Numerical Simulation

Full closure of the inlet and exit valves require annihilation of the elements trapped in between the valves and the top wall, requiring re-meshing of the computational domain. Re-meshing is avoided by allowing a gap in between the valves and the top wall. This gap was g=0.025 L at closed-valve position.

  

Figure 5.17: Spectral element mesh used for the discretization of the micro-pump system at ejection-stage (top, membrane is moving up). The discretization of the flow domain during suction stage is shown at the bottom (membrane is moving down). The bottom figure also shows elemental discretization obtained by 7th order modal expansions used.

The mesh used in this study is presented in figure (5.17) at various time instants, corresponding to suction and ejection stages. The flow domain in represented with 222 triangular (unstructured) spectral elements. Each element utilized 7^th-order polynomial modal expansions. We used successive p-refinements to verify convergence. The results at 5^th-order and 7^th-order modal expansions show that we are resolved well beyond the scientific accuracy of $1 \%$. A summary of the simulation parameters is given in table 5.6.

 

Parameter Value

Dimension 2d

Re 0.3, 3 and 30

$$\Delta t$$ 0.0025

N-Range 5,6 and 7

K_Tri 222

Method ALE Galerkin

  

Figure 5.18: Non-dimensional volumetric flow rate variation with in a period of the micro-pump, as a function of Reynolds number, Re=a*a*omega/nu, a/L = 1/10, a/h=1/3.

In figure 5.18, we present the instantaneous flow rate variation for two different Reynolds numbers (Re= 3, 30). The volumetric flow rate of fluid entering through the inlet valve is taken as positive and leaving flow rate is taken as negative. The sum of the two is the rate of change of the control volume due to the oscillation of the membrane

The membrane's motion is periodic. Therefore, the net amount of fluid displaced by the membrane in a period is zero

In other words, $\dot{\overline{Q}}_{in} = \dot{\overline{Q}}_{out}=\dot{\overline{Q}}$. Numerically integrating the curves under inlet and exit valves in figure 5.18, we determined the effective flow rate in our micro-pump for g=0.025 L. The ratio of the numerical values of the flow rate to the maximum flow rate given by equation (5.17) defines the efficiency ($\eta$) of the micro-pump. The results are presented in table 5.7. It is clearly seen that the efficiency of the pump decreases with increasing the Reynolds number ($Re=\frac{\omega a^2}{\nu}$). The average flow rate of the pump increases with the Reynolds number, as predicted by equation (5.17). This is either due to the increase in the size of the pump (increase in a), or it is due to increase in the frequency $\omega$. For fabrication of our conceptual design, actual-dimensions of the micro-pump can be determined by either selecting the admissible amplitude of vibration a or the actuation frequency $\omega$. For example, if we select a frequency range $1 kHz \le \omega \le 100 kHz$, we can determine the values of the amplitude a and hence, the rest of the pump-dimensions can be determined.

 

Re 0.3 3 30

$$\frac{\dot{\overline{Q}}}{W} \left[\frac{m^2}{s} \right]_{THEORETICAL} 3.82 \times 10^{-5} 3.82 \times 10^{-4} 3.82 \times 10^{-3}$$

$$\frac{\dot{\overline{Q}}}{W} \left[\frac{m^2}{s} \right]_{NUMERICAL} 3.51 \times 10^{-5} 3.36 \times 10^{-4} 2.86\times 10^{-3}$$

$$\eta 92 \% 88 \% 75 \%$$

 

There is a dominant vortical structure near the inlet valve through out the simulations, although its strength varies during the cycle.

  

Figure 5.19: Vorticity contours for Re=30 simulation. Top figure at tau omega = 0.28, corresponds to the beginning of the suction stage. Start-up vortices due to the motion of the inlet valve can be identified. Middle figure is at tau omega =0.72, corresponding to the end of the suction stage. A vortex jet pair is visible in the pump cavity. Bottom figure tau omega = 84, corresponding to early ejection stage. Further evolution of the vortex jet and the start-up vortex of the exit valve can be identified.

 

Figure 5.20: Close up of the vorticity contours for Re=30 simulation at the left valve (meshes shown on right side). tau omega = 0.28, corresponds to the beginning of the suction stage. Start-up vortices due to the motion of the inlet valve can be identified.

 

Figure 5.21: Close up of the vorticity contours for Re=30 simulation at the left valve (meshes shown on right side). tau omega =0.72, corresponding to the end of the suction stage. A Vortex jet pair is visible in the pump cavity.

  

Figure 5.22: Close up of the vorticity contours for Re=30 simulation at the left valve (meshes shown on right side). tau omega = 84, corresponding to early ejection stage. Further evolution of the vortex jet and the start-up vortex of the exit valve can be identified.

The vorticity contours are presented for Re=30 flow at various time instants in figure 5.19. The top figure is at $\tau \omega = 0.28$. This corresponds to the beginning of the suction stage. Start-up vortices due to the opening of the inlet valve can be identified as a vortex pair just at the top of the inlet valve. The middle figure corresponds to the end of the suction stage (at $\tau \omega =0.72$). A vortex jet pair is visible in the pump cavity. The flow pattern at $\tau \omega = 84$, corresponding to early ejection stage is given in the bottom figure. The exit valve has just open, and the start-up vortex due to its motion is visible at the top of the exit valve. The vortex pair in the pump cavity has evolved further. The negative vortex is trapped in the pump cavity, and the positive vortex jet hits the membrane and bounces back, towards the middle of the pump cavity. Presence of the vortex pair also creates strong vorticity on the membrane. In figure 5.22 we have repeated these time slices but showing a close up of the left valve of the pump and the mesh there. When the valve is closed we can see that the mesh is very tightly packed yet the fields still look convincingly smooth around this area.

  

Figure 5.23: Three-dimensional micropump simulation Re=3. Top: An instant during the suction stage. Bottom: An instant during the ejection stage.

After demonstrating the performance of the micro-pump with two-dimensional simulations, we also tested the design with three-dimensional simulations. We have extended the two-dimensional geometry of the micro-pump uniformly, using W=0.4L. We observed boundary layer growth on the side walls, which reduces the flow rate slightly. Nevertheless, the micro-pump performance is also acceptable in three-dimensions. The stream-wise velocity contours during the suction and the ejection stages are presented in figure 5.23.

The algorithm is fast enough to test two-dimensional conceptual design on a work station. Low Reynolds number ($Re \le 5$)simulations require about 222 triangular spectral elements with 5^th order expansions. The simulation takes about 0.6 CPU seconds per time step on a 195 MHz Silicon Graphics Onyx2 work station. Three-dimensional simulations however require large memory and CPU times. A parallel version of the algorithm is used to test three-dimensional effects on several design cases. For three-dimensional runs, we have used 444 spectral elements prisms, utilizing 5^th order polynomial expansions per element. There are 90 calculation points per element. The computational domain is divided into 8 sub-domains and the parallel simulations are performed on the IBM SP2 (thin-2nodes) at Brown University, Center for Fluid Mechanics. We have observed 18 CPU seconds per time-step for the parallel run.