Wall Boundary Conditions Research Articles

A high order moving boundary treatment for convection-diffusion equations

In this paper, a new type of inverse Lax-Wendroff boundary treatment is designed for high order finite difference schemes for solving general convection-diffusion equations on time-varying domain. This new method can achieve high order accuracy on Dirichlet boundary conditions with moving boundary. To ensure stability of the boundary treatment, we give a convex combination of the boundary treatments for the diffusion-dominated and the convection-dominated cases. A group convex combination of weights is carefully designed to avoid zero denominator, resulting in a unified algorithm for pure convection, convection-dominated, convection-diffusion, diffusion-dominated and pure diffusion cases. In order to match the time levels when constructing values of ghost points in the two intermediate stages of the third order Runge-Kutta method, we propose a new approximation to the mixed derivatives at the boundaries to ensure high order accuracy and to improve computational efficiency. In particular, we extend the boundary treatment to the compressible Navier-Stokes equations, which satisfies the isothermal no-slip wall boundary condition at any Reynolds number. We provide numerical tests for one- and two-dimensional problems involving both scalar equations and systems, demonstrating that our boundary treatment is high order accurate for problems with smooth solutions and also performs well for problems involving interactions between viscous shocks and moving rigid bodies.

Aerodynamics simulations of three-dimensional inviscid flow using curvilinear discontinuous Galerkin method on unstructured meshes

Over the last decades, the discontinuous Galerkin (DG) method has demonstrated its excellence in accurate, higher-order numerical simulations for a wide range of applications in aerodynamics simulations. However, the development of practical, computationally accurate flow solvers for industrial applications is still in the focus of active research, and applicable boundary conditions and fluxes are also very important parts. Based on curvilinear DG method, we have developed a flow solver that can be used for solving the three-dimensional subsonic, transonic and hypersonic inviscid flows on unstructured meshes. The development covers the geometrical transformation from the real curved element to the rectilinear reference element with the hierarchical basis functions and their gradient operation in reference coordinates up to full third order. The implementation of solid wall boundary conditions is derived by the contravariant velocities, and an enhanced algorithms of Harten-Lax-van Leer with contact (HLLC) flux based on curved element is suggested. These new techniques do not require a complex geometric boundary information and are easy to implement. The simulation of subsonic, transonic and hypersonic flows shows that the linear treatment can limit the accuracy at high order and demonstrates how the boundary treatment involving curved element overcomes this restriction. In addition, such a flow solver is stable on a reasonably coarse meshes and finer ones, and has good robustness for three-dimensional flows with various geometries and velocities. For engineering practice, a reasonable accuracy can be obtained at reasonably coarse unstructured meshes.

Application of Gas-Kinetic Scheme for Continuum and Near-Continuum Flow on Unstructured Mesh

A gas-kinetic scheme (GKS) with kinetic boundary condition based on unstructured mesh is present here. In the GKS method, the solid wall boundary conditions can be constructed by virtue of the gas distribution function, which is similar to the diffuse-scattering rule used in the other kinetic schemes. The kinetic boundary condition has a concise form and easy to implement. The use of unstructured mesh expands the adaptability of GKS to simulate the flows with complex geometry. The kinetic boundary condition can recover to the non-slip boundary condition in the continuum regime. In the slip regime, the slip velocity can be accurately predicted by kinetic boundary condition, which turns into the slip boundary condition. The use of kinetic boundary condition improves the calculation results of GKS in near-continuum flow. A series of test cases, from incompressible to compressible flow with a wide range of Knudsen number, are investigated to demonstrate the performance of kinetic boundary condition in near-continuum flow, which can provide a reference for the construction and optimisation for GKS-based multi-scale hybrid algorithms.

Influence of Gas-to-Wall Temperature Ratio on the Leakage Flow and Cooling Performance of a Turbine Squealer Tip

In high-pressure turbines, there is a large difference in temperature between the mainstream and the turbine blade surface. Most of the turbine blade tip heat transfer studies were conducted under the assumption that the Over-Tip-Leakage (OTL) flow field is independent of the wall thermal condition. Recent numerical and experimental studies have revealed that the two-way coupling effect between aerodynamics and heat transfer should not be neglected. The heat transfer coefficient obtained by the conventional method is not able to match the realistic engine condition accurately. This study investigates the impact of the wall thermal boundary condition on the tip cooling performance of squealer turbine blades. The RANS CFD result was validated against experimental tip heat transfer data obtained from a high-speed test rig with the effect of high-speed relative casing motion. The aerothermal performance for both uncooled and cooled squealer tips was studied at two different gas-to-wall temperature ratios, 1.7 and 1.1; the reference temperature is 204 K. It was found that the location and strength of cavity vortices varied with different wall thermal boundary conditions, leading to different signatures in tip heat transfer and cooling performance. It is recommended that the experimental heat transfer data and film cooling effectiveness obtained at the near-adiabatic wall boundary condition should be corrected before their application to the tip cooling design process. It would be more reliable to match the wall-to-gas temperature ratio during the tip experimental study.

Impacts of solid wall boundary conditions in the lattice Boltzmann method on turbulent outdoor flow: A case study of a single 1:1:2 building model

Solid wall boundary condition is an important topic in outdoor wind environment simulation represented by single building based on lattice Boltzmann method (LBM), but there is a lack of quantitative comparative analysis of simulation results under these boundary conditions. This paper conducted LBM-based large-eddy simulation (LBM-LES) of turbulent flow around a 1:1:2 single building, applying both half-way bounce-back and interpolation-based bounce-back (the Bouzidi–Firdaouss–Lallemand scheme, BFL) to solid wall boundaries. The simulation accuracy of different solid wall boundary conditions was examed with consideration of grid resolutions and relative positions between grid points and solid boundaries. Data obtained via experiments and the finite volume method-based large-eddy simulation (FVM-LES) technique were used to examine the flow structure and distribution of the time-averaged velocity and turbulent kinetic energy (TKE). The computational costs were also discussed. Results demonstrated that when the all the boundaries fell on the grid and orthogonal to the grid, the half-way bounce-back scheme attained a high accuracy. On the contrary, the BFL scheme achieved a higher accuracy if the solid wall boundaries did not fall on the grid. Under both boundary schemes, the grid resolutions of b/32 and b/16 (b: building width) could satisfy the acceptable range, but the b/32 grid resolution facilitated a better performance near the wall. The BFL scheme was more sensitive to the grid resolution and the accuracy improved faster with increasing grid resolution than half-way. As for the calculation costs, the calculation time of BFL is more than twice that of half-way bounce-back. However, BFL is still recommended if the wall boundaries are curved and complex in the actual built environment.

Comparison of the simulated seismic behavior of small- and large-scale cripple walls: PEER-CEA Project test programs

As part of the Pacific Earthquake Engineering Research Center (PEER)-California Earthquake Authority (CEA) project, experimental programs were performed at UC San Diego and UC Berkeley, with the former focusing on small-component testing and the latter focusing on large-component testing. During the small-component testing program, a suite of 28 cripple wall-only specimens was tested under quasi-static reverse cyclic lateral loading with variations in exterior finishes, retrofit condition, cripple wall height, anchorage condition, boundary conditions, vertical load, and loading protocol. In the large-component testing program, cripple wall-first story specimens, load path connections, and combined materials in occupied stories were tested. The more easily performed small-component specimens facilitate comparison with the large-component specimens to allow the assessment of their potential to emulate large-scale specimen behavior, with a particular interest in cross-comparing their global hysteresis and physical damage evolution. This cross-comparison among small- and large-scale specimen test results is intended to offer best practices for the future design of cripple wall seismic retrofits in residential housing, with a particular interest in: (1) supporting the realistic design of small-component specimens that may capture the response of large-component specimens and (2) to qualitatively determine where the small-component tests fall in the range of lower- to upper-bound estimation of strength and deformation capacity for the purposes of numerical modeling. Data from both test programs were utilized in the PEER-CEA project to assist with validating numerical modeling tools, which in turn were used to generate seismic loss models capable of quantifying the reduction of loss achieved by applying state-of-practice retrofit methods as identified in FEMA P-1100 Vulnerability-Base Seismic Assessment and Retrofit of One- and Two-Family Dwellings. The objective of this article is to present an overview of the small- and large-component testing programs and key results pertaining to exterior finishes, boundary conditions, the FEMA P-1100 retrofit, and cripple wall height.

NATURAL CONVECTION ON CORRUGATED PLATES: A NUMERICAL CASE STUDY ABOUT MESHES, BOUNDARY CONDITIONS AND PHYSICAL DOMAIN DETERMINATION

The transfer of heat and mass by natural convection is present in the most diverse physical and chemical phenomena of nature and engineering equipment. In the last decades, the number of research on natural convection has grown dramatically, highlighting studies in physical-mathematical modeling and numerical solutions, experimental analysis and design and optimization techniques for fluid-thermal systems. This case study analyzed the influence of several numerical parameters in physical-mathematical modeling and numerical solution of natural convection heat transfer problems on isothermal plates with square waves in turbulent conditions of high Rayleigh number. The numerical parameters analyzed were the mesh refinement degree, wall boundary conditions (with or without wall functions implemented in the turbulent parameters) and computational physical domain influence. Free and open-source computational numerical tools were exclusively used in the construction of this work. Meshes with wall functions implemented in turbulent parameters presented greater accuracy and required less computational effort and simulation time, besides enabling the use of a lower degree of mesh refinement. The best numerical configuration of the physical model for the situation problem studied were defined from the criteria of accuracy, computational effort demanded, and stability and numerical convergence of the solution.

Deep reinforcement learning for large-eddy simulation modeling in wall-bounded turbulence

The development of a reliable subgrid-scale (SGS) model for large-eddy simulation (LES) is of great importance for many scientific and engineering applications. Recently, deep learning approaches have been tested for this purpose using high-fidelity data such as direct numerical simulation (DNS) in a supervised learning process. However, such data are generally not available in practice. Deep reinforcement learning (DRL) using only limited target statistics can be an alternative algorithm in which the training and testing of the model are conducted in the same LES environment. The DRL of turbulence modeling remains challenging owing to its chaotic nature, high dimensionality of the action space, and large computational cost. In this study, we propose a physics-constrained DRL framework that can develop a deep neural network-based SGS model for LES of turbulent channel flow. The DRL models that produce the SGS stress were trained based on the local gradient of the filtered velocities. The developed SGS model automatically satisfies the reflectional invariance and wall boundary conditions without an extra training process so that DRL can quickly find the optimal policy. Furthermore, direct accumulation of reward, spatially and temporally correlated exploration, and the pre-training process are applied for efficient and effective learning. In various environments, our DRL could discover SGS models that produce the viscous and Reynolds stress statistics perfectly consistent with the filtered DNS. By comparing various statistics obtained by the trained models and conventional SGS models, we present a possible interpretation of better performance of the DRL model.

Comparison of turbulent inflow conditions for neutral stratified atmospheric boundary layer flow

In the present work, measurements of the atmospheric boundary layer (ABL) flow, reproduced in a boundary layer wind tunnel, and large eddy simulation (LES) of the full scale ABL are presented for two ground roughness lengths. In the numerical part the ABL is simulated by utilizing the cyclic boundary conditions in downstream direction and artificial turbulence specified at the inlet. Five synthetic turbulence generation techniques are used with reviewing two input methods for the Reynolds stress. Convergence of mean velocity and second order moments is discussed for various horizontal and vertical grid refinement as well as for different roughness and wall boundary conditions of the ground. The transition of mean velocity, second order moments, stream-wise velocity spectrum and longitudinal turbulent length scale in downstream direction is analyzed. The applicability of the synthetic turbulence condition was confirmed for the LES of ABL simulations. In this way, the inflow domain length for simulations can significantly be reduced.

Prandtl number effects on the hydrodynamic stability of compressible boundary layers: flow–thermodynamics interactions

Hydrodynamic stability of compressible boundary layers is strongly influenced by the Mach number ( $M$ ), Prandtl number ( $Pr$ ) and thermal wall boundary condition. These effects manifest on the flow stability via the flow–thermodynamics interactions. Comprehensive understanding of stability flow physics is of fundamental interest and important for developing predictive tools and closure models for integrated transition-to-turbulence computations. The flow–thermodynamics interactions are examined using linear analysis and direct numerical simulations in the following parameter regime: $0.5 \leq M \leq 8$ ; and $0.5 \leq Pr \leq 1.3$ . For the adiabatic wall boundary condition, increasing Prandtl number has a destabilizing effect. In this work, we characterize the behaviour of production, pressure–strain correlation and pressure dilatation as functions of the Mach and Prandtl numbers. First and second instability modes exhibit similar stability trends but the underlying flow physics is shown to be diametrically opposite. The Prandtl number influence on instability is explicated in terms of the base flow profile with respect to the different perturbation mode shapes.

Off-Gas System Scale-Up of HIsarna Iron-Making Process: A CFD-Based Approach

For all industrial applications, predicting system characteristics and behavior plays a vital role before constructing costly and complex multi-physic systems. Correct and reliable predictions become even more important once the aim is to go from small- to large-scale processes to establish an industrial demonstrations. In this study, a CFD-based scale-up of HIsarna off-gas system based on the Eulerian–Lagrangian approach is investigated and detailed step in scale-up procedure is discussed. A three-dimensional CFD model is developed and validated based on the available pilot scale data and used to design and scale up the post-combustion chamber (also known as reflux chamber). Detailed kinetics for volumetric and gas–solid reactions are incorporated in validated CFD model with a special attention to the wall boundary condition and modeling. The effect of reflux chamber geometry, oxygen injection ports, oxygen injection flowrate, isolation wall thickness, and inlet flue gas composition on different system characteristics such as heat loss through the wall, CO–H2–carbon mixture conversion, flue gas, and wall temperature are investigated. The aim of the scaled up geometry, like pilot scale, is to achieve full combustion of unwanted species inside the reflux chamber to assure zero emissions from the off-gas system. Compared to the pilot scale, the scaled up reflux chamber is capable of handling and removing higher amount of unwanted species coming from the main reactor and therefore lower CO–H2 and carbon particle emissions, mainly due to a larger size which provides larger volume and residence time for volumetric and gas–solid reaction to proceed.

The Influence of Various Types of Bracing on Force Distribution in Braced Barrel Vaults

In this paper the authors analysed four single-layered braced barrel vaults with different types of bracing. Each braced barrel vault covered the area of 20 m × 28 m and was made from steel hollow sections. The static-strength analyses of the structures were conducted using the Autodesk Robot Structural Analysis program, taking into account self-weight, glass cover, snow load and wind load. In the case of wind load, wind pressure perpendicular to the longitudinal wall was determined in accordance with the EN 1991-1- 4 standard. However, the standard does not specify how to calculate wind pressure perpendicular to a gable wall for braced barrel vaults. For this reason, two variants suggested by the authors of this article were analysed for this direction of the wind. The influence of the type of bracing on force distribution in a braced barrel vault and on material consumption was analysed. The impact of the gable wall boundary conditions on force distribution in the braced barrel vault was also evaluated. Both the bracing type and the boundary conditions had an impact on the force distribution in the analysed braced barrel vault.

Entropy stability for the compressible Navier-Stokes equations with strong imposition of the no-slip boundary condition

We consider the compressible Navier-Stokes equations subject to no-slip adiabatic wall boundary conditions. The main goal is to investigate stability properties of schemes imposing the no-slip condition strongly (injection) and the temperature condition weakly by a simultaneous approximation term. To this end, we propose a low-order summation-by-parts scheme. By verifying the complete linearisation procedure, we prove linear stability for the scheme. In addition, and assuming that the interior scheme is entropy stable, we also prove entropy stability for the full scheme including the boundary treatment. Furthermore, we propose a linearly stable 3rd-order scheme with the same imposition of the wall conditions. However, the 3rd-order scheme is not provably non-linearly stable. A number of simulations show that the boundary procedure is robust for both schemes.

Generalized and efficient wall boundary condition treatment in GPU-accelerated smoothed particle hydrodynamics

This paper presents a generalized and efficient wall boundary treatment in the smoothed particle hydrodynamics (SPH) method for 3-D complex and arbitrary geometries with single- and multi-phase flows to be executed on graphics processing units (GPUs). Using a force balance between the wall and fluid particles with a novel penalty method, a pressure boundary condition is applied on the wall dummy particles which effectively prevents non-physical particle penetration into the wall also in highly violent impacts and multi-phase flows with high density ratios. A new density reinitialization scheme is also presented to enhance the accuracy and robustness. The proposed method is simple in comparison with previous wall boundary formulations on GPUs and enforces no additional memory caching and thus is well suited for massively parallel architecture of GPUs. The method is validated in various test cases involving inviscid violent single- and multi-phase flows, demonstrating very good robustness, accuracy and performance. The new wall boundary treatment is able to improve the high accuracy of its previous version by Adami et al. [1] also in 3-D and multi-phase problems, while it is efficiently executable on GPUs. Therefore, the method can be a reliable solution for the long-lasting wall boundary condition challenge in SPH for a broad range of problems.

Comparative study of the single and double specularity coefficients on two-fluid modeling of a pseudo-2D gas–solid fluidized bed

The specularity coefficient is an unmeasurable parameter in the most popular wall boundary model during the two-fluid modeling of dense gas–solid flows. Using multiphaseEulerFoam solver, the influence of different specularity coefficient setting strategies on the gas–solid flow inside a pseudo-2D fluidized bed has been explored. It is found that the single specularity coefficient plays a regulatory role in the quantitative prediction. Increasing the specularity coefficient would cause a fluidization transition from freely bubbling to slugging, and the bed characteristics such as pressure drop and bed expansion present monotonic nonlinear changes. The double specularity coefficients approach is shown to significantly improve the predictive accuracy through verifying with the measured particle velocities, bubble diameter and rise velocity. In addition, the lognormal bubble size distribution and Gaussian bubble rise velocity distribution are observed. The specularity coefficient for walls in thickness direction is crucial and its different effects are unignorable. Overall, the present study provides a practical strategy of double specularity coefficients for the solid wall boundary conditions during two-fluid modeling.

Validation and application of numerical modeling for in-vessel melt retention in corium pools

In a hypothetical severe accident scenario in light water reactors (LWR), after the loss of primary coolant, the melted core moves to the lower plenum of the reactor pressure vessel and accumulate there. The melted core cautiously releases the decay heat which forms a pool of melted core, called corium, and turbulent natural convection starts. The decay heat transferred by the corium to the vessel may result in vessel failure due to the focusing effect in the absence of an effective cooling system. A numerical analysis is carried out using STAR-CCM+ commercial software to investigate the ability of the existing turbulence models and Algebraic Heat Flux Model (AHFM) to simulate the natural convective heat transfer phenomena in the corium pools. Efforts have been made to predict the BALI experimental results which are designed to simulate the heat transfer in corium pools, in the framework of severe accident studies. For BALI experiments various test campaigns are run varying the internal Rayleigh number, the viscosity of the simulant fluid, and test facility height. Various such experimental cases are verified and compared with numerical simulations. Temperature profiles along a vertical line, surface heat flux along the curved surface, average Nusselt number on the top, and curved surfaces are estimated. The effect of wall boundary conditions on heat transfer is analyzed. The numerical analysis is extended to Pressurized Water Reactor (PWR) and Boiling Water Reactor (BWR) geometries. The CFD analysis predicts the stratified zone and upper mixing zone very well, and the temperature profiles and average heat transfer values are in agreement with the experiments.

Comparative appraisal of nanofluid flows in a vertical channel with constant wall temperatures: an application to the rocket engine nozzle

ABSTRACT This study aims to investigate the entropy generation of fully developed nanofluid flow in a spongy medium between two fixed absorbent vertical plates with a transpiration cooling mechanism. For heat transfer analysis exponential space-dependent heat source and thermal dependent heat source along with linear radiation are employed. Three distinct scenarios for thermal wall boundary conditions are investigated: i) Constant walls temperature ii) Convective boundary condition on the left wall of the channel and iii) convective boundary conditions on both walls of the channel. Two nanofluids combinations i.e. copper (cylindrical shape) nanoparticles immersed in water and silver (platelet shape) nanoparticles in water are used. It is also noted that studies on the transpiration of nanofluid in rocket engines for cooling purposes are scrutinized when walls are stretchy. But the walls of the rocket engine are fixed. Keeping this in mind, we analyzed the cooling process of the rocket nozzle with a fixed wall channel. The numerical solution of the model equations is attained using the bvp4c method in MATLAB. Findings indicate that the copper/water nanofluid combination performs better than silver/water as for as cooling the nozzle of a rocket engine is concerned. Moreover, convective boundary conditions lower the fluid temperature.

Turbulent flow simulations of the common research model on Cartesian grids using recursive fitting approach

In this study, we propose a new method that uses the recursive fitting method and wall function for the three-dimensional turbulent flow simulation. The recursive fitting method based on the Cartesian grid method is employed for the automatic generation around two- and three-dimensional geometries. The grid approximately represents the concave and convex features, i.e., a wing–fuselage juncture and wing trailing edges. The wall boundary conditions can be imposed on the wall faces; thus, the conservation laws are strictly satisfied. Meanwhile, using the wall function allows turbulent boundary layers to be accurately reproduced. The turbulent flow simulation using the wall function is straightforward because the flow variables at the first cell centroid are employed for friction velocity estimation. However, the distance between the body surface and the first cell centroid varies since cells with different shapes are generated. Therefore, the simulation using the proposed method is validated through turbulent flow simulations in two dimensions. The results show the accurate predictions of the turbulent boundary layers. To investigate the capacity of the proposed method, three-dimensional simulation around aircraft at a transonic cruise condition is conducted. The automatically generated grids by the proposed method approximately represent the geometric features. The results of flow simulation show that the surface pressure and skin friction coefficient distributions agree with those of the reference simulation on a conventional body-fitted grid. Further, predicted pressure and viscous drag coefficients are reasonable compared with those of the reference simulation.

Arctic curves of the 6V model with partial DWBC and double Aztec rectangles

Previous numerical studies have shown that in the disordered and anti-ferroelectric phases the six-vertex (6V) model with partial domain wall boundary conditions (DWBC) exhibits an arctic curve whose exact shape is unknown. The model is defined on a s × n square lattice (s ⩽ n). In this paper, we derive the analytic expression of the arctic curve, for a = b = 1 and (Δ = 0), while keeping the ratio s/n ∈ [0, 1] as a free parameter. The computation relies on the tangent method. We also consider domino tilings of double Aztec rectangles and show via the tangent method that, for particular parameters, the arctic curve is identical to that of the 6V model with partial DWBC. Our results are confirmed by extensive numerical simulations.

Enstrophy evolution during head-on wall interaction of premixed flames within turbulent boundary layers

The statistical behaviors of mean enstrophy and its evolution during head-on interaction of premixed flames propagating toward a chemically inert flat wall across the turbulent boundary layer have been analyzed using direct numerical simulations for a friction velocity-based Reynolds number of Reτ=110. The enstrophy dynamics have been analyzed for both isothermal and adiabatic thermal wall boundary conditions. The contributions of vortex-stretching and viscous dissipation are found to be leading order source and sink, respectively, to the mean enstrophy transport in both non-reacting and reacting flows irrespective of the wall boundary condition. However, the contributions due to dilatation rate and baroclinic torque play important roles in addition to the leading order contributions of the vortex-stretching and viscous dissipation terms in the enstrophy transport in turbulent premixed flames. The thermal boundary condition has been demonstrated to affect the near-wall behavior of the enstrophy transport contribution due to dilatation rate, which also affects the near-wall distribution of the enstrophy. The magnitudes of the leading order contributors to the enstrophy transport decrease with the progress of head-on interaction for both wall boundary conditions. Moreover, the overall sink contributions to the enstrophy transport dominate over the source contributions, giving rise to a drop in the mean enstrophy with the progress of head-on interaction. The enstrophy distribution changes significantly during flame-wall interaction, which gives rise to a modification of the relative proportion of the coherent structures in the reacting flow turbulent boundary layer compared to the corresponding non-reacting flow features.