∆t≤ αim
0.2 max|V|P2 g
. (3.2)
Equation (3.2) has been used as a guideline for identifying a stable time step. Using the second order method in the velocity correction scheme presented above one finds,αim≈0.15.
The order of basis functions used areP = 10 and the smallest element side length is around g = 0.1. Doing some initial simulations at stable time step it was found that the highest speed of the fluid was below max|V|= 2. Plugging these values into 3.2 yields a time step value of,
∆t≈0.00035. (3.3)
Based on this estimate a set of experiments were then performed to investigate the stable time step for the highest resolution mesh, closest to the wall and at Reynolds number, Re = 300. Through this investigation it was found that the time step of ∆t = 0.0005 was the largest stable time step.
As a side note this discussion of choosing a stable time step highlights a crucial weakness of the SEM once again. For the classical FEM, P = 1, which means that the allowed time step for a given mesh is much larger. However the number of elements needed when using the classical FEM to obtain the same accuracy as for the SEM can be impossibly big which makes solving the equation at each time step extremely expensive.
3.2 The Discrete Domain
Another necessary step before simulations can be performed is to determine a suitable size of the domain of interest and how to discretize it. As the model is of an open flow problem a large domain is needed to assure that blockage is not introduced by the far-field boundary conditions17.
Domain Size: In order to avoid blockage the domain for the cylinder in free flow has been chosen to extend over the range: x×y = [−10D,30D]×[−15D,15D] where Dis the cylinder diameter. This choice was made based on an investigation performed by Huang and Sung [8] which showed that no significant change in the solution could be observed if the domain was increased further.
For similar domain parameters Huang and Sung found that both the Strouhal number,St, and base pressure coefficient,Cbp, agree well with previous simulation results presented in [10]
and [7]. A series of simulations validating the choice of domain size have been performed, and the results are presented in chapter 7.2.
Based on the choice of domain for the cylinder in free flow, the domain for the cylinder near a moving wall is chosen to extend over x×y = [−10D,30D]×[0,20D] to assure that the half open flow is simulated accurately.
17Blockage is a term for the fact that the far-field boundary conditions introduce artificial changes in the flow which a real world open flow would not. Another approach is to construct BC’s which allow a smaller domain, however this has need been investigated here.
Mesh Construction: The last step is to create a mesh which captures the features of the chosen domain on which to solve the discretized model problems.
As stated above the open flow property dictates either sophisticated boundary conditions or a large domain which may lead to a huge number of DOF’s. This makes the problem very expensive to solve. However initial investigations showed that in most of the domain upstream of the cylinder both the pressure and velocity are almost constant in time. For the free stream problem this is also true both above and below the cylinder. When introducing the wall it remains true above the cylinder away from the wall. The areas of almost constant pressure and velocity are illustrated on figure 3.1,
Figure 3.1: Domain for open flow problem with areas of almost constant velocity and pressure highlighted.
The fact that the pressure and velocity are almost constant in these areas means that far fewer DOF’s are needed to capture the solution with good accuracy. This fact lead the author to construct hybrid meshes consisting of both quadrilateral and triangular elements in order to save as many DOF’s as possible in areas where the solutions only vary slightly.
The choice of using hybrid meshes makes it easy to vary the DOF’s throughout the domain to get a good distribution which captures the solution without wasting calculation time.
Illustrations of hybrid meshes used in the simulations are presented in figure 3.2.
(a) (b)
Figure 3.2: Examples of the hybrid meshes used. (a) full domain. (b) zoom close to cylinder and the interesting part of the domain.
3.2 The Discrete Domain 3 DISCRETIZING THE PROBLEM
From figure 3.2 it is seen that the area around and behind the cylinder is densely meshed while the area away from the cylinder contains only few elements. This design choice saves DOF’s while assuring high resolution in areas where the solution changes rapidly.
Curvilinear Elements: The Nektar++ framework supports curvilinear elements (i.e.
elements with curved edges) to allow for better representations of curved geometries with fewer elements. This is important for maintaining the spectral accuracy of the methods along curved geometry. If only linear polynomials were used to resolve the cylinder a massive amount of elements would be needed to approximate the shape of the cylinder accurately.
However introducing a lot of elements defeats the purpose of using the SEM. Thus in order to represent the geometry of the cylinder with high accuracy without a large number of elements, curvilinear elements are used. All meshes generated have therefore been modified to use curvilinear elements of appropriate order along the cylinder.
Figure 3.3 illustrates the improvement in resolving the cylinder shape obtained from using third order polynomials compared to using first order polynomials, with the same number of elements.
(a) First order polynomials. (b) Third order polynomials.
Figure 3.3: Illustration of advantage of using higher order polynomials to mesh a circular object.
From the figure it is easy to see the improvement in the circular shape as the polynomial order is increased.
Based on the considerations stated above all simulations have been performed using tenth order curvilinear elements around the cylinder to assure a highly accurate resolution of the cylinder.
Example of Underresolved Geometry: A small illustration of the problem with un-derresolving the geometry is presented here. Figure 3.4 illustrates how unun-derresolving the cylinder by using low order polynomials to mesh it causes problems. The figure shows a close up of the cylinder surface downstream of the flow. In figure 3.4a third order polynomials have been used to approximate the cylinder. Here small vortices are seen to have formed along the cylinder. These vortices are formed at the intersection between elements along the
cylinder. This is due to the errors in resolving the problem domain to sufficient accuracy introduced by the low order polynomial approximation of the cylinder.
(a) Cylinder resolved using third order polynomials. Small vortices are clearly seen to have formed on the upper half of the cylinders backside.
(b) Cylinder resolved using tenth order polynomials. No small vor-tices are observed.
Figure 3.4: Illustration of vorticity contours at an instant in time for flow on a zoom on the domain behind the cylinder. The parameters used for both simulations are Re= 280,D/G= 5/3.
Figure 3.4b shows the same extract from the domain at approximately the same point in time. For this simulation tenth order polynomials have been used to mesh the cylinder. Here it is seen that no small vortices are observed as the error of using polynomials to approximate the cylinder surface has become insignificant.
4 SOFTWARE
4 Software
This chapter provides an overview of the software used for the pre processing, simulations and post processing of simulation data preparing it for final data analysis. This includes both software written by the author and open source software. The chapter is included to provide the reader with enough information to understand and use the developed software and can be skipped without loss with respect to understanding the results presented in the following sections.
Section 4.1 provides an overview of the software used in the pre processing phase. Section 4.2 gives a brief description of the frameworkNektar++used to perform the CFD calculations, as well as to calculate the needed derived quantities and norms. Lastly section 4.3 describes the different pieces of software written for post processing the data and performing the UQ analysis.