Transport and Mixing in Idealized 3D flow Models: (work with: H. Huntley, B. Lipphardt, A.D. Kirwan):

• 3D Lagrangian Coherent Structures (LCS) in ABC flow:

• The steady ABC flow is defined by the equations:
  • u = A sin(z) + C cos(y),
  • v = B sin(x) + A cos(z),
  • w = C sin(y) + B cos(x).
  
  
• Case I: T = 10, A varies from A = 0.1, to A = 1.1, and B = C = 0.8 fixed.:


• Case II: T = 10, A varies from A = 0.1, to A = 1.1, and B = C = 0.1 fixed.:


• Case III: T = 10, A varies from A = 0.1, to A = 1.1, and B = C = 0.5 fixed.


• Case IV: T = 50, A varies from A = 0.1, to A=1.1, and B=C=0.1:


• Case V: A is fixed while B = C changes from 0.1 to 1.1: A = 0.1, B = C varies from B = C= 0.1 to B = C = 1.1.

• The steady quadrupole flow:
• u = -[(π/L_y) A + (π/L_x)B_z] sin(πx/L_x) cos(πy/L_y)
• v = [(π/L_x) A - (π/L_y)B_z] cos(πx/L_x)sin(πy/L_y)
• w = π²[L_x^-2 + L_y ^-2]B cos(πx/L_x) cos(πy/L_y)


• 3D vs 2D Finite-time Lyapunov Exponents (FTLEs): Three cases have been studied


• Nonzero (u,v) Vertical shear with zero vertical velocity ( u_z, v_z ≠ 0 and w=0).


• Zero (u,v) Vertical shear with nonzero vertical velocity ( u_z, v_z = 0 and w ≠ 0).


• Nonzero (u,v) Vertical shear with nonzero vertical velocity ( u_z, v_z ≠ 0 and w ≠ 0).