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Alibhoy and Hui-Ling Chiang",authors:[{id:"147077",title:"Dr.",name:"Hui-Ling",middleName:null,surname:"Chiang",fullName:"Hui-Ling Chiang",slug:"hui-ling-chiang"},{id:"151519",title:"Mr.",name:"Abbas",middleName:null,surname:"Alibhoy",fullName:"Abbas Alibhoy",slug:"abbas-alibhoy"}]},{id:"37710",title:"The Role of Endocytosis in the Creation of the Cortical Division Zone in Plants",slug:"the-role-of-endocytosis-in-the-creation-of-the-cortical-division-zone-in-plants",signatures:"Ichirou Karahara, L. Andrew Staehelin and Yoshinobu Mineyuki",authors:[{id:"146804",title:"Dr.",name:"Yoshinobu",middleName:null,surname:"Mineyuki",fullName:"Yoshinobu Mineyuki",slug:"yoshinobu-mineyuki"},{id:"148613",title:"Dr.",name:"Ichirou",middleName:null,surname:"Karahara",fullName:"Ichirou Karahara",slug:"ichirou-karahara"},{id:"148615",title:"Prof.",name:"Andrew",middleName:null,surname:"Staehelin",fullName:"Andrew Staehelin",slug:"andrew-staehelin"}]},{id:"37711",title:"Roles of Cellular Redox Factors in Pathogen and Toxin Entry in the Endocytic Pathways",slug:"roles-of-cellular-redox-factors-in-pathogen-and-toxin-entry-in-the-endocytic-pathways",signatures:"Jianjun Sun",authors:[{id:"142839",title:"Dr.",name:"Jianjun",middleName:null,surname:"Sun",fullName:"Jianjun Sun",slug:"jianjun-sun"}]},{id:"37712",title:"Advanced Optical Imaging of Endocytosis",slug:"advanced-optical-imaging-of-endocytosis",signatures:"Jesse S. Aaron and Jerilyn A. Timlin",authors:[{id:"145101",title:"Dr.",name:"Jerilyn",middleName:null,surname:"Timlin",fullName:"Jerilyn Timlin",slug:"jerilyn-timlin"},{id:"149098",title:"Dr.",name:"Jesse",middleName:null,surname:"Aaron",fullName:"Jesse Aaron",slug:"jesse-aaron"}]},{id:"37713",title:"Imaging of Endocytosis in Paramecium by Confocal Microscopy",slug:"imaging-of-endocytosis-in-paramecium-by-confocal-microscopy",signatures:"Paola Ramoino, Alberto Diaspro, Marco Fato and Cesare Usai",authors:[{id:"145158",title:"Dr.",name:"Paola",middleName:null,surname:"Ramoino",fullName:"Paola Ramoino",slug:"paola-ramoino"},{id:"148812",title:"Prof.",name:"Alberto",middleName:null,surname:"Diaspro",fullName:"Alberto Diaspro",slug:"alberto-diaspro"},{id:"148814",title:"Prof.",name:"Marco",middleName:null,surname:"Fato",fullName:"Marco Fato",slug:"marco-fato"},{id:"148816",title:"Dr.",name:"Cesare",middleName:null,surname:"Usai",fullName:"Cesare Usai",slug:"cesare-usai"}]},{id:"37720",title:"Caveolae-Dependent Endocytosis in Viral Infection",slug:"caveolae-dependent-endocytosis-in-viral-infection",signatures:"Norica Branza-Nichita, Alina Macovei and Catalin Lazar",authors:[{id:"145513",title:"Dr.",name:"Norica",middleName:null,surname:"Branza Nichita",fullName:"Norica Branza Nichita",slug:"norica-branza-nichita"}]},{id:"37721",title:"Clathrin-Associated Endocytosis as a Route of Entry into Cells for Parvoviruses",slug:"clathrin-associated-endocytosis-as-a-route-of-entry-into-cells-for-parvoviruses",signatures:"F. Brent Johnson and Enkhmart Dudleenamjil",authors:[{id:"145118",title:"Dr.",name:"F. Brent",middleName:null,surname:"Johnson",fullName:"F. Brent Johnson",slug:"f.-brent-johnson"}]},{id:"37737",title:"Endocytosis of Non-Enveloped DNA Viruses",slug:"endocytosis-of-non-enveloped-dna-viruses",signatures:"Maude Boisvert and Peter Tijssen",authors:[{id:"144619",title:"Prof.",name:"Peter",middleName:null,surname:"Tijssen",fullName:"Peter Tijssen",slug:"peter-tijssen"},{id:"149965",title:"BSc.",name:"Maude",middleName:null,surname:"Boisvert",fullName:"Maude Boisvert",slug:"maude-boisvert"}]},{id:"37724",title:"Pathogen and Toxin Entry - How Pathogens and Toxins Induce and Harness Endocytotic Mechanisms",slug:"pathogen-and-toxin-entry-how-pathogens-and-toxins-induce-and-harness-endocytotic-mechanisms",signatures:"Thorsten Eierhoff, Bahne Stechmann and Winfried Römer",authors:[{id:"145373",title:"Prof.",name:"Winfried",middleName:null,surname:"Römer",fullName:"Winfried Römer",slug:"winfried-romer"},{id:"145376",title:"Dr.",name:"Thorsten",middleName:null,surname:"Eierhoff",fullName:"Thorsten Eierhoff",slug:"thorsten-eierhoff"},{id:"145613",title:"Dr.",name:"Bahne",middleName:null,surname:"Stechmann",fullName:"Bahne Stechmann",slug:"bahne-stechmann"}]},{id:"37725",title:"The Unique Endosomal/Lysosomal System of Giardia lamblia",slug:"the-unique-endosomal-lysosomal-system-of-giardia-lamblia",signatures:"Maria C. Touz",authors:[{id:"142741",title:"Dr.",name:"Maria",middleName:"Carolina",surname:"Touz",fullName:"Maria Touz",slug:"maria-touz"}]},{id:"37727",title:"Mutual Regulation of Receptor-Mediated Cell Signalling and Endocytosis: EGF Receptor System as an Example",slug:"mutual-regulation-of-receptor-mediated-cell-signalling-and-endocytosis-egf-receptor-system-as-an-exa",signatures:"Zhixiang Wang",authors:[{id:"146796",title:"Dr.",name:"Zhixiang",middleName:null,surname:"Wang",fullName:"Zhixiang Wang",slug:"zhixiang-wang"}]},{id:"37728",title:"Endocytosis in Notch Signaling Activation",slug:"endocytosis-in-notch-signaling-activation",signatures:"Elisa Sala, Luca Ruggiero, Giuseppina Di Giacomo and Ottavio Cremona",authors:[{id:"149522",title:"Prof.",name:"Ottavio",middleName:null,surname:"Cremona",fullName:"Ottavio Cremona",slug:"ottavio-cremona"},{id:"149524",title:"Dr.",name:"Giuseppina",middleName:null,surname:"Di Giacomo",fullName:"Giuseppina Di Giacomo",slug:"giuseppina-di-giacomo"},{id:"149526",title:"Dr.",name:"Elisa",middleName:null,surname:"Sala",fullName:"Elisa Sala",slug:"elisa-sala"},{id:"155377",title:"Dr.",name:"Luca",middleName:null,surname:"Ruggiero",fullName:"Luca Ruggiero",slug:"luca-ruggiero"}]},{id:"37731",title:"Hyaluronan Endocytosis: Mechanisms of Uptake and Biological Functions",slug:"hyaluronan-endocytosis-mechanisms-of-uptake-and-biological-functions",signatures:"Ronny Racine and Mark E. Mummert",authors:[{id:"147098",title:"PhD.",name:"Mark",middleName:null,surname:"Mummert",fullName:"Mark Mummert",slug:"mark-mummert"},{id:"147100",title:"Mr.",name:"Ronny",middleName:null,surname:"Racine",fullName:"Ronny Racine",slug:"ronny-racine"}]},{id:"37732",title:"Identification of Ubiquitin System Factors in Growth Hormone Receptor Transport",slug:"identification-of-ubiquitin-system-factors-in-growth-hormone-receptor-transport",signatures:"Johan A. Slotman, Peter van Kerkhof, Gerco Hassink, Hendrik J. Kuiken and Ger J. Strous",authors:[{id:"144795",title:"Prof.",name:"Ger",middleName:null,surname:"Strous",fullName:"Ger Strous",slug:"ger-strous"}]},{id:"37733",title:"Endocytosis of Particle Formulations by Macrophages and Its Application to Clinical Treatment",slug:"endocytosis-of-particle-formulations-by-macrophages-and-its-application-to-clinical-treatment",signatures:"Keiji Hirota and Hiroshi Terada",authors:[{id:"147552",title:"Prof.",name:"Hiroshi",middleName:null,surname:"Terada",fullName:"Hiroshi Terada",slug:"hiroshi-terada"}]},{id:"37734",title:"Endosomal Escape Pathways for Non-Viral Nucleic Acid Delivery Systems",slug:"endosomal-escape-pathways-for-non-viral-nucleic-acid-delivery-systems",signatures:"Wanling Liang and Jenny K. W. Lam",authors:[{id:"143095",title:"Dr.",name:"Jenny Ka Wing",middleName:null,surname:"Lam",fullName:"Jenny Ka Wing Lam",slug:"jenny-ka-wing-lam"},{id:"146268",title:"MSc.",name:"Wanling",middleName:null,surname:"Liang",fullName:"Wanling Liang",slug:"wanling-liang"}]}]}]},onlineFirst:{chapter:{type:"chapter",id:"72271",title:"Extracting Coherent Structures in Near-Wall Turbulence Based on Wavelet Analysis",doi:"10.5772/intechopen.92015",slug:"extracting-coherent-structures-in-near-wall-turbulence-based-on-wavelet-analysis",body:'\nTurbulence is a commonly seen but very complicated phenomenon in nature. Numerous tests have proven that turbulence is not a pure random process but contains different scales of fluctuations called coherent structures [1, 2, 3]. These structures significantly contribute to fluid entrainment and mass, momentum, and heat transfer [4, 5]. Therefore, investigating the coherent structures is of great significance to undercover the physics and to realize flow control.
\nAmong the techniques of turbulence analysis, wavelet transform has been proven feasible and power to detect and extract the coherent structures in turbulence [6, 7, 8, 9]. Early works are based on continuous wavelet transform (CWT). Liandrant [10] and Jiang [11, 12] proposed the maximum energy principle, which considered the signal at the maximum energy scale as the burst events in turbulence. Kim [13] identified the coherent structure around a vibrating cantilever based on CWT. However, a drawback of CWT is that it is unable to reconstruct the signal if the mother wavelet is not orthogonal [14, 15, 16]. To solve this problem, Longo [17] used the multiresolution analysis technique based on the discrete wavelet transform (DWT) and extracted the structures in turbulence. DWT has evident advantages compared with CWT since it is invertible and multi-scaled scales can be analyzed. Kadoch [18] combined DWT and direct numerical simulation (DNS), whose results proved that coherent structures preserve the vortical structures with only about 4\n
In this work, measurement of the turbulent boundary layer is carried out using hot-film anemometer in a gravitational low-speed water tunnel. A procedure based on the WT and correlation analysis is proposed to extract and verify the coherent and incoherent structure in turbulence.
\nA gravitational low-speed water tunnel was constructed for the experiment. The gravity generated by the water level difference drives the water flow in the tunnel, and the flow can be tested in the experimental section (\nFigure 1\n). A maximum water speed of 2.0 \n
Sketch (left) and photo (right) of the gravitational low-speed water tunnel. (1) Water tank; (2) stabilization section; (3) contraction section; (4) experimental section; (5) electromagnetism flowmeter; (6) water level observation section; (7) water pump; (8) water storage basin; (9) switch valve.
Sketch of experimental setups. (1) PC; (2) hot-film anemometer; (3) coordinate frame; (4) experimental plate; (5) hot-film probe.
By using the experimental setups in \nFigure 2\n, the flow velocity of the turbulence boundary layer was measured at a series of positions in the vertical direction. The mean velocity profile and the turbulence intensity distribution at the water speed 0.4 \n
The profile of mean velocity at the water speed 0.4 \n\nm\n/\ns\n\n.
Turbulence intensity at the water speed 0.4 \n\nm\n/\ns\n\n.
WT is a mapping of a time function, in a one-dimensional case, to the two dimensional time-scale joint representation. The temporal aspect of the signal can be preserved. The wavelet transform provides multiresolution analysis with dilated windows. The high-frequency part of the signal is analyzed using narrow windows, and the low-frequency part is done using wide windows. WT decomposes the signal into different frequency components and then studies each component with a resolution matched to its scale. It has advantages over traditional Fourier methods in analyzing physics where the signal contains discontinuities and sharp spikes.
\nWT of a signal \n
where \n
Scale a and position b should be discretized for applications. Usually we choose \n
The corresponding DWT can be expressed as:
\nThe orthogonality of \n
By choosing the scale \n
where \n
For turbulence, the fluctuating velocity of turbulence can be normally divided into two subparts:
\nwhere \n
By adopting the multiresolution analysis (\nFigure 5\n), the turbulence signal \n
Sketch of the multiresolution analysis.
where \n
To extract the coherent structures in turbulence, the signals at the central area of turbulence should be selected. According to previous studies [20, 21, 22], the formation of the coherent structures in turbulence is formed in the area of \n
Fluctuating velocity signals at three positions (a) \n\n\ny\n+\n\n=\n20.8\n\n, (b) \n\n\ny\n+\n\n=\n33.5\n\n, (c) \n\n\ny\n+\n\n=\n42.6\n\n.
For preliminary evaluations of the coherent structures, CWT is first utilized for the analysis. CWT is a mathematical mapping similar to the Fourier transform [23, 24]. It is linear, invertible, and orthogonal. However, the Fourier transform uses basis functions, including the sines and cosines, which extend to infinity in time, while wavelet basis functions drop towards zero outside a finite domain (compact support). This allows for an effective localization in both time and frequency. CWT uses inner products to measure the similarity between the turbulence signal and the wavelet function, which defines a mapping between the two. CWT compares the turbulence signal to shifted and compressed/stretched versions of the wavelet function. Compressing/stretching is also referred to as dilation or scaling and corresponds to the physical notion of scale. By continuously varying the values of the scale parameter, \n
In the work, the 5th order of Daubechies wavelet was selected as the basis function, whose central frequency fc is 0.6667 \n
Continuous wavelet transform coefficients at (a) \n\n\ny\n+\n\n=\n20.8\n\n, (b) \n\n\ny\n+\n\n=\n33.5\n\n, (c) \n\n\ny\n+\n\n=\n42.6\n\n.
To obtain the frequency range of coherent structures in turbulence, power spectrum densities of the three selected signals were calculated in \nFigure 8\n. The centralized frequencies of the coherent structures are found in the range 0 \n
Power spectrum densities at the three positions (a) \n\n\ny\n+\n\n=\n20.8\n\n, (b) \n\n\ny\n+\n\n=\n33.5\n\n, (c) \n\n\ny\n+\n\n=\n42.6\n\n.
WT of a signal is equivalent to local cross-correlation analysis between the signal and wavelet function. OWT carries out the multi-resolution analysis for both decomposition and reconstruction of the original turbulence signal. It is thought of the wavelet coefficients as digital filters as which the original signal is passed through low-pass filters to decompose into low-frequency components and passed through high-pass filters to analyze into high-frequency components.
\nUsing the multiresolution analysis of OWT, the turbulence signal was split into seven scales as in \nTable 1\n, which eliminates most of the redundant signals. The frequency range of the approximate signal is mainly in the range 0 \n
Signal | \nFrequency/\n | \nEnergy/\n | \n||
---|---|---|---|---|
\n | \n | \n\n | \n\n\n | \n\n\n | \n
\n\n | \n0 \n | \n100 | \n100 | \n100 | \n
\n\n | \n0 \n | \n85.6498 | \n77.0677 | \n81.3847 | \n
\n\n | \n260 \n | \n0.0147 | \n0.0200 | \n0.0259 | \n
\n\n | \n520 \n | \n0.0062 | \n0.0086 | \n0.0129 | \n
\n\n | \n1042 \n | \n0.0019 | \n0.0036 | \n0.0042 | \n
\n\n | \n2083 \n | \n0.0464 | \n0.0703 | \n0.0868 | \n
\n\n | \n4167 \n | \n0.3957 | \n0.8099 | \n0.8505 | \n
\n\n | \n83,334 \n | \n3.5993 | \n5.5974 | \n5.2322 | \n
\n\n | \n16,668 \n | \n10.2411 | \n16.4225 | \n12.4028 | \n
Frequency and energy distribution of seven level decompositions.
The extracted signals of each level are shown in \nFigure 9\n, where “A7” is the approximate signal, i.e., the coherent structures; where “sD7” is the incoherent structures, which is calculated by:
\nExtracted signals in turbulence. (a) Incoherent structure (\n\n\ny\n+\n\n=\n20.8\n\n); (b) coherent structure (\n\n\ny\n+\n\n=\n20.8\n\n); (c) incoherent structure (\n\n\ny\n+\n\n=\n33.5\n\n); (d) coherent structure (\n\n\ny\n+\n\n=\n33.5\n\n); (e) incoherent structure (\n\n\ny\n+\n\n=\n42.6\n\n); (f) coherent structure (\n\n\ny\n+\n\n=\n42.6\n\n).
and where “s” is the original signal. “D1 \n
To characterize the properties of the extracted signals, the probability density functions (PDFs) were analyzed in \nFigure 10\n. It can be observed that the incoherent structures are approximately Gaussian, demonstrating isotropic characteristics. The PDFs of coherent structures deviate from the Gaussian distribution, presenting strong anisotropic characteristics. And the PDFs of the coherent structures resemble that of the original turbulence signals. This means that coherent structures contribute the most to turbulence entrainment.
\nProbability density functions at three testing positions (a) \n\n\ny\n+\n\n=\n20.8\n\n, (b) \n\n\ny\n+\n\n=\n33.5\n\n, (c) \n\n\ny\n+\n\n=\n42.6\n\n.
For further validation of the extracted coherent and incoherent structures, correlation analysis was carried out here. A correlation parameter \n
where \n
The flow field of the turbulence boundary layer was measured using hot-film anemometer in a gravitational low-speed water tunnel. The coherent and incoherent structures in turbulence were separated successfully with an extraction method based on WT. With CWT, the turbulent structures can be observed in various scales. With DWT, multiresolution analysis can be carried out for the decomposition and reconstruction of vortical structures in different scales. The PDF of the incoherent structures was found to obey the Gaussian distribution, while that of the coherent structures deviate from it. The similarity of the PDFs of the coherent structures and the original turbulence signal demonstrate that the coherent structures make most contributions to turbulence. A correlation parameter between coherent and incoherent structures was defined, which proves the successful separation of coherent structure from turbulence.
\nThe authors acknowledge the support from the National Natural Science Foundation of China (Grant No. 51879218, 51679203) and Fundamental Research Funds for the Central Universities (Grant No. 3102018gxc007, 3102020HHZY030004).
\nThe authors declare no conflict of interest.
The continuous wavelet transform (CWT) has the drawback of redundancy. As the dilation parameter a and the shift parameter b take continuous values, the resulting CWT is a very redundant representation. Therefore, the discrete wavelet transform was proposed to overcome this problem by setting the scale and shift parameters on a discrete set of basis functions. Their discretization is performed by:
\nwhere \n
and the discrete wavelet decomposition of a signal \n
where \n
The basis function set \n
The advantage of the DWT is the multi-resolution analysis ability. Although the standard DWT is powerful, it has three major disadvantages that undermine its applications: shift sensitivity, poor directionality, and absence of phase information.
\nComplex wavelet transform can be used to overcome these drawbacks. It uses complex-valued filtering and decomposes the signal into real and imaginary parts, which can be used to calculate the amplitude and phase information.
\nFor turbulence analysis, the complex wavelet transform should be used since the modulus of the wavelet coefficients allows characterizing the evolution of the turbulent energy in both the time and frequency domains. The real-valued wavelets will make it difficult to sort out the features of the signal or the wavelet. On the contrary, the complex-valued wavelets can eliminate these spurious oscillations. The complex extension of a real signal \n
where \n
The complex wavelet transform is able to remove the redundancy for turbulence analysis where the directionality and phase information play important roles.
\nReprinted (adapted) with permission from Chinese Physics B, 2013, 22(7): 074703.
\nWT | wavelet transform |
PSD | power spectrum density |
OWT | orthogonal wavelet transform |
probability density function | |
CWT | continuous wavelet transform |
DWT | discrete wavelet transform |
DNS | direct numerical simulation |
\n\n\ns\n\nt\n\n\n\n | original turbulence signal |
\n\n\n\nφ\n\nm\n,\nn\n\n\n\nt\n\n\n\n | scaling function |
\n\n\na\n\n\n | scale parameter |
\n\n\nb\n\n\n | position parameter |
\n\n\n\nm\n0\n\n\n\n | critical scale |
\n\n\n\nψ\n\nm\n,\nn\n\n\n\nt\n\n\n\n | wavelet function |
\n\n\n\ns\n˜\n\n\n\n | coherent part of the signal |
\n\n\n\ns\n′\n\n\n\n | incoherent part of the signal |
\n\n\nf\n\n\n | frequency |
\n\n\n\nf\ns\n\n\n\n | sampling frequency |
\n\n\n\nf\nc\n\n\n\n | central frequency of particular wavelet basis |
\n\n\n\ny\n+\n\n\n\n | dimensionless wall distance |
\n\n\nA\n7\n,\n\nD\n1\n∼\nD\n7\n\n\n | detailed signal of each level |
\n\n\nv\n\n\n | fluctuating velocity signal |
\n\n\nβ\n\n\n | correlation parameter |
Supporting women in scientific research and encouraging more women to pursue careers in STEM fields has been an issue on the global agenda for many years. But there is still much to be done. And IntechOpen wants to help.
",metaTitle:"IntechOpen Women in Science Program",metaDescription:"Supporting women in scientific research and encouraging more women to pursue careers in STEM fields has been an issue on the global agenda for many years. But there is still much to be done. And IntechOpen wants to help.",metaKeywords:null,canonicalURL:null,contentRaw:'[{"type":"htmlEditorComponent","content":"At IntechOpen, we’re laying the foundations for the future by publishing the best research by women in STEM – Open Access and available to all. Our Women in Science program already includes six books in progress by award-winning women scientists on topics ranging from physics to robotics, medicine to environmental science. Our editors come from all over the globe and include L’Oreal–UNESCO For Women in Science award-winners and National Science Foundation and European Commission grant recipients.
\\n\\nWe aim to publish 100 books in our Women in Science program over the next three years. We are looking for books written, edited, or co-edited by women. Contributing chapters by men are welcome. As always, the quality of the research we publish is paramount.
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\\n\\nAdvantages of Publishing with IntechOpen
\\n\\n\\n"}]'},components:[{type:"htmlEditorComponent",content:'At IntechOpen, we’re laying the foundations for the future by publishing the best research by women in STEM – Open Access and available to all. Our Women in Science program already includes six books in progress by award-winning women scientists on topics ranging from physics to robotics, medicine to environmental science. Our editors come from all over the globe and include L’Oreal–UNESCO For Women in Science award-winners and National Science Foundation and European Commission grant recipients.
\n\nWe aim to publish 100 books in our Women in Science program over the next three years. We are looking for books written, edited, or co-edited by women. Contributing chapters by men are welcome. As always, the quality of the research we publish is paramount.
\n\nAll project proposals go through a two-stage peer review process and are selected based on the following criteria:
\n\nPlus, we want this project to have an impact beyond scientific circles. We will publicize the research in the Women in Science program for a wider general audience through:
\n\nInterested? If you have an idea for an edited volume or a monograph, we’d love to hear from you! Contact Ana Pantar at book.idea@intechopen.com.
\n\n“My scientific path has given me the opportunity to work with colleagues all over Europe, including Germany, France, and Norway. Editing the book Graph Theory: Advanced Algorithms and Applications with IntechOpen emphasized for me the importance of providing valuable, Open Access literature to our scientific colleagues around the world. So I am highly enthusiastic about the Women in Science book collection, which will highlight the outstanding accomplishments of women scientists and encourage others to walk the challenging path to becoming a recognized scientist." Beril Sirmacek, TU Delft, The Netherlands
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