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# The development of kinetic models and simulation methods to study molecular fluctuations, modal response, and shock-laminar separation bubble instabilities

#### Sawant, Saurabh Satish

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https://hdl.handle.net/2142/115321

## Description

- Title
- The development of kinetic models and simulation methods to study molecular fluctuations, modal response, and shock-laminar separation bubble instabilities
- Author(s)
- Sawant, Saurabh Satish
- Issue Date
- 2022-01-05
- Director of Research (if dissertation) or Advisor (if thesis)
- Levin, Deborah A
- Doctoral Committee Chair(s)
- Levin, Deborah A
- Committee Member(s)
- Chew, Huck Beng
- Goza, Andres
- Curreli, Davide
- Theofilis, Vassilis

- Department of Study
- Aerospace Engineering
- Discipline
- Aerospace Engineering
- Degree Granting Institution
- University of Illinois at Urbana-Champaign
- Degree Name
- Ph.D.
- Degree Level
- Dissertation
- Keyword(s)
- Hypersonics, Shocks, Instabilities, Noncontinuum effects, Laminar separation bubble, molecular fluctuations, linear stability theory, Direct Simulation Monte Carlo, Adaptive Mesh Refinement, High Performance Computing
- Abstract
- The study of laminar-to-turbulent transition of hypersonic flows typically includes investigation of instabilities generated in the boundary-layer and laminar separation bubble (LSB) induced by shock-wave/boundary-layer interactions (SBLIs), without reference to the internal structure of shocks or instabilities potentially generated in them. However, the predictions of shock structure based on the Navier-Stokes equations increasingly deviate from those obtained from kinetic theory as the degree of rarefaction in shock's internal nonequilibrium region increases with the Mach number. Therefore, a kinetic approach is required to resolve the internal structure of shocks, such as the Direct Simulation Monte Carlo (DSMC) method that approximates the solution of the more-general Boltzmann equation of transport. Kinetic modeling requires resolution at molecular length and time scales, which vary by order of magnitude from the freestream to the vicinity of compression surfaces inducing SBLIs. A three-dimensional highly scalable DSMC solver is developed to overcome these computational challenges, known as the Scalable Unstructured Gas-dynamic Adaptive mesh-Refinement (SUGAR), which employs strategies that have demonstrated 87% weak scaling efficiency for 8192 processors and 24 billion computational particles. In this work, the SUGAR solver is used to investigate the role of finite-thickness shock layers in the dynamics of complex three-dimensional SBLIs, characterize molecular fluctuations in the shock's internal nonequilibrium region, and obtain fully-resolved one-dimensional steady shock layers for their linear temporal analysis. Linear three-dimensional instability is analyzed in the shock layer and laminar separation bubble induced by SBLIs in a Mach 7 nitrogen flow at a freestream unit Reynolds number Re=5.2x10^4 over a 30°-55° double-wedge configuration. DSMC simulations have been fully resolved using 60 billion computational particles and 4.5 billion computational cells to capture the flow evolution from the inception of three-dimensionality, through linear growth of instabilities, to the early stages of nonlinear saturation. It is shown that the laminar separation bubble sustains self-excited, small-amplitude perturbations that originate past the primary separation line and lead to spanwise-periodic wall striations inside the bubble and downstream of the primary reattachment line, as known from earlier experiments, simulations, and instability analyses. A spanwise-periodic instability, synchronized with that in the separation zone, is identified herein for the first time, which exists in the internal structure of the separation and detached shock layers, and manifests itself as spanwise-periodic cats-eyes patterns in the amplitude functions of all linear flow perturbations. Investigation of molecular fluctuations in the internal nonequilibrium region of monatomic shock layers has revealed two orders of magnitude lower frequencies than the flow upstream. The presence of low-frequency fluctuations is shown to be a direct consequence of the bimodal nature of the probability density function (PDF) of gas particles in the shock, as opposed to their Maxwellian form in the upstream. A novel two-energy-bin Lotka-Volterra-type model is developed to describe the reduced-order dynamics of a large number of collision interactions of gas particles, which correctly predicts the differences in fluctuation frequencies in the shock versus those in the upstream and is consistent with the small-amplitude fluctuations obtained from DSMC computations. The PDFs of particle energies in the upstream and downstream equilibrium regions, as well as inside shocks, are analytically derived and shown for the first time to have the form of the non-central chi-squared (NCCS) distributions. The variation in dominant low-frequency fluctuations could be described by a Strouhal number St ~ O(0.01), as well as through linear correlations established between the average frequencies and the analytical bimodal NCCS PDFs, for a broad range of Mach number, upstream temperature, and number density. Finally, a framework for studying temporal analysis of finite-thickness normal shock layers has been developed and used to study the stability of Ma=1.2, 3, and 5 shocks obtained from DSMC subjected to two-dimensional, small amplitude, harmonic perturbations. Two sets of constitutive equations are used to obtain the closure of the compressible governing equations used: the traditional Navier-Stokes-Fourier (NSF) equations, applicable only at the lowest Mach number, and equations based on the Mott-Smith's bimodal velocity distribution function, which are better than the former in approximating the DSMC-derived transport flow quantities at high Mach numbers. Eigenspectra obtained from the two formulations are compared to quantify the effect of the resolution of internal shock structure on the stability of shock layers. The temporal analysis performed by Duck and Balakumar three decades ago of one-dimensional shock layers satisfying the Navier-Stokes equations revealed only stable continuous branches in the eigenspectra. However, it was inconclusive regarding what parameters govern the temporal decay rates of `continuum' modes or the asymptotic form of eigenfunctions in the far field of the shock layer. In this work, we find the least stable continuous branches that have both stationary and traveling continuum modes and show for the first time that their eigenfunctions decay monotonically and sinusoidally in the far field, respectively, while exhibiting an order of magnitude lower spatial decay rate in the upstream than the downstream. These characteristics of eigenfunctions are governed by only one physically possible root of a septic characteristic equation derived by analytically solving the normal mode equations in the far field. By deriving an approximate equation for this root, it is shown that the least stable continuous branch is always stable but becomes less stable by the decrease in Reynolds number defined based on the shock-thickness Re, increase in Mach number Ma, decrease in the specific gas constant γ, and increase in the degree of translational nonequilibrium. This is consistent with the observations from the linear stability solver that the noncontinuum effects lead to a less stable branch and that the prediction of decay rates strongly depends on how well the constitutive relations resolve the true translational nonequilibrium region in the shock layer.
- Graduation Semester
- 2022-05
- Type of Resource
- Thesis
- Copyright and License Information
- Copyright 2021 Saurabh Satish Sawant

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