Experimental and Numerical Study of the Scalar Turbulent/Non-Turbulent Interface Layer in a Jet Flow

Based on two large-eddy simulations (LES) of a non-reacting turbulent round jet with a nozzle based Reynolds number of 8,610 with the same configuration as the one that has recently been investigated experimentally (Gampert et al., 2012; J Fluid Mech, 2012; J Fluid Mech 724:337, 2013), we examine the scalar turbulent/non-turbulent (T/NT) interface layer in the mixture fraction field of the jet flow between ten and thirty nozzle diameters downstream. To this end, the LES—one with a coarse grid and one with a fine grid—are in a first step validated against the experimental data using the axial decay of the mean velocity and the mean mixture fraction as well as based on radial self-similar profiles of mean and root mean square values of these two quantities. Then, probability density functions (pdf) of the mixture fraction at various axial and radial positions are compared and the quality of the LES is discussed. In general, the LES results are consistent with the experimental data. However, in the flow region where the imprint of the T/NT interface layer is dominant in the mixture fraction pdf, discrepancies are observed. In a next step, statistics of the T/NT interface layer are studied, where a satisfactory agreement for the pdf of the location of the interface layer from the higher resolved LES with the experimental data is observed, while the one with the coarse grid exhibits considerable deviations. Finally, the mixture fraction profile across the interface is investigated where the same trend as for the pdf of the location is present. In particular, it is found that the sharp interface that is present in experimental studies (Gampert et al., J Fluid Mech, 2013; Westerweel et al., J Fluid Mech 631:199, 2009) is less distinct in the LES results and rather diffused in radial direction outside of the T/NT interface layer.     

On Near-Wall Treatment in (U)RANS-Based Closure Models

The present work is concerned with computational evaluation of a recently formulated near-wall relationship providing the value of the dissipation rate ε of the kinetic energy of turbulence k through its exact dependence on the Taylor microscale λ: ε = 10νk/λ ², (Jakirli? and Jovanovi?, J. Fluid Mech. 656:530–539, 2010). Dissipation rate determination benefits from the asymptotic behavior of the Taylor microscale resulting in its linear variation in terms of the wall distance (λ?∝?y) being valid throughout entire viscous sublayer. Accordingly, it can be applied as a unified near-wall treatment in all computational frameworks relying on a RANS-based model of turbulence (including also hybrid LES/RANS schemes) independent of modeling level—both main modeling concepts eddy-viscosity and Reynolds stress models can be employed. Presently, the feasibility of the proposed formulation was demonstrated by applying a conventional near-wall second-moment closure model based on the homogeneous dissipation rate ε _h ( ${\varepsilon_h =\varepsilon -0.5\partial \left( {{\nu \partial k}/ {\partial x_j }} \right)} / {\partial x_j }$ ; Jakirli? and Hanjali?, J. Fluid Mech. 539:139–166, 2002) and its instability-sensitive version, modeled in terms of the inverse turbulent time scale ω _h (ω _h ?=?ε _h /k; Maduta and Jakirli?, 2011), to a fully-developed channel flow with both flat walls and periodic hill-shaped constrictions mounted on the bottom wall in a Reynolds number range. The latter configuration is subjected to boundary layer separation from a continuous curved wall. The influence of the near-wall resolution lowering with respect to the location of the wall-closest computational node, coarsened even up to the viscous sublayer edge situated at $y_P^+ \approx 5$ in equilibrium flows, is analyzed. The results obtained follow closely those pertinent to the conventional near-wall integration with the wall-next node positioned at $y_P^+ \le 0.5$ .     

A New Divergence Free Synthetic Eddy Method for the Reproduction of Inlet Flow Conditions for LES

This paper describes a recent development of the Synthetic Eddy Method (SEM) proposed by Jarrin et al. (Int J Heat Fluid Flow 30(3):435–442, 2009) for generation of synthetic turbulence. The present scheme is designed to produce a divergence-free turbulence field that can reproduce almost all possible states of Reynolds stress anisotropy. This improved representation, when used to provide inlet conditions for an LES, leads to reduced near-inlet pressure fluctuations in the LES and to a reduced development length, both of which lead to lower computer resource requirements. An advantage of this method with respect to forcing approaches (which require an iterative approach) is the suitability for direct usage with embedded LES. Results for a turbulent channel flow are reported here and compared to those from the original SEM, and other direct approaches such as the VORTEX method of Sergent (2002) and the Synthesized Turbulence approach of Davidson and Billson (Int J Heat Fluid Flow 27(6):1028–1042, 2006), showing overall improved performance and a more accurate representation of turbulence structures immediately downstream of the inlet.     

A Hybrid-Filter Approach to Turbulence Simulation

Germano (Theor Comput Fluid Dyn 17:225–331, 2004) proposed a hybrid-filter approach, which additively combines an LES-like filter operator (F) and a RANS-like statistical operator (E) using a blending function k: H?=?kF?+?(1???k)E. Using turbulent channel flow as an example, we first conducted a priori tests in order to gain some insights into this hybrid-filter approach, and then performed full simulations to further assess the approach in actual simulations. For a priori tests, two separate simulations, RANS (E) and LES (F), were performed using the same grid in order to construct a hybrid-filtered field (H). It was shown that the extra terms arising out of the hybrid-filtered Navier–Stokes (HFNS) equations provided additional energy transfer from the RANS region to the LES region, thus alleviating the need for the ad hoc forcing term that has been used by some investigators. The complexity of the governing equations necessitated several modifications in order to render it suitable for a full numerical simulation. Despite some issues associated with the numerical implementation, good results were obtained for the mean velocity and skin friction coefficient. The mean velocity profile did not have an overshoot in the logarithmic region for most blending functions, confirming that proper energy transfer from the RANS to the LES region was a key to successful hybrid models. It is shown that Germano’s hybrid-filter approach is a viable and mathematically more appealing approach to simulate high Reynolds number turbulent flows.     

Dynamics of Isotropic Homogeneous Turbulence with Linear Forcing Using a Lattice Boltzmann Method

The Lattice Boltzmann Method (LBM) has proved to be a promising approach to solve the Navier–Stokes equations, especially for incompressible and isothermal cases. For turbulent flows, the quality of the predictions has been previously studied considering standard spectral forced (ten Cate et al., Comput Fluids 35:1239–1251, 2006) statistically homogeneous isotropic turbulence. In the present contribution, a recently proposed linear forcing scheme working in physical space (Lundgren 2003; Rosales and Meneveau, Phys Fluids 17(9):095106–1,8, 2005) has been integrated in a three-dimensional fifteen-velocity LBM formulation. Results have been analyzed, with special attention to the dynamics of the flow through the invariants of the velocity tensor. This topic had not been studied yet for the linear forcing, regardless of the nature (spectral or LBM) of the numerical method. Results fully agree with standard pseudo-spectral direct numerical simulations, results proving the validity of the LBM with linear forcing in real space to study this kind of turbulent flows.     

Back to the Future? A Re-examination of the Aerodynamics of Flettner-Thom Rotors for Maritime Propulsion

The paper examines, by way of a 3-dimensional, unsteady-RANS analysis, the high-Reynolds-number turbulent flow normal to a cylinder rotating about its axis, thus continuing the re-evaluation of this flow configuration as a potential means of providing low-cost ship propulsion. Comparisons are made between the available experimental and LES data both for the bare cylinder, as employed in Flettner’s notable Atlantic crossing, and the variant advocated by Thom in which close-packed discs are distributed along the cylinder. Our results display close agreement with available experimental and LES data other than the results of Thom (1934). We conclude that the addition of discs by the latter led, at the relatively low Reynolds numbers of his experiments, to the boundary layer on the cylinder being thinned to a point at which the boundary layers became laminar or transitional, thus leading to higher lift coefficients than pertain in turbulent flow.     

Recent improvements in the Zonal Detached Eddy Simulation (ZDES) formulation

An efficient generalized Zonal Detached Eddy Simulation method (ZDES) is presented, which aims at performing hybrid Reynolds Averaged Numerical Simulation (RANS)/Large Eddy Simulation (LES) calculations for both internal and external aerodynamics problems. It is based on a zonal formulation of the hybrid length scale that allows to combine the zonal approach with the best features of Delayed Detached Eddy Simulation (DDES) (Spalart et?al. Theor Comput Fluid Dyn 20:181–195, 2006). In other words, the presumed weak point of a zonal approach, namely that the location of separation has to be known in advance, is now overcome. What is more, the problem of slow LES content development in mixing layers when they are treated neither in RANS nor in LES mode is investigated. It is argued that the subgrid length scale Δ_max?=?max(Δx, Δy, Δz) entering DDES is physically justified to shield the boundary layer but is definitely not a good subgrid length scale in LES mode. Remedies are proposed based on new zonal subgrid length scales that depend not only on the grid spacing but also on the flow solution and especially on the local vorticity vector. The method is validated on a spatially developing mixing layer as well as in a backward facing step flow and then applied to a three-element airfoil. It is argued in this latter case that a precise control of the RANS mode thanks to a zonal approach is essential. More generally, in all simulated cases in this study, ZDES has proven to be very efficient as regards the behavior in LES mode while retaining the strongest asset of DDES, namely the treatment of the attached boundary layer in RANS mode. The issue of zonal or non-zonal treatment of turbulent flows is also briefly discussed.     

PIV experiments in rough-wall, laminar-to-turbulent, oscillatory boundary-layer flows

Exploratory measurements of oscillatory boundary layers were conducted over a smooth and two different rough beds spanning the laminar, transitional and turbulent flow regimes using a multi-camera 2D-PIV system in a small oscillatory-flow tunnel (Admiraal et al. in J Hydraul Res 44(4):437–450, 2006). Results show how the phase lag between bed shear stress and free-stream velocity is better defined when the integral of the momentum equation is used to estimate the bed shear stress. Observed differences in bed shear stress and phase lag between bed shear stress and free-stream velocity are highly sensitive to the definition of the bed position (y = b). The underestimation of turbulent stresses close to the wall is found to explain such differences when using the addition of Reynolds and viscous stresses to define both the bed shear stress and the phase lag. Regardless of the flow regime, in all experiments, boundary-layer thickness reached its maximum value at a phase near the flow reversal at the wall. Friction factors in smooth walls are better estimated using a theoretical equation first proposed by Batchelor (An introduction to fluid dynamics. Cambridge University Press, Cambridge, 1967) while the more recent empirical predictor of Pedocchi and Garcia (J Hydraul Res 47(4):438–444, 2009a) was found to be appropriate for estimating friction coefficients in the laminar-to-turbulent transition regime.     

A spectral chart method for estimating the mean turbulent kinetic energy dissipation rate

We present an empirical but simple and practical spectral chart method for determining the mean turbulent kinetic energy dissipation rate $ \left\langle \varepsilon \right\rangle $ in a variety of turbulent flows. The method relies on the validity of the first similarity hypothesis of Kolmogorov (C R (Doklady) Acad Sci R R SS, NS 30:301–305, 1941) (or K41) which implies that spectra of velocity fluctuations scale on the kinematic viscosity ν and $ \left\langle \varepsilon \right\rangle $ at large Reynolds numbers. However, the evidence, based on the DNS spectra, points to this scaling being also valid at small Reynolds numbers, provided effects due to inhomogeneities in the flow are negligible. The methods avoid the difficulty associated with estimating time or spatial derivatives of the velocity fluctuations. It also avoids using the second hypothesis of K41, which implies the existence of a ?5/3 inertial subrange only when the Taylor microscale Reynods number R _λ is sufficiently large. The method is in fact applied to the lower wavenumber end of the dissipative range thus avoiding most of the problems due to inadequate spatial resolution of the velocity sensors and noise associated with the higher wavenumber end of this range.The use of spectral data (30?≤?R _λ?≤?400) in both passive and active grid turbulence, a turbulent mixing layer and the turbulent wake of a circular cylinder indicates that the method is robust and should lead to reliable estimates of $ \left\langle \varepsilon \right\rangle $ in flows or flow regions where the first similarity hypothesis should hold; this would exclude, for example, the region near a wall.     

Numerical Simulation of Single-Stream Jets from a Serrated Nozzle

Hybrid large-eddy type simulations for cold jet flows from a serrated nozzle are performed at an acoustic Mach number Ma _ac ?=?0.9 and Re?=?1.03×10⁶. Since the solver being used tends towards having dissipative qualities, the subgrid scale (SGS) model is omitted, giving a numerical type LES (NLES) or implicit LES (ILES) reminiscent procedure. To overcome near wall streak resolution problems a near wall RANS (Reynolds averaged Navier-Stokes) model is smoothly blended to the LES making a hybrid RANS-ILES. The geometric complexity of the serrated nozzle is fully considered without simplification or emulation. An improved but still modest hexahedral multi-block grid with circa 20 million grid points (with respect to 12.5 million in Xia et al., Int J Heat Fluid Flow 30:1067–1079, 2009) is used. Despite the modest grid size, encouraging and improved results are obtained. Directly resolved mean and second-order fluctuating quantities along the jet centerline and in the jet shear layer compare favorably with measurements. The radiated far-field sound predicted using the Ffowcs Williams and Hawkings (FW-H) surface integral method shows good agreement with the measurements in directivity and sound spectra.     

Analytical Validation of a Continuum Model for Epitaxial Growth with Elasticity on Vicinal Surfaces

Within the context of heteroepitaxial growth of a film onto a substrate, terraces and steps self-organize according to misfit elasticity forces. Discrete models of this behavior were developed by Duport et al. (J Phys I 5:1317–1350, 1995) and Tersoff et al. (Phys Rev Lett 75:2730–2733, 1995). A continuum limit of these was in turn derived by Xiang (SIAM J Appl Math 63:241–258, 2002) (see also the work of Xiang and Weinan Phys Rev B 69:035409-1–035409-16, 2004; Xu and Xiang SIAM J Appl Math 69:1393–1414, 2009). In this paper we formulate a notion of weak solution to Xiang’s continuum model in terms of a variational inequality that is satisfied by strong solutions. Then we prove the existence of a weak solution.     

On the Rigorous Derivation of the 3D Cubic Nonlinear Schrödinger Equation with a Quadratic Trap

We consider the dynamics of the three-dimensional N-body Schrödinger equation in the presence of a quadratic trap. We assume the pair interaction potential is N ^3β-1 V(N ^β x). We justify the mean-field approximation and offer a rigorous derivation of the three-dimensional cubic nonlinear Schrödinger equation (NLS) with a quadratic trap. We establish the space-time bound conjectured by Klainerman and Machedon (Commun Math Phys 279:169–185, 2008) for ${\beta \in (0, 2/7]}$ by adapting and simplifying an argument in Chen and Pavlovi? (Annales Henri Poincaré, 2013) which solves the problem for ${\beta \in (0, 1/4)}$ in the absence of a trap.     

On the Formulation of Continuum Thermodynamic Models for Solids as General Equations for Non-equilibrium Reversible-Irreversible Coupling

The purpose of this work is the comparison of some aspects of the formulation of material models in the context of continuum thermodynamics (e.g., ?ilhavý in The mechanics and thermodynamics of continuous media, Springer, Berlin, 1997) with their formulation in the form of a General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC: e.g., Grmela and Öttinger in Phys. Rev. E 56: 6620–6632, 1997; Öttinger and Grmela in Phys. Rev. E 56: 6633–6655, 1997; Öttinger in Beyond equilibrium thermodynamics, Wiley, New York, 2005; Grmela in J. Non-Newton. Fluid Mech. 165: 980–998, 2010). A GENERIC represents a generalization of the Ginzburg-Landau model for the approach of non-equilibrium systems to thermodynamic equilibrium. Originally developed to formulate non-equilibrium thermodynamic models for complex fluids, it has recently been applied to anisotropic inelastic solids in a Eulerian setting (Hütter and Tervoort in J. Non-Newton. Fluid Mech. 152: 45–52, 2008; 53–65, 2008; Adv. Appl. Mech. 42: 254–317, 2009) as well as to damage mechanics (Hütter and Tervoort in Acta Mech. 201: 297–312, 2008). In the current work, attention is focused for simplicity on the case of thermoelastic solids with heat conduction and viscosity in a Lagrangian setting (e.g., ?ilhavý in The mechanics and thermodynamics of continuous media, Springer, Berlin, 1997, Chaps. 9–12). In the process, the relation of the two formulations to each other is investigated in detail. A particular point in this regard is the concept of dissipation and its model representation in both contexts.     

Numerical study on supersonic mixing and combustion with hydrogen injection upstream of a cavity flameholder

Characteristics of supersonic mixing and combustion with hydrogen injection upstream of a cavity flameholder are investigated numerically using hybrid RANS/LES (Reynolds-Averaged Navier–Stokes/Large-Eddy Simulation) method. Two types of inflow boundary layer are considered. One is a laminar-like boundary layer with inflow thickness of $\delta_{\inf } = 0.0$ and the other is a turbulent boundary layer with inflow thickness of $\delta_{\inf } = 2.5\,{\text{mm}}$ . The hybrid RANS/LES method acts as a DES (Detached Eddy Simulation) model for the laminar-like inflow condition and a wall-modeled LES for the turbulent inflow condition where the recycling/rescaling method is adopted. Although the turbulent inflow seems to have just minor influences on the supersonic cavity flow without fuel injection, its effects on the mixing and combustion processes are great. It is found that the unsteady turbulent structures in upstream incoming boundary layer interact with the injection jet, resulting in fluctuations of the upstream recirculation region and bow shock, and induce quick dispersion of the hydrogen fuel jet, which enhances the mixing as well as subsequent combustion.     

Analysis of Algebraic Closures of the Mean Scalar Dissipation Rate of the Progress Variable Applied to Stagnating Turbulent Flames

In the turbulent premixed reactive flows considered in this study, i.e. large Damköhler and Reynolds numbers, the flamelet regime of turbulent combustion applies and the scalar dissipation rate and mean reaction rate are inter related. In this situation various algebraic models for the mean chemical rate that are obtained from an equilibrium of the dominant terms of the transport equation for the scalar dissipation rate, are evaluated through their application to flames stabilized in a turbulent stagnating flow. An asymptotic analysis is first performed and results obtained through the resulting one-dimensional calculation are compared with the experimental data of Li et al. (Proc Combust Inst 25:1207–1214, 1994). Eventually, three-dimensional CFD calculations including suited algebraic closures to represent the turbulent transport terms are carried out. Results are satisfactorily compared to the experimental data of Cho et al. (Proc Combust Inst 22:739–745, 1988). As a first outcome, the analysis confirms the interest and the relevance of the corresponding algebraic closures to deal with turbulent premixed combustion in such conditions. In the search of a satisfactory representation of such premixed impinging flames, the computational results also clearly emphasize the strong intertwinment that exits between the mean reaction rate, i.e. scalar dissipation rate or micro-mixing taking place at the smallest scale of the reactive flowfield, and the Reynolds fluxes modelling, i.e. turbulent macro-mixing.     

Measurements of an unsteady liquid metal flow during spin-up driven by a rotating magnetic field

A cylindrical cavity with an aspect ratio of unity is filled with liquid metal and suddenly exposed to an azimuthal body force generated by a rotating magnetic field (RMF). This experimental study is concerned with the secondary meridional flow during the time, if the fluid spins up from rest. Vertical profiles of the axial velocity have been measured by means of the ultrasound Doppler velocimetry. The flow measurements confirm the spin-up concept by Ungarish (J Fluid Mech 347:105–118, 1997) and the continuative study by Nikrityuk et al. (Phys Fluids 17:067101, 2005) who suggested the existence of two stages during the RMF-driven spin-up, in particular the so-called initial adjustment phase followed by an inertial phase which is dominated by inertial oscillations of the secondary flow. Evolving instabilities of the double-vortex structure of the secondary flow have been detected at a Taylor number of 1.24 × 10⁵ verifying the predictions of Grants and Gerbeth (J Fluid Mech 463:229–240, 2002). Perturbations in form of Taylor–Görtler vortices have been observed just above the instability threshold.     

Testing an Improved Artificial Viscosity Advection Scheme to Minimise Wiggles in Large Eddy Simulation of Buoyant Jet in Crossflow

Obtaining accurate and wiggle free Large Eddy Simulation (LES) results of high Re configurations with obstacles is a challenge, especially when the resolution is moderate. This study focusses on LES of buoyant jet in crossflow (JICF). The zone in front of the jet is sensitive for wiggle formation because the jet acts as an obstacle. Only 10 grid cells over the diameter of the jet at outflow are used in order to be able to simulate very large mixing areas with limited CPU power. The resolution increases rapidly to 30–50 cells over the diameter of the bend over jet further downstream. This study tests an artificial viscosity advection scheme with sixth order dissipation, called AV6, which dissipates wiggles adequately with almost no dissipation on physical relevant scales. This desirable behaviour is demonstrated by a Fourier analysis of the Advection-Diffusion equation and turbulent flow simulations. AV6 is a mix of, and improvement over, the artificial viscosity scheme of Jameson et al. (1981) with fourth order dissipation, here called AV4, and a fifth order upwind scheme (UPW5) of Wicker and Skamarock (Mon Weather Rev 130:2088, 2002). AV6 is a robust, simple and easy to implement advection scheme and the total computational time of a simulation with AV6 is only a few percent more than with the second order central scheme (CDS2). Three realistic turbulent flow problems, relevant for buoyant JICF, are used to compare the performance of AV6 with CDS2, AV4 and UPW5 with each other and with experiments. Different grid resolutions and sub-grid scale models are used. The three test cases are a non-buoyant JICF, a buoyant jet in weak coflow, and a buoyant JICF. Of all tested advection schemes, AV6 produces best results and is preferred for LES of buoyant JICF.     

The Two-Dimensional Euler Equations on Singular Domains

We establish the existence of global weak solutions of the two-dimensional incompressible Euler equations for a large class of non-smooth open sets. Loosely, these open sets are the complements (in a simply connected domain) of a finite number of obstacles with positive Sobolev capacity. Existence of weak solutions with L ^p vorticity is deduced from a property of domain continuity for the Euler equations that relates to the so-called γ-convergence of open sets. Our results complete those obtained for convex domains in Taylor (Progress in Nonlinear Differential Equations and their Applications, Vol. 42, 2000), or for domains with asymptotically small holes (Iftimie et al. in Commun Partial Differ Equ 28(1–2), 349–379, 2003; Lopes Filho in SIAM J Math Anal 39(2), 422–436, 2007).