LECTURE NOTES ON INTERMEDIATE FLUID MECHANICS, Exams of Fluid Mechanics

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LECTURE NOTES ON

INTERMEDIATE FLUID MECHANICS

Joseph M. Powers

Department of Aerospace and Mechanical Engineering

University of Notre Dame

Notre Dame, Indiana 46556-

USA

updated

18 March 2013, 10:06am

• Preface
• 1 Governing equations
  • 1.1 Philosophy of rational continuum mechanics
    • 1.1.1 Approaches to fluid mechanics
    • 1.1.2 Mechanics
    • 1.1.3 Continuum mechanics
    • 1.1.4 Rational continuum mechanics
    • 1.1.5 Notions from Newtonian continuum mechanics
  • 1.2 Some necessary mathematics
    • 1.2.1 Vectors and Cartesian tensors
      • 1.2.1.1 Gibbs and Cartesian Index notation
      • 1.2.1.2 Rotation of axes
      • 1.2.1.3 Vectors
      • 1.2.1.4 Tensors
        • 1.2.1.4.1 Definition
        • 1.2.1.4.2 Alternating unit tensor
        • 1.2.1.4.3 Some secondary definitions
          • 1.2.1.4.3.1 Transpose
          • 1.2.1.4.3.2 Symmetric
          • 1.2.1.4.3.3 Antisymmetric
          • 1.2.1.4.3.4 Decomposition
        • 1.2.1.4.4 Tensor inner product
        • 1.2.1.4.5 Dual vector of a tensor
        • 1.2.1.4.6 Tensor product: two tensors
        • 1.2.1.4.7 Vector product: vector and tensor
          • 1.2.1.4.7.1 Pre-multiplication
          • 1.2.1.4.7.2 Post-multiplication
        • 1.2.1.4.8 Dyadic product: two vectors
        • 1.2.1.4.9 Contraction
        • 1.2.1.4.10 Vector cross product
        • 1.2.1.4.11 Vector associated with a plane
  • 1.2.2 Eigenvalues and eigenvectors 4 CONTENTS
  • 1.2.3 Grad, div, curl, etc.
    • 1.2.3.1 Gradient operator
    • 1.2.3.2 Divergence operator
    • 1.2.3.3 Curl operator
    • 1.2.3.4 Laplacian operator
    • 1.2.3.5 Relevant theorems
      • 1.2.3.5.1 Fundamental theorem of calculus
      • 1.2.3.5.2 Gauss’s theorem
      • 1.2.3.5.3 Stokes’ theorem
      • 1.2.3.5.4 Kinetic energy divergence identity
      • 1.2.3.5.5 Leibniz’s rule
      • 1.2.3.5.6 Reynolds transport theorem
• 1.3 Kinematics
  • 1.3.1 Lagrangian description
  • 1.3.2 Eulerian description
  • 1.3.3 Material derivatives
  • 1.3.4 Streamlines
  • 1.3.5 Pathlines
  • 1.3.6 Streaklines
  • 1.3.7 Kinematic decomposition of motion
    • 1.3.7.1 Translation
    • 1.3.7.2 Solid body rotation and straining
      • 1.3.7.2.1 Solid body rotation
      • 1.3.7.2.2 Straining
        • 1.3.7.2.2.1 Extensional straining
        • 1.3.7.2.2.2 Shear straining
        • 1.3.7.2.2.3 Principal axes of strain rate
  • 1.3.8 Expansion rate
  • 1.3.9 Invariants of the strain rate tensor
  • 1.3.10 Invariants of the velocity gradient tensor
  • 1.3.11 Two-dimensional kinematics
    • 1.3.11.1 General two-dimensional flows
      • 1.3.11.1.1 Rotation
      • 1.3.11.1.2 Extension
      • 1.3.11.1.3 Shear
      • 1.3.11.1.4 Expansion
    • 1.3.11.2 Relative motion along 1 axis
    • 1.3.11.3 Relative motion along 2 axis
    • 1.3.11.4 Uniform flow
    • 1.3.11.5 Pure rigid body rotation
    • 1.3.11.6 Pure extensional motion (a compressible flow)
• CONTENTS - 1.3.11.7 Pure shear straining - 1.3.11.8 Couette flow: shear + rotation - 1.3.11.9 Ideal point vortex: extension+shear - 1.3.12 Kinematics as a dynamical system
  • 1.4 Conservation axioms
    • 1.4.1 Mass
    • 1.4.2 Linear momenta
      • 1.4.2.1 Statement of the principle
      • 1.4.2.2 Surface forces
      • 1.4.2.3 Final form of linear momenta equation
    • 1.4.3 Angular momenta
    • 1.4.4 Energy
      • 1.4.4.1 Total energy term
      • 1.4.4.2 Work term
      • 1.4.4.3 Heat transfer term
      • 1.4.4.4 Conservative form of the energy equation
      • 1.4.4.5 Secondary forms of the energy equation
        • 1.4.4.5.1 Mechanical energy equation
        • 1.4.4.5.2 Thermal energy equation
        • 1.4.4.5.3 Non-conservative energy equation
        • 1.4.4.5.4 Energy equation in terms of enthalpy
        • 1.4.4.5.5 Energy equation in terms of entropy
    • 1.4.5 Entropy inequality
    • 1.4.6 Summary of axioms in differential form
      • 1.4.6.1 Conservative form
        • 1.4.6.1.1 Cartesian index form
        • 1.4.6.1.2 Gibbs form
        • 1.4.6.1.3 Non-orthogonal index form
      • 1.4.6.2 Non-conservative form
        • 1.4.6.2.1 Cartesian index form
        • 1.4.6.2.2 Gibbs form
        • 1.4.6.2.3 Non-orthogonal index form
      • 1.4.6.3 Physical interpretations
    • 1.4.7 Complete system of equations?
    • 1.4.8 Integral forms
      • 1.4.8.1 Mass
        • 1.4.8.1.1 Fixed region
        • 1.4.8.1.2 Material region
        • 1.4.8.1.3 Moving solid enclosure with holes
      • 1.4.8.2 Linear momenta
      • 1.4.8.3 Energy
      • 1.4.8.4 General expression
• 1.5 Constitutive equations 6 CONTENTS
  • 1.5.1 Frame and material indifference
  • 1.5.2 Second law restrictions and Onsager relations
    • 1.5.2.1 Weak form of the Clausius-Duhem inequality
      • 1.5.2.1.1 Non-physical motivating example
      • 1.5.2.1.2 Real physical effects.
    • 1.5.2.2 Strong form of the Clausius-Duhem inequality
  • 1.5.3 Fourier’s law
  • 1.5.4 Stress-strain rate relation for a Newtonian fluid
    • 1.5.4.1 Underlying experiments
    • 1.5.4.2 Analysis for isotropic Newtonian fluid
      • 1.5.4.2.1 Diagonal component
      • 1.5.4.2.2 Off-diagonal component
    • 1.5.4.3 Stokes’ assumption
    • 1.5.4.4 Second law restrictions
      • 1.5.4.4.1 One dimensional systems
      • 1.5.4.4.2 Two dimensional systems
      • 1.5.4.4.3 Three dimensional systems
  • 1.5.5 Equations of state
• 1.6 Boundary and interface conditions
• 1.7 Complete set of compressible Navier-Stokes equations - 1.7.0.1 Conservative form - 1.7.0.1.1 Cartesian index form - 1.7.0.1.2 Gibbs form - 1.7.0.2 Non-conservative form - 1.7.0.2.1 Cartesian index form - 1.7.0.2.2 Gibbs form
• 1.8 Incompressible Navier-Stokes equations with constant properties
  • 1.8.1 Mass
  • 1.8.2 Linear momenta
  • 1.8.3 Energy
  • 1.8.4 Summary of incompressible constant property equations
  • 1.8.5 Limits for one-dimensional diffusion
• 1.9 Dimensionless compressible Navier-Stokes equations
  • 1.9.1 Mass
  • 1.9.2 Linear momenta
  • 1.9.3 Energy
  • 1.9.4 Thermal state equation
  • 1.9.5 Caloric state equation
  • 1.9.6 Upstream conditions
  • 1.9.7 Reduction in parameters
• 1.10 First integrals of linear momentum
• CONTENTS - 1.10.1 Bernoulli’s equation - 1.10.1.1 Irrotational case - 1.10.1.2 Steady case - 1.10.1.2.1 Streamline integration - 1.10.1.2.2 Lamb surfaces - 1.10.1.3 Irrotational, steady, incompressible case - 1.10.2 Crocco’s theorem - 1.10.2.1 Stagnation enthalpy variation - 1.10.2.2 Extended Crocco’s theorem - 1.10.2.3 Traditional Crocco’s theorem
• 2 One-dimensional compressible flow
  • 2.1 Generalized one-dimensional equations
    • 2.1.1 Mass
    • 2.1.2 Linear momentum
    • 2.1.3 Energy
    • 2.1.4 Summary of equations
      • 2.1.4.1 Unsteady conservative form
      • 2.1.4.2 Unsteady non-conservative form
      • 2.1.4.3 Steady conservative form
      • 2.1.4.4 Steady non-conservative form
    • 2.1.5 Influence coefficients
  • 2.2 Flow with area change
    • 2.2.1 Isentropic Mach number relations
    • 2.2.2 Sonic properties
    • 2.2.3 Effect of area change
    • 2.2.4 Choking
  • 2.3 Normal shock waves
    • 2.3.1 Rankine-Hugoniot equations
    • 2.3.2 Rayleigh line
    • 2.3.3 Hugoniot curve
    • 2.3.4 Solution procedure for general equations of state
    • 2.3.5 Calorically perfect ideal gas solutions
    • 2.3.6 Acoustic limit
  • 2.4 Flow with area change and normal shocks
    • 2.4.1 Converging nozzle
    • 2.4.2 Converging-diverging nozzle
  • 2.5 Rarefactions and the method of characteristics
    • 2.5.1 Inviscid one-dimensional equations
    • 2.5.2 Homeoentropic flow of an ideal gas
    • 2.5.3 Simple waves
    • 2.5.4 Prandtl-Meyer rarefaction
    • 2.5.5 Simple compression 8 CONTENTS
    • 2.5.6 Two interacting expansions
    • 2.5.7 Wall interactions
    • 2.5.8 Shock tube
    • 2.5.9 Final note on method of characteristics
• 3 Potential flow
  • 3.1 Stream functions and velocity potentials
  • 3.2 Mathematics of complex variables
    • 3.2.1 Euler’s formula
    • 3.2.2 Polar and Cartesian representations
    • 3.2.3 Cauchy-Riemann equations
  • 3.3 Elementary complex potentials
    • 3.3.1 Uniform flow
    • 3.3.2 Sources and sinks
    • 3.3.3 Point vortices
    • 3.3.4 Superposition of sources
    • 3.3.5 Flow in corners
    • 3.3.6 Doublets
    • 3.3.7 Rankine half body
    • 3.3.8 Flow over a cylinder
  • 3.4 More complex variable theory
    • 3.4.1 Contour integrals
      • 3.4.1.1 Simple pole
      • 3.4.1.2 Constant potential
      • 3.4.1.3 Uniform flow
      • 3.4.1.4 Quadrapole
    • 3.4.2 Laurent series
  • 3.5 Pressure distribution for steady flow
  • 3.6 Blasius force theorem
  • 3.7 Kutta-Zhukovsky lift theorem
  • 3.8 Conformal mapping
• 4 Viscous incompressible laminar flow
  • 4.1 Fully developed, one dimensional solutions
    • 4.1.1 Pressure gradient driven flow in a slot
    • 4.1.2 Couette flow with pressure gradient
  • 4.2 Similarity solutions
    • 4.2.1 Stokes’ first problem
    • 4.2.2 Blasius boundary layer
• Bibliography

10 CONTENTS

Chapter 1

Governing equations

see Panton, Chapters 1-6, see Yih, Chapters 1-3, Appendix 1-2, see Aris.

1.1 Philosophy of rational continuum mechanics

1.1.1 Approaches to fluid mechanics

We seek here to present an approach to fluid mechanics founded on the principles of rational continuum mechanics. There are many paths to understanding fluid mechanics, and good arguments can be made for each. A typical first undergraduate class will combine a mix of basic equations, coupled with strong physical motivations, and allows the student to develop a knowledge which is of great practical value and driven strongly by intuition. Such an approach works well within the confines of the intuition we develop in everyday life. It often fails when the engineer moves in to unfamiliar territory. For example, lack of fundamental understanding of high Mach number flows led to many aircraft and rocket failures in the 1950’s. In such cases, a return to the formalism of a careful theory, one which clearly exposes the strengths and weaknesses of all assumptions, is invaluable in both understanding the true fluid physics, and applying that knowledge to engineering design. Probably the most formal of approaches is that of the school of thought advocated most clearly by Truesdell,^1 sometimes known as Rational Continuum Mechanics. Truesdell de- veloped a broadly based theory which encompassed all materials which could be regarded as continua, including solids, liquids, and gases, in the limit when averaging volumes were sufficiently large so that the micro- and nanoscopic structure of these materials was unimpor- tant. For fluids (both liquid and gas), such length scales are often on the order of microns, while for solids, it may be somewhat smaller, depending on the type of crystalline structure. The difficulty of the Truesdellian approach is that it is burdened with a difficult notation

(^1) Clifford Ambrose Truesdell, III, 1919-2000, American continuum mechanician and natural philosopher. Taught at Indiana and Johns Hopkins Universities.

11

1.1. PHILOSOPHY OF RATIONAL CONTINUUM MECHANICS 13

• motion is the consequence of forces,
• study of forces on a body is dynamics.

There are many subsets of mechanics, e.g. statistical mechanics, quantum mechanics, continuum mechanics, fluid mechanics, solid mechanics. Auto mechanics, while a legitimate topic for study, does not generally fall into the class of mechanics we consider here, though the intersection of the two sets is not the empty set.

1.1.3 Continuum mechanics

Early mechanicians, such as Newton, dealt primarily with point mass and finite collections of particles. In one sense this is because such systems are the easiest to study, and it makes more sense to grasp the simple before the complex. External motivation was also present in the 18th century, which had a martial need to understand the motion of cannonballs and a theological need to understand the motion of planets. The discipline which considers systems of this type is often referred to as classical mechanics. Mathematically, such systems are generally characterized by a finite number of ordinary differential equations, and the properties of each particle (e.g. position, velocity) are taken to be functions of time only. Continuum mechanics, generally attributed to Euler,^6 considers instead an infinite num- ber of particles. In continuum mechanics every physical property (e.g. velocity, density, pressure) is taken to be functions of both time and space. There is an infinitesimal property variation from point to point in space. While variations are generally continuous, finite num- bers of surfaces of discontinuous property variation are allowed. This models, for example, the contact between one continuous body and another. Point discontinuities are not allowed, however. Finite valued material properties are required. Mathematically, such systems are characterized by a finite number of partial differential equations in which the properties of the continuum material are functions of both space and time. It is possible to show that a partial differential equation can be thought of as an infinite number of ordinary differen- tial equations, so this is consistent with our model of a continuum as an infinite number of particles.

1.1.4 Rational continuum mechanics

The modifier “rational” was first applied by Truesdell to continuum mechanics to distin- guish the formal approach advocated by his school, from less formal, though mainly not irrational, approaches to continuum mechanics. Rational continuum mechanics is developed in a manner similar to that which Euclid^7 used for his geometry: formal definitions, axioms, and theorems, all accompanied by careful language and proofs. This course will generally

(^6) Leonhard Euler, 1707-1783, Swiss-born mathematician and physicist who served in the court of Cather- ine I of Russia in St. Petersburg, regarded by many as one of the greatest mechanicians. (^7) Euclid, Greek geometer of profound influence who taught in Alexandria, Egypt, during the reign of Ptolemy I Soter, who ruled 323-283 BC.

14 CHAPTER 1. GOVERNING EQUATIONS

follow the less formal, albeit still rigorous, approach of Panton’s text, including the adoption of much of Panton’s notation.

1.1.5 Notions from Newtonian continuum mechanics

The following are useful notions from Newtonian continuum mechanics. Here we use New- tonian to distinguish our mechanics from Einsteinian or relativistic mechanics.

• Space is three-dimensional and independent of time.
• An inertial frame is a reference frame in which the laws of physics are invariant.
• A Galilean transformation specifies how to transform from an inertial frame to a frame of reference moving at constant velocity. If the inertial frame has zero velocity and the moving frame has constant velocity v (^) o = u (^) o i + vo j + w (^) o k, the Galilean transformation (x, y, z, t) → (x′^ , y ′^ , z ′^ , t′^ ) is as follows

x′^ = x − u (^) o t, (1.1) y ′^ = y − vo t, (1.2) z ′^ = z − w (^) o t, (1.3) t′^ = t. (1.4)

• Control volumes are useful; we will study three varieties:
  • Fixed: constant in space,
  • Material: no flux of mass through boundaries, can deform,
  • Arbitrary: can move, can deform, can have different fluid contained within.
• Control surfaces enclose control volumes; they have the same three varieties:
  • Fixed,
  • Material,
  • Arbitrary.
• Density is a material property, not used in classical mechanics, which only considers point masses. We can define density ρ as

ρ = lim V → 0

∑N

i=1 mi V

Here V is the volume of the space considered, N is the number of particles contained within the volume, and mi is the mass of the ith particle. We can define a length scale

16 CHAPTER 1. GOVERNING EQUATIONS

10 -8^10 -6^10 -4^ 0.01 1

x (m)

1

100

ρ (kg/m^3 )

variation on the continuum scale

variation on the sub-continuum molecular scale

Figure 1.1: Sketch of possible density variation of a gas near atmospheric pressure.

Example 1. Find the variation of mean free path with density for air.

We turn to Vincenti and Kruger for numerical parameter values, which are seen to be M =

28. 9 kg/kmole, N = 6. 02252 × 10 23 molecule/mole, d = 3. 7 × 10 −^10 m. Thus,

λ =

(

28. (^9) kmolekg

) ( 10001 kmole mole^ ) √ 2 π

(

6. 02252 × 10 23 molecule mole

) ρ (3. 7 × 10 −^10 m) 2

, (1.7)

= 7.^8895 ×^10

− (^8) molecule mkg 2 ρ.^ (1.8) Note that the unit molecule is not really a dimension, but really is literally a “unit,” which may well be thought of as dimensionless. Thus, we can safely say

λ = 7.^8895 ×^10

− (^8) mkg 2 ρ.^ (1.9) A plot of the variation of mean free path λ as a function of ρ is given in Fig. 1.2. Vincenti and Kruger go on to consider an atmosphere with density of ρ = 1. 288 kg/m 3. For this density

λ = 7.^8895 ×^10

− (^8) mkg 2

1. (^288) mkg 3

, (1.10)

= 6. 125 × 10 −^8 m, (1.11) = 6. 125 × 10 −^2 μm. (1.12) Vincenti and Kruger also show the mean molecular speed under these conditions is roughly c = 500 m/s, so the mean time between collisions, τ , is

τ ∼ λ c =^6.^125 ×^10

− (^8) m 500 m s^ = 1.^225 ×^10

− (^10) s. (1.13)

1.1. PHILOSOPHY OF RATIONAL CONTINUUM MECHANICS 17

0.01 0. 1 1 10

10 -

10 -

10 -

10 -

10 -80.

λ (m)

ρ (kg/m 3 )

ρatm ∼ 1.288 kg/m^3

λatm ~ 6.125 x 10-8^ m

Figure 1.2: Mean free path length, λ, as a function of density, ρ, for air.

Density is an example of a scalar property. We shall have more to say later about scalars. For now we say that a scalar property associates a single number with each point in time and space. We can think of this by writing the usual notation ρ(x, y, z, t), which indicates ρ has functional variation with position and time.

• Other properties are not scalar, but are vector properties. For example the velocity vector v(x, y, z, t) = u(x, y, z, t)i + v(x, y, z, t)j + w(x, y, z, t)k, (1.14)

associates three scalars u, v, w with each point in space and time. We will see that a vector can be characterized as a scalar associated with a particular direction in space. Here we use a boldfaced notation for a vector. This is known as Gibbs^9 notation. We will soon study an alternate notation, developed by Einstein, and known as Cartesian^10 index notation.

• Other properties are not scalar or vector, but are what is know as tensorial. The relevant properties are called tensors. The best known example is the stress tensor. One can think of a tensor as a quantity which associates a vector with a plane inclined at a selected angle passing through a given point in space. An example is the viscous stress (^9) Josiah Willard Gibbs, 1839-1903, American physicist and chemist with a lifelong association with Yale University who made fundamental contributions to vector analysis, statistical mechanics, thermodynamics, and chemistry. Studied in Europe in the 1860s. Probably one of the few great American scientists of the nineteenth century. (^10) Ren´e Descartes, 1596-1650, French mathematician and philosopher of great influence. A great doubter of existence who nevertheless concluded, “I think, therefore I am.” Developed analytic geometry.

1.2. SOME NECESSARY MATHEMATICS 19

Here we adopt a convention for the Gibbs notation, which we will find at times conflicts with other conventions, in which italics font (a) indicates a scalar, bold font (a) indicates a vector, upper case sans serif (A) indicates a second order tensor, over-lined upper case sans

seri (A) indicates a third order tensor, double over-lined upper case sans serif (A) indicates a fourth order tensor. In Cartesian index notation, their is no need to use anything except italics, as all terms are thought of as scalar components of a more expansive structure, with the structure indicated by the presence of subscripts. The essence of the Cartesian index notation is as follows. We can represent a three dimensional vector a as a linear combination of scalars and orthonormal basis vectors:

a = ax i + ay j + az k. (1.16)

We choose now to associate the subscript 1 with the x direction, the subscript 2 with the y direction, and the subscript 3 with the z direction. Further, we replace the orthonormal basis vectors i, j, and k, by e 1 , e 2 , and e 3. Then the vector a is represented by

a = a 1 e 1 + a 2 e 2 + a 3 e 3 =

∑^3

i=

ai ei = ai ei = ai =

a 1 a 2 a 3

Following Einstein, we have adopted the convention that a summation is understood to exist when two indices, known as dummy indices, are repeated, and have further left the explicit representation of basis vectors out of our final version of the notation. We have also included a representation of a as a 3 × 1 column vector. We adopt the standard that all vectors can be thought of as column vectors. Often in matrix operations, we will need row vectors. They will be formed by taking the transpose, indicated by a superscript T , of a column vector. In the interest of clarity, full consistency with notions from matrix algebra, as well as transparent translation to the conventions of necessarily meticulous (as well as popular) software tools such as Matlab, we will scrupulously use the transpose notation. This comes at the expense of a more cluttered set of equations at times. We also note that most authors do not explicitly use the transpose notation, but its use is implicit.

1.2.1.2 Rotation of axes

The Cartesian index notation is developed to be valid under transformations from one Carte- sian coordinate system to another Cartesian coordinate system. It is not applicable to either general orthogonal systems (such as cylindrical or spherical) or non-orthogonal systems. It is straightforward, but tedious, to develop a more general system to handle generalized co- ordinate transformations, and Einstein did just that as well. For our purposes however, the simpler Cartesian index notation will suffice. We will consider a coordinate transformation which is a simple rotation of axes. This transformation preserves all angles; hence, right angles in the original Cartesian system will be right angles in the rotated, but still Cartesian system. We will require, ultimately, that whatever theory we develop must generate results in which physically relevant quantities such

20 CHAPTER 1. GOVERNING EQUATIONS

x* 1 ’ = x* cos 1 α + x* cos 2 β

β β β

α

α

α

x 1

x’ 1

x 2

x’ 2

x* 2

x* 1

P

x* 1 ’

Figure 1.3: Sketch of coordinate transformation which is a rotation of axes

as temperature, pressure, density, and velocity magnitude, are independent of the particular set of coordinates with which we choose to describe the system. To motivate this, let us consider a two-dimensional rotation from an unprimed system to a primed system. So, we seek a transformation which maps (x 1 , x 2 ) T^ → (x′ 1 , x′ 2 ) T^. We will rotate the unprimed system counterclockwise through an angle α to achieve the primed system. The rotation is sketched in Figure 1.3. Note that it is easy to show that the angle β = π/ 2 − α. Here a point P is identified by a particular set of coordinates (x∗ 1 , x∗ 2 ). One of the keys to all of continuum mechanics is realizing that while the location (or velocity, or stress, ...) of P may be represented differently in various coordinate systems, ultimately it must represent the same physical reality. Straightforward geometry shows the following relation between the primed and unprimed coordinate systems for x′ 1

x∗

′ 1 =^ x

∗ 1 cos^ α^ +^ x

∗ 2 cos^ β.^ (1.18)

More generally, we can say for an arbitrary point that

x′ 1 = x 1 cos α + x 2 cos β. (1.19)

We adopt the following notation

• (x 1 , x′ 1 ) denotes the angle between the x 1 and x′ 1 axes,