Low-Prandtl Forced Convection over a Flat Plate

Overview

This case verifies the solver's forced-convection heat transfer prediction for the textbook problem of laminar flow over an isothermal flat plate at a low Prandtl number representative of a liquid metal ($Pr=0.01$). For laminar, constant-property forced convection the local Nusselt number is given for all Prandtl numbers by the Churchill–Ozoe similarity correlation $N{u}_x=0.3387R{e}_x^{1/2}P{r}^{1/3}/[1+(0.0468/Pr{)}^{2/3}{]}^{1/4}$. In the low-Prandtl limit, where the thermal boundary layer is much thicker than the momentum layer, it approaches the compact slug-flow asymptote $N{u}_x\to 0.564\sqrt{R{e}_{x}Pr}=0.564\sqrt{P{e}_x}$. Reproducing the local Nusselt-number distribution along the plate exercises the coupled momentum and energy solution for an incompressible, constant-property fluid in the low-Prandtl regime, where the temperature field extends far beyond the velocity field.

Problem Setup

A uniform stream enters at the leading edge of a flat isothermal plate that forms the lower wall of a tall quasi-two-dimensional domain. The plate length is $L=1$ m and the free-stream velocity is $U_{\infty }=1$ m/s; with density $\rho =1$ kg/m³ and dynamic viscosity $\mu ={10}^{-5}$ Pa·s the plate Reynolds number is $R{e}_L=\rho U_{\infty }L/\mu ={10}^5$, so the boundary layer stays laminar over the whole plate. The fluid is incompressible (constant density) with the energy equation active, a Prandtl number $Pr=0.01$, and constant properties.

The plate is held at a fixed temperature $T_w=310$ K while the inlet stream is uniform at $T_{\infty }=300$ K, a modest $\Delta T=10$ K that keeps the constant-property assumption valid. The inlet imposes the uniform velocity and temperature, the downstream outlet is a static- pressure boundary, and the spanwise faces are symmetry planes. The top boundary is also a static- pressure outlet rather than a symmetry plane, so the flow displaced by the growing boundary layer escapes through the top and the edge velocity stays at $U_{\infty }$ (a symmetry top would instead accelerate the free stream by the displacement thickness and bias the comparison). Because the low-Prandtl thermal boundary layer is roughly $P{r}^{-1/2}\approx 10\times$ thicker than the momentum layer, the domain is made tall enough to contain it, and the wall-normal nodes are clustered toward the heated wall to resolve the near-wall temperature gradient that sets the heat flux.

Quantities of Interest

The primary quantity is the local Nusselt number $N{u}_x=h_{x}x/k$, where the convective coefficient $h_x=q_w^{″}(x)/(T_w-T_{\infty })$ is formed from the local wall heat flux $q_w^{″}(x)$, the imposed wall-to-free-stream temperature difference, and the thermal conductivity $k=\mu c_p/Pr$. The wall heat flux is read directly from the heated-wall boundary solution, and the resulting $N{u}_x(x)$ is compared with the laminar Churchill–Ozoe similarity solution at $Pr=0.01$. The distribution is sampled away from the singular leading edge; the developed-region mean is reported alongside the similarity value on the grid-converged solution.