MIT2_017JF09_ch06

# MIT2_017JF09_ch06 - x y z w w G w v G v v u u G u G y G x G...

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Unformatted text preview: x y z w w + G w v + G v v u u + G u G y G x G z 6 WATER WAVES 35 6 WATER WAVES Surface waves in water are a superb example of a stationary and ergodic random process. The model of waves as a nearly linear superposition of harmonic components, at random phase, is confirmed by measurements at sea, as well as by the linear theory of waves, the subject of this section. We will skip some elements of ﬂuid mechanics where appropriate, and move quickly to the cases of two-dimensional, inviscid and irrotational ﬂow. These are the major assumptions that enable the linear wave model. 6.1 Constitutive and Governing Relations First, we know that near the sea surface, water can be considered as incompressible, and that the density ρ is nearly uniform. In this case, a simple form of conservation of mass will hold: ∂u ∂v ∂w + + = 0 , ∂x ∂y ∂z where the Cartesian space is [ x, y, z ], with respective particle velocity vectors [ u, v, w ]. In words, the above equation says that net ﬂow into a differential volume has to equal net ﬂow out of it. Considering a box of dimensions [ δx, δy, δz ], we see that any δu across the x-dimension, has to be accounted for by δv and δw : δuδyδz + δvδxδz + δwδxδy = 0 . Next, we invoke Newton’s law, in the three directions: ∂u ∂u ∂u ∂u ∂p ∂ 2 u ∂ 2 u ∂ 2 u ρ ∂t + u ∂x + v ∂y + w ∂z = − ∂x + µ ∂x 2 + ∂y 2 + ∂z 2 ; ∂v ∂v ∂v ∂v ∂p ∂ 2 v ∂ 2 v ∂ 2 v ρ ∂t + v ∂x + w ∂y + u ∂z = − ∂y + µ ∂x 2 + ∂y 2 + ∂z 2 ; ∂w ∂w ∂w ∂w ∂p ∂ 2 w ∂ 2 w ∂ 2 w ρ ∂t + w ∂x + u ∂y + v ∂z = − ∂z + µ ∂x 2 + ∂y 2 + ∂z 2 − ρg. T T T T T x x x x y y y y I#!#0T#2#5 I#!#T#2#5 I#!#0T I#!#3 6 WATER WAVES 36 Here the left-hand side of each equation is the acceleration of the ﬂuid particle, as it moves ∂u through the differential volume. The terms such as u ∂x capture the fact that the force balance is for a moving particle; the chain rule expansion goes like this in the x-direction: du ∂u ∂u ∂x ∂u ∂y ∂u ∂z = + + + , dt ∂t ∂x ∂t ∂y ∂t ∂z ∂t where u = ∂x/∂t and so on. On the right side of the three force balance equations above, the differential pressure clearly acts to slow the particle (hence the negative sign), and viscous friction is applied through absolute viscosity µ . The third equation also has a term for gravity, leading in the case of zero velocities to the familiar relation p ( z ) = − ρgz , where z is taken positive upward from the mean free surface. 6.2 Rotation and Viscous Effects In a ﬂuid, unlike for rigid bodies, rotation angle is taken as the average of the angular deﬂections of the faces. Hence, a net rotation only occurs if the deﬂection of each face is additive. If they are opposing, then we have only a shearing of the element, with no rotation....
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MIT2_017JF09_ch06 - x y z w w G w v G v v u u G u G y G x G...

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