Flow Over Heated Plate

Classical conjugate heat transfer validation case with a heated flat plate (Vynnycky et al.).

Repository path: tutorials/conjugateHeatTransfer/flowOverHeatedPlate

Reference: Alkafri et al. (2024), multiRegionFoam: A Unified Multiphysics Framework, Sect. 7.1

Overview

An incompressible laminar flow over a flat plate of finite thickness is considered. A fluid of uniform temperature $T_\infty = 300\,\text{K}$ and velocity $U_\infty = 1\,\text{m/s}$ flows over a plate of length $L = 1\,\text{m}$ held at a constant bottom temperature $T_s = 310\,\text{K}$.

The aspect ratio is fixed at $\lambda = a/L = 0.25$. Results are validated against the numerical and analytical reference solutions from Vynnycky et al., computing the dimensionless conjugate boundary temperature

$$\theta = \frac{T - T_\infty}{T_s - T_\infty}.$$

Computational domain

Boundary conditions

Table: General boundary conditions

Boundary	Thermal	Velocity
Fluid
inlet	300 K	(1 0 0)ᵀ m/s
bottom (no-slip, plate)	coupled	(0 0 0)ᵀ m/s
slip-bottom (before plate)	zeroGradient	zeroGradient
outlet, top	zeroGradient	zeroGradient
Solid
top	coupled	–
bottom	310 K	–
left, right	zeroGradient	–

Table: Coupled thermal boundary conditions at the fluid–solid interface

Region	Boundary	Partitioned	Monolithic
Fluid	bottom	`regionCoupledTemperatureJump`	`monolithicTemperature`
Solid	top	`regionCoupledHeatFlux`	`monolithicTemperature`

Material properties

Table: Thermophysical properties of fluid and solid

Property	Symbol	Unit	Solid	Fluid
Density	ρ	kg/m³	1	1
Dynamic viscosity	μ	kg/ms	–	ρ_f U∞ L / Re
Thermal conductivity	k	W/m·K	100	k_s / k
Specific heat capacity	c_p	J/kg·K	100	k_f Pr / μ

Mesh

Parameter study

Table: Simulated parameter combinations

Re	Pr	k
500	0.01	1, 5, 20
10 000	0.01	1, 5, 20
500	100	1, 5, 20

Simulations are run for 10 s with a time step $\Delta t = 0.01\,\text{s}$.

Numerical schemes

Table: Numerical schemes

Scheme	Setting
`ddtScheme`	`backward`
`gradScheme`	`leastSquares`
`divScheme div(phi,U)`	`Gauss upwind`
`divScheme div(phi,T)`	`Gauss linearUpwind Gauss linear`
`laplacianScheme`	`Gauss linear corrected`
`interpolationScheme`	`linear`
`snGradScheme`	`corrected`

Results summary

Fig. 9 — Simulation results for different Pr, Re, and k values. (a) θ for Pr=0.01, Re=500. (b) θ for Pr=0.01, Re=10⁴. (c) θ for Pr=100, Re=500. (d) Average coupling time for partitioned (Aitken) and monolithic coupling.

Both monolithic and partitioned (Aitken relaxation) coupling reproduce the reference solutions from Vynnycky et al. for all Pr, Re, and k combinations. Monolithic coupling exhibits lower or equal average coupling time compared to partitioned coupling across all cases, with the most notable advantage at Pr = 100 where monolithic coupling time is nearly constant.

Running the case

cd $FOAM_RUN/../multiPhysicsFoam/tutorials/conjugateHeatTransfer/flowOverHeatedPlate

# Partitioned coupling — serial
./Allrun --partitioned --serial

# Partitioned coupling — parallel
./Allrun --partitioned --parallel

Note

Monolithic coupling is not yet supported for this case via the Allrun script.