@misc{StarickSchmidt, author = {Starick, Tommy and Schmidt, Heiko}, title = {Investigation of turbulent mixing using a stochastic hierarchical parcel swapping mixing model}, series = {92nd Annual Meeting of GAMM}, journal = {92nd Annual Meeting of GAMM}, address = {Aachen, Germany}, pages = {1}, language = {en} } @misc{StarickSchmidt, author = {Starick, Tommy and Schmidt, Heiko}, title = {Investigation of turbulent mixing using a stochastic hierarchical parcel swapping mixing model}, series = {Proceedings in applied mathematics and mechanics : PAMM}, volume = {22}, journal = {Proceedings in applied mathematics and mechanics : PAMM}, number = {1}, issn = {1617-7061}, doi = {10.1002/pamm.202200283}, abstract = {Turbulent mixing plays an important role in a variety of applications ranging from astrophysics to combustion and even pollutant dispersion. The Direct Numerical Simulation (DNS) that resolves all scales is not feasible for most engineering applications since the flow has a wide range of length and time scales, which yields extremely high resolution requirements. Large Eddy Simulations (LES) overcome this limitation by modeling the sub-grid scale effects. In transported Probability Density Function (PDF) methods, the key challenge is to develop an accurate mixing model. At this point, the Hierarchical Parcel-Swapping (HiPS) model, introduced by A.R. Kerstein [J. Stat. Phys. 153, 142-161 (2013)], is an attractive candidate. It is characterized by a computationally efficient representation of the effects of turbulence on a time-evolving structure of 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 the most common 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 prevailing turbulent time scales associated with the sub-trees. The mixing of adjacent fluid parcels is done either instantaneously or at rates consistent with the corresponding diffusion time scales. In this work, HiPS is detailed for the simulation of passive scalar mixing first. Preliminary results for the scalar power spectra, mean square displacement and scalar dissipation rate are shown and reveal a reasonable agreement with experimental findings. Furthermore, the integrated binary tree structure allows to satisfy a large number of criteria for a good mixing model. Considering the reduced order and computational efficiency, HiPS is an attractive candidate for modeling the mixing in transported PDF methods.}, language = {en} } @misc{XieMathiazhaganBarkowskietal., author = {Xie, Tianxiao and Mathiazhagan, Akilan and Barkowski, Daniel and Starick, Tommy and Berg, Heinz Peter and H{\"o}schler, Klaus}, title = {Comparison of convective heat transfer in metal foam-filled channels of three different cross-sections}, series = {Numerical Heat Transfer, Part A: Applications}, volume = {85}, journal = {Numerical Heat Transfer, Part A: Applications}, number = {2}, issn = {1521-0634}, doi = {10.1080/10407782.2023.2181892}, pages = {222 -- 236}, abstract = {This work introduces a new approach of analyzing convective heat transfer in porous medium by considering the foam structure as a type of fin. It provides the resulting heat transfer characteristics for the design of a longitudinally flowed tube bundle reformer used for the Micro Gas Turbine Solid Oxide Fuel Cell (MGT-SOFC) hybrid process. Owing to a limited experimental database available in literature for the above-mentioned situation, a physical model is initially introduced for a channel flow configuration between two large flat plates using a commercial PDE solver. This model is then validated with experimental results available in literature. A comparison with theoretical solutions is also conducted. Later, this model is modified/adapted for a pipe flow configuration. The physical model for a channel with representative cross-section shape of a longitudinally flowed tube bundle is more complex and is therefore built in a commercial CFD-Solver. A comparative study of the heat transfer behavior in channels of different cross-sections is performed based on a new dimensionless correlation, whose physical coherence with fin efficiency is explained and mathematically proved. The applicability of the heat transfer correlation from one cross-sectional shape to the other are discussed. The proposed new treatment of the porous medium as a fin structure considerably simplifies the heat transfer analysis in porous medium by the clear physical meaning behind fin efficiency and Biot number. This relationship contributes to a better understanding of the heat heat transfer characteristics in porous media in contrast to the correlation between Nusselt number and Reynolds number. Furthermore, this correlation enables a direct comparison between foam structures of different parameters because the fin efficiency is always between 0 and 1. The strong physical background of new correlations also enhances the reliability and plausibility at characterizing and designing the metal foam for heat transfer enhancement.}, language = {en} } @misc{StarickSchmidt, author = {Starick, Tommy and Schmidt, Heiko}, title = {Stochastic modeling of turbulent mixing based on a hierarchical swapping of fluid parcels}, series = {Proceedings in Applied Mathematics and Mechanics}, volume = {23}, journal = {Proceedings in Applied Mathematics and Mechanics}, number = {4}, issn = {1617-7061}, doi = {10.1002/pamm.202300280}, pages = {1 -- 8}, abstract = {Turbulent mixing is an omnipresent phenomenon that constantly affects our everyday life and plays an important role in a variety of industrial applications. The simulation of turbulent mixing poses great challenges, since the full resolution of all relevant length and time scales is associated with an immense computational effort. This limitation can be overcome by only resolving the large-scale effects and completely model the sub-grid scales. The development of an accurate sub-grid mixing model is therefore a key challenge to capture all interactions in the sub-grid scales. At this place, the hierarchical parcel-swapping (HiPS) model formulated by A.R. Kerstein [J. Stat. Phys. 153, 142-161 (2013)] represents a computationally efficient and scale-resolving turbulent mixing model. HiPS mimics the effects of turbulence on time-evolving, diffusive scalar fields. In HiPS, the diffusive scalar fields or a state space is interpreted as a binary tree structure, which is an alternative approach compared to the most common mixing models. Every level of the tree represents a specific length and time scale, which is based on turbulence inertial range scaling. The state variables are only located at the base of the tree and are treated 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 only takes places between adjacent fluid parcels and at rates consistent with the prevailing diffusion time scales. In this work, the HiPS model formulation for the simulation of passive scalar mixing is detailed first. Preliminary results for the mean square displacement, passive scalar probability density function (PDF) and scalar dissipation rate are given and reveal the strengths of the HiPS model considering the reduced order and computational efficiency. These model investigations are an important step of further HiPS advancements. The integrated auxiliary binary tree structure allows HiPS to satisfy a large number of criteria for a good mixing model. From this point of view, HiPS is an attractive candidate for modeling the mixing in transported PDF methods.}, language = {en} } @misc{StarickSchmidt, author = {Starick, Tommy and Schmidt, Heiko}, title = {Stochastic map-based modeling of a lifted methane/air jet flame in a vitiated coflow}, series = {Proceedings of the Thirteenth International Symposium on Turbulence and Shear Flow Phenomena (TSFP13) (2024) Montreal}, volume = {2024}, journal = {Proceedings of the Thirteenth International Symposium on Turbulence and Shear Flow Phenomena (TSFP13) (2024) Montreal}, pages = {6}, language = {en} } @misc{LignellBehrangKersteinetal., author = {Lignell, David O. and Behrang, Masoomeh and Kerstein, Alan R. and Wheeler, Isaac and Starick, Tommy and Schmidt, Heiko}, title = {Investigation of turbulent mixing of scalars with arbitrary Schmidt numbers using the stochastic Hierarchical Parcel Swapping Model}, series = {77th Annual Meeting of the Division of Fluid Dynamics, November 24-26, 2024; Salt Lake City, Utah}, journal = {77th Annual Meeting of the Division of Fluid Dynamics, November 24-26, 2024; Salt Lake City, Utah}, publisher = {American Physical Society}, abstract = {Hierarchical Parcel Swapping (HiPS) is a stochastic model of turbulent mixing. HiPS is based on a binary tree structure consisting of nodes emanating from the top of the tree and terminating in parcels at the base of the tree containing fluid properties. Length scales decrease geometrically with increasing tree level, and corresponding time scales follow inertial range scaling. Turbulent mixing is modeled by swapping subtrees at different tree levels. Swaps involving single parcels result in micromixing that changes scalar states. Swaps are implemented as a Poisson process at rates corresponding to level time scales. HiPS is extended to simulation of multiple scalars with arbitrary diffusivities, considering transport in the inertial, viscous-advective, and inertial-diffusive ranges. Fundamental analysis of particle dispersion is presented with comparisons to theoretical results and DNS data in the inertial and viscous ranges. Scalar energy spectra are analysed in the three ranges and reproduce known scaling exponents. Scalar dissipation statistics are analysed and reproduce the experimental and theoretical lognormal distribution with negative skewness represented by a stretched-exponential function. DNS data are used to evaluate empirical coefficients, facilitating quantitative applications. The physical fidelity demonstrated with HiPS suggests its use as a low-cost subgrid model for coarse-grained flow simulation, for which parcel-pair mixing is a common treatment.}, language = {en} } @misc{KleinZenkerStaricketal., author = {Klein, Marten and Zenker, Christian and Starick, Tommy and Schmidt, Heiko}, title = {Stochastic modeling of multi-stream mixing based on one-dimensional turbulence}, series = {77th Annual Meeting of the Division of Fluid Dynamics}, journal = {77th Annual Meeting of the Division of Fluid Dynamics}, publisher = {American Physical Society}, abstract = {Measurements of multiple scalar mixing in a turbulent jet show a strong location dependence of the scalar fluctuations and mixing processes. Mixing is quantitatively described by the state space of scalar fluctuations in terms of a joint probability density function (JPDF). The JPDF evolves in the downstream and radial directions and has non-Gaussian shape which is a burden for mixing modeling since factoring into marginal distribution functions is not permissible. Stochastic simulations based on one-dimensional turbulence (ODT) are able to reasonably reproduce the JPDF and its spatial evolution by a parabolic marching problem that circumvents constraints of the underlying elliptic problem. The model reproduces the inertial-advective range (exponent -5/3) and predicts the emergence of the viscous-advective range (exponent -1) at higher wavenumbers as the Schmidt number increases. The model offers full-scale resolution at affordable cost providing means to reasonably capture state-space statistics of turbulent mixing.}, language = {en} } @misc{KleinZenkerStaricketal., author = {Klein, Marten and Zenker, Christian and Starick, Tommy and Schmidt, Heiko}, title = {Stochastic modeling of multiple scalar mixing in a three-stream concentric coaxial jet based on one-dimensional turbulence}, series = {International Journal of Heat and Fluid Flow}, volume = {104}, journal = {International Journal of Heat and Fluid Flow}, issn = {0142-727X}, doi = {10.1016/j.ijheatfluidflow.2023.109235}, pages = {1 -- 17}, abstract = {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.}, language = {en} } @misc{KleinStarickZenkeretal., author = {Klein, Marten and Starick, Tommy and Zenker, Christian and Medina M{\´e}ndez, Juan Al{\´i} and Schmidt, Heiko}, title = {Reduced order stochastic modeling of turbulent mixing based on conservative baker's maps}, series = {Proceedings of the 14th International ERCOFTAC Symposium on Engineering Turbulence Modelling and Measurements (ETMM-14)}, journal = {Proceedings of the 14th International ERCOFTAC Symposium on Engineering Turbulence Modelling and Measurements (ETMM-14)}, publisher = {ERCOFTAC}, address = {Barcelona, Spain}, pages = {613 -- 618}, abstract = {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.}, language = {en} } @misc{StarickBehrangLignelletal., author = {Starick, Tommy and Behrang, Masoomeh and Lignell, David O. and Schmidt, Heiko and Kerstein, Alan R.}, title = {Turbulent mixing simulation using the Hierarchical Parcel-Swapping (HiPS) model}, series = {Technische Mechanik}, volume = {43}, journal = {Technische Mechanik}, number = {1}, issn = {0232-3869}, doi = {10.24352/UB.OVGU-2023-044}, pages = {49 -- 58}, abstract = {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.}, language = {en} } @misc{BehrangStarickWheeleretal., author = {Behrang, Masoomeh and Starick, Tommy and Wheeler, Isaac and Schmidt, Heiko and Kerstein, Alan and Lignell, David}, title = {Hierarchical parcel-swapping representation of turbulent mixing : part 4 : extension to the viscous range and to mixing of scalars with non-unity Schmidt numbers}, series = {Journal of fluid mechanics}, volume = {1020}, journal = {Journal of fluid mechanics}, publisher = {Cambridge University Press}, address = {Cambridge}, issn = {0022-1120}, doi = {doi:10.1017/jfm.2025.10512}, pages = {1 -- 39}, abstract = {Hierarchical parcel swapping (HiPS) is a multiscale stochastic model of turbulent mixing based on a binary tree. Length scales decrease geometrically with increasing tree level, and corresponding time scales follow inertial range scaling. Turbulent eddies are represented by swapping subtrees. Lowest-level swaps change fluid parcel pairings, with new pairings instantly mixed. This formulation suitable for unity Schmidt number Sc is extended to non-unity Sc. For high Sc, the tree is extended to the Batchelor level, assigning the same time scale (governing the rate of swap occurrences) to the added levels as the time scale at the base of the Sc=3 tree. For low Sc, a swap at the Obukhov-Corrsin level mixes all parcels within corresponding subtrees. Well-defined model analogues of turbulent diffusivity, and mean scalar-variance production and dissipation rates are identified. Simulations idealising stationary homogeneous turbulence with an imposed scalar gradient reproduce various statistical properties of viscous-range and inertial-range pair dispersion, and of the scalar power spectrum in the inertial-advective, inertial-diffusive and viscous-advective regimes. The viscous-range probability density functions of pair separation and scalar dissipation agree with applicable theory, including the stretched-exponential tail shape associated with viscous-range scalar intermittency. Previous observation of that tail shape for Sc=1, heretofore not modelled or explained, is reproduced. Comparisons to direct numerical simulation allow evaluation of empirical coefficients, facilitating quantitative applications. Parcel-pair mixing is a common mixing treatment, e.g. in subgrid closures for coarse-grained flow simulation, so HiPS can improve model physics simply by smarter (yet nearly cost-free) selection of pairs to be mixed.}, language = {en} } @misc{StarickSchmidt, author = {Starick, Tommy and Schmidt, Heiko}, title = {Numerical investigation of a lifted methane/air jet flame using stochastic map-based turbulence modeling}, series = {Proceedings of the Conference on Modelling Fluid Flow CMFF'25}, journal = {Proceedings of the Conference on Modelling Fluid Flow CMFF'25}, publisher = {Department of Fluid Mechanics, Faculty of Mechanical Engineering, Budapest University of Technology and Economics}, address = {Budapest}, isbn = {978-615-112-002-6}, pages = {1 -- 8}, abstract = {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.}, language = {en} } @misc{BehrangStarickSchmidtetal., author = {Behrang, Masoomeh and Starick, Tommy and Schmidt, Heiko and Lignell, David O.}, title = {A C++ library for turbulent mixing simulation using Hierarchical Parcel Swapping (HiPS)}, series = {SoftwareX}, volume = {31}, journal = {SoftwareX}, publisher = {Elsevier BV}, address = {Amsterdam}, issn = {2352-7110}, doi = {10.1016/j.softx.2025.102331}, pages = {1 -- 7}, abstract = {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.}, language = {en} }