Sharp vorticity gradients in two-dimensional turbulence and the energy spectrum

Formation of sharp vorticity gradients in two-dimensional (2D) hydrodynamic turbulence and their influence on the turbulent spectra are considered. The analog of the vortex line representation as a transformation to the curvilinear system of coordinates moving together with the di-vorticity lines is developed and compressibility of this mapping appears as the main reason for the formation of the sharp vorticity gradients at high Reynolds numbers. In the case of strong anisotropy the sharp vorticity gradients can generate spectra which fall off as k ^?3 at large k, which appear to take the same form as the Kraichnan spectrum for the enstrophy cascade. For turbulence with weak anisotropy the k dependence of the spectrum due to the sharp gradients coincides with the Saffman spectrum: E(k) ~ k ^?4. Numerical investigations of decaying turbulence reveal exponential growth of di-vorticity with a spatial distributed along straight lines. Thus, indicating strong anisotropy and accordingly the spectrum is close to the k ^?3-spectrum.     

An application of the k-ɛ model with variable Prandtl number to heat transfer computations in air flows

The two-equation `low Reynolds number' k-? model of turbulence with a set of universal constants suggested by Launder and Sharma is modified in the present paper. The variability of the turbulent Prandtl number Pr_t in the energy equation is assumed along with a change of a constant in the dissipation term of the turbulent kinetic energy equation. The turbulent heat transfer is computed for an air flow in a circular pipe for the Reynolds number within the range of 10⁴?4. The modification considerably improves the agreement between the numerical results and the experiment data published in the available literature.     

The Littlewood–Paley Spectrum in Two-Dimensional Turbulence

We introduce the Littlewood–Paley energy spectrum and prove that k ⁻³ is an upper bound for it in two dimensions. We also show that certain details of the spectrum of the driving forces can be recovered from the energy spectrum. Received 5 October 1996 and accepted 7 March 1997     

Decaying turbulence in a passive mean scalar gradient

We consider the generation of passive scalar fluctuations by decaying isotropic turbulence in the presence of a uniform mean scalar gradient. At high Reynolds numbers, two distinct similarity states may be established depending on the form of the energy spectrum at low wavenumber magnitude (k). In the first similarity state characterized by a low wavenumber magnitude energy spectrum proportional tok ², the mean-square scalar fluctuation grows liket ^4/5, while in the second similarity state characterized by a spectrum proportional tok ⁴, the mean-square scalar fluctuation grows approximately liket ^4/7. These two high Reynolds number asymptotic similarity states have been subsequently confirmed by large-eddy numerical simulations. As a consequence of the decreasing flow Reynolds number as the turbulence decays, these similarity states do not continue indefinitely. At very long times, a final period of decay of the turbulence occurs, and in this final period, the mean-square scalar fluctuation in the first state continues to grow liket ^1/2, while that in the second state ultimately decays liket ^–1/2.     

Simulations of Three-Dimensional Turbulent Mixing for Schmidt Numbers of the Order 1000

We report basic results from new numerical simulations of passive scalar mixing at Schmidt numbers (Sc) of the order of 1000 in isotropic turbulence. The required high grid-resolution is made possible by simulating turbulence at very low Reynolds numbers, which nevertheless possesses universality in dissipative scales of motion. The results obtained are qualitatively consistent with those based on another study (Yeung et al., Phys. Fluids 14 (2002) 4178-4191) with a less extended Schmidt number range and a higher Reynolds number. In the stationary state maintained by a uniform mean scalar gradient, the scalar variance increases slightly with Sc but scalar dissipation is nearly constant. As the Schmidt number increases, there is an increasing trend towards k ^?1 scaling predicted by Batchelor (Batchelor, J. Fluid Mech. 5 (1959) 113-133) for the viscous-convective range of the scalar spectrum; the scalar gradient skewness approaches zero; and the intermittency measured by the scalar gradient flatness approaches its asymptotic state. However, the value of Sc needed for the asymptotic behavior to emerge appears to increase with decreasing Reynolds number of the turbulence. In the viscous-diffusive range, the scalar spectrum is in better agreement with Kraichnan's (Kraichnan., Phys. Fluids 11 (1968) 945-953) result than with Batchelor's.     

A modified Lin equation for the energy balance in isotropic turbulence

At sufficiently large Reynolds numbers, turbulence is expected to exhibit scale-invariance in an intermediate (“inertial”) range of wavenumbers, as shown by power law behavior of the energy spectrum and also by a constant rate of energy transfer through wavenumber. However, there is an apparent contradiction between the definition of the energy flux (i.e., the integral of the transfer spectrum) and the observed behavior of the transfer spectrum itself. This is because the transfer spectrum T(k) is invariably found to have a zero-crossing at a single point (at k = k_*), implying that the corresponding energy flux cannot have an extended plateau but must instead have a maximum value at k = k_*. This behavior was formulated as a paradox and resolved by the introduction of filtered/partitioned transfer spectra, which exploited the symmetries of the triadic interactions (J. Phys. A: Math. Theor., 2008). In this paper we consider the more general implications of that procedure for the spectral energy balance equation, also known as the Lin equation. It is argued that the resulting modified Lin equations (and their corresponding Navier–Stokes equations) offer a new starting point for both numerical and theoretical methods, which may lead to a better understanding of the underlying energy transfer processes in turbulence. In particular the filtered partitioned transfer spectra could provide a basis for a hybrid approach to the statistical closure problem, with the different spectra being tackled using different methods.     

Direct numerical simulation of decaying two‐dimensional turbulence in a no‐slip square box using smoothed particle hydrodynamics

This paper explores the application of SPH to a DNS of decaying turbulence in a two‐dimensional no‐slip wall‐bounded domain. In this bounded domain, the inverse energy cascade, and a net torque exerted by the boundary, results in a spontaneous spin‐up of the fluid, leading to a typical end state of a large monopole vortex that fills the domain. The SPH simulations were compared against published results using a high‐accuracy pseudo‐spectral code. Ensemble averages of the kinetic energy, enstrophy and average vortex wavenumber compared well against the pseudo‐spectral results, as did the evolution of the total angular momentum of the fluid. However, although the pseudo‐spectral results emphasised the importance of the no‐slip boundaries as generators of long‐lived coherent vortices in the flow, no such generation was seen in the SPH results. Vorticity filaments produced at the boundary were always dissipated by the flow shortly after separating from the boundary layer. The kinetic energy spectrum of the SPH results was calculated using an SPH Fourier transform that operates directly on the disordered particles. The ensemble kinetic energy spectrum showed the expected k^?3 scaling over most of the inertial range. However, the spectrum flattened at smaller length scales (initially less than 7.5 particle spacings and growing in size over time), indicating an excess of small‐scale kinetic energy.Copyright © 2011 John Wiley & Sons, Ltd.     

Reynolds number, thickness and camber effects on flapping airfoil propulsion

The effect of varying airfoil thickness and camber on plunging and combined pitching and plunging airfoil propulsion at Reynolds number Re=200, 2000, 20 000 and 2×10⁶ was studied by numerical simulations for fully laminar and fully turbulent flow regimes. The thickness study was performed on 2-D NACA symmetric airfoils with 6-50% thick sections undergoing pure plunging motion at reduced frequency k=2 and amplitudes h=0.25 and 0.5, and for combined pitching and plunging motion at k=2, h=0.5, phase ?=90°, pitch angle θ_o=15° and 30° and the pitch axis was located at 1/3 of chord from leading edge. At Re=200 for motions where positive thrust is generated, thin airfoils outperform thick airfoils. At higher Re significant gains could be achieved both in thrust generation and propulsive efficiency by using a thicker airfoil section for plunging and combined motion with low pitch amplitude. The camber study was performed on 2-D NACA airfoils with varying camber locations undergoing pure plunging motion at k=2, h=0.5 and Re=20 000. Little variation in thrust performance was found with camber. The underlying physics behind the alteration in propulsive performance between low and high Reynolds numbers has been explored by comparing viscous Navier-Stokes and inviscid panel method results. The role of leading edge vortices was found to be key to the observed performance variation.     

Global Attractor for a Nonlinear Oscillator Coupled to the Klein–Gordon Field

The long-time asymptotics is analyzed for all finite energy solutions to a model\(\mathbf{U}(1)\)-invariant nonlinear Klein–Gordon equation in one dimension, with the nonlinearity concentrated at a single point: each finite energy solution converges as t→ ± ∞ to the set of all “nonlinear eigenfunctions” of the form ψ(x)e^{?iω t}. The global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation.We justify this mechanism by the following novel strategy based on inflation of spectrum by the nonlinearity. We show that any omega-limit trajectory has the time spectrum in the spectral gap [ ? m,m] and satisfies the original equation. This equation implies the key spectral inclusion for spectrum of the nonlinear term. Then the application of the Titchmarsh convolution theorem reduces the spectrum of each omega-limit trajectory to a single harmonic \(\omega\in[-m,m]\).The research is inspired by Bohr’s postulate on quantum transitions and Schrödinger’s identification of the quantum stationary states to the nonlinear eigenfunctions of the coupled\(\mathbf{U}(1)\)-invariant Maxwell–Schrödinger and Maxwell–Dirac equations.     

A Variational Model for Dislocations in the Line Tension Limit

We study the interaction of a singularly-perturbed multiwell energy (with an anisotropic nonlocal regularizing term of H ^1/2 type) and a pinning condition. This functional arises in a phase field model for dislocations which was recently proposed by Koslowski, Cuitiño and Ortiz, but it is also of broader mathematical interest. In the context of the dislocation model we identify the Γ-limit of the energy in all scaling regimes for the number N _? of obstacles. The most interesting regime is N _? ≈|ln ?|/?, where ? is a nondimensional length scale related to the size of the crystal lattice. In this case the limiting model is of line tension type. One important feature of our model is that the set of energy wells is periodic, and hence not compact. Thus a key ingredient in the proof is a compactness estimate (up to a single translation) for finite energy sequences, which generalizes earlier results from Alberti, Bouchitté and Seppecher for the two-well problem with a H ^1/2 regularization.     

A continuum description of partially coherent interfaces

A continuum model of a two-phase crystal-crystal system is constructed in which the structure of the interface between the phases is determined by energy minimization, rather than by being specified a priori. The interfacial structure is parameterized by a variable? corresponding to the jump in the surface deformation gradient (or strain) at the interface, so that coherence is defined locally by the condition? = 0. The energy of the system is taken to be the sum of the bulk and interfacial energies, where the interfacial energy densityf ^xs depends on?. In order to explore how the equilibrium interfacial structure depends on the functionf ^xs (?), a model system consisting of an elastic film on a rigid substrate is studied, and the interfacial energy density is taken to be nonconvex with a sharp minimum associated with coherence. In this case, it can be shown that the energy of the system is driven to its infimum by separating the interface into coherent and incoherent regions, which may be viewed as a continuum analog to a partially coherent interface. Further, this solution only appears above a certain critical thickness of the film, in agreement with misfit dislocation models of partially coherent interfaces.     

Uniqueness of Specific Interfacial Area–Capillary Pressure–Saturation Relationship Under Non-Equilibrium Conditions in Two-Phase Porous Media Flow

The capillary pressure?Csaturation (P ^c?CS ^w) relationship is one of the central constitutive relationships used in two-phase flow simulations. There are two major concerns regarding this relation. These concerns are partially studied in a hypothetical porous medium using a dynamic pore-network model called DYPOSIT, which has been employed and extended for this study: (a) P ^c?CS ^w relationship is measured empirically under equilibrium conditions. It is then used in Darcy-based simulations for all dynamic conditions. This is only valid if there is a guarantee that this relationship is unique for a given flow process (drainage or imbibition) independent of dynamic conditions; (b) It is also known that P ^c?CS ^w relationship is flow process dependent. Depending on drainage and imbibition, different curves can be achieved, which are referred to as ??hysteresis??. A thermodynamically derived theory (Hassanizadeh and Gray, Water Resour Res 29: 3389?C3904, 1993a) suggests that, by introducing a new state variable, called the specific interfacial area (a ^nw, defined as the ratio of fluid?Cfluid interfacial area to the total volume of the domain), it is possible to define a unique relation between capillary pressure, saturation, and interfacial area. This study investigates these two aspects of capillary pressure?Csaturation relationship using a dynamic pore-network model. The simulation results imply that P ^c?CS ^w relation not only depends on flow process (drainage and imbibition) but also on dynamic conditions for a given flow process. Moreover, this study attempts to obtain the first preliminary insights into the global functionality of capillary pressure?Csaturation?Cinterfacial area relationship under equilibrium and non-equilibrium conditions and the uniqueness of P ^c?CS ^w?Ca ^nw relationship.     

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.     

Kinetics of the interaction of a liquid diffusing into another one as studied by reflection spectroscopy (ATR)

An approach is offered for determining the reaction rate constant (k) between two liquid substances, the one penetrating into the other. The procedure is based on the experimental measurement of the diffusion coefficient (D). As model reaction the isotopic exchange process in the ketone octanone-2 molecule is chosen, whose active hydrogen atoms undergo deuteration by the strong base trioctyl-methyl-ammonium deuteroxide (TOMAOD). The diffusion coefficient of the penetrating TOMA-OD, when this reaction takes place, and the rate constant of the latter are determined by an attenuated total reflection (ATR) spectroscopic method [1] on the grounds of appropriate mathematical modeling [2]. The application of this simple and comparatively rapid approach results in thek-value of 1.04 × 10^?2 sec^?1 for the monomolecular interaction mentioned above. The reasons for such an assumption arise from the only initial process stage treatment, where the reactant (TOMA-OD) particles enter the substrate (the ketone) surrounded by an excess of its own molecules. This further allows an analytical solution of the resulting diffusion problem.     

Experimental and Theoretical Sensitivity Analysis of Turbulent Flow Past a Square Cylinder

The Siemens SGT-800 3^rd generation DLE burner fitted to an atmospheric combustion rig has been numerically investigated. Pure methane and methane enriched by 80 vol% hydrogen flames have been considered. A URANS (Unsteady Reynolds Averaged Navier-Stokes) approach was used in this study along with the k ? ω SST and the k ? ω SST-SAS models for the turbulence transport. The chemistry is coupled to the turbulent flow simulations by the use of a laminar flamelet library combined with a presumed PDF. The effect of the mesh density in the mixing and the flame region and the effect of the turbulence model and reaction rate model constant are first investigated for the methane/air flame case. The results from the k ? ω SST-SAS along with flamelet libraries are shown to be in excellent agreement with experimental data, whereas the k ? ω SST model is too dissipative and cannot capture the unsteady motion of the flame. The k ? ω SST-SAS model is used for simulation of the 80 vol% hydrogen enriched flame case without further adjusting the model constants. The global features of the hydrogen enrichment are very well captured in the simulations using the SST-SAS model. With the hydrogen enrichment the time averaged flame front location moves upstream towards the burner exit nozzle. The results are consistent with the experimental observations. The model captures the three dominant low frequency unsteady motion observed in the experiments, indicating that the URANS/LES hybrid model indeed is capable of capturing complex, time dependent, features such as an interaction between a PVC and the flame front.     

Gross-Pitaevskii Equation as the Mean Field Limit of Weakly Coupled Bosons

We consider the dynamics of N boson systems interacting through a pair potential N ^?1 V _a (x _i ?x _j ) where V _a (x)=a ^?3 V(x/a). We denote the solution to the N-particle Schrödinger equation by Ψ _{N, t} . Recall that the Gross-Pitaevskii (GP) equation is a nonlinear Schrödinger equation and the GP hierarchy is an infinite BBGKY hierarchy of equations so that if u _t solves the GP equation, then the family of k-particle density matrices solves the GP hierarchy. Under the assumption that a=N ^?? for 0N→∞ the limit points of the k-particle density matrices of Ψ _{N, t} are solutions of the GP hierarchy with the coupling constant in the nonlinear term of the GP equation given by ∫V(x)dx. The uniqueness of the solutions of this hierarchy remains an open question.     

Flow past a cylinder: shear layer instability and drag crisis

Flow past a circular cylinder for Re=100 to 10⁷ is studied numerically by solving the unsteady incompressible two‐dimensional Navier–Stokes equations via a stabilized finite element formulation. It is well known that beyond Re ～ 200 the flow develops significant three‐dimensional features. Therefore, two‐dimensional computations are expected to fall well short of predicting the flow accurately at high Re. It is fairly well accepted that the shear layer instability is primarily a two‐dimensional phenomenon. The frequency of the shear layer vortices, from the present computations, agree quite well with the Re^0.67 variation observed by other researchers from experimental measurements. The main objective of this paper is to investigate a possible relationship between the drag crisis (sudden loss of drag at Re ～ 2 × 10⁵) and the instability of the separated shear layer. As Re is increased the transition point of shear layer, beyond which it is unstable, moves upstream. At the critical Reynolds number the transition point is located very close to the point of flow separation. As a result, the shear layer eddies cause mixing of the flow in the boundary layer. This energizes the boundary layer and leads to its reattachment. The delay in flow separation is associated with narrowing of wake, increase in Reynolds shear stress near the shoulder of the cylinder and a significant reduction in the drag and base suction coefficients. The spatial and temporal power spectra for the kinetic energy of the Re=10⁶ flow are computed. As in two‐dimensional isotropic turbulence, E(k) varies as k^?5/3 for wavenumbers higher than energy injection scale and as k^?3 for lower wavenumbers. The present computations suggest that the shear layer vortices play a major role in the transition of boundary layer from laminar to turbulent state. Copyright © 2004 John Wiley & Sons, Ltd.     

Numerical investigations into the asymmetric effects on the aerodynamic response of a pitching airfoil

The effects of asymmetric sinusoidal motion on pitching airfoil aerodynamics were studied by numerical simulations for 2-D flow around a NACA0012 airfoil at Re=1.35×10⁵. Various unsteady parameters (amplitude of oscillation, d; reduced frequency, k) were applied to investigate the effect of asymmetry parameter S on the instantaneous force coefficients and flow patterns. The results reveal that S has a noticeable effect on the aerodynamic performance, as it affects the instantaneous force coefficient, maximum lift and drag coefficient, hysteresis loops and the flow structures.     

Computational and experimental analysis of supersonic air ejector: Turbulence modeling and assessment of 3D effects

Numerical and experimental analyses are performed on a supersonic air ejector to evaluate the effectiveness of commonly-used computational techniques when predicting ejector flow characteristics. Three series of experimental curves at different operating conditions are compared with 2D and 3D simulations using RANS, steady, wall-resolved models. Four different turbulence models are tested: k–ε, k–ε realizable, k–ω SST, and the stress–ω Reynolds Stress Model. An extensive analysis is performed to interpret the differences between numerical and experimental results. The results show that while differences between turbulence models are typically small with respect to the prediction of global parameters such as ejector inlet mass flow rates and Mass Entrainment Ratio (MER), the k–ω SST model generally performs best whereas ε-based models are more accurate at low motive pressures. Good agreement is found across all 2D and 3D models at on-design conditions. However, prediction at off-design conditions is only acceptable with 3D models, making 3D simulations mandatory to correctly predict the critical pressure and achieve reasonable results at off-design conditions. This may partly depend on the specific geometry under consideration, which in the present study has a rectangular cross section with low aspect ratio.