Multi‐scale time integration for transient conjugate heat transfer

The quasi‐steady assumption is commonly adopted in existing transient fluid–solid‐coupled convection–conduction (conjugate) heat transfer simulations, which may cause non‐negligible errors in certain cases of practical interest. In the present work, we adopt a new multi‐scale framework for the fluid domain formulated in a triple‐timing form. The slow‐varying temporal gradient corresponding to the time scales in the solid domain has been effectively included in the fluid equations as a source term, whilst short‐scale unsteadiness of the fluid domain is captured by a local time integration at a given ‘frozen’ large scale time instant. For concept proof, validation and demonstration purposes, the proposed methodology has been implemented in a loosely coupled procedure in conjunction with a hybrid interfacing treatment for coupling efficiency and accuracy. The present results indicate that a much enhanced applicability can be achieved with relatively small modifications of existing transient conjugate heat transfer methods at little extra cost. Copyright © 2016 John Wiley & Sons, Ltd.     

Fluid‐structure coupling of linear elastic model with compressible flow models

Cavitation erosion is caused in solids exposed to strong pressure waves developing in an adjacent fluid field. The knowledge of the transient distribution of stresses in the solid is important to understand the cause of damaging by comparisons with breaking points of the material. The modeling of this problem requires the coupling of the models for the fluid and the solid. For this purpose, we use a strategy based on the solution of coupled Riemann problems that has been originally developed for the coupling of 2 fluids. This concept is exemplified for the coupling of a linear elastic structure with an ideal gas. The coupling procedure relies on the solution of a nonlinear equation. Existence and uniqueness of the solution is proven. The coupling conditions are validated by means of quasi‐1D problems for which an explicit solution can be determined. For a more realistic scenario, a 2D application is considered where in a compressible single fluid, a hot gas bubble at low pressure collapses in a cold gas at high pressure near an adjacent structure.     

Combined interface boundary condition method for coupled thermal simulations

A new procedure for modeling the conjugate heat‐transfer process between fluid and structure subdomains is presented. The procedure relies on higher‐order combined interface boundary conditions (CIBC) for improved accuracy and stability. Traditionally, continuity of temperature and heat flux along interfaces is satisfied through algebraic jump conditions in a staggered fashion. More specifically, Dirichlet temperature conditions are usually imposed on the fluid side and Neumann heat‐flux conditions are imposed on the solid side for the stability of conventional sequential staggered procedure. In this type of treatment, the interface introduces additional stability constraints to the coupled thermal simulations. By utilizing the CIBC technique on the Dirichlet boundary conditions, a staggered procedure can be constructed with the same order of accuracy and stability as those of standalone computations. Using the Godunov–Ryabenkii normal‐mode analysis, a range of values of the coupling parameter is found that yields a stable and accurate interface discretization. The effectiveness of the method is investigated by presenting and discussing performance evaluation data using a 1D finite‐difference formulation for each subdomain. Copyright © 2007 John Wiley & Sons, Ltd.     

Nonlinear acceleration of coupled fluid–structure transient thermal problems by Anderson mixing

Conjugate heat‐transfer problems are typically solved using partitioned methods where fluid and solid subdomains are evaluated separately by dedicated solvers coupled through a common boundary. Strongly coupled schemes for transient analysis require fluid and solid problems to be solved many times each time step until convergence to a steady state. In many practical situations, a fairly simple and frequently employed fixed‐point iteration process is rather ineffective; it leads to a large number of iterations per time step and consequently to long simulation times. In this article, Anderson mixing is proposed as a fixed‐point convergence acceleration technique to reduce computational cost of thermal coupled fluid–solid problems. A number of other recently published methods with applications to similar fluid–structure interaction problems are also reviewed and analyzed. Numerical experiments are presented to illustrate relative performance of these methods on a test problem of rotating pre‐swirl cavity air flow interacting with a turbine disk. It is observed that performance of Anderson mixing method is superior to that of other algorithms in terms of total iteration counts. Additional computational savings are demonstrated by reusing information from previously solved time steps. Copyright © All rights reserved 2012 Rolls‐Royce plc.     

Flow simulation on moving boundary‐fitted grids and application to fluid–structure interaction problems

We present a method for the parallel numerical simulation of transient three‐dimensional fluid–structure interaction problems. Here, we consider the interaction of incompressible flow in the fluid domain and linear elastic deformation in the solid domain. The coupled problem is tackled by an approach based on the classical alternating Schwarz method with non‐overlapping subdomains, the subproblems are solved alternatingly and the coupling conditions are realized via the exchange of boundary conditions. The elasticity problem is solved by a standard linear finite element method. A main issue is that the flow solver has to be able to handle time‐dependent domains. To this end, we present a technique to solve the incompressible Navier–Stokes equation in three‐dimensional domains with moving boundaries. This numerical method is a generalization of a finite volume discretization using curvilinear coordinates to time‐dependent coordinate transformations. It corresponds to a discretization of the arbitrary Lagrangian–Eulerian formulation of the Navier–Stokes equations. Here the grid velocity is treated in such a way that the so‐called Geometric Conservation Law is implicitly satisfied. Altogether, our approach results in a scheme which is an extension of the well‐known MAC‐method to a staggered mesh in moving boundary‐fitted coordinates which uses grid‐dependent velocity components as the primary variables. To validate our method, we present some numerical results which show that second‐order convergence in space is obtained on moving grids. Finally, we give the results of a fully coupled fluid–structure interaction problem. It turns out that already a simple explicit coupling with one iteration of the Schwarz method, i.e. one solution of the fluid problem and one solution of the elasticity problem per time step, yields a convergent, simple, yet efficient overall method for fluid–structure interaction problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Fractional four-step finite element method for analysis of thermally coupled fluid-solid interaction problems

An integrated fluid-thermal-structural analysis approach is presented. In this approach, the heat conduction in a solid is coupled with the heat convection in the viscous flow of the fluid resulting in the thermal stress in the solid. The fractional four-step finite element method and the streamline upwind Petrov-Galerkin (SUPG) method are used to analyze the viscous thermal flow in the fluid. Analyses of the heat transfer and the thermal stress in the solid are performed by the Galerkin method. The second-order semiimplicit Crank-Nicolson scheme is used for the time integration. The resulting nonlinear equations are linearized to improve the computational efficiency. The integrated analysis method uses a three-node triangular element with equal-order interpolation functions for the fluid velocity components, the pressure, the temperature, and the solid displacements to simplify the overall finite element formulation. The main advantage of the present method is to consistently couple the heat transfer along the fluid-solid interface. Results of several tested problems show effiectiveness of the present finite element method, which provides insight into the integrated fluid-thermal-structural interaction phenomena.     

Accuracy and stability analysis of a second‐order time‐accurate loosely coupled partitioned algorithm for transient conjugate heat transfer problems

In this paper, a second‐order time‐accurate loosely coupled partitioned algorithm is presented for solving transient thermal coupling of solids and fluids, also referred to by conjugate heat transfer. The Crank–Nicolson scheme is used for time integration. The accuracy and stability of the loosely coupled solution algorithm are analyzed analytically. Based on the accuracy analysis, the design order of the time integration scheme is preserved by following a predictor (implicit)–corrector (explicit) approach. Hence, the need to perform an additional implicit solve (a subiteration) at each time step is avoided. The analytical stability analysis shows that by using the Crank–Nicolson scheme for time integration, the partitioned algorithm is unstable for large Fourier numbers, unlike the monolithic approach. Accordingly, using the stability analysis, a stability criterion is obtained for the Crank–Nicolson scheme that imposes restriction on Δt given the material properties and mesh spacings of the coupled domains. As the ratio of the thermal effusivities of the coupled domains reaches unity, the stability of the algorithm reduces. To demonstrate the applicability of the algorithm, a numerical example is considered (an unsteady conjugate natural convection in an enclosure). Copyright © 2013 John Wiley & Sons, Ltd.     

A computational method for geometric optimization of enhanced heat transfer devices based upon entropy generation minimization

A computational fluid dynamics‐based optimization methodology is developed, appropriate for the geometric optimization of enhanced heat transfer devices based upon the principle of entropy generation minimization, in which the objective function is evaluated from a flow field obtained by computational simulation. A quasi‐Newton optimization procedure is employed, with computation of the objective function gradients based upon a finite difference approach. The optimization procedure is developed to be general with regard to the choice of objective function, the details of the problem under consideration, and the computational methodology employed in solving the fluid flow and heat transfer problems. A novel implementation of a Taylor series‐based procedure for the fast solution of nearby problems is presented, which is found to greatly benefit the efficiency of the present methodology. Finally, a numerical experiment is presented, illustrating the use of the present method in the geometric optimization of a practical enhanced heat transfer device on the basis of the criterion of entropy generation minimization. The optimization of the fin spacing of a simple plate fin heat sink is considered, and a comparison of the computational results with results obtained by analytical optimization based upon empirical friction factor and Nusselt number correlations is given. Copyright © 2012 John Wiley & Sons, Ltd.     

A numerical model for thermal systems with helically-coiled tubes and its experimental verification

 A conjugate numerical model proposed by Nakayama et al. for the steady problem of cooling a fluid flowing through a coiled tube, has been successfully extended to investigate two distinctive thermal problems, namely, the transient cooling processes associated with a beer dispenser, and the transient processes of heat storage and recovery associated with a packed bed saturated with a molten salt. An axisymmetric numerical procedure is adopted for determining the velocity and temperature fields within the chilled water bath of the beer dispenser. A simplified one-dimensional heat transfer model is introduced for coupling the tube flow with the recirculating flow in the bath. A similar axisymmetric finite difference procedure is applied for the heat transfer analysis of the packed bed saturated with a molten salt. For the heat recovery process, a one-dimensional heat balance equation for the two-phase flow with a helically-coiled tube is introduced to update the wall surface temperatures, which are needed to calculate the temperature field in the saturated packed bed. The numerical results for both thermal systems associated with coiled tubes agree very well with the corresponding velocity and temperature data obtained from the experiments. Received on 28 August 2000 / Published online: 29 November 2001     

Streamline upwind finite element method for conjugate heat transfer problems

This paper presents a combined finite element method for solving conjugate heat transfer problems where heat conduction in a solid is coupled with heat convection in viscous fluid flow. The streamline upwind finite element method is used for the analysis of thermal viscous flow in the fluid region, whereas the analysis of heat conduction in solid region is performed by the Galerkin method. The method uses the three-node triangular element with equal-order interpolation functions for all the variables of the velocity components, the pressure and the temperature. The main advantage of the proposed method is to consistently couple heat transfer along the fluid-solid interface. Three test cases, i.e. conjugate Couette flow problem in parallel plate channel, counter-flow in heat exchanger, and conjugate natural convection in a square cavity with a conducting wall, are selected to evaluate the efficiency of the present method. The English text was polished byYunming Chen.     

Free Convection Heat Transfer over a Vertical Cylinder in a Saturated Porous Medium Using a Local Thermal Non-equilibrium Model

In this article, free convection heat transfer over a vertical cylinder with variable surface temperature distributions in a porous medium is analyzed. It is assumed that the fluid and solid phases are not in local thermal equilibrium and, therefore, a two-temperature model of heat transfer is applied. The coupled momentum and energy equations are presented and then they are transformed into ordinary differential equations. The similarity equations are solved numerically. The resulting velocity, streamlines, temperature distributions for fluid and solid phases are shown for different values of parameters entering into the problem. The calculated values of the local Nusselt numbers for both solid and fluid phases are also shown.     

Response of thermal storage units to periodic operating conditions

This paper aims to provide an exact solution to the problem of sensible heat transfer and storage in fluid-solid systems. Periodically fluctuating operating conditions are considered in detail, and a simple solution procedure is proposed in order to obtain the fluid and solid temperature distributions for any periodic thermal energy supply. Some significant examples are solved explicitly, and the relevant results are reported and briefly discussed in order to examine directly the storage unit's capability of smoothing out time fluctuations of the power source.     

Formulation of thermo-hydro-mechanical coupling behavior of unsaturated soils based on hybrid mixture theory

Thermo-Hydro-Mechanical （THM） coupling pro- cesses in unsaturated soils are very important in both theoretical researches and engineering applications. A coupled formulation based on hybrid mixture theory is derived to model the THM coupling behavior of unsaturated soils. The free-energy and dissipative functions for different phases are derived from Taylor＇s series expansions. Constitutive relations for THM coupled behaviors of unsaturated soils, which include deformation, entropy change, fluid flow, heat conduction, and dynamic compatibility conditions on the interfaces, are then established. The number of field equations is shown to be equal to the number of unknown variables; thus, a closure of this coupling problem is established. In addition to modifications of the physical conservation equations with coupling effect terms, the constitutive equations, which consider the coupling between elastoplastic deformation of the soil skeleton, fluid flow, and heat transfer, are also derived.     

On higher‐order mixed FEM for low Mach number flows: application to a natural convection benchmark problem

We consider higher‐order mixed finite elements with continuous pressures for the computation of stationary compressible flows at low Mach number. The proposed approach is based on a fully coupled treatment of the governing equations and therefore, for steady‐state calculations, does not rely on time‐stepping techniques. The non‐linear problem is solved by means of a quasi‐Newton iteration. The strongly coupled system resulting from higher‐order discretization of the linearized equations requires adequate solvers. We propose a new scheme based on multigrid methods with varying FEM ansatz orders on the grid hierarchy as well as multiplicative smoothers based on blocking techniques. Computational results are described for a benchmark configuration including a flow with heat transfer in the low Mach number regime. Furthermore, the issue of anisotropic grids is addressed in that context. Copyright © 2003 John Wiley & Sons, Ltd.     

Coupling of finite volume and finite element subdomains using different time integrators

This paper proposes a coupling strategy that can be used for transient fluid–structure interaction. The objective of this paper is to propose a time integrator coupling strategy, which ensures good properties to couple typical solid and typical fluid time integrators in linear cases. It is evaluated on a 1‐D toy problem only dedicated to the study of the quality of time integrators coupling. The structure is discretized by the linear finite element method and solved in time by the Newmark scheme, whereas the finite difference and finite volume methods are used for the fluid subdomain. By projecting the fluid equations on the eigenvector, we obtain a compatibility relation that corresponds to the characteristic line that transfers the fluid information from the inside to the fluid–structure interface. With this appropriate compatibility relation and the solid equations, the interface status is predicted for the next time step, ensuring zero interface energy. Hence, the order of accuracy and the stability are preserved to the minimum level of the two parts, the structure and the fluid. Furthermore, the coupling strategy allows incompatible time steps. Some numerical results are obtained for the 1‐D linear problem, and a good agreement with the analytical solution has been found. Copyright © 2013 John Wiley & Sons, Ltd.     

A two‐dimensional finite volume morphodynamic model on unstructured triangular grids

We discuss the application of a finite volume method to morphodynamic models on unstructured triangular meshes. The model is based on coupling the shallow water equations for the hydrodynamics with a sediment transport equation for the morphodynamics. The finite volume method is formulated for the quasi‐steady approach and the coupled approach. In the first approach, the steady hydrodynamic state is calculated first and the corresponding water velocity is used in the sediment transport equation to be solved subsequently. The second approach solves the coupled hydrodynamics and sediment transport system within the same time step. The gradient fluxes are discretized using a modified Roe's scheme incorporating the sign of the Jacobian matrix in the morphodynamic system. A well‐balanced discretization is used for the treatment of source terms. We also describe an adaptive procedure in the finite volume method by monitoring the bed–load in the computational domain during its transport process. The method uses unstructured meshes, incorporates upwinded numerical fluxes and slope limiters to provide sharp resolution of steep bed gradients that may form in the approximate solution. Numerical results are shown for a test problem in the evolution of an initially hump‐shaped bed in a squared channel. For the considered morphodynamical regimes, the obtained results point out that the coupled approach performs better than the quasi‐steady approach only when the bed–load rapidly interacts with the hydrodynamics. Copyright © 2009 John Wiley & Sons, Ltd.     

Prediction of transient heat transfer coefficients in thick-walled pressure components of cylindrical geometry

This paper presents a procedure for determining the transient heat transfer coefficient in cylindrical, thick-walled pressure parts. From theoretical considerations, the temperatures can be predicted at discrete locations throughout the wall, when input data such as thermocouple responses are known at one or several interior locations.Special emphasis is placed on the dynamic response of the thermometer, which measures the temperature of the inside fluid, to enable exact determination of both heat transfer coefficient and fluid temperature. The transient response of a thermocouple in a convectional thermowell (pocket) is described by the first-order convective heat transfer model in which the rate of thermoelement temperature change is proportional to the instantaneous difference between the thermoelement and fluid temperatures.     

Numerical simulation of glass fogging and defogging

A numerical procedure for the prediction of fogging and defogging phenomena is presented. The simulation involves the solution of an air flow field along a cold solid surface, the evaluation of the unsteady conduction through the solid itself, and a model for the heat and mass transfer within the thin water layer on the fogged surface. A suite of routines for the unsteady simulation of the water layer evolution is coupled with an equal order finite element Navier Stokes solver and a finite volume conduction code. The procedure is fully independent of the numerical details of the solid and fluid domain solvers. Two different coupling approaches may be followed: A loose one, where the Navier Stokes solution is used only for a steady state estimate of the heat transfer coefficient, or a close one, where the Navier Stokes, conduction and water layer codes are iterated simultaneously. The latter is required for the problem of natural convection, where temperature (and thus the energy balance of the water layer) and flow field are coupled. The water layer is modelled as a collection of closely packed tiny droplets, leaving a portion of dry area among them. The effect of the contact angle is taken into account, and physical assumptions allow to define the local ratio between wet and dry surface for both the fogging and defogging process. As a case study, a comparison with experimental data for a complete fogging and defogging cycle of a glass lens in natural convection is presented.     

Monolithic approach to thermal fluid-structure interaction with nonconforming interfaces

This paper presents a monolithic approach to the thermal fluid-structureinteraction(FSI) with nonconforming interfaces.The thermal viscous flow is governedby the Boussinesq approximation and the incompressible Navier-Stokes equations.Themotion of the fluid domain is accounted for by an arbitrary Lagrangian-Eulerian(ALE)strategy.A pseudo-solid formulation is used to manage the deformation of the fluid do-main.The structure is described by the geometrically nonlinear thermoelastic dynamics.An efficient data transfer strategy based on the Gauss points is proposed to guarantee theequilibrium of the stresses and heat along the interface.The resulting strongly coupledset of nonlinear equations for the fluid,structure,and heat is solved by a monolithicsolution procedure.A numerical example is presented to demonstrate the robustness andefficiency of the methodology.