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Numerical Modeling of Turbulent/Transitional Natural Convection within Tilted Rectangular Cuboids

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Numerical Modeling of Turbulent/Transitional Natural Convection within Tilted Rectangular Cuboids Using Three RANS Based Turbulence Models
Abstract
Three-dimensional numerical analyses of turbulent/transitional natural convection in a rectangular enclosure at tilt angles of -30°, 0° and +30° and at a moderately high Rayleigh number of Ra=5×108 have been carried out using three RANS based turbulence models. The enclosure is heated from left wall and cooled from right wall and the other sides of enclosure are at adiabatic condition. The enclosures are filled by air (Pr=0.71) and the flow regime is assumed to deal with transitional to turbulent condition. Two fully turbulent models of RNGk-ε, SSTk-⍵ and one turbulence/transitional model named as Reθ–γ transition model, or SST-transition model, are utilized for computations and their predictions are compared with each other. The flow pattern, iso-surfaces of temperature, variation of heat transfer and skin friction coefficients along the heated wall and some turbulence quantities such as turbulent kinetic energy and turbulent viscosity ratio are presented in details. The results reveal that the tilt angle plays as an effective role on the flow structure and temperature distributions. Moreover, the comparisons show that the transition/turbulence model of the Reθ–γ predicted more accurate results for flow and temperature fields than two other turbulence models.
Keywords: Natural convection; Turbulence/transition models; Tilted enclosure; Three-dimensional flow

  1. Introduction

Natural convection in a rectangular enclosure is an interesting fundamental problem and has been extensively studied both experimentally and numerically by investigators. The issue of natural convection within the enclosures has wide applications at industries, such as: cooling of electrical and electronic equipment, aeronautics, civil engineering, nuclear energy, solar collector, food industry [1-5].

Nomenclature
gi 
Gravitation acceleration ,
ms-2
H
Height of cavity ,
m
D
Depth of cavity,
m
L
Width of cavity ,
m
K
Thermal conductivity ,
Wm-1K-1
Cf
Skin friction coefficient
h
Heat transfer coefficient,
Wm-2k-1
Pr
Prandtl number
Ra
Rayleigh number,
Ra=gβ(TH-TC)H3/να
T
Temperature ,
K
TR
Reference temperature,
K
P
Pressure,
Pa
uiuj¯
Turbulent stress
T’uj¯
Turbulent heat flux
Reθe
Critical momentum Reynolds number
Reθt̃
Local transition onset momentum thickness
Reynolds number
Pθt
Pressure related to transition onset
k
Turbulent kinetic energy
ω
Specific dissipation rate
σk, σω 
Model constant
αk
Inverse effective Prandtl number for
k
αε
Inverse effective Prandtl number for
ε

Production rate

Destruction term
TH
Temperature of hot wall,
K
TC
Temperature of cold wall,
K
xi
Direction components,
m
ui
Velocity components,
ms-1
x, y, z
Cartesian coordinates ,
m
X, Y, Z
Dimensionless coordinates (
xW,
yD,
zD)
Greek symbols
α
Thermal diffusivity ,
m2s-1(k/ρCp)
μ
Dynamic viscosity ,
Nsm-2
μt
Turbulent viscosity,
Nsm-2
ν
Kinematic viscosity ,
m2s-1(μ/ρ)
ρ
Density ,
kgm-3
β
Thermal expansion coefficient,
k-1
Ω
Absolute value of vorticity
ε
Dissipation rate of
k

The two and three dimensional investigation of natural convection in a rectangular enclosure has been carried out in last decades. For example, Taylor and Ijam [6] performed a two-dimensional investigation of steady state free convection inside an enclosed cavity. They used the finite element method to prediction the temperature gradient. A three-dimensional numerical analysis on natural convection in a differentially heated cubic enclosure is presented by Fusegi et al [7]. As well as, Tric and Betrouni [8] done a three-dimensional study and developed accurate solutions for a cubic cavity heated differentially from side walls. Wakashima and Saitoh [9] presented a new benchmark solution for natural convection in a cubic cavity. Garoosi and Rashidi [10] carried out a numerical  two phase simulation of natural and mixed convection in a square enclosure filled with nanofluid with several pairs of heat source-sinks. They showed that there is an optimal volume fraction of the nanoparticles for each Rayleigh number and Richardson number that the maximum heat transfer rate occurs. Also, Malekshah and Salari [11] experimentally investigated the effect of height of fluid in a rectangular enclosure filled with two immiscible fluids (water-Air) that heated partially from side wall and cooled from top and bottom walls.
Moreover, there were many researchers which have a numerical study of natural convection within non- orthogonal shape of enclosure that have special applications. For example, the numerical study of natural convection heat transfer and entropy generation of  Water-Alumina nanofluid  in a Baffled L-shaped cavity were carried out by Armaghani et al [12]. Also, Triveni and Panua [13] conducted a numerical simulation of natural convection  in a triangular cavity with wavy bottom wall and showed that this carve shape has a positive effect on heat transfer enhancement than flat wall. Also, Sheikholeslami and Rokni [14] studied on modeling of nanofluid natural convection in a semi annulus that has been influenced from external magnetic field. As well as, Makulati et al [15] investigated the natural convection of water-alumina nanofluid in a inclined C-shaped enclosure under  the effect of magnetic field.
Recently study of 3D natural convection and the relevant entropy generation within a rectangular cuboid filled by two immiscible fluids of air and a nanoliquid is investigated numerically by Salari et al. [16]. Experimental and numerical studies of laminar and turbulent natural convection in a rectangular enclosure have also been carried out by numerous researchers. Malekshah and Salari [17] conducted a comprehensive numerical and experimental investigation on the natural convection heat transfer within a 3D enclosure filled with two immiscible fluids of water and air. Hsieh and Yang [18] investigated experimentally transient natural convection inside a rectangular enclosure filled with silicone oil in a Rayleigh number range of
6.9×107

4.12×108
. An experimentally study of low turbulence natural convection within an air filled square enclosure that heated from left wall and cooled from opposite  wall have been carried out by Tian and Karayiannis [19]. They showed that the much smaller vortices were created at the hot top and cold bottom corners, after that Ampofo [20] conducted an experimental investigation on turbulent natural convection of air in a square cavity with partitioned wall. Betts and Bokhari [21] performed an experiment to investigate the turbulent natural convection in a rectangular enclosure for two Rayleigh numbers of
0.86×106and 1.43×106.
They claimed that their results can be as an improved benchmark for the testing of turbulence models in low turbulence Reynolds numbers. A differentially heated air-filled cavity has been investigated experimentally and numerically by Salat et al. [22] for
Ra=1.5×109
. Zhang et al. [23] presented PIV measurements for turbulent thermal natural convection in a large enclosure with small heat source and simulated using three turbulence models of
k-ε,standard, RNGk-εandV2F
. They demonstrated that the numerical results of the
RNGk-ε
turbulence model have better consistency with experimental data at the centerline of the cavity than others.
Thanks to the advances in computer memories and CPU-technologies, recently many researchers studied numerically 2D and 3D turbulent natural convection in enclosures. Markatos and Pericleous [24] presented a numerical method to solve the laminar/turbulent flow and heat transfer within a square cavity that heated from side walls for a range of
103
to
1016
of Rayleigh numbers. They used the
 k-ε
turbulence model for the Rayleigh numbers greater than
106
. Hsieh and Lien [25] studied a buoyancy-driven turbulence flow within an enclosure using the unsteady RANS approach and a Low-Re number
k-ε
model. The conjugate turbulent natural convection and surface radiation in a rectangular enclosure filled with air was carried out by Sharma et al [26]. They used the standard
k-ε
  model for the range of Rayleigh number
108
to
1012 .
Also, Ridouance et al [27] investigated the turbulent natural convection in an air-filled isosceles triangular enclosure using low-Re number
k-ε
model. Peng and Davidson [28] studied on turbulent buoyancy flows in an enclosure differentially heated from side walls with low-Re number
k-⍵
models. They founded that the buoyancy source term for turbulence kinetic energy performed strong grid sensitivity and thereafter proposed a damping function to eliminate the refereed grid-dependency. Altaç and Uğurlubilek [29] assessed six RANS turbulence models of standard k-, RNG-k-, realizable K-, K-, SST-K-and RSM in natural convection from 2D and 3D rectangular enclosures for various aspect ratios while Rayleigh numbers had been varied from
108
to
1013
.
Also, there are some limited 3D numerical studies that have used large-eddy simulation (LES) and direct numerical simulation (DNS) to analysis turbulent natural convection in different heated cavities. For example, A DNS investigation of  turbulent flow in an air-filled rectangular enclosure at
Ra=4.5×1010
has been carried out by Trias et al. [30]. They expressed that their DNS results have showed that the transition of the vertical boundary layer occurs at more downstream positions than those observed in experimental and previous numerical works. Also,  Zhang et al. [31]  used the LES numerical method for investigation turbulent natural convection in a differentially heated square cavity and showed that their LES method has a good agreement with the existing numerical and experimental data.
There are some research studies on turbulent natural convection in an enclosure for various aspect ratios and titled angles. Choudhary and Subudhi [32]carried out an experimental study  turbulent natural convection in an enclosure filled with AL2O3-Water nanofluid and investigated the effect of different aspect ratio, Rayleigh numbers and volume concentration on enhancement or deterioration heat transfer. Cooper et al [33] performed an experimental investigation of buoyancy flows and focused on the effect of tilt angle on buoyancy flows inside tall, rectangular, differentially heated cavities. Baїri [34] performed a numerical and experimental study on natural convection in titled square air filled cavity and presented Nusselt-Rayleigh correlation for range of Rayleigh number
10 to1010
and different tilt angles
0°to 360°
.
For most practical buoyancy flows of engineering application, the Rayleigh number is larger than
108
, therefore, the turbulence models must be used to study these phenomena. The turbulence modeling of natural convection flows is still complex. Therefore, it is necessary to choice a suitable RANS model to have a better predicted of the turbulence phenomenon. In the present study, the capability of three RANS turbulence/transition models, two fully turbulence model  of
RNGk-ε , SSTk-⍵
and one recently developed turbulence/transition model of
SST transition
are compared in transitional natural convection in 3-D rectangular tilted enclosure (
H/L=2
) with three tilt angles of
-30°, 0, 30°
. To observe the transitional characteristics of natural convection flow, the specific Rayleigh number of
Ra=5×108 is
employed.
2. Physical Description and Mathematical Formulations
The 3D investigation of transitional natural convection in a tilted rectangular enclosure filled with air has been carried out in this study. According to Fig.1, the enclosures has been heated from left wall and cooled from opposite wall and the other sides of enclosure assumed to be adiabatic. The aspect ratio of enclosures is
HL=HD=2
and tiltangles are considered to be
-30°, 0° and+30°
degrees.
Dear professor, the revised of turbulence article done by Emad.

Main 2D Main 3D
Fig. 1. schematic of physical model, the front view of enclosure (left) and 3D tilted enclosure (left).

2.1. Mathematical formulation
The fluid flow patterns are obtained through solving of the Reynolds-Averaged Navier-Stokes (RANS) of momentum, continuity and energy equations. The fluid flow is presumed to be steady state and Newtonian with variable thermo-physical properties as a function of temperature. The governing equations are written as follow:
Mass:
∂∂xiρui=0
(1)
Momentum:
∂∂xjρujui=-∂P∂xi+∂∂xjτij-ρúiúj̅-ρgiβT-Tref
(2)
Energy:
∂∂xjρcp ujT=∂∂xjk∂T∂xj-ρcpújT’̅+ϕ
(3)
Where the density, viscous stress tensor and dissipation are defined as follows respectively:
ρ=ρT,P
(State equation),
τij=μ∂ui∂xj+∂uj∂xi-23μ∂uk∂xkδij
and
ϕ=τij ∂ui∂xj+τ́ij∂uí∂xj̅
.
As well as, the
xi
an d
xj
are the Cartesian coordinates of the system,
uj, P, T
are the mean components of the velocity, dynamic pressure and temperature, respectively. Moreover, the g, μ, α and β are the gravitational acceleration, density, dynamic viscosity, thermal diffusivity and thermal expansion of the fluid, respectively. Also,
Tref
is a reference temperature that usually taken as the averaged values of hot and cold wall temperature. It can be seen that the governing equations contain the Reynolds stress terms and the correlations of the fluctuating velocity and temperature, which they should be determined or modeled firstly.  Most applied models are generally based on the concept of Prandtl-Kolmogorov’s turbulent viscosity which is proposed in the form of the high Reynolds number. Therefore, the turbulent Reynolds stress and the correlation of the fluctuating values of velocity and temperature component are determined from the following algebraic relations (Boussinesq approximation):
Rij=-ρuíuj́̅=μt∂ui∂xj+∂uj∂xi-23ρδijk
(4)
-ρuíT́̅=μtσT∂T∂xi
(5)
where the
μt
and
σT
are turbulent viscosityand the turbulent Prandtl number, respectively.
2.2. Turbulence models
In this study, three turbulence models are assessed: the renormalized
k-ε
model (RNG
k-ε
), the shear stress
k-⍵
model (SST
k-⍵
) and the newly developed model of “Transition SST” model.
2.2.1 RNG
k-εmodel
The RNG
k-ε
model was derived by Yakhot and Orszag [35]and using renormalization group theory and is a modification of the standard
k-ε
model such as additional term in its ε equation that significantly improves the accuracy for rapidly strained flows. RNG techniques are used to develop a theory for the large scales where the effects of the small scales are represented by modified transport coefficients. The form of RNG model is summarized as follow:
∂∂tρk+∂∂xiρkui=∂∂xjαkμeff∂k∂xj-ρε+Rij∂ui∂xj+Gk
(6)
∂∂tρε+∂∂xiρεui=∂∂xjαεμeff∂ε∂xj+Cε1-RNGεkRij∂ui̅∂xj+Gk-C*ε2ρε2k
(7)
Where:
C*ε2=Cε2-RNG+Cμ-RNGη3(1-ηη0)1+αη
and
η=2D(u)2kε
,
η0=4.38 , α=0.012
.
That
αk and αε
are the inverse effective Prandtl number for k and ε, respectively; and
Cε1-RNG=1.44, Cε2-RNG=1.92 and Cμ-RNG=0.0845
Where
Du=12∇u+∇tu
is the mean strain rate tensor.
At this model, the effective viscosity is calculated by a differential equation as:
dρ2kεμ=1.72μ̂μ̂-1+Cμ̂dμ̂
(8)
2.2.2 Shear-stress transport
SSTk-ω
The SST k-w model is a two- equation eddy-viscosity turbulence model which developed by Menter [36]and it emphasized on an engineering predictive. The principle of the SST approach is to use the
k-ω
formulation in the inner near wall region and the free stream independence of the
k-ε
model in the outer part of the boundary layer. To achieve this, the standard
k-ε
model has been transformed into equation based on
kandω
which leads to the introduction of a cross-diffusion term in dissipation rate equation. Therefore, the formalism of SST model is:
DρkDt=τij∂ui∂xj-β*ρωk+∂∂xj(μ+σkμt)∂k∂xj
(10)
DρωDt=γvtτij∂ui∂xj-βρω2+∂∂xj(μ+σωμt)∂ω∂xj+2(1-F1)ρσω21ω∂k∂xj∂ω∂xj
(11)
The diversity of the SST formulation and the original
k-ω
model is that an additional cross-diffusion term that appears in the ⍵ equation and the modeling constants ϕ are different with the following relation:
∅=F∅1+(1-F)∅2
(12)
Where:
F=tanh⁡(A2)
(13)
And
A=max⁡(2k0.09ωy;500vy2ω)
(14)
∅1
and
∅2
represent any constant in the original
k-ω
and in the transformed
k-ε
model, respectively.
2.2.3. Transition/turbulence model of Re θ–γ-Transition shear-stress transport (SST-Transition)
A correlation-based transition model connected with SST turbulence model was developed by Langtry [37]. In this transition model, two transport equations for the intermittency coefficient and transition criterion of Reθ was developed and is capable to be used in the complex three-dimensional shear flows. One of the important transport equations of this transition model, the intermittency equation, is as follows:
∂(ργ)∂t+∂(ρujγ)∂xj=Pγ-Eγ+∂∂xjμ+μtσγ∂γ∂xj
(15)
Where,
Pγ=FlengthCalρSγFonset1/2(1-Celγ)
is the production term with two empirical correlations of
Fonset
for the transition onset, as well as
Flength
for the length of the transition region.
E γ
is the destruction term enabling the re-laminarization process of the boundary layer which is as follows:
Eγ=Ca2ρΩγFturbCe2γ-1
(16)
Where,
Ω=(2ΩijΩij)1/2
is the absolute value of the vorticity.
The
Fonset
term depends on the critical momentum Reynolds number
Reθc
. This critical Reynolds number is determined by the local transition onset momentum thickness Reynolds number
Rẽθt
obtained from following transportation equation:
∂(ρRẽθt)∂t+∂(ρujRẽθt)∂xj=Pθt+∂∂xjσθt(μ+μt)∂Rẽθt∂xj
(17)
2.3 Numerical Solution and Validation
The equations are solved based on the finite volume method that requires integration of the mean and turbulent transport equations over all of the discretized cells. For the velocity component, the staggered meshes were used in order to prevent the development off checkerboard instabilities overall the pressure field. Since the flow is steady in average, the SIMPLE algorithm is applied for pressure-velocity coupling and the power law scheme is used for the interpolation process. The solution procedure has been validated with Ampofo and Karayiannis [38]study that have an experimental benchmark data for turbulent natural convection in an air filled rectangular cavity with different turbulence models. It can be seen a reasonable agreement between results of the present study with Ref [38] investigation. As well as, it is illustrated from Fig. 2 that the transition/turbulence model of “SST Transition” has a better agreement with experiment data than other turbulence models, especially for the velocity field.

main Vy validation mainT validation
Fig. 2. Comparison of present study and Ref for different turbulence models.

A non-uniform structured mesh with gradually expanding grids clustered toward the walls for boundary-layer resolution is used in the computation. In order to decide upon a required grid resolution for grid-independent computations, different grid non-uniform grids are employed and calculated the heat transfer coefficient and skin friction on the hot wall. According Table.2, a non-uniform grid of 150
×
125
×
100 is chosen for the present simulations.
Table 1.Grid independency for
α=-30°
and all turbulence/transition models

Grid Number 75 
×75
×75
100 
×75
×75
125 
×100
×75
150 
×125
×75
175 
×150
×75
k-ε RNG 0.000449 0.000459 0.000464 0.000478 0.000473
Skin frictioncoefficient k-⍵ SST 0.000415 0.000423 0.000428 0.000441 0.000437
SST Transition 0.000392 0.0004 0.000404 0.000417 0.000413
Heat transfercoefficient k-ε RNG 0.8241 0.8459 0.8546 0.8721 0.8633
k-⍵ SST 0.7147 0.7320 0.7396 0.7547 0.7471
SST Transition 0.7432 0.7589 0.7667 0.7824 0.7745

3. Results and discussions
Turbulent/transitional natural convection within 3D tilted enclosures have been carried out using two well-known turbulence models (shear- stress transport k-⍵ and RNG k-ε) and one newly developed transition/turbulence model (Transition SST). This numerical analysis plans to evaluate the capability of these models for prediction of heat transfer and fluid flow fields of natural convection within three tilted enclosures. Also, the enclosure filled with air (
pr=0.71
) and the Rayleigh number is
Ra=5×108
to reach the longer transitional area near the side walls.

  1. Fluid flow Structure and Temperature Distribution

A tilted enclosure has a different flow pattern, temperature distribution and heat transfer characteristics compared a vertical enclosure. Figures 3-5 illustrated flow structure and temperature distribution in three state: vertical enclosure and tilted enclosures with positive and negative angles of 30 degrees. Vertical velocity structure has been showed in Fig.2 for different turbulence/transition models and tilt angles. The vertical velocity is chosen to demonstrate the boundary layer clearly vicinity of vertical walls. As expected, with dealing the air flow to the hot wall (left wall), the temperature of air become enhances and subsequently reduced density and rise up from side wall due to buoyancy force. With the arrival the heated air-flow to the cold wall (right wall), the heat transfer occurs between air-flow and cold wall and reduced temperature of air-flow and enhanced the density. Therefore, the air-flow will falls down from near the right wall. So, the buoyancy force, gravity field and temperature difference have an important role in creating this circulation. According Fig.3, although all turbulence models have good prediction of vertical velocity component iso-surfaces vicinity of vertical walls, but there are different in prediction of the flow pattern. The continuity of vertical velocity component iso-surfaces of transition/turbulence model, SST-Transition, is more than of both SST K-⍵ and RNG K-ε turbulence models. It is obvious from Fig.3 that the vertical velocities increase with increases of tilt angle from -30° to 30°. The velocity vectors of air-flow obtained from numerical results are shown in Fig. 4 for different tilt angles and turbulence models. For
α=-30°
, the separation flow accrue in up corner of hot wall (left wall) and this phenomenon was predicted by SST transition and K-⍵ SST models, while this area unforeseen by K-ε model. As well as, Fig. 4 shows that with increase of tilt angle from positive angle to negative angle (from -30° to +30°) the fluid flow vicinity of side walls become stronger and it was predicted by all turbulence models.

k-ε RNG SST Transition k-⍵ SST
-30 S- fluid strusture KE -30 S -flow structure -30 Trans S -flow structure
0 S-non orginal structure flow KE 0 S-flow structure 0 transition E^9t flow structure kwsst for 5e^8
+30 S-Flow structure +30 KE S -flow structure Tran +30 S-flow structure KWSST +30
Fig. 3. Vertical velocity structure for different turbulence models and tilt angle.
k-ε RNG SST Transition k-⍵ SST
-30 Vector KE -30 Vector Transition -30 Vector KWSST -30
0 Vector0 kE Vector Transition Vector KWSST
30 Vector KE +30 vector Tran +30 new Vector KWSST +30
Fig. 4. Velocity vector for different turbulence models and tilt angles.

The iso-surface temperature at different tilt angles by various turbulence models is shown in Fig. 5. It can be understand from parallel isotherms of air-flow that are extremely well stratified for all tilt angles at the center of enclosure that the dominate mechanism of heat transfer is conduction at this region and the fluid flow are not strong enough to transfer the heat energy at the center of the enclosure but, because of the temperature difference between air-flow and side walls, the convection become dominate mechanism. For this reason, the isotherms are convex at the near of side walls. Also, the convex of isotherms become larger at the top of hot wall (left wall) and bottom of cold wall (right wall). By changing the angle of enclosure from -30 to +30, the disorders and convexes of isotherms increase that demonstrated that the tilt angle has a positive effect on natural convection within enclosure. The reason of this enhance of natural convection is that with increases of tilt angles, the enclosure reaches to the Rayleigh-Benard convection that provide maximum natural convection mechanism and minimum conduction mechanism. The important point is that the K-⍵ SST turbulence model has a fewer isothermal layer than SST Transition and K-ε models.

k-ε RNG SST Transition k-⍵ SST
-30 Iso surface temperature Iso surface temperature -30 Trans iso surface temperature KWSST -30
0 iso surface KE iso surface temperature Transition 0 Isosurface temperature KWSST 0
30 iso-surface temperature +30 KE Iso surface temperature Tran +30 Iso surface temperature KWSST +30
Fig. 5.iso-surface temperature for different tilt angles and turbulence models.
  1. Turbulent kinetic energy

Fig. 6 illustrates the turbulent kinetic energy of air-flow within enclosure at the mid-surface of enclosure
Z=-0.5
for all tilt angles and different turbulence/transition models. The main purposes of this figure are to compare the ability of different turbulence models to predict the different region of boundary layer vicinity of walls and other part of enclosure. Also, the impact of tilt angle on turbulent kinetic energy can be investigated from Fig. 6. At the
α=-30°
, the turbulence kinetic energy are presented by three turbulence models. The K-ε RNG and K-⍵ SST turbulence models predicted that the maximum of turbulence kinetic energy of air-flow occurs at the adjacent of side walls and a region at near of top insulated wall. Also, the start point of turbulence boundary layer of K-ε RNG model is lower than the K-⍵ SST model at the vicinity of side walls. At the other hand, the transition SST turbulence model predicted that the high turbulence kinetic energy occur just at the near of hot and cold walls. The turbulence kinetic energy contours for the vertical state are similar to case
α=-30°
while the all of them are thicker and become more stretched. For
α=30°
, both K-ε RNG and SST transition transition/turbulence models show that the higher turbulence kinetic energy of air-flow located at the top of left wall and bottom of right wall. In fact, the reason for this is that the vertical velocity component becomes stronger at this tilt angle that the velocity vector of Fig. 4 and vertical velocity structure of Fig. 3 shows that the effect of tilt angle on turbulence kinetic energy is another important issue. It can be comprehend from Fig. 6 that the tilt angle has a remarkable effect on turbulence kinetic energy and their distribution within enclosure. So that, with increase of tilt angle from
-30°
to
+30°
, the turbulence kinetic energy enhanced and the wide region of enclosure are affected by turbulence region. As well as, the inception point of turbulence region moves to near of the top of hot wall and bottom of cold wall and extend to the part of isolated walls.

RNG k-ε SST Transition SST k-⍵
-30 main Turbulent Viscosity ratio KE -30 main 2D turbulent energy trans main 2D turbulent energy turbulenc energy -30
0 main turbulent energy for 5e^8 ke rng -0 main turbulent energy for e^8 transition rng -0 main turbulent energy for 5e^8 kw sst -0 turbulenc energy 0
30 1111 main turbulenc energy trans main 2D turbulent kinetic energy KWSST +30 turbulenc energy +30
Fig. 6.turbulent kinetic energy for different tilt angles and turbulence models.

3.3.Turbulent viscosity ratio
The turbulent viscosity ratio or the ratio of eddy viscosity to molecular viscosity, for air-flow within enclosures at vertical standing position,
α=0°
, for different turbulence models at the mid-depth (
Z=-0.5
) are illustrated in Fig. 7. The turbulent viscosity ratio is a parameter that specifies the laminar, turbulent or transitional regions of boundary layer. The lowest value of turbulent viscosity ratio which predicted by SST-Transition model occurred at top and bottom regions of enclosure, where the fluid flow is nearly laminar, whereas other models have shown a turbulent flow at the vicinity of these regions which are specified with a rectangular region on the figures. All of these turbulence/transition models show that the turbulent viscosity ratios are enhanced from down to up on the heated walland vice versa for the cooled wall.It is seem that the transition/turbulence model of SST-Transition predicted a wide range of turbulent viscosity ratio than other fully turbulence models. The different regions of boundary layer can be show from oval area which has shown in Fig 7. It can be elicited that the SST-Transition model has an elongation transitional/turbulent region than two other models. Indeed, it is a capability of SST-Transition model which determine the transitional/turbulent region than two fully turbulence models.

modified 0,ke, E^9 modified 0,transition, E^9 modified 0,kw sst, E^9
Fig. 7.turbulent kinetic energy for different tilt angles and turbulence models.

 
 
 

  1. Inner and outer boundary layer structures

The temperature profile of air-flow at the mid-height of side walls of enclosure for different tilt angles and all transition/turbulence models are presented in Fig. 6. It can be founded from this that the tilt angle is one of the effective parameters on temperature distribution within enclosure. All of transition/turbulence models almost have a similar pattern for prediction the temperature profile at the near of side walls. Of course, with deviation of enclosure from vertical state, the different transition/turbulence models have a variant temperature profileat the vicinity of side walls. The SST Transition model presents a different results than the RNG k- and the K-⍵ SST model respect to temperature profile at the
α=-30
.

T kol -30 teta kol 0t T kol +30
Fig. 6. Ttemperature profile at the Y = 0.5 for different configuration and turbulence models.

The velocity profiles at the mid-height of side wallsareillustrated in Fig.7 for different tilt angles and turbulence models. It is obtained that all transition/turbulence models have same prediction of velocity profile at the wide area of enclosure and outer of boundary layer. The effect of tilt angle on velocity profile is also visible. So that, the thickness of velocity boundary layer are thicker and thinner respectively for case
α=-30°
and
α=+30°
. The reason of this is that the start point of turbulent region of enclosure with
α=+30°
began upper than other stats. In fact, the turbulent boundary layer of tilted enclosure with
α=+30°
has been not formed at the mid-height of side walls.

Vy kol -30 Vy kol 0 modified Vy kol +30
Fig. 7. Velocity profile at the Y = 0.5 for different configuration and turbulence models.

The predicted vertical velocity profile within the vertical boundary layer vicinity of the hot wall (left wall) at severalY location (y/H) by variant turbulence models at the vertical state ( = 0°) has been shown in Fig 8. The vertical velocity is scaled by the buoyancy velocity Vb, given by
gβHTh-Tc1/2
. The Y positions from 0.1 to 0.5 are solid lines and more than of
Y=0.5
are dash lines to compare the velocity profile before and after the mid line of enclosure. The main aim of Fig. 8 is that these profiles depict the transition of the boundary layer from laminar near the bottom wall to fully turbulent near the top wall. From these figures and velocity boundary layers, we can see the different regions of velocity boundary layer, whereas, the laminar region has a thin boundary layer thickness than turbulent region. It can be approximated of Fig. 8 that the profiles for
Y=0.1
to
Y=0.4
are in the laminar boundary layer, while the profiles for
Y=0.8
and
Y=0.9
are in the fully turbulent boundary layer.As accepted, with increase the Y position, the thickness of velocity profile become thicker. Therefore, both SST transition and K-⍵ SST turbulence models have better simulation than K-ε RNG model at the fully turbulent region near the top wall that the thickness. From vertical velocity predicted by SST transition model, we can conclude that the profiles
Y=0.6
and
Y=0.7
are in the transitional boundary layer, because the thickness of boundary layer at these positions have suddenly grown.

main KE rng Vy for various Y main Transition vertical velocity in various Y main KW SST vertical velocity in various Y
Fig. 8.dimensionless vertical velocity at different Y positions for various turbulence models.

 
 
3.5. Skin friction coefficient and heat transfer coefficient
The prediction of shear stress and convection heat transfer that represented by skin friction coefficient and heat transfer coefficient, respectively,along the hot wall at different tilt angles are performed by several transition/turbulence models and shown in Fig. 9. The analysis of these coefficients at the cold wall are similar with opposite wall, therefore, these coefficient just have been studied at the hot wall.  From the hot wall, the heat transfer coefficients have a growth spurt close
Y=0
and then, decreased along the hot wall gradually. Also, the tilted enclosure with tilt angle
α = +30°
has a maximum peak of heat transfer coefficient. It is observed that three transition/turbulence models have a similar prediction of heat transfer coefficient at the top and bottom of hot wall. Also, the skin friction coefficient along the hot wall is shown in Fig. 9. The skin friction coefficient passes through a maximum point and then decreased with increase of Y position and the location of peak of skin friction coefficient points shift to upward as increase of tilt angle. The locations of these points are in conformity in with the location of transition point. Also, the concave of skin friction coefficients of tilted enclosure with
α = +30°
are lower than other configurations. As well as, the skin friction coefficient calculated by transition SST turbulence model has the lowest value which has different trend with respect to other results as well.

 = -30°  = 0°  = 30°
Cf and h left wall -30 modified Left h and Cf for al model0 modified+30 h and Cf left wall for all models+30
Fig. 9. Heat transfer coefficient (solid) and skin friction coefficient (dashed) respect to the hot wall (left wall) for different tilt angle  = -30°(a), 0° (b) and +30° (c) for various turbulence models.
  1. Conclusion

The assessment of three transition/turbulence models (k-ε RNG, k-⍵ SST and Transition SST) on the transition natural convection within three-dimensional rectangular enclosure filled with air and different tilt angle of enclosure (-30°, 0°,+30°)  at the specific Rayleigh number of
Ra=5×108
has been carried out in this study. The main aim of this analysis is to compare the obtained results of two traditional turbulence model of k-ε RNG and k-⍵ SST with the results of a newly developed transition/turbulence model of SST-transition. The following conclusions can be presented:

  1. The flow structure and temperature distribution predicted from three turbulence model are similar at the large area in the middle of enclosure. But, there were different prediction at the near of side vertical walls. So that, the k-⍵ SST model has a similar results to Transition SST model than k-ε RNG model in term of vertical velocity and temperature distribution vicinity of side walls.
  2. The turbulent kinetic energy of air-flow enhanced with increase of tilt angle.
  3. The transition/turbulent region is better shown by SST Transition model.
  4. At theα=+30°, the inception point of turbulent region moves to near of the top of hot wall and bottom of cold wall and extend to the part of isolated walls.
  5. The value of heat transfer coefficient is the highest at the tilt angle withα=+30°.
  6. The positive tilt angle is an effective parameter respect to enhance of heat transfer.

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