We correct a data processing error in the article "Construction of explicit and implicit dynamic finite difference schemes and application to the large-eddy simulation of the Taylor-Green vortex" by Dieter Fauconnier, Chris De Langhe and Erik Dick published in the Journal of Computational Physics 228 (2009), pp. 8053-8084.@en

We present well-resolved RANS simulations of two generic asymmetrically heated cooling channel configurations, a high aspect ratio cooling duct operated with liquid water at (Formula Presented) and a cryogenic transcritical channel operated with methane at (Formula Presented). The former setup serves to investigate the interaction of turbulence-induced secondary flow and heat transfer, and the latter to investigate the influence of strong non-linear thermodynamic property variations in the vicinity of the critical point on the flow field and heat transfer. To assess the accuracy of the RANS simulations for both setups, well-resolved implicit LES simulations using the adaptive local deconvolution method as subgrid-scale turbulence model serve as comparison databases. The investigation focuses on the prediction capabilities of RANS turbulence models for the flow as well as the temperature field and turbulent heat transfer with a special focus on the turbulent heat flux closure influence.@en

We investigate a reacting shock–bubble interaction through three-dimensional numerical simulations with detailed chemistry. The convex shape of the bubble focuses the shock and generates regions of high pressure and temperature, which are sufficient to ignite the diluted stoichiometric H2−O2 gas mixture inside the bubble. We study the interaction between hydrodynamic instabilities and shock-induced reaction waves at a shock Mach number of Ma=2.83. The chosen shock strength ignites the gas mixture before the shock-focusing point, followed by a detonation wave, which propagates through the entire bubble gas. The reaction wave has a significant influence on the spatial and temporal evolution of the bubble. The misalignment of density and pressure gradients at the bubble interface, caused by the initial shock wave and the subsequent detonation wave, induces Richtmyer–Meshkov and Kelvin–Helmholtz instabilities. The growth of the instabilities is highly affected by the reaction wave, which significantly reduces mixing compared to an inert shock–bubble interaction. A comparison with two-dimensional simulations reveals the influence of three-dimensional effects on the bubble evolution, especially during the late stages. The numerical results reproduce experimental data in terms of ignition delay time, reaction wave speed and spatial expansion rate of the bubble gas. We observe only a slight divergence of the spatial expansion in the long-term evolution.@en

We analyse the low-frequency dynamics of a high Reynolds number impinging shock-wave/turbulent boundary-layer interaction (SWBLI) with strong mean-flow separation. The flow configuration for our grid-converged large-eddy simulations (LES) reproduces recent experiments for the interaction of a Mach 3 turbulent boundary layer with an impinging shock that nominally deflects the incoming flow by. The Reynolds number based on the incoming boundary-layer thickness of is considerably higher than in previous LES studies. The very long integration time of allows for an accurate analysis of low-frequency unsteady effects. Experimental wall-pressure measurements are in good agreement with the LES data. Both datasets exhibit the distinct plateau within the separated-flow region of a strong SWBLI. The filtered three-dimensional flow field shows clear evidence of counter-rotating streamwise vortices originating in the proximity of the bubble apex. Contrary to previous numerical results on compression ramp configurations, these Görtler-like vortices are not fixed at a specific spanwise position, but rather undergo a slow motion coupled to the separation-bubble dynamics. Consistent with experimental data, power spectral densities (PSD) of wall-pressure probes exhibit a broadband and very energetic low-frequency component associated with the separation-shock unsteadiness. Sparsity-promoting dynamic mode decompositions (SPDMD) for both spanwise-averaged data and wall-plane snapshots yield a classical and well-known low-frequency breathing mode of the separation bubble, as well as a medium-frequency shedding mode responsible for reflected and reattachment shock corrugation. SPDMD of the two-dimensional skin-friction coefficient further identifies streamwise streaks at low frequencies that cause large-scale flapping of the reattachment line. The PSD and SPDMD results of our impinging SWBLI support the theory that an intrinsic mechanism of the interaction zone is responsible for the low-frequency unsteadiness, in which Görtler-like vortices might be seen as a continuous (coherent) forcing for strong SWBLI.@en

We present well-resolved large-eddy simulations of turbulent flow through a straight, high aspect ratio cooling duct operated with water at a bulk Reynolds number of Reb = 110 × 103 and an average Nusselt number of Nuxz = 371. The geometry and boundary conditions follow an experimental reference case and good agreement with the experimental results is achieved. The current investigation focuses on the influence of asymmetric wall heating on the duct flow field, specifically on the interaction of turbulence-induced secondary flow and turbulent heat transfer, and the associated spatial development of the thermal boundary layer and the inferred viscosity variation. The viscosity reduction towards the heated wall causes a decrease in turbulent mixing, turbulent length scales and turbulence anisotropy as well as a weakening of turbulent ejections. Overall, the secondary flow strength becomes increasingly less intense along the length of the spatially resolved heated duct as compared to an adiabatic duct. Furthermore, we show that the assumption of a constant turbulent Prandtl number is invalid for turbulent heat transfer in an asymmetrically heated duct.@en

We investigate the shock-induced turbulent mixing between a light and a heavy gas, where a Richtmyer-Meshkov instability (RMI) is initiated by a shock wave with Mach number Ma = 1.5. The prescribed initial conditions define a deterministic multimode interface perturbation between the gases, which can be imposed exactly for different simulation codes and resolutions to allow for quantitative comparison. Wellresolved large-eddy simulations are performed using two different and independently developed numerical methods with the objective of assessing turbulence structures, prediction uncertainties and convergence behaviour. The two numerical methods differ fundamentally with respect to the employed subgrid-scale regularisation, each representing state-of-the-art approaches to RMI. Unlike previous studies, the focus of the present investigation is to quantify the uncertainties introduced by the numerical method, as there is strong evidence that subgrid-scale regularisation and truncation errors may have a significant effect on the linear and nonlinear stages of the RMI evolution. Fourier diagnostics reveal that the larger energy-containing scales converge rapidly with increasing mesh resolution and thus are in excellent agreement for the two numerical methods. Spectra of gradient-dependent quantities, such as enstrophy and scalar dissipation rate, show stronger dependences on the small-scale flow field structures as a consequence of truncation error effects, which for one numerical method are dominantly dissipative and for the other dominantly dispersive. Additionally, the study reveals details of various stages of RMI, as the flow transitions from large-scale nonlinear entrainment to fully developed turbulent mixing. The growth rates of the mixing zone widths as obtained by the two numerical methods are ~t7/12 before re-shock and ~(t - t0)2/7 long after re-shock. The decay rate of turbulence kinetic energy is consistently ~(t-t0)-10/7 at late times, where the molecular mixing fraction approaches an asymptotic limit ≈ 0.85. The anisotropy measure.@en

In this paper, we investigate the high-speed dynamics of symmetric and asymmetric cavitation bubble-collapse. For this purpose, a sharp-interface numerical model is employed, that includes a numerically efficient evaporation/condensation model. The underlying assumption is that phase change occurs in thermal non-equilibrium and that the associated timescale is much larger than that of the wave-dynamics described by the interfacial Riemann problem. The sharp-interface model allows for an accurate tracking of the interface evolution throughout collapse and rebound. With a first set of simulations, we investigate the influence of the non-equilibrium on the relaxation behaviour of an oscillating vapour bubble. We observe that a good prediction of the phase-change rate is essential. Of high practical interest is the collapse of cavitation bubbles near walls under high ambient-pressure conditions. We investigate the differences in collapse evolution for detached and attached bubbles. It is shown that the maximum wall pressure strongly depends on the symmetry of the collapse mechanisms, and regions with a high probability of bubble rebound are identified. Asymmetric attached bubbles lead to significantly different topology changes during collapse than symmetric bubbles but exhibit roughly the same range of maximum pressures.@en

A new approach for the construction of implicit subgrid-scale models for large-eddy simulation based on adaptive local deconvolution is proposed. An approximation of the unfiltered solution is obtained from a quasi-linear combination of local interpolation polynomials. The physical flux function is modeled by a suitable numerical flux function. The effective subgrid-scale model can be determined by a modified-differential equation analysis. Discretization parameters which determine the behavior of the implicit model in regions of developed turbulence can be adjusted so that a given explicit subgrid-scale model is recovered to leading order in filter width. Alternatively, improved discretization parameters can be found directly by evolutionary optimization. Computational results for stochastically forced and decaying Burgers turbulence are provided. An assessment of the computational experiments shows that results for a given explicit subgrid-scale model can be matched by computations with an implicit representation. A considerable improvement can be achieved if instead of the parameters matching an explicit model discretization parameters determined by evolutionary optimization are used.@en

Further development of large-eddy simulation (LES) faces as major obstacles the strong coupling between subgrid-scale (SGS) modeling and the truncation error of the numerical discretization. One can exploit this link by developing discretization methods where the truncation error itself functions as an implicit SGS model. The adaptive local deconvolution method (ALDM) is an approach to LES of turbulent flows that represents a full coupling of SGS model and discretization scheme. To provide evidence for the validity of this new SGS model, well resolved large-eddy simulations of a fully turbulent flat-plate boundary-layer flow subjected to a constant adverse pressure gradient are conducted. Flow parameters are adapted to an available experiment. The Reynolds number based on the local free-stream velocity and momentum thickness is 670 at the inflow and 5100 at the separation point. Clauser's pressure-gradient parameter increases monotonically from 0 up to approximately 100 since a constant pressure gradient is prescribed. The adverse pressure gradient leads to a highly unsteady and massive separation of the boundary layer. The numerical predictions agree well with theory and experimental data.@en

Further development of Large Eddy Simulation (LES) faces as major obstacle the strong coupling between subgrid-scale (SGS) model and the truncation error of the numerical discretization. Recent analyzes indicate that for certain discretizations and certain flow configurations the truncation error itself can act as implicit SGS model. In this paper, we explore how implicit SGS models can be derived systematically and propose a procedure for design, analysis, and optimization of nonlinear discretizations. Implicit LES can be made rigorous by requiring that the numerical dissipation approximates the SGS dissipation obtained from the analysis of nonlinear interactions in turbulence.@en

The success of Large-Eddy Simulations (LES) of wall-bounded turbulence depends strongly on an accurate representation of the flow near the boundaries. Since in implicit LES the truncation error of the numerical discretization itself functions as SGS model, the order of accuracy of the discretization should be maintained near the boundary. In this paper, we analyze the performance of implicit LES for predicting turbulent flows along complex geometries. For the representation of wall-boundaries two different approaches are compared: (A) the case of body-fitted grids, and (B) a novel second-order accurate Conservative Immersed Interface Method (CIIM) for representing boundaries of arbitrary shape on Cartesian grids. The need for mapping domain boundaries on the grid raises the question whether a comparable accuracy with body-fitted grids can be achieved. In both (A) and (B) we employ the Adaptive Local Deconvolution Method (ALDM) for implicit LES. The assessment is carried out on the example of the flow over a circular cylinder at Re=3900. Two simulations with comparable numerical resolution are performed and compared to experimental reference data. Both methods result in similar accuracy for turbulent statistics and mean integral quantities. This demonstrates that ALDM on Cartesian grids is a feasible and efficient approach to simulate turbulent flows on complex domains.@en

We propose a conservative, second-order accurate immersed interface method for representing incompressible fluid flows over complex three dimensional solid obstacles on a staggered Cartesian grid. The method is based on a finite-volume discretization of the incompressible Navier-Stokes equations which is modified locally in cells that are cut by the interface in such a way that accuracy and conservativity are maintained. A level-set technique is used for description and tracking of the interface geometry, so that an extension of the method to moving boundaries and flexible walls is straightforward. Numerical stability is ensured for small cells by a conservative mixing procedure. Discrete conservation and sharp representation of the fluid-solid interface render the method particularly suitable for Large-Eddy Simulations of high-Reynolds number flows. Accuracy, second-order grid convergence and robustness of the method is demonstrated for several test cases: inclined channel flow at Re= 20, flow over a square cylinder at Re= 100, flow over a circular cylinder at Re= 40, Re= 100 and Re= 3900, as well as turbulent channel flow with periodic constrictions at Re= 10,595.@en

The unsteady behaviour in shockwave turbulent boundary layer interaction is investigated by analysing results from a large eddy simulation of a supersonic turbulent boundary layer over a compressiona-expansion ramp. The interaction leads to a very-low-frequency motion near the foot of the shock, with a characteristic frequency that is three orders of magnitude lower than the typical frequency of the incoming boundary layer. Wall pressure data are first analysed by means of Fourier analysis, highlighting the low-frequency phenomenon in the interaction region. Furthermore, the flow dynamics are analysed by a dynamic mode decomposition which shows the presence of a low-frequency mode associated with the pulsation of the separation bubble and accompanied by a forwarda-backward motion of the shock.@en

We investigate a passive flow-control technique for the interaction of an oblique shock generated by an 8.8° wedge with a turbulent boundary-layer at a free-stream Mach number of Ma∞=2.3 and a Reynolds number based on the incoming boundary-layer thickness of Reδ0=60.5×103 by means of large-eddy simulation (LES). The compressible Navier-Stokes equations in conservative form are solved using the adaptive local deconvolution method (ALDM) for physically consistent subgrid scale modeling. Emphasis is placed on the correct description of turbulent inflow boundary conditions, which do not artificially force low-frequency periodic motion of the reflected shock. The control configuration combines suction inside the separation zone and blowing upstream of the interaction region by a pressure feedback through a duct embedded in the wall. We vary the suction location within the recirculation zone while the injection position is kept constant. Suction reduces the size of the separation zone with strongest effect when applied in the rear part of the separation bubble. The analysis of wall-pressure spectra reveals that all control configurations shift the high-energy low-frequency range to higher frequencies, while the energy level is significantly reduced only if suction acts in the rear part of the separated zone. In that case also turbulence production within the interaction region is significantly reduced as a consequence of mitigated reflected shock dynamics and near-wall flow acceleration.@en

The adaptive local deconvolution method (ALDM) is proposed as a new nonlinear discretization scheme designed for implicit large-eddy simulation (ILES) of turbulent flows. In ILES the truncation error of the discretization of the convective terms functions as a subgrid-scale model. Therefore, the model is implicitly contained within the discretization, and an explicit computation of model terms becomes unnecessary. The discretization is based on a solution-adaptive deconvolution operator which allows to control the truncation error. Deconvolution parameters are determined by an analysis of the spectral numerical viscosity. An automatic optimization based on an evolutionary algorithm is employed to obtain a set of parameters which results in an optimum spectral match for the numerical viscosity with theoretical predictions for isotropic turbulence. Simulations of large-scale forced and decaying three-dimensional homogeneous isotropic turbulence show an excellent agreement with theory and experimental data and demonstrate the good performance of the implicit model. As an example for transitional flows, instability and breakdown of the three-dimensional Taylor-Green vortex are considered. The implicit model correctly predicts instability growth and transition to developed turbulence. It is shown that the implicit model performs at least as well as established explicit models.@en

The subgrid-scale (SGS) model in a large-eddy simulation (LES) operates on a range of scales which is marginally resolved by discretization schemes. Accordingly, the discretization scheme and the subgrid-scale model are linked. One can exploit this link by developing discretization methods from subgrid-scale models, or the converse. Approaches where SGS models and numerical discretizations are fully merged are called implicit LES (ILES). Recently, we have proposed a systematic framework for the design, analysis, and optimization of nonlinear discretization schemes for implicit LES. In this framework parameters inherent to the discretization scheme are determined in such a way that the numerical truncation error acts as a physically motivated SGS model. The resulting so-called adaptive local deconvolution method (ALDM) for implicit LES allows for reliable predictions of isotropic forced and decaying turbulence and of unbounded transitional flows for a wide range of Reynolds numbers. In the present paper, ALDM is evaluated for the separated flow through a channel with streamwise-periodic constrictions at two Reynolds numbers Re = 2,808 and Re = 10,595. We demonstrate that, although model parameters of ALDM have been determined for isotropic turbulence at infinite Reynolds number, it successfully predicts mean flow and turbulence statistics in the considered physically complex, anisotropic, and inhomogeneous flow regime. It is shown that the implicit model performs at least as well as an established explicit model.@en

Results of a large-eddy simulation (LES) of a supersonic turbulent boundary layer flow along a compression-expansion ramp configuration are presented. The numerical simulation is directly compared with an available experiment at the same flow conditions. The compression-expansion ramp has a deflection angle of β=25°. The flow is characterized by a free-stream Mach number of Ma∞=2.88 and the Reynolds number based on the incoming boundary layer thickness is Reδ0=132840. The Navier Stokes equations for compressible flows are solved on a cartesian collocated grid. About 32.5×106 grid points are used to discretize the computational domain. Subgrid scale effects are modeled implicitly by the adaptive local deconvolution method (ALDM). A synthetic inflow-turbulence technique is used, which does not introduce any low frequency into the domain, therefore avoiding any possible interference with the shock/boundary layer interaction system. Statistical samples are gathered over 800 characteristic time scales δ0/U∞. The numerical data are in good agreement with the experiment in terms of mean surface-pressure distribution, skin-friction, mean velocity profiles, velocity and density fluctuations. For the first time the full compression-expansion ramp configuration was taken into account. The computational results confirm theoretical and experimental findings on fluctuation-amplification across the shockwave/boundary layer interaction region and on turbulence damping through the interaction with rarefaction waves. The LES provide evidence of the existence of Görtler-like structures originating from the recirculation region and traveling downstream along the ramp. An analysis of the wall pressure field clearly shows the presence of a low frequency motion of the shock and strong influence of the Görtler-like vortices on the wall pressure spectra.@en

We propose and analyze a wall model based on the turbulent boundary layer equations (TBLE) for implicit large-eddy simulation (LES) of high Reynolds number wall-bounded flows in conjunction with a conservative immersed-interface method for mapping complex boundaries onto Cartesian meshes. Both implicit subgrid-scale model and immersed-interface treatment of boundaries offer high computational efficiency for complex flow configurations. The wall model operates directly on the Cartesian computational mesh without the need for a dual boundary-conforming mesh. The combination of wall model and implicit LES is investigated in detail for turbulent channel flow at friction Reynolds numbers from Re τ = 395 up to Re τ =100,000 on very coarse meshes. The TBLE wall model with implicit LES gives results of better quality than current explicit LES based on eddy viscosity subgrid-scale models with similar wall models. A straightforward formulation of the wall model performs well at moderately large Reynolds numbers. A logarithmic-layer mismatch, observed only at very large Reynolds numbers, is removed by introducing a new structure-based damping function. The performance of the overall approach is assessed for two generic configurations with flow separation: the backward-facing step at Re h = 5,000 and the periodic hill at Re H = 10,595 and Re H = 37,000 on very coarse meshes. The results confirm the observations made for the channel flow with respect to the good prediction quality and indicate that the combination of implicit LES, immersed-interface method, and TBLE-based wall modeling is a viable approach for simulating complex aerodynamic flows at high Reynolds numbers. They also reflect the limitations of TBLE-based wall models.@en

Well-resolved Large-Eddy Simulations (LES) of a pseudo-shock system in the divergent part of a Laval nozzle with rectangular cross section are conducted and compared with experimental results. The LES matches the parameter set of a reference experiment. Details of the experiment, such as planar side walls, are taken into account, all wall boundary layers are well-resolved and no wall model is used. The Adaptive Local Deconvolution Method (ALDM) with shock sensor is employed for subgrid-scale turbulence modeling and shock capturing. The LES results are validated against experimental wall-pressure measurements and schlieren pictures. A detailed discussion of the complex flow phenomena of three-dimensional shock-wave-boundary-layer interaction, including corner vortices and recirculation zones, is presented. Limitations of RANS approaches are discussed with reference to the LES results.@en