CE 576 Environmental Flows
Spring 2012
Homework 1
These exercises are meant to help you practice the concepts from lecture. Only the exercises
with starred numbers are mandatory. You will submit the mandatory exercises in your
homework groups; remember that only one recorder records the solutions. The other
exercises are optional; I will give you some extra credit if you submit them. As always,
feel free to ask me if you have any questions.
*1. We will work through some of the paper by Cormack et al. (1974). For now, consider
the abstract and sections 1 and 2.
a. In general, what do you as a reader want to see in the abstract of a journal article?
b. Critique the abstract of Cormack et al. (1974). What would you change? Offer
constructive criticism.
c. In general, what do you as a reader want to see in the introduction of a journal
article?
d. Critique the introduction of Cormack et al. (1974). What would you change?
Offer constructive criticism.
e. Why are time, the spatial coordinates, and the velocity components written with
primes in equations (1) through (5)?
f. List the dimensions of the Grashof number Gr, Prandtl number P r, and aspect
ratio A, which are defined on p. 212.
*2. Show that the stagnation point flow (u = Γx, v = −Γy) satisfies conservation of mass
by computing the fluxes in and out of the box with vertices at (1,1), (1,3), (3,3), and
(3,1).
y
3
2
1
0
1
2
3
x
3. Consider the flow u = Ωy, v = −Ωx, where Ω is a constant.
a. Show that it satisfies conservation of mass by substituting the velocity components into
∂u ∂v
+
= 0.
∂x ∂y
b. Describe the flow, as we did in class for the stagnation point flow.
4. Show that in three dimensions, the general statement of conservation of mass is
∂
∂
∂
∂ρ
+
(ρu) +
(ρv) +
(ρw) = 0
∂t
∂x
∂y
∂z
and the continuity equation for incompressible flow is
∂u ∂v ∂w
+
+
= 0.
∂x ∂y
∂z
5. A two-dimensional flow has velocity components
y
Γ
2
2π x + y 2
Γ
x
v=
,
2
2π x + y 2
u=−
where Γ is a constant. Is the flow incompressible?
6. A piston compresses a gas in a chamber in one dimension. The initial density, before
the piston starts moving, is ρ0 , and the piston velocity is V . The velocity u of the air
varies linearly between V at the piston and zero at the wall. At t = 0, the chamber
length is L0 . Compute the density as a function of time. Provide three reasons why
your result is plausible. (Hint: Remember that the length L of the chamber is a
function of time.)
V
u(x,t)
ρ(t)
x=0
x = L(t)
*7. Conservation laws can be used to determine whether a given flow is valid. If the
amplitude a of two-dimensional surface waves is small, then the velocity components
are
cosh k(z + H)
cos (kx − ωt)
u = aω
sinh kH
sinh k(z + H)
sin (kx − ωt)
w = aω
sinh kH
where ω = 2π/T is the frequency of the waves, T is the period, k = 2π/L is the
wavenumber, L is the wavelength, H is the water depth, and t is time. The hyperbolic
sine and cosine are defined by
and cosh y = 12 ey + e−y .
sinh y = 12 ey − e−y
a. Show that this velocity field satisfies conservation of mass for an incompressible
fluid.
b. Another way to check a result is to compare with observations. Look at the two
panels of the page (on the CE 576 website) that shows paths of particles under
waves. (Ignore the other two and the references in the caption to other panels
on another page.) Explain why the velocities above make sense in terms of the
particle paths. Discuss the sizes and shapes of the paths and the variation with
depth.
c. What does it mean for the amplitude of the waves to be small?
L
a
z=0
z
H
x
z = -H xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
8. Environmental flows can have varying density but still be incompressible.
a. Suppose the density is given by
ρ = ρ0 + ρa sin(mz − ωt),
where ρ0 , ρa , m, and ω are constants. What velocity field will make this flow
incompressible?
b. True or false: All incompressible flows have constant density.
*9. Conservation laws can be used to determine one velocity component given the others.
Lake Osoboxy is a rectangular lake whose background temperature profile is linear.
It has a length L of 750 m and a depth H of 20 m. When wind blows on the lake,
long, standing internal waves, or seiches, can form. For two-dimensional waves, the
vertical velocity is
w = ζ0 ω cos kx sin mz sin ωt,
where ζ0 is the amplitude of the displacement of isotherms (or lines of constant temperature), ω is the frequency, and k and m are wavenumbers.
a. Compute the horizontal velocity u.
b. Suppose the water surface is flat. Show that the wavenumbers must be k = nx π/L
and m = nz π/H, where the mode numbers nx and nz can take the values 1, 2,
3, etc.
c. The expression for the horizontal velocity can be used to estimate sediment resuspension from the bottom of a lake. Compute the maximum velocity at the
bottom (i.e., z = 0) of Lake Osoboxy. You will need the frequency
ω=N
k
,
(k 2 + m2 )1/2
which depends on the buoyancy frequency N . Take N = 0.05 rad/s and ζ0 = 1 m.
For Lake Osoboxy, how strongly does the maximum bottom velocity depend on
the mode numbers? Describe lakes in which the mode numbers might matter for
the bottom velocity.