Chapter%208%20Incompressible%20waves

Chapter%208%20Incompressible%20waves - VIII. Waves in...

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VIII. Waves in Incompressible Fluids 8.1 Basic Equations Consider an inviscid liquid (water) bounded above by an inviscid gas (air) and below by a rigid impermeable solid wall, as shown in Fig. 8.1 below. The air-water interface is located by material boundary F(x, z, t) = y - η (x, z, t) while the water-solid interface is the material boundary y = -h(x, z). Gravity is in the vertical direction. For simplicity, we assume a. fluids are incompressible and ρ i = constant is taken; b. flows are inviscid and irrotational in the regions D 1 and D 2 . Thus the governing equation is 2 Φ i = 0 (8.1) The boundary conditions are as follows. At the rigid bottom , y=-h(x, z), ∇ Φ 2 n = 0 or + = - . (8.2) On the air-water interface , kinematic & dynamic boundary conditions: y x z η = η (x, z, t) D 1 D 2 ρ 1 ρ 2 g Fig. 8.1 Wave in incompressible fluids. y=-h(x, z)
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a. Kinematic boundary condition on F = y - η = 0, = 0 or + u F = 0 because the interface is a material boundary consisting of the same fluid elements, i.e. + u + v + w = 0 or = + + for i=1 & 2. (8.3) b. Dynamic boundary condition: Shear stress condition is not needed for inviscid flows. The normal stress condition is [p] p 1 (x, η , z, t) - p 2 (x, η , z, t) = ακ = α (+ ) on y = η (x, z, t) (8.4) wherep 1 = pressure on the gas side, p 2 = pressure on the liquid side, α = surface tension coefficient, κ = curvature of the gas-liquid interface, R 1 2 = principal radii of curvature. On the interface y = η , we further have the Bernoulli equation, + ( ∇Φ ) 2 + + g η = c(t) for both gas & liquid (8.5) In summary, equations (8.1-5) govern the wave motion in both fluids. Although equations (8.1-5) are derived under some simplifications, the solution to (8.1-5) remains intractable analytically. The principle sources of difficulty are the following. a. Nonlinear terms appear in the boundary conditions. b. Location of the interface on which to apply the BC’s is unknown. c. Time is implicit in field equations (8.1) but explicit in the BC’s. To circumvent these difficulties, we resort to the method of 2
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regular perturbation . a. We first assume the existence of a " reference state ", i.e. a fluid motion for which the solution to the given field equation and boundary conditions is known. The reference state in the present case will be that of static equilibrium for which η (x, z, t) = 0 , Φ = 0 , p = - ρ gy. (8.6) b. We assume all fluid motions represent "small" departure from the reference state. c. All fluid motions are represented by referring them to the reference state. This is accomplished by expressing dependent fluctuations as power series expansions whose leading order terms are reference state solution. 8.2
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Chapter%208%20Incompressible%20waves - VIII. Waves in...

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