The transport of φ through a given pipe with no sources is governed by the following equation for steady-state convection and diffusion.
We saw in assignment #1 that this equation can be solved directly using an appropriate discretization scheme and matrix solver. This “steady-state” problem can also be solved by marching the following transient equation in time until the solution converges.
a) Derive the interior-node and boundary-node discretized equations for the transient convection-diffusion of φ (equation 2) for the following approximation schemes:
Scheme 1: Upwind in space, Explicit Euler in time.
Scheme 2: Upwind in space, Implicit Euler in time.
Scheme 3: Upwind in space, Trapezoidal in time. (complete for extra credit)
b) Using the above three schemes, calculate the distribution of φ(x, t) using the following given conditions and K = 0.2, 2.0, and 20.0 (for use in equation 3). Graph φ(x) at n = 0, 4, 16, 64 and 256.
Given
Pipe length = 1.0 m
= 1.0 kg/m3 (constant)L = 0.1 kg-s/m (constant)
Dirichlet boundary conditions, L = 100, R = 50 Initial conditions, (x, 0) = 50
ux = 2.5 m/s (constant) 20 control volumes
c) At n = 256, compare your transient solutions to the steady-state solution (numerical) from the problem set#1 by calculating the norm using the following equation:
