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ME3560 – Fluid Mechanics

Chapter IV. Fluid Kinematics

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ME 3560 Fluid Mechanics




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4.1 The Velocity Field• One of the most important parameters that need to be monitored whena fluid is flowing is the velocity.•In general the flow parameters are described in terms of the motion offluid particles rather than individual molecules.•Thus, this motion can be described in terms of the velocity andacceleration of the fluid particles.•In order to describe the flow parameters it is sought to provide at agiven instant in time, a description of any fluid property (such as , p, V,and a) as a function of the fluid's location.•This representation of fluid parameters as functions of the spatialcoordinates is termed a field representation of the flow.•The specific field representation may be different at different times,thus. Thus, to completely specify the velocity, V, in a process, thevelocity field must be expressed as: V = V (x, y, z, t).

ktzyxwjtzyxvitzyxuV ˆ),,,(ˆ),,,(ˆ),,,(



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• u, v, and w are the x, y, and zcomponents of the velocity vector.•The velocity of a particle is the timerate of change of the position vectorfor that particle.•The position of particle A relative tothe coordinate system is given by itsposition vector, rA, which (if theparticle is moving) is a function oftime.•The velocity is a vector therefore it has both a direction and amagnitude, V = |V| = (u2 + v2 + w2)1/2

•A change in velocity results in an acceleration. This acceleration maybe due to a change in speed and/or direction.

kyzyxwjyzyxviyzyxuV ˆ),,,(ˆ),,,(ˆ),,,(



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4.1.1 Eulerian and Lagragian Flow Descriptions• There are two general approaches in analyzing fluid mechanicsproblems:•Eulerian method. Measures the flow parameters at fixed locations tothen generate the parameters flowfield.

•Lagrangian method, involves following individual fluid particles asthey move about and determining how the fluid properties associatedwith these particles change as a function of time.

4.1.2 One-, Two-, and Three-Dimensional Flows•In general, a fluid flow is a rather complex three-dimensional, time-dependent phenomenon.

ktzyxwjtzyxvitzyxuV ˆ),,,(ˆ),,,(ˆ),,,(



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• However, sometimes it is possible to make simplifying assumptionsthat allow a much easier understanding of the problem withoutsacrificing needed accuracy:•Assume the flow is two–dimensional or one–dimensional.•Assume the flow is incompressible (even when dealing with gases)•One of these simplifications involves approximating a real flow as asimpler one– or two–dimensional flow.



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4.1.3 Steady and Unsteady Flows• Steady flow. The velocity (or any other parameter: T, p, , etc.) at agiven point in space does not vary with time V/t = 0.•Unsteady flows. Flow field parameters are time–dependent V/t0.•Among the various types of unsteady flows are nonperiodic flow,periodic flow, and truly random flow.

4.1.4 Streamlines, Streaklines, and Pathlines• Streamline is a line that is everywhere tangentto the velocity field.• If the flow is steady, nothing at a fixed point(including the velocity direction) changes withtime, so the streamlines are fixed lines in space.• For unsteady flows the streamlines may changeshape with time.

uv

dxdy

wz

vy

ux



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• Streakline consists of all particles in a flow that have previouslypassed through a common point.• Streaklines can be generated by taking instantaneous photographs ofmarked particles that all passed through a given location in the flowfield at some earlier time. Such a line can be produced by continuouslyinjecting marked fluid (neutrally buoyant smoke in air, or dye in water)at a given location.



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• Pathline is the line traced out by a given particle as it flows from onepoint to another.• The pathline is a Lagrangian concept that can be produced in thelaboratory by marking a fluid particle (dying a small fluid element) andtaking a time exposure photograph of its motion.• Pathlines, Streamlines, and Streaklines are the same for steady flows.• For unsteady flows none of these three types of lines need to be thesame.



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4.2 The Acceleration Field•A flow can be studied by either (1) following individual particles(Lagrangian description) or (2) remaining fixed in space and observingdifferent particles as they pass by (Eulerian description).• In either case, to apply Newton's second law (F = m a) it is necessaryto describe the particle acceleration in an appropriate fashion.•For the infrequently used Lagrangian method, we describe the fluidacceleration just as is done in solid body dynamics—a = a(t) for eachparticle.•For the Eulerian description we describe the acceleration field as afunction of position and time without actually following any particularparticle, a = a(x, y, z, t).• The acceleration of a particle is the time rate of change of its velocity.• For unsteady flows the velocity at a given point in space varies withtime. Also, a fluid particle may accelerate because its velocity changesas it flows from one point to another in space.



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4.2.1 The Material Derivative

zww

ywv

xwu

twa

zvw

yvv

xvu

tva

zuw

yuv

xuu

tua

zVw

yVv

xVu

tVa

zw

yv

xu

tdtD

dtVDa

)()()()()(



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• D( )/dt is termed the material derivative or substantial derivative.•A shorthand notation for the material derivative operator is

))(()()(

Vtdt

D

4.2.2 Unsteady Effects• The material derivative formula contains two types of terms:• Terms involving the time derivative ()/t: Local Derivative• Terms involving spatial derivatives ()/x, ()/y, and ()/z.• ()/t represents the effects of the unsteadiness of the flow.•V/t is termed the local acceleration. For steady flow: ()/t = 0•Physically, there is no change in flow parameters at a fixed point inspace if the flow is steady.•There may be a change of those parameters for a fluid particle as itmoves about.•If a flow is unsteady, its parameter values (V, T, , etc.) at any locationmay change with time



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4.2.3 Convective Effects• The portion of the material derivative represented by the spatialderivatives is termed the convective derivative.

• The convective derivative represents the fact that a flow propertyassociated with a fluid particle may vary because of the motion of theparticle from one point in space to another point in space.

• This contribution to the time rate of change of the parameter for theparticle can occur whether the flow is steady or unsteady.

• It is due to the convection, or motion, of the particle through space inwhich there is a gradient [( )=( )/x î + ( )/x ĵ + ( )/z k]

•The portion of a due to the term (V ꞏ )V is the convective acceleration



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4.3 Control Volume and System Representations•A system is a collection of matter of fixed identity (always the sameatoms or fluid particles), which may move, flow, and interact with itssurroundings.

•A system is a specific, identifiable quantity of matter. It may consist of arelatively large amount of mass or it may be an infinitesimal size.

•A system may interact with its surroundings by various means (by thetransfer of heat or the exertion of a pressure force, for example).

• A system may continually change size and shape, but it alwayscontains the same mass.



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•A control volume, is a volume in space (a geometric entity, independentof mass) through which fluid may flow.

•In fluid mechanics, it is difficult to identify and keep track of a specificquantity of matter.

• In several cases, the main interest is in determining the forces put on adevice rather than in the information obtained by following a givenportion of the air (a system) as it flows along.

• For these situations it is more adequate to use the control volumeapproach.

•Identify a specific volume in space (a volume associated with the deviceof interest) and analyze the fluid flow within, through, or around thatvolume.



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• In general, the control volume can be a moving volume, although formost situations we will use only fixed, nondeformable control volumes.• The matter within a control volume may change with time as the fluidflows through it.• The amount of mass within the volume may change with time.•The control volume itself is a specific geometric entity, independent ofthe flowing fluid.



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•All of the laws governing the motion of a fluid are stated in their basicform in terms of a system approach.•For example, “the mass of a system remains constant,” or “the time rateof change of momentum of a system is equal to the sum of all the forcesacting on the system.”



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4.4 The Reynolds Transport Theorem• The Reynolds transport theorem provides a mathematical way to relatethe properties of a flow between a control volume and a system.

• Extensive Property, is a property whose value is directly proportionalto the amount of the mass being considered.

-Mass -Energy

-Momentum• Intensive Property, is a property whose value is independent of theamount of mass.

-Density-Temperature

-Velocity-Pressure



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• An extensive property B is related to its corresponding intensiveproperty b by B = mb

• In general, for a system, an extensive property B can be determined as:

•The rate of change of an extensive property in a system is expressed as

sys

sys VdbB

B = m b = 1B = mV b = VB = E b = e

dt

Vdbd

dtdB syssys



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• In a similar way, the rate of change of an extensive property in a controlvolume is expressed as

dt

Vdbd

dtdB cvcv

4.4.1 Derivation of the Reynolds Transport Theorem• Consider two instants intime: t and t+t.•Assume that at t thecontrol volume and thesystem coincide, thus

)()( tBtB cvsys



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• Then, at a time t+t

•The change in the amount ofB in the system in the timeinterval δt divided by thistime interval is given by

)()()()( ttBttBttBttB IIIcvsys

ttBttB

tB syssyssys

)()(

ttBttBttBttB

tB sysIIIcvsys

)()()()(



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• Since at the initial time t we have Bsys(t) = Bcv(t),

• For δt→ 0, the left-hand side this equation is DBsys/Dt.• DBsys/Dt represents the time rate of change of property B associatedwith a system (a given portion of fluid) as it moves along.• For δt → 0, the first term on the right-hand side of this equation is thetime rate of change of the amount of B within the control volume

tttB

tttB

ttBttB

tB IIIcvcvsys

)()()()(

t

Vbd

tB

ttBttB cvcvcvcv

t

)()(lim0



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• The term

•Represents the amount of the extensive parameter B flowing out of thecontrol volume, across the control surface.•Thus, the rate at which this property flows from the control volume is:

tVAbVbttB IIII 222222 ))(()(

22220

)(lim bVAtttBB II

tout



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• The term

• Represents the amount of the extensive parameter B flowing into thecontrol volume, across the control surface•Thus, the rate at which this property flows from the control volume is:

tVAbVbttB II 111111 ))(()(

11110

)(lim bVAtttBB I

tin



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• By combining the previous equations, a relation between the time rateof change of B for the system and that for the control volume is given by

11112222 bVAbVAtB

DtDB

BBtB

DtDB

cvsys

inoutcvsys

• A generalization of the previous equation is given by the ReynoldsTransport Theorem

cscv

sys dAnVbVdbtDt

DBˆ



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Selection of a Control Volume• CV is typically fixed and non–deforming.• CV includes all relevant inlets and outlets where information isavailable or required.• If possible CS should be perpendicular to the velocity vector to simplify

•Contributions to the surface integrals must be simple and relevant. In CSother than inlets and outlets, select solid boundaries where V = 0.

nV ˆ

Sign of nV ˆ

Vin Vout

n n

inin VnV ˆ

outout VnV ˆ

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