Fluid Mechanics 1 Notes

Department of Mechanical Engineering The University of Hong Kong

Mechanics of Fluids MECH2008 (2008 – 2009)

Lecturer: Dr. C.O. Ng (office: HW7-1; phone: 28592622; email: [email protected]) Required Text: Fundamentals of Fluid Mechanics 5th Ed., B.R. Munson, D.F. Young & T.H.

Okiishi, Wiley Asia Student Edition. References: 1) Fluid Mechanics: Fundamentals and Applications, Y.A. Cengel & J.M.

Cimbala, McGraw-Hill. 2) Fluid Mechanics 6th Ed., F.M.White, McGraw-Hill. 3) Multi-Media Fluid Mechanics CD-ROM, Cambridge University Press. Assessment: In-course continuous assessment 10%

• a mid-term test to be announced later Examination in December 90% Topics Covered:

1. Flow kinematics with differential vector calculus 2. Differential equations of motion

3. Unidirectional viscous flow and hydrodynamic lubrication

4. Potential flow and stream function

5. Boundary layer and drag

6. Open-channel flow and fluid machines

Prerequisites:- This is a Level-II Mechanics of Fluids course demanding the knowledge you acquired in the first-year fluid and mathematics courses. In particular, the following topics are relevant and should be reviewed if you have already forgotten the stuffs:- 1) Properties of fluid (density, viscosity); 2) Principles of fluid statics; 3) Fluid dynamics by control volume approach

continuity equation energy equation (or Bernoulli equation) momentum equation head, head loss

4) Differentiation and integration; 5) Vector differential calculus (grad, div, curl, Gauss theorem, etc);

1

Lecture Notes and Worked Examples (The corresponding section numbers in the textbook or other references are noted wherever appropriate.) (I) DIFFERENTIAL ANALYSIS OF FLUID FLOW A. Description of Fluid Motion (Section 4.1.1)

• Lagrangian description: fluid particles are “tagged” or identified; rate of change of flow properties as observed by following a fixed particle; variables are functions of the initial position of particles and time.

• Eulerian description: fluid properties and variables are field variables, which are functions of position in space (with respect to a fixed frame of reference) and time. The Eulerian description, which is comparable to the data recorded by a measuring device fixed in position, is more convenient to use in fluid mechanics.

Eulerian and Lagrangian descriptionsof temperature of a fluid discharging from a smoke stack

ual particles

In the Lagrangian description, one must keep track of the position andvelocity of individ

• Rectangular (Cartesian) coordinates:

( )( )

( )

1 2 3

1 2 3

1 2 3

( , , ) ( , , ) 1, 2,3

( , , ) ( , , ) 1, 2,3

, , , , 1, 2,3

e.g.,

, , ( , , )

i

i

i

i

i

x y z x x x x i

u v w u u u u i

ix y z x x x x

uu v wu v wx y z x y z x

= = = =

= = = =

⎛ ⎞⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂ ∂= = = =⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

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

x

V

V =i i

∇

∇

z

V y

O

x

• Primitive variables: pressure ( , ) - scalar (0th order tensor)velocity ( , ) - vector (1st order tensor)

p tt

xV x

Deduced variable stre ss ( , ) - 2nd order tensortτ x

2

In the Eulerian description, one

defines field variables, such as the pressure field and the velocity field, at any location and instant in time.

B. Kinematics (Sections 4.2 and 6.1) • Total (a.k.a. material, substantial) derivative = local rate of change + convective (or

advective) rate of change = the rate of change as observed following a particle of fixed identity. It is an operator that can be applied to any scalar or vector quantity.

( ) ( ) ( )( )

( ) ( ) ( ) ( )

( )

local rate convective rate of changeof change

e.g., local acceleration =

convective acceleration =

ddt t

u v wt x y z

t

u v wx y z

∂= +

∂∂ ∂ ∂ ∂

= + + +∂ ∂ ∂ ∂

∂∂

∂ ∂ ∂+ +

∂ ∂ ∂

i

i

V

V

V VV V =

∇

∇V

- The local rate of change, also called the unsteady term, vanishes identically for a steady flow. Therefore a flow is steady if and only if / 0t∂ ∂ ≡ .

- The quantity ( )iV ∇ is a scalar convective operator that determines the time rate of change of any property (e.g., velocity, density, concentration, temperature) of a particle by reason of the fact that the particle moves from a place where the property has one value to another place where it has a different value.

The total derivative is defined by following a fluid particle as it moves throughout the flow field. In this illustration, the fluid particle is accelerating to the right as it moves up and to the right.

A velocity field with respect to a fixed

frame of reference (x, y). A point fixed in space is occupied by different fluid particles at different time.

3

•

Translation+ (rigid body motion)

RotationGeneral motion = +

Dilatation (change in volume) +Angular deformation (change in shape)

⎧ ⎫⎪⎪⎬⎪⎪⎪ ⎭⎪

⎨⎪⎪⎪⎪⎩

As illustrated by:-

The various modes of deformation can be expressed in terms of the velocity gradients.

• Divergence of velocity is the volumetric strain/dilatation rate (rate of change of volume per unit volume)

u v wx y z

∂ ∂ ∂≡ + +

∂ ∂ ∂V∇i

where u x∂ ∂ , v y∂ ∂ and w z∂ ∂ are the components of the volumetric strain rate due to elongation of a fluid element in the x-, y-, and z-directions, respectively. Consider a small element of dimensions x y zδ δ δ× × :

4

Because of the velocity differential uδ over a distance xδ , the element is lengthened in the x-direction by u tδ δ⋅ over a small period of time tδ . The corresponding change in volume is therefore

xV u t y zδ δ δ δ δ= ⋅ ⋅ ⋅ , and the volume strain rate (change in volume per volume per time) is

as , 0xV u t y z u u x tV t x y z t x xδ δ δ δ δ δ δ δ

δ δ δ δ δ δ⋅ ⋅ ⋅ ∂

= = =⋅ ⋅ ⋅ ⋅ ∂

→ .

Similarly, for the lengthening of the element in the y- and z-directions

, as , , 0y zV Vv w y z t

V t y V t zδ δ δ δ δ

δ δ∂ ∂

= =⋅ ∂ ⋅ ∂

→

The total volume strain rate is hence given by the divergence of the velocity.

as , , , 0V u v w x y z tV t x y zδ δ δ δ δ

δ∂ ∂ ∂

= + + →⋅ ∂ ∂ ∂

In this incompressible flow, in which the velocity divergence is identically zero, an initially square parcel of marked fluid will deform into a long thin shape (stretch in x-direction, but shrink in the y-direction) in the course of movement shown in the figure. The flow is irrotational is this case.

• Any shear deformation can be decomposed into rigid body rotation and angular deformation. Consider a small element undergoing shear deformation

5

Because of the velocity differential vδ over a distance xδ , the face OA rotates counterclockwise by an angle /v t xδα δ δ δ= ⋅ over a small period of time tδ . Therefore the angular velocity of OA is

as , 0d v x tdt xαα δ δ∂

= = →∂

Similarly, the face OB rotates clockwise at an angular velocity given by

as , 0d u y tdt yββ δ δ∂

= = →∂

x

y

O α

β

The deformation can be decomposed into a rigid body rotation at an angular velocity

( )1 12 2

v ux y

ω α β⎛ ∂ ∂

= − = −⎜ ∂ ∂⎝ ⎠

⎞⎟ , where counterclockwise rotation is taken to be positive,

x

y

O ( )1

2α β−

( )12

α β−

and an angular deformation, where the corner angle decreases at a rate given by v ux y

γ α β ∂ ∂= + = +

∂ ∂, where a positive rate means a decreasing angle,

x

y

O

( )12

( )12

α β+

α β+

γ

6

• Rate of angular deformation of a 2-D fluid element moving in the x-y plane (angular deformation is considered to be positive if it is to decrease the original right angle) is hence defined to be

0limxy t

v ut xδ y

δα δβγδ→

+ ∂ ∂= = +

∂ ∂.

For a 3-D element in general, the rate of change of the corner angle that is initially a right angle between the i-j axes

( ) jiij

j i

uu i jx x

γ∂∂

= + ≠∂ ∂

,

which is symmetric, i.e., ij jiγ γ= .

• Rotation of a fluid element (about an axis which is perpendicular to the plane of the fluid motion) is the average of the angular velocities of the two mutually perpendicular sides of the element, where counterclockwise rotation is considered to be positive:

1rotation about -axis: 2



z

x

y

v uzx y

w vxy z

u wyz x

ω

ω

ω

⎛ ⎞∂ ∂= −⎜ ⎟∂ ∂⎝ ⎠

⎛ ⎞∂ ∂= −⎜ ⎟∂ ∂⎝ ⎠

∂ ∂⎛ ⎞= −⎜ ⎟∂ ∂⎝ ⎠

Rotation (or angular velocity) vector ( ) x y z, ,ω ω ω ω=

Note that for a 2-D flow in the x-y plane, xω and yω vanish identically; hence the rotation vector is always perpendicular to the x-y plane.

• To generalize, we may define

1 1 1shear rate tensor angular deformation rate , and2 2 2

jiij ij

j i

uuex x

γ⎛ ⎞∂∂

= + = =⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠

1rate of rotation vorticity, where2

vorticity (curl of velocity)

22 yx

i j k

x y zu v w

w v u wi jy z z x

ωω

=

= ×

∂ ∂ ∂=

∂ ∂ ∂

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

ω

ζ V∇

2 z

v u kx y

ω

⎛ ⎞∂−⎜ ⎟∂ ∂⎝ ⎠

7

2=ζ ω The direction of a vector cross product is

determined by the right-hand rule. The vorticity vector is equal to twice the angular velocity vector of a rotating fluid particle.

The difference between a rotational and irrotational flow: fluid elements in a rotational region of the flow rotate about their own axis, but those in an irrotational region of the flow do not.

In this incompressible and rotational flow, an initially square fluid parcel will not only elongate, but also rotate about its axis as it moves over the time periods shown in the figure.

8

C. The Reynolds Transport Theorem (Section 4.4)

Define:

The Reynolds transport theorem (RTT) provides a link between the system approach and the control volume approach.

Two approaches of analyzing a problem. (a) System approach: follow the fluid as it moves and deforms; no mass crosses the boundary. (b) Control volume approach: consider the changes in a certain fixed volume; mass crosses the boundary.

- Material Volume: a volume that contains the same fluid as it moves and deforms following the motion of the fluid

- Material Surface: enclosing surface of a material volume; by definition no fluid particles can cross it.

- Control Volume: a volume of fluid in a flow field, usually fixed in space, to be occupied by different fluid particles at different times.

- Control Surface: imaginary or physical enclosing surface of a control volume. - Flux: amount of property (e.g., mass, momentum, energy) crossing a unit area of a

surface per unit time. We state without proof the Reynolds transport theorem, which provides a basis for developing differential equations for the various conservation laws:

rate of change of the local rate of change of theproperty within property within the fixedthe material volume control volume that happens

to coincide with the materialvolum

MV CV

d bdV bdVdt t

ρ ρ∂=

∂∫∫∫ ∫∫∫net out-flux of theproperty across theentire control surface

e at that instant

CSb d Aρ+ ∫∫ iV n

where

density of fluidan intensive property of fluid (property per unit mass)

material volume that happens to coincide with at time tb BMV CV

ρ ==

=

9

control volume (fixed in space)control surface

unit outward normal to

CVCS

CS

==

=n

The integral of over the control surface gives the net amount of the property B flowing out of the control volume (into the control volume if it is negative) per unit time.

A moving system (hatched region) and a fixed control volume (shaded region) in a diverging portion of a flow field at times t and

bV ndAiρ

t t+ . D. Conservation of Mass (Section 6.2)

If the property is mass, then b = 1, and

( )L.H.S. mass in 0

(by definition of , which always contains the same fluid)MV

d ddV MVdt dt

MV

ρ = =∫∫∫

( )by Gauss theorem is stationary

R.H.S.

=

CV CS

CV CV

CV

dV dAt

dV dVt

ρ ρ

ρ ρ

∂+

∂∂

+∂

∫∫∫ ∫∫

∫∫∫ ∫∫∫

V n

V

i

i∇

Equating L.H.S. and R.H.S., and removing the volume integral since CV is arbitrary, we get the differential form of Continuity Equation

10

( )

( )

( )

0

or, using the identity

0

or in index form 0 , 1, 2,3, summation over repeated indexi

i

t

ddt

ud i jdt x

ρ ρ

ρ ρ ρρ ρ

ρ ρ

∂+ =

∂=

+ =

∂+ = =

∂

V

V V + V,

V

i

i i i

i

∇

∇ ∇ ∇

∇

INCOMPRESSIBLE FLOW is defined as one in which the density of a fluid particle

is invariant with time 0ddtρ

⇔ = , which implies

0

(ie, divergence of velocity is zero for incompressible flow)In Cartesian coordinates, the continuity equation for inc

V =i ∇

ompressible flow reads

0u v wx y z

∂ ∂ ∂+ + =

∂ ∂ ∂

Note that a flow with constant density is always incompressible, but an incompressible flow does not necessarily have a constant density (e.g., flow in a stratified sea).

E. Applied Forces

• Body force due to gravity on a small fluid element = dVρ g • Surface stress =s τ ni , where n is the unit outward normal vector to the surface, and

( ) , 1, 2, 3xx xy xz

yx yy yz ij

zx zy zz

i jτ τ ττ τ τ ττ τ τ

⎡ ⎤⎢ ⎥= = =⎢ ⎥⎢ ⎥⎣ ⎦

τ

are the stress components on an infinitesimal cubic fluid element.

ijτ is a second order tensor, where the first index i denotes the face (on which the stress acts) being normal to ix , and the second index j denotes the stress component being in the jx direction.

11

In the textbook, the normal stress is denoted by iiσ in order to distinguish it from the shear stress ( ) ij i jτ ≠ . It can be shown that ijτ is symmetric, ie, ij jiτ τ= . Therefore there are only 6 independent stress components.

F. Conservation of Linear Momentum (Section 6.3) Apply Newton’s second law of motion to a material volume of fluid:

= +

rate of change surface bodyof momentum stress force

MV MS MV

d dV dA dVdt

ρ ρ∫∫∫ ∫∫ ∫∫∫V s g

The L.H.S. can be converted, using the transport theorem and the continuity equation,

into MV MV

d ddV dVdt dt

ρ ρ=∫∫∫ ∫∫∫VV .

The first term on the R.H.S. is MS MS MV

dA dA dV= =∫∫ ∫∫ ∫∫∫s τ n τi i∇ on using Gauss theorem.

Plugging these terms back, and removing the volume integral since the volume is arbitrary, we get the differential form of momentum equation

ddt

ρ ρ= +V τ gi∇

The left hand term is a total derivative, which can be expanded into the Eulerian form:

or iji ij i

j j

u uu gt t

τx x

ρ ρ ρ⎛ ⎞ ∂∂ ∂∂⎛ ⎞+ = + + = +⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

V V V τ gi i∇ ∇ ρ∂ ∂

By now, there are more unknowns than equations. To close the problem, we need to introduce CONSTITUTIVE (stress vs strain-rate) relations to relate the stress and the kinematics. If the fluid is Newtonian, a linear relationship is followed

1 for where , dynamic viscosity coefficient

0 for

jiij ij

j i

ij

uupx x

i ji j

τ δ μ

δ μ

⎛ ⎞∂∂= − + +⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠

=⎧= =⎨ ≠⎩

Finally, on substituting the above relationship, we obtain the Navier-Stokes equations

12

( )

2

2

2

2

2

or in index form,

, 1, 2,3, summation over repeated index

1or

(I) (II)

i i ij i

j i j

i i ij i

j i j

pt

u u upu g i jt x x x

u u upu gt x x x

ρ ρ μ

ρ ρ μ

νρ

∂⎛ ⎞+ = − + +⎜ ⎟∂⎝ ⎠

⎛ ⎞∂ ∂ ∂∂+ = − + + =⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

∂ ∂ ∂∂+ = − + +

∂ ∂ ∂ ∂

V V V g Vi∇ ∇ ∇

(III) (IV) (V)

where is the kinematic viscosityν μ ρ= /

Meanings of the five terms:- (I) – local acceleration; (II) – convective acceleration (inertia), nonlinear term of the equation; (III) – pressure gradient; (IV) – gravity; (V) – viscous diffusion of momentum owing to molecular viscosity of the fluid. Now, we have 4 equations (1 continuity + 3 components of momentum) for the four variables as functions of space and time , , , and x y zu u u p ( ), , ,x y z t . Note that it is the pressure gradient, rather than the pressure itself that drives the flow.

13

The Equations of Motion for an Incompressible Newtonian Fluid

In Rectangular Coordinates (x, y, z)

2 2 2

2 2 2

2 2 2

2 2

Continuity: 0

1-component:

1-component:

yx z

x x x x x x xx y z

y y y y y y yx y z

uu ux y z

u u u u u u upxx u u u g

t x y z x x y z

u u u u u u upy u u ut x y z y x y z

νρ

νρ

∂∂ ∂+ + =

∂ ∂ ∂

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

∂ ∂ ∂ ∂ ∂ ∂ ∂∂+ + + = − + + +

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ 2

2 2 2

2 2 2

1-component:

y

z z z z z z zx y z

g

u u u u p u u uz u u ut x y z z x y z

νρ

⎛ ⎞+⎜ ⎟⎜ ⎟

⎝ ⎠

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

zg

( , , ) are the components of the acceleration due to gravity in the , , and directions.If, say, and are horizontal axes and is positive upward, then 0, and .Also, the gravity

x y z

x y z

g g g x y zx y z g g= = = −

( )can be combined implicitly with the pressure term by introducing

* .x y zp p p g x g y g zρ ρ≡ − − + +g x =i

g g

In Cylindrical Coordinates (r, θ, z)

( )

( )

2

2 2

2 2 2 2

1 1Continuity: 0

1-component:

1 1 2

r z

r r r rr z

r rr

ru u ur r r z

u uu u u u pr u ut r r r z r

uu uru gr r r r r z

θ

θ θ

θ

θ

θ ρ

νθ θ

∂ ∂ ∂+ + =

∂ ∂ ∂

∂ ∂ ∂ ∂ ∂+ + − + = −

∂ ∂ ∂ ∂ ∂

⎡ ⎤∂∂ ∂ ∂ ∂⎛ ⎞+ + − +⎜ ⎟⎢ ⎥∂ ∂ ∂ ∂ ∂⎝ ⎠⎣ ⎦+

( )2 2

2 2 2 2

1-component:

1 1 2

-component:

r

rr z

r

z z zr

u u u u u u u pu ut r r r z r

u uuru gr r r r r z

uu u uz u ut r r

θ θ θ θ θ θ

θ θθ θ

θ

θθ ρ θ

νθ θ

θ

∂ ∂ ∂ ∂ ∂+ + + + = −

∂ ∂ ∂ ∂ ∂

⎡ ⎤∂ ∂∂ ∂ ∂⎛ ⎞+ + + +⎜ ⎟⎢ ⎥∂ ∂ ∂ ∂ ∂⎝ ⎠⎣ ⎦

∂ ∂ ∂+ + +

∂ ∂ ∂

+

2 2

2 2 2

1

1 1

zz

z z zz

u pz z

u u ur gr r r r z

ρ

νθ

∂ ∂= −

∂ ∂

⎡ ⎤∂ ∂ ∂ ∂⎛ ⎞+ + + +⎜ ⎟⎢ ⎥∂ ∂ ∂ ∂⎝ ⎠⎣ ⎦

14

G. Scaling and Approximation • Because of the inertia terms (convective acceleration), the Navier-Stokes (NS)

equations are non-linear equations. • Except for simple flow geometry, analytical solutions do not exist in general. • Fortunately, for many practical applications, not all terms in the equations are equally

important, and therefore some subdominant terms can be dropped in favor of a first approximation of the problem. The approximate equations can then be solved (analytically or numerically) with much greater ease than the full-blown ones.

• It is important to judge, for a particular problem, the relative significance of the individual terms in the NS equations, which can be reflected from the magnitude of the corresponding non-dimensional parameters.

For illustration, consider incompressible unsteady flow past a body:

Body

L

U

Characteristic scales: Length (L); Time scale of unsteadiness (T); Velocity (U); Pressure (P) Introduce dimensionless variables (distinguished by *):

/ , * / , / , * / , /U t t T L p p P g= = = = =V* V x* x g* gthe normalized NS can be expressed as

2

2 2**

L P gLpUT t U U UL

νρ

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

V * V * V* ∗g* Vi∇ ∇ *∇

The scales have been chosen to be representative of the variables so that all the dimensionless terms are order unity. Now, the importance of each term (relative to the inertia) is carried by its bracketed coefficient.

2

2

temporal accelerationStrouhal number (St) convective accelertion

pressure forceEuler number (E) inertia

ineritaReynolds number (Re) viscous forceineritaFroude number (Fr)

gravi

LUT

PU

UL

UgL

ρ

ν

= =

= =

= =

= =ty force

15

Possible Cases of Simplification:- Large Re Re 1 negligible viscous effect,

1NS reduces to Euler equations pt ρ

⇒

∂+ = − +

∂V V V gi∇ ∇

Small St St 1 negligible unsteady effect quasi-steady flow,

The local (temporal) acceleration term can be dropped.⇒ ⇒

∴ Small Re Re 1 negligible inertia effect (good news!)Viscous force is significant, and is to be balanced by pressure gradient.For slow and viscous flow and negligible gravity, the flow is called

⇒

Creeping Flow

21 0

Nonlinear inertia terms are now gone, analytical solutions are possibleif the flow geometry is simple enough.

p νρ

= − + V∇ ∇

Spatial Dimension Also, it is often the case that the flow varies only in one or two spatial dimensions, and therefore the problem can be reduced to a one- or two-dimensional problem, for which only one or two velocity components need to be solved. Some common cases of one-dimensional flow:

• fully developed pipe or channel flow: axial velocity as a function of radial distance from center of pipe ( )u u r= , or longitudinal velocity as a function of distance from the bottom of channel ( )u u y= ;

• axi-symmetrical flow: velocity is symmetrical about an axis (e.g., point source/sink, vortex).

16

(II) SIMPLE (EXACTLY OR NEARLY ONE-DIMENSIONAL) VISCOUS FLOW (Section 6.9) A. Mathematical Formulation for a Fluid Dynamics Problem Assumptions:

• constant fluid properties (density ρ , viscosity μ ) • Newtonian fluid (linear, isotropic and purely viscous material)

Basic Variables:

( )( )

Velocity ( , , ) , , , (3 variables)

Pressure , , , (1 variable)

u v w x y z t

p p x y z t

= =

=

V V Basic Governing Equations:

2

Continuity 0 (1 equation)1Navier-Stokes (3 equations)p

tν

ρ∂

+ = − + +∂

V =V V V g V

i

i

∇

∇ ∇ ∇

Other derived variables:

( )Stress ( , , , ) , isotropic tensor

with stress components (see the definition on page 11):

2 , (no

T

xx

x y z t p

upx

μ

τ μ

⎡ ⎤= − ∇ ∇ =⎣ ⎦

∂= − +

∂

τ I + V + V I

rmal stress)

2 , (normal stress)

(shear stress)

yy

xy yx

vpy

u vy x

τ μ

τ τ μ

∂= − +

∂

⎛ ⎞∂ ∂= = +⎜ ⎟∂ ∂⎝ ⎠

etc.

Vorticity w v u w v ui j ky z z x x y

⎛ ⎞ ⎛∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞= × − + − + −⎜ ⎟ ⎜⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠ ⎝ζ V = ⎞

⎟⎠

∇ Boundary Conditions:

• No-slip boundary condition: the velocity of a fluid in contact with a solid impermeable wall must equal that of the wall

fluid solid along a fluid-solid interface=V V If in particular the wall is stationary, the fluid adjacent to the wall must have zero velocity.

27

The development of velocity profiles due to the no-slip condition as a fluid flows past a blunt nose and a flat plate.

• Interface boundary condition between two fluids: when fluid A and fluid B meets at an interface, the velocity and stress must match between the two fluids at the interface

A B A B, along a fluid-fluid interface = =V V τ τ

If, say, the interface is flat (along x-direction) and the fluids are moving parallel to the interface, the continuity of stress implies the continuity of pressure and shear stress at the interface

A B A BA B

, du dup pdy dy

μ μ= =

• Free-surface boundary condition: a degenerate form of the above interface boundary condition occurs at the free-surface of a liquid, meaning that fluid A is a liquid (say, water, oil) and fluid B is a gas (usually air). By virtue of the fact air liquidμ μ , the shear stress at the air-liquid interface is negligibly small, and it is reasonable to approximate the shear stress to be at the interface, which is hence called a free surface,

liquid atmosphere liquidliquid

, 0 along the free surfacedup pdy

μ= =

28

• Other boundary conditions, such as inlet condition, outlet condition, periodic condition and symmetry, may also apply to certain types of boundaries, depending on the problem.

Boundary conditions along a plane of symmetry are defined so as to ensure that the flow field on one side of the symmetry plane is a mirror image of that on the other side, as shown above for a horizontal symmetry plane.

Initial Condition If the problem is time dependent (i.e., unsteady), an initial condition also needs to be specified. ************************************************************************* Let us consider in the following sections a few applications of the Navier-Stokes equations, in which the flow configuration is simple enough for analytical solutions (exact or approximate) to be deduced. The assumptions are that the flow is steady ( / 0t∴ ∂ ∂ = ), laminar, and incompressible and the fluid is Newtonian. B. Plane Poiseuille-Couette Flow

Note that this is a unidirectional flow ( ), 0u u y v= = . Therefore there is no dependence on x for all variables: . / 0x∂ ∂ = The flow is driven by three forcings: (1) motion of the upper plate; (2) pressure gradient in the x-direction, / a constantp x∂ ∂ = ; (3) gravity, if x is not in a horizontal direction. Recall the momentum equations:

x

Upper plate moving at a constant speed U

Lower fixed plate

y

u(y)

y = h

y = 0

29

-component: uxt

∂∂

uux∂

+∂

uvy∂

+∂

2

2

1 p ux x

νρ∂ ∂

= − +∂ ∂

2

2

2

2

1 ....................... (1)

-component:

x

x

u gy

u p gy x

vyt

νρ

⎛ ⎞∂+ +⎜ ⎟⎜ ⎟∂⎝ ⎠

∂ ∂⇒ = −

∂ ∂

∂∂

vux∂

+∂

vvy∂

+∂

2

2

1 p vy x

νρ∂ ∂

= − +∂ ∂

2

2

vy∂

+∂

1 0 ....................... (2)

y

y

g

p gyρ

⎛ ⎞+⎜ ⎟⎜ ⎟

⎝ ⎠∂

⇒ − + =∂

Note that the inertia terms are identically zero, which is true for all unidirectional flows irrespective of the Reynolds number. Equation (2) simply gives that the pressure ( ) yp p x g yρ= + . The R.H.S. of equation (1) is constant, so the equation can be integrated twice with respect to y, giving

2

1 2( )2x

p yu y g C y Cx

ρμ

∂⎛ ⎞= − + +⎜ ⎟∂⎝ ⎠

where and are integration constants that can be determined using the boundary conditions that

1C 2C

( 0) 0 (no slip at the lower plate), and ( ) (speed of the upper plate).

u yu y h U

= == =

Solving for these constants, we obtain the solution for the velocity profile (see Fig. 6.31 below):

22

( )2

Couette FlowPoiseuille Flow

xp h y y yu y g Ux h h h

ρμ⎡ ⎤∂⎛ ⎞ ⎛ ⎞= − + − +⎢ ⎥⎜ ⎟ ⎜ ⎟∂⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

Couette flow is caused by the motion of a boundary wall moving in its own plane, while Poiseuille flow is caused by axial pressure gradient or gravity in the direction of flow.

30

The shear stress in the flow is

constant stresslinear stress distribution due to due to Couette flowPoiseuille flow,zero stress at the centerline

( )2xy x

du y p h Ug ydy x h

τ μ ρ μ∂⎛ ⎞ ⎡ ⎤= = − + − +⎜ ⎟ ⎢ ⎥∂⎝ ⎠ ⎣ ⎦

The discharge (flow-rate) per unit width of channel is given by

3

0 12 2h

xp h hUQ udy gx

ρμ

∂⎛ ⎞= = − + +⎜ ⎟∂⎝ ⎠∫

The volume flow averaged (mean) velocity 2

/12 2x

p h Uu Q h gx

ρμ

∂⎛ ⎞= = − + +⎜ ⎟∂⎝ ⎠

It is left as an exercise for you to show the following Given that /p x−∂ ∂ is a positive constant and 0xg = , determine the location of the

maximum velocity. It is also the point where the shear stress vanishes (why?). Hence, find the minimum value of U such that the shear stress will not vanish throughout the flow.

C. Circular Poiseuille Flow

We now consider laminar flow through a circular tube: • The objective to find the relationship between volumetric flow rate and pressure

change along a pipe of circular section. • Examples include blood flow in capillaries, air flow in lung alveoli, where the

Reynolds number is not high enough for the flow to become turbulent. • Navier-Stokes equations in cylindrical coordinates are to be used, where

/ 0θ∂ ∂ = , since the flow is axially-symmetric (i.e., no dependence on angular position in a cross-section of the flow).

• We have seen that the gravity can be combined with the pressure gradient in a trivial manner, so let us ignore gravity in the following analysis.

Again, this is a unidirectional flow: 0, 0ru u uθ

r

( )zu r

Circular pipe of radius R

z

z= = ≠ is driven by a constant and steady pressure gradient dp/dz in the axial direction.

31

( )The continuity equation reduces to

1 rrur r∂∂

1 ur

θ

θ∂

+∂

0 must not depend on , ( )

The -component momentum equation is simplified to

zz z

z

u u z uz

z

ut

∂+ = ⇒ ∴ =∂

∂∂

zu r

zr

uur

∂+

∂zu u

rθ

θ∂

+∂

zz

uuz

∂+

∂2

2 2

1

1 1 z z

pz

u urr r r r

ρ

νθ

∂= −

∂

∂ ∂ ∂⎛ ⎞+ +⎜ ⎟∂ ∂ ∂⎝ ⎠

2

2zu

z∂

+∂ zg

⎡ ⎤+⎢ ⎥

⎢ ⎥⎣ ⎦

2

1 2

, which can be integrated twice with respect to to give

( ) ln4

z

z

d du r dpr rdr dr dz

r dpu r C r Cdz

μ

μ

⎛ ⎞⇒ =⎜ ⎟⎝ ⎠

= + +

The two integration constants C and can be determined using the boundary conditions:

1 2C

12

2

( 0) is finite 0

( ) 0 (no slip at boundary wall)4

z

z

u r CR dpu r R C

dzμ

= ⇒ =

= = ⇒ = −

Plugging back, we get the expression for the velocity profile

( )2 2

214z

dp R ru rdz Rμ

⎡ ⎤= − −⎢ ⎥

⎣ ⎦

which is a parabolic distribution with the maximum at the center: 2

max ( 0)4zR dpu u r

dzμ= = = −

The flow-rate is 4

02

8R

z zA

R dpQ u dA u rdrdz

= = = −∫ ∫ππμ

The mean velocity is half the maximum velocity

4

2max

28/ ..................... (1)

8 2

R dpuR dpdzu Q A R dz

πμ

π μ

−= = = − =

The shear stress at wall is given by

42

zw

r R

du R dp udr dz R

τ μ μ=

= − = − = For a given length L of the pipe, the pressure drop is ( )/p dp dzΔ = − L

g and the head loss

due to friction is h p /f ρ= Δ . Hence we may obtain from equation (1) the Darcy-Weisbach equation

32

2

( 2 diameter of pipe)2

64where the Darcy friction factor , and Re is the Reynolds number.Re

fL uh f D RD g

Duf ρμ

= = =

= =

Recall that the Moody diagram (attached below) provides the graphical functional dependence of the friction factor on the Reynolds number and the relative wall roughness. The above relation is represented by the straight line near the left end of the figure, where the flow is laminar, or the Reynolds number Re < 2,000. Note that for laminar flow, the friction loss is not affected by wall roughness ε .

The pipe flow becomes turbulent when Re > 4,000, for which the friction factor is given by the empirical Colebrook formula

101 / 2.512.0 log

3.7 ReD

f fε⎛ ⎞

= − +⎜ ⎟⎜ ⎟⎝ ⎠

,

which is graphically represented in the Moody diagram above.

33

D. Nearly One-Dimensional Flow by Lubrication Approximation Lubrication in a Slider Bearing A slider bearing is designed as a thrust bearing to support very large loads. To carry these loads, the fluid film between the solid surfaces must develop normal stresses, so we are interested in predicting the pressure distribution and thus the load-carrying capacity of the bearing. Typical examples of slider bearings are found in the shafts of screw-propelled ships and in the high-speed turbines of electricity-generating stations. For example, the thrust of the ship’s propellers may be transmitted through a series of pads (see the figure below) to the hull of the ship. Each pad (slider) may be tilted slightly to account for the relative effects of pressure, speed and viscosity, and thus maintain the fluid film between the two surfaces (the slider and its guide) which are in relative motion, and thereby reduce friction.

In most lubrication problems the relevant Reynolds number is so small that viscous terms in the Navier-Stokes equation dominate completely. The reason is not necessarily that the coefficient of viscosity is large; it is more due to the fact that the thickness of the film is extremely small compared to the lateral dimensions of the bearing. The Reynolds number may be defined as

slider speed film thickness 1Re ρμ

× ×=

Let us now formulate a model of the slider bearing with a planar face, with the further assumptions that 1. The lubricant is an incompressible Newtonian viscous fluid with constant viscosity. 2. The bearing has infinite length into the paper, and the bearing guide is flat. The gap

height h(x) between the slider and its guide varies so gently that the flow is nearly one-dimensional through a section of the bearing.

3. Gravity can be ignored. 4. The flow has settled down and we need consider only the steady problem.

46

The local film thickness is

2 11( ) h hh x h x

L−⎛ ⎞= + ⎜ ⎟

⎝ ⎠ (1)

where so that the film is extremely thin. 1 2h h L< The flow is quasi-one-dimensional, and we may recall the equation for discharge for combined Poiseuille-Couette flow (on page 31):

3

12 2dp h hUQdx μ

= − −

where the gravity has been ignored and U is now in the negative direction. By conservation of mass, Q must be a constant. If there were Couette flow alone, the discharge would decrease down the slider as the gap height decreases from to . Therefore in order to balance the flow, the pressure gradient must not be zero. Rearranging the terms the above equation gives

2h 1h

3

122

dp hU Qdx h

μ ⎛= − +⎜⎝ ⎠

⎞⎟ (2)

We further suppose that both ends of the bearing are exposed to surrounding lubricant, or to the atmosphere. Then we have

0( ) (0)p L p p= = as the boundary condition for the pressure. It follows that integrating (2) from 0 to L is equal to zero

0

2 30

( ) (0) 0

02

L

L

dp dx p L pdx

U Q dxh h

= − =

⎛ ⎞⇒ − + =⎜ ⎟⎝ ⎠

∫

∫

47

Now, on substituting (1) for h(x), the above integral can be carried out to give

( )

1 2

1 2

h hQ Uh h−

=+

We may put Q back into (2), which is integrated again, but the upper limit is now a general position “x”

0 2 30 0 ( ) 12

2x xdp U Qdx p x p dx

dx h hμ ⎛ ⎞= − = − +⎜ ⎟

⎝ ⎠∫ ∫

After some algebra, we get

( )(0 12 2 22 1

6( )( )

UL )2p x p h h h hh h h

μ= − − −

−

Note that sinc h 0e , 1 ( )h h x≤ ≤ 2 ( )p x p≥ or a positive pressure distribution is established within the bearing fluid to support the normal load. It can be readily shown that, by setting dp/dx = 0 in (2), the pressure reaches a maximum

( )( ) ( )

2 1 1max 0

1 2 1 2 1 2

3 at

2UL h h h Lp p x

h h h h h hμ −

− = =+ +

It may be shown that in the left-hand section of the bearing, the pressure gradient is positive so that it drives fluid in the flow direction, and in the right-hand section the pressure gradient is negative so that it drives fluid against the flow direction. In this way, the Poiseuille flow will balance the Couette flow to result in a constant discharge throughout the pad.

Lubrication performance is found to be favorable, since by using orders of magnitude we may estimate that

02

0

Drag shear stress / 1Bearing load pressure /

L

L

dx U h hUL h Lpdx

τ μμ

∫∫

∼ ∼ ∼ ∼

By virtue of the small film thickness, the bearing can support a large load with only small frictional resistance.

48

(III) INVISCID AND POTENTIAL FLOWS (Sections 6.4-6.6) Analysis can be considerably simplified if the flow under consideration can be regarded as INVISCID and IRROTATIONAL. A. Inviscid (Nonviscous) Flow (Section 6.4)

• Flow of an ideal fluid with zero viscosity ( )0μ = would be inviscid exactly. • In practice, flow is approximately inviscid when the effects of shear stresses on the

motion are small as compared to other influences. One guiding condition is that the Reynolds number Re must be very large:

viscous force 1 1 , where Reinertia force Re

VLρμ

=∼

• Many flows involving water or air, whose viscosity is small, can practically be considered as inviscid as long as the viscous effects are not dominant (e.g., far from a wall).

• When the viscous force becomes negligible, the Navier-Stokes equations reduce to Euler’s equations

nonlinear termis still here

1 (viscous term is missing here)pt ρ

∂+ = − +

∂V V V g∇ ∇i

• For incompressible flow, Euler’s equations of motion can be integrated along a streamline to yield the Bernoulli equation (which you learnt already in Year I; read Section 6.4.2 for a review)

2

constant along a streamline2

p zg gρ

+ + =V

• It is remarkable that the Bernoulli equation provides an algebraic (rather than vector differential) relationship between pressure, velocity and position in the earth’s gravitational field.

B. Irrotational (Potential) Flow (Section 6.4.3)

• Recall that vorticity (curl of velocity) is twice the rotation (angular velocity) of a fluid element.

• A fluid element will acquire vorticity when acted upon by a couple to cause it to rotate. One source of rotation is unbalanced shear stresses acting on its periphery. When shear stresses are absent, it is possible that the flow is irrotational.

• A flow field is irrotational if, at every point, the vorticity vanishes or 0∇×V = .

• It can be shown that the flow of an inviscid fluid which is irrotational at a particular instant of time remains irrotational for all subsequent times. That means, the motion of an inviscid fluid which is started from rest is always irrotational (provided the flow lies outside a boundary layer).

• This result is known as the Persistence of Irrotational Motion of an inviscid fluid. It is because the setting up of a rotation would require forces tangential to the boundary; and such forces, which arise through the viscous properties of the fluid, are non-existent in the inviscid fluid model.

49

• The constant in the Bernoulli equation becomes universal (i.e., not specific to a streamline) when the flow is irrotational (Section 6.4.4). Therefore, for incompressible irrotational flow, the Bernoulli equation can be applied between any two points in the flow field:

2 21 1 2 2

1 22 2p V p Vz zg g g gρ ρ

+ + = + +

• The procedures of finding a solution for an irrotational flow field are typically: o Firstly, solve for the kinematics (velocity components) from an equation

derived from the condition of zero vorticity, which is the subject matter of the following sections;

o Secondly, find the pressure from the Bernoulli equation. You should appreciate that solving irrotational flow equations is usually much simpler than solving the full Navier-Stokes equations.

• You are cautioned that irrotationality fails to apply to a boundary layer, which is a thin layer that develops next to a solid wall owing to no-slip of the flow at the wall. No matter how small its viscosity is, a real fluid cannot “slide” past a solid boundary. The flow in a boundary layer is always viscous and highly rotational (a rapid change in velocity from zero at wall to the free stream value over a short distance); real fluid behavior must be accounted for in a boundary layer (Chapter 9).

50

C. The Velocity Potential (Section 6.4.5) • For any scalar field φ , curl(gradφ ) = 0 is an identity. See a proof below.

• Alternatively speaking, a velocity field V is irrotational or curl V = 0 if and only if

there exists a scalar field φ such that V = grad φ . • The scalar function is called velocity potential

Cartesian coordinates: , ,

1Cylindrical coordinates: , , r z

u v wx y z

u u ur rθ

φ

φ φ φ

φ φθ

≡ ∇

∂ ∂= = =

∂ ∂

∂ ∂= =

∂ ∂

V

∂∂

zφ∂

=∂

Irrotational flow is therefore also called potential flow.

• The velocity potential satisfies Laplace’s equation on substituting the above relation into the continuity equation:

2 2 2

2 2 2

2 2

2 2 2

0 0, or 0

Cartesian coordinates: 0

1 1Cylindrical coordinates: 0

x y z

rr r r r z

φ φ

φ φ φ

φ φ φθ

= ⇒ = =

∂ ∂ ∂+ + =

∂ ∂ ∂

∂ ∂ ∂ ∂⎛ ⎞ + + =⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

i iV 2 ∇ ∇ ∇ ∇

• The immediate upshot is: for irrotational flow, one only needs to solve for a scalar

function (instead of a vector with 2 or 3 components) from one single equation in order to determine the kinematics (good news!). However, the differential equation for the scalar function is one order higher than that for the vector function (no free

51

lunch!). Once the potential is found, its spatial gradients will give the velocity components.

D. Equipotential Lines and Streamlines (Sections 6.2.3 and 6.5) • A two-dimensional potential flow field can be graphically represented using a flow

net composed of equipotential lines and streamlines. • Equipotential lines are (contour) lines of constant velocity potential, while streamlines

are lines in the flow field that are everywhere tangent to the velocity. It can be shown that these two sets of lines are orthogonal (i.e., they intersect each other at right angles).

• You may recall the following mathematical statement:

It follows that: equipotential lines

streamlines equipotential linesφ ≡ ⊥

⇒ ⊥V∇

The grad of a scalar function, say φ∇ , gives the maximum rate of spatial change of the function, and is in a direction normal to the local line along which the function is constant..

52

Figure 6.15 shows a flow net for a 90o bend. A flow net is useful in the visualization of a flow pattern. To further understand what information a flow net can provide, we need to know something about stream function.

Stream Function • For 2-D incompressible flow, another scalar function, viz stream function can be

introduced to identically satisfy the continuity equation.

( ) ( )A stream function , or , is defined such that

, for 2-D flow in Cartesian coordinates,

which satisfie

x y r

u vy x

ψ ψ θ

ψ ψ∂ ∂= = −

∂ ∂

( )

s 0 identically.

1 , for 2-D flow in Polar coordinates,

which satisfies 0 identical

r

r

u vx y

u ur r

urur

θ

θ

ψ ψθ

θ

∂ ∂+ =

∂ ∂

∂ ∂= = −

∂ ∂∂∂

+ =∂ ∂

ly.

Note that the stream function is introduced based on kinematics consideration only. It is definable for any two-dimensional incompressible flow fields, irrespective of the flow being inviscid or not.

• Physically, ψ is constant along a streamline since

V ψ is constant

d dx dy vdx udyx yψ ψψ ∂ ∂

= + = − +∂ ∂

That means, a line of constant ψ (along which 0dψ = ) will have its slope in the same direction of flow: . This is nothing but the defining property for a streamline.

/ /dy dx v u=

Note that a solid boundary is always a streamline. At a particular instant of time, there is no fluid crossing any streamline, and distinct streamlines cannot cross.

53

• Given any two points in space whose stream function values are known, then the volume flow rate across any line joining these two points is equal to the difference in values of their stream functions.

2

12 1

One can readiy see from the above figure that

dq udy vdx dy dx dy x

q dψ

ψ

ψ ψ ψ

ψ ψ ψ

∂ ∂= − = + =

∂ ∂

∴ = = −∫

• If the 2-D flow is irrotational, the stream function also satisfies Laplace’s equation, since

2 2

2 2

0 0

0

0

v ux y

x x y y

x y

ψ ψ

ψ ψ

∂ ∂× = ⇒ − =

∂ ∂

⎛ ⎞∂ ∂ ∂ ∂⎛ ⎞⇒ − − =⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠∂ ∂

⇒ + =∂ ∂

V∇

Therefore, for two-dimensional irrotational flow, both the velocity potential φ and the stream function ψ satisfy Laplace’s equation. They are called harmonic functions, and they are harmonic conjugates of each other. These functions are related, but their origins are different: – The stream function is defined by continuity; the Laplace equation for ψ results

from irrotationality. – The velocity potential is defined by irrotationality; the Laplace equation for φ

results from continuity. By now, referring back to Figure 6.15, you should understand that in a flow net the velocity is roughly given by

Vn sφ ψΔ Δ

≈ ≈Δ Δ

where is the spacing between two adjacent equipotential lines, and is the spacing between two adjacent streamlines. Therefore, the velocity is higher in a region where the mesh is finer, and lower where the mesh is coarser.

nΔ sΔ

54

E. Some Simple Plane Potential Flows (Sections 6.5.1-6.5.4) 1) Uniform Flow with constant velocity U

For case (a) where the flow is purely in the x-direction:

( )

( )

Velocity potential cos equipotential lines are parallel to the -axis

Stream function sin streamlines are parallel to the -axis

Ux Ury

Uy Urx

φ θ

ψ θ

= =

∴

= =

∴

Can you write down the corresponding φ and ψ for case (b) where the flow is at an angle α with the x-axis?

2) Source and Sink A 2-D source is a line (from a mathematical perspective) that runs perpendicular to the plane of flow and injects fluid equally in all directions. The figure shows the flow field of a source at the origin, from which fluid particles emerge and follow radial pathlines. The strength of a source, denoted by m, is the volume rate of flow emanating from unit length of the line.

/m V L=

55

By conservation of mass, 2 rm ruπ= for any radial distance r from the source located at the

origin. Hence, 12mu

r r1, 0r u

r r rθφ ψ φ ψ

θ∂ ∂

= = = = − =∂ ∂θ π

∂ ∂=

∂ ∂. On integrating,

( )

( )

Velocity potential ln 2

equipotential lines are concentric circles centered on the origin

Stream function 2

streamlines are radial lines

m r

m

φπ

ψ θπ

=

∴

=

∴

The radial and tangential velocities are:

02rmu u

r θπ= =

o when 0m > , the flow is radially outward, the origin is a SOURCE o when 0m < , the flow is radially inward, the origin is a SINK o the origin is a singularity where ru → ∞ o conservation of mass is satisfied everywhere except the origin

3) Vortex In contrast to a source, a vortex has the pathlines being circles centered on the origin, and fluid particles move along these circles. The vortex can be used to model the flow round the plughole in a bathtub. An irrotational vortex is called a free vortex. The strength of a vortex is measured by the circulation around a closed curve C

that encloses the center of the vortex. Hence, C

dΓ = ∫V si

1 12r r r r r rθ0, u uφ ψ φ ψ Γ

θ θ π∂ ∂ ∂ ∂

= = = = = − =∂ ∂ ∂ ∂

.

On integrating,

( )

( )

Velocity potential 2

equipotential lines are radial lines

Stream function ln 2

streamlines are concentric circles centered on the origin

r

φ θπ

ψπ

Γ=

∴

Γ= −

∴

The radial and tangential velocities for a free vortex are

0 2ru u

rθ πΓ

= =

56

o the flow is not defined at the origin o the vorticity curl V = 0, except at r = 0, where V is not defined o free vortex (a) is irrotational flow, tangential velocity decreases radially 1u rθ

−∝ o forced vortex (b) is rotational flow, tangential velocity increases radially u rθ ∝

4) Doublet Consider a combination of a source and a sink of equal strength m and separated at a distance 2a (left figure):

If the source and sink are moved indefinitely closer together ( )0a → in such a way that the product 2am (distance apart × strength) is kept finite and constant, then we obtain a doublet. The streamline pattern for a doublet is shown in the right figure above. The line joining the source to the sink is called the axis of the doublet, and is taken to be positive in the direction from sink to source. The strength of the doublet is /K ma π= .

57

( )

cosVelocity potential

equipotential lines are circles through the origin tangent to the -axis

sin Stream function

streamlines are circles thro

Kr

y

Kr

θφ

θψ

=

∴

= −

∴( )ugh the origin tangent to the -axisx

The basic potential flows that have been discussed so far are more mathematical constructions than physically realistic entities (although a source/sink may represent the flow field of an injection/withdrawing well, and so on). However a combination of these basic potential flows may provide a representation of some flow fields of practical interest. This is the subject matter for the next section.

F. Superposition of Basic Potential Flows (Sections 6.6) Let us first be reminded that for inviscid flow, a solid boundary is a streamline, and conversely, a streamline can be considered as a solid boundary. The kinematic conditions along the two are the same: normal velocity = 0. In fact, we may replace any streamline in a flow field by an impermeable surface without disturbing the flow. Since the governing equation (Laplace’s equation) for potential flow is linear, superposition of solutions gives the solution to the combined effect. In the following

58

examples, you will see how ideal flows can be described by a combination of basic solutions. The key thing is to locate the dividing streamline. The general procedures are as follows. (1) Sketch some streamlines for the combined flow. (2) Find the location of a stagnation point where the velocity vanishes (you may expect that when flow past a body, there is a point somewhere on the body surface where the flow velocity is zero). (3) Evaluate the stream function at the stagnation point stagnationψ . (4) The dividing streamline, which passes through the stagnation point, can be determined by letting the stream function be equal to the stagnation stream function ( ),rψ θ = stagnationψ .

1) Source + Uniform Flow = Flow Past a Half Body

It is more convenient to use polar coordinates ( ),r θ where r is the radial distance from the source. It is along the negative x-axis where the flow due the uniform flow is directly opposite to that due the source. At a point x b= − the velocities due to the two flows cancel each other, and this is identified as the stagnation point.

( ) uniform flow velocity radial outward flow due to source

2 2

rU um mU b

b Uπ π

r b∴ = =

⇒ = ⇒ =

( ) uniform flow sourceCombined stream function ,

2

rmUy

ψ θ ψ ψ

θπ

= +

= +

sin2mUr θ θπ

= +

The stream function at the stagnation point has the value

( )stagnation , 2mr b bUψ ψ θ π π= = = = =

Therefore the dividing streamline that passes through the stagnation point is given by

( )

( )

( )

stagnation ,

or sin2 2

or sin

or

r

m mUr

br

y b

ψ θ ψ

θ θπ

π θθ

π θ

=

+ =

−=

= −

59

This streamline, which has the shape shown above, can be considered as a solid boundary of a half-body that extends from x b= − to x → +∞ . The flow exterior to this streamline represents the flow past a half-body, whose thickness at large x can be estimated to be 2 bπ , since

0

as 2

by

bπ

θπ π

⎧ ⎧→ →⎨ ⎨−⎩ ⎩

Note that the every fluid particle emanated from the source is completely enclosed within the dividing streamline. The flow pattern around the half-body is described by streamlines stagnationψ ψ> . The velocity components and the pressure can then be determined as described in earlier sections.

2) Source + Sink + Uniform Flow = Flow Past a Rankine Oval

The source and the sink are of the same strength: any mass of fluid injected by the source is eventually drawn into the sink. The dividing streamline is now a closed curve. This finite body, called Rankine Oval, has two stagnation points, one at the front end and the other at the rear end of its boundary.

3) Doublet + Uniform Flow = Flow Past a Circular Cylinder As the source and the sink combine to become a doublet, the Rankine Oval becomes a circular cylinder. As the flow past a circular cylinder is of fundamental interest, let us examine the flow in some detail.

60

The combined stream function is sinsin

( strength of doublet)

KUrr

K

θψ θ= −

=

The radial velocity is 2

1 cosrKu U

r rψ θθ

∂ ⎛ ⎞= = −⎜ ⎟∂ ⎝ ⎠

Obviously, the radial velocity vanishes at a circular surface with the radius

( )1/ 2/r a K U= =

This defines the dividing streamline, which represents the surface of a circular cylinder of radius a. Substituting a for K, the stream function can be written as

2

21 sinaUrr

ψ θ⎛ ⎞

= −⎜ ⎟⎝ ⎠

from which we obtain the velocity components

2

2

2

2

1 1 cos

1 sin

rau U

r r

au Ur rθ

ψ θθ

ψ θ

⎛ ⎞∂= = −⎜ ⎟∂ ⎝ ⎠

⎛ ⎞∂= − = − +⎜ ⎟∂ ⎝ ⎠

On the cylinder surface the tangential velocity is ,r a= 2 sinsu Uθ θ= − . As expected, there are 2 stagnation points, at 0, θ π= . The pressure distribution on the cylinder surface can be found from the Bernoulli equation

( )

2 20

2 20

1 1 2 2

1 1 4sin2

s s

s

p U p u

p p U

θρ ρ

ρ θ

+ = +

⇒ − = −

where 0p is the far upstream pressure. It is remarkable that the pressure distribution is symmetrical about the horizontal and the vertical diameters. Therefore there is no net force arising from the pressure distribution around the cylinder in both streamwise and lateral directions. In other words, both drag and lift forces are exactly zero, as predicted from the potential flow theory. This zero drag prediction is contrary to what has been observed in reality. There is always a significant drag developed on a cylinder when it is placed in a stream of moving fluid. This discrepancy is called d’Alembert’s Paradox, which was not explained until the concepts of boundary layer and flow separation were developed. A comparison between the inviscid and the real pressure distributions is shown above.

61

http://en.wikipedia.org/wiki/D'Alembert's_paradox

4) Free Vortex + Doublet + Uniform Flow = Flow Past a Rotating Circular Cylinder The effect of adding a vortex is to upset the symmetry of flow about the horizontal diameter. Therefore, the pressure in the upper half of the cylinder is not balanced by the pressure in the lower half. This results in a net lift force acting laterally on the cylinder.

5) Sink + Free Vortex = Spiral Flow

6) Two separated sources of equal strength = source flow with a neighboring wall

62

(IV) FLOW PAST A BODY AND BOUNDARY LAYER THEORY (Chapter 9)

A. Introduction (Section 9.1.2)

In 1904, Prandtl developed the concept of the boundary layer, which provides an important link between ideal-fluid flow (inviscid irrotational flow) and real-fluid flow (viscous rotational flow). It was accepted that for fluids with relatively small viscosity (or more exactly, flow with a high Reynolds number), the effect of internal friction in the fluid is appreciable only in a narrow region surrounding the fluid boundaries. Therefore the flow sufficiently far away from the solid boundaries may be considered as ideal flow (in which effects of viscosity are neglected). However, flow near the boundaries suffers retardation by the boundary shear forces and at the boundaries the velocity is zero (no-slip condition). A steep velocity gradient is therefore resulted in a thin layer adjacent to the boundaries, which is known as the boundary layer. It is of great significance when behavior of real fluid is considered. For example, it explains the d’Alembert’s paradox – the drag force experienced by a cylinder in stream that cannot be predicted with a potential theory.

Flow of a uniform stream parallel to a flat plate. The larger the Reynolds number, the thinner the boundary layer along the plate at a given x-location.

75

Flow past a circular cylinder; the boundary layer separates from the surface of the body in the wake for large Reynolds number.

B. Description of the Boundary Layer (Section 9.2.1) (1) Development of the Boundary Layer

nominal limit of boundary layer u = 0.99 U

y

76

• On-coming flow is irrotationa• The boundary layer starts out

move in smooth layers and thflow moves on, the continualfluid particles, causing the bodownstream from the leadingthickness.

• The flow within the boundaryforces. The velocity gradient wall, and decreases with distawith the main stream flow. Rthe boundary layer, but is irro

• As the thickness of laminar beddying commences. These ctransition zone.

• It finally transforms into a turhaphazard paths. Due to the tuniform than that in the laminplate continues indefinitely bnegligible roughness size), lasub-layer in immediate contapart of the velocity change oc

Coturbbou(lefby thicvar

U

l and has a uniform velocity U. as a laminar boundary layer, in which fluid particles e velocity distribution is approximately parabolic. As the action of shear stress tends to slow down additional undary layer thickness to increase with distance edge. See below for a definition of the boundary layer

layer is subject to wall shear, and dominated by viscous (hence the rotation of fluid particles) is the largest at the nce away from the wall, and tends to zero on matching

oughly speaking, the flow is said to be rotational within tational outside the boundary layer. oundary layer increases, it becomes unstable and some hanges take place over a short length known as the

bulent boundary layer, in which particles move in urbulent mixing, the velocity distribution is much more ar boundary layer. The increase of thickness along the

ut with a diminishing rate. If the plate is smooth (i.e., minar flow persists in a very thin film called the viscous ct with the plate and it is in this sub-layer that the greater curs.

mparison of laminar and ulent flat plate ndary layer profiles t: non-dimensionalized the boundary layer kness; right: in physical iables).

77

(2) Thicknesses of the Boundary Layer i) Boundary Layer Thickness δ The velocity within the boundary layer increases to the velocity of the main stream asymptotically. It is conventional to define the boundary layer thickness δ as the distance from the boundary at which the velocity is 99% of the main stream velocity.

There are other ‘thicknesses’, precisely defined by mathematical expressions, which are measures of the effect of the boundary layer on the flow.

ii) Displacement Thickness δ∗ It is defined by

*

01 u dy

Uδ

∞ = − ∫

δ∗ is the distance by which the boundary surface would have to be shifted outward if the fluid were frictionless and carried at the same mass flowrate as the actual viscous flow. It also represents the outward displacement of the streamlines caused by the viscous effects on the plate. Conceptually one may ‘add’ this displacement thickness to the actual wall and treat the flow over the ‘thickened’ body as an inviscid flow. iii) Momentum Thickness θ It is defined by

0

1u u dyU U

θ∞ = −

∫ θ is the thickness of a layer of the main stream whose flux of momentum equals the deficiency in the boundary layer, equivalent to the loss of momentum flux per unit width divided by 2Uρ due to the presence of the growing boundary layer. The momentum thickness is often used when determining the drag on an object. Note that when evaluating the above integrals for *δ and θ, the upper integration limit can practically be replaced by δ.

78

C. Laminar Boundary Layer Over a Flat Plate • Heuristic Analysis

x

δ(x) U leading edge

of plate

Consider steady flow past a flat plate at zero incidence. The effect of viscosity is to diffuse momentum normal to the plate. Consider a fluid element that is close enough to the wall to be influenced by viscosity. In travelling a distance x, it has been influenced by viscosity for a time . The influence of viscosity will have spread laterally to a distance

/t x U∼

( )

( )

1/ 21/ 2

1/ 21/ 2

or Rex

xtU

x Ux

νδ ν

δ ν −

≡

∼ ∼

∼

The above analysis is rather crude, and does not yield a full equation for the growth of the boundary layer thickness. It however correctly describes one important relationship for the laminar boundary layer: where ( ) 1/ 2/ Rexxδ −∝ Re /x Ux ν≡

x

is the local Reynolds number in terms of the distance from the leading edge x. This relationship is found to be valid at a distance far behind the leading edge: / 1.δ The heuristic analysis can be further carried on to find relations for the wall stress:

1/ 23

0

or ww

y

u Uy x

τ ντ ρν νρ δ=

∂= ∂

∼ ∼ U The wall shear stress wτ decreases with increase of x until the boundary layer turns turbulent. The local friction coefficient, which is defined as follows, is given by

( ) 1/ 21 22

2 Rewf xC

Uτρ

−≡ ∼ While the numerical factor of 2 is far from the true value, the functional dependence of Cf on Rex is correctly predicted.

79

• Exact Solution by Blasius (Section 9.2.2) A more rigorous analysis, using the technique of similarity solution, was developed by Blasius for the laminar boundary layer over a flat plate. While the details of the analysis are beyond the scope of this course, it is important to note the following results derived from Blasius’ solution.

( )( )

( )

( )

1/ 2

1 122

1 122

5boundary layer thickness Re

0.664friction coefficient Re

1.328drag coefficient Re

x

wf

x

DL

xx

CU

CU L

δ

τρ

ρ

=

≡ =

≡ =D

/ 2

/ 2

1

where D is the skin friction drag force on unit width of a plate of length L:

. 0

L

wdxτ= ∫D

D. The Boundary Layer Momentum-Integral Equation (Section 9.2.3) By virtue of the property that the boundary layer thickness δ is much smaller than the streamwise length scale (say, L): / Lδ , one may simplify the Navier-Stokes equations to obtain the boundary-layer approximation:

2

2

continuity 0

1-momentum

1-momentum 0

u vx y

u u px u v ux y x

pyy

νρ

ρ

∂ ∂ + = ∂ ∂

y∂ ∂ ∂

+ = − +∂

∂ ∂ ∂ ∂

= −∂

∂

with the boundary conditions:

( , ) (0,0) at 0

, as (where , are the velocity and pressure of the inviscid flow just outside the boundary layer)

u v yu U p P y

U P

= == = → ∞

From the y-momentum equation, it is clear that the pressure in the boundary layer is constant laterally across the layer and equal to the near-wall pressure of the inviscid flow outside the boundary layer. On integrating the x-momentum equation with respect to y from y = 0 to y = δ, and after some algebra including the use of the continuity equation, one may obtain the Karman momentum integral equation

80

( )2 *

where wall shear stress density near-wall velocity of the outer inviscid flow

moment

w

w

d dUU Udx dx

U

τ ρ θ ρ δ

τρ

θ

= +

===

=0

*

0

um thickness 1

displacement thickness 1

boundary layer thickness

u u dyU U

u dyU

δ

δδ

δ

= − = =

=

∫

∫ −

0

This momentum integral equation is applicable to laminar, transitional or turbulent boundary layer. In particular, in the absence of pressure gradient (e.g., flow over a flat plate), the free stream velocity U = constant and dU , and therefore the momentum integral equation reduces to / dx =

2w

dUdxθτ ρ=

by which the skin friction drag and drag coefficient are simply given by

2

2 21 1 20 02 2

2, 2L L L L

w L D U LUddx U dx U C

dx U L Lρρ θ θθτ ρ ρ θρ

= = = = = =∫ ∫D

D

where Lθ is the momentum thickness at x L= . (1) Laminar Boundary Layer Over a Flat Plate Revisited – approximate solution by

momentum integral equation

It is remarkable that approximate solutions, which are reasonably close to the exact ones, can be obtained for the boundary layer thickness and drag coefficients from the momentum integral equation on adopting an assumed velocity profile

( )

( )

where is the -coordinate normalized with respect to the local boundary

layer thickness.

u fU

y yx

η

ηδ

=

=

The steps are as follows:- a) Find the relation between θ and δ

( ) (1

0 01 1 , is a numerical constantu u dy f f d a a

U Ua

δθ δ η δ = − = − =

∫ ∫ )

81

b) Find the wall shear stress from Newton’s law of viscosity

( )

0 0

0

//

is another numerical constant

wy y

b

du U u Udy y

U df Ub bd η

τ ρν ρνδ δ

ρνρνδ η δ

= =

=

∂= =

∂

= =

c) Substitute θ and wτ into the momentum integral equation

2

2

1 2

U db a Udx

b d daU dx dx

ρν δρδ

ν δ δδ

=

⇒ = =

Integrating the above equation with respect to x, assuming that δ = 0 at x = 0:

( )

2

1/ 2

2

2 / Rex

b xaU

b ax

νδ

δ

=

⇒ =

Furthermore,

( )

( )

1 122

1 122

2friction coefficient Re

2 2drag coefficient Re

wf

x

DL

abCU

abCU L

τρ

ρ

≡ =

≡ =D

/ 2

/ 2

)

It turns out that the values of a and b are rather insensitive to the choice of the approximate velocity profile u U/ (f η= as long as it is a reasonable one satisfying the boundary conditions. Some assumed velocity profiles are

( )

2

3

parabolic

cubic2 2

sine2

η ηη

πη

−

23

sin

f η η

= −

which satisfy

( )( )( )

0 0, (no-slip at 0)

1 1, ( at )

' 1 0 (no stress at )

f y

f u U y

f y

δ

δ

= =

= = =

= =

.

82

(2) Turbulent Boundary Layer Over a Flat Plate (Section 9.2.5) The one-seven-power law, suggested by Prandtl, is used for the velocity profile in the turbulent boundary layer with zero pressure gradient:

( )1/ 7

1/ 7 or u y fU

η ηδ

= =

by which the momentum thickness is

( ) ( )1 1 1/ 7 1/ 7

0 0

71 172

f f d dθ δ η δ η η η= − = − =∫ ∫ δ The one-seven-power law fails to describe the velocity profile at , where

. The following empirical formula obtained for pipe flow can be adopted here:

0y =/u y∂ ∂ → ∞

1/ 4

20.0225w UUντ ρδ

=

Substituting θ and wτ into the momentum integral equation, and integrating with respect to x:

1/ 4

5/ 44 72 0.0225 constant5 7

xUνδ × = +

It is assumed that the turbulent boundary layer starts from 0.x = (This is a contradiction to the fact that the boundary layer starts out as a laminar one, but this assumption has given good results.) Therefore, the constant = 0. Further simplification yields

( ) ( ) ( )1/5 1/5 1/52

0.370 0.0288 0.072, , Re Re Re

wD

x x

Cx U

L

τδρ

= = = .

These results are valid for smooth flat plates with 5 1 . 5 70 Re 10L× < < Note that for the turbulent boundary layer flow the boundary layer thickness increases with x as 4/5xδ ∼ and the shear stress decreases as 1/5

w xτ −∼ . For laminar flow these dependencies are 1/ 2x and 1/ 2x− , respectively.

83

E. Effect of Pressure Gradient (Section 9.2.6) The pressure in the streamwise direction (i.e., along the body surface) will not be constant if the body is not a flat plate. Consequently, the free stream velocity at the edge of the boundary layer U is also not a constant but a function of x. Whether the free-stream flow is accelerating or decelerating along the body surface will have dramatically different effects on the development of the boundary layer. Let us re-examine flow past a circular cylinder, and find out what causes d’Alembert’s paradox. You may recall that inviscid flow past a circular cylinder has a symmetrical pressure distribution around the surface of the cylinder about the vertical axis. This results in a zero pressure drag, which is however not true in reality for any fluid with a finite viscosity. Such discrepancy is now referred to as d’Alembert’s paradox. Despite the discrepancy, the potential theory helps to reveal that the pressure and hence the free-stream velocity Ufs on the cylinder’s surface are not constant. From A to C, the pressure gradient is negative and the flow is accelerating, and from C to F, the opposite is true.

84

The real fluid flow past a circular cylinder is like this:

Flow Past A-B-C

the streamlines are converging, i.e., flow is accelerating, and the free-stream velocity U reaches a maximum at C.

•

• the pressure is decreasing along the cylinder surface, i.e., / 0p x∂ ∂ < , net pressure force is in forward direction, and the pressure gradient is said to be ‘favorable’. the accelerating flow tends to offset the ‘slowing down’ effect of the boundary on the fluid. Therefore, the rate of boundary layer thickening decreases and flow remains stable.

•

Flow Past C-D

the streamlines are diverging, and the flow is retarding. • • the pressure is increasing along the cylinder surface, i.e., / 0p x∂ ∂ > , net pressure

force opposes the flow, and the pressure gradient is said to be ‘adverse’ or ‘unfavorable’.

•

y

it reduces the energy and forward momentum of the fluid particles in proximity to the surface, causing the thickness to increase sharply and fluid near the surface be brought to a standstill ( /u∂ ∂ at the surface is zero) at D. See figure (b).

85

Flow Past D-E-F flow close to the cylinder surface starts to reverse at D (separate point), i.e., fluid no longer to follow the contour of the surface. The phenomenon is termed separation.

•

•

•

large irregular eddies formed in the reverse flow (the wake), in which much energy is lost to heat. the pressure in the wake remains approximately the same as at the separation point D, and is therefore lower than that predicted by the inviscid theory (see figure c). This lowering of pressure behind the cylinder resulting from flow separation leads to a net pressure drag on the cylinder. This explain d’Alembert’s paradox. Note that the wider the wake, the larger the pressure drag, and vice versa.

Influence of the pressure gradient

86

(a) (b)

Influence of a strong pressure gradient on a turbulent flow: (a) flow is relaminarized by a negative (favorable) pressure gradient; (b) the boundary layer is thickened by a positive (unfavorable) pressure gradient.

Further remarks about flow separation •

•

upper)

separation can occur only under an adverse pressure gradient and when the fluid is viscous. separation occurs with both laminar and turbulent boundary layers. Laminar boundary layer is more prone to separation than turbulent boundary layer. Thus, as shown in figure (c) on page 85, the turbulent boundary layer can flow farther around the cylinder before it separates than can the laminar boundary layer. Therefore the wake size will be narrower if the flow is turbulent at the separation point than if it is laminar. This explains why it is desirable to have dimples on a golf ball, which can effectively reduce the drag by inducing a narrower turbulent wake behind the ball.

Turbulent boundary layers are more resistant to flow separation than are laminar boundary layers exposed to the same adverse pressure gradient. The laminar boundary layer (cannot negotiate the sharp turn of 20o, and separates at the corner (flow is from left to right). The turbulent boundary layer (lower) on the other hand manages to remain attached around the sharp corner.

87

F. Drag (Section 9.3) Any object moving through a fluid (or a stationary object immersed in a viscous flow) will experience a drag, – a net force in the direction of flow due to the pressure and shear forces on the surface of the object.

D

Drag = Pressure Drag + Skin Friction Drag where Pressure Drag = resultant force arising from the non-uniform and asymmetrical pressure

distribution around the surface the body. It is also called form drag as it depends on the form or the shape of the body.

Skin Friction Drag = resultant force due to fluid shear stress on the surface of the object.

cos wp dA sin dAθ τ θ= +∫ ∫D

The drag coefficient CD is given by the ratio of the total drag force to the dynamic force

1 22

DCU Aρ

=D

where U = relative velocity of fluid far upstream of the object, A = frontal area – the projected area of the object when viewed from a direction

parallel to the oncoming flow if it is a blunt (or bluff) object (e.g., a cylinder); or the planform area – the projected area of the object when viewed from above it if it is a streamlined object (e.g., a flat plate).

drag

88

Typically the drag coefficient depends on (i) the shape of the object, (ii) orientation of the object with the flow (e.g., a flat plate normal to flow has a

different CD than a flat plate parallel to flow), (iii) the Reynolds number Re /UD ν= where D is a characteristic dimension of the

object, (iv) surface roughness if the drag is dominated by skin friction and the boundary

layer is turbulent. Flow Past a Flat Plate When a flat plate is held normal to flow, the flow is separated upon past over the plate. A region of eddying motion (wake) is formed at the rear of the plate, the pressure there being much reduced. Therefore the pressure drag is dominant, and the plate is a bluff body in this position. The drag shows little dependence on the Reynolds number. When a flat plate is held parallel to flow, formation of the boundary layer over the plate is appreciable and flow separation is negligible. Therefore the skin friction drag is significant. The plate is a streamlined body in this position. The drag coefficient increases when the boundary layer becomes turbulent. Flow Past a Circular Cylinder/Sphere

89

• Re ≤ 1

– creeping flow – no flow separation – DC decreases with increasing Re ( 64 / ReDC = for a sphere) (Note that a decrease in the drag coefficient with Re does not necessarily imply a corresponding decrease in drag. The drag force is proportional to the square of the velocity, and the increase in velocity at higher Re will usually more than offset the decrease in the drag coefficient.)

• Re = 10 – separation starts occurring on the rear of the body forming a pair of vortex bubbles

there – vortex shedding begins at Re ≅ 90, leading to an oscillating Karman vortex street

wake (see next page) – region of separation increases with increasing Re – DC continues to decrease with increasing Re until Re = 103, at which pressure

drag dominates • 3 510 Re 10< <

– DC remains relatively constant, which is a characteristic behavior of blunt bodies – flow in the boundary layer is laminar, but the flow in the separated region is

highly turbulent, thereby a wide turbulent wake • 5 610 Re 10< <

– a sudden drop in DC somewhere within this range of Re – this large reduction in DC is due to the flow in the boundary layer becoming

turbulent, which moves the separation point further on the rear of the body, reducing the size of the wake and hence the magnitude of the pressure drag. This is in sharp contrast to streamlined bodies, which experience an increase in the drag coefficient (mostly due to skin friction drag) when the boundary layer turns turbulent.

90

Laminar boundary layer separation with a turbulent wake for flow past a circular cylinder at Re = 2000.

(a) (b)

low over (a) a smooth sphere at Re = 15,000, and (b) a sphere at Re = 30,000 with a trip ire; the delay of boundary layer separation is clearly seen by comparing these two hotographs. The delay of separation in turbulent flow is caused by the rapid fluctuations f the fluid in the transverse direction, which enables the turbulent boundary layer to avel farther along the surface before separation occurs, resulting in a narrower wake and smaller pressure drag. Recall also that turbulent flow has a fuller velocity profile as ompared to the laminar case, and thus it requires a stronger adverse pressure gradient to vercome the additional momentum close to the wall.

Fwpotraco Karman Vortex Streets T

91

he Karman vortex street is one of the best-known vortex patterns in fluid mechanics.

al to

res.

eriodic flow is that the forces on the ecause the flow

ually

The vortex street is just a special type of unsteady separation over bluff bodies such as a cylinder. The vortex street is highly periodic having a frequency which is proportionU/D, where D is the length of the bluff body measured transverse to the flow and U is the incoming flow speed. This periodicity is responsible for the "singing" of telephone wiIn fact, vortex streets are almost always involved when the wind generates a fairly pure tone as it blows over obstacles.

A practical consequence of the regular, pbody are also periodic. Bis asymmetric fore and aft as well as in the direction transverse to the flow, the body will experience both an oscillating drag and lift. If the frequency of the shedding is close to a structural frequency, resonance can occur, uswith unpleasant results.

(V) OPEN-CHANNEL FLOW (Chapter 10)

A. Introduction • Studies of open channel flow are important for design and planning of river control,

inland navigation, surface drainage, irrigation, water supply and urban sanitation. • An open channel is a conduit in which the liquid flows with a free surface subjected to

atmospheric pressure. It includes river channels (natural water courses), constructed channels (e.g., canals, flumes) and enclosed conduits (e.g., sewers, culverts) operating partially full.

• Open channels normally have a very small slope 0.01S < , and the pressure variation with depth is nearly hydrostatic. The hydraulic grade lines for all the streamtubes are the same and coincide with the free surface of the flow. Open channel flow is essentially caused by slope of the channel and self-weight of the liquid.

B. Types of Flow 1. Uniform and Non-Uniform Flow Uniform flow – mean velocity does not change (in both magnitude and direction) from

one section to another along the channel. With a free surface this implies a constant cross section and flow depth, which is called the normal depth. Hence the liquid surface is parallel to the base of the channel. Uniform flow results from an exact balance between the gravity and frictional effects. (NB: uniform channel flow however does not require uniformity of velocity across any one section of the flow.)

Non-uniform (varied) flow – the mean velocity V and the fluid depth y change with

distance x along the channel. Non-uniform channel flow can be classified into gradually, or rapidly varying flow when the flow depth changes slowly ( )/dy dx 1 , or rapidly ( )/dy dx ∼1 with distance along the channel.

2. Steady and Unsteady Flow

The flow is unsteady or steady depending on whether or not the depth at a given location changes with time. Surface waves propagating on a channel is in fact unsteady, but may appear steady to an observer who travels at the same speed as the waves.

3. Laminar and Turbulent Flow The flow is laminar or turbulent depending on the magnitude of the Reynolds number, which for open channels can be defined as Re /hVR ν≡ , where hR is the hydraulic radius

103

equal to the ratio of the cross section area to the wetted perimeter. Open-channel flows in rivers, culverts, surface drainage, etc typically involve water as the fluid (with fairly small viscosity) and have relatively large dimensions so that the flows are invariably turbulent. For turbulent channel flow, the velocity profile is rather uniform (except close to the bottom and lateral boundaries) across a section of the flow, and therefore it is a common practice to use the section-averaged velocity V = V(x) as the primary variable in open channel flow.

4. Tranquil (Sub-critical) and Rapid (Super-critical) Flow Using the momentum principle, one may show that a small-amplitude surface wave (which is, say, caused by some small disturbance to the flow) will travel in a shallow pool of liquid at the speed c g= y , where y is the local fluid depth. An open channel flow may have sharply different behaviors when its flow velocity V is smaller or larger than this wave speed. The ratio of these two velocities is known as the Froude number, the magnitude of which corresponds to the following types of flow:

1 s⎧ ub-critical flow Fr 1 critical flow

1 super-critical flow

Vgy

<⎪≡ =⎨⎪>⎩

Briefly speaking, a sub-critical flow is so low in speed (thereby called tranquil) that a disturbance to the flow may send waves both upstream and downstream of the channel. In sharp contrast, a super-critical flow is a high speed flow (thereby called rapid) and a wave cannot be transmitted upstream.

C. Energy Considerations

For channels under consideration, the bottom slope is very small ( 0.001). Streamlines are therefore virtually straight and parallel, and pressure variation is hydrostatic. This implies

∼

Free surface = hydraulic grade line

Datum

Streamline

Energy line

Channel bed

Lh

2 / 2V g

/p gρ

z

104

that a point with pressure p is at a depth /p gρ below the free surface, so the sum of the pressure and elevation heads ( )/p g zρ + represents the height of the free surface above the datum level, or the hydraulic grade line coincides with the free surface.

l

Referring to the above figure, the energy equation may be applied to the two points (1) and (2) on the channel bed, giving

2 2

1 1 2 21 22 2 L

p V p Vz zg g g gρ ρ+ + = + + + h

where is the head loss due to frictional effects between the two sections. Clearly, Lh

1 1/p g yρ = , 2 2/p g yρ = , , and h1 2 oz z S− = l lL fS= , where S and 0 fS are respectively the slopes of the channel bed and the energy line. fS is also called the energy gradient, as it is the rate at which energy head is lost to friction. On substituting, the energy equation becomes

( ) ( )

( )

2 22 1

1 2 0

1 2 0

2

or

f

f

V Vy y S

g

E E S S l

−− = + −

= + −

S l

where the specific energy, E, is defined as

2

2VE y

g= + .

For uniform flow, the flow is invariant with distance along the channel, so 1 2y y= and

, and the energy equation gives 1V V= 2

0 (for uniform flow)fS S=

That is, the energy gradient is exactly equal to the geometrical gradient of the channel when the flow is uniform.

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D. Uniform Flow: Chezy-Manning Equation

Define: Normal depth ny = depth of uniform flow

Hydraulic radius hR = flow sectional area wetted perimeter

AP

=

There is no change in momentum along the channel when the flow is uniform, and therefore the net force acting on a control volume, as shown above, must be zero. By uniformity, the two end forces are equal . Consequently, the downward gravity force is exactly balanced by the bottom friction:

1F F= 2

( ) 0

0

sinw

w h

Pl W gAl SgR S

τ θ ρτ ρ

= =

⇒ =

By analogy with pipe flow, the wall stress wτ can be expressed as

2

8wf Vτ ρ=

where f is the friction factor, which for complete turbulent flow depends only on the roughness of channel. Combining these equations, we get the so-called Chezy equation for uniform flow 0hV C R S=

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where 8 /C g= f is the Chezy coefficient. This coefficient is found to be not a constant, but depends on the size and shape of the channel section, and on the roughness of the boundaries. The simplest and most widely used relation for C is the Manning formula:

1/ 6

hRCn

= where n is the Manning roughness coefficient. On substituting Manning formula into the Chezy equation, we get the so-called Chezy-Manning equation:

2/3 1/ 22 /3 1/ 20 h fh R SR SV

n n= =

for the velocity of a uniform channel flow. It follows that the flow rate is given by

2/3 1/ 2 5/3 1/ 2

0 02/3

hAR S A SQ VAn P

= = =n

. Note that the Manning roughness coefficient is NOT dimensionless. When SI units are used, it has the following values for different types of channels.

For a channel of rectangular cross section with a width b, the hydraulic radius

2 2 /h

yb yR1y b y b

= =+ +

In the limiting case of a very wide rectangular channel such that b , then y hR y≈ , or the hydraulic radius is approximately equal to the flow depth. In a very wide channel, the uniform flow velocity and the discharge per unit width of channel are therefore

2/3 1/ 2 5/3 1/ 22/3 1/ 2 5/3 1/ 20 0, n f n fn n

n

y S yy S y SQV q Vyn n b n n

= = = = = =S

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E. Specific Energy: Alternative Depth of Flow Let us recall the specific energy that has been introduced in Section C. The specific energy, E, is the sum of the pressure and velocity heads

2

2VE y

g= + .

Specific energy is also the energy head referred to the base of the channel. It is a very important concept in open-channel flow. While the specific energy is constant along the channel when the flow is uniform, it may increase or decrease down the channel when the flow is non-uniform. For the convenience of discussion, let us consider only rectangular cross section from here onward. The results can be extended to an arbitrary cross section, but will not be considered here. 1. Variation of Specific Energy with Flow Depth Since the velocity is related to the unit width discharge by /V q y= , we may write the specific energy as

2

22qE ygy

= +

Here the equation consists of three variables: E, y, q. It would be of interest to examine the cases: i) q is constant, and E varies with y; and ii) E is constant, and q varies with y. If q is kept constant, E varies with y in the following manner.

This is known as the specific energy diagram, in which the following points are notable.

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• For 0, , (asymptotic to the axis)qy V E Ey

→ = →∞ ∴ →∞ .

• For , 0, (asymptotic to the line )qy V E h y E= . y

→∞ = → ∴ →

• Between these two extremes, E declines to a minimum minE at the critical point: o depth and velocity at this point are called the critical depth cy and critical

velocity cV ; o critical depth represents the least possible specific energy with which the fixed

discharge q is able to flow in the channel of given slope. • For each other value of E greater than the minimum, there are two possible values of y,

one greater and one less than cy . The two corresponding depths for a given value minE E> are known as the alternative depths.

• Flow can be classified into

, flow is tranquil or sub-critical, flow is critical, flow is rapid or super-critical

c c

c c

c c

y y V Vy y V Vy y V V

> <= =< >

2. Criterion for Minimum Specific Energy Since the minimum E occurs when dE/dy = 0, we may readily obtain, using the relation for E given above, the following condition for critical flow

2

3

2 22

3

1 0

Fr 1

dE qdy gy

q Vgy gy

= − =

⇒ ≡ = =

Being consistent with our earlier discussion in Section B.4, the flow is critical, sub-critical or super-critical depending on the value of the Froude number. Combining all of the above,

min

min

min

1 , sub-critical flow, Fr 1 , critical flow,

1 , super-critical flow,

c c

c c

c c

y y V V E EV y y V V E Egy y y V V E E

< > < >⎧⎪≡ = = = =⎨⎪> < > >⎩

Also, the critical flow depth cy is given by

1/32 2

23Fr 1 cc

q qygy g

⎛ ⎞≡ = ⇒ = ⎜ ⎟

⎝ ⎠,

and hence the minimum specific energy is 2

min 2

32 2c c

c

qE ygy

= + = y .

It is remarkable that, for a given discharge, the critical flow occurs when the specific energy is the lowest possible value . minE

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3. Criterion for Maximum Discharge The equation for the specific energy may be rearranged into

( )1/ 22q g y E y= − If E is held constant, q varies with y as follows.

y

sub-critical flow

yc

super-critical flow

qqmax

critical flow

0

The discharge per unit width, q, reaches a maximum value at a particular depth. For each other value of , there are two possible values of flow depth y. Again, the criterion for this maximum is given by

maxq

maxq q<q / 0dq dy = , which after some algebra leads to

2max

3 1qgy

= ,

which is identical to the critical flow condition. The corresponding flow depth is therefore the critical depth cy . Hence, we may make another remark that for a given specific energy E, the critical flow occurs when the discharge is the largest possible value . maxq 4. Significance of Bed Slope The bed slope which is required to produce uniform flow in a channel operating at the critical depth is called the critical slope . This critical slope can be found by using the Chezy-Manning equation and the critical flow condition

0cS

2/3 1/ 2 2

00 4/3 hc c c

c c chc

R S gV gy Sn R

= = ⇒ =y n

)

It can be seen that the critical slope depends on the discharge and the boundary roughness.

If uniform slope occurs in a channel with bed slope less than the critical , the flow must be tranquil (sub-critical), and the slope is said to be mild (or sub-critical).

( 0 0cS S<

Likewise, with , the flow must be rapid (super-critical) and the slope is said to be steep (or super-critical).

0 0cS S>

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F. Gradually Varied Flow 1. Introduction • Uniform flow, in which a uniform flow depth is maintained, requires constancy of all

channel characteristics (i.e., cross-sectional shape, bed slope, roughness) throughout the flow. In natural streams, this condition is hardly attained and flow is invariably non-uniform in nature. Even for artificial channels with uniform cross section, etc., uniform flow is only a condition to be approached asymptotically. The surface of a varied (i.e., non-uniform) flow is not parallel to the bed and takes the form of a curve.

• Steady varied flow is broadly divided into two kinds:- o Gradually varied flow, in which changes of depth and velocity take place over a

long distance and degree of non-uniformity is very slight. Boundary friction is significant and is to be accounted for.

o Rapidly varied flow, in which the sectional area of flow changes abruptly within a short distance. Turbulent eddying loss is more important than boundary friction in this case. Hydraulic jump, which is a typical example of rapidly varied flow, will be examined in the next section.

• Gradually varied flow may result from o a change in cross-sectional shape, bed slope, boundary roughness of the channel. o the installation of control structures (e.g., sluice gate, weirs, etc.).

2. General Equation for Gradually Varied Flow When the flow is non-uniform, the specific energy is no longer a constant, but varies with distance, x, along the channel. It has been derived in Section C that

( ) ( ) ( )0 fE x x E x S S+ Δ − = − Δx where and 0S fS are respectively the slopes of the channel bed and the energy grade line. This relation implies that the specific energy gradient is equal to the difference between these two slopes.

0 fdE S Sdx

= −

On the other hand, by the definition of the specific energy,

( )2 2 2

22 31 1 F

2 2dE d V d q q dy dyy ydx dx g dx gy gy dx dx

⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + = + = − = −⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠r

Combining the two equations above, one may get the general equation governing the free surface gradient in a gradually varied flow:

. 0

21 FrfS S−dy

dx=

− Note that fS and Fr are functions of x, and the geometrical bed slope S may or may not change with x depending on construction. The equation above may be integrated numerically to yield the surface profile for any type of gradually varied flow. This kind of profile evaluation is normally carried out by a commercial package nowadays and will not be discussed here.

0

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3. Profile Classification The general surface profile equation can be utilized in establishing the various forms of varied flow profile. The equation may be rearranged into

00 2

1 /1 Fr

fS Sdy Sdx

−⎛ ⎞= ⎜ ⎟−⎝ ⎠

.

Let us consider a very wide rectangular channel. As noted earlier in Section D, when the flow is uniform,

2

0 5/3fn

qnS Sy

⎛ ⎞= = ⎜ ⎟

⎝ ⎠

where ny is the normal depth.

An assumption is made here that for gradually varied flow the energy gradient fS can be related to the local flow depth y in the same manner as in the above formula for uniform flow. Therefore, when the flow is gradually varied

( )2 2

10/30 05/3 5/3, / /f f

n

qn qnS S S S yy y

⎛ ⎞ ⎛ ⎞= = ⇒ =⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠n y .

On the other hand, the Froude number can be written as 32 2

23 3 3Fr 1c

c

yq qgy y gy

⎛ ⎞= = =⎜ ⎟

⎝ ⎠∵ .

Putting these relations back into the surface profile equation

( )( )

10/3

0 3

1 /1 /

n

c

y ydy Sdx y y

−=

−.

With the equation above, we may outline the various forms of varied flow profiles, with a classification based on the following features. Type of Bed Slope

Mild Slope (M): 0 0 , c nS S y y< > c

c

c

∞

∞

Steep Slope (S): 0 0 , c nS S y y> <

Critical Slope (C): 0 0 , c nS S y y= =

Horizontal Slope (H): 0 0, nS y= =

Adverse Slope (A): 0 0, nS y< =

( )y Relative to Normal Depth ( )ny and Critical Depth ( )cy Flow Depth

Type 1: , and n cy y y y> > 0 Backwater curvedydx

> ⇒

Type 2: is between and n cy y y 0 Dropdown curvedydx

< ⇒

Type 3: , and n cy y y y< < 0 Backwater curvedydx

> ⇒

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Hence, a flow occurring on a mild slope has an M1 curve when its depth is greater than the normal depth, and has an M3 curve when its depth is less than the critical depth, and so on. Further notes:- The critical and normal depth lines (CDL, NDL) and the channel bed form the boundaries of 3 zones. The curves of gradually varied flow surface profile approach each of these zone boundaries in a specific manner:

a) Upper limits of depth – the curves tend to become asymptotic to a horizontal water line.

b) Normal depth line – approached asymptotically (except for C curves).

c) Critical depth line – intersected at right angles (except for C curves).

d) Bed – intersected at right angles.

There are altogether 12 possible profiles of gradually varied flow, as depicted below. Channel

slope Depth

Relations dydx

Type of Profile Symbol Type of

Flow Form of Profile

n cy y y> > + Backwater 1M Sub-critical

n cy y y> > - Dropdown 2M Sub-critical

Mild

n cy y> > y + Backwater 3M Super-critical

c ny y y> = + Backwater 1C Sub-critical

c ny y y= = Parallel to bed 2C Uniform,

Critical Critical

c ny y y= > + Backwater 3C Super-critical

c ny y y> > + Backwater 1S Sub-critical

c ny y y> > - Dropdown 2S Super-critical Steep

c ny y y> > + Backwater 3S Super-critical

cy y> - Dropdown 2H Sub-critical Horizontal

cy y> + Backwater 3H Super-critical

cy y> - Dropdown 2A Sub-critical

Adverse cy y> + Backwater 3A Super-

critical

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G. Hydraulic Jump 1. Introduction

When a change from rapid (super-critical) to tranquil (sub-critical) flow occurs in open channel, a hydraulic jump appears, through which the depth increases abruptly in the direction of flow. In engineering practice, the hydraulic jump frequently appears downstream from overflow structures (spillways) or underflow structures (sluice gated) where velocities are high. It may be used as an effective dissipation of kinetic energy (and thus prevent scour) of channel bottom) or as a mixing device in water or sewage treatment designs where chemicals are added to the flow.

2. General Equation of Hydraulic Jump

In spite of the complex appearance of a hydraulic jump with its turbulence and air entrainment, it may be analyzed by application of the momentum equation. Consider flow in a rectangular channel, and apply momentum equation for the control volume of unit width between sections 1 and 2:

( )1 2 2 1F F q V Vρ− = − Substituting the following relations

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( )2 21 1 2 2

1 1, hydrostatic pressure variation with depth2 2

F gy F gyρ ρ= = ∵ ,

( )1 21 2

, by continuityq qV Vy y

= = ,

and after rearranging

2 22 21 2

1 22

2 21 2 1 2

2 22 22 2

1 31 1

221 1

22 2

2 2

2 0

2Fr 0 where Fr

or 2Fr 0

y yq qgy gy

qy y y yg

y y qy y g

y yy y

+ = +

⇒ + − =

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⇒ + − = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞+ − =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

y

1

which can be solved for 2 /y y and 1 / 2y y respectively

( )

( )

221

1

212

2

1 1 1 8Fr2

and1 1 1 8Fr2

yy

yy

= − + +

= − + +

These are the general equations for a hydraulic jump. Notes:-

• The depths 1y and 2y on either side of a jump are called conjugate depths for the jump.

• It can be shown by noting the Froude numbers that the upstream and downstream flow of a jump is always rapid (super-critical) and tranquil (sub-critical) respectively. From the equations above

( ) ( ) ( )

( ) ( ) ( )

1/ 222 1 2 1

1 2 1

1/ 221 2 1 2

2 1 2

/ /Fr 1

2

/ /Fr 1

2

y y y yy y

y y y yy y

⎡ ⎤+= >⎢ ⎥⎢ ⎥⎣ ⎦

⎡ ⎤+= <⎢ ⎥⎢ ⎥⎣ ⎦

∵

∵

>

<

• A super-critical flow may turn into sub-critical only through an abrupt change in the form of a hydraulic jump. Conversely, a sub-critical flow may turn into super-critical through a gradual and smooth transition.

3. Energy Loss in Hydraulic Jump As an excellent energy dissipator, the energy lost or power dissipated in a hydraulic jump needs to be quantified.

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If the head loss is , the balance of energy heads before and after the jump gives Lh

( )

( )

2 21 2

1 2

2 21 2 1 2

2

1 2 2 21 2

2 2 22 21 2 1 2

1 2 1 2 1 22 21 2

32 1

1 2

2 21

2

1 1 2

1 1 2 04

4

L

L

V Vy y hg g

h y y V Vg

qy yg y y

y y y y qy y y y y yy y g

y yy y

+ = + +

⇒ = − + −

⎛ ⎞= − + −⎜ ⎟

⎝ ⎠⎛ ⎞⎛ ⎞ ⎛ ⎞+

= − + − + − =⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠⎝ ⎠

−=

∵

which also equals the difference in specific energy across the jump. To ensure a positive magnitude, 2 1y y> and not vice versa. Therefore, a hydraulic jump is irreversible. The power dissipated per width of channel is LP gqhρ= .

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H. Some Examples of Composite-Flow Profiles • Downstream of a sluice gate

• A change of bed slope from steep to mild: the hydraulic jump may be formed either on the steep slope or on the mild slope depending on whether the downstream conjugate depth 2y is smaller or greater than the normal depth on the mild slope.

• Other types of change of bed slope

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• Free fall at the end of a channel

• Flow over a bump (bottom friction is ignored): the free surface over the bump is depressed or elevated when the flow is sub-critical or super-critical, respectively.

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When the flow before the bump is sub-critical (state 1a), the flow depth y2 decreases

s the bump height is increased, point 2 continues shifting to the left on the

(state 2a). Since the decrease in flow depth is always greater than the bump height (why?), the free surface will be suppressed. But if the flow before the bump is super-critical (state 1b), the flow depth rises over the bump (state 2b), creating a bigger bump over the free surface. The situation is reversed if the channel has a depression in its bed: the flow depth increases if the approach flow is sub-critical and decreases if it is super-critical. A bzΔ

m (specific energy diagra while point 1 remains unaffected), until finally reaching the critical point at which the specific energy is the minimum, and the flow over the bump is critical. This critical height of a bump is given by the difference between the original and the minimum specific energy levels 1 minbcz E EΔ = − . Since the specific energy has already reached the minimum level at flow over the bump will only remain critical even when the bump height is further increased. To overcome a bump of height greater than bcz

this state, the

Δ , the approach flow must adapt itself (say, either to increase the upstream energy level 1E or to reduce the flow rate so that

minE is decreased) so that the bump height 1 minE E= − is always satisfied. When this ens, the flow is said to be choked.

happ

he fact that flow over a sufficiently high obstruction in an open channel is always Tcritical is the working principle of weirs (broad-crested or sharp-crested), which are used to measure the volume flow rate in open channels.

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Fluid Mechanics 1 Notes

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