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MECH 221 FLUID MECHANICS (Fall 06/07) Chapter 4: FLUID KINETMATICS

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MECH 221 FLUID MECHANICS (Fall 06/07) Chapter 4: FLUID KINETMATICS Instructor: Professor C. T. HSU – PowerPoint PPT presentation

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Title: MECH 221 FLUID MECHANICS (Fall 06/07) Chapter 4: FLUID KINETMATICS


1
MECH 221 FLUID MECHANICS(Fall 06/07)Chapter 4
FLUID KINETMATICS
  • Instructor Professor C. T. HSU

2
4. FLUID KINETMATICS
  • Fluid kinematics concerns the motion of fluid
    element. As the fluid flows, a fluid particle
    (element) can translate, rotate, and deform
    linearly and angularly

Translation Rotation
Linear deformation Circulation
Dilation Viscous stress
Angular deformation
3
4.1. Translation
  • The translation considers mainly the velocity and
    acceleration along the trajectory of fluid
    element in linear motion

4
4.1. Translation
  • For the fluid element moving along the trajectory
    r(t), the velocity is simply given by v dr/dt
    (u,v,w). As the description is basically
    Lagrangian, the acceleration a is given by
  • which, for steady flows, reduces to

5
4.2. Linear Deformation (Strain)
  • Deformation change of shape of fluid element
  • For easily understanding, we illustrate here in
    two-dimensions. The results then can be easily
    extended to 3-dimensions. Consider the
    rectangular fluid element at the initial time
    instant given in the following picture

6
4.2. Linear Deformation (Strain)
  • The initial distance between points A and B is ?x
    and between A and C is ?y. After a short time of
    ?t, the distances then become ?x?Lx and ?y?Ly
    due to different velocities at B and C from A

7
4.2. Linear Deformation (Strain)
  • The linear strain rate in x and y directions are
    then given by
  • Similarly, for 3-D flows we have in the
    z-direction,

8
4.3. Dilation
  • Volumetric expansion contraction
  • The fluid dilation is defined as the change of
    volume per unit volume. We are more interested in
    the rate of dilation that determines the
    compressibility of fluids. For 2-D flows,

9
4.3. Dilation
  • Then, the rate of dilatation becomes,
  • It is easy to generalize this dilation rate for
    3-D flows and to reach
  • For incompressible flow, the rate of dilation is
    zero,

for 2-D flows
10
4.4. Angular Deformation (Strain)
  • Now consider the deformation between A and B
    caused by the change in velocity v, and the
    deformation between A and C by change in u

11
4.4. Angular Deformation (Strain)
  • For , the counter clockwise rotation
    of AB is equal to clockwise rotation of AC
    therefore, the fluid element is in pure angular
    strain without net rotation and the angular
    strain is equal to either or .
    However, if ? , the strain then is
    equal to
  • . The rate of angular strain
    is then given by

12
4.4. Angular Deformation (Strain)
  • Similarly, we can extend to other planes y-z and
    z-x to obtain

13
4.5. Rotation
  • If then the fluid element is under
    rigid body rotation on the x-y plane. No angular
    strain is experienced, i.e.,

14
4.5. Rotation
  • When ? , the rotation of fluid element
    in x-y plane is the average rotation of the two
    mutually perpendicular lines AB and AC
    therefore,
  • where a counter clockwise rotation is chosen as
    positive and the rotation axis is in the z
    direction

15
4.5. Rotation
  • Rotation is a vector quantity for fluid elements
    in 3-D motion. A fluid particle moving in a
    general 3-D flow field may rotate about all three
    coordinate axes, thus

16
4.5. Rotation
  • The vorticity of a flow field is defined as

17
4.5. Rotation
  • Therefore,
  • The flow vorticity is twice the rotation
  • In 2-D flow, ?/?z0 and w0 (or const.), so there
    is only one component of vorticity,
  • Irrotational flow is defined as having

18
4.5. Rotation
  • A fluid particle moving, without rotation, in a
    flow field cannot develop a rotation under the
    action of a body force or normal surface force.
    If fluid is initially without rotation, the
    development of rotation requires the action of
    shear stresses. The presence of viscous forces
    implies the flow is rotational
  • The condition of irrotationality can be a valid
    assumption only when the viscous forces are
    negligible. (as example, for flow at very high
    Reynolds number, Re, but not near a solid
    boundary)

19
4.6. Circulation
  • Consider the flow field as shown below
  • The circulation, , is defined as the line
    integral of the tangential velocity about a
    closed curve fixed in the flow,

20
4.6. Circulation
  • Where is the line-element vector tangent to the
    closed loop C of the integral. It is possible to
    decompose the integral loop C into the sum of
    small sub-loops, i.e.,
  • Without loss of generality, each sub-loop can be
    a rectangular grid as illustrated below.

21
4.6. Circulation
  • Therefore,
  • As a result, we have

where A is the area enclosed the contour
22
4.6. Circulation
  • Stokes' theorem in 2-D
  • The circulation around a closed contour (loop)
    is the sum of the vorticity (flux) passing
    through the loop
  • This is an expression to illustrate the Greens
    Theorem. In fact, the surface A can be a curved
    surface

23
4.6. Circulation
  • Then for each sub-loop on the surface, we have
    locally
  • where is the vorticity normal to the surface
    enclosed by the small increment loop C

24
4.7. Viscous Stresses
  • The strain rate tensor S is a symmetric tensor
    that measures the rate of linear and angular
    deformations of fluid element. The strain rate
    tensor is expressed as
  • where the superscript T represents the
    transpose

25
4.7. Viscous Stresses
  • In term of a Cartesian coordinate system, they
    are expressed as

and
26
4.7. Viscous Stresses
  • Following the Stokes hypothesis, the viscous
    stress tensor is linearly related to the rate
    of dilation and the strain rate tensor by

where I represents the unit tensor, i.e.,
27
4.7. Viscous Stresses
  • The proportional constants of the above
    linear relation are the volume viscosity and
    shear viscosity of the fluid respectively. It is
    seen that the fluid viscosity leads to additional
    normal stresses, as well as shear stresses. Note
    that is a symmetric tensor, i.e.,
  • Total stress is given by
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