2D conservation of momentum contd.

1
2-D conservation of momentum (contd.)
Or, in cartesian tensor notation,
Where repeated subscripts imply Einsteins
summation convention, i.e.,
2
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
3
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
4
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)
5
Conservation of energy and species
The additional governing equations for
conservation of energy and species are
(4)
(5)
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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)

7
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
8

• 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)
9

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

y or r
x
x
h
10

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

11

• Or,
• and
• etc.

12

• 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

13

• Applying the LBI method yielded
• ?
• or,

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• 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

15

• 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.
16

• At this point, we have a choice
• We can solve the full system as is, i.e. an
  (MN)x(MN) linear sparse system.
• Or
• We can split the operator and apply the
  Alternating Direction Implicit (ADI) method to
  reduce the 2-D operator to a product of 1-D
  operators in each of the coordinate directions
  and solve each alternately, at each time step.

17
Douglas-Gunn ADI scheme

• Recall that after Crank-Nicolson differencing in
  time, linearization, and discretization of the
  spatial derivatives, we have
• Split the operator by implicit factorization,
  approximate to either order (Dt) or (Dt)2 as in
  the original discretization errors.
• ?

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• Note that
• Thus, operator splitting yields
• Defining , we have

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• Note that now, the solution of
• and
• is identical to solving two equivalent quasi 1-D
  problems in each of the coordinate directions x
  and h.
• The Douglas-Gunn ADI scheme after implicit
  factorization can be shown to be unconditionally
  stable in 3-D as well as long as the convective
  term is absent, but is conditionally stable
  with the convective term present.

20

• The conditional stability of this scheme worsens
  and may vanish if there are periodic boundary
  conditions.
• A virtue of the Douglas-Gunn ADI approach is that
  the same boundary conditions used for
• can be be used for W. This is a result of
  consistent splitting of the operator. Other
  operator splitting schemes exist that are
  inconsistent - the same BCs used for
• cannot be used for W.

21

• In the present case study problem, our governing
  equations are of the form
• Applying the LBI method to this equation yields
• The Douglas-Gunn operator splitting then yields

22

• Still, no matrix inversion is required.
• The solution procedure can be implemented as
  follows
• Set
• ? ? solve for W in xinv
• Next, solve
• for fn1 - fn in rinv

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2-D LBI Code with Douglas-Gunn operator splitting
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Sample solutions for 2-D LBI (x-y geometry)
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Key features

• 10,000 time steps at Dt 10-4 (non-dimensional)
• then for 100,000 time steps at Dt 2x10-3
  (non-dimensional).
• Lref 1 cm. Pref 1.013 x 105 Pa Tref 300 K
• 260 x 50 grid (x r)
• Artificial Dissipation or