# Limits of Fluid Modeling for High Pressure Flow Simulations

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Fluid Properties

#### 2.2. Computational Fluid Dynamics

## 3. Results

#### 3.1. Property Representation Error

#### 3.1.1. Density Errors

#### 3.1.2. Heat Capacity Errors

#### Heat Transfer

#### Pseudo Boiling

#### 3.1.3. 2D Heat Transfer

#### 3.2. Sampling Error

#### 3.2.1. Monte Carlo

- 1
- Within bounds defined by the boundary conditions, states can be randomly sampled from the physical fluid properties to obtain information about how numerical properties are reconstructed on a discrete representation.
- 2
- Extremal values of fluid properties that minimize transport need to be taken into account as they act as bottle necks.

- 1
- Prescribe a mesh resolution n and the boundary conditions ${T}_{\mathrm{min}}$ and ${T}_{\mathrm{max}}$.
- 2
- Determine $n-1$ random temperatures (uniform distribution) in the interval $[{T}_{\mathrm{min}},{T}_{\mathrm{max}}]$. Together with the boundary conditions ${T}_{\mathrm{min}}$ and ${T}_{\mathrm{max}}$, n temperature intervals are thus identified.
- 3
- For each interval, perform the analysis illustrated in Figure 11.
- 4
- Save the maximum error obtained across all intervals $[{T}_{\mathrm{min}},{T}_{\mathrm{max}}]$.
- 5
- Repeat the above steps N times to analyze a distribution of the error for a given resolution n.
- 6
- Repeat the above steps for different n to study the impact of the resolution.

#### 3.2.2. 1D Heat Transfer

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CFD | Computational Fluid Dynamics |

EOS | Equation Of State |

HEOS | Helmholtz EOS |

SRK | Soave Redlich Kwong EOS |

PR | Peng Robinson EOS |

## References

- Villermaux, E. Mixing and spray formation in coaxial jets. J. Propuls. Power
**1998**, 14, 807–817. [Google Scholar] [CrossRef] - Yang, V.; Anderson, W. (Eds.) Liquid Rocket Engine Combustion Instability; AIAA: Washington, DC, USA, 1995. [Google Scholar]
- Candel, S.; Herding, G.; Synder, R.; Scouflaire, P.; Rolon, C.; Vingert, L.; Habiballah, M.; Grisch, F.; Péalat, M.; Bouchardy, P.; et al. Experimental Investigation of Shear Coaxial Cryogenic Jet Flames. J. Propuls. Power
**1998**, 14, 826–834. [Google Scholar] [CrossRef] - Delplanque, J.P.; Sirignano, W. Numerical study of the transient vaporization of an oxygen droplet at sub- and super-critical conditions. Int. J. Heat Mass Transf.
**1993**, 36, 303–314. [Google Scholar] [CrossRef] - Yang, V.; Lin, N.; Shuen, J. Vaporization of Liquid Oxygen (LOX) Droplets in Supercritical Hydrogen Environments. Combust. Sci. Technol.
**1994**, 97, 247–270. [Google Scholar] [CrossRef] - Sirignano, W.; Delplanque, J.P. Transcritical vaporization of liquid fuels and propellants. J. Propuls. Power
**1999**, 15, 896–902. [Google Scholar] [CrossRef] - Mayer, W.; Tamura, H. Propellant Injection in a Liquid Oxygen/Gaseous Hydrogen Rocket Engine. J. Propul. Power
**1996**, 12, 1137–1147. [Google Scholar] [CrossRef] - Mayer, W.; Ivancic, B.; Schik, A.; Hornung, U. Propellant Atomization and Ignition Phenomena in Liquid Oxygen/Gaseous Hydrogen Rocket Combustors. J. Propuls. Power
**2001**, 17, 794–799. [Google Scholar] [CrossRef] - Oschwald, M.; Schik, A. Supercritical nitrogen free jet investigated by spontaneous Raman scattering. Exp. Fluids
**1999**, 27, 497–506. [Google Scholar] [CrossRef] - Oschwald, M.; Smith, J.J.; Branam, R.; Hussong, J.; Schik, A.; Chehroudi, B.; Talley, D. Injection of Fluids into Supercritical Environments. Combust. Sci. Technol.
**2006**, 178, 49–100. [Google Scholar] [CrossRef] - Habiballah, M.; Orain, M.; Grisch, F.; Vingert, L.; Gicquel, P. Experimental studies of high-pressure cryogenic flames on the mascotte facility. Combust. Sci. Technol.
**2006**, 178, 101–128. [Google Scholar] [CrossRef] - Candel, S.; Juniper, M.; Singla, G.; Scouflaire, P.; Rolon, C. Structures and dynamics of cryogenic flames at supercritical pressure. Combust. Sci. Technol.
**2006**, 178, 161–192. [Google Scholar] [CrossRef] - Chehroudi, B.; Talley, D.; Coy, E. Initial growth rate and visual characteristics of a round jet into sub- and supercritical environment of relevance to rocket, gas turbine, and Diesel engines. In Proceedings of the 37th Aerospace Sciences Meeting and Exhibit, Reno, NV, USA, 11 January–14 January 1999. [Google Scholar]
- Chehroudi, B.; Talley, D.; Coy, E. Visual characteristics and initial growth rates of round cryogenic jets at subcritical and supercritical pressures. Phys. Fluids
**2002**, 14, 850–861. [Google Scholar] [CrossRef] - Davis, D.; Chehroudi, B. The effects of pressure and acoustic field on a cryogenic coaxial jet. In Proceedings of the 42nd Aerospace Sciences Meeting and Exhibit, Reno, NV, USA, 5 January–8 January 2004. [Google Scholar]
- Davis, D.; Chehroudi, B. Measurements in an acoustically driven coaxial jet under sub-, near-, and supercritical conditions. J. Propuls. Power
**2007**, 23, 364–374. [Google Scholar] [CrossRef] - Banuti, D.T. Thermodynamic Analysis and Numerical Modeling of Supercritical Injection. Ph.D. Thesis, University of Stuttgart, Stuttgart, Germany, 2015. [Google Scholar]
- Habiballah, M.; Zurbach, S. Test case RCM-3—Mascotte single injector 60 bar. In Proceedings of the 2nd International Workshop Rocket Combustion Modeling—Atomization, Combustion and Heat Transfer, DLR, Lampoldshausen, Germany, 25–27 March 2001. [Google Scholar]
- Vingert, L.; Nicole, A.; Habiballah, M. Test Case RCM-2, Mascotte single injector. In Proceedings of the 3rd International Workshop on Rocket Combustion Modeling, Vernon, France, 13–15 March 2006. [Google Scholar]
- Oefelein, J.C.; Dahms, R.N.; Lacaze, G.; Manin, J.L.; Pickett, L.M. Effects of pressure on fundamental physics of fuel injection in Diesel engines. In Proceedings of the ICLASS, Heidelberg, Germany, 2–6 September 2012. [Google Scholar]
- Banuti, D.T. A thermodynamic look at injection in aerospace propulsion systems. In Proceedings of the AIAA Scitech 2020 Forum, Orlando, FL, USA, 6–10 January 2020. [Google Scholar]
- Banuti, D.T.; Hannemann, V.; Hannemann, K.; Weigand, B. An efficient multi-fluid-mixing model for real gas reacting flows in liquid propellant rocket engines. Combust. Flame
**2016**, 168, 98–112. [Google Scholar] [CrossRef] - Oefelein, J.C.; Yang, V. Modeling High-Pressure Mixing and Combustion Processes in Liquid Rocket Engines. J. Propul. Power
**1998**, 14, 843–857. [Google Scholar] [CrossRef] - Ely, J.F.; Hanley, H. Prediction of transport properties. 1. Viscosity of fluids and mixtures. Ind. Eng. Chem. Res.
**1981**, 20, 323–332. [Google Scholar] [CrossRef] - Ely, J.F.; Hanley, H. Prediction of transport properties. 2. Thermal conductivity of pure fluids and mixtures. Ind. Eng. Chem. Res.
**1983**, 22, 90–97. [Google Scholar] [CrossRef] - Benedict, M.; Webb, G.; Rubin, L. An Empirical Equation for Thermodynamic Properties of Light Hydrocarbons and Their Mixtures: I. Methane, Ethane, Propane, and n-Butane. J. Chem. Phys.
**1940**, 8, 334–345. [Google Scholar] [CrossRef] - Peng, D.Y.; Robinson, D.B. A new two-constant equation of state. Ind. Eng. Chem. Res.
**1976**, 15, 59–64. [Google Scholar] [CrossRef] - Soave, G. Equilibrium Constants from a Modified Redlich-Kwong Equation of State. Chem. Eng. Sci.
**1972**, 27, 1197–1203. [Google Scholar] [CrossRef] - Banuti, D.T. Crossing the Widom-line—Supercritical pseudo-boiling. J. Supercrit. Fluids
**2015**, 98, 12–16. [Google Scholar] [CrossRef] - Banuti, D.T. The latent heat of supercritical fluids. Period. Polytech. Chem. Eng.
**2019**, 63, 270–275. [Google Scholar] [CrossRef][Green Version] - Banuti, D.; Raju, M.; Ihme, M. Between supercritical liquids and gases–reconciling dynamic and thermodynamic state transitions. J. Supercrit. Fluids
**2020**, 165, 104895. [Google Scholar] [CrossRef] - Banuti, D.T.; Hannemann, K. The absence of a dense potential core in supercritical injection: A thermal break-up mechanism. Phys. Fluids.
**2016**, 28, 035103. [Google Scholar] [CrossRef] - Longmire, N.; Banuti, D. Onset of heat transfer deterioration caused by pseudo-boiling in CO
_{2}laminar boundary layers. Int. J. Heat Mass Transf.**2022**, 193, 122957. [Google Scholar] [CrossRef] - Bell, I.H.; Wronski, J.; Quoilin, S.; Lemort, V. Pure and Pseudo-pure Fluid Thermophysical Property Evaluation and the Open-Source Thermophysical Property Library CoolProp. Ind. Eng. Chem. Res.
**2014**, 53, 2498–2508. [Google Scholar] [CrossRef][Green Version] - Ghosal, S. Analysis of Discretization Errors in LES; Center for Turbulence Research Annual Research Briefs: Moffett Field, CA, USA, 1995. [Google Scholar]
- Chow, F.K.; Moin, P. A further study of numerical errors in large-eddy simulations. J. Comput. Phys.
**2003**, 184, 366–380. [Google Scholar] [CrossRef] - Lacaze, G.; Oefelein, J.C. A non-premixed combustion model based on flame structure analysis at supercritical pressures. Combust. Flame
**2012**, 159, 2087–2103. [Google Scholar] [CrossRef][Green Version] - Linstrom, P.J.; Mallard, W.G. NIST Chemistry Webbook, NIST Standard Reference Database Number 69; National Institute of Standards and Technology: Gaithersburg, MD, USA, 2001. Available online: http://webbook.nist.gov/chemistry (accessed on 1 March 2021).
- Prausnitz, J.M.; Lichtenthaler, R.N.; de Azevedo, E.G. Molecular Thermodynamics of Fluid-Phase Equilibria, 2nd ed.; Prentice-Hall: Hoboken, NJ, USA, 1985. [Google Scholar]
- van der Waals, J. Over de Continuiteit van den Gas- en Vloeistoftoestand. Ph.D. Thesis, University of Leiden, Leiden, The Netherlands, 1873. [Google Scholar]
- Banuti, D.T. A critical assessment of adaptive tabulation for fluid properties using neural networks. In Proceedings of the AIAA Aerospace Sciences Meeting, Virtual Event, 11–15, 19–21 January 2021. [Google Scholar]
- Harstad, K.G.; Miller, R.S.; Bellan, J. Efficient high-pressure state equations. AIChE J.
**1997**, 43, 1605–1610. [Google Scholar] [CrossRef] - Reid, R.C.; Prausnitz, J.M.; Poling, B.E. The Properties of Gases and Liquids, 4th ed.; McGraw Hill: New York, NY, USA, 1987. [Google Scholar]
- Pini, M.; Vitale, S.; Colonna, P.; Gori, G.; Guardone, A.; Economon, T.; Alonso, J.; Palacios, F. SU2: The Open-Source Software for Non-ideal Compressible Flows. J. Phys. Conf. Ser.
**2017**, 821, 012013. [Google Scholar] [CrossRef] - Economon, T.D. Simulation and Adjoint-Based Design for Variable Density Incompressible Flows with Heat Transfer. AIAA J.
**2020**, 58, 757–769. [Google Scholar] [CrossRef] - Longmire, N.P.; Banuti, D. Extension of SU2 using neural networks for thermo-fluids modeling. In Proceedings of the AIAA Propulsion and Energy 2021 Forum, Virtual Event, 9–11 August 2021. [Google Scholar] [CrossRef]
- Longmire, N.; Banuti, D.T. Modeling of the supercritical boiling curve by forced convection for supercritical fluids in relation to regenerative cooling. In Proceedings of the AIAA Aerospace Sciences Meeting, Virtual Event, 11–15, 19–21 January 2021. [Google Scholar]
- Schmidt, E.; Eckert, E.; Grigull, E. Jahrbuch der Deutschen Luftfahrtforschung Bd. II, AAF Translation Nr. 527, Air Material Command; Wright Field: Dayton, OH, USA, 1939; p. 53158. [Google Scholar]
- Schmidt, E. Wärmetransport durch natürliche Konvektion in Stoffen bei kritischem Zustand. Int. J. Heat Mass Transf.
**1960**, 1, 92–101. [Google Scholar] [CrossRef] - Banuti, D.T.; Ma, P.C.; Hickey, J.P.; Ihme, M. Thermodynamic structure of supercritical LOX–GH2 diffusion flames. Combust. Flame
**2018**, 196, 364–376. [Google Scholar] [CrossRef]

**Figure 1.**Near-critical fluid properties, oxygen data from CoolProp [34].

**Figure 2.**Isobaric specific heat capacity ${c}_{p}$ (

**a**) and Prandtl number Pr (

**b**) sampled in computational mesh; data from Lacaze and Oefelein [37]. Actual ${c}_{p}$ peak value as horizontal lines from reference data (NIST): dashed; Peng-Robinson: dotted.

**Figure 3.**Comparison of NIST data and ANN fit of CO${}_{2}$ properties. From top to bottom: density $\rho $, isobaric specific heat capacity ${c}_{p}$, viscosity $\mu $, thermal conductivity k, and isochoric specific heat capacity ${c}_{v}$. From left to right: $p=\{10,12,20\}$ MPa. The ANN was fit for $T=\{250,700\}$ K, data are shown to 200 K to demonstrate ANN behavior outside of the fitted interval. The pseudo boiling temperature ${T}_{pb}$ at which ${c}_{p}$ reaches a maximum increases with rising pressure. From [47].

**Figure 4.**Comparison of Peng–Robinson and Soave–Redlich–Kwong EOS to reference data for density $\rho $ of, from top to bottom, methane, oxygen, carbon dioxide, water; at three different pressures from left to right $p=[6,10,20]$ MPa.

**Figure 5.**Comparison of Peng–Robinson and Soave–Redlich–Kwong EOS to reference data for isobaric specific heat capacity ${c}_{p}$ of, from top to bottom, methane, oxygen, carbon dioxide, water; at three different pressures from left to right $p=[6,10,20]$ MPa.

**Figure 6.**Pseudo boiling properties compared between PR EOS (dashed) and reference data (solid). Left: The pseudo boiling line as the locus of ${c}_{p}$ peaks shows pronounced differences between both data sets. (Right) the pseudo boiling strength ${B}_{2}$ [29] is underpredicted for reduced pressures up to 3; beyond is well matched for carbon dioxide, but not for the other species.

**Figure 7.**Density contour plots at 12 MPa for wall temperature ${T}_{w}=390$ K. The figure shows the density distribution for a dense/cool flow coming from the left, and passing over a heated plate at the bottom. At the heated wall, a low-density vapor-like layer forms.

**Figure 8.**Fluid properties in the boundary layer for a wall temperature of ${T}_{w}=390$ at a pressure of $p=12$ MPa comparing the ANN (blue) simulation to the Peng–Robinson (red) simulation. Left: density profile; Right: heat capacity profile; Right: average heat flux along the flat plate for heated flat plate of different wall temperatures.

**Figure 9.**Shear layer with anchored flame behind coaxial injector. LOX post thickness $\delta $, mesh resolution d, LOX temperature ${T}_{\mathrm{LOX}},\mathrm{in}$, adiabatic flame temperature ${T}_{\mathrm{ad}}$, pseudo boiling temperature ${T}_{\mathrm{pb}}$.

**Figure 10.**Comparison of interpolation to physical data for ideal gas. Data from NIST database [38] for oxygen at 0.01 MPa, i.e., at ideal gas conditions. (

**a**) Comparison of isobaric heat capacity. (

**b**) Comparison of thermal conductivity.

**Figure 11.**Illustration of error from underresolved sampling of non-monotonous fluid properties. (

**a**) isobaric specific heat capacity ${c}_{p}$; (

**b**) thermal conductivity k. The black solid line is NIST reference data; the orange circles in ‘interpolation’ mark sample positions in a mesh, where the middle value $\overline{\varphi}$ is the interpolation of the available mesh values $\overline{\varphi}=\frac{1}{2}(\varphi \left({T}^{-}\right)+\varphi \left({T}^{+}\right))$; the purple square ‘NIST’ marks the evaluation of the property $\widehat{\varphi}$ from NIST data at the average temperature, i.e., $\widehat{\varphi}=\varphi \left(\frac{1}{2}({T}^{-}+{T}^{+})\right)$; the red circle marks the local maximum (${c}_{p}$) or minimum (k), the interpolation error (dashed red) is the difference between $\overline{\varphi}$ and $\widehat{\varphi}$, the max error (dashed blue) marks the difference between $\overline{\varphi}$ and the maximum (${c}_{p}$) or minimum (k) value ${\varphi}_{\mathrm{ext}}$.

**Figure 12.**Results of Monte Carlo analysis over single random interval $[{T}_{\mathrm{min}},{T}_{\mathrm{max}}]$ for 10,000 samples. (

**a**) isobaric specific heat capacity ${c}_{p}$; (

**b**) thermal conductivity k. ‘Direct’ is ratio $\varphi /\overline{\varphi}$; ‘max/min’ are ratios ${\varphi}_{\mathrm{ext}}/\overline{\varphi}$. A ratio of ${10}^{0}$ means that the physical value is exactly sampled.

**Figure 13.**Histograms of maximum error ratio between interpolated and maximum value of ${c}_{p}$. From top to bottom: 4, 10, 20, and 50 cells resolution. From left to right columns: 100, 1000, and 5000 samples. Note the change in x scale.

**Figure 14.**Histograms of maximum error ratio between interpolated and maximum value of ${c}_{p}$. From top to bottom: 100, 200, 400, and 800 cells resolution. From left to right columns: 100, 1000, and 5000 samples. Note the change in x scale.

**Figure 15.**Plots of the maximum heat capacity versus time for O${}_{2}$ at $7.0$ MPa in 1D mixing simulation using Peng–Robinson fluid properties (

**a**) and ANN fluid properties (

**b**).

**Figure 16.**Plot of the maximum error ratio between simulation value and NIST value for thermal conductivity versus time for O${}_{2}$ at $7.0$ MPa.

**Figure 17.**Plot of the temperature along the domain of the mesh at different time steps, (

**a**–

**f**) corresponding to $t=\{0,0.25,0.5,1,2,3\}$ (s). The red line is a simulation using ANN fluid property models on a coarse mesh of $n=10$ elements, and the black line is a case on a fine mesh of $n=800$ elements using ANN fluid property models.

**Figure 18.**Plot of the movement of the temperature curves. The left column of plots tracks the x position of a certain lower temperature threshold, from top to bottom the temperature thresholds were $T=\{100.1,101,120\}$. The middle column of plots tracks the x position where a certain high temperature threshold, from top to bottom the temperature thresholds were $T=\{299.9,299,280\}$. The far right column is then the difference between the lower and higher temperature thresholds. The plots were scaled using the initial value so that the plots start at the 0.

**Figure 19.**Plot of the temperature along the domain of the mesh at different time steps, (

**a**–

**f**) corresponding to $t=\{0,0.25,0.5,1,2,3\}$ s. The red line is a simulation using ANN fluid property models on a coarse mesh of $n=10$ elements, the light blue line is a case on a coarse mesh of $n=10$ elements using ANN fluid property models but the isobaric heat capacity fix was applied, the purple line is a case on a coarse mesh of $n=10$ elements but the thermal conductivity fix shown in Equation (3), the blue line is a case on a coarse mesh of $n=10$ elements but the thermal conductivity fix shown in Equation (4), the dark blue line is a case on a coarse mesh of $n=10$ elements but the thermal conductivity fix shown in Equation (5) and the black line is a case on a fine mesh of $n=800$ elements using ANN fluid property models.

**Figure 20.**Plot of the movement of the temperature curves now including the results where fixes for the thermal conductivity and isobaric heat capacity were used. The left column of plots tracks the x position of a certain lower temperature threshold, from top to bottom the temperature thresholds were $T=\{100.1,101,120\}$. The middle column of plots tracks the x position where a certain high temperature threshold, from top to bottom the temperature thresholds were $T=\{299.9,299,280\}$. The far right column is then the difference between the lower and higher temperature thresholds. The plots were scaled using the initial value so that the plots start at the 0.

Pressure in MPa | ${\mathit{c}}_{\mathit{p},\mathbf{max}}^{\mathbf{ref}}$ J/kg/K | ${\mathit{c}}_{\mathit{p},\mathbf{max}}^{\mathbf{CFD}}$ J/kg/K | Ratio | Rel. Error | Pr${}_{\mathbf{max}}^{\mathbf{ref}}$ | Pr${}_{\mathbf{max}}^{\mathbf{CFD}}$ | Ratio | Rel. Error |
---|---|---|---|---|---|---|---|---|

60 | 15.6 | 6.46 | 2.41 | 58.59% | 9.38 | 4.24 | 2.21 | 54.83% |

70 | 7.78 | 5.40 | 1.44 | 30.59% | 5.13 | 3.47 | 1.48 | 32.29% |

80 | 5.42 | 4.84 | 1.12 | 10.73% | 3.76 | 3.14 | 1.20 | 16.52% |

90 | 4.31 | 4.00 | 1.08 | 7.19 % | 3.10 | 2.58 | 1.20 | 16.69% |

# Elements | Spacing d in mm |
---|---|

10 | 1.0 |

20 | 0.5 |

40 | 0.25 |

80 | 0.125 |

100 | 0.1 |

800 | 0.0125 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Longmire, N.P.; Banuti, D.T. Limits of Fluid Modeling for High Pressure Flow Simulations. *Aerospace* **2022**, *9*, 643.
https://doi.org/10.3390/aerospace9110643

**AMA Style**

Longmire NP, Banuti DT. Limits of Fluid Modeling for High Pressure Flow Simulations. *Aerospace*. 2022; 9(11):643.
https://doi.org/10.3390/aerospace9110643

**Chicago/Turabian Style**

Longmire, Nelson P., and Daniel T. Banuti. 2022. "Limits of Fluid Modeling for High Pressure Flow Simulations" *Aerospace* 9, no. 11: 643.
https://doi.org/10.3390/aerospace9110643