183 7.8 Comparison of local variation of Nusselt numbers between irreversible . and variable density low Mach number model for natural convection turbulent flow coupled to gas radiation at Ra = 1010 and τ = 1.0. a) Variation of the convective Nusselt number at the hot wall, (b) variation of the convective Nusselt number at the cold wall, (c) variation of the radiative Nusselt number at the hot wall, (d) variation of the radiative Nusselt number at the wall cold . 117 5.6 Comparison of average values of Nusselt number (convection, radiation and total) in the heated cylinder using Algorithms A, B and C with the work of Mezrhab et al.
Introduction to the Fundamental Radiation Terminology
- Solid angle
- Emissive power
- Total hemispherical emissive power
- Intensity
- Irradiation
It is the rate at which energy is emitted per unit area at all wavelengths and directions. The intensity is the rate at which radiant energy is emitted at wavelength λ in the (θ, φ) direction, per unit area of the emitting surface perpendicular to this direction, solid angle and wavelength interval dλ.
Introduction to Radiative Exchange in a Participating Medium
Attenuation by absorption and out-scattering
The loss of the radiant energy as it propagates through the medium is due to absorption and scattering. The absorption of radiant energy by the medium is simply converted to internal energy, while scattering is basically change in the direction of the radiant energy to another direction.
Augmentation by emission and in-scattering
Introduction to Radiative Transfer Equation and Solution Methods . 8
- Numerical methods for radiative heat transfer
- Unstructured grid FVM formulation for radiation
- Coupled natural convection with thermal radiation
- Non-Boussinesq models for coupled convection with radiation 15
Most industrial problems involving radiative heat transfer require calculations on irregular complex geometries. 103] proposed wall models for large eddy simulations by allowing for the influence of heat transfer by thermal radiation.
Motivation
Studies on the influence of variable properties (density, thermal conductivity, viscosity) in a turbulent natural convective heat transfer scenario are rare and performed by few researchers [108-111]. Zamora and Kaiser [108–110] studied the effects of radiant heat transfer with the influence of variable properties on turbulent buoyancy driven flows in square and open cavities.
Research Objectives
The developed numerical model is used to study the influence of partitions on buoyancy-induced convection with thermal radiation in a differentially heated cubic enclosure. The computational framework is extended to consider buoyancy-driven turbulent natural convection with and without the influence of thermal radiation in a differentially heated square cavity at large temperature difference.
Overview of the Thesis
A class of the SIMPLE algorithm uses a face-based data structure in conjunction with a semi-implicit method that follows the work of Dalal et al. The numerical model for the simulation of RTE in unstructured FVM is consistent with the work of Murthy and Mathur [114].
Governing Equations
Governing equations for laminar flow
However, several applications such as the short-pulsed laser, pulsed laser in turbid media, material processing and living tissue require the solution of transient radiation heat transfer. The first term on the RHS of Eq. 2.4) represents the increase of radiant energy by emission, while the second and third terms represent the attenuation of energy by absorption and scattering, respectively.
Governing equations for turbulent flow
Along similar lines, the influence of turbulent fluctuations on the radiative heat transfer was neglected by Zhang et al. Therefore, the influence of turbulent fluctuations on radiative heat transfer is neglected in the present study.
Low-Mach Number Formulation
Below are the governing equations for turbulent flows in dimensional form according to the low Mach number formula. I(r,si)Φ (si,s) dΩi (2.54) The dimensionless form of the governing equations for buoyancy-driven turbulent flow is represented as.
Discretization Procedure
Discretization of convection fluxes
The discretization of the convective term is defined as Z. Here, φf is the value of φ at the center of the face f. However, the subscripts 'C' and 'D' correspond to upwind and downwind cells, respectively, which are determined by the flow direction.
Diffusive flux treatment
It should be noted that in the current unstructured grid framework the subscripts 'P' and 'N' correspond to the owner and neighbor cells of a face determined by the direction of the face normal as shown in Fig. Projecting the cell centers over the surface normally results in two more points P′ and N′. Furthermore, the discretization of the diffusion term can be written as.
Discretization procedure for RTE
It requires the spatial and angular discrete of Eq. 2.81) on the control volume and control angles as shown in Fig. For each discrete direction 'm', integrating Eq. 2.81) on the control volume and control angles results in,.
Pressure Gradient Calculation
This approach is referred to as Modified Green Gauss (MGG) and is used to treat pressure and gravity terms in this work. The above gradient calculation ensures that the recovery of velocities using centroidal pressure gradients is consistent with the discretization of the pressure Poisson equation, which requires normal pressure gradients at the surfaces of a cell.
Source Term Discretization
Radiative heat flux
Divergence of radiative heat flux
This relates the net heat flux in the radiant energy from the volume element to the radiated energy and the absorbed radiation within the volume. Interestingly, scattering does not appear in the Eq. 2.106) since it only redirects the photons and does not affect the net energy content of the volume.
Handling of dP dt term
Pressure Velocity Coupling
The first term on the RHS of Eq. 2.110) represents the auxiliary flux calculated from the auxiliary velocity (F0f∗ = u∗f ·Sf) by solving the Navier-Stokes equation by subtracting the terms for pressure and buoyancy Eq. After the pressure field is obtained, the fluxes corrected to the planes using momentum interpolation are defined as.
Incompressible and Compressible Models
Incompressible model
Compressible approach (OpenFOAM)
Implementation of Boundary Conditions
- Boundary condition for Navier-Stokes and energy equation
- Boundary condition for radiative heat transfer
- Two-dimensional boundary treatment
- Flux specified boundary condition
In this case, the two boundaries perpendicular to the third direction are treated as symmetry (flow-free boundaries) and the intensity boundary conditions are implemented by treating the boundaries as mirror boundaries. For each incident ray I on the wall, the reflected ray R can be calculated from vector algebra as
Order of Accuracy Test
Discretization error for convective fluxes
In the present study, three different convective schemes are used which are first-order upwind (FOU), a mixture of first-order upwind (FOU) and central difference scheme (CDS) (Khosla-Rubin scheme with a delayed constant of 0.5), and a second-order high-resolution scheme referred to as CUBISTA [117]. In all cases, the errors show a decreasing trend with grid refinement, with the slopes indicating the order of accuracy.
Discretization error for diffusive fluxes
The CUBISTA scheme shows the lowest errors where the rate of error decay shows a slope of 2 (on the log-log scale). 2.5, it can be seen that the discretization error decreases at a rate of 2, showing a second-order accuracy of the diffusion term.
Discretization error for Navier-Stokes equation
The problem is simulated for Re= 100 using the CUBISTA scheme for handling convective fluxes, CDS for diffusive fluxes, and a second-order back-difference temporal scheme for discretizing the time term. Along similar lines using a fixed spatial resolution of ∆x=0.005, simulations are performed to estimate the temporal discretization error using different values of the time step of.
Closure
The laminar low Mach number solver is validated against standard works for natural convection flows within a large temperature difference enclosure. Additionally, the convection coupled thermal radiation solver is validated against two- and three-dimensional problems available in the literature.
Validation of Pure Radiation Problems
- Surface to surface radiation
- Infinitely long parallel plates
- Infinitely long co-centric cylinders
- Rectangular enclosure with opening
- Quadrilateral enclosure containing absorbing-emitting medium 63
- Radiative heat transfer in a rectangular idealized furnace
- Three-dimensional kidney shaped combustion chamber
- Enclosure filled with scattering medium
- Enclosure filled with absorbing emitting and scattering medium 69
For κa=10.0 m−1 the radiant heat flux is close to unity, this is mainly due to the large value of the intensities falling on the wall from the adjacent hot environment. Further, from the isotherms, it is observed that for a lower extinction coefficient β = 0.1 m−1 the temperature is higher and more uniformly distributed.
Variable Density Flows at Large Temperature Difference Coupled
The grid resolution of 300×300 and angular resolution of 1×14 were chosen following the work of Darbandi and Abrar [33]. The average values of Nusselt number (convection, radiation and total) at the hot and cold walls are tabulated in table 3.5 and the values are compared with the literature.
Turbulent Natural Convection Inside the Enclosure
This study consists of an experimental validation with turbulent natural convection in a cavity of dimensions m performed by Ampofo and Karayiannis [80]. This study consists of a numerical validation of turbulent natural convection in a differentially heated cavity performed by Henkes and Hoogendoorn [84].
Closure
In this chapter, investigations are carried out to check the validity of the Boussinesq approximation, which is commonly used in radiative-convective flows. In this chapter, the investigations aim at quantifying the deviations in the results when the Boussinesq approximation is violated.
Validity of Incompressible Model for Radiative-Convective Flows
Convective-radiative flows at small temperature difference
The vertical walls of the cabinet are isothermal cold walls, while the horizontal top and bottom walls are adiabatic, and the cylinder is heated at a constant temperature. From these results, it is seen that an incompressible model is able to capture the correct flow physics at a small temperature difference with moderate influence of radiation.
Convective-radiative flows at large temperature difference
This weakens the buoyancy currents in the upper half of the cavity above the heated cylinder. In the case of surface-to-surface radiation (τ = 0), two primary vortices are formed as shown in Fig.
Convective-radiative flows with significant influence of radiation 92
However, the isotherms obtained using non-Boussinesq models show a sharp drop in temperature in the region just above the heated cylinder due to the formation of secondary vortices, as evidenced by the streamline variation in Fig. Qualitative similarities in the temperature and speed variation can be seen in the area under the heated cylinder.
Solution Algorithm
- Algorithm A
- Algorithm B
- Algorithm C
- Discrete mass conservation error
- Discrete energy conservation error
- Implications of mass and energy conservation errors in a com-
The discrete form of velocity divergence condition in Eq. 5.2 Solution algorithm 105 Table 5.1: Three algorithms for coupling radiation with flow. Rearranging further results in Eq. Ffn−Ffn+1) (5.23) Equation (5.23) represents the discrete form of energy conservation equation with the non-zero terms in the RHS representing the energy conservation error E(t) in a given finite cell.
Results and Discussion
Small temperature difference natural convection in a parti-
However, from the time step required to obtain stable simulations in Table 5.3, it is seen that Algorithm B requires the use of a value of time step two order, and Algorithm C requires one order less than that required for Algorithm A. Implications of the time step requirement for convergence to steady state can be realized by analyzing the rate of convergence depicted in fig.
Large temperature difference natural convection coupled with
Combined convection with surface radiation in a square cavity
However, the time step required to reach a steady state, as shown in Table 5.7, suggests that algorithm B requires a two-order value of time step, and algorithm C requires a smaller order than that required by algorithm A, comparable to the observations in section 5.3.2. . The implications of the time step requirement can be understood from the comparison of the convergence speed for the three algorithms as shown in Fig.
The Balanced Force Algorithm
Balanced vs. unbalanced approach: spurious currents arising
In particular, by using a fixed temperature and density profile within the enclosure, there is only one way of coupling between temperature and velocity, i.e., the velocity cannot affect the temperature and thus the density. In fact, there have been studies in the past that have considered modifications to the moment interpolation approach when body forces are included (See Refs [127–129]).
Three-dimensional buoyancy driven flow coupled with gas ra-
These results are in excellent agreement with the published results of Kumar and Eswaran [31] and demonstrate the ability of the algorithm to accurately predict combined natural convection with gas radiation in three-dimensional enclosures. The finding that the balanced formulation was necessary for the accurate calculation of the turbulent natural convection problems described in Chapter 7 is indirect evidence of the ability of force balancing to design a robust and accurate algorithm for solving non-Boussinesq fluid flows.
Closure
In this chapter, the developed numerical framework is applied to investigate the influence of partition walls on buoyancy-induced convection with thermal radiation in a differentially heated cubic shell. The content of this chapter was published as Parmananda, M., Dalal, A., and Natarajan, G., 2018, "The influence of partition walls on predicting heat transfer through the combined effects of convection and thermal radiation in cubic shells", International Journal of Heat and Mass Transfer, (available online, DOI: https://doi.org/10.1016/j.ijheatmasstransfer.
Problem Description
Grid independence study
Figure 6.2 (a) and (b) represent the non-uniform gratings on the X−Y and Y −Z planes, respectively, for the cubic enclosure with symmetrical partitions along the top and bottom walls. The point distribution in the grid ensures sufficient clustering near the walls as shown in fig.
Results and Discussion
Influence of inline and offset configuration of top-bottom par-
However, a secondary vortex forms next to the upper partition at the center core of the cavity. In the case of offset partitions, clustering of lower radiation values can be observed Nusselt number at the bottom of the cold wall as shown in fig.
Influence of inline and offset configuration of front-back par-
These observations are consistent with the Nusselt number variations for partitions along the top and bottom walls. 6.14 (b) and (d) show grouping of lower values of the radiation Nusselt number near Z=1 for cold wall and Z=0 for the warm wall.
Entropy Generation
The dominance of entropy generation by heat transfer is also recognized by the average and maximum values of entropy production as seen from tables (6.5, 6.6). The radiative entropy generation isosurface reveals the occurrence of maximum entropy generation near the isothermal hot wall at X=0.
The Importance of Three-Dimensional Non-Boussinesq Analysis for
Furthermore, the radiative Nusselt number reveals enhanced heat transfer due to radiation at the hot wall. 6.19 (a, b) represents the local variation of the convective Nusselt number at the cold and warm wall, respectively.
Closure
The simulation results reveal that heat transfer irreversibility is the dominant entropy production mechanism. The results of this study reveal that the inclusion of surface and gas radiation leads to an asymmetry in the flow and heat transfer.
Problem Description
Mesh independence study
Results and Discussion
Influence of surface and gas radiation on turbulent non-Boussinesq
Influence of surface and gas radiation on temperature and ve-
Influence of surface and gas radiation on turbulent viscosity
Influence of surface and gas radiation on Nusselt number vari-
Influence of variable temperature dependent thermo-physical
Closure
Scope for Future Work
Radiation emission from a surface of area A 1 along the direction s
Schematics of absorption, emission, in-scattering and out-scattering
Illustration of non-orthogonality in the grids
Representation of angular domain
Symmetry Boundary
Order of accuracy test for different convective schemes
Order of accuracy for diffusive scheme
Instantaneous contours of velocity and pressure. Comparison between
Computational domain for parallel plates
Computational domain for co-centric cylinders
Surface radiation inside a room
Representation of idealized combustion chamber
Radiative heat flux along (a) the cold wall, (b) the hot wall
Heat flux at top wall
