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A numerical study of the turbulent flow of molten steel in a domain with a phase-change boundary
Abstract
A numerical algorithm, based on the Galerkin finite element method and the enthalpy formulation, is developed for solving the coupled turbulent fluid flow and heat transfer problem in a domain with a moving phase-change boundary. The governing equations consist of the continuity equation, the Navier–Stokes equations, the energy equation and the modified K–ε equations. The formulation of the method is cast into the framework of the Bubnov–Galerkin finite element method. A numerical study shows that the developed numerical algorithm is stable and capable of capturing the rapid change of temperature and velocity near the phase-change boundary.
... Numerical modeling of solidification during the continuous casting process has had widespread application as an effective tool for better understanding the transporting and solidifying features. However, most of mathematical models were focused on the analysis of fluid flow, heat transfer and solidification in the conventional slab/billet mold without EMBR [1][2][3][4][5][6][7][8][9][10][11][12] or with EMBR. [13][14][15] In terms of numerical analysis of thin-slab casting process, [16][17][18][19][20][21][22][23] O'Connor et al. 16) analyzed thermo-mechanical state in the funnel-type mold of CSP thin-slab caster. ...
... 7,8,13) 3.1.3. Turbulent Model For calculating the turbulent viscosity m t in Eq. (2), various turbulent models, such as Prandtl mixing-length-type model, 2) one-equation model, 1) low-Reynolds [7][8][9][10]14) and high-Reynolds [3][4][5]11,13) number two-equation k-e models, and large eddy simulation model 15) have been proposed in the modeling of the continuous casting solidification process. Although some investigation suggested that the numerical treatments of turbulent models and its suppression approach affected the mushy zone, 14,27,28) a realizable k-e turbulent model equations, 12,29) which possibly better describes the mold flow with higher turbulent Reynolds number and predicts the spreading rate of SEN impinging jets in a thin slab casting at higher casting speed, is used in this study. ...
... ................... (11) Where |u s | is the casting speed, ( , ) stands for inner product. n z is a vector along the casting direction and n f is normal vector of the curved funnel surface. ...
A mathematical model of three-dimensional fluid flow, heat transfer and solidification in thin slab casting is developed based on the enthalpy-porosity approach in a single-phase framework and used to analyze and compare the fluid flow and thermal behaviors in the funnel-type molds of Compact Strip Production (CSP®) and Flexible Thin Slab Casting (FTSC®) caster. An Electromagnetic Brake (EMBR) model is also incorporated into this coupled model to see the effects of EMBR on fluid flow and solidification in a CSP caster with EMBR system. The numerical treatments specific to the complex funnel-shaped mold boundaries are discussed. The obtained numerical results are partly validated by those available in literature and theoretical analyses. The fluid flow and thermal features of thin slab in CSP mold and FTSC mold are analyzed and their comparisons between them are made. It is found that the flow pattern, level fluctuation, superheat dissipation, temperature distribution and shell growth in both molds present different features. The structure of SEN strongly affects the flow pattern of the impinging jet from SEN ports, and then the dissipation mode of the melt superheat and the uniform growth of solidification shell. A four-port SEN® with two small upper ports in FTSC mold is helpful to activate the meniscus flow and increase the meniscus temperature, but high level fluctuation may occur. The EMBR in a CSP mold can both suppress the meniscus flow and increase the meniscus temperature. A complete flow-control system including the design of SEN, EMBR and operational conditions is essential for thin slab production considering their individual flow and thermal features in the funnel-type molds of CSP and FTSC casters.
... To achieve high production rate and ensure high quality of the casting products, intensive studies have been carried out worldwide over the last few decades to model various aspects of the continuous casting process, in particular the heat transfer -solidification process [10,27], the flow phenomena [22,25], the electromagnetic stirring [12,14,15,21] and formation of oscillation marks [11]. However, as analyzed by Thomas [23], the continuous casting process involves a staggering complexity of at least 18 interacting phenomena at the mechanic level, and due to the complexity, previous work mainly focus on each individual phenomena or interaction of two or three phenomena only. ...
... In this work, we further develop our previous works [25,29] in various aspects. In [25], an enthalpy finite element method was developed to model the interacting phenomena of heat transfer, solidification and turbulence in the casting process. ...
... In this work, we further develop our previous works [25,29] in various aspects. In [25], an enthalpy finite element method was developed to model the interacting phenomena of heat transfer, solidification and turbulence in the casting process. The work of [29] extend the model of [25] by incorporating the effect of the electromagnetic field in the coupled heat transfer -turbulence flow model. ...
This paper presents a mathematical model and numerical technique for simulating the two-fluid flow and the meniscus interface movement in the electromagnetic continuous steel casting process. The governing equations include the continuity equation, the momentum equations, the energy equation, the level set equation and two transport equations for the electromagnetic field derived from the Maxwell’s equations. The level set finite element method is applied to trace the movement of the interface between different fluids. In an attempt to optimize the casting process, the technique is then applied to study the influences of the imposed electromagnetic field and the mould oscillation pattern on the fluid flow, the meniscus shape and temperature distribution.
... Some attempts have also been made to derive alternative formulae for the determination of the slip length Yang and Zhu, (2006). Although exact and numerical solutions to various flow problems of Newtonian fluids under the no-slip assumption have been obtained and are available in literature (Slattery, 1999;Wu, B. Wiwatanapataphee, 2007;Wiwatanapataphee, et al., 2004;Wiwatanapataphee, et al., 2004), very few exact solutions for the slip case are available in literature. Recently, some steady state and transient slip solutions for the flows through a pipe, a channel and an annulus have been obtained (Yang, K.Q. ...
... Some attempts have also been made to derive alternative formulae for the determination of the slip length Yang and Zhu, (2006). Although exact and numerical solutions to various flow problems of Newtonian fluids under the no-slip assumption have been obtained and are available in literature (Slattery, 1999;Wu, B. Wiwatanapataphee, 2007;Wiwatanapataphee, et al., 2004;Wiwatanapataphee, et al., 2004), very few exact solutions for the slip case are available in literature. Recently, some steady state and transient slip solutions for the flows through a pipe, a channel and an annulus have been obtained (Yang, K.Q. ...
In this paper, we present analytical solutions for transient flow of Newtonian fluid through micro channels with Navier slip boundary. The derivation of the solutions is based on Fourier series expansion in space. We then investigate the influence of the slip parameters on the phenomena of the transient flow and show the transient pressure field and velocity profile at various instants of time.
... Some attempts have also been made to derive alternative formulae for the determination of the slip length Yang and Zhu, (2006). Although exact and numerical solutions to various flow problems of Newtonian fluids under the no-slip assumption have been obtained and are available in literature (Slattery, 1999;Wu, B. Wiwatanapataphee, 2007;Wiwatanapataphee, et al., 2004;Wiwatanapataphee, et al., 2004), very few exact solutions for the slip case are available in literature. Recently, some steady state and transient slip solutions for the flows through a pipe, a channel and an annulus have been obtained (Yang, K.Q. ...
... Some attempts have also been made to derive alternative formulae for the determination of the slip length Yang and Zhu, (2006). Although exact and numerical solutions to various flow problems of Newtonian fluids under the no-slip assumption have been obtained and are available in literature (Slattery, 1999;Wu, B. Wiwatanapataphee, 2007;Wiwatanapataphee, et al., 2004;Wiwatanapataphee, et al., 2004), very few exact solutions for the slip case are available in literature. Recently, some steady state and transient slip solutions for the flows through a pipe, a channel and an annulus have been obtained (Yang, K.Q. ...
In this paper, we present analytical solutions for transient flow of Newtonian fluid through
micro channels with Navier slip boundary. The derivation of the solutions is based on Fourier series
expansion in space. We then investigate the influence of the slip parameters on the phenomena of the
transient flow and show the transient pressure field and velocity profile at various instants of time.
... Some attempts have also been made to derive alternative formulae for the determination of the slip length Yang and Zhu, (2006). Although exact and numerical solutions to various flow problems of Newtonian fluids under the no-slip assumption have been obtained and are available in literature (Slattery, 1999;Wu, B. Wiwatanapataphee, 2007;Wiwatanapataphee, et al., 2004;Wiwatanapataphee, et al., 2004), very few exact solutions for the slip case are available in literature. Recently, some steady state and transient slip solutions for the flows through a pipe, a channel and an annulus have been obtained (Yang, K.Q. ...
... Some attempts have also been made to derive alternative formulae for the determination of the slip length Yang and Zhu, (2006). Although exact and numerical solutions to various flow problems of Newtonian fluids under the no-slip assumption have been obtained and are available in literature (Slattery, 1999;Wu, B. Wiwatanapataphee, 2007;Wiwatanapataphee, et al., 2004;Wiwatanapataphee, et al., 2004), very few exact solutions for the slip case are available in literature. Recently, some steady state and transient slip solutions for the flows through a pipe, a channel and an annulus have been obtained (Yang, K.Q. ...
We develop a mathematical model and numerical technique to study the coupled turbulent flow and heat transfer process in continuous steel casting under an electromagnetic force. The complete set of field equations is solved numerically. We examine the influence of the electromagnetic field on flow patterns of molten steel, on the temperature field and on the steel solidification.
... Thus, turbulence kinetics is critical to the accurate prediction of casting quality when simulating [6]. According to [7], improper control of turbulence and heat transfer of molten metals would result into surface erosion and entrapment of slag particles which subsequently lead to poor quality in castings. Based on this, Khan et al. [8], developed a multiphase flow model to reduce the effect of turbulence of molten metal during pouring process using different Tundish system. ...
The study investigates the effects of turbulence flow of molten metal on the walls of the runner and the mould cavity during pouring via simulation approach. Effects of temperature, solidification time, shrinkage porosity, thermal modulus, hot spot formation and interfacial heat transfer coefficient were simulated under turbulent flow condition. Turbulence kinetics and flow properties which include velocity and viscosity were equally simulated. The result of temperature simulation showed an ideal variation of temperature distribution during flow and in the cavity. Molten metal at liquid state was observed to vary from 654.7-636.0 oC and became solid at 473.3-492.0 oC for a total simulation time of 169.5838 seconds. Consequently, at higher pouring temperature, diffusion of heat into the walls of the mould will occur due to the momentum of flow thereby leading to erosion of the mould content. In addition to this, velocity and viscosity of the molten metal was found to have effect on the turbulent kinetics. Thus, this simulation technique will help the foundry industry in improving the gating system design by studying the defects associated with turbulence flow and incorporating filters to remove the inclusions in the gating system
... Detailed discussions on microstructure formation and mathematical modelling of transport phenomenon during solidification of binary systems can be found in 5 the reviews of Rappaz (1989) and Viskanta (1990). In the last few decades intensive studies have been made to model various problems, for example: to solve radiative transfer problem in triangular meshes, Feldheim & Lybaert (2004) used discrete transfer method (DTM can be see in the work of Lockwood & Shah (1981)), Galerkin finite element method was used by Wiwatanapataphee et al. (2004) and Tryggvason et al. (2005) to study the turbulent fluid flow and heat transfer problems in a domain with moving phase-change boundary and Dimova et al. (1998) also used Galerkin finite element method to solve nonlinear phenomena. Finite volume method for the calculation of solute transport in directional solidification has been studied and validated by Lan & Chen (1996). ...
... The results indicated that the electromagnetic force reduced the speed of the molten steel near the mould wall. Wu and Wiwatanapataphee [29], [30] proposed a mathematical model to solve the coupled turbulent two-fluid flow and heat transfer with solidification. However, many phenomena such as the heat transfer process, the formation of oscillation marks and the meniscus behavior, have not been fully understood nor well controlled. ...
This paper aims to study the effect of turbulence on the flow of two fluids and the heat transfer - solidification process in electromagnetic continuous steel casting. The complete set of field equations is established. The flow pattern of the fluids, the meniscus shape and temperature field as well as solidification profiles obtained from the model with and with no turbulence effect are presented. The results show that the model with turbulence gives a large circulation zone above the jet, much larger variation of the meniscus geometry, a slow solidification rate and higher temperature in the top part of the strand region.
... Nonlinear turbulence models often arise from scientific research, modelling of nonlinear phenomena, and optimal control of complex systems [14,15]. Nonlinear turbulence model can be also used in the study of heat transfer [16] and phase-change process [17]. The nonlinear solutions are more accurate in the solution of complex phenomena. ...
Most of the RANS turbulence models solve the Reynolds stress by linear hypothesis with isotropic model. They can not capture all kinds of vortexes in the turbomachineries. In this paper, an improved nonlinear k-ε turbulence model is proposed, which is modified from the RNG k-ε turbulence model and Wilcox's k-ω turbulence model. The Reynolds stresses are solved by nonlinear methods. The nonlinear k-ε turbulence model can calculate the near wall region without the use of wall functions. The improved nonlinear k-ε turbulence model is used to simulate the flow field in a curved rectangular duct. The results based on the improved nonlinear k-ε turbulence model agree well with the experimental results. The calculation results prove that the nonlinear k-ε turbulence model is available for high pressure gradient flows and large curvature flows, and it can be used to capture complex vortexes in a turbomachinery.
... Although analytical and numerical solutions to many flow problems of Newtonian fluids have been established under the no-slip assumption [11][12][13], very few exact solutions for time-dependent problems have been obtained for the slip case [14,15]. Hence, we investigate, in this paper, the time periodic electroosmotic flows in a microchannel under the Navier slip assumption. ...
Recent research confirms that slip of a fluid on the solid surface occurs at micrometer scale. Slip on solid surface may cause the change of interior material deformation which consequently leads to the change of velocity profile and stress field. This paper concerns the time
periodic electroosmotic flow in a channel with slip boundary driven by an alternating electric field, which arises from the study of particle manipulation and separation such as flow pumping and mixing enhancement. Although exact solutions to various flow problems of electroosmotic flows under the no-slip condition have been obtained, exact solutions for problems under slip boundary conditions have seldom been addressed. In this paper, an exact solution is derived for the time periodic electroosmotic flow in two-dimensional straight channels under slip boundary conditions.
... For many fluid flow problems, under the nonslip assumption, exact and numerical solutions have been obtained and can be found in the literature [6,[16][17][18]. Steady state solutions under slip conditions have also been established for flows of Newtonian fluids through pipes, channels, and annulus [19,20]. ...
We study the slip flow of fluids driven by the combined effect of electrical force and pressure gradient. The underlying boundary value problem is solved through the use of Fourier series expansion in time and Bessel function in space. The exact solutions and numerical investigations show that the slip length and electrical field parameters have significant effects on the velocity profile. By varying these system parameters, one can achieve smooth velocity profiles or wave form profiles with different wave amplitude and frequency. This opens the way for optimizing the flow by choosing the slip length, the electrical field, and electrolyte solutions.
... As industry development is moving toward casting thin steel plates, these problems, especially the problem of molten steel breakouts, are expected to become more and more critical to the success of the process. Over the last few decades, extensive studies have been carried out to model various aspects of the continuous casting process, in particular the heat transfer and steel solidification process (Hill and Wu, 1994; Wu et al., 1994; Wu and Wiwatanapataphee, 2004), the electromagnetic stirring (Archapitak et al., 2004; Jenkins and Hoog, 1996; Kim et al., 2002; Li and Tsukihashi, 2001; Sivesson et al., 1998), the flow phenomena (Thomas, 1990, Wiwatanapataphee, 1998; and Wiwatanapataphee et al., 2004) and the formation of oscillation marks (Hill et al., 1999). However, as analyzed by Thomas (2001), the continuous casting process involves a staggering complexity of at least eighteen interacting phenomena at the mechanistic level. ...
In the continuous steel casting, an electromagnetic field, generated from the source current through the coil, is imposed to the system to control the fluid flow pattern and the steel solidification process. In this paper, we develop a mathematical model and numerical technique to study the coupled turbulent flow and heat transfer process in continuous steel casting under electromagnetic force. The complete set of field equations are established and solved numerically within the framework of the Galerkin finite element method. The influences of electromagnetic field on flow pattern of molten steel and temperature field as well as steel solidification are presented in the paper. It is found that the introduction of an electromagnetic field affects the flow field and temperature filed. In terms of the thickness of the solidified steel shell, the influence of electromagnetic field is very significant.
... The validity of the Navier slip boundary condition is supported by many experimental results131415. For many fluid flow problems, under the nonslip assumption , exact and numerical solutions have been obtained and can be found in the literature [6,161718. Steady state solutions under slip conditions have also been established for flows of Newtonian fluids through pipes, channels, and annulus [19, 20]. More recently, various analytical solutions for pressure-driven time-dependent slip flows of Newtonian fluids through microtubes and microannulus were derived [16, 18]. ...
We study the slip flow of fluids driven by the combined effect of electrical force and pressure gradient. The underlying boundary value problem is solved through the use of Fourier series expansion in time and Bessel function in space. The exact solutions and numerical investigations show that the slip length and electrical field parameters have significant effects on the velocity profile. By varying these system parameters, one can achieve smooth velocity profiles or wave form profiles with different wave amplitude and frequency. This opens the way for optimizing the flow by choosing the slip length, the electrical field, and electrolyte solutions.
... Consequently, many studies have been performed on the early solidification stage in order to aid in understanding the heat transfer mechanisms during steel melt solidification. The mostly extensively studied aspect is the analysis of heat transfer according to flow patterns in the steel melt [5][6][7][8][9][10][11]. Many studies have also been performed on the behavior of the mold flux, which is a major barrier to heat transfer from the steel melt to the copper mold, and an important factor in lubrication between the steel shell and the copper mold [12][13][14][15][16][17][18]. ...
In this study, we establish a 3-D numerical analysis model to analyze the thermal behavior and to obtain the detailed heat
transfer coefficient of a copper mold in a continuous casting system. This heat transfer coefficient changes according to
variations in the mold geometry or cooling system. For increased flow speeds of the cooling water, the heat transfer coefficient
also increases, but the rate of increase for the coefficient diminishes at higher flow speeds. As the thickness of the mold
between the melt and the cooling water slots increases, the mold’s heat transfer coefficient decreases. However, the uniformity
of the heat transfer coefficient improves with greater thickness. The effect of distance between cooling water slots on the
mold’s heat transfer coefficient is also observed. Calculations show that greater distances between cooling water slots decrease
the heat transfer coefficient.
... Detailed discussions on microstructure formation and mathematical modelling of transport phenomenon during solidification of binary systems can be found in 5 the reviews of Rappaz (1989) and Viskanta (1990). In the last few decades intensive studies have been made to model various problems, for example: to solve radiative transfer problem in triangular meshes, Feldheim & Lybaert (2004) used discrete transfer method (DTM can be see in the work of Lockwood & Shah (1981)), Galerkin finite element method was used by Wiwatanapataphee et al. (2004) and Tryggvason et al. (2005) to study the turbulent fluid flow and heat transfer problems in a domain with moving phase-change boundary and Dimova et al. (1998) also used Galerkin finite element method to solve nonlinear phenomena. Finite volume method for the calculation of solute transport in directional solidification has been studied and validated by Lan & Chen (1996). ...
... Over the last three decades, extensive studies have been carried out to model many aspects of the CC process [1,4,5,6,7,8,10]. Little work has been done to solve the coupled turbulent two-fluid flow and heat transfer with solidification [11,12,13]. These studies have resulted in a basic understanding of the physics of the CC process and provided some basic guidelines for the design of the process. ...
In this paper, we develop a mathematical model to study the coupled turbulent two-fluid flow and heat transfer process in continuous steel casting. The complete set of field equations are established. The turbulence effect on a flow pattern of molten steel and lubricant oil, meniscus shape and temperature field as well as solidifi-cation are presented in the paper.
... It should be addressed here that the various coefficient matrices related to system (20), which are not presented here for the sake of brevity, are the integration of combinations of the interpolation functions and their spatial derivatives over the surface area or volume of the element and are evaluated by using an invertible parametric coordinate transformation between the actual shape of an arbitrary element and a master element of very simple shape. The time integration method for numerical solution involves a generalized iterative scheme [24], [26] based on the backward Euler differentiation method. The convergence criteria used for the iterative scheme is given by ...
This paper aims to present a mathematical model and numerical technique to study the three-dimensional fluid flow and heat transfer in the electromagnetic steel casting process. The effect of source current density on the electromagnetic force as well as on the flow and temperature fields is in-vestigated. The numerical results indicate that the source current density of different magnitudes have a considerable effect on the flow field of molten steel and a higher source current generates a lower speed of molten steel near the mould wall. Consequently, the recirculation zone moves further away from the mould wall, and the speed of molten steel decreases while the heat transfer rate increases significantly.
... For an overview on unstructured mesh techniques see for example [1]. In the last few decades intensive studies have been done to model various problems, for example: to solve radiative transfer problem in triangular meshes [2] using discrete transfer method (DTM [3]), Galerkin finite element method used to study the turbulent fluid flow and heat transfer problems in a domain with moving phase-change boundary [4, 5] and also to solve nonlinear phenomena [6]. Finite volume (FV) method for the calculation of solute transport in directional solidification has been studied and validated in [7]. ...
In most issues representing physical problems, the complex geometry cannot be represented by a Cartesian grid. The multi-block
grid technique allows artificially reducing the complexity of the geometry by breaking down the real domain into a number
of sub-domains with simpler geometry. The main aim of this article is to show the usefulness of simple solvers in complex
geometry problems, when using curvilinear coordinates combined with multi-block grids. This requires adapted solvers to a
nine nodes computational cell instead of the five nodes computational cell used with Cartesian coordinates for two-dimensional
cases. These developments are presented for the simple iterative methods Jacobi and Gauss-Seidel and also for the incomplete
factorization method strongly implicit procedure (SIP). These adapted solvers are tested in two cases: a simple geometry (heat
transfer in a circular cross-section) and a complex geometry (solidification case). Results of the simple geometry case show
that all the adapted solvers have good performance with a slight advantage for the SIP solver. For increasing the complexity
of the geometry, the results showed that Jacobi and Gauss-Seidel solvers are not suitable. However, the SIP method has a reasonable
performance. A conclusion could be drawn that the SIP method could be used in complex geometry problems using multi-block
grid technique when high precision results are not required.
KeywordsCurvilinear coordinates-Heat transfer-Multi-block grid-Strongly implicit procedure
... Detailed discussions on microstructure formation and mathematical modelling of transport phenomenon during solidification of binary systems can be found in 5 the reviews of Rappaz (1989) and Viskanta (1990). In the last few decades intensive studies have been made to model various problems, for example: to solve radiative transfer problem in triangular meshes, Feldheim & Lybaert (2004) used discrete transfer method (DTM can be see in the work of Lockwood & Shah (1981)), Galerkin finite element method was used by Wiwatanapataphee et al. (2004) and Tryggvason et al. (2005) to study the turbulent fluid flow and heat transfer problems in a domain with moving phase-change boundary and Dimova et al. (1998) also used Galerkin finite element method to solve nonlinear phenomena. Finite volume method for the calculation of solute transport in directional solidification has been studied and validated by Lan & Chen (1996). ...
... Some attempts have also been made to derive alternative formulae for the determination of the slip length[25]. Although exact and numerical solutions to various flow problems of Newtonian fluids under the no-slip assumption have been obtained and are available in literature[22,30,28,29], very few exact solutions for the slip case are available in literature. Recently, some steady state slip solutions for the flows through a pipe, a channel and an annulus have been obtained[32,15]. ...
Recent advances in microscale experiments and molecular simulations confirm that slip of fluid on solid surface occurs at small scale, and thus the traditional no-slip boundary condition in fluid mechanics cannot be applied to flow in micrometer and nanometer scale tubes and channels. On the other hand, there is an urgent need to understand fluid flow in micrometer scale due to the emergence of biochemical lab-on-the-chip system and micro-electromechanical system fabrication technologies. In this paper, we study the pressure driven transient flow of an incompressible Newtonian fluid in microtubes with a Navier slip boundary condition. An exact solution is derived and is shown to include some existing known results as special cases. Through analysis of the derived solution, it is found that the influences of boundary slip on the flow behaviour are qualitatively different for different types of pressure fields driving the flow. For pressure fields with a constant pressure gradient, the boundary slip does not alter the interior material deformation and stress field; while, for pressure fields with a wave form pressure gradient, the boundary slip causes the change of interior material deformation and consequently the velocity profile and stress field. We also derive asymptotic expressions for the exact solution through which a parameter β is identified to dominate the behaviour of the flow driven by the wave form pressure gradient, and an explicit formulae for the critical slip parameter leading to the maximum transient flow rate is established.
Laser powder bed fusion technique has been considered as one of the best metal additive manufacturing processes considering the part quality and dimensional accuracy. However, the high cost of metal powders makes it difficult to afford for all kinds of applications. Reusability of the powders can overcome this issue, but the effect on various material properties is still in research stage. This paper analyzes the morphology and powder size distribution of 15-5 precipitation hardening stainless steel powders which were reused up to ten times. The powder size distribution was found to be slightly different for ten times reused powder as compared to raw powder and consisted of finer as well as agglomerated particles. The oxygen content was found to be more for the one time and five time recycled powder; however, for the ten time recycled powder, the oxygen content was found to be less.
The impact pad is a reinforced zone of the refractory lining of the bottom part of the steel ladle designed to resist against thermo-mechanical loads due to the liquid steel impact during filling. Results of an inquiry performed on a steel plant are first presented. Degradations observed on different kinds of impact pad materials have been indentified and the influence of process parameters on their life span has been evaluated. Mechanical experiments (compression tests and three-point bend tests) have been performed up to 1500°C on different alumina-magnesia refractories. They allow for the characterization of the thermomechanical behaviour and the strains due to phase-transformations at high temperatures. A thermo-elasto-viscoplastic model taking into account the phasetransformations is proposed. The last part deals with numerical simulations carried out with ABAQUS and FLUENT softwares. These computations aim at describing the thermal and mechanical behaviour of the impact pad during the steel making process, and more precisely during the impact of the liquid steel. Combined with micro-structural observations and refining process data, these simulations enabled us to figure out how the different types of loads might be responsible for the degradations observed on worn impact pads
In this paper, two dimensional unsteady flow and energy equations are employed for simulating the fluid flow, heat transfer and solidification during direct chill continuous casting of Al-Mg alloy billet. In these processes, the formation of some macro defects such as thermal cracking, hot tearing, surface cracking, etc, has been found to initiate during the starting phase of the operation. In this paper, a numerical study of these developing fluid flow and thermal phenomena from the beginning of casting operation to a steady-state is presented. The computations are based on an iterative deforming finite volume method and a single-domain enthalpy formulation to solve the growth of metal being solidified and the accompanying dynamic fluid flow and thermal behavior. The effect of phase change on convection is accounted for using a Darcy's law-type porous media treatment. The results indicate that the fluid flow and temperature fields change drastically at the initial stage but evolve slowly afterwards.
There has been a growing research interest in the melting and solidification technology among mathematicians and engineers. The topic has obvious practical importance in a wide range of applications. Natural convection may play a significant role in heat transfer and hence affect the progress of the solidification. A fixed-grid finite volume numerical approach is developed and used to simulate physical details of convection flow in the solidification problems. This approach is based on the enthalpy–porosity method that is traditionally used to track the motion of the liquid–solid front and to obtain the temperature distribution and the velocity profiles in the liquid phase. The enthalpy–porosity model treats the mushy region as a porous medium.
In this paper, the numerical and experimental studies of unsteady natural convection during solidification of cylindrical ingots are presented. The aim of the study consists of the numerical determination of the fluid flow, the temperature evolution, and the solidification front versus time. To validate the numerical model, an experimental study of the simple casting of cylindrical ingots was undertaken within the laboratory of metallurgy. The measured temperature was compared with values calculated numerically. A good agreement was obtained.
The present review covers the heat transfer literature published in 2004 in English language, including some translations of foreign language papers. Though extensive, some selection is necessary. Only articles published by a process of peer review in archival journals are reviewed. Papers are grouped into subject-oriented sections and further divided into sub-fields. Many papers deal with the fundamental science of heat transfer, including experimental, numerical and analytical work; others relate to applications or natural systems. In addition to reviewing journal articles, this Review also takes note of important conferences and meetings on heat transfer and related areas, major awards presented in 2004, and relevant books published in 2004.
As an expansion of our former work, a series of numerical simulations of fluid flow, heat transfer, and macrosolidification
under the control of both the conventional and the new type electromagnetic brake (EMBr) were carried out. In the present
work, the physical models and parameters remain the same as those measures in our former research, and the solution processes
of the flow field and electromagnetic field are similar. The specific heat method is used to treat the latent heat of solidification,
and the Darcy’s Law-type porous media treatment is used to account for the effect of phase change on convection. User-defined
functions (UDFs) are compiled for the following two aspects: One is the Darcy source term, and the other is the thermal conductivity,
which varies in mushy zones. From the comparisons of two kinds of EMBr produced by two type magnets separately, the new type
EMBr gives its superiorities of the effects on the flow field, heat transfer, and solidification with lower cost of electrical
energy.
In this paper, we study the pressure gradient driven transient flow of an incompressible Newtonian liquid in micro-annuals under a Navier slip boundary condition. By using the Fourier series expansion in time and Bessel functions in space, an exact solution is derived and is shown to include some existing known results as special cases. By analysing the exact solution, it is found that the influences of boundary slip on the flow behaviour are qualitatively different for different types of pressure fields driving the flow. For pressure fields with a constant pressure gradient, the flow rate increases with the increase in the slip parameter l almost linearly when l is large; while, for pressure fields with a wave form pressure gradient within a certain frequency range, as the slip parameter l increases, the amplitude of the flow rate increases first and then approaches a constant value when l becomes sufficiently large. It is also found that to achieve a given flow rate, one could have different designs and the graphs for the design are presented and discussed in this paper.
A steady-state three-dimensional heat flow model based on the concept of artificial effective thermal conductivity has been developed. On the basis of available literature information, boundary conditions to the governing heat flow equation have been applied, and the equation was solved via the control-volume based finite difference procedure. The model is sufficiently general and can be applied to various geometrical shapes of relevance to continuous casting of steel. Sensitivity of the predicted results to various numerical approximations including grid configurations, as well as to other modelling parameters such as axial conduction, mushy zone modelling procedure, choice of value of K(eff) have been extensively studied. It has been shown that assumptions and numerical procedures influence the computed results significantly. Finally, numerical predictions have been compared with three sets of experimental measurements reported in literature on shell thickness in industrial casters. In contrast to some earlier claims, these indicated only poor to moderate agreement between model predictions and experimental results.
The paper presents numerical predictions of various turbulent shear flows in which the structure of the viscous sublayer exerts appreciable influence on the flow. The model of turbulence employed is one where the turbulence energy and its dissipation rate are calculated by way of transport equations which are solved simultaneously with the conservation equations for the mean flow. The flows considered include isothermal low Reynolds number pipe flows, and wall boundary layers with streamwise pressure gradient and wall injection; the predictions span both natural transition and laminarisation. Although complete agreement with experiment is not yet achieved in every case, it is argued that only a turbulence model of (at least) this level of complexity will permit a universal modelling of the near-wall turbulence structures commonly found in thermal power equipment.
The paper reviews the problem of making numerical predictions of turbulent flow. It advocates that computational economy, range of applicability and physical realism are best served at present by turbulence models in which the magnitudes of two turbulence quantities, the turbulence kinetic energy k and its dissipation rate ϵ, are calculated from transport equations solved simultaneously with those governing the mean flow behaviour. The width of applicability of the model is demonstrated by reference to numerical computations of nine substantially different kinds of turbulent flow.
The superheat contained in liquid steel must be removed before it can solidify. This heat represents a significant fraction of the total heat removed by the mold. Its dissipation has a great influence on growth of the solidifying shell and later on development of the microstructure. To investigate this, a finite-element model has been applied to compute the fluid flow and temperature distribution within the liquid pool and heat transfer to the inside of the solidifying shell of a continuously-cast steel slab. It includes separate models of the nozzle and mold cavity using the K-ε turbulence model in FIDAP. Calculated velocities have been compared with experiments using a plexiglass water model. Heat transfer predictions are compared with available liquid temperature measurements. Results indicate that the maximum in heat input to the shell occurs in the vicinity of the impingement point on narrow face, and confirms that most superheat is dissipated in or just below the mold. The model is then used to predict the effect of casting variables, superheat temperature difference, casting speed, nozzle jet angle, mold width, and submergence depth on flow pattern and heat flux to the narrow face shell. Superheat temperature difference and casting speed have the most important effects on heat flux. Calculated heat flux profiles are then input to a 1D solidification model to show their effect on growth of the narrow face shell.
The causes of sticker breakout during continuous casting have been investigated by (i) a review of published literature, (ii) quality control testing of mould powder supplies, (iii) on-plant monitoring of various casting parameters, (iv) metallographic examination of both the shells removed from the mould following a breakout and the slag films attached to the shell, and (v) simulation experiments. The evidence obtained supports the view that sticker breakouts occur when a carbonaceous agglomerate blocks the flow of slag into the mould/strand gap and a low melting point high carbon meniscus is formed that does not heal during negative strip. It was observed that some breakouts are preceded by the formation of longitudinal grooves on the surface of the skelp which were attributed to the effects of a CO gas bubble. It was concluded that the carbonaceous agglomerate responsible for the sticker breakout could be formed by either unmelted casting powder or erosion of the refractories used for submerged entry nozzles and stopper rods.
A steady state, two-dimensional mathematical model for continuous billet casting operations has been developed. Towards this, governing equations of fluid flow and heat transfer together with their appropriate set of boundary conditions were derived and solved numerically via a control volume based implicit finite difference procedure (e.g., SIMPLE). The effect of various assumptions and procedures applied to modelling of turbulence phenomena, thermal buoyancy, flow through the mushy zone, free surface conditions etc., on the sensitivity of the computed results was investigated computationally. Of all these, modelling of heat and fluid flow phenomena in the mushy region was found to have relatively more effect on the predicted results. In addition to these, a set of three different billet casting operations reported in literature were simulated mathematically and direct comparisons were made between predicted and observed solidified shell profiles. Such comparisons demonstrated reasonable to excellent agreement between theory and experiments.
This paper is concerned with finite element methods of least-squares type for the approximate numerical solution of incompressible, viscous flow problems. Our main focus is on issues that are critical for the success of the finite element methods, such as decomposition of the Navier-Stokes equations into equivalent first-order systems, mathematical prerequisites for the optimality of the methods, and the use of mesh-dependent norms. In conclusion we present a novel application of least-squares principles involving an optimal boundary control problem for fluid flows. 1
The paper summarizes current strategies for computing heat transfer coefficients in complex turbulent flows based on numerical solution of the averaged equations for momentum and enthalpy and corresponding equations for averaged properties of the turbulent flow field. It argues that, for accuracy and width of applicability, a fine-grid low-Reynolds-number treatment should be employed near the wall in place of wall functions, despite the attractive simplicity of the latter approach. Several examples are provided that bring out the benefit from adopting second-moment closures, in which attention is focused on the turbulent stresses and heat fluxes themselves rather than on effective viscosities and thermal diffusivities. Directions for future research are briefly discussed, an important contribution to this effort being the direct numerical simulation of the near-wall dynamic and thermal turbulence field.
An improved k-epsilon turbulence model for predicting wall turbulence is presented. The model was developed in conjunction with an accurate calculation of near-wall and low-Reynolds-number flows to meet the requirements of the Evaluation Committee report of the 1980-1981 Stanford Conference on Complex Turbulent Flows. The proposed model was tested by application to turbulent pipe and channel flows, a flat plate boundary layer, a relaminarizing flow, and a diffuser flow. In all cases, the predicted values of turbulent ...
Semi-empirical laws and microscopic descriptions of transport behavior have been integrated with principles of classical mixture theory to obtain a set of continuum conservation equations for binary, solid-liquid phase change systems. For a restricted, yet frequently encountered, class of phase change systems, the continuum equations have been cast into forms amenable to clear physical interpretation and solution by conventional numerical procedures.RésuméDes lois semi-empiriques et des descriptions microscopiques du mécanisme de transfert ont été intégrées à la theorie classique des mélanges pour obtenir un système d'équations de bilan continu pour des systèmes à changement de phase solide-liquide. Pour une classe limitée mais fréquemment rencontrée, de systèmes les équations ont été mise sous des formes permettant une interprétation physique claire et une résolution par des procédures numériques conventionnelles.ZusammenfassungHalbempirische Gesetze und mikroskopische Beschreibungen des Transportverhaltens wurden mit den Prinzipien der klassischen Mischungstheorie verbunden. Es ergibt sich ein Satz von Erhaltungsgleichungen für binäre Fest-flüssig-Phasenwechsel-Systeme. Für eine bestimmte, häufig auftretende, Gruppe von Phasenwechselsystemen, wurden die Gleichungen in eine Form gebracht, die eine klare physikalische Interpretation ermöglicht und auβerdem mit herkömmlichen numerischen Verfahren lösbar ist.РефератДля пoлyчeния cиcтeмы фeиoмeиoлoгичecкич ypaвнeний пepeнoca для бинapиыч cиcтeм твepдoe тeлo-жидкocть пpи фaзoвыч измeиeиияч выпoлнeиo oбьeдинeниe пoлyзмпиpичecкич зaкoнoв и микpocкoиичecкич oпиcaинй пpoцeccoв пepeнoca c зaкoиaми тeopии cмeceй. Для oгpaиичeииoгo, oдиaкo, чacтo вcтpeчaюшeгocя клacca cиcтeм пpи фaзoвыч пepeчoдaч пpивoдятcя ypaвнeиия цepeиoca в вндe, yдoбиoм для ич фнзичecкoй иитepпpeтaции и peщeиия c пoмoщью oбычиыч чиcлeииыч мeтoдoв.
The high Reynolds number form of the k-ε model is extended and tested by application to fully developed pipe flow. It is established that the model is valid throughout the fully turbulent, semilaminar and laminar regions of the flow. Unlike many previously proposed forms of the k-ε model, the present form does not have to be used in conjunction with empirical wall function formulas and does not include additional terms in the k and ε equations. Comparison between predicted and measured dissipation rate in the important wall region is also possible.
Recent turbulence closure models have been successfully employed in the
prediction of wall-bounded shear flows. Certain difficulties can,
however, arise in certain situations, including cases involving low
Reynolds numbers. In recent years, a number of proposals have been made
for model extensions which would allow an application of the turbulence
models in the case of low Reynolds numbers or in situations involving
near-wall flow. Unfortunately, as the results provided by each of the
obtained models were compared for different flows, it is not clear which
of the many proposed models can be used with confidence. For this
reason, the present study is concerned with a systematic evaluation of
existing two-equation, 'low-Reynolds number' turbulence models. The
obtained results indicate clearly the relative performance of the
various models proposed to describe near-wall flows. The performance of
the models is discussed along with approaches for performance
improvement.
An assessment is made of the state-of-the-art in incompressible
turbulent flow simulation over six levels of increasing completeness and
complexity: (1) correlations, (2) integral methods, (3) one-point
average methods, (4) two-point average methods, (5) large eddy
simulation, and (6) full turbulence simulation. Higher level model
computations are generally less sensitive to the quality of the
turbulence model than lower level ones; they also allow a greater range
of simulable flows, and require a greater amount of laboratory data
detail to furnish initial and/or boundary conditions. Attention is in
turn given to the various turbulence models, the treatment of geometric
complexity, the need for the conservation of important variables, and
discretization methods.
A new turbulence model that is valid right down to the solid wall has
been developed. Although the general approach is similar to that of
Jones and Launder, the detailed proposals are quite different. The
Taylor series expansion technique has been used to systematically study
the proper behavior of the turbulent shear stress, kinetic energy and
its rate of dissipation near a solid wall. The model was applied to the
problem of a fully developed turbulent channel flow and to a turbulent
boundary layer flow over a flat plate. Results were compared with the
theory of Jones and Launder and with the experimental data. For the
cases considered, present theory not only yields better predictions of
the peak turbulent kinetic energy, but requires about one tenth of the
computer time of the theory of Jones and Launder.
A new turbulence model that is valid down to the solid wall has been developed. The Taylor series expansion technique has been used to systematically study the proper behavior of the turbulent shear stress and the kinetic energy and its rate of dissipation near a solid wall. The model was applied to the problem of a fully developed turbulent channel flow and to a turbulent boundary-layer flow over a flat plate.
In the continuous casting of steel, many problems, such as surface cracks in solidified steel and breakouts of molten steel from the bottom of moulds, frequently occur in practice. It is believed that the occurrence of these problems is directly related to the events in the mould, especially the transfer of heat from the strand surface across the lubricating mould powder and its interface with the mould wall to the mould cooling-water. However, as far as the authors are aware, there is no published work dealing with heat transfer across both the lubricating layer and the interface. Generally, a parameter representing the average overall heat transfer coefficient between the strand surface and the mould cooling-water is employed, instead of including the lubricating layer, the mould wall and their interface in the computation region. The existing treatment consequently does not permit analysis of some of the more important phenomena, such as the effect of mould powder properties and interface thermal contact resistance on the solidification of steel. In this paper, a novel finite element model is developed and the heat transfer across the interface between the lubricating layer and the mould wall is simulated by introducing a new type of element, referred to as the thermal contact element. The proposed model is used to investigate the effect of various casting parameters on heat transfer from the molten steel to the cooling-water. The results indicate that the thermal contact resistance between the mould wall and the mould powder is a key factor which dominates the thickness of the solidified steel shell and the heat extraction rate from the mould wall.
An extended κ–ϵ model (to include low-Reynolds-number regions) employing weighting functions is presented. Wall functions for the near-wall zones are developed giving correct boundary values for the Shear stress and κ–ϵ. A finite element model using a penalty formulation for incompressible turbulent flow is applied to Solve a flow between two plates. Results with mesh boundaries situated in the near-wall region and a: the wall are compared with measured values.
Numerical simulation of forming processes is studied using the Navier-Stokes equations and the energy equation governing viscous, incompressible fluids. A review of various important aspects of forming processes is also reviewed for completeness. A penalty finite element model of the equations is developed. The finite element model is used to study the effectiveness of various constitutive models developed for metal forming and polymer processing. The element-by-element (EBE) data structure with efficient iterative solution methods is used to solve the large systems of equations that arise in the analysis of complex 3-D problems. Numerical results for problems representative of various forming processes (e.g. extrusion and solidification) are presented.
A number of solution strategies for heat-flow models of continuous casting processes are developed and compared for interfacing
with optimization algorithms. These two-dimensional (2-D) slice models include nonlinear thermodynamic and transport properties.
As a result of the compari-son, a number of modifications are applied to enhance the accuracy of the simulation as well as
the efficiency of the solution. Here, we found that by applying the Kirchhoff transformation to the heat-flow equations and
by iterative adjustment of temperature-dependent properties, fewer calculations are required per time-step and larger time-steps
can be taken. Consequently, this leads to approximately a twentyfold reduction in computation time when using 2-D slice models
to determine three-dimensional, steady-state temperature fields. As a result, a complex, 2-D heat-flow model for a cast strand
can be solved in 2 to 5 minutes on a Micro VAX II and is thus suitable for incorporation into a systematic optimization procedure
and for process simu-lation in real time.
A fully coupled fluid flow, heat, and solute transport model was developed to analyze turbulent flow, solidification, and
evolution of macrosegregation in a continuous billet caster. Transport equations of total mass, momentum, energy, and species
for a binary iron-carbon alloy system were solved using a continuum model, wherein the equations are valid for the solid,
liquid, and mushy zones in the casting. A modified version of the low-Reynolds numberk-ε model was adopted to incorporate turbulence effects on transport processes in the system. A control-volume-based finite-difference
procedure was employed to solve the conservation equations associated with appropriate boundary conditions. Because of high
nonlinearity in the system of equations, a number of techniques were used to accelerate the convergence process. The effects
of the parameters such as casting speed, steel grade, nozzle configuration on flow pattern, solidification profile, and carbon
segregation were investigated. From the computed flow pattern, the trajectory of inclusion particles, as well as the density
distribution of the particles, was calculated. Some of the computed results were compared with available experimental measurements,
and reasonable agreements were obtained.
To investigate superheat dissipation in a continuous slab casting machine, mathematical models have been developed to compute
fluid flow velocities, temperature distribution within the liquid pool, heat transfer to the inside of the solidifying shell,
and its effect on growth of the shell. Three-dimensional (3-D) velocity and heat-transfer predictions compare reasonably with
pre-vious experimental measurements and two-dimensional (2-D) calculations. The results indicate that the maximum heat input
to the shell occurs near the impingement point on the narrow face and confirm that most of the superheat is dissipated in
or just below the mold. Superheat tem-perature and casting speed have the most important and direct influence on heat flux.
The effects of other variables, including mold width, nozzle jet angle, and submergence depth, are also investigated. Calculated
heat flux profiles are then input to a one-dimensional (1-D) solidifi-cation model to calculate growth of the shell. Shell
thickness profiles down the wide and narrow faces are compared with the predictions of conventional heat conduction models
and available measurements.
The continuous casting process employed in the steel industry is a many faceted big industrial problem which has given rise to many sub-problems. Here, we examine the problem involving the determination of the solid-liquid steel interface and we develop and extend a previously proposed model, which incorporates heat transfer through two layers of solid and liquid mould powder and the interface between the solid powder and the mould wall. The problem simplifies to the classical Stefan problem except that the condition on the boundary is nonlinear. Integral formulation procedures are used to establish the normalized pseudo steady state temperature as an upper bound to the normalized actual temperature. The pseudo steady state approximation yields an upper bound on the interface position, which an independent numerical enthalpy scheme confirms to be an extremely accurate approximation for the parameter values occurring in practice. The present work is important since it provides a simple method for the prediction of the solid-liquid steel interface and a bounding procedure which can be used to validate other estimates.
The surface of continuously cast slabs is characterized by the presence of oscillation marks. Direct linkage of the continuous
casting process and hot rolling process requires that cast slabs should be free of surface defects. In the present work, a
mechanical model has been developed for the prediction of the depth of oscillation marks of the depression type. It is based
on the beam bending theory and on viscoplastic material behavior. The downward movement of the strand is taken correctly into
account, which has not been done in previous models. Auxiliary parts of the model are the models for the determination of
the temperatre field and of the fluid flow and pressure in the meniscus region and in the gap between strand and mold. The
deflection of the shell is computed as a function of time and distance from the shell tip. The retained deflection, which
corresponds to the depth of oscillation marks observed on the slab surface, is determined for different values of stroke,
frequency, and casting velocity. The theoretical data are compared with the measured data as available in the literature.
An enthalpy formulation based fixed grid methodology is developed for the numerical solution of convection-diffusion controlled mushy region phase-change problems. The basic feature of the proposed method lies in the representation of the latent heat of evolution, and of the flow in the solid-liquid mushy zone, by suitably chosen sources. There is complete freedom within the methodology for the definition of such sources so that a variety of phase-change situations can be modelled. A test problem of freezing in a thermal cavity under natural convection is used to demonstrate an application of the method.
With a numerical code for solving the boundary-layer equations, the performance of different turbulence models for the natural convection boundary layer for air along a heated vertical plate is tested. The algebraic Cebeci-Smith model, the standard k-ε model with wall functions for k and ε and different low-Reynolds number k-ε models are tested. The Cebeci-Smith model calculates a too low wall-heat transfer and turbulent viscosity. The standard k-ε model with wall functions gives a too high wall-heat transfer, but the velocity and temperature profiles agree reasonably with experiments. Accurate wall-heat transfer results require the use of low-Reynolds number k-ε models; the models of Lam and Bremhorst, Chien, and Jones and Launder perform best up to a Grashof number of 1011. For larger Grashof numbers the Jones and Launder model is best. A sensitivity study shows that the wall-heat transfer with the standard k-ε model largely depends on the choice of the wall functions for k and ε. Replacing these wall functions by zero wall conditions for k and ε and adding the functions D and f{hook}μ of the Chien model to the standard k-ε model gives a simple, but accurate low-Reynolds number k-ε model for the natural convection boundary layer.
Continuous ingot casting is an important processing technology for many materials. Under most practical circumstances, turbulence plays a critical role which, along with transport mechanisms such as buoyancy, surface tension, and phase change, is responsible for the quality of the end products. A modified turbulence model based on the standard k-epsilon two-equation closure is proposed and applied to predict the phase change and convection-diffusion characteristics during titanium alloy ingot casting in an electron beam melting process. In conjunction with an adaptive grid computational technique, solutions of the coupled mass continuity, momentum, energy, and turbulence transport have been obtained in the context of the enthalpy formulation. Effects of casting speed and gravity on solidification and convection characteristics have been investigated and compared to ones obtained previously with a simple zero-equation turbulence model. The present turbulence model predicts that the mushy zone is generally of substantial thickness as a result of the convection effect, that the solidus line has a high curvature, and that the temperature gradient close to the solidus line is higher than elsewhere. Under all conditions, the turbulence structure largely reflects the combined influence of convection and energy input by the electron beam and the superheated feeding material from the top surface. The numerical results have been compared with an experimentally determined pool profile from a casting ingot.
The Flux of Flux in a Continuous Steel Caster
- N Fowkes
- A Woods
N. Fowkes, A. Woods, The Flux of Flux in a Continuous Steel Caster, Department of Mathematics, University of Western Australia, 1989.
Continuous Casting: Heat Flow, Solidiÿcation and Crack Formation
- J K Brimacombe
- I V Samarasekera
- J E Laid
J.K. Brimacombe, I.V. Samarasekera, J.E. Laid, Continuous Casting: Heat Flow, Solidiÿcation and Crack Formation, Vol. 2, Iron and Steel Society, AIME, New York, 1984.
On turbulence ow near a wall
- E R V Driest
E.R.V. Driest, On turbulence ow near a wall, J. Aeronaut. Sci. 23 (1996) 1007-1011.
Mathematical Theory of Elasticity, Robert E
- I S Sokolniko
I.S. Sokolniko, Mathematical Theory of Elasticity, Robert E. Krieger Publishing Company, Malabar, FL, 1986.
Turbulence models for near-wall and low Reynolds number flows
- Patel