chapter 6 differential analysis of fluid flow student

Offered by: Hang-Suin Yang

Chapter 6

Chapter 6 Differential Analysis of Fluid Flow

Outline

6.1 Fluid Element Kinematics

6.2 Conservation of Mass

6.3 The Linear Momentum Equation

6.4 Inviscid Flow

6.5 Some Basic, Plane Potential Flows

Outline

6.6 Superposition of Basic, Plane Potential Flows

6.7 Other Aspects of Potential Flow Analysis

6.8 Viscous Flow

6.9 Some Simple Solutions for Laminar, Viscous, Incompressible Flows

6.10 Other Aspects of Differential Analysis

We can break the element's complex motion into four components: translation, rotation, linear deformation, and angular deformation.

6.1.1 Velocity and Acceleration Fields Revisited

i j kV u v w

This method of describing the fluid motion is calledthe Eulerian method. It is also convenient to expressthe velocity in terms of three rectangular componentsso that

i j kx y z

V V V V Va u v w

t t x y z

u u u u v v v vu v w u v w

t x y z t x y z

w w w wu v w

t x y z

The acceleration of a fluid particle can be expressed as

material derivative

or substantial derivative

Du v w V

t t x y z t

6.1.2 Linear Motion and Deformation

u x tx ( V)

u ud u x t y u t y x y t

( V) Vu u

d x y t tx x

the rate at which the volume is changing per unit volume due to the gradient ∂u/∂x is

1 ( V) lim

d u t u

dt x t x

If velocity gradients ∂v/∂y and ∂w/∂z are also present, then

1 ( V)

d u v wV

dt x y z

Volumetric dilatation rate

For an incompressible flow

6.1 Fluid Element Kinematics6.1.3 Angular Motion and Deformation

Rotation

Deformation

Rotation + Deformation

v t vv x t

uu y t

OA t t tv

x t xx

Similarly

v uvx yx

u v uy x y

Similarly

1 1 1 , ,

2 2 2x y z

w v u w v u

y z z x x y

1i j k i j k

w v u w v u

y z z x x y

The three components, ωx, ωy, and ωz can be combined to give the rotation vector, ω, in the form

The vorticity, ζ, is defined as a vector that is twice the rotation vector; that is,

z zm z

CVin out

x x y y z z

tm m m

Conservation of mass

xx x x x x

yy y y y y

zz z z z z

m um u y z m m x m x y z

x xm v

m v x z m m y m x y zy y

m wm w x y m m z m x y z

( )CVmx y z x y z

CVin out

u v wx y z x y z

t x y z

u v wV

t x y z t

u v wu v w

t x y z x y z

Continuity Equation

6.2 Conservation of Mass6.2.2 Cylindrical Polar Coordinates

Div in cylindrical coordinates

( )( ) ( )1

( )( ) ( )1 1

vr v r vV

vr v v

r r r z

( , , )p r z

Continuity equation

( )( ) ( )1 10r zvr v v

Vt t r r r z

6.2.3 The Stream Function

Steady, incompressible, two-dimensional flow field can be expressed in terms of a stream function.

u v wu v w

t x y z x y z

Continuity Equation

Velocity components in a two-dimensional flow field can be expressed in terms of a stream function.

0 0u v

x y x y y x

ψ(x, y) called the stream function

, u vy x

0 constantd dx dy vdx udyx y

1( , )x y c

The change in the value of the stream function is related to the volume rate of flow.

d dx dy vdx udy dqx y

the volume rate of flow, q, betweentwo streamlines such as ψ1 and ψ2 canbe determined by

2 1q d

In cylindrical coordinates the continuity equation for incompressible, plane, two-dimensional flow reduces to

( )( ) ( )1 10r zvr v v

Vt t r r r z

( )1 1 0r vrv

( )1 1 1 1 r vrv

r r r r r r r

( )1 1 1 1 0r vrv

r r r r r r r

r rrv v

v vr r

6.3 The Linear Momentum EquationTo develop the differential momentum equations we can start with the linear momentum equation (Newton's Second Law)

VdV V V dAt

Conservation of momentum

ii i i

inlet outletports ports

mvmv mv F

z zm z

mvmv mv

6.3 The Linear Momentum EquationIn x-direction

u y z u u y z uu y z u x

v x z u

v x z uv x z u y

w x y u

w x y uw x y u z

i( )( )CVmv

x y z ut t

uu x y z

Newton`s second law of motion

Momentum carried by flow

inlet at x

inlet at y

inlet at z

mv u y z u u y z

mv v x z u uv x z

mv w x y u uw x y

v x z u

v x z uv x z u y

w x y u

w x y uw x y u z

inlet at x x inlet at x inlet at x

inlet at y y inlet at y inlet at y

inlet at z z inlet at z inlet at z

mv mv mv xx

u y z u x y zx

mv mv mv yy

uv x z uv x y zy

mv mv mv zz

uw x y uwz

) x y z

2i i ( ) ( ) ( )

mv mv u x y z uv x y z uw x y zx y z

u v wu x y z

u u v w x y zx y z

u u uu v w x y z

ii i i

mvmv mv F

u u v wu u u u v w

t t x y z x y z

u u uu v w F

u Du V F

6.3 The Linear Momentum Equation6.3.1 Description of Forces Acting on the Differential Element

In x-direction Body force: (Ex. gravity)

,body x x xF x y z g g x y z

Surface force: (Ex. pressure, viscous force)

,xx xy xz

Direction offorce

Direction ofsurface

Normal stress Shear stress

Surface forces:

sur x xx

sur y yx

sur z zx

sur x xxsur x x sur x xx

sur y yxsur y y sur y yx

sur z zxsur z z sur z zx

FF F x y z x y z

F F y x z x y zy y

FF F z x y x y z

xx y z xx

xx x y zx

yx x z

yxyx y x z

xx y z xx

xx x y zx

zx x y

zxzx z x y

,i , , , , , ,sur sur x sur x x sur y sur y y sur z sur z z

yxxx zx

F F F F F F F

x y zx y z

xx y z xx

xx x y zx

yx x z

yxyx y x z

xx y z xx

xx x y zx

zx x y

zxzx z x y

Dyxxx zx

u u u u uu v w g

t t x y z x y z

In x-direction

In y-direction

In z-direction

Dxy yy zy

v v v v vu v w g

t t x y z x y z

Dyzxz zz

w w w w wu v w g

t t x y z x y z

6.3.2 Equations of Motion

6.4 Inviscid Flow

The shearing stresses are assumed to be negligible are said to be inviscid, nonviscous, or frictionless.

The negative sign is used so that a compressive normal stress (which is what we expect in a fluid)

will give a positive value for p.

xx yy zz p

0yx zx xy zy xz yz

6.4 Inviscid Flow6.4.1 Euler's Equations of Motion

u u u u u pu v w g

t t x y z x

In x-direction

In y-direction

In z-direction

v v v v v pu v w g

t t x y z y

w w w w w pu v w g

t t x y z z

6.4 Inviscid Flow

u u u u u pu v w g

t t x y z x

v v v v v pu v w g

t t x y z y

w w w w w pu v w g

t t x y z z

V V g pt

Euler's equations of motion

6.4 Inviscid Flow6.4.2 The Bernoulli Equation

We will restrict our attention to steady flow so Euler's equation in vector form becomes

V V g pt

Body force

kg g g z

6.4 Inviscid Flow

( )V V g p

Euler's equations of motion

1( ) ( ) ( )

2V V V V V V

1( ) ( )

pV V g z V V

6.4 Inviscid Flow

1( ) ( ) 0

pV V g z ds V V ds

Take the dot product of each term with adifferential length ds along a streamline.Thus,

( )V V ds

where i j kds dx dy dz

6.4 Inviscid Flow1

( ) 02

pV V g z ds

where1

i j k ( i j k)

p p p pds dx dy dz

p p p dpdx dy dz

similarly

2 21 1 1( ) ( )

2 2 2V V ds V ds dV

g z ds gdz

6.4 Inviscid Flow

1 ( ) 0

1 1 0 0

1 constant along a streamline

pV V g z ds

dp pdV gdz d V gz

Bernoulli equation

6.4 Inviscid Flow

6.4.3 Irrotational Flow

The vorticity is zero in an irrotational flow field.

If we make one additional assumption—that the flow is irrotational—the analysis of inviscid flow problems is further simplified.

, , w v u w v u

y z z x x y

6.4 Inviscid Flow

flow in the entrance toa pipe may be uniform(if the entrance isstreamlined) and thuswill be irrotational

6.4 Inviscid Flow6.4.4 The Bernoulli Equation for Irrotational Flow

The Bernoulli equation can be applied between any two points in an irrotational flow field.

1( ) ( ) 0

pV V g z ds V V ds

21 constant is not limited along a streamline

6.4 Inviscid Flow6.4.5 The Velocity Potential

For an irrotational flow 0V

We know that ( ) 0

So we can let

i j k i j kV u v wx y z

ϕ is called the velocity potential

6.4 Inviscid Flow

y z y z z y

z x z x x zv u

x y x y y x

, , u v wx y z

The velocity components

6.4 Inviscid Flow

For inviscid, incompressible,irrotational flow fields it followsthat

Laplace's equation

This type of flow is commonly called a potential flow.

2 22 2

Vx y z x y

Laplace's equation

Similarly, for the streamlines in an irrotational flow

0 i jV Vy x

6.4 Inviscid Flow

Laplace's equation in cylindrical coordinates

1r zv v v

r θ z r θ ze e e e e e

1 , , r zv v v

r r r r z

Velocity components in cylindrical coordinates

Cartesian coordinates

u ux y

Cylindrical coordinates

r rv vr ror

v vr r

=conatant

0 along

dy ud dx dy udx vdy

x y dx v

=conatant

0 along

dy vd dx dy vdx udy

x y dx u

Along = constant (called equipotential lines)

Along = constant (called streamlines)

=conatant =conatant

1along along

dy dy u v

dx dx v u

For any potential flow field a “flow net” can be drawn thatconsists of a family of streamlines and equipotential lines.

6.5 Some Basic, Plane Potential Flows6.5.1 Uniform Flow

C is an arbitrary constant, which can be set equal to zero

( cos sin )sin

( cos sin )

U x yv U

6.5 Some Basic, Plane Potential Flows6.5.2 Source and Sink

Source

Let m be the volume rate of flow

2 2r r

mm rv v

m mv v

r r r ror

v vr r

2 ln1 2

mr r r

Sources and sinks do not really exist in real flowfields, and the line representing the source or sink is amathematical singularity in the flow field.

mv v as r

If m is positive, the flow is a source flow.

If m is negative, the flow is a sink flow.

The flowrate, m, is the strength of the source or sink.

6.5 Some Basic, Plane Potential Flows6.5.3 Vortex

vr K r

where K is a constantK

6.5 Some Basic, Plane Potential FlowsIrrotational flow Rotational flow

Both sticks are rotating, theaverage angular velocity of thetwo sticks is zero and the flow isirrotational

The rotational vortex iscommonly called a forced vortex,whereas the irrotational vortex isusually called a free vortex.

A mathematical concept commonly associated with vortex motion is that of circulation.

0 0 0V ds ds d

For an irrotational flow

V ds v rd rd Kr

6.5 Some Basic, Plane Potential Flows6.5.4 Doublet

A doublet is formed by an appropriate source-sink pair.

Laplace's equation

is linear2 2

2 21 2

Superposition of is allowed

1 2 1 2 1 2( )2 2 2

1 21 2

tan tan2tan tan( )

1 tan tanm

sin sintan , tan

cos cos

r a r a

12 2 2 2

2 2 sin 2 sintan tan

ar m ar

m r a r a

The so-called doublet is formed byletting the source and sink approachone another (a → 0) while increasingthe strength m (m → ∞) so that theproduct ma/π remains constant.

sin where

12 2 2 20 0

2 sin 2 sin sinlim tan lim

2 2a a

m ar m ar ma

r a r a r

6.6 Superposition of Basic, Plane Potential Flows6.6.1 Source in a Uniform Stream—Half-Body

sin2 2

ln cos ln2 2

uniform flow source

m mUy Ur

m mUx r Ur r

0 , rv at r b

1 1sin

0 cos 2 2

m mU b

sin sin2 2 2stagnation

m m mUr Ub bU

stagnation

mbU Ur

sin ( )y r b

sin sin2

mv Ur U

6.6 Superposition of Basic, Plane Potential FlowsApplying the Bernoulli equation between a point farfrom the body, where the pressure is p0 and the velocityis U, and some arbitrary point with pressure p andvelocity v, it follows that

22 2 2 2 2

1 1 1( ) cos sin

1 1 2 cos

bUp U p v v p U U

b bp U

An important point to be noted is that the velocity tangent to the surface of the body is not zero; that is, the fluid “slips” by the boundary

6.6 Superposition of Basic, Plane Potential Flows6.6.2 Rankine Ovals

12 2 2

2 sin tan

uniform flow source

m arUy

m ayUy

12 2 2

2 2 2 2 2 2 4

12 2 2

2 2 2 2 2 2

2 ( ) 2 ( )

2 ( ) 4

m ayu Uy

y y x y a

m a x y aU

x y a x y a

m ayv Uy

x x x y a

x y a a y

0 , 0u v at x l y

2 2 2 2 2 2 4 2 2

2 ( 0 )0

2 ( 0 ) 2 ( 0 ) ( )

m a l a m au U U

l a l a l a

12 2 2

2tan 0 , 0

2tagnation

m ayUy at x l y

0 0 , tagnation at x y h

12 2 2

2 tan 0

m ahUh

h a Uhh

1.A large variety of body shapes with differentlength to width ratios can be obtained by usingdifferent values of Ua/m

2.The potential solution for the Rankine ovalswill give a reasonable approximation of thevelocity outside the thin, viscous boundarylayer and the pressure distribution on the frontpart of the body only.

6.6 Superposition of Basic, Plane Potential Flows6.6.3 Flow around a Circular Cylinder

sin sin sin

cos cos cos

uniform flow doublet

K KUy Ur

K KUx Ur

The stream function for flow around a circular cylinder

0 , 0, rv at r a

1 1 sinsin

cos 0 K

U K a Ua

2 21 sin and 1+ cos

a aUr Ur

11 cos

1+ sin

0 rv at r a

2 sin sv U at r a

The pressure distribution on the cylinder surface is obtained from the Bernoulli equation.

2 2 20

1 1( )

2 sin2

p U p v v

p v p U

1 (1 4sin )

2sp p U

6.6 Superposition of Basic, Plane Potential FlowsThe resultant force (per unit length) developed on the cylinder can be determined by integrating the pressure over the surface.

F p a d

升力

阻力

Potential theory incorrectly predicts that the drag on a cylinder is zero.Why?

An additional, interestingpotential flow can bedeveloped by adding afree vortex to the streamfunction or velocitypotential for the flowaround a cylinder. In thiscase

2 21 sin ln or 1 cos

a aUr r Ur

21+ sin 2 sin

av U v U

r r r a

The tangential velocity, vθ, on the surface of the cylinder (r = a) now becomes

6.6 Superposition of Basic, Plane Potential FlowsAt stagnation point, vθs = 0

2 sin 0 sin2 4s stag stagv U

The pressure distribution on the cylinder surface is obtained from the Bernoulli equation.

1 1 12 sin

2 2 2 2s sp U p v p Ua

1 2 sin 1 4sin

2 4sp p UaU a

F p a d

F p a d U

The development of this lift on rotating bodies is called the Magnus effect.

1.The method of superposition of basic potentials andstream functions has been used to obtain detaileddescriptions of irrotational flow around certain bodyshapes immersed in a uniform stream.

2.It is possible to extend the idea of superposition byconsidering a distribution of sources and sinks, vortexes,or doublets, which when combined with a uniform flowcan describe the flow around bodies of arbitrary shape.

3.Potential flow solutions are always approximatebecause the fluid is assumed to be frictionless.

4.An important point to remember is that regardless ofthe particular technique used to obtain a solution to apotential flow problem, the solution remainsapproximate because of the fundamental assumption ofa frictionless fluid.

Outer flowNeglect viscosity Vorticity = 0 Inner flow

Viscosity is importantVorticity generated

Wake regionViscosity is not importantVorticity ≠ 0

6.8 Viscous Flow

For incompressible Newtonian fluids it is known that the stressesare linearly related to the rates of deformation and can beexpressed in Cartesian coordinates as (for normal stresses)

zxzy 2

u u up p

v v vp p

w w wp p

stresses are linearly related to the rate of strain

6.8.1 Stress-Deformation Relationships

(for shearing stresses)

6.8 Viscous Flow

In cylindrical polar coordinates the stresses for incompressible Newtonian fluids are expressed as

r zzr rz

6.8 Viscous Flow

In x-direction

In y-direction

In z-direction

Dyxxx zx

u u u u uu v w g

t t x y z x y z

Dxy yy zy

v v v v vu v w g

t t x y z x y z

Dyzxz zz

w w w w wu v w g

t t x y z x y z

6.8.2 The Navier-Stokes Equations

6.8 Viscous FlowIn x-direction

In y-direction

In z-direction

The Navier-Stokes equations are the basic differential equationsdescribing the flow of Newtonian fluids.

u u u u u p u u uu v w g

t t x y z x x y z

v v v v v p v v vu v w g

t t x y z y x y z

w w w w w p w w wu v w g

t t x y z z x y z

6.8 Viscous Flow

Navier-Stokes equation in cylindrical coordinates

2 2 2 2

( )1 1 2r r r r r r rr z r

v v vv v v v rv v vpv v g

t r r r z r r r r r r z

r-direction:

θ-direction:

z-direction:

2 2 2 2

( )1 1 1 2

v v v v v v vv v

t r r r z

rv v vvpg

r r r r r r z

1 1z z z z z z zr z z

vv v v v v v vpv v g r

t r r z z r r r r z

1. A principal difficulty in solving the Navier-Stokes equations is their nonlinearity arising from the convective acceleration terms.

2. The Navier-Stokes equations apply to both laminar and turbulent flow, but for turbulent flow each velocity component fluctuates in an apparently random fashion, with a very short time scale, and this added complication makes an analytical solution intractable.

6.9.1 Steady, Laminar Flow between Fixed Parallel Plates

u p u u ug

t x x y z

v p v v vg

t y x y z

w p w w wg

t z x y z

1 1 22

0 ( , )

pp p x y

g p gy f xy

p u u p py c u y c y c

x y y x x

21 2 1

B.C. 0 at

ph c h c c

c hph c h c x

2 21 ( )

pu y h

The volume rate of flow, q, passing between the plates (for a unit width in the z direction) is obtained from the relationship

32 21 2

( )2 3

p h pq udy y h dy

The pressure gradient ∂p/∂x is negative, since thepressure decreases in the direction of flow.

If we let Δp represent the pressure drop between twopoints a distance ℓ apart, then

3 32 2

h p h pq

The mean velocity V=q/2h, then

The maximum velocity, umax, occurs at y = 0

1 3( )

p h pu y h u V

The pressure gradient in the x direction is constant, then

1 0 0( ) ( 0)p p

f x p x p xx x

0 at 0p p x y where is reference pressure

1 0( )p

p gy f x gy p xx

h p qq p p gy x

6.9.2 Couette Flow (g = 0)0 at 0

B.C. at

u U y b

u p u u ug

t x x y z

v p v v vg

t y x y z

w p w w wg

t z x y z

( )p p x

0 at 0B.C.

u U y b

p u pu y c y c

20 at 0 1B.C. ( )

u y y pu U y by

u U y b b x

In dimensionless form2

u y b p y y

U b U x b b

The simplest type of Couette flow is one for which thepressure gradient is zero; that is, the fluid motion iscaused by the fluid being dragged along by themoving boundary.

u y b p y y y

U b U x b b b

This situation would be approximated bythe flow between closely spacedconcentric cylinders in which onecylinder is fixed and the other cylinderrotates with a constant angularvelocity, ω.

The flow in an unloaded journal bearing might beapproximated by this simple Couette flow if the gapwidth is very small.

6.9.3 Steady, Laminar Flow in Circular Tubes

cosrg g

Continuity equation

( )D 1 1( ) 0 0

Dr zvrv v

V Vt r r r z

The flow is parallel to the walls, then

0 0 ( , ) ( )zr z z z

vv v v v r v r

Navier-Stokes equations

2 2 2 2

D ( )1 1 2

D ( )1 1 1 2

r r r rr

z z z zz

vv rv v vpg

t r r r r r r z

v rv v vvpg

t r r r r r r z

v v v vpg r

t z r r r r z

1 0 cos

z r r r

Navier-Stokes equations reduced to

sin ( , )p gr f z

sin ( , )p gr h r z

1( )f zy

1( ) constant

Suppose that the pressure gradient is constant

v vp pr r r c

z r r r r z

pv r c r c

This type of flow is commonlycalled Hagen-Poiseuille flow, orsimply Poiseuille flow.

0finite at 0

B.C. 10 at

pc Rv r R

2 21 ( )

pv r R

The volume rate of flow, Q, passing through the tube and the pressure gradient.

1 2 ( )

Q v rdrd

p R pr R rdr

where 8

R p p pQ

Poiseuille's law relates pressure drop and flowrate for steady, laminar flow in circular tubes

The mean velocity

R p R pR V Q V

The maximum velocity, vmax occurs at r = 0

2 22 2

1(0 ) 2

p R p R pv R V

In dimensionless form2

1( ) 1

vp rv r R

6.9.4 Steady, Axial, Laminar Flow in an Annulus

10 zvp

rz r r r

0 at B.C.

0 at z i

v vp pr r r c

z r r r r z

pv r c r c

0 at B.C.

0 at z i

2 22 21

ln4 ln( / )

i oz o

r rp rv r r

z r r r

The volume rate of flow Q is2 2 2

2 2 24 4

8 ln( / )

( ) where

8 ln( / )

z o io i

o io i

r rpQ v rdrd r r

r rp p pr r

l r r z l

The maximum velocity, vmax occurs at r = rm that

0 at zm

One can obtain

1/22 2

2ln( / )o i

These results for flow through an annulus are valid only if the flow is laminar.

For tube cross sections other than simple circular tubes it is common practice to use an “effective” diameter, termed the hydraulic diameter, Dh, which is defined as

4 cross-sectional area

wetted perimeterhD

For circular tube with diameter D24 ( /4)

For an annulus2 24 cross-sectional area 4 ( )

2( )wetted perimeter 2 2

o ih o i

r rD r r

In terms of the hydraulic diameter, the Reynolds number is

Re hD V

mean velocity

Re < 2100, the flow will be laminar.

( ) 0D

Continuity equation

Navier-Stokes equation

2DVp g V

Very few practical fluid flow problems can be solved using an exact analytical approach.

6.10.1 Numerical Methods

( ) 0D

DVp g V

Thanks for your attention

chapter 6 differential analysis of fluid flow student

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