Laminar Flow Through a Channel with Heated Walls

In this application, AcuSolve is used to simulate high Peclet number laminar flow through a channel with heated walls. AcuSolve results are compared with analytical results adapted from Hua and Pillai (2010). The close agreement of AcuSolve results with analytical results validates the ability of AcuSolve to model cases involving heat transfer to a moving fluid with a high Peclet number.

Problem Description

The problem consists of water at 20 °C flowing through a channel of infinite width with top and bottom walls heated to 75 °C. The channel is 0.2 m high and 0.8 m long, as shown in the following image, which is not drawn to scale. A centrally located slice 0.02 m wide is modeled with slip boundary conditions so that side-wall influences can be ignored. Water enters the channel with an average velocity of 0.003 m/s. As the fluid flows through the channel, it is heated by the top and bottom plates.

The simulation was performed as a two dimensional problem by restricting flow in the out-of-plane direction through the use of a mesh that is one element thick. In addition, the symmetry of the geometry in the height direction is exploited to allow for modeling only half of the geometry. These characteristics allow for accurate simulation of flow while minimizing computational time.

AcuSolve Results

The AcuSolve solution converged to a steady state and the results reflect the mean flow conditions. As the cool water enters the channel at temperature less than that of the walls, heat is transferred to the water by conduction. As the fluid travels along the channel, the temperature differential between the wall and the adjacent water decreases. The thermal boundary layer within the water develops as the flow propagates downstream.

Temperature at a specified vertical position in the channel increases along the length of the channel. The nature of that temperature change is dependent on proximity to the heated wall. Water in the center of the flow will have the least temperature change. Since this problem was solved for symmetrical flow, only the top half of the flow region was modeled. The vertical position for results shown in the following figure are based on the center of flow having a Y-position of 0.

Summary

The AcuSolve results compare well with the analytical results for the development of a thermal boundary layer in a heated channel. In this application, the model is set up to yield large gradients as the flow convects away from the inlet. The boundary conditions and mesh size were chosen specifically to yield a high Peclet number. The element Peclet number in the flow direction in this case is Pe=30.0, as calculated from the following equation.

(1)

P e = \frac{ρ C p | v | h e}{k}

$P e = \frac{ρ C p | v | h e}{k}$

where $ρ$ $ρ$ is the density, $C P$ $C P$ the heat capacity, ν the velocity, $h e$ $h e$ the length, and k the thermal conductivity.

This example shows the robustness of the stabilized technique in AcuSolve when element Peclet number is high.

Note: Standard Galerkin finite element formulations become unstable when the element Peclet number is greater than 1.0.

Simulation Settings for Laminar Flow Through a Channel with Heated Walls

SimLab database file: <your working directory>\channel_laminar_heat\channel_laminar_heat.slb

Global

Problem Description
- Flow - Steady State
- Temperature equation - Advective Diffusive
- Turbulence equation - Laminar
Auto Solution Strategy
- Relaxation factor - 0.4
Material Model
- Fluid_Material
  - Density - 1000.0 kg/m3
  - Specific Heat - 1000 J/kg-K
  - Viscosity - 1.0e-12 kg/m-sec
  - Conductivity - 1.0 W/m-K
Model
Volumes
- Fluid
  - Element Set
    - Material model - Fluid_Material
Surfaces
- Inlet
  - Simple Boundary Condition
    - Type - Inflow
    - Inflow type - Velocity
    - X velocity - 0.003 m/sec
    - Temperature - 293.15 K
- Outlet
  - Simple Boundary Condition
    - Type - Outflow
- Symm_MaxZ
  - Simple Boundary Condition
    - Type - Slip
- Symm_MinY
  - Simple Boundary Condition
    - Type - Slip
- Symm_MinZ
  - Simple Boundary Condition
    - Type - Slip
- Wall
  - Simple Boundary Condition
    - Type - Wall
    - Temperature BC type - Value
    - Temperature - 348.15 K

References

Hua Tan and K. M. Pillai. "Numerical Simulation of Reactive Flow in Liquid Composite Molding Using Flux-Corrected Transport (FCT) Based Finite Element/Control Volume (FE/CV) Method". International Journal of Heat and Mass Transfer. 53:2256-2271, 2010.