ME3560 – Fluid Mechanics
Summer 2016
ME 3560 Fluid Mechanics
Chapter IV. Fluid Kinematics
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Chapter IV. Fluid Kinematics
ME3560 – Fluid Mechanics
Summer 2016
4.1 The Velocity Field
• One of the most important parameters that need to be monitored when
a fluid is flowing is the velocity.
•In general the flow parameters are described in terms of the motion of
fluid particles rather than individual molecules.
•Thus, this motion can be described in terms of the velocity and
acceleration of the fluid particles.
•In order to describe the flow parameters it is sought to provide at a
given instant in time, a description of any fluid property (such as ρ, p, V,
and a) as a function of the fluid's location.
•This representation of fluid parameters as functions of the spatial
coordinates is termed a field representation of the flow.
•The specific field representation may be different at different times,
thus. Thus, to completely specify the velocity, V, in a process, the
velocity field must be expressed as: V = V (x, y, z, t).
r
V = u ( x, y, z , t )iˆ + v( x, y, z , t ) ˆj + w( x, y, z , t )kˆ
Chapter IV. Fluid Kinematics
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ME3560 – Fluid Mechanics
Summer 2016
r
V = u ( x, y, z , y )iˆ + v( x, y, z , y ) ˆj + w( x, y, z , y )kˆ
• u, v, and w are the x, y, and z
components of the velocity vector.
•The velocity of a particle is the time
rate of change of the position vector
for that particle.
•The position of particle A relative to
the coordinate system is given by its
position vector, rA, which (if the
particle is moving) is a function of
time.
•The velocity is a vector therefore it has both a direction and a
magnitude, V = |V| = (u2 + v2 + w2)1/2
•A change in velocity results in an acceleration. This acceleration may
be due to a change in speed and/or direction.
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Chapter IV. Fluid Kinematics
ME3560 – Fluid Mechanics
Summer 2016
4.1.1 Eulerian and Lagragian Flow Descriptions
• There are two general approaches in analyzing fluid mechanics
problems:
•Eulerian method. Measures the flow parameters at fixed locations to
then generate the parameters flowfield.
•Lagrangian method, involves following individual fluid particles as
they move about and determining how the fluid properties associated
with these particles change as a function of time.
4.1.2 One-, Two-, and Three-Dimensional Flows
•In general, a fluid flow is a rather complex three-dimensional, timedependent phenomenon.
r
V = u(x, y, z,t)iˆ +v(x, y, z,t) ˆj + w(x, y, z,t)kˆ
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Chapter IV. Fluid Kinematics
ME3560 – Fluid Mechanics
Summer 2016
• However, sometimes it is possible to make simplifying assumptions
that allow a much easier understanding of the problem without
sacrificing needed accuracy:
•Assume the flow is two–dimensional or one–dimensional.
•Assume the flow is incompressible (even when dealing with gases)
•One of these simplifications involves approximating a real flow as a
simpler one– or two–dimensional flow.
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Chapter IV. Fluid Kinematics
Summer 2016
ME3560 – Fluid Mechanics
4.1.3 Steady and Unsteady Flows
• Steady flow. The velocity (or any other parameter: T, p, ρ, etc.) at a
given point in space does not vary with time ∂V/∂t = 0.
•Unsteady flows. Flow field parameters are time–dependent ∂V/∂t≠0.
•Among the various types of unsteady flows are nonperiodic flow,
periodic flow, and truly random flow.
4.1.4 Streamlines, Streaklines, and Pathlines
• Streamline is a line that is everywhere tangent
to the velocity field.
• If the flow is steady, nothing at a fixed point
(including the velocity direction) changes with
time, so the streamlines are fixed lines in space.
• For unsteady flows the streamlines may change
shape with time.
∂x ∂ y ∂z
Chapter IV. Fluid Kinematics
u
=
v
=
w
dy v
=
dx u
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ME3560 – Fluid Mechanics
Summer 2016
• Streakline consists of all particles in a flow that have previously
passed through a common point.
• Streaklines can be generated by taking instantaneous photographs of
marked particles that all passed through a given location in the flow
field at some earlier time. Such a line can be produced by continuously
injecting marked fluid (neutrally buoyant smoke in air, or dye in water)
at a given location.
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Chapter IV. Fluid Kinematics
ME3560 – Fluid Mechanics
Summer 2016
• Pathline is the line traced out by a given particle as it flows from one
point to another.
• The pathline is a Lagrangian concept that can be produced in the
laboratory by marking a fluid particle (dying a small fluid element) and
taking a time exposure photograph of its motion.
• Pathlines, Streamlines, and Streaklines are the same for steady flows.
• For unsteady flows none of these three types of lines need to be the
same.
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Chapter IV. Fluid Kinematics
ME3560 – Fluid Mechanics
Summer 2016
4.2 The Acceleration Field
•A flow can be studied by either (1) following individual particles
(Lagrangian description) or (2) remaining fixed in space and observing
different particles as they pass by (Eulerian description).
• In either case, to apply Newton's second law (F = m a) it is necessary
to describe the particle acceleration in an appropriate fashion.
•For the infrequently used Lagrangian method, we describe the fluid
acceleration just as is done in solid body dynamics—a = a(t) for each
particle.
•For the Eulerian description we describe the acceleration field as a
function of position and time without actually following any particular
particle, a = a(x, y, z, t).
• The acceleration of a particle is the time rate of change of its velocity.
• For unsteady flows the velocity at a given point in space varies with
time. Also, a fluid particle may accelerate because its velocity changes
as it flows from one point to another in space.
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Chapter IV. Fluid Kinematics
Summer 2016
ME3560 – Fluid Mechanics
4.2.1 The Material Derivative
r
r
r
r
r ∂V
∂V
∂V
∂V
a=
+u
+v
+w
∂z
∂t
∂x
∂y
∂u
∂u
∂u
∂u
a=
+u
+v
+w
∂t
∂x
∂y
∂z
∂v
∂v
∂v
∂v
a=
+u
+v
+w
∂t
∂x
∂y
∂z
∂w
∂w
∂w
∂w
a=
+u
+v
+w
∂t
∂x
∂y
∂z
r
r DV
a=
dt
D( ) ∂ ( )
∂()
∂()
∂()
=
+u
+v
+w
dt
∂t
∂x
∂y
∂z
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Chapter IV. Fluid Kinematics
Summer 2016
ME3560 – Fluid Mechanics
• D( )/dt is termed the material derivative or substantial derivative.
•A shorthand notation for the material derivative operator is
D( ) ∂ ( )
=
+ (V ⋅ ∇)( )
dt
∂t
4.2.2 Unsteady Effects
• The material derivative formula contains two types of terms:
• Terms involving the time derivative ∂()/∂t: Local Derivative
• Terms involving spatial derivatives ∂()/∂x, ∂()/∂y, and ∂()/∂z.
• ∂()/∂t represents the effects of the unsteadiness of the flow.
•∂V/∂t is termed the local acceleration. For steady flow: ∂()/∂t = 0
•Physically, there is no change in flow parameters at a fixed point in
space if the flow is steady.
•There may be a change of those parameters for a fluid particle as it
moves about.
•If a flow is unsteady, its parameter values (V, T, ρ, etc.) at any location
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may change with time
Chapter IV. Fluid Kinematics
ME3560 – Fluid Mechanics
Summer 2016
4.2.3 Convective Effects
• The portion of the material derivative represented by the spatial
derivatives is termed the convective derivative.
• The convective derivative represents the fact that a flow property
associated with a fluid particle may vary because of the motion of the
particle from one point in space to another point in space.
• This contribution to the time rate of change of the parameter for the
particle can occur whether the flow is steady or unsteady.
• It is due to the convection, or motion, of the particle through space in
which there is a gradient [∇( )=∂( )/∂x î + ∂( )/∂x ĵ + ∂( )/∂z k]
•The portion of a due to the term (V · ∇)V is the convective acceleration
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Chapter IV. Fluid Kinematics
ME3560 – Fluid Mechanics
Summer 2016
4.3 Control Volume and System Representations
•A 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.
•A system is a specific, identifiable quantity of matter. It may consist of a
relatively large amount of mass or it may be an infinitesimal size.
•A system may interact with its surroundings by various means (by the
transfer of heat or the exertion of a pressure force, for example).
• A system may continually change size and shape, but it always
contains the same mass.
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Chapter IV. Fluid Kinematics
ME3560 – Fluid Mechanics
Summer 2016
•A control volume, is a volume in space (a geometric entity, independent
of mass) through which fluid may flow.
•In fluid mechanics, it is difficult to identify and keep track of a specific
quantity of matter.
• In several cases, the main interest is in determining the forces put on a
device rather than in the information obtained by following a given
portion of the air (a system) as it flows along.
• For these situations it is more adequate to use the control volume
approach.
•Identify a specific volume in space (a volume associated with the device
of interest) and analyze the fluid flow within, through, or around that
volume.
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Chapter IV. Fluid Kinematics
ME3560 – Fluid Mechanics
Summer 2016
• In general, the control volume can be a moving volume, although for
most situations we will use only fixed, nondeformable control volumes.
• The matter within a control volume may change with time as the fluid
flows through it.
• The amount of mass within the volume may change with time.
•The control volume itself is a specific geometric entity, independent of
the flowing fluid.
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Chapter IV. Fluid Kinematics
ME3560 – Fluid Mechanics
Summer 2016
•All of the laws governing the motion of a fluid are stated in their basic
form in terms of a system approach.
•For example, “the mass of a system remains constant,” or “the time rate
of change of momentum of a system is equal to the sum of all the forces
acting on the system.”
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Chapter IV. Fluid Kinematics
ME3560 – Fluid Mechanics
Summer 2016
4.4 The Reynolds Transport Theorem
• The Reynolds transport theorem provides a mathematical way to relate
the properties of a flow between a control volume and a system.
• Extensive Property, is a property whose value is directly proportional
to the amount of the mass being considered.
-Mass
-Energy
-Momentum
• Intensive Property, is a property whose value is independent of the
amount of mass.
-Density
-Temperature
-Velocity
-Pressure
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Chapter IV. Fluid Kinematics
Summer 2016
ME3560 – Fluid Mechanics
B=m
b=1
B = mV
b=V
B=E
b=e
• An extensive property B is related to its corresponding intensive
property b by
B = mb
• In general, for a system, an extensive property B can be determined as:
Bsys = ∫ ρ b dV
sys
•The rate of change of an extensive property in a system is expressed as


d  ∫ ρ b dV 


dBsys
sys


=
dt
dt
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Chapter IV. Fluid Kinematics
ME3560 – Fluid Mechanics
Summer 2016
• In a similar way, the rate of change of an extensive property in a control
volume is expressed as


d  ∫ ρ b dV 
dBcv
cv


=
dt
dt
4.4.1 Derivation of the Reynolds Transport Theorem
• Consider two instants in
time: t and t+δt.
•Assume that at t the
control volume and the
system coincide, thus
Bsys (t ) = Bcv (t )
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Chapter IV. Fluid Kinematics
ME3560 – Fluid Mechanics
Summer 2016
• Then, at a time t+δt
Bsys (t + δt ) = Bcv (t + δt ) − BI (t + δt ) + BII (t + δt )
•The change in the amount of
B in the system in the time
interval δt divided by this
time interval is given by
δ Bsys Bsys (t + δt ) − Bsys (t )
=
δt
δt
δ Bsys Bcv (t + δt ) − BI (t + δt ) + BII (t + δt ) − Bsys (t )
=
δt
δt
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Chapter IV. Fluid Kinematics
ME3560 – Fluid Mechanics
Summer 2016
• Since at the initial time t we have Bsys(t) = Bcv(t),
δ Bsys Bcv (t + δt ) − Bcv (t ) BI (t + δt ) BII (t + δt )
=
−
+
δt
δt
δt
δt
• For δt → 0, the left-hand side this equation is DBsys/Dt.
• DBsys/Dt represents the time rate of change of property B associated
with a system (a given portion of fluid) as it moves along.
• For δt → 0, the first term on the right-hand side of this equation is the
time rate of change of the amount of B within the control volume
Bcv (t + δt ) − Bcv (t ) ∂ Bcv ∂ ∫cv ρ bd V
lim
=
=
δ t →0
∂t
∂t
δt
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Chapter IV. Fluid Kinematics
ME3560 – Fluid Mechanics
• The term
Summer 2016
BII (t + δt ) = ( ρ 2 b2 )(δ VII ) = ρ 2 b2 A2V2δt
•Represents the amount of the extensive parameter B flowing out of the
control volume, across the control surface.
•Thus, the rate at which this property flows from the control volume is:
BII (t + δ t )
&
Bout = lim
= ρ 2 A2V2 b2
δ t →0
δt
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Chapter IV. Fluid Kinematics
ME3560 – Fluid Mechanics
• The term
Summer 2016
BI (t + δt ) = ( ρ1b1 )(δ VI ) = ρ1b1 A1V1δt
• Represents the amount of the extensive parameter B flowing into the
control volume, across the control surface
•Thus, the rate at which this property flows from the control volume is:
B I (t + δ t )
&
Bin = lim
= ρ1 A1V1b1
δ t →0
δt
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Chapter IV. Fluid Kinematics
Summer 2016
ME3560 – Fluid Mechanics
• By combining the previous equations, a relation between the time rate
of change of B for the system and that for the control volume is given by
∂ Bcv &
=
+ Bout − B& in
∂t
DB sys
Dt
∂ Bcv
=
+ ρ 2 A2V2 b2 − ρ1 A1V1b1
∂t
DB sys
Dt
• A generalization of the previous equation is given by the Reynolds
Transport Theorem
DB sys
Dt
∂
=
∂t
∫
cv
ρ b dV + ∫
cs
r
ρ b V ⋅ nˆ dA
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Chapter IV. Fluid Kinematics
Summer 2016
ME3560 – Fluid Mechanics
Selection of a Control Volume
• CV is typically fixed and non–deforming.
• CV includes all relevant inlets and outlets where information is
available or required.
• If possible CS should be perpendicular to the velocity vector to simplify
r
V ⋅ nˆ
•Contributions to the surface integrals must be simple and relevant. In CS
other than inlets and outlets, select solid boundaries where V = 0.
Sign of
r
V ⋅ nˆ
n
n
Vin
Vout
r
Vin ⋅ nˆ = −Vin
r
Vout ⋅ nˆ = Vout
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Chapter IV. Fluid Kinematics