Ch-6 Differential Analysis of Fluid Flow

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ME 311: Fluid MechanicsDifferential 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 continuity

equations 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 encouragedNo question is stupidYour 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 governing

differential equations.• We look to simplifying assumptions to solve the

equations.• Numerical methods provide another avenue for

solution (Computational Fluid Dynamics)

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Kinematic Velocity Field

Continuum 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 Field

Lagrangian 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 dependencespatial 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

ˆˆˆ()

,

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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 v velocity:

“Simplest” form of motion— the element moves as a solid body. Unlikely to be the only affect as we see velocity gradients in the 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 CoordinatesSystem: 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:

Control Volume:

We could apply either approach to find the differential form. It turns out the System approach is better as we don’t 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, we would 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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Looking at the various sides of the differential element, we must use subscripts 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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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 for aForce Terms

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Inviscid FlowAn 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:

Euler’s 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 Euler’s 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 Euler’s 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 give Navier-Stokes Equations:

French Mathematician, L. M. H. Navier (1758-1836) and English 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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Viscous Flows: Navier-Stokes EquationsThe governing equations can be written in cylindrical coordinates as well:

(r-direction)

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Viscous Flows: Navier-Stokes Equations

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 Plates2. Couette Flow3. Steady, Laminar Flow in Circular Tubes4. Steady, Axial Laminar Flow in an Annulus

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Viscous Flows: Exact Solutions/Parallel Plate Flow

Assumptions:1. Plates are infinite and parallel/horizontal2. The flow is steady and laminar3. Fluid flows 2D, in the x-direction only u=u(y) only, v and w = 04. Fully develop 5. Incompressible

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Assumptions:1. Plates are infinite and parallel/horizontal2. The flow is steady and laminar3. Fluid flows 2D, in the x-direction only u=u(y) only, v and w = 04. Fully develop 5. 5. Incompressible

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2

21

3

3

333

33 3

33 4

3

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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

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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

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Solution of flow between two flat plates (Couette flow)Solution of flow between two flat plates (Couette flow)

The differential equation may be solved by integration

2

2

1d u dpdy dy

dy dx

Hence 1du dpy A

dy dx

And a further integration wrt y yields21

2

dp yu Ay B

dx

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Boundary conditionsBoundary 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=0

Then the second b.c. gives21

02

dp cAc

dx

1

2

dp cA

dx

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Quadratic velocity profile for flow in a channelQuadratic velocity profile for flow in a channel

Substituting the values for A and B into the previous equation gives the quadratic equation:

21

2

dpu y yc

dx

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

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Graph of velocity profileGraph of velocity profile

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

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Volume flow rateVolume flow rate

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

y=0

y=c

dy

dq udy 2

02

cpq yc y dy

l

3

12

c pq

l

per unit width (z direction)

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Maximum and mean velocityMaximum and mean velocity

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

2

max 8

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 Flow

Again 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-directionvz = f(r)

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Navier-StokesNavier-Stokes

• 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 CoordinatesCylindrical Coordinates

• In cylindrical coordinates (r,,z) the continuity equation is:

01)(1

z

vv

rr

rv

rzr

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

rzrr

rz

rrr

r

gz

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

vv

r

vvv

r

v

r

vv

t

v

r

zr

r

2

2

22

2

2

21)(11

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

zzzz

zz

zzr

z

gz

vv

rr

vr

rrz

p

z

vv

v

r

v

r

vv

t

v

2

2

2

2

2

11

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• We will return to the pipe flow problem from the start of the 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

gz

vv

r

v

rr

rv

rrr

p

z

vv

r

vv

r

v

r

vv

t

v

z

rzrr

rz

rrr

r

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• -direction Navier-Stokes:

0

21)(112

2

22

2

2

p

gz

vv

r

v

rr

rv

rr

p

r

z

vv

r

vvv

r

v

r

vv

t

v

r

zr

r

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• z-direction Navier-Stokes

zz

zzzz

zz

zzr

z

gr

vr

rrz

p

gz

vv

rr

vr

rrz

p

z

vv

v

r

v

r

vv

t

v

2

2

2

2

2

11

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• Integrate:

2

2

1

1

1

2

4

00

2

2

cr

gL

ppv

ratfiniter

vkeeptoc

r

v

r

crg

L

pp

r

vrc

rg

L

pp

gr

vr

rrz

p

zoL

z

z

zz

oL

zz

oL

zz

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Viscous Flows: Exact Solutions/Pipe Flow

Solving the equations with the no slip conditions applied at r = R (the walls of the pipe).

“Parabolic Velocity Profile”

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Viscous Flows: Exact Solutions/Pipe FlowThe volumetric flow rate:

The mean velocity:

Pressure drop per length of pipe:

The maximum velocity:

Non-Dimensional velocity profile:

For Laminar Flow:

Substituing Q,

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Conservation of Energy

The energy equation is developed similar to the momentum equation for an infinitesimal 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, Fourier’s 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, Fourier’s 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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We 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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Now, 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, let’s 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 a perfect 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 Fourier’s Law(5) There are no internal heat sources.

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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 momentum simultaneously. 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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In, 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 flow can be solved independent of the Heat Transfer”

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Some 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 and continuity 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 its

solution.

“Viscous Flow Equations”

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Summary 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 Conditions

Types 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 Boundary

Moving 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 method2. Finite element (finite volume) method3. Boundary element method.

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

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Computational Fluid Dynamics: Finite Element

These 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 equations or more that are solved.

83

Computational Fluid Dynamics: Finite DifferenceThese methods discretize the domain of the flow of interest as well (Finite Difference 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.

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Inviscid Flow: Bernoulli Equation

Daniel Bernoulli(1700-1782)

Earlier, we derived the Bernoulli Equation from a direct application of Newton’s 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 V are parrallel, , is perpendicular to V, and thus to ds.

We note,

Now, combining the terms:

Integrate:

Then,1) Inviscid flow2) Steady flow3) Incompressible flow4) Along a streamline

87

Inviscid Flow: Irrotational Flow

Irrotational 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.

88

Inviscid Flow: Irrotational Flow

Example flows, where inviscid flow theory can be used:

Viscous RegionInviscid Region

89

Inviscid Flow: Bernoulli Irrotational Flow

Recall, in the Bernoulli derivation,

However, for irrotational flow, .

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

Then,

1) Inviscid flow2) Steady flow3) Incompressible flow4) Irrotational Flow

90

Potential Flow: Velocity Potential

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 Laplace’s Equation.

91

Potential Flow: Velocity Potential

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 be specified.

92

Potential Flow: Plane Potential Flows

Laplace’s Equation is a Linear Partial Differential Equation, thus there are know theories for solving these equations.

Furthermore, linear superposition of solutions is allowed:

where and

are solutions to Laplace’s 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, Laplace’s Equation

For plane, irrotational flow, we use either the potential or the stream function, which both must satisfy Laplace’s 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

94


Lines of constant are called equipotential lines.

The equipotential lines are orthogonal to lines of constant , streamlines where 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.

95

Potential Flow: Uniform Flow

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 similar to above.

96

Potential Flow: Uniform Flow

For Uniform Flow in an Arbitrary direction,

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Potential Flow: Source and Sink Flow

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 (per unit 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:

98

Potential Flow: Source and Sink Flow

0

Now, obtain the stream function for the flow:

Then, integrate to obtain the solution:

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

lines

lines

99

Potential Flow: Vortex FlowIn 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

100

Potential Flow: Vortex FlowHow 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 Vortex Rotational 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 the flow.

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 rotating tank filled with fluid

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Potential Flow: Vortex Flow

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 line integral of the tangential component of the velocity taken around a closed curve in the flow field.

For irrotational flow the circulation is generally zero.

102

Potential Flow: Vortex Flow

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:

103

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

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:

104

Potential Flow: Doublet Flow

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:

105

Potential Flow: Summary of Basic Flows

106

Potential Flow: Superposition of Basic Flows

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 around the body in detail.

107

Superposition of Potential Flows: Rankine Half-Body

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 where the 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

108


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

Now, we note that m/2 = bU, so following this constant streamline gives the 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:

109


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 Bernoulli Equation:

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

110


Notes on this type of flow:

• 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.

111

Superposition of Potential Flows: Rankine Oval

Rankine Ovals are the combination a source, a sink and a uniform flow, producing a closed body.

Some equations describing the flow: The body half-length

The body half-width

“Iterative”

Potential and Stream Function

112

Superposition of Potential Flows: Rankine Oval

Notes on this type of flow:

• 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.

113

Superposition of Potential Flows: Flow Around a Circular Cylinder

Combines a uniform flow and a doublet flow:

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:

114


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

The maximum velocity occurs at the top and bottom of the cylinder, magnitude of 2U.

115


Pressure distribution on a circular cylinder found with the Bernoulli equation

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.

116


The resultant force per unit force acting on the cylinder can be determined 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 (d’Alembert’s Paradox, 1717-1783).

Viscosity in real flows is the Culprit Again!

Jean le Rond d’Alembert (1717-1783)

Ch-6 Differential Analysis of Fluid Flow

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