Linearized Block Implicit (LBI) Method applied to Quasi-1D Flow

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Linearized Block Implicit (LBI) Method applied to
Quasi-1D Flow

• The Linearized Block Implicit Method (LBI) method
  was developed by Briley McDonald to solve the
  3-D compressible Navier-Stokes equations by
  time-marching.
• The LBI method is particularly well suited to
  handle large systems of coupled PDEs (such as
  those that occur in reacting flows and plasmas),
  resulting from many simultaneous elementary
  processes.

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• The LBI method is best illustrated via the
  governing equations of quasi-1D flow.
• The basic steps of the LBI method are
• Crank-Nicolson difference in time,
• Central difference in space,
• Linearize using chain-rule differentiation, and
• solve the resulting block tridiagonal matrix

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Non-dimensionalized governing equations
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• Evaluate equations at time level (n1/2), with
  the intent of relating quantities at (n1) to
  their respective values at n. This is done at
  each point i, but the i is dropped in the
  difference equations below, for convenience.
• Continuity or conservation of mass

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• Next, use Taylors series to evaluate the time
  derivative
• Subtracting,
• ?

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• Next, evaluate the spatial derivative
• Similarly,
• Adding,

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• ?
• Thus, the conservation of mass equation becomes
• Now,

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• Using the chain rule for partial differentiation,
  we have
• discretizing the time derivative yields

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• Next, apply central differencing for the spatial
  derivative
• and
• from which we have

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• Therefore,

Thus, the conservation of mass equation
becomes
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• Defining , ,
• , , and
• , the continuity equation becomes

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• Next, apply the LBI method to conservation of
  momentum
• In this equation, the first and third terms are
  handled exactly as in the case of conservation of
  mass. The second term is handled using the same
  procedure, but let us look at it in detail

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• As before,
• Using the chain rule for partial differentiation,
  we have
• discretizing the time derivative yields

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• Next, apply central differencing for the spatial
  derivative

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• Thus, the conservation of momentum equation
  becomes

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• In the same manner, the energy equation becomes

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• Before describing how these equations can be
  solved, let us modify them to add artificial or
  numerical dissipation for stability. As before,
  we will add a term , to the right hand side of
  the continuity equation, , to the right hand
  side of the momentum equation, and , to the
  right hand side of the energy equation.

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• Each of these terms would have to be evaluated at
  time level n1/2. So,
• and
• ?

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• ?
• or,
• ?
• and

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• Thus,
• similarly,
• and

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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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• Note that the elements of matrix Bi are the same
  as the elements of Ai, except that the index
  (i1) is replaced by (i-1), and the elements of
  Bi are negative of the elements of Ai (except for
  the numerical dissipation terms).

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

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• Where is the vector of unknowns. Note that
  in order to have an NxN matrix, Y0, YN1, B1, and
  AN must be eliminated using boundary conditions.
  This is done by re-defining D1, A1, BN, and DN
• and
• Now, Y0 can be related to Y1 and/or Y2 via
  boundary conditions
• ? and

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• Thus, the first row of the block tri-diagonal
  matrix is then replaced with and , respectively.
  In a similar manner YN1 can also be related to
  YN and/or YN-1 via boundary conditions
• ? and
• General guidance for formulating boundary
  conditions can be obtained using the method of
  characteristics, or by physical intuition.
  Different types of boundary conditions are
  discussed next.