9867

Solution of the Laplace Equation Using Coordinates Fitted to the Boundary Conditions

This Demonstration solves the 2D steady-state heat conduction equation in a physical domain that does not conform to an orthogonal coordinate system. The physical domain is mapped onto a unit square using boundary-fitted coordinates. The transformed heat conduction equation in the computational domain is then solved using a central-difference finite-difference scheme. You can vary the number of grid points in the and directions of the computational domain as well as the Biot number parameter for heat transfer from the upper surface. The resulting isotherms are shown for both the physical and computational domains. The mapping of the grid points from the - computational domain to the - points in the physical domain is also shown.

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Background
Undergraduate students are often exposed to various numerical methods for solving partial differential equations. For example, in a sophomore engineering heat-transfer course, the finite-difference method is introduced to solve steady-state heat conduction problems, in which the computational domain conforms to one of the traditional orthogonal coordinate systems (i.e., rectangular, cylindrical, or spherical). In this Demonstration, we consider a 2D steady-state heat conduction problem in a physical domain based on a nonstandard orthogonal coordinate system:
We show how this problem can be solved using a traditional central-difference method with boundary-fitted coordinates.
The steady-state temperature distribution in the cavity is described by the following boundary-value problem (BVP) with four boundary conditions (BC1–BC4):
,
BC1: for and ,
BC2: for and ,
BC3: for and ,
BC4: for and .
The upper boundary is specified as an arc of a circle, given by , and has unit normal , .
Dimensionless Variables
It is desirable to make the BVP dimensionless before attempting to solve it numerically. We introduce the following dimensionless variables:
, , and .
The dimensionless form of the equations becomes:
,
BC1: for and ,
BC2: for and ,
BC3: for and ,
BC4: for and ,
where is the Biot number.
The interface equation becomes:
.
For notational convenience, we suppress the hat () notation and use the above formulation in what follows.
Boundary-Fitted Coordinates
The physical domain is defined by the coordinates and .
: , .
We now suppose that we have the following mapping that maps the physical domain into a unit square in the computational domain:
, ,
: , .
Thus the dimensionless temperature profile in the computational domain is defined by the mapping: . We can use the chain rule for differentiation to determine the explicit form for the Laplacian in the transformed coordinates (ξ, η).
Thus
.
Clearly when s uniform (flat) then and we get:
,
which is the Laplacian in rectangular coordinates with scaled with the height .
Finite Difference Solution
To generate our finite-difference formula, we use the following grid template:
and use central differences for all derivatives. Thus at node , we have the following approximation for the function :
,
,
,
.
Thus the discretized version of the Laplacian of ψ at an interior node 0 is given by
,
where the coefficients are given by
, ,
, .
Once the grid is specified, then the coefficients can be computed at the given node, for a given .
Boundary Conditions at the Interface
At the interface, the heat flux is given by
at .
We will use a central-difference scheme for this BC:
and ,
which introduces a fictitious node that lies outside the domain:
,
where
and .
Solving for the fictitious node, we get
.
References
[1] C. Chen and J. M. Floryan, "Numerical Simulation of Nonisothermal Capillary Interfaces," Journal of Computational Physics, 111(1), 1994 pp. 183–193. doi:10.1006/jcph.1994.1053.
[2] M. Hamed and J. M. Floryan, "Marangoni Convection. Part 1. A Cavity with Differentially Heated Sidewalls," Journal of Fluid Mechanics, 405, 2000 pp. 79–110. journals.cambridge.org/abstract_S002211209900734X.
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