1. Date of download: 7/9/2016 Copyright © ASME. All rights reserved. From: Obtaining Time-Varying Pulsatile Gas Flow Rates With the Help of Dynamic Pressure Difference and Other Measurements for an Orifice-Plate Meter J. Fluids Eng. 2013;135(4):041101-041101-19. doi:10.1115/1.4023195 Proposed assembly of the orifice meter with instruments Figure Legend:

2. Date of download: 7/9/2016 Copyright © ASME. All rights reserved. From: Obtaining Time-Varying Pulsatile Gas Flow Rates With the Help of Dynamic Pressure Difference and Other Measurements for an Orifice-Plate Meter J. Fluids Eng. 2013;135(4):041101-041101-19. doi:10.1115/1.4023195 A representative example of pressure-difference and mass flow rate values encountered in the pulsatile flows of interest Figure Legend:

3. Date of download: 7/9/2016 Copyright © ASME. All rights reserved. From: Obtaining Time-Varying Pulsatile Gas Flow Rates With the Help of Dynamic Pressure Difference and Other Measurements for an Orifice-Plate Meter J. Fluids Eng. 2013;135(4):041101-041101-19. doi:10.1115/1.4023195 The schematic shows small amplitude of density variations at a point in the flow field. The amplitude of ∂ρ/∂t (represented by the slope of the dotted line) is significant over the small time scale Δt c ≡ 1/fP of interest but is negligible if averaged over large times T ≫ 1/fP. Figure Legend:

4. Date of download: 7/9/2016 Copyright © ASME. All rights reserved. From: Obtaining Time-Varying Pulsatile Gas Flow Rates With the Help of Dynamic Pressure Difference and Other Measurements for an Orifice-Plate Meter J. Fluids Eng. 2013;135(4):041101-041101-19. doi:10.1115/1.4023195 The geometry of a specific orifice meter. For the hardware, l OM1 = 127 mm. This is also the geometry of OM1 in Fig. 6. Figure Legend:

5. Date of download: 7/9/2016 Copyright © ASME. All rights reserved. From: Obtaining Time-Varying Pulsatile Gas Flow Rates With the Help of Dynamic Pressure Difference and Other Measurements for an Orifice-Plate Meter J. Fluids Eng. 2013;135(4):041101-041101-19. doi:10.1115/1.4023195 (a) Five steady and one unsteady pressure-difference history curves are shown. These are used for assessing the adequacy of the models proposed in Refs. [3–7]. (b) For the unsteady pressure-difference history (over time interval Δt Unsteady ) marked in (a), the model Δpom(t)=k·Q(t)2 should lead to Q(t) values that lie on parabolic curves (such as the solid black curve for a representative k value). Instead reasonable estimates of the flow rate Q(t) values (obtained by CFD) plotted against the unsteady pressure-difference values lead to the nonparabolic dotted curve. For steady pressure-difference histories shown in (a), however, the corresponding steady flow rates (reasonably estimated by CFD) are adequately described by the solid black curve given by: Δpom- steady(t)=k·Qsteady2 (where k = 3.26847·10 10 Pa·s 2 /m 6 and ρ 0 = 19 kg/m 3 ). (c) For the unsteady pressure-difference history (over time interval Δt Unsteady ) marked in (a), the relationship between Δpom(t)-k·Q(t)2 and dQ(t)/dt should be approximately linear if the model Δpom(t)=k·Q(t)2+L·dQ(t)/dt is reasonable. Instead, reasonable estimates of the flow rate Q(t) and its derivative dQ(t)/dt values (obtained by CFD) are such that the resulting dotted curve is far from the best linear fit (with k taken from the steady curve fit of (b) and L = −39,100 Pa · s 2 /m 4 being the slope of the solid line in (c)). Figure Legend:

6. Date of download: 7/9/2016 Copyright © ASME. All rights reserved. From: Obtaining Time-Varying Pulsatile Gas Flow Rates With the Help of Dynamic Pressure Difference and Other Measurements for an Orifice-Plate Meter J. Fluids Eng. 2013;135(4):041101-041101-19. doi:10.1115/1.4023195 Experimental flow loop Figure Legend:

7. Date of download: 7/9/2016 Copyright © ASME. All rights reserved. From: Obtaining Time-Varying Pulsatile Gas Flow Rates With the Help of Dynamic Pressure Difference and Other Measurements for an Orifice-Plate Meter J. Fluids Eng. 2013;135(4):041101-041101-19. doi:10.1115/1.4023195 The geometry of the orifice meter OM2 used in Fig. 6 Figure Legend:

8. Date of download: 7/9/2016 Copyright © ASME. All rights reserved. From: Obtaining Time-Varying Pulsatile Gas Flow Rates With the Help of Dynamic Pressure Difference and Other Measurements for an Orifice-Plate Meter J. Fluids Eng. 2013;135(4):041101-041101-19. doi:10.1115/1.4023195 Time-varying values of measured pressure differences (across OM1 and OM2 in Fig. 6) ΔpOM1(t) and ΔpOM2(t) for run 3 in Tables 1 and 2 Figure Legend:

9. Date of download: 7/9/2016 Copyright © ASME. All rights reserved. From: Obtaining Time-Varying Pulsatile Gas Flow Rates With the Help of Dynamic Pressure Difference and Other Measurements for an Orifice-Plate Meter J. Fluids Eng. 2013;135(4):041101-041101-19. doi:10.1115/1.4023195 The fast Fourier transform representation of the differential-pressure data Δp OM1 (t) and Δp OM2 (t) in Fig. 8 Figure Legend:

10. Date of download: 7/9/2016 Copyright © ASME. All rights reserved. From: Obtaining Time-Varying Pulsatile Gas Flow Rates With the Help of Dynamic Pressure Difference and Other Measurements for an Orifice-Plate Meter J. Fluids Eng. 2013;135(4):041101-041101-19. doi:10.1115/1.4023195 (a) Velocity V x (r, x, t) profiles as a function of r at x M′ = 0.12 m in Fig. 4 and time t = 0.8235 s for three different representative mesh sizes (with mesh 1, mesh 2, and mesh 3 described in the Appendix) associated with the quadrilateral elements used in the CFD solver. The incompressible CFD results are for run 3 in Tables 1 and 2. (b) Velocity V x (r, x, t) profiles as a function of r at x M′ = 0.12 m in Fig. 4 and time t = 0.8235 s arrived at for four different time steps (with TS1, TS2, TS3, and TS4 described in the Appendix) used in the CFD solver. Figure Legend:

11. Date of download: 7/9/2016 Copyright © ASME. All rights reserved. From: Obtaining Time-Varying Pulsatile Gas Flow Rates With the Help of Dynamic Pressure Difference and Other Measurements for an Orifice-Plate Meter J. Fluids Eng. 2013;135(4):041101-041101-19. doi:10.1115/1.4023195 (a) For run 3 in Table 1 and time t = t P marked on Fig. 8, the plot shows the streamline pattern within OM1 (see Fig. 4). Only half of the geometry (0 ≤ r ≤ R = 8.51 mm) that is symmetric with respect to the x-axis in Fig. 4 is shown. (b) At the cross-section location x B = 0.068 m in Fig. 4, the pressure variations p CFD (x, r, t) as a function of radius r are obtained from incompressible CFD simulations. As shown for three different time instants t = 1.0 s, 1.025 s, and 1.05 s, these variations are found to be nonuniform. (c) At the cross-section location x M′ = 0.12 m in Fig. 4, the pressure variations p CFD (x, r, t) as a function of radius, r, are obtained from incompressible CFD simulations. As shown for three different time instants t = 1.0 s, 1.025 s, and 1.05 s, these variations are found to be uniform. Figure Legend:

12. Date of download: 7/9/2016 Copyright © ASME. All rights reserved. From: Obtaining Time-Varying Pulsatile Gas Flow Rates With the Help of Dynamic Pressure Difference and Other Measurements for an Orifice-Plate Meter J. Fluids Eng. 2013;135(4):041101-041101-19. doi:10.1115/1.4023195 For the representative solution associated with the pressure differences in Fig. 8 (run 3 in Tables 1 and 2), the values of m·CFD- I|OM1(t) and m·CFD-I|OM2(t) are shown above Figure Legend:

13. Date of download: 7/9/2016 Copyright © ASME. All rights reserved. From: Obtaining Time-Varying Pulsatile Gas Flow Rates With the Help of Dynamic Pressure Difference and Other Measurements for an Orifice-Plate Meter J. Fluids Eng. 2013;135(4):041101-041101-19. doi:10.1115/1.4023195 Plots of m·CFD-I|OM1(t) for two different turbulence models (mod-1 and mod-2) are shown for run 3 in Tables 1 and 2 Figure Legend:

14. Date of download: 7/9/2016 Copyright © ASME. All rights reserved. From: Obtaining Time-Varying Pulsatile Gas Flow Rates With the Help of Dynamic Pressure Difference and Other Measurements for an Orifice-Plate Meter J. Fluids Eng. 2013;135(4):041101-041101-19. doi:10.1115/1.4023195 For the representative values of m·CFD-I|OM1(t) and m·CFD-I|OM2(t) in Fig. 12, the figure above shows the respective empirically corrected values m·Inc|OM1(t)=α·m·CFD-I|OM1(t) and m·Inc|OM2(t)=α·m·CFD-I|OM2(t) for turbulence model mod 1. In addition, the figure also shows m·Inc|OM1(t) for turbulence model mod 2. Figure Legend:

15. Date of download: 7/9/2016 Copyright © ASME. All rights reserved. From: Obtaining Time-Varying Pulsatile Gas Flow Rates With the Help of Dynamic Pressure Difference and Other Measurements for an Orifice-Plate Meter J. Fluids Eng. 2013;135(4):041101-041101-19. doi:10.1115/1.4023195 The plot above shows FFT of the empirically corrected mass flow rates values m·Inc|OM1(t) and m·Inc|OM2(t) in Fig. 14 for turbulence model mod 1 Figure Legend:

16. Date of download: 7/9/2016 Copyright © ASME. All rights reserved. From: Obtaining Time-Varying Pulsatile Gas Flow Rates With the Help of Dynamic Pressure Difference and Other Measurements for an Orifice-Plate Meter J. Fluids Eng. 2013;135(4):041101-041101-19. doi:10.1115/1.4023195 For the representative run 3 in Tables 1 and 2, the plot above shows computed values of the nondimensional integral NI(t) Figure Legend:

17. Date of download: 7/9/2016 Copyright © ASME. All rights reserved. From: Obtaining Time-Varying Pulsatile Gas Flow Rates With the Help of Dynamic Pressure Difference and Other Measurements for an Orifice-Plate Meter J. Fluids Eng. 2013;135(4):041101-041101-19. doi:10.1115/1.4023195 (a) This figure shows the time-varying values of pressure fluctuation at cross sections located by point L (pL'(t)≡pL(t)-p¯L) and M (pM'(t)≡pL'(t)-ΔpOM1(t)). These data are for run 4 of Tables 1 and 2. (b) The plot shows the FFT of pL'(t) and pM'(t) in (a). Figure Legend:

18. Date of download: 7/9/2016 Copyright © ASME. All rights reserved. From: Obtaining Time-Varying Pulsatile Gas Flow Rates With the Help of Dynamic Pressure Difference and Other Measurements for an Orifice-Plate Meter J. Fluids Eng. 2013;135(4):041101-041101-19. doi:10.1115/1.4023195 The incompressible mass flow rate values of m·Inc|OM1(t) for run 6 and its compressibility corrected values m·L(t) and m·M(t), respectively associated with points L and M in Fig. 4, are shown Figure Legend:

19. Date of download: 7/9/2016 Copyright © ASME. All rights reserved. From: Obtaining Time-Varying Pulsatile Gas Flow Rates With the Help of Dynamic Pressure Difference and Other Measurements for an Orifice-Plate Meter J. Fluids Eng. 2013;135(4):041101-041101-19. doi:10.1115/1.4023195 (a) The plot shows the mass flow rate m·M(t) at point M of OM1 in Fig. 4 for run 3 of Tables 1 and 2. These values are obtained from compressible flow models for two different thermal boundary conditions for the orifice-meter walls. One is an isothermal assumption and the other is an adiabatic assumption. (b) This plot shows, for run 3 in Tables 1 and 2, the comparison between m·stored(t) values obtained from the compressible flow CFD model and their values obtained from the proposed compressibility correction theory (Eq. (21)) for the incompressible CFD model (which has m·stored(t)=0). (c) This plot shows, for run 3 in Tables 1 and 2, the comparison between mass flow rate m·M-Comp(t) obtained from the compressible flow CFD model and m·M(t) obtained from a compressibility correction on the m·Inc|OM1(t) values—which are obtained from an incompressible CFD model. Figure Legend:

20. Date of download: 7/9/2016 Copyright © ASME. All rights reserved. From: Obtaining Time-Varying Pulsatile Gas Flow Rates With the Help of Dynamic Pressure Difference and Other Measurements for an Orifice-Plate Meter J. Fluids Eng. 2013;135(4):041101-041101-19. doi:10.1115/1.4023195 This figure shows the geometry for the two devices that are merged together, with proper instruments, and placed between points L and O of Fig. 6 Figure Legend: