57401301 Ch 6 Differential Analysis of Fluid Flow

7/29/2019 57401301 Ch 6 Differential Analysis of Fluid Flow

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ME 311: Fluid Mechanics

Differential Analysis


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Course Outline Navier-Stokes equations for Laminar Flow

Characterization of Laminar and Turbulent Flow Reynold Stresses

Boundary layer theory

Flow over flat plate and in pipes

Lift and Drag Forces

Applying energy, momentum and continuityequations of Thermofluids to turbo-machinery,

Performance of Turbo-Machines.


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Recommended Text Books

1 Fundamentals of Fluid

Mechanics

Munson, Young

& Okiishi

2 Mechanics of Fluids B. S. Massey

3. Fluid Mechanics Victor L. Streeter

and E. Benjamin

Wylie

4. Mechanics of Fluids Merle C. Potter;

David C. Wiggert

5 Introduction to Fluid

Mechanics

Robert W. Fox ;

Alan T.

McDonald


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General Classroom Rules

Mutual respect (golden rule)

Punctuality

Minimal disturbance to fellow students and teacher

Turn off your cell phone

No chewing /tobacco

Questions are encouraged

No question is stupid

Your question is valuable to others in learning


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My Preference

Learning happens both inside and

outside the classroom

Inside classroom: interactive,participation

Outside classroom: Any time

Welcome feedback anytime during the

quarter (class format/materials/pace)

you are welcome to come to see me /

call me any time any where.


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Outline

Introduction

Kinematics Review Conservation of Mass

Stream Function

Linear Momentum Inviscid Flow

Viscous Flows

Navier-Stokes Equations

Exact Solutions

Intro. to Computational Fluid Dynamics

Examples


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Differential Analysis: Introduction

Some problems require more detailed analysis.

We apply the analysis to an infinitesimal control

volume or at a point. The governing equations are differential equations

and provide detailed analysis.

Around only 80 exact solutions to the governingdifferential equations.

We look to simplifying assumptions to solve theequations.

Numerical methods provide another avenue forsolution (Computational Fluid Dynamics)


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Kinematic Velocity FieldContinuum Hypothesis: the flow is made of tightly packed fluid particles that

interact with each other. Each particle consists of numerous molecules, and we

can describe velocity, acceleration, pressure, and density of these particles at a

given time.


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Kinematic Acceleration FieldLagrangian Frame:

Eulerian Frame: we describe the acceleration in terms of position and time

without following an individual particle. This is analogous to describing the

velocity field in terms of space and time.

A fluid particle can accelerate due to a change in velocity in time (unsteady)

or in space (moving to a place with a greater velocity).


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Kinematic Acceleration Field: Material (Substantial) Derivative

time dependence spatial dependence

We note:

Then, substituting:

The above is good for any fluid particle, so we drop A:


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Kinematic Acceleration Field: Material (Substantial) Derivative

Writing out these terms in vector components:

x-direction:

y-direction:

z-direction:

Writing these results in short-hand:

where,

kz

jy

ix

()

,


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Kinematics: Deformation of a Fluid Element

General deformation of fluid element is rather complex, however, we can

break the different types of deformation or movement into a

superposition of each type.

Linear Motion Rotational Motion

Linear deformation Angular Deformation

General

Motion


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Kinematics: Linear Motion and DeformationLinear Motion/Translation due to u and vvelocity:

Simplest form of motion the element

moves as a solid body. Unlikely to be the

only affect as we see velocity gradients inthe fluid.

Deformation: Velocity gradients can cause deformation, stretching

resulting in a change in volume of the fluid element.

Rate of Change for one direction:

For all 3 directions: The shape does not change, linear deformation

The linear deformation is zero for incompressible fluids.

= 0


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Kinematics:Angular Motion and DeformationAngular Motion/Rotation:

Angular Motion results from

cross derivatives.


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Kinematics:Angular Motion and Deformation

The rotation of the element about the z-axis is the average of the angular

velocities :

Likewise, about the y-axis, and the x-axis:

Counterclockwise rotation is considered positive.

and

The three components gives the rotation vector:

Using vector identities, we note, the rotation vector is one-half the curl of the

velocity vector:


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The definition, then of the vector operation is the following:

The vorticity is twice the angular rotation:

Vorticity is used to describe the rotational characteristics of a fluid.The fluid only rotates as and undeformed block when ,

otherwise, the rotation also deforms the body.

If , then there is no rotation, and the flow is said to be irrotational.


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Angular deformation:

The associated rotation gives rise to angular deformation, which results in the

change in shape of the element

Shearing Strain:

Rate of Shearing Strain:

If , the rate of shearing strain is zero.

The rate of angular deformation is related to the shear stress.


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Conservation of Mass: Cartesian Coordinates

System: Control Volume:

Now apply to an infinitesimal control volume:

For an infinitesimal control volume:

Now, we look at the mass flux in the x-direction:

Out: In:


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Conservation of Mass: Cartesian Coordinates

Net rate of mass in the outflow y-direction:

Net rate of mass in the outflow z-direction:

Net rate of mass in the outflow x-direction:

Net rate of mass flow for all directions:

+

Now, combining the two parts for the infinitesimal control volume:

= 0

Divide out


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Conservation of Mass: Cartesian CoordinatesFinally, the differential form of the equation for Conservation of Mass:

a.k.a. The Continuity Equation

In vector notation, the equation is the following:

If the flow is steady and compressible:

If the flow is steady and incompressible:


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Conservation of Mass: Cylindrical-Polar Coordinates

If the flow is steady and compressible:

If the flow is steady and incompressible:


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Conservation of Mass: Stream Functions

Stream Functions are defined for steady, incompressible, two-dimensional flow.

Continuity:

Then, we define the stream functions as follows:

Now, substitute the stream function into continuity:

It satisfies the continuity condition.

The slope at any point along a streamline:


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Streamlines are constant, thus d = 0:

Now, calculate the volumetric flow rate between streamlines:


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In cylindrical coordinates:


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Conservation of Linear Momentum

P is linear momentum,System:

ControlVolume:

We could apply either approach to find the differential form. It turns out the System

approach is better as we dont bound the mass, and allow a differential mass.

By system approach,m is constant.

If we apply the control volume approach to an infinitesimal control volume, wewould end up with the same result.


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Conservation of Linear Momentum: Forces Descriptions

Body forces or surface forces act on the differential element: surface forces act

on the surface of the element while body forces are distributed throughout the

element (weight is the only body force we are concerned with).

Body Forces:

Surface Forces: Normal Stress:

Shear Stress:


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Conservation of Linear Momentum: Forces Descriptions

Looking at the various sides of the differential element, we must usesubscripts to indicate the shear and normal stresses (shown for an x-face).

The first subscript indicates the plane on which the stress acts

and the second subscript the direction


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Conservation of Linear Momentum: Forces

Descriptions


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Now, the surface forces acting on a small cubicle element in each

direction.

Then the total forces:


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Conservation of Linear Momentum: Equations of Motion

Now, we both sides of the equation in the system approach:

In components:


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Conservation of Linear Momentum: Equations of Motion

Writing out the terms for the Generalize Equation of Motion:

The motion is rather complex.

Material derivative foraForce Terms


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Inviscid Flow

An inviscid flow is a flow in which viscosity effects or shearing effects become

negligible.

If this is the case,

And, we define

A compressive force give a positive pressure.

The equations of motion for this type of flow then becomes the following:

Eulers

Equations


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Inviscid Flow: Euler s Equations

Leonhard Euler

(1707 1783)

Famous Swiss mathematician who pioneered work on the

relationship between pressure and flow.

In vector notation Eulers Equation:

The above equation, though simpler than the generalized equations, are still

highly non-linear partial differential equations:

There is no general method of solving these equations for an analytical solution.

The Eulers equation, for special situations can lead to some useful information

about inviscid flow fields.


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Surface Stress Terms for a General Newtonian Fluid

General Stress Elements:

Normal Stresses:

Shear Stresses:

Note, and is known as the second viscosity coefficient

is the viscosity of the fluid and for the general form is allowed to be non-constant.

xxxx p

yyyy p

zzzz p


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Viscous Flows: Surface Stress Terms

Now, we allow viscosity effects for an incompressible Newtonian Fluid:

Normal Stresses:

Shear Stresses:

Cartesian

Coordinates:


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Viscous Flows: Surface Stress Terms

Normal Stresses:

Shear Stresses:

Cylindrical

Coordinates:


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Viscous Flows: Navier-Stokes Equations

Now plugging the stresses into the differential

equations of motion for incompressible flow giveNavier-Stokes Equations:

French Mathematician, L. M. H. Navier (1758-1836) andEnglish Mathematician Sir G. G. Stokes (1819-1903)

formulated the Navier-Stokes Equations by including

viscous effects in the equations of motion.

L. M. H. Navier(1758-1836)

Sir G. G. Stokes(1819-1903)

(x direction)


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Viscous Flows: Navier Stokes Equations

Local Acceleration Advective Acceleration

(non-linear terms)

Pressure term Weight term

Viscous terms

Terms in the x-direction:


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The governing equations can be written in cylindrical coordinates as well:

(r-direction)


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There are very few exact solutions to Navier-Stokes Equations, maybe a

total of 80 that fall into 8 categories. The Navier-Stokes equations are

highly non-linear and are difficult to solve.

Some simple exact solutions presented in the text are the following:

1. Steady, Laminar Flow Between Fixed Parallel Plates

2. Couette Flow3. Steady, Laminar Flow in Circular Tubes

4. Steady, Axial Laminar Flow in an Annulus


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4

Assumptions:

1. Plates are infinite and parallel/horizontal

2. The flow is steady and laminar3. Fluid flows 2D, in the x-direction only u=u(y)

only, v and w = 0

4. Fully develop

5. 5. Incompressible

32

2

21

3

3

333

33 3

33 4

3


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Viscous Flows: Exact Solutions/Parallel Plate Flow

Navier-Stokes Equations Simplify Considerably:

Applying Boundary conditions (no-slip conditions at y = h) and solve:

The pressure gradient must be specified and is

typically constant in this flow! The sign is

negative.

(Integrate Twice)

3 3


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Solution is Parabolic:

Can determine Volumetric Flow Rate:


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Navier-Stokes Equations Simplify Considerably:

Applying Boundary conditions (no-slip conditions at y = h) and solve:

The pressure gradient must be specified and is

typically constant in this flow! The sign is

negative.

(Integrate Twice)

3 3

Solution of flow between two flat plates (Couette flow)


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Solution of flow between two flat plates (Couette flow)

The differential equation may be solved by integration

2

2

1d u dpdy dydy dx

Hence 1du dp y Ady dx

And a further integration wrt y yields21

2

dp yu Ay B

dx

Boundary conditions


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Boundary conditions

Due to molecular bonding between the fluid and the wall

it may be assumed that the fluid velocity on the wall is zero

u=0 at y=0u=0 at y=c

This is known as the no-slip condition.

To satisfy the first boundary condition, B=0Then the second b.c. gives

210

2

dpcAc

dx

1

2

dp cA

dx

Quadratic velocity profile for flow in a channel


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Quadratic velocity profile for f low in a channel

Substituting the values for A and B into the previous

equation gives the quadratic equation:

212

dpu y ycdx

For a long, straight channel, of length l, p decreases

with length at a constant rate, so

dp p

dx l

21

2

pu y yc

l

Graph of velocity profile


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G ap o e oc y p o e

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

y

u

Volume flow rate


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Volume flow rate

To calculate the volume flow rate, integrate from y=0 to y=c

y=0

y=c

dy

dq udy 202

cp

q yc y dy

l

3

12

c pq

l

per unit width (z direction)

Maximum and mean velocity


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y

Max velocity occurs at y=c/2, the centre of the channel

2

max8

c pu l

Mean velocity is gained by dividing the flow rate by the

channel width

/u q c

2

max

2

12 3

c pu u

l


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Viscous Flows: Exact Solutions/Couette FlowAgain we simplify Navier-Stokes Equations:

Same assumptions as before except the no-slip

condition at the upper boundary is u(b) = U.

Solving,

If there is no Pressure Gradient:

The termDetermines effects of

pressure gradient

Dimensionless,


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Viscous Flows: Exact Solutions/Pipe Flow

Assumptions:

Steady Flow and Laminar FlowFlow is only in the z-direction

vz = f(r)

Navier-Stokes


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Note that exactly the same result for the velocity

distribution could be derived by solving the Navier-

Stokes equations in radial coordinates.


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Cylindrical Coordinates


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The Navier-Stokes equation in the r-direction is:

rzrr

r

z

rr

r

r

g

z

vv

r

v

rr

rv

rrr

p

z

vv

r

vv

r

v

r

vv

t

v

2

2

22

2

2

2

21)(1



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The Navier-Stokes equation in the -direction is:

gz

vv

r

v

rr

rv

rr

p

r

z

v

vr

vvv

r

v

r

v

vt

v

r

z

r

r

2

2

22

2

2

21)(11



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y

The Navier-Stokes equation in the z-direction is:

zzzz

z

z

zz

r

z

gz

vv

rr

vr

rrz

p

z

v

v

v

r

v

r

v

vt

v

2

2

2

2

2

11


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We will return to the pipe flow problem from the start ofthe lecture and solve it using the Navier-Stokes

equations.

Continuity:

0

0

v

vr

01)(1

z

vv

rr

rv

rzr

0 0


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r-direction Navier-Stokes:

0

21)(1

2

2

2

2

22

2

2

2

z

v

r

p

gzvv

rv

rrrv

rrrp

zvv

rvv

rv

rvv

tv

z

rzrr

rz

rrr

r


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-direction Navier-Stokes:

0

21)(112

2

22

2

2

p

gzvv

rv

rrrv

rrp

r

z

vv

r

vvv

r

v

r

vv

t

v

r

zr

r

z direction Navier Stokes


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z-direction Navier-Stokes

zz

zzzz

zz

zzr

z

gr

vr

rrz

p

gzvv

rrvr

rrzp

zvvv

rv

rvv

tv

2

2

2

2

211

Integrate:


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Integrate:

2

2

1

1

1

2

4

00

2

2

cr

gL

ppv

ratfinite

r

vkeeptoc

r

v

r

crgL

pp

rvrcrg

Lpp

gr

vr

rrz

p

zoL

z

z

zzoL

zz

oL

zz


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Solving the equations with the no slip

conditions applied at r = R (the walls of the

pipe).

Parabolic Velocity Profile



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The volumetric flow rate:

The mean velocity:

Pressure drop per length of pipe:

The maximum velocity:

Non-Dimensional velocity profile:

For Laminar Flow:

Substituing Q,

Conservation of Energy


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gy

The energy equation is developed similar to the momentum equation for aninfinitesimal control volume.



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The energy equation is developed similar to the momentum equation for an

infinitesimal control volume.

(Heat and Work)

Internal Kinetic Potential

(Time rate of change

following the particle)

Differentiate:To get the L.H.S:

Now for the R.H.S., define the fluid properties of Heat and Work:

Heat Conduction into the element, Fouriers Law

Heat per Unit Area

Heat:


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Now, we do a control volume analysis on our control element:

Heat Flow into the left x-face of the element

Heat Flow out of the right x-face of the element

The above can be written for all six faces of the cube with the net result

between the in and out:

The net heat flow is transferred to the element, neglecting production terms

Heat: Heat Conduction into the element, Fouriers Law

Heat per Unit Area


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Conservation of EnergyWork: Work is done on the element per unit area.

on the left x-face

on the right x-face

We can do the same for the other faces, and the net rate of work done is:

In condensed form:


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Conservation of EnergyWe can rewrite the equation using and identity:

We note, then, that from the momentum equation:

Now, the rate of change of work is the following:

Kinetic Potential

=

Now, when we substitute work and heat back into the governing equation:

We note potential and kinetic energy portions cancelled on each side!


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Conservation of EnergyNow, we can split the stress tensor into pressure and viscous terms:

Using continuity, we can rewrite the pressure term:

Now, rewriting the Conservation of Energy:

Noting, the definition of fluid enthalpy:

And, defining the dissipation function:

This term always takes energy from the flow!


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Conservation of EnergyWriting out the terms of Viscous Dissipation for a Newtonian Fluid:

Now, with the substitutions, the energy equation take the following form:

We note,


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Now, lets assume the flow is incompressible:

Enthalpy:

Then,

If the flow velocity is low relative to Heat Transfer then terms of order

U disappear.

is the thermal expansion coefficient, for aper fect gas the second term goes to zero!

If, we assume constant thermal conductivity:

Heat Convection Equation


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Summary of Governing Equations

Mass:

Momentum:

Energy:

Most General forms of the Equations:

Only Assumptions:

(1) The fluid is a continuum

(2) the particles are essentially in thermodynamics equilibrium

(3) Only body forces are gravity

(4) The Heat conduction follows Fouriers Law

(5) There are no internal heat sources.


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Summary of Governing Equations

Some general comments on the general form of the governing equations:

1. They are a coupled system of non-linear partial differential equations

you must solve energy, continuity, and linear momentumsimultaneously. No closed form solution exists!

2. For Newtonian flow, the shear and normal stresses can be written in

terms of the velocity gradients introducing no new unknowns.

3. There appear to be five equations and nine unknowns in the system of

equations: , k, p, u, v, w, h, and T.4. However, we note the following:

5. Now, we have five unknowns and five equations

),(),,(

),(),,(

TpkkTphh

TpTp


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Summary of Governing EquationsIn, general in Fluid Mechanics/CFD we often work with a simplified form of the

equations known as the Navier-Stokes Equations:

Additional Assumptions:

(1) The fluid is Newtonian(2) Incompressible

(3) Constant properties (k, )

where,

uncoupled equations: The fluid flowcan be solved independent of the Heat

Transfer


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Summary of Governing EquationsSome general comments on the Navier-Stokes governing equations:

1. They are non-linear partial differential equations which are uncoupled

in energy, and linear momentum. We can solve linear momentum andcontinuity equations separately for the flow field without knowledge of

the Temperature field (4 Equations, 4 unknowns, u, v, w, p).

2. For Newtonian flow, the shear and normal stresses can be written in

terms of the velocity gradients introducing no new unknowns.

3. There appear to be five equations and 5 unknowns in the system of

equations: p, u, v, w, and T.

4. If the convective term disappears we have a linear solution.

5. If the convective term remains we have a non-linear solution.

6. The Energy equations relies on the solution of the flow field for itssolution.

Viscous Flow Equations


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Summary of Governing EquationsSummary of the Euler form of the governing equations: Inviscid Flow Equations

Linear Momentum:

Continuity and Energy are the same as for Navier-Stokes Equations

Some general remarks:

(1) The system of equations have five unknowns and five equations (same as

Navier-Stokes)

(2) Flow is Inviscid (frictionless), Pressure is the only normal stress, and

there are no shear stresses.

(3) A specialized case of inviscid flow is irrotational flow.(4) The energy and momentum equations are also uncoupled in this set of

equations.


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Physical Boundary ConditionsTypes of Boundary Conditions: Fluid/Gas-Solid Interface

Fluid-Fluid Interface

Gas-Fluid Interface


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Physical Boundary Conditions

No Slip Condition:

At the fluid-boundary interface the velocities must be equal. If the boundary

is stationary, then u, v, w = 0.

The temperature of the fluid has to equal the temperature

of boundary at the interface.

Heat Flux in the fluid must equal the heat flux of the solid at the interface

At a solid boundary:

No Temperature Jump:

Equality of Heat Flux:

Examples:

Stationary Solid BoundaryMoving Boundary:


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Computational Fluid Dynamics: Differential AnalysisGoverning Equations:

Navier-Stokes:

Continuity:The above equations can not be solved for most practical problems with analytical

methods so Computational Fluid Dynamics or experimental methods are

employed.

The numerical methods employed are the following:1. Finite difference method

2. Finite element (finite volume) method

3. Boundary element method.

These methods provide a way of writing the governing equations in discreteform that can be analyzed with a digital computer.


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Computational Fluid Dynamics: Finite ElementThese methods discretize the domain of the flow of interest (Finite Element

Method Shown):

The discrete governing equations are solved in every element. This

method often leads to 1000 to 10,000 elements with 50,000 equationsor more that are solved.


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Computational Fluid Dynamics: Finite DifferenceThese methods discretize the domain of the flow of interest as well (FiniteDifference Method Shown):

Finite Difference Mesh:

Comparison between

Experiment and CFD

Analysis:


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Computational Fluid Dynamics: Pitfalls

Numerical Solutions can diverge or exhibit unstable wiggles.

Finer grids may cause instability in the solution rather than better

results.

Large flow domains can be computationally intensive.

Turbulent flows have yet to be well described with CFD.

Inviscid Flow: Bernoull i Equation


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Daniel Bernoulli

(1700-1782)

Earlier, we derived the Bernoulli Equation from a direct

application of Newtons Second Law applied to a fluid particle

along a streamline.

Now, we derive the equation from the Euler Equation

First assume steady state:

Select, the vertical direction as up, opposite gravity:

Use the vector identity:

Now, rewriting the Euler Equation:

Rearrange:


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Inviscid Flow: Bernoulli EquationNow, take the dot product with the differential length ds along a streamline:

ds and Vare parrallel, , is perpendicular to V, and thus to ds.

We note,

Now, combining the terms:

Integrate:

Then,1) Inviscid flow

2) Steady f low

3) Incompressible flow

4) Along a streamline

I i id Fl I t ti l Fl


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Inviscid Flow: Irrotational FlowIrrotational Flow: the vorticity of an irrotational flow is zero.

= 0

For a flow to be irrotational, each of the vorticity vector components must be

equal to zero.

The z-component:

The x-component lead to a similar result:

The y-component lead to a similar result:

Uniform flow will satisfy these conditions:

There are no shear forces in irrotational flow.

Inviscid Flow: Irrotational Flow


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Example flows, where inviscid flow theory can be used:

Viscous RegionInviscid Region

Inviscid Flow: Bernoull i Irrotational Flow


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Recall, in the Bernoulli derivation,

However, for irrotational flow, .

Thus, for irrotational flow, we do not have to follow a streamline.

Then,

1) Inviscid flow

2) Steady flow

3) Incompressible flow4) Irrotational Flow

Potential Flow: Velocity Potential


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For irrotational flow there exists a velocity potential:

Take one component of vorticity to show that the velocity potential is irrotational:

Substitute u and v components:

02

1 22

xyyx

we could do this to show all vorticity components are zero.

Then, rewriting the u,v, and w components as a vector:

For an incompressible flow:

Then for incompressible irrotational flow:

And, the above equation is known as Laplaces Equation.

Potential Flow: Velocity Potential


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Laplacian Operator in Cartesian Coordinates:

Laplacian Operator in Cylindrical Coordinates:

Where the gradient in cylindrical coordinates, the gradient operator,

Then,

May choose cylindrical

coordinates based on the

geometry of the flow problem,

i.e. pipe flow.

If a Potential Flow exists, with

appropriate boundary

conditions, the entire velocity

and pressure field can bespecified.

Potential Flow: Plane Potential Flows


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Laplaces Equation is a Linear Partial Differential Equation, thus there are

know theories for solving these equations.

Furthermore, linear superposition of solutions is allowed:

where andare solutions to Laplaces equation

For simplicity, we consider 2D (planar) flows:

Cartesian:

Cylindrical:

We note that the stream functions also exist for 2D planar flows

Cartesian:

Cylindrical:



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For irrotational, planar flow:

Now substitute the stream function:

Then, Laplaces Equation

For plane, irrotational flow, we use either the potential or the stream function,

which both must satisfy Laplaces equations in two dimensions.

Lines of constant are streamlines:

Now, the change of from one point (x, y) to a nearby point (x + dx, y + dy):

Along lines of constant we have d = 0,

0



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Lines of constant are called equipotential lines.

The equipotential lines are orthogonal to lines of constant , streamlineswhere they intersect.

The flow net consists of a family of streamlines and equipotential lines.

The combination of streamlines and equipotential lines are used to visualize a

graphical flow situation.

The velocity is inversely proportional

to the spacing between streamlines.

Velocity increases

along this streamline.

Velocity decreases

along this streamline.

Potential Flow: Uniform Flow


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The simplest plane potential flow is a uniform flow in which the streamlines

are all parallel to each other.

Consider a uniform flow in the x-direction:Integrate the two equations:

= Ux + f(y) + C

= f(x) + C

Matching the solution

C is an arbitrary constant, can be set to zero:

Now for the stream function solution:

Integrating the two equations similarto above.

Potential Flow: Uniform Flow


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For Uniform Flow in an Arbitrary direction,

Potential Flow: Source and Sink Flow

S Fl


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Source/Sink Flow is a purely radial flow.

Fluid is flowing radially from a line through

the origin perpendicular to the x-y plane.

Let m be the volume rate emanating from the line (perunit length.

Then, to satisfy mass conservation:

Since the flow is purely radial:

Now, the velocity potential can be obtained:

Integrate

0If m is positive, the flow is radially outward, source flow.

If m is negative, the flow is radially inward, sink flow.

m is the strength of the source or sink!

This potential flow does not exist at r = 0, the origin, because it is not a real flow, but can

approximate flows.

Source Flow:

Potential Flow: Source and Sink Flow


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0

Now, obtain the stream function for the flow:

Then, integrate to obtain the solution:

The streamlines are radial lines and the equipotentiallines are concentric circles centered about the origin:

lines

lines

Potential Flow: Vortex Flow

In vortex flow the streamlines are concentric circles and the equipotential


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In vortex flow the streamlines are concentric circles, and the equipotential

lines are radial lines.

where K is a constant.

Solution:

The sign of K determines whether the flow rotates

clockwise or counterclockwise.

In this case, ,

The tangential velocity varies inversely with the distance from the origin. At the

origin it encounters a singularity becoming infinite.

lines

lines

Potential Flow: Vortex FlowHow can a vortex flow be irrotational?


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How can a vortex flow be irrotational?

Rotation refers to the orientation of a fluid element and not the path

followed by the element.

Irrotational Flow: Free VortexRotational Flow: Forced Vortex

Traveling from A to B, consider two sticks

Initially, sticks aligned, one in the flow direction, and the

other perpendicular to the flow.

As they move from A to B the perpendicular-aligned

stick rotates clockwise, while the flow-aligned stick

rotates counter clockwise.

The average angular velocities cancel each other, thus, the

flow is irrotational.

Irrotational Flow:

Velocity

increases

inward.

Velocity

increases

outward.

Rotational Flow: Rigid Body RotationInitially, sticks aligned, one in the flow

direction, and the other perpendicular to theflow.

As they move from A to B they sticks move

in a rigid body motion, and thus the flow is

rotational.

i.e., water

draining from

a bathtub

i.e., a rotatingtank filled with

fluid


A bi d t fl i i hi h th i f d t t th d


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A combined vortex flow is one in which there is a forced vortex at the core, and

a free vortex outside the core.

A Hurricane is

approximately a

combined vortex

Circulation is a quantity associated with vortex flow. It is defined as the lineintegral of the tangential component of the velocity taken around a closed

curve in the flow field.

For irrotational flow the

circulation is generally

zero.



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However, if there are singularities in the flow, the circulation is not zero if the

closed curve includes the singularity.

For the free vortex:

The circulation is non-zero and constant for the free vortex:

The velocity potential and the stream function can be rewritten in terms of the

circulation:

An example in which the closed surface circulation will be zero:

Beaker Vortex:

Potential Flow: Doublet FlowCombination of a Equal Source and Sink Pair:


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Rearrange and take tangent,

Note, the following:

Substituting the above expressions,

and

Then,

If a is small, then tangent of angle is approximated by the angle:

Potential Flow: Doublet Flow


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Now, we obtain the doublet flow by letting the source and sink approach one

another, and letting the strength increase.

K is the strength of the doublet, and is

equal to ma/is then constant.

The corresponding velocity potential then is the following:

Streamlines of a Doublet:

Potential Flow: Summary of Basic Flows


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Potential Flow: Superposition of Basic Flows

Beca se Potential Flo s are go erned b linear partial differential eq ations


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Because Potential Flows are governed by linear partial differential equations,

the solutions can be combined in superposition.

Any streamline in an inviscid flow acts as solid boundary, such that there is no

flow through the boundary or streamline.

Thus, some of the basic velocity potentials or stream functions can be

combined to yield a streamline that represents a particular body shape.

The superposition representing a body can lead to describing the flow aroundthe body in detail.

Superposition of Potential Flows: Rankine Half-Body

The Rankine Half-Body is a combination of a source and a uniform flow.


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The Rankine Half Body is a combination of a source and a uniform flow.

Stream Function (cylindrical coordinates):

Potential Function (cylindrical coordinates):

There will be a stagnation point, somewhere along the negative x-axis wherethe source and uniform flow cancel (

For the source: For the uniform flow:

Evaluate the radial velocity:

cosUvr

For Uvr

Then for a stagnation point, at some r = -b, = :

2

mvr and


Now, the stagnation streamline can be defined by evaluating at r = b, and


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Now, the stagnation streamline can be defined by evaluating at r b, and = .

Now, we note that m/2 = bU, so following this constant streamline givesthe outline of the body:

Then, describes the half-body outline.

So, the source and uniform can be used to describe an aerodynamic body.

The other streamlines can be obtained by setting constant and plotting:

Half-Body:


The width of the half-body:


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The width of the half-body:

Total width then,

The magnitude of the velocity at any point in the flow:

Noting,

and

Knowing, the velocity we can now determine the pressure field using the BernoulliEquation:

Po and U are at a point far away from the body and are known.


Notes on this type of flow:


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Provides useful information about the flow in the front part of streamlined body.

A practical example is a bridge pier or a strut placed in a uniform stream

In a potential flow the tangent velocity is not zero at a boundary, it slips The flow slips due to a lack of viscosity (an approximation result).

At the boundary, the flow is not properly represented for a real flow.

Outside the boundary layer, the flow is a reasonable representation.

The pressure at the boundary is reasonably approximated with potential flow.

The boundary layer is to thin to cause much pressure variation.

Superposition of Potential Flows: Rankine Oval

Rankine Ovals are the combination a source, a sink and a uniform flow,


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producing a closed body.

Some equations describing the flow: The body half-length

The body half-width

Iterative

Potential and Stream Function

Superposition of Potential Flows: Rankine Oval



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yp

Provides useful information about the flow about a streamlined body.

At the boundary, the flow is not properly represented for a real flow.

Outside the boundary layer, the flow is a reasonable representation. The pressure at the boundary is reasonably approximated with potential flow.

Only the pressure on the front of the body is accurate though.

Pressure outside the boundary is reasonably approximated.

Superposit ion of Potential Flows: Flow Around a Circular Cylinder

Combines a uniform flow and a doublet flow:


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and

Then require that the stream function is constant for r = a, where a is the

radius of the circular cylinder:

K = Ua2

Then, and

Then the velocity components:


At the surface of the cylinder (r = a):


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y ( )

The maximum velocity occurs at the top and bottom of the cylinder,

magnitude of 2U.


Pressure distribution on a circular cylinder found with the Bernoulli equation


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Then substituting for the surface velocity:

Theoretical and experimental agree

well on the front of the cylinder.

Flow separation on the back-half in the

real flow due to viscous effects causes

differences between the theory and

experiment.


The resultant force per unit force acting on the cylinder can be determined


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by integrating the pressure over the surface (equate to lift and drag).

(Drag)

(Lift)

Substituting,

Evaluating the integrals:

Both drag and lift are predicted to be zero on fixed cylinder in a uniform flow?

Mathematically, this makes sense since the pressure distribution is

symmetric about cylinder, ahowever, in practice/experiment we see

substantial drag on a circular cylinder (dAlemberts Paradox, 1717-

1783).Viscosity in real flows is the Culprit Again!

Jean le Rond

dAlembert

(1717-1783)

57401301 Ch 6 Differential Analysis of Fluid Flow

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Transcript of 57401301 Ch 6 Differential Analysis of Fluid Flow