20. Conjugate Heat Transfer

A resolved fluid and solid occupy separate domains. Their common boundary has one temperature and equal, opposite outward heat flow. The solid is stationary and conducts heat; the fluid retains its own energy equation. The finite-volume solid formulation below uses constant isotropic properties within each material region and perfect thermal contact between regions.

20.1. Solid Energy Balance

With density \(\rho_s\), specific heat \(c_s\), conductivity \(\kappa_s\) and volumetric heat source \(S_s\), solid temperature satisfies

(20.1.1)\[\rho_s c_s\frac{\partial T_s}{\partial t} =\nabla\cdot(\kappa_s\nabla T_s)+S_s\]

Properties are intrinsic solid values. Define the cell heat capacity \(C_P=\rho_{s,P}c_{s,P}V_P\), in J/K. Integration over cell \(P\) and backward-Euler time discretization give

(20.1.2)\[C_P\frac{T_P^{n+1}-T_P^n}{\Delta t} +\sum_{f\in\partial P}\sigma_{Pf}Q_f^{n+1}=S_{s,P}V_P\]

\(Q_f\) is integrated conductive power in W, positive along the stored face normal. The incidence sign \(\sigma_{Pf}\) converts it to outward power for the cell, following Loci Face Orientation and Cell Incidence. The source is positive for heating. Summing over cells cancels every interior-face contribution.

20.2. Conduction Across a Face

Let \(L=\mathrm{cl}(f)\) and \(R=\mathrm{cr}(f)\) for an interior face. Its area is \(A_f\) and unit normal \(\hat{\vec n}_f\) points from \(L\) to \(R\). Decompose the two center-to-face displacements into normal and tangential parts:

(20.2.1)\[\begin{split}\begin{aligned} \vec x_f-\vec x_L&=d_L\hat{\vec n}_f+\vec t_L,\\ \vec x_R-\vec x_f&=d_R\hat{\vec n}_f+\vec t_R. \end{aligned}\end{split}\]

Both normal distances must be positive. Continuity of temperature and normal heat flow through the two material segments gives the face conductance

(20.2.2)\[G_f=\frac{A_f}{d_L/\kappa_L+d_R/\kappa_R}\]

This resistance sum retains the conductivity jump. With reconstructed cell gradients \(\vec g_L\) and \(\vec g_R\), the face power is approximated by

(20.2.3)\[Q_f=G_f\left(T_L-T_R+\vec g_L\cdot\vec t_L+\vec g_R\cdot\vec t_R\right)\]

The temperature difference is implicit. Tangential terms are evaluated from the current iterate and updated until the assembled energy residual converges. The face contributes \(+G_f\) to both diagonals and \(-G_f\) to both neighbor coefficients. Its tangential correction is subtracted from the left-cell source and added to the right-cell source.

For a boundary face, let \(\vec x_f-\vec x_P=d_P\hat{\vec n}_f+\vec t_P\) and \(G_b=\kappa_PA_f/d_P\). A prescribed boundary temperature \(T_b\) produces

(20.2.4)\[Q_b=G_b\left(T_P+\vec g_P\cdot\vec t_P-T_b\right)\]

It adds \(G_b\) to the diagonal and \(G_b(T_b-\vec g_P\cdot\vec t_P)\) to the source. A prescribed outward flux \(q_b\) instead gives \(Q_b=q_bA_f\), which is subtracted from the source. Adiabatic boundaries have \(Q_b=0\).

At a coupled wall the fluid supplies inward power \(Q_{\mathrm{in}}=-Q_b\). The solid applies it to the cell energy balance and returns

(20.2.5)\[T_b=T_P+\vec g_P\cdot\vec t_P+\frac{Q_{\mathrm{in}}}{G_b}\]

Thus the interface temperature includes the thermal resistance between the solid cell center and its surface. The boundary gradient is lagged during local correction iterations; acceptance also checks consistency of this lagged tangential reconstruction with the current gradient.

20.3. Assembled Solid Equation

Combining storage, interior conduction and prescribed-temperature boundaries produces the row

(20.3.1)\[\begin{split}\begin{aligned} a_PT_P^{n+1}-\sum_N G_{PN}T_N^{n+1}&=b_P,\\ a_P&=\frac{C_P}{\Delta t}+\sum_N G_{PN}+\sum_{b,T}G_b,\\ b_P&=\frac{C_P}{\Delta t}T_P^n+S_{s,P}V_P +\sum_{b,T}G_bT_b-\sum_{b,q}Q_b+b_{P,\mathrm{tan}}. \end{aligned}\end{split}\]

\(b_{P,\mathrm{tan}}\) collects the deferred tangential terms above. The positive storage term makes a transient, fully adiabatic solid well posed. The cell residual is measured in W. Local convergence requires \(|r_P|\leq\epsilon_{\mathrm{abs}}+\epsilon_{\mathrm{rel}}|b_P|\) for every cell, successful linear convergence and a complete positive temperature field.

20.4. Fluid–Solid Interface Iteration

Each physical step starts from immutable accepted fluid and solid states. At exchange iteration \(k\), the fluid solves its step using trial wall temperature \(T_b^{(k)}\) and returns its assembled outward wall power. The matching map transfers that integrated power into the solid. The solid solves from its accepted temperature at time \(n\) and returns \(\widehat T_b^{(k)}\). Neither participant advances its accepted history during these trials.

The temperature residual and next trial are

(20.4.1)\[\begin{split}\begin{aligned} R_T^{(k)}&=\max_f\left|\widehat T_{b,f}^{(k)}-T_{b,f}^{(k)}\right|,\\ T_b^{(k+1)}&=T_b^{(k)}+\omega\left(\widehat T_b^{(k)}-T_b^{(k)}\right). \end{aligned}\end{split}\]

Acceptance uses the unrelaxed residual, successful participant solves, per-face agreement between returned fluid power and applied solid power, and their global heat balance. Underrelaxation changes only the next trial. The accepted result is the final evaluated pair. Both domains advance to \(n+1\) together after acceptance; a rejected exchange retains the states at \(n\).