Feedback control of the Rayleigh–Bénard convection by means of mode reduction

An optimal feedback control is synthesized for the Rayleigh–Bénard convection by means of empirical reduction of modes. The Boussinesq equation is reduced to a minimal set of ordinary differential equations by using the Karhunen–Loève Galerkin procedure. The state feedback control synthesis, that drives the intensity of convection to a preset trajectory by adjusting heat flux at the bottom of the system, is constructed using this low‐dimensional dynamic model by first performing an extended Kalman filter estimate of the velocity and temperature fields and then developing the optimal feedback law by means of the linear regulator theory. The present technique allows for the practical implementation of modern control concepts to the natural convection and is found to yield satisfactory results. Copyright © 2002 John Wiley & Sons, Ltd.     

Role of the inclination of an inverted‐V upper plate on the heat and flow behavior of trapped gases in a modified Rayleigh–Bénard cavity

Thermal buoyant air inside a modified Rayleigh–Bénard (RB) cavity bounded by a lower flat plate and an inverted‐V upper plate has been investigated numerically using the finite‐volume method. The second‐order‐accurate QUICK and SIMPLE schemes were used for the discretization of the convective terms and the pressure–velocity coupling in the set of conservation equations, respectively. The problem under study is controlled by two parameters: (1) the Rayleigh number ranging from 10³ to 10⁶ and (2) the relative height of the vertical sidewalls d. In reference to the latter, it varies from one limiting case corresponding to the standard RB cavity (a rectangle with d = 1) to another limiting case represented by an isosceles triangular cavity where d = 0. The numerical results for the velocity and temperature fields are presented in terms of streamlines, isotherms, local and mean heat fluxes. An additional effort was devoted to determine the critical Ra values characterizing the transition from symmetrical to asymmetrical buoyant airflow responsive to incremental changes in Ra. For purposes of engineering design, a general correlation equation for the Nusselt number in terms of the pertinent Ra and d was constructed using nonlinear multiple regression theory. Copyright © 2007 John Wiley & Sons, Ltd.     

Higher‐order‐accurate numerical method for temporal stability simulations of Rayleigh‐Bénard‐Poiseuille flows

For Rayleigh‐Bénard‐Poiseuille flows, thermal stratification resulting from a wall‐normal temperature gradient together with an opposing gravitational field can lead to buoyancy‐driven instability. Moreover, for sufficiently large Reynolds numbers, viscosity‐driven instability can occur. Two higher‐order‐accurate methods based on the full and linearized Navier‐Stokes equations were developed for investigating the temporal stability of such flows. The new methods employ a spectral discretization in the homogeneous directions. In the wall‐normal direction, the convective and viscous terms are discretized with fifth‐order‐accurate biased and fourth‐order‐accurate central compact finite differences. A fourth‐order‐accurate explicit Runge‐Kutta method is employed for time integration. To validate the methods, the primary instability was investigated for different combinations of the Reynolds and Rayleigh number. The results from these primary stability investigations are consistent with linear stability theory results from the literature with respect to both the onset of the instability and the dependence of the temporal growth rate on the wave angle. For the cases with buoyancy‐driven instability, strong linear growth is observed for a broad range of spanwise wavenumbers. The largest growth rates are obtained for a wave angle of 90°. For the cases with viscosity‐driven instability, the linear growth rates are lower and the first mode to experience nonlinear growth is a higher harmonic with half the wavelength of the fundamental.     

A finite-element study of the bénard problem using parameter-stepping and bifurcation search

The problem of fluid motion in a cavity with rigid sidewalls that is heated uniformly from below is studied by the finite-element method. The techniques of parameter-stepping and monitoring the determinant of the Jacobian matrix to find bifurcations are used. Results are presented for width-to-height ratios in the range 1 to 4, and for three different boundary conditions on the horizontal surfaces, namely both rigid, both free, and rigid bottom with free top. The non-linear branches above the critical Rayleigh number are examined. Extensions to non-Boussinesq flow are trivial.     

A class of higher order compact schemes for the unsteady two‐dimensional convection–diffusion equation with variable convection coefficients

A class of higher order compact (HOC) schemes has been developed with weighted time discretization for the two‐dimensional unsteady convection–diffusion equation with variable convection coefficients. The schemes are second or lower order accurate in time depending on the choice of the weighted average parameter μ and fourth order accurate in space. For 0.5?μ?1, the schemes are unconditionally stable. Unlike usual HOC schemes, these schemes are capable of using a grid aspect ratio other than unity. They efficiently capture both transient and steady solutions of linear and nonlinear convection–diffusion equations with Dirichlet as well as Neumann boundary condition. They are applied to one linear convection–diffusion problem and three flows of varying complexities governed by the two‐dimensional incompressible Navier–Stokes equations. Results obtained are in excellent agreement with analytical and established numerical results. Overall the schemes are found to be robust, efficient and accurate. Copyright © 2002 John Wiley & Sons, Ltd.     

Remarks on the links between low‐order DG methods and some finite‐difference schemes for the Stokes problem

In this paper we demonstrate that some well‐known finite‐difference schemes can be interpreted within the framework of the local discontinuous Galerkin (LDG) methods using the low‐order piecewise solenoidal discrete spaces introduced in (SIAM J. Numer. Anal. 1990; 27 (6): 1466–1485). In particular, it appears that it is possible to derive the well‐known MAC scheme using a first‐order Nédélec approximation on rectangular cells. It has been recently interpreted within the framework of the Raviart–Thomas approximation by Kanschat (Int. J. Numer. Meth. Fluids 2007; published online). The two approximations are algebraically equivalent to the MAC scheme, however, they have to be applied on grids that are staggered on a distance h/2 in each direction. This paper also demonstrates that both discretizations allow for the construction of a divergence‐free basis, which yields a linear system with a ‘biharmonic’ conditioning. Both this paper and Kanschat (Int. J. Numer. Meth. Fluids 2007; published online) demonstrate that the LDG framework can be used to generalize some popular finite‐difference schemes to grids that are not parallel to the coordinate axes or that are unstructured. Copyright © 2008 John Wiley & Sons, Ltd.     

A two‐phase flow interface capturing finite element method

A new interface capturing algorithm is proposed for the finite element simulation of two‐phase flows. It relies on the solution of an advection equation for the interface between the two phases by a streamline upwind Petrov–Galerkin (SUPG) scheme combined with an adaptive mesh refinement procedure and a filtering technique. This method is illustrated in the case of a Rayleigh–Taylor two‐phase flow problem governed by the Stokes equations. Copyright © 2006 John Wiley & Sons, Ltd.     

A new characteristic approach for incompressible thermo‐flow in Cartesian and non‐Cartesian grids

A virtual‐characteristic approach is developed for thermo‐flow with finite‐volume methodology in which a multidimensional characteristic (MC) scheme is applied along with artificial compressibility. To obtain compatibility equations and pseudo‐characteristics, energy equation is taken into account in the MC scheme. With this inherent upwinding of convective fluxes, no artificial viscosity is required even at high Reynolds numbers. Another remarkable advantage of the MC scheme lies in its faster convergence rate with respect to the averaging scheme that is found to exhibit substantial delays in convergence. As benchmarks, forced and mixed convections in a cavity and in flow over cylinder and between parallel plates are examined for a wide range of Reynolds, Grashof, and Prandtl numbers. The MC and averaging schemes are applied for simulation purposes. Results show the better performance of the MC scheme in forced and mixed convections. Results confirm the robustness of the MC scheme in terms of accuracy and convergence. Copyright © 2015 John Wiley & Sons, Ltd.     

A hybrid Padé ADI scheme of higher‐order for convection–diffusion problems

A high‐order Padé alternating direction implicit (ADI) scheme is proposed for solving unsteady convection–diffusion problems. The scheme employs standard high‐order Padé approximations for spatial first and second derivatives in the convection‐diffusion equation. Linear multistep (LM) methods combined with the approximate factorization introduced by Beam and Warming (J. Comput. Phys. 1976; 22 : 87–110) are applied for the time integration. The approximate factorization imposes a second‐order temporal accuracy limitation on the ADI scheme independent of the accuracy of the LM method chosen for the time integration. To achieve a higher‐order temporal accuracy, we introduce a correction term that reduces the splitting error. The resulting scheme is carried out by repeatedly solving a series of pentadiagonal linear systems producing a computationally cost effective solver. The effects of the approximate factorization and the correction term on the stability of the scheme are examined. A modified wave number analysis is performed to examine the dispersive and dissipative properties of the scheme. In contrast to the HOC‐based schemes in which the phase and amplitude characteristics of a solution are altered by the variation of cell Reynolds number, the present scheme retains the characteristics of the modified wave numbers for spatial derivatives regardless of the magnitude of cell Reynolds number. The superiority of the proposed scheme compared with other high‐order ADI schemes for solving unsteady convection‐diffusion problems is discussed. A comparison of different time discretizations based on LM methods is given. Copyright © 2009 John Wiley & Sons, Ltd.     

A fourth‐order compact finite difference scheme for the steady stream function–vorticity formulation of the Navier–Stokes/Boussinesq equations

A fourth‐order compact finite difference scheme on the nine‐point 2D stencil is formulated for solving the steady‐state Navier–Stokes/Boussinesq equations for two‐dimensional, incompressible fluid flow and heat transfer using the stream function–vorticity formulation. The main feature of the new fourth‐order compact scheme is that it allows point‐successive overrelaxation (SOR) or point‐successive underrelaxation iteration for all Rayleigh numbers Ra of physical interest and all Prandtl numbers Pr attempted. Numerical solutions are obtained for the model problem of natural convection in a square cavity with benchmark solutions and compared with some of the accurate results available in the literature. Copyright © 2003 John Wiley & Sons, Ltd.     

Time‐accurate stabilized finite‐element model for weakly nonlinear and weakly dispersive water waves

Introduction of a time‐accurate stabilized finite‐element approximation for the numerical investigation of weakly nonlinear and weakly dispersive water waves is presented in this paper. To make the time approximation match the order of accuracy of the spatial representation of the linear triangular elements by the Galerkin finite‐element method, the fourth‐order time integration of implicit multistage Padé method is used for the development of the numerical scheme. The streamline‐upwind Petrov–Galerkin (SUPG) method with crosswind diffusion is employed to stabilize the scheme and suppress the spurious oscillations, usually common in the numerical computation of convection‐dominated flow problems. The performance of numerical stabilization and accuracy is addressed. Treatments of various boundary conditions, including the open boundary conditions, the perfect reflecting boundary conditions along boundaries with irregular geometry, are also described. Numerical results showing the comparisons with analytical solutions, experimental measurements, and other published numerical results are presented and discussed. Copyright © 2007 John Wiley & Sons, Ltd.     

A stabilized Nitsche‐type extended embedding mesh approach for 3D low‐ and high‐Reynolds‐number flows

This paper presents a stabilized extended finite element method (XFEM) based fluid formulation to embed arbitrary fluid patches into a fixed background fluid mesh. The new approach is highly beneficial when it comes to computational grid generation for complex domains, as it allows locally increased resolutions independent from size and structure of the background mesh. Motivating applications for such a domain decomposition technique are complex fluid‐structure interaction problems, where an additional boundary layer mesh is used to accurately capture the flow around the structure. The objective of this work is to provide an accurate and robust XFEM‐based coupling for low‐ as well as high‐Reynolds‐number flows. Our formulation is built from the following essential ingredients: Coupling conditions on the embedded interface are imposed weakly using Nitsche's method supported by extra terms to guarantee mass conservation and to control the convective mass transport across the interface for transient viscous‐dominated and convection‐dominated flows. Residual‐based fluid stabilizations in the interior of the fluid subdomains and accompanying face‐oriented fluid and ghost‐penalty stabilizations in the interface zone stabilize the formulation in the entire fluid domain. A detailed numerical study of our stabilized embedded fluid formulation, including an investigation of variants of Nitsche's method for viscous flows, shows optimal error convergence for viscous‐dominated and convection‐dominated flow problems independent of the interface position. Challenging two‐dimensional and three‐dimensional numerical examples highlight the robustness of our approach in all flow regimes: benchmark computations for laminar flow around a cylinder, a turbulent driven cavity flow at Re = 10000 and the flow interacting with a three‐dimensional flexible wall. Copyright © 2016 John Wiley & Sons, Ltd.     

A two-dimensional effective model describing fluid–structure interaction in blood flow: analysis, simulation and experimental validation

We derive a closed system of effective equations describing a time-dependent flow of a viscous incompressible Newtonian fluid through a long and narrow elastic tube. The 3D axially symmetric incompressible Navier–Stokes equations are used to model the flow. Two models are used to describe the tube wall: the linear membrane shell model and the linearly elastic membrane and the curved, linearly elastic Koiter shell model. We study the behavior of the coupled fluid–structure interaction problem in the limit when the ratio between the radius and the length of the tube, , tends to zero. We obtain the reduced equations that are of Biot type with memory. An interesting feature of the reduced equations is that the memory term explicitly captures the viscoelastic nature of the coupled problem. Our model provides significant improvement over the standard 1D approximations of the fluid–structure interaction problem, all of which assume an ad hoc closure assumption for the velocity profile. We performed experimental validation of the reduced model using a mock circulatory flow loop assembled at the Cardiovascular Research Laboratory at the Texas Heart Institute. Experimental results show excellent agreement with the numerically calculated solution. Major applications include blood flow through large human arteries. To cite this article: S. Čanić et al., C. R. Mecanique 333 (2005).     

A 5‐equation,transient, hyperbolic, 1‐dimensional model for slug capturing in pipes

A novel numerical scheme for slug capturing in pipes using a 1‐dimensional transient hyperbolic 5‐equation 2‐fluid model is presented. Previous work has shown that 1‐dimensional 2‐fluid models are able to capture slug flow automatically. In this work, a similar approach is further developed using a new numerical scheme, applied to a hyperbolic 5‐equation 2‐fluid model. Starting from a finite volume discretisation of a 5‐equation 2‐fluid hyperbolic model and adding appropriate closure relations, a second‐order code is implemented and applied to air‐water flows in horizontal pipes, simulating the 2‐phase to 1‐phase flow process. The code is evaluated in some common standard test cases. A slug capturing application is also discussed. We show, in an air/water horizontal pipe, slug initiation, growth, and development. Moreover, a grid refinement analysis is performed showing that the method is grid independent and we show the code capability to take into account eventual surface tension effects, through the instantaneous pressure relaxation process. Finally, a prediction of flow regime transitions is shown and compared with a well‐known theoretical flow pattern map in addition to a preliminary comparison of computed slug characteristics against well‐known empirical correlations.     

A fluid solver based on vorticity–helical density equations with application to a natural convection in a cubic cavity

We study numerically a recently introduced formulation of incompressible Newtonian fluid equations in vorticity–helical density and velocity–Bernoulli pressure variables. Unlike most numerical methods based on vorticity equations, the current approach provides discrete solutions with mass conservation, divergence‐free vorticity, and accurate kinetic energy balance in a simple and natural way. The method is applied to compute buoyancy‐driven flows in a differentially heated cubic enclosure in the Boussinesq approximation for Ra ∈ {10⁴,10⁵,10⁶}. The numerical solutions on a finer grid are of benchmark quality. The computed helical density allows quantification of the three‐dimensional nature of the flow. Copyright © 2011 John Wiley & Sons, Ltd.     

Transition and heat transfer predictions in a turbine cascade at various free‐stream turbulence intensities through a one‐equation turbulence model

The one‐equation Spalart–Allmaras turbulence model, coupled with criteria for the prediction of the transition onset, is employed for the numerical prediction of the heat transfer along the nozzle guide vanes of a high‐pressure turbine, at various operating conditions. Emphasis is put on how to overcome a known shortcoming of the Spalart–Allmaras model, i.e. its insensitivity to free‐stream turbulence. For this purpose, an extra viscosity coefficient is defined and used in the mean flow equations. This extra viscosity is proportional to the free‐stream turbulence with a damping in the boundary layer. Its use is adequate to circumvent the aforementioned weakness of the Spalart–Allmaras model, without any other intervention in the model itself. For the prediction of the onset of transition, the Abu‐Ghannam and Shaw and the Mayle criteria are used, depending on the level of free‐stream turbulence. Both yield very satisfactory predictions in a wide range of Reynolds numbers and/or turbulence intensities. From a numerical point of view, this paper proposes techniques for the implementation of the solution method on unstructured grids with triangular elements and reconfirms findings of previous works, like the suitability of the containment‐circle tessellation in highly stretched grids. Copyright © 2002 John Wiley & Sons, Ltd.     

A study on the correlation between the thermal contact conductance and effective factors in fin–tube heat exchangers with 9.52 mm tube

The thermal contact resistance is a principal parameter interfering with heat transfer in a fin–tube heat exchanger. However, the thermal contact resistance in the interface between tubes and fins has not been clearly investigated. The objective of the present study is to examine the thermal contact conductance for various fin–tube heat exchangers with tube diameter of 9.52 mm and to find a correlation between the thermal contact conductance and effective factors such as expansion ratio, fin type, fin spacing and hydrophilic coating. In this study, experiments have been conducted only to measure heat transfer rate between hot and cold water. To minimize heat loss to the ambient air by the natural convection fin–tube heat exchangers have been placed in an insulated vacuum chamber. Also, a numerical scheme has been employed to calculate the thermal contact conductance with the experimental data. As a result, a new correlation including the influences of expansion ratio, slit of fin and fin coating has been introduced, and the portion of each thermal resistance has been estimated in the fin–tube heat exchangers with 9.52 mm tube.