Wolfram Demonstrations Project

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.

Contributed by: Brian G. Higgins and Housam Binous

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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

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.