Abstract
This paper describes two new solution algorithms for steady recirculating flows that use a penalty formulation to eliminate the pressure from the finite difference form of the governing equations. One algorithm uses successive substitution to linearize the equations, while the other employs the Newton-Raphson linearization. In both cases, the equations are solved in a fully coupled manner using a sparse matrix form of LU decomposition. The D'Yakonov iteration is used to avoid unnecessary factorizations of the coefficient matrix, significantly improving the computational efficiency. The Newton-Raphson linearization leads to faster convergence, but the execution times of the two methods are comparable. The algorithms converge rapidly and are robust to changes in grid size and Reynolds number. In a number of laminar two-dimensional flows, the new methods proved to be two to ten times faster than some conventional iterative methods.
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Abbreviations
- A :
-
coefficient matrix
- A e :
-
area of east control-volume face
- a :
-
coefficient in the discretization equations
- â :
-
coefficients in the modified momentum equations in the penalty formulation, Eqs. (13)
- b :
-
constant term in the discretization equations
- F :
-
symbolic form of nonlinear system of equations, F (ϕ)=A ϕ−b=0
- H :
-
characteristic length
- J :
-
Jacobian matrix
- n :
-
iteration number
- P :
-
dimensionless pressure
- p :
-
pressure
- Re:
-
Reynolds number
- U :
-
dimensionless u velocity
- u :
-
x-direction velocity
- V :
-
dimensionless v velocity
- v :
-
y-direction velocity
- X, Y :
-
dimensionless coordinates, X=x/H, Y=y/H
- x, y :
-
physical coordinates
- Δx :
-
x-direction width of the control volume
- ε:
-
normalized error in the unconverged solution, Eqs. (19)
- Λ:
-
dimensionless penalty parameter, Eq. (7)
- λ:
-
dimensional penalty parameter, Eq. (10)
- μ:
-
viscosity
- ρ:
-
density
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Communicated by S. N. Atluri, January 1, 1989
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Braaten, M.E., Patankar, S.V. Fully coupled solution of the equations for incompressible recirculating flows using a penalty-function finite-difference formulation. Computational Mechanics 6, 143–155 (1990). https://doi.org/10.1007/BF00350519
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DOI: https://doi.org/10.1007/BF00350519