On the construction of discrete filters for symmetry-preserving regularization models

Since direct numerical simulations cannot be computed at high Reynolds numbers, a dynamically less complex formulation is sought. In the quest for such a formulation, we consider regularizations of the convective term that preserve the symmetry and conservation properties exactly. This requirement yielded a novel class of regularizations [Verstappen R. On restraining the production of small scales of motion in a turbulent channel flow. Comput Fluids 2008;37:887–97.] that restrains the convective production of smaller and smaller scales of motion in an unconditionally stable manner, meaning that the velocity cannot blow up in the energy-norm (in 2D also: enstrophy-norm). The numerical algorithm used to solve the governing equations must preserve the symmetry and conservation properties too. To do so, one of the most critical issues is the discrete filtering. The method requires a list of properties that, in general, is not preserved by classical filters for LES unless they are imposed a posteriori. In the present paper, we propose a novel class of discrete filters that preserves such properties per se. They are based on polynomial functions of the discrete diffusive operator, , with the general form . Then, the coefficients, d_m, follow from the requirement that, at the smallest grid scale k_c, the amount by which the interactions between the wavevector-triples (k_c, k_c − q, q) are damped must become virtually independent of the qth Fourier-mode. This allows an optimal control of the subtle balance between convection and diffusion at the smallest grid scale to stop the vortex-stretching. Finally, the resulting filters are successfully tested for the Burgers’ equation.     

Parameter-free symmetry-preserving regularization modeling of a turbulent differentially heated cavity

Since direct numerical simulations of buoyancy driven flows cannot be computed at high Rayleigh numbers, a dynamically less complex mathematical formulation is sought. In the quest for such a formulation, we consider regularizations (smooth approximations) of the non-linearity: the convective term is altered to reduce the production of small scales of motion by means of vortex stretching. In doing so, we propose to preserve the symmetry and conservation properties of the convective terms exactly. This requirement yielded a novel class of regularizations [Comput Fluids 2008;37:887] that restrain the convective production of smaller and smaller scales of motion in an unconditionally stable manner, meaning that the velocity cannot blow up in the energy-norm (in 2D also: enstrophy-norm). The numerical algorithm used to solve the governing equations preserves the symmetry and conservation properties too. In the present work, a criterion to determine dynamically the regularization parameter (local filter length) is proposed: it is based on the requirement that the vortex stretching must stop at the scale set by the grid. Therefore, the proposed method constitutes a parameter-free turbulence model. The resulting regularization method is tested for a 3D natural convection flow in an air-filled (Pr = 0.71) differentially heated cavity of height aspect ratio 4. Direct comparison with DNS results at Rayleigh number 6.4 × 10⁸ ? Ra ? 10¹¹ shows fairly good agreement even for very coarse grids. Finally, the robustness of the method is tested by performing simulations with Ra up to 10¹⁷. A 2/7 scaling law of Nusselt number has been obtained for the investigated range of Ra.     

The impacts of the ALE and hydrostatic-pressure approaches on the energy budget of unsteady free-surface flows

This paper focuses on the energy budget in the calculation of unsteady free-surface flows on moving grids with and without using the ‘arbitrary Lagrangian-Eulerian’ (ALE) formulation or hydrostatic-pressure assumption. The numerical tool is an in-house general-purpose solver for the unsteady, incompressible and homogeneous Navier-Stokes equations in a Cartesian domain. An explicit fractional-step method and co-located finite-volume method are used for the second-order accurate integrations in time and space. The test cases are nonlinear and linear irrotational standing waves, which allow to characterise the impacts of an ALE or Eulerian formulation with moving grids by comparison with the anticipated energy conservation. The study is also extended to viscous waves for varying wave-height-to-water-depth and basin aspect ratios. The Eulerian viewpoint produces marked overdamping as early as in the first wave period for the range of relative wave heights η₀/h > 0.01, where η₀ is the wave semi-amplitude and h is the undisturbed water depth. The hydrostatic calculations misrepresent the evolution of the potential and kinetic energies for h/L > 0.1, where L is the basin length, with spurious modes arising from different initial conditions.     

Direct numerical simulations of transverse and spanwise magnetic field effects on turbulent flow in a 2:1 aspect ratio rectangular duct

Magnetic fields are used extensively to direct liquid metal flows in material processing. Continuous casting of steel uses different configurations of magnetic fields to optimize turbulent flows in rectangular cross-sections to minimize defects in the solidified steel product. Realizing the importance of a magnetic field on turbulent flows in rectangular cross-sections, the present work is aimed at understanding the effect of a magnetic field on the turbulent metal flow at a nominal bulk Reynolds number of ∼5300 (based upon full duct height) (Re_τ = 170, based upon half duct height) and Hartmann numbers (based upon half duct height) of 0, 6.0 and 8.25 in a 2:1 aspect ratio rectangular duct. Direct numerical simulations in a non-MHD 2:1 aspect ratio duct followed by simulations with transverse and span-wise magnetic fields have been performed with 224 × 120 × 512 cells (∼13.7 million cells). The fractional step method with second order space and time discretization schemes has been used to solve the coupled Navier-Stokes-MHD equations. Instantaneous and time-averaged natures of the flow have been examined through distribution of velocities, various turbulence parameters and budget terms. Spanwise (horizontal) magnetic field reorganizes and suppresses secondary flows more strongly. Turbulence suppression and velocity flattening effects are stronger with transverse (vertical) magnetic field.     

Remote sensing of evapotranspiration and carbon uptake at Harvard Forest

A land surface vegetation index, defined as the difference of microwave land surface emissivity at 19 and 37 GHz, was calculated for a heavily forested area in north central Massachusetts. The microwave emissivity difference vegetation index (EDVI) was estimated from satellite SSM/I measurements at the defined wavelengths and used to estimate land surface turbulent fluxes. Narrowband visible and infrared measurements and broadband solar radiation observations were used in the EDVI retrievals and turbulent flux estimations. The EDVI values represent physical properties of crown vegetation such as vegetation water content of crown canopies. The collocated land surface turbulent and radiative fluxes were empirically linked together by the EDVI values. The EDVI values are statistically sensitive to evapotranspiration fractions (EF) with a correlation coefficient (R) greater than 0.79 under all-sky conditions. For clear skies, EDVI estimates exhibit a stronger relationship with EF than normalized difference vegetation index (NDVI). Furthermore, the products of EDVI and input energy (solar and photosynthetically active radiation) are statistically significantly correlated to evapotranspiration (R = 0.95) and CO₂ uptake flux (R = 0.74), respectively.     

Numerical investigation of supersonic flow for axisymmetric cones

An existing curvilinear finite-volume code with robust shock-capturing scheme was modified to allow for simulations of supersonic flow for axisymmetric cone geometries. It is shown how for an axisymmetric coordinate system the convective and viscous flux derivatives in the circumferential direction reduce to a y-momentum equation source term. The advantage of this approach over an axisymmetric code is that the governing equations and the discretization do not need to be changed. This paper provides a detailed derivation of the axisymmetric source terms from the full Navier-Stokes equations. Results are shown for a sharp and a blunt cone for approach flow Mach numbers of M = 3.5 and M = 7.99.     

Pharmacokinetics of insulin lispro in type 2 diabetes during closed-loop insulin delivery

Insulin pharmacokinetics is not well understood during continuous subcutaneous insulin infusion in type 2 diabetes (T2D). We analyzed data collected in 11 subjects with T2D [6 male, 9 white European and two of Indian ethnicity; age 59.7(12.1) years, BMI 30.1(3.9) kg/m², fasting C-peptide 1002.2(365.8) pmol/l, fasting plasma glucose 9.6(2.2) mmol/l, diabetes duration 8.0(6.2) years and HbA1c 8.3(0.8)%; mean(SD)] who underwent a 24-h study investigating closed-loop insulin delivery at the Wellcome Trust Clinical Research Facility, Cambridge, UK. Subcutaneous delivery of insulin lispro was modulated every 15 min according to a model predictive control algorithm. Two complementary insulin assays facilitated discrimination between exogenous (lispro) and endogenous plasma insulin concentrations measured every 15–60 min. Lispro pharmacokinetics was represented by a linear two-compartment model whilst parameters were estimated using a Bayesian approach applying a closed-form model solution. The time-to-peak of lispro absorption (t_max) was 109.6 (75.5–120.5) min [median (interquartile range)] and the metabolic clearance rate (MCR_I) 1.26 (0.87–1.56) × 10⁻² l/kg/min. MCR_I was negatively correlated with fasting C-peptide (r_s = −0.84; P = .001) and with fasting plasma insulin concentration (r_s = −0.79; P = .004). In conclusion, compartmental modelling adequately represents lispro kinetics during continuous subcutaneous insulin infusion in T2D. Fasting plasma C-peptide or fasting insulin may be predictive of lispro metabolic clearance rate in T2D but further investigations are warranted.     

Dynamic hybridization of MILES and RANS for predicting airfoil stall

Hybridization comprised of an algebraic turbulence model based on the Reynolds average Navier-Stokes (RANS) equations with a monotonically integrated large eddy simulation (MILES) is proposed to simulate transient fluid motion during separation and vortex shedding at high Reynolds numbers. The proposed hybridization utilizes the Baldwin-Lomax model with the Degani-Schiff modification as the RANS model in the near-wall region and a MILES far from the wall. Although the hybridization is assumed to be a MILES with wall modeling, the transition line between the RANS and the MILES modes is determined by the turbulent intensity that is dominated by the large eddies in the grid-scale. This hybrid model is applied to the flows past three different types of airfoils (NACA63₃-018, NACA63₁-012 and NACA64A-006) near stall, at a chord Reynolds number of Re = 5.8 × 10⁶. These airfoils are classified as trailing-edge-stall, leading-edge-stall and thin-airfoil-stall airfoils, respectively. The computed results are compared with wind tunnel experiments. The hybrid model successfully demonstrates accurate stall angle and surface pressure distribution predictions near the stall for each type of airfoil. The airfoil simulation results confirmed that the hybrid model provides a better prediction than the RANS model for unsteady turbulent flows with separation and vortex shedding simulations.     

Large-eddy simulations of film cooling flows

Large-eddy simulations (LES) of a jet in a cross-flow (JICF) problem are carried out to investigate the turbulent flow structure and the vortex dynamics in gas turbine blade film cooling. A turbulent flat plate boundary layer at a Reynolds number of Re_∞ = 400,000 interacts with a jet issued from a pipe. To study the effect of the jet inclination angle α on the flow field, two angles are chosen, the perpendicular injection at 90° and the streamwise inclined injection at 30°. For the normal injection case a small blowing ratio of the jet velocity to the cross-stream velocity R = 0.1 is examined. For the streamwise inclined injection case two blowing ratios R = 0.1 and R = 0.48 are investigated to check the impact of the jet velocity on the cooling performance. The time-dependent turbulent inflow information for the cross-flow is provided by a simultaneously performed LES of a spatially developing turbulent boundary layer. Whereas in the perpendicular injection case a rather large separation region is found at the leading edge of the jet hole, in the streamwise inclined injection cases no separation is observed. Compared with the normal injection case at the same blowing ratio, the streamwise inclination weakens the jet-cross-flow interaction significantly. Thus, the first appearance of the counter-rotating vortex pair (CVP) is shifted downstream and its strength is reduced. The increase of the blowing ratio leads to a stronger penetration of the jet into the cross-flow, resulting in a more upstream located and more pronounced CVP. Downstream of the jet exit the streamwise vortices are so large that besides the jet fluid also the cross-stream is partially entrained into this zone, which yields the worst cooling performance.     

Two-step simulation of slat noise

The flow field and the acoustic field of a high-lift configuration consisting of a slat and a main wing are numerically investigated by a hybrid method. In a first step, the unsteady flow field is computed via a large-eddy simulation (LES) and in a second step, the acoustic field is determined by solving the acoustic perturbation equations (APE). The mean flow field is compared to experimental findings followed by an investigation of the turbulent structures which are visualized by λ₂ contours. The analysis of the acoustic field shows that at the main wing trailing edge acoustic pressure fluctuations of approximately 5 kHz are generated. Correlations between the noise sources and the acoustic pressure identify the slat-gap region to be responsible for the mixture of broadband and tonal noise between 1 and 3 kHz. The decay of the pressure spectrum is found to be approximately f⁻² which is in agreement with the literature.     

GeM software package for computation of symmetries and conservation laws of differential equations

We present a recently developed Maple-based “GeM” software package for automated symmetry and conservation law analysis of systems of partial and ordinary differential equations (DE). The package contains a collection of powerful easy-to-use routines for mathematicians and applied researchers. A standard program that employs “GeM” routines for symmetry, adjoint symmetry or conservation law analysis of any given DE system occupies several lines of Maple code, and produces output in the canonical form. Classification of symmetries and conservation laws with respect to constitutive functions and parameters present in the given DE system is implemented. The “GeM” package is being successfully used in ongoing research. Run examples include classical and new results.

Program summary

Title of program: GeMCatalogue identifier: ADYK_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADYK_v1_0Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions: noneComputers: PC-compatible running Maple on MS Windows or Linux; SUN systems running Maple for Unix on OS SolarisOperating systems under which the program has been tested: Windows 2000, Windows XP, Linux, SolarisProgramming language used: Maple 9.5Memory required to execute with typical data: below 100 MegabytesNo. of lines in distributed program, including test data, etc.: 4939No. of bytes in distributed program, including test data, etc.: 166 906Distribution format: tar.gzNature of physical problem: Any physical model containing linear or nonlinear partial or ordinary differential equations.Method of solution: Symbolic computation of Lie, higher and approximate symmetries by Lie's algorithm. Symbolic computation of conservation laws and adjoint symmetries by using multipliers and Euler operator properties. High performance is achieved by using an efficient representation of the system under consideration and resulting symmetry/conservation law determining equations: all dependent variables and derivatives are represented as symbols rather than functions or expressions.Restrictions on the complexity of the problem: The GeM module routines are normally able to handle ODE/PDE systems of high orders (up to order seven and possibly higher), depending on the nature of the problem. Classification of symmetries/conservation laws with respect to one or more arbitrary constitutive functions of one or two arguments is normally accomplished successfully.Typical running time: 1-20 seconds for problems that do not involve classification; 5-1000 seconds for problems that involve classification, depending on complexity.     

Oriented growth of A2Te3 (A = Sb, Bi) films and their devices with enhanced thermoelectric performance

Oriented thermoelectric (TE) p-Sb₂Te₃ and n-Bi₂Te₃ thin films with special nanostructures have been synthesized by a simple vacuum thermal evaporation technique. The composition and microstructure of the films were studied by X-ray diffraction (XRD) and scanning electron microscopy (SEM), presenting a well preferential crystal growth with dense slender columnar grains grown perpendicular to the substrate, and energy dispersive X-ray spectrum (EDX) indicating the compositions distribution in the films. The electric transport properties, i.e., conductivity and Seebeck coefficient, and the thermal transportation of the oriented films show optimized properties. Prototype devices were built up by p and n elements in series and parallel circuits. The largest output power and cooling could be achieved in Sb₂Te₃ parallel device with P_max = 6.51 μW at ΔT = 106 K, and cooling of 4.1 K at 2 V. The 24-pair p-n couples series device could output maximum voltage of 313 mV at ΔT = 102 K. The power generation and the cooling of the devices show times enhanced TE performances than those consisting of common films. The results proved that introducing nanostructures into films is an effective choice to obtain high-efficient micro TE device.     

On wave harmonics of heated circular cylinders

A hybrid (FV + FE) acoustic damping method, which is investigated and optimized in terms of wave harmonics behavior of the method, is utilized at Ma = 0.01 for heated circular cylinders. Discretization of nonlinear convective terms is a modified approximate Riemann solver. Modification is realized through multiplication of dissipation term with an acoustic damping matrix. This is necessary to avoid drawbacks of standard algorithm and to improve accuracy of results, if low speed applications are concerned. Later, harmonics of velocity and temperature fields behind a heated circular cylinder are investigated numerically for moderate Reynolds numbers between 70 and 110. A parametric study of the first harmonics is carried out precisely by increasing temperature ratios, T^∗ = T_wall/T_∞ from 1.03 to 1.8. Results agree well for high temperature ratios, T^∗ = 1.5, 1.8 and Re ? 100, with the key issues stated in the experimental work of Ezersky et al. [Ezersky AB, Lecordier JC, Paranthoën P, Soustov PL, Structure of vortices in a Karman street behind a heated cylinder. Phys Rev E 2000;61:2107]. Moreover, it is found that decrease in frequency of oscillations can be explained as heated cylinders have larger vortex formation region as a result of heat flux from walls in comparison to non-heated cylinders. At lower temperature ratios, T^∗ = 1.03, 1.1, temperature can be taken as passive scalar field.     

Embedding meshes into twisted-cubes

The n-dimensional twisted-cube, TN_n, is a variation of the hypercube. In this paper, we study embedding of meshes into TN_n. We prove three major results in this paper: (1) For any integer n ? 1, a 2 × 2ⁿ⁻¹ mesh can be embedded into TN_n with dilation 1 and expansion 1. (2) For any integer n ? 4, an m × k(m ? 3, k ? 3) mesh cannot be embedded into TN_n with dilation 1. (3) For any integer n ? 4, two node-disjoint 4 × 2ⁿ⁻³ meshes can be embedded into TN_n with dilation 2 and expansion 1.     

Asymptotic behavior and blow up of solutions for semilinear parabolic equations at critical energy level

In this paper we study the initial boundary value problem of semilinear parabolic equations with semilinear term f(u). By using the family of potential wells method we prove that if f(u) satisfies some conditions, J(u₀) ≤ d and I(u₀) > 0, then the solution decays to zero exponentially as t → ∞. On the other hand, if J(u₀) ≤ d, I(u₀) < 0, then the solution blows up in finite time.     

Direct numerical simulation of turbulent channel flows using a stabilized finite element method

Direct numerical simulations (DNS) of incompressible turbulent channel flows at Re_τ = 180 and 395 (i.e., Reynolds number, based on the friction velocity and channel half-width) were performed using a stabilized finite element method (FEM). These simulations have been motivated by the fact that the use of stabilized finite element methods for DNS and LES is fairly recent and thus the question of how accurately these methods capture the wide range of scales in a turbulent flow remains open. To help address this question, we present converged results of turbulent channel flows under statistical equilibrium in terms of mean velocity, mean shear stresses, root mean square velocity fluctuations, autocorrelation coefficients, one-dimensional energy spectra and balances of the transport equation for turbulent kinetic energy. These results are consistent with previously published DNS results based on a pseudo-spectral method, thereby demonstrating the accuracy of the stabilized FEM for turbulence simulations.     

Fault-tolerant edge-pancyclicity of locally twisted cubes

The n-dimensional locally twisted cube LTQ_n is a new variant of the hypercube, which possesses some properties superior to the hypercube. This paper investigates the fault-tolerant edge-pancyclicity of LTQ_n, and shows that if LTQ_n (n ? 3) contains at most n − 3 faulty vertices and/or edges then, for any fault-free edge e and any integer ? with 6 ? ? ? 2ⁿ − f_v, there is a fault-free cycle of length ? containing the edge e, where f_v is the number of faulty vertices. The result is optimal in some senses. The proof is based on the recursive structure of LTQ_n.     

Influence of combustion parameters on NOx production in an industrial boiler

NO_x formation during the combustion process occurs mainly through the oxidation of nitrogen in the combustion air (thermal NO_x) and through oxidation of nitrogen with the fuel (prompt NO_x). The present study aims to investigate numerically the problem of NO_x pollution using a model furnace of an industrial boiler utilizing fuel gas. The importance of this problem is mainly due to its relation to the pollutants produced by large boiler furnaces used widely in thermal industrial plants. Governing conservation equations of mass, momentum and energy, and equations representing the transport of species concentrations, turbulence, combustion and radiation modeling in addition to NO modeling equations were solved together to present temperature and NO distribution inside the radiation and convection sections of the boiler. The boiler under investigation is a 160 MW, water-tube boiler, gas fired with natural gas and having two vertically aligned burners.The simulation study provided the NO distribution in the combustion chamber and in the exhaust gas at various operating conditions of fuel to air ratio with varying either the fuel or air mass flow rate, inlet air temperature and combustion primary air swirl angle. In particular, the simulation provided more insight on the correlation between the maximum furnace temperature and furnace average temperatures and the thermal NO concentration. The results have shown that the furnace average temperature and NO concentration decrease as the excess air factor λ increases for a given air mass flow rate. When considering a fixed value of mass flow rate of fuel, the results show that increasing λ results in a maximum value of thermal NO concentration at the exit of the boiler at λ = 1.2. As the combustion air temperature increases, furnace temperature increases and the thermal NO concentration increases sharply. The results also show that NO concentration at exit of the boiler exhibits a minimum value at around swirl angle of 45°.     

Electrical and CO gas sensing properties of nanostructured La1−xCexCoO3 perovskite prepared by activated reactive synthesis

A series of nanostructred La_1−xCe_xCoO₃ perovskite-type (x ranging from 0 to 0.2), with a crystallite size of around 10 nm and a specific surface area of up to 55 m²/g were prepared using the activated reactive grinding method. XRD results showed that Ce segregates as CeO₂ when the addition level exceeds 10 at%. CO was chosen as a typical reducing gas and its interaction with surface oxygen was investigated. TPD-O₂ was used to investigate the effect of Ce-doping on total surface oxygen. The experimental results confirmed a positive effect of Ce-doping of up to 10 at% on total surface oxygen (α-O₂). TPD-CO and XPS analyses were performed to find the total carbon adsorption (i.e. related to the adsorption of CO) on the surface of the synthesized samples. Both methods confirmed that more carbon adsorbs on the surface of doped formulations compared to the pure LaCoO₃. Ce-doping increases the surface oxygen, thereby facilitating the adsorption and oxidation processes. CO gas sensing properties of thick La_1−xCe_xCoO₃ films were performed. La_0.9Ce_0.1CoO₃ showed the highest conductivity and the lowest activation energy. The optimum CO sensing temperature for doped formulation was found to be 100 °C compared to 130 °C for pure perovskite. Ce-doped samples showed a maximum response ratio of 240% with respect to 100 ppm CO in air compared to 60% obtained with pure LaCoO₃.