Differential Equations (Simplified)

8/9/2019 Differential Equations (Simplified)

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Chapter 4: Fluid KinematicsME33 : Fluid Flow 1

Differential Analysis of Fluid FlowO!ecti"es:

C#$ or inte%ral$ forms of e&uations are useful for determinin% overall effects

'owe"er$ we cannot otain detailed (nowled%e aout the flow field inside the C# ⇒ moti"ation for differential analysis


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Chapter 4: Fluid KinematicsME33 : Fluid Flow )

Acceleration Field

( ) ( ) ( )( ), , particle particle particle particleV V x t y t z t =

particle particle particle

particle

dx dy dz V dt V V V a

t dt x dt y dt z dt

∂ ∂ ∂ ∂= + + +

∂ ∂ ∂ ∂r

, , particle particle particledx dy dz

u v wdt dt dt

= = =

particleV V V V a u v wt x y z

∂ ∂ ∂ ∂= + + +∂ ∂ ∂ ∂r


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Material Deri"ati"e

/he total deri"ati"e operator d0dt is call the materialderivative and is often %i"en special notation$ D0Dt,

Other names for the material deri"ati"e include: total,particle, Lagrangian, Eulerian, and substantial deri"ati"e,

( )

DV dV V

V V Dt dt t

∂

= = + ∇∂

r r r

g


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Chapter 4: Fluid KinematicsME33 : Fluid Flow

2rolem

Flow throu%h the con"er%in% no++le can e approimated y

the onedimensional "elocity distriution

.a5 Find a %eneral epression for the fluid acceleration in the

no++le, .5 For the specific case #6 7 16 ft0s and 8 7 9 in$compute the acceleration$ in %s$ at the entrance and at the eit,


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Kinematic Description

*n fluid mechanics$ an elementmay under%o four fundamentaltypes of motion,a5 /ranslation

5 ;otation

c5 8inear strain

d5 -hear strain


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Chapter 4: Fluid KinematicsME33 : Fluid Flow >

8inear -train ;ate

Linear Strain Rate is defined as the rate of increase in len%th per unitlen%th,

*n Cartesian coordinates

#olumetric strain rate in Cartesian coordinates

-ince the "olume of a fluid element is constant for an incompressileflow$ the "olumetric strain rate must e +ero,

, , xx yy zz u v w

x y z ε ε ε

∂ ∂ ∂= = =∂ ∂ ∂

1 xx yy zz

DV u v w

V Dt x y z ε ε ε

∂ ∂ ∂= + + = + +

∂ ∂ ∂


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Chapter 4: Fluid KinematicsME33 : Fluid Flow ?

-hear -train ;ate

Shear Strain Rate at a point is defined as halfof the rate of decrease of the angle between two

initially perpendicular lines that intersect at a

point ,

-hear strain rate can e epressed in Cartesian

coordinates as:

1 1 1

, ,2 2 2 xy zx yz

u v w u v w

y x x z z yε ε ε

∂ ∂ ∂ ∂ ∂ ∂ = + = + = + ÷ ÷ ÷∂ ∂ ∂ ∂ ∂ ∂


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-hear -train ;ate

@e can comine linear strain rate and shear strainrate into one symmetric secondorder tensor called

the strain-rate tensor.

1 1

2 2

1 1

2 21 1

2 2

xx xy xz

ij yx yy yz

zx zy zz

u u v u w

x y x z x

v u v v w

x y y z yw u w v w

x z y z z

ε ε ε

ε ε ε ε

ε ε ε

∂ ∂ ∂ ∂ ∂ + + ÷ ÷ ÷∂ ∂ ∂ ∂ ∂ ÷ ÷ ∂ ∂ ∂ ∂ ∂ ÷ ÷= = + + ÷ ÷ ÷∂ ∂ ∂ ∂ ∂ ÷ ÷ ÷ ∂ ∂ ∂ ∂ ∂ ÷+ + ÷ ÷ ÷∂ ∂ ∂ ∂ ∂


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2rolem

k Cyj xi 03 ++

xyτ

Given the steady, incompressible velocity

distribution V=

where C is a constant, estimate the shear stressat the point (x,y,z).


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Chapter 4: Fluid KinematicsME33 : Fluid Flow 1)

#orticity and ;otationality

/he vorticity vector is defined as the curl of the "elocity"ector

#orticity is e&ual to twice the an%ular "elocity of a fluidparticle,Cartesian coordinates

Cylindrical coordinates

*n re%ions where ζ 7 6$ the flow is called irrotational.Elsewhere$ the flow is called rotational.

V ζ = ∇ ×

2ζ ω =

w v u w v ui j k y z z x x y

ζ ∂ ∂ ∂ ∂ ∂ ∂ = − + − + − ÷ ÷ ÷∂ ∂ ∂ ∂ ∂ ∂ rr r r

( )1 z r z r r z

ruuu u u ue e e

r z z r r

θ θ θ ζ

θ θ

∂∂∂ ∂ ∂ ∂ = − + − + − ÷ ÷ ÷∂ ∂ ∂ ∂ ∂ ∂

r r r r


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#orticity and ;otationality


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Comparison of /wo Circular Flows

-pecial case: consider two flows with circular streamlines

( ) ( )20,

1 10 2

r

r z z z

u u r r ru u

e e er r r r

θ

θ

ω ω

ζ ω θ

= = ∂ ∂ ∂ ÷= − = − = ÷ ÷∂ ∂ ∂

r r r r ( ) ( )

0,

1 10 0

r

r z z z

K u u r

ru K ue e e

r r r r

θ

θ ζ θ

= =

∂ ∂∂= − = − = ÷ ÷∂ ∂ ∂

r r r r


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2rolem

A "elocity field is %i"en yV 7 .3y)3)5 i Cy j 6 k

Determine the "alue of the constant C if the flow is to

e irrotational,


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Conser"ation of Mass

;ecall C# form from ;eynolds /ransport/heorem .;//5

@ell eamine two methods to deri"e

differential form of conser"ation of mass

Di"er%ence .Bausss5 /heorem

Differential C# and /aylor series epansions


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Chapter 4: Fluid KinematicsME33 : Fluid Flow 1=

Conser"ation of MassDivergence Theorem

Di"er%ence theorem allows us totransform a "olume inte%ral of the

di"er%ence of a "ector into an area

inte%ral o"er the surface that defines the"olume,


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Chapter 4: Fluid KinematicsME33 : Fluid Flow 1>

Conser"ation of MassDivergence Theorem

;ewrite conser"ation of mass

sin% di"er%ence theorem$ replace area inte%ralwith "olume inte%ral and collect terms

*nte%ral holds for ANY C#$ therefore:


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Chapter 4: Fluid KinematicsME33 : Fluid Flow 1?

Conser"ation of MassDifferential CV and Taylor series

First$ define aninfinitesimal control

"olume dx dy dz

et$ we approimate the

mass flow rate into or out

of each of the 9 faces

usin% /aylor series

epansions around the

center point$ e,%,$ at the

ri%ht face *%nore terms hi%her than order dx


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Chapter 4: Fluid KinematicsME33 : Fluid Flow )6


*nfinitesimal control "olume

of dimensions dx, dy, dz Area of ri%ht

face 7 dy dz

Mass flow rate throu%h

the ri%ht face of the

control "olume


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ow$ sum up the mass flow rates into and out ofthe 9 faces of the C#

2lu% into inte%ral conser"ation of mass e&uation

et mass flow rate into C#:

et mass flow rate out of C#:


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Chapter 4: Fluid KinematicsME33 : Fluid Flow ))


After sustitution$

Di"idin% throu%h y "olume dxdydz

Or$ if we apply the definition of the di"er%ence of a "ector


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Conser"ation of Mass lternative form

se product rule on di"er%ence term


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Conser"ation of MassCylindrical coordinates

/here are many prolems which are simpler to sol"e ifthe e&uations are written in cylindricalpolar coordinates

Easiest way to con"ert from Cartesian is to use "ectorform and definition of di"er%ence operator in cylindricalcoordinates


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Chapter 4: Fluid KinematicsME33 : Fluid Flow )

Conser"ation of MassCylindrical coordinates


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Conser"ation of Mass!pecial Cases

-teady compressile flow

Cartesian

Cylindrical


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Chapter 4: Fluid KinematicsME33 : Fluid Flow )=

Conser"ation of Mass!pecial Cases

*ncompressile flow

Cartesian

Cylindrical

and ρ 7 constant


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Chapter 4: Fluid KinematicsME33 : Fluid Flow )>

Conser"ation of Mass

*n %eneral$ continuity e&uation cannot eused y itself to sol"e for flow field$

howe"er it can e used to

1, Determine if "elocity field is incompressile), Find missin% "elocity component


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Chapter 4: Fluid KinematicsME33 : Fluid Flow )?

2rolem

A "elocity field is %i"en yV 7 .3y)3)5 i Cy j 6 k

Determine the "alue of the constant C if the flow is to

e incompressile,


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-treamlines

A Streamline is a cur"e that is

e"erywhere tan%ent to theinstantaneous local "elocity

"ector,

Consider an arc len%th

must e parallel to the local

"elocity "ector

Beometric ar%uments results

in the e&uation for a streamline

dr dxi dyj dzk = + +dr

V ui vj wk = + +

dr dx dy dz

V u v w= = =


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-treamlines

A-CA; surface pressure contours

and streamlines

Airplane surface pressure contours$

"olume streamlines$ and surfacestreamlines


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2athlines

A Pathline is the actual pathtra"eled y an indi"idual fluidparticle o"er some time period,-ame as the fluid particlesmaterial position "ector

2article location at time t:

2article *ma%e #elocimetry

.2*#5 is a modern eperimentaltechni&ue to measure "elocityfield o"er a plane in the flowfield,

( ) ( ) ( )( ), , particle particle particle x t y t z t

start

t

start

t

x x Vdt = + ∫ rr r


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-trea(lines

A Streakline is thelocus of fluid particles

that ha"e passed

se&uentially throu%h a

prescried point in theflow,

Easy to %enerate in

eperiments: dye in a

water flow$ or smo(e

in an airflow,


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Comparisons

For steady flow$ streamlines$ pathlines$ andstrea(lines are identical,

For unsteady flow$ they can e "ery different,

-treamlines are an instantaneous picture of the flow

field

2athlines and -trea(lines are flow patterns that ha"e

a time history associated with them,

-trea(line: instantaneous snapshot of a time

inte%rated flow pattern,

2athline: timeeposed flow path of an indi"idual

particle,


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/imelines

A imeline is thelocus of fluid particles

that ha"e passed

se&uentially throu%h a

prescried point in theflow,

/imelines can e

%enerated usin% a

hydro%en ule wire,


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/he -tream Function

Consider the continuity e&uation for anincompressile )D flow

-ustitutin% the cle"er transformation

Bi"es

/his is true for any smooth

function ψ .$y5


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/he -tream Function

@hy do this-in%le "ariale ψ replaces .u,v 5, Once ψ is(nown$ .u,v 5 can e computed,

2hysical si%nificance1, Cur"es of constant ψ are streamlines of the flow

), Difference in ψ etween streamlines is e&ual to"olume flow rate etween streamlines


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/he -tream Function"hysical !ignificance

Alon% a streamline

∴ Chan%e in ψ alon%streamline is +ero


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2rolem solution:

/h - F i


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/he -tream Function"hysical !ignificance

Difference in ψ etweenstreamlines is e&ual to

"olume flow rate etween

streamlinesdH7.#•n5dA7.idψ 0dy!dψ 0d5•.idy0ds!d0ds5ds.1577 dψ 0d ddψ 0dy dy 7 dψ


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2rolem

An incompressile stream function is defined y

where # and $ are positive constants% #se this stream function to find the volume flow &

passing through the rectan%ular surface whose corners are defined y . x, y, z' (

)*$, +, +', )*$, +, b', )+, $, b', and )+, $, +'% !how the direction of &%


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Conser"ation of 8inear Momentum

;ecall C# form

sin% the di"er%ence theorem to con"ert area

inte%rals


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-ustitutin% "olume inte%rals %i"es$

;eco%ni+in% that this holds for any C#$the inte%ral may e dropped

This is Cauchys -.uation

Can also e deri"ed usin% infinitesimal C# and ewtons )nd 8aw .see tet5


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Alternate form of the Cauchy E&uation can ederi"ed y introducin%

*nsertin% these into Cauchy E&uation and

rearran%in% %i"es

.Chain ;ule5


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nfortunately$ this e&uation is not "eryuseful

16 un(nowns

-tress tensor$ σ ij

: 9 independent components

Density ρ

#elocity$ V : 3 independent components

4 e&uations .continuity momentum5

9 more e&uations re&uired to close prolemI


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a"ier-to(es E&uation

First step is to separate σ ij into pressure and"iscous stresses

-ituation not yet impro"ed9 un(nowns in σ ij ⇒ 9 un(nowns in τ ij / 1 in ",which means that weve added 01

σ ij =

σ xx σ xy σ xz

σ yx σ yy σ yzσ zx σ zy σ zz

=

− p 0 00

− p 0

0 0 − p

+

τ xx τ xy τ xz

τ yx τ yy τ yzτ zx τ zy τ zz

#iscous .De"iatoric5

-tress /ensor


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.toothpaste5

.paint5

.&uic(sand5

;eduction in thenumer of "ariales is

achie"ed y relatin%

shear stress to strain

rate tensor,For ewtonian fluid

with constant

properties

ewtonian fluid includes most common

fluids: air$ other %ases$ water$ %asolineewtonian closure is analo%ous

to 'oo(es 8aw for elastic solids

-


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Chapter 4: Fluid KinematicsME33 : Fluid Flow 4?


-ustitutin% ewtonian closure into stresstensor %i"es

sin% the definition of ε ij :

i - ( E i


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-ustitutin% σ ij into Cauchys e&uation %i"es thea"ier-to(es e&uations

/his results in a closed system of e&uationsI

4 e&uations .continuity and momentum e&uations5

4 un(nowns .$ #$ @$ p5

*ncompressile -E

written in "ector form


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' t l


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'ow to sol"e

-tep Analytical Fluid Dynamics Computational Fluid Dynamics

1 -etup prolem and %eometry$ identify all dimensions andparameters

) 8ist all assumptions$ approimations$ simplifications$ oundaryconditions

3 -implify 2DEs .partial differentiale&uations5


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Chapter 4: Fluid KinematicsME33 : Fluid Flow

Eact -olutions of the -E

-olutions can also e

classified y type or

%eometry

1, Couette shear flows

), -teady duct0pipe flows

3, nsteady duct0pipe flows

4, Flows with mo"in%

oundaries

, -imilarity solutions

9, Asymptotic suction flows=, @inddri"en E(man flows

/here are aout >6(nown eact solutions

to the -E

/he can e classified

as:8inear solutions where

the con"ecti"e

term is +ero

onlinear solutionswhere con"ecti"e

term is not +ero

E t - l ti f th -E


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Eact -olutions of the -E

1,-et up the prolem and %eometry$ identifyin% allrele"ant dimensions and parameters

),8ist all appropriate assumptions$ approimations$simplifications$ and oundary conditions

3,-implify the differential e&uations as much aspossile

4,*nte%rate the e&uations,Apply


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li d diti


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Chapter 4: Fluid KinematicsME33 : Fluid Flow >

oslip oundary condition

For a fluid in contactwith a solid wall$ the

"elocity of the fluid

must e&ual that of the

wall

* t f d diti


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Chapter 4: Fluid KinematicsME33 : Fluid Flow ?

*nterface oundary condition

@hen two fluids meet at

an interface$ the "elocity

and shear stress must e

the same on oth sides

*f surface tension effects

are ne%li%ile and the

surface is nearly flat



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De%enerate case of the interface


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2rolem

Consider a steady$ twodimensional$ incompressile flow of a newtonian fluid with the

"elocity field u 7 N)y$ " 7 y) N )$ and w 7 6, .a5 Does this flow satisfy conser"ation of

mass .5 Find the pressure field p.$ y5 if the pressure at point . 7 6$ y 7 65 is e&ual to

2a,

2rolem


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2rolem

*f + is up$P what are the conditions on constants a and

for which the "elocity field u 7 ay$ Q 7 $ w 7 6 is an

eact solution to the continuity and a"ier-to(es

e&uations for incompressile flow

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Differential Equations (Simplified)