EG501V Computational Fluid Dynamics

SESSION 2016-17

00 December 2016

Problem 1 (a: 2 marks; b: 15 marks; c: 8 marks; total 25 marks)

We use dimensionless quantities throughout this problem.

Figure. Left: flow geometry and streamlines; right: discretization and numbering of unknowns.

Consider the two-dimensional flow in a sharp 90o bend as sketched in the left panel of the figure. This flow can be described by means of a stream function φ(x, y) that obeys the following PDE: The right panel of the figure defines the flow geometry and boundary conditions: = 0 at the inlet (bottom); = 0  at the outlet

(right) ; φ = 0  on the right and lower wall; φ = 1 on the left and upper wall. The figure also defines the discretization, with spacing Δ = 1 . We realize that the problem is symmetric with respect to the dashed line in the left panel. This implies that we only need to solve φ in the numbered points in the right panel of the figure.

a.   What type of PDE (parabolic, elliptic, hyperbolic) are we dealing with?

b.   From a discretization of the PDE, and from the boundary conditions determine the 9×9

matrix [A] and the 9-dimensional vector b such that the 9-dimensional vector φ

containing φk , k = 1…9 satisfies [A]φ = b . Number the unknowns φk as indicated in the figure.

c.   The fluid velocity in x andy-direction ( ux and uy ) is related to the stream function φ

according to ux = and uy = − .   The solution to [ A b ]φ = b is φ
= [0.6266, 0.4347, 0.3737, 0.3594, 0.9093, 0.8186, 0.7385, 0.7006, 0.6901]. Given this solution, determine ux and uy in points 1, 3, 4, and 5 based on central differences approximations.

Problem 2 (a: 7 marks; b: 9 marks; c: 9 marks; total: 25 marks)

Figure. Catalytic layer & discretization

In steady state, the concentration c of a chemical species that is being consumed in a layer (thickness d) of solid catalytic material can be described by the following reaction-diffusion

equation: with Γ the diffusion coefficient, and k the reaction rate constant. At the

left side of the layer (at x=0) the concentration c is maintained at c0, at the right side (at x=d) at c=0. In dimensionless form the parameters of this problem are: d=4,  Γ = 1 , k=1, and c0  = 1 .

In order to solve for c as a function of x, c is discretized to ci , i = 1…3 , with a constant spacing Δx = 1 between the points, see the figure.

a.   First discretize the differential equation: write it as a linear algebraic equation in terms of

ci . Then set up a linear system of equations in matrix-vector form  [A] c(→) = b(→) with  c(→) the

vector containing the three unknown concentrations ci , [A]a 3x3 matrix, and b a three-

dimensional vector. Determine [A] and b .

The rest of this question is about solving  [A] c(→) = b . If you do not have an answer under Question a., assume and (these [A] and b are not the correct answer for a.).

b.   Solve the system  [A] c(→) = b by means of Gaussian elimination; show all the steps that lead to your solution.

c.   Perform. two (2) Gauss-Seidel iterations on the system [A] c = b. Take as the

starting vector of the iteration process.

Problem 3 (a: 5 marks; b: 5 marks; c: 15 marks; total: 25 marks)

We are dealing with a turbulent flow in which the chemical reaction A + B → P takes place. The reaction is of second order which means that the number of moles of product P being produced    per unit volume and per unit time (symbol rP, unit [mol/(m3.s)]) is rP = kPcAcB with cA and cB the concentration [mol/m3] of species A and B respectively, and kP the reaction rate constant

[m3/(mol.s)]. We assume all species have the same diffusion coefficient  Γ  [m2/s].

a.   For a non-reacting system, the transport equations for chemical species A and B read

respectively. Argue that for the reacting system these equations become and

b.   What is the transport equation for the concentration cP [mol/m3] of chemical species P?

c.   Derive through a Reynolds decomposition of the equation

an equation for the time-average concentration c .

Assume a two dimensional situation. Identify the terms in the equation that need closure.

Problem 4 (25 marks)

We use dimensionless quantities throughout this problem.

The discrete version of the pressure-correction equation reads:

Given the 3x3 mesh of control volumes as in the figure, given the boundary conditions for

pressure correction π which is  ∂π∂n = 0 on all four boundaries, and the preliminary velocity

values ux(*)  and uy(*)  as given in the figure, determine the matrix-vector system  [A = b(→) that

needs to be solved in order to  in the same order as the pressure points numbered in the figure).

Further given: hx = 1, hy = 0.5, ρ △t = 1 .

Figure. Staggered mesh. Pressure defined in the centre of each control volume (dots); velocity at the vertices of each control volume (arrows). All values given are velocity values.