Marine Hydrodynamics (Lecture 3)

2.20 – Marine Hydrodynamics Lecture 3 1.7 Stress Tensor 1.7.1 Stress Tensor τij The stress (force per unit area) at a point in a fluid needs nine components to be completely specified, since each component of the stress must be defined not only by the direction in which it acts but also the orientation of the surface upon which it is acting. The first index i specifies the direction in which the stress component acts, and the second index j identifies the orientation of the surface upon which it is acting. Therefore, the ith component of the force acting on a surface whose outward normal points in the j th direction is τij . X2 22 12 32 21 23 11 13 31 X1 33 X3 Figure 1: Shear stresses on an infinitesimal cube whose surfaces are parallel to the coordinate system. 1 X2 2 P A1 n̂ 1 A3 Q X1 3 R area A0 X3 A2 Figure 2: Infinitesimal body with surface PQR that is not perpendicular to any of the Cartesian axis. Consider an infinitesimal body at rest with a surface PQR that is not perpendicular to any of the Cartesian axis. The unit normal vector to the surface PQR is n̂ = n1 x̂1 +n2 x̂2 +n3 x̂3 . The area of the surface = A0 , and the area of each surface perpendicular to Xi is Ai = A0 ni , for i = 1, 2, 3.  Newton’s law: Fi = (volume force)i for i = 1, 2, 3 on all 4 faces Note: If δ is the typical dimension of the body : surface forces ∼ δ 2 : volume forces ∼ δ 3 An example of surface forces is the shear force and an example of volumetric forces is the gravity force. At equilibrium, the surface forces and volumetric forces are in balance. As the body gets smaller, the mass of the body goes to zero, which makes the volumetric forces  equal to zero and leaving the sum of the surface forces equal zero. So, as δ → 0, all4f aces Fi = 0 for i = 1, 2, 3 and ∴ τi A0 = τi1 A1 + τi2 A2 + τi3 A3 = τij Aj . But the area of each  surface ⊥ to Xi is Ai = A0 ni . Therefore τi A0 = τij Aj = τij (A0 nj ), where τij Aj is the notation (represents the sum of all components). Thus τi = τij nj for i = 1, 2, 3, where τi is the component of stress in the ith direction on a surface with a normal n . We call τ i the stress vector and we call τij the stress matrix or tensor. 2 1.7.2 Example: Pascal’s Law for Hydrostatics In a static fluid, the stress vector cannot be different for different directions of the surface normal since there is no preferred direction in the fluid. Therefore, at any point in the fluid, the stress vector must have the same direction as the normal vector n and the same magnitude for all directions of n . Pascal’s Law for hydrostatics: τij no summation    = − (pi ) (δij ) ⎡ ⎤ 0 −p1 0 τ = ⎣ 0 −p2 0 ⎦ 0 0 −p3 where pi is the pressure acting perpendicular to the ith surface. If p0 is the pressure acting perpendicular to the surface PQR, then τi = −ni p0 , but: τi = τij nj = −(pi )δij nj = −(pi )(ni ) Therefore po = pi , i = 1, 2, 3 and n is arbitrary. 3 1.7.3 Symmetry of the Stress Tensor To prove the symmetry of the stress tensor we follow the steps: j ij ji ji i o ij Figure 3: Material element under tangential stress.  1. The of surface forces = body forces + mass× acceleration. Assume no symmetry. Balance of the forces in the ith direction gives: (δ)(τij )T OP − (δ)(τij )BOT T OM = O(δ 2 ), since surface forces are ∼ δ 2 , where the O(δ 2 ) terms include the body forces per unit depth. Then, as δ → 0, (τij )T OP = (τij )BOT T OM .  2. The of surface torque = body moment + angular acceleration. Assume no sym­ metry. Balance of moments about o gives: (τji δ)δ − (τij δ)δ = O(δ 3 ), since the body moment is proportional to δ 3 . As δ → 0 , τij = τji . 4 1.8 Mass and Momentum Conservation Consider a material volume ϑm and recall that a material volume is a fixed mass of mate­ rial. A material volume always encloses the same fluid particles despite a change in size, position, volume or surface area over time. 1.8.1 Mass Conservation The mass inside the material volume is: M (ϑm ) = ρdϑ ϑm(t) Sm(t) ϑm ( t ) Figure 4: Material volume ϑm (t) with surface Sm (t). Therefore the time rate of increase of mass inside the material volume is: d d M (ϑm ) = dt dt ρdϑ = 0, ϑm (t) which is the integral form of mass conservation for the material volume ϑm . 5 1.8.2 Momentum Conservation The fluid velocity inside the material volume in the ith direction is denoted as ui . Linear momentum of the material volume in the ith direction is ρui dϑ ϑm(t) Newton’s law of motion: The time rate of change of momentum of the fluid in the material control volume must equal the sum of all the forces acting on the fluid in that volume. Thus: d (momentum)i =(body force)i + (surface force)i dt d ρui dϑ = Fi dϑ + τij nj dS  dt ϑm (t) Divergence Theorems ϑm (t) Sm (t) τi v .n̂ dS ∇· vdϑ = ⊂⊃  For vectors: ϑ For tensors: ϑ ∂vj ∂xj S vj nj ∂τij dϑ = ⊂⊃ τij nj dS ∂xj S Using the divergence theorems we obtain d dt ρui dϑ = ϑm(t) ϑm(t) ∂τij Fi + ∂xj  dϑ which is the integral form of momentum conservation for the material volume ϑm . 6 1.8.3 Kinematic Transport Theorems Consider a flow through some moving control volume ϑ(t) during a small time interval Δt. Let f (x, t) be any (Eulerian) fluid property per unit volume of fluid (e.g. mass, momentum, etc.). Consider the integral I(t): I(t) = f (x, t) dϑ ϑ(t) According to the definition of the derivative, we can write d I(t + Δt) − I(t) I(t) = lim Δt→0 dt ⎧ Δt ⎪ 1 ⎨ f (x, t + Δt)dϑ − = lim Δt→0 Δt ⎪ ⎩ ϑ(t+Δt) ϑ(t) ⎫ ⎪ ⎬ f (x, t)dϑ ⎪ ⎭ S(t+Δt) ϑ ( t + Δt ) ϑ( t ) S(t) Figure 5: Control volume ϑ and its bounding surface S at instants t and t + Δt. 7 Next, we consider the steps 1. Taylor series expansion of f (x, t + Δt) about (x, t). f (x, t + Δt) = f (x, t) + Δt 2. dϑ = ϑ(t+Δt) dϑ + [Un (x, t)Δt] dS and Un (x, t) is the normal velocity of S(t). dϑ = Δϑ dϑ Δϑ ϑ(t) where, ∂f (x, t) + O((Δt)2 ) ∂t S(t) S(t+Δt) S(t) v U n ( x, t )Δt + O( Δt ) 2 dS Figure 6: Element of the surface S at instants t and t + Δt. Putting everything together: ⎧ ⎪ ⎨ d 1 I(t) = lim Δt→0 ⎪ dt Δt ⎩ dϑf + Δt ϑ(t) dϑ ∂f + Δt ∂t ϑ(t) S(t) 8 dϑf + O(Δt)2 dSUn f − ϑ(t) ⎫ ⎪ ⎬ ⎪ ⎭ (1) From Equation (1) we obtain the Kinematic Transport Theorem (KTT), which is equivalent to Leibnitz rule in 3D. d dt ∂f (x, t) dϑ + ∂t f (x, t)dϑ = ϑ(t) ϑ(t) f (x, t)Un (x, t)dS S(t) For the special case that the control volume is a material volume it is ϑ(t) = ϑm (t) and Un = v · n ˆ, where v is the fluid particle velocity. The Kinematic Transport Theorem (KTT), then takes the form d dt ∂f (x, t) dϑ + ∂t f (x, t)dϑ = ϑm (t) ϑm (t) Sm (t) f (x, t)(v · n ˆ)dS    f (vi ni ) (Einstein Notation) Using the divergence theorem, dϑ = ⊂⊃  α  ·n ˆdS  ∇· α ϑ ∂ α ∂xi i S αi ni we obtain the 1st Kinematic Transport Theorem (KTT) d dt  f (x, t) dϑ = ϑm (t) ϑm (t) where f is some fluid property per unit volume. 9  ∂f (x, t) + ∇ · (fv ) dϑ,    ∂t ∂ (f vi ) ∂xi 1.8.4 Continuity Equation for Incompressible Flow • Differential form of conservation of mass for all fluids Let the fluid property per unit volume that appears in the 1st KTT be mass per unit volume ( f = ρ): 0 = ↑ conservation of mass d dt  ρdϑ ϑm (t) = ↑ 1st KTT ϑm (t)  ∂ρ + ∇ · (ρv ) dϑ ∂t But since ϑm is arbitrary the integrand must be ≡ 0 everywhere. Therefore: ∂ρ + ∇ · (ρv ) = 0 ∂t ∂ρ + [v · ∇ρ + ρ∇ · v ] = 0 ∂t   Dρ Dt Leading to the differential form of Conservation of Mass: Dρ + ρ∇ · v = 0 Dt 10 • Continuity equation ≡ Conservation of mass for incompressible flow In general it is ρ = ρ(p, T, . . .), but we consider the special case of an incompressible flow, i.e. Dρ = 0 (Lecture 2). Dt Note: For a flow to be incompressible, the density of the entire flow need not be constant (ρ(x, t) = const). As an example consider a flow of more than one incom­ pressible fluids, like water and oil, as illustrated in the picture below. Constant ρ fluid particle ρ2 fluid particle ρ1 oil water Figure 7: Interface of two fluids (oil-water) Since for incompressible flows Dρ = 0, substituting into the differential form of the Dt conservation of mass we obtain the Continuity Equation: ∇ · v  ≡  ∂vi ∂xi =  rate of volume dilatation 11 0 1.8.5 Euler’s Equation (Differential Form of Conservation of Momentum) • 2nd Kinematic Transport Theorem ≡ 1st KTT + differential form of conservation of mass for all fluids. If G = fluid property per unit mass, then ρG = fluid property per unit volume d dt  ρGdϑ ϑm (t) = ↑ 1st KT T ϑm (t) = ϑm (t)  ∂ (ρG) + ∇ · (ρGv ) dϑ ∂t ⎡ ⎢ ⎢ ⎢G ⎢ ⎣  ∂ρ + ∇ · ρv ∂t    =0 from mass conservation ⎤ ⎥ ⎥ ∂G +ρ + v · ∇G ⎥ ⎥ dϑ ∂t   ⎦ = DG Dt The 2nd Kinematic Transport Theorem (KTT) follows: d dt ρGdϑ = ϑm ρ ϑm DG dϑ Dt Note: The 2nd KTT is obtained from the 1st KTT (mathematical identity) and the only assumption used is that mass is conserved. 12 • Euler’s Equation We consider G as the ith momentum per unit mass (vi ). Then, ϑm (t) ∂τij Fi + ∂xj  dϑ = ↑ conservation of momentum d dt ρvi dϑ ϑm (t) = ↑ 2nd KTT ϑm (t) ρ Dvi dϑ Dt But ϑm (t) is an arbitrary material volume, therefore the integral identity gives Euler’s equation: ⎛ ρ ⎞ ⎟ ⎜ ∂vi Dvi ∂τij = Fi + ≡ ρ⎜ + v · ∇vi ⎟ ⎝ ⎠    Dt ∂t ∂xj ∂v vj ∂xi j And in vector tensor form: Dv ρ ≡ρ Dt ∂v + v · ∇v ∂t 13  = F + ∇ · τ