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Radiative Heat Transfer (RHT) in Laminar BuoyancyDriven Cavity
Written by  for Version  Revised by  Revision date  Revised version 

@rsanfer  7.0.2  @rsanfer  20200206  7.0.2 
Solver: 

Uses: 

Prerequisites: 
Laminar Buoyancydriven Cavity 
Complexity: 
Basic 
Goals
This tutorial couples SU2’s incompressible fluid solver with a oneequation Radiative Heat Transfer (RHT) model. Upon its completion, the user will be familiar with the following capabilities of SU2:
 Steady, 2D, laminar, incompressible, NavierStokes equations with an FDS convective scheme
 P1 model: 1equation Radiative Heat Transfer model
 Twoway coupling of convective and radiative heat transfer
In this tutorial, we use a very similar problem definition as for the Laminar Buoyancydriven Cavity tutorial, a 1x1 m cavity in 2D with opposing hot and cold vertical walls and insulated horizontal walls. In this case, the media is participating, which means it absorbs part of the energy emitted by the hot and cold walls in form of radiation, and this absorption will have an impact on the overall behaviour of the flow. The properties of the test case are shown next:
Resources
You can find the resources for this tutorial in the folder multiphysics/radiation of the Tutorials repository. Please download the config file and the mesh file.
Background
SU2 adopts a P1 model for the simulation of Radiative Heat Transfer. The P1 model focuses on the integral magnitudes of the infinitedimensional RTE equation, and works under the assumption that the energy distribution is linearly isotropic\(^1\). In residual form, the P1 equation computes the radiative energy \(E\) as
\[\mathscr{R}(E) = \nabla \cdot \mathbf{F}^{r}(E) + \kappa (E  \langle I_b \rangle) = 0,\]where the radiative flux is
\[\mathbf{F}^{r}(E) = \left( \frac{1}{3(\kappa + \sigma_s)} \nabla E \right),\]\(\kappa\) and \(\sigma_s\) are, respectively, the absorption and scattering coefficient, and \(\langle I_b \rangle\) is the first moment of the blackbody intensity in an absorbing and emitting gray medium\(^2\).
Mesh Description
The cavity is discretized with an structured mesh using 50 nodes in the horizontal and vertical boundaries, for a total of 2500 rectangular elements. The nodes are concentrated towards the boundary regions to adequately capture the boundary layer. The boundary conditions are as follows:
Configuration File Options
We start the tutorial by definining the problem as an incompressible, Navier Stokes simulation
SOLVER = INC_NAVIER_STOKES
and we set the properties for the flow as defined in the goals section of this tutorial. More detail can be found in the Laminar Buoyancydriven Cavity tutorial.
INC_DENSITY_MODEL= VARIABLE
INC_ENERGY_EQUATION = YES
INC_DENSITY_INIT= 0.00597782417156
INC_VELOCITY_INIT= ( 1.0, 0.0, 0.0 )
INC_TEMPERATURE_INIT= 288.15
INC_NONDIM = DIMENSIONAL
FLUID_MODEL= INC_IDEAL_GAS
SPECIFIC_HEAT_CP= 1004.703
MOLECULAR_WEIGHT= 28.96
VISCOSITY_MODEL= CONSTANT_VISCOSITY
MU_CONSTANT= 1.716e5
CONDUCTIVITY_MODEL= CONSTANT_CONDUCTIVITY
KT_CONSTANT= 0.0246295028571
BODY_FORCE= YES
BODY_FORCE_VECTOR= ( 0.0, 9.81, 0.0 )
MARKER_HEATFLUX= ( upper, 0.0, lower, 0.0 )
MARKER_ISOTHERMAL= ( left, 461.04, right, 115.26 )
This tutorial focuses on the incorporation of Radiative effects to the incompressible NavierStokes solver in SU2. We first need to define the radiative model of choice. At the time of writing, the only available model is the P1 1equation model, but the structure of SU2 has been defined to facilitate the implementation of new models.
RADIATION_MODEL = P1
Then, the properties of the model are set. For this example, there is no scattering defined, while the absorption coefficient is 0.1
ABSORPTION_COEFF = 0.1
SCATTERING_COEFF = 0.0
Next, we set the emissivity of the boundaries, where only the vertical walls have been considered to be emissive:
MARKER_EMISSIVITY = ( left, 1.0, right, 1.0 )
The last step is defining the maximum CFL for the diffussive P1 equation, which does not necessarily have to be the same as for the flow equations. In this case, the P1 equation is stable for a CFL = 1E5
CFL_NUMBER_RAD = 1E5
It only remains to set the solution method for the flow equations, where the CFL number is limited to 100 for stability
NUM_METHOD_GRAD= WEIGHTED_LEAST_SQUARES
CONV_NUM_METHOD_FLOW= FDS
MUSCL_FLOW= YES
SLOPE_LIMITER_FLOW= NONE
TIME_DISCRE_FLOW= EULER_IMPLICIT
CFL_NUMBER= 100
The convergence of the problem is controlled using
INNER_ITER= 2000
CONV_CRITERIA = RESIDUAL
CONV_FIELD = RMS_PRESSURE, RMS_VELOCITYX, RMS_TEMPERATURE
CONV_RESIDUAL_MINVAL = 8
And, finally, the output of the problem is set. We can also output the convergence of the P1 equation using the keyword RMS_RAD_ENERGY
SCREEN_OUTPUT = (INNER_ITER, RMS_PRESSURE, RMS_VELOCITYX, RMS_TEMPERATURE, RMS_RAD_ENERGY)
OUTPUT_FILES = (RESTART, PARAVIEW)
SOLUTION_FILENAME = solution_rad
RESTART_FILENAME = restart_rad
VOLUME_FILENAME = radiation_tutorial
TABULAR_FORMAT = CSV
CONV_FILENAME= history
Running SU2
Follow the links provided to download the config and mesh files.
Execute the code with the standard command
$ SU2_CFD config_radiation.cfg
which will show the following convergence history:
Simulation Run using the Singlezone Driver
++
 Inner_Iter rms[P] rms[U] rms[T] rms[E_Rad]
++
 0 4.566528 19.960693 0.498633 1.150738
 1 4.802575 4.203141 0.590760 0.916037
 2 4.802134 4.671948 0.371239 0.841595
 3 4.860542 5.097286 0.290456 0.766761
 4 5.125438 5.076119 0.212441 0.717057
 5 5.414242 5.426825 0.115330 0.682917
 6 5.644717 5.533804 0.188125 0.655063
 7 5.591365 5.450852 0.476124 0.634548
 8 5.659326 5.447890 0.654348 0.617676
 9 5.808779 5.522011 0.809558 0.605703
 10 5.913957 5.579328 0.984125 0.594388
...
 1176 13.565946 13.985971 7.947333 5.654825
 1177 13.571295 13.991320 7.952680 5.660154
 1178 13.576645 13.996671 7.958032 5.665530
 1179 13.581994 14.002019 7.963378 5.670843
 1180 13.587344 14.007371 7.968731 5.676236
 1181 13.592692 14.012719 7.974075 5.681530
 1182 13.598042 14.018070 7.979430 5.686941
 1183 13.603390 14.023418 7.984772 5.692217
 1184 13.608740 14.028770 7.990128 5.697647
 1185 13.614087 14.034116 7.995468 5.702903
 1186 13.619438 14.039469 8.000826 5.708353
The code is stopped as soon as the values of rms[P]
, rms[U]
and rms[T]
are below the convergence criteria set in the config file.
All convergence criteria satisfied.
++
 Convergence Field  Value  Criterion  Converged 
++
 rms[P] 13.6194 < 8 Yes
 rms[U] 14.0395 < 8 Yes
 rms[T] 8.00083 < 8 Yes
From the convergence, it can be observed that the convective part of the problem converges quickly, however the energy equation is more stiff due to the quartic dependence of the radiative energy on the flow temperature. The resultant radiative energy field is shown next
Assessing the radiation effects
We can easily turn off the radiation model to assess the effects of incorporating the RHT effects to the calculation. It is only necessary to select NONE
as the radiation model
RADIATION_MODEL = NONE
Running the case now, the convergence to the required level is faster
++
 Inner_Iter rms[P] rms[U] rms[T]
++
 0 4.566528 19.960693 0.498633
 1 4.802575 4.203141 0.590593
 2 4.802249 4.671488 0.370967
 3 4.860553 5.096879 0.291130
 4 5.129303 5.068037 0.205752
 5 5.423720 5.416774 0.099057
 6 5.623784 5.512461 0.198612
 7 5.475529 5.390245 0.456552
 8 5.502776 5.326692 0.630527
 9 5.665078 5.403656 0.805833
 10 5.772144 5.469751 0.987757
...
 747 13.526578 14.055709 7.916489
 748 13.535008 14.064139 7.924919
 749 13.543438 14.072569 7.933349
 750 13.551869 14.080999 7.941779
 751 13.560299 14.089429 7.950209
 752 13.568729 14.097859 7.958640
 753 13.577160 14.106289 7.967070
 754 13.585590 14.114719 7.975500
 755 13.594020 14.123149 7.983930
 756 13.602450 14.131579 7.992360
 757 13.610881 14.140010 8.000790
We can compare the temperature field in the case with radiation (left) versus the case without radiation (right). The latter corresponds to the Laminar Buoyancydriven Cavity for Ra = 1.0E06
. It can be observed how the radiated case has an average temperature which is notably higher that in the nonabsorbing case.
The temperature field has also an impact in the velocity fields, which are compared next for the cases with (left) and without radiation (right).
References
\(^1\) Frank, M., et al. (2006), Partial moment entropy approximation to radiative heat transfer. Journal of Computational Physics 218(1), 1–18. DOI: 10.1016/j.jcp.2006.01.038
\(^2\) Jensen, K., et al. (2007), On various modeling approachesto radiative heat transfer in pool fires. Combustion and Flame 148(4), 263–279. DOI: 10.1016/j.combustflame.2006.09.008
Attribution
If you are using this content for your research, please kindly cite the following reference in your derived works:
Sanchez, R. et al. (2020), Adjointbased sensitivity analysis in hightemperaturefluid flows with participating media, (Submitted to) Modeling, Simulation and Optimization in the Health and EnergySector, SEMA SIMAI SPRINGER SERIES

This work is licensed under a Creative Commons Attribution 4.0 International License