in drag coefficient
Consider a plate moving with constant speed U∞ in a viscous fluid that is sta-
tionary at infinity. The work required to move the plate against the drag D =< ∫plate−surfacedS(−n ⋅ up + νω × n ⋅ u) >, has to balance the dissipation rate of the
kinetic energy of the fluid, ∫ ρ(x)dV , where the volume integral is over all fluid- space (assumed infinite), the brackets< ⋅ ⋅ ⋅ > represent a time average and (x) is the time-averaged kinetic energy dissipation rate per unit mass, i.e. (x) = ∑3i=1∑3j=1ν < (∂ui/∂xj)2 >. In other words, if energy is being used to move the plate through the
2.5. A THEORETICAL ARGUMENT TO EXPLAIN THE CHANGE IN DRAG COEFFICIENT
fluid, it has to be dissipated somewhere. Note that in this balance, we have neglected a surface integral term representing energy dissipation rate by surface friction, as skin friction drag is negligible for blunt objects such as plates normal to the flow. Hence we see that:
DU∞= ∫ ρ(x)dV (2.6)
The vast majority of the kinetic energy dissipation resides inside the turbulent wake, specifically very close to the wake generator, and the volume of this wake, Vwake, can therefore be used to define an average dissipation by:
Vwake¯= ∫ (x)dV (2.7)
where the integral is over the volume of the wake. The dissipation rate coefficient is defined as:
¯
= C¯U∞3/` (2.8)
where ` is a characteristic length of the object that is creating the wake, defined as `=√A here. By combining equations 2.6 and 2.7 we then note that:
DU∞= ρVwake¯ (2.9) hence: DU∞ = ρVwakeC¯U 3 ∞ ` D U2 ∞ = ρVwakeC¯ ` 1 2CDρ` 2 = ρVwakeC¯ ` CD = CVC¯ (2.10)
frontal area of the plate and the characteristic length ` being defined as √A. A similar relationship can be found for a planar wake, where equation 2.10 becomes CD = CAC¯, with CA= Awake/A.
It is now apparent with (2.10) that the drag of a blunt object can be controlled by either changing the dissipation in the wake, or by keeping the dissipation constant but somehow changing the size of the wake or most likely a combination of the two. This also poses the question whether the volume of the wake can be changed without changing the area of the plate. The results in the following three chapters may suggest that it is possible to achieve this, thus manipulating the drag, by introducing a range of length-scales on the edge of the plate without changing the area.
Chapter 3
Turbulent wake properties of fractal
plates
3.1
Introduction
In the previous chapter, a theoretical argument was presented as a framework where to couch a possible explanation for the change in the drag coefficient of bluff bodies whereby the drag coefficient is a product of the volume of the wake and the average rate of turbulent dissipation within the wake. One of these values, the drag coeffi- cient, was measured directly with the use of a six axis load cell and so our attention now turns to obtaining measurements which may eventually lead to estimates of the volume of the wake and how best to approach the measurement of the dissipation within the wake. In this chapter, we focus on the turbulent properties within the wake that will hopefully lead to future researchers having a better idea of how best to measure the dissipation in, and the volume of, the wake.
Compared to the numerous studies that have investigated the turbulent wake properties of disks, our understanding of the square plate wake is comparatively sparse. Fail et al. [1957] is one of the few studies to look at any form of wake characteristics, however even their study focused mainly on the mean flow charac- teristics. In a similar manner, the vortex shedding and the drag coefficient of square plates have been well documented in the past, however there is a gap in our knowl-
edge of how turbulence parameters, such as turbulence intensity, kinetic energy and Reynolds stresses, vary across the wake. As the square plate is used as our main comparison case, it was necessary to gain this understanding first and so in order to proceed, we must first document these properties for both the fractal and square plates and comment on any differences that arise.
As well as understanding the distribution of turbulence characteristics across the wake, another key question that needs to be addressed is whether or not an axisymmetric wake can only be generated, as Pope [2000] puts it in his text book, by “round bodies”. One of the parameters that controls the rate of spreading of the wake are the Reynolds stresses < u′iu′j >. Townsend [1956] argues that in a non- axisymmetric wake, there would be an imbalance of Reynolds stresses with the values being highest where the wake is at its narrowest owing to the large velocity gradients that are present. The larger Reynolds stresses would create a faster spreading rate than in other parts of the wake causing the wake to expand. This would imply that at some point downstream the wake must reach a state of equilibrium where the distribution of Reynolds stresses are equal throughout the wake, resulting in an axisymmetric wake. Hence, it is entirely possible that a non-round body could create an axisymmetric wake.