Notes and Exercises Accompanying the Lecture Series Advanced Fluid Mechanics b56a

176 Pages·2003·2.13 MB·English

Notes and Exercises
Accompanying the Lecture Series
Advanced Fluid Mechanics b56a
F.T.M. Nieuwstadt
Lab. Aero- en Hydrodynamics
Leeghwaterstraat 21
2628 CA Delft
Preface
In this lecture series we discuss basic concepts of fluid dynamics from a fundamental point of
view. As such, this lecture series forms the point of departure for all further advanced courses
on fluid mechanics.
The lecture series is based on the book of G.K. Batchelor, which is generally accepted to
be “the” book on the basics of fluid dynamics and it is highly recommended to anyone who is
going to be involved in fluid-mechanics research.
G.K. Batchelor
An Introduction to Fluid Dynamics
Cambridge University Press, 1967.
These notes should therefore be considered as a collection of comments and annotations for
Batchelor’s book, to which we also refer for a further discussions and explanation. In this
course we shall not treat the whole book of Batchelor. Below, we give the sections of the book,
of which the material will be discussed and which are therefore highly recommended to be
studied.
1.1 1.2 2.1 appendix 2
2.2 2.3
2.4 2.5 2.6
3.1 3.2 3.3 (1.3, 1.4)
3.3 (1.9) 2.7 6.1
6.2 6.3 6.8 (6.4, 6.10)
6.5 (2.7) 6.6
3.4 (1.5) 3.5 4.1
4.2 4.3 4.5 (4.6)
4.7 4.8 4.9
4.12 5.1 (5.2) 5.3 5.4
5.5 5.7 5.8 (5.9) 5.10 5.11 5.12
Besides, attheendofeach chapterorsection thereareproblemsgiven whichcanbestudied
in order to practice the material discussed. Note, that these problem are in general not easy
and they require in some case extensive calculations. They are therefore also to be considered
as material to extend the theory treated in each chapter.
Moreover, as additional material to consult during the study of this course, we mention
below several alternative books. In particular the book of Prandtl and Tietjens is highly
recommended.
Books that may be consulted when studying this material
L.PrandtlandO.G.Tietjens. FundamentalsofHydro-andAerodynamics,DoverPublications,
Inc., New York, 1957.
D.J. Acheson. Elementary Fluid Mechanics. Oxford Applied Mathematics and Computing
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Science Series.
L.D. Landau and E.M. Lifshitz. Fluid Mechanics. Vol. 6. Course of Theoretical Physics.
Pergamon Press, 1984.
L.M. Milne-Thomson. Theoretical Hydrodynamics. Mac-Millan, 1974.
R.L. Panton Incompressible Flow. John Wiley, 1984.
I. Shames. Mechanics of Fluids. McGraw-Hill, 1962.
Furthermore, it should be mentioned that during preparation of the English version of
these lectures, I received help from Emile Coyajee, Rene Delfos and Chiara Tesauro, which I
gratefully acknowledge.
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Contents
1 Introduction 6
1.1 Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Coordinate-systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Material derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Kinematics 14
2.1 Conservation of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Stream function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Velocity field, local analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.1 The rate-of-deformation tensor, e . . . . . . . . . . . . . . . . . . . . . 19
ij
2.3.2 The rotation tensor, ξ . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
ij
2.3.3 Local description of the velocity field . . . . . . . . . . . . . . . . . . . . 21
2.4 Velocity field, global analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4.1 Given volume expansion, ∆. . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4.2 Given vorticity distribution, ω . . . . . . . . . . . . . . . . . . . . . . . 27
i
3 Dynamics 34
3.1 Conservation of momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1.1 Material integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1.2 Conservation of momentum in differential form . . . . . . . . . . . . . . 35
3.1.3 Conservation of momentum in integral form . . . . . . . . . . . . . . . . 38
3.2 The stress tensor, σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
ij
3.2.1 Stress in a fluid at rest . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.2 Stress in a fluid in motion . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 Equation of motion and boundary conditions . . . . . . . . . . . . . . . . . . . 43
3.3.1 Equation of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3.2 Initial and boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 44
3.3.3 Non-inertial coordinate system . . . . . . . . . . . . . . . . . . . . . . . 46
3.4 Vorticity dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4 Non-viscous fluids: potential flow 53
4.1 Euler-equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 Rotation-free flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2.1 Potential flows: general properties . . . . . . . . . . . . . . . . . . . . . 56
4.3 Three-dimensional potential flows . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3.1 Flow from a container . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
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4.3.2 Parallel flow around a sphere . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3.3 A sphere, moving in an infinite medium . . . . . . . . . . . . . . . . . . 64
4.3.4 Influence of boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.4 Two-dimensional potential flows . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.4.1 Parallel flow around a cylinder . . . . . . . . . . . . . . . . . . . . . . . 72
4.4.2 Flow around a cylinder with circulation . . . . . . . . . . . . . . . . . . 74
4.5 Waves on a free surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5 Thermodynamics 82
5.1 Conservation of energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.2 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.3 Energy integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6 Viscous Flows 88
6.1 The Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.2 Exact solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.2.1 One-dimensional flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.2.2 Circular flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.2.3 Other exact solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.3 The Reynolds number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.4 R (cid:1) 1, Stokes flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.4.1 Lubrication theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.4.2 Hele-Shaw flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.4.3 Percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.5 Stokes-flow around a sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.6 The Oseen approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.7 R (cid:2) 1, boundary layers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.7.1 Boundary layers near a flat plate . . . . . . . . . . . . . . . . . . . . . . 124
A Notations and computational rules 135
B Curvilinear coordinates. 138
B.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
B.2 Cylinder-coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
B.3 Spherical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
C Flow field for a given volume expansion and vorticity 146
C.1 Flow field for a given volume expansion . . . . . . . . . . . . . . . . . . . . . . 146
C.2 Flow field for a given vorticity, ω. . . . . . . . . . . . . . . . . . . . . . . . . . . 147
D General properties of potential flows 150
D.1 The solution for the potential Φ is single valued . . . . . . . . . . . . . . . . . . 150
D.2 The integral of the kinetic energy . . . . . . . . . . . . . . . . . . . . . . . . . . 151
D.3 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
D.4 Minimal energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
D.5 Maximum value of the potential. . . . . . . . . . . . . . . . . . . . . . . . . . . 152
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E The theory of complex functions for two-dimensional potential flows 153
E.1 Analytical function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
E.2 Complex potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
E.3 Blasius theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
E.4 Cauchy integral theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
E.5 Conformal transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
E.6 Flow around a flat plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
F Crocco’s theorm 164
G Stokes flow around a sphere 165
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Chapter 1
Introduction
1.1 Fluids
sect:fluida
Fluids comprise a large number of flowing media, which play an important role in many
applications and circumstances. First we have the so-called standard fluids. These are gases
such as air or liquids such as water. Applications for these standard fluids are for instance
found in industries such as aircraft, energy production and petro-chemical but these fluids are
also relevant forenvironmental applications such as meteorology, hydrology andoceanography.
Besides the standard fluids we also distinguish other, non-standard fluids which are usually
characterized by special properties. Examples are biological fluids such as blood, oils that may
occur in heat exchangers or bearings, liquid metals which are used during casting processes,
liquid stone as we may encounter in the mantle of the earth and finally plasmas, which are
considered in astrophysics and fusion physics.
How should we in general characterize a fluid? A suitable characterization would be that
a fluid does not have a fixed form or shape in contrast with a solid. A more quantitative
definition would be that in order to deform a solid we need a continuous force (in time), at
least if we neglect plastic deformations. An example is an elastic solid, of which the form is
restored as soon as the force is removed. Fluids do not have a fixed form so we do not need
to impose a force to keep a fluid (in rest) in an arbitrary shape. In other words fluids takes
the form imposed by the reservoir, in which the fluid is kept (an exception must here be made
for fluids that can form a free surface). However, a fluid offers resistance to an imposed force
when due this force its form changes or when the fluid starts to flow. This definition will be
used later in the description of a fluid in terms of change of form (or alternatively in terms of
the rate of deformation) under the influence of forces imposed on the fluid. This description
is known as the constitutive law for the fluid.
The so-called simple fluids are those, which have the property that forces (apart from an
isotropic pressure force) are linearly proportional to the rate of deformation. It is said that
these fluids are Newtonian. The equivalent for a solid would be the case of an elastic material,
for which the force depends linearly on deformation and which is also known as Hooke’s law
There are some fluids, which combine the properties of an elastic solid and a simple fluid.
These are called a visco-elastic fluid, which reacts as a solid to fast deformations and as a fluid
to slow deformations, where fast or slow is defined with respect to a characteristic time scale
of the material. These fluids are also characterized as complex in contrast to the simple fluids
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u
local
velocity
(cid:1)3 (cid:2)V
L3
Figure 1.1: Continuum hypothesis
introduces above. Other examples of complex fluids are fluids, of which the forces depend
non-linearly on the rate of deformation, such as shear-thinning or shear-thickening fluids. The
behaviour of these complex fluids is studied in the field of rheology.
The term fluid has been used above for both gases and liquids. This suggests that both
media,ofwhichthestructureonamolecularscaledifferswidely, canbedescribedbyanalogous
methodsandequations. Thebackgroundforthisanalogyistheso-calledcontinuumhypothesis.
This hypothesis states that every property of a fluid is described as the average over a small
volume δV. As an example let us take the velocity component u in the x-direction. At each
position (x,y,z) the velocity is defined as the integral over the volume δV centred around
(x,y,z) according to
(cid:1) (cid:1) (cid:1)
(cid:2) (cid:3)
1
(cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1)
u(x,y,x) = u x,y ,z dxdy dz . (1.1)
δV
δV
(cid:1) (cid:1) (cid:1)
where the x, y and z run over the volume δV. The behaviour of u as function of δV is
illustrated in figure 1.1.
Above we mentioned a small volume δV. Small in this case means that the a characteristic
length related to the volume is small with respect to the length scale (L) of the fluid motion,
but large with respect to the scale (λ) of the molecular structure of the material. For example,
the characteristic molecular scale λ can be taken equal to the mean free path in a gas or to
the inter-molecular distance between m(cid:2)olec(cid:3)ules for a liquid.
When δV is too small, say δV ≈ O λ3 , figure 1.1 shows large variations in the value of u
as a result of statistical fluctuations, which become(cid:2)do(cid:3)minant when the number of molecules
in the averaging volume is small. When δV ≈ O L3 , i.e. of the order of the geometry of
the flow, u also varies as a function of δV as a result of spatial variations in the velocity.
When λ ≪ L, there exists in between these two limits a range of values for δV, for which u
approximately constant. The continuum hypothesis now defines the velocity at each point in
space as the average over a fluid volume surrounding this point, with a size which lies in the
region where the average is no function of the size of the averaging volume. This small volume
of fluid which actually represents the fluid properties at its centre point is usually denoted as
a fluid element.
By applying this continuum hypothesis any effect of the molecular structure on the def-
inition of velocity has been eliminated. Moreover, by this method fluid properties, such as
velocity can be defined in each point in space and it can be shown that this results in a con-
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tinuous and differentiable function. Therefore, the invoking of the continuum hypothesis lies
at the basis of continuum mechanics of which fluid mechanics is a subdivision.
Nevertheless, it will be clear that gases and liquids must have different properties as a
result of their different molecular structure. A gas is for instance compressible while a liquid
can be considered as almost incompressible. Compressible means here that the density ρ of
the fluid is a function of the pressure p. Thermodynamics tells us that all materials, including
gases and liquids, are compressible and the relation between density and pressure is described
by an equation of state. As mentioned above, for liquids the change in density as a result of
a pressure increase is in practical circumstances negligible and in this case we can speak of an
incompressible fluid (in section 2.1 we will learn a another definition of incompressibility in
terms of flow behaviour). However, we can note already here that for an incompressible fluid
the pressure can no longer be a thermodynamic variable and must be defined differently. This
will be discussed in section 3.2.
The molecular structure of the fluid becomes also important when exchange or transport
processes are considered, such as exchange of heat or exchange of momentum. The latter
processformsthebackgroundoffriction. Inagasmoleculesarefarapartandinteractprimarily
bymeans of collisions. Inaliquidmolecules aremuchcloser andalthough they can move freely
with respect to each other, they nevertheless influence each other by means of inter-molecular
forces. This difference in structure has for instance a direct influence on the magnitude of the
exchange coefficients for heat and momentum. In addition this difference in structure causes
also a different behaviour a function of temperature. When the temperature increases in a
gas, the momentum or velocity of the molecules becomes larger. As a result, collisions become
more frequent and energy exchange during collisions becomes larger with as effect that the
exchange coefficients for heat and momentum become larger. Incontrast, a larger temperature
in a fluid increases the vibrations of the molecules so that they move to a larger distance with
respect to each other. Inter-molecular forces at these larger distances become weaker with as
result that coefficients for heat and momentum exchange decrease.
Problem 1.1 The continuum hypothesis fails when we consider a flow, of which the charac-
teristic length scale is of the same order of magnitude as the molecular structure. Give some
examples of such flows.
Problem 1.2 Is there a characteristic time scale, below which the continuum hypothesis is
no longer valid? Estimate the order of magnitude of this time scale.
1.2 Coordinate-systems
Let us introduce the term kinematics, by which we mean the description of the velocity field
of a flow as a function of the space and time coordinates. There are basically two methods to
describe the flow: the Eulerian and Lagrangian method.
In the case of the Lagrangian description we start with a fluid element, which we follow
on its way through the flow. Suppose we label this fluid element with a label a. The vector
X(t;a) then gives the position of this particular fluid element (i.e. labelled by a) as a function
of time. The X(t;a) thus traces a curve in 3D-space as function of time and this curve is
generally called a trajectory or a fluid-element path. The velocity of the fluid element is then
by definition equal to
dX(t;a)
V = . (1.2)
dt
8
We can in principle do this for every fluid element assuming that each element can be
distinguished by means of a different and unique label a. The vector a can be chosen at will
but in most cases one takes the initial position of the fluid element, i.e. the position of the
element at t = t0 or a = X(t0). If we now consider that initially, i.e. t = t0, every position
in space occupied by single fluid element then all these elements can be uniquely labelled by
their initial position. The X(t;a) for all labels a then gives a description of the complete flow
field as a function of time and by means of (1.2) we can compute the velocity at every position
and at every time.
In this Lagrangian description the independentcoordinates are the a and t. The trajectory
X(t;a) is the dependent variable. The capital letter X is used here to make the distinction
between the trajectory of a fluid element and the position vector in a Cartesian coordinate
system, which is indicated by x = (x,y,z) ≡ (x1,x2,x3).
In the Eulerian description the velocity vector u = (u,v,w) ≡ (u1,u2,u3) is defined at
each position and each time in the form of a vector field: u(x,t). In this case x and t are the
independent variables and the velocity field is the dependent variable.
The relationship between the Lagrangian and Eulerian description follows from the fact
that the velocity at position x and time t must be equal to the velocity of the fluid particle
which is at this position and at this particular time. In the form of an equation this implies
dX(t;a)
= u(x = X,t). (1.3)
dt
From a practical point of view the Eulerian description is the easier one to use. If we
put, for instance, a measuring device at a fixed position in a flow, we measure an Eulerian
variable. The Lagrangian description has advantages mainly from a theoretical point of view.
Forinstancetheequations,whichgovern afluidmotion, canbeveryconveniently andelegantly
formulated in terms of the Lagrangian description. The Lagrangian description has also its
advantages whenwe want to study the motion of individualfluidelements. An example of this
latter application is the dispersion of contaminants.
Coupled to the Eulerian description formulated in some given coordinate system x, t, we
can introduce some useful concepts. These are:
• Stationary
In this case all dependent flow variables, e.g. the velocity, are not a function of time.
This means that the velocity field is given by u(x), from which time t has disappeared.
In the opposite case when the velocity u depends on time, we call the flow instationary
or non-stationary.
• Streamline
A streamline is defined as the line, which is everywhere tangent to the velocity vector
u(x,t) at a given time t. Note that in an instationary flow the stream-line pattern might
varyforeach timeinstant. Basedonthisdefinitiontheequation, fromwhichastreamline
can be computed, is given by
dx1 dx2 dx3
= = . (1.4)
u1(x1,x2,x3,t) u2(x1,x2,x3,t) u3(x1,x2,x3,t)
In general a streamline is not equal to the trajectory X(t) but they are identical for a
stationary flow.
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