Publication Cover
Numerical Heat Transfer, Part A: Applications
An International Journal of Computation and Methodology
Volume 69, 2016 - Issue 12
210
Views
9
CrossRef citations to date
0
Altmetric
Original Articles

Local RBF meshless scheme for coupled radiative and conductive heat transfer

, &
Pages 1390-1404 | Received 25 Jul 2015, Accepted 23 Nov 2015, Published online: 02 May 2016
 

ABSTRACT

A local radial basis function meshless (LRBFM) method is developed to solve coupled radiative and conductive heat transfer problems in multidimensional participating media, in which compact support radial basis functions (RBFs) augmented on a polynomial basis are employed to construct the trial function, and the radiative transfer equation (RTE) and energy conservation equation are discretized directly at nodes by the collocation method. LRBFM belongs to a class of truly meshless methods which require no mesh or grid, and can be readily implemented in a set of uniform or irregular node distributions with no node connectivity. Performances of the LRBFM is compared to numerical results reported in the literature via a variety of coupled radiative and conductive heat transfer problems in 1D and 2D geometries. It is demonstrated that the local radial basis function meshless method provides high accuracy and great efficiency to solve coupled radiative and conductive heat transfer problems in multidimensional participating media with uniform and irregular node distribution, especially for coupled heat transfer problems in irregular geometry with Cartesian coordinates. In addition, it is extremely simple to implement.

Nomenclature

ai, bj=

coefficients for RBF approximation

a, b=

vector of coefficient

d=

radius of the support domain, m

I=

radiation intensity, W/m2 sr

i, j, k=

general spatial indices

M=

number of discrete directions

Nsol=

total number of solution nodes

n=

unit normal vector

G=

interpolation matrix

D,B,M,N=

tool matrix

p=

vector of Legendre polynomial

pj=

Legendre polynomial of jth order

q=

radiative heat flux, W/m2

r=

distance between points x and xi, m

S=

source term of the RTE

s=

unit vector in a given direction

G=

incident radiation

T=

temperature, K

R=

solution domain

w=

weight function

x=

vector of location

β=

extinction coefficient, m−1

μm, ηm, ξm=

direction cosine in direction m

κa=

absorption coefficient, m−1

κs=

scattering coefficient, m−1

ε=

wall emissivity

αsup=

dimensionless parameter

σ=

Stefan–Boltzmann constant, W/m2 K4

τ=

optical thickness

Φ=

scattering phase function

Ω=

vector of radiation transfer direction

Ω=

solid angle, sr

ω=

single scatting albedo

Subscripts=
b=

black body

i, j=

node index

m=

direction index

j, k=

Legendre polynomial order index

w=

wall

Superscript=
m, m=

direction index

T=

transposition

Nomenclature

ai, bj=

coefficients for RBF approximation

a, b=

vector of coefficient

d=

radius of the support domain, m

I=

radiation intensity, W/m2 sr

i, j, k=

general spatial indices

M=

number of discrete directions

Nsol=

total number of solution nodes

n=

unit normal vector

G=

interpolation matrix

D,B,M,N=

tool matrix

p=

vector of Legendre polynomial

pj=

Legendre polynomial of jth order

q=

radiative heat flux, W/m2

r=

distance between points x and xi, m

S=

source term of the RTE

s=

unit vector in a given direction

G=

incident radiation

T=

temperature, K

R=

solution domain

w=

weight function

x=

vector of location

β=

extinction coefficient, m−1

μm, ηm, ξm=

direction cosine in direction m

κa=

absorption coefficient, m−1

κs=

scattering coefficient, m−1

ε=

wall emissivity

αsup=

dimensionless parameter

σ=

Stefan–Boltzmann constant, W/m2 K4

τ=

optical thickness

Φ=

scattering phase function

Ω=

vector of radiation transfer direction

Ω=

solid angle, sr

ω=

single scatting albedo

Subscripts=
b=

black body

i, j=

node index

m=

direction index

j, k=

Legendre polynomial order index

w=

wall

Superscript=
m, m=

direction index

T=

transposition

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.