@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{TsaiSchmidtKlein, author = {Tsai, Pei-Yun and Schmidt, Heiko and Klein, Marten}, title = {Modeling simultaneous momentum and passive scalar transfer in turbulent annular Poiseuille flow}, 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.202200272}, abstract = {Simultaneous momentum and passive scalar transfer in weakly heated pressure-driven turbulent concentric annular pipe flow is numerically investigated using the cylindrical formulation of the stochastic one-dimensional turbulence (ODT) model,which is utilized here as standalone tool. In the present study, we focus on the model calibration for heated annular pipes based on recent reference direct numerical simulations (DNS) from Bagheri and Wang (Int. J. Heat Fluid Flow 86, 108725,2020; Phys. Fluids 33, 055131, 2021). It is shown that the model is able to individually capture scalar and momentum transfer, but not both equally well at the same time. We attribute this to less dissimilar scalar and momentum transport in the model at the low Reynolds number investigated. It is argued that the model prefers a fully developed turbulent state due to its construction. Nevertheless, it is demonstrated that ODT is able to reasonably capture the radial inner-outer asymmetry of the scalar and momentum boundary layers which yields better predictive capabilities than wall-function-based approaches.}, language = {en} } @misc{GaoSchmidtKleinetal., author = {Gao, Tianyun and Schmidt, Heiko and Klein, Marten and Liang, Jianhan and Sun, Mingbo and Chen, Chongpei and Guan, Qingdi}, title = {One-dimensional turbulence modeling of compressible flows. I. Conservative Eulerian formulation and application to supersonic channel flow}, series = {Physics of Fluids}, volume = {35}, journal = {Physics of Fluids}, number = {3}, issn = {1089-7666}, doi = {10.1063/5.0125514}, abstract = {Accurate but economical modeling of supersonic turbulent boundary layers is a standing challenge due to the intricate entanglement of temperature, density, and velocity fluctuations on top of the mean-field variation. Application of the van Driest transformation may describe well the mean state but cannot provide detailed flow information. This lack-in modeling coarse and fine-scale variability is addressed by the present study using a stochastic one-dimensional turbulence (ODT) model. ODT is a simulation methodology that represents the evolution of turbulent flow in a low-dimensional stochastic way. In this study, ODT is extended to fully compressible flows. An Eulerian framework and a conservative form of the governing equations serve as the basis of the compressible ODT model. Computational methods for statistical properties based on ODT realizations are also extended to compressible flows, and a comprehensive way of turbulent kinetic energy budget calculation based on compressible ODT is put forward for the first time. Two canonical direct numerical simulation cases of supersonic isothermal-wall channel flow at Mach numbers 1.5 and 3.0 with bulk Reynolds numbers 3000 and 4880, respectively, are used to validate the extended model. A rigorous numerical validation is presented, including the first-order mean statistics, the second-order root mean square statistics, and higher-order turbulent fluctuation statistics. In ODT results, both mean and root mean square profiles are accurately captured in the near-wall region. Near-wall temperature spectra reveal that temperature fluctuations are amplified at all turbulent scales as the effects of compressibility increase. This phenomenon is caused by intensified viscous heating at a higher Mach number, which is indicated by the steeper profiles of viscous turbulent kinetic energy budget terms in the very near-wall region. The low computational cost and predictive capabilities of ODT suggest that it is a promising approach for detailed modeling of highly turbulent compressible boundary layers. Furthermore, it is found that the ODT model requires a Mach-number-dependent increase in a viscous penalty parameter Z in wall-bounded turbulent flows to enable accurate capture of the buffer layer.}, language = {en} } @misc{GaoSchmidtKleinetal., author = {Gao, Tianyun and Schmidt, Heiko and Klein, Marten and Liang, Jianhan and Sun, Mingbo and Chen, Chongpei and Guan, Qingdi}, title = {One-dimensional turbulence modeling of compressible flows: II. Full compressible modification and application to shock-turbulence interaction}, series = {Physics of Fluids}, volume = {35}, journal = {Physics of Fluids}, number = {3}, issn = {1089-7666}, doi = {10.1063/5.0137435}, abstract = {One-dimensional turbulence (ODT) is a simulation methodology that represents the essential physics of three-dimensional turbulence through stochastic resolution of the full range of length and time scales on a one-dimensional domain. In the present study, full compressible modifications are incorporated into ODT methodology, based on an Eulerian framework and a conservative form of the governing equations. In the deterministic part of this approach, a shock capturing scheme is introduced for the first time. In the stochastic part, one-dimensional eddy events are modeled and sampled according to standard methods for compressible flow simulation. Time advancement adjustments are made to balance comparable time steps between the deterministic and stochastic parts in compressible flows. Canonical shock-turbulence interaction cases involving Richtmyer-Meshkov instability at Mach numbers 1.24, 1.5, and 1.98 are simulated to validate the extended model. The ODT results are compared with available reference data from large eddy simulations and laboratory experiments. The introduction of a shock capturing scheme significantly improves the performance of the ODT method, and the results for turbulent kinetic energy are qualitatively improved compared with those of a previous compressible Lagrangian ODT method [Jozefik et al., "Simulation of shock-turbulence interaction in non-reactive flow and in turbulent deflagration and detonation regimes using one-dimensional turbulence," Combust. Flame 164, 53 (2016)]. For the time evolution of profiles of the turbulent mixing zone width, ensemble-averaged density, and specific heat ratio, the new model also yields good to reasonable results. Furthermore, it is found that the viscous penalty parameter Z of the ODT model is insensitive to compressibility effects in turbulent flows without wall effects. A small value of Z is appropriate for turbulent flows with weak wall effects, and the parameter Z serves to suppress extremely small eddy events that would be dissipated instantly by viscosity.}, language = {en} } @misc{KleinSchmidt, author = {Klein, Marten and Schmidt, Heiko}, title = {Exploring stratification effects in stable Ekman boundary layers using a stochastic one-dimensional turbulence model}, series = {Advances in Science and Research}, volume = {19/2022}, journal = {Advances in Science and Research}, issn = {1992-0636}, doi = {10.5194/asr-19-117-2022}, pages = {117 -- 136}, abstract = {Small-scale processes in atmospheric boundary layers are typically not resolved due to cost constraints but modeled based on physical relations with the resolved scales, neglecting expensive backscatter. This lack in modeling is addressed in the present study with the aid of the one-dimensional turbulence (ODT) model. ODT is applied as stand-alone column model to numerically investigate stratification effects in long-lived transient Ekman flows as canonical example of polar boundary layers by resolving turbulent winds and fluctuating temperature profiles on all relevant scales of the flow. We first calibrate the adjustable model parameters for neutral cases based on the surface drag law which yields slightly different optimal model set-ups for finite low and moderate Reynolds numbers. For the stably stratified cases, previously calibrated parameters are kept fixed and the model predictions are compared with various reference numerical simulations and also observations by an exploitation of boundary layer similarity. ODT reasonably captures the temporally developing flow for various prescribed stratification profiles, but fails to fully capture the near-surface laminarization by remaining longer in a fully developed turbulent state, which suggests preferential applicability to high-Reynolds-number flow regimes. Nevertheless, the model suggests that large near-surface turbulence scales are primarily affected by the developing stratification due to scale-selective buoyancy damping which agrees with the literature. The variability of the wind-turning angle represented by the ensemble of stratified cases simulated covers a wider range than reference reanalysis data. The present study suggests that the vertical-column ODT formulation that is highly resolved in space and time can help to accurately represent multi-physics boundary-layer and subgrid-scale processes, offering new opportunities for analysis of very stable polar boundary layer and atmospheric chemistry applications.}, language = {en} } @misc{MedinaMendezKleinSchmidt, author = {Medina M{\´e}ndez, Juan Ali and Klein, Marten and Schmidt, Heiko}, title = {The One-Dimensional Turbulence Aspects of Internal Forced Convective Flows}, series = {14th WCCM-ECCOMAS Congress 2020}, journal = {14th WCCM-ECCOMAS Congress 2020}, publisher = {Scipedia}, doi = {10.23967/wccm-eccomas.2020.338}, pages = {1 -- 12}, abstract = {We present an overview of issues for the modeling of internal forced convective flows with the One-Dimensional Turbulence (ODT) model. Results of recent research as well as prospective research issues are presented for statistically streamwise homogeneous flows and streamwise inhomogeneous mixed convective flows. The results illustrate the capabilities of the model to evaluate and bring insight into a wide range of physical phenomena in the field of convective flows. Nonetheless, as a model, ODT is best suited for the evaluation of asymptotically turbulent flows, i.e., away from laminar regimes.}, language = {en} } @misc{KleinSchmidt, author = {Klein, Marten and Schmidt, Heiko}, title = {Towards a stochastic model for electrohydrodynamic turbulence with application to electrolytes}, series = {91st Annual Meeting of GAMM 2020@21}, journal = {91st Annual Meeting of GAMM 2020@21}, pages = {1}, abstract = {We investigate turbulent electrohydrodynamic (EHD) Couette flows of dilute electrolytes and how they are affected by a prescribed electric field. In this canonical problem, molecular diffusion and electric drift currents can interact with turbulence which yields intricate dynamics down to the Kolmogorov and Batchelor scales that need to be resolved. The electrolytes considered have neutral bulk charge and consist of two independent, positive and negative, ion species with the same valence and mobility. The top wall of the set-up is moving and held at a different voltage relative to the bottom one. Resolution requirements and numerical feasibility are addressed by utilizing the stochastic one-dimensional turbulence (ODT) model as stand-alone tool in order to resolve all relevant scales of the flow for a dimensionally reduced setting. Deterministic diffusion and charge-carrier drift are directly resolved, whereas the effects of turbulent advection and pressure fluctuations are modeled by a stochastic process that operates along the wall-normal ODT domain. For the hydrodynamic and low Schmidt number EHD regime, ODT reasonably captures and extrapolates relevant leading-order boundary-layer properties of reference direct numerical simulations (DNS). For the high Schmidt number EHD regime, the model predicts notable interactions between turbulence and elektrokinetics only for large enough Reynolds numbers that manifests itself by a significant increase of the turbulent drag. Present ODT results suggests that the origin of this effect is related to the time-scale separation of convective versus electric drift and molecular transport processes transport across the boundary layer. In the talk, we will address the model formulation and its application to EHD Couette flow. Additionally, we will comment on the representation of electrokinetics and hydro-dynamics for the selected set-up. Finally, we will discuss the flow regimes in terms of skin friction drag and flow profiles with an eye also on electric variables and time scales.}, language = {en} } @misc{KleinSchmidt, author = {Klein, Marten and Schmidt, Heiko}, title = {Investigating Rayleigh-B{\´e}nard convection at low Prandtl numbers using one-dimensional turbulence modeling}, series = {Proc. of the 11th International Symposium on Turbulence and Shear Flow Phenomena (TSFP11), Southampton, UK, July 30 to August 2, 2019}, journal = {Proc. of the 11th International Symposium on Turbulence and Shear Flow Phenomena (TSFP11), Southampton, UK, July 30 to August 2, 2019}, pages = {1 -- 3}, abstract = {We numerically investigate the heat transfer in turbulent Rayleigh-Bénard convection at two Prandtl numbers, Pr = 0.021 and 0.7, respectively. Small-scale resolving simulations up to the Rayleigh numbers Ra = 10^(13) (Pr = 0.021) and 10^(16) (Pr = 0.7) are made feasible by utilizing the stochastic, one-dimensional turbulence (ODT) model. Present ODT simulations exhibit effective Nusselt number Nu scalings of the form Nu ∼ Ra^γ. At low Rayleigh numbers, ODT yields a scaling exponent of γ = 0.29 (Pr = 0.021) and 0.32 (Pr = 0.7), respectively. Both values are systematically, but just slightly, overestimating available reference data. At high Rayleigh numbers, present ODT results exhibit an increase of the exponent to γ = 0.32 (Pr = 0.021) and 0.36 (Pr = 0.7), respectively. Our preliminary results suggest that ODT might be able to capture a transition from the classical to the ultimate state of convection in terms of (i) critical Rayleigh number and (ii) increase of γ.}, language = {en} } @misc{KleinSchmidtLignell, author = {Klein, Marten and Schmidt, Heiko and Lignell, David O.}, title = {Map-based modeling of high-Ra turbulent convection in planar and spherical geometries}, series = {Conference on Modelling Fluid Flow 2018 (CMFF'18)}, journal = {Conference on Modelling Fluid Flow 2018 (CMFF'18)}, pages = {1}, abstract = {Turbulent convection is important in many technological and geophysical applications. A model problem for such flows is Rayleigh-B{\´e}nard (RB) convection. The classical RB setup is a fluid- filled box with a heated bottom and cooled top. For geophysical applications, the spherical geometry of the confinement is sometimes important (e.g. in mantle convection). This is addressed by a spherical annulus configuration in which fluid is confined between an inner hot and an outer cold sphere. In this case, the gravity field is radial and its strength can also vary with the radius. Numerical simulations of RB convection are challenging because of the high Rayleigh numbers (Ra) observed in applications. 3-D direct simulations have been performed up to Ra ~ 10^(12), but even larger values of Ra are relevant. Hence modeling is needed if one wishes to increase the accessible Rayleigh number limit within the considerable future. The difficulty is that gradient-diffusion approaches do not allow for scale interactions, which can be crucial for the dynamics of the flow and the resulting heat transfer. In order to make such simulations feasible we make use of a different modeling strategy, the so-called One-Dimensional Turbulence (ODT). ODT resolves all scales of the flow along a notional line of sight, but reduces cost by assuming statistical homogeneity of the flow in the off-line directions. Along the line, turbulent advection is modeled by discrete mapping events, which mimic the effect of turbulent stirring. These events are stochastically sampled with highest probability where shear and buoyancy yield net available energy in analogy to real turbulence. In the talk, we evaluate ODT results against available reference data (e.g. flow statistics, heat transfer) using a new and fully adaptive version of ODT. This new version allows to simulate turbulent convection in spherical geometry. We address this by discussing the effects of radius ratio and radius-dependent gravity.}, language = {en} } @misc{KleinZenkerHerthaetal., author = {Klein, Marten and Zenker, Christian and Hertha, Katja and Schmidt, Heiko}, title = {Modeling One and Two Passive Scalar Mixing in Turbulent Jets Using One-Dimensional Turbulence}, series = {14th WCCM-ECCOMAS Congress 2020}, journal = {14th WCCM-ECCOMAS Congress 2020}, publisher = {Scipedia}, doi = {10.23967/wccm-eccomas.2020.205}, pages = {1 -- 12}, abstract = {Turbulent mixing of two passive scalars is investigated in a constant-property jets using stochastic one-dimensional turbulence (ODT). Scalars are separately injected by a central round and a surrounding annular jet that issue into a uniform co-flow of low velocity. These scalars are transported downstream and dispersed in radial direction by turbulent advection and molecular diffusion. The jet as well as the turbulent inflow are numerically simulated with ODT as stand-alone tool using a temporal (T-ODT) and spatial (S-ODT) formulation. We show that ODT captures key properties of the turbulent mixing for one scalar by performing individual scalar statistics and for two scalars by computation of joint probabilities. Some limitations of the one-dimensional modeling approach are also discussed.}, language = {en} } @misc{KleinSchmidt, author = {Klein, Marten and Schmidt, Heiko}, title = {Stochastic modeling of transient neutral and stably-stratified Ekman boundary layers}, series = {91st Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM) - PAMM, Proceedings in Applied Mathematics and Mechanics}, volume = {20}, journal = {91st Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM) - PAMM, Proceedings in Applied Mathematics and Mechanics}, number = {1}, issn = {1617-7061}, doi = {10.1002/pamm.202000127}, pages = {3}, abstract = {Turbulence is a transient phenomenon in atmospheric boundary layers. These transients occur often due to surface temperature variations (e.g. due to diurnal forcing) that directly influence the near-surface flow by local stratification effects. Relevant dynamical and transport processes occur on a scale of meters near the surface which is a standing challenge for numerical weather and climate prediction. Here we investigate neutral and stably-stratified Ekman flows as a canonical problem for the night-time atmospheric boundary layer over flat terrain. The set-up used consists of an incompressible fluid over a smooth horizontal no-slip wall in a rotating frame of reference. The bulk flow is in geostrophic balance and acts as momentum source. In the case of stable stratification, temperature is prescribed as sudden cooling on a fully-developed turbulent neutrally-stratified Ekman boundary layer. When the stratification is weak, the temperature behaves like a passive scalar, but when it is strong, turbulence may locally disappear. Transient simulations across a relevant range of Reynolds and Froude numbers are made feasible by utilizing the stochastic one-dimensional turbulence (ODT) model. ODT aims to resolve vertical (wall-normal) transport processes on all relevant scales for a one-dimensional domain. Deterministic molecular diffusion and Coriolis forces are directly resolved, whereas turbulent advection is modeled by a stochastic process. The model obeys several relevant physical principles as, for example, Richardson's 1/4 law of stratified turbulence. Preliminary results suggest that the stand-alone model generally captures Reynolds (turbulence) and Froude number (stratification) effects when stratification is weak. For low Froude number (strong stratification), these results indicate that the model tends to overestimate turbulence effects near the surface unless stratification becomes so strong that near-surface turbulence is energetically prohibited. In the talk, we will address the model formulation and its application to Ekman flow. We will show and discuss model results for surface fluxes, boundary-layer profiles, and corresponding fluctuation statistics. In addition, we will discuss stratification effects and comment on their representation in the model.}, language = {en} } @misc{KleinSchmidtKerstein, author = {Klein, Marten and Schmidt, Heiko and Kerstein, Alan R.}, title = {Transition to the ultimate regime in a stochastic model for radiatively driven turbulent convection}, series = {Verhandlungen der Deutschen Physikalischen Gesellschaft - BPCPPDYSOE21}, journal = {Verhandlungen der Deutschen Physikalischen Gesellschaft - BPCPPDYSOE21}, language = {en} }