityto avoid some limitations related to the ground environment that adversely affect the experimentalstudy of this problem (in particular, the aforementioned buoyancy-driven convection that in manycircumstances overshadows this kind of convection). Moreover, it is worth highlighting that in zero-gconditions it is possible to form very large floating liquid volumes with extended liquid/gas interfacesthat facilitate significantly the development and ensuing study of these flows; in fact, during recentyears, the availability of sounding rockets, orbiting laboratories such as the Spacelab, and especiallythe ISS, has made possible microgravity experiments with large free surfaces, which could not beperformed on Earth under normal-gravity conditions.

Prior to the space program, these phenomena had been ignored in investigations of materials process-ing on Earth. Microgravity has allowed convection driven by gradients of surface tension to becomeobvious. As anticipated, once it became recognized, it was found to be significant in some Earth-basedprocesses as well (semiconductor crystal growth first of all, but also other important technological pro-cesses and instances in nature, see, again Lappa, 2004). It was learned that this surface-tension-drivenconvection could not only be vigorous, but could also become asymmetric, oscillatory and even turbulent(see, e.g. Lappa, 2010).


1.6 Rotating Systems: The Coriolis and Centrifugal Forces

Purely ‘inertial’ reference frames (all inertial frames which are theoretically in a state of constant,rectilinear motion with respect to one another) represent a purely ideal condition. Every object onthe surface of a planet will rotate with the planet and, therefore, rigorously speaking, will experiencea nonrectilinear motion. The same concept applies to orbiting platforms. Even if in such conditions(weightlessness) gravity is no longer influent, effects induced by platform motion along a curvedtrajectory, however, will be still there.

These simple considerations lead to the conclusion that a rigorous treatment of both natural andindustrial fluid-dynamic processes cannot leave aside a proper consideration of effects of noninertialorigin.

The starting point for such a treatment is the realization that when the balance equations derived inthe preceding sections in the ideal case of an inertial system are transformed to a rotating frame ofreference, the so-called Coriolis and centrifugal forces appear.

Both forces are proportional to the mass of the considered object, i.e. they are body forces, just likegravity considered in Section 1.4.

From a historical standpoint, a clear distinction between the Coriolis and centrifugal forces wasoriginally introduced by Gaspard-Gustave Coriolis in a couple of landmark studies focusing on thesupplementary forces that are detected in rotating systems of reference (Coriolis, 1832, 1835). Hedivided these supplementary forces into two categories, with a category containing the force that arisesfrom the cross product of the angular velocity of the coordinate system and the projection of the particle’svelocity into a plane perpendicular to the system’s axis of rotation, and the other one simply representingthe classical centrifugal force. The force pertaining to the former category is now universally knownas the Coriolis force.

The related mathematical expression per unit volume reads:

Fco = −2ρ ∧ V (1.68)

where, as usual, ρ and V are the density and velocity of the considered fluid particle, respectively, and the constant angular velocity of the rotating frame of reference in which the fluid is considered.

It needs no demonstration that such a force satisfies the following three fundamental properties:

• It becomes zero if the considered fluid particle is stationary in the rotating frame.• It acts to deflect moving particles at right angles to their direction of travel (this being a simple

consequence of the intrinsic property of the vector product).• From an energetic standpoint it does no work on a fluid particle (because it is perpendicular to the

velocity, i.e. V · ( ∧ V ) = 0).

It is also worth pointing out that the factor 2 appearing in Equation 1.68 has a precise physicalmeaning, which deserves some additional discussion. The acceleration entering the Coriolis force, infact, can be seen as the effect of two sources of change in velocity resulting from rotation.

In practice, the first is the change of the velocity of a fluid particle in time. The same particle velocity(as seen in an inertial frame of reference) would be seen as different velocities at different times in arotating frame of reference. As a logical consequence, the apparent acceleration must be proportionalto the angular velocity of the reference frame (the rate at which the coordinate axes change direction),as well as to the component of velocity of the particle in a plane perpendicular to the axis of rotation.This leads to a term − ∧ V (the minus sign arises from the traditional definition of the cross product,and from the sign convention for angular velocity vectors).


The second contribution is the change of velocity in space. Different positions in a rotating frame ofreference have different velocities (as seen from an inertial frame of reference). In order for an objectto move in a straight line it must, therefore, be accelerated so that its velocity changes from point topoint by the same amount as the velocities of the frame of reference. The effect is proportional to theangular velocity (which determines the relative speed of two different points in the rotating frame ofreference), and to the component of the velocity of the particle in a plane perpendicular to the axisof rotation (which determines how quickly it moves between those points). This also gives a term− ∧ V , which explains the factor 2 in Equation 1.68.

The derivation of a mathematical expression for the other force, i.e. the centrifugal contribution, isless complex. It simply reads:

Fce = ρ2r (1.69)

where r = rir is the radial vector (r being the perpendicular distance from the axis of rotation and irthe related unit vector). Introducing a scalar potential defined as

�ce = − (2r2)

2= −

∣∣ ∧ r∣∣2

2(1.70)

Equation 1.69 can be also written in condensed form as

Fce = −ρ∇�ce (1.71)

Additional insights into the similarities and differences between such forces can be obtained by obser-vation and cross-comparison of Equations 1.68 and 1.69.

The reader will immediately realize that the Coriolis force acts in a direction perpendicular to therotation axis and to the velocity of the particle in the rotating frame, and is proportional to the particle’sspeed in the rotating frame. While the Coriolis force is proportional to the rotation rate, the centrifugalforce is proportional to its square. Moreover, it acts outwards in the radial direction and is proportionalto the distance of the fluid particle from the axis of the rotating frame.

Both forces are of an inertial nature and can be regarded as ‘fictitious’ forces or ‘pseudo’ forces (tointroduce a clear distinction with respect to gravity, which is a real force, i.e. a force not dependent onthe adoption of a rotating reference frame).

The consideration of them in the momentum equation in the framework of the incompressible-fluidapproximation defined in Section 1.3, leads to the following expression in dimensional form:

ρ∂V

∂t+ ρV · ∇V + ∇p = μ∇2V + ρgig + ρ2r − 2ρ ∧ V (1.72)

where, V is the velocity of the generic fluid particle in the rotating frame and as usual, p the pressure,g the gravity acceleration, μ the dynamic viscosity.

1.6.1 Generalized Gravity

Notably, as the centrifugal contribution expressed by Equation 1.69 is a function of the relative positiononly, it can be combined with the gravity term to give a generalized gravitational force defined as

g = g

(ig + 2r

gir

)(1.73)


On the basis of Equation 1.70 such a force, in turn, can be expressed as the gradient of a generalizedscalar potential as follows:

g = −∇� = −∇[gz − (2r2)

2

](1.74)

z being the vertical coordinate (see, e.g. Figure 1.4).As also illustrated in Section 1.2.4, taking the curl of Equation 1.72 leads to the vorticity (ζ ) equation,

which has proven to play a very useful role in the analysis of rotating flows and is, therefore, reportedhere. Taking into account that the following vector identity holds

∇ ∧ ( ∧ V ) = ∇ · V − V ∇ · + V · ∇ − · ∇V (1.75)

that ∇ · V = 0 (incompressible fluid), that is a constant vector (constant magnitude and direction)and also considering the vector identities Equations 1.44 and 1.45, the vorticity equation reduces to:

ρ∂ζ

∂t+ ρV · ∇ζ = ρ

(ζ + 2

)· ∇V + μ∇2ζ + ∇ ∧ (ρg) (1.76)

Such an equation can be further simplified recalling the irrotational nature of generalized gravity g

(as shown by Equation 1.74, g can be expressed as the gradient of a scalar potential function and thecurl of the gradient of a scalar quantity is always zero). Introducing the Boussinesq approximation(by which variations of density are neglected everywhere with the exception of the buoyancy term; theproblem here is formally equivalent to that treated earlier in Section 1.4) and using the vector identity∇ ∧ (ρg) = ∇ρ ∧ g + ρ∇ ∧ g, the vorticity equation can be finally cast in compact form as:

ρ∂ζ

∂t+ ρV · ∇ζ = ρ

(ζ + 2

)· ∇V + μ∇2ζ + ∇ρ ∧ g (1.77)

1.6.2 The Coriolis, Taylor and Rossby Numbers

As for classical buoyancy and Marangoni flows, here it is convenient to introduce characteristic numbersto measure the relative importance of rotation with respect to other effects.

Relevant nondimendional parameters are the Coriolis and Taylor numbers. Another significant param-eter is the so-called Rossby number (with its variants).

The Coriolis number or nondimensional rotation rate is generally introduced as:

τ = d2

ν(1.78)

The Taylor number can be defined as:

Ta = 42

ν2ξ 4 (1.79)

with ξ simply assumed to be equal to d or ξ = (d5r /d)1/4 (the first definition being traditionally used in

the context of studies devoted to the Rayleigh–Benard problem for which the Taylor number becomesdirectly proportional to the square of the Coriolis number, the latter for the baroclinic instability), d anddr being reference extensions of the considered system in the axial and radial directions, respectively(they depend on the considered geometry). Regardless of the expression used for ξ , which is selectedaccording to the category of phenomena considered, Equation 1.79 represents the (squared) ratio ofCoriolis to viscous forces (in the literature the Ekman number Ek, can be also encountered, generallydefined as the ratio of the viscous to the Coriolis forces, i.e. Ek ∝ Ta−1).


As anticipated at the beginning of this section, several variants of the so-called Rossby numbercan be introduced. The ‘convective’ Rossby number Roc (generally used in the literature for theRayleigh–Benard problem), in particular, is a very useful dimensionless number potentially characteris-ing the respective importance of buoyancy and rotation. It can be introduced directly as a combinationof Ra, τ (or Ta) and Pr:

Roc =√

Ra

Ta Pr= 1

2τ

√Ra

Pr= 1

2

√gβT �T

d(1.80)

The ‘thermal’ Rossby number RoT, providing again a measure of the ratio of buoyancy to Coriolisforces (it is traditionally used in the framework of studies dealing with the baroclinic problem), reads:

RoT = gβT �T d

2d2r

(1.81)

As the product of such a number with the Taylor number will give the ratio of buoyancy to viscousforces, it follows that such a product is directly proportional to the Grashof number defined by Equation1.58 (for a fixed geometry only two of the Gr, Ta and RoT numbers are independent).

Another nondimensional parameter also introduced in the literature as a measure of the relativeimportance of Coriolis and buoyancy forces (see, e.g. Philips, 1963; Grimshaw, 1975) is the ratioof the related characteristic frequencies, i.e. (f /ωBV), where ωBV = (

gβT∂T∂z

)1/2is the well-known

Brunt–Vaisala frequency (also often referred to as buoyancy frequency) and f = 2sin(ϕ) is theCoriolis parameter defined in Section 1.2 (which reduces to 2 in the case of geometries with cylindricalsymmetry and planar horizontal boundaries rotating around their own symmetry axis).

A yet related, but more general nondimensional parameter used to account for the relative importanceof the Coriolis force with respect to inertial forces is the classical Rossby number defined as

Ro = U

Lf(1.82)

where U and L are, respectively, the characteristic velocity and horizontal length scales of the phe-nomenon considered. This characteristic number is often regarded as an indicator of how close the flowis to solid body rotation (as illustrated in Section 1.2.2, a flow in solid body rotation has vorticity thatis directly proportional to its angular velocity; the Rossby number is a measure of the departure of thevorticity from that of solid body rotation). Obviously, if the Rossby number is much less than one, thesystem will be strongly affected by the Coriolis force.

1.6.3 The Geostrophic Flow Approximation

In regions where the Coriolis force greatly exceeds viscous and other inertial forces, Equations 1.72and 1.77 reduce to:

∇p ∼= ρg − 2ρ ∧ V (1.83)

2ρ · ∇V ∼= −∇ρ ∧ g (1.84)

which means for such a case there will be a very close balance between generalized gravity and thecomponent of the pressure gradient along the same direction (this condition is generally known ashydrostatic balance). More importantly, there will be also a close balance between the horizontalcomponent of the pressure gradient and the horizontal component of the Coriolis force.


Such flows, which occur commonly in the atmosphere and oceans where the Rossby number is small(mainly because the scale of motion is large), are called ‘geostrophic’ (this word has its origin in twoGreek roots: geo (Earth) + strophe (part of an ancient Greek choral ode sung by the chorus whenmoving from right to left , hence indicating movement performed by the chorus during the singing ofthis part)).

Equation 1.84, traditionally referred to in the literature as the ‘thermal wind equation’, links verticalshear with tilted surfaces of constant density (if ρ varies in the horizontal then the geostrophic currentwill vary in the vertical) and is generally considered as a cardinal starting point for the interpretationof many phenomena of both natural (atmospheric or oceanic) and technological interest.

1.6.4 The Taylor–Proudman Theorem

Another important consequence of the dominance of Coriolis forces is the so-called Taylor–Proudmantheorem (Proudman, 1916; Taylor, 1917, 1922, 1923a). In the lowest-order approximation , the vorticityEquation 1.84 can be written as

2 · ∇V ∼= 0 (1.85)

This is equivalent to the statement that the relative velocity field does not vary in the direction of therotational axis. As a consequence, the flow will tend to be two dimensional in planes perpendicularto the rotation axis (vortex tubes will tend to remain parallel to the rotation axis, resisting bending,shrinking or stretching).

We will come back to these concepts via practical examples developed in the next chapters.

1.6.5 Centrifugal and Stratification Effects: The Froude Number

Resorting to the reference quantities defined in Section 1.3 (the conventional scalings), replacing thedensity in the buoyancy term with ρ ∼= ρ0 [1 − βT (T − TREF)] and simply with ρ0 in the other terms,the momentum equation in nondimensional form can be written as:

∂V

∂t+ V · ∇V + ∇p = Pr ∇2V − Pr RaT

(ig + Fr

χrir

)− 2 Pr τ(i ∧ V ) (1.86)

Ra being the usual characteristic numbers for thermogravitational convection defined by Equation 1.56and Fr the so-called centrifugal Froude number defined as:

Fr = 2dr

g(1.87)

Moreover, χ is the nondimensional ratio of the characteristic lengths in radial and vertical directions,respectively, i.e. χ = dr /d .

The centrifugal Froude number measures the relative importance of centrifugal effects with respectto gravitational ones (Fr 1 representing, therefore, a situation in which thermal centrifugal effectscan be neglected in comparison to the thermal gravitational ones).

Following the same process for the vorticity Equation 1.77 gives:

Dζ

∂t= ∂ζ

∂t+ V · ∇ζ =

(ζ + 2 Pr τ i

)· ∇V + Pr ∇2ζ + Pr Ra

(ig + Fr

χrir

)∧ ∇T (1.88)


From Equation 1.88 the reader will also immediately gather the well-known result that a state ofhydrostatic equilibrium (V = 0 everywhere) is impossible unless density (temperature) variations areabsent on the level surfaces perpendicular to the generalized gravity, i.e. unless the following conditionholds everywhere:

∇T ∧ g = 0 (1.89)

From a mathematical point of view Equation 1.89 must be regarded as conditions necessary, butnot sufficient for hydrostatic equilibrium, which, in practice, means that the fluid will be in stagnantconditions only if the system satisfies additional specific constraints (the characteristic number must notexceed a given threshold).

For Fr 1 the necessary condition for hydrostatic equilibrium reduces to the well- known situationof parallelism between gravity and ∇T .

In particular, when gravity is opposite to ∇T , g has no effect (does not induce convective flows) forany value of its magnitude (this effect is well known: for instance, stably stratified fluids uniformlyheated from above do not exhibit spontaneous convective motion); when gravity is concurrent to ∇T

(fluid heated from below) then convection arises only if the critical conditions for the onset of convectionare exceeded, i.e. if the Rayleigh number is larger than a critical value Ra = Racr, so that instabilitysets in (obviously, for Fr � 1 the other limit is attained in which the necessary condition for hydrostaticequilibrium reduces to the parallelism between the centrifugal force and ∇T ).

When ∇T ∧ g �= 0 and Fr 1 (the typical situation considered in canonical studies on this specificcase is that of ∇T perpendicular to g; the related problem is generally referred to as convectionin systems with lateral heating), flow can emerge for any value of the temperature gradient (thereis no threshold). The horizontal component of such flows typically tends to develop stable thermalstratification (bottom cold, top hot). For such cases a relevant characteristic parameter is the so-calledstratification Froude number defined as:

Fr = U

ωBV d(1.90)

where U is a characteristic velocity in the horizontal direction, ωBV the Brunt–Vaisala frequency relatedto the (positive) gradient of temperature in the vertical direction (already introduced in Section 1.6.2)and d is now the vertical scale of the motion (which does not necessarily coincide with the systemvertical extension).

1.6.6 The Rossby Deformation Radius

While rotation favours the formation of ‘deep’ flows having weak variations along the axis of the fluid’srotation (as a result of the Taylor–Proudman theorem introduced in Section 1.6.4, which ‘forbids’ motionalong the rotational axis in the limit of inviscid fluid), in general, stratification promotes the formationof ‘shallow’ convective structures having strong variations across stratification surfaces and motionessentially parallel to these surfaces.

Specific information on the type of flow regime established can be obtained by combiningthe stratification Froude number with the classical Rossby number defined by Equation 1.82 Inparticular, if

Fr2

Ro< Ro (1.91)


thermal stratification will oppose vertical fluid displacement (density stratification will create, in fact, arestoring force driving a fluid particle back to its density level). Vice versa, for

Fr2

Ro> Ro (1.92)

rotation will be the dominant process, opposing changes in the component of velocity parallel to therotation axis (this discussion, among other things, also provides evidence that both thermal stratificationand rotation imply the existence of a certain direction along which motions are ‘restricted’).

When

Fr2

Ro∼= Ro (1.93)

i.e. the Rossby number and stratification Froude number have the same order of magnitude, in particular,stratification and rotation will exert their influence on the resulting flow with comparable importance.Such a case occurs when:

Ld = ωBV d

f(1.94)

This equation provides a precise relationship between the typical scales of the flow in the horizontal(Ld ) and vertical (d ) directions. Ld is generally known as the ‘Rossby deformation radius’, a usefulreference quantity in the characterization of many types of flow.

When the characteristic horizontal scale of fluid motion becomes ∼= Ld , the two opposite tendenciestowards deep or shallow flow discussed above can have antagonistic consequences in determiningtypical dynamics (e.g. of the Earth’s atmosphere and oceans for which the typical depth scale of thefluid is about 7 km and the full ocean depth, respectively, while for typical atmospheric and oceanicvalues of ωBV and f , the Rossby radius Ld

∼= 1000 and 50 km, respectively, a scale which is muchsmaller than the Earth’s radius).

Before embarking in an exhaustive treatment of such phenomena (following chapters), the finalsection of this chapter is entirely devoted to some propaedeutical concepts that must be necessarilyprovided to help the reader in the understanding of the descriptions and elaborations given later. Evenif most of such notions have their origin in atmospheric science, it will be shown in this book how theyhave also extensive background application to typical problems of technological interest.

1.7 Some Elementary Effects due to Rotation

1.7.1 The Origin of Cyclonic and Anticyclonic flows

For the convenience of the reader who is not familiar with such concepts the terms ‘cyclone’ and‘anticyclone’ are introduced in Figure 1.5.

In meteorology, a cyclone is an area of closed, circular fluid motion rotating in the same directionas the Earth. This is usually characterized by spiralling winds that rotate counter-clockwise in thenorthern hemisphere and clockwise in the southern hemisphere; an anticyclone, on the other hand,rotates clockwise in the northern hemisphere, and counter-clockwise in the southern hemisphere.

In the former case, the air, usually warmer at the surface, rises due to buoyancy, in a column.By continuity surrounding surface air tends to flow inward. Owing to the Coriolis force that pulls


Figure 1.5 Genesis of cyclonic and anticyclonic flows in the northern hemisphere: when warmer air at the surface risesdue to buoyancy in a column, by continuity, surrounding surface air tends to flow inward; owing to the Coriolis forcethat pulls winds to the right of their initial path in the northern hemisphere, such flowing air will be deflected to the righthence initiating a counter-clockwise rotation. The opposite occurs for anticyclones as they are typically initiated by colddescending air.

winds to the right of their initial path in the northern hemisphere (and to the left in the southernhemisphere), such flowing air will be deflected to the right in the northern hemisphere (to the left inthe southern hemisphere) thus initiating a counter-clockwise (clockwise) rotation. The opposite occursfor anticyclones as they are typically initiated by descending air.

A related term often used in the realm of meteorology is ‘cyclogenesis’. It is used as a generalheading for several processes, all of which lead to the formation of a structure with the properties of acyclone. It can occur at various scales, from the microscale to the synoptic scale. There are four mainscales, or sizes of systems in meteorology: the macroscale, the synoptic scale, the mesoscale and themicroscale. The macroscale refers to systems with global size. Synoptic scale systems cover a portionof the Earth’s surface with dimensions of O(103) km across. The mesoscale is the next smaller scale,and often is split into two sub-ranges: meso-alpha phenomena range from O(102) to O(103) km across(e.g. the tropical cyclone), while meso-beta phenomena range from O(10) to O(102) km across (thescale of the mesocyclone). The microscale is the smallest of the meteorological scales, with O(1) km(the scale of tornadoes and waterspouts).

Notably, as anticipated at the end of Section 1.6, it will be illustrated in the following chapters (inparticular, Chapters 2 and 4) how the concepts of cyclone and anticyclone also apply to the laboratoryscale (typically ranging from O(1) to O(102) cm) which is relevant to technological problems andindustrial applications.

1.7.2 The Ekman Layer

Effects of viscosity can be very substantial in rotating systems. As an example, the Taylor–Proudmantheorem discussed in Section 1.6.4 (where it was derived in the case of inviscid fluids) is no longervalid in boundary layers ( BLs) for any value of the Rossby number .

For fundamental information about the important concept of ‘BL’ in non-rotating systems the readeris referred to Lappa (2010). The boundary layer can be generically defined as a particular region of thefluid in which some specific effects, ignored in the remaining part of the domain occupied by the fluid,cannot be neglected, and/or as a region where extremely steep gradients of velocity and/or temperatureare present.


It is known, in fact, that such BLs can develop with respect to the distribution of velocity ormomentum (and in such a case they are referred to as kinetic or viscous BLs), with respect to thedistribution of energy (referred to as thermal BLs accordingly) or both (if the force driving the flowis relatively strong). In general, for small-Pr fluids [Pr < O(1)] these layers appear primarily withrespect to the distribution of momentum (velocity) and the thermal BL, if present, is thicker than thekinetic one (i.e. the kinetic BL is nested in the thermal one) or it is of a scale comparable to theoverall extension of the geometry considered. By contrast, in high-Pr regimes [Pr ≥ O(1)] the oppositebehaviour holds, with BLs appearing primarily with respect to the distribution of temperature (typically,the thickness of the kinetic BL tends to be considerably larger than the thickness of the thermal BL,i.e. the latter is nested in the former).

For the viscous BL, in particular, when non-rotating flows are considered, the dominant balance ofmomentum is between convective and viscous terms.

For problems in which the effects of rotation are significant, however, the region in which viscouseffects become important is typically defined by a balance between Coriolis and viscous or stress terms.The resulting BL is known as an ‘Ekman layer’ (named after Swedish oceanographer V. W. Ekman,see Ekman, 1905).

In determining the related properties, generally, it is assumed that the nonlinear and time-dependentterms in the momentum equation are negligible, that the Boussinesq approximation is applicable and thathydrostatic balance holds in the vertical direction (buoyancy is constant, not varying in the horizontal).Moreover, within the layer, frictional terms are important, whereas geostrophic behaviour governed byEquation 1.83 is assumed beyond it.

Accordingly, the related frictional-geostrophic balance is written as:

1

ρo

∇H p = −2 ∧ V H + ν∇2V H (1.95)

where the subscript H indicates that the operator ∇ is limited to the horizontal, V H is the horizontalvector component of velocity and the application of the laplacian, in general, is considered solely forthe vertical direction, i.e.

∇2V H∼= ∂2

∂z2V H (1.96)

The kinematic viscosity may be eventually include a turbulent contribution (ν = ν + νT ) due, e.g.,to the presence of atmospheric small-scale turbulent motion.

Indicating with the superscript ‘G’ the flow satisfying the non-viscous (inviscid) geostrophic balance

1

ρo

∇H p ∼= −2 ∧ V GH (1.97)

Equation 1.95 can be rewritten as:

2 ∧ (V H − V G

H

) = ν∂2

∂z2V H (1.98)

These equations can be solved to determine the viscous departure of the velocity from geostrophicbalance as a function of height in the BL.

Related boundary conditions are V = 0 on the solid boundary and V → V G as the verticalcoordinate → ∞.


υ/uG

u/uG

gz dimensionless height0.50

0.25

0.000.0 0.2 0.4 0.6 0.8 1.0

2p/3

p/3p/6

p/2

Figure 1.6 Sketch of the Ekman spiral describing the turning of winds with height in the atmospheric boundary layer asthe effects of friction diminish with height (u/uG = 1 and v/uG = 0 representing the limit case of purely geostrophic wind).

1.7.3 Ekman Spiral

For Earth’s atmosphere the above equations yield for V G = (uG, 0) the following well-known solution:

u = uG[1 − e−γ z cos(γ z)

](1.99a)

v = uG[e−γ z sin(γ z)

](1.99b)

where γ = (f/2ν)1/2 and f is the usual Coriolis parameter defined in Section 1.2.

This solution (see Figure 1.6), which is valid in the northern hemisphere, is known as the Ekmanspiral (the hodograph of the velocity against height is a clockwise spiral converging to (uG, 0)).

Its most remarkable property is that the velocity not only changes with the considered distance fromthe wall, but also changes direction (in other words it describes the turning of the winds with height inthe BL as the effects of friction diminish with height).

It is also remarkable how this solution leads to the more or less immediate realization that a charac-teristic nondimensional length scale can be introduced for the BLs as δEk = Ta−1/4.

1.7.4 Ekman Pumping

As an additional useful and pedagogical example of the role potentially played by viscous effects inrotating fluids, it is worth considering fluid motion induced by impulsively started rotation in an opencylinder (a tank) containing an isothermal liquid (thermal buoyancy absent).

As will be illustrated in this section, when such a cylinder begins to rotate abruptly, the fluid at thebottom forms an Ekman layer; in such a layer the fluid typically moves with the cylinder, though thebulk of the fluid remains stationary until a final condition of overall solid body rotation is reached.

This canonical problem related to the genesis, evolution and general properties of the Ekman layerin the simplest case of finite-sized cavities with geometrical axisymmetry was originally examined byGreenspan and Howard (1963).

On the basis of the results reported in such a classical article, it is worth beginning the discussionfrom the remark that although the initial and final states of rigid rotation are simple, the transient


behaviour is a non-trivial process as it involves the time-dependent motion of a coupled diffusive anddispersive mechanical system .

Three complex distinct stages of evolution, in fact, can be clearly identified and discussed:

• the development of the Ekman layer,• secondary flow and inviscid fluid spin-up,• the viscous decay of small residual modal oscillations.

A detailed description of such steps is elaborated in the following, introducing as necessary linkswith other related concepts given in the preceding sections.

Concerning the first stage, it is obvious that the initial impulsive change in the angular velocityimmediately produces a shear layer at the solid bottom, which then starts to thicken by viscous diffusion.In practice, within a few revolutions a quasi steady Ekman BL develops. In addition, at this stageoscillations at twice the rotation frequency are developed, whose existence can be justified on the basisof nonlinear effects (in general, however, they have a very small amplitude).

The growing layer is characterized by fluid moving in the same direction as the bottom (azimuthally),which gives rise to the so-called primary circulation indicated in Figure 1.7. As a consequence of theCoriolis force (let us recall that this force is given by the vector product of local velocity and , seeEquation 1.68) acting on the primary flow, however, an outward radial secondary flow also develops.

Figure 1.7 Sketch of primary and secondary circulations induced by the presence of a growing Ekman layer in a cylinderwith impulsively impressed angular velocity starting from conditions of quiescent fluid (the symbol x indicates flowperpendicular to the page).


Such radial transport tends to be balanced by a small flow towards the BL from the essentiallygeostrophic (inviscid) upper region. Due to continuity, in turn, such a vertical flow is typically main-tained through the establishment of an equally small radially inward flow in the region above the Ekmanlayer (in other words, the convergence of fluid into the Ekman layer, together with the constraints ofthe geometrical configuration, produce a weak radial convection in the inviscid region in order to con-serve mass). As such fluid is essentially inviscid, the angular momentum of a ring of fluid movinginward to replace the fluid entering the Ekman layer is conserved and, hence, the ring acquires anincreased azimuthal velocity. This means that the Ekman layer tends to behave as a sink for lowangular momentum fluid in the interior, the fluid being replaced by higher angular momentum fluiddrawn from larger radii.

As the condition of the external fluid approaches the values appropriate to the final steady state, theEkman layer decays (stage 3). This happens in a dimensional time of O(τ 1/2/) (τ being the Coriolischaracteristic number d2/ν where d is the depth of the liquid).

The aforementioned small oscillations persist until they are finally destroyed by viscosity at a dimen-sional time of O(τ /).

At this time there is no longer any inviscid region. Viscous forces are then important at all positionsand act to eliminate the residual modal oscillations. The final state is represented by fluid in solidbody rotation with each fluid particle experiencing a vorticity two times the tank angular velocity (i.e.f = 2, as explained in Section 1.2.2).

The transient process described above could be also summarized by just pointing out that the viscousforce between the Ekman layer and the stationary fluid causes fluid adjacent to the BL to accelerateand that the Coriolis force propels it outward (the result being a flow circulating along the BLs anddownward along the centre of the cylinder, as shown in Figure 1.7). This mechanism is called Ekmanpumping or Ekman suction and is often important when studying the spin-up of a fluid.

Its most notable consequence is that owing to the downward flow over the central region, requiredto supply the radial motion in the Ekman layer, the angular velocity of the inviscid fluid tends to beincreased to the new value of the tank in a much shorter time that would be expected from the ordinaryviscous diffusion mechanism .

The foregoing discussion applies to the case of an isothermal fluid initially at rest. For usefulconsiderations about the development of BLs in nonisothermal rotating systems and in the presence ofnatural convection, the reader is referred to the recent study of King et al. (2009, and all referencestherein).

These researchers identified possible transitions between two different regimes controlled by therelative thicknesses of the thermal (nonrotating) and Ekman (rotating) BLs. They formulated a predictivedescription of the transition between the two regimes on the basis of the competition between the relatedtwo BLs. This transition scaling theory unifies the disparate results of an extensive array of previousexperiments, and is broadly applicable to natural convection systems.

1.7.5 The Stewartson Layer

As illustrated in the preceding section, an Ekman layer tends to form on the bottom (and also on thetop if it is a solid wall) of a rotating cavity. This, in turn, results in the emergence of a flow as shownin Figure 1.7. Since these currents of fluid must form a closed loop in a finite-size geometry, it is verylikely that a BL will be present also on the sidewall. Such BLs, which, indeed, are known to effectivelyemerge and affect several cases of practical interest, are known as Stewartson layers (Stewartson, 1957,1966; Moore and Saffman, 1969).


For simplicity here we consider a closed cylindrical container bounded by nonslip walls. As inthe previous section, the bulk is geostrophic and Ekman layers occur near the top and bottom plates.Moreover, we assume that the bulk geostrophic flow has a negative constant vorticity. Because of theEkman suction, there will be, therefore, a net radial outward flow found in each of the Ekman layers.

In practice, these flows converge into the Stewartson layer on the sidewall, which hence must containa vertical flow. The derivation of the general properties of these BLs is significantly complex and forthis reason it is omitted here. Rather, we limit ourselves to mentioning (the reader being referred toStewartson (1957, 1966), Smith (1987) and Vooren (1993) for an exhaustive treatment) that a layer ofthickness δ1 ∝ Ta−1/8 can carry the required vertical flow and match to the bulk. This layer, however,cannot subsequently connect to the sidewall. An additional region, therefore, must exist within thezone with thickness ∝ Ta−1/8. This layer has a thickness δ2 ∝ Ta−1/6 from the sidewall. The flowsinside these two regions close the circulation set up by the Ekman layers (while Ekman layers havea nondimensional thickness δEk ∝ Ta−1/4, the Stewartson layer at the sidewall, hence, has a sandwichstructure consisting of a thicker outer layer of nondimensional thickness δ1 ∝ Ta−1/8 and a thinner innerlayer of nondimensional thickness δ2 ∝ Ta−1/6).

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1 Equations, General Concepts and Nondimensional Numbers