Two new upwind difference schemes for a coupled system of convection–diffusion equations arising from the steady MHD duct flow problems

In this paper, we develop two new upwind difference schemes for solving a coupled system of convection–diffusion equations arising from the steady incompressible MHD duct flow problem with a transverse magnetic field at high Hartmann numbers. Such an MHD duct flow is convection-dominated and its solution may exhibit localized phenomena such as boundary layers, namely, narrow boundary regions where the solution changes rapidly. Most conventional numerical schemes cannot efficiently solve the layer problems because they are lacking in either stability or accuracy. In contrast, the newly proposed upwind difference schemes can achieve a reasonable accuracy with a high stability, and they are capable of resolving high gradients near the layer regions without refining the grid. The accuracy of the first new upwind scheme is O(h + k) and the second one improves the accuracy to O(ε²(h + k) + ε(h² + k²) + (h³ + k³)), where 0 < ε ? 1/M ? 1 and M is the high Hartmann number. Numerical examples are provided to illustrate the performance of the newly proposed upwind difference schemes.     

Verified predictions of shape sensitivities in wall-bounded turbulent flows by an adaptive finite-element method

A Continuous Sensitivity Equation (CSE) method is presented for shape parameters in turbulent wall-bounded flows modeled with the standard k–? turbulence model with wall functions. Differentiation of boundary conditions and their complex dependencies on shape parameters, including the two-velocity scale wall functions, is presented in details along with the appropriate methodology required for the CSE method. To ensure accuracy, grid convergence and to reduce computational time, an adaptive finite-element method driven by asymptotically exact error estimations is used. The adaptive process is controlled by error estimates on both flow and sensitivity solutions. Firstly, the proposed approach is applied on a problem with a closed-form solution, derived using the Method of the Manufactured Solution to perform Code Verification. Results from adaptive grid refinement studies show Verification of flow and sensitivity solvers, error estimators and the adaptive strategy. Secondly, we consider turbulent flows around a square cross-section cylinder in proximity of a solid wall. We examine the quality of the numerical solutions by performing Solution Verification and Validation. Then, Sensitivity Analysis of these turbulent flows is performed to investigate the ability of the method to deal with non-trivial geometrical changes. Sensitivity information is used to estimate uncertainties in the flow solution caused by uncertainties in the shape parameter and to perform fast evaluation of flows on nearby configurations.     

Adjoint-based error estimation and adaptive mesh refinement for the RANS and k–ω turbulence model equations

In this article we present the extension of the a posteriori error estimation and goal-oriented mesh refinement approach from laminar to turbulent flows, which are governed by the Reynolds-averaged Navier–Stokes and k–ω turbulence model (RANS-kω) equations. In particular, we consider a discontinuous Galerkin discretization of the RANS-kω equations and use it within an adjoint-based error estimation and adaptive mesh refinement algorithm that targets the reduction of the discretization error in single as well as in multiple aerodynamic force coefficients. The accuracy of the error estimation and the performance of the goal-oriented mesh refinement algorithm is demonstrated for various test cases, including a two-dimensional turbulent flow around a three-element high lift configuration and a three-dimensional turbulent flow around a wing-body configuration.     

Layout optimization methodology of piezoelectric transducers in energy-recycling semi-active vibration control systems

An optimization methodology is proposed for the piezoelectric transducer (PZT) layout of an energy-recycling semi-active vibration control (ERSAVC) system for a space structure composed of trusses. Based on numerical optimization techniques, we intend to generate optimal location of PZTs under the constraint for the total length of PZTs. The design variables are set as the length of the PZT on each truss based on the concept of the ground structure approach. The transient problems of the mechanical and electrical vibrations based on the ERSAVC theory are considered as the equations of state. The objective is to minimize the integration of the square of all displacement over the whole analysis time domain. The sensitivity of the objective function is derived based on the adjoint variable method. Based on these formulations, an optimization algorithm is constructed using the fourth-order Runge–Kutta method and the method of moving asymptotes. Numerical examples are provided to illustrate the validity and utility of the proposed methodology. Using the proposed methodology, the optimal location of PZTs for the vibration suppression for multi-modal vibration is studied, which can be benchmark results of further study in the context of ERSAVC systems.     

Iterative optimization based on an objective functional in frequency-space with application to jet-noise cancellation

In many technical applications, like supersonic jets, noise with a characteristic spectrum including certain dominant frequencies (e.g. jet-screech) is prevalent, and the elimination of sharp peaks in the acoustic spectrum is the aim of active or passive flow/noise control efforts. A mathematical framework for the optimization of control strategies is introduced that uses a cost objective in frequency-space coupled to constraints in form of partial differential equations in the time domain. An iterative optimization scheme based on direct and adjoint equations arises, which has been validated on two examples, the one-dimensional Burgers equation and the two-dimensional compressible Navier–Stokes equations. In both cases, the iterative scheme has proven effective and efficient in targeting and removing specified frequency bands in the acoustic spectrum. It is expected that this technique will find use in acoustic and other applications where the elimination or suppression of distinct frequency components is desirable.     

DYNAMIC RESPONSE OF WIND-EXCITED BUILDING USING CFD

Computational wind engineering as a new branch of computational fluid dynamics (CFD) has been developed recently to evaluate the interaction between wind and buildings numerically. In the present study, a systematic examination of wind effects on tall buildings and flow condition around buildings has been carried out using commercially available CFD software FLUENT 5. Both renormalization group (RNG) k-ε method and large eddy simulation (LES) with the Smagorinsky model are adopted as turbulence models and the results are compared with the wind-tunnel measurements. The weighted amplitude wave superposition (WAWS) method is used to generate atmospheric wind turbulence. The RNG k-ε method can predict the vortex shedding phenomenon well when compared with experiments for uniform flow input, but fails to predict the shedding frequency accurately for fluctuating incoming flow. On the other hand, the LES model shows reasonably good agreement with experiment in predicting vortex-shedding phenomenon for both uniform and fluctuating flows at inlet. Random-vibration based theory is employed for estimating r.m.s. response of tall buildings and the results compared well with the experimental results for a square building.     

A numerical analysis on flow field and heat transfer by interaction between a pair of vortices in rectangular channel flow

This paper is a numerical study of the effect of flow field and heat transfer created by interactions between a pair of vortices generated by a vortex generator in a rectangular channel flow. In order to analyze the vortices produced by the vortex generator, the pseudo-compressibility method is introduced into the Navier–Strokes (NS) equation of a three-dimensional unsteady, incompressible viscous flow. A two-layer k–ε turbulence model is used on the flat plate three-dimensional turbulence boundary to predict the turbulence characteristics of the vortices. The computational results accurately predict the vortex characteristics, which are related to Reynolds stress, turbulent kinetic energy, and flow field. Also, in the prediction of thermal boundary layers, skin friction characteristics, and heat transfers, the present results are reasonably close to the experimental results obtained by other researchers.     

Autocorrelation function formulations and the turbulence dissipation rate: Application to dispersion models

The classical statistical diffusion theory and the binomial autocorrelation function are used to obtain a new formulation for the turbulence dissipation rate ε. The approach employs the Maclaurin series expansion of a logarithm function contained in the dispersion parameter formulation. The numerical coefficient of this new relation for ε is 100% larger than the numerical coefficient of the classical relation derived from the exponential autocorrelation function. A similar approach shows that the dispersion parameter obtained from the even exponential autocorrelation function does not result in a relation for ε and, therefore, is not suitable for application in dispersion models. In addition, a statistical comparison to experimental ground-level concentration data demonstrates that this newly derived relation for ε as well as other formulations for the turbulence dissipation rate are suitable for application in Lagrangian stochastic dispersion models. Therefore, the analysis shows that there is an uncertainty regarding the turbulence dissipation rate function form and the autocorrelation function form.     

Continuous and Discrete Adjoint Approach Based on Lattice Boltzmann Method in Aerodynamic Optimization Part I: Mathematical Derivation of Adjoint Lattice Boltzmann Equations

The significance of flow optimization utilizing the lattice Boltzmann (LB) method becomes obvious regarding its advantages as a novel flow field solution method compared to the other conventional computational fluid dynamics techniques. These unique characteristics of the LB method form the main idea of its application to optimization problems. In this research, for the first time, both continuous and discrete adjoint equations were extracted based on the LB method using a general procedure with low implementation cost. The proposed approach could be performed similarly for any optimization problem with the corresponding cost function and design variables vector. Moreover, this approach was not limited to flow fields and could be employed for steady as well as unsteady flows. Initially, the continuous and discrete adjoint LB equations and the cost function gradient vector were derived mathematically in detail using the continuous and discrete LB equations in space and time, respectively. Meanwhile, new adjoint concepts in lattice space were introduced. Finally, the analytical evaluation of the adjoint distribution functions and the cost function gradients was carried out.     

Adjoint-based h–p adaptive discontinuous Galerkin methods for the 2D compressible Euler equations

In this paper, we investigate and present an adaptive Discontinuous Galerkin algorithm driven by an adjoint-based error estimation technique for the inviscid compressible Euler equations. This approach requires the numerical approximations for the flow (i.e. primal) problem and the adjoint (i.e. dual) problem which corresponds to a particular simulation objective output of interest. The convergence of these two problems is accelerated by an hp-multigrid solver which makes use of an element Gauss–Seidel smoother on each level of the multigrid sequence. The error estimation of the output functional results in a spatial error distribution, which is used to drive an adaptive refinement strategy, which may include local mesh subdivision (h-refinement), local modification of discretization orders (p-enrichment) and the combination of both approaches known as hp-refinement. The selection between h- and p-refinement in the hp-adaptation approach is made based on a smoothness indicator applied to the most recently available flow solution values. Numerical results for the inviscid compressible flow over an idealized four-element airfoil geometry demonstrate that both pure h-refinement and pure p-enrichment algorithms achieve equivalent error reductions at each adaptation cycle compared to a uniform refinement approach, but requiring fewer degrees of freedom. The proposed hp-adaptive refinement strategy is capable of obtaining exponential error convergence in terms of degrees of freedom, and results in significant savings in computational cost. A high-speed flow test case is used to demonstrate the ability of the hp-refinement approach for capturing strong shocks or discontinuities while improving functional accuracy.     

Surface pressure characteristics of a highly loaded turbine blade at design and off-design conditions; a CFD methodology

The flow field passing through a highly loaded low pressure (LP) turbine cascade is numerically investigated at design and off-design conditions. The Field Operation And Manipulation (OpenFOAM) platform is used as the computational Fluid Dynamics (CFD) tool. In this regard, the influences of grid resolution on the results of k-ε, k-ω, and large-eddy simulation (LES) turbulence models are investigated and compared with those of experimental measurements. A numerical pressure undershoot is appeared near the end of blade pressure surface which is sensitive to grid resolution and flow turbulence modeling. The LES model is able to resolve separation on both coarse and fine grid resolutions. In addition, the off-design flow condition is modeled by negative and positive inflow incidence angles. The numerical experiments show that a separation bubble generated on blade pressure side is predicted by LES. The total pressure drop has also been calculated at incidence angles between -20° and +8°. The minimum total pressure drop is obtained by k-ω and LES at design point.     

Aerodynamic design optimization by using a continuous adjoint method

This paper presents the fundamentals of a continuous adjoint method and the applications of this method to the aerodynamic design optimization of both external and internal flows.General formulation of the continuous adjoint equations and the corresponding boundary conditions are derived.With the adjoint method,the complete gradient information needed in the design optimization can be obtained by solving the governing flow equations and the corresponding adjoint equations only once for each cost function,regardless of the number of design parameters.An inverse design of airfoil is firstly performed to study the accuracy of the adjoint gradient and the effectiveness of the adjoint method as an inverse design method.Then the method is used to perform a series of single and multiple point design optimization problems involving the drag reduction of airfoil,wing,and wing-body configuration,and the aerodynamic performance improvement of turbine and compressor blade rows.The results demonstrate that the continuous adjoint method can efficiently and significantly improve the aerodynamic performance of the design in a shape optimization problem.     

An adjoint method for shape optimization in unsteady viscous flows

A new method for shape optimization for unsteady viscous flows is presented. It is based on the continuous adjoint approach using a time accurate method and is capable of handling both inverse and direct objective functions. The objective function is minimized or maximized subject to the satisfaction of flow equations. The shape of the body is parametrized via a Non-Uniform Rational B-Splines (NURBS) curve and is updated by using the gradients obtained from solving the flow and adjoint equations. A finite element method based on streamline-upwind Petrov/Galerkin (SUPG) and pressure stabilized Petrov/Galerkin (PSPG) stabilization techniques is used to solve both the flow and adjoint equations. The method has been implemented and tested for the design of airfoils, based on enhancing its time-averaged aerodynamic coefficients. Interesting shapes are obtained, especially when the objective is to produce high performance airfoils. The effect of the extent of the window of time integration of flow and adjoint equations on the design process is studied. It is found that when the window of time integration is insufficient, the gradients are most likely to be erroneous.     

Local-in-time adjoint-based method for design optimization of unsteady flows

We present a new local-in-time discrete adjoint-based methodology for solving design optimization problems arising in unsteady aerodynamic applications. The new methodology circumvents storage requirements associated with the straightforward implementation of a global adjoint-based optimization method that stores the entire flow solution history for all time levels. This storage cost may quickly become prohibitive for large-scale applications. The key idea of the local-in-time method is to divide the entire time interval into several subintervals and to approximate the solution of the unsteady adjoint equations and the sensitivity derivative as a combination of the corresponding local quantities computed on each time subinterval. Since each subinterval contains relatively few time levels, the storage cost of the local-in-time method is much lower than that of the global methods, thus making the time-dependent adjoint optimization feasible for practical applications. Another attractive feature of the new technique is that the converged solution obtained with the local-in-time method is a local extremum of the original optimization problem. The new method carries no computational overhead as compared with the global implementation of adjoint-based methods. The paper presents a detailed comparison of the global- and local-in-time adjoint-based methods for design optimization problems governed by the unsteady compressible 2-D Euler equations.     

Condensation de l'acide sulfurique dans un conduit d'evacuation des gaz de combustion

A method for computing the condensation of water vapour and sulphuric acid in a removal gas conduct is proposed. It utilizes a theoretical approach to determine the condensate production during the phase change from the numerically computed thermal and dynamic properties of the steam in the conduct. The temperature and velocity distributions are given from a k — ε model in one-phase incompressible flow, taking into account the initial temperature and flow rate in the conduct. The quality of the model is tested on a full-scale experimentation pilot equipped with thermocouples and collectors of liquid condensate. The deposit production is obtained for different conditions of temperature and concentration, and the model proves to be satisfactory in domestic boiler conditions.     

A wavelet-based multiresolution approach to large-eddy simulation of turbulence

The wavelet-based multiresolution analysis (MRA) technique is used to develop a modelling approach to large-eddy simulation (LES) and its associated subgrid closure problem. The LES equations are derived by projecting the Navier–Stokes (N–S) equations onto a hierarchy of wavelet spaces. A numerical framework is then developed for the solution of the large and the small-scale equations. This is done in one dimension, for the Burgers equation, and in three dimensions, for the N–S problem. The proposed methodology is assessed in a priori tests on an atmospheric turbulent time series and on data from direct numerical simulation. A posteriori (dynamic) tests are also carried out for decaying and force-driven Burgers turbulence.     

Output-based space–time mesh adaptation for the compressible Navier–Stokes equations

This paper presents an output-based adaptive algorithm for unsteady simulations of convection-dominated flows. A space–time discontinuous Galerkin discretization is used in which the spatial meshes remain static in both position and resolution, and in which all elements advance by the same time step. Error estimates are computed using an adjoint-weighted residual, where the discrete adjoint is computed on a finer space obtained by order enrichment of the primal space. An iterative method based on an approximate factorization is used to solve both the forward and adjoint problems. The output error estimate drives a fixed-growth adaptive strategy that employs hanging-node refinement in the spatial domain and slab bisection in the temporal domain. Detection of space–time anisotropy in the localization of the output error is found to be important for efficiency of the adaptive algorithm, and two anisotropy measures are presented: one based on inter-element solution jumps, and one based on projection of the adjoint. Adaptive results are shown for several two-dimensional convection-dominated flows, including the compressible Navier–Stokes equations. For sufficiently-low accuracy levels, output-based adaptation is shown to be advantageous in terms of degrees of freedom when compared to uniform refinement and to adaptive indicators based on approximation error and the unweighted residual. Time integral quantities are used for the outputs of interest, but entire time histories of the integrands are also compared and found to converge rapidly under the proposed scheme. In addition, the final output-adapted space–time meshes are shown to be relatively insensitive to the starting mesh.     

Improved Migdal recursion formula for the Ising model in two dimensions on a triangular lattice

The ε perturbation around the Migdal transformation proposed by Martinelli and Parisi has been computed up to second order for the Ising model in two dimensions on a triangular lattice. Perturbation around the value ε = 2 has also been introduced to evaluate higher orders in the series. The condition of null derivatives around ε = 1 for the critical values is analyzed. The critical temperature and the critical index ν are obtained with a satisfactory accuracy of a few percent.     

Use of Lagrangian statistics for the analysis of the scale separation hypothesis in turbulent channel flow

Turbulence models often involve Reynolds averaging, with a closure providing the Reynolds stress tensor as function of mean velocity gradients, through a turbulence constitutive equation. The main limitation of this linear closure is that it rests on an analogy with kinetic theory. For this analogy to be valid there has to be a scale separation between the mean velocity variations and the turbulent Lagrangian free path whose mean value is the turbulent mixing length. The aim of this work is to better understand this hypothesis from a microscopic point of view. Therefore, fluid elements are tracked in a turbulent channel flow. The flow is resolved by direct numerical simulation (DNS). Statistics on particle trajectories ending on a certain distance y₀ from the wall are computed, leading to estimations of the turbulent mixing length scale and the Knudsen number. Comparing the computed values to the Knudsen number in the case of scale separation, we may know in which region of the flow and to what extent the turbulence constitutive equation is not verified. Finally, a new non-local formulation for predicting the Reynolds stress is proposed.     

ENHANCEMENT OF FORCED CONVECTIVE HEAT TRANSFER IN NARROW CONCENTRIC ANNULI BY TURBULENCE PROMOTERS

Abstract

Forced convective heat transfer in a narrow concentric annulus was enhanced by turbulence promoters to improve the heat removal from a high-temperature gas-cooled reactor, a gas-cooled fusion reactor, and other narrow flow passages. The present experiments, which differed from those performed in conventional research, were carried out to examine the effect of turbulence promoters on the inner insulated wall opposite the outer smooth heated wall. This was achieved by changing the ratio of the pitch and the height P/ε, the ratio of the height and the space ε/ε¹, and the type of turbulence promoters used. Experimental results were examined for the local heat transfer coefficient distribution on the smooth outer tube, the average heat transfer coefficient, the friction factor, and the thermal performance. Five kinds of evaluations for thermal performance were carried out.P24