Fluid Kinematics
Velocity Field •
Continuum hypothesis: – fluid is made up of fluid particles; – each particle contains numerous molecules; – infinitesimal particles of a fluid are tightly packed together
•
Thus, motion of a fluid is described in terms of fluid particles rather than individual molecules.
•
This motion can be described in terms of the velocity and acceleration of the fluid particles
•
At a given instant of time, description of any fluid property may be given as a function of fluid location
•
Representation of fluid parameters as function of spatial coordinates is termed a field representation of the flow
•
Fluid parameters are functions of position ant time. For example, temperature in the room is completely specified by temperature field
T T x, y , z , t
Velocity Field Velocity of a particle
VA
d rA dt
Velocity magnitude
V V u 2 v 2 w2 Velocity field
Particle location in terms of its position vector
V u x, y , z , t ˆi v x, y , z , t ˆj w x, y , z , t kˆ V V x, y , z , t
Eulerian and Lagrangian Flow Description There are two approaches in analyzing fluid mechanics problem Eulerian method uses field concept Lagrangian method involves following individual particle moving through the flow Lagrangian information can be derived from the Eulerian data – and vice versa Most fluid mechanics considerations involve the Eulerian method.
Eulerian and Lagrangian descriptions of temperature of a flowing fluid
One-, Two, and Three-Dimensional Flows. Steady and Unsteady Flows •
Steady flow – the velocity at a given point in space does not vary with time, otherwise, flow is unsteady
•
In general, fluid flow is three-dimensional and unsteady
•
In many situations, flow can be simplified to steady, two- or one-dimensional flow in order to make solution easier without loss of accuracy
Flow visualization of the complex threedimensional flow past a model airfoil
Streamlines, Srteaklines and Pathlines •
Streamlines, streaklines, and pathlines are used for flow visualization
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Streamline is used in analytical work while the streakline and pathline are used in experimental work
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Streamline is a line, that is everywhere tangent to the velocity field
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Streamlines are obtained by integrating differential equation of streamline. For twodimensional flow dy/dx = v/u
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If flow is steady, streamlines are fixed lines in space
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Streakline consist of all particles in a flow that have previously passed through the common point.
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Pathline is the line traced out by a given particle as it flows from one point to another
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For steady flow streamlines, streaklines, and pathlines are the same
Example 4.3
Solution (a) Streamline is given by solution of v0 dy v dx u u0 sin t y v0 Integration gives u0 v0 cos t y v0 v0 x C At t = 0, C = u0v0/ω , and equation of streamline is x
u0
y 1 v 0
cos
At t = π/2ω , C = 0, and equation of streamline is x
y u0 sin v 0
These two streamlines are not the same because flow is unsteady At the origin
V v0 j
at t 0
V u0 i v0 j
at t 2
(b) Pathline is obtained from velocity field dx y u0 sin t dt v0 Integration gives y v0t C1
and
C1 x u0 sin t C2 v 0
For the particle that was at the origin at time t 0, the pathline is x0
and
y v0t
For the particle that was at the origin at time t 2 , the pathline is
x u0 t and 2
dy v0 dt
and
y v0 t 2
(c) Discuss the shape of the streakline that passes through the origin
Acceleration Field •
For Eulerian description one describes the acceleration field as a function of position and time
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Acceleration is the time rate of change of velocity of a given particle
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For unsteady flow the velocity at a given point in space (occupied by different particles) may vary with time, giving rise to a portion of the fluid acceleration
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In addition, a fluid particle may experience an acceleration because its velocity changes as it flows from one point to another in space
Acceleration Field VA VA rA , t VA x A t , y A t , z A t
a
V V V V u v w t x y z
Velocity and position of particle A at time t
or
a
DV Dt
Material Derivative Operator
D Dt
u v w
t
x
y
z
is termed the material derivative or substantial derivative In vector notation:
D Dt
t
Vg
Material derivative of any variable is the rate at which that variable changes with time for a given particle (as seen by one moving along with the fluid – Lagrangian description. Material derivative is also called comoving derivative) For example, the time rate of change of temperature of a fluid particle (particle A) as it moves through the temperature field T = T(x,y,z,t) is given by
DT T T T T T u v w VgT Dt t x y z t
Material Derivative. Unsteady Effects Portion of material derivative represented by time derivative is termed the local derivative Local derivative is the result of the unsteadiness of the flow
Uniform, unsteady flow in a constant diameter pipe
Material Derivative. Convective Effects Portion of the material derivative represented by the spatial derivative is termed the convective derivative Convective derivative is a result of the spatial variation of the flow
Steady state operation of a water heater
Material Derivative. Convective Effects Portion of the material derivative represented by the spatial derivative is termed the convective derivative Convective derivative is a result of the spatial variation of the flow
Uniform, steady flow in a variable area pipe
Control Volume and System Representation • •
System is a collection of matter of fixed identity (always the same atoms or fluid particles), which may move, flow, and interact with its surroundings Control volume is a volume in space (geometric entity, independent of mass) through which fluid may flow
Typical control volumes: (a) fixed control volume, (b) fixed or moving control volume, (c) deforming control volume
Control Volume and System Representation •
Both, control volume and system concepts can be used to describe fluid flow
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Governing laws of fluid motion are stated in terms of fluid systems, not control volume
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To shift from one representation to the other Reynolds transport theorem is used
Reynolds Transport Theorem Physical laws are stated in terms of physical parameters (velocity, acceleration, mass, temperature, momentum etc.) Let B represent fluid parameter and b represent amount of that parameter per unit mass. Then B mb Parameter B is termed an extensive property, and the parameter b is termed an intensive property Amount of extensive property that system possesses at a given instant is Bsys bdV sys
Time rate of change of extensive property of a system dBsys dt
d
sys
bdV dt
Reynolds Transport Theorem. Derivation Control volume and system for flow through a variable area pipe
Simplified version of the Reynolds transport theorem for fixed control volume with one inlet and one outlet having uniform properties (density, velocity, and the parameter b) across the inlet and outlet with the velocity normal to sections (1) and (2) is DBsys Dt
Bcv 2 A2V2b2 1 AV 1 1b1 t
Reynolds Transport Theorem. General Form General form for of the Reynolds transport theorem for a fixed, nondeforming control volume is given by (details) DBsys Dt
bdV b V gnˆ dA cv cs t
Control volume and system for flow through an arbitrary, fixed control volume
Physical Interpretation DBsys Dt
bdV b V gnˆ dA cv cs t
Possible velocity configurations on portions of the control surface: (a) inflow, (b) no flow across the surface, (c) outflow
Reynolds Transport Theorem. Moving CV
Example of a moving control volume
Reynolds Transport Theorem. Moving CV
V W Vcv DBsys Dt
bdV b Wgnˆ dA cv cs t
Selection of a Control Volume
END OF CHAPTER
Supplementary slides
Outflow across a typical portion of the control surface
Outflow across a typical portion of the control surface
B b V b V cos t A
bV cos t A bV cos A b V B&out lim lim t 0 t 0 t t B&out
csout
dB&out
csout
bV cos A
B&out
csout
b V gnˆ dA
Inflow across a typical portion of the control surface
B&in
csin
bV cos dA b V gnˆ dA csin
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