T5 nozzle computation with specified throat-condition history

H. G. Hornung, 2023-07-26

The computation of the nozzle flow in T5 is normally performed with a block-marching scheme which takes much less time than a time-accurate method. However, if one wants to study the time-dependence of the flow in the test section, it is necessary to use the latter. For that case, it is most convenient to start from a known throat flow state, so that the whole of the flow is supersonic. Fortunately, it is a good approximation to compute the throat state with a one-dimensional equilibrium method in the usual way, from the initial shock-tube state, the shock speed and the reservoir pressure. Since the reservoir pressure varies with time in a known (measured) fashion, repeated applications of this method yield the flow state at the throat at discrete times. The software package Eilmer4 provides the feature of specifying an inflow boundary condition that depends on time, through a file that gives the flow state as a function of time. In this report I describe how this method is used to compute time traces of the nozzle exit conditions for two T5 runs.

1. Computation of the transient throat state

For this task I use the one-dimensional nozzle-flow code of the gas-dynamics toolkit of the Centre for Hypersonics of the University of Queensland which is called nenzf1d. The input to this code consists of the initial shock tube state (gas composition, temperature and pressure), the shock speed and the reservoir pressure, as well as the specification of the equilibrium behavior of the gas in the form of the files called gas.lua and cea-air5species-gas-model.lua, and the relaxation behavior of the gas in the form of the files called chem.lua and kinetics.lua. In my case the gas is air and the model is the Bose/Candler 5-species 6-reaction 2-temperature model. In addition nenzf1d requires the nozzle shape to be provided in the form of a few Bezier points. In the case at hand the part of nenzf1d that computes the one-dimensional flow through the nozzle is not needed, and only the condition at the throat that it supplies is used. All the named files as well as an example input file t5-air-2T.yaml and the corresponding output file 47 accompany this report.

# Sample input file for nenzf1d is a YAML 1.1 file.
# t5-air-2T.yaml
# Data for T5 shot 2946
# HGH 2023-5-29
title: "T5-2916 2T-air with contoured nozzle."        # Any string will do.

species: ['N2', 'O2', 'N', 'O', 'NO']         # List
molef: {'N2': 0.79, 'O2': 0.21}               # Map of nonzero values will suffice.
# Gas model and reactions files need to be consistent with the species above.
# Gas model 1 is usually a CEAGas model file.
# Gas model 2 is a thermally-perfect gas model for the finite-rate chemistry.
gas-model-1: cea-air5species-gas-model.lua
gas-model-2: air-5sp-2T.lua
reactions: air-5sp-6r-2T.lua
reactions-file2: air-energy-exchange.lua

# Observed parameter values for shock-tube operation from Table 1 in Appendix A.
T1: 300         # K
p1: 45.0e3     # Pa
Vs: 4071.0      # m/s
pe: 47.0e6     # Pa
ar: 106.0       # contoured nozzle
# pp_ps: 0.0105   # From Figure 8.

meq_throat: 1.03  # To get supersonic condition with frozen-gas sound speed
C: 0.94           # estimate of Rayleigh_Pitot/(rho*V^2) for frozen gas at exit

# Define the expanding part of the nozzle as a schedule of diameters with position.
xi: [0.0000, 0.1, 0.2, 0.4, 0.6, 0.8, 1.0]
di: [0.030, 0.0625, 0.13,   0.21,   0.26,   0.290,   0.306]

# Optionally, we can adjust the stepping parameters for the supersonic expansion.
# x_end: 1.0
# t_final: 1.0e-3
# t_inc: 1.0e-10
# t_inc_factor: 1.0001
# t_inc_max: 1.0e-7
NENZF1D: 1D analysis of shock-tunnel with nonequilibrium nozzle flow.
  Revision-id: 61c8fa55
  Revision-date: Mon Jun 27 20:46:03 2022 +1000
  Compiler-name: ldc2
  Build-date: Mon Aug 15 09:45:22 AM PDT 2022
  T5-2916 2T-air with contoured nozzle.
Initial gas in shock tube (state 1).
  pressure    45 kPa
  density     0.52048 kg/m^3
  temperature 300 K
  H1          0.00187165 MJ/kg
Incident-shock process to state 2.
  V2          439.594 m/s
  Vg          3631.41 m/s
  pressure    7739.5 kPa
  density     4.82007 kg/m^3
  temperature 5040.27 K
Reflected-shock process to state 5.
Isentropic relaxation to state 5s.
  entropy     10095.8 J/kg/K
  H5s         15.8715 MJ/kg
  H5s-H1      15.8696 MJ/kg
  pressure    47000 kPa
  density     16.285 kg/m^3
  temperature 8117.31 K
Isentropic flow to throat to state 6 (Mach 1).
  V6          1774.71 m/s
  mflux6      17924.6
  pressure    26538 kPa
  density     10.1 kg/m^3
  temperature 7500.82 K
Isentropic flow to slightly-expanded state 6e (Mach 1.03).
  V6e         1823.13 m/s
  mflux6e     17909.3
  ar6e        1.00085
  pressure    25670.5 kPa
  density     9.8234 kg/m^3
  temperature 7465.81 K
Isentropic expansion to nozzle exit of given area (state 7).
  area_ratio  106
  V7          4913.36 m/s
  pressure    29.7586 kPa
  density     0.034416 kg/m^3
  temperature 2910.7 K
  mflux7      17924.4
  pitot7      814.62 kPa
End of part A: shock-tube and frozen/eq nozzle analysis.
Initializing gas state using CEA saved data
Begin part B: continue supersonic expansion with finite-rate chemistry.
Start position:
  x           3.93087e-05 m
  area-ratio  1.00085
Start condition:
  velocity    1859.98 m/s
  sound-speed 1858.13 m/s
  (v-V6e)/V6e 0.0202148
  pressure    25670.5 kPa
  density     9.81517 kg/m^3
  temperature 7465.81 K
  T_modes[0]  7465.81 K
  massf[N2]   0.683111
  massf[O2]   0.00864601
  massf[N]    0.048463
  massf[O]    0.18372
  massf[NO]   0.0760601
Exception thrown in nenzf1d.run!
  Exception message: Coefficients for CEA curve could not be determined.

In the output file 47 the part of interest is the flow state at the throat. This is given just after the line that states: Begin part B: continue supersonic expansion with finite-rate chemistry. This part of the output gives velocity, speed of sound, pressure, temperature, vibrational temperature and composition. The first step is to choose a point on the reservoir-pressure trace, see figure 1, noting the time and the pressure. Then this pressure is used in the input file to nenzf1d. The throat flow state in the resulting output is then transcribed together with the time into a line in the file "throat-file.dat" (an example of which accompanies this report). This is repeated for other time/pressure pairs with suitably densely spaced times along the trace, till the period of interest is covered. Since the boundary layer in the nozzle is partially turbulent, the throat file needs an additional parameter for the turbulence model, which in my case is the Spalart-Allmaras model.

Figure 1. Reservoir-pressure trace for T5-2946.

2. Nozzle flow computation

The next step is to run the nozzle-start program in Eilmer4. (See accompanying file t5ns.lua.) This requires as input the same gas files, that are now actually used, as well as the nozzle profile which is specified as a spline through many coordinate points. In t5ns.lua the inflow boundary condition is the file throat-file.dat which is invoked by west=InFlowBC_Transient:new{fileName = "throat-file.dat"} in the specification of the boundary conditions. Running this computation with MPI and 48 processors takes about one day with the resolution I have chosen (2560x92 cells), heavily clustered towards the nozzle wall for good resolution of the boundary layer.

With a partially turbulent boundary layer it is necessary to specify the transition point. Since the nozzle exit pressure is sensitive to the choice of the transition point, I chose the transition point by tuning it to give an exit pressure that matches the exit pressure measured at the mid point of the useful test time.

3. Results

The two conditions considered here were T5-2946 and T5-3029. Figure 2 shows the distribution of flow variables along the radius in the nozzle exit plane of T5-2946 at t=1.5ms. The best location of the transition point in this case was x=0.425m. The corresponding pressure traces are shown in figure 3.

2946 exit
Figure 2. Distribution of flow variables in the exit plane in the case of T5-2946 at t=1.5ms.
2946 ptrace
Figure 3. Pressure traces at different radii in the case of T5-2946.

Figure 4 shows the distribution of flow variables along the radius in the nozzle exit plane of T5-3029 at t=1.9ms. The best location of the transition point in this case was x=0.4m.

3029 exit
Figure 4. Distribution of flow variables in the exit plane in the case of T5-3029 at t=1.9ms.
3029 ptrace
Figure 5. Pressure traces at different radii in the case of T5-3029.

4. Conclusions

A procedure is described by which it is possible to compute the time-dependent behavior of the test section conditions in T5. The procedure is applied to two particular T5 runs. All the files needed for this procedure accompany this report, in the hope that one of the graduate students can make the procedure less laborious and thus provide a useful tool for future use in T5 research.