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Transformation of circular into triangular craters as a function of parameters in a train of laser pulses
Pergamon PII: SOO42-207X(96)00221-7
Vacuum/volume 48lnumber l/pages 51 to 8211997 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0042-207x/97 $17.00+.00
Transformation of circular into triangular craters as a function of parameters in a train of laser pulses S Lugomer” and M StipanEiC, b “Ruder BoSkoviC Institute, BijeniEka c. 54, 70001, Zagreb Croatia; bElectrotechnical Faculty, lstarska 3, Osijek, Croatia received 18 May 7996
In the pulsed laser-material interaction with the pulse-repetition rate, v, in the KHz range, and the scan velocity v = 25.4- 127 mm/s f I-5”/s), the morphology of the laser trace changes drastically. The trace morphology arises from few simultaneous processes: material vaporization and crater formation with surface hydrodynamic flow. Depending on v and v, the crater form changes from circular to triangular, passing through a generation of coherent hydrodynamic and chaotic structures finishing with pairing or merging of three vortices to form a triangular one; the laser trace changes from the discontinuous to continuous. The phenomena observed are important from the theoretical aspect for the study of interaction of 2,3 or more vortices on a short time scale, and the applied aspect is important for deep laser scribing and surface modification uses. Copyright 0 1996 Elsevier Science Ltd
Introduction
In the laser surface processing of metals, ceramics and other materials as well as in (deep) scribing by short laser pulses, very complex dynamical phenomena (which affect the process quality), take place.‘,2,3,4In a deep scribing by the high repetition pulsed laser, the complexity of physical phenomena that strongly depend on the laser parameters and on the experimental conditions, is very impressive. Such phenomena appear in the vacuum processing as well as in the processing in air at pressures ranging from very low up to the atmospheric ones. Details in the laser-trace morphology may differ in vacuum and in the air. The underlying physical phenomena depend on the conditions of deep laser vapourization and therefore on laser parameters. However, if the scribing or the surface processing are realized at very high pulse repetition rate, the additional parameters such as the pulse repetition frequency (v) and the scan velocity (v), play a very important role in the trace morphology. If the pulse repetition rate is in the KHz range, the trace-morphology difference between vacuum and the air processing seems to diminish. Namely, the remnants of the vapour cloud generated by the nth pulse in a train are still present above the surface, when the (n+ I)th pulse is fired. For this reason a kind of gaseous atmosphere is present in the interaction space, even in the case of vacuum processing. We have studied the change of morphology of the laser trace in a deep scribing on metal surfaces, in the air at the atmospheric pressure which is based on the process of crater formation along the laser trace. The trace morphology is determined by two simultaneous processes:
l
l
the crater formation which contains a vortex of liquid metal, generated by the pulse and the shear flow of molten surface layer around the crater,
both of which are based on nonlinear hydrodynamics. The crater morphology shows a reach spectrum of patterns which changes from the single circular, to irregular, and finally to triangular crater, when the pulse repetition frequency (for a given scan velocity) changes in the KHz range. The laser trace itself changes from discontinuous to a more or less regular continuous one. Transition from circular to triangular craters occurs through gradual change of surface hydrodynamics which includes the coherent hydrodynamic structures.‘,‘,‘,’ Coherent hydrodynamic structures represent the microscale vortex pairing in the laser trace. The experiments have shown that transition from regular into chaotic formations occurs with the change of pulse repetition frequency (v), and/or scan velocity (v). Vorticity is concentrated on the centerline of the laser trace. By the change of laser and/or experimental parameters, the hydrodynamic structures change (are going to be disturbed). These structures develop for the characteristic evolution time which is determined by characteristic values of v and v. The structures found in the area of the spot size, stay frozen by the fast cooling after the train of pulses has passed the area of the spot size on the surface. Experimental
Mechanically polished plates of Ta and MO of 1 mm thickness were exposed to the pulsed treatment of a Q-switched Nd:YAG 51
S Lugomer and M StipanEid: Transformation
laser (i, = 1.06pm) teristics: l l l l
KORAD-44,
with
of circular into triangular craters
the following
charac-
laser peak power/pulse: 5 kW power density (adjustable up to): lo9 W/cm’ energy density (adjustable up to): 100 J/cm’ pulse width: x 200 ns.
A special KORAD optical head for work with metals and ceramics, with 4.4 cm focal distance was used; the focal beam size was 90 pm at the begining and 50 pm in later experiments. Pulse repetition frequency was varied from 5 KHz to IOKHz, and the scan velocity ranged from 25.4-127 mm/s (l-5 “is). Since the system is equipped with a DEC PDPl 1 minicomputer, the experimental parameters have been varied by the variation of the input parameters. The surface morphology was studied by means of the optical microscope and by the LEITZ 200 scanning electron microscope.
Results and discussion Laser trace on Ta, MO and W plates shows the presence of surface melting and cracking for the beam parameters and the experimental parameters in the range v = 5-15 KHz, u = 25.4170 mm/s, and for the beam diameter 2d = 90 jlrn. When the parameters v and u (which play the role of the control parameters) change, the cracking mode of the surface response vanishes and the new modes like the ‘cratering mode’ set in. The transition is not abrupt but rather smooth and gradual, so that for both parameters a window of values exists in which the cracking and cratering modes coexist simultaneously. ’ Changing the parameters further on, the cracking mode vanishes completely, and the cratering mode remains only. This cratering mode generates a sharp and deep laser trace, as can be seen for the MO and Ta surface in Figure 1. Figure l(a) shows the four scribing lines, of which the upper two reveal only the morphology
Figure 1. Laser traces on MO and Ta surface seen in metallurgical microscope. The traces have been generated at the scan velocity II - 127mm/s, and variable pulse repetition frequency. Beam diameter = 9Opm. (a), (b), (c), (0, and (g) MO surface. v = 5 KHz; at M = 20 x ; 50 x ; SOx ; 200 x ; 100 x , respectively. (d), (e), (g) and (h) T asurface;v -5KHzatM=50~;50~;100x; 100x.
52
S Lugomer and M StipanEid: Transformation
of circular into triangular
craters
of ripples in the molten surface layer without craters. This is more clearly seen on larger magnification in Figure 1(b) and 1(f). Decreasing the beam diameter to 50 pm, the profile of the laser trace changes, and the trace actually consists of the overlapping craters. (Figure l(a), the lower two lines). Their morphology is not better seen even at larger magnification in the optical microscope. (Figure l(c) and l(g)). However, the scribing on Ta surface under the same conditions is more dramatic and irregular. (Figure l(d) and l(e)). The laser trace has a larger depth and width and the saw-tooth-like side profile. Figure l(h). We have performed a detail study of the change of the crater morphology and of the trace morphology, starting from v and 21, such that the pulses do not overlap so that individual craters are generated. Then, by gradually changing parameters, the transition from discontinuous to continues trace is studied.
surrounded by the heat affected zone (HAZ) of molten metal as shown in Figure 2(a). Central rotating vapour jet causes also the rotation of the liquid layer on the crater’s wall, as schematically shown in Figure 2(b). Thus, the density of liquid metal, because of radial temperature difference AT, and because of crater’s geometry, is not horizontal but tilted with respect to the horizontal line as shown in Figure 2(c). In principle, the baroclinic vorticity may cause the instability in the annular flow.‘-* (See later). The analysis of the trace morphology, and consequently of dynamics of the molten metal, can be divided into two parts:
trace. Every pulse in a train is associated with generation of vapour which leaves the crater in the spiral motion and with the vortex of liquid metal which forms the crater’s walls. Vorticity along the trace is ‘seeded’, i.e. generated by the central part of the Gaussian beam, which induces the surface temperature T = TB, and therefore the vapourization in the zone of radius ~a,
When the parameters v and 21are such that well separated, noninteracting vortices are formed by the series of pulses, the craters formed by evaporation have inverse conical topology (see Figure 3(a)). Every crater is formed by the jet of vapourized metal which travels very fast (~10’ cm/s). The surface edge of the crater indicates the rotation of vapour, and therefore the spiral motions of the vapour jet. There is no indication of eventual vortex bifurcation into secondary (daughter) vortices. In addition there is no indication of any self-organization of molten metal, nor the generation of coherent structures on the surface. When the parameters v and v change in such a way that the neighbour vortices formed by the (two) successive pulses become closer, the crater topology does not change. However, on the surface, (where the shear layer of molten metal is formed), the coherent ‘S’ type structures appear between the neighbour craters. This may be also reached by keeping the pulse repetition frequency constant (v = 5 KHz), and changing the scan velocity (see Figure 3(b)). This figure indicates two important things:
Discontinuous
(i) The rotational flow inside the crater together with the evaporating jet (in the center of crater), (ii) The shear flow of the molten surface layer around the crater in the heat affected zone (HAZ).
l
HA2
/-
6 ‘, / /
T= P,/
4,
(a)
4/
‘,
/
P4’
Cc)
-
(b) Figure 2. Generation of crater and the spiral annular flow by the Gaussian pulse shown schematically. (a) Gaussian pulse and the corresponding crater. The heat affected zone (HAZ) is of larger radius than the crater and represents the molten surface layer. (b) Spiral rotation: central, vapour jet, and rotational annular flow. (c) The lines of equal density are tilted with respect to the horizontal line under angle v, which is the reason for baroclinic instability (see text).
l
The vortex is still ‘alive’ when the new vortex is formed. (Only in this case the formation of coherent structures is possible, i.e. the coherent structures can be formed among the active vortices, only.) The lifetime of the vortices increases when two (or more) laser spots overlap, because of prolonged heating of the HAZ.
When the parameters v and Y change so that nearest vortices come closer, the liquid metal rotating around the crater edge on the surface is no longer arranged in a synchronized way into coherent structures. The coherent structures are lost, and the vortices touch each other. The shape of the craters is no longer circular, but irregular (see Figure 3(c)). Finally, when the two vortices generated by laser pulses come even closer, they start to rotate together, forming a single large vortex. This gives rise to a quasi continuous laser trace with side oscillations the frequency of which correspond to the pulse repetition frequency v. The studies of behaviour of two interacting vortices have been performed at larger spatial scale in laboratory conditions, as well as by the numerical simulations. Denoting the vortex diameter by d, and the vortex-vortex distance by, /, the control parameter of the vortex behavior is given by the ratio d/d.’ According to calculations of Rabinowitz7 for 0.7
S Lugomer
and M StipanEiC: Transformation
of circular into triangular craters
without merging is shown in Figure 4(a), (b) and (c), respectively. The trajectory of the liquid particles in the field of the vortex pair is given in Figure 4(d). The morphology of the shear layer along the laser trace seeded by vortices (around vapourization spots), is similar to that of Green-Taylor (G-T) vortices.6 Brochet et ~1.~ have studied a small-scale structure of three-dimensional Green-Taylor vortex and transition into turbulence. They used flow visualization in two-dimensional sections and three-dimensional perspective plots, at Re = 1600. They found that initial quasi-invisced evolution leads to the formation around t = 4 of vortex sheets and daughter vortices, and substantial roll-up of the vortex through t = 6. At t = 7, a violent topological change occurs6 Coherent structures are clearly seen for t = 5 and t = 6 and are shown in Figure 5. Brachet et ul. conclude that transition to spatial chaos, controlled by time parameter, manifests at high Re numbers. We can not compare the complete scenario in the vapourizationcaused-turbulence on the surface with experiments of Brachet et ~i.,~ in all segments. One can only say that topological characteristics of the hydrodynamic structures formed by the train of pulses on the surface at given scan velocity show the same spatial characteristics as the G-T vortices. However, the viscosity effects in laser experiments, which may be assumed small in the beginning. become significant because of cooling associated with crossing of the laser beam over the surface. Therefore, the surface morphology is typical for the turbulent structures associated with G-T vortices. An argument in favour is Figure 3(b) which can be compared with Figure 5(b) and (g) of Brachet et u/.~
Figure 3. SEM micrograph showing the change of laser trace morphology from discontinuous to quasicontinuous at constant pulse repetition frequency (V = 5 KHz) and variable scan velocity. Beam diameter = 50pm. (a) o = 6”/s (GZ152,4mm/s). Two circular craters formed by the laser pulses on Ta surface. (b) a = 5 “is (= 127 mm/s). Coherent hydrodynamic structures formed on the surface of Ta between two circular craters. (c) r = 4 “is ( z 101.6 mm/s). Coherent structures have vanished, and irregular (chaotic) are formed between two (joined) craters. (See text).
54
The particle trajectory for pairing of two vortices. The last figure (Figure 4(d)) shows trajectory of the liquid particle in the case of pairing of two vortices and indicates the transition from discontinuous to the quasi-continuous laser trace. Hydrodynamic pictures (Figures 4 and 5) indicate that the origin of the trace irregularity lies in the pairing process itself. Topological countervnrt is a picture of attractor of particle’s trajectory which indicates transition from order to chaos. Assuming the representation of the particle’s trajectory (the ‘eight type’ phase portrait) as suggested by Newmark and Landa’ for a general class of such a type of nonlinear phenomena, one can represent the transition from order to chaos, as given in Figure 6. It shows the ‘eight type’ phase portrait of the point mapping of plane-into-plane with the fixed seddle point 0, and the two loops of separations S = S+. For the small perturbation of mapping, the ‘eight type’ phase portrait is disturbed and the touch or the crossection of the invariant curves S- and S+ may appear (Figure 6(b)). It is quite obvious that this transition in our case (for given material parameters) is dependent in the first case on r or 1’. In other words, by changing these parameters in fine steps one can study the transition from sinchronicity into chaos, through the gradual change of crater morphology, and of the trace morphology. Continuous trace If the parameters v and r change in such a way that the overlapping (covering) rate of successive pulses is large, then one and the same area receives a few laser pulses. This may happen either by increasing the pulse repetition frequency at constant scan velocity, or at constant pulse repetition frequency, and by decreasing the scan velocity. Of course, there is a third possibility, namely that both parameters change simultaneously. We have
S Lugomer and M StipanEiC: Transformation
of circular into triangular
craters d/l =0.8
24.9
(b)
(a)
Cc)
(4
Figure 4. Pairing of of two vortices without merging (a) and (b). (c) Merging of two coherent structures with homogeneous distribution of vorticity. (d) Trajectories of liquid particles in the field of the vortex pair. The chaotic motion of particles occurs in the shaded area. ({ = nondimensional
coordinate of the particle. From Ref. 7.)
taken the first way, i.e. we generated the laser trace by gradually increasing v from 5 KHz to 10 KHz, at constant scan velocity of u = 127 mm/s (5 “is). Therefore, not two but three vortices came into interaction, generating very irregular trace morphology of a complex shape. Such traces are shown in Figure 7(aHf). An insight into morphology of craters in these traces was obtained by magnification of selected characteristic segments of every trace. Characteristics of craters corresponding to Figure 7(aF(f) are given in Figure 8(a)-(f). Starting with Figure S(a), which shows one sided saw-tooth shape, and continuing to Figure 8(b) and (c), the crater’s morphology changes into a more and more triangular one. Finally, the triangular crater appears at v = 8 KHz, (Figure 8(d)). A well defined triangular shape remains up to v = 10 KHz, as shown in Figure 8(f). The triangular crater is, strictly speaking, an inverse thetraedar type crater. Table 1 indicates that by increasing the pulse repetition frequency from 5 KHz to 10 KHz (Figure 8(a)-(f)), the pulse-topulse distance decreases from 25pm to 12.5pm. In the same time measured crater-to-crater distance increases from 25 pm to 45 pm, as well as the crater diameter from 15 pm to E 33 pm. The number of overlapping pulses inside the spot of diameter 2R
(N = (2Rv/v)), increases from N = 2 to N = 4, while the corresponding pulse-to-pulse covering rate S (S = (2R/N)) decreases from 0.50 to 0.25. These data clearly indicate that craters are formed by interaction and merging of two or three vortices, every one of which is generated by the single pulse in the series. Strictly speaking, Figure 8(a), (b) and (c), are formed by two interacting vortices, while the craters in Figure 8(d), (e) and (f) are formed by three interacting vortices the central part of which is vapourized. Therefore, the triangular craters are the three jointed (merged) craters. Speaking in terms of hydrodynamics, the triangular crater is a fingerprint (bed) of the three interacting and coalescing (merging) axial vortices of circular crossection. This may be compared with the ‘triangular’ vortex and the vortex tripole.‘” When two circular vortices start to interact they begin the rotation around the common centre, and moving out of the line of the laser trace, a zig-zag crater-line inside the laser trace is formed; this also occurs with three interacting vortices. Consequently, the vortices of the triangle are rotated left and right with respect to the laser trace, as a direct consequence of the tripolar vortex rotation. It is interesting to note that the triangular crater in Figure 8(d)
Table 1. Parameters corresponding to Figure 8
_
Figure 8
V (mm/s)
v (KHz)
Step d (pm)
Crater-to-crater (pm)
A B C D E F
127 127 127 127 127 127
5 6 7 8 9 10
25.0 20.8 17.8 15.6 13.8 12.5
25 30 33 35 40 45
distance Crater 15 20 22 26 28 33
diameter
(pm)
N number pulses 2.0 2.4 2.8 3.2 3.6 4.0
of covering
S pulse covering rate 0.50 0.41 0.36 0.31 0.27 0.25
55
S Lugomerand
M StipanEid:Transformation
of circular
into triangular
craters
I
(b) 5
00
Cd) 0)
Figure 5. Coherent structures produced in the viscous evolution of the Taylor-Green vortex (V = (1600)-r), (b)-(d) and (gt(i) in planes !: = 0, $3, n/4, respectively at t = 5 and t = 6. From Ref. 6.
(v = 8 KHz) is empty, containing no grains inside. However, the crater formed at v = 9KHz (Figure S(e)) contains a large polygonal grain (almost a regular one). Finally, the triangular crater formed at v = 10 KHz (Figure 8(f)) contains three sharp irregular grains. Formation of one or more grains inside the triangular crater is a consequence of specific and very complex flow of molten metal during interaction of vortices. The type of flow that establishes such a crater in laser pulsed operation is similar to the flow that establishes triangular vortex, studied by Carnevale and Kloosterziel.” An example of a triangular vortex they studied is shown in Figure 9. The time behaviour (evolution) from generation to transition into other types of vortices (vortex pair) is shown in Figure 10. 56
give contours
of vorticity
The particle trajectory for merging of three vortices. Similar to the previous case of flow associated with merging of two vortices, we look for the phase portrait of the particle trajectory in the merging of three vortices. Synchronism of the phase portrait of point transformation is shown in Figure 11(a).” The points 0,, O,,... 0, and O;, O;,... 06 under transformation T are cyclically transformed one into another. All the points 0,, 0, ,... 0, and O;, 0; ,... 0; are fixed multiples and represent the seddle points.’ Synchronism relates not only to the point transformation but to differential equations too. In this case the phase portrait is extended to the variants shown in Figure 1 l(b) and (c),~ where the points 0,, 02,... 0, are the seddle points of equilibrium. The
S Lugomer and M StipanEiC: Transformation
of circular into triangular
craters
The problems involved are very complex and have been studied experimentally and theoretically. The most general consideration of the problem for the ‘shallow’ and ‘deep’ layers was conducted by Ponomaryev.”
(b)
Figure 6. Transition from regular to chaotic trajectory of the liquid particle. (a) Regular trajectory of particle in joined vortices. (b) Chaotic trajectory. From Ref. 9.
phase portraits of differential equations are different. This relates to the fact that the curves S+ and S- have a different orientation
with respect to the seddle points in Figure 1l(b) and (c). In the case of the phase portrait in Figure 1I(b) the stable equilibrium is established inside each of the loops, while for the Figure 11(c), the equilibrium is unstable.’ It follows from above that in the vicinity of the fixed points O,, 02,... 0, and the invariant curves, some complex seddle invariant manifestations may occur. In the case of the complex invariant set, if the phase point stays or move in the vicinity of the invariant set, the new possibility appears. This possibility is chaotic change of the phase differences and in this connection of the amplitude oscillations. This type of sinchronicity is called the stochastic sinchronism. Stochasticity appears on the phase portrait in the small motions of the phase point in the vicinity of the points O,, 02,... 0, and of the curves S- and S+ as shown in Figure 11(d). For more detail description of the problem in the phase space, see Newmark and Landa.’ Topological analogy of flow in the crater with other physical phenomena In contrast to hydrodynamic structures on the surface which may be compared with two-dimensional Green-Taylor vortices, the inner structure of craters may be compared with the structures formed in axially symmetric flow. The topological approach we used above enables one to establish the analogy among the axial flows in different kind of experiments, which at first sight have nothing in common. Generally speaking, the axial flows which lead to circular and triangular craters (observed in pulsed laser experiments) may be observed in two different types of vortex dynamics: l
l
in the pairing of 2, or 3 initially independent vortices which come into critical distance, and in the vortex bifurcation into 2,3,4 or more daughter vortices. Examples of such phenomena can be found in the baroclinic instability of annular flo~,‘.~.‘.~formation of the Rosby soliton in magnetohydrodynamic’ and in thermohydraulic experiments.
Baroclinic instability in the rotating fluid. Let us consider a situation involving both density variation and rotation, which is a basic state. We are concerned with instabilities that arise from an inclination v of constant density surfaces p (isopycnals) as the fluid is stably stratified, but there is also a horizontal density gradient, which is the general situation.’ Now, we have to specify the geometry of flow which is axially symmetric, which brings us to the problem known as baroclinic instability in the rotating annulus of fluid subjected to differential heating along the radius5 namely, in the Gaussian spot, the central vapourizing part which rotates (spiral rising) playing the role of inner rotating cylinder. The outer, nonmolten part of the spot plays the role of fixed, axially symmetric cylinder. Molten metal between them is annular fluid put in the forced rotation, across which is imposed a temperature difference AT. The instability that develops has a triangular, or quadratic form, or some polygonal form, depending on parameters like the thermal Rosby number 8? 8 = ga(AT)H R2L2 ’ and the Taylor number Tu: ’ 4!A2L5 Ta = x where g = acceleration due to gravity, CI= thermal density coefficient, H = fluid thickness, R = angular velocity, v = kinematic viscosity. Vortex bifurcation in magnetohydrodynamic experiments (MHD). Many analogies may be found in the literature relating to experimental and theoretical work, for example the work of Obuhov* and Nezlin and Snezkin.” They show that this type of flow belongs to the special case of Kolmogorov flo~.~,*,” They all consider turbulent flow in the presence of an external periodic field. For example, 0buhov8 has performed the experimental and theoretical modelling of the Kolmogorov flow in axially symmetric container in the field of external periodic force. The external periodic force was experimentally realized by the conducting edges of the container which played the role of electrodes and by the axial magnetic field depending on the container radius. Dependence of the azimutal force on radius of the container was close to sinusoidal (to be exact, it was a Bessel function of the zero order). When the rotating fluid reaches conditions for the vortex bifurcation, a series of new phenomena set in. This occurs when the rotating frequency reaches the first, second, third,... critical value: Q = Szc, if = R,,... Then, the circular vortex is divided into 3 smaller vortices as seen in Figure 12(a), or into 4 small vortices (Figure 12(b)), etc. Formation of such a vortex is known as a Rosby soliton.5,6,7.8Under small perturbation, the circular motion started with the largest velocity gradient in the center line. The stability analysis for a given perturbation shows bifurcation into smaller vortices, depending on the Reynolds number. The symmetry index of bifurcation may be theoretically derived, and strongly depends on the flow geometry. Obuhov’ has presented the theoretical results of Ponomaryev” general theory of flow in an axially sinmetric container for any type of force, as well as the 57
S Lugomer and M StipanCiC: Transformation
of circular into triangular craters
F‘igure 7. Morphology of the laser trace on Ta surface generated at constant scan velocity u = 5 “is ( = 127mm/s), flrequency. (a) v = 5 KHz, (b) v = 6 KHz, (c) Y = 7 KHz, (d) v = 8 KHz, (e) v = 9 KHz, (f) Y = 10 KHz.
58
and variable
pulse repetii :ion
Sl .ugomer
and M StipanEiC: Transformation
of circular into triangular
craters
Fig ure 8. Characteristic crater morphology indicating triangular vortices (inverse thetraedar crater) on Ta surface, for the traces seen in Figure 7. at conlstant scan velocity r = 5 “is (z 127 mm/s). (a) v = 5 KHz, (b) v = 6 KHz, (c) v = 7 KHz, (d) v = 8 KHz, (e) v = 8 KHz, (f) 1:= 10 KHz.
59
S Lugomer and M StipanEik: Transformation
of circular into triangular craters
Figure 9. Photograph of the triangle vortex visualized by dye. This vortex was produced in the rotating-tank by stirring the water in a hollow cylinder with three equally spaced vertical indentations and then removing the cylinder vertically. From Ref. 10.
theoretical map of the flow pattern (izoclines of flow), obtained by the Galerkin’s method for Re/Re,,,, = 3.8 Obuhov’ and others5X7.” found that the Rayleigh number of the liquid they used was
z 1600 although the other authors located the mentioned phenomena in the broad range of the Rayleigh numbers ranging
from Re = 103-10’. Comparing this with our problem it should be underlined that it is of the same nature. However, the kinematic viscosity of molten metals, especially of refractive metals, is a very difficult problem, because of their high melting point Tk 3000 K and must be estimated. In the previous cases, the assumption was Vz = 0, i.e. there was no vertical flow in the container. However, discussing the evolution of the vortex structure, as the self-excited soliton, Krasnov13 has elaborated also the case Vz # 0. Thus, vortex and vortex bifurcation appear equally in the cases with Vz = 0 and V,#O, which is similar to our experimental situation. One can say that formation of circular and triangular vortices occurs in all the mentioned cases. Only, in the laser-generated case, the phenomenon occurs on the micron scale. The same phenomenon may appear also on meso- and mega-scale (atmospheric and oceanographic phenomena). An example is the tornado formation. Thermohydraulic and magnetohydrodynamic experiments produce a phenomenon which shows the topological universality, and which equally appears on all spatial scales. The driving forces show the same axial and radial simmetry and therefore similar flow, only the nature of the driving forces is different. Conclusion Study of the laser trace morphology has shown that it is determined by the crater morphology. Since every crater is generated
Figure 10. Plan-view photographs showing the characteristic evolution ot a triangular vortex in a rotating fluid. The experiment was carried out in a large (125cm diameter) tank placed on a turntable. The working depth of the fluid was I1 cm and the horizontal scale of the photographs is 80cm. Increasing time is from left to right, top to bottom. The photographs were taken in the rotating frame at times (a) t = O.OT, (b) 2.2T, (c) 8.9T, (d) 11.1T, (e) 12.27T. (f) 13.3T. The evolution is visualized by adding dye to the vortex. The triangle vortex that forms rotates in a clockwise sense. The phases of the evolution are discussed in the text. From Ref. 9. 60
S Lugomer and M StipanEiC: Transformation
of circular into triangular
craters
(a)
S-
4i!i3 0,
*02
s‘
s+ o; cl
S’
01
0
s-
S+
Oi
Cc)
(b)
-
01
Cd) Figure 11. Merging of three vortices. (a) Phase portrait of point transformation. (b) Phase portrait of differential equations trajectories. (c) Another type of phase portrait of differential equation describing particle trajectory. (See text). From Ref. 9.
describing
particle
61
S Lugomer and M StipanEiC: Transformation
of circular into triangular
craters by the laser pulse together with the molten metal inside it, the crater’s shape is determined by the characteristic annular vortex dynamics. On the other hand, this flow directly depends on the pulse repetition frequency (in the KHz range), and on the scan velocity. Therefore, the crater morphology changes from the circular one (characteristic for individual, well separated craters which do not interact), to the crater morphology with coherent hydrodynamic structures. They are formed in vortex-vortex interaction including joining (merging) of two, or even three vortices. This last case results in formation of characteristic triangular craters. References
Figure 12. Generation of the Rosby soliton in MHD experiment. (a) Generation of three vortices. (b) Generation of four vortices. (From Ref.
8)
62
I. Lugomer, S., Vacuum, 1992,43,443. 2. Lugomer, S., Vacuum, 1995,46, 1325. 3. Lugomer. S., Laser Techno1og.v - Laser Driven Processes. Prentice Hall. New Jersey, 1990. 4. Rikalin, N. N., Laser and E-Beam Material Processing. Machinostroenie, Moscow, 1985 (in Russian). 5. Triton, D. J. and Davies, P. A., in Hydrodynamics Instabilities and the Transilion to Turbulence, eds H. L. Swinney and J. P. Golub. Springer Verlag, Berlin, New York, 1985, p. 249. 6. Brachet, M. E., Merion, D. I., Orszag, S. A., Nickel, B. G., Morf, R. H. and Frisch, V., J. Fluid Mech., 1983, 130,411. I. Rabinowitz. M. I. and Suschik. M. H.. in Nonlinear Waves - Self Organization, eds A. B. Gaponov-Grehov and M. I. Rabinowitz. Nauka, Moscow, 1983, p. 76 (in Russian). 8. Obuhov, A. M., in Problems of Nonlinear and Turbulenr Processes in Physics, Vol. 2, eds A. S. Davidov and V. M. Chernushenko. Naukova dumka, Kiev, 1985, p. 192 (in Russian). 9. Netimark, Y. 1. and Landa, P. S., Stochastic and Chaotic Oscillations. Nauka, Moscow, 1987 (in Russian). 10. Carnevale, G. F. and Kloosterziel, R. C., J. Fluid Mech., 1994, 259, 305. 1 I. Nezlyn, M. V. and Snyezhkin, E. N., in Nonlinear Waves- Structure and Bifurcations, eds A. B. Gaponov-Grehov and M. 1. Robinowitz. Nauka, Moscow, 1987, p. 67. 12. Ponomarjev, P. M., Mech. Liquids and Gases, 1, 1980, 3 (in Russian). 13. Krasnow, Y. K., J. Fluid Mech., 1983, 130, 174.
Vacuum/volume 48lnumber l/pages 51 to 8211997 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0042-207x/97 $17.00+.00
Transformation of circular into triangular craters as a function of parameters in a train of laser pulses S Lugomer” and M StipanEiC, b “Ruder BoSkoviC Institute, BijeniEka c. 54, 70001, Zagreb Croatia; bElectrotechnical Faculty, lstarska 3, Osijek, Croatia received 18 May 7996
In the pulsed laser-material interaction with the pulse-repetition rate, v, in the KHz range, and the scan velocity v = 25.4- 127 mm/s f I-5”/s), the morphology of the laser trace changes drastically. The trace morphology arises from few simultaneous processes: material vaporization and crater formation with surface hydrodynamic flow. Depending on v and v, the crater form changes from circular to triangular, passing through a generation of coherent hydrodynamic and chaotic structures finishing with pairing or merging of three vortices to form a triangular one; the laser trace changes from the discontinuous to continuous. The phenomena observed are important from the theoretical aspect for the study of interaction of 2,3 or more vortices on a short time scale, and the applied aspect is important for deep laser scribing and surface modification uses. Copyright 0 1996 Elsevier Science Ltd
Introduction
In the laser surface processing of metals, ceramics and other materials as well as in (deep) scribing by short laser pulses, very complex dynamical phenomena (which affect the process quality), take place.‘,2,3,4In a deep scribing by the high repetition pulsed laser, the complexity of physical phenomena that strongly depend on the laser parameters and on the experimental conditions, is very impressive. Such phenomena appear in the vacuum processing as well as in the processing in air at pressures ranging from very low up to the atmospheric ones. Details in the laser-trace morphology may differ in vacuum and in the air. The underlying physical phenomena depend on the conditions of deep laser vapourization and therefore on laser parameters. However, if the scribing or the surface processing are realized at very high pulse repetition rate, the additional parameters such as the pulse repetition frequency (v) and the scan velocity (v), play a very important role in the trace morphology. If the pulse repetition rate is in the KHz range, the trace-morphology difference between vacuum and the air processing seems to diminish. Namely, the remnants of the vapour cloud generated by the nth pulse in a train are still present above the surface, when the (n+ I)th pulse is fired. For this reason a kind of gaseous atmosphere is present in the interaction space, even in the case of vacuum processing. We have studied the change of morphology of the laser trace in a deep scribing on metal surfaces, in the air at the atmospheric pressure which is based on the process of crater formation along the laser trace. The trace morphology is determined by two simultaneous processes:
l
l
the crater formation which contains a vortex of liquid metal, generated by the pulse and the shear flow of molten surface layer around the crater,
both of which are based on nonlinear hydrodynamics. The crater morphology shows a reach spectrum of patterns which changes from the single circular, to irregular, and finally to triangular crater, when the pulse repetition frequency (for a given scan velocity) changes in the KHz range. The laser trace itself changes from discontinuous to a more or less regular continuous one. Transition from circular to triangular craters occurs through gradual change of surface hydrodynamics which includes the coherent hydrodynamic structures.‘,‘,‘,’ Coherent hydrodynamic structures represent the microscale vortex pairing in the laser trace. The experiments have shown that transition from regular into chaotic formations occurs with the change of pulse repetition frequency (v), and/or scan velocity (v). Vorticity is concentrated on the centerline of the laser trace. By the change of laser and/or experimental parameters, the hydrodynamic structures change (are going to be disturbed). These structures develop for the characteristic evolution time which is determined by characteristic values of v and v. The structures found in the area of the spot size, stay frozen by the fast cooling after the train of pulses has passed the area of the spot size on the surface. Experimental
Mechanically polished plates of Ta and MO of 1 mm thickness were exposed to the pulsed treatment of a Q-switched Nd:YAG 51
S Lugomer and M StipanEid: Transformation
laser (i, = 1.06pm) teristics: l l l l
KORAD-44,
with
of circular into triangular craters
the following
charac-
laser peak power/pulse: 5 kW power density (adjustable up to): lo9 W/cm’ energy density (adjustable up to): 100 J/cm’ pulse width: x 200 ns.
A special KORAD optical head for work with metals and ceramics, with 4.4 cm focal distance was used; the focal beam size was 90 pm at the begining and 50 pm in later experiments. Pulse repetition frequency was varied from 5 KHz to IOKHz, and the scan velocity ranged from 25.4-127 mm/s (l-5 “is). Since the system is equipped with a DEC PDPl 1 minicomputer, the experimental parameters have been varied by the variation of the input parameters. The surface morphology was studied by means of the optical microscope and by the LEITZ 200 scanning electron microscope.
Results and discussion Laser trace on Ta, MO and W plates shows the presence of surface melting and cracking for the beam parameters and the experimental parameters in the range v = 5-15 KHz, u = 25.4170 mm/s, and for the beam diameter 2d = 90 jlrn. When the parameters v and u (which play the role of the control parameters) change, the cracking mode of the surface response vanishes and the new modes like the ‘cratering mode’ set in. The transition is not abrupt but rather smooth and gradual, so that for both parameters a window of values exists in which the cracking and cratering modes coexist simultaneously. ’ Changing the parameters further on, the cracking mode vanishes completely, and the cratering mode remains only. This cratering mode generates a sharp and deep laser trace, as can be seen for the MO and Ta surface in Figure 1. Figure l(a) shows the four scribing lines, of which the upper two reveal only the morphology
Figure 1. Laser traces on MO and Ta surface seen in metallurgical microscope. The traces have been generated at the scan velocity II - 127mm/s, and variable pulse repetition frequency. Beam diameter = 9Opm. (a), (b), (c), (0, and (g) MO surface. v = 5 KHz; at M = 20 x ; 50 x ; SOx ; 200 x ; 100 x , respectively. (d), (e), (g) and (h) T asurface;v -5KHzatM=50~;50~;100x; 100x.
52
S Lugomer and M StipanEid: Transformation
of circular into triangular
craters
of ripples in the molten surface layer without craters. This is more clearly seen on larger magnification in Figure 1(b) and 1(f). Decreasing the beam diameter to 50 pm, the profile of the laser trace changes, and the trace actually consists of the overlapping craters. (Figure l(a), the lower two lines). Their morphology is not better seen even at larger magnification in the optical microscope. (Figure l(c) and l(g)). However, the scribing on Ta surface under the same conditions is more dramatic and irregular. (Figure l(d) and l(e)). The laser trace has a larger depth and width and the saw-tooth-like side profile. Figure l(h). We have performed a detail study of the change of the crater morphology and of the trace morphology, starting from v and 21, such that the pulses do not overlap so that individual craters are generated. Then, by gradually changing parameters, the transition from discontinuous to continues trace is studied.
surrounded by the heat affected zone (HAZ) of molten metal as shown in Figure 2(a). Central rotating vapour jet causes also the rotation of the liquid layer on the crater’s wall, as schematically shown in Figure 2(b). Thus, the density of liquid metal, because of radial temperature difference AT, and because of crater’s geometry, is not horizontal but tilted with respect to the horizontal line as shown in Figure 2(c). In principle, the baroclinic vorticity may cause the instability in the annular flow.‘-* (See later). The analysis of the trace morphology, and consequently of dynamics of the molten metal, can be divided into two parts:
trace. Every pulse in a train is associated with generation of vapour which leaves the crater in the spiral motion and with the vortex of liquid metal which forms the crater’s walls. Vorticity along the trace is ‘seeded’, i.e. generated by the central part of the Gaussian beam, which induces the surface temperature T = TB, and therefore the vapourization in the zone of radius ~a,
When the parameters v and 21are such that well separated, noninteracting vortices are formed by the series of pulses, the craters formed by evaporation have inverse conical topology (see Figure 3(a)). Every crater is formed by the jet of vapourized metal which travels very fast (~10’ cm/s). The surface edge of the crater indicates the rotation of vapour, and therefore the spiral motions of the vapour jet. There is no indication of eventual vortex bifurcation into secondary (daughter) vortices. In addition there is no indication of any self-organization of molten metal, nor the generation of coherent structures on the surface. When the parameters v and v change in such a way that the neighbour vortices formed by the (two) successive pulses become closer, the crater topology does not change. However, on the surface, (where the shear layer of molten metal is formed), the coherent ‘S’ type structures appear between the neighbour craters. This may be also reached by keeping the pulse repetition frequency constant (v = 5 KHz), and changing the scan velocity (see Figure 3(b)). This figure indicates two important things:
Discontinuous
(i) The rotational flow inside the crater together with the evaporating jet (in the center of crater), (ii) The shear flow of the molten surface layer around the crater in the heat affected zone (HAZ).
l
HA2
/-
6 ‘, / /
T= P,/
4,
(a)
4/
‘,
/
P4’
Cc)
-
(b) Figure 2. Generation of crater and the spiral annular flow by the Gaussian pulse shown schematically. (a) Gaussian pulse and the corresponding crater. The heat affected zone (HAZ) is of larger radius than the crater and represents the molten surface layer. (b) Spiral rotation: central, vapour jet, and rotational annular flow. (c) The lines of equal density are tilted with respect to the horizontal line under angle v, which is the reason for baroclinic instability (see text).
l
The vortex is still ‘alive’ when the new vortex is formed. (Only in this case the formation of coherent structures is possible, i.e. the coherent structures can be formed among the active vortices, only.) The lifetime of the vortices increases when two (or more) laser spots overlap, because of prolonged heating of the HAZ.
When the parameters v and Y change so that nearest vortices come closer, the liquid metal rotating around the crater edge on the surface is no longer arranged in a synchronized way into coherent structures. The coherent structures are lost, and the vortices touch each other. The shape of the craters is no longer circular, but irregular (see Figure 3(c)). Finally, when the two vortices generated by laser pulses come even closer, they start to rotate together, forming a single large vortex. This gives rise to a quasi continuous laser trace with side oscillations the frequency of which correspond to the pulse repetition frequency v. The studies of behaviour of two interacting vortices have been performed at larger spatial scale in laboratory conditions, as well as by the numerical simulations. Denoting the vortex diameter by d, and the vortex-vortex distance by, /, the control parameter of the vortex behavior is given by the ratio d/d.’ According to calculations of Rabinowitz7 for 0.7
S Lugomer
and M StipanEiC: Transformation
of circular into triangular craters
without merging is shown in Figure 4(a), (b) and (c), respectively. The trajectory of the liquid particles in the field of the vortex pair is given in Figure 4(d). The morphology of the shear layer along the laser trace seeded by vortices (around vapourization spots), is similar to that of Green-Taylor (G-T) vortices.6 Brochet et ~1.~ have studied a small-scale structure of three-dimensional Green-Taylor vortex and transition into turbulence. They used flow visualization in two-dimensional sections and three-dimensional perspective plots, at Re = 1600. They found that initial quasi-invisced evolution leads to the formation around t = 4 of vortex sheets and daughter vortices, and substantial roll-up of the vortex through t = 6. At t = 7, a violent topological change occurs6 Coherent structures are clearly seen for t = 5 and t = 6 and are shown in Figure 5. Brachet et ul. conclude that transition to spatial chaos, controlled by time parameter, manifests at high Re numbers. We can not compare the complete scenario in the vapourizationcaused-turbulence on the surface with experiments of Brachet et ~i.,~ in all segments. One can only say that topological characteristics of the hydrodynamic structures formed by the train of pulses on the surface at given scan velocity show the same spatial characteristics as the G-T vortices. However, the viscosity effects in laser experiments, which may be assumed small in the beginning. become significant because of cooling associated with crossing of the laser beam over the surface. Therefore, the surface morphology is typical for the turbulent structures associated with G-T vortices. An argument in favour is Figure 3(b) which can be compared with Figure 5(b) and (g) of Brachet et u/.~
Figure 3. SEM micrograph showing the change of laser trace morphology from discontinuous to quasicontinuous at constant pulse repetition frequency (V = 5 KHz) and variable scan velocity. Beam diameter = 50pm. (a) o = 6”/s (GZ152,4mm/s). Two circular craters formed by the laser pulses on Ta surface. (b) a = 5 “is (= 127 mm/s). Coherent hydrodynamic structures formed on the surface of Ta between two circular craters. (c) r = 4 “is ( z 101.6 mm/s). Coherent structures have vanished, and irregular (chaotic) are formed between two (joined) craters. (See text).
54
The particle trajectory for pairing of two vortices. The last figure (Figure 4(d)) shows trajectory of the liquid particle in the case of pairing of two vortices and indicates the transition from discontinuous to the quasi-continuous laser trace. Hydrodynamic pictures (Figures 4 and 5) indicate that the origin of the trace irregularity lies in the pairing process itself. Topological countervnrt is a picture of attractor of particle’s trajectory which indicates transition from order to chaos. Assuming the representation of the particle’s trajectory (the ‘eight type’ phase portrait) as suggested by Newmark and Landa’ for a general class of such a type of nonlinear phenomena, one can represent the transition from order to chaos, as given in Figure 6. It shows the ‘eight type’ phase portrait of the point mapping of plane-into-plane with the fixed seddle point 0, and the two loops of separations S = S+. For the small perturbation of mapping, the ‘eight type’ phase portrait is disturbed and the touch or the crossection of the invariant curves S- and S+ may appear (Figure 6(b)). It is quite obvious that this transition in our case (for given material parameters) is dependent in the first case on r or 1’. In other words, by changing these parameters in fine steps one can study the transition from sinchronicity into chaos, through the gradual change of crater morphology, and of the trace morphology. Continuous trace If the parameters v and r change in such a way that the overlapping (covering) rate of successive pulses is large, then one and the same area receives a few laser pulses. This may happen either by increasing the pulse repetition frequency at constant scan velocity, or at constant pulse repetition frequency, and by decreasing the scan velocity. Of course, there is a third possibility, namely that both parameters change simultaneously. We have
S Lugomer and M StipanEiC: Transformation
of circular into triangular
craters d/l =0.8
24.9
(b)
(a)
Cc)
(4
Figure 4. Pairing of of two vortices without merging (a) and (b). (c) Merging of two coherent structures with homogeneous distribution of vorticity. (d) Trajectories of liquid particles in the field of the vortex pair. The chaotic motion of particles occurs in the shaded area. ({ = nondimensional
coordinate of the particle. From Ref. 7.)
taken the first way, i.e. we generated the laser trace by gradually increasing v from 5 KHz to 10 KHz, at constant scan velocity of u = 127 mm/s (5 “is). Therefore, not two but three vortices came into interaction, generating very irregular trace morphology of a complex shape. Such traces are shown in Figure 7(aHf). An insight into morphology of craters in these traces was obtained by magnification of selected characteristic segments of every trace. Characteristics of craters corresponding to Figure 7(aF(f) are given in Figure 8(a)-(f). Starting with Figure S(a), which shows one sided saw-tooth shape, and continuing to Figure 8(b) and (c), the crater’s morphology changes into a more and more triangular one. Finally, the triangular crater appears at v = 8 KHz, (Figure 8(d)). A well defined triangular shape remains up to v = 10 KHz, as shown in Figure 8(f). The triangular crater is, strictly speaking, an inverse thetraedar type crater. Table 1 indicates that by increasing the pulse repetition frequency from 5 KHz to 10 KHz (Figure 8(a)-(f)), the pulse-topulse distance decreases from 25pm to 12.5pm. In the same time measured crater-to-crater distance increases from 25 pm to 45 pm, as well as the crater diameter from 15 pm to E 33 pm. The number of overlapping pulses inside the spot of diameter 2R
(N = (2Rv/v)), increases from N = 2 to N = 4, while the corresponding pulse-to-pulse covering rate S (S = (2R/N)) decreases from 0.50 to 0.25. These data clearly indicate that craters are formed by interaction and merging of two or three vortices, every one of which is generated by the single pulse in the series. Strictly speaking, Figure 8(a), (b) and (c), are formed by two interacting vortices, while the craters in Figure 8(d), (e) and (f) are formed by three interacting vortices the central part of which is vapourized. Therefore, the triangular craters are the three jointed (merged) craters. Speaking in terms of hydrodynamics, the triangular crater is a fingerprint (bed) of the three interacting and coalescing (merging) axial vortices of circular crossection. This may be compared with the ‘triangular’ vortex and the vortex tripole.‘” When two circular vortices start to interact they begin the rotation around the common centre, and moving out of the line of the laser trace, a zig-zag crater-line inside the laser trace is formed; this also occurs with three interacting vortices. Consequently, the vortices of the triangle are rotated left and right with respect to the laser trace, as a direct consequence of the tripolar vortex rotation. It is interesting to note that the triangular crater in Figure 8(d)
Table 1. Parameters corresponding to Figure 8
_
Figure 8
V (mm/s)
v (KHz)
Step d (pm)
Crater-to-crater (pm)
A B C D E F
127 127 127 127 127 127
5 6 7 8 9 10
25.0 20.8 17.8 15.6 13.8 12.5
25 30 33 35 40 45
distance Crater 15 20 22 26 28 33
diameter
(pm)
N number pulses 2.0 2.4 2.8 3.2 3.6 4.0
of covering
S pulse covering rate 0.50 0.41 0.36 0.31 0.27 0.25
55
S Lugomerand
M StipanEid:Transformation
of circular
into triangular
craters
I
(b) 5
00
Cd) 0)
Figure 5. Coherent structures produced in the viscous evolution of the Taylor-Green vortex (V = (1600)-r), (b)-(d) and (gt(i) in planes !: = 0, $3, n/4, respectively at t = 5 and t = 6. From Ref. 6.
(v = 8 KHz) is empty, containing no grains inside. However, the crater formed at v = 9KHz (Figure S(e)) contains a large polygonal grain (almost a regular one). Finally, the triangular crater formed at v = 10 KHz (Figure 8(f)) contains three sharp irregular grains. Formation of one or more grains inside the triangular crater is a consequence of specific and very complex flow of molten metal during interaction of vortices. The type of flow that establishes such a crater in laser pulsed operation is similar to the flow that establishes triangular vortex, studied by Carnevale and Kloosterziel.” An example of a triangular vortex they studied is shown in Figure 9. The time behaviour (evolution) from generation to transition into other types of vortices (vortex pair) is shown in Figure 10. 56
give contours
of vorticity
The particle trajectory for merging of three vortices. Similar to the previous case of flow associated with merging of two vortices, we look for the phase portrait of the particle trajectory in the merging of three vortices. Synchronism of the phase portrait of point transformation is shown in Figure 11(a).” The points 0,, O,,... 0, and O;, O;,... 06 under transformation T are cyclically transformed one into another. All the points 0,, 0, ,... 0, and O;, 0; ,... 0; are fixed multiples and represent the seddle points.’ Synchronism relates not only to the point transformation but to differential equations too. In this case the phase portrait is extended to the variants shown in Figure 1 l(b) and (c),~ where the points 0,, 02,... 0, are the seddle points of equilibrium. The
S Lugomer and M StipanEiC: Transformation
of circular into triangular
craters
The problems involved are very complex and have been studied experimentally and theoretically. The most general consideration of the problem for the ‘shallow’ and ‘deep’ layers was conducted by Ponomaryev.”
(b)
Figure 6. Transition from regular to chaotic trajectory of the liquid particle. (a) Regular trajectory of particle in joined vortices. (b) Chaotic trajectory. From Ref. 9.
phase portraits of differential equations are different. This relates to the fact that the curves S+ and S- have a different orientation
with respect to the seddle points in Figure 1l(b) and (c). In the case of the phase portrait in Figure 1I(b) the stable equilibrium is established inside each of the loops, while for the Figure 11(c), the equilibrium is unstable.’ It follows from above that in the vicinity of the fixed points O,, 02,... 0, and the invariant curves, some complex seddle invariant manifestations may occur. In the case of the complex invariant set, if the phase point stays or move in the vicinity of the invariant set, the new possibility appears. This possibility is chaotic change of the phase differences and in this connection of the amplitude oscillations. This type of sinchronicity is called the stochastic sinchronism. Stochasticity appears on the phase portrait in the small motions of the phase point in the vicinity of the points O,, 02,... 0, and of the curves S- and S+ as shown in Figure 11(d). For more detail description of the problem in the phase space, see Newmark and Landa.’ Topological analogy of flow in the crater with other physical phenomena In contrast to hydrodynamic structures on the surface which may be compared with two-dimensional Green-Taylor vortices, the inner structure of craters may be compared with the structures formed in axially symmetric flow. The topological approach we used above enables one to establish the analogy among the axial flows in different kind of experiments, which at first sight have nothing in common. Generally speaking, the axial flows which lead to circular and triangular craters (observed in pulsed laser experiments) may be observed in two different types of vortex dynamics: l
l
in the pairing of 2, or 3 initially independent vortices which come into critical distance, and in the vortex bifurcation into 2,3,4 or more daughter vortices. Examples of such phenomena can be found in the baroclinic instability of annular flo~,‘.~.‘.~formation of the Rosby soliton in magnetohydrodynamic’ and in thermohydraulic experiments.
Baroclinic instability in the rotating fluid. Let us consider a situation involving both density variation and rotation, which is a basic state. We are concerned with instabilities that arise from an inclination v of constant density surfaces p (isopycnals) as the fluid is stably stratified, but there is also a horizontal density gradient, which is the general situation.’ Now, we have to specify the geometry of flow which is axially symmetric, which brings us to the problem known as baroclinic instability in the rotating annulus of fluid subjected to differential heating along the radius5 namely, in the Gaussian spot, the central vapourizing part which rotates (spiral rising) playing the role of inner rotating cylinder. The outer, nonmolten part of the spot plays the role of fixed, axially symmetric cylinder. Molten metal between them is annular fluid put in the forced rotation, across which is imposed a temperature difference AT. The instability that develops has a triangular, or quadratic form, or some polygonal form, depending on parameters like the thermal Rosby number 8? 8 = ga(AT)H R2L2 ’ and the Taylor number Tu: ’ 4!A2L5 Ta = x where g = acceleration due to gravity, CI= thermal density coefficient, H = fluid thickness, R = angular velocity, v = kinematic viscosity. Vortex bifurcation in magnetohydrodynamic experiments (MHD). Many analogies may be found in the literature relating to experimental and theoretical work, for example the work of Obuhov* and Nezlin and Snezkin.” They show that this type of flow belongs to the special case of Kolmogorov flo~.~,*,” They all consider turbulent flow in the presence of an external periodic field. For example, 0buhov8 has performed the experimental and theoretical modelling of the Kolmogorov flow in axially symmetric container in the field of external periodic force. The external periodic force was experimentally realized by the conducting edges of the container which played the role of electrodes and by the axial magnetic field depending on the container radius. Dependence of the azimutal force on radius of the container was close to sinusoidal (to be exact, it was a Bessel function of the zero order). When the rotating fluid reaches conditions for the vortex bifurcation, a series of new phenomena set in. This occurs when the rotating frequency reaches the first, second, third,... critical value: Q = Szc, if = R,,... Then, the circular vortex is divided into 3 smaller vortices as seen in Figure 12(a), or into 4 small vortices (Figure 12(b)), etc. Formation of such a vortex is known as a Rosby soliton.5,6,7.8Under small perturbation, the circular motion started with the largest velocity gradient in the center line. The stability analysis for a given perturbation shows bifurcation into smaller vortices, depending on the Reynolds number. The symmetry index of bifurcation may be theoretically derived, and strongly depends on the flow geometry. Obuhov’ has presented the theoretical results of Ponomaryev” general theory of flow in an axially sinmetric container for any type of force, as well as the 57
S Lugomer and M StipanCiC: Transformation
of circular into triangular craters
F‘igure 7. Morphology of the laser trace on Ta surface generated at constant scan velocity u = 5 “is ( = 127mm/s), flrequency. (a) v = 5 KHz, (b) v = 6 KHz, (c) Y = 7 KHz, (d) v = 8 KHz, (e) v = 9 KHz, (f) Y = 10 KHz.
58
and variable
pulse repetii :ion
Sl .ugomer
and M StipanEiC: Transformation
of circular into triangular
craters
Fig ure 8. Characteristic crater morphology indicating triangular vortices (inverse thetraedar crater) on Ta surface, for the traces seen in Figure 7. at conlstant scan velocity r = 5 “is (z 127 mm/s). (a) v = 5 KHz, (b) v = 6 KHz, (c) v = 7 KHz, (d) v = 8 KHz, (e) v = 8 KHz, (f) 1:= 10 KHz.
59
S Lugomer and M StipanEik: Transformation
of circular into triangular craters
Figure 9. Photograph of the triangle vortex visualized by dye. This vortex was produced in the rotating-tank by stirring the water in a hollow cylinder with three equally spaced vertical indentations and then removing the cylinder vertically. From Ref. 10.
theoretical map of the flow pattern (izoclines of flow), obtained by the Galerkin’s method for Re/Re,,,, = 3.8 Obuhov’ and others5X7.” found that the Rayleigh number of the liquid they used was
z 1600 although the other authors located the mentioned phenomena in the broad range of the Rayleigh numbers ranging
from Re = 103-10’. Comparing this with our problem it should be underlined that it is of the same nature. However, the kinematic viscosity of molten metals, especially of refractive metals, is a very difficult problem, because of their high melting point Tk 3000 K and must be estimated. In the previous cases, the assumption was Vz = 0, i.e. there was no vertical flow in the container. However, discussing the evolution of the vortex structure, as the self-excited soliton, Krasnov13 has elaborated also the case Vz # 0. Thus, vortex and vortex bifurcation appear equally in the cases with Vz = 0 and V,#O, which is similar to our experimental situation. One can say that formation of circular and triangular vortices occurs in all the mentioned cases. Only, in the laser-generated case, the phenomenon occurs on the micron scale. The same phenomenon may appear also on meso- and mega-scale (atmospheric and oceanographic phenomena). An example is the tornado formation. Thermohydraulic and magnetohydrodynamic experiments produce a phenomenon which shows the topological universality, and which equally appears on all spatial scales. The driving forces show the same axial and radial simmetry and therefore similar flow, only the nature of the driving forces is different. Conclusion Study of the laser trace morphology has shown that it is determined by the crater morphology. Since every crater is generated
Figure 10. Plan-view photographs showing the characteristic evolution ot a triangular vortex in a rotating fluid. The experiment was carried out in a large (125cm diameter) tank placed on a turntable. The working depth of the fluid was I1 cm and the horizontal scale of the photographs is 80cm. Increasing time is from left to right, top to bottom. The photographs were taken in the rotating frame at times (a) t = O.OT, (b) 2.2T, (c) 8.9T, (d) 11.1T, (e) 12.27T. (f) 13.3T. The evolution is visualized by adding dye to the vortex. The triangle vortex that forms rotates in a clockwise sense. The phases of the evolution are discussed in the text. From Ref. 9. 60
S Lugomer and M StipanEiC: Transformation
of circular into triangular
craters
(a)
S-
4i!i3 0,
*02
s‘
s+ o; cl
S’
01
0
s-
S+
Oi
Cc)
(b)
-
01
Cd) Figure 11. Merging of three vortices. (a) Phase portrait of point transformation. (b) Phase portrait of differential equations trajectories. (c) Another type of phase portrait of differential equation describing particle trajectory. (See text). From Ref. 9.
describing
particle
61
S Lugomer and M StipanEiC: Transformation
of circular into triangular
craters by the laser pulse together with the molten metal inside it, the crater’s shape is determined by the characteristic annular vortex dynamics. On the other hand, this flow directly depends on the pulse repetition frequency (in the KHz range), and on the scan velocity. Therefore, the crater morphology changes from the circular one (characteristic for individual, well separated craters which do not interact), to the crater morphology with coherent hydrodynamic structures. They are formed in vortex-vortex interaction including joining (merging) of two, or even three vortices. This last case results in formation of characteristic triangular craters. References
Figure 12. Generation of the Rosby soliton in MHD experiment. (a) Generation of three vortices. (b) Generation of four vortices. (From Ref.
8)
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I. Lugomer, S., Vacuum, 1992,43,443. 2. Lugomer, S., Vacuum, 1995,46, 1325. 3. Lugomer. S., Laser Techno1og.v - Laser Driven Processes. Prentice Hall. New Jersey, 1990. 4. Rikalin, N. N., Laser and E-Beam Material Processing. Machinostroenie, Moscow, 1985 (in Russian). 5. Triton, D. J. and Davies, P. A., in Hydrodynamics Instabilities and the Transilion to Turbulence, eds H. L. Swinney and J. P. Golub. Springer Verlag, Berlin, New York, 1985, p. 249. 6. Brachet, M. E., Merion, D. I., Orszag, S. A., Nickel, B. G., Morf, R. H. and Frisch, V., J. Fluid Mech., 1983, 130,411. I. Rabinowitz. M. I. and Suschik. M. H.. in Nonlinear Waves - Self Organization, eds A. B. Gaponov-Grehov and M. I. Rabinowitz. Nauka, Moscow, 1983, p. 76 (in Russian). 8. Obuhov, A. M., in Problems of Nonlinear and Turbulenr Processes in Physics, Vol. 2, eds A. S. Davidov and V. M. Chernushenko. Naukova dumka, Kiev, 1985, p. 192 (in Russian). 9. Netimark, Y. 1. and Landa, P. S., Stochastic and Chaotic Oscillations. Nauka, Moscow, 1987 (in Russian). 10. Carnevale, G. F. and Kloosterziel, R. C., J. Fluid Mech., 1994, 259, 305. 1 I. Nezlyn, M. V. and Snyezhkin, E. N., in Nonlinear Waves- Structure and Bifurcations, eds A. B. Gaponov-Grehov and M. 1. Robinowitz. Nauka, Moscow, 1987, p. 67. 12. Ponomarjev, P. M., Mech. Liquids and Gases, 1, 1980, 3 (in Russian). 13. Krasnow, Y. K., J. Fluid Mech., 1983, 130, 174.