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








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







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







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.


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.


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)


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


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.


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?


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.


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?


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


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







Piston Example

Coke Bottle Example

NOTES: Comments

22.581 module 2: conservation of mass -...

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