Coursework 1, CEGEM068/CEGEG068 Finite-Element Modelling
and Numerical Methods
(due-in date: Mon 12 March 2018, noon)
Problem 1 (convective heat flow in a tapered fin):
The temperature distribution T in a tapered fin is governed by
− ddx
parenleftbigg
xdTdx
parenrightbigg
+α(T −T∞) = 0, (0 ≤x≤L),
where α is a constant, T∞ is the ambient temperature and L is the length of the fin. We
consider the case where the temperature is held constant at the right end (x = L) and the heat
flow per unit time is constant at the left end (x = 0). Thus the boundary conditions are
bracketleftbigg
xdTdx
bracketrightbigg
x=0
= 0, T(L) = T0.
We use the following data (in appropriate units): L = 4, T0 = 250, T∞ = 75, α = 0.4168, while
the thickness of the fin at the base at x = L is 0.2 (see figure).
Derive the variational principle for this problem and use this to derive and solve the FE equation
for a uniform. mesh of 4 2-node linear elements. Obtain the temperatures at the nodes and the
heat flow per unit time at x = L.
Give some discussion to the computed solution (its physical sense, accuracy, ...). How would
you improve the accuracy of the solution?
1
Problem 2 (fluid flow around a long cylinder):
A long cylinder is positioned transversely in a flow field between two flat plates as shown below
(figure left). Because the cylinder is assumed to be long, end effects may be neglected and hence
the flow field is two dimensional and can be studied in a plane perpendicular to the axis of the
cylinder. The fluid is assumed to be incompressible. The flow is assumed to be irrotational and
can therefore be described by a scalar stream function ψ satisfying Laplace’s equation
∂2ψ
∂x2 +
∂2ψ
∂y2 = 0.
Because of biaxial symmetry we need to consider only a quarter of the full domain. On this
domain, ψ satisfies the boundary conditions given below (figure right), where u0 is the constant
inlet velocity and a is half the distance between the plates.
Using the Galerkin method, derive and solve the FE equation for the mesh shown below. Use
the global node numbering indicated. For the solution obtained, plot the contour lines in the
full domain of the flow.
Give some discussion to the computed solution (its physical sense, accuracy, ...). How would
you improve the accuracy of the solution?