EG501V Computational Fluid Dynamics
SESSION 2017-18
00 December 2017
Problem 1 [40 marks]
We use dimensionless quantities throughout this problem.
Figure. Left: solution domain and boundary conditions; right: discretization and numbering of unknowns.
We want to numerically solve a steady convection-diffusion equation in the variable φ in a two- dimensional square domain with side length L=1. The equation reads ▽.(φu) = Γ▽2φ . The velocity vector u is constant in the entire domain and has components ux = 1, uy = 1 ; the diffusion coefficient Γ =1 . The boundary conditions are φ = 0 on the west boundary, φ = 1 on the south boundary, ∂φ∂x = 0 on the east boundary, ∂φ ∂y = 0 on the north boundary; see the left panel of the figure. The square domain is discretized according to the right panel of the figure. The dots are equally spaced.
a. Give the discrete equations for points 1, 3, and 9 based on finite difference discretization with a central scheme for the convective term. [16 marks]
b. Give the discrete equations for points 1, 3, and 9 based on finite volume discretization with an upwind scheme for the convective term and the volume centres coinciding with the numbered dots. [16 marks]
A control volume based Peclet number can be defined as the spacing between points.
c. Determine Pecv and decide if you want to apply the central or upwind scheme. [8 marks]
Problem 2 [20 marks]
We use dimensionless quantities throughout this problem.
Figure: flow geometry, boundary conditions and discretization.
Two-dimensional fluid flow can be described by means of a stream function φ(x, y) that obeys
the following elliptic PDE: = 0 . Consider the two-dimensional geometry as shown in
the figure. It defines the flow geometry and boundary conditions: = 0 at the inlet (left) and at
the outlet (right); φ = 0 on the lower wall; φ = 1 on the entire upper wall. The figure also defines the discretization.
a. From the discretization (with Δx = 1 and Δy = 0.5 ) of the PDE, and from the boundary
conditions determine the 10×10 matrix [A] and the 10-dimensional vector b such that the
10-dimensional vector φ containing φk , k = 1…10 satisfies [A]φ = b . Number the unknowns φk as indicated in the figure. [12 marks]
b. The fluid velocity in x andy-direction ( ux and uy ) is related to the stream function according to ux = and uy = − . The solution to [A] φ = b is φ = [0.3284 , 0.3200 ,
0.2767 , 0.2641 , 0.6611 , 0.6487 , 0.5458 , 0.5219 , 0.7951 , 0.7678]. Given this solution, determine ux in points 1, 5 and 8, and determine uy in points 6 and 9 based on central
differences approximations. Is the overall flow from left to right (in positive x-direction)
or from right to left (in negative x-direction)? [8 marks]
Problem 3 [20 marks]
Water (density ρ = 1000 kg/m3 ; dynamic viscosity μ = 0.001 Pa.s) flows steadily through a horizontal, straight pipe with circular cross section of diameter D=0.2 m. The volumetric flow rate is φv = 0.01 m3/s.
a. Argue that this is turbulent flow. [4 marks]
Pressure drop in the pipe is due to friction. The pressure drop per unit length can be written as with U the average velocity in the pipe and f the friction factor. Given the pipe wall roughness f can be considered a constant: f = 0.015 .
b. What is the average energy dissipation rate (symbol ε ) in the pipe? [4 marks]
c. Argue − based on a dimensional analysis − that the expression for the Kolmogorov length scale (the micro scale of turbulence) reads l K = (ν 3ε)14 with ν = μρ the kinematic viscosity. Determine lK based on the average dissipation rate in the pipe. [6 marks]
The shear stress at the inner pipe surface follows from a force balance in the streamwise direction over the liquid in the pipe: The wall shear velocity u* relates to the wall shear
stress: We want to simulate the flow in the pipe with help of standard wall
functions. For this the distance of the first grid point next to the inner pipe wall needs to satisfy the condition 30 < y+ <100 .
d. Given this condition, what is the allowed range of distances of the first grid point from the inner pipe wall? [6 marks]
Problem 4 (20 marks)
Consider the linear system
a. Solve the system [A] x(→) = b by means of Gaussian elimination; show all the steps that lead to your solution. [10 marks]
b. Perform. two (2) Gauss-Seidel iterations on the system [A] x(→) = b(→) . Take as the starting vector of the iteration process. For each of the two Gauss-Seidel iterations determine the normalized residual with (n) = b(→) −[A](n) and || || indicating the length (norm) of the vector. [10 marks]