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Figure 5. Time evolution of the interface in the two-dimensional KHI problem for the density ratio of P7/p,=2 at various Ri numbers: (a) t* =0.5; (b) t* = 1.0; (c) t* =1.5; and (d) t* =2.0 (a=0.001).

Figure 5 Time evolution of the interface in the two-dimensional KHI problem for the density ratio of P7/p,=2 at various Ri numbers: (a) t* =0.5; (b) t* = 1.0; (c) t* =1.5; and (d) t* =2.0 (a=0.001).

Related Figures (11)

Figure 1. Configuration of Kelvin-Helmholtz instability at initial time, t=0.
Figure 1. Configuration of Kelvin-Helmholtz instability at initial time, t=0.
‘igure 2. The initial (a) t=Os and the final (b) t=1s shapes of the square-droplet problem with thin interface.
‘igure 2. The initial (a) t=Os and the final (b) t=1s shapes of the square-droplet problem with thin interface.
Figure 3. Interface curvature versus droplet radius and the form of the Dirac delta function at the final time (t=1s) for the square-droplet deformation problem: (a) thick interface and (b) thin interface.
Figure 3. Interface curvature versus droplet radius and the form of the Dirac delta function at the final time (t=1s) for the square-droplet deformation problem: (a) thick interface and (b) thin interface.
Figure 4. Time evolution of the interface in the two-dimensional KHI problem for the density ratio of (p2/p,=2), and «=0.001 at Ri =0.01, which is given between dimensionless time t=0.25 and t=4.0 with a time interval of At=0.25. The time step increment is from left-to-right for each row.
Figure 4. Time evolution of the interface in the two-dimensional KHI problem for the density ratio of (p2/p,=2), and «=0.001 at Ri =0.01, which is given between dimensionless time t=0.25 and t=4.0 with a time interval of At=0.25. The time step increment is from left-to-right for each row.
Figure 6. Time evolution of the interface in the two-dimensional KHI problem for the density ratio of P7/p,=5 at various Ri numbers: (a) t* =0.5; (b) t* = 1.0; (c) t*=1.5; and (d) t* =2.0 (a=0.001).
Figure 6. Time evolution of the interface in the two-dimensional KHI problem for the density ratio of P7/p,=5 at various Ri numbers: (a) t* =0.5; (b) t* = 1.0; (c) t*=1.5; and (d) t* =2.0 (a=0.001).
Figure 7. Time evolution of the interface in the two-dimensional KHI problem for the density ratio of P>/p,=10 at various Ri numbers: (a) t*=0.5; (b) f* = 1.0; (c) t*=1.5; and (d) t* =2.0 («=0.001).
Figure 7. Time evolution of the interface in the two-dimensional KHI problem for the density ratio of P>/p,=10 at various Ri numbers: (a) t*=0.5; (b) f* = 1.0; (c) t*=1.5; and (d) t* =2.0 («=0.001).
Figure 8. Growth rate G of the KHI in the linear regime for various Ri numbers and density ratios (Ri numbers are based on surface tension; «=0.001).
Figure 8. Growth rate G of the KHI in the linear regime for various Ri numbers and density ratios (Ri numbers are based on surface tension; «=0.001).
Figure 9. Effect of stabilizing forces on the growth rate G of the KHI in the linear regime (p/p, = 10; «=0.01).
Figure 9. Effect of stabilizing forces on the growth rate G of the KHI in the linear regime (p/p, = 10; «=0.01).
Figure 10. Effect of artificial viscosity on the growth rate G of the KHI in the linear regime (p/p, = 10)
Figure 10. Effect of artificial viscosity on the growth rate G of the KHI in the linear regime (p/p, = 10)
Figure 11. Time evolution of the interface in the two-dimensional KHI problem for the density ratio of (p2/p,=10), and «=0.01 at Ri=0.01, which is given between dimensionless time t=0.25 and t=6.0 with a time interval of At=0.25. The time step increment is from left-to-right for each row.
Figure 11. Time evolution of the interface in the two-dimensional KHI problem for the density ratio of (p2/p,=10), and «=0.01 at Ri=0.01, which is given between dimensionless time t=0.25 and t=6.0 with a time interval of At=0.25. The time step increment is from left-to-right for each row.
Figure 12. Snapshots of the interface in two-dimensional KHI problem for different density ratios at Ri =0 (both surface tension and body forces are set to zero): (a) t* =0.25; (b) f* =1.5; and (c) t* =2.75. The artificial force due to the improper treatment of the density gradient in the standard SPH discretization scheme is repulsive in nature; hence, resulting in a gap formation along the interface as reported in [22, 23, 33]. The existence of the artificial force damps the initial perturbation of the interface thereby suppressing the onset of the instability, and hindering the mixing of two fluids, which was also discussed in [23] referring to the entropy of mixing of a thermodynamic system. To be able to show that the SPH discretization scheme relying on the utilization of the particle number density does not create any significant artificial surface tension effect which can suppress the onset and the growth of the KHI, Figure 12 presents the results of simulations with density ratios of 2, 5, and 10, for which both the body and the surface tension forces are set equal to zero (which corresponds to the Ri =0).
Figure 12. Snapshots of the interface in two-dimensional KHI problem for different density ratios at Ri =0 (both surface tension and body forces are set to zero): (a) t* =0.25; (b) f* =1.5; and (c) t* =2.75. The artificial force due to the improper treatment of the density gradient in the standard SPH discretization scheme is repulsive in nature; hence, resulting in a gap formation along the interface as reported in [22, 23, 33]. The existence of the artificial force damps the initial perturbation of the interface thereby suppressing the onset of the instability, and hindering the mixing of two fluids, which was also discussed in [23] referring to the entropy of mixing of a thermodynamic system. To be able to show that the SPH discretization scheme relying on the utilization of the particle number density does not create any significant artificial surface tension effect which can suppress the onset and the growth of the KHI, Figure 12 presents the results of simulations with density ratios of 2, 5, and 10, for which both the body and the surface tension forces are set equal to zero (which corresponds to the Ri =0).
Related topics:
Numerical SimulationsComputational Fluid MechanicsInterfacial PhenomenaTwo Phase FlowAerodynamicsNumerical AnalysisLES and DNS
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