# Statistical mechanics-based reduced-order modeling of turbulence in reactor systems

New system-level codes are being developed for advanced reactors for safety analysis and licensing purposes. Thermal-hydraulics of advanced reactors is a challenging problem due to complex flow scenarios assisted by free jets and stratified flows that lead to turbulent mixing. For these reasons, the 0D or 1D models used for reactor plena in traditional safety analysis codes like RELAP cannot capture the physics accurately and introduce a large degree of modeling uncertainty. System-level calculation codes based on the advection-diffusion equation neglect turbulent fluctuations. These fluctuations are extremely important as they introduce higher-order moments, which are responsible for vortex stretching and the passage of energy to smaller scales. Alternatively, extremely detailed simulations with velocity coupling from the Navier-Stokes equations are able to capture turbulence effects accurately using DNS. These solutions are accurate because they resolve the flow into the smallest possible length and time scales (Kolmogorov scale) important to the flow, which makes DNS computationally expensive for simple geometries and impossible at the system level.

The flow field can be described through a reduced-order model using the principles of statistical mechanics. Statistical mechanics-based methods provide a method for extracting statistics from data and modeling that data using easily represented differential equations. The Kramers-Moyal (KM) expansion method can be used as a subgrid-scale (SGS) closure for solving the momentum equation. The stochastic Burgers equation is solved using DNS, and the DNS solutions are used to calculate the KM coefficients, which are then implemented as an SGS closure model. The KM method outperforms traditional methods in capturing the multi-scale behavior of Burgers turbulence. The functional dependencies of the KM coefficients are also uniform for several boundary conditions, meaning the closure model can be extended to multiple flow scenarios.

For the case of the Navier-Stokes equations, each particle trajectory tends to follow some scaling law. Kolmogorov hypothesized that the flow velocity field follows a -5/3 scaling in the inertial region where Markovian characteristics can be invoked to model the interaction between eddies of adjacent sizes. This law holds true in the inertial region where the flow is Markovian. For scalar turbulence, the scaling laws are affected by thermal diffusion. If a fluid has a Prandtl number close to one, the thermal behavior is dominated by momentum, so the spectra for velocity and temperature are similar. For small Prandtl number fluids, such as liquid metals, the thermal diffusion dominates the lower scales and the slope of the spectrum shifts from the -5/3 slope to a -3 slope, also called the Batchelor region. System-level thermal hydraulics codes need to be able to capture these behaviors for a range of Prandtl number fluids. The KM-based model can also be used as a surrogate for velocity or temperature fluctuations in scalar turbulence. Using DNS solutions for turbulent channel flow, the KM model is used to provide a surrogate for temperature and velocity signals at different wall locations in the channel for Pr = 0.004, Pr = 0.025, and Pr = 0.71. The KM surrogate matches well for all wall locations, but is not able to capture the viscous dissipation in the velocity signal, or the thermal dissipation in the low Prandtl number cases. The dissipation can be captured by implementing a Gaussian filter.

Statistical mechanics-based methods are not limited to modeling turbulence in a reactor. Renewable power generation, such as wind, can be modeled using the Ornstein-Uhlenbeck (OU) method, which allows the long-term trends and short-term fluctuations of wind power to be decoupled. This allows for large fluctuations in wind power to be scaled down to a level that a reactor can accommodate safely.

Since statistical mechanics methods are based in physics, the calculated coefficients provide some information about the inputted signal. In a high-temperature gas-cooled reactor, strong heating can cause flow that is expected to be turbulent to show laminar characteristics. This laminarization results in reduced heat removal. The KM coefficients can be used to classify the laminarization from probed velocity signals more effectively than traditional statistical analyses.

## Funding

### United States Nuclear Regulatory Commission Grant No. 31310021M0044

## History

## Degree Type

- Doctor of Philosophy

## Department

- Nuclear Engineering

## Campus location

- West Lafayette