Measurement Method of Nanofluids Average Velocity Based on Laser Speckle Image

Abstract

In this paper, based on the optical properties of speckle, the correlation between laser speckle image and nanoparticles is analyzed, and a method to measure the flow velocity of nanofluids using speckle image is proposed. The influence of nanofluids type, temperature and concentration on speckle image was investigated. On the basis of grating spatial filtering, based on PIV (Particle Image Velocimetry) cross-correlation algorithm and optical flow field algorithm, the principle of gray conservation and interpretation window are introduced to extract the spectral characteristic curve of speckle image signal. Considering the interference of the environment noise, the wavelet packet decomposition method is proposed to separate the low-frequency features and obtain the peak frequency of the signal. The static and dynamic speckle images are analyzed by establishing a circular tube flow measurement system with laser speckle circulation to verify the rationality and feasibility of the velocity measurement method. The experimental results show that the speckle image of CuO nanofluids in static experiment is better than that of Al₂O₃ nanofluids, and increasing temperature has no obvious effect of improving image quality. The average relative error between the calculated results and the measured results is 4.9 and 4.5%, which proves that this method is reasonable and feasible.

SUPPLEMENTARY MATERIALS WAVELET PACKET TRANSFORM DETAILS

Firstly, the coefficients of orthogonal low-pass filter and high pass filter are defined as l_s and h_s, respectively, and the following conditions must be met:

$$\begin{gathered} \sum {{l}_{{s - 2}}}{{l}_{{s - 2k}}} = {{\delta }_{{j,k}}},\quad \sum {{l}_{s}} = \sqrt 2 , \\ {{h}_{k}} = {{( - 1)}^{k}}{{h}_{{1 - k}}}. \\ \end{gathered} $$

(1)

If the sequence of wavelet packets is defined as: $\{ {{T}_{s}}(t){\text{|}}s \in {{N}_{1}}\}$, then there are:

$$\left\{ \begin{gathered} {{T}_{{2s}}}(t) = \sqrt 2 \sum {{l}_{k}}{{T}_{s}}(2t - k), \hfill \\ {{T}_{{2s + 1}}}(t) = \sqrt 2 \sum {{h}_{k}}{{T}_{s}}(2t - k). \hfill \\ \end{gathered} \right.$$

(2)

The scaling function of wavelet packet function is:

$${{T}_{0}}(t) = {{F}^{{ - 1}}}\left[ {\prod {{H}_{0}}\left( {\frac{\omega }{{{{2}^{j}}}}} \right)} \right].$$

(3)

Where the frequency response of l_s is H₀(w), as follows:

$${{H}_{0}}({{\omega }}) = \frac{1}{{\sqrt 2 }}\mathop \sum \limits_k {{l}_{k}}{{e}^{{ - jk{{\omega }}}}}.$$

(4)

Then, from formula (2), we can get:

$$\begin{gathered} {{T}_{s}}(t - k) = \frac{1}{{\sqrt 2 }}\sum\limits_i {{{l}_{{k - 2i}}}} {{T}_{{2s}}}\left( {\frac{t}{2} - i} \right) \\ + \,\,\frac{1}{{\sqrt 2 }}\sum\limits_i {{{h}_{{k - 2i}}}} {{T}_{{2s + 1}}}\left( {\frac{t}{2} - i} \right). \\ \end{gathered} $$

(5)

It can be seen from the above formula that on the left side of the formula, it constitutes a set of normal orthogonal bases of L²(R).

The continuous time signal of discrete sampling signal is given as follows:

$$f(t) = \sum\limits_k^{} {{{e}_{{0,0,k}}}{{T}_{0}}(t - k)} $$

(6)

where e_0.0,k represents the k-th component of the discrete sampling signal. Formula (6) is similar to Shannon’s theorem. Then from formula (5), we can get:

$${{T}_{0}}(t\, - \,k)\, = \,\frac{1}{{\sqrt 2 }}\sum\limits_t^{} {{{l}_{{k - 2i}}}{{T}_{0}}\left( {\frac{t}{2}\, - \,i} \right)\, + \,\frac{1}{{\sqrt 2 }}\sum\limits_t^{} {{{h}_{{k - 2i}}}{{T}_{t}}\left( {\frac{t}{2}\, - \,i} \right).} } $$

(7)

Formula (7) is combined with formula (6), and the dissolving formula (6) is as follows:

$$\begin{gathered} f(t) = \sum\limits_k^{} {{{e}_{{0,0,k}}}\left[ {\left. {\frac{1}{{\sqrt 2 }}} \right)\sum\limits_i^{} {{{l}_{{k - 2i}}}{{T}_{0}}\left( {\frac{t}{2} - i} \right)} } \right.} \\ \left. { + \frac{1}{{\sqrt 2 }}\sum\limits_i^{} {{{h}_{{k - 2i}}}{{T}_{1}}\left( {\frac{t}{2} - i} \right)} } \right] \\ \end{gathered} $$

$$\begin{gathered} = \sum\limits_i^{} {\frac{1}{{\sqrt 2 }}\left( {\sum\limits_k^{} {{{e}_{{0,0,k}}}{{l}_{{k - 2i}}}} } \right){{T}_{0}}\left( {\frac{t}{2} - i} \right)} \\ + \,\sum\limits_i^{} {\frac{1}{{\sqrt 2 }}\left( {\sum\limits_k^{} {{{e}_{{0,0,k}}}{{l}_{{k - 2i}}}} } \right){{T}_{0}}\left( {\frac{t}{2} - i} \right)} . \\ \end{gathered} $$

(8)

In the above formula, let e_1,0,k be:

$${{e}_{{1,0,k}}} = \sum\limits_g^t {{{e}_{{0,0,g}}}{{l}_{{g - 2k}}},\quad {{e}_{{1,1,k}}} = \sum\limits_g^{} {{{e}_{{0,0,g}}}{{h}_{{g - 2k}}}} } $$

(9)

The formula (9) is further substituted into (8) to simplify:

$$f(t) = \sum\limits_k^{} {\frac{1}{{\sqrt 2 }}{{e}_{{1,0,k}}}{{T}_{0}}\left( {\frac{t}{2} - k} \right) + \sum\limits_k^{} {\frac{1}{{\sqrt 2 }}{{e}_{{1,1,k}}}\left( {\frac{t}{2} - k} \right)} } $$

(10)

Substitute the following formula into formula (8):

$${{T}_{0}}\left( {\frac{t}{2} - i} \right),\quad {{T}_{1}}\left( {\frac{t}{2} - i} \right).$$

(11)

There are:

$$f(t) = \sum\limits_{s = 0}^{{{2}^{J}} - 1} {\sum\limits_{k \in Z} {{{w}_{{J,s,k}}}} } {{2}^{{ - J/2}}}{{T}_{s}}({{2}^{{ - J}}} - k);$$

(12)

$$\left\{ \begin{gathered} {{w}_{{j,2s,k}}} = \mathop \sum \limits_m {{w}_{{j - 1,s,g}}}{{l}_{{g - 2k}}}, \hfill \\ {{w}_{{j,2s + 1,k}}} = \mathop \sum \limits_m {{w}_{{j - 1,s,g}}}{{h}_{{g - 2k}}}. \hfill \\ \end{gathered} \right.$$

(13)

The above algorithm is called tower algorithm, in which $j,s$ can be expressed as the following formula:

$$\begin{gathered} j = 1,2,3, \ldots ,J, \\ s = 0,1, \ldots {{,2}^{j}} - 1. \\ \end{gathered} $$

(14)

Then, the corresponding formula is as follows:

$${{{\mathbf{w}}}_{{j - 1,s,k}}} = \sum\limits_g {{{l}_{{k - 2g}}}} {{{\mathbf{w}}}_{{j,2s,g}}} + \sum\limits_s {{{h}_{{k - 2g}}}} {{{\mathbf{w}}}_{{j,2s + 1.g}}}.$$

(15)

From the above deduction, we can see that formula (13) is the wavelet packet decomposition algorithm, and formula (15) is the wavelet packet recovery method.