Solid Subdomains Research Articles

Numerical Simulation of the Conjugate Heat Transfer of a “Fluid–Solid Body” System on an Unmatched Grid Interface

Recently, when modeling transient problems of conjugate heat transfer, the independent construction of grid models for fluid and solid subdomains is increasingly being used. Such grid models, as a rule, are unmatched and require the development of special grid interfaces that match the heat fluxes at the interface. Currently, the most common sequential approach to modeling problems of conjugate heat transfer requires the iterative matching of boundary conditions, which can significantly slow down the process of the convergence of the solution in the case of modeling transient problems with fast processes. The present study is devoted to the development of a direct method for solving conjugate heat transfer problems on grid models consisting of inconsistent grid fragments on adjacent boundaries in which, in the general case, the number and location of nodes do not coincide. A conservative method for the discretization of the heat transfer equation by the direct method in the region of inconsistent interface boundaries between liquid and solid bodies is proposed. The proposed method for matching heat fluxes at mismatched boundaries is based on the principle of forming matched virtual boundaries, proposed in the GGI (General Grid Interface) method. A description of a numerical scheme is presented, which takes into account the different scales of cells and the sharply different thermophysical properties at the interface between liquid and solid media. An algorithm for constructing a conjugate matrix, the form of matrix coefficients responsible for conjugate heat transfer, and methods for calculating them are described. The operability of the presented method is demonstrated by the example of calculating conjugate heat transfer problems, the grid models of which consist of inconsistent grid fragments. The use of the direct conjugation method makes it possible to effectively solve both stationary and non-stationary problems using inconsistent meshes, without the need to modify them in the conjugation region within a single CFD solver.

Integral Transforms with Single Domain Formulation for Transient Three-Dimensional Conjugated Heat Transfer

An analysis based on integral transforms is undertaken for transient three-dimensional conjugated conduction-convection heat transfer, with a focus on mini or micro channels-based devices. The numerical-analytical approach Generalized Integral Transform Technique (GITT) is combined with a single domain reformulation, providing accurate, robust, and cost-effective simulations for determining temperature distributions within the domain. The fluid and solid subdomains are represented as one single region, while the integral transformation is carried out using a three-dimensional eigenvalue problem encompassing the thermophysical properties and velocity field abrupt spatial variations. The steady state problem solution is employed as a filter and solved through an integral transformation based on the corresponding two-dimensional eigenvalue problem defined for the channel cross section. The transformed ordinary differential systems for both the steady and homogeneous transient problems are handled analytically, requiring only the numerical solution of the associated matrix eigensystem analysis. Converged numerical results for dimensionless temperature distributions are then critically compared with a finite element solution and a previously proposed GITT solution that implements a partial transformation scheme under a pseudo-transient formulation, both for the conjugated problem with thermally developing laminar flow in a rectangular channel.

Unsteady conjugate heat transfer characteristics in hexagonal cavity equipped with a multi-blade dynamic modulator

PurposeThe transient analysis of the thermal response and frictional loss characteristics for flow-modulated conjugate heat transfer phenomena has been investigated in the present study. The flow domain is a partitioned cavity of a hexagonal structure equipped with a multi-blade flow modulator. The clockwise rotating blade is adiabatic and stirrers the internal flow along with the natural convection caused by the bottom heated floor of uniform heat flux. The conjugate behavior is introduced through the solid subdomains consisting of two brick-made partitions and one glass partition of uniform thickness. The material of the partition wall reflects the physical aspects of industrial applications. ApproachThe two-dimensional unsteady continuity, momentum, and energy equations are expressed in a non-dimensional form where the buoyant force is modeled through the Boussinesq approximation. The Arbitrary Lagrangian Euler (ALE) finite element is adopted to solve the moving mesh problem by formulating a free triangular discretization scheme. Parametric computational investigations are carried out for air as the working fluid (Pr = 0.71) and 3 different configurations of the rotating modulator while varying the other parameters, i.e., Reynolds number (Re) and Rayleigh number (Ra) for a fixed Biot number (Bi = 104). This dynamic mesh problem encompasses a wide range of parameters, i.e., (100 ≤Re≤ 103), and (104≤Ra≤ 106) for Bi = 104 to evaluate the thermodynamic response of the present thermo-fluid system. Various thermo-fluid system responses are visualized through the spatially average Nusselt number evaluated on the heated surface, system effectiveness, average thermal storage capacity, and frictional power loss of the flow domain. The thermal response is fragmented into individual responses in terms of component signal frequency by the Fast Fourier Transform (FFT) analysis. FindingsAccording to the current analysis, increasing the number of blades increases the rate of heat transfer by a negligible amount. On the other hand, among three different types of modulators, the three-blade modulator shows the lowest power loss coefficient and the lowest power instability. Thus, a three-blade modulator qualifies as the optimum design to impart external motion to the working fluid to introduce forced convection. LimitationsThis research has some scope for improvement. Future studies can be conducted on the improvement of blade designs, thickness, and size of the blades. Practical ImplicationsThe present study can shed light on the modeling of a better storage system where heat transfer and moisture control are integrated for the proper functioning of the storage, such as food storage, cold storage, garment inventory, etc. OriginalityThis study is original, and no previous research has been conducted considering the hexagonal domain and variation of the number of blades. ConclusionThe present study reveals that a modulator best serves its purpose at low Rayleigh and high Reynolds numbers. Moreover, the three-blade modulator has the lowest power loss coefficient and lowest power instability, making it the most suitable modulator to impart external excitation to the working fluid.

Thermo-mechanics aspects of isochoric cryopreservation: A new modeling approach and comparison with experimental data

A new mathematical model is proposed for the analysis of thermo-mechanics effects during isochoric cryopreservation. In that process, some ice crystallization in a fixed-volume container drives pressure elevation, which may be favorable to the preservation of biological material when it resides in the unfrozen portion of the same container. The proposed model is comprehensive, integrating for the first time concepts from the disparate fields of thermodynamics, heat transfer, fluid mechanics, and solid mechanics. The novelty in this study is in treating the cryopreserved material as having a pseudo-viscoelastic behavior over a very narrow temperature range, without affecting the mechanical behavior of the material in the rest of the domain. This unique approach permits treating the domain as a continuum, while avoiding the need to trace freezing fronts and sperate the analysis to liquid and solid subdomains. Consistent with the continuum approach, the heat transfer problem is solved using the enthalpy approach. The presented analysis focusses on isochoric cooling of pure water between standard atmospheric conditions and the triple point of liquid water, ice Ih, and ice III (-22°C and 207.4 MPa). The proposed model is also applicable to isochoric vitrification, by substituting the pseudo-viscoelastic material model with the real viscosity model of the vitrifying material. Results of this study display good agreement with phase-diagram data at steady state, and with experimental data from the literature. Furthermore, this study provides a venue to discussing experimentation aspects of isochoric cryopreservation. The proposed model is further demonstrated on a 3D problem, while discussing scale considerations, crystallization conditions, and transient effects. Notably, the new model can be used to bridge the gap between limited pressure and temperature measurements during cryopreservation and the analysis of the continuum. Arguably, this study presents the most advanced thermo-mechanics model to solve practical problems relating to isochoric cryopreservation.

Variational coupling of non-matching discretizations across finitely deforming fluid-structure interfaces.

This paper presents a stabilized monolithic method for coupling incompressible viscous fluids with finitely deforming elastic solids across non-matching interfacial meshes. Governing equations for the solid are written in the finite deformation Lagrangian frame using velocity field as the primary unknown, while the governing equations for the fluid are written in an Arbitrary Lagrangian-Eulerian (ALE) frame to accommodate large motions of the fluid-solid interfaces. Interface coupling terms are derived by embedding Discontinuous Galerkin (DG) ideas in the Variational Multiscale (VMS) framework and locally resolving the fine-scale variational equations from the fluid and the solid subdomains along the common interface. The unique attribute of the proposed Variational Multiscale Discontinuous Galerkin (VMDG) method is a systematic procedure for deriving analytical expression for the traction Lagrange multiplier at the fluid-solid interface. The structure of the interface stabilization tensor emerges naturally and is shown to be a function of the boundary operators that are associated with the domain interior operators from the fluid and the solid subdomains. The derived stabilization tensor possesses the features of area-averaging and stress-averaging and evolves spatially and temporally with the evolving nonlinear fields at the interface. An analysis of the mathematical properties of the interface stabilization tensor is presented and subsequently verified numerically. The method is implemented using four-node tetrahedral elements for the fluid as well as the solid subdomains while employing equal-order interpolations for the various interacting fields. Benchmark problems are presented for the verification of the method for finitely deforming fluid-structure interfaces.

Blood–artery interaction in calcified aortas and abdominal aortic aneurysms

A stabilized Fluid–Structure Interaction (FSI) method is presented for coupling shear-rate dependent model of blood with finitely deforming anisotropic hyperelastic model of arteries. The blood–artery coupling conditions are weakly enforced to accommodate non-matching blood–artery meshes which provides great flexibility in independent discretization of the fluid and solid subdomains in the patient-specific geometric models. The variationally derived interface coupling terms play an important role in the concurrent solution of the nonlinear mixed-field problems on non-matching discretizations. Two test cases are presented that investigate the effect of the growth of aortic aneurysm on the local changes in the blood flow patterns, and stress concentrations in calcified arteries under pulsating flows to highlight the clinical relevance of the proposed method for cardiovascular applications.

A hybrid partitioned deep learning methodology for moving interface and fluid–structure interaction

In this work, we present a hybrid partitioned deep learning framework for the reduced-order modeling of moving interfaces and predicting fluid–structure interaction. Using the discretized Navier–Stokes in the arbitrary Lagrangian–Eulerian reference frame, we generate the full-order flow snapshots and point cloud displacements as target physical data for the learning and inference of coupled fluid–structure dynamics. The hybrid operation of this methodology comes by combining two separate data-driven models for fluid and solid subdomains via deep learning-based reduced-order models (DL-ROMs). The proposed multi-level framework comprises the partitioned data-driven drivers for unsteady flow and the moving point cloud displacements. At the fluid–structure interface, the force information is exchanged synchronously between the two partitioned subdomain solvers. The first component of our proposed framework relies on the proper orthogonal decomposition-based recurrent neural network (POD-RNN) as a DL-ROM procedure to infer the point cloud with a moving interface. This model utilizes the POD basis modes to reduce dimensionality and evolve them in time via long short-term memory-based recurrent neural networks (LSTM-RNNs). The second component employs the convolution-based recurrent autoencoder network (CRAN) as a self-supervised DL-ROM procedure to infer the nonlinear flow dynamics at static Eulerian probes. We introduce these probes as spatially structured query nodes in the moving point cloud to treat the Lagrangian-to-Eulerian conflict together with convenience in training the CRAN driver. To determine these Eulerian probes, we construct a novel snapshot-field transfer and load recovery algorithm. They are chosen in such a way that the two components (i.e., POD-RNN and CRAN) are constrained at the interface to recover the bulk force quantities. These DL-ROM-based data-driven drivers rely on the LSTM-RNNs to evolve the low-dimensional states. A popular prototypical fluid–structure interaction problem of flow past a freely oscillating cylinder is considered to assess the efficacy of the proposed methodology for a different set of reduced velocities that lead to vortex-induced vibrations. The proposed framework tracks the interface description with acceptable accuracy and predicts the nonlinear wake dynamics over the chosen test data range. The proposed framework aligns with the development of partitioned digital twin of engineering systems, especially those involving moving boundaries and fluid–structure interactions.

A partitioned framework for coupling LBM and FEM through an implicit IBM allowing non-conforming time-steps: Application to fluid-structure interaction in biomechanics

This paper presents a partitioned framework for the numerical simulation of fluid-structure interactions by coupling the lattice Boltzmann method (LBM) and the finite element method (FEM). The two numerical methods LBM and FEM are coupled with an implicit immersed boundary method (IBM) in a strong way, which ensures exactly the no-slip condition and the continuities of velocity and stress at the fluid-solid interface and each instant in time. In the proposed partitioned coupling procedure, the coupling system of equations are first established and then condensed to the interface. By solving the condensed coupling system of equations, the interface force field is obtained and sent to both solvers to accomplish time integrations in each sub-domain. In addition, two strategies based on linear interpolation in time are proposed to handle the cases with non-conforming time-steps in the fluid and solid sub-domains. Through several 2D and 3D numerical test-cases on the mechanical heart valve, the fluid-induced vibration of a deformable solid beam, the flapping flag, the proposed coupling framework is validated with good agreements with references. Finally, a test-case on the interaction between the blood flow and the aortic valve is carried out, showing the applicability of the present framework in realistic biomechanical applications.

A sharp interface Lagrangian-Eulerian method for rigid-body fluid-structure interaction

This paper introduces a sharp interface method to simulate fluid-structure interaction (FSI) involving rigid bodies immersed in viscous incompressible fluids. The capabilities of this methodology are benchmarked using a range of test cases and demonstrated using large-scale models of biomedical FSI. The numerical approach developed herein, which we refer to as an immersed Lagrangian-Eulerian (ILE) method, integrates aspects of partitioned and immersed FSI formulations by solving separate momentum equations for the fluid and solid subdomains, as in a partitioned formulation, while also using non-conforming discretizations of the dynamic fluid and structure regions, as in an immersed formulation. A simple Dirichlet-Neumann coupling scheme is used, in which the motion of the immersed solid is driven by fluid traction forces evaluated along the fluid-structure interface, and the motion of the fluid along that interface is constrained to match the solid velocity and thereby satisfy the no-slip condition. To develop a practical numerical method, we adopt a penalty approach that approximately imposes the no-slip condition along the fluid-structure interface. In the coupling strategy, a separate discretization of the fluid-structure interface is tethered to the volumetric solid mesh via stiff spring-like penalty forces. Our fluid-structure coupling scheme relies on an immersed interface method (IIM) for discrete geometries, which enables the accurate determination of both velocities and stresses along complex internal interfaces. Numerical methods for FSI can suffer from instabilities related to the added mass effect, but computational tests indicate that the methodology introduced here remains stable for selected test cases across a broad range of solid-fluid mass density ratios, including extremely small, nearly equal, equal, and large density ratios. Biomedical FSI demonstration cases include results obtained using this method to simulate the dynamics of a bileaflet mechanical heart valve in a pulse duplicator, and to model transport of blood clots in a patient-averaged anatomical model of the inferior vena cava.

An enhanced explicit time-marching procedure to analyse acoustic–elastodynamic coupled models

This paper proposes an effective approach to analyse wave propagation through coupled fluid–structure models considering explicit time-marching formulations. In this context, different explicit time-domain algorithms are applied to analyse the acoustic and elastodynamic subdomains of the model, allowing greater time-step values to be considered for the coupled analysis, improving the performance of the explicit solution. Since explicit algorithms are considered, separated analyses of the acoustic and elastodynamic subdomains may be easily carried out, with the interaction between the different subdomains of the model being accomplished by interface forces. In this work, these interface forces are computed considering an implicit formulation, allowing the stability limits of the explicit solution to be governed by the decoupled subdomains of the model; thus, enabling greater time-steps to be applied if enhanced time-marching procedures are considered within the solid subdomains (which are usually the most restrictive subdomains of the coupled model). The proposed methodology provides adaptive algorithmic dissipation in the higher modes and it is highly accurate, as well as it is very simple to implement. Numerical results are presented at the end of the manuscript, illustrating the performance and effectiveness of the new approach.

Numerical modeling of strongly coupled microscale multiphase flow and solid deformation

SummaryWhen fluid flows in porous media under subsurface conditions, significant deformation can occur. Such deformation is dependent on structural and phase characteristics. In this paper, we investigate the effect of multiphase flow on the deformation of porous media at the pore scale by implementing a strongly coupled partitioned solver discretized with finite volume (FV) technique. Specifically, the role of capillary forces on grain deformation in porous media is investigated. The fluid and solid subdomains are meshed using unstructured independent grids. The model is applied for solving multiphase coupled equations and is capable of capturing pore scale physics during primary drainage by solving the Navier‐Stokes equation and advecting fluid indicator function using volume of fluid (VOF) while the fluid is interacting with a nonlinear elastic solid matrix. The convergence of the coupled solver is accelerated by Aitken underrelaxation. We also reproduce geomechanical stress conditions, at the pore scale, by applying uniaxial stress on the solid while simultaneously solving the multiphase fluid‐solid interaction problem to investigate the effect of external stress on fluid occupancy, velocity‐field distribution, and relative permeability. We observe that the solid matrix exhibits elasto‐capillary behavior during the drainage sequence. Relative permeability endpoints are shifted on the basis of the external stress exerted.

Thermo-coupled FSI analysis of LN2 jet impinging on hot dry rock

Liquid nitrogen (LN2) jet is a potential drilling rate improvement method in hot dry rock geothermal. LN2 jet impinging on rock is a complex multi-physics coupling issue that involves thermo-coupled fluid-structure interaction (FSI). To address such issue, we adopted a loosely coupled method which uses separate and independent techniques for the fluid and solid subdomains and exchanges data along the fluid–solid interfaces via a fixed-point iteration scheme. Nonlinear k-ε turbulence model was adopted to predict the flow field and heat transfer in the fluid subdomain. Damage plasticity model was employed in the solid subdomain to capture the dynamic damage behavior under the joint action of jet impact and heat transfer. Results indicate that the convective heat transfer rate and the dynamic responses of rock are closely related to the near-wall flow of LN2 jet. Both maximum heat transfer rate and damage occur at 1D (nozzle exit diameter) away from the stagnation point, which is just between the velocity peak and the turbulence intensity peak positions. Thermal stresses and corresponding irreversible deformation dominate the failure of hot dry rock. Increasing initial rock temperature contributes to enhancing the heat transfer rate, aggravating rock damage and shortening the starting time for damage.

Topology optimization of heat and mass transfer problems in two fluids—one solid domains

Among various topology optimization methods used in fluid flow problems, density approach has gained more interest compared to other techniques as level set approach, topological derivative technique, and phase field method. The key part of density approach is the penalized interpolation function, which forces progressively porous cells made of fluid and solid simultaneously to belong discretely to fluid or solid sub-domains. However this type of problem was only solved in mono-fluid domains, in which the method accounts for the distribution of a single fluid and a single solid. The actual work aims to extend topology optimization in fluid flow problems to bi-fluid domain. A new interpolation function was developed for this purpose. Furthermore a penalization function was integrated in the multiobjective function, which ensure that each fluid takes its own path in the device, while maintaining a minimal required solid thickness between the channels of different fluids. The results showed the capacity of the proposed method to deal with multiple fluid phases in minimizing the pressure drop while maximizing heat exchange between different flows. The main conclusion is the potential of density approach to be applied on optimization of heat exchangers.

Acoustic waves scattered by elastic waveguides using a spectral approach with a 2.5D coupled boundary-finite element method

This work presents a two-and-a-half dimensional (2.5D) spectral formulation based on the finite element method (FEM) and the boundary element method (BEM) to study wave propagation in acoustic and elastic waveguides. The analysis involved superposing two dimensional (2D) problems with different longitudinal wavenumbers. A spectral finite element (SFEM) is proposed to represent waveguides in solids with arbitrary cross-section. Moreover, the BEM is extended to its spectral formulation (SBEM) to study unbounded fluid media and acoustic enclosures. Both approaches use Lagrange polynomials as element shape functions at the Legendre–Gauss–Lobatto (LGL) points. The fluid and solid subdomains are coupled by applying the appropriate boundary conditions at the limiting interface. The proposed method is verified by means of two benchmark problems: wave propagation in an unbounded acoustic medium and the scattering of waves by an elastic inclusion. The convergence and the computational effort are evaluated for different h−p strategies. Numerical results show good agreement with the reference solution. Finally, the proposed method is used to study the pressure field generated by an array of elastic fluid-filled scatterers immersed in an acoustic medium.

Solid Geometry Processing on Deconstructed Domains

AbstractMany tasks in geometry processing are modelled as variational problems solved numerically using the finite element method. For solid shapes, this requires a volumetric discretization, such as a boundary conforming tetrahedral mesh. Unfortunately, tetrahedral meshing remains an open challenge and existing methods either struggle to conform to complex boundary surfaces or require manual intervention to prevent failure. Rather than create a single volumetric mesh for the entire shape, we advocate for solid geometry processing on deconstructed domains, where a large and complex shape is composed of overlapping solid subdomains. As each smaller and simpler part is now easier to tetrahedralize, the question becomes how to account for overlaps during problem modelling and how to couple solutions on each subdomain together algebraically. We explore how and why previous coupling methods fail, and propose a method that couples solid domains only along their boundary surfaces. We demonstrate the superiority of this method through empirical convergence tests and qualitative applications to solid geometry processing on a variety of popular second‐order and fourth‐order partial differential equations.

A unified continuum and variational multiscale formulation for fluids, solids, and fluid–structure interaction

We develop a unified continuum modeling framework using the Gibbs free energy as the thermodynamic potential. This framework naturally leads to a pressure primitive variable formulation for the continuum body, which is well-behaved in both compressible and incompressible regimes. Our derivation also provides a rational justification of the isochoric–volumetric additive split of free energies in nonlinear elasticity. The variational multiscale analysis is performed for the continuum model to construct a foundation for numerical discretization. We first consider the continuum body instantiated as a hyperelastic material and develop a variational multiscale formulation for the hyper-elastodynamic problem. The generalized-α method is applied for temporal discretization. A segregated algorithm for the nonlinear solver, based on the original idea introduced in Scovazzi et al. (2016), is carefully analyzed. Second, we apply the new formulation to construct a novel unified formulation for fluid–solid coupled problems. The variational multiscale formulation is utilized for spatial discretization in both fluid and solid subdomains. The generalized-α method is applied for the whole continuum body, and optimal high-frequency dissipation is achieved in both fluid and solid subproblems. A new predictor multi-corrector algorithm is developed based on the segregated algorithm. The efficacy of the new formulations is examined in several benchmark problems. The results indicate that the proposed modeling and numerical methodologies constitute a promising technology for biomedical and engineering applications, particularly those necessitating incompressible models.

A coupled BEM/FEM formulation for drop interaction in Stokes flows with flexible and slip confining boundaries

In this work, fluid-fluid-solid coupled models are analysed, considering the interaction of boundary and finite element techniques. In this context, the paper focuses on the study of deforming drops through bulk fluids bounded by flexible walls. Here, the fluid subdomains are assumed to be viscous and incompressible, and they are modelled by the BEM. The solid subdomains are assumed to be elastic, and they are modelled by the FEM. Both discontinuity of tractions on the fluid-fluid common boundaries and discontinuity of velocities on the fluid-solid interfaces are considered. For the discontinuity of velocities, a formulation based on nonlinear slip boundary conditions is adopted, which is treated employing a relaxed iterative approach. A Lagrangian representation is considered and remeshing is applied on the fluid-fluid interfaces, reducing the appearance of numerical problems. Numerical results are presented to illustrate the performance and potentialities of the proposed techniques.

A non-intrusive partitioned approach to couple smoothed particle hydrodynamics and finite element methods for transient fluid-structure interaction problems with large interface motion

We propose a non-intrusive numerical coupling method for transient fluid-structure interaction (FSI) problems simulated by means of different discretization methods: smoothed particle hydrodynamics (SPH) and finite element (FE) methods for the fluid and the solid sub-domains, respectively. As a partitioned coupling method, the present algorithm can ensure a zero interface energy during the whole period of numerical simulation, even in the presence of large interface motion. In other words, the time integrations of the two sub-domains (second order Runge---Kutta scheme for fluid and Newmark integrator for solid) are synchronized. Thanks to this energy-conserving feature, one can preserve the minimal order of accuracy in time and the numerical stability of the FSI simulations, which are validated with a 1D and a 2D trivial numerical test cases. Additionally, some other 2D FSI simulations involving large interface motion have also been carried out with the proposed SPH---FE coupling method. Finally, an example of aquaplaning problem is given in order to show the feasibility of such coupling method in multi-dimensional applications with complicated structural geometries.

2.5D coupled BEM-FEM used to model fluid and solid scattering wave

SUMMARY We describe a hybrid method to study fluid and solid interaction problems in the frequency domain. The numerical method is based on subdomain decomposition. The BEM is used to model unbounded solid mediums, whereas the confined subdomains, both fluid and solid, are represented by the FEM. The analysis is carried out by superposing two-and-a-half dimension (2.5D) problems for different longitudinal wave numbers. A novel 2.5D FEM formulation for inviscid fluids is proposed, which include the energy lost at the fluid boundary enclosure. The fluid and solid subdomains are coupled, and appropriate boundary conditions are imposed at the interfaces. The proposed technique is verified from analytical solutions. A cylindrical cavity located in an unbounded solid medium excited by a dilatational point source is studied. Computed results are in good agreement with the analytical solution. Later, noise and vibration in a concrete tunnel due to an internal pressure load is analysed with the proposed methodology. Results show that tunnel and soil displacements increase with the load speed, as did the air pressure inside the tunnel, according with the travelling ranges defined by the wave propagation velocities in each medium. Copyright © 2014 John Wiley & Sons, Ltd.

Analysis and application of high order implicit Runge–Kutta schemes for unsteady conjugate heat transfer: A strongly-coupled approach

Thermal interaction of fluids and solids, or conjugate heat transfer (CHT), is encountered in many engineering applications. Since time-accurate computations of unsteady CHT can be computationally demanding, we consider the use of high order implicit time integration schemes which have the potential to be more efficient relative to the commonly used second order implicit schemes. We present a strongly-coupled solution algorithm where the high order L-stable explicit first-stage singly diagonally implicit Runge–Kutta (ESDIRK) schemes are used to advance the solution in time within each separate fluid and solid subdomains. Furthermore, the stability and rate of convergence of performing (Gauss–Seidel) subiterations at each stage of the ESDIRK schemes are analyzed. The results from solving a numerical example (an unsteady conjugate natural convection in an enclosure) show good agreement with the performed analytical stability analysis. In addition, the (computational) work-(temporal) precision character of several schemes in solving a strongly coupled CHT problem is compared over a range of accuracy requirements. From the efficiency investigation, it is observed that performing subiterations with the strongly-coupled ESDIRK algorithm is more efficient than lowering time-step size using a high order loosely-coupled IMEX algorithm. In addition, by using the ESDIRK schemes, gain in computational efficiency relative to Crank–Nicolson is observed for time-accurate solutions (a factor of 1.4 using the fourth order ESDIRK). The computational gain is higher for smaller tolerances.