hw2
From: Chris Nkinthorn, 2020-01-23
For: Prof. A. Hirsa for Incompressible Flow
Problem 1: Gravity Wave
For a gravity wave with surface elevation $\eta(x,t) = a e ^{i(kx-\omega t)}$ find the pressure $p$ at the bottom $y = -h$. Use linear theory and the same coordinate system used in class.
Begin with the unsteady bernoulli equation : $P-P{0}=-\rho \phi{t}-P \frac{|\nabla \phi|^{2}}{2}-\rho g y$
rearrange and factor evaluating
now we have functional which we need to solve all the terms outside of the parenthesises sum or factor to the LHS
relationship in the potential
time derivative
Problem 2: Fluid Boundary
Two immiscible liquids have an interface at $y=0$ and are confined between the boundaries $y = h$ and bottom $y = -h$ If he upper liquid has density $\rho_1$ and lower liquid has density $\rho_2$, show that for the case where $h \rightarrow \infty$ the wave velocity $c$ for gravity waves is given by $c = \sqrt{\dfrac{g(\rho_2-\rho_1)}{k(\rho_1+\rho_2)}}$.
Problem 3: Small Dispersion
Small aptitude gravity waves propagate along the surface of a fluid of depth $h$ moving in the plus $x$ direction with uniform velocity $V$. Consequently, the potential $\phi$ may be written in the form
Also formulate the surface condition required to establish a dispersion relation. Find the dispersion relation.
Elevation Effect
Determine the form of $F(y)$.
Process of Substitution
Surface Condition
Needs to be a traction-free surface at the fluid-to-fluid boundary. The equation which describes this dynamic boundary is the previously used Dynamic Bernoulli Equation where there is where the fluid pressure matches the atmospheric.
Applying incompressible flow, both the pressure gradient and density terms factor out.
see the kinematic relationship from the
Note that the second vertical component $V = \phi_{y}=\frac{D\eta}{D t} = \eta_t + \phi_x \eta_x$ which we linearize as
$\phi_t + g\eta = 0$ linearize dynamic free surface
$\phi_y = \eta_t$ linearize kinematic free surface
which combined become
$\underline{\phi_{tt} + g\phi_y =0}$
Dispersion Relationship
solve the infinite wave train equation
$\phi_{,ii} = 0$ for $y \in [-h,0]$
bc
$\phi_y = 0 $ at $y = -h$
$\phi_{tt} + g\phi_y = 0$ for y = 0 instead of small amplitude $\eta$ as unconstrained
solution is the separation of variable: $\phi = A(t)e^{(k(y\pm ix))}$ which for a positive traveling wave
where the following terms are
k
\omega
c
A
substitute into combined linear equation at the surface
typically leads to $\omega^{2}=g k \tanh k h$ but will be different for this one
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