2-D Conservation of Mass

1
2-D Conservation of Mass
ru
(x,y)
dy
dx
rv
2
2-D Conservation of Momentum
3
2-D Conservation of Momentum (contd.)Surface
forces
tyx(dx)(1)
(x,y)
dy
txy(dy)(1)
dx
4
2-D Conservation of Momentum (contd.)
Similarly,
The body forces are expressed as
where is the body force per unit volume.
For example,
5
2-D conservation of momentum (contd.)
Or, in cartesian tensor notation,
Where repeated subscripts imply Einsteins
summation convention, i.e.,
6
Conservation of momentum (contd.)
The shear stress tij is related to the rate of
strain (i.e., spatial derivatives of velocity
components) via the following constitutive
equation (which holds for Newtonian fluids),
where m is called the coefficient of dynamic
viscosity (a measure of internal friction within
a fluid)
Deduction of this constitutive equation is beyond
the scope of this class. Substituting for tij in
the momentum conservation equations yields
7
Navier-Stokes equations for 2-D, compressible flow
The conservation of mass and momentum equations
for a Newtonian fluid are known as the
Navier-Stokes equations. In 2-D, they are
8
Navier-Stokes equations for 2-D, compressible
flow in Conservative Form
The Navier-Stokes equations can be re-written
using the chain-rule for differentiation and the
conservation of mass equation, as
(1)
(2)
(3)
9
Conservation of energy and species
The additional governing equations for
conservation of energy and species are
(4)
(5)
10
Summary for 2-D compressible flow

• UNKNOWNS r, u, v, T, P, ni N5, for N species
• EQUATIONS
• Navier-Stokes equations (3 equations
  conservation of mass and conservation of
  momentum in x and y directions)
• Conservation of Energy (1 equation)
• Conservation of Species ((N-1) equations for n
  species)
• Ideal gas equation of state (1 equation)
• Definition of density (1 equation)

11
Recapitulation

• So far,
• Formulation of case studyquasi 1-D compressible
  flow
• Numerical solution techniques
• Steady vs. Time-marching to steady state
• Finite differences (FD). Time marching to steady
  state
• (a) Explicit schemes (McCormack, FTBS)
• Easier to program
• Restricted to small Dt for stiff problems
• May not yield a solution at all for really stiff
  systems.

12
Recapitulation (contd.)

• (b)Implicit schemes (LBI)
• Harder to program
• Allows use of larger Dt even for stiff problems
• May be the only way to find a solution for really
  stiff systems
• Finite elements (FE), time marching to steady
  state
• (a)Linearization same as for LBI FD method
• (b)well-suited for complex geometries.

13
Recapitulation (contd.)

• Both FD and FE techniques ultimately require
  solution of linear equations Mx f
• In the LBI method, M is a block tri-diagonal
  matrix
• Solution of systems such as Mx f using PETSc
  allows you to explore parallel solution vs.
  serial solution.
• ? implications for performance
• Iterative methods (ex. Conjugate gradient) are
  well-suited to parallelization.

14
Extension of LBI method to 2-D flows

• Non-dimensionalize the 2-D governing equations
  exactly as we did the quasi 1-D governing
  equations.
• Take geometry into account. For example,

Outer Body
Center Body
15

• Let ri(x) represent the inner boundary, where x
  is measured along the flow direction.
• Let ro(x) represent the outer boundary, where x
  is along the flow direction.

ri(x)
ro(x)
16

• The real domain
• is then transformed into a rectangular
  computational domain, using coordinate
  transformation

y or r
x
x
h
17

• The coordinate transformation is given by
• The governing equations are then transformed

18

• Or,
• and
• etc.

19

• This will result in a PDE with h and x as the
  independent variables for example,
• Recall that for quasi 1-D flow, we had equations
  of the form

20

• Applying the LBI method yielded
• ?
• or,

21

• Applying the same procedure to our transformed
  2-D problem would yield
• Recall that after linearization of the quasi 1-D
  problem, the resulting matrix system was

22

• Now, in 2-D, the linearization procedure will
  result in

Where each Fi, Gi, Hi are themselves block
tri-diagonal systems as in the quasi 1-D problem.
In other words, etc.