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.
where is the density,
the heat capacity,
ν the velocity, 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.