Chapter 2 Defferential Equation @ Potential Flow

CHAPTER 2 : POTENTIAL FLOW

(DIFFENTIAL EQUATIONS FOR FLUID FLOW)

Siti Mariam Basharie

JKLA, FKMP - OCT 2013

Overview o

Flow Field : • Velocity and Acceleration Field • Material Derivatives Linear Motion and Deformation Angular Motion and Deformation • Rotational and Irrotational Flow • Vorticity Stream Function Velocity Potential Continuity Equation Momentum equation • The Navier-Stokes Equation - for Viscous Flow • The Euler Equation - for Inviscid Flow Potential Flow (uniform flow, source and sink, and vortex )

• In general, the fluids flow is a net of motion

of molecules from one point to another point

as a function of time.

• A portion of fluid contains so many

molecules and it becomes unrealistic for us

to attempt to account for the motion of

individual molecules.

• An assumption had been made to make it

possible to produce model for the flow of

fluids in and around solid boundaries.

• Assumption: The fluids are inviscid,

incompressible and irrotational. This type of

fluid flow that governed by Laplace’s

equation are called Potential Flows.

• Since each particle contains numerous

molecules, we can describe the flow of a

fluid in terms of motion of fluid particles (rather than individuals molecules).

Flow Field

• The flow field is a representation of fluid parameters or variables as function of spatial coordinates (location) and as function of time.

• It is a vector field which is used to describe the motion of a fluid mathematically.

• Let say, we have a fluid

element which flow from point 1 at time t1 to point 2 at time t2.

Contd…Flow Field

• Each of variables involved in the motion of this fluid element can be given a field representation.

• For example, we have velocity field V(x,y,z,t), acceleration field a(x,y,z,t), temperature field T(x,y,z,t), pressure field, p(x,y,z,t) and density field ρ(x,y,z,t).

• In the study of fluid dynamics, the velocity and acceleration fields are two basic and important field variables which are the focus in this chapter/topic.

• Both the velocity and acceleration equations will be presented in Eulerian* viewpoint and also in Cartesian coordinates only.

*Eulerian method – we observe the characteristics of fluid elaments passing fixed

point

Velocity Field

• Velocity is a vector quantity (i.e it has a direction along with a magnitude).

• Velocity field implies a distribution of velocity in a given region and denoted in a functional form as V(x,y,z,t) meaning that velocity is a function of both the location and time.

Contd…Velocity Field • The fluid flow generally is unsteady and

in three dimensional. • So, velocity vector can be expressed in

Cartesian coordinates as : (2.1)

• Velocity may have three components,

u,v and w, each in x, y and z directions respectively. It is usual written as :

(2.2)

• It is clear that each of u,v and w also can be functions of x,y,z and t. Thus

(2.3)

Acceleration Field

• Another important parameter in the study of fluid in motion is the acceleration. Acceleration is related to the velocity, and it can be determined once the velocity field is known.

• The acceleration is the change in velocity, dV, over the change in time, δt,

• But it is not just a simple derivative since the velocity is a function of time, AND space (x,y,z).

• The change in velocity must be track in both time and space.

• Using the chain rule of calculus, the change in velocity is,

dt

dVa

ttztytxVV ,,,

dt

dz

z

V

dt

dy

y

V

dt

dx

x

V

t

V

dt

dVa

Contd… Acceleration Field

• This can simplified using u, v, and w, the velocity magnitudes in the three coordinate directions. In cartesian coordinates, the acceleration field is,

• Or in component form

dt

dzwand

dt

dyv

dt

dxu ,

Material Derivative

• The time and space derivative used to determine the acceleration field from the velocity is so common in fluid mechanics and it has a special name.

• It is called the Material or Substantial or Total Derivative and has a special symbol, D( )/Dt.

• For cartesian coordinates, it is

• or in vector form,

where is the del or gradient operator.

Contd…Material Derivative

• From the equation, it can be seen that the material derivative consists of two terms.

• The first term ∂t( )/∂t is referred to as the local rate of change, and it represents the effect of unsteadiness. For steady flow, the local derivative ∂t( )/∂t = 0)

• The second term, , is referred to as the convective rate of change, and it represents the variation due to the change in the

position of the fluid particle, as it moves through a field with a gradient. If there is no gradient (no spatial change) then

is zero so there is no convective change. • As an example, the acceleration field equation can be written

as,

Go for In Class Practice

Conservation of Mass

The law of conservation of mass state, “ Any change of mass within the control volume is equal to the net gain of mass flowing into the volume through the control surface.” Or it can be written as,

Or ,

Or it can be rewrite as,

Continuity Equation

According to the principle of conservation of mass, it is known that mass is conserved for a system. Consider a small cubical element of fluid as shown in the figure. Let the density and velocities in the x-, y-, and z-directions at the center of the element be ρ, u, v and w, respectively.

Contd…

Contd…

The net mass flow rate through the control surface of the small differential cubical element is given by the difference between each parallel face, or Hence, the net mass flow rate through the control surface is given by,

Contd…

• While the change of mass within the element is given by;

• By applying the conservation of mass to the element, we have

The element volume cancels out of all terms, leaving the partial differential equation involving the derivatives of density and velocity.

Or in compact vector notation,

This equation is the unsteady, three-dimensional continuity equation at a point in a compressible fluid.

For steady flows, the density is not a function of time (dρ/dt = 0), and thus the conservation of mass equation reduces to If the flow is incompressible (i.e. constant density), then the conservation of mass equation becomes the continuity equation : or

Go for In Class Practice

Fluid Motion and Deformation

• Fluid element motion consists of translation, linear deformation, rotation and angular deformation.

• A small cubical fluid element which is initially in one position will move to another position during a short time interval δt as illustrated in figure.

• Because of the generally complex velocity variation within the field, we expect the element not only to translate from one position but also have its volume changed (linear deformation), to rotate and to undergo a change in shape (angular deformation).

Linear Motion and Deformation

• The simplest motion that fluid element can undergo is translation.

• In translation, the fluid particle move from origin O to new origin, O’ with small interval time, δt.

• If we consider of velocity gradient during the translation, the element generally be deformed & rotated as it move.

• Let x-component velocity at point O & B is u, then x-component velocity at point A & C can be expressed as;

• The difference in velocity cause stretching of the element volume by amount of ;

• The change in original volume

xx

uu

txx

u

tzyxx

uV

22

• The rate at which the volume δV changing/unit volume due to gradient ; • If the velocity gradient and are also present; • This rate of change of the volume per unit volume is called the volumetric dilatation rate. • For incompressible flow, the volumetric dilatation rate = 0 since the element volume cannot change without a change in fluid density.

xu /

x

u

t

txu

dt

Vd

V t

/lim

1

0

yv / zw /

V

z

w

y

v

x

u

dt

Vd

V.

1

23

Angular Motion and Deformation • Derivatives , and simply cause a linear deformation.

• Cross derivatives and will cause the element to rotate and

generally to undergo an angular deformation.

• Let assume in very short time, δt, the line segments OA & OB rotate

through the angle of δβ and δα to the new position OA’ & OB’. The angular

velocity of line OA;

xu / zw /yv /

yu / xv /

ttOA

0lim

• For small angle; , therefore;

• So that;

• Similar with angular velocity of the line OB;

• The rotation of the element about z-axis, ωz is defined as the average of the

angular velocities ωOA and ωOB of the two mutually perpendicular lines OA

and OB.

• Similar for other two axis (x & y axis),

24

tan tx

v

x

vt

t

xv

tOA

/lim

0

y

ut

t

yu

tOB

/lim

0

y

u

x

vz

2

1

z

v

y

wx

2

1

x

w

z

uy

2

1

What happen if y

u

x

v

* In fluid conventions: CCW is +ve & CW is -ve

25

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

• The rotation vector, ω is equal to one-half the curl of velocity vector.

• Vorticity, is defined as a vector that twice the rotation vector.

VVcurl 2

1

2

1ω

kji zyxˆˆˆ ω

wvu

zyx

kji ˆˆˆ

2

1k

y

u

x

vj

x

w

z

ui

z

v

y

w ˆ2

1ˆ2

1ˆ2

1

ζ

V 2A flow is said to be irrotational when the curl of the velocity or vorticity vanishes, or 0 V

Rotational and irrotational flow

• If the vorticity at a point in a flow field is nonzero, the fluid particle that happens to occupy that point in space is rotating; the flow in that region is called rotational.

• Likewise, if the vorticity in a region of the flow is zero

• (or negligibly small), fluid particles there are not rotating; the flow in that region is called irrotational.

• Physically, fluid particles in a rotational region of flow rotate end over end as they move along in the flow.

Contd…

• For example, fluid particles within the viscous boundary layer near a solid wall are rotational (and thus have nonzero vorticity), while fluid particles outside the boundary layer are irrotational (and their vorticity is zero).

Contd…

• Rotation of fluid elements is associated with wakes, boundary layers, flow through turbomachinery (fans, turbines, compressors, etc.), and flow with heat transfer.

• The vorticity of a fluid element cannot change except through the action of viscosity, nonuniform heating (temperature gradients), or other nonuniform phenomena. Thus if a flow originates in an irrotational region, it remains irrotational until some nonuniform process alters it.

29

Example 1 : Vorticity Determine the vorticity for the following velocity field :

Is it the flow field rotational or irrotational?

jxyiyxV ˆ2ˆ22

Determine an expression for the vorticity of the flow field for the given velocity component below. Is it a rotational flow or an irrotational flow?

jyixyV ˆˆ 43

Exercise 1

30

Exercise 2 For a certain two-dimensional flow field the velocity vector is given by;

Is this flow irrotational?

jyxixyV ˆ2ˆ4 22

Exercise 3 The three components of velocity in a flow field are given by;

Determine an expression for the rotation vector, Is this an irrotational flow field?

42

32

2

222

zxzw

zyzxyv

zyxu

Streamline • A streamline is a curve that is everywhere tangent to the velocity

vector. • Streamlines are useful as indicators of the instantaneous direction of

fluid motion throughout the flow field. • Every streamline in a flow has a unique streamline function, ᴪ,

associated with it (i.e. the stream function, ψ, is constant). • As such, no flows across any streamline are possible → flows are

“fenced-up” from one another by streamlines.

Stream Function & Velocity Potential

Stream Function,

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

• We still have two variables, u and v to deal with, so the stream function, ψ was defined so that the number of unknowns can be reduced from two to one.

• The stream function, ψ is defined as ;

ᴪ

0

y

v

x

u

Contd…

• The continuity equation then can be rewrite as;

• The stream function, ψ, was just mathematically defined so that the number of unknowns (i.e., u and v) are reduced from two to one.

• This makes solving the 2D continuity equation easier, but it is now a second order differential equation.

022

xyyxxyyx

Contd…

• Another particular advantage of using the stream function is related to the fact that lines along which ψ is constant are streamlines. Recall that streamlines are lines in the flow field that are everywhere tangent to the velocities, It follows from the definition of the streamline that the slope at any point along a streamline is given by

• But for any smooth function ψ of two variables x and y, we know by the chain rule of mathematics that the total change of ψ from point (x, y) to another point (x + dx, y + dy) is given by,

35

Example 1 : Stream Function The velocity vector of a two-dimensional flow field is given by;

Find the corresponding stream function.

jyxiyxV ˆˆ2 2

Example 2 : Stream Function The components of velocity in a flow field are given as;

a) Is this 1D, 2D or 3D flow?

b) Is it incompressible?

c) Find stream function if possible.

zyw

zyv

u

2

3

3

4

0

36

Velocity potential, Φ.

• In vector form, the above equations can simplified as;

• The velocity potential is a consequence of the irrotationality. Therefore, for

incompressible, irrotational flow (with ), it follow that;

where is the Laplacian operator. Therefore the equation can be

rewrite as;

xu

yv

zw

V

V

02

.2

02

2

2

2

2

2

zyx

• Velocity potential lines are perpendicular to streamlines.

• Similar to the stream function, it can be shown that velocity potential function can be written as;

Example 1 : Velocity potential

If the expression for stream function is described by (x,y) = x2 – y2,

determine whether flow is rotational or irrotational. If the flow is

irrotational, then determine the velocity potential, and velocity vector,

V.

Example 2 : Velocity potential

The velocity components in steady, incompressible, two dimensional flow are;

u = 2y

v = 4x

Determine the corresponding stream function and velocity potential.

Exercise 1: Velocity Potential

Exercise 2 : Velocity Potential If the expression for stream function is described by (x,y) = x3 –

3xy2, determine whether flow is rotational or irrotational. If the flow is

irrotational, then determine the velocity potential, and velocity vector,

V.

The flow field is represented by the potential function = x2 – y2,

i) Verify that it is an incompressible flow.

ii) Determine the corresponding stream function.

Forces acting on the Differential Element

• The differential form of the linear momentum equation (also known as the Navier-Stokes equations) will be derived by applying the Newton’s Second Law, ΣF= ma.

• There are two types of forces acting on the fluid element: body force (FB) and surface force (FS).

• The only body force considered is the gravitational force (weight) of the fluid element while for surface forces the forces that are considered include pressure forces and viscous forces.

• Generally, gravity only acts in one direction, but since the coordinate system is not set, all three terms are included for the general case.

• Therefore ΣF = ΣFB + ΣFS

Conservation of Momentum

Consider a cubical fluid element

For 3D control volume, all forces act in 3D direction and flow stresses act on all six cubical faces.

However, for simplicity, only x direction will be considered first.

There are two types of stresses applied on the surface: normal stress and shear stress.

Normal stress acts perpendicular to the surface while shear stress is tangential to the surface.

For stress τij, the subscript i refers to the axis normal to the surface, and the subscript j represents the direction of the stress.

Body Force

The weight or gravitational force on the fluid element is simply given by,

Surface forces – pressure forces

Surface forces – Viscous forces

Contd…

• For moving fluid element, the net surface force in x direction is;

• By canceling terms, the total net force in the x direction Fx, is given by the sum of body force and surface forces in x direction or;

Contd…

• By applying Newton’s Second Law, Fx = max, with

• we obtain;

• and dividing with dxdydz, we obtain the momentum equations (for x direction) in non-conservation form as;

z

uw

y

uv

x

uu

t

udxdydzmax

z

uw

y

uv

x

uu

t

u

zyxx

pg zxyxxx

x

dxdydzz

uw

y

uv

x

uu

t

udxdydz

zyxx

pdxdydzg zxyxxx

x

Contd…

• Or in the form of;

• And the y and z components can be obtained as;

zyxx

pg

z

uw

y

uv

x

uu

t

u zxyxxxx

zyxy

pg

z

vw

y

vv

x

vu

t

v zyyyxy

y

zyxx

pg

z

ww

y

wv

x

wu

t

w zzyzxzz

Contd…

• The non-conservation of is not very useful to us as it contains many unknown (ρ, u, v, w, τxx, τxy, τxz, τyy, τyz, and τzz) compare to the only four equations (include continuity eqn.) that we have.

• Of course, to be mathematically solvable, the number of

equations must equal the number of unknowns, and thus we need six more equations.

• To reduce the number of unknowns, we have to convert the annoying stresses terms into the velocity and pressure terms.

• We know that, for the Newtonian fluid, the shear stress is proportional to the shear strain rate or rate of deformation.

Contd…

• The normal and shear stresses are given as;

• Where μ is the molecular viscosity coefficient, and λ is the second viscosity coefficient which is equal to -2/3 μ (Stokes hypothesis).

Contd…

• For incompressible flow, the term .V is zero based on the continuity equation.

• The stresses elements then can be write in matrix form as;

Contd…

• By replacing the stresses terms, we obtain the linear momentum equation (for x direction) in conservation form;

• We note that as long as the velocity components are smooth functions of x, y, and z, the order of differentiation is irrelevant. For example,

Contd…

• After some rearrangement of the viscous terms, we obtain;

• The term in parentheses is zero because of the continuity equation for incompressible flow . We also recognize the last three terms as the Laplacian of velocity component u.Thus, we write the x-component of the momentum equation as;

Contd…

• In a similar fashion we can write the y and z components of the momentum equation as;

• These equations is called as Navier-Stokes equations. It is for incompressible flow with constant viscosity.

Have to remember all these equations !!!

Contd…

• The Navier-Stokes equations is the core fluid mechanics equations and is an unsteady, nonlinear, 2nd order, partial differential equation for viscous flow.

• It is extremely hard to solve, and only simple 2D problems can be solved.

• Computational Fluid Dynamics (CFD) is most often used to solve the Navier-Stokes equations.

Euler Equation – for inviscid flow

Some common fluids such as air and water have a small viscosity and may be able to neglect the effect of viscosity (thus, no shear stress considered).

Fluid fields in which the shear stresses are assumed to be negligible are know as inviscid, nonviscous or frictionless flow.

For inviscid flow, all shear stresses are zero , and

the Navier-Stokes equation in x direction reduces to;

• The above equations (also have in y and z directions) are known as Euler's equations.

z

uw

y

uv

x

uu

t

u

x

pgx

Contd…

Note that the equations governing inviscid flow have been simplified tremendously compared to the Navier-Stokes equations.

However, they still cannot be solved analytically due to the complexity of the nonlinear terms (i.e., u ∂u/∂x, v ∂u/∂y, w ∂u/∂z, etc.).

Hence, in the study of fluid mechanics, numerical methods such as the finite element and finite difference methods (along with the use of computers) are often used to approximate the fluid flow problems.

Euler's equations can be simplified further to obtain Bernoulli's equation, which is applicable to steady, incompressible, inviscid flow along a streamline.

Bernoulli Equation

Euler's equations can be simplified further to obtain Bernoulli's equation, which is applicable to steady, incompressible, inviscid flow along a streamline.

58

Example 1 : Find pressure field (Cengel E.g 9-13)

Consider the steady, two-dimensional velocity field of ;

Calculate the pressure as a function of x and y.

jcxayibaxV ˆˆ

59

Flow through the converging nozzle can be approximated by one-dimensional velocity distribution;

a) Find a general expression for the fluid acceleration in the nozzle.

b) For the specific case V0 = 3.048 m/s and L = 15.24 cm, calculate the acceleration at the entrance and at the exit of the nozzle.

Example 2 : Find acceleration field

0 2

10

w 0v

L

xVu

60

1. The velocity field for a steady, two-dimensional incompressible

Newtonian fluid flow is given by the following relation ;

By neglecting the effect of gravity, obtain the relation representing

the pressure field p in terms of the fluid density, ρ and the coordinate x

and y if,

a) the pressure at the origin (0,0) is Patm

b) the Patm is set to be zero.

jxyixyV ˆˆ2 22

Exercise (have to be submitted)

61

2. A frictionless, two-dimensional incompressible steady flow is given by

a) Determine if this field is a valid solution to continuity equation, thus Navier-Stokes equation

b) Check whether it is rotational or not. If yes, calculate the vorticity.

c) By neglecting the gravitational effect, find the expression for the pressure gradient in the x direction.

jyixyV ˆˆ2 2

Contd...Exercise (have to be submitted)

62

3. Discuss the following basic flow patterns and give an example of each

flow;

a) Source and sink

b) Vortex

i. Free vortex

ii. Forced vortex

Contd...Exercise (have to be submitted)

• Constant velocity magnitude with straight

parallel streamlines.

• Since

• Also similar for velocity potential;

• For uniform flow at some arbitrary angle;

Basic Potential Flows

0

xU

y

Uy

Uy

Ux

Ux

sincos xyU

sincos yxU

i) Uniform Flows

• Fluid flowing radially outwards (source) or inwards (sink) from a single point.

• Source/sink strength → m, K, Q etc. The

strength are flow rate of the fluids outwards /

inwards from the single point.

• Conservation of mass indicate that;

ii) Source / Sink

QVr r 2

r

QVr

2 * There are no tangential

velocity, Vϴ for source and sink.

• The stream function can be derived:

• Similar for velocity potential:

22

1 Q

r

Q

r

rQ

r

Q

rln

22

• Fluid rotating about a single point. • 2 types ; 1. Free vortex 2. Forced vortex • Free vortex (an irrotational vortex)– e.g: swirling motion of the water as it drains from a sink, tornados, waterspouts, and hurricanes. • Forced vortex (a rotational vortex) –one which has non-zero vorticity away from the core – can be maintained indefinitely in that state only through the application of some extra force, that is not generated by the fluid motion itself.

65

iii) Vortex

• Vortex strength → K = Γ/2π

• For standard math. conversion., +ve Γ represent ccw and vise versa.

• For azimuthal velocity component;

66

rrr

ln22

Contd…Vortex

rr

KV

2

* There are no radial velocity, Vr

for free vortex.

22

1

rr• The stream function can be derived:

• Similar for velocity potential:

Extra Informations

Types of Motion or Deformation of Fluid Elements

a) Translation

b) Rotation

c) Linear deformation

d) Angular deformation

The curl

• The curl is a vector operator that describes the fluid element rotation of a 3-dimensional vector field. The curl is a form of differentiation for vector fields.

• A vector field whose curl is zero is called irrotational.

• The del operator and the cross product, is used for curl.

• Unlike the gradient and divergence, curl does not generalize as simply to other dimensions; some generalizations are possible, but only in three dimensions is the geometrically defined curl of a vector field again a vector field. This is a similar phenomenon as in the 3 dimensional cross product, and the connection is reflected in the notation ∇ × for the curl.

Forces acting on the Differential Element

* Body forces – forces that act throughout the entire body of CV (e.g : gravitational force, electric and magnetic forces )

* Surface forces – forces that act on the control surface

(e.g : pressure force, viscous forces and reaction forces at points of contact).

The Navier-Stokes equations

• The Navier–Stokes equations are mathematical equations that describe the motion of fluids. The equations are named after Claude-Louis Navier and George Gabriel Stokes. The equations result from applying Newton's second law to fluid motion, with the belief that the fluid stress is the sum of a diffusing viscous term (in relation to the gradient of velocity), plus a pressure term.

• They are very useful because they describe the physics of many things of academic and economic interest. They may be used to model the weather, ocean currents, water flow in a pipe, the air's flow around a wing, and motion of stars inside a galaxy. The Navier–Stokes equations in their full and simplified forms help with the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of pollution, and many other things.

• The Navier–Stokes equations are also of great interest in a purely mathematical sense. Somewhat surprisingly, given their wide range of practical uses, mathematicians have not yet proven that in three dimensions solutions always exist (existence), or that if they do exist, then they do not contain any singularity (or infinity or discontinuity) (smoothness).