Advection

1
Advection

• The differential of the multivariable function,
  T(x,y,z,t) labeled dT is an infinitesimal change
  in the value of the function.

2
Advection

• As an example, let the scalar function represents
  a temperature field, T. The differential of the
  temperature field is
• We can use the above differential to take the
  time derivative of the temperature field

3
Advection

• If we represent a time dependent curve in space
  as
• So that
• Then the time derivative of temperature becomes
• Where

4
Advection

• The time derivative of temperature is due to two
  contributions
• 1 A partial derivative with respect to time
  (local change with respect to time)
• 2 Change in temperature due to the transport of
  air caused by the flow field along with
  variations of temperature (advection of the
  temperature field).

5
Advection

• As a matter of notation convention, we wish for
  advection to be defined in term of the transport
  of higher value quantities to lower values. Thus
  we must define the advection as
• Now positive advection of temperature indicates
  warmer air being transported into colder regions.
  
• Note this is just a notation convention.

6
Advection

• Let us examine the advection term geometrically
• Figure showing how the angle q relates to the
  vector flow field and the gradient of the
  temperature field. The dashed lines represent
  isotherms.

7
Advection

• Let us examine the advection term geometrically
• The above equation shows us that intensity of
  advection is caused by three contributions
• 1. The magnitude of the flow field
• 2. The intensity of the gradient of the
  temperature field
• 3. The relative orientation of the flow field
  with the gradient of the temperature field

8
Exercise

• Calculate the temperature advection at the
  specified point

9
Lagrangian framework

• The Lagrangian framework describes the fluid
  motion by tracing the motion or trajectory of all
  the parcels as they move throughout the fluid
  domain.
• For initial position, , the Lagrangian
  fluid velocity field is mathematically defined as
• Classical physics analogies would be a cannonball
  trajectory or a Quarterback throwing a football.
  

10
Lagrangian framework

• The flow field depends on the
  initial position and time.
• In terms of acceleration (Newtons second law),
  we only need to consider variations with respect
  to time (there is no spatial dependence in the
  above expression)

11
Eulerian framework

• The Eulerian description considers all points in
  space and time as independent variables. In this
  framework, one examines what happens at a given
  spatial point in the fluid field and observes how
  the system evolves over time. There is no
  reference to the initial position in the Eulerian
  framework.
• Mathematically, a flow field is define in the
  Eulerian framework as
• The Eulerian framework is representative of
  placing a set of fixed current meters at every
  point in the fluid domain and recording the flow
  field measurements over time. It provides a
  global description of the flow at a given time.

12
Eulerian framework

• Due to the dependence on time of the position
  vectors (curves). We end up with two
  terms when finding the acceleration in the
  Eulerian description
• In index notation, the above equation takes the
  form
• Notice that i is a free index and j is a dummy or
  repeating index

13
Exercise Flow description

• Which is a better description of the flow the
  Eulerian or Lagrangian framework?
• (Video)

14
The Material Derivative

• Let us determine how to equate the acceleration
  in both frameworks. Assume that for a specific
  position, measured at time, , in the
  fluid, that the Eulerian and Lagrangian velocity
  field are equal
• The notation is used to indicate that the
  derivative is taken in the Eulerian framework

15
The Material Derivative

• Figure 3 Picture showing the relationship
  between the Lagrangian and Eulerian description
  of a parcel velocity at position and time
  . The velocity is the same in both systems at
  this point. The initial position of the parcel
  is shown to indicate the concept of how the
  history of the parcel relates to its position at
  and time .

16
The Material Derivative

• The equivalence of the two frameworks show us
  that the material derivative allows us to observe
  the changes of a fluid parcel while moving with
  the flow.

17
The Material Derivative

• As is the case with all vector objects, the
  application of the material derivative is not
  unique to the velocity field. Thus we can
  identify a general material derivative operator
• Is the material derivative a vector or scalar
  operator?
• Memorize this operator.

18
Exercise

• Given the following velocity and temperature flow
  field, find the material time rate of change of
  the temperature field at x50m y 50m.
• Find