A rational method for determining intermittency in the transitional boundary layer

In the study of turbulence, the intermittency factor \((\gamma )\) at a given point in the flow field is defined as the fraction of time the flow remains turbulent. The value of the intermittency factor varies from 0 to 1, where a zero value represents a fully laminar region and the value of one represents a fully turbulent regime. The initial idea of intermittency was proposed by Corrsin (1943) to differentiate the turbulent and non-turbulent patterns in axisymmetric turbulent jets. The first intermittency measurements were made by Townnsend (1948) in a turbulent wake. Though the idea of intermittency is derived from fully turbulent flows, its use became prominent in transitional boundary layer research since the discovery of turbulent spots in the transition zone by Emmons (1951). In the case of transition modelling, the intermittency distribution plays an essential role in capturing the length of the transition zone; however, the actual measurement of intermittency remains a challenge. Numerous techniques have been developed by many researchers (Hedley and Keffer 1974; Falco and Gendrich 1990; Kuan and Wang 1990; Schneider 1995; Zhang et al. 1996; Canepa et al. 2002; Fransson et al. 2005; Imayama et al. 2012) for the measurement of \(\gamma\); but the methods proposed in the literature do not seem to work for all type of flows, resulting in no universal procedure for determining \(\gamma\).

In the early days of transition research, the procedure followed for intermittency measurement was quite complex (Schubauer and Klebanoff 1956). It involved photographing the hot-wire signal from the oscilloscope screen and identifying the number and duration of bursts (occurrence of abrupt high-frequency peaks in the signal, considered to be the result of turbulent spots) in a given period of time. Subsequently, the intermittency was obtained by taking the ratio of the aggregate duration of the bursts to the total time of the acquired signal. Later, the intermittency was deduced using computer-based methods. Hedley and Keffer (1974) conceived a generic procedure for measuring the intermittency in a fully turbulent boundary layer, which is still used as a reference in transitional boundary layer research by many investigators. In their method, intermittency is determined by following a procedure consisting of four sequential steps that involve the selection and application of a detector function, a criterion function, a threshold value and an indicator function. These functions and the associated steps are explained as follows:This is the general procedure followed in many algorithms developed by several researchers for determining the intermittency using the time history signals obtained from hot-wires/hot-films. Despite there being many algorithms for intermittency calculation, none of the methods is found to be robust. This is because of the subjectivities arising from the choice of detector function, the time interval in the criterion function and arbitrariness in the selection of the threshold value (Canepa et al. 2002). Among these, it is found that the selection of the threshold value, \(T^h\) possesses the highest level of subjectivity as it strongly influences the intermittency determination. Some of the well-known intermittency calculation methods in transitional research are the TERA, MTERA, dual-slope and Probability Density Function (PDF) methods. In the TERA (Falco and Gendrich 1990) and MTERA (Zhang et al. 1995) methods, \(T^h\) is determined by an arbitrary percentage of the root mean square (rms) value of the criterion function. Due to the arbitrariness in choosing the threshold percentage, this method has not been well received by other investigators. Kuan and Wang (1990) proposed the dual-slope method; this method involves a graphical search to identify the \(T^h\) value. In this method, if the cumulative frequency distribution of the sensitised transitional signal is plotted against the detector function, then the distribution displays two different slopes and the point where the two different slopes intersect is considered to be the threshold. There are mixed reviews available for this method. Some researchers (Ramesh et al. 1996; Jahanmiri et al. 1991) succeeded in using this method, while others (Canepa et al. 2002; Fransson et al. 2005) have faced difficulty in identifying the two different slopes. Also, because of the graphical approach involved in this method, there is a certain level of subjectivity that exists in selecting the threshold value. However, at present, this method is widely used for various transitional flows. Next, in the PDF method (Solomon 1996), the \(T^h\) value is selected from the intersection of two conditional PDFs obtained from the selected laminar and turbulent portions of the detector function measured at a given point. This method is easy to employ on the signals acquired from surface hot-films, since the surface films are free from the contamination of edge intermittency (Narasimha 1985) and their sensitivity is generally higher than that of hot-wires. Conversely, in this method, an initial, arbitrary threshold value has to be assumed for segregating the laminar and turbulent portion of the signal. Also, this method would not produce good results near the onset and breakdown of transition due to the absence of an intersection between the conditional PDFs of laminar and turbulent signal (Canepa et al. 2002). In another PDF method, the threshold value is chosen corresponding to the valley between the laminar and turbulent peaks observed in the PDF of the entire signal (Schneider 1995). This method tends to fail especially near the transition onset region where the existence of the valley becomes ambiguous. Further suggestions (Canepa et al. 2002) have been made to overcome the difficulties encountered in the above method, however, such modifications result in a time-intensive process to determine the intermittency. In the case of direct numerical simulations, Nolan and Zaki (2013) and Marxen and Zaki (2019) have employed an image analysis technique (so called Ostu’s method, Otsu 1979) to arriving at a threshold value.

From the literature, it is clear that the proposed methods include a significant level of subjectivity in deciding the threshold value; also, the threshold values have to be re-defined for each station to determine the intermittency distribution. Further, the threshold values are not easily reproduced by other investigators using the same data or with the same experimental setup. Motivated by these problems, a new method is proposed in the present work using a more physical approach to determine the intermittency in the transitional zone. The advantages of the proposed approach when compared with those reported in the literature are as follows: (1) the method is simple, rational and the choice of threshold value is automatic, (2) a single threshold value can be used for the whole transition region, thus reducing analysis time, and (3) threshold values and intermittency distributions should be quantitatively reproducible by other investigators with the same data or under similar experimental conditions.

The proposed methodology is evaluated on an experimental arrangement involving the interaction between an aerofoil wake and a downstream flat plate boundary layer. Here, the aerofoil wake acts as a source of free-stream disturbance, which triggers laminar–turbulent transition on the flat plate boundary layer. This type of transition is generally referred as wake-induced transition, commonly seen in multi-element aerofoils (during aircraft take-off and landing phases) and turbomachinery flows. The rig allows the interaction to be controlled by simply changing the proximity of the aerofoil to the flat plate, which in turn presents a range of test cases to check the robustness of the proposed method. To this end, the streamwise intermittency distribution in the downstream, transitional boundary layer is studied in detail by applying the proposed approach to hot-wire measurements. Furthermore, the consistency of the new technique reveals that some interesting scaling characteristics between the proximity of the aerofoil and transition region are reported. It is worth to mention at this point, to the best of our knowledge, this kind of experimental setup, particularly focusing on transition research, was not previously reported in the literature.

All the experiments reported in this paper were performed in the low-turbulence wind tunnel at City, University of London. This is a closed-circuit wind tunnel with test section dimensions 0.924 \(\times\) 0.915 \(\times\) 3.66 m. The section width increases from 0.924 to 0.940 m, over the length of 3.66 m in the downstream direction. The operating velocity inside the test section can vary from 0.5 to 25 m/s; for the present experiment, a free-stream velocity \((U_0)\) of 20 m/s was considered.

An aluminium flat plate with dimensions 2082 \(\times\) 915 \(\times\) 10 mm was used in the present experiment. The flat plate was mounted vertically in the test section and it was fitted with an elliptical leading edge of about 63 mm in length, which was made up of an asymmetric, wooden biconvex surface. To maintain the stagnation point on the measurement side of the leading edge and to maintain a zero pressure gradient, a trailing-edge flap of 110 mm length was employed. Eventually, overall length of the flat plate (l) considered for the present experiment was, 2082 + 63 + 110 = 2255 mm.

Ahead of and above the leading edge of the flat plate, a NACA 0014 aerofoil, with a chord length (c) of 250 mm, was placed at zero degree angle of attack, shown schematically in Fig. 1. The vertical separation between the aerofoil and the flat plate is denoted as the ‘height’ \((h_w)\) and the horizontal separation is denoted as the ‘overlap’ \((x_w)\). The streamwise and wall-normal measurement stations are defined using a coordinate system, x–y, having its origin at the leading edge of the flat plate. The aerofoil spanned the entire test section, both ends meeting the tunnel walls. Furthermore, the aerofoil was tripped at around 25% of the chord using a sandpaper roughness strip (average roughness height = 425 \(\upmu\)m) to avoid vortex shedding. With the flat plate and aerofoil mounted in the tunnel, the free-stream turbulence intensity in the streamwise direction, at 20 m/s, was measured as 0.015% (in the frequency range of 5 Hz–5 kHz). This compares with 0.007% when the tunnel is empty.

All velocity measurements were carried out using a Dantec DISA 55M01, Constant Temperature Anemometry (CTA) unit. The hot-wire probe (boundary layer type) featured a single tungsten wire (diameter 5 \(\upmu\)m, length 1.25 mm). The probe was calibrated in the wind tunnel using the velocity obtained from the pitot-static tube. During the calibration process, the upstream aerofoil was removed to avoid wake interference. The raw signals acquired from the probe were sampled at a frequency of 10 kHz and passed through an analogue filter (Krohn-Hite, model 3360 series), where the signal was band-pass filtered between 5 Hz and 5 kHz. The lower bound of 5 Hz was chosen to remove the DC component and electronic noise; while the upper bound of 5 kHz was chosen to avoid aliasing. In all measurements, the signal was acquired for a total duration of 30 s to obtain converged steady state. A National Instruments DAQ card was used to digitise the acquired signal and a user interface was developed in LabVIEW for controlling the wind tunnel speed and the traverse mechanism for the boundary layer probe. The laboratory was air-conditioned but the temperature inside the (closed-circuit) wind tunnel increased by up to 3 \(^{\circ }\)C when compared with the ambient temperature at which the calibration was conducted. This was compensated for by performing a temperature correction on the voltage signal. For the present experiments, the individual uncertainties in calibration, linearization (curve fit) and analogue to digital conversion were found to be 0.06, 0.0095 and 0.0035 m/s respectively. The combined standard uncertainty for the hot-wire velocity was, thus, calculated as 0.3% of free-stream speed. To traverse the probe inside the test section, a 3-axis system was used with independent movement in the streamwise (x), wall-normal (y) and spanwise (z) directions. The smallest step size in the streamwise and spanwise directions was 10 \(\upmu\)m, while that in the wall-normal direction was 1.25 \(\upmu\)m. A laser positioning system was employed to avoid accidental contact between the probe and the surface of the flat plate during near-wall boundary layer measurements.

As highlighted in Sect. 1, the selection of threshold value \((T^h)\) plays a vital role in intermittency determination and it has a direct effect on the intermittency distribution. For instance, selection of a lower \(T^h\) will indicate greater intermittency in the laminar regime and earlier transition onset in the flow. On the other hand, the selection of a higher \(T^h\) might indicate that a fully turbulent flow regime is still transitional. Therefore, it is crucial to have an optimum \(T^h\) to accurately determine the intermittency. As discussed in Sect. 1, most of the intermittency determination methods in the literature involve an arbitrary constant or a subjective approach to determine the threshold value.

To overcome the problem with subjective selection of the threshold value, an objective approach is introduced in this paper. The basic idea behind this approach is illustrated in Fig. 5 using the streamwise fluctuating velocity signals obtained at four different points on the flat plate, designated as 1, 2, 3 and 4, whose fluctuating velocity signals are shown by \(u_1(t)\), \(u_2(t)\), \(u_3(t)\) and \(u_4(t)\), respectively. Points 1 and 2 are chosen in the upstream region where the flow tends to be laminar, and the points 3 and 4 are taken from the downstream region of the plate, generally falling in the transition region. It is known that, as we move downstream in the boundary layer, the magnitude of the fluctuations in the flow also increases. This is clearly demonstrated in Fig. 5, especially from points 1 to 3, where the magnitude of the fluctuation velocity increases in the downstream distance, until we reach point 3 and observe turbulent spots (spiky signals). By observing the signals at the transition onset (point 3) and the middle of the transition zone (point 4), it is easy to distinguish the perturbations due to turbulent spots from the laminar perturbations. An interesting point is that the laminar perturbations (within the red dashed line) do not seem to increase in magnitude between the transition onset point (point 3) and the middle of the transitional zone (point 4), in contrast to the increasing amplitude of these perturbations in the laminar region (points 1&2). Such a characteristic was observed in all of the experiments conducted in this work. This observation prompts an assumption that the magnitude of the laminar perturbations remains constant throughout the transition region. In this regard, it is proposed to choose the magnitude of the laminar perturbation at the transition onset point as a threshold value for intermittency determination. Such a threshold value would retain only the turbulent fluctuations from the transitional signals, eliminating the laminar perturbations. This approach has been applied to the present measurements to determine the intermittency. It will be demonstrated later in this paper that the obtained results look sensible and that the derived intermittency matches well with a detailed visual analysis of the fluctuating signal.

Most of the intermittency estimation methods presented in the literature are subjective, with some level of arbitrariness in choosing the threshold value. In addition, those methods require a different threshold value to be chosen for each streamwise station, making the procedure laborious and time consuming. In the present work, a rational and objective technique is proposed to address the shortcomings involved in the other methods. The underlying idea is to detect the maximum laminar perturbation in the transitional flow and adopt this as the threshold for determining intermittency. It has been shown that using the threshold value obtained from the laminar perturbations at or around the onset point captures the transition zone very well and matches with the commonly used dual-slope method. These results in turn justify the assumption that the magnitude of the laminar perturbations remains constant in the transition zone, which might be an inherent physical characteristic of the transitional boundary layer.

The specific advantages of the proposed method when compared with the other existing methods are as follows: (1) a single threshold value is sufficient for the streamwise intermittency determination in the entire flow, thus reducing the analysis time when compared with other methods where a threshold value has to be chosen for each streamwise station. (2) The threshold value is selected in an objective way, which can be quantitatively reproducible by other investigators with the same data or under similar experimental conditions. In other methods, there is a certain level of subjectivity in determining the threshold value, for example graphical approaches (dual-slope method) or constants involved in the criterion function (TERA/MTERA), which make quantitative comparisons somewhat arbitrary. (3) This method is simple, rational, and the threshold value is effectively self-selecting. In other methods, as discussed in the introduction, there can be difficulty in determining the threshold value near the onset and breakdown points. The principal limitation of the present method is that the proposed approach cannot be used for determining intermittency distribution in the wall-normal direction, \(\gamma (y)\).

Additionally, an interesting scaling characteristic of the transitional boundary layer is observed in the present experiment. The skewness and intermittency distributions obtained for various heights of the aerofoil scale well with the onset location. Further, it was found that the length of the transition zone also increases linearly with respect to the aerofoil height above the flat plate.

The robustness of the present intermittency determination method needs to be verified by application to other types of flows, which would also help to substantiate the assumptions involved in the present approach. On the other hand, it is clear from the results presented here that, in the case of wake–boundary layer interactions, the present approach works very well and should be valuable for transitional boundary layer research in turbomachinery and other multi-element flows.