# Turbulent Flow Through a Heated Periodic Channel

In this application, AcuSolve is used to solve for the flow and temperature field within a channel containing a heated wall. The wall is maintained at a constant temperature, inducing heat flux into the fluid, to predict the thermal law of the wall. The non-dimensional temperature versus the non-dimensional height above the wall is compared to the analytical correlation provided by Kader.

## Problem Description

The problem consists of a fluid having material properties close to air with a density of 1.0 kg/m3, molecular viscosity of 0.0001 kg/m-s, specific heat of 1005.0 J/kg-K and thermal conductivity of 0.139 W/m-K. The properties are specified to obtain a Prandtl number of 0.72. The velocity is defined as periodic and is driven by a constant acceleration body force equal to 2.0 m/s2. The temperature is specified as periodic+unknown ratio allowing it to develop until it reaches a steady solution. The simulation description is shown in the following image, which is not drawn to scale. The model is simulated using the one equation Spalart Allmaras turbulence model along with the advective-diffusive temperature equation. The thermal wall distribution is validated against correlation data published by Kader 1981.

The simulation was performed as a two-dimensional problem by constructing a volume mesh that contains a single layer of elements extruded in the cross-stream direction, normal to the flow plane and by imposing periodic boundary conditions on the extruded planes. The streamwise direction contains two elements allowing the flow and temperature solution to develop. Only the lower half of the channel is modeled, assuming the solution is mirror across the top slip plane.

## AcuSolve Results

The AcuSolve solution converged to a steady state and the results reflect the mean flow conditions within the channel. The simulation results demonstrate that the channel wall induces a thermal flux into the flow field, producing a temperature distribution dependent on the distance from the wall. The thermal law of the wall computed from the AcuSolve results is compared with correlation data for the corresponding Prandtl number, as demonstrated in Kader 1981. The plot shown below gives the non-dimensional value of T+ as a function of Y+, where T+ and Y+ are computed per the following relationships:(1)
$T+=\frac{\left(Tw-T\right)\rho Cpu\tau }{\stackrel{.}{Q}}T$
(2)
$Y+=\frac{yu\tau }{v}$
(3)
$u\tau =\sqrt{\tau w/\rho }$

Where $T+$ , $Tw$ is the fixed wall temperature, T is the temperature away from the wall, $\rho$ is the fluid density, $Cp$ is the fluid specific heat, $uT$ is computed from the magnitude of the shear stress and ${\stackrel{˙}{Q}}_{i}$ is the local surface heat flux.

The image below shows black circles representing the correlation data and a solid red line for the AcuSolve results. The results demonstrate that the temperature distribution away from the wall is resolved properly and compares well with the correlation. Since an empirical relationship for T+ is used for comparison, there will be minor discrepancies compared with the simulation results. Additional comparisons have been made against DNS results for T+ from Kawamura et al. 1998.

## Summary

The AcuSolve solution compares well with the correlation data for turbulent flow within a heated channel. In this application, the constant temperature wall induces a surface heat flux, giving rise to a temperature gradient within the channel. AcuSolve can capture the correct temperature gradient at any location above the wall if an appropriate first layer height is selected to resolve a Y+ value of 1.0. The good agreement with correlation data for T+ demonstrates that AcuSolve can predict the locally varying temperature distribution within a turbulent channel.

## Simulation Settings for Turbulent Flow Through a Heated Periodic Channel

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

Global

• Problem Description
• Flow type - Steady State
• Temperature equation - Advective Diffusive
• Turbulence equation - Spalart Allmaras
• Auto Solution Strategy
• Max time steps - 50
• Convergence tolerance - 0.0001
• Relaxation Factor - 0.4
• Temperature - On
• Material Model
• Fluid
• Type - Constant
• Density - 1.0 kg/m3
• Viscosity - 1.0e-4 kg/m*sec
• Specific Heat - 1005.0 J/kg*K
• Conductivity - 0.139 W/m*K
• Body Force
• BodyForce
• Gravity
• X-component - 2.0 m/sec2

Model

• Volumes
• Fluid
• Element Set
• Material model - Fluid
• Body force - BodyForce
• Surfaces
• Symmetry_1
• Simple Boundary Condition - (disabled to allow for periodic conditions to be set)
• Symmetry_2
• Simple Boundary Condition - (disabled to allow for periodic conditions to be set)
• Inflow
• Simple Boundary Condition - (combination of integrated BC and periodic BC set)
• Integrated Boundary Conditions - Temperature - 300.0 K
• Outflow
• Simple Boundary Condition - (disabled to allow for periodic conditions to be set)
• Slip_surface
• Simple Boundary Condition
• Type- Slip
• Wall_surface
• Simple Boundary Condition
• Type- Wall
• Temperature BC type - Value
• Temperature - 350.0 K
• Periodics
• 2D
• Periodic Boundary Conditions
• Type - Periodic
• Flow
• Periodic Boundary Conditions
• Type - Periodic
• Temperature
• Individual Periodic BCs
• Temperature: Type - Single Unknown Ratio
• Nodes
• Zero_z-velocity
• Z-Velocity: Type - Zero

## References

B.A. Kader, "Temperature and concentration profiles in fully turbulent boundary layers", International Journal of Heat and Mass Transfer, Volume 24, Issue 9, 1981, Pages 1541-1544.

H. Kawamura, K. Ohsaka, H. Abe and K. Yamamoto, "DNS of turbulent heat transfer in channel flow with low to medium-high Prandtl number fluid", International Journal of Heat and Mass Transfer, 19:482-491, 1998.