CE%20150%20Fluid%20Mechanics

1
CE 150Fluid Mechanics

• G.A. Kallio
• Dept. of Mechanical Engineering, Mechatronic
  Engineering Manufacturing Technology
• California State University, Chico

2
Fluid Kinematics

• Reading Munson, et al., Chapter 4

3
Introduction

• In this chapter we consider fluid kinematics,
  which addresses the behavior of fluids while they
  are flowing without concern of the actual forces
  necessary to produce the motion
• Specifically, we will address
• fluid velocity
• fluid acceleration
• flow pattern description and visualization

4
Fluid Models

• Continuum model fluids are a collection of fluid
  particles that interact with each other and
  surroundings each particle contains a sufficient
  number of molecules such that fluid properties
  (e.g., velocity) can be defined.
• Molecular model the motions of individual fluid
  molecules are accounted for not a practical
  model unless fluid density is very small or flow
  over very small objects are considered.

5
Flow Descriptions

• Lagrangian description properties of individual
  fluid particles are defined as a function of time
  as they move through the fluid the overall fluid
  motion is found by solving the EOMs for all fluid
  particles.
• Eulerian description properties are defined at
  fixed points in space as the fluid flows past
  these points this is the most common description
  and yields the field representation of fluid flow.

6
The Velocity Field

• Consider an array of sensors that can
  simultaneously measure the magnitude and
  direction of fluid velocity at many fixed points
  within the flow as a function of time in the
  limit of measuring velocity at all points within
  the flow, we would have sufficient information to
  define the velocity vector field

7
The Velocity Field

• u, v, and w are the x, y, and z components of the
  velocity vector
• The magnitude of the velocity, or speed, is
  denoted by V as
• Velocity field may be one- (u), two- (u,v)or
  three- (u,v,w) dimensional
• Steady vs. unsteady flows

8
Visualization of Fluid Flow

• Three basic types of lines used to illustrate
  fluid flow patterns
• Streamline a line that is everywhere tangent to
  the local velocity vector at a given instant.
• Pathline a line that represents the actual path
  traversed by a single fluid particle.
• Streakline a line that represents the locus of
  fluid particles at a given instant that have
  earlier passed through a prescribed point.

9
Streamlines

• Streamlines are useful in fluid flow analysis,
  but are difficult to observe experimentally for
  unsteady flows
• For 2-D flows, the streamline equation can be
  determined by integrating the slope equation
• The resulting equation is normally written in
  terms of the stream function ? (x,y) constant

10
Pathlines Streaklines

• The pathline is a Lagrangian concept that can be
  visualized in the laboratory by marking a fluid
  particle and taking a time exposure photograph of
  its trajectory
• The streakline can be visualized in the
  laboratory by continuously marking all fluid
  particles passing through a fixed point and
  taking an instantaneous photograph
• Streamlines, pathlines, and streak-lines are
  identical for steady flows

11
Acceleration Field

• Acceleration is the time rate of change of
  velocity
• Using the Eulerian description, we note that the
  total derivative of each velocity component will
  consist of four terms, e.g.,

12
Acceleration Field

• Collecting derivative terms from all velocity
  components,
• The operator is termed the material, or
  substantial, derivative it represents the rate
  at which a variable (V in this case) changes with
  time for a given fluid particle moving through
  the flow field

13
Acceleration Field

• The term is called the local
  acceleration it represents the unsteadiness of
  the fluid velocity and is zero for steady flows.
• The terms are
  called convective accelerations they represent
  the fact that the velocity of the fluid particle
  may vary due to the motion of the particle from
  one point in space to another it can occur for
  both steady and unsteady flows.

14
The Control Volume

• A control volume is a volume in space through
  which fluid may flow in some cases, the volume
  may move or deform
• The control volume has a boundary which separates
  it from the surroun-dings and defines a control
  surface
• In the study of fluid dynamics, the control
  volume approach is used to analyze fluid flow and
  fluid machinery
• The control volume approach is consistent with
  the Eulerian description

15
The Reynolds Transport Theorem

• The basic laws governing the motion of a fluid
  (e.g., conservation of mass, momentum, and
  energy) are usually written in terms of a fixed
  quantity of mass, or system
• Because a control volume does not always have
  constant mass, the basic laws must be rephrased
• The Reynolds Transport Theorem is a tool that
  allows one to shift from a system viewpoint
  (fixed mass) to a control volume viewpoint
• In thermodynamics, a system is defined more
  generally as a fixed mass or control volume

16
The Reynolds Transport Theorem

• Let B any fluid parameter, such as mass,
  velocity, temperature, momentum, etc.
• Let b B/m, a fluid parameter per unit mass
• The mass m may be that contained in a system or a
  control volume

17
The Reynolds Transport Theorem

• Example 4.7 (B m, b 1)

18
The Reynolds Transport Theorem
19
The Reynolds Transport Theorem

• Reynolds Transport Theorem (RTT) for fixed
  control volume with one inlet, one exit and
  uniform properties
• LHS term is Lagrangian
• RHS terms are Eulerian

20
The Reynolds Transport Theorem

• A general control volume may have multiple inlets
  and outlets, three-dimensional flow, and
  nonuniform properties the general form of the
  RTT is
• for a control volume moving at constant velocity
  Vcv, replace V by V-Vcv

21
Physical Interpretation

• The RTT allows one to translate the time rate of
  change of some parameter B of the system in terms
  of the time rate of change of B of the control
  volume and the net flow rate of B across the
  control surface
• A material derivative is used because the
  translation consists of an unsteady term ?( )/?t
  and convective effects associated with the flow
  of the system across the control surface

22
Steady Flow

• For steady flow,
• For B m (mass), the LHS is zero since the mass
  of a system is constant
• For B V (velocity), the LHS is nonzero in
  general
• For B T (temperature), the LHS is also nonzero
  in general