A turbulent boundary layer (TBL) over a flat rigid surface can be seen in many industrial and environmental flows. Therefore, for the prediction and control of turbulence and design of better industrial apparatus, it is of great importance to elucidate the spatiotemporal structure of the TBL. In TBLs, it is well known that coherent motions associated with the bursting events exist near the wall. In the outer turbulent/nonturbulent (potential) interface of the TBL, large-scale coherent motions exist, which have the size of order of the boundary-layer thickness. The latter structures are called `bulge' and `valley' and are unsteady in space and time (Robinson 1991). Therefore, it is of importance to clarify the spatiotemporal structures of these coherent motions in order to elucidate the structure of the TBL. However, it is difficult to study the entire structure of the TBL by a single-point measurement using an I- or X-type hot-wire probe. In addition, previous studies (Ichijo & Kobashi 1982, Kobashi et al. 1984, Thomas & Bull 1983) have suggested that these coherent motions are closely associated with wall static pressure fluctuations. Therefore, the simultaneous measurement of multipoint instantaneous velocities and wall static pressure fluctuations is appropriate for clarifying the spatiotemporal structure of the TBL.
In this chapter, we will show the experimental results on the relationships between the bursting events occurring near the wall and the large-scale bulge and valley structures in the outer turbulent/nonturbulent interface of the TBL, and the spatiotemporal structures of the coherent motions in a zero-pressure-gradient TBL. With regard to the experiments, multipoint instantaneous streamwise velocities and instantaneous wall static pressures are simultaneously measured using a combination of a rake consisting of 23 I-type hot-wire probes, which covers the entire TBL, and a microphone pressure sensor. The velocity signals are analysed by the KL (Karhunen-Loéve) expansion (also known as the proper orthogonal decomposition (POD) method). The flow field is reconstructed by using either lower- or higher-order modes to investigate the coherent motions in the TBL. The bursting events are detected by applying the VITA (Variable Interval Time Average) technique to the instantaneous velocity signals in the original flow. The large-scale bulge and valley structures are detected by using the newly proposed conditional sampling method, which is applied to both the original flow and the reconstructed flow by using the KL expansion. Further, to show the future research direction, the experimental data on the statistical properties of TBL affected by the disturbances in the outer mainstream are demonstrated. The disturbances are generated using the bi-plane square grid. The effect of the disturbance on the velocity statistics is discussed.
2. Analytical methods
2.1. VITA technique
The VITA (Variable Interval Time Average) technique (Blackwelder & Kaplan 1976, Blackwelder & Haritonidis 1983, Chen & Blackwelder 1978, Osaka et al. 1986) is one of the conditional mean methods used to detect coherent motions in turbulent flows. The VITA technique is appropriate for detecting the bursting events in a TBL since the bursting events accompany the large velocity fluctuations. Following the general VITA technique (Blackwelder & Haritonidis 1983), we define the variable-interval time average of the fluctuating streamwise velocity as
where is the averaging time. Note that approaches the conventional time-averaged result as increases. The detection criterion is established by introducing a threshold value for the VITA variance signal. Thus, the detection function is defined as (Blackwelder & Haritonidis 1983, Chen & Blackwelder 1978, Osaka et al. 1986)
where is the time-averaged mean velocity, is the rms (root mean square) value of velocity fluctuations, and is the threshold value, which is constant. In general, the selection of and is important for detecting the bursting events using the VITA technique. In this study, we chose and, on the basis of a previous study (Osaka et al. 1986), where is the friction velocity and is the kinematic viscosity. We have carefully verified that the detected burst frequency is in a good agreement with previously published data (Blackwelder & Haritonidis 1983, Osaka et al. 1986).
2.2. Methods for detecting large-scale coherent motions
The large-scale coherent motions in the outer turbulent/nonturbulent interface of the TBL are detected using the following method. The first probe from the top of the rake, which is placed near the turbulent/nonturbulent interface of the TBL (; and are the distance from the wall and the thickness of the boundary layer, respectively), is used to detect the large-scale coherent motions since these motions pass through this probe. The detection of the motions depends on both and at. Note that the direction of the rotation is based on the configuration shown in Figs. 1 and 2; the flow is from the left to the right along the -direction. Therefore, the mean spanwise vorticity is in the clockwise direction.
2.2.1. Detection of clockwise large-scale coherent motions
The large velocity fluctuation at with
where is the threshold value, which is constant. In this study, from a careful inspection of the detected signals, has been chosen equal to 3.
2.2.2. Detection of anticlockwise large-scale coherent motions
The large velocity fluctuation at with
It has been verified that the value of is appropriate for detecting the valley structures.
2.3. KL expansion
The KL（Karhunen-Loéve）expansion, which is also known as the proper orthogonal decomposition (POD) method and is an extension of the general Fourier series (Davenport & Root 1958, Lumley 1967, 1981), is the representation of a stochastic process as a linear combination of orthogonal functions. The basis functions of the expansion are defined such that the coefficients are uncorrelated (independent) with each other and are arranged in the descending order according to the magnitude of their eigenvalues. The sum of all the eigenvalues corresponds to the sum of the mean square values in the entire region where the KL expansion is applied. Here, the KL expansion is applied for the simultaneous measurement of the streamwise velocities at 23 points in the TBL. Therefore, there are 23 modes () in the KL expansion. The lower modes are expected to extract the turbulence structures in the near-wall region (i.e., the inner region), where turbulence energy is larger, and the higher modes are expected to extract the turbulence structures from the outer region to the outside of boundary layer, where turbulence energy is smaller. In this study, the sum of the first three modes (lower modes) retains 65% of the total energy; therefore, the flow field reconstructed by using the lower modes () is considered to represent the energetic coherent motions. The modes from 4 to 23, which contain the rest of the 35% energy, are regarded as the higher modes in this work.
3.1. Experimental apparatus
Experiments are carried out in an Eiffel-type wind tunnel. The length, height, and span of the test section of this tunnel are 2,600, 300, and 520 mm, respectively. The TBL is developed along a smooth flat plate (Fig. 1) placed horizontally in the wind tunnel. The plate is 2,100 mm in length, 8 mm in thickness, and 520 mm in span and is made of Bakelite in order to reduce the heat loss from the hot-wire probe in near-wall measurements. The leading edge of the plate has a parabolic shape. To generate a fully developed TBL in a short distance from the leading edge, transition was promoted by using a tripping wire of 1.0 mm diameter placed 50 mm downstream from the leading edge of the plate, as shown in Fig. 1. The zero-pressure-gradient condition is achieved by adjusting the height of the test section. The coordinate system is as follows: the
3.2. Velocity and wall static pressure measurements
A constant-temperature hot-wire anemometer (CTA) is used for the velocity measurement. The rake consists of 23 I-type probes, which covers the entire TBL at mm; the sensors of the rake cover a length of mm. The wires are made of tungsten. The length of the sensor is mm and its diameter is μm, giving an aspect ratio of. By comparing the data obtained from a single I-type probe with those from the rake of I-type probes, it has been confirmed that there is no interference between the probes, which are placed with 2.2 mm apart from each other.
To measure the instantaneous static pressure at the wall, the 1/8 inch microphone pressure sensor (Brüel & Kjær Type 4138) is used. The sensor is mounted flush to the wall surface at the measurement position (mm) through a pressure hole of mm diameter. The frequency response of the pressure sensor has been checked; both the phase difference and the magnitude ratio between the input signal and the output signal from the sensor exhibit flat responses up to 7 kHz. These results suggest that the frequency response limit of the pressure sensor is 7 kHz. Note that nondimensional diameter of the sensing area in this study is. Previous studies (Gravante et al. 1998, Schewe 1983) have shown that when, the spatial resolution is sufficient to resolve the essential structures of the static pressure fluctuations at the wall. Therefore, in addition to the frequency response, the spatial resolution is sufficient to measure the static pressure fluctuations at the wall. In fact, we have confirmed that when, the power spectra of wall static pressure fluctuations are identical.
3.3. Experimental conditions and data sampling method
The mean speed of the uniform flow is set at m/s. The unit Reynolds number is m-1, and the Reynolds number based on the momentum thickness at the measurement point (mm) is. Note that the spatial resolution of the hot-wire probe in the near-wall measurement is sufficient for (Robinson 1991). The and at the measurement point (mm) are 41 mm and 0.453 m/s, respectively. We have confirmed that there is no pressure gradient in the streamwise direction, and the mean flow is two-dimensional. The sampling frequency and sample size are 10 kHz and 229,376, respectively, and are sufficient to obtain reliable statistics: these parameters are based on the performance of the A/D converter (Microscience ADM-688PCI) and the computer (Dell Dimension 8200) used. The output voltage signals from the hot-wire anemometer and the pressure sensor are converted into 12-bit digital data by the A/D converter and recorded on the hard disc of the computer. The A/D converter and the digital output board are PCI interface boards and are synchronized through master-slave connections. The time lag between the output of the clock signal from the master unit and the input of the signal into the slave unit is only 125 ns; therefore, very fast multichannel sampling is achieved. To confirm this point, we input the same wave pattern generated by the function generator into the whole channels simultaneously, and compare the output signals. The result showed that the output signals are the same and no discernible time lag is observed. Statistical processing of the digitized data was carried out using the computer.
4. Results and discussion
4.1. Reliability of measurements
The vertical profile of streamwise mean velocity is shown in Fig. 3, where
4.2. Relationships between bursting events and large-scale coherent motions
The clockwise and anticlockwise large-scale coherent motions are detected using the method proposed in the subsection 2.2, and the results are shown in Figs. 6 and 7, respectively. In Figs. 6 and 7, the upper and lower graphs show the conditionally averaged time histories of the normalized velocity, , and the conditionally averaged time histories of normalized wall static pressure, , respectively. Here, and are the ensemble-averaged values of conditionally sampled velocity and wall static pressure, respectively, and and are the ordinary (non-conditional) time-averaged velocity and wall static pressure, respectively. In Figs. 6 and 7, is the time lag from the detection time at which the instantaneous velocity signal takes the maximum or minimum value at the detection point of, and is normalized by using
On the other hand, Fig. 7 shows that when the anticlockwise large-scale coherent motion passes, the velocity decreases outside the TBL () and slightly increases inside the TBL (). The detected signals suggest the valley, which exists at the edge of the bulges. It has been reported (Robinson 1991, Thomas & Bull 1983) that there is an inflow from the free-stream flow into the turbulent region, and the inflow has an instantaneous velocity vector that is against the mean flow. Therefore, it is expected that the decrease in the velocity in the outer region in Fig. 7 is associated with the inflow from the free-stream flow. The wall static pressure increases when the anticlockwise large-scale coherent motion passes. These results are in agreement with the model for the structure of a TBL proposed by Thomas & Bull (1983).
The VITA technique is applied to the raw signals to detect the bursting events. Figure 8 shows the ensemble-averaged time histories of conditionally sampled velocity
and of conditionally sampled wall static pressure. Here, and are the conditionally averaged velocity and wall static pressure by the VITA technique, respectively. The solid and broken lines in Fig. 8 denote and, respectively. is the time lag from the detection time at which the instantaneous velocity is the same as the ordinary (non-conditional) time-averaged velocity at each measuring point. Figure 8 shows the cycle of events of ejection () at, which is the upward motion of a low-speed fluid lump, and sweep () at, which is the downward motion of a high-speed fluid lump (Osaka et al. 1986). The wall static pressure exhibits a rapid increase when the burst occurs above the measurement point (Kobashi et al. 1984). Considering the results shown in Figs. 6 and 7 together with the result in Fig. 8, it is concluded that the bulge structure, which is accompanied by the pressure decrease, cannot coexist with the bursting event, which is accompanied by the pressure increase. On the other hand, the valley structure, which is accompanied by the pressure increase, can coexist with the bursting event.
4.3. Flow field reconstructed by using lower modes (
n=1 ~ 3)
Using the method proposed in the subsection 2.2, the clockwise and anticlockwise large-scale coherent motions are detected in the flow field reconstructed by using the lower modes () of the KL expansion, and the results are shown in Figs. 9 and 10, respectively. Figures 9 and 10 show that both the clockwise large-scale coherent motion (i.e. bulge), which is accompanied by the pressure decrease, and the anticlockwise large-scale coherent motion (i.e. valley), which is accompanied by the pressure increase, exist in the flow field reconstructed by using the lower modes. Comparing these results with those shown in Figs. 6 and 7, which are obtained by applying the method to the original flow field (), it is found that these structures are more clearly observed in the flow field reconstructed by using the lower modes and they are observed even near the wall.
Figure 11 shows the ensemble-averaged time histories of conditionally sampled velocity and wall static pressure when the velocity is reconstructed by the lower modes () and the detection by the VITA technique is applied to the original flow field () at. It is observed from Fig. 11 that the velocity inside the TBL (in the reconstructed flow) increases and the velocity outside the TBL (in the reconstructed flow) decreases when the bursting event occurs. These velocity patterns are similar to those during the anticlockwise large-scale coherent motion (i.e. valley). Thus, the results suggest that the bursting event and valley can coexist at the same longitudinal location in the flow reconstructed by using the lower modes ().
4.4. Flow field reconstructed by using higher modes (
n=4 ~ 23)
Using the method proposed in the subsection 2.2, the clockwise and anticlockwise large-scale coherent motions are detected in the flow field reconstructed by using the higher modes () of the KL expansion, and the results are shown in Figs. 12 and 13, respectively. Figure 12 (for the bulge) shows a velocity variation near the wall. However, the velocity hardly changes near the wall in Fig. 13 (for the valley). These results suggest that in the flow field reconstructed by using the higher modes, only the clockwise large-scale coherent motion (i.e. bulge) exists near the wall and the anticlockwise large-scale coherent motion (i.e. valley) does not influence the flow field near the wall. These results also suggest that the turbulent kinetic energy in the bulge structure is larger than that in the valley structure. This result is consistent with the fact that the valley is generated by the bulge structures.
Figure 14 shows the ensemble-averaged time histories of conditionally sampled velocity and wall static pressure when the velocity is reconstructed by the higher modes () and the detection by the VITA technique is applied to the original flow field () at. Figure 14 shows that when the bursting events are detected, the velocity in the outer region does not change in the reconstructed flow. This result suggests that in the flow field reconstructed by using the higher modes, the large-scale coherent motions are weakly associated with the bursting events.
4.5. Effect of freestream turbulence on the Turbulent Boundary Layer
In the industrial flows such as those in turbomachines and heat exchangers, inflow streams often contain the turbulent eddies generated by the upstream structures. These free stream turbulence influences the large scale coherent structure and the turbulence characteristics of the TBL. In this subsection, some experimental data on the statistical properties of TBL affected by the freestream turbulence generated by the square grid are demonstrated.
The experiments are performed by using the experimental apparatus shown in the subsection 3.1. The turbulence-generating grid is installed at 100mm upstream from the leading edge of the plate. The square-mesh biplane grid
The vertical profiles of y+ dU+/dy+ in both cases with and without the grid are shown in Fig.15, where the result by the Direct Numerical Simulation (DNS) in the case of Re
Multipoint instantaneous streamwise velocities and instantaneous wall static pressures have been simultaneously measured by using a combination of a rake consisting of 23 I-type hot-wire probes and a microphone pressure sensor in a fully developed, zero-pressure-gradient turbulent boundary layer (TBL) in a wind tunnel. The KL（Karhunen-Loéve）expansion, the VITA technique and the newly proposed method for the detection of large-scale coherent motions were applied to examine the relationships between large-scale coherent motions and the bursting events. Further, the statistical properties of the TBL affected by the freestream turbulence generated by the square grid were investigated.
The main results obtained in this chapter are summarized as follows:
The structure of the TBL inferred from the present experiment is in agreement with the TBL model proposed by Thomas & Bull (1983).
The bursting events and valley structure (both accompanied by an increase in wall static pressure) can coexist at the same longitudinal location, whereas the bursting events and bulge structure cannot coexist with each other.
In the flow field reconstructed by using the lower modes, both the bulge structure, which is accompanied by bursting events, and the valley structure, which is not accompanied by the bursting events, exist. On the other hand, in the flow field reconstructed by using the higher modes, the large-scale coherent structures are weakly associated with the bursting events.
The grid turbulence introduced into the outer mainstream has the effect to enlarge the logarithmic region to the outer side of turbulent boundary layer.
This study was partially supported by the Japanese Ministry of Education, Culture, Sports, Science and Technology through Grants-in-Aid (Nos. 22360076 and 22360077).