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2: Introduction and Fundamental Concepts in Mechanics and Fluid Mechanics. Mechanics Force-Body-Motion Fluid Mechanics Fluid as A Continuum / Continuum Assumption Fluid as A Continuum Property at A Point Property Fields

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2 introduction and fundamental concepts in mechanics and fluid mechanics
2: Introduction and Fundamental Concepts in Mechanics and Fluid Mechanics
  • Mechanics
    • Force-Body-Motion
  • Fluid Mechanics
    • Fluid as A Continuum / Continuum Assumption
      • Fluid as A Continuum
      • Property at A Point
      • Property Fields
  • Methods of Description of Motion: Lagrangian VS Eulerian Descriptions
      • Scalar, Vector, and Tensor Fields
  • Classification of Fields
    • Steady VS Unsteady Fields [Does f at any one point change with time t?]
    • Uniform VS Non-Uniform Fields [Does f at any one time t change with the spatial location in the region R?]
  • Classification of Fluid Flows
fundamental concept mechanics force body motion
Fundamental ConceptMechanics: Force-Body-Motion
  • Mechanics: Three main components: Force - Body - Motion.
  • System-Surroundings-Interactions:

Investigate the resulting effects of the mechanical interactions between the system and its surroundings.

    • Interactions: Forces (in FBD) and Work
    • Effects: Motion (and other related quantities)

Mechanics

Motion of aBody under the action of Forces.

(Effects of Forces on the Motion of a Body.)

slide4

Thermodynamics

Thermal Energy and Thermodynamic Properties of Substance

Force

Body

Motion

Linear Motion

Translation

Angular Motion

(Rigid-Body-Like) Rotation

Deformation

  • Force: An effortto move a body against itsinertia.

(in Newton’s second law aspect.)

fundamental concept three efforts of force three types of body and three types of motion
Fundamental ConceptThree Efforts of Force, Three Types of Body, and Three Types of Motion

Forces (3 Efforts)

Body (3 Types):

(~ according to the degree of

idealization of permitted motions.)

Motion (3 Components):

Particle

Linear Motion (Translation)

Force

Rigid Body

Angular Motion (Rigid-Body-Like Rotation)

Moment of Force

Deformable Body

Deformation

Intensity of Force

force body motion in equations of mechanics
Force-Body-Motion in Equations of Mechanics

1. Force and linear motion

(translation).

2. (Moment of) Force and angular motion (rotation).

3. (Intensity of) Force and deformation.

(Constitutive Relation)

Strain

Stress

Hooke’s law

Newton’s law of viscosity

fundamental concept fluid mechanics
Fundamental Concept: Fluid Mechanics

Fluid Mechanics

= Mechanics (Force and Motion) and Thermodynamics (Energy and Energy Transfer)

of Fluid Motion

slide9

Heater

Exit

Inlet

air

Mass flowrate

Density profile

Mass flowrate

Density profile

Velocity profile

Velocity profile

Temperature profile

Temperature profile

Fluid Mechanics

= Mechanics (Force and Motion) and Thermodynamics (Energy and Energy Transfer)

of Fluid Motion

System (Control Volume): Fluid stream only, excluding the solid heater

Given: At inlet,

Question: At exit,

fundamental concept definition of fluids

a

a (t)

Solid

finite deformation

under constant shear

Fluid

continuous deformation

under constant shear

no matter how small shear is

Fundamental Concept: Definition of Fluids

Simple models for simple solid and fluid.

Definition of Fluid:

A fluid is a substance that deforms continuouslyunder the application of a shear (tangential) stress no matter how small the shear stress may be. (Fox, et al., 2004)

fundamental concept fluid as a continuum continuum assumption property at a point

d m, dV

Continuum Assumption

y

x

z

Fundamental ConceptFluid as A Continuum: Continuum Assumption, Property at A Point

dV

dV’

  • Fluctuation due to random molecular motion
  • Continuum assumption breaks down.
  • Macroscopic spatial variation

Random motion of molecules has little effect on macroscopic mean density.

Random motion of molecules significantly affects macroscopic mean density.

  • Above this limit:
    • Continuum:A fluid is assumed to be a continuum.
    • Property at a point: Its property is assumed at a point, e.g., density at a point.
    • Field Function: A fluid property f is assumed as a continuous field function of space and time:
  • Below this limit, the continuum assumption breaks down, and the molecular motion of molecules must be taken into account.
fundamental concept two methods of description of fluid motion lagrangian vs eulerian descriptions

Eulerian Description

Lagrangian Description

y

y

y

x

x

x

z

z

Reference time to

Current time t

z

Current time t

Time

‘Particle name’

Time

Spatial position

Time evolution of a property fat a fixed point in space is given by

Time evolution of a property f of a material particle is given by

Fundamental ConceptTwo Methods of Description of Fluid Motion: Lagrangian VS Eulerian Descriptions

Eulerian Description

Lagrangian Description

slide14

Lagrangian Description

reads the current (at time t) value of the property of the material particle is equal to .

Eulerian Description

reads the value of the property at position and time is equal to .

fundamental concept lagrangian vs eulerian views
Fundamental Concept: Lagrangian VS Eulerian Views
  • To put simply

Eulerian view:

      • We watch an identified region in space of interest, and
      • see what happens in the region.

[Here, we watch a block in the street and see New York cabs passing in and out of our block.]

Lagrangian view:

      • We watch/follow the identified mass of interest, and
      • see what happens to the identified mass.

[Here, we watch and follow, say, one taxicab wherever it goes.]

eulerian field descriptions scalar vector and tensor fields velocity and property fields

Space-time point

Timet

Eulerian/Field Descriptions Scalar, Vector, and Tensor Fields: Velocity and Property Fields
  • Common Property Fields:

Scalar Field: density:

pressure:

temperature:

Vector Field: velocity:

Note that it is customary to use (u, v, w) for the (x,y,z) components of velocity, respectively.

Tensor Field: stress tensor:

steady and uniform property field f steadiness does f at any one point change with time t

t

t

t+dt

t+dt

Steady and Uniform Property Field fSteadiness:Doesf at any one point change with time t ?
  • Steadiness: Doesfat any one point change with time t ?

Shading represents the value of the property f , say light = high value, dark = low value

changes

with time t

does not change

with time t

slide18

t

t

does not change

with spatial location

changes

with spatial location

Steady and Uniform Property Field fUniformity in a region R: Doesf at any one time t change with spatial location ?
  • Uniformity in a region R: Doesfat any one time t change with spatial location ?
example steadiness and uniformity

t

t

t+dt

t+dt

t

t+dt

t

t+dt

Example: Steadiness and Uniformity

Steady ?

Uniform at time t?

Uniform at time t + dt?

? ? ?

classification of fluid flows

Vorticity (related to angular velocity)

Classification of Fluid Flows

r = constant

  • M < 1 - Subsonic
  • M = 1 - Sonic;
  • M ~ 1 - Transonic
  • M > 1 - Supersonic
  • M > ~ 5 - Hypersonic

m = 0, or effect of viscous stress can be neglected.

The number of spatial coordinates x, y, z, that is required to specify the velocity field.

Laminar: Smooth and orderly (non-random), can be steady or unsteady.

Turbulent Flow: Random fluctuation of velocity field, inherently unsteady.

Internal: Flow that is bounded by solid surfaces.

External: Flow over body immersed in unbounded fluid.

laminar transition turbulence subsonic jet

Laminar: orderly motion

Turbulent: random motion

Laminar-Transition-Turbulence: Subsonic Jet

Transition from laminar flow to turbulent flow via instability in subsonic jet.

From Van Dyke, M., 1982, An Album of Fluid Motion, Parabolic Press.

laminar transition turbulence subsonic jet1
Laminar-Transition-Turbulence: Subsonic Jet

Transition from laminar flow to turbulent flow via instability in subsonic jet.

From Van Dyke, M., 1982, An Album of Fluid Motion, Parabolic Press.

example classification and velocity field
Example: Classification and Velocity Field

Question:

1. Classify the following velocity fields by stating whether it is

    • steady or unsteady?
      • if unsteady, also find
    • 1-, 2-, or 3-dimensional?
    • Then, write down the functional form of the field, e.g.,
  • Also find
    • the divergence of the velocity field
      • NOTE: The divergence of the velocity field is related to the compressibility of the flow.
    • the vorticity, which is defined as the curl of the velocity field
      • NOTE: The vorticity is a measure of the angular velocity of a fluid element.
example classification and velocity field1
Example: Classification and Velocity Field
  • In addition, using MLtT as the set of primary dimensions, also state the dimension of the constant a, b, c, etc.
representation of a vector field a vector plot
Representation of A Vector Field: A Vector Plot

Let a velocity field be given by

where the field is defined in the region and , x and y are given in meters, and a = 1 s-1 and b = -1 s-1. Sketch a vector plot for this field.

example a vector plot
Example: A Vector Plot
  • Sketch a vector plot for the following velocity fields. x and y are given in meters.