Spatial composition and formation process of wave dissipative structure

Since the theory of dissipative structures has beenproposed, many researches on them were performed. However, these studiesmostly focused on the thermodynamic theory, laws and stability ofdissipative structures, few studies were carried out on the basicstructure-cells and their interaction in the dissipative structures. Since1970's, different dissipative structures (patterns) have been found incrystal growth system, such as spoke patterns, rotating spoke patterns andwave patterns. In this paper a famous wave dissipative structure wasinvestigated, observed in the melt thermal convections in Czockralski (Cz)crystal growth system. The wave patterns with different wave numbers n areobtained numerically in a Cz oxide melt thermal convection system withcrucible radius r_c=100mm and crystal disc radius r_s=50\,mm. The oxidemelt is filled in a rest crucible, whose aspect ratio is r_c:h (radius:height). The motion of the oxide melt is induced by sidewall heating of thecrucible and a rotating disc. The disc has a common axis with the crucibleand just contacts with the free surface of the oxide melt. The rotating rateof the disc is \Omega_s. The governing equations of LiNbO_3 meltflows were solved by a block- structured boundary-fitted-coordinate method. Toensure the correct coupling of pressure and velocity fields, the well-knownmomentum interpolation technique of Rhie and Chow was applied. For theconvective term, QUICK scheme was applied. A pressure - correction equationis derived according to SIMPLE algorithm. By changing r:h and\Omega_s, numericalsimulations are conducted to obtain stable n-folded wave patterns. Properorthogonal decomposition (POD) is applied to extract the basic modes of then-folded wave patterns (n=2,3,4 in the present paper). According to POD, thebasic modes are optimal structures to form the parental n-folded wavepatterns and consist of many small-scale vortexes in general. In the presentpaper some interesting findings are achieved: (1) n-folded wave dissipativestructure is made up of many small-scale basic modes; (2) basic modes appearin groups; (3) each group has n similar basic modes but of different phaseangle; (4) with the increase of group order, the number of the vortex in thebasic modes increases by twice. The contribution of the basic modes to formthe parental dissipative structure is different and time-dependent. Themacroscale wave dissipative structure is found to be formed by thealternative appearance and disappearance of the basic modes. These resultsenrich the knowledge of dissipative structure.