Image-preprocessing method for near-wall particle image velocimetry (PIV) image interrogation with very large in-plane displacement

Assume that the intensity distribution in the window overlapping the interface can be decomposed into four parts:

$$\begin{eqnarray} &&I = I_{f}\cdot S+I_{o}\cdot \bar{S}+k_{B} \cdot S +k_{R}\cdot T . \end{eqnarray} \tag{ 5 }$$

Here, I_f and I_o stand for the intensity distributions in the flow and wall regions, S and T are step and top hat functions representing the signal truncation and reflection across the interface. k_B and k_R are scaling parameters for the background noise and the reflection. Thus, the cross-correlation between I_a and I_b is

$$\begin{eqnarray} I_{fa}\ast I_{fb} &=&(I_{fa}\cdot S+I _{oa} \cdot \bar{S}+k_{B} \cdot S + k _{R} \cdot T)\nonumber\\ &&\ast\, (I _{fb}\cdot \bar{S} + I_{ob} \cdot \bar{S}+k_{B} \cdot S+k_{R}\cdot T)\nonumber\\ &=&(I_{fa}\cdot S)\ast (I_{fb}\cdot S) +(I_{oa} \cdot \bar{ S}) \ast (I_{ob} \cdot \bar{S}) \nonumber\\ &&+k_{B}^{2}\cdot (S\ast S) +k_{R}^{2} \cdot (\bar{S} \ast \bar{ S})\nonumber\\ &&+(I_{fa}\cdot S)\ast (I_{ob} \cdot \bar{S}+ k_{B} \cdot S +k_{R}\cdot T)\nonumber\\ && +(I_{oa}\cdot \bar{S})\ast (I_{fb}\cdot S+k_{B} \cdot S +k_{R}\cdot T)\nonumber\\ &&+(k_{B} \cdot S)\ast (I_{fb}\cdot S+I_{ob}\cdot \bar{S}+k_{R}\cdot T)\nonumber\\ && +(k_{R} \cdot T)\ast (I_{fb}\cdot S+I_{ob}\cdot \bar{S}+k_{S}\cdot B). \end{eqnarray} \tag{ 6 }$$

For conciseness, the effect of the cross-term can be neglected. Therefore, the dominant terms in the cross-correlation can be written as

$$\begin{eqnarray} &&I_{fa} \ast I_{fb} =(I_{fa}\cdot S)\ast (I_{fb}\cdot S) +(I_{oa} \cdot \bar{ S}) \ast (I_{ob} \cdot \bar{S}) \nonumber\\ &&+\, k_{B}^{2} \cdot (S\ast S) +k_{R}^{2} \cdot (\bar{S} \ast \bar{ S}) . \end{eqnarray} \tag{ 7 }$$

Here, (I_fa · S)*(I_fb · S) and $(I_{oa} \cdot \bar{ S}) \ast (I_{ob} \cdot \bar{S})$ represent the cross correlations in the flow and wall regions, $k_{B}^{2} \cdot (S\ast S)$ and $k_{R}^{2} \cdot (\bar{S} \ast \bar{ S})$ represent the interference due to the background noise and reflections. The last two terms can be neglected in the absence of noise and reflections.

Figure 2 shows the correlation map near the wall with the wall region preprocessed in different ways. A displacement profile with a gradient is imposed in the flow region. As shown by the black arrow, $(I_{oa} \cdot \bar{ S}) \ast (I_{ob} \cdot \bar{S})$ creates an artificial peak value R_o at the origin. As shown by the white arrow, (I_fa · S)*(I_fb · S) creates a correlation peak R_f, which indicates the mean displacement of the flow. R_f stretches parallel to the wall and divides into small discrete peaks due to the displacement gradient. Moreover, if the wall region is masked out (MO, figure 2(d)), a wide region of high-intensity noise is generated by the intensity truncation around the origin. In comparison, this noise is almost removed completely when the wall region is filled with stationary particles (SP, figure 2(d)), so the SNR is improved. As the window centroid approaches the interface, R_o will increase from zero while R_f decreases. Then, the result will be determined by the more intense one of the two peaks. The image-preprocessing implementation is described in the next section.

Zoom In Zoom Out Reset image size

Attention is focused on the initial interrogation step with a relative coarse window size. Suppose that the window is in the shape of a square with size W₀ × W₀. To achieve the non-slip condition, R_o is assigned to be approximately equal to R_f when the centroid of the interrogation window is located at the wall:

$$\begin{equation} R_{o}\approx R_{f}. \end{equation} \tag{ 8 }$$

In the absence of the velocity gradient, the background noise and the out-of-plane loss of pairs, R_f can be estimated from the auto-correlation peak value in the flow region, i.e.,

$$\begin{equation} R_{f}\approx \int \!\!\!\int I_f^2(x,y)\cdot {\rm d}S=\frac{1}{2}\cdot \overline{I_f^2(x,y)}\cdot W_0^2. \end{equation} \tag{ 9 }$$

Similarly, R_o can be estimated as

$$\begin{equation} R_{o}= \int \!\!\!\int I_o^2(x,y)\cdot {\rm d}S=\frac{1}{2}\cdot \overline{I_o^2(x,y)}\cdot W_0^2. \end{equation} \tag{ 10 }$$

Therefore, the condition to fulfil equation (8) is

$$\begin{equation} \overline{I_o^2(x,y)}=\overline{I_f^2(x,y)}. \end{equation} \tag{ 11 }$$

In this work, the stationary particle images are distributed uniformly in the wall region with a constant density D_s (figure 2(e)). The intensity of each artificial particle image obeys the Gaussian function:

$$\begin{equation} I_{{\rm sp}}=I_{\rm spm}(x,y)\cdot \exp \left(\frac{(-(x-x_{0})^{2}-(y-y_{0})^{2})}{(d_{p})^{2}}\right). \end{equation} \tag{ 12 }$$

Each particle image has a constant diameter d_p. Thus, the only parameter to be determined is I_spm(x, y). Two modes are presented in this paper. For the first mode (termed SP), I_m(x, y) is uniform within the interrogation window. Thus,

$$\begin{equation} \overline{I_o^2(x,y)}=D_s \cdot \int \!\!\!\int I_{{\rm sp}}^2 {\rm d}x\, {\rm d}y=\frac{\pi }{4} d_p^2 I_{{\rm sm}}^2 D_s. \end{equation} \tag{ 13 }$$

From equations (11) and (13), I_spm can be estimated as:

$$\begin{equation} I_{\rm spm}\simeq \left(\frac{4\overline{I_f^2(x,y)}}{\pi d_p^2 D_s }\right)^{1/2} . \end{equation} \tag{ 14 }$$

The second mode can be considered to be an optimized SP(termed OSP). The existence of displacement gradient and out-of-plane loss of pairs reduce R_f. Therefore, for the SP mode, the evaluation result biases towards zero. To reduce this bias, the square of I_ospm(x, y) in the OSP mode increases linearly with respect to the distance to the interface d (figure 2(c)):

$$\begin{equation} I_{\rm osp}=I_{\rm ospw}(2d/W_0)^\frac{1}{2}. \end{equation} \tag{ 15 }$$

Moreover, the effect of the stationary particles will diminish as the window size decreases. The SP and OSP modes will be compared in the next section. If the wall is inclined to the window at an angle θ (figure 3), $\overline{I_o^2(x,y)}$ in the wall region (indicated by the blue colour) is

$$\begin{eqnarray} \overline{I_o^2(x,y)}&=&\frac{\int \!\!\!\int I_o^2(x,y)\,{\rm d}x\, {\rm d}y}{W_0^2/2}\nonumber \\ &=& f(\theta ) \frac{\pi }{8} d_p^2 D_s I_{\rm ospw}^2 \end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray} f(\theta )&=&\left\lbrace \begin{array}{@{}cc@{}}\displaystyle\frac{3\tan ^2\theta +1}{3\tan \theta \sqrt{1+\tan ^2\theta }} &{\rm for} \quad \theta \in [45^{\circ },90^{\circ })\\ \displaystyle\frac{\tan ^2\theta +3}{3\sqrt{1+\tan ^2\theta }} &{\rm for} \quad \theta \in [0, 45^{\circ }).\end{array}\right. \end{eqnarray} \tag{ 17 }$$

f(θ) ranges from 0.94 to 1 when θ varies between 0° and 90°. Suppose

$$\begin{equation} R_o=R_f, \ \ \ \ {\rm when}\quad f(\theta )=1. \end{equation} \tag{ 18 }$$

Then from equations (11) and (18), I_ospw can be estimated by

$$\begin{equation} I_{\rm ospw}\simeq \left(\frac{8\overline{I_f^2(x,y)}}{\pi d_p^2 D_s }\right)^{1/2}. \end{equation} \tag{ 19 }$$

Thus,

$$\begin{equation} I_{\rm osp}=4\cdot \left(\frac{{\rm d}\overline{I_f^2(x,y)}}{\pi d_p^2 D_s W_0}\right)^{1/2}. \end{equation} \tag{ 20 }$$

Zoom In Zoom Out Reset image size

For OSP, local intensity compensation is necessary to eliminate the intensity truncation at the interface. The mean intensity in the flow region near the interface is assessed in the first step interrogation window as

$$\begin{equation} i_f= \overline{I_f(x,y)} . \end{equation} \tag{ 21 }$$

The local intensity compensation for each pixel is

$$\begin{equation} i_{c}=\left\lbrace \begin{array}{@{}ll@{}}i_f\cdot (1-2d/w)& {{\rm for}}\ d < w/2\\ 0 &{{\rm for}}\ d \ge w/2. \end{array}\right. \end{equation} \tag{ 22 }$$

The image preprocessing for the present method includes

• (i)  
  identifying the wall–flow interface,
• (ii)  
  subtracting the background noise,
• (iii)  
  estimating the local distribution parameters in the flow region near the interface,
• (iv)  
  adding the stationary particles in the wall region and
• (v)  
  compensating the intensity at the wall.

The interface location is identified by the symmetric line of the corresponding image pairs mirrored at the wall (Kähler et al 2006). Then the background noise is removed by subtraction of the minimum value over an ensemble of PIV images at each pixel (Wereley et al 2002). The third step is to estimate the local $\overline{I_f^2(x,y,t)}$ and i_f(x, y, t) near the interface in the flow region. If the particle seeding is quasi-steady, the parameters can be obtained by the average of a time series. The fourth step adds the stationary particles in the wall region. The particles are distributed uniformly with constant density and diameter. Two intensity arrangement modes (SP and OSP) are presented in this paper. In the last step, intensity at the wall is adjusted to remove the truncation.

The effect of the improvement on near-wall PIV evaluations will be assessed by Monte Carlo simulations in the next section.

This section presents an analysis of a synthetic image set to assess the effects of the different methods.

The flow region has a particle density of 0.05 particles per pixel². The particles are distributed randomly in a light sheet centred at z = 0. The light sheet intensity distribution is Gaussian expressed as

$$\begin{equation} I_{{\rm ls}}(z)=I_{\rm max}\cdot \exp \left(\frac{-z^{2}}{(\Delta z)^{2}/8}\right). \end{equation} \tag{ 23 }$$

Here, Δz is the light sheet thickness and I_max is the maximum intensity level of 4095, corresponding to a 16 bit sensor. The particle image centred at (x₀, y₀) reflects light according to the Gaussian function:

$$\begin{equation} I_{f}(x,y)=I_{ls}(z)\cdot \exp \left(\frac{-(x-x_{0})^{2}-(y-y_{0})^{2}}{d_f^{2}}\right). \end{equation} \tag{ 24 }$$

The particle images are integrated at each pixel with fill factor 1. A background noise is added with average of 16% and fluctuations of 5% of the maximum intensity. 500 pairs of particle images are generated. The size of images is 128 pixels × 128 pixels. The wall is placed horizontally at y = 64 pixels. The displacement profile of the boundary layer obeys to the Blasius solution with the thickness of 55 pixels.

Images are preprocessed in three ways, as shown in figure 2. The first way is the conventional masking-out process shown in figure 2(a) (termed MO). The second is the SP mode in figure 2(b). The third is OSP in figure 2(c). The WIDIM scheme has a one pass interrogation with a 64 pixels × 64 pixels window and three passes with a 32 pixels × 32 pixels window using the PID technique. A PTV technique proposed by Theunissen et al (2004) is used in the last step. The relocation of vector method (termed RV) is also used in the MO evaluation. The vectors are relocated to the centroids of the fluid region instead of the centroids of the interrogation window, when the windows overlap the wall boundary. Then vectors on the grid are obtained by interpolations.

3.2.1. WIDIM for MO for various displacement gradients

The WIDIM results with the MO method are shown in figure 4 for two displacement gradients. For a displacement of 8 pixels, the final result coincides well with the imposed displacement profile. Although the RV method was not used in the interrogation, the estimated displacements at the wall seem to approach zero. For a displacement of 24 pixels, the evaluations are distorted seriously for both MO and $\mathbf {MO\&RV}$. Large random uncertainties are created by the velocity gradient and intensity truncation. Smoothing provides only limited reduction of the uncertainties because the process is only applicable to the flow side. Meanwhile, as shown in figure 6(c), the MO interrogation without RV has a non-zero displacement at the interface. RV indeed eliminates this non-zero displacement at the interface and reduces the uncertainties to some extent. However, the relocated data are significantly biased towards zero near the interface because of the in-plane loss of pairs for the larger displacement.

Zoom In Zoom Out Reset image size

The evaluation errors are shown in figure 5 for various displacement gradients using WIDIM for MO. The term 'mean bias' stands for the bias of the evaluated results' mean value from the imposed value. The term 'RMS uncertainties' stands for the RMS of the evaluated results' deviations from their mean value.

Zoom In Zoom Out Reset image size

Three different particle diameters, d_f, were investigated for the particles in the flow region. Each curve has a critical point (CP). For d_f = 3 pixels, CP is 1.1 pixels/pixels. For d_f = 6 or 9 pixels, CP is a little larger. Errors are rather small when the displacement gradient is smaller than CP but both the mean bias and the RMS uncertainties increase dramatically above CP and the estimated displacement eventually cannot converge to the correct result. Such results show that the WIDIM scheme is not effective on MO PIV images with very large displacement gradient near the wall.

3.2.2. WIDIM and PTV for SP and OSP with very large displacement gradients

The smoothed initial estimates for SP, OSP, MO and $\mathbf {MO\&RV}$ are compared in figure 6. The free stream had a displacement of 24 pixels, corresponding to about 2.1 pixels/pixels near the wall. The addition of stationary particles introduces a reliable no-slip condition at/beyond the wall that is not possible with MO. The uncertainties, therefore, can be reduced by smoothing from both sides of the shear layer. For SP and OSP, the RMS uncertainties decrease to zero when approaching the wall. The maximum occurs in the middle of the shear layer, with a value about 50% that of MO and $\mathbf {MO\&RV}$. The initial estimates give a proper prediction of the displacement fields and deform the PIP in the correct direction.

Zoom In Zoom Out Reset image size

The mean bias and RMS uncertainties are further reduced in the following interrogation result as shown in figure 7. The evaluations for all besides MO groups gradually converge to the imposed displacement profile. $\mathbf {MO\& RV}$ still have considerable mean bias error and RMS uncertainties in the middle position of the shear layer. As shown in figure 8, spurious vectors exist near the wall for MO and $\mathbf {MO\&RV}$ which interfere displacement gradient estimation and mislead the PID. For SP and OSP, the displacement velocity field is more regular with small spatial variations in the X direction for SP and smaller for OSP.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Figure 9 shows the estimated displacement using the PTV evaluation. The results are converted to a grid by a second-order polynomial fit. For SP and OSP, the mean evaluated data coincide precisely with the imposed displacement profile. The RMS uncertainties are one order of magnitude less than for MO (figure 10(a)). At some points for OSP, the RMS uncertainties approach 1% of the free stream displacement. The mean bias errors are less than 5% of the free stream displacement (figure 10(b)). Small errors still exist within 3 pixels of the interface.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

3.2.3. Comparison of SP and OSP

Figure 11 compares the initial estimates near the wall for SP and OSP. Free stream displacement of 24 pixels is imposed in the flow region. As the window centroid approaches the wall, R_o (indicated by white arrow) increases while R_f decays (indicated by black arrow). From equation (11), the particle intensity is uniform across the interface. However, the very large displacement gradient in the flow region reduces R_f. With SP, R_o becomes much stronger than R_f even though only 1/3 of the window overlaps the wall region as shown in figure 11(d). Thus, the point d in figure 11(a) at this location is estimated to be near zero, far biased from the imposed value. By comparison, the growth of R_o is much slower in OSP. Estimate for point g in figure 11(b) at the same location is near the imposed value. Therefore, a mean bias towards zero exists on the smoothed data for SP. For OSP, the bias error is reduced to some extent, making the evaluation close to the imposed profile. In this method, the evaluation is postulated to be insensitive to the selection of parameters such as D_s. Figure 12 presents the evaluation errors of the WIDIM scheme with SP and OSP for different D_s. One interesting phenomenon is that as the displacement gradient increases, the RMS uncertainties near the wall increase somewhat but the mean bias errors decrease. For OSP, the evaluation errors are nearly stable for each value of displacement gradient. For SP, the evaluation errors are stable only when displacement gradient ⩽ 1.62 pixel/pixel; otherwise, large variations occur in the evaluation results. The mean bias and RMS uncertainties will increase when the selection of D_s is less than normal. OSP is shown to be more stable than SP when evaluating PIV images with large displacement gradients. In general, the numerical simulations show that OSP provides the best evaluations of PIV images with large displacements and displacement gradients near the wall. However, the simulations have not taken into account the out-of-plane loss of pairs, surface curvature, boundary reflection and other phenomena which are encountered in real experimental conditions. The next section further evaluates the method with the experimental data.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

3.3.1. Computation cost

3.3.2. Wall detection uncertainties

In order for the proposed technique to work, the detection of the wall is necessary. The effect of wall detection accuracy is necessary to be considered.

In general, the evaluation algorithm presented in this paper combines PIV with PTV. The latter one is free of the wall–fluid boundary. If the PIV prediction error near the wall is within half the distance between the flow particles, the PTV evaluation is always available. For the OSP preprocessing method, the main effect of the wall detection uncertainty is on the initial PIV evaluation. The sample points of PIV evaluation are at stationary discrete locations with the distance d = W₀ · OV. Here, OV is the overlapping factor of interrogation window. For W₀ = 64 pixels and OV = 75%, d = 16 pixels. Figure 13 simply describes a variation of R_f and R_o on the correlation map with the window centroidâs location. P_c is the critical location where R_f = R_o. Suppose that the detected wall location moves by δ_w. Correspondingly, P_c moves by δ_c with δ_c < δ_w. If δ_w < d/4 = 4 pixels, then δ_c < 4 pixels. Thus, no more than one evaluation sample switches its value and the created bias error is able to approach zero in the following PIV evaluation with PID. The wall detection technique presented in the paper follows the idea of Kähler et al (2006), which identifies the wall location by the symmetric line of the corresponding image pairs mirrored at the wall. The accuracy of the technique depends on the determination of particle position, the curvature of wall boundary, magnification in the optical path and so on. In the experiments, the derivation of wall location might be 1–3 pixels.

Zoom In Zoom Out Reset image size

So far, the wall surface discussed in the paper should satisfy the condition R_c ≫ W₀, with R_c being the curvature radius in pixels. The condition is applicable for the case on a relative flat surface or using high magnification imaging system. Walls with irregular features, such as a leading edge with very large curvature or small-scaled bluntness elements on the surface, are at present not considered. Such special situation is expected to be studied in the following investigation.

3.3.3. Optimum of artificial particle density and size

The first experiment data set is used to verify the capability of the proposed method to evaluate mean results. The experiment was performed in a small quiet supersonic wind tunnel located at the National University of Defense Technology, Hunan. The facility had a rectangular cross-section of 100 mm × 120 mm. The light sheet for illuminating the particles was 0.5 mm thick. The camera resolution was 2048 pixels × 2048 pixels, corresponding to a 60 mm × 60 mm field of view. The PIV images were obtained with a 492 m s⁻¹ free stream velocity, which corresponds to a Reynolds number Re = 9 × 10⁶ and Mach number Ma = 3. (The local sound speed is 164 m s⁻¹.) Nanometre size titanium dioxide particles were used to sample the flow. The particle displacement in the freestream was nearly 90 pixels. The window-correlation evaluations used 256 pixels × 256 pixels as the coarse sample, 64 pixels × 64 pixels as the medium resolution sample and PTV as the final high spatial resolution evaluation. The results of 255 pairs of recordings were averaged to give the mean result. Figure 14 shows the final PTV results with different preprocessing methods, normalized by the friction velocity u_τ. The evaluated profiles for three cases in the logarithmic region are nearly the same. The wall-shear stress can be estimated from the logarithmic region of the boundary layer profile using the Clauser method (Clauser 1956) as

$$\begin{equation} \frac{u(y)}{u_{\tau }}=\frac{1}{K}\ln \left(\frac{y\cdot u_{\tau }}{\nu }\right)+B. \end{equation} \tag{ 25 }$$

Here, K = 0.41 and B = 5.1. The u_τ was estimated as 9.76 m s⁻¹. Not surprisingly, the evaluations differ in the viscous sub-layer. The MO mode has non-zero displacements near the interface. The OSP evaluation gives the best description of the profile near the interface, although there are still small errors near the interface (figure 14).

Zoom In Zoom Out Reset image size

The second experimental data set is used to test the ability to detect velocity fluctuations in the flow near the wall. The aim of the experiment was to investigate the origin of asymmetric vortex formation over a slender body of revolution at high angle of attack. Previous studies indicated that this phenomenon is related to the asymmetric development of disturbances in the boundary layer with large adverse pressure gradients in the vicinity of the nose tip (Ericsson and Reding 1992, Bridges 2006).

The experiment was performed in the low turbulence water tunnel located at Peking University, Beijing. The facility had a rectangular cross-section of 400 mm × 400 mm. The slender body of revolution model was made of organic glass with a fineness ratio 3:1 tangent-ogive nose. The model diameter was 25 mm. The angle of attack was set at 45°. The model surface had been polished carefully by toothpaste to reduce the surface diffused reflection. The light sheet was placed perpendicular to the model axis. The light sheet thickness was estimated to be 0.2 mm. The measured cross-section was 18 mm from the model tip. The flow measurements used a histar8G type high speed CMOS camera with the Nikkor Micro 200 mm lens and a 200 mm long extension ring. The camera resolution was 1024 pixels × 1024 pixels, corresponding to a 9 mm × 9 mm field of view. The recording rate was set at 1000 Hz. The results presented here were taken at a 0.2 m s⁻¹ free stream velocity, which corresponds to a Reynolds number based on the diameter of the model of Re = 4940. A tiny artificial disturbance was placed on the port side in the vicinity of the tip. The downstream vortex was then shed first on the port side.

A high concentration of glass beads was seeded upstream to ensure sufficient particle density in the images. The diameter of the particles is about 1–10 μm. The camera recording mode was single frame with a time interval of nearly 1 ms. Thus, the displacement of the free stream particle images in the frames was about 25 pixels. The evaluations were conducted with a 64 pixels × 64 pixels coarse sample and 32 pixels × 32 pixels medium resolution sample and finally PTV to achieve high spatial resolution.

Figure 15 shows the mean velocity profile at azimuth 45° on the port side, summarized from 100 instantaneous results. The behaviours of different cases were the same as in the first ensemble. The OSP evaluation gives the best description of the profile near the interface. There are some differences from the simulations (Moore 1951), because some conditions such as angle of attack and the pressure gradient, were different.

Zoom In Zoom Out Reset image size

High speed PIV system was used to enhance the instantaneous measurements accuracy. The instantaneous results were summarized by every ten results so that some interferences, e.g. the local unsteady and non-uniform particle image distributions, were reduced. Figure 16 shows the velocity fluctuations' distribution, which are mainly in the thin shear layer near the interface with the maximum value less than 0.015 U. The fluctuations on the port side are larger than on the starboard side. As mentioned before, an artificial disturbance was placed on the port side ahead of the measured cross-section. The evaluated result is in agreement with the reality.

Zoom In Zoom Out Reset image size

The third experimental data set is used to test the capability of the proposed method to explore small flow structures near the interface. In the recent years, the existence of second-mode waves and their significance to the transition of hypersonic boundary layer (figure 17) have been investigated theoretically, experimentally and numerically by many researchers (Mack 1984, Schneider 2004, Zhong and Wang 2012). However, the direct instantaneous PIV measurement is still a challenge. The third ensemble of experimental data was obtained in the newly established Mach 6 quiet wind tunnel (M6QT) of Peking University. The free stream velocity was around 800 m s⁻¹, corresponding to unit Reynolds number Re = 5.8 × 10⁶ /m and Mach number Mach = 6. The PIV time interval was more than 1 μs which made particle displacements between frames as high as 2 mm, the same order of the wave length of second-mode waves. The corresponding displacements on the recordings were about 100 pixels. Meanwhile, the ratio of in-plane particle pair was less than 50%, which led to a small SNR in correlation map.

Zoom In Zoom Out Reset image size

The images were rotated horizontally at first and preprocessed by the OSP method. Then they were evaluated in sum-of-correlation mode. The evaluations were conducted with a 256 pixels × 256 pixels coarse sample and then 128 pixels × 128 pixels medium resolution sample to capture the mean displacements. Based on the mean result, the instantaneous recordings were then evaluated with 64 pixels × 64 pixels small sample. Smaller sample is unavailable because of the low SNR. Figure 18 compares the results between the $\mathbf {MO\& RV}$ and the OSP method. It can be seen that the new method has successfully extracted the small flow structures of the second-mode wave in the hypersonic boundary layer from PIV images with low SNR when the traditional method was not successful. When the transition occurred, a series of small vortex was created in the hypersonic boundary layer by the wave–vorticity interaction. The inner structure of the vortex is shown in figure 18(b). The measured wavelength coincides with the flow visualization and pressure measurement (Zhang et al 2013). Such result is obtained for the first time, which proves the capability of the current method to explore the near-wall small flow structures in the high speed flow with low SNR.

Zoom In Zoom Out Reset image size