# A Temporal Slow Growth Formulation for Direct Numerical Simulation of Compressible Wall--Bounded Flows

## TL;DR

This paper introduces a novel temporal slow growth formulation for DNS of wall-bounded turbulent flows, enabling efficient simulations with periodic boundary conditions and aiding RANS model calibration.

## Contribution

The new formulation extends slow growth modeling to complex boundary layers using temporal homogenization and Fourier methods, simplifying calibration and validation of RANS turbulence models.

## Key findings

- Effective for incompressible boundary layers
- Applicable to transonic flows with cooled walls
- Enables efficient, periodic DNS simulations

## Abstract

A new slow growth formulation for DNS of wall-bounded turbulent flow is developed and demonstrated to enable extension of slow growth modeling concepts to complex boundary layer flows. As in previous slow growth approaches, the formulation assumes scale separation between the fast scales of turbulence and the slow evolution of statistics such as the mean flow. This separation enables the development of approaches where the fast scales of turbulence are directly simulated while the forcing provided by the slow evolution is modeled. The resulting model admits periodic boundary conditions in the streamwise direction, which avoids the need for extremely long domains and complex inflow conditions that typically accompany spatially developing simulations. Further, it enables the use of efficient Fourier numerics. Unlike previous approaches, the present approach is based on a temporally evolving boundary layer and is specifically tailored to give results for calibration and validation of RANS turbulence models. The use of a temporal homogenization simplifies the modeling, enabling straightforward extension to flows with complicating features, including cold and blowing walls. To generate data useful for calibration and validation of RANS models, special care is taken to ensure that the mean slow growth forcing is closed in terms of the mean and other quantities that appear in standard RANS models, ensuring that there is no confounding between typical RANS closures and additional closures required for the slow growth problem. The performance of the method is demonstrated on two problems: an essentially incompressible, zero-pressure-gradient boundary layer and a transonic boundary layer over a cooled wall with wall transpiration.