fundamentals of fluid mechanics chapter 6 flow analysis … · control...

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FUNDAMENTALS OFFUNDAMENTALS OFFLUID MECHANICSFLUID MECHANICS

Chapter 6 Flow Analysis Chapter 6 Flow Analysis Using Differential MethodsUsing Differential Methods

JyhJyh--CherngCherng ShiehShiehDepartment of BioDepartment of Bio--Industrial Industrial MechatronicsMechatronics Engineering Engineering

National Taiwan UniversityNational Taiwan University11/16/200911/16/2009

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MAIN TOPICSMAIN TOPICS

Fluid Element KinematicsFluid Element KinematicsConservation of MassConservation of MassConservation of Linear MomentumConservation of Linear Momentum InviscidInviscid FlowFlowSome Basic, Plane Potential FlowSome Basic, Plane Potential FlowViscous flowViscous flow

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Chapter 5 vs. Chapter 6Chapter 5 vs. Chapter 6

透過透過Chapter 5Chapter 5的的finite control volume approachfinite control volume approach，，無法獲得細部的、詳細的無法獲得細部的、詳細的informationinformation。。

有些時候，細部的、詳細的資訊是必要的。因此，有有些時候，細部的、詳細的資訊是必要的。因此，有必要發展新的方法，該方法有別於必要發展新的方法，該方法有別於Chapter 5Chapter 5的的finite finite control volumecontrol volume，而是採用，而是採用infinitesimal control infinitesimal control volumevolume。。

新的方法，稱為新的方法，稱為differential analysisdifferential analysis。以。以infinitesimal control volumeinfinitesimal control volume為基礎為基礎，，導出的方程導出的方程是是differential equationdifferential equation，不是，不是Chapter 5Chapter 5的的integral equationintegral equation。。

。

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Learning ObjectiveLearning Objective

Describing the fluid element motion.Describing the fluid element motion.Deriving differential form of continuity equation, Deriving differential form of continuity equation,

linear momentum equation, and equation of motion.linear momentum equation, and equation of motion.Backing to Backing to inviscidinviscid flow and flow and irrotationalirrotational flow flow

Recheck BERNOULLI EQUATION.Recheck BERNOULLI EQUATION.Viscous flowViscous flow NavierNavier--Stokes coordinates Stokes coordinates

Difficulty to solve (Beyond the scope of this Difficulty to solve (Beyond the scope of this course)course) Simple case Simple case Simple solutions.Simple solutions.

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Motion of a Fluid ElementMotion of a Fluid Element

Fluid Translation: The element moves from one point to another.Fluid Translation: The element moves from one point to another. Fluid Rotation: The element rotates about any or all of the x,y,Fluid Rotation: The element rotates about any or all of the x,y,z axesz axes.. Fluid Deformation:Fluid Deformation:

Angular Deformation:The elementAngular Deformation:The element’’s angles between the sides s angles between the sides change.change.

Linear Deformation:The elementLinear Deformation:The element’’s sides stretch or contract.s sides stretch or contract.

從Fluid element的觀點說明流體的運動

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Fluid Translation Fluid Translation velocity and accelerationvelocity and acceleration

The velocity of a fluid The velocity of a fluid particle particle can be expressedcan be expressed

The The total accelerationtotal acceleration of the particle is given byof the particle is given bykwjviu)t,z,y,x(VV

zVw

yVv

xVu

tV

DtVDa

wdtdz,v

dtdy,u

dtdx

dtdz

zV

dtdy

yV

dtdx

xV

tV

DtVDa

tDVDa

is called the is called the material , or substantial derivativematerial , or substantial derivative..

Acceleration field

Velocity field

基於field representation角度來描述particle的速度與加速度

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Physical SignificancePhysical Significance

tV

zVw

yVv

xVu

tDVDa

TotalAcceleration Of a particle Convective

Acceleration

LocalLocalAccelerationAcceleration

tVV)V(

tDVDa

第四章已經討論過了……

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Scalar ComponentScalar Component

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

z

y

x

Rectangular Rectangular coordinates systemcoordinates system

zVVV

rV

rVV

tVa

zVV

rVVV

rV

rVV

tVa

zVV

rVV

rV

rVV

tVa

zz

zzr

zz

zr

r

rz

2rr

rr

r

Cylindrical Cylindrical coordinates systemcoordinates system

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Linear TranslationLinear Translation

All points in the element have All points in the element have the same velocity (which is only the same velocity (which is only true if there are o velocity true if there are o velocity gradients), then the element will gradients), then the element will simply translate from one simply translate from one position to another.position to another.

從一個位置移動到另一位置，過程中element的所有points速度都一樣。

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Linear Deformation Linear Deformation 1/21/2

The shape of the fluid element, described by the angles at The shape of the fluid element, described by the angles at its vertices, remains unchanged, since its vertices, remains unchanged, since all right angles all right angles continue to be right anglescontinue to be right angles..

A change in the x dimension requires a nonzero value of A change in the x dimension requires a nonzero value of

A A ……………………………… y y A A ……………………………… z z z/w

y/v x/u

Element的vertices處的角度不變，但邊長改變了！ Vertices處維持直角關係

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Linear Deformation Linear Deformation 2/22/2

The change in length of the sides may produce change in The change in length of the sides may produce change in volume of the element.volume of the element.

The change inThe change in )t)(zy(xxuV

The rate at which the The rate at which the V is changing per V is changing per unit volume due to gradient unit volume due to gradient u/ u/ xx

xu

dtVd

V1

If If v/ v/ y and y and w/ w/ z are involvedz are involved

Volumetric dilatation rateVolumetric dilatation rate V

zw

yv

xu

dtVd

V1

邊長改變，體積也改變了！

體積改變率

因u/ u/ xx造成的體積改變或擴張率造成的體積改變或擴張率

當其他邊的邊長改變也納入考慮

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Angular Motion Angular Motion -- Rotation Rotation 1/41/4

δtδαω lim

0δtOA

The angular velocity of line OA

txv

x

txxv

tan

For small angles

xv

OA

yu

OB

CWCW

CCWCCW

“-” for CW

邊線OA的角速度

當轉動角度很小

角度改變量帶入

邊線OB的角速度

好好解讀v/v/xx v/v/yy

好好解讀u/u/xx u/u/yy

隊伍前進的角度

NOTENOTE：轉動方向：轉動方向

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Angular Rotation Angular Rotation 2/42/4

yu

xv

21

21

OBOAz

The rotation of the element about the zThe rotation of the element about the z--axis is defined as the axis is defined as the average of the angular velocities average of the angular velocities OAOA and and OBOB of the two of the two mutually perpendicular lines OA and OB.mutually perpendicular lines OA and OB.

zv

yw

21

x

xw

zu

21

y

kji zyx

In vector formIn vector form

繞Z軸旋轉，取CCW為”+”

Element繞Z軸旋轉的角速度為邊OA與邊OB角速度的平均值

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kyu

xvj

xw

zui

zv

yw

21

xw

zu

21

zv

yw

21

yx

yu

xv

21

z

kyu

xv

21j

xw

zu

21i

zv

yw

21V

21Vcurl

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Defining Defining vorticityvorticity V2

Defining Defining irrotationirrotation 0V

表達

定義：渦旋度

渦旋度為角速度的2倍

何謂『無旋性』渦旋度 = 0

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kyu

xv

21j

xw

zu

21i

zv

yw

21

wvuzyx

kji

21V

21Vcurl

21

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VorticityVorticity

Defining Defining VorticityVorticity ζζ whichwhich is a measurement of the rotation of a is a measurement of the rotation of a fluid elementfluid element as it moves in the flow field:as it moves in the flow field:

In cylindrical coordinates systemIn cylindrical coordinates system::

V21k

yu

xvj

xw

zui

zv

yw

21

VVcurl2

rz

zrzr

Vr1

rrV

r1e

rV

zVe

zVV

r1eV

渦旋度為角速度的2倍

用來衡量流體元素在流場移動過程中的轉動情形

VorticityVorticity in a flow field is in a flow field is related to fluid particle rotationrelated to fluid particle rotation

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Angular Deformation Angular Deformation 1/21/2

Angular deformation of a particle is given by the sum of the twoAngular deformation of a particle is given by the sum of the twoangular deformationangular deformation

tyyututy

yuutx

xvtvtx

xvv

x/

y

uxv...

t

tyy

yut

xx

xv

limt

lim0t0t

ξ（Xi）η（Eta）

Rate of shearing strain or the rate of angular deformationRate of shearing strain or the rate of angular deformation

txv

x

txxv

tan

總的角度變形

y/

xy-plane，繞z軸

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Angular Deformation Angular Deformation 2/22/2

The rate of angular deformation in xy plane

The rate of angular deformation in yz plane

The rate of angular deformation in zx plane

yu

xv

zv

yw

zu

xw

xy-plane，繞z軸

yz-plane，繞x軸

zx-plane，繞y軸Shear deformation

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Example 6.1 Example 6.1 VorticityVorticity

For a certain twoFor a certain two--dimensional flow field dimensional flow field thth evelocityevelocity is given by is given by

Is this flow Is this flow irrotationalirrotational? ?

j)yx(2ixy4V 22

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Example 6.1 Example 6.1 SolutionSolution

0wyxvxy4u 22

0xw

zu

21

0zv

yw

21

y

x

0yu

xv

21

z

This flow is This flow is irrotationalirrotational

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Conservation of Mass Conservation of Mass 1/51/5

To derive the differential equation for conservation of To derive the differential equation for conservation of mass in rectangular and in cylindrical coordinate system.mass in rectangular and in cylindrical coordinate system.

The derivation is carried out by applying conservation of The derivation is carried out by applying conservation of mass to a differential control volume.mass to a differential control volume.之前所談的是finite CV，現在要談的是differential CV

目標

如何做？

With the control volume representation of the conservation of maWith the control volume representation of the conservation of massss

0dAnVVdt CSCV

The differential form of continuity equation???The differential form of continuity equation???

Based on CV method，描述質量守恆原理的governing equation

微分型式的連續方程式？

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The CV chosen is an infinitesimal cube with sides of length The CV chosen is an infinitesimal cube with sides of length x, x, y, y, and and z.z.

zyxt

Vdt CV

2x

xuu|u

2dxx

2x

xuu|u

2xx

CV是一個極微小的立方體

PART I

PART II

直角座標系

0dAnVVdt CSCV

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Net rate of mass Net rate of mass Outflow in xOutflow in x--directiondirection

zyxxuzy

2x

xuuzy

2x

xuu

Net rate of mass Net rate of mass Outflow in yOutflow in y--directiondirection

zyxyv

Net rate of mass Net rate of mass Outflow in zOutflow in z--directiondirection

zyxzw

X方向的淨質量流出率

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Net rate of mass Outflow

zyxzw

yv

xu

The differential equation for conservation of massThe differential equation for conservation of mass

0Vtz

wyv

xu

t

Continuity equationContinuity equation

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Incompressible fluidIncompressible fluid

Steady flowSteady flow

0Vzw

yv

xu

0Vz

)w(y

)v(x

)u(

Special case

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Example 6.2 Continuity EquationExample 6.2 Continuity Equation

The velocity components for a certain incompressible, steady floThe velocity components for a certain incompressible, steady flow w field arefield are

Determine the form of the z component, w, required to satisDetermine the form of the z component, w, required to satisfy the fy the continuity equation.continuity equation.

?wzyzxyvzyxu 222

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Example 6.2 Example 6.2 SolutionSolution0

zw

yv

xu

The continuity equationThe continuity equation

)y,x(f2zxz3w

zx3)zx(x2zw

zxyv

z2xu

2

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Conservation of MassConservation of MassCylindrical Coordinate System Cylindrical Coordinate System 1/31/3

The CV chosen is an infinitesimal cube with sides of The CV chosen is an infinitesimal cube with sides of length length drdr, rd, rdθθ, and , and dzdz..

The net rate of mass flux out through the control surfaceThe net rate of mass flux out through the control surface

zrzVrV

rVrV zrr

The rate of change of mass The rate of change of mass inside the control volumeinside the control volume

drdzrdt

圓柱座標系

r)Vr(

r1 r

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Conservation of MassConservation of MassCylindrical Coordinate System Cylindrical Coordinate System 2/32/3

The continuity equationThe continuity equation

By By ““DelDel”” operatoroperator

The continuity equation becomesThe continuity equation becomes

0z

)V()V(r1

r)Vr(

r1

tzr

zk

r1e

rer

0Vt

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Conservation of MassConservation of MassCylindrical Coordinate SystemCylindrical Coordinate System 3/33/3

Incompressible fluidIncompressible fluid

Steady flowSteady flow

0Vz

)V()V(r1

r)rV(

r1 zr

0Vz

)V()V(r1

r)Vr(

r1 zr

Special case

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Stream Function Stream Function 1/61/6

Streamlines ?Streamlines ? Lines tangent to the instantaneous velocity vectors at Lines tangent to the instantaneous velocity vectors at every point.every point.

Stream function Stream function ΨΨ(x,y)(x,y) [Psi] ? Used to represent the velocity [Psi] ? Used to represent the velocity component u(x,y,t) and v(x,y,t) of a twocomponent u(x,y,t) and v(x,y,t) of a two--dimensional dimensional incompressible flow.incompressible flow.

Define a function Define a function ΨΨ(x,y), called the stream function, which relates (x,y), called the stream function, which relates the velocities shown by the figure in the margin asthe velocities shown by the figure in the margin as

xv

yu

何謂streamline？與速度相切的線

叫它stream function。

Stream function用來表示兩維、不可壓縮流的速度

就是如此定義，無庸問『為什麼』

質量守恆下的產物質量守恆下的產物

想要用來表達速度的函數

不如此定義不能滿足連續方程式

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The stream function The stream function ΨΨ(x,y) (x,y) satisfies the twosatisfies the two--dimensional form of dimensional form of the incompressible continuity equationthe incompressible continuity equation

ΨΨ(x,y) (x,y) ?? Still unknown for a particular problem, but at least we Still unknown for a particular problem, but at least we have simplify the analysis by having to determine only one have simplify the analysis by having to determine only one unknown, unknown, ΨΨ(x,y)(x,y) , rather than the two function u(x,y) and v(x,y)., rather than the two function u(x,y) and v(x,y).

0xyyx

0yv

xu 22

因為有之前的定義方式，才有『stream

function滿足twotwo--dimensional form of dimensional form of the incompressible continuity the incompressible continuity equationequation』這個結果

Ψ(x,y)?莫宰羊，雖然書裡頭講「寧可不知道一個stream function，也不要不知道兩個函數－u(x,y)與v(x,y)」，但說穿了還是與streamline、stream coordinate一樣，為了吹捧一個開創出來的名詞罷了！？？？

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Another advantage of using stream function is related to the facAnother advantage of using stream function is related to the fact that t that line along which line along which ΨΨ(x,y) =constant(x,y) =constant are streamlines.are streamlines.

How to prove ? From the definition of the streamline that the slHow to prove ? From the definition of the streamline that the slope ope at any point along a streamline is given byat any point along a streamline is given by

uv

dxdy

streamline

Velocity and velocity Velocity and velocity component along a streamlinecomponent along a streamline

Stream function另一個好處： ΨΨ(x,y(x,y) =constant) =constant 代表一條代表一條streamlinestreamline

如何證明ΨΨ(x,y(x,y) =constant) =constant代表streamlines？

依據依據streamlinestreamline的定義，的定義，沿著沿著streamlinestreamline的任意的任意點，其斜率為點，其斜率為v/uv/u。。

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The change of The change of ΨΨ(x,y)(x,y) as we move from one point (x,y) as we move from one point (x,y) to a nearly point (to a nearly point (x+dx,y+dyx+dx,y+dy) is given by) is given by

0udyvdx0d

udyvdxdyy

dxx

d

uv

dxdy

streamline

Along a line of constant Along a line of constant ΨΨ

This is the definition for a streamline. Thus, if we know the fuThis is the definition for a streamline. Thus, if we know the function nction ΨΨ(x,y) we can (x,y) we can plot lines of constant plot lines of constant ΨΨto provide the family of streamlines that are helpful in to provide the family of streamlines that are helpful in visualizing the pattern of flow. There are an infinite number ofvisualizing the pattern of flow. There are an infinite number of streamlines that make up streamlines that make up a particular flow field, since for each constant value assigned a particular flow field, since for each constant value assigned to to ΨΨa streamline can be a streamline can be drawn.drawn.

由一點移到另一點的過程中由一點移到另一點的過程中stream functionstream function的改變的改變ddΨΨ可以寫成可以寫成

只要ΨΨ(x,y(x,y))已知已知，，賦予不同的常數賦予不同的常數，，然後把然後把它畫出來它畫出來，，就會得到一條就會得到一條streamlinefamilyof streamlines讓flow可被看得到。

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The actual numerical value associated with a particular streamliThe actual numerical value associated with a particular streamline is ne is not of particular significance, but the change in the value of not of particular significance, but the change in the value of ΨΨ is is related to the volume rate of flow.related to the volume rate of flow.

For a unit depth, the flow rate across AB isFor a unit depth, the flow rate across AB is

For a unit depth, the flow rate across BC isFor a unit depth, the flow rate across BC is

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y

y

y

y

2

1

2

1

2

1

ddyy

udyq

12xx

xx

1

2

2

1

2

1ddx

xvdxq

賦予特定常數得到特定賦予特定常數得到特定streamline，本身沒有什麼物理意義，但不同常數所得到不同streamline間的差，則與volume rate of flow有關。

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Thus the volume flow rate between any two Thus the volume flow rate between any two streamlines can be written as the difference streamlines can be written as the difference between the constant values of between the constant values of ΨΨ defining defining two streamlines.two streamlines.

The velocity will be relatively high wherever The velocity will be relatively high wherever the streamlines are close together, and the streamlines are close together, and relatively low wherever the streamlines are relatively low wherever the streamlines are far apart.far apart.兩條streamline 間的Volume flow rate =代表這兩條streamline兩個constant values of constant values of ΨΨ的差。

streamlines間隔越緊密，則表示流體速度較高。

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Stream Function Stream Function Cylindrical Coordinate SystemCylindrical Coordinate System

For a For a twotwo--dimensional, incompressible flowdimensional, incompressible flow in the rin the rθθplane, conservation of mass can be written as:plane, conservation of mass can be written as:

The velocity components can be related to the stream The velocity components can be related to the stream function, function, ΨΨ(r,(r,θθ) through the equation) through the equation

rvand

r1vr

0vr

)rv( r

圓柱座標系統：Stream function與速度的關係

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Example 6.3 Stream FunctionExample 6.3 Stream Function

The velocity component in a steady, incompressible, two The velocity component in a steady, incompressible, two dimensional flow field aredimensional flow field are

Determine the corresponding stream function and show on a skDetermine the corresponding stream function and show on a sketch etch several streamlines. Indicate the direction of glow along the several streamlines. Indicate the direction of glow along the streamlines.streamlines.

4xv2yu

39


(y)fx2(x)fy 22

12

Cyx2 22

From the definition of the stream functionFrom the definition of the stream function

x4x

vy2y

u

For simplicity, we set C=0For simplicity, we set C=0

22 yx2 ΨΨ=0=0

ΨΨ≠≠0012/

xy 22

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Conservation of Linear MomentumConservation of Linear Momentum

Applying NewtonApplying Newton’’s second law to control volumes second law to control volume

amDt

VDm

zVw

yVv

xVu

tVm

tDmVDF

SYSDt

PDF

For a For a infinitesimal system of mass dminfinitesimal system of mass dm, what, what’’s the s the The The differential form of linear momentum equation?differential form of linear momentum equation?

VdVdmVP)system(V)system(Msystem

Newton第二定律

Based on SYSTEM METHODBased on SYSTEM METHOD

微分型式的線動量方程式？

直接把牛頓第二定律用到fluid element

41

Forces Acting on Element Forces Acting on Element 1/21/2

The forces acting on a fluid element may be classified as body forces and surface forces; surface forces include normal forces and normal forces and tangentialtangential (shear) forces.

kFjFiF

kFjFiF

FFF

bzbybx

szsysx

BS

Surface forces acting on a fluid Surface forces acting on a fluid element can be described in terms element can be described in terms of normal and shearing stresses.of normal and shearing stresses.

AFlim n

0tn

A

Flim 10t1

A

Flim 20t2

作用在極微小的fluid element

作用在某一平面上的Surface force可以分成

作用在fluid element上的surface force

42

Double Subscript Notation for StressesDouble Subscript Notation for Stresses

xy

The direction of the The direction of the normal to the plane normal to the plane on which the stress on which the stress actsacts

The direction of the stressThe direction of the stress

Stress作用的平面的法線方向

Stress本身所指的方向

作用在與x軸垂直的平面，且指向y

作用在某一平面上的stress

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Forces Acting on Element Forces Acting on Element 2/22/2

zyxgFzyxgFzyxgF

zyxzyx

F

zyxzyx

F

zyxzyx

F

zbz

yby

xbx

zzyzxzsz

zyyyxysy

zxyxxxsx

Equation of MotionEquation of Motion

參照材料力學

x方向的surface force

從fluid element的角度

44

Equation of MotionEquation of Motion

These are the differential equations of motion for anyThese are the differential equations of motion for anyfluid fluid satisfying the continuum assumptionsatisfying the continuum assumption..How to solve u,v,w ?How to solve u,v,w ?

zww

ywv

xwu

tw

zyxg

zvw

yvv

xvu

tv

zyxg

zuw

yuv

xuu

tu

zyxg

zzyzxzz

zyyyxyy

zxyxxxx

zzyyxx maFmaFmaF 微分型式的運動方程式

非線性方程式

簡化

注意：力平衡

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InviscidInviscid FlowFlow

Shearing stresses develop in a moving fluid because of the viscoShearing stresses develop in a moving fluid because of the viscosity sity of the fluid.of the fluid.

For some common fluid, such as air and water, the viscosity is sFor some common fluid, such as air and water, the viscosity is small, mall, and therefore it seems reasonable to assume that under some and therefore it seems reasonable to assume that under some circumstances we may be able to simply neglect the effect of circumstances we may be able to simply neglect the effect of viscosity.viscosity.

Flow fields in which the shearing stresses are assumed to be Flow fields in which the shearing stresses are assumed to be negligible are said to be negligible are said to be inviscidinviscid, , nonviscousnonviscous, or frictionless., or frictionless.

zzyyxxp Define the pressure, p, as the negative of the normal stressDefine the pressure, p, as the negative of the normal stress

Special case

忽略其中的shear stresses，只剩下normal stresses

Shearing stress來自流體的黏性

忽略黏度效應，假設該流體的shearing stress~0

各種說法

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EulerEuler’’s Equation of Motion s Equation of Motion

Under frictionless conditionfrictionless condition, the equations of motion are reduced to EulerEuler’’s Equation:s Equation:

zww

ywv

xwu

tw

zpg

zvw

yvv

xvu

tv

ypg

zuw

yuv

xuu

tu

xpg

z

y

x

V)V(tVpg

在沒有在沒有shear stressshear stress下下，運動方程式得以，運動方程式得以簡化，但仍然是非線簡化，但仍然是非線性方程式，稱之為性方程式，稱之為EulerEuler’’s equations equation

47

Bernoulli Equation Bernoulli Equation 1/31/3

EulerEuler’’s equation for s equation for steady flow steady flow along a streamline isalong a streamline is

V)V(pg

VVVV21VV

zgg

)VV()VV(2

pzg

Selecting the coordinate system with the zSelecting the coordinate system with the z--axis vertical so that axis vertical so that the acceleration of gravity vector can be expressed asthe acceleration of gravity vector can be expressed as

Vector identity Vector identity ……..

進一步假設：STEADY FLOW

設定座標系統：設定座標系統：zz軸垂直向上軸垂直向上

整理

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sdVVsdzgsdV21sdp 2

VV perpendicular to perpendicular to

V

VVzgV21p 2

sd

With With kdzjdyidxsd

dpdzzpdy

ypdx

xpsdp

在streamline的分量：力功

VV 既然與垂直，自然會與垂直，其結果當然為零。

sd

ALONG a streamline

49


0gdzVd21dp 2

ttanconsgz2

Vdp 2

ttanconsgz2

Vp 2

0sdzgsdV21sdp 2

Integrating Integrating ……

For For steadysteady, , inviscidinviscid, , incompressible incompressible fluid ( commonly called fluid ( commonly called ideal fluids) ideal fluids) along a streamlinealong a streamline

Bernoulli equationBernoulli equation

注意：從注意：從general equation of general equation of motion motion EulerEuler’’s equations equation加了那些條件？加了那些條件？

回憶CHAPTER 3

50

Bernoulli Equation Along a StreamlineBernoulli Equation Along a Streamline

For the special case of For the special case of incompressible flowincompressible flow

Restrictions : Steady flow.Restrictions : Steady flow.Incompressible flow.Incompressible flow.Frictionless flow.Frictionless flow.Flow along a streamline.Flow along a streamline.

ttanconsz2

Vp2

BERNOULLI EQUATIONBERNOULLI EQUATION

CgzV21dp 2

不可壓縮流體不可壓縮流體

一再提醒，每一個結論（推導出來的方程式），都有它背後假設一再提醒，每一個結論（推導出來的方程式），都有它背後假設條件，即一路走來，是基於這些假設才有如此結果。條件，即一路走來，是基於這些假設才有如此結果。

The Bernoulli equation is a very The Bernoulli equation is a very powerful tool in fluid mechanics, powerful tool in fluid mechanics, published by Daniel Bernoulli published by Daniel Bernoulli (1700~1782) in 1738.(1700~1782) in 1738.NONO 剪力剪力

51

IrrotationalIrrotational Flow Flow 1/51/5

Irrotation ? The irrotational condition is

In rectangular coordinates system

In cylindrical coordinates system

0V

0xw

zu

zv

yw

yu

xv

0Vr1

rrV

r1

rV

zV

zVV

r1 rzrz

如果流體為無旋性，則…

The vorticity is zero in an irrotational flow field.

52


A general flow field would not be A general flow field would not be irrotationalirrotational flow.flow.A special uniform flow field is an example of an A special uniform flow field is an example of an irrotationirrotation

flowflow

一般流場不會是無旋性

Special case

53


Flow fields involving Flow fields involving real fluids often include real fluids often include both regions of both regions of negligible shearing negligible shearing stressesstresses and regions of and regions of significant shearing significant shearing stressesstresses.. 流場可以區分成兩區塊…

Boundary layer

54


A general flow field A solid body is placed in a uniform stream of fluid. Far away frA solid body is placed in a uniform stream of fluid. Far away from om

the body remain uniform, and in this far region the flow is the body remain uniform, and in this far region the flow is irrotationalirrotational. .

The flow around the body remains The flow around the body remains irrotationalirrotational except very near the except very near the boundary.boundary.

Near the boundary the Near the boundary the velocity changes rapidly from velocity changes rapidly from zero at the boundary (nozero at the boundary (no--slip slip condition) to some relatively condition) to some relatively large value in a short distance large value in a short distance from the boundary.from the boundary.

只有遠離solid body的區域尚可假設為無旋性

將solid body放在一均勻流

靠近solid body部分，在短距離內速度由零上升

Boundary layerBoundary layer

55


A general flow field Flow from a large reservoir enters a pipe through a streamlined Flow from a large reservoir enters a pipe through a streamlined

entrance where the velocity distribution is essentially uniform.entrance where the velocity distribution is essentially uniform. Thus, Thus, at entrance the flow is at entrance the flow is irrotationalirrotational. (b). (b)

In the central core of the pipe the flow remains In the central core of the pipe the flow remains irrotationalirrotational for some for some distance.distance.

The The boundary layerboundary layer will develop along the wall and grow in will develop along the wall and grow in thickness until it fills the pipe.thickness until it fills the pipe. Viscous forces are dominantViscous forces are dominant

Chapter 8Chapter 8

流體從一個大型貯存器流入管子

還可以算是均勻流

慢慢發展開來直到充滿整個pipe尚可稱為irrotational flow

56

Bernoulli Equation for Bernoulli Equation for IrrotationalIrrotational Flow Flow 1/31/3

The Bernoulli equation forThe Bernoulli equation for steady, incompressible, and steady, incompressible, and inviscidinviscidflowflow isis

The equation can be applied betweenThe equation can be applied between any two points on the same any two points on the same streamlinestreamline. . In general,In general, the value of the constant will vary from the value of the constant will vary from streamline to streamlinestreamline to streamline..

Under additionalUnder additional irrotationalirrotational conditioncondition, , the Bernoulli equation ?the Bernoulli equation ?Starting with EulerStarting with Euler’’s equation in vector forms equation in vector form

VVVV21kgp1V)V(

ttanconsgz2

Vp 2

ZERO Regardless of the direction of ZERO Regardless of the direction of dsds

再加上irrotational condition

無庸沿streamline即為零

57


With irrotaionalirrotaional condition

kgp1V21VV

21 2

0V

rd

0gdzVd21dpgdzdpVd

21

rdkgrdp1rdV21

22

2

VVVV21kgp1V)V(

無庸沿著streamline

任意方向都可以任意方向都可以

力力功功

58


Integrating for incompressible flowIntegrating for incompressible flow

This equation is valid between any two points in a steady, This equation is valid between any two points in a steady, incompressible, incompressible, inviscidinviscid, and , and irrotationalirrotational flow.flow.

ttanconsgz2

Vp 2

ttancongz

2Vdp 2

2

222

1

211 z

g2Vpz

g2Vp

適用於任意兩點

關鍵：irrotational flow

59

Velocity Potential Velocity Potential ΦΦ((x,y,z,t)x,y,z,t) 1/41/4

The stream function for twoThe stream function for two--dimensional dimensional incompressibleincompressibleflow isflow is ΨΨ(x,y) which satisfies the continuous conditions(x,y) which satisfies the continuous conditions

For an For an irrotationalirrotational flowflow, the velocity components can be , the velocity components can be expressed in terms of a scalar function expressed in terms of a scalar function ψψ((x,y,z,tx,y,z,t)) asas

where where ψψ((x,y,z,tx,y,z,t)) is called the velocity potential.is called the velocity potential.

zw

yv

xu

Stream functionStream function：：2D2D＋＋continuous conditioncontinuous conditionVelocity potentialVelocity potential：：3D3D＋＋irrotationalirrotational conditioncondition

就如此定義velocity potential與速度的關係為了滿足irrotationalcondition

回憶：streamfunction與連續方程式

60


In vector formIn vector form

For an incompressible flow For an incompressible flow V

LaplacianLaplacian operatoroperator

0V

0zyx

VV 22

2

2

2

22

For incompressible, For incompressible, irrotationalirrotational flowflow

Laplace’s equation

called a potential flowcalled a potential flow

加上『不可壓縮』，則velocity potential滿足Laplace’sequation

61


InviscidInviscid, incompressible, , incompressible, irrotationalirrotational fields are governed fields are governed by Laplaceby Laplace’’s equation.s equation.

This type flow is commonly called This type flow is commonly called a potential flowa potential flow..To complete the mathematical formulation of a given To complete the mathematical formulation of a given

problem, boundary conditions have to be specified. These problem, boundary conditions have to be specified. These are usually velocities specified on the boundaries of the are usually velocities specified on the boundaries of the flow field of interest.flow field of interest.

注意『條件』

可用Laplace’s equation描述的flow，稱為potential flow

在使用Laplace’s equation描述potential flow之餘，邊界條件也要講清楚，才有得解

62


In In cylindrical coordinatecylindrical coordinate, r, , r, θθ, and z, and z

zv

r1v

rv zr

0zr

1r

rrr

12

2

2

2

2

63

LaplaceLaplace’’ss equation and Potential flowequation and Potential flow

InviscidInviscid, incompressible, , incompressible, irrotationalirrotational fields are fields are governed by governed by LaplaceLaplace’’ssequation.equation.This type flow is commonly This type flow is commonly

called a potential flow.called a potential flow.

rotational flow

64

Potential Flow TheoryPotential Flow Theory

Velocity potential ψψ ((x,y,z,tx,y,z,t)) exists only for irrotationalflow.

Irrotationality may be a valid assumption for those regions of a flow in which viscous forces are negligible.

In an irrotational flow, the velocity field may be defined by the potential function, ψψ((x,y,z,tx,y,z,t)), the theory is often referred to as potential flow theory.關於potential flow 的theory 稱為potential flow theory

僅存在irrotational flow

Irrotationality在viscous forces可以忽略的區域是可以接受的假設

65

LaplaceLaplace’’s Equation s Equation 1/21/2

For two-dimensional, incompressible flow

For two-dimensional, irrotational flow

)1(x

vy

u

(1)(1) + + irrotationalirrotational condition condition ……

(2) + continuity equation (2) + continuity equation ……

0yx

0xv

yu

2

2

2

2

)2(y

vx

u

0yx

0yv

xu

2

2

2

2

66

LaplaceLaplace’’s Equation s Equation 2/22/2

For a two-dimensional incompressible flow, we can define a stream function Ψ; if the flow is also irrotational, Ψwill satisfy Laplace’s equation.

For an irrotational flow, we can define a velocity potential Φ; if the flow is also incompressible, Φ will satisfy Laplace’s equation.

Any function ψψ or Ψ that satisfies Laplace’s equation represents a possible two-dimensional, incompressible, irrotational flow field. 當stream function與velocity potential都滿

足Laplace’s equation時，所代表的意義(1) two-dimensional, (2) incompressible,(3) irrotational flow field。

67

Example 6.4 Velocity Potential and Example 6.4 Velocity Potential and InviscidInviscid Flow Pressure Flow Pressure 1/21/2

The twoThe two--dimensional flow of a dimensional flow of a nonviscousnonviscous, incompressible fluid in , incompressible fluid in the vicinity of the 90the vicinity of the 90°° corner of Figure E6.4a is described by the corner of Figure E6.4a is described by the stream functionstream function

Where Where ΨΨ has units of mhas units of m22/s when r is in meters. (a) Determine, if /s when r is in meters. (a) Determine, if possible, the corresponding velocity potential. (b) If the presspossible, the corresponding velocity potential. (b) If the pressure at ure at point (1) on the wall is 30 point (1) on the wall is 30 kPakPa, what is the pressure at point (2)? , what is the pressure at point (2)? Assume the fluid density is 10Assume the fluid density is 1033 kg/mkg/m33 and the xand the x--y plane is y plane is horizontal horizontal –– that is, there is no difference in elevation between that is, there is no difference in elevation between points (1) and (2).points (1) and (2).

2sinr2 2

68

Example 6.4 Velocity Potential and Example 6.4 Velocity Potential and InviscidInviscid Flow Pressure Flow Pressure 2/22/2

69

Example 6.4 Example 6.4 SolutionSolution1/21/2

2sinr4rv

2cosr4r1vr

)(f2cosr22cosr4r

v 12

r

)r(f2cosr22sinr4r1v 2

2

C2cosr2 2 Let C=0

2cosr2 2

70

Example 6.4 Example 6.4 SolutionSolution2/22/2Bernoulli equation between points (1) and (2) with no elevation change

)VV(2

ppg2

Vpg2

Vp 22

2112

222

211

2222

2221

22r

2

s/m4...V

s/m16...V

vvV

kPa36...p2

xy4sincosr42cosr2 22

71

Some Basic, Plane Potential FlowSome Basic, Plane Potential Flow

Since LaplaceSince Laplace’’s equation is linear, various solutions can be added to s equation is linear, various solutions can be added to obtain other solution obtain other solution –– that is , if that is , if ψψ11(x,y,z)(x,y,z) and and ψψ22(x,y,z)(x,y,z) are two are two solutions to Laplacesolutions to Laplace’’s equation, then s equation, then ψψ== ψψ11 + + ψψ22 is also solution.is also solution.

The practical implication of this result is that if we have certThe practical implication of this result is that if we have certain ain basic solution we can combine them to obtain more complicated anbasic solution we can combine them to obtain more complicated and d interesting solutions.interesting solutions.

Several basic velocity potentials, which describe some relativelSeveral basic velocity potentials, which describe some relatively y simple flows, will be determined.simple flows, will be determined.

These basic velocity potential will be combined to represent These basic velocity potential will be combined to represent complicate flows.complicate flows.

對象： (1) two-dimensional, (2) incompressible,(3) irrotational flow field。

Laplace’s equation是Linear equation，個別solutions加起來也是該Laplace’s equation的solutions

Basic flow＋basic flow Complicated flow

描述basic flow的stream function？Velocity potential？

72

ΦΦ, , ΨΨ , u, v, , u, v, vvrr, v, vθθ

r1vand

rv

rvand

r1v

r

r

yvand

xu

xvand

yu

73

ΦΦ and and ΨΨ 1/21/2

ForFor ΨΨ=constant=constant, , ddΨΨ =0=0 andand

The slope of a streamline The slope of a streamline –– a line of constanta line of constant ΨΨ

Along a line of constantAlong a line of constant ψψ, , ddψψ =0=0 andand

The slope of a potential line The slope of a potential line –– a line of constanta line of constant ψψ

0dyy

dxx

d

uv

y/x/

dxdy

0dyy

dxx

d

vu

y/x/

dxdy

Line of constant Ψ and constant ψψ are orthogonal.

相互正交的線

74

ΦΦ and and ΨΨ 2/22/2

Flow net for 90° bend

The lines of constant The lines of constant ψψ (called (called equipotentialequipotential lines) are orthogonal to lines lines) are orthogonal to lines of constant of constant ΨΨ (streamlines) at all points (streamlines) at all points where they intersect..where they intersect..

For any For any potential flowpotential flow a a ““flow netflow net”” can can be drawn that consists of a be drawn that consists of a family of family of streamlines and streamlines and equipotentialequipotential lineslines. .

Velocities can be estimated from the flow Velocities can be estimated from the flow net, since the velocity is inversely net, since the velocity is inversely proportional to the streamline spacingproportional to the streamline spacing速度與streamline間隔成『反比』

Equipotential lines + streamline

75

Uniform Flow Uniform Flow 1/21/2

A uniform flow is a simplest plane flow for which the A uniform flow is a simplest plane flow for which the streamlines are all straight and parallel, and the magnitude streamlines are all straight and parallel, and the magnitude of the velocity is constant.of the velocity is constant.

CUx0y

,Ux

u=U and v=0u=U and v=0

Uy0x

,Uy

一種我們稱為Basic flow的flow

描述uniform flow的stream function與velocity potential

76

Uniform Flow Uniform Flow 2/22/2

For a uniform flow of constant velocity V, inclined to an For a uniform flow of constant velocity V, inclined to an angle angle αα to the xto the x--axis.axis.

x)cosV(y)sinV(x)sinV(y)cosV(

77

Source and Sink Source and Sink 1/21/2

For a For a sourcesource flow ( from origin flow ( from origin radiallyradially) with volume flow ) with volume flow rate per unit depth m rate per unit depth m (m=2(m=2 r r vvrr ))

rln2m

2m0v

r2mv r

r1vand

rv

rvand

r1v

r

r

78

Source and Sink Source and Sink 2/22/2

For a For a sinksink flow (toward origin flow (toward origin radiallyradially) with volume flow ) with volume flow rate per unit depth rate per unit depth mm

rln2m

2m0v

r2mvr

r1vand

rv

rvand

r1v

r

r

79

Example 6.5 Potential Flow Example 6.5 Potential Flow -- SinkSink

A A nonviscousnonviscous, incompressible fluid flows between wedge, incompressible fluid flows between wedge--shaped shaped walls into a small opening as shown in Figure E6.5. The velocitywalls into a small opening as shown in Figure E6.5. The velocitypotential (in ftpotential (in ft22/s), which approximately describes this flow is/s), which approximately describes this flow is

Determine the volume rate Determine the volume rate of flow (per unit length) into of flow (per unit length) into the opening.the opening.

rln2

80


The components of velocity The components of velocity

0r1v

r2

rvr

s/ft05.13

...Rdvq 26/

0r

The The flowrateflowrate per unit widthper unit width

81

VortexVortex

A vortex represents a flow in which the streamlines are A vortex represents a flow in which the streamlines are concentric circles.concentric circles.

Vortex motion can be either rotational or Vortex motion can be either rotational or irrotationalirrotational..For an For an irrotationalirrotational vortex (vortex (ccwccw, center at origin) with , center at origin) with

vortex strength vortex strength KK

KrlnKrKv0v r

At r=0, the velocity At r=0, the velocity becomes infinite.becomes infinite.

singularity

Streamlines是同心圓

Vortex motion可以是旋性或非旋性

82

Free Vortex Free Vortex 1/21/2

Free (Free (IrrotationalIrrotational) vortex (a) is that rotation refers to the ) vortex (a) is that rotation refers to the orientation of a fluid element and not the path followed by orientation of a fluid element and not the path followed by the element.the element.

A pair of small sticks were A pair of small sticks were placed in the flow field at placed in the flow field at location A, the sticks would location A, the sticks would rotate as they as they move to rotate as they as they move to location B. location B.

Free vortex：是fluid element在旋轉。

水缸底部排水口所形成的渦流

把一對棒狀物放到流場中，該棒狀物由AB，出現旋轉。

83

Free Vortex Free Vortex 2/22/2

One of the sticks, the one that is aligned the streamline, One of the sticks, the one that is aligned the streamline, would follow a circular path and rotate in a would follow a circular path and rotate in a counterclockwise directioncounterclockwise direction

The other rotates in a clockwise The other rotates in a clockwise direction due to the nature of the direction due to the nature of the flow field flow field –– that is, the part of the that is, the part of the stick nearest the origin moves stick nearest the origin moves faster than the opposite end.faster than the opposite end.

The The average velocity of the average velocity of the two sticks is zero.two sticks is zero.

該對棒狀物中的一根，順著streamline，CCW旋轉；另一根則因靠近圓心的一端走得快，另一端走得慢，而CW旋轉。

84

Forced VortexForced Vortex

If the flow were rotating as a If the flow were rotating as a rigid body, such that rigid body, such that vvθθ=K=K11r r where Kwhere K11 is a constant.is a constant.

Force vortex is rotational and Force vortex is rotational and cannot be described with a cannot be described with a velocity potential.velocity potential.

Force vortex is commonly Force vortex is commonly called a rotational vortex.called a rotational vortex.

旋性，流體像一個剛體在旋轉，無法以velocity potential來描述

稱為rotational vortex

離心幫浦的旋轉葉面上所造成的渦流

85

Combined VortexCombined Vortex

A combine vortex is one with a forced vortex as a central A combine vortex is one with a forced vortex as a central core and a velocity distribution corresponding to that of a core and a velocity distribution corresponding to that of a free vortex outside the core.free vortex outside the core.

0

0

rrrv

rrrKv

Core內部為forced vortex，外部為free vortex

Vortex in a beaker

86

Circulation Circulation 1/31/3

Circulation Circulation ΓΓ is defined as theis defined as the line integral of the line integral of the tangential velocity component about any closed curvetangential velocity component about any closed curvefixed in the flow:fixed in the flow:

A ZA Zc dA)V(dA2sdV

sdwhere the is an element vector tangent to the curve where the is an element vector tangent to the curve

and having length and having length dsds of the element of arc. Itof the element of arc. It’’s positive s positive corresponds to a c.c.w. direction of integration around the corresponds to a c.c.w. direction of integration around the curve.curve.

把tangential velocity componenttangential velocity component沿著closed curve線積分

CCW：”+”

87


For For irrotationalirrotational flow , flow , ΓΓ =0=0

0dsddA)V(sdVccAc

For For irrotationalirrotational flow , flow , ΓΓ =0=0

The circulation around any path that does not include the singular point at the origin will be zero.

The circulation around any path that does not include the singular point at the origin will be zero.

Closed curve不圍住single point

88


For free vortexFor free vortex

The circulation around any path that encloses singularities will be nozero.The circulation around any path that encloses singularities will be nozero.

/2K

K2)rd(rK2

0

2

rln2

rKv

Free vortex之圓心為singularity

89

Example 6.6 Potential Flow Example 6.6 Potential Flow –– Free VortexFree Vortex

A liquid drains from a large tank through a small opening as A liquid drains from a large tank through a small opening as illustrated in Figure E6.6. A vortex forms whose velocity illustrated in Figure E6.6. A vortex forms whose velocity distribution away from the tank opening can be approximated as tdistribution away from the tank opening can be approximated as that hat of a free vortex having a velocity potentialof a free vortex having a velocity potential

2

Determine an expression Determine an expression relating the surface shape relating the surface shape to the strength of the to the strength of the vortex as specified by the vortex as specified by the circulation circulation ΓΓ..

90


0pp 21

Since the free vortex represents an Since the free vortex represents an irrotationalirrotational flow field, the flow field, the Bernoulli equationBernoulli equation

2

222

1

211 z

g2Vpz

g2Vp

At free surface

s

22

21 z

g2V

g2V

r2r1v

Far from the origin at point (1), V1=vθ=0

gr2z 22

2

s

91

Doublet Doublet 1/21/2

For a doublet ( produced mathematically by allowing a For a doublet ( produced mathematically by allowing a source and a sink of numerically equal strength to merge) source and a sink of numerically equal strength to merge) with a strength mwith a strength m

212m

The combined stream function for the pair isThe combined stream function for the pair is

21

2121 tantan1

tantantanm

2tan

22

122

21

arsinar2tan

2m

arsinar2

m2tan

acosrsinrtanand

acosrsinrtan

92

Doublet Doublet 2/22/2

2222 arsinmar

arsinar2

2m

aa→→0 0

The soThe so--called doublet is formed by letting acalled doublet is formed by letting a→→00，，mm→∞→∞

r1

arr

22

rsinK

maK

rcosK

93

Streamlines for a DoubletStreamlines for a Doublet

Plots of lines of constant Plots of lines of constant ΨΨ reveal that the streamlines for reveal that the streamlines for a doublet are circles through the origin tangent to the x a doublet are circles through the origin tangent to the x axis.axis.

95

Superposition of Elementary Plane Superposition of Elementary Plane Flows Flows 1/21/2Potential flows are governed by Laplace’s equation, which

is a linear partial differential equation.Various basic velocity potentials and stream function, ψψ

and Ψ, can be combined to form new potentials and stream functions.

213213

把basic flow加疊起來所得到的complicated flow，可以由個別basic flow的stream function與velocity potential加起來所得到的stream function與velocity potential來描述它。

96

Superposition of Elementary Plane Superposition of Elementary Plane Flows Flows 2/22/2Any streamline in an inviscid flow field can be considered

as a solid boundary, since the conditions along a solid boundary and as streamline are the same – that is, there is no flow through the boundary or the streamline.

We can combine some of the basic stream functions to yield a streamline that corresponds to a particular body shape of interest, that combination can be used to describe in detail the flow around that body.

Methods of superpositionMethods of superposition

在inviscid flow field中streamline可以看成是solid boundary

97

HalfHalf--Body : Uniform Stream + Source Body : Uniform Stream + Source 1/41/4

rln2mcosUr

2msinUr

sourceflowuniform

sourceflowuniform

sinUrv

r2mcosU

r1vr

The stagnation point occurs at x = The stagnation point occurs at x = --bb

U2mb0

b2mUvr

The combination of a uniform flow and a The combination of a uniform flow and a source can be used to describe the flow source can be used to describe the flow around a streamlined body placed in a around a streamlined body placed in a uniform stream.uniform stream.

98


2m

stagnation

sin)(br

bUsinUrbU

bU2m

The value of the stream function at the stagnation point can be The value of the stream function at the stagnation point can be obtained by evaluating obtained by evaluating ΨΨ at r=b at r=b θθ==ππ

The equation of the streamline passing through the stagnation The equation of the streamline passing through the stagnation point ispoint is

The streamline can be replaced by a solid boundary. The streamline can be replaced by a solid boundary. The body is open at the downstream end, and thus is The body is open at the downstream end, and thus is called a HALFcalled a HALF--BODY. The combination of a uniform BODY. The combination of a uniform flow and a source can be used to describe the flow flow and a source can be used to describe the flow around a streamlined body placed in a uniform stream.around a streamlined body placed in a uniform stream.

99


)(bysin

)(br

2

22

2222

r2

rbcos

rb21U

r2m

rcosUmUvvV

As As θθ0 or 0 or θθ= = ππ the halfthe half--width approaches width approaches ±±bbππ.. The width The width of the halfof the half--body asymptotically approach 2body asymptotically approach 2ππb.b.The velocity components at any pointThe velocity components at any point

sinUrv

r2mcosU

r1vr

U2mb

100


With the velocity known, the pressure at any point can be determined from the Bernoulli equation

Where elevation change have been neglected.

220 V2

1pU21p

Far from the bodyFar from the body

101

Example 6.7 Potential Flow Example 6.7 Potential Flow –– HalfHalf--BodyBody

The shape of a hill arising from a plain can be approximated witThe shape of a hill arising from a plain can be approximated with h the top section of a halfthe top section of a half--body as is illustrated in Figure E6.7a. The body as is illustrated in Figure E6.7a. The height of the hill approaches 200 ft as shown. (a) When a 40 mi/height of the hill approaches 200 ft as shown. (a) When a 40 mi/hr hr wind blows toward the hill, what is the magnitude of the air velwind blows toward the hill, what is the magnitude of the air velocity ocity at a point on the hill directly above the origin [point (2)]? (bat a point on the hill directly above the origin [point (2)]? (b) What ) What is the elevation of point (2) above the plain and what is the is the elevation of point (2) above the plain and what is the difference in pressure between point (1) on the plain far from tdifference in pressure between point (1) on the plain far from the hill he hill and point (2)? Assume an air density of 0.00238 slugs/ftand point (2)? Assume an air density of 0.00238 slugs/ft33??

102


2

222

rbcos

rb21UV

The velocity isThe velocity is

At point (2), At point (2), θθ==ππ/2/2

2b

sin)(br

hr/mi4.47...41U)2/b(

b1UV 22

2

222

The elevation at (2) above the plain isThe elevation at (2) above the plain is ft1002

ft2002by2

103


From the Bernoulli equationFrom the Bernoulli equation

psi0647.0ft/lb31.9...)yy()VV(2

pp

zg2

Vpzg2

Vp

212

21

2221

2

222

1

211

s/ft5.69hr/s3600mi/ft5280)hr/mi4.47(V

s/ft7.58hr/s3600mi/ft5280)hr/mi40(V

2

1

104

RankineRankine Oval: Uniform Stream + Doublet Oval: Uniform Stream + Doublet 1/41/4

21

21

rlnrln2mcosUr

2msinUr

2221

221

ayxay2tan

2mUy

arsinar2tan

2msinUr

The corresponding streamlines for this flow field are obtained bThe corresponding streamlines for this flow field are obtained by y setting setting ΨΨ=constant. It is discovered that the streamline forms a =constant. It is discovered that the streamline forms a closed body of length 2closed body of length 2 and width 2h.and width 2h. RankineRankine ovalsovals

105


2/12/12 1

Uam

aa

Uma

The stagnation points occur at the upstream and downstream ends The stagnation points occur at the upstream and downstream ends of of the body. These points can be located by determining where alongthe body. These points can be located by determining where alongthe x axis the velocity is zero.the x axis the velocity is zero.

The stagnation points correspond to the points where the uniformThe stagnation points correspond to the points where the uniformvelocity, the source velocity, and the sink velocity all combinevelocity, the source velocity, and the sink velocity all combine to to give a zero velocity. give a zero velocity.

The locations of the stagnation points depend on the value of a,The locations of the stagnation points depend on the value of a, m, m, and U.and U.

The body halfThe body half--length 2length 2 DimensionlessDimensionless

106


ah

mUa2tan1

ah

21

ah

mUh2tan

a2ahh

222

The body halfThe body half--width, h, can be obtained by determining the value of width, h, can be obtained by determining the value of y where the y axis intersects the y where the y axis intersects the ΨΨ=0 streamline. Thus, with =0 streamline. Thus, with ΨΨ=0, =0, x=0, and y=h.x=0, and y=h.

The body halfThe body half--width 2hwidth 2h

DimensionlessDimensionless

107


108

Flow around a Circular Cylinder Flow around a Circular Cylinder 1/41/4

rcosKcosUr

rsinKsinUr

When the distance between the sourceWhen the distance between the source--sink pair approaches zero, the sink pair approaches zero, the shape of the shape of the rankinerankine oval becomes more blunt and in fact oval becomes more blunt and in fact approaches a circular shape. approaches a circular shape.

cosra1Ur

sinra1Ur

2

2

2

2

Ψ=constant for r=aΨ=0 for r=a K=Ua2

In order for the stream function to represent flow around a circular cylinder

109


The velocity componentsThe velocity components

sinra1U

rr1v

cosra1U

r1

rv

2

2

2

2

r

On the surface of the cylinder (r=a)

sinU2v0vr

The pressure distribution on the cylinder surface can be obtaineThe pressure distribution on the cylinder surface can be obtained d from the Bernoulli equationfrom the Bernoulli equation

22ss

20 v2

1pU21p

Far from the bodyFar from the body

)sin41(U21pp 220s

sinU2v

110


On the upstream part of the On the upstream part of the cylinder, there is approximate cylinder, there is approximate agreement between the potential agreement between the potential flow and the experimental results. flow and the experimental results. Because of the viscous boundary Because of the viscous boundary layer that develops on the cylinder, layer that develops on the cylinder, the main flow separates from the the main flow separates from the surface of the cylinder, leading to surface of the cylinder, leading to the large difference between the the large difference between the theoretical, frictionless solution theoretical, frictionless solution and the experimental results on the and the experimental results on the downstream side of the cylinder.downstream side of the cylinder.

111


2

0sy

2

0sx

adsinpF

adcospF

112

Flow around a Flow around a ……

Circular cylinderCircular cylinder

Ellipse

Circular cylinderWith separation

Potential and Potential and viscous flowviscous flow

Potential flowPotential flow

113

Flow around a Circular Cylinder + Free Flow around a Circular Cylinder + Free Vortex Vortex 1/41/4

2cos

ra1Urrln

2sin

ra1Ur 2

2

2

2

Adding a free vortex to the stream function or velocity Adding a free vortex to the stream function or velocity potential for the flow around a cylinder. potential for the flow around a cylinder.

The circle rThe circle r==a will still be a streamline, since the a will still be a streamline, since the streamlines for the added free vortex are all circular.streamlines for the added free vortex are all circular.

a2sinU2

rv

ar

The tangential velocity The tangential velocity on the surface of the on the surface of the cylindercylinder

114


This type of flow could be approximately created by This type of flow could be approximately created by placing a rotating cylinder in a uniform stream. placing a rotating cylinder in a uniform stream.

Because of the presence of viscosity in any real fluid, the Because of the presence of viscosity in any real fluid, the fluid in contacting with the rotating cylinder would rotate fluid in contacting with the rotating cylinder would rotate with the same velocity as the cylinder, and the resulting with the same velocity as the cylinder, and the resulting flow field would resemble that developed by the flow field would resemble that developed by the combination of a uniform flow past a cylinder and a free combination of a uniform flow past a cylinder and a free vortex.vortex.

115


A variety of streamline patterns A variety of streamline patterns can be developed, depending can be developed, depending on the vortex strength on the vortex strength ГГ..

The location of stagnation The location of stagnation points on a circular cylinder (a) points on a circular cylinder (a) without circulation; (b, c, d) without circulation; (b, c, d) with circulation.with circulation.

Ua4sin stag

stag0v

116


For the cylinder with circulation, the surface pressure, For the cylinder with circulation, the surface pressure, ppss, is , is obtained from the Bernoulli equationobtained from the Bernoulli equation

222

222

0s

2

s2

0

Ua4aUsin2sin41U

21pp

a2sinU2

21pU

21p

UTadsinpF

0adcospF2

0 sy

2

0 sx

DragDrag

LiftLift

117

Example 6.8 Potential Flow Example 6.8 Potential Flow –– Cylinder Cylinder 1/21/2

When a circular cylinder is placed in a uniform stream, a stagnaWhen a circular cylinder is placed in a uniform stream, a stagnation tion point is created on the cylinder as is shown in Figure E6.8a. Ifpoint is created on the cylinder as is shown in Figure E6.8a. If a a small hole is located at this point, the stagnation pressure, small hole is located at this point, the stagnation pressure, ppstagstag, can , can be measured and used to determine the approach velocity, U. (a) be measured and used to determine the approach velocity, U. (a) Show how Show how ppstagstag and U are related. (b) If the cylinder is misaligned and U are related. (b) If the cylinder is misaligned by an angle by an angle αα (Figure E6.8b), but the measured pressure still (Figure E6.8b), but the measured pressure still interpreted as the stagnation pressure, determine an expression interpreted as the stagnation pressure, determine an expression for for the ratio of the true velocity, U, to the predicted velocity, Uthe ratio of the true velocity, U, to the predicted velocity, U’’. Plot . Plot this ratio as a function of this ratio as a function of αα for the range for the range --20 20 °≦α≦°≦α≦2020°°..

118

Example 6.8 Potential Flow Example 6.8 Potential Flow –– Cylinder Cylinder 2/22/2

119


The Bernoulli equation between a point on the stagnation streamlThe Bernoulli equation between a point on the stagnation streamline ine upstream from the cylinder and the stagnation pointupstream from the cylinder and the stagnation point

2/1

0stag

stag2

0

)pp(2U

pg2

Up

If the cylinder is misaligned by an angle, If the cylinder is misaligned by an angle, αα, the pressure actually , the pressure actually measured, pmeasured, paa, will be different from the stagnation pressure., will be different from the stagnation pressure.

The difference between the pressure at the The difference between the pressure at the stagnation point and the upstream pressurestagnation point and the upstream pressure

1/2

0a

0stag2/1

0a pppp

)(predictedU'U(true))pp(2'U

120


On the surface of the cylinder (r=a)On the surface of the cylinder (r=a) sinU2v

2a

20 )sinU2(2

1pU21p

2/12

20stag

220a

)sin41('U

U

U21pp

)sin41(U21pp

The Bernoulli equation between a point upstream if the cylinder The Bernoulli equation between a point upstream if the cylinder and and the point on the cylinder where r=a, the point on the cylinder where r=a, θθ==αα. .

121

Viscous FlowViscous Flow

To incorporate viscous effects into the differential analysis of fluid motion

StressStress--Deformation Relationship Deformation Relationship

zww

ywv

xwu

tw

zyxg

zvw

yvv

xvu

tv

zyxg

zuw

yuv

xuu

tu

zyxg

zzyzxzz

zyyyxyy

zxyxxxx

General equation of motionGeneral equation of motion再將viscous effect納入

122

StressStress--Deformation Relationship Deformation Relationship 1/21/2

The stresses must be The stresses must be expressed in terms of the expressed in terms of the velocity and pressure velocity and pressure field.field.

zv

yw

zu

xw

yu

xv

zw2V

32p

yv2V

32p

xu2V

32p

zyyz

zxxz

yxxy

zz

yy

xx

Cartesian coordinates

123

StressStress--Deformation Relationship Deformation Relationship 2/22/2

rv

zv

vr1

zv

vr1

rv

rr

zv2p

rvv

r12p

rv2p

zrzrrz

zzz

rrr

zzz

r

rrr

Introduced into the differentialIntroduced into the differentialequation of motionequation of motion……..

Cylindrical polar coordinates

代回general equation of motion

124

The The NavierNavier--Stokes Equations Stokes Equations 1/51/5

These obtained equations of motion are called the These obtained equations of motion are called the NavierNavier--Stokes Equations.Stokes Equations.

V32

zw2

zyw

zv

yzxu

xw

xzpg

DtDw

yw

zv

zV

32

yv2

yxv

yu

xypg

DtDv

zu

xw

zxv

yu

yV

32

xu2

xxpg

DtDu

z

y

x

Cartesian coordinatesCartesian coordinates非常有名的Navier-Stokes equation

125

2z

2

2z

2

2z

z

zz

zzr

z

2

2r

22

2

22

zr

r

2r

2

22r

2

22rr

r

rz

2rr

rr

zvv

r1

rvr

rr1g

zP

zvvv

rv

rvv

tv

zvv

r2v

r1

rv

rvr

rr1gp

r1

zvv

rvvv

rv

rvv

tv

zvv

r2v

r1

rv

rvr

rr1g

rp

zvv

rvv

rv

rvv

tv


Cylindrical polar coordinatesCylindrical polar coordinates

126


UnderUnder incompressible flow with constant viscosity incompressible flow with constant viscosity conditionsconditions, , the the NavierNavier--Stokes equations are reduced to:Stokes equations are reduced to:

2

2

2

2

2

2

z

2

2

2

2

2

2

y

2

2

2

2

2

2

x

zw

yw

xwg

zp

zww

ywv

xwu

tw

zv

yv

xvg

yp

zvw

yvv

xvu

tv

zu

yu

xug

xp

zuw

yuv

xuu

tu

再來一步步透過假設，簡化Navier-Stokes equations

假設不可壓縮且黏度是constant

127


UndeUnder r frictionless conditionfrictionless condition, , the equations of motion are the equations of motion are reduced toreduced to EulerEuler’’s Equations Equation::

z

y

x

gzp

zww

ywv

xwu

tw

gyp

zvw

yvv

xvu

tv

gxp

zuw

yuv

xuu

tu

pgDt

VD

假設沒有摩擦Euler’s equation

128


The The NavierNavier--Stokes equations apply to both laminar and Stokes equations apply to both laminar and turbulent flow, but for turbulent flow each velocity turbulent flow, but for turbulent flow each velocity component fluctuates randomly with respect to time and component fluctuates randomly with respect to time and this added complication makes an analytical solution this added complication makes an analytical solution intractable.intractable.

The exact solutions referred to are for laminar flows in The exact solutions referred to are for laminar flows in which the velocity is either independent of time (steady which the velocity is either independent of time (steady flow) or dependent on time (unsteady flow) in a wellflow) or dependent on time (unsteady flow) in a well--defined manner.defined manner.

Navier-Stokes equation適用於Laminar and turbulent flow

若是turbulent flow…解析上非常棘手

129

Some Simple Solutions for Viscous, Some Simple Solutions for Viscous, Incompressible FluidsIncompressible FluidsA principal difficulty in solving the A principal difficulty in solving the NavierNavier--Stokes Stokes

equations is because of their nonlinearity arising from the equations is because of their nonlinearity arising from the convective acceleration termsconvective acceleration terms..

There are no general analytical schemes for solving There are no general analytical schemes for solving nonlinear partial differential equations.nonlinear partial differential equations.

There are a few special cases for which the convective There are a few special cases for which the convective acceleration vanishes. In these cases exact solution are acceleration vanishes. In these cases exact solution are often possible.often possible.

以Navier-Stokes equations（非線性，無general analytical schemesgeneral analytical schemes） special case Exact solution可能性提高

130

Steady, Laminar Flow between Fixed Steady, Laminar Flow between Fixed Parallel Plates Parallel Plates 1/61/6

Consider flow between the two horizontal, infinite Consider flow between the two horizontal, infinite parallel plate.parallel plate.For this geometry the fluid particle move in the x For this geometry the fluid particle move in the x

direction parallel to the pates, and there is no velocity direction parallel to the pates, and there is no velocity in the y or z direction in the y or z direction –– that is, that is, v=0 and w=0v=0 and w=0..

131


From the continuity equation that From the continuity equation that ∂∂u/u/∂∂xx=0.=0.There would be no variation of u in the z direction for There would be no variation of u in the z direction for

infinite plates, and for steady flow so that u=infinite plates, and for steady flow so that u=u(yu(y).).The The NavierNavier--Stokes equations reduce toStokes equations reduce to

zp0g

yp0

yu

xp0 2

2

由continuity condition得到

u = u(y,z) u= u(y)

簡化

132


2

2

yu

xp0

zp0

gyp0

xfgyp 1IntegratingIntegrating

?c?ccycyxp

21u 2121

2

IntegratingIntegrating

133


With the boundary conditions u=0 at y=With the boundary conditions u=0 at y=--h u=0 at y=hh u=0 at y=h

212 hx

p21c,0c

22 hyxp

21u

Velocity distributionVelocity distribution

No slip boundarycondition Liquid-liquid no-slip

134

Steady, Laminar Flow between Fixed Steady, Laminar Flow b

fundamentals of fluid mechanics chapter 6 flow analysis … · control...

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