2.4.1 1-D Nozzle Calculation
2.5 Run Conditions and Uncertainty Estimates
2.5.1 Overview
The shock speed is measured by two time of arrival pressure transducers with a known physical displacement along the axial direction of the shock tube, as discussed in Section 2.2. These transducers have an approximate measurement uncertainty of 8×10−6 s. The uncertainty in the shock speed measurement thus increases as the measured time of arrival difference decreases, based on the time scale that the data acquisition system can resolve. At a shock speed of 3000 m/s, typical for the
present study, the uncertainty is ∼30 m/s. The shock tube fill pressure uncertainty is ∼0.25 kPa, and the measured reservoir pressure uncertainty, based upon recorded pressure traces such as those presented in Figure 2.8, is typically ∼4 MPa. The measured uncertainties are presented in Table 2.1.
Uncertainties on the calculated quantities are estimated by perturbing Cantera Shock and Detonation Toolbox condition computations, similar to those described in Section 2.3, within the range of the uncertainties on the measured shock speed, reservoir pressure, and initial shock tube pressure. Only experiments with measured shock speeds that fall within the uncertainty for the adjusted shock speed curve1 predicted by the shock jump conditions from the primary diaphragm burst pressure, driver gas composition, and initial shock tube conditions are included in the present data set.
There are a number of other potential sources of measurement error, including nonideal gas behavior in the reservoir due to the high pressure, the extrapolation of the shock speed (which decays as it propagates down the shock tube) to the end wall, nonuniformity of reservoir conditions due to nonideal shock reflection, and the method of correcting flow conditions from the ideal reflected-shock pressure to measured reser- voir pressure using an isentropic expansion. Furthermore, the one-dimensional con- toured nozzle computation, described in Section 2.4.1, does not account for bound- ary layer growth within the nozzle, off-design operation conditions that lead to flow nonuniformity, or vibration-translation nonequilibrium and freezing within the noz- zle, which is particularly significant for the N2 cases. However, the axisymmetric nozzle computations described in Section 2.4.2, which provide the input conditions for boundary layer analysis, do include these nozzle effects.
Run conditions and uncertainty estimates for three typical T5 conditions taken from low enthalpy (2649), mid-range enthalpy (2645), and high enthalpy (2788) shots
1See Figure 3.13 for an example from the present study.
Measurement Symbol Uncertainty Units
Shock Speed Us ±13–53 m/s
Shock Tube Fill Pressure P1 ±0.25 kPa
Reservoir Pressure Pres ±2–4 MPa
Table 2.1: Estimated uncertainty of measured quantities, all shots.
are made below. Computed boundary layer edge condition uncertainties are estimated at the edge of the cone boundary layer using a Taylor-Maccoll solution from the nozzle exit conditions.
The fluid properties of greatest interest in the present work are typically those near the surface of the conical test article, after the conical shock at the boundary layer edge (see Tables 2.4, 2.7, and 2.10), since those quantities define the boundary layer’s properties. For typical conditions from a wide range of enthalpies, the greatest uncertainty at the boundary layer edge in percentage terms is found to be the edge pressure. This is a result of the relatively uncertain measurement of reservoir condi- tions at the end of the shock tube. The smallest uncertainty is found in the Mach number and edge velocity.
2.5.2 Uncertainty Estimates (Low Enthalpy, Shot 2649)
Experiment 2649 had a computed enthalpy of 4.78 MJ/kg. Uncertainty values for the measured tunnel quantities, computed thermal quantities, and computed boundary layer edge quantities are presented in Tables 2.2, 2.3, and 2.4, respectively.
Measurement Symbol Value Uncertainty Units Percent
Shock Speed Us 2256 ±17 m/s 0.8
Shock Tube Fill Pressure P1 90.0 ±0.25 kPa 0.3
Reservoir Pressure Pres 22.2 ±2 MPa 9.1
Table 2.2: Estimated uncertainty of measured quantities, shot 2649.
Computed Quantity Symbol Value Uncertainty Units Percent
Reservoir Enthalpy hres 4.78 ±0.14 MJ/kg 3.0
Reservoir Temperature Tres 3785 ±78 K 2.0
Reservoir Density ρres 20.1 ±1.5 kg/m3 7.4
Freestream Temperature T∞ 604 ±23 K 3.8
Freestream Density ρ∞ 0.0438 ±0.0066 kg/m3 8.5
Freestream Pressure P∞ 7.63 ±1.7 kPa 11.3
Freestream Velocity U∞ 2923 ±40 m/s 1.4
Freestream Mach Number M∞ 5.97 ±0.03 - 0.5
Table 2.3: Estimated uncertainty of computed thermal quantities, shot 2649.
Computed Quantity Symbol Value Uncertainty Units Percent
Edge Temperature Te 678 ±25 K 3.6
Edge Density ρe 0.0596 ±0.0041 kg/m3 6.9
Edge Pressure Pe 11.7 ±1.1 kPa 9.3
Edge Velocity Ue 2895 ±39 m/s 1.4
Edge Mach Number Me 5.58 ±0.02 - 0.4
Table 2.4: Estimated uncertainty of computed boundary layer edge quantities, shot 2649.
2.5.3 Uncertainty Estimates (Mid-Range Enthalpy, Shot 2645)
Experiment 2645 had a computed enthalpy of 9.96 MJ/kg. Uncertainty values for the measured tunnel quantities, computed thermal quantities, and computed boundary layer edge quantities are presented in Tables 2.5, 2.6, and 2.7, respectively.
Measurement Symbol Value Uncertainty Units Percent
Shock Speed Us 3209 ±35 m/s 1.1
Shock Tube Fill Pressure P1 85.35 ±0.25 kPa 0.3
Reservoir Pressure Pres 54.3 ±4 MPa 7.4
Table 2.5: Estimated uncertainty of measured quantities, shot 2645.
Computed Quantity Symbol Value Uncertainty Units Percent
Reservoir Enthalpy hres 9.96 ±0.32 MJ/kg 3.3
Reservoir Temperature Tres 6166 ±150 K 2.5
Reservoir Density ρres 27.5 ±2.2 kg/m3 7.9
Freestream Temperature T∞ 1485 ±61 K 4.1
Freestream Density ρ∞ 0.0588 ±0.0043 kg/m3 7.4
Freestream Pressure P∞ 25.7 ±2.5 kPa 9.9
Freestream Velocity U∞ 4043 ±62 m/s 1.5
Freestream Mach Number M∞ 5.33 ±0.03 - 0.5
Table 2.6: Estimated uncertainty of computed thermal quantities, shot 2645.
Computed Quantity Symbol Value Uncertainty Units Percent
Edge Temperature Te 1616 ±65 K 5.2
Edge Density ρe 0.0768 ±0.0056 kg/m3 7.4
Edge Pressure Pe 36.4 ±3.5 kPa 9.6
Edge Velocity Ue 4002 ±61 m/s 1.5
Edge Mach Number Me 5.06 ±0.02 - 0.5
Table 2.7: Estimated uncertainty of computed boundary layer edge quantities, shot 2645.
2.5.4 Uncertainty Estimates (High Enthalpy, Shot 2788)
Experiment 2788 had a computed enthalpy of 13.1 MJ/kg. Uncertainty values for the measured tunnel quantities, computed thermal quantities, and computed boundary layer edge quantities are presented in Tables 2.8, 2.9, and 2.10, respectively.
Measurement Symbol Value Uncertainty Units Percent
Shock Speed Us 3707 ±46 m/s 1.3
Shock Tube Fill Pressure P1 65.0 ±0.25 kPa 0.4
Reservoir Pressure Pres 54.7 ±4 MPa 7.3
Table 2.8: Estimated uncertainty of measured quantities, shot 2788.
Computed Quantity Symbol Value Uncertainty Units Percent
Reservoir Enthalpy hres 13.1 ±0.46 MJ/kg 3.5
Reservoir Temperature Tres 7375 ±180 K 2.4
Reservoir Density ρres 21.9 ±1.8 kg/m3 8.2
Freestream Temperature T∞ 1932 ±79 K 4.1
Freestream Density ρ∞ 0.0469 ±0.0039 kg/m3 8.4
Freestream Pressure P∞ 27.3 ±2.5 kPa 9.4
Freestream Velocity U∞ 4550 ±73 m/s 1.6
Freestream Mach Number M∞ 5.19 ±0.03 - 0.5
Table 2.9: Estimated uncertainty of computed thermal quantities, shot 2788.
Computed Quantity Symbol Value Uncertainty Units Percent
Edge Temperature Te 2095 ±84 K 4.0
Edge Density ρe 0.0606 ±0.0050 kg/m3 8.3
Edge Pressure Pe 38.4 ±3.5 kPa 9.1
Edge Velocity Ue 4504 ±72 m/s 1.6
Edge Mach Number Me 4.94 ±0.02 - 0.5
Table 2.10: Estimated uncertainty of computed boundary layer edge quantities, shot 2788.