Boundary Conditions for Linearized Block Implicit (LBI) Method applied for Quasi-1D Flow

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Boundary Conditions for Linearized Block Implicit
(LBI) Method applied for Quasi-1D Flow

• The method of characteristics applied to the
  quasi-1D problem yields three characteristics u,
  u-a, and ua, where
• Inlet BCs
• case (i) subsonic inlet
• 2 conditions can be prescribed, e. g.
  Po and To

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Boundary Conditions (contd.)

• case (ii) supersonic inlet
• 3 conditions can be prescribed, e. g.
  Po, To, and
• Minlet.
• Exit BCs
• case (i) subsonic exit
• 1 condition can be
• prescribed, usually P.

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Boundary Conditions (contd.)

• case (i) supersonic exit
• NO conditions can be
• prescribed.

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Boundary Conditions(contd.)

• We now illustrate for the subsonic inlet, how the
  boundary conditions are brought into the final
  block tri-diagonal matrix, as discussed in slides
  27 and 28 of the last lecture.
• The inlet is assumed to be related to the
  reservoir or supply for the gas, for which Po and
  To are specified and constant.
• The flow from the reservoir to the nozzle inlet
  is assumed to be loss-free or isentropic.

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Boundary Conditions(contd.)

• For the flow between reservoir and inlet, we can
  write from cons. Of energy.
• In addition, for an isentropic flow,
  thermodynamics tells us that P and T are related
  via
• In addition, we have the ideal gas equation of
  state PrRT

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Boundary Conditions(contd.)

• These equations can be non-dimensionalized (as
  before) to yield
• (1)
• (2)
• PrT (3)

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Boundary Conditions(contd.)

• Substituting (3) into (2),
• ? (4)
• In addition, we have (5)

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Boundary Conditions(contd.)

• The quantities and
• at the inlet location i0 can now be
  evaluated. This is equivalent to differentiating
  with respect to time.
• Therefore, defining as the difference
  operator which is equivalent to differentiation,
  equations (4) and (5) become
• and

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Boundary Conditions(contd.)

• The quantity ?T(0) has now been related to ?u(0),
  and ?r(0) has also been related to ?u(0)

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Boundary Conditions(contd.)

• ?u(0) is determined by implicit extrapolation
  from knowledge of ?u(1) and ?u(2), i.e. using
  central differences
• or, and
• ?

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• Recall the linear system of equations for all is
  resulting from the LBI method is a Nx(N2) block
  tri-diagonal matrix, where there are N interior
  points

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• The three discretized conservation equations
  including the artificial dissipation terms, can
  be written as
• where Bi, Di, and Ai are 3x3 matrices for each i,
  and Fi is a 3x1 column vector. They are given
  by

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• The boundary condition at i0 dictates that the
  D1 and A1 elements of the matrix be re-defined in
  order to eliminate B1 from the first row. The
  first row is the equation
• which for quasi-1D flow can be written more
  explicitly as
• for each governing equation.

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Elimination of B1 from the Block Tri-diagonal
Matrix

• Recall from slide 10 of this lecture that ?u(0)
  can be expressed in terms of ?u(1) and ?u(2)
  using implicit extrapolation ( ).
• ?r(0) can then be obtained from
• ?T(0) can then be obtained from

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Elimination of B1 from the Block Tri-diagonal
Matrix (contd.)

• Thus,

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Elimination of B1 from the Block Tri-diagonal
Matrix (contd.)

• Next, using implicit extrapolation,
• we have
• collecting terms, it can be seen that D1Du and
  A1Du can be re-defined in terms of B1.

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Elimination of B1 from the Block Tri-diagonal
Matrix (contd.)

• Thus,
• and

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• The boundary conditions at the exit, nj1, can be
  handled in a similar manner, with conditions at
  iN1 related to those at iN and iN-1
• except that here, the Bs and Ds are modified by
  eliminating the As.