After the validation process was completed a large number of simulations were performed for the cylinder near the moving wall. As detailed in section 1.2 this problem has three parameters of interest. These are time,t, the Reynolds number,Re, and the relation between the cylinder diameter, D, and the distance between the cylinder and wall, G, i.e. D/G.
Obviously time is not a parameters which is varied like the Reynolds number or D/G.
Instead after periodic shedding has been achieved, time is allowed to run over more then one full period of shedding and the period noted for each simulation. Then for each simulation, data about the flow pressure, velocity, vorticity and the first and second derivatives of the vorticity are stored at a sequence of time steps in order to allow investigation of the flow in time.
6.5 Uncertainty Quantification 6 SIMULATIONS
Re
20 60 140 220 300
D/G
5 4 5
3 5
5 2
10
G/D 0.8
0.6 0.4
0.2 0.1
Figure 6.4: Illustration of the 2 parameter space (Re, D/G) with parameter values at which simulations were performed marked by grey and red dots. Note that the scaling of the D/G axis is linear if we consider the quantity G/D i.e.
the gab distance for a cylinder with D= 1.
Parameter Domain: The investigation of the structure of the vortices etc, is done over the range (Re×D/G)∈({20,300} × {10,54}). This forms a 2 dimensional parameters space which is illustrated in figure 6.4.
The illustration in figure 6.4 shows the full parameter space along with all points at which a simulation has been performed. The gray large circular points mark the initial simulations performed to gain an overview of the flow structures in the domain of interest. The smaller red points are all the simulations performed to investigate phenomena identified during the initial simulations.
6.5 Uncertainty Quantification
Three separate UQ investigations have been performed. In each case the Reynolds number was taken to be the uncertain parameter,X=Re.
Cylinder in Free flow: The first two investigations were performed for a cylinder in free flow with focus on comparing results obtained using UQ to known published results and to results obtained during simulations for this work. The quantities of interest here areStand CDp. One investigation was performed assuming that the Re was uniformly distributed in the interval Re ∈ U[50,600]. The other was performed assuming a normal distribution for Re,N(300,50).
Note: Technically it is wrong to use normal distribution for theReas,Re≥0 is dictated by the physics. However usingN(300,50)forReyields a probability of obtaining an unphysical
value, i.e. Re < 0, given by R−∞0 fRe,G(x)dx ≈ 10−9, which is deemed acceptable for the present test.
For both investigations three different basis orders have been used for the gPC expansions.
In both cases these are a first, a second and a fourth order expansion.
Cylinder Near Moving Wall: The third investigation was performed for the cylinder near the moving wall at the diameter to gap ratio D/G = 10. Here the application of UQ was used to perform an investigation of the variation in the path travelled by a vortex downstream of the cylinder depending onRe. Here the Reynolds number was again assumed to be uniformly distributed, this time on the intervalRe∈ U[150,240]. For this investigation three different basis orders were also used. In this case these are a third, a fifth and a seventh order basis.
7 VALIDATION
7 Validation
For any investigation of a physical system using computer simulations it is very important to assure the results of the simulations may be trusted. The best way of doing this is of course to validate the simulations against real world experiments. It is however outside the scope of this thesis to make experiments which may validate the flow structures and physical quantities investigated. Therefore other methods of validations have been used. These are,
• Validation of the Nektar++ framework and incompressible Navier-Stokes solver by application to problems with known solutions.
• Validation of solution accuracy through convergence testing measured inL2-norm.
• Validation against published work.
Section 7.1 provides the results of a convergence test for the quantities used in the analysis process. Section 7.2 presents comparisons of results obtained in this project and published results. Finally appendix A.4 contains a set of tests of the Nektar++ framework and its incompressible Navier-Stokes solver IncNavierStokesSolver validating it against known solutions. These tests were performed by the author prior to this project are may therefore be found in the appendix.
7.1 Convergence
Given that the problem to be solved is well posed the SEM promises that the numerical solution converge to a unique solution for the 2D-Incompressible Navier-Stokes problem as the number of elements and/or the polynomial expansion order are increased.
That the solution converge to a unique solution makes is possible to investigate when the do-main is well enough resolved by observing changes inL2 norm for each of the field quantities of interest. When theL2norm has converged to a fixed value this indicates that the solution has converged. In order to assure sufficient resolution a convergence tests with increasing polynomial order on the SEM basis functions and an increasing number of elements has been performed26.
The tests were performed by simulating the flow using the boundary and initial conditions presented in section 1.3 from the initial timet= 0 to the final timet= 40 using a time step length of ∆t= 5·10−4.
Table 7.1 contains the parameters used for polynomial order and number of elements in the mesh and the corresponding approximate number of DOF’s for the system.
26The test has been performed for all fields of interest using the build inL2-norm calculator inNektar++.
Simulation number Number of elements Polynomial order ≈ DOF
1 1500 6 54000
2 1500 8 96000
3 1500 10 150000
4 1800 6 64800
5 1800 8 115200
6 1800 10 180000
7 2400 6 86400
8 2400 8 153600
9 2400 10 240000
10 3000 6 108000
11 3000 8 192000
12 3000 10 300000
Table 7.1: Table displaying the number of elements and polynomial order used for each simulation in the convergence test. The Reynolds number used: Re= 300 and the gap to diameter ratio D/G= 5.
Tables 7.2 and 7.3 show the results of the tests.
1 2 3 4 5 6
u: 28.541 28.540 28.540 28.541 28.540 28.540 v: 3.6269 3.6285 3.6282 3.6282 3.6282 3.6282 p: 2.8721 2.8726 2.8738 2.8756 2.8730 2.8733 ω: 13.926 13.925 13.925 13.924 13.925 13.925 ωx: 119.09 118.88 118.75 118.80 118.70 118.71 ωy: 191.47 191.50 191.51 191.62 191.51 191.51 ωx,x: 1990.1 1997.1 1997.1 1984.4 1985.1 1979.4 ωx,y: 1661.6 1657.0 1653.5 1658.2 1654.0 1652.4 ωy,x: 1661.6 1657.0 1653.5 1658.2 1654.0 1652.4 ωy,y: 4442.4 4335.3 4329.6 4370.4 4333.4 4329.7
Table 7.2: TheL2-norm calculations for each of the 10 fields of interest obtained from the first 6 simulations, for which parameters are specified in table 7.1. Bold entries highlight that the field has converged to four significant digits.