Review of fluid dynamics

Chapter 2 Review of fluid dynamics 2.1 Preliminaries ome basic concepts: A fluid is a substance that deforms continuously under stress. A Material olume 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 Control olume 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 Lagrangian 1 description, where properties are considered following a particle, and the Eulerian 2 description, where properties are described as a function of space and time. The convective derivative is defined by D Dt + u ; (2.1) t it gives the rate of change of a quantity following a fluid parcel. We shall need the following two theorems of vector calculus. In all surface integrals surrounding closed volumes, the normal vector n is taken to point out of the volume, and d n d. Divergence or Gauss s Theorem: v d = v d, (2.2) 1 Joseph Louis Lagrange (1736 1813): French mathematician. 2 Leonhard Euler (1707 1783): wiss mathematician. The leading mathematician of his time. First worked out the the equations of motion for fluids. 29

30 CHAPTER 2. REIEW OF FLUID DYNAMIC where is a closed surface enclosing the volume. tokes Theorem: C v dl = where C is the curve bounding the open surface. v d, (2.3) Exercise 2.1 Derive the vector forms of these theorems, namely (in suffix form) v v d i = d (2.4) x i and C v dl i = ɛ ijk v x j d k. (2.5) 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 dt = ηρ d + ηρu d. (2.6) t The partial derivative on the right-hand side of (2.6) could equally well be a total derivative: the integral 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 ( u = (u, v) = ψ y, ψ ), (2.7) 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 x 0 can be examined by expanding the velocity in a Taylor series. In suffix notation, u i (x) = u i (x 0 ) + (x j x 0j ) u i x j +. (2.8) x0

2.3. GOERNING LAW 31 Hence the flow close to the point x 0 is made up of uniform flow plus a linear term which may be decomposed into straining and vorticity contributions: where the symmetrized rate of strain tensor, u i x j = e ij + ɛ ikj ω k, (2.9) e ij = 1 2 ( ui x j + u j x i ), (2.10) corresponds to purely straining motion, while the antisymmetric part corresponds 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 tokes theorem with the line integral of velocity around a close curve bounding the surface: this is the circulation along a fluid circuit, defined by Γ u dl = ω d. (2.12) 2.3 Governing laws C 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 Conservation of mass Consider an arbitrary control volume. The rate of change of mass in the volume is given by the rate of inflow of mass into. This is a statement in words of (2.6) for η = 1. Hence d dt ρ d = ρu d, (2.13) Using the divergence theorem on the right-hand side, we can rewrite (2.13) as the volume integral 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. REIEW OF FLUID DYNAMIC 2.3.2 Incompressible flow Incompressible flow corresponds to a flow in which the volume of a fluid element does not change. Consequently, u d = 0, (2.15) where is a surface enclosing a fixed volume. Then by (2.2), u = 0. (2.16) Consequently, and the mass conservation equation becomes 2.3.3 Conservation of momentum u = 0, (2.17) Dρ/Dt = 0. (2.18) Newton s second law for a system in an inertial frame is dp dt = F, (2.19) 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 F b + F s = ρu d + ρu(u d). (2.20) t 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 τ ij. Hence, in mixed notation, t ρu i d = ρu i (u d) + τ ij d j + ρf i d. (2.21) Using the vector divergence theorem and the continuity equation, we obtain u i t + u u i j = 1 τ ij + F i. (2.22) x j ρ x j The form of the stress tensor τ ij needs to be investigated. In what follows, the body force F i is usually gravity g.

2.4. TRE TENOR 33 Aside 2.1 urface forces. By definition, the surface force on a control volume acts on the surface of the volume, and hence can be expressed as a surface integral. Both the surface force, F s, and the normal to the surface, d, 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 df si = τ ij d j, (2.23) where τ ij 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 tress tensor The form of the stress tensor depends on the nature of the fluid. For many substances, ranging from honey through water and oil to ideal gases, 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 pressure forces acting. The pressure on an infinitesimal element of surface is perpendicular to the surface. Mathematically this corresponds to pressure being the isotropic portion of the stress tensor. Hence τ ij = pδ ij + d ij. (2.24) There is a minus sign since fluid flows from high pressure regions to low pressure regions. The tensor d ij 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 d ij = λδ ij u k x k + µ ( ui x j + u j x i ). (2.25) 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 tokes equations ubstituting (2.25) into (2.22) and assuming incompressibility gives the Navier tokes 3 equations: ρ Du Dt = p + ρf + µ 2 u. (2.26) 3 Charles Navier (1785 1836): French hydrologist. George Gabriel tokes (1830 1905): Irish mathematician. First set tokes theorem as an examination question.

34 CHAPTER 2. REIEW OF FLUID DYNAMIC These may also be written The quantity ν is the kinematic viscosity. Du Dt = 1 ρ p + F + ν 2 u. (2.27) 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 τ ij = pδ ij, (2.28) and the Navier tokes equations become the Euler equations Du Dt = 1 p. (2.29) 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 boundary: u = u B. (2.30) An inviscid fluid can slip, but cannot flow through a solid boundary. Then u n = u B n. (2.31) 2.8 Equation of state All the gases that we deal with satisfy the ideal gas 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 salinity. The effect of pressure can be removed by considering the potential density as in the case of a gas discussed in 1.2.2. The variation with salinity and temperature can be represented by a Taylor series about a refernce density ρ 0 = ρ(t 0, 0 ) in the form

2.9. ENERGY EQUATION 35 where ρ(t, ) = ρ 0 (1 α(t T 0 ) + β( 0 )), (2.33) and α = 1 ρ 0 ρ T (2.34) β = 1 ρ 0 ρ, (2.35) are the coefficients of expansion for temperature and salinity, respectively. The negative sign for the temperature coefficient results from the fact that the density decreases with increasing temperature. 2.9 Energy equation The first law of thermodynamics in integral form is Q Ẇs Ẇshear Ẇother = eρ d + [U +pv+ 1 t 2 u 2 + P ]ρu d, (2.36) where the energy per unit mass of the system is e U + u 2 /2 + P, with U the internal energy of the fluid, p its pressure, v 1/ρ its specific volume and P its potential energy (per unit mass). For a gravitational field, the last term becomes P = 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 orticity and irrotational flow For constant density flows, the vorticity equation takes the form Dω Dt = ω u + ν 2 ω. (2.37) The first term on the right-hand side is a stretching term: vorticity can be produced by the stretching of fluid elements. This is just conservation of angular 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 gradients - 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. REIEW OF FLUID DYNAMIC Helmholtz s 4 theorem: vortex lines, which may be defined analogously to streamlines, move with the flow. Kelvin s 5 theorem: the circulation around a curve is constant in time: d 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 vector calculus, the fluid velocity may then be expressed using a velocity potential φ: u = φ. (2.39) Hence the velocity potential in an incompressible fluid satisfies Laplace s equation: 2 φ = 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 H p ρ + 1 2 u2 + Ω (2.41) is constant along a streamline, i.e. u H = 0. For an irrotational flow in such a fluid, with velocity potential φ, the quantity B(t) φ t + p ρ + 1 2 u2 + Ω (2.42) 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 4 Hermann von Helmholtz (1821 1894): German physician and physicist. Carried out pioneering work in acoustics, fluid dynamics and electricity. 5 ir William Thomson, Lord Kelvin (1824 1907): cottish mathematician and physicist. Devised the Kelvin scale of temperature.

2.11. BERNOULLI THEOREM 37 Interpret this result physically. Dc = 0, (2.43) Dt 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 t + 1 2 φ 2 + gη = 0. (2.44) Can you find another boundary condition at the surface?

38 CHAPTER 2. REIEW OF FLUID DYNAMIC