Team
Principal Investigators:
- Martin Oberlack,
Technical University of Darmstadt, Institute for Fluid Dynamics - Christian Rohde,
University of Stuttgart, Institute of Applied Analysis and Numerical Simulation
Project staff:
- Simon Görtz (TU Darmstadt)
- Qian Huang (University of Stuttgart)
Abstract
We consider turbulent flow fields that are governed by the incompressible Navier-Stokes system and focus on convection-dominated scenarios when nonlinear hyperbolic transport starts to prevail. This low-viscosity regime gives rise to the dissipative anomaly which is classically expressed in the form of statistical scaling laws connecting the energy dissipation rate with the mean variation of randomly forced flow fields. Most of these famous scaling laws cannot be linked rigorously to the underlying hydromechanical equations but still remain hypotheses. To describe the dynamics of ensemble averages for turbulent observables on rigorous grounds, statistical turbulence aims at deriving equations for the associated probability density distributions. This leads to the Lundgren-Monin-Novikov (LMN) hierarchy, i.e. an infinite family of linear statistical conservation laws in the form of kinetic Fokker-Planck equations. To analyze the LMN hierarchy we pursue in this project two approaches exploiting the mutual relations between them.
The first approach exploits statistical symmetries to get a less complex hierarchy, and aims then to approximate the statistical conservation laws via tailored finite-dimensional ansatz spaces. More precisely, the LMN hierarchy for isotropic turbulence is reduced and projected via a product/sum ansatz to a finite-dimensional nonlinear eigenvalue problem for which we expect anomalous scaling. For vanishing viscosity, a singular asymptotic presumably leads to dissipation as an invariant. A corresponding approach for the log-law of shear flows gives rise to an analogous nonlinear eigenvalue problem, where the von Kármán constant appears as an eigenvalue.
The second approach targets at a hybrid numerical approximation method. It relies on truncating the LMN hierarchy for one-point correlations which results in an unclosed statistical conservation law. Based on a structure-preserving discretisation for this linear equation the unclosed transport terms are determined via precision-controlled uncertainty quantification methods for the underlying hyperbolically dominated evolution equations. Scalar viscous balance laws provide a testing ground for the entire method. To address the the Navier-Stokes system we use reduced hierarchies for isotropic turbulence and new hierarchies derived from hyperbolic relaxation systems approximating the flow equations.
Both new approaches to the LMN hierarchy will be validated, and then jointly used to verify or falsify selected information from scaling laws in low-viscosity regimes.
Publications:
2026
Yangyang Cao, Qian Huang, Julian Koellermeier, Alexander Kurganov, Yongle Liu: "Flux Globalization Based Well-Balanced Path-Conservative Central-Upwind Schemes for Shallow Water Linearized Moment Equations", Computers & Fluids, vol. 311, p. 107036, https://doi.org/10.1016/j.compfluid.2026.107036 (2026)
Keim, J., Schwarz, A., Kopper, P., Blind, M., Rohde, C., Beck, A.: Entropy stable high-order discontinuous Galerkin spectral-element methods on curvilinear, hybrid meshes. J. Comput. Phys. 114829 (2026). https://doi.org/https://doi.org/10.1016/j.jcp.2026.114829
Barthwal, R., Rohde, C., Sen, A.: Existence and stability of the Riemann solutions for a non-symmetric Keyfitz--Kranzer type model. Nonlinearity. 39, Paper No. 035006 (2026). https://doi.org/10.1088/1361-6544/ae4afc
Barthwal, R., Rohde, C., Wang, Y.: A Generalized Riemann Problem Solver for a Hyperbolic Model of Two-Layer Thin Film Flow. J. Sci. Comput. 106, Paper No. 25 (2026). https://doi.org/10.1007/s10915-025-03151-1
Rahul Barthwal, Philipp Öffner, Christian Rohde: "Global Existence for a Class of Keyfitz--Kranzer Systems with Application to Thin-Film Flows", arXiv:2604.16000 (2026)
Qian Huang, Simon Görtz, Paul Hollmann, Johannes Conrad, Christian Rohde, Martin Oberlack: "A data-driven approach for 2D vorticity PDF equations by a new conditional average estimation" arXiv:2604.15551 (2026)
Q Huang, C Rohde, R Zhang. Convergence of a two-parameter hyperbolic relaxation system toward the incompressible Navier-Stokes equations. 2026, arXiv:2601.19846
Jan Giesselmann, Jens Keim, Fabio Leotta, Christian Rohde: "Justification of a Relaxation Approximation for the Navier-Stokes-Cahn-Hilliard System", arxiv.2601.18463 (2026)
2025
Qiang Huang, Christian Rohde, Wen-An Yong, Ruixi Zhang : "A hyperbolic relaxation approximation of the incompressible Navier-Stokes equations with artificial compressibility", J. Differential Equations. 438, 113339 (2025). https://doi.org/10.1016/j.jde.2025.113339
Keim, Jens; Konan, Hasel; Rohde, Christian, A Note on Hyperbolic Relaxation of the Navier-Stokes-Cahn-Hilliard system for incompressible two-phase flow, ESAIM: ProcS Volume 78 (2025), https://doi.org/10.1051/proc/202578188
Christian Rohde, Florian Wendt: Mathematical Justification of a Baer-Nunziato Model for a Compressible Viscous Fluid with Phase Transition, 2025, arXiv:2504.10161
Nilias Chaudhuri, Christian Rohde, Florian Wendt: Weak-Strong Uniqueness and Relaxation Limit for a Navier-Stokes-Korteweg Model, 2025, arXiv:2512.09719
Christian Rohde, Florian Wendt: Effective Equations for a Compressible Liquid-Vapor Flow Model with Highly Oscillating Initial Density, 2025, arXiv:2512.15535
Rahul Barthwal, Firas Dhaouadi, Christian Rohde.: On hyperbolic approximations for a class of dispersive and diffusive-dispersive equations, 2025, arXiv:2512.04882
Keim, Jens; Schwarz, Anna; Kopper, Patrick; Blind, Marcel; Rohde, Christian; Beck, Andrea: Entropy stable high-order discontinuous Galerkin spectral-element methods on curvilinear, hybrid meshes, 2025, arXiv:2507.04334
Schwarz, Anna; Keim, Jens; Rohde Chrisitan; Beck, Andrea: Entropy stable shock capturing for high-order DGSEM on moving meshes, 2025, arXiv:2503.23237
Qian Huang, Christian Rohde: "Numerical approximations to statistical conservation laws for scalar hyperbolic equations", arXiv:2509.05039 (2025)
2024
Mária Lukácová-Medvid’ová, Christian Rohde: Mathematical Challenges for the Theory of Hyperbolic Balance Laws in Fluid Mechanics: Complexity, Scales, Randomness, Jahresbericht der Deutschen Mathematiker-Vereinigung 2024, https://doi.org/10.1365/s13291-024-00290-6
V. Stegmayer, S. Görtz, S. Akbari, and M. Oberlack. “On the Lundgren hierarchy of helically symmetric turbulence”. In: Fluid Dynamics Research (2024), p. 041402