Immersed Boundary Method for Boltzmann Model Kinetic Equations

Graduate Student: Cem Pekardan

Predicting the flow properties and forces around unsteady and large displacement motion in a number of rarefied flow applications including the microbeams and membranes in radio-frequency MEMS switches and filters requires transient solution of Boltzmann equation with moving boundaries.

In this project, we develop and apply the immersed boundary schemes to solve the Boltzmann equation with the ES-BGK collision operator to study the effects of fluid damping on the motion of the complex immersed boundaries ([1] and [2]). A new immersed boundary method for Boltzmann model kinetic equations (Interrelaxation method-[1]) was developed and verified with ESBGK-Finite Volume Method solver and implemented into MEMOSA software.

A schematic of Interrelaxation method is given in the following figure. Scheme can be summarized as follows;

Step 1: Distribution function and equilibrium distribution function values are interpolated from only fluid cells to IB faces,

Step 2: The outgoing function values from the fluid cells are calculated using the relaxation procedure (space homogenous relaxation procedure),

Step 3: Conservation of mass flux algorithm is applied on the solid face to determine the incoming velocity distribution function values,

Step 4: All the incoming IB face values are then calculated from relaxation of incoming solid distribution function in in each direction with the relaxation procedure.

Figure 1. Immersed boundary method representation

As an example, pressure and temperature animations the 2D simulations of a cross section of a sinusoidally excited platinum microbeam are shown. The beam has a 120 μm width and 8 μm thickness. The microbeam is excited with a maximum velocity of 1 m/s which is simulated for two cycles. The oscillation frequency is 16 μs. The initial gap between the microbeam and the substrate is 2 μm with the beam forced to initially move towards the substrate.

In the figure below, circles represent damping force with the quasi-steady approximation and IBM results are given for different time steps. These values are smaller than the unsteady damping force for the given time instants. Compression and expansion under the beam can result in significant increase in the damping force by the flow inertia when the quasi-steady approximation is not valid.

Figure 2. Steady and unsteady damping force comparison for the oscillating beam

As a second example, Figure 1 represents the pressure contours and streamlines for a micro-beam which is accelerated towards the substrate with 10^6 g acceleration. The width of the beam is 5 μm and the thickness is 0.3 μm and motion starts from rest. Results are shown for various times ranging from 200 ns to 600 ns and simulation is stopped when the gap becomes 200 nm. It can be seen at the last time instant (600 ns-Figure 1) that pressure in the gap increases to as high as 1.98 atm and unsteady effects start to affect the flow field.

Figure 3. Pressure contours and streamlines for t=200 ns (upper left), t=400 ns (upper right) and t=600 ns (bottom)

Ref.[1] C. Pekardan, L. Sun, A. Alexeenko, “Immersed Boundary Method for Boltzmann Model Kinetic Equations,” 28th International Symposium on Rarefied Gas Dynamics-2012, AIP Conference Proceedings, 1501,358.

Ref.[2] C. Pekardan, A. Venkattraman, A. Alexeenko, “Applications of Immersed Boundary Method with ES-BGK model Boltzmann Equation,” AIAA Thermophysics Conference-2013, San Diego, CA, 2900.

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