.. include:: /images.rst PIC06: Rayleigh-Taylor Instability ---------------------------------- .. _description-pic06: Description ~~~~~~~~~~~ The simulation of Rayleigh-Taylor instability using PIC methodology follows the work of Snider :cite:Snider2001. The domain is initialized with a lighter phase at the bottom and a heavier phase at the top. When the simulation begins, the phases invert, and the growth of a mixing layer is recorded as a function of time. Researchers in the past have proposed the following functional form for the development of the mixing layer, .. math:: :label: pic06eq1 h = \alpha A g t^{2} where the non-dimensional parameter, :math:A, used to characterize the system is Atwood number: .. math:: :label: pic06eq2 A = \frac{\rho_{s}-\rho_{g}}{\rho_{s}+\rho_{g}} and the value of :math:\alpha is between 0.05 and 0.07 (Youngs :cite:Youngs1984; Linden et al. :cite:Linden1994; Snider and Andrews :cite:Snider1996). A rectangular domain (0.1 m X 0.6 m X 0.1 m) is chosen for simulating this system. The values for fluid and particle properties are borrowed from the work of Snider :cite:Snider2001. A larger value of particle diameter is used and the interphase drag coefficient (:math:\propto 1/d_{p}) is scaled accordingly. The list of parameters used in this exercise are summarized in :numref:pic06table2. .. _pic06table1: .. csv-table:: Material properties used in Rayleigh-Taylor Instability simulations :widths: auto :header: "","Case 1", "Case 2", "Case 3" "Particle diameter (m) ","2 X 10-6","2 X 10-6","2 X 10-6" "Particle density (kg/m3)","3 ","5 ","7 " "Fluid density (kg/m3) ","1 ","1 ","1 " "Fluid viscosity (Pa-s) ","0.001 ","0.001 ","0.001 " "Atwood number ","0.1667 ","0.2857 ","0.4737 " "Drag coefficient, :math:\beta $$kg·m\ :sup:-3 ·s\ :sup:-1$$","100 :math:\rho_{s} \epsilon_{s}","100 :math:\rho_{s} \epsilon_{s}","100 :math:\rho_{s} \epsilon_{s}" .. _setup-pic06: Setup ~~~~~ .. _pic06table2: .. csv-table:: PIC-06 Setup, Initial and Boundary Conditions. :widths: auto :header: "Computational/Physical model", " ", " " "3D, Transient", " ", " " "Multiphase", " ", " " "Gravity", " ", " " "Thermal energy equation is not solved", " ", " " "Turbulence equations are not solved (Laminar)", " ", " " "Uniform mesh", " ", " " "First order upwind discritization scheme", " ", " " " ", " ", " " "**Geometry**", " ", " " "Coordinate system", "Cartesian", " ", "Grid partitions" "x-length", "0.10", "$$m$$", "40" "y-length", "0.60", "$$m$$", "240" "z-length", "0.10", "$$m$$", "40" " ", " ", " " "**Material**", " ", " " "Gas density, :math:\rho_{g}", "1.0", "(kg·m\ :sup:-3)" "Gas viscosity, :math:\mu_{g}", "1.8E-5", "(Pa·s)" " ", " ", " " "**Solids Type**", "PIC", " " "Diameter, :math:d_{p}", "0.001", "$$mm$$" "Density, :math:\rho_{s}", ":numref:pic06table1", "(kg·m\ :sup:-3)" " ", " ", " " "**Solids Properties (PIC)**", " ", " " "Pressure linear scale factor, :math:P_{s}", "1.0", "(Pa)" "Exponential scale factor, :math:\gamma", "4.0", "(-)" "Statistical weight", "7.2E+08", "(-)" " ", " ", " " "**Initial Conditions**", " ", " ", " " "x-velocity, :math:u_{g}", "0.0", "(m·s\ :sup:-1)" "y-velocity, :math:v_{g}", "0.0", "(m·s\ :sup:-1)" "z-velocity, :math:w_{g}", "0.0", "(m·s\ :sup:-1)" "Gas volume fraction, :math:\epsilon_{g}", "0.80", "(-)" "Gas volume fraction at packing, :math:\epsilon_{g}^{*}", "0.4", "(-)" "Pressure, :math:P_{g}", "101,325", "(Pa)" " ", " ", " " "**Boundary Conditions**", " ", " ", " " "Top boundary", " ", " ", "Pressue outflow" "All other boundaries", " ", " ", "Free-slip walls" .. _results-pic06: Results ~~~~~~~ The contour plots :numref:pic06fig1 show the evolution of volume fraction fields at the end of 1 second. The instability is triggered by a non-homogenous solids concentration field due to inherent randomness in generating the parcels. The instability is more pronounced at higher values of A. :numref:pic06fig2 shows the time evolution of the mixing layer, where the coordinates used by Snider :cite:Snider2001 are used. The results are consistent with the work of Snider :cite:Snider2001. The analytical value for the slope of this curve based on :eq:pic06eq1 is :math:\sqrt{\alpha}, which is matched reasonably well by MFiX-PIC. As A increases, the particles reach the bottom of the domain sooner resulting in the associate curve reaching a plateau. .. _pic06fig1: .. figure:: ../media/pic06-1.png :align: center :width: 320px :height: 400px Sectional view of volume fraction contour of the lighter phase at t = 0.8s; A = 0.1667, 0.2857, and 0.4737 (left to right) .. _pic06fig2: .. figure:: ../media/pic06-2.png :align: center :width: 400px :height: 280px Evolution of mixing layer for A = 0.1667, 0.2857, and 0.4737. The dashed line is the theoretical solution, :eq:pic06eq1.