ME 540

Numerical Solutions Applied To Heat Transfer And Fluid Mechanics Problems

Steps in the finite difference solution of a problem

Determine the appropriate mathematical model for the problem.

• Conservation equations
• Boundary conditions
• Initial conditions

Example: Transient Heat Conduction

Energy conservation equation
Boundary Conditions
At a surface of a conducting structure either the temperature or the heat flux is specified. A fixed temperature does not represent of any real physical configuration, but can be an appropriate approximation for a structure adjacent to a region with very high conductivity and total heat capacity. Surface heat flux within the structure is expressed in terms of the normal derivative of the temperature. Useful flux boundary conditions include:
Constant surface heat flux,
Convection,
For problems with conduction through two regions with two different materials, fluxes must match at the boundary between the regions,
Initial Conditions
T(x, y, z, t=0) = f(x,y,z)

Example: Fluid Flow with Heat Transfer (2D - Steady State)

Conservation equations:
Continuity
Momentum Equations (u is "x" component, v is "y" component)

Energy Equation
Boundary Conditions
 Inlet x = 0, u = Uin, v = 0, T = Tin Walls y = b, u = 0, v = 0, T = Tw Symmetry y = 0, v = 0, Outlet ?

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Created by  Frank Schmidt
Maintained by John Mahaffy : jhm@psu.edu