Statistics and Modelling of Turbulent Kinetic Energy Transport in Different Regimes of Premixed Combustion

The statistical behaviour of turbulent kinetic energy transport in turbulent premixed flames is analysed using data from three-dimensional Direct Numerical Simulation (DNS) of freely propagating turbulent premixed flames under decaying turbulence. For flames within the corrugated flamelets regime, it is observed that turbulent kinetic energy is generated within the flame brush. By contrast, for flames within the thin reaction zones regime it has been found that the turbulent kinetic energy decays monotonically through the flame brush. Similar trends are observed also for the dissipation rate of turbulent kinetic energy. Within the corrugated flamelets regime, it is demonstrated that the effects of the mean pressure gradient and pressure dilatation within the flame are sufficient to overcome the effects of viscous dissipation and are responsible for the observed augmentation of turbulent kinetic energy in the flame brush. In the thin reaction zones regime, the effects of the mean pressure gradient and pressure dilatation terms are relatively much weaker than those of viscous dissipation, resulting in a monotonic decay of turbulent kinetic energy across the flame brush. The modelling of the various unclosed terms of the turbulent kinetic energy transport equation has been analysed in detail. The predictions of existing models are compared with corresponding quantities extracted from DNS data. Based on this a-priori DNS assessment, either appropriate models are identified or new models are proposed where necessary. It is shown that the turbulent flux of turbulent kinetic energy exhibits counter-gradient (gradient) transport wherever the turbulent scalar flux is counter-gradient (gradient) in nature. A new model has been proposed for the turbulent flux of turbulent kinetic energy, and is found to capture the qualitative and quantitative behaviour obtained from DNS data for both the corrugated flamelets and thin reaction zones regimes without the need to adjust any of the model constants.     

A DNS Study of Closure Relations for Convection Flux Term in Transport Equation for Mean Reaction Rate in Turbulent Flow

The present work aims at modeling the entire convection flux \(\overline {\rho \mathbf {u}W}\) in the transport equation for a mean reaction rate \(\overline {\rho W}\) in a turbulent flow, which (equation) was recently put forward by the present authors. In order to model the flux, several simple closure relations are developed by introducing flow velocity conditioned to reaction zone and interpolating this velocity between two limit expressions suggested for the leading and trailing edges of the mean flame brush. Subsequently, the proposed simple closure relations for \(\overline {\rho \mathbf {u}W}\) are assessed by processing two sets of data obtained in earlier 3D Direct Numerical Simulation (DNS) studies of adiabatic, statistically planar, turbulent, premixed, single-step-chemistry flames characterized by unity Lewis number. One dataset consists of three cases characterized by different density ratios and is associated with the flamelet regime of premixed turbulent combustion. Another dataset consists of four cases characterized by different low Damköhler and large Karlovitz numbers. Accordingly, this dataset is associated with the thin reaction zone regime of premixed turbulent combustion. Under conditions of the former DNS, difference in the entire, \(\overline {\rho {u}W}\), and mean, \(\tilde {u}\overline {\rho W}\), convection fluxes is well pronounced, with the turbulent flux, \(\overline {\rho u^{\prime \prime }W^{\prime \prime }}\), showing countergradient behavior in a large part of the mean flame brush. Accordingly, the gradient diffusion closure of the turbulent flux is not valid under such conditions, but some proposed simple closure relations allow us to predict the entire flux \(\overline {\rho \mathbf {u}W}\) reasonably well. Under conditions of the latter DNS, the difference in the entire and mean convection fluxes is less pronounced, with the aforementioned simple closure relations still resulting in sufficiently good agreement with the DNS data.     

Application of Reynolds stress transport turbulence closure models to flows affected by Lorentz and buoyancy forces

The effect of magnetic field strength and orientation on two types of electromagnetically influenced turbulent flows was studied numerically under the Reynolds averaged Navier–Stokes (RANS) framework. Previous work (Wilson et al., 2014) used an electromagnetically extended linear eddy-viscosity model, whilst the current paper focuses on the performance of a more advanced Reynolds stress transport type model both with and without electromagnetic modifications proposed by Kenjereš et al. (2004). First, a fully-developed 2D channel flow is considered with a magnetic field imposed in either the wall-normal or streamwise direction. Both forms of the RSM gave good agreement with the DNS data for the wall-normal magnetic field across the range of Hartmann numbers with the additional electromagnetic terms providing a small, but noticeable, difference. For the streamwise magnetic field, where electromagnetic influence is only through the turbulence, the electromagnetically extended RSM performed well at moderate Hartmann numbers but returned laminar flow at the highest Hartmann number considered, contrary to the DNS. The RSM results were, however, significantly better than the previous eddy-viscosity model predictions. The second case is that of unsteady 3D Rayleigh–Bénard convection with a magnetic field imposed in either a horizontal or vertical direction. Results revealed that a significant reorganization of the flow structures is predicted to occur. For a vertically oriented magnetic field, the plume structures increase in number and become thinner and elongated along the magnetic field lines, leading to an increase in thermal mixing within the core in agreement with Hanjalić and Kenjereš (2000). With a horizontal magnetic field, the structures become two-dimensional and a striking realignment of the roll cells’ axes with the magnetic field lines occurs. The results demonstrate the capability of the Reynolds stress transport approach in modelling MHD flows that are relevant to industry and offer potential for those wishing to control levels of turbulence, heat transfer or concentration without recourse to mechanical means.     

Natural convection flow under a magnetic field in an inclined square enclosure differentialy heated on adjacent walls

Steady, laminar, natural-convection flow in the presence of a magnetic field in an inclined square enclosure differentially heated along the bottom and left vertical walls while the other walls are kept isothermal was considered. The governing equations were solved numerically for the stream function, vorticity and temperature ratio using the differential quadrature method for various Grashof and Hartmann numbers, inclination angle of the enclosure and direction of the magnetic field. The orientation of the enclosure changes the temperature gradient inside and has a significant effect on the flow pattern. Magnetic field suppresses the convective flow and its direction also influences the flow pattern, causing the appearance of inner loops and multiple eddies. The surface heat flux along the bottom wall is slightly increased by clockwise inclination and reduced by half by the counterclockwise inclination. The surface heat flux along the upper portion of the left side wall is reversed by the rise of warmer fluids due to the convection currents for no inclination and clockwise inclination of the enclosure.     

流向磁场作用下圆柱绕流的直接数值模拟

绕流是托卡马克装置中液态包层内常见的流动形态,对流场与热量分布有着重要的影响.本文通过直接数值模拟(DNS),研究了不同磁场强度下$Re=3900$的圆柱绕流,分析了磁场强度对于湍流尾迹的影响.无磁场情况下,直接数值模拟的结果与前人的实验及模拟结果吻合很好.圆柱下游的尾迹中,随着流向距离的增大, 流向速度剖面逐渐从U型进化呈V型, 并慢慢趋于平缓,这表明尾迹中的流动结构受圆柱影响逐渐减小.圆柱后方两侧的剪切层中,由于Kelvin-Helmholtz不稳定性的影响,可以清晰地看到小尺度剪切层涡的脱落.通过对无磁场的计算结果施加流向磁场,本文计算了哈特曼数($Ha$)分别为20, 40和80的工况,以研究磁场效应对于湍流的影响.结果表明磁场较弱时,流动依然呈三维湍流状态.随着磁场增强, 近圆柱尾流区受磁场抑制明显,回流区被拉长,剪切层失稳位置向下游转移.圆柱后方的涡结构由于受到竖直方向洛伦兹力的挤压作用,随着哈特曼数的增加尾迹区域逐渐变窄.相比于无磁场情况的涡结构,由于磁场的耗散作用,相应的涡结构尺度变小.该研究不仅扩展了现有磁场下湍流运动的参数范围,对于液态包层的设计及安全运行同样具有重要的理论指导意义和工程应用价值.      

Buoyancy modelling with incompressible SPH for laminar and turbulent flows

This work aims to model buoyant, laminar or turbulent flows, using a two‐dimensional incompressible smoothed particle hydrodynamics model with accurate wall boundary conditions. The buoyancy effects are modelled through the Boussinesq approximation coupled to a heat equation, which makes it possible to apply an incompressible algorithm to compute the pressure field from a Poisson equation. Based on our previous work [1], we extend the unified semi‐analytical wall boundary conditions to the present model. The latter is also combined to a Reynolds‐averaged Navier–Stokes approach to treat turbulent flows. The k ? ? turbulence model is used, where buoyancy is modelled through an additional term in the k ? ? equations like in mesh‐based methods. We propose a unified framework to prescribe isothermal (Dirichlet) or to impose heat flux (Neumann) wall boundary conditions in incompressible smoothed particle hydrodynamics. To illustrate this, a theoretical case is presented (laminar heated Poiseuille flow), where excellent agreement with the theoretical solution is obtained. Several benchmark cases are then proposed: a lock‐exchange flow, two laminar and one turbulent flow in differentially heated cavities, and finally a turbulent heated Poiseuille flow. Comparisons are provided with a finite volume approach using an open‐source industrial code. Copyright © 2015 John Wiley & Sons, Ltd.     

DIRECT NUMERICAL SIMULATIONS ON THE TURBULENT FLOW PAST A CONFINED CIRCULAR CYLINDER WITH THE INFLUENCE OF THE STREAMWISE MAGNETIC FIELDS1)

绕流是托卡马克装置中液态包层内常见的流动形态,对流场与热量分布有着重要的影响.本文通过直接数值模拟(DNS),研究了不同磁场强度下$Re=3900$的圆柱绕流,分析了磁场强度对于湍流尾迹的影响.无磁场情况下,直接数值模拟的结果与前人的实验及模拟结果吻合很好.圆柱下游的尾迹中,随着流向距离的增大, 流向速度剖面逐渐从U型进化呈V型, 并慢慢趋于平缓,这表明尾迹中的流动结构受圆柱影响逐渐减小.圆柱后方两侧的剪切层中,由于Kelvin-Helmholtz不稳定性的影响,可以清晰地看到小尺度剪切层涡的脱落.通过对无磁场的计算结果施加流向磁场,本文计算了哈特曼数($Ha$)分别为20, 40和80的工况,以研究磁场效应对于湍流的影响.结果表明磁场较弱时,流动依然呈三维湍流状态.随着磁场增强, 近圆柱尾流区受磁场抑制明显,回流区被拉长,剪切层失稳位置向下游转移.圆柱后方的涡结构由于受到竖直方向洛伦兹力的挤压作用,随着哈特曼数的增加尾迹区域逐渐变窄.相比于无磁场情况的涡结构,由于磁场的耗散作用,相应的涡结构尺度变小.该研究不仅扩展了现有磁场下湍流运动的参数范围,对于液态包层的设计及安全运行同样具有重要的理论指导意义和工程应用价值.     

Modeling plane turbulent Couette flow

Among the salient features of shear-driven plane Couette flow is the constancy of the total shear stress (viscous and turbulent) across the flow. This constancy gives rise to a quasi-homogenous core region, which makes the bulk of the flow substantially different from pressure-driven Poiseuille flow. The present second-moment closure study addresses the conflicting hypotheses relating to turbulent Couette flow. The inclusion of a new wall-proximity function in the wall-reflection part of the pressure-strain model seems mandatory, and the greement with recent experimental and direct numerical simulation (DNS) results is encouraging. Analysis of model computations in the range 750 ≤ Re ≤ 35,000 and comparisons with low-Re DNS data suggest that plane Couette flow exhibits a local-equilibrium core region, in which anisotropic, homogeneous turbulence prevails. However, the associated variation of the mean velocity in the core, as obtained by the model, conflicts with the intuitively appealing assumption of homogeneous mean shear. The constancy of the velocity gradient exhibited by the DNS therefore signals a deficiency in the modeled transport equation for the energy dissipation rate.     

Experimental investigation of the resistance to the flow of a conducting fluid in flat insulated ducts in the presence of a transverse magnetic field with allowance for fringe effects and wall roughness

The distribution of pressure, velocity, and electrical potential has been investigated for a mercury flow in insulated rectangular ducts with a large side ratio (Hartmann-type flow). The ranges of variation of the Reynolds, Hartmann, and Stewart numbers were 7·10²R5·10⁵, 0H490, and 0N24, respectively. Special attention is given to the sections of the channel where the flow enters and leaves the magnetic field. In these zones the pressure is sharply nonuniform and the velocity profiles in a plane perpendicular to the field acquire an M shape. A relation is established between the length of the entrance section, where the flow is three-dimensional, and the MHD similarity criteria. It is shown that ducts which are hydraulically smooth in the absence of a magnetic field become increasingly rough as the field grows stronger. Data are obtained on the resistance coefficient for a stabilized flow measured in a magnetic field and on the dependence of the critical Reynolds number on the Hartmann number.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 10–21, July–August, 1971.     

Spontaneous rotation in the exact solution of magnetohydrodynamic equations for flow between two stationary impermeable disks

Magnetohydrodynamic (MHD) flow of a viscous electrically conducting incompressible fluid between two stationary impermeable disks is considered. A homogeneous electric current density vector normal to the surface is specified on the upper disk, and the lower disk is nonconducting. The exact von Karman solution of the complete system of MHD equations is studied in which the axial velocity and the magnetic field depend only on the axial coordinate. The problem contains two dimensionless parameters: the electric current density on the upper plate Y and the Batchelor number (magnetic Prandtl number). It is assumed that there is no external source that produces an axial magnetic field. The problem is solved for a Batchelor number of 0–2. Fluid flow is caused by the electric current. It is shown that for small values of Y, the fluid velocity vector has only axial and radial components. The velocity of motion increases with increasing Y, and at a critical value of Y, there is a bifurcation of the new steady flow regime with fluid rotation, while the flow without rotation becomes unstable. A feature of the obtained new exact solution is the absence of an axial magnetic field necessary for the occurrence of an azimuthal component of the ponderomotive force, as is the case in the MHD dynamo. A new mechanism for the bifurcation of rotation in MHD flow is found.     

'T-RANS' Simulation of Deterministic Eddy Structure in Flows Driven by Thermal Buoyancy and Lorentz Force

The paper reports on the application of the Time-dependent Reynolds-Averaged Navier–Stokes (T-RANS) approach to analysing the effects of magnetic force and bottom-wall configuration on the reorganisation of a large coherent structure and its role in the transport processes in Rayleigh–Bénard convection. The large-scale deterministic motion is fully resolved in time and space, whereas the unresolved stochastic motion is modelled by a `subscale' model for which the conventional algebraic stress/flux expressions were used, closed with the low-Re number (k)-(ε)-(θ²) three-equation model. The applied method reproduces long-term averaged mean flow properties, turbulence second moments, and all major features of the coherent roll/cell structure in classic Rayleigh–Bénard convection in excellent agreement with the available DNS and experimental results. Application of the T-RANS approach to Rayleigh–Bénard convection with wavy bottom walls and a superimposed magnetic field yielded the expected effects on there organisation of the eddy structure and consequent modifications of the mean and turbulence parameters and wall heat transfer. This revised version was published online in August 2006 with corrections to the Cover Date.     

Finite element method for unsteady MHD flow through pipes with arbitrary wall conductivity

A finite element method is given to obtain the numerical solution of the coupled equations in velocity and magnetic field for unsteady MHD flow through a pipe having arbitrarily conducting walls. Pipes of rectangular, circular and triangular sections have been taken for illustration. Computations have been carried out for different Hartmann numbers and wall conductivity at various time levels. It is found that if the wall conductivity increases, the flux through a section decreases. The same is the effect of increasing the Hartmann number. It is also observed that the steady state is approached at a faster rate for larger Hartmann numbers or larger wall conductivity. Selected graphs are given showing the behaviour of velocity, induced magnetic field and flux across a section.     

Buoyant flow in long vertical enclosures in the presence of a strong horizontal magnetic field. Part 2. Finite enclosures

Numerical computations and experiments were carried out for a buoyant flow of liquid metal (mercury in the experiments) in a long vertical enclosure of square cross-section, in the presence of a uniform horizontal magnetic field. A strong emphasis is put on the case of a magnetic field perpendicular to the applied temperature gradient for two reasons: (1) the MHD damping is smaller than with any other orientation, and (2) the quasi-two-dimensionality of the flow in this case yields a quite efficient velocity measurement technique. The enclosure is heated by a thermally controlled flow of water from one of the vertical walls and cooled by a similar technique from the facing wall. Those two walls are good thermal conductors (thick copper plates in the experiments), whereas the four other walls are thermally insulating. All walls are electrically insulated from the fluid. In this paper, as well as in the companion paper by Tagawa et al. (Eur. J. Mech. B Fluids 21 (4) (2002) 383–398), we model analytically the Hartmann layers present along the walls perpendicular to the magnetic field. This modeling, which yields boundary conditions for the core flow without any meshing of the thin layers, is quite accurate when Hartmann layers are stable. The numerical results are in fairly good agreement with the experimental data. They namely reveal how the heat flux and the fluid flow organization depend on the magnetic field.     

The structure of parallel layers in steady two-dimensional magnetohydrodynamic flows in sudden duct expansions and contractions

The structure of non-linear, steady, two-dimensional parallel layers at high values of the Hartmann number, Ha, the Reynolds number, Re, and the interaction parameter, N, for duct expansions and contractions has been investigated. The magnetic field is transverse to the flow. For the flow regime, in which the electromagnetic force balances the inertial force in the layer, a viscous sublayer at the solid wall parallel to the magnetic field has been obtained, in which the flow is driven by the pressure gradient induced in the outer, inviscid layer.     

Analytical investigation of Jeffery-Hamel flow with high magnetic field and nanoparticle by Adomian decomposition method

In this study, the effects of magnetic field and nanoparticle on the Jeffery-Hamel flow are studied using a powerful analytical method called the Adomian decomposition method (ADM). The traditional Navier-Stokes equation of fluid mechanics and Maxwell’s electromagnetism governing equations are reduced to nonlinear ordinary differential equations to model the problem. The obtained results are well agreed with that of the Runge-Kutta method. The present plots confirm that the method has high accuracy for different α, Ha, and Re numbers. The flow field inside the divergent channel is studied for various values of Hartmann number and angle of channel. The effect of nanoparticle volume fraction in the absence of magnetic field is investigated.     

Effects of magnetic field on nanofluid forced convection in a partially heated microchannel

This paper numerically examines the laminar forced convection of a water–Al₂O₃ nanofluid flowing through a horizontal microchannel. The middle section of the microchannel is heated with a constant and uniform heat flux. The middle section is also influenced by a transverse magnetic field with a uniform strength. The effects of pertinent parameters such as the Reynolds number (0≤Re≤1000), the solid volume fraction (0≤?≤0.04) and the Hartmann number (0≤Ha≤100) on the flow and temperature fields and the heat transfer performance of the microchannel are examined against numerical predictions. The results show that the microchannel performs better heat transfers at higher values of the Reynolds and Hartmann numbers. For all values of the Reynolds and Hartmann numbers considered in this study, the average Nusselt number on the middle section surface of the microchannel increases as the solid volume fraction increases. The rate of this increase is considerably more at higher values of the Reynolds number and at lower values of the Hartmann number.     

Hydromagnetic free convection flow with induced magnetic field effects

An exact solution is presented for the hydromagnetic natural convection boundary layer flow past an infinite vertical flat plate under the influence of a transverse magnetic field with magnetic induction effects included. The transformed ordinary differential equations are solved exactly, under physically appropriate boundary conditions. Closed-form expressions are obtained for the non-dimensional velocity (u), non-dimensional induced magnetic field component (B _x ) and wall frictional shearing stress i.e. skin friction function (τ _x ) as functions of dimensionless transverse coordinate (η), Grashof free convection number (G _r ) and the Hartmann number (M). The bulk temperature in the boundary layer (Θ) is also evaluated and shown to be purely a function of M. The Rayleigh flow distribution (R) is derived and found to be a function of both Hartmann number (M) and the buoyant diffusivity parameter (ϑ ^*). The influence of Grashof number on velocity, induced magnetic field and wall shear stress profiles is computed. The response of Rayleigh flow distribution to Grashof numbers ranging from 2 to 200 is also discussed as is the influence of Hartmann number on the bulk temperature. Rayleigh flow is demonstrated to become stable with respect to the width of the boundary layer region and intensifies with greater magnetic field i.e. larger Hartman number M, for constant buoyant diffusivity parameter ϑ ^*. The induced magnetic field (B _x ), is elevated in the vicinity of the plate surface with a rise in free convection (buoyancy) parameter G _r , but is reduced over the central zone of the boundary layer regime. Applications of the study include laminar magneto-aerodynamics, materials processing and MHD propulsion thermo-fluid dynamics.     

Effect of magnetic field on 3D flow and heat transfer during solidification from a melt

We present the effect of a magnetic field on three-dimensional fluid flow and heat transfer during solidification from a melt in a cubic enclosure. The walls of the enclosure are considered perfectly electrically conducting and the magnetic field is applied separately in three directions. The finite-volume method with enthalpy formulation is used to solve the mathematical model in the solid and liquid phases. The results obtained by our computer code are compared with the numerical and experimental data found in the literature. For Gr = 5 × 10⁵ and Ha = 0, 25, 50, 75, and 100 (where Gr and Ha are the Grashof and Hartmann numbers, respectively), the effects of magnetic field on flow and thermal fields, and on solid/liquid interface shape are presented and discussed. The interface is localized with and without magnetic field. The results show a strong dependence between the interface shape and the intensity and orientation of magnetic field. When the magnetic field is applied along the X-direction, the magnetic stability diagrams (V_max–Ha) and (Nu_avg–Ha) show the strongest stabilization of the flow field and heat transfer.     

Mixed convection stagnation-point flow on vertical stretching sheet with external magnetic field

The problem of steady laminar magnetohydrodynamic （MHD） mixed con- vection stagnation-point flow of an incompressible viscous fluid over a vertical stretch- ing sheet is studied. The effect of an externally magnetic field is taken into account. The transformed boundary layer equations are solved numerically by using an implicit finite-difference scheme. Numerical results are obtained for various values of the mixed convection parameter, Hartmann number, and Prandtl number. The effects of an exter- nally magnetic field on the skin friction coefficient, local Nusselt number, velocity, and temperature profiles for both A 〉 1 and A ~ 1, where A is the velocity ratio parameter, are presented graphically and discussed in detail. Both assisting and opposing flows are considered, and it is found that dual solutions exist for the opposing flow.     

Effect of induced magnetic field on natural convection in vertical concentric annuli

In the present paper,we have considered the steady fully developed laminar natural convective flow in open ended vertical concentric annuli in the presence of a radial magnetic field.The induced magnetic field produced by the motion of an electrically conducting fluid is taken into account.The transport equations concerned with the considered model are first recast in the non-dimensional form and then unified analytical solutions for the velocity,induced magnetic field and temperature field are obtained for the cases of isothermal and constant heat flux on the inner cylinder of concentric annuli.The effects of the various physical parameters appearing into the model are demonstrated through graphs and tables.It is found that the magnitude of maximum value of the fluid velocity as well as induced magnetic field is greater in the case of isothermal condition compared with the constant heat flux case when the gap between the cylinders is less or equal to 1.70 times the radius of inner cylinder,while reverse trend occurs when the gap between the cylinders is greater than 1.71 times the radius of inner cylinder.These fields are almost the same when the gap between the cylinders is equal to 1.71 times the radius of inner cylinder for both the cases.It is also found that as the Hartmann number increases,there is a flattening tendency for both the velocity and the induced magnetic field.The influence of the induced magnetic field is to increase the velocity profiles.