Numerical study of nonlinear heat transfer
from a wavy surface to a high
permeability medium with pseudo-spectral
and smoothed particle methods
Beg, OA, Motsa, SS, Beg, TA, Abbas, AJ, Kadir, A and Sohail, A
http://dx.doi.org/10.1007/s40819-017-0318-4
Title Numerical study of nonlinear heat transfer from a wavy surface to a high
permeability medium with pseudo-spectral and smoothed particle methods
Authors Beg, OA, Motsa, SS, Beg, TA, Abbas, AJ, Kadir, A and Sohail, A
Publication title International Journal of Applied and Computational Mathematics
Publisher Springer
Type Article
USIR URL This version is available at: http://usir.salford.ac.uk/id/eprint/41345/
Published Date 2017
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1
INTERNATIONAL JOURNAL OF APPLIED AND COMPUTATIONAL MATHEMATICS
ISSN: 2349-5103 (print version)
ISSN: 2199-5796 (electronic version)
PUBLISHER: SPRINGER
ACCEPTED FEBRUARY 1
ST
2017
NUMERICAL STUDY OF NONLINEAR HEAT TRANSFER FROM A WAVY SURFACE TO A HIGH
PERMEABILITY MEDIUM WITH PSEUDO-SPECTRAL AND SMOOTHED PARTICLE METHODS
O. Anwar Bég *
Fluid Mechanics and Propulsion, Mechanical and Aeronautical Engineering, School of Computing,
Science and Engineering, UG17 Newton Building, University of Salford, M5 4WT, UK.
S.S. Motsa
School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag
X01, Scottsville 3209, Pietermaritzburg, South Africa.
T.A. Bég
Renewable Energy and Geodynamics, Israfil House, Dickenson Rd., Manchester, M13, England, UK.
A. J. Abbas and A. Kadir
Spray Research Group, Petroleum and Gas Engineering, School of Computing, Science and
Engineering, Room G82 Newton Building, University of Salford, M5 4WT, UK.
Ayesha Sohail
Applied Mathematics, Mathematics Department, COMSATS-Lahore, Pakistan.
Abstract
Motivated by petro-chemical geological systems, we consider the natural convection boundary layer
flow from a vertical isothermal wavy surface adjacent to a saturated non-Darcian high permeability
porous medium. High permeability is considered to represent geologically sparsely packed porous
media. Both Darcian drag and Forchheimer inertial drag terms are included in the velocity boundary
layer equation. A high permeability medium is considered. We employ a sinusoidal relation for the
wavy surface. Using a set of transformations, the momentum and heat conservation equations are
converted from an (x,y) coordinate system to an (x,
) dimensionless system. The two-point boundary
value problem is then solved numerically with a pseudo-spectral method based on combining the
Bellman-Kalaba Quasi Linearization Method with the Chebyschev Spectral collocation technique
(SQLM). The SQLM computations are demonstrated to achieve excellent correlation with smoothed
particle hydrodynamic (SPH) Lagrangian solutions. We study the effect of Darcy number (Da),
Forchheimer number (Fs), amplitude wavelength (A) and Prandtl number (Pr) on the velocity and
temperature distributions in the regime. Local Nusselt number is also computed for selected cases.
The study finds important applications in petroleum engineering and also energy systems exploiting
porous media and undulating (wavy) surface geometry. The SQLM algorithm is shown to be
exceptionally robust and achieves fast convergence and excellent accuracy in nonlinear heat transfer
simulations.
Key words: Natural convection; wave amplitude; porous media; Darcian drag; Forchheimer drag; Nusselt
number; Spectral Quasi Linearization Method (SQLM), Smooth Particle Hydrodynamics (SPH); petroleum
and geological flows.
* Corresponding Author; email: O.A.Beg@salford.ac.uk
1. INTRODUCTION
The analysis of fluid flow and heat transfer past wavy surfaces and in wavy channels is of fundamental
importance in many areas of chemical, petroleum and mechanical engineering. Wavy walls which are
2
a form of surface re-structuring (e.g. riblets on internal surfaces in contact with conveyed fluids) have
been implemented in various systems in petroleum and energy engineering since reduction of viscous
drag in a flow through ducts reduces the pressure losses which in turn decreases costs associated with
gas compression [1-3].
Many fundamental studies of flow from wavy surfaces have been
communicated. For example,
Caponi et al. [4] studied the laminar viscous boundary layer flow over a
moving wavy wall.
Another key area of application for wavy surfaces is heat transfer. A number of numerical and
experimental studies in this area have been reported. Yao [5] considered the free convection heat
transfer past a wavy vertical wall. Moulic and Yao [6] studied the free convection flow past a wavy
wall in the presence of uniform heat flux at the wall. Chow and Oosthuizen studied experimentally the
natural convection from short cylinders with wavy surfaces. Rees and Pop [7] studied the boundary
layer flow and heat transfer on a moving plane numerically. They analyzed the effect of spatially
stationary surface waves on the forced convection regime for the case where the Reynolds number,
assuming that surface waves have order of unity amplitude and wavelength. Numerical solutions for
the local skin friction coefficient and the local Nusselt number for both the cases of a constant wall
temperature and a constant wall heat flux were obtained using the Keller box finite difference method.
More recently Hossain et al. [8] presented numerical solutions for free convection past a vertical wavy
surface with variable viscosity effects. Tashtoush and Abu-Irshaid [9] numerically investigated the
heat transfer along a wavy surface with a variable prescribed heat flux. Solutions were obtained for
various values of the heat flux exponent, amplitude and Prandtl number. The wavelength of the local
Nusselt number and surface temperature variation was shown to be equal to the wavy surface, whereas
the wavelength of the average Nusselt number was shown to be half of the wavy surface. Abbassi and
Tajik [10] considered the laminar free convection over an inclined wavy surface with constant heat
flux using a computational method. Hossain and Islam [11] studied computationally the transient heat
transfer in wavy channels. Shalini & Kumar [12] used the finite element method to analyze the natural
convection from a wavy vertical wall to a thermally stratified porous enclosure under non-Darcian
assumptions has been analyzed numerically by the finite element method. The effect of inertial forces
due to non-Darcian Forchheimer term, thermal stratification level, vertical wavy wall amplitude, wave
phase, roughness parameter, and Rayleigh number on the convection process was discussed in detail.
The computations revealed multiple circulation zones in flow field, thermal boundary layer
development along the lower segment of wavy wall and the occurrence of a “cold region” with large
stratification in the temperature field. It was also shown that the Nusselt number could be altered by as
much as 90% by varying the phase of the wavy surface. In the presence of thermal stratification, the
influence of non-Darcian effects was shown to be maximized when wave phase of the wavy wall was
approximately 300°. Rahman and Badr [13] presented an experimental study of the natural convection
heat and mass transfer from a vertical wavy surface in a non-Darcian porous medium Mass-transfer
3
coefficients were obtained experimentally for vertical wavy surfaces of varying amplitude-to-
wavelength ratio.
Thus far, to the knowledge of the authors’, the free convection heat flow past a sinusoidal
wavy vertical surface to a Newtonian fluid in a non-Darcian isotropic porous medium has received
limited attention in the literature. This regime is of relevance to petro-chemical (geological) transport
phenomena e.g. flows of oils near boundaries of ducts with filters [1-3], solar collectors [14-17] and
also petro-chemical materials processing operations [18-20]. In the present study we employ a pseudo-
spectral numerical algorithm which combines the Bellmann-Kalaba quasi-linearization method [21]
and the Chebyschev spectral collocation method [22] to obtain accelerated convergence and improved
stability for nonlinear boundary value problems. Validation is attained with a smoothed particle
hydrodynamic (SPH) [23] code. Details of numerical formulations for both methods employed are
described in detail
2. MATHEMATICAL MODEL
Consider the steady, laminar free convection flow past a vertical wavy surface embedded in a
saturated non-Darcian porous medium. The physical regime is shown in figure 1 below. The wavy
surface is maintained at a constant wall temperature, T
w.
The porous medium is modeled as an
isotropic, homogenous system. For a general 3-dimensional anisotropic medium in a generalized
Cartesian coordinate system (X, Y, Z), Coussy [24] has shown that the permeability tensor, K
P
is
second order tensor of the form:
K
p
=
ZZ
p
ZY
p
ZX
p
YZ
p
YY
p
YX
p
XZ
p
XY
p
XX
p
KKK
KKK
KKK
(1)
Generally the coefficients are not all unique so that the permeability tensor is symmetric in three-
dimensional flows and it is therefore assumed that, K
p
XX
= K
p
YX
, K
p
YZ
, K
p
ZY
and K
p
ZX
= K
p
XZ
. The
permeability coefficients K
p
XX
, K
p
YY
and K
p
ZZ
define the permeability in the X, Y and Z directions and
therefore only three values are needed to simulate a general three-dimensional anisotropic porous
medium. Clearly for the special case of isotropic flow in three dimensions, all three permeabilities will
be identical i.e. K
p
XX
= K
p
YY
= K
p
ZZ
= K
p
. For two- dimensional porous isotropic flow, as considered in
this paper, this simplification still holds and we simulate the hydraulic conductivity of the porous
medium with a single permeability, K
p
. Under this assumption, for a Darcy-Forchheimer porous
medium, the pressure gradient is related to the porous drag forces, following Vafai and Tien [25], as:
VVV-V-P
PP
K
b
K
(2)
4
where P is pressure,
is density of the non-Newtonian fluid,
is the dynamic viscosity of the fluid, K
p
is the permeability of the porous medium, V is velocity vector and b is the Forchheimer inertial
parameter which is related to the geometry of the porous medium. The geometry of the wavy surface
is defined by the following relation:
*)*sin(**)(* XKaXY
(3)
where
* is a geometrical function, a* designates the wavy surface amplitude, K* is the wave number
(=2/L), L is the characteristic length associated with the wavy surface. The surface is therefore
arbitrary and specific characteristics can be prescribed by assigning particular values to a* and K*,
which are appropriate for the heat transfer system being considered. We assume that in the free
convection flow, density differences are only incorporated in the buoyancy force i.e. following the
Oberbeck-Boussinesq approximation, and implement expressions (2) and (3) above. Additionally
viscous dissipation and thermal dispersion effects in the porous medium are ignored as are pressure
work effects. Under these approximations, the governing boundary layer equations for the transport of
mass, heat and momentum can be shown to take the form:
Mass Conservation
0
*
*
*
*
Y
V
X
U
(4)
X*-direction Momentum
2
**)(
*
*
)
*
*
*
*
*
*( U
K
b
U
K
TTg
X
P
Y
U
V
X
U
U
PP
(5)
Y*-direction Momentum
2
**
*
*
)
*
*
*
*
*
*( V
K
b
V
K
Y
P
Y
V
V
X
V
U
PP
(6)
Energy
]
**
[
*
*
*
*
2
2
2
2
Y
T
X
T
Y
T
V
X
T
U
(7)
where U* and V* are the X* and Y* direction velocities,
is the coefficient of volume expansion, g
denotes gravitational acceleration, T
is the free stream temperature (far from wavy surface), T is
temperature in the fluid,
is the thermal diffusivity and all other parameters have been defined
previously. The corresponding boundary conditions are prescribed at the wall and in the free stream as
follows:
w
TTVUXYAt ,0**:)(*:0*
(8)
**;;0*;0*:* PPTTVUYAs
(9)