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 Control Volume 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 function of space and time. The convective derivative 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 calculus. In all surface integrals surrounding closed volumes, the normal vector n is taken to point out of the volume, and dS ≡ n dS. • Divergence 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 FLUID DYNAMICS 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 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 ∂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 series. 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 vorticity 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 line integral 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 Conservation of mass 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 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. 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 momentum 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 surface integral. 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 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 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.