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Chapter 4

FLUID KINEMATICS

Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Fluid Mechanics: Fundamentals and Applications

2nd EDITION IN SI UNITSYunus A. Cengel, John M. Cimbala

McGraw-Hill, 2010

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Objectives

Understand the role of the material derivative intransforming between Lagrangian and Euleriandescriptions

Distinguish between various types of flowvisualizations and methods of plotting the

characteristics of a fluid flow

Appreciate the many ways that fluids move anddeform

Distinguish between rotational and irrotational

regions of flow based on the flow property vorticity

Understand the usefulness of the Reynoldstransport theorem

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41 LAGRANGIAN AND EULERIAN DESCRIPTIONS

Kinematics: The study of motion.Fluid kinematics: The study of how fluids flow and how to describe fluid motion.

With a small number of objects, such

as billiard balls on a pool table,individual objects can be tracked.

In the Lagrangian description, one

must keep track of the position andvelocity of individual particles.

There are two distinct ways to describe motion: Lagrangian and Eulerian

Lagrangian description: To follow the path of individual objects.

This method requires us to track the position and velocity of each individualfluid parcel (fluid particle) and take to be a parcel of fixed identity.

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A more common method is Eulerian description of fluid motion.

In the Eulerian description of fluid flow, a finite volume called a flow domainor control volume is defined, through which fluid flows in and out.

Instead of tracking individual fluid particles, we define field variables,functions of space and time, within the control volume.

The field variable at a particular location at a particular time is the value of

the variable for whichever fluid particle happens to occupy that location atthat time.

For example, the pressure field is a scalar field variable. We define thevelocity field as a vector field variable.

Collectively, these (and other) field variables define the flow field. Thevelocity field can be expanded in Cartesian coordinates as

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In the Eulerian description, onedefines field variables, such asthe pressure field and the

velocity field, at any locationand instant in time.

In the Eulerian description wedont really care what happens toindividual fluid particles; rather weare concerned with the pressure,velocity, acceleration, etc., of

whichever fluid particle happensto be at the location of interest atthe time of interest.

While there are many occasions in

which the Lagrangian descriptionis useful, the Eulerian descriptionis often more convenient for fluidmechanics applications.

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A Steady Two-Dimensional Velocity Field

Velocity vectors for the velocity field of Example 41. The scale is shown bythe top arrow, and the solid black curves represent the approximate shapesof some streamlines, based on the calculated velocity vectors. Thestagnation point is indicated by the circle. The shaded region represents aportion of the flow field that can approximate flow into an inlet.

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Acceleration Field

Newtons second law applied to a fluidparticle; the acceleration vector (gray arrow)

is in the same direction as the force vector(black arrow), but the velocity vector (red

arrow) may act in a different direction.

The equations of motion for fluid flow

(such as Newtons second law) arewritten for a fluid particle, which wealso call a material particle.

If we were to follow a particular fluid

particle as it moves around in theflow, we would be employing theLagrangian description, and theequations of motion would be directlyapplicable.

For example, we would define theparticles location in space in termsof a material position vector(xparticle(t), yparticle(t), zparticle(t)).

e.g:

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Local acceleration(nonzero for unsteady state,

zero for steady state)

Advective (convective)acceleration (nonzero for

steady and unsteady state

Velocity field

Given:

Therefore,

HOW?

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The components of the acceleration vector in cartesian coordinates:

Flow of water through the nozzle of agarden hose illustrates that fluidparticles may accelerate, even in asteady flow. In this example, the exitspeed of the water is much higher thanthe water speed in the hose, implyingthat fluid particles have acceleratedeven though the flow is steady.

Advectiveacceleration

Localacceleration

Figure 4.8

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Material Derivative

The total derivative operator d/dt in this equation is given a special name, thematerial derivative; it is assigned a special notation, D/Dt, in order toemphasize that it is formed by following a fluid particle as it moves through

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

The material derivative D/Dt is defined by followinga fluid particle as it moves throughout the flow field.In this illustration, the fluid particle is accelerating to

the right as it moves up and to the right.

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The material derivative D/Dt iscomposed of a local or unsteady partand a convective or advective part.

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Note: From Example 4.1

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Homework

Please read and understand about:

Flow Patterns and Flow Visualization Streamlines and Streamtubes, Pathlines,

Streaklines, Timelines

Refractive Flow Visualization Techniques

Surface Flow Visualization Techniques

Plots of Fluid Flow Data

Profile Plots, Vector Plots, Contour Plots

..to do assignment

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45 VORTICITY AND ROTATIONALITY

Another kinematic property of great importance to the analysis of fluid flows isthe vorticity vector, defined mathematically as the curl of the velocity vector

The direction

of a vectorcross productis determinedby the right-hand rule.

The vorticity vector is equal totwice the angular velocity vectorof a rotating fluid particle.

Vorticity is equal to twice

the angular velocity of a fluidparticle

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If the vorticity at a point in a flow field is nonzero, thefluid particle that happens to occupy that point inspace is rotating; the flow in that region is called

rotational ( 0 ; rotational).

Likewise, if the vorticity in a region of the flow is zero(or negligibly small), fluid particles there are notrotating; the flow in that region is called irrotational.( = 0 ; irrotational).

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

The difference betweenrotational and irrotationalflow: fluid elements in a

rotational region of theflow rotate, but those inan irrotational region ofthe flow do not.

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For a two-dimensional flow in the xy-plane, the vorticity vectoralways points in the z-direction or z-direction. In thisillustration, the flag-shaped fluid particle rotates in thecounterclockwise direction as it moves in the xy-plane;

its vorticity points in the positive z-direction as shown.

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For a two-dimensional flow in the r-plane, the vorticity vector alwayspoints in the z (or -z) direction. Inthis illustration, the flag-shaped fluidparticle rotates in the clockwise

direction as it moves in the r-plane;its vorticity points in the -z-directionas shown.

Note: rotate clockwise = vorticity

points in -z direction.counterclockwise = z direction

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Comparison of Two Circular Flows

Streamlines and velocity profiles for (a) flow A, solid-body rotation and (b) flow B, a linevortex. Flow A is rotational, but flow B is irrotational everywhere except at the origin.

Not all flows with circular streamlines are rotational

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A simple analogy: (a) rotationalcircular flow is analogous to aroundabout, while (b) irrotationalcircular flow is analogous to aFerris wheel.

A simple analogy can be madebetween flow A and a merry-go-

round or roundabout, and flow Band a Ferris wheel.

As children revolve around aroundabout, they also rotate atthe same angular velocity as thatof the ride itself. This is analogousto a rotational flow.

In contrast, children on a Ferriswheel always remain oriented in

an upright position as they traceout their circular path. This isanalogous to an irrotational flow.

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46 THE REYNOLDS TRANSPORT THEOREM

Two methods of analyzing the spraying ofdeodorant from a spray can:(a) We follow the fluid as it moves anddeforms. This is the system approachnomass crosses the boundary, and the total

mass of the system remains fixed.(b) We consider a fixed interior volume of thecan. This is the control volume approachmass crosses the boundary.

The Reynolds transport theorem(RTT) provides a link betweenthe system approach and the

control volume approach.

The relationshipbetween the time rates

of change of anextensive property for asystem and for a controlvolume is expressed bythe Reynolds transport

theorem (RTT).

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A moving system (hatched region) anda fixed control volume (shaded region)in a diverging portion of a flow field attimes t and t+t. The upper and lower

bounds are streamlines of the flow.

The time rate of change of theproperty Bof the system is equal tothe time rate of change of Bof thecontrol volume plus the net flux of Bout of the control volume by masscrossing the control surface.

This equation applies at any instantin time, where it is assumed that

the system and the control volumeoccupy the same space at thatparticular instant in time.

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Relative velocity crossing a controlsurface is found by vector addition ofthe absolute velocity of the fluid andthe negative of the local velocity of

the control surface.

Reynolds transporttheorem applied to a

control volume movingat constant velocity.

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An example control volume in whichthere is one well-defined inlet (1) and

two well-defined outlets (2 and 3). Insuch cases, the control surface integralin the RTT can be more convenientlywritten in terms of the average values offluid properties crossing each inlet

and outlet.

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Alternate Derivation of the Reynolds Transport Theorem

A more elegant mathematical derivation of

the Reynolds transport theorem is possiblethrough use of the Leibniz theorem

The Leibniz theorem takes into account thechange of limits a(t) and b(t) with respect totime, as well as the unsteady changes ofintegrand G(x, t) with time.

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The material volume (system)and control volume occupy thesame space at time t (the blue

shaded area), but move anddeform differently. At a later time

they are not coincident.

The equation above is validfor arbitrarily shaped,moving and/or deforming CV

at time t.

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Relationship between Material Derivative and RTT

The Reynolds transport theorem for finitevolumes (integral analysis) is analogous to thematerial derivative for infinitesimal volumes

(differential analysis). In both cases, wetransform from a Lagrangian or systemviewpoint to an Eulerian or control volumeviewpoint.

While the Reynolds transporttheorem deals with finite-sizecontrol volumes and thematerial derivative deals withinfinitesimal fluid particles, the

same fundamental physicalinterpretation applies to both.

Just as the material derivativecan be applied to any fluidproperty, scalar or vector, theReynolds transport theoremcan be applied to any scalar orvector property as well.