Exam 1 ME 363 SP 2017
RULES
1) Generic calculators.
2) NO notes.
3) Please put your answers INSIDE the box provided for each problem. Answers outside
the box may not receive full credit.
4) Your name goes on the last page ONLY.
5) Writing in the back page
Internal Flow
Shaughnessy, Chapter 13 section 1 & 2
Flows confined to pipes or other conduits
1
Entrance Length= length required to achieve fullydeveloped conditions
We are interested in predicting the length necessary for fullydeveloped flow to occur
Momentum:
Differential Balance on
Pipe Flow
9
NavierStokes in cylindrical coordinates (r, ,z ), the vector operations become more complicated, for
instance (with u = (ur ,u ,uz )
2
2
u
2u
u
u ur
u
u 2
P
1 ur
1 ur ur ur
2 u
(rmom) r + ur r +
+ uz r
Reynolds Transport
Theorem and Mass
Conservation
1
Fundamental Concepts
Material region (m)
Defined as a region with a fixed mass or a fixed number of fluid
particles.
Illustration
m
@t=3t
m @t=0
m @t=t
Control Volume
Defined as a prescribed region in spa
Particle Path
Consider an infinitesimal fluid element. This is
also referred to as a fluid particle;
The fluid particle is by definition
composed of the same fluid
surrounding it. It is also moving at the
local fluid velocity.
Particle
Fluid
Element
Defin
Introduction to
Flow Field Description
Ref: Chapter 6 from Shaughnessy
1
In fluid mechanics, we are generally interested in the flow of material and in the resulting
forces this has on surrounding materials. To describe both of these effects, we need a
de
Example 2
If we are given the density field of a helium microdevice, can we obtain the velocity field ?
Assume 1D.
Consider 1d x = cos x
x 0, , and u x = 0,t = 3 m/s
4
( )
( )
(
)
First, this is NOT an incompressible fluid.
By applying the RTT in 1D
Background Mathematics. B
11
Tensors
If T is a tensor, for purposes of this class T acts as a linear mapping from ! n to ! n . (n=2 or 3)
( e.g.
For instance, given u ! 3
= u1e1 + u2 e 2 + u3e 3 )
T i u = v ! 3
In this fashion T i u is simply a matrixve
Introduction to ME 363
Fall 2015
Instructor: Prof. Mario F. Trujillo
1
Background
Fluid Dynamics or Fluid Mechanics
Credit hours: 3
My notes will be posted on [email protected]
Reference texts:
Basis for HW & Exams.
E.J. Shaughnessy, I.M. Katz, and J.P. Schaffer
External flow over
bluff bodies
Shaughnessy et al. Chapter 14
1
Common fluid mechanics calculations and/or experiments concern the fluid forces exerted
on an object. These can be categorized as drag forces or lift forces, and they can be calculated
direct
Example 7
Compute h(t) based on the given information.
Water is the working fluid.

AT
dM C
=
dV + b n dV = ( u b ) n ds
dt
t
C
CS
CS
( x,t )
t
h(t)
= 0 at ( x,t ) because of incompressibility ; b = 0 (stationary frame)
EXCEPT AT THE TOP SURFACE, wher
Hydrostatics
Shaughnessy, Sections 5.1 through 5.5
1
Hydro (water) Statics (notmoving)
Start with the NavierStokes equation
u
+ u i u = p + 2 u + g
t
z
g
For the case of u = 0, this gives
0 = p + g
with g = ( 0,0,g )
p = g
y
p p
p
i+
j+
k = gk
x
y
z
HW#10
ME363 Fall 2015 Prof. Trujillo
Due date: Dec. 7th 11:59 PM
1. Consider the analysis of the boundary layer as discussed in the notes.
(a) What are the governing equations for a boundary layer in reduced form ?
(b) In terms of the function f (), where
Bernoulli Equation
and its applications
Chapter 8 Shaughnessy
1
From mass conservation ( = const ) :
+ i ( u) = 0 =
+ u. + i u i u = 0
t
t
and momentum ( Navier  Stokes) :
(Continuity )
u
+ u i u = P + 2 u + g
t
Vorticity is defined as
u = and is dire
Dimensional Analysis and Similitude
1
Dimensional Analysis and Similitude
Beside the
i) C (integral) analysis
ii) Differential analysis
Another type of analysis is dimensional. It doesn't provide specific answers
on the velocity field, pressure field, or
ME 363 HW #5 solution
Problem 1:
Consider the flow field through a duct as shown in the picture below. The velocity is given by
2
2
7 y x
u ( x,t ) = ( 0,0,w ( x, y ) , where w ( x, y ) = 1 1 . Compute the viscous term from
4 b a
the NavierStokes equat
u
The shear stress will be = , where u is the streamwise component
y
and y is the coordinate normal to the surface.
The direction of the shear force depends on the object being considered.
Specifically:
(a) It is in the direction of the flow if we are
Momentum:
Differential Balance on
Pipe Flow
9
NavierStokes in cylindrical coordinates (r, ,z ), the vector operations become more complicated, for
instance (with u = (ur ,u ,uz )
2
2
u
2u
u
u ur
u
u 2
P
1 ur
1 ur ur ur
2 u
(rmom) r + ur r +
+ uz r
ME 363 HW #4 Solution
Problem 1:
Show that
w
d
w
w
w
+u
+v
+w
k
w ( X(t),Y (t), Z(t),t ) k =
t
dt
x
y
z
From the lecture notes:
d
w ( X(t),Y (t), Z(t),t ) k =
dt
w dX ( t ) w dY ( t ) w dZ ( t ) w dt 1
=
+
+
+
k
Y dt
Z dt
t dt
X dt
w w
w
w
=
ME 363 Solutions HW #3
Problem 1
Let u = u ( x, y ) i + v(x, y)j. Calculate the xcomponent of u i u considering
that the problem is 3D.

u i u =
(
u
v
w
x
x
x
u
v
w
u v w
y
y
y =
u
v
w
z
z
z
)
(
u
v
0
x
x
u v 0 u
v
y
y 0
0
0
0
u i u = u u x + v u
Boundary Layer
Over a Flat Plate
Shaughnessy et al. Chapter 14
Section 2
1
Moving up from the (y = 0), u(x, y) increases
BOUNDARY LAYER ALONG A FLAT PLATE
from 0 until it begins to approach U . The
smallest value of y, where u(x, y) is sufficiently
close
Momentum:
Integral Balance
Chapter 7 Shaughnessy, Jr.
1
Integral Balance
From RTT:
dEsys d
=
d =
d + (u i n)ds
dt
dt (t )
t
(t )
( t )
m
m
m
Let = u, Esys = L sys (momentum) =
m (t )
ud units:
kg m 3
m
m = kg
3
s
m s
( u )
d
d
L SYS =
ud =
d +
Background Mathematics. A
1
Scalar&Vectors
Scalars are real number, i.e. member of !, for instance , 1.23, 2.0, 1013 ,etc.
As such only the magnitude of the scalar is apparent.
Vectors carry additional information to their magnitude, they convey directi
Calculating the Force on a Fluid
1
Body and Surface forces
Our work thus far has allowed us to write the Lagrangian derivative of velocity and equate
this to derivatives of velocity fixed in some laboratory frame.
a=
(
Du X f (t),t
Dt
) = u
t
+ u ( x,t )
Lagrangian (following the fluid particle)
Eulerian (fixed in space)
For more material: Look at relevant sections from Chapter 6
Shaughnessy et al. However, these notes are intended to be
selfcontained.
1
Lagrangian Description
The Lagrangian description
Fluid Particle Acceleration
1
Expressions for the calculation of both body and surface forces have been developed and
presented. This allows us to calculate surface forces among objects. There is a second very useful
purpose for the stress tensor : the c
Momentum:
Differential Balance
Chapter 11 Shaughnessy et al.
u
+ u u = P + 2 u + g
t
NavierStokes equations
These equations govern the flow behavior in microdevices, engines, lakes, our circulatory
system, oceans, atmosphere, etc. Please memorize them
HW#6
ME363 Fall 2015 Prof. Trujillo
Due date: October 29th 11:59 PM
1. Flow through a vertical conical nozzle is illustrated in figure below. The fluid enters
the nozzle through a circular port of area 50 cm2 and leaves the nozzle through a port
of area 1
HW#9
ME363 Fall 2015 Prof. Trujillo
Due date: November 27th 11:59PM
1. The momentum equation that we derivedin class reads u
t + B = u ; where the
V 2
Bernoulli function is B = P + 2 + gz . If the flow is incompressible, obtained a reduced form of
this eq
HW#5
ME363 Fall 2015 Prof. Trujillo
Due date: Oct. 16th 11:59 PM
1. Consider Lake Mendota on an August evening with zero fluid acceleration everywhere.
Take the stress tensor to be defined by = P I, where P is the pressure and I is the
identity tensor.
(a