Numerical Solution of Natural Convection on a Heated Vertical Plate at Low Diffusivity Ratios

Technical Summary: Numerical Solution of Natural Convection on a Heated Vertical Plate at Low Diffusivity Ratios

Problem Statement
This study investigates the transient natural convection flow along an isothermal vertical plate under conditions of low diffusivity ratios, specifically targeting fluids where thermal diffusivity significantly exceeds momentum diffusivity (low Prandtl numbers, $Pr \ll 1$). While natural convection over vertical plates is a well-established benchmark, the transient evolution and boundary-layer structures in low-$Pr$ regimes—characteristic of liquid metals and molten salts—differ substantially from moderate or high-$Pr$ fluids (like air or oils). The paper addresses a gap in the literature regarding the specific transient behavior of thermal boundary layers and heat transfer mechanisms in these highly diffusive conditions, where thermal diffusion dominates buoyancy-driven momentum transport.

Methodology
The research employs a numerical approach based on the Finite Difference Method (FDM) to solve the governing dimensionless equations derived from the Boussinesq approximation. The system of equations includes the continuity equation, momentum equations, and the energy equation.

• Discretization: The spatial domain utilizes a structured uniform Cartesian grid. Diffusive terms are approximated using second-order central finite differences to preserve spatial accuracy, while nonlinear convective terms are discretized using a first-order upwind scheme to suppress spurious numerical oscillations in regions with steep gradients.
• Time Integration: An explicit Euler scheme is used for time advancement. To ensure numerical stability, the time step ($\Delta\tau$) strictly adheres to the Courant–Friedrichs–Lewy (CFL) condition, which is particularly restrictive for low-$Pr$ fluids due to enhanced thermal diffusivity.
• Validation: The numerical methodology is validated through a grid-independence study (converging on a $100 \times 100$ mesh) and by comparing results against established classical correlations and recent literature data for low-$Pr$ convection. The simulations cover a Prandtl number range of $0.01$ to $0.71$, with fixed Reynolds ($Re=10^2$) and Rayleigh ($Ra=10^5$) numbers.

Key Results
The numerical simulations reveal distinct physical behaviors in low-$Pr$ regimes compared to conventional convection:

1. Boundary Layer Structure: As the Prandtl number decreases, the thermal boundary layer becomes significantly thicker than the hydrodynamic (velocity) boundary layer. For extremely low $Pr$(e.g.,$Pr=0.01$), the temperature field penetrates deeply into the fluid domain, resulting in a very gradual thermal decay.
2. Velocity and Temperature Profiles: In low-$Pr$ fluids, the peak velocity shifts away from the heated wall, and the momentum boundary layer broadens. This occurs because buoyancy forces are distributed over a larger thermal region, leading to weaker local acceleration near the wall. Conversely, higher $Pr$ fluids exhibit sharper thermal gradients and velocity peaks closer to the surface.
3. Heat Transfer Rates: The local Nusselt number ($Nu_x$) decreases as the Prandtl number drops. The dominance of thermal diffusion spreads thermal energy rapidly into the fluid, reducing the wall-normal temperature gradient and consequently lowering convective heat transfer rates.
4. Transient Evolution: Low-$Pr$ convection exhibits a delayed thermal response. Fluids with $Pr=0.01$ show slow asymptotic convergence toward thermal equilibrium, whereas moderate and high-$Pr$ fluids stabilize much more rapidly.
5. Scaling Laws: The results confirm a scaling relationship where the thermal boundary-layer thickness ($\delta_t$) is inversely proportional to the square root of the Prandtl number ($\delta_t \sim Pr^{-1/2}$), consistent with theoretical predictions for diffusion-dominated regimes.

Significance and Claims
The paper claims to provide a robust numerical framework capable of accurately capturing the strongly diffusive thermal transport mechanisms inherent to low-$Pr$ natural convection. By utilizing an explicit FDM with specific stabilization strategies (upwinding and strict CFL adherence), the study demonstrates that it is possible to resolve the complex coupling between velocity and temperature fields in these regimes.

The authors assert that their findings offer physical insights relevant to engineering applications involving liquid-metal cooling systems, nuclear reactor thermal management, and advanced thermal energy technologies. The work emphasizes that understanding the transient evolution of thermal fields in low-diffusivity fluids is critical for designing passive cooling systems and heat exchangers where thermal diffusion dictates the transport behavior. The study concludes that the proposed numerical model is reliable for investigating these phenomena, as evidenced by its agreement with benchmark data and its ability to reproduce expected physical trends, such as the thickening of thermal layers and the reduction of heat transfer coefficients in low-$Pr$ conditions.