of vortex formation. The vortex formation around cables and pylons in bridge construction and the drag and vibrations caused by these vortices. The vortices formed around ship hulls as they travel through the oceans. The vortices formed behind race cars etc. If a better understanding of the influence of the vortices can be obtained it may be possible to using this knowledge to design structures which are more resilient and ships and cars which are capable of moving faster and more efficient due to reduced drag etc.
The use of numerical methods for obtaining this insight is very valuable as the numerical approach is a highly cost efficient way of performing experiments to obtain better under-standing of physical processes. If the same knowledge was to be obtained through physical experiments, multiple and in some cases huge set-up’s would have to be constructed which quickly becomes both very time consuming and very expensive.
1.2 The Model Problems
The main problem under consideration is that of an incompressible fluid flowing around an infinitely long cylinder oriented along the z-axis positioned near a moving wall modelled in two dimensions. The problem is investigated in the low Reynolds number regime (Re ∈ [20,300]) after the initial transient flow has disappeared. In this context the Reynolds number is defined as Re = ρU∞µD where ρ is the density of the fluid, U∞ is the far field velocity of the fluid, Dis the cylinder diameter and µis the dynamic viscosity of the fluid.
The fluid in the far field and the wall moves with the same velocity, U∞ = (u∞,0) while the cylinder remains stationary. This leads to no slip boundary conditions on the cylinder of the form: (u, v) = (0,0) and likewise no slip on the wall of the form: (u, v) = (u∞,0). It also leads to a velocity condition at the outflow boundary of ∇(u, v)·n= 0 and a pressure condition of p = 0. The boundary conditions are elaborated on in section 1.3. The second model problem is also a 2D model of a cylinder but this time in free flow. This problem is considered in the Reynolds number regime,Re∈[100,600].
U∞
D
y x
(a)
U∞
U∞
D
G y
x
(b)
Figure 1.1: Schematics for 2D-domain containing a cross section of an infinitely long cylinder perpendicular to the flow. (a) Cylinder in free flow, (b) Cylinder near a wall.
First, the cylinder in free flow is considered in order to validate numerical solutions and establish a reference for the flow structure. This model problem is also used for part of the UQ investigations. Then the main model problem is considered by introducing the wall and
an investigation for a decreasing distance between cylinder and wall and varying Reynolds number Re is performed. Increasing the Reynolds number above a critical value changes the flow from stationary to instationary with a periodic behaviour and thus introduces a time parameter t. Below the critical value of the Reynolds number the flow is stationary and thus, based on the definition of a vortex provided in (1.20), no vortices exist. Schematic drawings of the two model problems are provided in figures 1.1a and 1.1b.
In figure 1.1 U∞= (u∞,0) is the inflow and far field velocity as well as the velocity of the wall. Dis the cylinder diameter andGthe size of the gap between the cylinder and wall. D andGare not of any interest as two separate parameters since the dynamics of the flow does not depend on them individually but only on their ratio. This fact is easily realised as D andGare the only geometric parameters in the system. Thus scalingDby a factorasimply corresponds to shrinkingGby the same factora. This means that the parameter of interest in the following is their ratio, chosen as DG. The choice of DG opposed to DG is made based on the same considerations as those given by Huang and Sung in [8]. The considerations are that it is easier to illustrate small gap heights more clearly on a plot and that with this choice the case of a cylinder in free flow corresponds to DG = 0 instead of DG → ∞.
From the present discussion three independent parameters for the problem have been iden-tified. Two of the three are input parameters, i.e. Re and DG. As the problem becomes instationary for increasing Reynolds number time emerges as the third parameter. In the following section where the physics and model equations are presented it is seen that these three parameters along with the boundary conditions are enough to describe the problem completely.
1.2 The Model Problems 1 THE PROBLEM AND UNDERLYING PHYSICS
Different flow regimes: As mentioned the problem of the cylinder in free flow has been investigated thoroughly in several studies. For Reynolds numbers below the maximal value of interest in this work four different regimes for the flow as a function of Reynolds number have been characterised. In [9] Brøns et. al. identify, describe and analyse the first three of these regimes. The regimes may briefly be summarised as,
• Attached Flow: Re . 5. In this regime the flow is steady and attached to the cylinder. That is, the fluid passes over the cylinder and continues downstream without any circulation. Here, the fluid has a point of attachment on the upstream side of the cylinder and a point of detachment on the downstream side of the cylinder. This flow is sketched in figure 1.2a.
• Steady-Seperation: 5.Re≤Recrit. In this regime the flow is still steady, however a bauble of recirculation has appeared behind the cylinder. That is, a steady recircu-lation of fluid is confined to the backside of the cylinder. Here, the fluid has a point of attachment on the upstream side of the cylinder and two points of detachment and a point of attachment on the downstream side of the cylinder. This flow is sketched in figure 1.2b.
• Periodic Shedding: Re > Recrit. In this regime the flow is no longer steady but instead it exhibits a periodic behaviour. This periodic behaviour consists of a periodic shedding of vortices from the backside of the cylinder. The vortices travel downstream creating a so-called vortex train behind the cylinder. The position of the points of detachment on the downstream side of the cylinder fluctuate in time and the point of attachment no longer exists. This flow is sketched in figure 1.2c.
• Breakdown of the Two Dimensionality in the Flow: For the cylinder in free flow, Henderson [10] identifies Recrit2 ≈190 to be the Reynolds number at which the two-dimensionality of the flow breaks down. Beyond Recrit2 the flow exhibits three-dimensional behaviour and the 2D simulations are no longer enough to capture the physics completely.
(a) Sketch of streamlines for flow in the
attached flow regime. Re.5. (b) Sketch of streamlines for flow in the steady separation flow regime. Fluid is circulating in two closed baubles behind the cylinder. 5.Re≤Recrit.
(c) Sketch of periodic shedding where vortices are shed from the downstream side of the cylinder.
Re > Recrit
Figure 1.2: Sketches of different regimes of fluid flow in the Reynolds number range, Re∈]0,600[ for a cylinder in free flow.
There is no reason to expect that the different regimes should not exist when introducing a moving wall near the cylinder, although the Re values at which the flow transitions from one regime to another may be expected to vary. This variation has been investigated for the critical value for periodic shedding, when introducing and moving the wall closer to the cylinder.
An important note is that from the definition of a vortex used in this work, see equation (1.20), no vortices exist in the recirculating flow in the steady-separation regime. Vortices first appear at the transition from the stationary to periodic regimes. As the vortices are the objects of interest in this work the main regime of interest is the periodic shedding regime where a vortex train has appeared behind the cylinder.
Another note is that the range of Reynolds numbers investigated in this work goes beyond the breakdown of the two-dimensionality of the flow. This means that the presented data will not mirror a true three-dimensional flow exactly. However, it is estimated that the maximal investigated Reynolds number is close enough to the two-dimensional breakdown for results to still be a reasonable approximation to the behavior of the actual three-dimensional flow.