TY - GEN A1 - Klein, Marten A1 - Zenker, Christian A1 - Starick, Tommy A1 - Schmidt, Heiko T1 - Stochastic modeling of multiple scalar mixing in a three-stream concentric coaxial jet based on one-dimensional turbulence T2 - International Journal of Heat and Fluid Flow N2 - Modeling turbulent mixing is a standing challenge for nonpremixed chemically reacting flows. Key complications arise from the requirement to capture all relevant scales of the flow and the necessity to distinguish between turbulent advective transport and molecular diffusive transport processes. In addition, anisotropic mean shear, variable advection time scales, and the coexistence of turbulent and nonturbulent regions need to be represented. The fundamental issues at stake are addressed by investigating multi-scalar mixing in a three-stream coaxial jet with a map-based stochastic one-dimensional turbulence model. ODT provides full-scale resolution at affordable costs by a radical reduction of complexity compared to high-fidelity Navier–Stokes solvers. The approach is partly justified by an application of the boundary-layer approximation, but neglects fluctuating axial pressure gradients. It is demonstrated that low-order scalar statistics are reasonably but not fully captured. Despite this shortcoming, it is shown that the model is able to reproduce experimental state-space statistics of multi-stream multi-scalar mixing. The model therefore offers physics-compatible improvements in multi-stream mixing modeling despite some fundamental limitations that remain from unjustified assumptions. KW - map-based stochastic advection modeling KW - multiple passive scalars KW - one-dimensional turbulence KW - turbulent jet KW - turbulent mixing Y1 - 2023 U6 - https://doi.org/10.1016/j.ijheatfluidflow.2023.109235 SN - 0142-727X N1 - This article is part of the "TSFP12 Special Issue". VL - 104 SP - 1 EP - 17 ER - TY - GEN A1 - Tsai, Pei-Yun A1 - Schmidt, Heiko A1 - Klein, Marten T1 - Investigating Reynolds number effects in turbulent concentric coaxial pipe flow using stochastic one-dimensional turbulence modeling T2 - Proceedings in Applied Mathematics and Mechanics N2 - The present study numerically investigates turbulent momentum transfer in concentric coaxial (annular) pipe flow with small radius ratios (𝜂 = 𝑅𝑖∕𝑅𝑜 = 0.1,0.04,0.02). To model the flow, a stochastic one-dimensional turbulence (ODT) model formulated for cylindrical geometry is used that provides full-scale resolution along a representative radial coordinate. The present investigation extends the model validation by Tsai et al. (PAMM,22:e202200272, 2023), to radius ratios smaller than 0.1 and addresses boundary layers with strong span-wise curvature effects. The focus is on the assessment and analysis of statistical flow features in the vicinity of the inner cylinder wall, particularly in cases with small radius ratios. Following Boersma & Breugem (Flow Turbul. Com-bust.,86:113–127, 2011), classical boundary-layer and mixing-length theory is utilized to analyze the model predictions. The results demonstrate that the ODT model captures leading-order curvature and mixing-length effects by its physics-compatible construction. Utilizing the model for extrapolation to high Reynolds numbers inaccessible to conventional high-fidelity numerical approaches shows that curvature effects persist and nonlocally affect the entire boundary layer. The model results provide support for a spanwise-curvature-modified wall function. KW - turbulent coaxial pipe flow KW - stochastic modeling KW - one-dimensional turbulence KW - spanwise curvature KW - turbulent boundary layer KW - high Reynolds number Y1 - 2023 U6 - https://doi.org/10.1002/pamm.202300167 SN - 1617-7061 N1 - Proceedings of the GAMM Annual Meeting 2023, Dresden, Germany VL - 23 IS - 4 SP - 1 EP - 8 ER - TY - GEN A1 - Klein, Marten A1 - Starick, Tommy A1 - Zenker, Christian A1 - Medina MĂ©ndez, Juan AlĂ­ A1 - Schmidt, Heiko T1 - Reduced order stochastic modeling of turbulent mixing based on conservative baker’s maps T2 - Proceedings of the 14th International ERCOFTAC Symposium on Engineering Turbulence Modelling and Measurements (ETMM-14) N2 - The detailed numerical representation of turbulent mixing processes is a standing challenge for non-premixed chemically reacting flows. The full range of relevant flow scales needs to be captured and it is also necessary to distinguish turbulent advective from molecular diffusive processes in order to represent Reynolds and Schmidt number effects. These requirements are addressed here by utilizing two different map-based stochastic turbulence modeling strategies. The one-dimensional turbulence (ODT) model utilizes event-based turbulence modeling, whereas the hierarchical parcel-swapping (HiPS) model is a fully event-based mixing model. ODT provides full-scale resolution at affordable costs by dimensional model reduction based on the boundary-layer approximation to shear flow. HiPS is far less costly than ODT but currently limited to locally homogeneous isotropic turbulence. The physics-compatible modeling capabilities with respect to phase-space representation of turbulent mixing are demonstrated for two canonical cases using standalone model formulations. KW - turbulent mixing KW - one-dimensional turbulence (ODT) KW - hierarchical parcel swapping (HiPS) KW - stochastic turbulence modeling KW - round jet KW - passive scalar Y1 - 2023 UR - https://etmm.ercoftac.org/etmm/program/conference-program/ UR - https://drive.google.com/file/d/1q2BDOO5bXfqq0Y4z4HCGndiFI033bPyg/view?usp=drive_link SP - 613 EP - 618 PB - ERCOFTAC CY - Barcelona, Spain ER - TY - GEN A1 - Vallem, Rishindra A1 - Klein, Marten A1 - Schmidt, Heiko T1 - Numerical modeling and simulation of two-phase internal flow instabilities using Smoothed Particle Hydrodynamics (SPH) T2 - STAB Jahresbericht 2023 KW - Smoothed Particle Hydrodynamics (SPH) KW - Kelvin-Helmholtz instability KW - linear stability analysis KW - two-phase flow Y1 - 2023 UR - https://www.dlr.de/as/Portaldata/5/Resources/dokumente/veranstaltungen/stab_workshop/Jahresbericht2023.pdf UR - https://www.dlr.de/as/desktopdefault.aspx/tabid-128/268_read-1678/ VL - 21/2023 SP - 158 EP - 159 PB - Deutsche Strömungsmechanische Arbeitsgemeinschaft (STAB) CY - Göttingen, Germany ER - TY - GEN A1 - Tsai, Pei-Yun A1 - Schmidt, Heiko A1 - Klein, Marten T1 - Stochastic modeling of asymmetric turbulent boundary layers in annular pipe flow Y1 - 2023 UR - https://www-docs.b-tu.de/fg-stroemungsmodellierung/public/Tsai_2023_poster_PragueSummerSchoolX.pdf U6 - https://doi.org/10.13140/RG.2.2.18789.78567 N1 - Stochastics in Fluids Summer School and Workshop 2023, Czech Academy of Sciences Institute of Mathematics, Prague, Czech Republic ER - TY - GEN A1 - Starick, Tommy A1 - Behrang, Masoomeh A1 - Lignell, David O. A1 - Schmidt, Heiko A1 - Kerstein, Alan R. T1 - Turbulent mixing simulation using the Hierarchical Parcel-Swapping (HiPS) model T2 - Technische Mechanik N2 - Turbulent mixing is an omnipresent phenomenon that permanently affects our everyday life. Mixing processes also plays an important role in many industrial applications. The full resolution of all relevant flow scales often poses a major challenge to the numerical simulation and requires a modeling of the small-scale effects. In transported Probability Density Function (PDF) methods, the simplified modeling of the molecular mixing is a known weak point. At this place, the Hierarchical Parcel-Swapping (HiPS) model developed by A.R. Kerstein [J. Stat. Phys. 153, 142-161 (2013)] represents a computationally efficient and novel turbulent mixing model. HiPS simulates the effects of turbulence on time-evolving, diffusive scalar fields. The interpretation of the diffusive scalar fields or a state space as a binary tree structure is an alternative approach compared to existing mixing models. The characteristic feature of HiPS is that every level of the tree corresponds to a specific length and time scale, which is based on turbulence inertial range scaling. The state variables only reside at the base of the tree and are understood as fluid parcels. The effects of turbulent advection are represented by stochastic swaps of sub-trees at rates determined by turbulent time scales associated with the sub-trees. The mixing of adjacent fluid parcels is done at rates consistent with the prevailing diffusion time scales. In this work, a standalone HiPS model formulation for the simulation of passive scalar mixing is detailed first. The generated scalar power spectra with forced turbulence shows the known scaling law of Kolmogorov turbulence. Furthermore, results for the PDF of the passive scalar, mean square displacement and scalar dissipation rate are shown and reveal a reasonable agreement with experimental findings. The described possibility to account for variable Schmidt number effects is an important next development step for the HiPS formulation. This enables the incorporation of differential diffusion, which represents an immense advantage compared to the established mixing models. Using a binary structure allows HiPS to satisfy a large number of criteria for a good mixing model. Considering the reduced order and associated computational efficiency, HiPS is an attractive mixing model, which can contribute to an improved representation of the molecular mixing in transported PDF methods. KW - differential diffusion KW - hierarchical parcel-swapping KW - HiPS KW - mixing model KW - scalar mixing Y1 - 2023 U6 - https://doi.org/10.24352/UB.OVGU-2023-044 SN - 0232-3869 VL - 43 IS - 1 SP - 49 EP - 58 ER - TY - GEN A1 - Klein, Marten A1 - Medina MĂ©ndez, Juan AlĂ­ A1 - Schmidt, Heiko T1 - Stochastic modeling of electrohydrodynamically enhanced drag in one-way and fully coupled turbulent Poiseuille and Couette flow T2 - Technische Mechanik N2 - Joint predictive modeling of hydrodynamics and electrokinetics is a standing numerical challenge but crucial for various applications in electrochemistry and power engineering. The present lack in modeling of electrohydrodynamic (EHD) turbulent flows lies in the treatment of small-scale processes and scale interactions. To overcome these limitations, a stochastic one-dimensional turbulence (ODT) model is utilized. The model aims to resolve all scales of the flow, but only on a notional line-of-sight, modeling turbulent advection by a stochastically sampled sequence of eddy events that punctuate deterministic molecular diffusive advancement. In this study, two canonical flow configurations are investigated that address different coupling strategies and flow physics. First, EHD effects in a variable-density vertical pipe flow of an ideal gas with an inner concentric electrode are investigated with a one-way coupled model formulation. Electric fields are generated by means of a corona discharge and the corresponding effect of a fixed ionic charge density field. Second, in order to reduce physical complexity, EHD effects the turbulent boundary layers in plane Couette flow of an isothermal univalent ionic liquid are investigated with a fully coupled model formulation. Both application cases demonstrate that ODT has predictive capabilities due to multiscale resolution of transport processes. Present results suggest that more expensive fully than one-way coupling of electrokinetics is crucial when charge relaxation times are significantly larger than the mean advection time scale. KW - turbulent drag enhancement KW - electrohydrodynamic turbulence KW - high Schmidt number KW - multiphysical boundary layers KW - one-dimensional turbulence Y1 - 2023 U6 - https://doi.org/10.24352/UB.OVGU-2023-049 SN - 0232-3869 N1 - This article is part of the "Special Issue for CMFF’22". VL - 43 IS - 1 SP - 111 EP - 127 ER - TY - GEN A1 - Klein, Marten A1 - Schmidt, Heiko T1 - Capturing features of turbulent Ekman–Stokes boundary layers with a stochastic modeling approach T2 - Advances in Science and Research N2 - Atmospheric boundary layers (ABLs) exhibit transient processes on various time scales that range from a few days down to seconds, with a scale separation of the large-scale forcing and the small-scale turbulent response. One of the standing challenges in modeling and simulation of ABLs is a physically based representation of complex multiscale boundary layer dynamics. In this study, an idealized time-dependent ABL, the so-called Ekman–Stokes boundary layer (ESBL), is considered as a simple model for the near-surface flow in the mid latitudes and polar regions. The ESBL is driven by a prescribed temporal modulation of the bulk–surface velocity difference. A stochastic one-dimensional turbulence (ODT) model is applied to the ESBL as standalone tool that aims to resolve all relevant scales of the flow along a representative vertical coordinate. It is demonstrated by comparison with reference data that ODT is able to capture relevant features of the time-dependent boundary layer flow. The model predicts a parametric enhancement of the bulk–surface coupling in the event of a boundary layer resonance when the flow is forced with the local Coriolis frequency. The latter reproduces leading order effects of the critical latitudes. The model results suggest that the bulk flow decouples from the surface for high forcing frequencies due to a relative increase in detached residual turbulence. KW - turbulent boundary layer KW - stochastic modeling KW - periodic forcing KW - one-dimensional turbulence KW - rotating flow Y1 - 2023 UR - https://asr.copernicus.org/articles/20/55/2023/ U6 - https://doi.org/10.5194/asr-20-55-2023 SN - 1992-0636 SN - 1992-0628 N1 - This article is part of the special issue “EMS Annual Meeting: European Conference for Applied Meteorology and Climatology 2022”. N1 - This research is supported by the German Federal Government, the Federal Ministry of Education and Research and the State of Brandenburg within the framework of the joint project EIZ: Energy Innovation Center with funds from the Structural Development Act (StrukturstĂ€rkungsgesetz) for coal-mining regions. VL - 20 SP - 55 EP - 64 ER - TY - GEN A1 - Medina MĂ©ndez, Juan Ali A1 - Schmidt, Heiko T1 - Towards the evaluation of heat and mass transfer in pipe flows with cocurrent falling films using One-Dimensional Turbulence T2 - Proceedings in Applied Mathematics and Mechanics Y1 - 2023 UR - https://onlinelibrary.wiley.com/doi/10.1002/pamm.202200271 U6 - https://doi.org/10.1002/pamm.202200271 VL - 23 IS - 1 CY - Aachen ER - TY - GEN A1 - Tsai, Pei‐Yun A1 - Schmidt, Heiko A1 - Klein, Marten T1 - Stochastic modeling and theoretical analysis of weakly heated concentric coaxial pipe flows at low prandtl number T2 - Proceedings in Applied Mathematics and Mechanics N2 - Heated concentric coaxial (annular) pipe flows are numerically investigated by a stochastic one‐dimensional turbulence (ODT) model. The main objective of this work is to more accurately predict the heat transfer in tubular heat exchangers at low Prandtl numbers by extending the analysis for weak temperature fluctuations as recently introduced in Tsai et al., Proceedings in Applied Mathematics and Mechanics, 23:e202300167, 2023. The ODT model offers the required predictive capabilities at affordable cost by providing full‐scale resolution of viscous, conductive, and turbulent advective transport processes along a representative radial coordinate. The Prandtl numbers and are considered for which the radius ratio of the annular pipe and the Reynolds number are varied. Numerical results demonstrate that the geometry (radius ratio) has a significant influence on the thermal boundary layer that emerges over the inner and outer curved wall, respectively. Application of boundary layer theory and mixing length arguments yield an analytical expression that includes both Reynolds number and curvature effects. Unknown closure coefficients are estimated with ODT, providing a physically based correlation for the Nusselt number. KW - Turbulent pipe flow KW - Coaxial heat exchanger KW - Low Prandtl number KW - Boundary layer theory KW - Mixing-length theory Y1 - 2025 U6 - https://doi.org/10.1002/pamm.70006 SN - 1617-7061 VL - 25 IS - 1 SP - 1 EP - 7 PB - Wiley CY - Weinheim ER - TY - GEN A1 - Marinković, Pavle A1 - Medina, Juan A. A1 - Schöps, Mark Simon A1 - Klein, Marten A1 - Schmidt, Heiko T1 - Experiences from the bottom‐up development of an object‐oriented CFD solver with prospective hybrid turbulence model applications T2 - Proceedings in Applied Mathematics and Mechanics N2 - In this work, we discuss recent experiences related to the development and enhancement of a hybrid stochastic computational fluid dynamics (CFD) solver, the C++ version of the Implicit/Explicit (IMEX) time‐advancement algorithm used in the one‐dimensional turbulence‐based (ODT) large eddy simulation (LES) model, abbreviated as ODTLES. After being ported from Fortran 90, the current capabilities of the C++ code are restricted to the reproducibility of turbulent channel flow simulations with respect to the former Fortran code version that was able to achieve reasonable agreement with available reference direct numerical simulation (DNS) for low to moderate Reynolds number turbulent channel flows. This is far from satisfactory so that current efforts are centered on improving the solver code structure through comprehensive refactoring, robust unit testing, and strict adherence to code style guides, following the principles of Clean Code. We focus the discussion on a methodology to balance unit, regression, and integration testing, here for the LES component of the code. The objective is to frame a starting point that is relevant also for other CFD codes, irrespective of whether they utilize conventional or novel discretization or flow modeling approaches. KW - Computational fluid dynamics (CFD) KW - Solver development KW - Software maintainabilit KW - Software dependability KW - Large-eddy simulation (LES) KW - Stochastic subgrid scale modeling Y1 - 2025 U6 - https://doi.org/10.1002/pamm.202400190 SN - 1617-7061 VL - 25 IS - 1 PB - Wiley ER - TY - GEN A1 - Polasanapalli, Sai Ravi Gupta A1 - Klein, Marten A1 - Schmidt, Heiko T1 - Effects of wall slip on large-scale flow in turbulent Rayleigh–BĂ©nard convection T2 - Proceedings of the Conference on Modelling Fluid Flow CMFF’25 N2 - The current study investigates the effects of surface boundary conditions—no-slip, free-slip, and finite-slip-on flow dynamics and heat transfer inturbulent Rayleigh–BĂ©nard (RB) convection for different Prandtl numbers. Using a three-dimensional lattice Boltzmann method (LBM) solver in direct numerical simulation (DNS) mode, simulations are performed for three Prandtl numbers Pr = 0.786, 4.38, 10 and two Rayleigh numbers Ra = 2 ×10^6 , 10^7 . The aim is to understand how surface conditions influence flow patterns, thermal mixing, and heat transfer efficiency in a cubic cavity with heated bottom and cooled top walls. Results show that free-slip conditions significantly enhance heat transfer, yielding higher Nusselt numbers due to thinner thermal boundary layers and stronger convective currents. In contrast, finite-slip conditions produce results similar to no-slip cases, indicating minimal impact for the slip lengths considered. The results demonstrate that surface boundary conditions play a role in modulating flow dynamics and heat transfer in RB convection. KW - Lattice Boltzmann method (LBM) KW - Navier-slip KW - Prandtl number effects KW - Rayleigh–BĂ©nard (RB) convection KW - Surface boundary conditions KW - Turbulent flows Y1 - 2025 UR - https://www.cmff.hu/papers25/CMFF25_Conference_Proceedings.pdf SN - 978-615-112-002-6 PB - Budapest University of Technology and Economics CY - Budapest ER - TY - GEN A1 - Starick, Tommy A1 - Schmidt, Heiko T1 - Numerical investigation of a lifted methane/air jet flame using stochastic map-based turbulence modeling T2 - Proceedings of the Conference on Modelling Fluid Flow CMFF’25 N2 - This numerical study investigates a lifted methane/air jet flame in a vitiated coflow by means of the map-based, stochastic one-dimensional turbulence (ODT) model. The dimensional reduction of ODT allows for simulations with affordable computational costs and provides nonetheless full-scale resolution along a notional line of sight crossing the turbulent flow field. The considered Cabra burner configuration consists of a jet flame issuing from a central nozzle into a vitiated coflow of hot combustion products. Radial and centerline profiles for mixture fraction, temperature and selected species mass fractions obtained from ODT using a reduced and detailed reaction mechanism are in appropriate agreement with the existing experimental measurements. A two-dimensional illustration of the autoignition index is given, which enables the distinction between autoignition and propagation driven reaction zones. Additionally, the sensitivity of the jet combustion to velocity and temperature variations is investigated. Considering the reduced order of ODT and the sensitivity of the subtle interactions of the hot coflow with the cold jet on the entire reaction process, ODT is able to predict the flow characteristics and reasonably matches the experimental data. As a consequence, ODT is an efficient and alternative model for turbulent reactive flow simulations. KW - Autoigniton KW - Lifted jet flame KW - Methane/air combustion KW - ODT KW - One-dimensional turbulence KW - Stochastic turbulence modeling Y1 - 2025 UR - https://www.cmff.hu/papers25/CMFF25_Final_Paper_PDF_63.pdf SN - 978-615-112-002-6 SP - 1 EP - 8 PB - Department of Fluid Mechanics, Faculty of Mechanical Engineering, Budapest University of Technology and Economics CY - Budapest ER - TY - GEN A1 - Behrang, Masoomeh A1 - Starick, Tommy A1 - Schmidt, Heiko A1 - Lignell, David O. T1 - A C++ library for turbulent mixing simulation using Hierarchical Parcel Swapping (HiPS) T2 - SoftwareX N2 - Turbulence models are crucial for simulating flows at all scales, capturing both large-scale structures and small-scale mixing. Software libraries that implement such models should support modular integration, customization, and scalability across different simulation frameworks. This paper presents Hierarchical Parcel Swapping (HiPS), a C++ library documented with Doxygen and available on GitHub. HiPS supports both mixing and reactions and can be used as a standalone model or as a subgrid model in CFD simulations. The code includes examples for users to run it as a standalone model. Additionally, considerations for using it as a subgrid model are provided. KW - Mixing KW - Reaction KW - Simulation KW - Turbulence Y1 - 2025 U6 - https://doi.org/10.1016/j.softx.2025.102331 SN - 2352-7110 VL - 31 SP - 1 EP - 7 PB - Elsevier BV CY - Amsterdam ER - TY - GEN A1 - Klein, Marten A1 - Haque, Zubaer A1 - Schmidt, Heiko T1 - Map-based stochastic turbulence modeling for utilization in wind engineering T2 - IFAC-PapersOnLine N2 - Aeroelastic simulation tools provide means for modeling wind energy systems and their response to control actions, but these tools are notably limited by the representation of the atmospheric turbulence variability under changing conditions. The objective of this study is to assess the potential for more realistic fluctuation modeling by utilizing stochastic one-dimensional turbulence (ODT). As preliminary step towards control-oriented applications, wind speed fuctuations generated by ODT are compared to those from a state-of-the-art model, such as TurbSim. It is demonstrated that weak excursions are statistically similar, whereas ODT offers a broader variability range and enhanced turbulence intermittency features. KW - Boundary element method KW - Computer-aided engineering KW - Control applications KW - Dynamic models KW - Stochastic modeling KW - Turbulence KW - Variability Y1 - 2025 U6 - https://doi.org/10.1016/j.ifacol.2025.12.046 SN - 2405-8963 VL - 59 IS - 26 SP - 271 EP - 276 PB - Elsevier BV CY - Amsterdam ER - TY - GEN A1 - Joshi, Abhishek A1 - Klein, Marten A1 - Schmidt, Heiko T1 - Non‐negligible influence of forcing mechanisms on turbulent mixing at low Reynolds numbers : a one‐dimensional turbulence study T2 - Proceedings in applied mathematics and mechanics : PAMM N2 - This study presents a numerical investigation of passive scalar mixing in homogeneous isotropic turbulence (HIT). Different volumetric forcing schemes have been used in the literature, but the side effects are rarely discussed, either because these are assumed irrelevant or because it is too costly to conduct such an analysis with a high‐fidelity model. In this study, we have used One‐Dimensional Turbulence (ODT) model to compare forcing schemes at low Reynolds numbers. Our analysis reveals critical flaws in the linear forcing model when applied to ODT. While both schemes exhibit spectral deviations from direct numerical simulation (DNS), the stochastic forcing scheme demonstrates superior dynamic fidelity, better capturing the turbulent energy cascade. In contrast, the linear forcing scheme suffers from a non‐physical energy deficit at large scales and is approximately 10 times more computationally expensive. These artefacts directly impact scalar mixing: The stochastic scheme produces classic, multi‐scale intermittency, whereas linear forcing generates extreme gradients confined only at the dissipative scales. These results demonstrate that the choice of forcing is a critical modelling decision in ODT, leading to fundamentally different model‐dependent artifacts in both turbulence dynamics and scalar mixing statistics, at least in low Reynolds number regimes. KW - Homogeneous isotropic turbulence (HIT) KW - Turbulent mixing KW - Stochastic modeling KW - One-dimensional turbulence KW - Forcing schemes Y1 - 2026 U6 - https://doi.org/https://doi.org/10.1002/pamm.70073 SN - 1617-7061 VL - 26 IS - 1 SP - 1 EP - 11 PB - Wiley CY - Weinheim ER -