Repeated Coupling Method

Theoretical Foundations of Iterative Coupling

Conjugate Heat Transfer and the Interface Conditions

Conjugate heat transfer (CHT) is the problem of solving convection in a fluid together with conduction in a solid, joined at their shared interface. Two conditions must hold simultaneously on the interface \( \Gamma \):

$$ T_f = T_s \quad (\text{temperature continuity}), \qquad -k_f \frac{\partial T_f}{\partial n} = -k_s \frac{\partial T_s}{\partial n} \quad (\text{heat flux continuity}) $$

How you enforce these two conditions is what separates the solution strategies. A monolithic method assembles fluid and solid into a single system and solves them at once; a partitioned method runs a fluid solver and a solid solver separately and exchanges information at the interface. Iterative coupling is a partitioned scheme in which the exchange between the two solvers is repeated until convergence within the same time instant (or within the same steady solution) — buying the flexibility of partitioning (reuse proven solvers as they are) together with the rigor of the monolithic approach (the interface conditions are actually converged).

How the Dirichlet-Neumann Split Works

Stability Is Decided by the Property Ratio

The stability of D-N iteration is governed by the ratio of thermal "strength" of the two media meeting at the interface. For steady problems the controlling parameter is the ratio of interfacial thermal resistances (conductance ratio); for transient problems it is the ratio of thermal effusivities \( b = \sqrt{k \rho c_p} \). The rule of thumb is that passing heat flux to the side that conducts heat well (usually the metal solid) and temperature to the side that conducts poorly (the fluid) is the stable arrangement, while the reverse tends to diverge. A combination with an extreme property ratio, such as metal against air, has a wide stability margin; where the effusivities are comparable, such as polymer against water, decay is slow in either direction and the relaxation and acceleration techniques below become mandatory.

Numerical Methods Behind the Algorithm

The Basic Iteration Loop and Convergence Criteria

For a time level \( t^{n+1} \) (or for the steady solution) the iterative coupling loop takes the following form.

1. Solve the fluid with interface temperature \( T_\Gamma^{(k)} \) and obtain the interface heat flux \( q_\Gamma^{(k)} \)
2. Solve the solid with \( q_\Gamma^{(k)} \) and obtain a new interface temperature \( \tilde{T}_\Gamma^{(k+1)} \)
3. Update with relaxation: \( T_\Gamma^{(k+1)} = (1-\omega) T_\Gamma^{(k)} + \omega \tilde{T}_\Gamma^{(k+1)} \)
4. Repeat until the interface residual \( \| T_\Gamma^{(k+1)} - T_\Gamma^{(k)} \| \) drops below the tolerance

The robust practice is to test not only the change in interface temperature but also agreement between the heat fluxes on the two sides (relative difference of the integrated values below 0.1–1%). If the temperature has stopped moving while the fluxes still disagree, that is stagnation, not convergence.

Relaxation and Acceleration — Fixed ω, Aitken, Quasi-Newton

Data Mapping Across Non-Matching Meshes

Interface mesh resolution normally differs greatly between fluid and solid, with the fluid side finer by an order of magnitude. Shape-function interpolation is adequate for an intensive quantity such as temperature, but transferring heat flux requires conservative mapping (weights that preserve the integrated heat), otherwise energy leaks at the interface. Mapping error is a systematic error that iteration cannot remove, so "the heat crossing the interface agrees on both sides" must always be measured as a verification item for the coupling setup.

Guidelines for Practical Application

Running a Steady CHT Case

For steady problems, investing in the initial state cuts the iteration count sharply: (1) partially converge the fluid alone first (with the wall adiabatic or at an estimated temperature), (2) pre-heat the solid using an estimated heat transfer coefficient, (3) start the coupling iteration from that state. Starting from a cold solid and a uniform fluid creates a large initial inconsistency that makes the iteration thrash. The coupling frequency — how many fluid iterations run between exchanges — is also a tuning parameter: running the fluid too long wastes effort per exchange, while running it too little hands over an unconverged heat flux and seeds oscillation.

Transient CHT and the Time-Scale Gap

The defining practical feature of CHT is that the time scales of fluid and solid differ by orders of magnitude (fluid convective time: milliseconds to seconds; solid thermal diffusion time: minutes to hours). If you want the temperature history of the solid but the time step is chained to the fluid CFL limit, the computation is bankrupt. The standard remedies are:

• Quasi-steady approach — re-solve the fluid as a steady analysis at every large solid time step (effective when the fluid responds much faster than the solid)
• Subcycling — advance the fluid many steps within one solid step and pass the time-averaged heat flux
• Iterate at every time level (strong coupling) — for comparable effusivities where weak coupling is unstable. Two to five iterations per time level bring the interface residual down

Verification Checklist

• Does the integrated interface heat flux agree between the fluid and solid sides (relative difference below 1%)?
• Does the overall energy balance close (heat in = heat out + stored heat)?
• Does the result stay unchanged when the coupling convergence tolerance is tightened by an order of magnitude?
• Is the near-wall resolution on the fluid side (y+ and wall treatment) adequate for the required heat transfer coefficient accuracy?
• For transient cases, does the solid temperature history stay unchanged when the time step and subcycle count are halved?

Support in Major Tools

Implementation by Tool

Built-in CHT or Partitioned Coupling?

When the solid region can be handled inside a single code (Fluent, STAR-CCM+, chtMultiRegionFoam), that is the first choice, because interface consistency is guaranteed by the code itself. Partitioned coupling (System Coupling, preCICE) is the right pick in one of three situations: (1) the solid side needs dedicated nonlinear structural or material capabilities, (2) an existing validated solid model must be reused, (3) fluid and solid are established with different teams and different tools. Partitioned coupling buys flexibility at the price of moving responsibility for the stability, mapping, and convergence management described in this article onto the user — choose it with that in mind.

Research Frontiers

Better Approximations of the Interface Jacobian

Work on quasi-Newton schemes (IQN-ILS, IQN-IMVJ and relatives) follows a pattern in which methods matured for fluid-structure interaction (FSI) are carried over to CHT. Multi-vector methods that reuse the history from several time steps, and preconditioners built from surface operators that mimic the interface physics, continue to push in one direction: keep the iteration count at a few regardless of the property ratio. The benefit is largest exactly where strong coupling is required, and it is reaching practice through the coupling libraries.

The Mathematics of Time-Scale Separation and Multi-Rate Methods

Multiple-time-scale methods that confront the fluid-solid time-scale gap head on (higher-order synchronization and interpolation between different time steps), and combinations of CHT with time-spectral methods that solve the periodic steady state directly (the rotation period of turbomachinery), are both under study. Industrial problems in engine and gas turbine thermal management — "second-scale fluid against hour-scale thermal inertia" — are the driving force.

Accelerating the Coupling with Machine-Learning Surrogates

Rather than running an expensive fluid analysis at every iteration, research is advancing on approximating the interface-temperature-to-interface-heat-flux response with a surrogate (GP or NN) to accelerate the iteration, leaving only the final check to the real solver. This is particularly effective in repetitive contexts such as CHT inside a design optimization loop. The key to deployment is detecting degradation once the state leaves the surrogate's valid range (a temperature range outside the training domain), and monitoring the predictive variance of a Kriging model is a natural fit for that role.

Troubleshooting

Symptoms, Causes, and Fixes

A Procedure for Isolating Instability

Related: all conjugate heat transfer articles, forced convection over a flat plate, checking and diagnosing solver convergence.