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Home , Incompressible flow, Total derivative, Volume integral

Chapter 2

Review of fluid dynamics

2.1 Preliminaries

Some basic concepts: • A fluid is a substance that deforms continuously under stress. • A Material Volume is a “tagged” region that moves with the fluid. Hence there can be no flow into and out of the volume. It can be likened to a closed system in thermodynamics. • A is an arbitrary volume of space through which fluid can flow. It can be likened to an open system in thermodynamics. • There are two fundamental descriptions of fluid motion: the Lagrangian1 description, where properties are considered following a particle, and the Eulerian2 description, where properties are described as a of space and time. The convective is defined by D ∂ ≡ + u · ∇; (2.1) Dt ∂t it gives the rate of change of a quantity following a fluid parcel. We shall need the following two theorems of vector . In all surface surrounding closed volumes, the normal vector n is taken to point out of the volume, and dS ≡ n dS. • or Gauss’s Theorem: Z Z v · dS = ∇ · v dV, (2.2) S V 1Joseph Louis Lagrange (1736–1813): French mathematician. 2Leonhard Euler (1707–1783): Swiss mathematician. The leading mathematician of his time. First worked out the the equations of motion for fluids.

29 30 CHAPTER 2. REVIEW OF

where S is a closed surface enclosing the volume V . • Stokes’ Theorem: Z Z v · dl = ∇ × v · dS, (2.3) C S where C is the curve bounding the open surface S.

Exercise 2.1 Derive the vector forms of these theorems, namely (in suffix form) Z Z ∂v v dSi = dV (2.4) S V ∂xi and Z Z ∂v v dli = ijk dSk. (2.5) C S ∂xj We now give a theorem that governs the rate of change of integrals over control volumes. • Reynolds Transport Theorem: Let N be an extensive property of the system, and η the corresponding intrinsic property (i.e. “N per unit mass”). Then the rate of change of N is given by dN ∂ Z Z = ηρ dV + ηρu · dS. (2.6) dt ∂t V S The on the right-hand side of (2.6) could equally well be a : the over the control volume depends only on time, and not on space.

2.2 Kinematics

A particle path is the trajectory described by a fluid particle. A streakline is the curve described by a substance released into the flow from a certain point. A streamline is a curve everywhere parallel to the flow. The three are the same for steady flows, i.e. where u does not depend on time. For incompressible flow in two dimensions, we may define a streamfunction ψ such that  ∂ψ ∂ψ  u = (u, v) = − , , (2.7) ∂y ∂x The streamfunction is arbitrary up to a constant. A similar function may be defined for three-dimensional axisymmetric flow. The motion of fluid close to a point x0 can be examined by expanding the velocity in a Taylor . In suffix notation,

∂ui ui(x) = ui(x0) + (xj − x0j) + ··· . (2.8) ∂x j x0 2.3. GOVERNING LAWS 31

Hence the flow close to the point x0 is made up of uniform flow plus a linear term which may be decomposed into straining and contributions:

∂ui = eij + ikjωk, (2.9) ∂xj where the symmetrized rate of strain tensor,   1 ∂ui ∂uj eij = + , (2.10) 2 ∂xj ∂xi corresponds to purely straining motion, while the antisymmetric part corre- sponds to the local rotational motion. The vorticity ω of a fluid is defined by ω ≡ ∇ × u. (2.11) As indicated above, it is related to the local angular velocity of the fluid. The flux of vorticity through an open surface can be identified by Stokes’ theorem with the of velocity around a close curve bounding the surface: this is the circulation along a fluid circuit, defined by Z Z Γ ≡ u · dl = ω · dS. (2.12) C S

2.3 Governing laws

The governing equations for the motion of fluids take the form of conservation laws and constitutive laws. The former relate the flux of some quantity into and out of a volume. The latter are related to the properties of matter. It is hence natural to obtain the conservation laws in integral form and then derive differential equations.

2.3.1 Consider an arbitrary control volume V . The rate of change of mass in the volume is given by the rate of inflow of mass into V . This is a statement in words of (2.6) for η = 1. Hence d Z Z ρ dV = − ρu · dS, (2.13) dt V S Using the on the right-hand side, we can rewrite (2.13) as the over an arbitrary volume of a quantity that vanishes identically. Hence the integrand is zero, i.e. ∂ρ + ∇ · (ρu) = 0. (2.14) ∂t The equation (2.14) is known as the continuity equation. The procedure used to pass from (2.13) to (2.14) will be used repeatedly. 32 CHAPTER 2. REVIEW OF FLUID DYNAMICS

2.3.2 Incompressible flow Incompressible flow corresponds to a flow in which the volume of a fluid ele- ment does not change. Consequently, Z u · dS = 0, (2.15) S where S is a surface enclosing a fixed volume V . Then by (2.2), Z ∇ · u = 0. (2.16) V Consequently, ∇ · u = 0, (2.17) and the mass conservation equation becomes

Dρ/Dt = 0. (2.18)

2.3.3 Conservation of Newton’s second law for a system in an inertial frame is dP = F , (2.19) dt where P is the linear momentum of the system. We may apply the Reynolds transport theorem for the momentum, which corresponds to η = u. The total force on the system is the sum of body and surface forces. Hence ∂ Z Z F b + F s = ρu dV + ρu(u · dS). (2.20) ∂t V S In words, the rate of change of total momentum in a control volume must be given by the flux of momentum into the volume plus the work done on the fluid on the boundary of the volume, as well as any body forces F acting in the volume. This is the form used in many engineering applications. We now replace force by force per unit mass, and write the surface force in terms of the stress tensor τij. Hence, in mixed notation, ∂ Z Z Z Z ρui dV = − ρui(u · dS) + τij dSj + ρFi dV. (2.21) ∂t V S S V Using the vector divergence theorem and the continuity equation, we obtain

∂ui ∂ui 1 ∂τij + uj = − + Fi. (2.22) ∂t ∂xj ρ ∂xj

The form of the stress tensor τij needs to be investigated. In what follows, the body force Fi is usually gravity g. 2.4. STRESS TENSOR 33

Aside 2.1 Surface forces. By definition, the surface force on a control volume acts on the surface of the volume, and hence can be expressed as a . Both the surface force, F s, and the normal to the surface, dS, are vectors. They do not necessarily point in the same direction, except in an ideal fluid. The mathematical way of relating two vectors is by a tensor: over a small surface area, the components of the infinitesimal force dF s are related to the components of the infinitesimal surface area by dFsi = τij dSj, (2.23) where τij is the stress tensor. Its units are just force per unit area, measured in Pascals. In the case where the force and surface normal are aligned, the stress corresponds to minus the usual pressure.

2.4 Stress tensor

The form of the stress tensor depends on the nature of the fluid. For many sub- stances, ranging from honey through water and oil to ideal , the stress is related in a linear and isotropic manner to the rate of strain (this is an example of a constitutive relation). These fluids are known as Newtonian fluids. In a fluid at rest, the stress tensor does not vanish: there may still be pres- sure forces acting. The pressure on an infinitesimal element of surface is per- pendicular to the surface. Mathematically this corresponds to pressure being the isotropic portion of the stress tensor. Hence

τij = −pδij + dij. (2.24)

There is a minus sign since fluid flows from high pressure regions to low pres- sure regions. The tensor dij is the deviatoric stress tensor. For a Newtonian fluid, the deviatoric stress tensor is related linearly and isotropically to the rate of strain tensor. Hence   ∂uk ∂ui ∂uj dij = λδij + µ + . (2.25) ∂xk ∂xj ∂xi This is a constitutive equation for a Newtonian fluid; λ is the bulk viscosity and µ is the dynamic shear viscosity. Note that in an incompressible fluid, the bulk viscosity is not relevant.

2.5 Navier–Stokes equations

Substituting (2.25) into (2.22) and assuming incompressibility gives the Navier– Stokes3 equations: Du ρ = −∇p + ρF + µ∇2u. (2.26) Dt 3Charles Navier (1785–1836): French hydrologist. George Gabriel Stokes (1830–1905): Irish mathematician. First set Stokes’ theorem as an examination question. 34 CHAPTER 2. REVIEW OF FLUID DYNAMICS

These may also be written Du 1 = − ∇p + F + ν∇2u. (2.27) Dt ρ The quantity ν is the kinematic viscosity.

2.6 Euler equations

An ideal or inviscid fluid (a fiction really) is a fluid with zero viscosity. The stress tensor then reduces to the form

τij = −pδij, (2.28) and the Navier–Stokes equations become the Euler equations Du 1 = − ∇p. (2.29) Dt ρ There are no real inviscid fluids, except maybe for superfluid helium, so any results obtained using the Euler equation must be an approximation at best. The Euler equations are very useful in certain situations, but not so useful in others.

2.7 Boundary conditions

A viscous fluid adjacent to a solid boundary satisfies the no-slip condition, which states that the velocity of the fluid is equal to the velocity of the bound- ary: u|S = uB. (2.30) An inviscid fluid can slip, but cannot flow through a solid boundary. Then

u · n|S = uB · n. (2.31)

2.8 Equation of state

All the gases that we deal with satisfy the ideal relation to a very good approximation, and so p = RT ρ, (2.32) where R is the ideal gas constant and T is the temperature in Kelvin. The density of sea water is a function of pressure, temperature T and salin- ity S. The effect of pressure can be removed by considering the potential den- sity as in the case of a gas discussed in § 1.2.2. The variation with salinity and temperature can be represented by a about a refernce density ρ0 = ρ(T0,S0) in the form 2.9. ENERGY EQUATION 35

ρ(T,S) = ρ0(1 − α(T − T0) + β(S − S0)), (2.33) where 1 ∂ρ α = − (2.34) ρ0 ∂T and 1 ∂ρ β = , (2.35) ρ0 ∂S are the coefficients of expansion for temperature and salinity, respectively. The negative sign for the temperature coefficient results from the fact that the den- sity decreases with increasing temperature.

2.9 Energy equation

The first law of thermodynamics in integral form is Z Z ˙ ˙ ˙ ˙ ∂ 1 2 Q−Ws −Wshear −Wother = eρ dV + [U +pv+ 2 |u| +VP ]ρu·dS, (2.36) ∂t V S

2 where the energy per unit mass of the system is e ≡ U + |u| /2 + VP , with U the internal energy of the fluid, p its pressure, v ≡ 1/ρ its specific volume and VP its potential energy (per unit mass). For a gravitational field, the last term becomes VP = gz. The combination U + pv is the enthalpy of the fluid. The terms of the left-hand side correspond to the heat transfer to the fluid and to the work done by shaft, shear and other forces (such as electromagnetic forces) respectively.

2.10 Vorticity and irrotational flow

For constant density flows, the takes the form Dω = ω · ∇u + ν∇2ω. (2.37) Dt The first term on the right-hand side is a stretching term: vorticity can be pro- duced by the stretching of fluid elements. This is just conservation of angu- lar momentum like an ice skater pulling his/her hands in and spinning faster. The last term corresponds to viscous diffusion of vorticity. For variable-density flows, there are extra terms on the right-hand side related to the production of vorticity by density - see § 3.5. For an ideal fluid, the viscous dissipation term vanishes. The resulting equation corresponds in fact to the conservation of the vector quantity ω. Under these conditions, the following theorems hold: 36 CHAPTER 2. REVIEW OF FLUID DYNAMICS

• Helmholtz’s4 theorem: vortex lines, which may be defined analogously to streamlines, move with the flow.

• Kelvin’s5 theorem: the circulation around a curve is constant in time: d Z u · dl = 0. (2.38) dt D

• Lagrange’s theorem: “a flow is irrotational if it was so at any earlier time”.

An irrotational flow has zero vorticity everywhere. By a theorem of , the fluid velocity may then be expressed using a φ:

u = ∇φ. (2.39)

Hence the velocity potential in an incompressible fluid satisfies Laplace’s equa- tion: ∇2φ = 0. (2.40)

2.11 Bernoulli’s theorem

Bernoulli’s theorem is a first integral of the momentum equation. It can also be thought of as an energy equation governing the change in head (energy) of the fluid as it flows. For an incompressible fluid in which body forces are conservative, i.e. F = −∇Ω, the quantity p 1 H ≡ + u2 + Ω (2.41) ρ 2 is constant along a streamline, i.e. u · ∇H = 0. For an irrotational flow in such a fluid, with velocity potential φ, the quan- tity ∂φ p 1 B(t) ≡ + + u2 + Ω (2.42) ∂t ρ 2 is a function of time only. Bernoulli’s equation is not the energy equation in general. For inviscid, incompressible flow, however, which is what is required to derive the Bernoulli equation, there are no contributions from heating and internal energy to the energy equation, so the two are the same.

Problem 2.1 Passive scalars. Consider a quantity such as a chemical species that is neither created nor destroyed, with concentration c (per unit volume). Using Reynolds transport theorem and the continuity equation show that

4Hermann von Helmholtz (1821–1894): German physician and physicist. Carried out pioneer- ing work in acoustics, fluid dynamics and electricity. 5Sir William Thomson, Lord Kelvin (1824–1907): Scottish mathematician and physicist. Devised the Kelvin scale of temperature. 2.11. BERNOULLI’S THEOREM 37

Dc = 0, (2.43) Dt Interpret this result physically.

Problem 2.2 Using the definition for the coefficient of expansion (2.34), show that for a perfect gas α = T −1.

Problem 2.3 A layer of water is initially at rest and can be approximated as an ideal fluid. Waves generated at a distance (say by a storm) arrive at the location of interest. Explain why the flow is irrotational. The undisturbed surface is z = 0. Use the unsteady form of Bernoulli’s equation (2.42) to show that on the surface z = η(x, y, t) ∂φ 1 + |∇φ|2 + gη = 0. (2.44) ∂t 2 Can you find another boundary condition at the surface? 38 CHAPTER 2. REVIEW OF FLUID DYNAMICS