22.581 Module 2: Conservation of Mass -...
Transcript of 22.581 Module 2: Conservation of Mass -...
22.581 Module 2: Conservation of Mass
D.J. Willis
Department of Mechanical EngineeringUniversity of Massachusetts, Lowell
22.581 Advanced Fluid DynamicsFall 2017
Tues 11th September 2017
Outline
1 Reference Frames
Lagrangian vs. Eulerian Description of Fluid Motion
Eulerian Description
2 Conservation of Mass
Integral Conservation Laws: Material Volume
Integral Conservation Laws: Control Volume
Conservation of Mass Examples
Classic Physics Question
Image from M. McCloskey, Intuitive Physics, Scientific American
248 (1983), pp. 122-130
Euler vs. Lagrange: Classical Mechanics
Two descriptive approaches in Classical Mechanics:1 The particle or Lagrangian description
Typically used where materials are/can be followedUsed commonly in solid mechanicsUsed occasionally in fluid dynamics
2 The field or Eulerian description
Commonly used in fluid dynamics settingsNeed to be able to get Lagrangian derivatives in an Eulerian setting
Material Volume: Lagrangian Description of FluidMotion
Material Volume (MV): A volume which contains the same, finite,set of particles at all times.
Most physical laws (eg. Conservation of Momentum) are easilyexpressed for Material Volumes.
Material Particle: Lagrangian Description
Keeping track of large numbers of particles is difficult
We shrink the material volume down to an infinitesimal sized fluidparticle
Time (t) and original position (~a) are taken as independent variablesto describe the particle.The position of the particle at a given time is ~r(~a, t).
Material Particle: Lagrangian Description
English: After some time t, a particle with initial position ~a is now
at a location ~r.
Continuing this generalization: A fluid property, F is expressed as
F(~a, t). Meaning, after some time t, the particle with original
location ~a has some property F.
Material Particle: Computing Velocity andAcceleration
The velocity and acceleration of a Lagrangian fluid particle are
simply the partial derivatives with respect to time.
~ua =∂~r∂t
~aa =∂~v∂t
=∂2~r∂t2
Dynamics 101: point position, velocity and acceleration!
Outline
1 Reference Frames
Lagrangian vs. Eulerian Description of Fluid Motion
Eulerian Description
2 Conservation of Mass
Integral Conservation Laws: Material Volume
Integral Conservation Laws: Control Volume
Conservation of Mass Examples
Eulerian Description
Eulerian description: examines fluid properties at an individual
stationary point
Independent variables: ~r′ position in space, t′ time.
A fluid property, F is expressed as F(~r′, t′)
Lagrangian Derivatives in an EulerianDescription(KC section 3.3)
Lagrangian: time rate of change of a property F(~a, t), is computed
as we follow a material point.How do we compute this same important rate of change in anEulerian or field description?
∂∂t [F(~r
′, t′)] 6= ∂∂t [F(~a, t)]
In the Lagrangian we are following the material particles, notstaying fixed in space.
So, (∂[F(~r′, t′)]
∂t
)a=?
Ie. What is the rate of change of a property if we follow a particle,
if we were to measure only at a fixed field point?
Outline
1 Reference Frames
Lagrangian vs. Eulerian Description of Fluid Motion
Eulerian Description
2 Conservation of Mass
Integral Conservation Laws: Material Volume
Integral Conservation Laws: Control Volume
Conservation of Mass Examples
Integral Conservation of Mass for a MaterialVolume
Recall:1 Mass (extensive) per unit volume→ density (intensive).2 Recall, by definition, the mass of fluid contained in a material
volume remains constant. As a result, any material volume bydefinition conserves mass.
Due to shear-stress→ rate of deformation of fluids, it is difficult to
follow material volumes.
Integral Conservation of Mass for a MaterialVolume
What does the conservation of mass statement look like for a
material volume?
Differential element of volume inside the MV has a mass:
dMass = ρdV
Therefore, we can sum (integrate) over all the volumes in the
material volume:
0 =dMass
dt|MV =
ddt
(∫∫∫MV
ρdV)
Since the integral, is equivalently the mass inside the material
volume, there can be no gain, or loss of mass in the control
volumedMass
dt |a = 0 by definition.
Outline
1 Reference Frames
Lagrangian vs. Eulerian Description of Fluid Motion
Eulerian Description
2 Conservation of Mass
Integral Conservation Laws: Material Volume
Integral Conservation Laws: Control Volume
Conservation of Mass Examples
Integral Conservation of Mass for a ControlVolume
We will progress as follows for control volumes:
Describe the conservation of mass for CVsDiscuss choice and notation for control volumesExample applications.
Integral Conservation of Mass for a ControlVolume
Control volume analyses relate to Eulerian or mixed
Eulerian-Lagrangian analyses.
A control volume (CV) is a defined imaginary volume
A control surface(CS) is the imaginary surface surrounding the
control volume
Some key points:Control volumes can:
Be fixed in size and stationary in the reference frame – typical ofundergraduate courses in fluid mechanics.Control volumes may change location, size and/or shape – we willsee this in this class.
Integral Conservation of Mass for a ControlVolume
Consider an arbitrary/general
control volume (CV):
A conservation of mass
statement implies:
Mass Collecting in CV + Mass leaving CV −Mass entering CV = 0
The normal vector points outwards from the control volume/surface.When fluid leaves a CV, it is a positive flux (~u · n̂ > 0)
Integral Conservation of Mass for a ControlVolume
We’ll start with each term: Accumulation in CV, and then Mass flux
across CS.
The mass of a small volume in the CV is::
dMCV = ρ(t, x, y, z)dV
The mass in the entire control volume is the summation (integral) of alldifferential elements of mass:
MCV =
∫∫∫CVρ(t, x, y, z)dV
Integral Conservation of Mass for a ControlVolume
What is of interest is not the mass, but the rate of change of mass w.r.t.time inside the CV
dMCV
dt=
ddt
(∫∫∫CVρdV
)If ρ is constant in the C.V =
ddt
(ρ · Vol)
The rate of change of mass w.r.t. time in the CV must match the flux ofmass across the CS by teh conservation of mass principle.
Integral Conservation of Mass for a ControlVolume
Mass entering or leaving the CV can be due to a combination oftwo phenomena:
1 The velocity of the fluid causing a flux in/out of the control volume(traditional).
2 The velocity of the C.S. – as the control volume changessize/shape, it may setup a flux through the moving control surface.
Consider first the mass leaving the C.V. due to the velocity of the
fluid.
Integral Conservation of Mass for a Fixed in SpaceControl Volume
How much flow leaves the CV because the fluid is flowing across
the CS?
Integral Conservation of Mass for a Fixed in SpaceControl Volume
If the control volume isfixed in space, the massper unit time that leavesthe C.V. is:
Mass of fluid leaving CV over time dt = ρ~u · n̂CS · dArea · dt
n̂CS: Is the unit normal vector to the control surface at the location
being considered.
Integral Conservation of Mass for a Fixed in SpaceControl Volume
Therefore, the rate of mass flux through an area dA for a fixed in
space control volume is:
Rate of Mass Leaving dA =
(∂Mass∂t
)flux
= ρ~u · n̂CV · dArea
To get the mass flux through the entire control surface due to flow
velocity, we must integrate the above expression over the entire
control surface:
Rate of Mass Leaving CV =
∫∫CSρ · (~u · n̂CS) dA
Integral Conservation of Mass for a MovingControl Volume in Stationary Fluid
Case 2:
Thought experiment: How does a CV moving relative to the fluidgenerate a mass flux into or out of the control volume?
Integral Conservation of Mass for a MovingControl Volume in Stationary Fluid
The fluid that leaves the
an area dA due to CV
motions (an enlarging
Control Volume) is:
Mass of fluid leaving CV over time dt =(∂Mass∂t
)flux
= −ρ~VCS · n̂CSdACS
Integral Conservation of Mass for a MovingControl Volume in Stationary Fluid
To determine the total mass flux through the control surface, we
must integrate the expression for an element dA (see previous
slide) over the entire control surface:
Total Mass Flux Through CS = −∫∫
CSρ~VCS · n̂CSdACS
The negative sign means: an enlarging control volume (positive
control volume normal velocity) increases mass flux into control
volume.
General Form of the Mass Flux through a CS
Putting the:1 Mass flux due to fluid velocity2 Mass flux due to control surface velocity/shape change
together into a single expression, we determine the conservation
of mass statement:
Mass flux out of CV =
∫∫CS(t)
ρ(~u− ~VCS
)︸ ︷︷ ︸~v rel.to CV
·n̂dS
=
∫∫CS(t)
ρ (urn) dS
Where, urn = (~u− ~VCS) · n̂CS is the relative, normal velocity of the
fluid to the CS.
Integral Conservation of Mass
By enforcing that the mass flux out of the control volume is
identically the change in mass inside of the control volume with
respect to time, the resulting conservation of mass statement is:
0 =ddt
∫∫∫CV(t)
ρdV︸ ︷︷ ︸Mass Collection in CV
+
∫∫CS(t)
ρ(~u− ~VCS) · n̂CSdA︸ ︷︷ ︸Mass Flux leaving CV through CS
Where, urn = (~u− ~VCS) · n̂CS.
Integral Conservation of Mass
Question: How does a material volume conservation of mass
analysis relate to a control volume analysis? Explain!
Control Volume Annotation, Selection and Hints
The following are considerations in the selection of a controlvolume:
What do you know and where do you know it?What do you want to know, and where do you want to know it?Is there a way to define a non-accumulating control volume?Is there a way to define a stationary control volume?Can I use reference frames to make my life easy?
Control Volume Annotation, Selection and Hints
Hints:
Hint: it is often easy (easier) to pick control volumes where the flowis normal to the C.S.Hint: Break your control surface into segments→ analyze eachseparatelyHint: Perform the volume integral over the control volume even ifyou think it is going to be zero valued.Hint: Be careful of velocity distributions – they require that youintegrate the flux.Be careful of flow compresibiliy!
Outline
1 Reference Frames
Lagrangian vs. Eulerian Description of Fluid Motion
Eulerian Description
2 Conservation of Mass
Integral Conservation Laws: Material Volume
Integral Conservation Laws: Control Volume
Conservation of Mass Examples
Piston Example
Coke Bottle Example
NOTES: Comments