Chapter 6 Differential Analysis of Fluid...

57:020 Mechanics of Fluids and Transport Processes Chapter 6 Professor Fred Stern Fall 2006

1

1

Chapter 6 Differential Analysis of Fluid Flow Fluid Element Kinematics Fluid element motion consists of translation, linear defor-mation, rotation, and angular deformation.

Types of motion and deformation for a fluid element.

Linear Motion and Deformation:

Translation of a fluid element

Linear deformation of a fluid element


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2

Change inδ∀ :

( )u x y z tx

δ δ δ δ δ∂⎛ ⎞∀ = ⎜ ⎟∂⎝ ⎠

the rate at which the volume δ∀ is changing per unit vo-lume due to the gradient ∂u/∂x is

( ) ( )0

1 limt

d u x t udt t xδ

δ δδ δ→

∀ ∂ ∂⎡ ⎤ ∂= =⎢ ⎥∀ ∂⎣ ⎦

If velocity gradients ∂v/∂y and ∂w/∂z are also present, then using a similar analysis it follows that, in the general case,

( )1 d u v wdt x y zδ

δ∀ ∂ ∂ ∂= + + = ∇ ⋅

∀ ∂ ∂ ∂V

This rate of change of the volume per unit volume is called the volumetric dilatation rate. Angular Motion and Deformation For simplicity we will consider motion in the x–y plane, but the results can be readily extended to the more general case.


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3

Angular motion and deformation of a fluid element

The angular velocity of line OA, ωOA, is

0limOA t tδ

δαωδ→

= For small angles

( )tanv x x t v t

x xδ δ

δα δα δδ

∂ ∂ ∂≈ = =∂

so that ( )

0limOA t

v x t vt xδ

δω

δ→

∂ ∂⎡ ⎤ ∂= =⎢ ⎥ ∂⎣ ⎦

Note that if ∂v/∂x is positive, ωOA will be counterclockwise. Similarly, the angular velocity of the line OB is

0limOB t

ut yδ

δβωδ→

∂= =∂

In this instance if ∂u/∂y is positive, ωOB will be clockwise.


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4

The rotation, ωz, of the element about the z axis is defined as the average of the angular velocities ωOA and ωOB of the two mutually perpendicular lines OA and OB. Thus, if counterclockwise rotation is considered to be positive, it follows that

12z

v ux y

ω ⎛ ⎞∂ ∂= −⎜ ⎟∂ ∂⎝ ⎠

Rotation of the field element about the other two coordinate axes can be obtained in a similar manner:

12x

w vy z

ω ⎛ ⎞∂ ∂= −⎜ ⎟∂ ∂⎝ ⎠

12y

u wz x

ω ∂ ∂⎛ ⎞= −⎜ ⎟∂ ∂⎝ ⎠

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

1 12 2x y z curlω ω ω= + + = = ∇ ×ω i j k V V

since

1 12 2 x y z

u v w

∂ ∂ ∂∇ × =∂ ∂ ∂

i j k

V

1 1 12 2 2

w v u w v uy z z x x y

⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞= − + − + −⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠ ⎝ ⎠i j k


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5

The vorticity, ζ, is defined as a vector that is twice the rota-tion vector; that is,

2ς = = ∇ ×ω V The use of the vorticity to describe the rotational characte-ristics of the fluid simply eliminates the (1/2) factor asso-ciated with the rotation vector. If 0∇ × =V , the flow is called irrotational. In addition to the rotation associated with the derivatives ∂u/∂y and ∂v/∂x, these derivatives can cause the fluid ele-ment to undergo an angular deformation, which results in a change in shape of the element. The change in the original right angle formed by the lines OA and OB is termed the shearing strain, δγ,

δγ δα δβ= + The rate of change of δγ is called the rate of shearing strain or the rate of angular deformation:

lim lim ⁄ ⁄

Similarly,

The rate of angular deformation is related to a correspond-ing shearing stress which causes the fluid element to change in shape.


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The Continuity Equation in Differential Form The governing equations can be expressed in both integral and differential form. Integral form is useful for large-scale control volume analysis, whereas the differential form is useful for relatively small-scale point analysis. Application of RTT to a fixed elemental control volume yields the differential form of the governing equations. For example for conservation of mass

∑ ∫ ∂ρ∂−=⋅ρ

CS CVVd

tAV

net outflow of mass = rate of decrease across CS of mass within CV


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7

( ) dydzdxux

u ⎥⎦⎤

⎢⎣⎡ ρ

∂∂+ρ

outlet mass flux

Consider a cubical element oriented so that its sides are ⎢⎢to the (x,y,z) axes

Taylor series expansion

retaining only first order term We assume that the element is infinitesimally small such that we can assume that the flow is approximately one di-mensional through each face. The mass flux terms occur on all six faces, three inlets, and three outlets. Consider the mass flux on the x faces

( )flux outflux influxx ρu ρu dx dydz ρudydz

x∂⎡ ⎤= + −⎢ ⎥∂⎣ ⎦

= dxdydz)u(x

ρ∂∂

V Similarly for the y and z faces

dxdydz)w(z

z

dxdydz)v(y

y

flux

flux

ρ∂∂=

ρ∂∂=

inlet mass flux ρudydz


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8

The total net mass outflux must balance the rate of decrease of mass within the CV which is

dxdydzt∂ρ∂−

Combining the above expressions yields the desired result

0)V(t

0)w(z

)v(y

)u(xt

0dxdydz)w(z

)v(y

)u(xt

=ρ⋅∇+∂ρ∂

=ρ∂∂+ρ

∂∂+ρ

∂∂+

∂ρ∂

=⎥⎦

⎤⎢⎣

⎡ ρ∂∂+ρ

∂∂+ρ

∂∂+

∂ρ∂

ρ∇⋅+⋅∇ρ VV

0VDtD =⋅∇ρ+ρ ∇⋅+

∂∂= VtDt

D

Nonlinear 1st order PDE; ( unless ρ = constant, then linear) Relates V to satisfy kinematic condition of mass conserva-tion Simplifications: 1. Steady flow: 0)V( =ρ⋅∇ 2. ρ = constant: 0V =⋅∇

dV

per unit V differential form of con-tinuity equations


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9

i.e., 0zw

yv

xu =

∂∂+

∂∂+

∂∂ 3D

0yv

xu =

∂∂+

∂∂ 2D

The continuity equation in Cylindrical Polar Coordinates

The velocity at some arbitrary point P can be expressed as

r r z zv v vθ θ= + +V e e e The continuity equation:

( ) ( ) ( )1 1 0r zr v v vt r r r z

θρ ρ ρρθ

∂ ∂ ∂∂ + + + =∂ ∂ ∂ ∂

For steady, compressible flow

( ) ( ) ( )1 1 0r zr v v vr r r z

θρ ρ ρθ

∂ ∂ ∂+ + =

∂ ∂ ∂ For incompressible fluids (for steady or unsteady flow)

( )1 1 0r zrv v vr r r z

θ

θ∂ ∂ ∂+ + =

∂ ∂ ∂


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The Stream Function Steady, incompressible, plane, two-dimensional flow represents one of the simplest types of flow of practical im-portance. By plane, two-dimensional flow we mean that there are only two velocity components, such as u and v, when the flow is considered to be in the x–y plane. For this flow the continuity equation reduces to

0yv

xu =

∂∂+

∂∂

We still have two variables, u and v, to deal with, but they must be related in a special way as indicated. This equation suggests that if we define a function ψ(x, y), called the stream function, which relates the velocities as

,u vy xψ ψ∂ ∂= = −

∂ ∂

then the continuity equation is identically satisfied: 2 2

0x y y x x y x y

ψ ψ ψ ψ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞+ − = − =⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠

Velocity and velocity components along a streamline


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Another particular advantage of using the stream function is related to the fact that lines along which ψ is constant are streamlines.The change in the value of ψ as we move from one point (x, y) to a nearby point (x + dx, y + dy) along a line of constant ψ is given by the relationship:

0d dx dy vdx udyx yψ ψψ ∂ ∂= + = − + =

∂ ∂

and, therefore, along a line of constant ψ dy vdx u

=

The flow between two streamlines

The actual numerical value associated with a particular streamline is not of particular significance, but the change in the value of ψ is related to the volume rate of flow. Let dq represent the volume rate of flow (per unit width per-pendicular to the x–y plane) passing between the two streamlines.

dq udy vdx dx dy dx yψ ψ ψ∂ ∂= − = + =

∂ ∂

Thus, the volume rate of flow, q, between two streamlines such as ψ1 and ψ2, can be determined by integrating to yield:


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12

2

12 1q d

ψ

ψψ ψ ψ= = −∫

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

( )1 1 0rrv vr r r

θ

θ∂ ∂+ =

∂ ∂

and the velocity components, vr and vθ, can be related to the stream function, ψ(r, θ), through the equations

1 ,rv vr rθ

ψ ψθ

∂ ∂= = −∂ ∂

Navier-Stokes Equations Differential form of momentum equation can be derived by applying control volume form to elemental control volume The differential equation of linear momentum: elemental fluid volume approach


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13

∑ V

(1) =

(2) =

= combining and making use of the continuity equation yields ∑

∑ or ∑f where ∑ ∑ ∑ ∑f ∑f ∑f


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14

Body forces are due to external fields such as gravity or magnetics. Here we only consider a gravitational field; that is, ∑ ρ== dxdydzgFdF gravbody and kgg −= for g↓ z↑ i.e., kgf body ρ−= Surface forces are due to the stresses that act on the sides of the control surfaces symmetric (σij = σji) σij = - pδij + τij 2nd order tensor normal pressure viscous stress = -p+τxx τxy τxz

τyx -p+τyy τyz

τzx τzy -p+τzz As shown before for p alone it is not the stresses them-selves that cause a net force but their gradients.

dFx,surf = ( ) ( ) ( ) dxdydzzyx xzxyxx ⎥

⎦

⎤⎢⎣

⎡ σ∂∂+σ

∂∂+σ

∂∂

= ( ) ( ) ( ) dxdydzzyxx

pxzxyxx ⎥⎦

⎤⎢⎣

⎡ τ∂∂+τ

∂∂+τ

∂∂+

∂∂−

δij = 1 i = j δij = 0 i ≠ j


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15

This can be put in a more compact form by defining vector stress on x-face kji xzxyxxx τ+τ+τ=τ and noting that

dFx,surf = dxdydzxp

x ⎥⎦⎤

⎢⎣⎡ τ⋅∇+

∂∂−

fx,surf = xxp τ⋅∇+

∂∂− per unit volume

similarly for y and z

fy,surf = yyp τ⋅∇+

∂∂− kji yzyyyxy τ+τ+τ=τ

fz,surf = zzp τ⋅∇+

∂∂− kji zzzyzxz τ+τ+τ=τ

finally if we define

kji zyxij τ+τ+τ=τ then

ijijsurf pf σ⋅∇=τ⋅∇+−∇= ijijij p τ+δ−=σ


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16

Putting together the above results

Dt

VDfff surfbody ρ=+∑ =

kgf body ρ−= ijsurface pf τ⋅∇+−∇=

DV Va V VDt t

∂= = + ⋅∇∂

ˆ

ija gk pρ ρ τ= − − ∇ + ∇ ⋅ inertia body force force surface surface force due to force due due to viscous gravity to p shear and normal

stresses


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17

For Newtonian fluid the shear stress is proportional to the rate of strain, which for incompressible flow can be written 2

where, = coefficient of viscosity

= rate of strain tensor

=

where,

Navier-Stokes Equation 0 Continuity Equation

Ex) 1-D flow


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18

Four equations in four unknowns: V and p Difficult to solve since 2nd order nonlinear PDE

x: y: z:

0zw

yv

xu =

∂∂+

∂∂+

∂∂

Navier-Stokes equations can also be written in other coor-dinate systems such as cylindrical, spherical, etc. There are about 80 exact solutions for simple geometries. For practical geometries, the equations are reduced to alge-braic form using finite differences and solved using com-puters.


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19

Ex) Exact solution for laminar incompressible steady flow in a circular pipe

Use cylindrical coordinates with assumptions 0 : Fully-developed flow 0 : Flow is parallel to the wall Continuity equation: 0 B.C. 0 0 ⇒ 0 i.e., 0


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20

Momentum equation:

or 0 sin (1) 0 cos (2) 0 (3) where, sin cos Equations (1) and (2) can be integrated to give sin

⇒ pressure is hydrostatic and ⁄ is not a func-tion of or


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21

Equation (3) can be written in the from 1 1

and integrated (using the fact that ⁄ = constant) to give 12

Integrating again we obtain 14 ln

B.C. 0 ∞ ⇒ 0 0 ⇒ 14

⇒ at any cross section the velocity distribution is parabolic


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22

1) Flow rate : 2 8

where, 2 If the pressure drops Δ over a length ℓ: ℓ Δ8 ℓ

2) Mean velocity : 1 Δ8 ℓ Δ8 ℓ

3) Maximum velocity : 0 4 Δ4 ℓ 2

⇒ 1

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u1ρu1 = ∑F

i.e.

from

dyduμ

u =

u =

=τ


uy1 ρ=Δ= u2

∑ ρ= uFx

ypΔ=

0dyd =τ

ddyd⎜⎝

⎛μ

dyud2

2μ

m mome

Cyu =

DyC +μ

=

u(0) =

u(t) = U

ytU=

μ=μdydu


yu2Δ

=⋅ρ AdV

ddp⎜

⎝⎛ +−

0

0dydu =⎟

⎠

⎞

0=

entum eq

D

0 ⇒ D

U ⇒ C =

=μtU con

ransport Proce

(ρ= uQ 2

yxdxdp Δ⎟

⎠⎞Δ

quation

= 0

= tUμ

nstant

sses

) =− 0u1

xy +Δτ−

0

dyd

⎜⎝

⎛ τ+τ+

xdyy

Δ⎟⎠

⎞τ =

Chapter 24

2

= 0

6

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dydu γ=μ

ransport Proce

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0xu =

∂∂

x0

∂∂−=

dxdhγ

rated twi

Aydxdh +

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flow wi

( zpx

γ+∂

h = p

plates

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dyudμ+

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ntal dxdz =

al dxdz =-1

uv

Chapter 25

2

essure

0=

1

u = u(y)v = o

0yp =

∂∂

6

25


26

26

BAy2y

dxdhu

2++γ=μ

now apply boundary conditions to determine A and B u(y = 0) = 0 ⇒ B = 0 u(y = t) = U

2t

dxdh

tUAAt

2t

dxdhU

2γ−μ=⇒+γ=μ

⎥⎦⎤

⎢⎣⎡ γ−μ

μ+

μγ=

2t

dxdh

tU1

2y

dxdh)y(u

2

= ( ) ytUyty

dxdh

22 +−

μγ−

This equation can be put in non-dimensional form:

ty

ty

ty1

dxdh

U2t

Uu 2

+⎟⎠⎞

⎜⎝⎛ −

μγ−=

define: P = non-dimensional pressure gradient

= dxdh

U2t2

μγ− zph +

γ=

Y = y/t ⎥⎦

⎤⎢⎣

⎡ +γμ

γ−=dxdz

dxdp1

U2z2

⇒ Y)Y1(YPUu +−⋅=

parabolic velocity profile

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u

q

=

=

=Uut

Uu =


Uu

U

tq

udy

t

0

t

0

∫==

∫=

∫ ⎢⎣⎡ −=

t

0y

tP

21

6P

⇒+=


tPy

Uu −=

[ ]t

dy

+− 22 y

tP

12tu =⇒

ransport Proce

ty

tPy

2

2+−

⎥⎦⎤+ dy

ty

dd

2t2

⎜⎝⎛ γ−

μ

sses

ty

=2Pt −

2U

dxdh +⎟

⎠⎞

2t

3Pt +−

Chapter 27

2

6

27


28

28

For laminar flow 1000tu <ν

Recrit ∼ 1000

The maximum velocity occurs at the value of y for which:

0dydu =

t1y

tP2

tP0

Uu

dyd

2 +−==⎟⎠⎞

⎜⎝⎛

( )P2t

2t1P

P2ty +=+=⇒ @ umax

( )P4

U2U

4UPyuu maxmax ++==∴

note: if U = 0: 32

4P

6P

uu

max==

The shape of the velocity profile u(y) depends on P:

1. If P > 0, i.e., 0dxdh < the pressure decreases in the

direction of flow (favorable pressure gradient) and the velocity is positive over the entire width

θγ−=⎟⎠

⎞⎜⎝

⎛ +γ

γ=γ sindxdpzp

dxd

dxdh

a) 0dxdp <

for U = 0, y = t/2


29

29

b) θγ< sindxdp

1. If P < 0, i.e., 0>dxdh the pressure increases in the di-

rection of flow (adverse pressure gradient) and the ve-locity over a portion of the width can become negative (backflow) near the stationary wall. In this case the dragging action of the faster layers exerted on the fluid particles near the stationary wall is insufficient to over-come the influence of the adverse pressure gradient.

0sindxdp >θγ−

θγ> sindxdp or

dxdpsin <θγ

2. If P = 0, i.e., 0dxdh = the velocity profile is linear

ytUu =

a) 0dxdp = and θ = 0

b) θγ= sindxdp

For U = 0 the form ( ) YY1PYUu +−= is not appropriate

u = UPY(1-Y)+UY

= ( ) UYY1Ydxdh

2t2

+−μ

γ−

Note: we derived this special case


30

30

Now let U = 0: ( )Y1Ydxdh

2tu

2−

μγ−=

3. Shear stress distribution Non-dimensional velocity distribution

( )* 1uu P Y Y YU

= = ⋅ − +

where * uuU

≡ is the non-dimensional velocity,

2

2t dhPU dx

γμ

≡ − is the non-dimensional pressure gradient

yYt

≡ is the non-dimensional coordinate. Shear stress

dudy

τ μ=

In order to see the effect of pressure gradient on shear stress using the non-dimensional velocity distribution, we define the non-dimensional shear stress:

*

212U

ττρ

=

Then

( )( )

**

2

1 212

Ud u U dutd y t Ut dYU

μτ μρρ

= =

( )2 2 1PY PUtμ

ρ= − + +

( )2 2 1PY PUtμ

ρ= − + +

( )2 1A PY P= − + +


31

31

where 2 0AUtμ

ρ≡ > is a positive constant.

So the shear stress always varies linearly with Y across any section. At the lower wall ( )0Y = :

( )* 1lw A Pτ = +

At the upper wall ( )1Y = :

( )* 1uw A Pτ = − For favorable pressure gradient, the lower wall shear stress is always positive: 1. For small favorable pressure gradient ( )0 1P< < : * 0lwτ > and * 0uwτ > 2. For large favorable pressure gradient ( )1P > : * 0lwτ > and * 0uwτ < ( )0 1P< < ( )1P > For adverse pressure gradient, the upper wall shear stress is always positive: 1. For small adverse pressure gradient ( )1 0P− < < : * 0lwτ > and * 0uwτ >

τ τ


32

32

2. For large adverse pressure gradient ( )1P < − : * 0lwτ < and * 0uwτ >


33

33

( )1 0P− < < ( )1P < − For 0U = , i.e., channel flow, the above non-dimensional form of velocity profile is not appropriate. Let’s use dimen-sional form:

( ) ( )2

12 2t dh dhu Y Y y t ydx dx

γ γμ μ

= − − = − −

Thus the fluid always flows in the direction of decreasing piezometric pressure or piezometric head because

0, 02

yγμ

> > and 0t y− > . So if dhdx is negative, u is posi-

tive; if dhdx is positive, u is negative.

Shear stress:

1

2 2du dh t ydy dx

γτ μ ⎛ ⎞= = − −⎜ ⎟⎝ ⎠

Since 1 02

t y⎛ ⎞− >⎜ ⎟⎝ ⎠

, the sign of shear stress τ is always oppo-

site to the sign of piezometric pressure gradient dhdx , and the magnitude of τ is always maximum at both walls and zero at centerline of the channel.

τ τ

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h do notion

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Chapter 34

3

0dhdx

>

τ

6

34


35

35

x-momentum ( ) 2

2

dyudzp

x0 μ+γ+

∂∂−=

y-momentum ( )⇒γ+∂∂−= zpy

0 hydrostatic pressure variation

0dxdp =⇒

θγ−=μ sindy

ud2

2

cysindydu +θ

μγ−=

DCy2ysinu

2++θ

μγ−=

dsinccdsin0dydu

dy

θμγ+=⇒+θ

μγ−==

=

u(0) = 0 ⇒ D = 0

dysin2ysinu

2θ

μγ+θ

μγ−=

= ( )yd2ysin2

−θμγ


36

36

u(y) = ( )yd2y2sing −

νθ

d

0

32

d

0 3ydysin

2udyq ⎥

⎦

⎤⎢⎣

⎡−θ

μγ=∫=

= θμγ sind

31 3

θν

=θμγ== sin

3gdsind

31

dqV

22

avg

in terms of the slope So = tan θ ∼ sin θ

ν

=3

SgdV o2

Exp. show Recrit ∼ 500, i.e., for Re > 500 the flow will be-come turbulent

θγ−=∂∂ cosyp

ν= dVRecrit ∼ 500

Cycosp +θγ−= ( ) Cdcospdp o +θγ−==

discharge per unit width


37

37

i.e., ( ) opydcosp +−θγ= * p(d) > po * if θ = 0 p = γ(d − y) + po

entire weight of fluid imposed if θ = π/2 p = po no pressure change through the fluid

Chapter 6 Differential Analysis of Fluid...

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Transcript of Chapter 6 Differential Analysis of Fluid...