CHAPTER 4: FLUID KINEMATICS

Introduction

Fluid Kinematics deals with the motion of fluidswithout considering the forces and moments whichcreate the motion.

Items discussed in this Chapter. Introduction Velocity Field Acceleration Field Control Volume and System Representation Reynolds Transport Theorem Examples

Introduction

• If a Fluids is subjected to shear force then it tends to flow

• Fluids that are subjected to pressure imbalance resulted influid flow

• In kinematics we are not concerned with the force, but themotion, thus, we are interested in flow visualization.

• We can learn a lot about flows from watching.

Velocity field-Continuum Hypothesis

Continuum Hypothesis: The flow is made of tightly packed fluidparticles that interact with each other. Each particle consists ofnumerous molecules, and we can describe the field variablesvelocity, acceleration, pressure, and density of these particles at agiven time.

Velocity field Different particles in fluid flow, which move at different

velocities and may be subjected to different accelerations.

The velocity and acceleration of a fluid particle may changeboth with respect to time and space.

In the study of fluid flow it is necessary to observe the motionof the fluid particles at various points in space and at asuccessive instants of time.

Velocity field - Lagarangian Vs Eulerian method

Generally there are two methods by which the motion of a fluidmay be described, Lagarangian and Eulerian method.

In the Lagarangian method any individual fluid particle is selected ,and observation is made about the behavior of this particle duringits course of motion through space (Control mass approach).

In the Eulerian method any point in the space occupied by the fluidis selected and observation is made of whatever changes ofvelocity, density and pressure which take place at that point (controlvolume approach).

Lagrangian Description

Lagrangian description of fluid flow tracks the position and velocityof individual particles.

Based upon Newton's laws of motion, it is difficult to use Lagrangiandescription for practical flow analysis as. Fluids are composed of billions of molecules. Interaction between molecules hard to describe/model.

However, useful for specialized applications in Sprays, particles, bubble dynamics, rarefied gases. Coupled Eulerian-Lagrangian methods.

Named after Italian mathematician Joseph Louis Lagrange (1736-1813).

Eulerian Description In a Eulerian description of fluid flow a flow domain or control volume is

defined by which fluid flows in and out. We define field variables which are functions of space and time.

Pressure field, P = P(x, y, z, t) Velocity field,

Acceleration field,

These (and other) field variables define the flow field.

Well suited for formulation of initial boundary-value problems (PDE's). Named after Swiss mathematician Leonhard Euler (1707-1783).

( ) ( ) ( ), , , , , , , , ,V u x y z t i v x y z t j w x y z t k= + +

( ) ( ) ( ), , , , , , , , ,x y za a x y z t i a x y z t j a x y z t k= + +

( ), , ,a a x y z t=

( ), , ,V V x y z t=

Lagarangian Vs Eulerian method

Lagrangian

Measurement of fluid temperature

Eulerian

Eulerian methods are commonlyused in fluid experiments oranalysis—a probe placed in aflow.

Example: Coupled Eulerian-Lagrangian Method

Simulation of micron-scaleairborne probes. The probepositions are tracked using aLagrangian particle modelembedded within a flow fieldcomputed using an EulerianCFD code.

http://www.ensco.com/products/atmospheric/gem/gem_ovr.htm

Velocity field- Uniform Vs Non-Uniform

Uniform flow: If the flow velocity is the same magnitude and directionat every point in the fluid at the given instant of time, it is said to beuniform.

Non-uniform: If at a given instant, the velocity is not the same atevery point the flow is non-uniform. (In practice, fluid that flows neara solid boundary will be non-uniform – as the fluid at the boundarymust take the speed of the boundary, usually zero.

0=∂∂

sV

0≠∂∂

sV

Velocity field- Steady Vs Unsteady

Steady Flow: The velocity at a given point in space does not vary with time.

Very often, we assume steady flow conditions for cases where there is only a slight time dependence, since the analysis is “easier”

Unsteady Flow: The velocity at a given point in space does vary with time.

Almost all flows have some unsteadiness. In addition, there areperiodic flows, non-periodic flows, and completely random flows.

Velocity field- Steady Vs Unsteady

Periodic flow: “fuel injectors” creating a periodic swirling in thecombustion chamber. Effect occurs time after time.

Random flow: “Turbulent”, gusts of wind, splashing of water in thesink

Steady or Unsteady only pertains to fixed measurements, i.e.exhaust temperature from a tail pipe is relatively constant “steady”;however, if we followed individual particles of exhaust they cool!

Velocity field- Steady Vs Uniform

Combining the above we can classify any flow into one of four type:

Steady uniform flow. Conditions do not change with position in thestream and with time at a point.

An example is the flow of water in a pipe of constant diameterat constant velocity.

Steady non-uniform flow. Conditions change from point to point in thestream but do not change with time at a point.

An example is flow in a tapering pipe with constant velocity atthe inlet - velocity will change as you move along the length ofthe pipe toward the exit.

Velocity field- Steady Vs Uniform

Unsteady uniform flow. At a given instant in time the conditions atevery point are the same, but will change with time.

Example : An example is a pipe of constant diameter connectedto a pump pumping at a constant rate which is then switched off.

4. Unsteady non-uniform flow. Every condition of the flow maychange from point to point and with time at every point.

For example waves in a channel.

Velocity Field- 1D, 2D, and 3D Flows

Most fluid flows are complex three dimensional, time-dependentphenomenon, however we can make simplifying assumptions allowingan easier analysis or understanding without sacrificing accuracy. Inmany cases we can treat the flow as 1D or 2D flow.

Three-Dimensional Flow: All three velocity components areimportant and of equal magnitude. Flow past a wing is complex 3Dflow, and simplifying by eliminating any of the three velocities wouldlead to severe errors.

Velocity Field- 1D, 2D, and 3D Flows

Two-Dimensional Flow : In many situations one of the velocitycomponents may be small relative to the other two, thus it isreasonable in this case to assume 2D flow.

One-Dimensional Flow: In some situations two of the velocitycomponents may be small relative to the other one, thus it isreasonable in this case to assume 1D flow. There are very fewflows that are truly 1D, but there are a number where it is areasonable approximation.

Flow Visualization

Flow visualization is the visual examination of flow-fieldfeatures.

Important for both physical experiments and numerical (CFD)solutions.

Numerous methods Streamlines and streamtubes Pathlines Streaklines Timelines Refractive techniques Surface flow techniques

Velocity field - Streamlines

A Streamline is an imaginary curvedrawn through the flowing fluid in such away that the tangent to it at any pointgives the direction of the velocity at thatpoint.

Because the fluid is moving in thesame direction as the streamlines,fluid can not cross a streamline.

Streamlines can not cross each other.If they were to cross this wouldindicate two different velocities at thesame point. This is not physicallypossible.

The above point implies that anyparticles of fluid starting on onestreamline will stay on that samestreamline throughout the fluid.

Streamlines

A Streamline is a curve that is everywhere tangent to the instantaneous local velocity vector.

Consider an arc length

must be parallel to the local velocity vector

Geometric arguments results in the equation for a streamline

dr dxi dyj dzk= + +

dr

V ui vj wk= + +

dr dx dy dzV u v w

= = =

Velocity field - Streamlines

NASCAR surface pressure contours and streamlines

Airplane surface pressure contours, volume streamlines, and surface streamlines

Velocity field - Pathlines

( ) ( ) ( )( ), ,particle particle particlex t y t z t

A Pathline is the actual pathtraveled by an individual fluidparticle over some time period.

Same as the fluid particle'smaterial position vector

Particle location at time t:

Particle Image Velocimetry (PIV) isa modern experimental techniqueto measure velocity field over aplane in the flow field.

start

t

startt

x x Vdt= + ∫

Velocity field - Streakline

A Streakline is the locus offluid particles that have passedsequentially through aprescribed point in the flow.

Easy to generate inexperiments: dye in a waterflow, or smoke in an airflow.

Comparisons

For steady flow, streamlines, pathlines, and streaklines areidentical.

For unsteady flow, they can be very different. Streamlines are an instantaneous picture of the flow field

Pathlines and Streaklines are flow patterns that have a timehistory associated with them.

Streakline: instantaneous snapshot of a time-integrated flowpattern.

Pathline: time-exposed flow path of an individual particle.

Plots of Data A Profile plot indicates how the value of a scalar property varies

along some desired direction in the flow field.

A Vector plot is an array of arrows indicating the magnitude anddirection of a vector property at an instant in time.

A Contour plot shows curves of constant values of a scalar propertyfor magnitude of a vector property at an instant in time.

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).

Acceleration Field: Material (Substantial) Derivative

time dependence spatial dependenceWe note:

Then, substituting:

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

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()∂∂

+∂∂

+∂∂

=∇ ,

Fluid flows experience fairlylarge accelerations ordecelerations, especiallyapproaching stagnationpoints.

Acceleration field

The time dependant term in the acceleration field is called the localacceleration and is nonzero only for unsteady flows.

The spacial dependant term in the acceleration field is called theadvective acceleration and accounts for the effect of the fluidparticle moving to a new location in the flow, where the velocity isdifferent.

The total derivative operator d/dt is called the material derivativeand is often given special notation, D/Dt.

Advective acceleration is nonlinear: source of many phenomenonand primary challenge in solving fluid flow problems.

Advective acceleration provides ``transformation'' betweenLagrangian and Eulerian frames.

Other names for the material derivative include: total, particle,Lagrangian, Eulerian, and substantial derivative.

Acceleration Field: Material (Substantial) Derivative Applied to the Temperature Field in a Flow:

The material derivative of any variable is the rate at which that variable changes with time for a given particle (as seen by one moving along with the fluid—Lagrangian description).

Acceleration Field: Unsteady EffectsIf the flow is unsteady, its paramater values at any location may change with time (velocity, temperature, density, etc.)

The local derivative represents the unsteady portion of the flow:

If we are talking about velocity, then the above term is local acceleration.

In steady flow, the above term goes to zero.

If we are talking about temperature, and V = 0, we still have heat transfer because of the following term:

0 0 0

=

Acceleration Field: Unsteady EffectsConsider flow in a constant diameter pipe, where the flow is assumed to be spatially uniform:

00 00 0

Acceleration Field: Convective EffectsThe portion of the material derivative represented by the spatial derivatives is termed the convective term or convective accleration:

It represents the fact the flow property associated with a fluid particle may vary due to the motion of the particle from one point in space to another.

Convective effects may exist whether the flow is steady or unsteady.

Example 1: Example 2:

Acceleration = Deceleration

Control Volume and System Representations

Systems of Fluid: a specific identifiable quantity of matter that may consist ofa relatively large amount of mass (the earth’s atmosphere) or a single fluidparticle. They are always the same fluid particles which may interact with theirsurroundings.

Control Volume: is a volume or space through which the fluid may flow,usually associated with the geometry.

Example: following a system the fluid passing through a compressor

We can apply the equations of motion to the fluid mass to describe theirbehavior, but in practice it is very difficult to follow a specific quantity of matter.

When we are most interested in determining the forces put on a fan, airplane,or automobile by the air flow past the object rather than following the fluid as itflows along past the object.

Identify the specific volume in space and analyze the fluid flow within,through, or around that volume.

Control Volume and System Representations

Fixed Control Volume:

Fixed or Moving Control Volume:

Deforming Control Volume:

Surface of the Pipe

Surface of the Fluid

Volume Around The Engine

Inflow

Outflow

Outflow Deforming Volume

Reynolds Transport Theorem: Preliminary ConceptsAll the laws of governing the motion of a fluid are stated in their basic form interms of a system approach, and not in terms of a control volume.

The Reynolds Transport Theorem allows us to shift from the systemapproach to the control volume approach, and back.

General Concepts:

B represents any of the fluid properties, m represent the mass, and brepresents the amount of the parameter per unit volume.

Examples:

Mass b = 1Kinetic Energy b = V2/2

Momentum b = V (vector)

B is termed an extensive property, and b is an intensive property. B isdirectly proportional to mass, and b is independent of mass.

Reynolds Transport Theorem: Preliminary ConceptsFor a System: The amount of an extensive property can be calculated by

adding up the amount associated with each fluid particle.

Now, the time rate of change of that system:

Now, for control volume:

For the control volume, we only integrate over the control volume, this isdifferent integrating over the system, though there are instance whenthey could be the same.

Reynolds Transport Theorem: DerivationConsider a 1D flow through a fixed control volume between (1) and (2):

CV, and system at t1

System at t2

System at t2

Writing equation in terms of the extensive parameter:Originally,

At time 2:

Divide by δt:

Reynolds Transport Theorem: DerivationNoting,

Let,

Time rate of change of mass within the control volume:

The rate at which the extensive property flows out of the control surface:

(1) (2) (3) (4)

(1)

(2)

(4)

Reynolds Transport Theorem: Derivation

The rate at which the extensive property flows into the control surface:

(3)

Now, collecting the terms:

or

Restrictions for the above Equation:1) Fixed control volume2) One inlet and one outlet3) Uniform properties4) Normal velocity to section (1) and (2)

Reynolds Transport Theorem: DerivationThe Reynolds Transport Theorem can be derived for more general conditions.

Result:

This form is for a fixed non-deforming control volume.

Reynolds Transport Theorem: Physical Interpretation

(1) (2) (3)

(1) The time rate of change of the extensive parameter of a system, mass, momentum, energy.

(2) The time rate of change of the extensive parameter within the control volume.

(3) The net flow rate of the extensive parameter across the entire control surface. “outflow across the surface”

“inflow across the surface”

“no flow across the surface”

Mass flow rate:

Reynolds Transport Theorem: Analogous to Material Derivative

Time dependant Portion Convective Portion

Steady Effects:

Unsteady Effects (inflow = outflow):

Kinematic Description In fluid mechanics, an element

may undergo four fundamentaltypes of motion.a) Translationb) Rotationc) Linear straind) Shear strain

Fluids motion and deformation isbest described in terms of ratesa) velocity: rate of translationb) angular velocity: rate of rotationc) linear strain rate: rate of linear

straind) shear strain rate: rate of shear

strain

Rate of Translation and Rotation

To be useful, these rates must be expressed in terms of velocity and derivatives of velocity

The rate of translation vector is described as the velocity vector. In Cartesian coordinates:

Rate of rotation at a point is defined as the average rotationrate of two initially perpendicular lines that intersect at thatpoint. The rate of rotation vector in Cartesian coordinates:

V ui vj wk= + +

1 1 12 2 2

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

ω ∂ ∂ ∂ ∂ ∂ ∂ = − + − + − ∂ ∂ ∂ ∂ ∂ ∂

Linear Strain Rate Linear Strain Rate is defined as the rate of increase in length per unit length.

In Cartesian coordinates

Volumetric strain rate in Cartesian coordinates

Since the volume of a fluid element is constant for an incompressible flow, the volumetric strain rate must be zero.

, ,xx yy zzu v wx y z

ε ε ε∂ ∂ ∂= = =∂ ∂ ∂

1xx yy zz

DV u v wV Dt x y z

ε ε ε ∂ ∂ ∂= + + = + +

∂ ∂ ∂

Shear Strain Rate

Shear Strain Rate at a point is defined as half of the rate ofdecrease of the angle between two initially perpendicular linesthat intersect at a point.

Shear strain rate can be expressed in Cartesian coordinatesas:

1 1 1, ,2 2 2xy zx yz

u v w u v wy x x z z y

ε ε ε ∂ ∂ ∂ ∂ ∂ ∂ = + = + = + ∂ ∂ ∂ ∂ ∂ ∂

Vorticity and Rotationality

The vorticity vector is defined as the curl of the velocity vector

Vorticity is equal to twice the angular velocity of a fluid particle.

Cartesian coordinates

In regions where ζ = 0, the flow is called irrotational. Elsewhere, the flow is called rotational.

Vζ = ∇×

2ζ ω=

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

ζ ∂ ∂ ∂ ∂ ∂ ∂ = − + − + − ∂ ∂ ∂ ∂ ∂ ∂

Vorticity and Rotationality