1. Consider the equation
∂c/∂t + u(∂c/∂x) = α(∂2c/∂x2) + q,
where c = c(x, t) is the concentration of a contaminant, u = 1 + 0.6 cos(πx) is a fixed velocity field, α = 0.01 is the diffusivity of the contaminant, and
q = exp {-3(t - 1)2 - 100(x - 0.3)2}
is the source of the contaminant. The equation is defined on a domain x ∈ [0, 2]. The initial condition is c(x, 0) = 0 and the boundary condition at the left is c(0, t) = 0. Solve this problem up to a final time t = 4.
(a) We have no information about the right boundary, so we have to decide for ourselves whether we even need a boundary condition there and, if so, what would be a reasonable choice. Explain your reasoning.
(b) Write down, in text, how you are solving this problem. Show your code.
(c) Explain how you ensure that your code is correctly implemented.
(d) Show what process you use to decide on a suitable grid-spacing Δx and a suitable time step Δt, and write clearly what your final chosen values are.
(e) Plot c vs x at times t = 1, 2 and 3 in the same plot.
(f) At every instant in time, the peak concentration is located at some xpeak(t). Plot this location vs t. Interpret the plot.
2. A turbulent flow being forced by a uniform pressure gradient between two infinite parallel plates (located at y = -1 and y = 1) can be described by
d/dy[(v+vt)du/dy] = -1, y ∈ [-1, 1].
We take the fluid viscosity v = 0.005, and we assume that the turbulent viscosity is
vt = 0.4y^[1 - exp(-y^/0.09)]2, y^ = min{y+1, 1-y}.
Note that vt depends on the distance to the nearest wall y^, not on y itself. For boundary conditions, we take the lower wall to be stationary (i.e., u(-1) = 0) but we assume that the upper wall is moving at speed 20 (i.e., u(1) = 20).
Solve this using a direct method, i.e., do not use a shooting method.
(a) Write down (in text, not Matlab code) how you are solving this problem. I want the numerical details on how you construct your matrix equation etc.
(h) Implement and solve in Matlab. Plot the solution.
(c) What is your best estimate of the velocity on the centerline, i.e., u(0)?
(d) How many grid-points do you need in order to get u(0) to within approximately 0.1 units of the true value? Both tell me your answer and explain how you reached this decision.
Note that there are multiple ways of solving this problem - if you get stuck, try to take a step back and think of some other way of solving it.