Symmetry Based Turbulence Theory of a Turbulent Round Jet

**Principal Investigator:**

Martin Oberlack

**Affiliation:**

TU Darmstadt

**Local Project ID:**

pn73fu

**HPC Platform used:**

SuperMUC-NG at LRZ

**Date published:**

Turbulence has been a topic of research for many decades and finds its applications in many aspects of life. Still, turbulence is not fully understood up until today. The Navier-Stokes equations, which are used to describe the motion of viscous fluids, do not have a general analytical solution. Consequently, many researchers work with specific canonical cases to understand turbulence better.

In the recent years as computers became increasingly powerful, more and more direct numerical simulations (DNS) have been conducted to solve turbulent flows. DNS solve the Navier-Stokes equations without explicitly modeling the turbulence. The main advantage of DNS over experiments is that any imaginable quantity can be generated at any point of the domain. However, with increasing Reynolds numbers *Re* the computational effort rises almost proportional to *Re³* which is why so far DNS is only used for fundamental research as opposed to in industrial applications. Therefore, DNS is not suited for complex geometries until many decades in the future e.g., for DNS of an airplane or a car. In industrial applications the quantities of interest usually only involve mean velocities instead of instantaneous velocity fields. Rather than solving highly complex infinite dimensional hierarchy of moment equations, simplified models are used. However, accuracy of the solution suffers from simplified models of the turbulence.

A method to model turbulence with the infinite dimensional hierarchy of moment equations using Lie symmetry groups has been developed by the PI and his research team. The Lie symmetry principle has been contemplated as a key feature of physics by Einstein’s seminal work on special relativity. With the advent of quantum mechanics in the 1920s, symmetries have been established as an axiomatic basis of physics in general. Today it acts as the foremost guiding principle to understand and mathematically model new physical laws to be discovered.

With the recent publications Hoyas et al (2022) and Oberlack et al (2022), we have been able to conduct large-scale DNS of a plane channel flow between two infinite parallel plates at *Re = 10,000 *on SuperMUC-NG to test new scaling laws which can be derived directly from theory without any further simplification or modeling. With Lie symmetry analysis, symmetry invariant solutions for arbitrary order of moments have been generated from the infinite dimensional hierarchy of moment equations. One of the key results show that the moments of the streamwise velocity in the channel center follows a power-law scaling with an exponent only dependent of the first and second order moment. The knowledge obtained shall be extended to turbulent round jet flows.

The goal of this project is to validate the scaling laws that have been derived for a turbulent round jet using Lie-symmetry analysis with numerical data. The study of turbulent jet flows is not only interesting as a canonical shear flow. Turbulent jet flow finds applications in acoustic control, mixing of fluids, combustion and even exhalation. It can be described as a stream of fluid that is spreading into an ambient medium through a nozzle. To model the turbulence of a jet flow, a great effort has been made to identify the relevant parameters with experimental and numerical data. Additionally, in jet flow specifically, large computational boxes are needed to capture the spreading of a turbulent round jet. Thus, the simulation of such a flow requires a vast amount of computational resources.

The CFD code Nek5000 has been used for the direct numerical simulation of a turbulent round jet with and without co-flow. The Nek5000 code is based on the spectral element method (SEM) and solves the Navier-Stokes equations on a hexahedral mesh. The code is well established in the scientific turbulence community and is highly scalable over a million ranks using Message Passing Interface (MPI) for parallelization. As a timestepping method the implicit second order backward differentiation formula (BDF2) scheme is used. The timestep is dynamically controlled and so the computational costs are further optimized. SEM features an exponentially growing accuracy with increase of the order of the polynomial basis functions used.

Currently, we are running two DNS cases on SuperMUC-NG of which one is a DNS of a turbulent round jet at *Re=3,500* without co-flow and one is a DNS of a turbulent round jet at *Re=3,500 *with a co-flow velocity ratio of *0.05*. Additionally, a passive scalar equation is solved with *Pr=0.71*. The simulation is carried out on two separate domains. On the first domain, a turbulent pipe flow is generated. Then, the velocity field of the turbulent pipe flow is interpolated onto the main computational box at each time step to generate a turbulent round jet. The main computation is carried out on a truncated cone box with a length of 75 jet nozzle diameters and a radius of 4 nozzle diameters at the inlet and 32 jet nozzle diameters at the outlet. The size of the computational box ensures the full capture of the spreading of the jet and to minimize the influence of the far-field boundary conditions. Altogether the box consists of ~ 90 million grid points, which amounts to ~60 million DOFs.

The figures show the results of the DNS of a turbulent round jet without co-flow. A cross section of the velocity magnitude and the passive scalar concentration of the DNS can be observed in Figure 1. The highly accurate data generated by the DNS allows us to validate the scaling laws that have been derived. The jet scaling laws predict that all the higher moments can be calculated from the exponent of the first and second moment. Figure 2 shows that the exponents of the third up to the tenth moment indeed show great agreement with the theory. The scaling laws of the passive scalar show a similar result in Figure 3.

The large-scale simulations are currently conducted on SuperMUC-NG. The highly resolved data generated by the simulations enables us to investigate self-similar behavior up to an axial distance of 75 diameters from the inlet nozzle. The Nek5000 code allows us to efficiently run the simulation on 512 nodes in parallel. When the high moment statistics of the co-flow DNS are converged, we will also hope to validate the scaling laws of co-flow jets derived with Lie symmetry analysis. Currently, both DNS are still running to extract statistics for the turbulent budgets. As soon as the turbulent round jet with and without co-flow are better understood, the research is planned to be extended onto different inlet conditions where we expect a higher influence of certain parameters in the scaling laws.

[2] Hoyas, Sergio, Oberlack, Martin, Alcántara-Ávila, Francisco, Kraheberger, Stefanie V. & Laux, Jonathan 2022 Wall turbulence at high friction Reynolds numbers. Phys. Rev. Fluids 7, 014602.

[3] Oberlack, Martin, Hoyas, Sergio, Kraheberger, Stefanie V., Alcántara-Ávila, Francisco, & Laux, Jonathan 2022 Turbulence Statistics of Arbitrary Moments of Wall-Bounded Shear Flows: A Symmetry Approach. Phys. Rev. Lett. 128, 024502.