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Chapter 4: Fluid Kinematics

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Items discussed in this Chapter. Material derivative and its relationship to Lagrangian and Eulerian descriptions of fluid flow. Flow visualization. – PowerPoint PPT presentation

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Title: Chapter 4: Fluid Kinematics


1
Chapter 4 Fluid Kinematics
Fundamentals of Fluid Mechanics
Department of Hydraulic Engineering - School of
Civil Engineering - Shandong University - 2007
2
Overview
  • Fluid Kinematics deals with the motion of fluids
    without necessarily considering the forces and
    moments which create the motion.
  • Items discussed in this Chapter.
  • Material derivative and its relationship to
    Lagrangian and Eulerian descriptions of fluid
    flow.
  • Flow visualization.
  • Plotting flow data.
  • Fundamental kinematic properties of fluid motion
    and deformation.
  • Reynolds Transport Theorem

3
Lagrangian Description
  • Two ways to describe motion are Lagrangian and
    Eulerian description
  • Lagrangian description of fluid flow tracks the
    position and velocity of individual particles.
    (eg. Brilliard ball on a pooltable.)
  • Motion is described based upon Newton's laws.
  • 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 applications
  • Sprays, particles, bubble dynamics, rarefied
    gases.
  • Coupled Eulerian-Lagrangian methods.
  • Named after Italian mathematician Joseph Louis
    Lagrange (1736-1813).

4
Eulerian Description
  • 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, PP(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).

5
Example Coupled Eulerian-Lagrangian Method
  • Global Environmental MEMS Sensors (GEMS)
  • Simulation of micron-scale airborne probes. The
    probe positions are tracked using a Lagrangian
    particle model embedded within a flow field
    computed using an Eulerian CFD code.

http//www.ensco.com/products/atmospheric/gem/gem_
ovr.htm
6
Example Coupled Eulerian-Lagrangian Method
  • Forensic analysis of Columbia accident
    simulation of shuttle debris trajectory using
    Eulerian CFD for flow field and Lagrangian method
    for the debris.

7
EXAMPLEL A A Steady Two-Dimensional
Velocity Field
  • A steady, incompressible, two-dimensional
    velocity field is given by
  • A stagnation point is defined as a point in
    the flow field where the velocity is identically
    zero. (a) Determine if there are any stagnation
    points in this flow field and, if so, where? (b)
    Sketch velocity vectors at several locations in
    the domain between x - 2 m to 2 m and y 0 m
    to 5 m qualitatively describe the flow field.

8
Acceleration Field
  • Consider a fluid particle and Newton's second
    law,
  • The acceleration of the particle is the time
    derivative of the particle's velocity.
  • However, particle velocity at a point at any
    instant in time t is the same as the fluid
    velocity,
  • To take the time derivative of, chain rule must
    be used.

,t)
9
Acceleration Field
Where ? is the partial derivative operator and d
is the total derivative operator.
  • Since
  • In vector form, the acceleration can be written
    as
  • First term is called the local acceleration and
    is nonzero only for unsteady flows.
  • Second term is called the advective acceleration
    and accounts for the effect of the fluid particle
    moving to a new location in the flow, where the
    velocity is different.

10
EXAMPLE Acceleration of a Fluid Particle through
a Nozzle
Nadeen is washing her car, using a nozzle.
The nozzle is 3.90 in (0.325 ft) long, with an
inlet diameter of 0.420 in (0.0350 ft) and an
outlet diameter of 0.182 in. The volume flow rate
through the garden hose (and through the nozzle)
is 0.841 gal/min (0.00187 ft3/s), and the flow is
steady. Estimate the magnitude of the
acceleration of a fluid particle moving down the
centerline of the nozzle.
11
Material Derivative
  • The total derivative operator d/dt is call the
    material derivative and is often given special
    notation, D/Dt.
  • Advective acceleration is nonlinear source of
    many phenomenon and primary challenge in solving
    fluid flow problems.
  • Provides transformation'' between Lagrangian
    and Eulerian frames.
  • Other names for the material derivative include
    total, particle, Lagrangian, Eulerian, and
    substantial derivative.

12
EXAMPLE B Material Acceleration of a
Steady Velocity Field
  • Consider the same velocity field of Example A.
    (a) Calculate the material acceleration at the
    point (x 2 m, y 3 m). (b) Sketch the material
    acceleration vectors at the same array of x- and
    y values as in Example A.

13
Flow Visualization
  • Flow visualization is the visual examination of
    flow-field features.
  • Important for both physical experiments and
    numerical (CFD) solutions.
  • Numerous methods
  • Streamlines and streamtubes
  • Pathlines
  • Streaklines
  • Timelines
  • Refractive techniques
  • Surface flow techniques

While quantitative study of fluid dynamics
requires advanced mathematics, much can be
learned from flow visualization
14
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

15
EXAMPLE C Streamlines in the xy
PlaneAn Analytical Solution
For the same velocity field of Example A, plot
several streamlines in the right half of the flow
(x gt 0) and compare to the velocity vectors.
where C is a constant of integration that can be
set to various values in order to plot the
streamlines.
16
Streamlines
Airplane surface pressure contours, volume
streamlines, and surface streamlines
NASCAR surface pressure contours and streamlines
17
Streamtube
  • A streamtube consists of a bundle of streamlines
    (Both are instantaneous quantities).
  • Fluid within a streamtube must remain there and
    cannot cross the boundary of the streamtube.
  • In an unsteady flow, the streamline pattern may
    change significantly with time.? the mass flow
    rate passing through any cross-sectional slice of
    a given streamtube must remain the same.

18
Pathlines
  • A Pathline is the actual path traveled by an
    individual fluid particle over some time period.
  • Same as the fluid particle's material position
    vector
  • Particle location at time t

19
Pathlines
A modern experimental technique called particle
image velocimetry (PIV) utilizes (tracer)
particle pathlines to measure the velocity field
over an entire plane in a flow (Adrian, 1991).
20
Pathlines
Flow over a cylinder
Top View
Side View
21
Streaklines
  • A Streakline is the locus of fluid particles that
    have passed sequentially through a prescribed
    point in the flow.
  • Easy to generate in experiments dye in a water
    flow, or smoke in an airflow.

22
Streaklines
23
Streaklines
Karman Vortex street
Cylinder
x/D
A smoke wire with mineral oil was heated to
generate a rake of Streaklines
24
Comparisons
  • For steady flow, streamlines, pathlines, and
    streaklines are identical.
  • 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 time history associated with them.
  • Streakline instantaneous snapshot of a
    time-integrated flow pattern.
  • Pathline time-exposed flow path of an
    individual particle.

25
Comparisons
26
Timelines
  • A Timeline is a set of adjacent fluid particles
    that were marked at the same (earlier) instant in
    time.
  • Timelines can be generated using a hydrogen
    bubble wire.

27
Timelines
Timelines produced by a hydrogen bubble wire are
used to visualize the boundary layer velocity
profile shape.
28
Refractive Flow Visualization Techniques
  • Based on the refractive property of light waves
    in fluids with different index of refraction, one
    can visualize the flow field shadowgraph
    technique and schlieren technique.

29
Plots of Flow Data
  • Flow data are the presentation of the flow
    properties varying in time and/or space.
  • 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 and direction of a vector property
    at an instant in time.
  • A Contour plot shows curves of constant values of
    a scalar property for the magnitude of a vector
    property at an instant in time.

30
Profile plot
Profile plots of the horizontal component of
velocity as a function of vertical distance flow
in the boundary layer growing along a horizontal
flat plate.
31
Vector plot
32
Contour plot
Contour plots of the pressure field due to flow
impinging on a block.
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