Validation/Conjugate heat transfer

Heat crossing a solid–fluid interface

Neither check needs a reference table. If both regions start at one temperature nothing may move; and with no heat source anywhere, no temperature may leave the range its own data started in.

VerificationIsothermal case holds to 5.5×10⁻⁴ K; heat crosses, undershooting by 0.03 K

Why this case

Conjugate heat transfer is a fluid and a solid solved together, exchanging heat across a shared wall. There is no closed-form answer for the geometry the product ships — but there are two statements that are exactly true of any such problem, and neither has a constant in it to argue about.

Isothermal invariance. If everything — both regions, every boundary — starts at one temperature, it must stay there, for ever. There is nothing to drive a change. Any drift is the coupled interface manufacturing heat.
The maximum principle. With the solid started hot and the fluid cold, and no source anywhere, no temperature in either region may leave the interval between the two starting values. That is a theorem about the heat equation, not a modelling assumption, and it holds at every instant on every mesh.

Together they cover the two ways a conjugate coupling fails: doing something when it should do nothing, and doing too much when it should do something. A third check covers the way this one actually failed, which was doing nothing when it should have done something.

Physics	Conjugate heat transfer, fluid and solid solved together
Solver	chtMultiRegionFoam, two regions split from one mesh
Geometry	The duct with a solid block in it that the application writes
Both regions start at	293.15 K
Case B starts the solid at	400 K
Run to	0.05 s (last write 0.0474966 s)
Agreement	Isothermal case holds to 5.5 × 10⁻⁴ K; heat crosses the interface and nothing leaves its starting interval
Last run	2026-08-27

As the application stages it, the duct carries 25.6 m/s at the inlet and is not isothermal — the flow accelerates around the block and the gas cools 2.35 K by expansion. That is compressible flow behaving correctly and it has nothing to say about the interface, so the inlet velocity is set to zero and nothing else is touched. What remains is the conjugate problem on its own.

What this case found

The coupled interface was silently resolving to an adiabatic wall

OpenFOAM matches a patch name against the wildcard entries of a boundaryField in reverse order of declaration: the last pattern that matches wins. The product wrote the coupled interface pattern "(region_to_.*)" first and the catch-all ".*" last, so the catch-all took the interface and turned it into zeroGradient. The two regions were then solved side by side with an adiabatic wall between them.

A solid at 400 K sat beside a fluid at 293.15 K for two seconds and ended at exactly 400.000000 K, minimum and maximum, with no heat crossing at all. Nothing reported anything. The case meshed, ran topoSet, split the regions, converged, wrote results, and checkMesh was content.

Reading the deck would have reported the intention and agreed with itself for ever. The written result records what was actually used, so the check reads the interface type back out of the solved field rather than out of the case setup. That is the first of the three checks and it runs before any temperature is looked at.

It is fixed — the two blocks are in the other order — and the fixture from before the fix is kept as the failing case the check has to reject.

The interface condition as the solver resolved it, before and after
Run	Fluid side	Solid side	Solid temperature range
Before the fix, 2 s	zeroGradient	zeroGradient	400.0 – 400.0 K
After the fix	compressible::turbulentTemperatureCoupledBaffleMixed	compressible::turbulentTemperatureCoupledBaffleMixed	399.9928 – 400.0 K

The same case, the same mesh, the same solver. Only the order of two blocks in the boundary-field dictionary differs.

A coupling that transfers nothing satisfies the maximum principle perfectly, which is exactly how this survived. So the check does not only ask whether the temperatures stayed inside their interval — it asks whether heat actually crossed, and refuses a run in which less than one per cent of the initial difference moved.

Result

Both cases, temperature range reached in each region
Case	Region	Minimum (K)	Maximum (K)	Samples
A — isothermal	Solid	293.15000	293.15000	64
A — isothermal	Fluid	293.14945	293.14998	576
B — solid at 400 K	Solid	399.99280	400.00000	64
B — solid at 400 K	Fluid	293.12178	312.82965	576

Case A must not move at all; case B must stay inside [293.15, 400] and must not stay still. Both regions in both cases are read from the written fields at 0.0474966 s.

In the isothermal case the solid is unchanged to every digit written and the fluid holds to 5.5 × 10⁻⁴ K of where it started, against the 0.01 K the check allows. The residual fluid motion is 2.2455 × 10⁻⁵ m/s — the flow has been stopped, and it stays stopped.

In the gradient case the fluid rises from 293.15 K to 312.82965 K in 0.0475 s while the solid falls to 399.9928 K. Heat crosses, in the right direction, and neither region exceeds 400 K.

The temperature range reached in the solid and the fluid in each of the two cases, against the interval the data started in — The full range each region reaches, in both cases, against the shaded interval its own data started in. The two isothermal bars are single lines at this scale, which is the result rather than a drawing artefact.

The fluid minimum sits 0.028 K below the interval, and we have not attributed it

In the gradient case the coldest fluid cell reads 293.12178 K, which is 0.028 K below the 293.15 K the data started at. The check allows 0.1 K and this is inside it, but it is a real number and it is not zero.

We do not know whether it is the compressible solver’s own pressure and density work in a region that is not exactly still, or discretisation of the mixed interface condition, and this study cannot separate the two. It is recorded rather than explained. What can be said is that it is far smaller than the rise from 293.15 K to 312.82965 K the same fluid makes in the same run, and that the isothermal case — which has neither driving temperature difference nor interface flux — drifts by 5.5 × 10⁻⁴ K over the same run.

What the check now catches

Three things, in this order, because the first makes the other two meaningful:

The interface condition the solver resolved, read from the written field rather than from the deck. A run whose interface came back as zeroGradient on either side is rejected before any temperature is examined.
Isothermal invariance, to 0.01 K in both regions. Any drift here is the coupling manufacturing heat — an area weighting that does not match on the two sides, or a sign, produces exactly this and produces it immediately.
The maximum principle, and the exchange it implies. Nothing may leave the starting interval by more than 0.1 K, and at least one per cent of the initial difference must have crossed. The second half of that is the part the defect would have failed; the first half it passed perfectly.

Repeat this yourself

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Other cases

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