Characteristic Length - People

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Background Review


Now that you have gathered what
fluid mechanics

is all about, let us present some
tools that we use investigate fluid flow behavior. Since fluids are mostly invisible, it
is difficult to vis
ualize what goes inside a fluid medium. Historically, the study of
fluid mechanics started even before people had even conceived of some modern flow
visualization tools.
Developing a solid mathematical basis can assist understanding
and measurements of var
ious flow quantities.


There are
two ways

to investigate fluid flow phenomena. If we consider a fluid
medium from a
macroscopic

standpoint,
fluid particles appear as a smooth medium

in which individual particles cannot be distinguished. On the other hand,

if we
investigate fluid mechanics from a
microscopic

standpoint, there is no continuous
medium at all,
and flow phenomena occur as a continuous interaction between
discrete fluid particles
. This latter approach is a lot more chaotic, as you are aware
from

physical chemistry and
Brownian motion
. It is considerably more difficult to
analyze also, since any such flow models could not be analyzed before the advent of
supercomputers. Therefore historically people chose the macroscopic approach to
analyze fluid
mechanics. In this class we shall adopt this approach also.


The benefits of the macroscopic model are several. It is simpler to model using
calculus and applied mathematics. The models have a rational means and therefore
are both verifiable and, in most
instances, completely solvable.


Before we develop the macroscopic outlook of fluids, let us consider when such an
outlook is justifiable. For very obvious reasons, this approach will fail whenever
fluid particles are not very closely packed. For example,
if we travel to a very high
altitude earth’s atmosphere is very thin. Air particles are very
lightly packed

there.
One may find only a few air particles in several cubic feet of air collected there. On
the other hand, if we look at a tankful of water at se
a level, the water medium
definitely can qualify to be
a smooth, continuous medium
. We shall commonly use
the term
characteristic length

to represent
a typical dimension in a fluid flow

medium. For example, if we are investigating
internal flows

of air in a room, any of
the length,
L
, width,
W
, or, height,
H
, can be chosen as a characteristic length
depending on the direction of the flow. Similarly, if we are exploring
external flows

over a circular cylinder, the diameter,
D
, or, the length,
L

can be chosen as the
length scale depending on the flow direction.


What allows us to treat fluid medium as a continuum
? Remember, the
continuum
requirement must be satisfied

to study fluid me
dia in the
macroscopic

view. Our
requirement also must be met from a mathematical standpoint. We simply want to
record changes in fluid media by creating derivatives and integrals of fluid flow
variables. As you recall from cal
culus,
any function must be treated continuous if we
wish to work with its derivatives.
Given below is a simple hypothesis that allows us
to treat fluid as a continuous medium.


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Continuum Hypothesis:

A fluid medium can be considered a continuum (with
all t
he associated mathematical benefits) if and only if the minimum characteristic
length, L in that medium can be considered to be much larger than the
m
ean
f
ree
p
ath (the distance a fluid particle travels freely before colliding with the next fluid
particle
in a Brownian motion), i.e.,
L >> m.f.p
.


Thus the above hypothesis can act as our crude looking glass through which any
fluid media can appear as a continuum. Note that
any time this hypothesis is
invalidated in an application, we may not be able to apply

any mathematical
equation that we develop in this class.
Fortunately most fluid flow applications can
be treated successfully because they do satisfy this requirement.


Once we can consider fluid as a continuum, we may think of every point in the fluid
me
dium occupied by a certain fluid particle.

This is a big advantage in our model
since
no part of our fluid medium can be considered empty any more
. Furthermore,
instead of writing motion of individual fluid particles, we’ll always analyze a
complete flow f
ield by writing all fluid properties as continuous functions of space
and time variables, e.g.,
r
(x,y,z,t),
V
(x,y,z,t),
T
(x,y,z,t), etc.
Simply by going to a
particular location of the flow field, we can determine what is the motion of a
specific fluid pa
rticle that is in that location of the flow field at some instant.


As with any study of mechanics, we begin our study of fluid mechanics with the
study of flow kinematics. Fluid flow motions are commonly described by two
popular approaches
-

(i) the
Lagra
ngian approach
, and, (ii) the
Eulerian approach
.


The first approach is based on tracking a specific fluid particle at all times. For
example, a
path line

tracks the locus of all points in the fluid medium a particle
travels through at different times. Si
milarly, a
streak line

tracks all particles that
pass through a specific location in the fluid medium at different times. This latter
Eulerian

approach

is the more popular one since it can focus on a control volume at
all times by passing fluid particles t
hrough it at different times. However the
Lagrangian approach
, or, the
system approach
, have become popular more
recently, with the advent of supercomputers. In this approach a collection of fluid
particles are studied at all times while they travel throug
h the fluid medium.
Although this collection of fluid particles can rotate, distort, or move relative to one
another, they are not allowed to leave the system boundaries.


Reynolds

was the
first to relate the fluid property changes in a fluid system
(Lagr
angian approach) to those in a fluid control volume (Eulerian approach
). The
theorem presented below is an extremely powerful tool to understand almost all
conservation laws in fluid mechanics.






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Reynolds Transport Theorem


This theorem was
developed i
n general terms

using
two fluid variables



慮d N
潮l礮yThis w慹Ⱐwe c潵ld define them t漠oepresent specific 癡vi慢les 慮d de癥l潰
different c潮ser癡vi潮 l慷s in fluid mech慮ics⸠Let

represent the
intensi癥

(m慳s
independent) pr潰ert礠慮d N⁲epresent the
c潲resp潮din朠
e硴ensi癥

(m慳s
dependent) pr潰ert礠 c潲resp潮din朠 t漠

in⁡⁣潮tr潬 癯vume⸠Acc潲din朠 t漠ohese
definiti潮sⰠN⁣慮 be⁲el慴ed t漠

慳 f潬l潷s: ⁆潲 愠am慬l m慳s dmⰠ

dN‽

dm‽



dV⸠.Ⱐ



With this definition, the
rate of change of N in th
e system can be related to the rate
of change of N in the control volume

as


The right hand side has two terms.
The
first term

represents the local rate of change
of the property N inside the control volume
, and
the
second term

represents the rate
of chang
e of property N inside the control volume due to the flux of N through the
control surfaces.


Note that the
control surface

of any control volume
represents the surface area
completely enclosing the control volume from all directions
. For example, the
cont
rol surface of a parallelepiped can be represented by 6 surfaces. As long as
there exist a net inflow and outflow through those surfaces, the net change of
property in the stationary control volume may be affected. The local rate of change
of N, represente
d by the first term in the above equation, is totally independent of
the second term
. This term may be set to zero for a steady flow, when all property
changes occur only due to influx and outflux of quantities, without any property
addition or reduction l
ocally.



With this single theorem given above, all basic conservation laws of fluid mechanics
may be derived. For example, if we wish to derive the mass
-
conservation law
(continuity equation), we set


=‱Ⱐ慮d theref潲e N =⁍⸠Simil慲l礠 f潲 the
m潭entum
equ慴i潮 慮d the⁥ner杹⁥qu慴i潮 we m慹aset


= VⰠ慮d


= e
specific
ener杹gⰠrespecti癥l礮yThe equ慴i潮s resultin朠fr潭 th潳e⁴w漠equ慴i潮s
require
慤diti潮慬 use ⁴he
Newton’s second law of motion
, and the fi
rst law of
thermodynamics to attain the final forms given below.

We summarize all
three
conservation laws

resulting from the use of the Reynolds Transport Theorem below.


Mass Conservation:


Momentum

Conservation:


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Energy Conservation:



In the above eq
uations, each conservation law has a
local component

and a
convective component
.

The left
-
hand sides of the second and the third equation
represent external influences on the control volume.

For example, in the momentum
equation, the external forces are of

the body force and surface force type,
represented by
F
B

and
F
S
. Similarly, on the left
-
hand side of the energy equation,
the external influences are the rate of heat addition to the control volume and the
rate of work done on the control volume.


Using
the control volume forms in problem solving is often called
the integral
approach

or,
the control volume approach (
since the laws represent summations of
local and convective effects on the control volume and the control surface). This
approach is extremel
y useful for engineers
whenever you need to determine the
overall effects of flow motions on complete systems
. It does not however provide us a
detailed point
-
by
-
point picture of the flow field. To be able to determine the forms
of the above equation at ea
ch point of the fluid flow
, we must shrink the size of the
control volume to an infinitesimally small size.
In the limit when this happens we are
able to obtain the differential equations of fluid motion (resulting in the differential
approach).


The solu
tion of the differential equations of fluid motions is the basis of many of the
successful flow
-
solving computer programs available in the market today. If you
need to understand the solution at every point in a flow field in these programs, you
must have
a thorough knowledge of the
differential approach

of fluid motion. The
assumptions and simplifications in this approach require working closely with
calculus and the fluid flow conservation laws. You should develop a very
syste
matic
approach to problem solving

if you wish to master this area. Solved problems in
these notes offer examples of this methodology.
Compare with your background
knowledge

and
add to it as you advance.


The
top
ics in the realm of fluid dynamics

studied in this course can be categorized
based upon some flow variables.

For example, if the flow is categorized based upon
density

Ⱐwe慹at慬k⁡ 潵t
inc潭pressible
癳⸠
c潭pressible

fl潷s⸠M潳t 潦⁴he
liquid fl潷s c慮 be⁴re慴ed 慳 inc潭pressible (i⹥⸬.


=⁣潮st慮t)⸠H潷e癥r s潭e
条ge潵s fl潷s m慹a慬s漠oe⁴re慴ed like⁩nc潭pressible fl潷s⸠F潲 e硡xpleⰠif⁴he fl潷
潦⁡ r is⁡
M慣h

N漮

M (=V/cⰠwhereⰠV =⁳peed 潦⁦l潷Ⱐc‽⁳peed 潦⁳潵nd in the⁡ r
medium) <‰⸳Ⱐ t m慹abe⁴re慴ed 慳 慮 inc潭pressible fl潷⸠Simil慲l礠 fl潷s m慹abe
c慴e杯gized 慳
irr潴慴i潮al

(


=‰ ⁶ ⸠
r潴慴i潮慬



in癩scid

(


=‰ 癳⸠
癩sc潵s



The stud礠潦⁩n
癩scid fl潷s is⁩mp潲t慮tⰠ since it 慤ds much t漠潵r underst慮din朠 潦
fluid d祮慭ics⸠
Th潵杨 m潳t 潦 the⁲e慬 fluid fl潷s 慲e⁶ sc潵sⰠ,here 慲e⁳潭e
癩sc潵s⁦l潷s th慴 m慹abe⁴re慴ed like⁡ in癩scid fl潷⸠
This will bec潭e cle慲⁡ ⁹潵
pr潧oess thr潵杨 th

fl潷 kinem慴ics
⸠T漠cl慳sif礠
癩sc潵s⁦l潷s
Ⱐwe use 愠a潮
-
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dimensional variable called the
Reynolds Number
,
Re

(=

VL/

Ⱐ,here V⁡ d L
represent fluid ch慲慣teristic speed⁡ d
len杴h sc慬e
)⸠If the⁦l
ow’s
Re > Re
Critical
, the
flow is considered a
turbulent flow
. Otherwise the flow is considered
laminar
. In this
class we shall consider applications from both kinds of flows. Likewise, you will see
applications of both
internal

and
external

flows.



Most
of our problems will involve multi
-
dimensional fluid flows. Thus it is
important to recall
what we mean by dimension

in a fluid flow. For example, if the
velocity vector in a continuous fluid medium can be expressed as V = 10 x i + 30 x
2

j
+ 5 x
3

k, is it
a 3
-
dimensional flow?
Although it has all three components of velocity
present (u = 10 x, v = 30 x
2
, and, w = 5 x
3
) it turns out to be a one
-
dimensional flow.

Thus dimensionality
does not depend on how many components of fluid velocity is
present, but what

are the total number of independent

(in this case,
Cartesian
) space
variables upon which the velocity vector depends
. Note that in this example, u(x),
v(x) and w(x) only,
without any explicit dependence on the vari
ables y and z.
Similarly, since the flow velocity vector does not explicitly involve the time variable,
this flow would be considered
steady
.


Let us begin with the task of deriving the mass conservation law (continuity
equation) in the
differential approach
.

Continue