1
2
Taylor polynomials of f (x) = e(1x ) around x0 = 1
p1 (x) = f (x0 ) + f 0 (x0 ) (x x0 )
p2 (x) = f (x0 ) + f 0 (x0 ) (x x0 ) +
f 00 (x0 )
(x x0 )2
2!
We need to calculate f (x0 ), f 0 (x0 ), f 00 (x0 ).
2
f (x) = e(1x ) ,
f (x0 ) = f (1) = 1,
2
f 0 (
Homework #4
MAE 122
Fall Quarter, 2014
Due: 11/19/2014
9:00am (via TED)
1. A region of the ocean can be considered as comprised of three layers of water with the
interfaces are tilted as shown in the figure below.
a. Using the displacements in the figure,
MAE122 Homework 5
December 8th, 2014
Problem 1 The Turbulent Boundary Layer
We may obtain a value for the friction velocity via the Shields parameter
p
u = (s 1)gd ,
(1)
where the threshold for initiation of sand motion is typically taken as 0.05, s = s /
MAE122 Homework 1
October 15th, 2014
Problem 1 Open Channel Flow
By the principle of conservation of mass we have
m
= U1 A1 = U2 A2 ,
(1)
dividing through by and noting that Ai = yi bi gives us the flow rate
Q = U1 (y1 b1 ) = U2 (y2 b2 ) .
(2)
If the flo
MAE122 Homework 6
December 12th, 2014
Problem 1 The Manning Formula
We assume a wide rectangular channel, so that b > y, where b is the width of the channel and y
is the flow depth. This leads to
P = 2y + b b ,
(1)
A
by
= y,
(2)
P
b
where P is the channel
Homework #1
MAE 122
Fall Quarter, 2014
Due: 10/15/2013 9:00am
1. Water flowing along a channel meets a constriction so that the water surface at the
constriction dips relative to the upstream level as shown in the figure below. Find the
flowrate Q in the
Homework #6
MAE 122
Fall Quarter, 2014
Due: Dec 12, 2014
9:00am (via TED)
1. Determine the water depth, h, and velocity, V, in uniform river flow obeying the
Manning formula (Ch 7, eq 32), each in term of the discharge Q. Assume a wide
rectangular channel
MAE122 Homework 3
November 10th, 2014
Problem 1 Reynolds Averaged Navier-Stokes Equations
Beginning with the x-momentum equation
u
u
u
u
1 p
+u
+v
+w
=
+
t
x
y
z
x
2u 2u 2u
+ 2 + 2
x2
y
z
,
(1)
use the Reynolds decomposition (i.e. u = u
+ u0 ) to derive
UNIVERSITY OF CALIFORNIA, SAN DIEGO
FLOW AND TRANSPORT IN THE ENVIRONMENT
MAE 122
Fall Quarter, 2014
Instructor: G. Pawlak
Midterm 1
Closed book, notes
Read carefully!
(You must show support for your answers!)
1. (24 pts) Tides in a (fictional) long, pr
Homework #4
MAE 122
Fall Quarter, 2015
Due: 11/11/2015
11:59pm (midnight; via TED)
1. Lake Wakatipu is a narrow, 75 km long lake in the Southern Alps on the South Island of
New Zealand. The water level at Queenstown, partway down the east side of the lake
Homework #5
MAE 122
Fall Quarter, 2014
Due: 12/8/2014
9:00am (via TED)
1. The threshold for initiation of sand motion under a steady current is determined via a
critical value for the Shields parameter:
u*2
=
(s 1)gd
where the critical value for is typica
Homework #3
MAE 122
Fall Quarter, 2014
Due: 11/10/2013
9:00am (via TED)
1. Beginning with the x-momentum equation:
u
u
u
u
1 p 2 u 2 u 2 u
+u +v +w
=
+
+
+
t
x
y
z
x x 2 y 2 z 2
use the Reynolds decomposition, (i.e. u = u + uwhere u represents the tim
1
%
%
%
%
%
%
%
MatLab
Objective: This script will plot the functions y1 = sin(x) + 5/4 in green
and y2 = x + exp(-x/2) in magenta over the range x=[0,3].
A title will be added and name will be placed in an appropriate location.
Input variables : None
Out
Homework #5 Solutions
1. Odefunction
function [dxdt]=odefunc(t,x);
%
% Objective: Computes the function, f(t,x), which is used in an ODE solver
% for ODE \dot x=f(t,x).
% input variables:
% t - time
% x - state
% output variables:
% f - f(t,x)
% functions
MAE122 Homework 1
October 6th, 2016
Problem 1 Richardson No.
The condition for mixing in a stratified shear layer is
Ri =
1
gh
,
2
4
0 Usurf
(1)
where h represents the height over which the vertical excursions are taking
place. Assuming that these vertic
MAE 107 HW 6 Solutions (MATLAB)
4. Your homework should include 4 pieces of MATLAB code: a function that evaluates sin(x)
cos(x), a function that computes the left endpoint integral estimate for a given n (or an array
of n), a function that computes the
MAE 107 HW 3 Solutions
1.
2 3 1 4
2x x +
x + x
6
6
1
2
)x 1)x + 2)x + 1
= ( x +
6
6
f (x) = 1 +
2
We do not need to perform a square root operation (the fastest code would just hardwire the
coefficients out to machine accuracy.) 4 additions and 4 multipli
MAE 122 Homework 3
October 31st, 2016
Problem 1
Set up the full solution for C(y, t) in terms of an integral of C(y, t = 0), where
C0
y
C(y, t = 0) =
1 + cos
0 < y < Ly ,
2
Ly
C(y, t = 0) = 0
y > Ly .
(1)
(2)
Following the discussion in Chapter 1 of Sok
Homework #2
MAE 122
Fall Quarter, 2014
Due: 10/24/2014
9:00 AM (via TED)
1. (a) At UCSD, how fast are you moving due to the rotation of the earth?
(b) What is the magnitude and direction of the centrifugal acceleration due to the rotation of
the earth and
MAE122 Homework 2
October 24th, 2014
Problem 1 A Rotating Reference Frame
(a) At UCSD, how fast are you moving due to the rotation of the Earth?
Let = 7.3 105 (rad/s) be the angular speed at which the Earth rotates. Linear velocity
and angular velocity ar
MAE122 Homework 4
November 19th, 2014
Problem 1 Baroclinic Stratification
(a) Calculate the pressure gradient along the bottom between A and B.
The pressure gradient is given by
dp
p
pB pA
=
=
.
dx
x
10h0
(1)
Hydrostatic pressure balance requires
Z
p=
z
Z
UNIVERSITY OF CALIFORNIA, SAN DIEGO
FLOW AND TRANSPORT IN THE ENVIRONMENT
MAE 122
Fall Quarter, 2014
Instructor: G. Pawlak
Midterm 2
(Read each problem carefully.
You must show support for your answers!)
1. (35 pts) Measurements from a weather balloon s
Homework #3
MAE 122
Fall Quarter, 2015
Due: 11/3/2013
11:59 pm (midnight, via TED)
1. Consider a wave train with a deep water wave height (2A) of 1 meter approaching the
shore. Calculate the depth at which conditions for the shallow water approximation is