ME 33, Fluid Flow Chapter 4: Fluid Kinematics - kau.edu.sa · PDF fileFundamental kinematic properties of ﬂuid motion and deformation. Reynolds Transport Theorem. ME 33, ... Chapter

ME 33, FluidFlow

Chapter 4:Fluid

Kinematics

Lagrangianand EulerianDescriptions

FlowVisualization

Plots of Data

KinematicDescriptions

Reynolds–TransportTheorem

ME 33, Fluid FlowChapter 4: Fluid Kinematics

Eric G. Paterson

Department of Mechanical and Nuclear EngineeringThe Pennsylvania State University

Spring 2005

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Overview

Fluid Kinematics deals with the motion of fluids withoutconsidering the forces and moments which create themotion.Items discussed in this Chapter.

Material derivative and its relationship to Lagrangianand Eulerian descriptions of fluid flow.Flow vizualization.Plotting flow data.Fundamental kinematic properties of fluid motion anddeformation.Reynolds Transport Theorem

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Lagrangian Description

Lagrangian description of fluid flow tracks the position~xA, ~xB, · · · , and velocity ~VA, ~VB, . . . of individualparticles.

Based upon Newton’s laws of motion.Difficult to use for practical flow analysis.

Fluids are composed of billions of molecules.Interaction between molecules hard to describe/model.

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

Named after Italian mathematician Joseph LouisLagrange (1736-1813).

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Eulerian Description

Eulerian description of fluid flow a flow domain orcontrol volume is defined, through which fluid flows inand out.We define field variables which are functions of spaceand time.

Pressure field, P = P(x , y , z, t)Velocity field, ~V = ~V (x , y , z, t)~V = (u, v , w) = u(x , y , z, t)~i+v(x , y , z, t)~j+w(x , y , z, t)~kAcceleration field, ~a = ~a(x , y , z, t)~a = (ax , ay , az) =

ax(x , y , z, t)~i + ay (x , y , z, t)~j + az(x , y , z, t)~kThese (and other) field variables define the flow field .

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

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Example: Coupled Eulerian-LagrangianMethod

Simulation of micron-scale airborne probes. The probepositions are tracked using a Lagrangian particle modelembedded within an Eulerian CFD code. GlobalEnvironmental MEMS Sensors (GEMS) website

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

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

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Example: Coupled Eulerian-LagrangianMethod

Forensic analysis of Columbia accident: simulation ofshuttle debris trajectory using Eulerian CFD for flow fieldand Lagragian method for the debris.

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Acceleration Field

Consider a fluid particle and Newton’s second law,~Fparticle = mparticle~aparticle

The acceleration of the particle is the time derivative of

the particle’s velocity. ~aparticle =d ~Vparticle

dt

However, particle velocity at a point is the same as thefluid velocity,~Vparticle = ~V

(xparticle(t), yparticle(t), zparticle(t), t

)To take the time derivative of ~V , chain rule must beused.

~aparticle =∂~V∂t

dtdt

+∂~V

∂xparticle

dxparticle

dt

+∂~V

∂yparticle

dyparticle

dt+

∂~V∂zparticle

dzparticle

dt

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Acceleration Field

Since dxparticledt = u, dyparticle

dt = v , and dzparticledt = w

~aparticle(x , y , z, t) =∂~V∂t

+ u∂~V∂x

+ v∂~V∂y

+ w∂~V∂z

In vector form, the acceleration can be written as

~a(x , y , z, t) =d ~Vdt

=∂~V∂t

+(

~V · ~∇)

~V

First term is called the local acceleration and isnonzero only for unsteady flows.

Second term is called the advective acceleration andaccounts for the effect of the fluid particle moving to anew location in the flow, where the velocity is different.

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Material Derivative

The total derivative operator ddt is call the material

derivative and is often given special notation, DDt .

D~VDt

=d ~Vdt

=∂~V∂t

+(

~V · ~∇)

~V

Advective acceleration is nonlinear: source of manyphenomenon and primary challenge in solving fluid flowproblems.

Provides “transformation” between Lagragian andEulerian frames.

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

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Flow Visualization

Flow visualization is the visual examination of flow–fieldfeatures.

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

Streamlines and streamtubesPathlinesStreaklinesTimelinesRefractive techniquesSurface flow techniques

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Streamlines

A Streamline is a curve that iseverywhere tangent to theinstantaneous local velocityvector.

Consider an arc lengthd~r = dx~i + dy~j + dz~k

d~r must be parallel to the localvelocity vector~V = u~i + v~j + w~k

Geometric arguments results inthe equation for a streamlinedrV = dx

u = dyv = dz

w

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Streamlines

NASCAR surface pressurecontours and streamlines

Airplane surface pressurecontours, volume streamlines,

and surface streamlines

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Pathlines

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

Same as the fluid particle’smaterial position vector(xparticle(t), yparticle(t), zparticle(t)

)Particle location at time t:

~x = ~xstart +

∫ t

tstart

~V dt

Particle Image Velocimetry(PIV) is a modern experimentaltechnique to measure velocityfield over a plane in the flowfield.

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PIV

Companion animation on ANGEL

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Streaklines

A Streakline is the locusof fluid particles thathave passed sequentiallythrough a prescribedpoint in the flow.

Easy to generate inexperiments: dye in awater flow, or smoke inan airflow.

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Streaklines

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Comparisons

For steady flow, streamlines, pathlines, and streaklinesare identical.For unsteady flow, they can be very different.

Streamlines are an instantaneous picture of the flowfieldPathlines and Streaklines are flow patterns that have atime history associated with them.Streakline: instantaneous snapshot of a time-integratedflow pattern.Pathline: time-exposed flow path of an individualparticle.

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Timelines

A Timeline is the locusof fluid particles thathave passed sequentiallythrough a prescribedpoint in the flow.

Timelines can begenerated using ahydrogen bubble wire.

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Plots of Data

A Profile plot indicates how the value of a scalarproperty varies along some desired direction in the flowfield.

A Vector plot is an array of arrows indicating themagnitude and direction of a vector property at aninstant in time.

A Contour plot shows curves of constant values of ascalar property for magnitude of a vector property at aninstant in time.

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Kinematic Description

In fluid mechanics, an element mayundergo four fundamental types ofmotion.

1 translation2 rotation3 linear strain4 shear strain

Because fluids are in constantmotion, motion and deformation isbest described in terms of rates

1 velocity: rate of translation2 angular velocity: rate of rotation3 linear strain rate: rate of linear strain4 shear strain rate: rate of shear

strain

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Rate of Translation and Rotation

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

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

~V = u~i + v~j + w~k

Rate of rotation at a point is defined as the averagerotation rate of two initially perpendicular lines thatintersect at that point. The rate of rotation vector inCartesian coordinates:

~ω = 12

(∂w∂y −

∂v∂z

)~i + 1

2

(∂u∂z −

∂w∂x

)~j + 1

2

(∂v∂x −

∂u∂y

)~k

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Linear Strain Rate

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

In Cartesian coordinates:

εxx = ∂u∂x εyy = ∂v

∂y εzz = ∂w∂z

Volumetric strain rate in Cartesian coordinates

1V

DVDt = εxx + εyy + εzz = ∂u

∂x + ∂v∂y + ∂w

∂z

Since the volume of a fluid element is constant for anincompressible flow, the volumetric strain rate must bezero.

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Shear Strain Rate

Shear Strain Rate at a point is defined as half of therate of decrease of the angle between two initiallyperpendicular lines that intersect at a point.

Shear strain rate can be expressed in Cartesiancoordinates as:

εxy = 12

(∂u∂y + ∂v

∂x

)εzx = 1

2

(∂w∂x + ∂u

∂z

)εyz =

12

(∂v∂z + ∂w

∂y

)

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Shear Strain Rate

We can combine linear strain rate and shear strain rate intoone symmetric second-order tensor called the strain–ratetensor .

εij =

εxx εxy εxz

εyx εyy εyz

εzx εzy εzz

=∂u∂x

12

(∂u∂y + ∂v

∂x

)12

(∂u∂z + ∂w

∂x

)12

(∂v∂x + ∂u

∂y

)∂v∂y

12

(∂v∂z + ∂w

∂y

)12

(∂w∂x + ∂u

∂z

) 12

(∂w∂y + ∂v

∂z

)∂w∂z

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Shear Strain Rate

Purpose of our discussion of fluid element kinematics:1 better appreciation of the inherent complexity of fluid

dynamics2 mathematical sophistication requried to fully describe

fluid motion

Strain–rate tensor is important for numerous reasons.For example,

1 Develop relationships between fluid stress and strainrate.

2 Feature extraction in CFD simulation.

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Shear Strain Rate

Example: Trailing-edge turbulence (Paterson and Peltier,2004).

Visualization of turbulent eddies using the intrinsic–swirl parameterwhich is computed using an eigen-analysis of the strain-rate tensor.

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Vorticity and Rotationality

The vorticity vector is defined as the curl of thevelocity vector ~ζ = ~∇× ~V

Vorticity is equal to twice the angular velocity of a fluidparticle. ~ζ = 2~ωCartesian coordinates~ζ =

(∂w∂y −

∂v∂z

)~i +

(∂u∂z −

∂w∂x

)~j +

(∂v∂x −

∂u∂y

)~k

Cylindrical coordinates ~ζ =(1r

∂uz∂θ − ∂uθ

∂z

)~er +

(∂ur∂z − ∂uz

∂r

)~eθ + 1

r

(∂(ruθ)

∂r − ∂ur∂θ

)~ez

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

Elsewhere, the flow is called rotational .

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Vorticity and Rotationality

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Comparison of Two Circular Flows

Special case: consider two flows with circular streamlines

ur = 0 and uθ = ωr~ζ = 1

r

(∂(ruθ)

∂r − ∂ur∂θ

)~ez =

1r

(∂(ωr2)

∂r − 0)

~ez = 2ω ~ez

ur = 0 and uθ = Kr

~ζ = 1r

(∂(ruθ)

∂r − ∂ur∂θ

)~ez =

1r

(∂(K )∂r − 0

)~ez = 0

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Comparison of Two Circular Flows

Special case: consider two flows with circular streamlines

ur = 0 and uθ = ωr~ζ = 1

r

(∂(ruθ)

∂r − ∂ur∂θ

)~ez =

1r

(∂(ωr2)

∂r − 0)

~ez = 2ω ~ez

ur = 0 and uθ = Kr

~ζ = 1r

(∂(ruθ)

∂r − ∂ur∂θ

)~ez =

1r

(∂(K )∂r − 0

)~ez = 0

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Reynolds–Transport Theorem (RTT)

Recall System vs. Control Volume arguments.A system is a quantity of matter of fixed identity. Nomass can cross a system boundary.A control volume is a region in space chosen forstudy. Mass can cross a control surface.The fundamental conservation laws (conservation ofmass, energy, and momentum) apply directly tosystems.However, in most fluid mechanics problems, controlvolume analysis is preferred over system analysis (forthe same reason that the Eulerian description is usuallypreferred over the Lagrangian description).Therefore, we need to transform the conservation lawsfrom a system to a control volume. This isaccomplished with the Reynolds transport theorem(RTT).

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There is a direct analogy between the transformation fromLagrangian to Eulerian descriptions (for differential analysisusing infinitesimally small fluid elements) and thetransformation from systems to control volumes (for integralanalysis using large, finite flow fields).

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Material derivative (differential analysis):

DbDt

=∂b∂t

+(

~V · ~∇)

b

General RTT, nonfixed CV (integral analysis):

dBsys

dt=

∫CV

∂

∂t(ρb) dV +

∫CS

ρb~V · ~n dA

B = Extensive properties: mass(m), momentum(m~V ),energy(E), angular momentum (~H)

b = Intensive properties = B/m: 1, ~V , e,(~r × ~V

)In Chaps 5 and 6, we will apply RTT to conservation ofmass, energy, linear momentum, and angularmomentum.

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Interpretation of the RTT:

Time rate of change of the property B of the system isequal to (Term 1) + (Term 2)

Term 1: the time rate of change of B of the controlvolume

Term 2: the net flux of B out of the control volume bymass crossing the control surface

dBsys

dt=

∫CV

∂

∂t(ρb) dV +

∫CS

ρb~V · ~n dA

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RTT: Special Cases

For moving and/or deforming control volumes,

dBsys

dt=

ddt

∫CV

ρb dV +

∫CS

ρb~Vr · ~n dA

Where the absolute velocity ~V in the second term isreplaced by the relative velocity ~Vr = ~V − ~VCS~Vr is the fluid velocity expressed relative to a coordinatesystem moving with the control volume.

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RTT: Special Cases

For steady flow, the time derivative drops out

dBsys

dt=

��:0∫CV

∂

∂t(ρb) dV +

∫CS

ρb~V · ~n dA =

∫CS

ρb~V · ~n dA

For control volumes with well-defined inlets and outlets

dBsys

dt=

ddt

∫CV

ρb dV +∑out

ρavgbavgVr ,avgA−∑

in

ρavgbavgVr ,avgA

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ME 33, Fluid Flow Chapter 4: Fluid Kinematics - kau.edu.sa · PDF fileFundamental kinematic properties of ﬂuid motion and deformation. Reynolds Transport Theorem. ME 33, ... Chapter