A Reduced-Order Modeling Approach to Enable Kinetic Simulations of Non-equilibrium
Hypersonic Flows
Marco Panesi
AFOSR YIP
Grant No: FA9550-15-1-0132 DEF
Department of Aerospace Engineering
University of Illinois at Urbana-Champaign
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Outline
• Motivation and Background • Master Equation Analysis • Coarse Grained Model
• Conclusions
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Standard Non-equilibrium Models
• Standard non-equilibrium models for hypersonic flows were mainly developed in the 1980’s and are correlation based:
E.g., dissociation model of Park Multi-temperature model: Average temperature for fictitious Arrhenius rate coefficient
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Motivation
• A large effort is underway at AIRFOCE which attempts to characterize the microscopic interaction of N2-N2, O2-O2 and N2-O2 from first-principles calculations.
• Ab initio calculations can provide the transition probabilities governing the transfer of energy between the flow and the internal energy modes of atoms and molecules in the gas.
• The large amount of information provided by ab initio calculations has great value, but it must be tailored to fulfill the needs of the problem that is being solved.
• Thus, it is imperative that reduced-order models be developed.
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Objective
• METHODOLOGY: Developing non-equilibrium models for hypersonic flows based on microscopic theory and applying them to macroscopic scale.
Work at the interface between computational chemistry, experimental data, and CFD.
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Background: State-to-State Kinetics
MT Models: Conventional methodologies rely on the assumption of Maxwell-Boltzmann distribution:
State to State Models: the internal states are treated as independent species governed by their own kinetics.
Boltzmann Plot
James Clerk Maxwell Ludwig Eduard Boltzmann
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High Fidelity Modeling: Roadmap
Objectives: To assess the fundamental assumptions adopted in the modeling of hypersonic plasma flows.
Key Relaxation Processes: 1. Energy Transfer: It is crucial to the understanding on the shock layer kinetics 2. Dissociation: critical process governing the redistribution of the kinetic
energy within the internal energy modes and chemistry. 3. Recombination: critical in the boundary layer area, and in the expansion
regions of the flow-field.
Dissociation N2 + N = N + N + N • Rotational equilibrium (T = Trot) • Landau-Teller VT relaxation model • Internal Energy – Chemistry relaxation coupling (e.g., VC) • Existence of a QSS rates • … Rovibrational State-to-state method
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Analysis of Dissociating and Recombining Flows
Test cases under consideration: 1. Master Equation 2. Flow Behind a normal shock wave 3. Quasi 1D nozzle flow
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A Novel Approach to the Modeling of Non-equilibrium Flows
First Principles Computation: 1. Quantum chemistry calculations to
generate realistic nuclear interaction potentials (PES)
2. Quasi classical trajectory (QCT) method for the reaction cross-sections
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Non-equilibrium Flow Behind a Normal Shock Wave
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Flow-field Quantities
• Rovibrational STS model predicts larger relaxation distance with respect to the vibrational STS model
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Post-Shock: Rovibrational Populations
• The distribution deviates from the Maxwell Boltzmann distribution • Distribution is dissected into multiple strands for different v
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1D Shock Tube Problem
Left: rotational and vibrational temperatures Right: population of the first vibrational levels
• Assumption of fast rotational equilibrium is questionable • Dissociation is better described by a unique temperature
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Master Equation Solution
1. Rate Coefficient is in EXCELLENT agreement with Appleton data 2. Exchange reaction is important for correct estimation of reaction rate constant. 3. In the high temperature region the QSS assumption FAILS!
QSS Rates Estimation
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Comparison MT and STS Models
Can we use QCT derived rate coefficients and relaxation parameters in the conventional MT models?
NO! Using QCT derived rates based on the QSS assumption (or Boltzmann) are unable to reproduce the STS results.
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Conclusions of the STS Analysis
MT modeling (QSS based) • Conventional MT models are unable to reproduce the STS results,
because of the invalidity of the QSS assumption.
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Model Reduction
State To State - RVC
State To State - VC
STS - EC
MT
Accuracy
Complexity
CGM
Increasing Number of Assumptions
T, TR,TV, TE
T, TR,TV
T, TR
T
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Coarse Grained Method
The methodology of reduction consists of two distinct steps: 1. Local Representation and Reconstruction. It relies on the lumping of the
internal energy levels in macroscopic energy groups and the reconstruction of the population of each grouped state, ni, using macroscopic quantities.
The coefficients and are retrieved using constraints based on the maximum entropy principle and a variational method.
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Coarse Grained Method
2. Macroscopic Moment Equations and Rate Coefficients. Macroscopic
governing equations (referred to as macroscopic moment equations) are obtained by taking moments of the master equations and by using the reconstructed local representation.
• Zero Order Moment: Uniform Grouping (piece-wise reconstruction).
• First Order Moment: Boltzmann Grouping (linear reconstruction).
Governing Equations
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Coarse Grained Method
• Novel lumping scheme obtained by sorting the levels by energy and grouping in a bin all levels with similar energies
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…reconstruction of the population of each grouped state, ni, using macroscopic quantities.
Coarse Grained Method
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Considerations
1. Grouping strategy is crucial • The choice of the grouping or grouping strategy should be guided by
the physical intuition. • For example, levels characterized by similar energies are likely to be in
equilibrium between each other and should be grouped together.
2. State to state models and Multi-temperature models are a particular case of coarse grained approach.
• For example, conventional TTv model can be obtained by grouping the
vibrational levels in a single bin and prescribing a Boltzmann distribution for each rotational level with T=TRot
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Conventional Model and Coarse Grain Modeling
MT is a particular case of coarse grain model (1 Group)
Boltzmann distribution (TVib, TRot) Conventional TTVib model if TRot=T
MT
VCR is a particular case of coarse grain model (n Group)
Boltzmann distribution (Trot) Conventional Vib. STS model
when TRot = T
VCR(n)
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Novel Grouping Strategy
Two groups in the vibrational energy structure
Two different “rotational” temperature for the two groups
VCR(i) Vibrational specific model
Hyb(2,2)
VCR Three energy groups Three internal energy equations MT is a special case (1 internal
temperature)
BC(3)
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Evolution of the Vibrational Distribution
MT models are unable to predict the distribution function
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Evolution of the Vibrational Distribution
BC(3) model is in good agreement with the STS model
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Evolution of the Vibrational Distribution
HyBVC shows excellent agreement with the STS model
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Evolution of the Vibrational Distribution
VCR2 shows excellent agreement with the STS model
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Technical Challenges Remaining
• Diatom-Diatom Reactions Given the large number of possible channels the derivation of the “exact” rovibrational STS model is not feasible.
• Analysis of Recombining Flows Challenges are due to the strong deviation from the equilibrium distribution in expanding flows.
• Application to CFD (e.g., US3D) and Validation Validation data should include spatially resolved population measurements of the (ro-) vibrational population and atomic densities. (E.g., S.Sharma, et al. JTHT, Vol. 7, No. 4 (1993), pp. 697-703. )
• Other systems, gas mixtures, higher order reconstruction
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Publications and Honors
Journal Publications 1. A., Munafo, Y. Liu, M. Panesi “Physics of dissociation and energy transfer in
shock heated nitrogen flows”, Physics of Fluids, Under Review, (2015) 2. Y. Liu, M. Panesi, A. Sahai and M. Vinokur “General multi-group macroscopic
modeling for thermo-chemical non-equilibrium gas mixtures” J. Chem. Phys. 142, 134109 (2015);
3. Panesi, M., Munafo, A., Magin, T. E., and Jaffe, R. L., ”Study of the non-equilibrium shock heated nitrogen flows using a rovibrational state-to-state method”, Phys. Rev. E, Vol. 90, 013009 (2014).
4. Panesi, M. Jaffe, R.L. Schwenke, D.W. Magin, T.E. 2013 Rovibrational internal energy transfer and dissociation of N2-N system in hypersonic flows. J. Chem. Phys. 138, 044312 (2013).
Research Honors 1. 2015 Air Force Summer Faculty Fellowship Program, California, USA. (2015) 2. 2015 Award on Physical Modelling (8th Symposium on Aerothermodynamics
for Space Vehicles - ESA) (2015) 3. 2015 Air Force Young Investigator Award (YIP) (2015)
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Conclusions
• Using the classical moment method, we introduced a general methodology for modeling thermal and chemical non-equilibrium processes. Based on the maximum entropy principle subject to a series of moment constraints, the logarithm of the distribution function in each energy group is expressed and reconstructed as a power series in internal energy.
• Conventional MT and STS models are “only” particular cases of the more general Coarse-Grain Method.
• These models have been applied to the study of rovibrational energy
excitation and dissociation processes behind strong one-dimensional shock waves in nitrogen flow.
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NEQRAD Group
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Acknowledgments
Special thanks to: UIUC
• Dr. A. Munafo (UIUC) • Dr. R. Macdonald (UIUC) - NDSEG • Dr. S. Venturi (UIUC)
NASA
• Dr. R.L. Jaffe (NASA Ames Research Center) • Dr. D.W. Schwenke (NASA Ames Research Center) • Dr. Y. Liu (NASA Ames Research Center) AIRFORCE • Dr. J.L. Cambier (USAF, AFOSR) • Dr. E. Josyula (USAF, AFRL Wright Patterson)
AFOSR YIP
Grant No: FA9550-15-1-0132 DEF
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Backup Slides
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Summary (and Conclusions)
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Summary (and Conclusions)
Significant reduction of the CPU time is obtained with the Bin model
N2-N System: CPU Time in function of the # of BINS
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Convective Heating
The MT model over-estimate the convective heating by 18 % if the parameters are calibrated using the RVC model
Park Model over-predict by a factor 2
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Detecting QSS Breakdown
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Coarse Grained Method
• Novel lumping scheme obtained by sorting the levels by energy and grouping in a bin all levels with similar energies
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Energy Transfer
