Interaction of a Cylindrical Roughness Element and a Two-Dimensional TS-Wave

Abstract

Roughness induced transition in the leading edge region of airfoils is one of the challenges in the design of highly efficient wind turbine blades especially with respect to various inflow turbulence levels. The present experimental investigation discusses, therefore, the interaction of a laminar boundary layer perturbed by a single, 2D disturbance mode with a cylindrical roughness element at medium height. High and low speed streaks in the roughness wake suggest the presence of a counter-rotating vortex pair. A significantly increased amplification of the disturbance mode downstream of the roughness in comparison to the zero roughness height case leads to the formation of 3D structures with a dominant spanwise length scale corresponding to the roughness diameter. However, these 3D structures are only very weakly distinctive far downstream.

1 Introduction

Boundary layer transition in the leading edge region of airfoils is usually inherent with a significant increase in aerodynamic drag and decrease in lift followed by a performance loss. For the design of highly efficient airfoils and wind turbine blades a detailed understanding of roughness induced transition especially with respect to different inflow turbulence levels is required, since the airfoil leading edge can easily be contaminated by insects or be prone to surface corrosion. In general, roughness elements are distinguished based on their shape and on their height. Depending on the roughness shape 2D roughness steps, 3D isolated roughness elements and distributed, 3D roughness elements can be separated [9].

Based on the height of the roughness element, a distinction into three different ranges is suggestive, since flow phenomena leading to transition downstream of the roughness alter significantly not only depending on the shape but also with increasing roughness height in relation to the boundary layer displacement thickness: For low roughness heights (i), base flow velocity variations caused by the roughness element are small. Nevertheless, such small-scale velocity variations as caused by small roughness elements can result in an energy transfer from free stream disturbances to boundary layer instability modes (Tollmien Schlichting (TS) waves) via receptivity as first shown in the experiments by Aizin and Polyakov [2] and also by Goldstein [6] using the triple deck analysis. Since the receptivity process is very sensitive to the frequency of the disturbance and the height of the roughness a linear relationship in-between roughness height and TS-wave amplitude is only present below a certain threshold for the roughness height [13].

For roughness heights in a medium range (ii), a distinct mean flow velocity distortion downstream of the roughness element results in a considerable change of the boundary layer stability characteristics. Moreover, the receptivity process at the roughness element reaches a non-linear stage [13]. Downstream of the roughness element the excited disturbance modes may undergo transient growth [5], which is, however, strongly dependent on the receptivity process at the roughness element. For 2D roughness elements the downstream evolving disturbances are at first 2D, before the flow develops a 3D nature in the process of transition [8]. In contrast, 3D roughness elements highly favor the development of different vortical structures (horse-shoe shaped vortex wrapped around the front side of the roughness and a trailing vortex rising vertically in the wake of the roughness) directly at the (cylindrical) roughness [10]. The particular vortical structures are, however, highly sensitive to the roughness shape.

For roughness heights close to or larger than the displacement thickness (iii), a strong mean flow distortion causes highly nonlinear effects. Hence, transition occurs close to the roughness element and is also referred to as bypass transition. Detailed investigations were conducted by Acarlar and Smith [1] and Klebanoff et al. [7] showing that different vortical structures appeared to lead to turbulence in the inner (close to the surface) and outer region of the boundary layer, which is hinted at by two inflection points in the mean flow velocity profile. Again, the vortical structures strongly depended on the particular roughness shape.

Within this context, the present investigation focuses on the interaction of a laminar boundary layer with a cylindrical roughness element located in stage (ii). The experiments are conducted in an airfoil leading edge region at a Reynolds number of $Re=6 \cdot 10^6$ to achieve a close alignment to operational conditions of wind turbines. Moreover, the model is equipped with a disturbance source upstream of the roughness element giving the possibility to model different inflow turbulence levels. In the present study the laminar boundary layer is, therefore in a first step, perturbed by a single, 2D TS-wave upstream of the roughness.

2 Experimental Setup and Data Acquisition

The experiments were conducted in the Laminar Wind Tunnel of the IAG, which is of Eiffel type. Due to several screens in the inlet section and an effective contraction ratio of 20:1 a very low longitudinal turbulence level of $Tu_x =0.02\,\%$ ($f=10-5000~ \mathrm {Hz}$, $u_{\infty }=30~\mathrm {m/s}$) is obtained. For the experiments a particularly designed airfoil section (BE72 [11]) was employed having a similar pressure gradient in the leading edge region as the NACA $64_3-418$ airfoil, which can be regarded a typical airfoil for wind turbine applications. However, the pressure distribution in the trailing edge section of the BE72 was significantly modified in order to reduce the model chord length ($c = 2.4~\mathrm {m}$). Therewith, experiments can be conducted at a typical Reynolds number for wind turbine applications ($Re = 6 \cdot 10^6$) and at low free stream velocity ($u_{\infty } = 20~\mathrm {m/s})$. The pressure gradient at the roughness position was close to zero corresponding to an angle of attack of $\alpha = 4.4^{\circ }$.

A cylindrical roughness element (diameter $d_r = 20~\mathrm {mm}$) is integrated into the model at a streamwise location of $x/c = 0.15$. Using a linear actuator with a minimal incremental motion smaller than $0.1 \upmu {\text{ m }}$ the roughness can be displaced in a range of $0<\delta _{1,ref}<1.5$. Additionally, the roughness height is measured with a laser-triangulation system with a resolution of $0.15~\mu \mathrm m$ corresponding to 0.2 % of the boundary layer displacement thickness. For the excitation of the disturbance modes upstream of the roughness element a very thin slit is integrated into the model surface at $x/c=0.1$ and via tubes connected to 128 individually controllable micro-loudspeakers. Employing a 128 channel signal generator [3] with a common quartz based clock, disturbance modes in a wide streamwise and spanwise wave number range can be excited.

For the hot-wire measurements the same clock as for the excitation of the disturbance modes is used enabling phase-locked measurements. The hot-wire signal is split into AC and DC part. The AC part is high-pass filtered and amplified. AC and DC part of the signal are acquired with a 18-Bit AD converter (NI USB-6289). For the spectral analysis the total of 32,768 samples for each point is split into eight blocks and ensemble-averaged in the temporal domain prior to a transformation into the spectral domain via Fast Fourier transform (FFT). For the evaluation of the spanwise wavenumber spectra via Complex FFT points are interpolated to an equidistant grid in a range $-7.5 < z/d_r < 7.5$. The spanwise wavenumber $\beta$ is non-dimensionalized by the roughness diameter, without considering $2\pi$. Hence, $\beta d_r =1$ corresponds to a wavelength equal to $d_r$. For further details on the experimental setup compare Plogmann et al. [12].

3 Results

3.1 Base Flow Characteristics

Experiments are conducted in a close to zero pressure gradient boundary layer as can be seen from the velocity at the boundary layer edge being almost constant (Fig. 1a). Hence, the shape factor is constant and close to the value of the Blasius boundary layer ($H_{12}=2.591$). The displacement thickness at the position of the roughness element is $\delta _{1,ref} = 0.72$. Both for the displacement thickness ($\delta _1$) and the shape factor ($H_{12}$) a good agreement in-between measured and calculated values is obvious. For further details on the base flow see [11].

Fig. 1

Comparison of measured (symbol) and calculated (line) base flow characteristics (a) and measured (symbol) and calculated (line, LST) TS-wave characteristics of mode $f_1 = 549\, \mathrm {Hz}$ (b) for $h/\delta _{1,ref}=0$

At the roughness position boundary layer disturbances are convectively amplified in a range $270<f<690~\mathrm {Hz}$ according to linear theory. In the present experiments a single, 2D TS-wave with frequency $f_1 = 549~\mathrm {Hz}$, which is in the range of the most amplified disturbances at the roughness position, was excited upstream of the roughness. Figure 1b shows the streamwise development of the TS-wave amplitude and phase for zero-roughness height in comparison to linear theory. Only minor variations in the spanwise distribution of the TS-wave amplitude and phase reveal a good two-dimensionality of the excited mode downstream of the roughness element for zero roughness height.

3.2 Mean Flow Distortion

The following results focus on the influence of the roughness element at $h/\delta _{1,ref}=0.5$ on the laminar boundary layer, which is perturbed by the single, 2D TS-wave mode $f_1$. At first, Fig. 2 shows the mean flow velocity distortion caused by the roughness element. Clearly, in the near wake of the roughness element the mean flow velocity is considerably reduced with the highest velocity distortion being present in the centerline region at a wall normal distance corresponding to the roughness height. Further downstream the mean flow distortion in the centerline region is considerably decreasing and the maximum is shifted off the wall to approximately $y/\delta _{1,ref}=1$. Therefore, at $s/d_r=7$ the undisturbed boundary layer profile is recovered on the centerline, as can be deduced from a mean flow distortion being in the order of 1–2 %.

Moreover, regions of increased or decreased velocity, so-called streaks, form out of center downstream of the roughness. Two high speed streaks are present at spanwise positions of $z/d_r = \pm 0.5$ corresponding to the edges of the roughness. The amplitude of the low speed streaks, which are located on the outer edges of the high speed streaks, respectively, is considerably lower compared to the amplitude of the high speed streaks. In the streamwise development the wall normal maximum of these streaks is shifted off the wall in compliance with the observations made in the centerline region. Starting at $s/d_r =4.75$ a considerable amplitude decrease of the high speed streaks can be observed. Only small scattering in the spanwise shape factor distribution at $s/d_r = 13.75$ shows that the undisturbed mean flow velocity profile ($H_{12}=2.59$) is nearly recovered in the entire spanwise domain, in which the mean flow was affected by the roughness.

Fig. 2

Mean flow distortion and shape factor (black line) for $h/\delta _{1,ref}=0.5$, (a) streamwise development downstream of the roughness element and (b) at $s/d_r=3.75$

In agreement with observations by Wang [14], the observed streaks persist at a constant spanwise scale in the streamwise evolution. From these high and low speed streaks the presence of two counter-rotating vortex pairs in the wake of the roughness can be inferred as indicated in Fig. 2b and also described by Legendre and Werle [10]. Hence, the trailing vortex pair shifts low momentum fluid away from the wall in the center (lift-up effect), while both vortex pairs (trailing and horse shoe vortex) contribute to the formation of the high speed streaks on the edges of the roughness, which entrain high-momentum fluid towards the wall. Finally, the low speed streaks on the outer edges of the high speed streaks are a consequence of the horse-shoe vortex pair. Clearly, low speed streaks result in an increase of the shape factor while the shape factor is considerably reduced in regions with high speed streaks.

3.3 Disturbance Roughness Interaction

Below, the influence of the cylindrical roughness element on the initially 2D disturbance mode with a frequency of $f_1=549~\mathrm {Hz}$ and a maximum wall-normal amplitude of $A/u_{\infty }= 0.01\,\%$ at the position of the roughness element is considered. Directly downstream of the roughness element a strong amplification at the edges of the roughness element leads to the formation of two peaks in the spanwise amplitude distribution with a wall normal maximum at $y/\delta _{1,ref} = 0.9$ (Fig. 3a). In the centerline region, however, the amplitude is considerably damped due to the significantly decreased mean flow velocity immediately downstream of the roughness element (Fig. 2). Hence, the interaction of the roughness element and the 2D disturbance mode results, at first, in the excitation of 3D modes in a wide spanwise wave number range predominately in the inner region of the boundary layer (compare Fig. 4a). The amplitude of these 3D modes is, however, considerably lower compared to the 2D mode.

Fig. 3

Streamwise amplitude development of the disturbance mode $f_1$ and shape factor (black line) (a) and amplitude distribution of $f_1$ in the wall-normal spanwise plane at $s/d_r=6.75$ (b)

In the downstream development the two amplitude peaks at the edges of the roughness element move towards the center before merging to a common peak at $s/d_r=2.75$ (Fig. 3a). At the same time local minima develop at spanwise positions of $z/d_r = \pm 0.5$ corresponding to the edges of the roughness element. From the comparison to the spanwise distribution of the shape factor it is obvious that these local minima in the spanwise amplitude distribution coincide with the high speed streaks caused by the two vortex pairs in the wake of the roughness element. Therefore, the most significant influence of the roughness element on the amplification of the excited mode $f_1$ is, at first, restricted to the spanwise extent of the roughness element. The strong amplification in the centerline region and the influence of the high speed streaks at the edges of the roughness element lead to the formation of a dominant 3D mode ($\beta d_r = 1$) out of the broadband nature of the 3D structures immediately downstream of the roughness element (Fig. 4). This mode ($\beta d_r = 1$) develops in the inner region of the boundary layer ($y/\delta _{1,ref}<2$) with its maximum located near $y/\delta _{1,ref} = 0.8$ and is considerably amplified in streamwise direction. The spanwise length scale corresponds to the roughness diameter and can, therewith, clearly be linked to the interaction with the roughness element. In the outer region of the boundary layer only 3D modes of broadband character without the presence of dominant peaks can be observed.

Fig. 4

Spanwise wavenumber spectra of the disturbance mode $f_1$ in the wall-normal spanwise plane at $s/d_r = 0.75, 1.75, 2.75, 3.75, 4.75, 5.75, 6.75, 8.75, 13.75$ (a–i)

Only further downstream ($s/d_r=4$) the minima at $z/d_r = \pm 0.5$ start to move outwards while a wavy type structure develops in-between these two minima (Fig. 3b). The increasing spanwise extent, in which the mode $f_1$ is affected by the interaction with the roughness element coincides with the amplitude decrease of the high speed streaks (compare also Fig. 2). Moreover, the amplitude of the mode $f_1$ reaches a maximum close to $s/d_r=5$ before being slightly damped as can also be deduced from the amplification on the centerline (Fig. 5). In contrast, the (undisturbed) mode $f_1$ is further growing in regions, which are not affected by the interaction with the roughness element. Hence, the 3D structures in spanwise direction are considerably damped as can also be seen in Fig. 3a. This observation is consistent with a strong damping of the most significant 3D mode ($\beta d_r = 1$) observed in the spanwise wavenumber spectra (see Fig. 4g–i). At $s/d_r=13.75$ only minor scattering in the spanwise wavenumber spectra is present showing that no significant 3D modes, especially with respect to the roughness element, persist far downstream. In agreement, De Paula et al. [4] found that 3D modes excited by a cylindrical roughness element are damped for low disturbance amplitudes at the roughness element. However, in comparison to present investigation ($A/u_{\infty }= 0.01$) the disturbance mode amplitude at the roughness position was considerably higher ($A/u_{\infty }= 0.45$).

Fig. 5

Amplification of mode $f_1$ and $2 f_1$ on roughness. centerline for $h/\delta _{1,ref} = 0$ and $h/\delta _{1,ref} = 0.5$

Finally Fig. 5 shows that the strong amplification of the fundamental mode $f_1$ downstream of the roughness element is connected with a transfer of energy into the harmonic mode $2 f_1$. Comparing the amplification of the fundamental and the harmonic mode shows that both modes experience a very similar growth behavior. However, the total amplitude of the harmonic mode is considerably lower as can also be seen from the comparison of Fig. 6a, b. Clearly, comparing the amplitude of the fundamental and the harmonic mode in the wall-normal spanwise plane supports the argument that the highest energy transfer from the fundamental to the harmonic mode is present in regions of maximum amplitude for the fundamental mode.

Fig. 6

Comparison of amplitude distribution for mode $f_1$ (a) and $2 f_1$ (b) in the wall-normal spanwise plane at a streamwise position of $s/d_r = 4.75$

4 Conclusion

The interaction of a laminar boundary layer perturbed by a 2D, TS-mode with a cylindrical roughness at $h/ \delta _{1,ref} =0.5$ was investigated. High and low speed streaks in the wake of the roughness reveal the presence of two counter-rotating vortex pairs. In the downstream development the streak amplitude is, at first, nearly constant before decreasing significantly. Far downstream the undisturbed boundary layer profile is, therefore, recovered in the entire spanwise domain.

The interaction of the 2D, low amplitude TS-mode with the cylindrical roughness element results, at first, in a strong amplification in the centerline region, which is accompanied by an energy transfer from the excited fundamental into the first harmonic mode. The high speed streaks at the edges of the roughness lead to the formation of spanwise minima in the disturbance amplitude. The evolving 3D structures show a dominant spanwise length scale, which can be attributed to the roughness diameter.

Further downstream, as the amplitude of the high speed streaks decreases, the spanwise extent, in which the disturbance mode is affected by the interaction with the roughness element, starts to spread considerably. At the same time a significant decrease in the amplitude of the 3D structures is observed. Therefore, no dominant 3D modes resulting from the influence of the roughness element on the low amplitude TS-mode persist far downstream.