ENVIRONMENTAL FLUID MECHANICS Mixing
Benoit Cushman-Roisin Thayer School of Engineering Dartmouth College
Example of stirring by turbulent flow
Pierre Welander (1955)
1
Taylor's theory of point dispersion
Consider the chaotic trajectory x (t ) of a point

ENGS-151 - Spring 2013 ENVIRONMENTAL FLUID MECHANICS Mid-Term Exam SOLUTIONS 1. (10 points) (Problem 2-4 in book) Shortly after a torrential rain, the water level in a reservoir behind a dam rises at the rate of 1 cm per hour while the water allowed to pa

ENGS-151 - Spring 2013 ENVIRONMENTAL FLUID MECHANICS Mid-Term Exam Assigned: Monday 4 February 2013 Due: 10am, Thursday 7 February 2013 1. (10 points) (Problem 2-4 in book) Shortly after a torrential rain, the water level in a reservoir behind a dam rises

ENGS-151 - Spring 2013 ENVIRONMENTAL FLUID MECHANICS HOMEWORK #7 SOLUTIONS
1. (10 points) Consider a stream in a frictionless rectangular channel which, by means of repeated constrictions, undergoes a series of passages into supercriticality and subsequen

ENGS-151 - Spring 2013 ENVIRONMENTAL FLUID MECHANICS HOMEWORK #6 SOLUTIONS
1. (10 points) Determine the water depth h and velocity u in uniform river flow obeying the Manning formula, each in term of the discharge Q. Assume a wide rectangular channel. The

ENGS-151 - Spring 2013
ENVIRONMENTAL FLUID MECHANICS
HOMEWORK #5
SOLUTIONS
1. (5 points) Show that in a non-buoyant jet, the local Reynolds number Re based on the
local center velocity and local jet diameter is invariant with distance. What does this
impl

ENGS-151 Spring 2013
ENVIRONMENTAL FLUID MECHANICS
HOMEWORK #4
SOLUTIONS
1. (10 points) The earth receives 1.75 x 1017 W from the sun, and we can assume that
half of this energy input is being dissipated in the atmosphere, with the rest going to land
and

ENGS-151 Winter 2013
ENVIRONMENTAL FLUID MECHANICS
HOMEWORK #3
SOLUTIONS
1. (5 points) In each of the two asymptotic limits surface gravity waves have their
wavelength proportional to a power of their period (i.e. Tn). What are then the values
of the expo

ENGS-151 Spring 2013 ENVIRONMENTAL FLUID MECHANICS HOMEWORK #2 SOLUTIONS
1. (10 points) (Problem 2-11 of book) In the lowest 10 km of the atmosphere, the temperature decreases vertically by about 1oC for every 100 meters of height. Using the hydrostatic b

ENGS-151 Winter 2013 ENVIRONMENTAL FLUID MECHANICS HOMEWORK #1 SOLUTIONS
Computer simulation of the salinity distribution in the the Fly River Estuary, Papa New Guinea
1. (5 points) (Problem 1-4 of book) The Fly River Estuary in Papua New Guinea has an av

ENGS-151 - Spring 2013 ENVIRONMENTAL FLUID MECHANICS Final Exam
Assigned: 12:30pm Monday 11 March 2013 Due: 5 pm Friday 15 March 2013
1. (10 points) At short wavelengths, surface gravity waves are affected by surface tension, and the theory leads to a mod

RIVERS:
RAPIDLY VARIED FLOW
Benoit Cushman-Roisin
Equations governing 1D hydraulics:
1. Mass conservation:
A
( Au ) 0
t
x
2. Momentum budget:
u
u
h
u2
u
g
gS C D
t
x
x
R
where
h water depth, A(h) cross - sectional area, u velocity
S slope, R hydraulic r

RIVERS: GRADUALLY VARIED FLOW
Benoit Cushman-Roisin Dartmouth College Hanover, NH USA
Equations governing 1D hydraulics:
1. Mass conservation:
A ( Au ) 0 t x
2. Momentum budget:
u2 u u h u g gS C D R t x x
where
h water depth, A(h) cross - sectional area,

RIVER HYDRAULICS
Benoit Cushman-Roisin Dartmouth College Hanover, New Hampshire, USA
River flow is 3D and unsteady (turbulent).
But:
length of river > width & depth
As a result: downstream velocity, aligned with the channel, dominates the flow. 1D assumpt

ENVIRONMENTAL FLUID MECHANICS
Plumes & Thermals
Why do these structures behave the way they do?
How much mixing do they accomplish?
Benoit Cushman-Roisin
Thayer School of Engineering
Dartmouth College
http:/thayer.dartmouth.edu/~cushman/books/EFM/chap10.p

ENVIRONMENTAL FLUID MECHANICS
Partial Differential Equations
governing the motion of environmental fluids
Benoit Cushman-Roisin
Thayer School of Engineering
Dartmouth College
Procedure
The theory proceeds with the consideration of an infinitesimal control

ENVIRONMENTAL FLUID MECHANICS
Lecture 1 Introduction
Why do we need to worry about
motion of air and water?
Benoit Cushman-Roisin
Thayer School of Engineering
Dartmouth College
Air is pure. Air is blue.
It lifts birds wings .
Water is strong. It lifts boa

ENVIRONMENTAL FLUID MECHANICS Instabilities
(http:/bb.nightskylive.net/asterisk/viewtopic.php?f=9&t=15332)
Benoit Cushman-Roisin Thayer School of Engineering Dartmouth College
http:/hmf.enseeiht.fr/travaux/CD0001/travaux/optmfn/hi/01pa/hyb72/kh/kh_theo.ht

ENVIRONMENTAL FLUID MECHANICS
Estuaries
Benoit Cushman-Roisin Thayer School of Engineering Dartmouth College
Estuaries are regions where rivers meet the coastal ocean.
They are biologically unique because these are the only water bodies where salinity val