A zonal RANS-LES method to determine the flow over a high-lift configuration

High Reynolds number flows are particularly challenging problems for large-eddy simulations (LES) since small-scale structures in thin and often transitional boundary layers are to be resolved. The range of the turbulent scales is enormous, especially when high-lift configuration flows are considered. For this reason, the prediction of high Reynolds number flow over the entire airfoil using LES requires huge computer resources. To remedy this problem a zonal RANS-LES method for the flow over an airfoil in high-lift configuration at Re_c=1.0×10⁶ is presented. In a first step, a 2D RANS solution is sought, from which boundary conditions are formulated for an embedded LES domain, which comprises the flap and a sub-part of the main airfoil. The turbulent fluctuations in the boundary layers at the inflow region of the LES domain are generated by controlled forcing terms, which use the turbulent shear stress profiles obtained from the RANS solution. The comparison with an LES solution for the full domain and with experimental data shows likewise results for the velocity profiles and wall pressure distributions. The zonal RANS-LES method reduces the computational effort of a full domain LES by approx. 50%.     

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.     

Large Eddy Simulation of a turbulent non-premixed propane-air reacting flame in a cylindrical combustor

Large Eddy Simulation (LES) is applied to investigate the turbulent non-premixed combustion flow, including species concentrations and temperature, in a cylindrical combustor. Gaseous propane (C₃H₈) is injected through a circular nozzle which is attached at the centre of the combustor inlet. Preheated air with a temperature of 773 K is supplied through the annulus surrounding of this fuel nozzle. In LES a spatial filtering is applied to the governing equations to separate the flow field into large-scale and small-scale eddies. The large-scale eddies which carry most of the turbulent energy are resolved explicitly, while the unresolved small-scale eddies are modelled using the Smagorinsky model with C_s = 0.1 as well as dynamically calibrated C_s. The filtered values of the species mass fraction, temperature and density, which are the functions of the mixture fraction (conserved scalar), are determined by integration over a beta probability density function (β-PDF). The computational results are compared with those of the experimental investigation conducted by Nishida and Mukohara [1]. According to this experiment, the overall equivalence ratio of 0.6, which is calculated from the ratio of the air flow rate supplied to the combustion chamber to that of the stoichiometric reaction, is kept constant so that the turbulent combustion at the fuel nozzle exit starts under the fuel-rich conditions.     

A new solution branch for the Blasius equation—A shrinking sheet problem

In this work, a similarity equation of the momentum boundary layer is studied for a moving flat plate with mass transfer in a stationary fluid. The solution is applicable to the practical problem of a shrinking sheet with a constant sheet velocity. Theoretical estimation of the solution domain is obtained. It is shown that the solution only exists with mass suction at the wall surface. The equation with the associated boundary conditions is solved using numerical techniques. Greatly different from the continuously stretching surface problem and the Blasius problem with a free stream, quite complicated behavior is observed in the results. It is seen that there are three different solution zones divided by two critical mass transfer parameters, f₀₁≈1.7028 and f₀₂≈1.7324. When f₀<f₀₁, there is no solution for this problem, multiple solutions for f₀₁<f₀≤f₀₂, and one solution when (f₀=f₀₁)(f₀>f₀₂). There is a terminating point for the solution domain and the terminating point corresponds to a special algebraically decaying solution for the current problem. The current results provide a new solution branch of the Blasius equation, which is greatly different from the previous study and provide more insight into the understanding of the Blasius equation.     

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.     

Application of Compact Schemes to Large Eddy Simulation of Turbulent Jets

We present 3-D large eddy simulation (LES) results for a turbulent Mach 0.9 isothermal round jet at a Reynolds number of 100,000 (based on jet nozzle exit conditions and nozzle diameter). Our LES code is part of a Computational Aeroacoustics (CAA) methodology that couples surface integral acoustics techniques such as Kirchhoff's method and the Ffowcs Williams– Hawkings method with LES for the far field noise estimation of turbulent jets. The LES code employs high-order accurate compact differencing together with implicit spatial filtering and state-of-the-art non-reflecting boundary conditions. A localized dynamic Smagorinsky subgrid-scale (SGS) model is used for representing the effects of the unresolved scales on the resolved scales. A computational grid consisting of 12 million points was used in the present simulation. Mean flow results obtained in our simulation are found to be in very good agreement with the available experimental data of jets at similar flow conditions. Furthermore, the near field data provided by the LES is coupled with the Ffowcs Williams–Hawkings method to compute the far field noise. Far field aeroacoustics results are also presented and comparisons are made with experimental measurements of jets at similar flow conditions. The aeroacoustics results are encouraging and suggest further investigation of the effects of inflow conditions on the jet acoustic field.     

An investigation of turbulent open channel flow with heat transfer by large eddy simulation

Large eddy simulation of fully developed turbulent open channel flow with heat transfer is performed. The three-dimensional filtered Navier-Stokes and energy equations are numerically solved using a fractional-step method. Dynamic subgrid-scale (SGS) models for the turbulent SGS stress and heat flux are employed to close the governing equations. Two typical temperature boundary conditions, i.e., constant temperature and constant heat flux being maintained at the free surface, respectively, are used. The objective of this study is to explore the behavior of heat transfer in the turbulent open channel flow for different temperature boundary conditions and to examine the reliability of the LES technique for predicting turbulent heat transfer at the free surface, in particular, for high Prandtl number. Calculated parameters are chosen as the Prandtl number (Pr) from 1 up to 100, the Reynolds number (Re_τ) 180 based on the wall friction velocity and the channel depth. Some typical quantities, including the mean velocity, temperature and their fluctuations, heat transfer coefficients, turbulent heat fluxes, and flow structures based on the velocity, vorticity and temperature fluctuations, are analyzed.     

Large Eddy Simulations of a spatially developing incompressible 3D mixing layer using the v-ω formulation

The LES of a 3D mixing layer spatially developing downstream of a flat plate has been conducted for a high Reynolds number (Re_θ=2835). To overcome the problem of the pressure condition on the free boundaries, use of the (v-ω) formulation of the Navier-Stokes equations has been preferred over the primitive (v-p) formulation. To deal with the difficult problem of the divergence free constraint on both the velocity and the vorticity field, a new and efficient numerical algorithm has been devised which turns out to be very attractive. The velocity components are computed as a solution of a Cauchy-Riemann problem using a fractional step method. As one of the main advantages of the vorticity-based formulation is the treatment of the free flow boundary conditions, special care has been devoted to these boundary conditions. An optimum approximation of the outflow boundary condition has been carried out which satisfies the conservation of mass, making the long time integration easier and more accurate. The numerical results are compared to a reference experiment [Appl. Sci. Res. 53 (1994) 263; J. Delville, PhD thesis, University of Poitiers, 1995] for a rather high Reynolds number. Using inlet perturbations and the mixed scale model, the LES results agree very well with the reference experiments. The validation of the numerical procedure is reviewed on the mean and fluctuating quantities. Good prediction of the spatial evolution is demonstrated for the distribution of the vorticity thickness as well as for the Reynolds stress profiles and spatial correlations. In order to estimate the quality of the spatio-temporal development, a spectral analysis is also reported on the space and time spectra, pointing out a highly 3D arrangement with length scales and frequencies in rather good agreement with the ones generally admitted.     

Primality testing with fewer random bits

In the usual formulations of the Miller-Rabin and Solovay-Strassen primality testing algorithms for a numbern, the algorithm chooses candidatesx ₁,x ₂, ...,x _k uniformly and independently at random from _n , and tests if any is a witness to the compositeness ofn. For either algorithm, the probabilty that it errs is at most 2^–k .In this paper, we study the error probabilities of these algorithms when the candidates are instead chosen asx, x+1, ..., x+k–1, wherex is chosen uniformly at random from _n . We prove that fork=[1/2log₂ n], the error probability of the Miller-Rabin test is no more thann ^–1/2+o(1), which improves on the boundn ^–1/4+o(1) previously obtained by Bach. We prove similar bounds for the Solovay-Strassen test, but they are not quite as strong; in particular, we only obtain a bound ofn ^–1/2+o(1) if the number of distinct prime factors ofn iso(logn/loglogn).     

A Parallel Overset Grid High-Order Flow Solver for Large Eddy Simulation

This work describes the development and validation of a parallel high-order compact finite difference Navier–Stokes solver for application to large-eddy simulation (LES) and direct numerical simulation. The implicit solver can employ up to sixth-order spatial formulations and tenth-order filtering. The parallelization of the solver is founded on the overset grid technique. LES were then performed for turbulent channel flow with Reynolds numbers ranging from Re _τ=180 to 590, and flow past a circular cylinder with a transitional wake at Re _D =3900. The channel flow solutions were obtained using both an implicit LES (ILES) approach and a dynamic sub-grid scale model. The ILES method obtained virtually identical solutions at half the computational cost. The original vector and new parallel solvers produce indistinguishable mean flow solutions for the circular cylinder. Repeating the cylinder simulation on a much finer mesh resulted in significantly better agreement with experimental data in the near wake than the coarse grid solution and other previous numerical studies.     

Horizontal Large Eddy Simulation of Stratified Mixing in a Lock-Exchange System

This paper presents analytical and numerical results for two new anisotropic modifications of the Rational and Clark-α LES models. The main difference from their standard form is that in this study horizontal (as opposed to isotropic) spatial filtering is used, which is appropriate for turbulent mixing in stratified flows. We present several mathematical results regarding the horizontal Rational and Clark-α LES models. We also present numerical experiments that support the analytical developments and show that both horizontal LES models perform better than their standard, isotropic counterparts in approximating mixing in a 3D lock-exchange problem at Reynolds number Re=10,000.     

Performance of a bend-diffuser system: Experimental and numerical studies

The turbulent flow inside a combined bend-diffuser configuration with a rectangular cross section is experimentally and numerically studied. The experimental study includes the outer and inner-wall-pressure measurements and the overall system/diffuser loss determination. Simulation is performed using the high-Reynolds number k-ε turbulence model improved by the low-Reynolds number k-ε turbulence model near the walls, because of its success to predict the flow with strong adverse pressure gradient. So the present paper provides a numerical procedure for the calculation of turbulent flow in a sequence curved, expanding passages, with emphasis on the bend-diffuser configuration system consisting of a 90° bend followed by a diffuser with different expanding angles ranges from 2θ = 6-30° at different inflow Reynolds numbers. Satisfied comparisons with reported experimental data in the literature as well as that carried out by the present authors at the heat engine laboratory of Menoufiya university show that the numerical method with the utilized closure turbulence model reproduces the essential features of upstream curved flow effects on the diffuser performance. The effect of spacer length (between the bend and diffuser) is also experimentally and numerically included. The results show that there is an optimum diffuser angle which depends on the inflow Reynolds number and produces the minimum pressure loss and hence good performance of such complex geometry is obtained.     

Wall modeling for LES of high Reynolds number channel flows: What turbulence information is retained?

A novel near-wall eddy-viscosity formulation for Large-eddy simulation (LES) has been used to compute high Reynolds number channel flows up to Re_τ = 1,000,000. These computations allow an insight into what turbulence information is retained when LES with a wall model is applied to such high Reynolds numbers. Detailed results are presented for the mean and rms velocities, as well as energy spectra. It is observed that, when an appropriate scaling is used, the rms velocities, energy spectra and the production of turbulence kinetic energy are weakly Reynolds number dependent at these high Reynolds numbers.     

The use of Richardson extrapolation for the numerical solution of low Mach number flow in confined regions

We use artificial compressibility together with Richardson extrapolation in the Mach numberM as a method for solving the time dependent Navier-Stokes equation for very low Mach number flow and for incompressible flow. The question of what boundary conditions one should use for low Mach number flow, especially at inflow and outflow boundaries, is investigated theoretically, and boundary layer suppressing boundary conditions are derived. For the case of linearization around a constant flow we show that the low Mach number solution will converge with the rateO(M²) to the true incompressible solution, provided that we choose the boundary conditions correctly. The results of numerical calculations for the time dependent, nonlinear equations and for flow situations with time dependent inflow velocity profiles are presented. The convergence rateM ² to incompressible solution is numerically confirmed. It is also shown that using Richardson extrapolation toM ²= 0 in order to derive a solution with very small divergence can with good result be carried through withM ² as large as 0.1 and 0.05. As the time step in numerical methods must be chosen approximately such thatt · (i/(M x)–v/x ²) is in the stability region of the time stepping method, and asM ²=0.05 is sufficiently small to yield good results, the restriction on the time step due to the Mach number is not serious. Therefore the equations can be integrated very fast by explicit time stepping methods. This method for solving very low Mach number flow and incompressible flow is well suited to parallel processing.     

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.     

Drag and lift coefficients of inclined finite circular cylinders at moderate Reynolds numbers

The flow around two-dimensional cylinders at moderate Reynolds numbers has been much studied, both for cylinders perpendicular to the flow and for cylinders yawed to the flow. In contrast, yawed finite aspect ratio cylinders have received little attention. In this article we describe computer simulations of cylinders with aspect ratios 2  L/D  20 yawed at angles 0°  α  90° relative to a free stream. The simulations were carried out for Reynolds numbers in the range 1  Re  40. The simulations show that the Independence Principle [Zdravkovich MM. Flow around circular cylinders, vol. 2: applications. New York: Oxford University Press; 2003[1]] is not accurate for α  45°. We have also found that for all aspect ratios, the ratio of the lift to drag force reaches a maximum for 40° < α < 50°. Finally, we present C_L and C_D relationships as best curve fits to computational data.     

Large-eddy simulation of streamwise-rotating turbulent channel flow

In many engineering and industrial applications the investigation of rotating turbulent flow is of great interest. Whereas some research has been done concerning channel flows with a spanwise rotation axis, only few investigations have been performed on channel flows with a rotation about the streamwise axis. In the present study an LES of a turbulent streamwise-rotating channel flow at Re_τ = 180 is performed using a moving grid method. The three-dimensional structures and the details of the secondary flow distribution are analyzed and compared with experimental data. The numerical-experimental comparison shows a convincing agreement as to the overall flow features. The results confirm the development of a secondary flow in the spanwise direction, which has been found to be correlated to the rotational speed. Furthermore, the findings show the distortion of the main flow velocity profile, the slight decrease of the streamwise Reynolds stresses in the vicinity of the walls, and the pronounced increase of the spanwise Reynolds stresses at higher rotation rates near the walls and particularly in the symmetry region. As to the numerical set-up it is shown that periodic boundary conditions in the spanwise direction suffice if the spanwise extent of the computational domain is larger than 10 times the channel half width.     

Persistence of turbulent flow in microchannels at very low Reynolds numbers

A fully resolved numerical simulation of a turbulent microchannel flow, with uniformly spaced two-dimensional obstruction elements mounted at the wall and normal to the flow direction, was carried out at a very low Reynolds number of Re ≃ 970 based on the centerline velocity and the microchannel height. Employing the lattice Boltzmann numerical technique, all energetic scales of turbulence were resolved with about 19 × 10⁶ grid points (1261 × 129 × 128 in the x ₁, x ₂, and x ₃ directions). The simulated results confirm the self-maintenance of turbulence at such a low Reynolds number. Turbulence persisted over more than 1,000 turnover times, which was sufficiently long to prove its self-maintenance. These findings support the conjecture that turbulence developing in microchannels having rough walls can not only be initiated but also maintained at very low Reynolds numbers.     

Parallel solution of recurrences on a tree machine

The recurrencex _o =a _o x _i =a _i+b _i x _i–1,i = 1, 2,...,n–1 requiresO(n) operations on a sequential computer. Elegant parallel solutions exist, however, that reduce the complexity toO(logN) usingNn processors. This paper discusses one such solution, designed for a tree-structured network of processors.A tree structure is ideal for solving recurrences. It takes exactly one sweep up and down the tree to solve any of several classes of recurrences, thus guaranteeing a solution inO(logN) time for a tree withNn leaf nodes. Ifn exceedsN, the algorithm efficiently pipelines the operation and solves the recurrence inO(n/N + logN) time.