Academia.eduAcademia.edu

Figure 3 – uploaded by Chihiro Matsuoka

See full PDFdownloadDownload figure
The linear solution {Egq. (18)] provides the initial conditions for solving the nonlinear interaction between the two interfaces. Following the case of the conventional single-layer RMI, we normalize the physical quantities in Eq. (16) by the wavenumber k and the initial velocity shear of a single interface vj,, which corre- sponds to the linear growth rate in the single-layer RMI.” From now on, the dimensionless variables for space kx, time kvjjnt, circulation of vortex sheet ikT;/v;, (i = 1, 2), and vortex sheet strength yj/vjj, are used as x, t, Tj, and y;. The dimensionless dis- tance kd between the interfaces is replaced with d below. Then, from Egs. (16) and (17), we have the normalized linear solution

Figure 3 The linear solution {Egq. (18)] provides the initial conditions for solving the nonlinear interaction between the two interfaces. Following the case of the conventional single-layer RMI, we normalize the physical quantities in Eq. (16) by the wavenumber k and the initial velocity shear of a single interface vj,, which corre- sponds to the linear growth rate in the single-layer RMI.” From now on, the dimensionless variables for space kx, time kvjjnt, circulation of vortex sheet ikT;/v;, (i = 1, 2), and vortex sheet strength yj/vjj, are used as x, t, Tj, and y;. The dimensionless dis- tance kd between the interfaces is replaced with d below. Then, from Egs. (16) and (17), we have the normalized linear solution

Related Figures (16)

We consider two-dimensional inviscid and incompressible flows in which two fluid interfaces I; and I, with density and tan- gential velocity jumps (Fig. 1). The fluids over I, (density po) and
We consider two-dimensional inviscid and incompressible flows in which two fluid interfaces I; and I, with density and tan- gential velocity jumps (Fig. 1). The fluids over I, (density po) and
We assume that the non-perturbative states of interfaces I; and I, are y = d/2 and y = —d/2 (d > 0), respectively (refer to Fig. 1), and consider small deviations from those states. Linearizing the Bernoulli equation (3) and the kinematic boundary condition (13) at the non-perturbative interfaces, we obtain the following linearized equations at I, (y = d/2): where the quantities with tilde denote small perturbations: |i] « 1 (i=1, 2) and |$;| « 1 (¢=0, 1, 2).
We assume that the non-perturbative states of interfaces I; and I, are y = d/2 and y = —d/2 (d > 0), respectively (refer to Fig. 1), and consider small deviations from those states. Linearizing the Bernoulli equation (3) and the kinematic boundary condition (13) at the non-perturbative interfaces, we obtain the following linearized equations at I, (y = d/2): where the quantities with tilde denote small perturbations: |i] « 1 (i=1, 2) and |$;| « 1 (¢=0, 1, 2).
where k is the wavenumber, and we can select the amplitudes a; and a independently. Using a; (i = 1, 2), the coefficients Bo, By, Biz, and B2 are given as Taking into account the Laplace field (2) and the periodicity in the x direction, we have the following linear solution to Eqs. (14) and (a r-\._
where k is the wavenumber, and we can select the amplitudes a; and a independently. Using a; (i = 1, 2), the coefficients Bo, By, Biz, and B2 are given as Taking into account the Laplace field (2) and the periodicity in the x direction, we have the following linear solution to Eqs. (14) and (a r-\._
FIG. 2. Interfacial structures for 5 = 0 with the colored scale of the vortex sheet strengths and the velocity fields for the (left) double-layer RMI and (right) single- layer RMI, where the Atwood numbers are (left) A; = —1 and Ap = 1 and (right) A= 1. Both panels show the last computed stage of the evolution for those Atwood numbers, where t = 0.12 for the left panel and t = 0.55 for the right panel. The initial distance d between /, and /, in the left panel is selected as d = zz.
FIG. 2. Interfacial structures for 5 = 0 with the colored scale of the vortex sheet strengths and the velocity fields for the (left) double-layer RMI and (right) single- layer RMI, where the Atwood numbers are (left) A; = —1 and Ap = 1 and (right) A= 1. Both panels show the last computed stage of the evolution for those Atwood numbers, where t = 0.12 for the left panel and t = 0.55 for the right panel. The initial distance d between /, and /, in the left panel is selected as d = zz.
FIG. 3. Fourier amplitudes for /, in the (a) double-layer interfaces and (b) single- layer interface, where the Atwood numbers are the same as in Fig. 2. The slope of the dotted line is —5/2. We show the growth of the Fourier amplitude V. Xi( t) +¥ (t)” for A; =—1and A = 1 and the corresponding Fourier amplitude of the single-layer RMI for A = 1 in Fig. 3. We depict the Fourier ampli- tude for I; in Fig. 3, but the amplitude for I, at the corresponding
FIG. 3. Fourier amplitudes for /, in the (a) double-layer interfaces and (b) single- layer interface, where the Atwood numbers are the same as in Fig. 2. The slope of the dotted line is —5/2. We show the growth of the Fourier amplitude V. Xi( t) +¥ (t)” for A; =—1and A = 1 and the corresponding Fourier amplitude of the single-layer RMI for A = 1 in Fig. 3. We depict the Fourier ampli- tude for I; in Fig. 3, but the amplitude for I, at the corresponding
FIG. 4. Curvatures of (a) /; (blue line) and /> (red line) in the double-layer interfaces at t = 0.12 and (b) single-layer interface at t = 0.55, where the Atwood number: are the same as in Figs. 2 and 4. Curvatures of J; (i = 1, 2) at t = 0.12 and the one of the single- layer interface at t = 0.55 are depicted as a function of the Lagrange parameter e in Fig. 4. Although the interfacial structures in Fig. 2 are sufficiently smooth, the corresponding curvatures in Fig. 4 have cusps at the spikes [e = 0 on I; and e = +7 on I in Fig. 4(a) and e = 0 in Fig. 4(b)]. We mention that the vortex sheet strength y; (y) in the neighborhood of the spike is not so large; however, the derivative Vi,e (Ye) is quite large because the sheet strength y; (y) changes its sign at the spike (refer to Fig. 2). The numerical calculations for these Atwood numbers break down immediately after (¢ = t, = +0.12 for the double-layer RMI and t = t, = 0.55 for the single-layer RMI) the appearance of the curvature singularity. From now on, we refer to t, as the break-down time.
FIG. 4. Curvatures of (a) /; (blue line) and /> (red line) in the double-layer interfaces at t = 0.12 and (b) single-layer interface at t = 0.55, where the Atwood number: are the same as in Figs. 2 and 4. Curvatures of J; (i = 1, 2) at t = 0.12 and the one of the single- layer interface at t = 0.55 are depicted as a function of the Lagrange parameter e in Fig. 4. Although the interfacial structures in Fig. 2 are sufficiently smooth, the corresponding curvatures in Fig. 4 have cusps at the spikes [e = 0 on I; and e = +7 on I in Fig. 4(a) and e = 0 in Fig. 4(b)]. We mention that the vortex sheet strength y; (y) in the neighborhood of the spike is not so large; however, the derivative Vi,e (Ye) is quite large because the sheet strength y; (y) changes its sign at the spike (refer to Fig. 2). The numerical calculations for these Atwood numbers break down immediately after (¢ = t, = +0.12 for the double-layer RMI and t = t, = 0.55 for the single-layer RMI) the appearance of the curvature singularity. From now on, we refer to t, as the break-down time.
FIG. 5. Interfacial structures for 5 = 0 with the colored scale of the vortex sheet strengths and the velocity fields for the (left) double-layer RMI and (right) single- layer RMI, where the Atwood numbers are (left) A; = —0.2 and A> = 0.2 and (right) A = 0.2. Both panels show the last computed stage of the evolution for those Atwood numbers, where ft = 1.0 for the left panel and t = 0.94 for the right panel. The initial distance d between /, and / in the left panel is selected as d = zz.
FIG. 5. Interfacial structures for 5 = 0 with the colored scale of the vortex sheet strengths and the velocity fields for the (left) double-layer RMI and (right) single- layer RMI, where the Atwood numbers are (left) A; = —0.2 and A> = 0.2 and (right) A = 0.2. Both panels show the last computed stage of the evolution for those Atwood numbers, where ft = 1.0 for the left panel and t = 0.94 for the right panel. The initial distance d between /, and / in the left panel is selected as d = zz.
FIG. 6. Fourier amplitudes for /, in the (a) double-layer interfaces and (b) single- layer interface, where the Atwood numbers are the same as in Fig. 5. The slope of the dotted line is —5/2.
FIG. 6. Fourier amplitudes for /, in the (a) double-layer interfaces and (b) single- layer interface, where the Atwood numbers are the same as in Fig. 5. The slope of the dotted line is —5/2.
FIG. 8. Interfacial structures for 6 = 0 with the colored scale of the vortex sheet strengths and the velocity fields of the double-layer RMI for the initial distance d = 7/4 between the two interfaces, where the Atwood numbers are the same as in Fig. 5. The left and right panels show the velocity fields at t = 1.0 and t = 2.0, respectively. FIG. 7. Curvatures of (a) /; (blue line) and /2 (red line) in the double-layer interfaces at t = 1.0 and (b) single-layer interface at t = 0.94, where the Atwood numbers are the same as in Figs. 5 and 6.
FIG. 8. Interfacial structures for 6 = 0 with the colored scale of the vortex sheet strengths and the velocity fields of the double-layer RMI for the initial distance d = 7/4 between the two interfaces, where the Atwood numbers are the same as in Fig. 5. The left and right panels show the velocity fields at t = 1.0 and t = 2.0, respectively. FIG. 7. Curvatures of (a) /; (blue line) and /2 (red line) in the double-layer interfaces at t = 1.0 and (b) single-layer interface at t = 0.94, where the Atwood numbers are the same as in Figs. 5 and 6.
FIG. 9. Fourier amplitude (upper panel) and the curvature at f = 2.0 (lower panel) in the double-layer interfaces with the initial distance d = 7/4. The Atwood numbers are the same as in Figs. 5-8. The slope of the dotted line in the upper panel is —§/2.
FIG. 9. Fourier amplitude (upper panel) and the curvature at f = 2.0 (lower panel) in the double-layer interfaces with the initial distance d = 7/4. The Atwood numbers are the same as in Figs. 5-8. The slope of the dotted line in the upper panel is —§/2.
FIG. 10. Temporal evolution of interfacial structures of the double-layer RMI for 6 = 0.15 with the colored scale of the vortex sheet strengths, where the initial dis- tance d = 7/2, the Atwood numbers A; = A = —0.2, and the linear amplitudes a, and a» satisfy the in-phase condition (24). The depicted times are t = (a) 2, (b) 4, (c) 6, and (d) 10. We show the temporal evolution of interfacial structures with the initial distance d = 2/2 and the linear amplitudes a; = a = 1 [in phase (24)] in Fig. 10, where the Atwood numbers are A; = Az = -0.2 [po = 2/3 and f2 = 3/2 in Eq. (22)]; that is, we consider the case of (23a). This is the natural extension of the conventional single- layer RMI such that the sum of two initial sheet strengths in Eq. (20) becomes equal to the initial sheet strength of the single-layer RMI y in the limit of d > 0: limg-o(yile, 0) + y2(e, 0)) = -2 sine = y(e, 0). The two interfaces gradually approach each other and merge, and finally, they behave like one interface. This merging occurs for all ini- tial distances d = 7 (refer to Fig. 12), 2/2, and 7/4 when 6 is finite. In particular, when the initial distance between two interfaces is small (d = n/2 and d = 7/4), the two interfaces finally merge including the bubbles (x = +7) and spikes (x = 0), as shown in Fig. 10. The sheet strengths y; and y2 concentrate at the vortex cores, ’ the center of the mushrooms, and the same-signed neighboring cores rotate around each other. The merging phenomenon is caused by the incompress- ibility of the system that the area-preserving condition must hold. In order to maintain the area of the initial rectangle region between two interfaces, the distance between the interfaces is needed to become
FIG. 10. Temporal evolution of interfacial structures of the double-layer RMI for 6 = 0.15 with the colored scale of the vortex sheet strengths, where the initial dis- tance d = 7/2, the Atwood numbers A; = A = —0.2, and the linear amplitudes a, and a» satisfy the in-phase condition (24). The depicted times are t = (a) 2, (b) 4, (c) 6, and (d) 10. We show the temporal evolution of interfacial structures with the initial distance d = 2/2 and the linear amplitudes a; = a = 1 [in phase (24)] in Fig. 10, where the Atwood numbers are A; = Az = -0.2 [po = 2/3 and f2 = 3/2 in Eq. (22)]; that is, we consider the case of (23a). This is the natural extension of the conventional single- layer RMI such that the sum of two initial sheet strengths in Eq. (20) becomes equal to the initial sheet strength of the single-layer RMI y in the limit of d > 0: limg-o(yile, 0) + y2(e, 0)) = -2 sine = y(e, 0). The two interfaces gradually approach each other and merge, and finally, they behave like one interface. This merging occurs for all ini- tial distances d = 7 (refer to Fig. 12), 2/2, and 7/4 when 6 is finite. In particular, when the initial distance between two interfaces is small (d = n/2 and d = 7/4), the two interfaces finally merge including the bubbles (x = +7) and spikes (x = 0), as shown in Fig. 10. The sheet strengths y; and y2 concentrate at the vortex cores, ’ the center of the mushrooms, and the same-signed neighboring cores rotate around each other. The merging phenomenon is caused by the incompress- ibility of the system that the area-preserving condition must hold. In order to maintain the area of the initial rectangle region between two interfaces, the distance between the interfaces is needed to become
FIG. 11. Temporal evolution of (the absolute value of) the growth rates of (a) bub bles and (b) spikes for 6 = 0.15, where the Atwood number of the single RM is A = 0.2 and those of the double-layer are the same as in Fig. 10. The blac! ine denotes the single-layer RMI, and the blue, green, and red lines indicate the growth rates of the double-layer RMI with initial distances d = 7/4, /2, and 7. Fo the double-layer RMI, the solid and dotted lines denote the growth rates of /, anc lp, respectively. Figure 11 shows the growth rates of (a) bubbles and (b) spikes for the single-layer RMI (black line) with the Atwood number A = 0.2 and the double-layer RMI (blue, green, and red lines) with the Atwood numbers A, = Az = —0.2, where the initial distances d are selected as d = 7/4 (blue lines), 7/2 (green lines), and z (red lines). The growth rates of I; and I; are depicted with solid and dotted lines, respectively. The bubbles and spikes for these Atwood num- bers and the linear amplitudes (a; = a2 = 1) appear at the same x coordinate on the two interfaces I; and Iz, where the former appear at x = +7 and the latter appear at x = 0. Since a) = ag, the initial sheet strength yj(e, 0) is equal to y2(e, 0) [refer to Eq. (20)]; therefore, the growth rates of the bubble and spike for I; coincide with those for In
FIG. 11. Temporal evolution of (the absolute value of) the growth rates of (a) bub bles and (b) spikes for 6 = 0.15, where the Atwood number of the single RM is A = 0.2 and those of the double-layer are the same as in Fig. 10. The blac! ine denotes the single-layer RMI, and the blue, green, and red lines indicate the growth rates of the double-layer RMI with initial distances d = 7/4, /2, and 7. Fo the double-layer RMI, the solid and dotted lines denote the growth rates of /, anc lp, respectively. Figure 11 shows the growth rates of (a) bubbles and (b) spikes for the single-layer RMI (black line) with the Atwood number A = 0.2 and the double-layer RMI (blue, green, and red lines) with the Atwood numbers A, = Az = —0.2, where the initial distances d are selected as d = 7/4 (blue lines), 7/2 (green lines), and z (red lines). The growth rates of I; and I; are depicted with solid and dotted lines, respectively. The bubbles and spikes for these Atwood num- bers and the linear amplitudes (a; = a2 = 1) appear at the same x coordinate on the two interfaces I; and Iz, where the former appear at x = +7 and the latter appear at x = 0. Since a) = ag, the initial sheet strength yj(e, 0) is equal to y2(e, 0) [refer to Eq. (20)]; therefore, the growth rates of the bubble and spike for I; coincide with those for In
FIG. 12. Comparison between the interfacial structures of the (left) double-layer RMI and (right) single-layer RMI at the same time t = 10, where the initial distance in the left panel is d = z and the Atwood numbers are the same as in Fig. 10. The Atwood number of the right panel is A = 0.2. We show the interfacial structures at t = 10 (the final stage of the calculation) for the double-layer RMI with the initial distance d = 1 (left) and the single-layer RMI for A = 0.2 (right) in Fig. 12, where the Atwood numbers of the left panel are the same as in Fig. 10. As we see from Fig. 11(b), the asymptotic behavior of the growth rate of the spike in the single-layer RMI (black line) is closest to that of the upper interface I; with the initial distance d = 7 (red solid line), although the interfacial shape, especially the structure in the neighborhood of vortex cores, is quite different from each other, as shown in Fig. 12. Unlike the interfaces with d = 7/2 in Fig. 10, the bubbles and spikes of the two interfaces with d = 7 keep at a distance even at t = 10, although the interfaces around the vortex cores merge. Secondary vortex cores with relatively strong sheet strengths appear in the neighborhood of the spike on I, and the bubble on J; at this time in the double-layer RMI.
FIG. 12. Comparison between the interfacial structures of the (left) double-layer RMI and (right) single-layer RMI at the same time t = 10, where the initial distance in the left panel is d = z and the Atwood numbers are the same as in Fig. 10. The Atwood number of the right panel is A = 0.2. We show the interfacial structures at t = 10 (the final stage of the calculation) for the double-layer RMI with the initial distance d = 1 (left) and the single-layer RMI for A = 0.2 (right) in Fig. 12, where the Atwood numbers of the left panel are the same as in Fig. 10. As we see from Fig. 11(b), the asymptotic behavior of the growth rate of the spike in the single-layer RMI (black line) is closest to that of the upper interface I; with the initial distance d = 7 (red solid line), although the interfacial shape, especially the structure in the neighborhood of vortex cores, is quite different from each other, as shown in Fig. 12. Unlike the interfaces with d = 7/2 in Fig. 10, the bubbles and spikes of the two interfaces with d = 7 keep at a distance even at t = 10, although the interfaces around the vortex cores merge. Secondary vortex cores with relatively strong sheet strengths appear in the neighborhood of the spike on I, and the bubble on J; at this time in the double-layer RMI.
FIG. 13. Interfacial structures at t = 10 with (a) in phase [Eq. (24), a; = a2 = 1] and (b) out of phase [Eq. (25), a; = 1 and a) = —1] initial conditions, where the Atwood numbers A; = —0.2 and Az = 0.2 and the initial distance d = 7. A period of two wavelengths is depicted in these figures.
FIG. 13. Interfacial structures at t = 10 with (a) in phase [Eq. (24), a; = a2 = 1] and (b) out of phase [Eq. (25), a; = 1 and a) = —1] initial conditions, where the Atwood numbers A; = —0.2 and Az = 0.2 and the initial distance d = 7. A period of two wavelengths is depicted in these figures.
FIG. 14. Growth rates of (upper) bubbles and (lower) spikes for a fixed initial dis- tance d = zr, where the Atwood numbers are (a) A; = Az = —0.2 and [(b) and (c)] A, = —0.2 and Ap = 0.2 for both panels. The linear amplitudes in (a) and (b) are selected as a; = a2 = 1 (in phase), and those in (c) are a; = 1 and a2 = —1 (out of phase). The black solid line denotes the growth rate of the single-layer RMI for A = 0.2. The growth rates depicted with black and red lines are the same as those in Fig. 11.
FIG. 14. Growth rates of (upper) bubbles and (lower) spikes for a fixed initial dis- tance d = zr, where the Atwood numbers are (a) A; = Az = —0.2 and [(b) and (c)] A, = —0.2 and Ap = 0.2 for both panels. The linear amplitudes in (a) and (b) are selected as a; = a2 = 1 (in phase), and those in (c) are a; = 1 and a2 = —1 (out of phase). The black solid line denotes the growth rate of the single-layer RMI for A = 0.2. The growth rates depicted with black and red lines are the same as those in Fig. 11.
Related topics:
EngineeringPhysicsMechanicsClassical MechanicsPhysics of FluidsVortex Sheet
580 California St., Suite 400
San Francisco, CA, 94104
© 2025 Academia. All rights reserved