Drag and near wake characteristics of flat plates normal to the flow with fractal edge geometries

Since the first basic flow visualizations of the wake structure behind a circular disc performed by Marshall and Stanton (1931), researchers have focused their attention on understanding this flow in more detail. To the best of our knowledge, Fail et al (1957) obtained some of the earliest measurements of the drag and shedding frequencies of circular discs as well as other plates of similar frontal area. For the circular disc, they found that the drag coefficient was 1.12, while the Strouhal number, St, based on the square root of the frontal area, was found to be 0.115. Carmody (1964) took hot-wire and pressure measurements behind a circular disc showing that 95% of the total transfer of energy from the mean motion to the turbulent motion takes place very close to the plate, usually within the first three plate diameters. Carmody (1964) also offered a C_D of 1.14.

So far, the characteristics of the plates had only been considered with a laminar freestream. However, in 1971 Bearman (1971) began looking at the effect that a turbulent freestream has on the drag force on plates. Bearman (1971) studied square and circular discs of different sizes and found the base pressure coefficient, (C_P )_b, to be −0.363, and that it was the same for both types of plates. With the addition of a turbulent freestream, the base pressure decreased and with it, the drag coefficient increased. The 8 inch square plate, for example, had a C_D of 1.152 with a laminar freestream, but this then increased with a turbulent freestream, typically by a few per cent.

The influence of a turbulent freestream on the drag of plates was further studied by Humphries and Vincent (1976a, 1976b), who found that the drag of a circular disc could be increased up to 1.28 as the turbulent intensity and integral length scale in the turbulent freestream are increased. These studies as well as that by Bearman (1971) suggested that the reason for the increase in drag was due to a delay of the shear layer's transition to turbulence, which would keep the base pressure low and hence increase the drag.

Some further progress in understanding the wake structure was made by Berger et al (1990), whose experiments focused on the near wake at relatively low Reynolds numbers and revealed various regimes with various associated shedding frequencies. Further studies by Lee and Bearman (1992), Miau et al (1997), Kiya et al (2001) and Shenoy and Kleinstreuer (2008) have added to the understanding of the wake of a circular disc; however, as stated by Kiya et al (2001): 'the wake of a square plate normal to the flow is not understood yet'.

Although the study of regular polygon plates has been dominated by circular discs as well as square plates, a few other shapes have also been considered. Fail et al (1957), besides studying these two plates, also studied an equilateral triangle plate and a tabbed plate—all four plates used by Fail et al (1957) are shown, scaled, in figures 4(k)–(n). What was rather surprising was that all these plates had similar drag coefficients between 1.12 and 1.15 and Strouhal numbers between 0.115 and 0.117 when the incoming flow was laminar. A similar C_D value of 1.18 was found by Humphries and Vincent (1976b) for the triangular plate with a laminar freestream.

Elliptical and rectangular plates were studied by Kiya and Abe (1999) and again by Kiya et al (2001), where the main finding was that the wake had the same shape as the plate itself; however, the alignment is rotated by 90° with the major axis of the plate aligned with the minor axis of the wake. Two distinct shedding frequencies are also observed, one for each axis; however, the Strouhal number is found to decrease with aspect ratio of the plate, from roughly 0.12 where the aspect ratio is 1, to roughly 0.06 for the major axis and 0.11 for the minor axis of an elliptical plate with an aspect ratio of 3.

From the literature, it is clear that both the drag coefficient and the Strouhal number of circular discs and regular polygon plates appear to fall within a very narrow band of values, with the difference between their C_D values being, at most, not more than 5%. Even the tabbed plate, which could have three defined length scales: one associated with the inner diameter, one with the outer diameter, i.e. including the tabs, and the tab itself, has a similar C_D and St. This plate is of particular interest, as one would expect the tabs to somehow interact with the shear layer that may change its transition point and with it the drag coefficient, as suggested by the arguments of Bearman (1971) and Humphries and Vincent (1976a, 1976b). This, however, is not the case, as Fail et al (1957) show that the circular plate and the tabbed plate have the same value of C_D = 1.12. The question then becomes: if more length scales are introduced, could that cause a change that would create a larger drag coefficient?

If an energy balance is done between the work required to move a plate through a fluid in the direction normal to its plane, and the dissipation of the turbulent kinetic energy that is produced in the fluid (see the appendix), then the following expression can be obtained:

$$\begin{equation} C_D = C_V C_{\bar{\epsilon}}, \end{equation} \tag{ 1 }$$

where the wake volume coefficient is defined as C_V  = V_wake/2A^1.5, V_wake being the volume filled by the wake, A is the frontal area of the plate and $C_{\bar {\epsilon }}$ is the coefficient of the average rate of dissipation of turbulent kinetic energy in the volume of the wake, as defined by

$$\begin{equation} \bar{\epsilon} = (C_{\bar{\epsilon}}{U_{\infty}^3})/{\sqrt{A}}. \end{equation} \tag{ 2 }$$

It is now apparent with (1) that the drag of a blunt object can be controlled either by changing the dissipation in the wake or by keeping constant the dissipation but somehow changing the size of the wake or even a combination of the two. This also poses the question of whether the volume of the wake can be changed without changing the area of the plate. Here we show 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. The purpose of this study is to investigate these phenomena. We use fractal geometries to introduce a variety of length scales to the edge of a square plate. In doing so, we follow recent works by, e.g., Seoud and Vassilicos (2007), Kang et al (2011) and Laizet and Vassilicos (2012), amongst others (see an exhaustive list of references in Gomes-Fernandes et al (2012)). In particular, Kang et al (2011) measured drag forces of Sierpinski carpets and triangles, similar to the shapes used by Abou El-Azm Aly et al (2010) and Nicolleau et al (2011), who investigated the pressure drop across fractal-shaped orifices. Laizet and Vassilicos (2012) showed that the pressure drop across a space-filling (D_f = 2) fractal square grid is much lower than that across a regular grid with the same blockage and the same effective mesh size.

The fractal dimension, D_f, is defined mathematically by equation (3) (see Mandelbrot (1983)) and measures how rough or smooth an object is. For a line in a 2D plane, a value of 1 would mean a smooth straight line, while any value greater than 1 would mean a rough shape, the maximum value of D_f being 2. The parameters defining the fractal edges of our plates are

$$\begin{equation} D_{\rm f} = {\log({d^n}) \over \log({{\lambda}\over{l_n}})}, \end{equation} \tag{ 3 }$$

$$\begin{equation} l_n = \lambda / r^n, \end{equation} \tag{ 4 }$$

$$\begin{equation} r = 4(\cos \alpha - 1) + d, \end{equation} \tag{ 5 }$$

$$\begin{equation} P_n = Sd^nl_n, \end{equation} \tag{ 6 }$$

where r is the ratio between successive lengths, P_n is the perimeter of the plate at iteration n and S is the number of sides on the base shape on which the fractal edges are applied (S = 4 for a base square plate). A simple way to understand the fractal shape of the edge is illustrated in figure 3. The base straight line of length λ is replaced by a number, d, of segments of length l₁ such that the area under the line remains the same. This is iteration n = 1. The second iteration n = 2 consists of replacing each individual segment of length l₁ by the same pattern of d segments, but smaller, i.e. the segments now have a length l₂ = l₁/r. And so on for as many iterations as required. A total of eight 'fractal plates' were manufactured, as well as a square and a cross plate. The fractal plates are based on a square plate of 256 mm × 256 mm × 5 mm size, 5 mm being the thickness of the plate, and were designed such that the area remained constant, achieved by a simple 'add on and take away' method as in the example of figure 3.

Zoom In Zoom Out Reset image size

Two patterns were used to generate the fractal plates, a square (α = 90°, d = 8 and D_f = 1.5) and a triangle pattern (α = 45°, d = 4 and D_f = 1.3), with the square pattern shown in figure 3. Consecutive iterations were obtained by applying the same pattern to each new length scale generated, repeated up to four times, giving four plates for each pattern as can be seen in figure 4. It can also be seen from this figure that the fractal plates no longer resemble a square, but more of a cross shape, hence a cross plate was also manufactured. The same figure also has scaled drawings of the plates used by Fail et al (1957).

Zoom In Zoom Out Reset image size

Although all the plates have the same frontal area, the perimeter (P_n) varied as can be seen in table 1, where the perimeter can be anything up to 16 times longer than the original square plate. It is by applying a fractal pattern that we are able to introduce a larger perimeter and different length scales into the flow without changing the area of the plate.

Figure 5 shows the drag coefficient for the plates, collected in the DC tunnel using both types of sensors, where we can deduce that the change in drag coefficient is not down to the resolution and accuracy of the iml sensor, within acceptable experimental error. Note that the maximum difference between the largest and the smallest drag force measurement is 1.57 N based on a free-stream velocity of 20 m s⁻¹, which is 250 times larger than the resolution of the ATI sensor, which has a resolution of 1/160 N. The largest discrepancy between the two sensors is obtained for the D_f = 1.5(4) plate for which we have no definite explanation for the time being; however, despite this discrepancy between the two sensors, conclusions can already be made from this one figure. Firstly, there is a clear effect of iteration within each fractal dimension, which is seen in figure 5 as an increase with increasing perimeter length, and there is an effect of fractal dimension itself. For the D_f = 1.3 family of plates, it is only at the last iteration that we see a noticeable increase in C_D compared to the square plate of roughly 3.5%, which is similar to the maximum difference between all the plates in figure 1. As the fractal dimension is increased, a more pronounced change in C_D is observed, with the D_f = 1.5(2) plate showing the biggest increase in C_D, roughly 7%. This particular plate can also be compared with the D_f = 1.3(4) plate as both sets of plates have the same perimeter, and this shows that by increasing the fractal dimension, one can increase the drag of the plate irrespective of the perimeter length and frontal area, implying that the fractal dimension is a controlling factor in the drag of these plates. The same can be said of the D_f = 1.3(2) and the D_f = 1.5(1) plates, which also have the same perimeter—see table 1. Special note must also be taken of the cross plate in figure 5, which has the same drag coefficient as the square plate, adding to the evidence that the drag coefficient of non-fractal plates does fall within a narrow band of values.

Zoom In Zoom Out Reset image size

Of particular interest in figure 5 is the last iteration of the D_f = 1.5 family of plates, where a sharp decline in C_D is observed. Hence, two issues must be addressed with respect to the D_f = 1.5 family of plates: the initial increase in C_D with fractal iteration, and the sudden drop of C_D as the iteration increases past a certain value.

Equation (1) shows that the coefficient of drag is independent of Reynolds number, $Re = U_{\infty }\sqrt {A} / \nu$, as Re → ∞ if in this limit $C_{\bar {\epsilon }}$ and C_V are both independent of Reynolds number or if they both depend on Reynolds number in a way that cancels out when one takes the product $C_{\bar {\epsilon }}C_V$. To check that C_D is indeed independent of Re in our measurements, the plates were subjected to a range of velocities in the DC tunnel, ranging between 5 m s⁻¹ ⩽ U_∞ ⩽ 20 m s⁻¹. A selection of the results can be seen in figure 6, although it should be noted that all the plates exhibited a Reynolds number independence for Re > 1.5 × 10⁵. The fact that the plates show a consistent trend in the drag coefficient across a range of Reynolds numbers strengthens the support for our conclusions in section 3.1. Even at the lowest Reynolds number the difference between the largest and the smallest force is 16 times larger than the resolution of the ATI sensor, implying that the changes in drag coefficient that we observed are real effects, particularly at higher Reynolds numbers.

Zoom In Zoom Out Reset image size

Using a Pitot-static probe, the axial pressure, i.e. along the x-direction on the centre line crossing the centre of the plate, was collected for all the plates from $x = \sqrt A$ to $20\sqrt A$ in the Honda tunnel. Our results, along with those of Carmody (1964) for the circular disc, are shown in figure 7, where the coefficient of pressure is defined as C_P  = P/(0.5ρU²_∞), with P being the dynamic pressure (i.e. total pressure minus static pressure) of the Pitot-static tube in the wake of the plates. Based on these results, it would appear that there is little difference between the square and the cross plate and that even the D_f = 1.3(4) has similar pressure coefficient values. The D_f = 1.5(4), however, shows a smaller pressure drop close to the plate and a very slow rate of pressure recovery; even at a distance of $20\sqrt A$ the wake has not recovered to the same levels as the other plates. Although only the fourth iteration of both fractal dimension families are shown, it should be noted that there is a decreasing pressure drop behind the plate for increasing fractal iteration and that further downstream there is a faster rate of pressure recovery as fractal iteration is increased. These results also suggest, judging from the position where C_P  = 0, that the length of the re-circulating region is roughly similar for all plates; however, no conclusions can be made about the overall volume of the bubble itself.

Zoom In Zoom Out Reset image size

The axial evolution of the turbulence intensity, T_i = u'/U_∞ where u' is the fluctuating velocity, is shown in figure 8 for the square, cross and the last iteration of both fractal plates where, as before, there is no noticeable difference between the T_i values for the cross and square plates. Here we observe that at the closest position the fractal plates have lower levels of turbulence intensity compared to the square and cross plates; however after $x \geqslant 6\sqrt {A}$, there is a noticeable change where the fractal plates actually have higher levels of turbulence intensity, and at a distance of $x = 20\sqrt A$, the D_f = 1.5(4) has the highest level of T_i of all the plates.

Zoom In Zoom Out Reset image size

As in figure 7, figure 8 shows the fourth iteration of both fractal dimension families, which had the lowest turbulence intensity close to the plate and the highest further downstream compared to the other fractal iterations, i.e. the rate of decay decreased as the fractal dimension and iteration increased.

It is clear that although one could use the axial measurements for direct comparison from one plate to another, the same cannot be said of a single traverse through the wake, as is often the practice with axisymmetric plates. Although the square and cross plates both have reflection and rotation symmetry, they are not axisymmetric and so one must take planar measurements to fully understand the development of the wake. This point is even more critical for the fractal plates which only have rotation symmetry (by 90°). Planar measurements of the pressure coefficient are taken in the DC tunnel.

In equation (1), the wake volume coefficient as well as the dissipation coefficient are defined over all fluid space which is assumed to be infinite; however, in practice, it is not possible to measure these values over an infinite domain of the wake. The volume coefficient in equation (1) can, however, be decomposed into the sum of 2D normal slices through the wake over several downstream locations; that is to say, $C_V = A^{-1.5}\int ^{\infty }_0 C_A \,{\mathrm { d}}x$, where C_A is now defined as

$$\begin{equation} C_A = \frac{A_{\rm wake}}{A}, \end{equation} \tag{ 7 }$$

where A_wake is the planar, cross-section area of the wake and is the only variable that is a function of x. This area can be approached experimentally in a number of ways; however, for this study, we shall define a length based on a certain fraction of the velocity deficit, U_d in the wake, where the velocity deficit is defined as U_d = U_∞ − 〈U(x,y,z)〉 and the corresponding length is the point where the deficit velocity is some fraction, ζ, of the centre-line velocity deficit; that is to say, U_d(x,±y_ζ,±z_ζ) = ζU_d(x,0,0) (U is the streamwise velocity component of the fluid velocity and the brackets 〈··· 〉 signify an average over time). As we are collecting pressure from the rake measurements, this can also be expressed as

$$\begin{equation} 1 - \sqrt{C_P(x,\pm y_{\zeta},\pm z_{\zeta})} = \zeta(1 - \sqrt{C_P(x,0,0)}). \end{equation} \tag{ 8 }$$

The value of the width was found by fitting a sixth order polynomial to the data and finding the point where the velocity deficit was a certain fraction, ζ, of the centre-line deficit velocity. Note that y_ζ and z_ζ are the vertical and spanwise distances from the centre line (see figure 2 for definition), where the pressure coefficient is a certain fraction ζ of the centre-line pressure coefficient. Finally, to obtain an estimate of the wake area, the area in plots such as those in figure 9 was integrated from the origin to newly created data points identified as the white circles in that figure.

Zoom In Zoom Out Reset image size

As already mentioned in section 2, the spacing between the Pitot probes on the rake was set to 50 mm; however, to determine whether this spacing was adequate to obtain the area of the wake a preliminary test was done. At a distance of $x=6\sqrt {A}$ the rake was traversed by 10 mm in the z-direction (see figure 2 for definition of the coordinates) thus obtaining 40 data points which were 10 mm apart. The width of the wake was found in the same way as described above for both sets of data, i.e. one with 10 data points and another with 40 data points, where it was found that both sets of data gave similar results. It is believed, however, that the accuracy of this method would be lower closer to the plate where the wake is much smaller and hence there are fewer data points available for the polynomial fit. Similarly, this method does not work when there are negative C_P values, which is why we show data from $x=4\sqrt {A}$ onwards.

In figure 9, we show the planar pressure coefficient measurements taken in one quarter of the wake for a selection of plates, where the corresponding coordinates of the detected wake edge can also be seen. These boundaries are based on a ζ value of 0.5, i.e. the point where the pressure reaches 50% of the centre-line pressure. From a quick inspection, it appears that the wake of the D_f = 1.5(4) is the smallest of all the plates—see figure 9(c), and that, just as was noted in figure 7, it has a very low pressure coefficient in the middle of the wake. We have, for a simple comparison, also plotted in figure 7 the centre-line evolution of the pressure coefficient of the square plate taken in the DC tunnel where good agreement between the two tunnels is observed.

As mentioned earlier a number of values for ζ were looked at, and it was observed that the trend of C_A from one plate to another was always consistent irrespective of the value of ζ, as can be seen in figure 10. This does suggest that most of the information in terms of getting a good value for C_A can actually be obtained from looking at the point where the pressure reaches 50% of the centre-line pressure. From these results, the impression seems to be that no matter what value of ζ is chosen, the area of wake for the square plate will be the largest, followed by the D_f = 1.3 family of plates, although only the last iteration is shown here for clarity. The smallest wake can be seen for the D_f = 1.5 family of plates, but it must be pointed out that these results are for one downstream location at $x = 8\sqrt {A}$.

Zoom In Zoom Out Reset image size

If we now look at figure 11, where the growth of the wake is shown, we see a clear pattern emerging between the fractal dimensions at $x/\sqrt {A} \geqslant 6$, with all of the D_f = 1.3 family of plates having a smaller coefficient of area for all positions compared to the square plate, but a larger C_A compared to the D_f = 1.5 fractal plates. This fact is particularly interesting, because if we use the drag coefficients from the DC tunnel, where the planar measurements are taken, then we note that the square, the first three iterations of the D_f = 1.3 plates and the last iteration of the D_f = 1.5 plates all have similar drag coefficients, but the pressure in the wake and the size of the wake are very different in each case. This means that there must be, as predicted in section 1.3, another effect causing the drag to stay the same even though the areas are different.

Zoom In Zoom Out Reset image size

Figure 11 gives an indication that the area coefficient changes with iteration, particularly for the D_f = 1.3 fractal plates where C_A drops for consecutive iterations before increasing again for the last iteration, particularly at the largest value of $x/\sqrt {A}$.

Consider a plate moving with constant speed U_∞ in a viscous fluid that is stationary at infinity. The work performed in moving the plate against the drag, $D = \langle \int _{\mathrm { plate-surface}} \,\mathrm {d}S(-\mathbf {n}\cdot \mathbf {u}p+\nu \omega \times \mathbf {n}\cdot \mathbf {u}) \rangle$, balances the dissipation rate of the kinetic energy of the fluid, $\int \rho \epsilon (\mathbf {x})\,\mathrm {d}V$, 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. $\epsilon (\mathbf {x}) = \sum ^{3}_{i=1}\sum ^{3}_{j=1}\nu \langle ({\partial u_i}/{\partial x_j})^2\rangle$. In this balance, we have neglected a surface integral term representing the energy dissipation rate by surface friction, as skin friction drag is negligible for blunt objects such as plates normal to the flow.

The vast majority of the kinetic energy dissipation resides inside the turbulent wake and the volume, V_wake, of this wake can therefore be used to define an average dissipation by $V_{\mathrm { wake}}{\bar {\epsilon }} = \int {\epsilon (\mathbf {x})}\, \mathrm {d}V$, where the integral is over the volume of the wake. We define the dissipation rate coefficient $C_{\bar {\epsilon }}$ as $\bar {\epsilon } = C_{\bar {\epsilon }}{U_{\infty }^3}/{\ell }$, where ℓ is a characteristic length of the object that is creating the wake, defined as $\ell = \sqrt {A}$ for our study. The balance between the work performed in moving the plate, which is DU_∞, and the dissipation rate $\int \rho \epsilon (\mathbf {x})\,\mathrm {d}V$ leads to

$$\begin{equation} C_D = C_V C_{\bar{\epsilon}}, \end{equation} \tag{ A.1 }$$

where the wake volume coefficient is defined as C_V  = V_wake/2A^1.5 with A being the frontal area of the plate and the characteristic length ℓ being defined as $\sqrt A$. A similar relationship can be found for a planar wake, where equation (1) becomes $C_D = C_A C_{\bar {\epsilon }}$, with C_A = A_wake/A.