Conjugate Heat Transfer Formulation

• Conjugate Heat Transfer (CHT) is a formulation that rigorously couples fluid energy transport with solid heat conduction via continuous temperature and flux conditions.
• Numerical methods, including immersed boundary, finite-element, and lattice Boltzmann approaches, are employed to accurately capture multiscale thermal interactions.
• Recent advancements emphasize multiscale upscaling, data-driven surrogates, and rigorous validation to improve simulations in applications such as electronic cooling and heat exchangers.

Conjugate heat transfer (CHT) refers to the direct coupling of heat transport in solids and fluids by enforcing interfacial continuity of both temperature and heat flux at the fluid–solid interface. CHT is fundamental to the analysis of thermal systems where advective and diffusive processes interact across material boundaries, such as in geothermal reservoirs, electronic cooling, turbomachinery, energy storage, and heat exchangers, including systems involving phase change, turbulent convection, multiphase flows, and complex geometries. The rigorous formulation and solution of CHT problems involve coupled partial differential equations (PDEs) for momentum and energy in the fluid, coupled to heat conduction in the solid, constrained by precise interface conditions. Variations in model fidelity, complexity, and algorithmic strategies are central to contemporary research across multiscale, multiphysics, and data-driven CHT simulation.

1. Governing Equations and Interface Conditions

The mathematical core of CHT is the coupled solution of energy (and, commonly, momentum) equations in each subdomain with interface conditions enforcing continuity of both temperature and normal heat flux.

Fluid Domain (General Form)

For incompressible flow with temperature-dependent material properties, the fluid equations are:

• Mass conservation

$\nabla\cdot\mathbf{u} = 0$

• Navier–Stokes (possibly Boussinesq/low Mach, with/without chemical reactions)

$$\rho\left(\frac{\partial\mathbf{u}}{\partial t} + \mathbf{u}\cdot\nabla\mathbf{u}\right) = -\nabla p + \nabla\cdot\boldsymbol{\tau} + \mathbf{F}$$

• Energy (convection–diffusion, possibly with phase change or reaction source, $Q$)

$$\rho c_p \left(\frac{\partial T}{\partial t} + \mathbf{u}\cdot\nabla T\right) = \nabla\cdot(k\nabla T) + Q$$

Solid Domain

Here, only conduction contributes to energy transport:

$$\rho_s c_{p,s} \frac{\partial T_s}{\partial t} = \nabla\cdot(k_s \nabla T_s) + Q_s$$

Interface Conditions (on $$\Gamma_{fs} = \partial\Omega_f \cap \partial\Omega_s$$)

1. Temperature continuity: $T_{\mathrm{fluid}} = T_{\mathrm{solid}}$
2. Heat-flux continuity:

$$k_f \frac{\partial T_f}{\partial n} = k_s \frac{\partial T_s}{\partial n}$$

Interface coupling must only involve the conductive component of the heat flux for physical consistency and Galilean invariance. Inclusion of advective (e.g., total enthalpy) fluxes in interface conditions leads to violation of Galilean invariance, as rigorously demonstrated (Hu, 10 Apr 2024).

2. Representative Formulations and Numerical Implementations

CTH models feature a range of numerical and theoretical approaches, from classical interface-resolved finite-volume/FEM to modern micro-continuum and spectral macroscopic upscaling.

Immersed Boundary and Level-Set Methods

Sharp-interface CHT immersed boundary methods enforce interface conditions at the mesh faces cut by $\Gamma_{fs}$, often using level-set fields to assign cut-cell volumes and faces (Crocker et al., 2014). Discrete fluxes and field values are rescaled according to geometric intersection metrics (volume/area fractions), maintaining second-order accuracy even for high conductivity ratios.

Micro-Continuum and Volume-Fraction Methods

Single-field/micro-continuum models (e.g., GeoChemFoam) average fluid and solid variables on each computational cell via local phase fractions $\epsilon_f$ and $\epsilon_s$, with effective parameters (e.g., permeability via a penalized Kozeny–Carman closure, conductivity via harmonic averaging) enforcing the correct sharp-interface limits in a “diffuse” sense (Maes et al., 2021).

Monolithic and Partitioned Schemes

Monolithic finite-element or finite-volume methods assemble all domain variables into a single global linear or nonlinear system, treating the interface as internal to the mesh (possibly with Lagrange multipliers for continuity constraints) (Chen et al., 18 Dec 2024, Ebbs-Picken et al., 2023). Partitioned (domain-decomposition) methods alternately solve for fluid and solid, enforcing interface exchange via iterative Dirichlet–Neumann, Robin/Steklov, or even optimization-based matching (Fang et al., 16 Mar 2025).

Lattice Boltzmann and Single-Domain Schemes

LBM-based CHT solvers use a unified, globally defined distribution function, with solid–fluid “painting” via a solid-fraction field, enabling automatic satisfaction of temperature and heat-flux continuity in both static and moving domains (Zhang et al., 31 Oct 2024, Zhao et al., 2019). Solid-fraction transport and careful treatment of material properties avoid spurious interface artifacts.

Upscaled and Multirate Models

In strongly heterogeneous systems, micro–macro upscaling (e.g., GMRT model) systematically derives a hierarchy of coupled equations for mobile and immobile (mobile/diffusive) regions, with transfer rates obtained via spectral decompositions and micro-cell solutions (Municchi et al., 2019).

3. Analytical and Asymptotic Reductions

CTH admits limiting reductions in specific physical regimes. In the thermal dunking problem, the CHT system, upon separation of scales and spatial uniformity assumptions for small Biot number, reduces to the classical lumped-capacitance model (LCM). Here, the error in this approximation is decomposed into “time homogenization,” “lumping,” and “empirical Biot-number substitution” errors, with explicit computable a priori bounds (Guo et al., 10 Nov 2025). For accurate boundary resistance description (finite interfacial resistance), the temperature jump is proportional to the interface heat flux and the (possibly spatially varying) contact resistance.

4. Interface Condition Controversies and Consistency

A crucial result from (Hu, 10 Apr 2024) is that the only physically consistent conjugate boundary condition is continuity of the conductive heat flux:

$$k_s\,\nabla T_s\cdot n_s = k_f\,\nabla T_f\cdot n_f$$

together with $T_s = T_f$. Proposals to include advective or total enthalpy flux in the interface condition (i.e., $[-k\nabla T+\rho c T \mathbf{u}]_s^f\cdot n = 0$) are shown to violate Galilean invariance and should not be used in inertial-frame physical models, except under special conditions (e.g., equal volumetric heat capacities).

5. Non-Dimensional Parameters and Physical Regimes

CTH is controlled by a set of dimensionless groups:

Symbol	Definition	Notes
$Re$	Reynolds number $U H/\nu$	Flow regime, inertia vs. viscosity
$Pr$	Prandtl number $\nu/\alpha$	Diffusive vs. advective heat transport strategy
$Ra$	Rayleigh number $\alpha g \Delta T H^3 / (\nu \kappa)$	Buoyancy-driven convection intensity
$Nu$	Nusselt number $Q/(k\Delta T/H)$	Non-dimensional heat transfer; computed from CHT simulations/data
$k_r$	Thermal conductivity ratio $k_s/k_f$	Interfacial discontinuity in material property
$\kappa_{ratio}$	Diffusivity ratio $\kappa_s/\kappa_f$	Governs transition from Dirichlet to Neumann temperature conditions in RB convection (Ettel et al., 28 Feb 2025)
$B$	Biot number $hL/k_s = r_2 \overline{Nu}$	Transition from spatial to lumped solid temperature (Guo et al., 10 Nov 2025)

As $k_r\to\infty$, the interface enforces nearly isothermal (Dirichlet) conditions; as $k_r\to0$, the interface enforces constant flux (Neumann) conditions. The full spectrum of physically realizable boundaries (and hence observable large-scale convection structures, Nusselt scaling, and boundary-layer thicknesses) can be explored via parametric variation of $k_r$ or $\kappa_{ratio}$ (Ettel et al., 28 Feb 2025).

6. Specialized CHT Formulations: Boiling, Phase Change, and Heterogeneous Media

Advanced CHT studies extend the basic model to multiphysics:

• Boiling/evaporation: Incorporation of phase-change models (e.g., VOF or LBM plus Clapeyron/EOS) introduces latent heat sources at the interface, and dynamic movement of vapor–liquid–solid contact lines, with interface-coupled heat and mass transfer solved simultaneously (Odumosu et al., 24 Nov 2024, Zhao et al., 2019).
• Reactive flows and combustion: Coupled species transport, Arrhenius kinetics, and heat release are solved in fluid, with solid–fluid conjugate coupling at heat-exposed bodies (Chen et al., 18 Dec 2024).
• Heterogeneous/porous media: Micro-continuum and generalized multi-rate transfer models address CHT in complex microstructures, with macroscopic transfer-rate closure performed via micro-cell spectral analysis (Maes et al., 2021, Municchi et al., 2019).

7. Numerical, Reduced-Order, and Data-Driven Models

Given the computational demands of high-fidelity CHT simulation, reduced-order surrogate models based on proper orthogonal decomposition (POD), convolutional neural networks, and hybrid deep-learning integrators are widely studied (Ebbs-Picken et al., 2023, Hajisharifi et al., 28 Aug 2025). Deep-learning-based ROMs accelerate design-space exploration and optimization, achieving accurate field reconstructions even with sparse or partial data sets, with speedups of several orders of magnitude relative to full-order models.

8. Scaling, Validation, and Practical Considerations

Rigorous validation employs mesh-converged, benchmarkable problems with analytical/Numerical solutions—e.g., co-annular cylinders, pipe flows, RB cells, or boiling curves—assessing both global (Nusselt number, heat-flux) and local (wall temperature, boundary layer thickness) quantities (Crocker et al., 2014, Ettel et al., 28 Feb 2025). Reported discretization schemes span collocated FV, FEM, FDM, multi-region interface coupling, monolithic and partitioned time integration, with explicit, implicit, or semi-implicit solver choices. For high-contrast or multiscale applications, acceleration and stability are improved with multiscale modal decomposition, e.g., via modal (spectral) projection plus local correction (Dreze et al., 30 Jul 2025).

9. Extensions: Error Estimation and Empirical Correlations

In practice, empirical correlations (e.g., Nusselt vs. Reynolds/Prandtl) undergird engineering Biot-number estimates. Recent work provides computable, geometry-specific error bounds on CHT-to-lumped-model approximations and data-driven procedures to transfer empirical correlations to noncanonical shapes by introducing learned characteristic scales (Guo et al., 10 Nov 2025). This enables reliable hierarchy-building from truth models (CHT) to reduced, tractable ones (e.g., solid-only Robin or lumped-capacitance ODEs) with rigorously justified error budgets.

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In summary, conjugate heat transfer formulations rigorously couple conduction and advection/diffusion across multi-material interfaces using physically consistent interface conditions—specifically, continuity of temperature and (conductive) heat flux. The domain admits a wide array of numerical, analytical, and surrogate modeling approaches, each with distinct scaling, accuracy, and computational properties, validated by thorough physical and mathematical analysis across canonical and application-driven scenarios. Advances in interface modeling, micro–macro upscaling, multiscale acceleration, and data-driven surrogate construction continue to extend the scope and impact of CHT frameworks in contemporary thermal science and engineering.