- Recent developments in the growth of CNTs
- Methods to modify the surfaces of CNTs and decorate their surfaces for specific applications
- Applications of CNTs in biocomposites such as in orthopedic bone cement
- Application of CNTs as chemical sensors
- CNTs for fuelcells
- Health related issues when using CNTs
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Brown",slug:"donald-e.-brown"}]},{id:"13284",title:"Development of a Classification Rule Mining Framework by Using Temporal Pattern Extraction",slug:"development-of-a-classification-rule-mining-framework-by-using-temporal-pattern-extraction",signatures:"Hidenao Abe",authors:[{id:"15499",title:"Dr.",name:"Hidenao",middleName:null,surname:"Abe",fullName:"Hidenao Abe",slug:"hidenao-abe"},{id:"16800",title:"Dr.",name:"Miho",middleName:null,surname:"Ohsaki",fullName:"Miho Ohsaki",slug:"miho-ohsaki"},{id:"16801",title:"Prof",name:"Takahira",middleName:null,surname:"Yamaguchi",fullName:"Takahira Yamaguchi",slug:"takahira-yamaguchi"}]},{id:"13285",title:"Evolutionary-Based Classification Technique",slug:"evolutionary-based-classification-technique",signatures:"Rasha Shaker Abdul-Wahab",authors:[{id:"17076",title:"Dr.",name:"Rasha",middleName:null,surname:"abdul-wahhab",fullName:"Rasha abdul-wahhab",slug:"rasha-abdul-wahhab"},{id:"17082",title:"Dr.",name:"Amer",middleName:null,surname:"Al-Roubaie",fullName:"Amer Al-Roubaie",slug:"amer-al-roubaie"}]},{id:"13286",title:"Multiobjective Design Exploration in Space Engineering",slug:"multiobjective-design-exploration-in-space-engineering",signatures:"Akira Oyama and Kozo Fujii",authors:[{id:"14100",title:"Dr.",name:"Akira",middleName:null,surname:"Oyama",fullName:"Akira Oyama",slug:"akira-oyama"},{id:"16345",title:"Prof.",name:"Kozo",middleName:null,surname:"Fujii",fullName:"Kozo Fujii",slug:"kozo-fujii"}]},{id:"13287",title:"Privacy Preserving Data Mining",slug:"privacy-preserving-data-mining",signatures:"Xinjing Ge and Jianming Zhu",authors:[{id:"14533",title:"Prof.",name:"Jianming",middleName:null,surname:"Zhu",fullName:"Jianming Zhu",slug:"jianming-zhu"}]},{id:"13288",title:"Using Markov Models to Mine Temporal and Spatial Data",slug:"using-markov-models-to-mine-temporal-and-spatial-data",signatures:"Jean-François Mari, Florence Le Ber, El Ghali Lazrak, Marc Benoît, Catherine Eng, Annabelle Thibessard and Pierre Leblond",authors:[{id:"15705",title:"Dr.",name:"Jean-Francois",middleName:null,surname:"Mari",fullName:"Jean-Francois Mari",slug:"jean-francois-mari"},{id:"16658",title:"Prof.",name:"Florence",middleName:null,surname:"Le Ber",fullName:"Florence Le Ber",slug:"florence-le-ber"},{id:"23895",title:"Dr.",name:"El Ghali",middleName:null,surname:"Lazrak",fullName:"El Ghali Lazrak",slug:"el-ghali-lazrak"},{id:"23896",title:"Prof",name:"Mard",middleName:null,surname:"Benoit",fullName:"Mard Benoit",slug:"mard-benoit"},{id:"23897",title:"Prof.",name:"Annabelle",middleName:null,surname:"Thibessard",fullName:"Annabelle Thibessard",slug:"annabelle-thibessard"},{id:"23898",title:"Dr.",name:"Catherine",middleName:null,surname:"Eng",fullName:"Catherine Eng",slug:"catherine-eng"},{id:"23899",title:"Prof.",name:"Pierre",middleName:null,surname:"Leblond",fullName:"Pierre Leblond",slug:"pierre-leblond"}]}]}]},onlineFirst:{chapter:{type:"chapter",id:"65415",title:"Leading Edge Receptivity at Subsonic and Moderately Supersonic Mach Numbers",doi:"10.5772/intechopen.83672",slug:"leading-edge-receptivity-at-subsonic-and-moderately-supersonic-mach-numbers",body:'\n
This chapter is concerned with the effect of unsteady free-stream disturbances on laminar to turbulent transition in boundary layer flows. The exact mechanism depends on the nature and intensity of the disturbances. Transition at high disturbance levels (say >1%) usually begins with the excitation of low frequency streaks in the boundary layer flow that eventually break down into turbulent spots. This phenomena was initially studied by Dryden [1] and much later for compressible flows by Marensi et al. [2]. But the focus of this chapter is on low free steam disturbances levels (say less than 1%) where the transition usually results from a series of events beginning with the generation of spatially growing instability waves by acoustic and/or vortical disturbances in the free-stream. This so-called receptivity phenomenon results in a boundary value problem and therefore differs from classical instability theory which results in an eigenvalue problem for the Rayleigh or Orr-Sommerfeld equations that only apply when the mean flow can be treated as being nearly parallel (see, for example, Reshotko, [3]). The relevant boundary conditions cannot be imposed on the Orr-Sommerfeld or Rayleigh equations in the infinite Reynolds number limit being considered here but the free-stream disturbances can produce unsteady boundary layer perturbations in regions of rapidly changing mean flow that eventually produce unstable Rayleigh or Orr-Sommerfeld equation eigensolutions further downstream. These regions of nonparallel flow can result from surface roughness elements [4, 5], blowing or suction effects [6] or from the nonparallel mean flow that occurs near the boundary layer leading edge [7, 8].
\nThe mechanism is similar in all cases but the simplest and arguably the most fundamental of these is the one resulting from the nonparallel leading edge flow and the focus here is, therefore, on that case. The initial studies were carried out for two dimensional incompressible flows. Ref. [7] used a low frequency parameter matched asymptotic expansion to show that there is an overlap domain where appropriate asymptotic solutions to the forced boundary layer equations (which apply near the edge) match onto the so-called Tollmien-Schlichting waves that satisfy the Orr-Sommerfeld equation in a region that lies somewhat further downstream. The coupling to the free-stream disturbances turns out to be fairly weak for the two dimensional incompressible flow considered in [7] due to the relatively large decay of boundary layer disturbances upstream of the Tollmien-Schlichting wave region where the Orr-Sommerfeld equation applies.
\nBut there can be a much stronger coupling in supersonic flows which can support a number of different instabilities [9]. The coupling mechanism can be either viscous or inviscid and the instability can either be of the viscous Tollmien-Schlichting type or can be purely inviscid when the mean boundary layer flow has a generalized inflection point. The inviscid coupling, which was first analyzed in [10], tends to be dominant when the obliqueness angle \n
Comparison of the Fedorov/Khokhlov solution with experiment [12].
Fedorov and Khokhlov [10] analyzed the generation of inviscid instabilities in a supersonic flat plate boundary layer by fast and slow acoustic disturbances in the free stream. They showed that the slow acoustic mode propagates downstream/upstream when the obliqueness angle \n
Smith [13] showed that viscous instabilities, which exhibit the same triple-deck structure as the subsonic Tollmien-Schlichting waves, can also occur at supersonic speeds when the obliqueness angles \n
The analysis of Ref. [7] was extended to compressible subsonic and supersonic flat plate boundary layer flows by Ricco and Wu [14] who showed that highly oblique vortical disturbances can generate a limiting form of the Smith instability [13]. They found that the instability wave lower branch lies further upstream at supersonic speeds than the subsonic lower branch and much further upstream than the incompressible lower branch considered in [7], which means that the instability wave/free-stream disturbance coupling is much greater at supersonic speeds than it is in the incompressible flow considered in [7]. Goldstein and Ricco [11] show that the instability does not possess an upper branch in this case and matches onto a low frequency (short streamwise wavenumber) Rayleigh instability (that can be identified with the 1st Mack mode) when the downstream distance is slightly smaller than the downstream distance where acoustically generated instability corresponding to the smallest possible \n
As noted above, the present chapter is concerned with the unsteady flow in a flat plate boundary layer generated by mildly oblique vortical disturbance and small \n
Low-sweep Aerion AS2 supersonic Bizjet. \n\n\nM\n∞\n\n≤\n1.5\n\n. Posted by Tim Brown on the Manufacturer Newsletter.
Since the boundary layer is believed to be convectively unstable, the receptivity phenomena are best illustrated by considering a small amplitude harmonic distortion with angular frequency \n
As noted above the phenomenon is analyzed by requiring the Reynolds number \n
The free-steam disturbances will be inviscid at the lowest order of approximation and, as is well known [15], can be decomposed into an acoustic component that carries no vorticity, and vortical and entropic components that produce no pressure fluctuations. But only the first two will be considered here.
\nThe vortical disturbance \n
where \n
but are otherwise arbitrary constants while the acoustic component is governed by the linear wave equation which has a fundamental plane wave solution
\nfor the velocity and pressure perturbation \n
and, as noted in Section 1, \n
The leading edge interaction will produce large scattered fields for \n
for the former disturbances and that
\nfor the latter, where the subscripts −/+ refer to the slow/fast acoustic modes. Eq. (7) shows that the slow mode wavenumber becomes infinite when the obliqueness angle is equal to the critical angle referred to in the introduction.
\nAs indicated above our interest here is in explaining how the incident harmonic distortions generate oblique instabilities at large downstream distances in the viscous boundary layer that forms on the surface of the plate. We begin by considering the fluctuations imposed on this flow by the free-stream vortical disturbance (2).
\nAs noted in the introduction, these disturbances will generate oblique Tollmien-Schlichting instability waves which are known to exhibit a triple-deck structure in the vicinity of their lower branch which lies at an \n
since the spanwise wavenumber must remain constant as the disturbances propagate downstream.
\nThe continuity condition (3) and the obliqueness restriction (6) will be satisfied if we put
\nThe vortical velocity (2) will then interact with the plate to produce an inviscid velocity field [12] that generates a slip velocity at the surface of the plate which must be brought to zero in a thin viscous boundary layer whose temperature, density and streamwise velocity, say \n
and are determined from the similarity equations given in Stewartson [16] and Ref. [14].
\nWe begin by considering the flow in the vicinity of the leading edge where the streamwise length scale is \n
where \n
while \n
The lowest order triple-deck solution will match onto the quasi-two dimensional solution \n
Prandtl [18], Glauert [19] and Lam and Rott [20] showed that
\nwhere
\nis an exact eigensolution of the two-dimensional linearized unsteady boundary layer equations that satisfies the homogeneous boundary conditions \n
Lam and Rott [20], [21] analyzed the two dimensional flat plate boundary layer and showed that the linearized equations possess asymptotic eigensolutions that satisfy a no-slip condition at the wall when \n
Ref. [14] showed that the Lam and Rott [20, 21] analysis also applies to compressible flows when the full compressible solution (14) and (15) is used in the outer region and the viscous wall layer solution is slightly modified to account for the temperature and viscosity variations. The function \n
where \n
The only difference from the Lam-Rott result is the \n
Fedorov and Khokhlov [10] used matched asymptotic expansions to analyze the generation of Mack mode instabilities by oblique acoustic waves of the form (4) where the wavenumbers \n
Ref. [10] shows that the boundary layer disturbance produced by diffraction of the slow acoustic wave by the nonparallel mean flow in the region where \n
of the obliqueness angle \n
takes on \n
The instability emerges from the downstream limit of the solution in this region. But as noted in the introduction this occurs too far downstream to be of practical interest when scaled up to actual flight conditions if \n
It follows from (7) that
\nwhere
\nwhen \n
This shows that \n
The spanwise wavenumber will equal the vortical spanwise wavenumber (8) when \n
Structure of diffraction region for \n\nΔ\nθ\n=\nO\n\n\nε\n\n2\n/\n3\n\n\n\n\n.
Ref. [11] shows that the smallest value of \n
The pressure in region 2 is of the form
\nwhere
\nand the surface pressure \n
where \n
and the integral and the derivative of the Airy function \n
which behaves like
\nas \n
Inserting (28) and (27) into (25) shows that
\nwhere
\nwhich is formally the same as the equation considered in [10] who showed that the solution behaves like
\nThe acoustically and vortically generated boundary layer disturbances considered in this section will eventually evolve into propagating eigensolutions in regions that lie further downstream. The resulting flow will have a triple-deck structure of the type considered in [13], [23] and [14] in the former (i.e., vortically generated) case. But the acoustically generated disturbance will only develop an eigensolution structure much further downstream. The minimum distance occurs when \n
Refs. [13, 14, 23] show that the linearized Navier-Stokes equations possess an eigensolution of the form
\nin the triple-deck region where \n
in the main boundary layer where \n
and
\nis a scaled transverse coordinate. The complex wavenumber \n
where the lowest order term in this expansion satisfies the following dispersion relation ([13, 14, 23])
\nand
\nwhose solution must satisfy the inequality
\nin order to insure that the eigensolution does not exhibit unphysical wall normal growth.
\nThis requirement will be satisfied for all \n
The dispersion relation (38) and (39) will be satisfied at small values of \n
The cross flow velocity \n
The dispersion relation (38) is expected to have at least one root corresponding to each of the infinitely many roots of (18). But only the lowest order \n
The real and negative imaginary parts of \n
\n\n\nRe\n\n\nκ\n0\n†\n\n\n\n as a function of \n\n\nx\n1\n†\n\n\n calculated from (38) together with the initial condition (41) for \n\n\nM\n∞\n\n=\n2\n,\n3\n,\n4\n\n (double dot dashed, dot dashed, and solid lines, respectively) and three values of \n\n\n\nβ\n¯\n\n†\n\n≥\n2\n\n. The dashed curve in the main graph is the rescaled large-\n\n\nx\n1\n†\n\n\n asymptote (49).
\n\n\n−\nIm\n\n\nκ\n0\n†\n\n\n\n as a function of \n\n\nx\n1\n†\n\n\n calculated from (38) together with the initial condition (41) for \n\n\nM\n∞\n\n=\n2\n,\n3\n,\n4\n\n (double dot dashed, dot dashed and solid lines, respectively) and three values of the frequency scaled transverse wavenumber.
The triple-deck eigensolution (33) (which contains the Lam-Rott solution as an upstream limit) can undergo a significant amount of damping before it turns into a spatially growing instability wave at the lower branch of the neutral stability curve.
\nThe exponential damping in Eq. (33) is proportional to \n
As noted above the acoustically driven solution will only match onto an eigensolution in the downstream region when \n
into (38), using (29), and taking the limit as \n
when the square root \n
It can then be shown by direct substitution that the solution \n
where \n
The pressure component of the resulting solution will then match onto the downstream limit (32) and (30) of the acoustically generated diffraction region solution when \n
Figure 6 is a plot of the scaled lowest order wavenumber \n
Scaled wavenumber \n\n\n\nκ\n¯\n\n0\n\n/\n\n\nβ\n¯\n\n¯\n\n=\n\nκ\n0\n\n/\n\nβ\n¯\n\n\n as a function of \n\n\n\n\n\n\nβ\n¯\n\n¯\n\n\nT\nw\n\n\n\n4\n\n\n\nx\n̂\n\n1\n\n/\n\nλ\n2\n\n=\n\n\n\n\nβ\n¯\n\n\nT\nw\n\n\n\n4\n\n\nx\n1\n\n/\n\nλ\n2\n\n\n for various values of \n\n\nM\n∞\n\n\n. The solid lines represent the numerical solution. Dashed lines are the asymptotic solution(45) .
Eqs. (29), (38) and (39) show that
\nwhen \n
when \n
The dashed curves in the main plot of Figure 4 represent the re-scaled large-\n
As noted in [11], the solution to the reduced dispersion relation (44) satisfies the rescaled version
\nof (48), which can be considered to be a special case of this result if we put
\nand allow r to be zero or 1/3.
\nThe expansion (37) then generalizes to [11]
\nwhere
\nand \n
Eq. (49) shows, among other things, that the lowest order wave number and streamwise growth rate approach zero but do not become negative as the disturbance propagates downstream. The boundary layer thickness which is \n
where it becomes of the order of the spanwise length scale, which remains constant at \n
Eqs. (37), (43), (51) and (52) show that the Tollmien-Schlichting wave becomes more oblique and
\nas \n
which means that the solution should be proportional to \n
in this stage of evolution. The solution should remain inviscid in the main boundary layer and the viscous wall layer (i.e., a Stokes layer) is expected to be completely passive.
\nThe scaled variable
\nwill be \n
where \n
whose solution must satisfy the following boundary conditions
\nMatching with the upstream solution (33) and (37) requires that \n
Inserting (10) and (57) into (59), using (60) and assuming the ideal gas law \n
which means that
\nwhere
\nEq. (64) clearly approaches zero when \n
where \n
Ref. [11] proved that (60)–(64) possess an asymptotic solution of the form \n
The Rayleigh eigenvalues \n
Parts (a) and (b) of Figure 7 are plots of the real and imaginary parts respectively of these eigenvalues as a function of \n
(a) \n\nRe\n\n\n\nα\n¯\n\n\n\n and (b) \n\n\n\nIm\n\n\nα\n¯\n\n\n\n\n\n vs. \n\n\nβ\n̂\n\n\n calculated from the modified Rayleigh solution. The red dashed curves are calculated from the asymptotic formula(56) . The red dashed lines in the inset are \n\n\n\nIm\n\n\n\nα\n¯\n\n\n\n\n=\nC\n\n\nβ\n̂\n\n\n, where \n\nC\n=\n36\n\n for \n\n\nM\n∞\n\n=\n2\nC\n=\n129.4\n\n for \n\n\nM\n∞\n\n=\n3\n\n and \n\nC\n=\n340.1\n\n for \n\n\nM\n∞\n\n=\n4\n\n.
This chapter uses high Reynolds number asymptotics to study the nonlocal behavior of boundary layer instabilities generated by small amplitude free-stream disturbances at subsonic and moderate supersonic Mach numbers. The appropriate small expansion parameter turns out to be \n
The free-stream vortical disturbances generate unsteady flows in the leading edge region that produce short spanwise wavelength instabilities in a viscous triple-deck region which lies at an \n
Fedorov and Khokhlov [10] used asymptotic methods to study the generation of inviscid instabilities in supersonic boundary layers by fast and slow acoustic disturbances in the free stream whose obliqueness angle \n
But, the inviscid instability, which first appears at an \n
This research was sponsored by NASA’s by the Transformative Aeronautics Concepts Program of the Aeronautics Research Mission Directorate under the Transformational Tools and Technologies (TTT) Project. PR was supported by the Air Force Office of Scientific Research award number AFOSR Grant FA9550-15-1-0248. We would also like to thank Dr. Meelan Choudhari for bringing the photograph in Figure 2 to our attention.
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",metaTitle:"Retraction and Correction Policy",metaDescription:"Retraction and Correction Policy",metaKeywords:null,canonicalURL:"/page/retraction-and-correction-policy",contentRaw:'[{"type":"htmlEditorComponent","content":"IntechOpen’s Retraction and Correction Policy has been developed in accordance with the Committee on Publication Ethics (COPE) publication guidelines relating to scientific misconduct and research ethics:
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\\n"}]'},components:[{type:"htmlEditorComponent",content:'IntechOpen’s Retraction and Correction Policy has been developed in accordance with the Committee on Publication Ethics (COPE) publication guidelines relating to scientific misconduct and research ethics:
\n\n1. RETRACTIONS
\n\nA Retraction of a Chapter will be issued by the Academic Editor, either following an Author’s request to do so or when there is a 3rd party report of scientific misconduct. Upon receipt of a report by a 3rd party, the Academic Editor will investigate any allegations of scientific misconduct, working in cooperation with the Author(s) and their institution(s).
\n\nA formal Retraction will be issued when there is clear and conclusive evidence of any of the following:
\n\nPublishing of a Retraction Notice will adhere to the following guidelines:
\n\n1.2. REMOVALS AND CANCELLATIONS
\n\n2. STATEMENTS OF CONCERN
\n\nA Statement of Concern detailing alleged misconduct will be issued by the Academic Editor or publisher following a 3rd party report of scientific misconduct when:
\n\nIntechOpen believes that the number of occasions on which a Statement of Concern is issued will be very few in number. In all cases when such a decision has been taken by the Academic Editor the decision will be reviewed by another editor to whom the author can make representations.
\n\n3. CORRECTIONS
\n\nA Correction will be issued by the Academic Editor when:
\n\n3.1. ERRATUM
\n\nAn Erratum will be issued by the Academic Editor when it is determined that a mistake in a Chapter originates from the production process handled by the publisher.
\n\nA published Erratum will adhere to the Retraction Notice publishing guidelines outlined above.
\n\n3.2. CORRIGENDUM
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I am also a member of the team in charge for the supervision of Ph.D. students in the fields of development of silicon based planar waveguide sensor devices, study of inelastic electron tunnelling in planar tunnelling nanostructures for sensing applications and development of organotellurium(IV) compounds for semiconductor applications. I am a specialist in data analysis techniques and nanosurface structure. 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After obtaining a Master's degree in Mechanical Engineering, he continued his PhD studies in Robotics at the Vienna University of Technology. Here he worked as a robotic researcher with the university's Intelligent Manufacturing Systems Group as well as a guest researcher at various European universities, including the Swiss Federal Institute of Technology Lausanne (EPFL). During this time he published more than 20 scientific papers, gave presentations, served as a reviewer for major robotic journals and conferences and most importantly he co-founded and built the International Journal of Advanced Robotic Systems- world's first Open Access journal in the field of robotics. Starting this journal was a pivotal point in his career, since it was a pathway to founding IntechOpen - Open Access publisher focused on addressing academic researchers needs. Alex is a personification of IntechOpen key values being trusted, open and entrepreneurial. 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