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CFD analysis for characterization of non-linear power law material in a channel driven cavity with a square cylinder by measuring variation in drag and lift forces
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Original Article
CFD analysis for characterization of non-linear power law material in a channel driven cavity with a square cylinder by measuring variation in drag and lift forces Rashid Mahmood a , S. Bilal a , Afraz Hussain Majeed a , Ilyas Khan b,∗ , Kottakkaran Sooppy Nisar c a
Department of Mathematics, Air University Sector E-9 AIR Complex, P. O. 44000, Islamabad, Pakistan Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, 72915, Vietnam c Department of Mathematics, College of Arts and Sciences, Prince Sattam bin Abdulaziz University, Wadi Aldawaser, 11991, Saudi Arabia b
a r t i c l e
i n f o
a b s t r a c t
Article history:
Current communication is manifested to investigate flowing features of power law mate-
Received 11 December 2019
rial in a newly proposed physical configuration namely the channel driven cavity. Since
Accepted 3 February 2020
power law fluid discloses the dynamical features of shear thinning, shear thickening and
Available online xxx
Newtonian materials so in present communication it is considered for depiction of flow
Keywords:
below the channel. The flow is induced with parabolic inlet velocity and a Neumann condi-
attributes. To achieve the desired outcomes from the work, a unit length cavity is placed Finite element method
tion is applied at the outlet, while no slip condition is set at all other boundaries. A square
Power law material
cylinder is placed in the channel with varying positions giving rise to three computational
Drag and lift coefficients
grids named as G1 , G2 and G3. Mathematical modelling is constructed by obliging funda-
Channel-driven cavity
mental conservation and rheological laws for power law fluid. Since the representative equations are complex in nature so an efficient computational procedure based on finite element method (FEM) is executed. A hybrid computational grid is generated at coarse level and then further refinement is done to improve the accuracy of the solution. The solution is approximated by adopting P2 − P1 element based on second and first order polynomial shape functions. Graphical trends against involved parametric variables are adorned. In addition for more physical insight of problem velocity and pressure plots and line graphs are added. Furthermore, the hydrodynamical benchmark quantities like pressure drop, drag and lift coefficients are evaluated in tabular form around the outer surface of obstacle. © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
∗
Corresponding author. E-mail: [email protected] (I. Khan). https://doi.org/10.1016/j.jmrt.2020.02.010 2238-7854/© 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: Mahmood R, et al. CFD analysis for characterization of non-linear power law material in a channel driven cavity with a square cylinder by measuring variation in drag and lift forces. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.02.010
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Nomenclature # EL # DOF p ıp pA pB ui
Number of Elements Number of degrees of freedom Dimensionless pressure Pressure drop Pressure at point A of obstacle Pressure at point B of obstacle Density of the fluid Dimensionless velocity components
Shear rate Mean inflow velocity Maximum inflow velocity Power law index Consistency index Drag coefficient Lift coefficient
•
Umean Umax n m CD CL
1.
Introduction
The flow in a space generated between two parallel surfaces with relative tangential velocity is called channel flow. Research concerning with dynamics of channel flow has attained superb attention of researchers due to its extraordinary applications in multiple disciplines. Such types of flows are utilized in hydraulics, aviation, chemical engineering, process engineering [1–6]. Flow in a channel is generated either by generating pressure gradient or by drag force. The fluid executed by effective pressure difference in a channel is termed as Poiseuille flow and was identified by Jean Poiseuille in 1938. Comprehensive work on Poiseuille flow was by various researchers which can be accessed through [7–11]. Later on researchers tried to work on other factors that create the fluid flow in a channel. A comprehensive literature review regarding fluid flow in a channel under various physical constraints is enclosed in references [12–16]. Interpretation about fluid flow in confined enclosures has intended fluid dynamists towards itself due to its technoscientific utility in real life processes. Aidun et al. [17] pointed the direct relevancy of cavity flows to coaters and melting spinning processes which are used to manufacture microcrystalline materials. Zumbrunnen et al. [18] disclosed the importance of eddies structures formation in cavity in drag reducing riblets and synthetization of fine polymer composites. Initially fluid experts thought that flow generation inside cavity is produced by moving the walls with prescribed velocity or making some thermal change through boundaries. For the sake of interest of readers few recent investigations regarding fluid flow in cavities are engrossed in Refs. [19–21]. Interaction of fluid with solid structures has received an overwhelming attraction during last few decades due to pervasive implications in heat exchanger tubes, cooling systems for nuclear power plants, transmission cables, bridges, high rise buildings, and electronic equipment. After getting view about these procedural applications experts are taking keen interest due to the fact that these are capitalized to mount the lift and to demise the drag. The square shaped obstacle
among these different structures of cylinders serve as fundamental component of design and structures. These structures often interact with fluids and experiences flow induced forces which lead to the failure under certain unfavourable circumstances. In this regard premier work on obstacle flow was done by Schaefer and Turek [22] in which they explored benchmark problem for incompressible flow around a cylinder and constructed comparison between findings made with different solutions approaches. Initially viscous flow around obstacles was considered but due to technological changes across the frontiers of industrial work, non-Newtonian fluids have gained considerable attention. Such fluids are defined on the basis of their rheological pattern. Fluid experts interprets that the models disobeying Newton law of viscosity and expressing non-linear relation between deformation rate and shear stress are called as non-Newtonian fluids. For better physical interpretation they characterize them into fundamental three classes are i) shear thinning (Pseudoplastic) ii) shear thickening (Dilatant) iii) thixotropic fluids. On the basis of this diversification various fluid models have been proposed. Among these Ostwald-de Waele model commonly known as power law fluid is one of the generalized model that expresses the properties of shear thinning and thickening fluids. Shear thinning and thickening materials in channel and cavities are interrogated abundantly due to their potential applications like multiphase mixtures, i.e., foams, suspensions, emulsions, etc. and high molecular weight polymeric systems, i.e., solutions, melts, blends, etc. exhibit shear-thinning(pseudoplastic fluids, n < 1) and/or shear-thickening (dilatant fluids, n > 1). Some recent available literature is discussed here. Computational finite element analysis emphasizing the roles of Reynolds number and power law index on the global and detailed flow characteristics of shear thinning fluid demarcated in flow regime across obstacle was elucidated by Coelho et al. [23]. Chhabra et al. [24] investigated steady and incompressible flow of power law fluid past a circular cylinder by implementing finite difference scheme for different magnitudes of Reynold number 1, 20 and 40. Sivakumar et al. [25] studied the flowing features of power-law fluid past an elliptic cylinder and examined the behaviour of shear-thinning (n < 1) and for shear-thickening (n > 1). Patnana et al. [26] investigated uplift in drag coefficient against fixed magnitude of Reynold number and power law index in incompressible generalized Newtonian fluid across an unconfined circular cylinder. Power law fluid across circular cylinder to check the dependence of individual drag coefficient on fluid model parameter was adumbrated by Soares et al. [27]. Chandra and Chhabra [28] examined the influence on flow separation around surface of cylinder immersed in power law fluid across a semi-cylinder and under transitional Reynolds numbers for the flow separation over the surface of cylinder. Sanyal and Amit [29] measured variation in drag and lift coefficients over the outer surface of cylinder at low Reynold number within range of 0 ≤ Re ≤ 50. The measurement about depending relationship of translational velocity with drag coefficient in case of power law fluid was studied by Yang et al. [30]. Noferesti et al [31] performed analysis on characteristics of generalized Newtonian fluid over a quadrilateral cylinder and found increment in drag coefficient by raising aspect ratio of cylinder. In recent years Paliwal et al. [32] have scrutinized flow of power law
Please cite this article in press as: Mahmood R, et al. CFD analysis for characterization of non-linear power law material in a channel driven cavity with a square cylinder by measuring variation in drag and lift forces. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.02.010
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liquids past a square cylinder confined between slip boundaries within the range of5 ≤ Re ≤ 40. Analysis on bifurcation of stream lines by considering shear thinning fluid model inside channel as well as through axisymmetric tube was executed by Shahzadi et al. [33]. They utilized methodology of dynamical systems to inspect the local bifurcation and changes in the topologies. Discussion about mixed convection peristaltic flow of shear thinning tangent hyperbolic fluid under the inclusion of nanoparticles in a curved channel was manifested by Shahzadi and Nadeem [34]. Nadeem and Shahzadi [35] investigated the impact of thermophoresis and Brownian movement on peristaltic motion of shear thinning hyperbolic fluid in a curved channel. Nadeem and Shahzadi [36] conducted theoretical analysis for peristaltic motion of water base nanofluid containing distinct types of the nanoparticles like Cu, TiO2 , and Al2 O3 and by modelling the equations under low Reynolds number as well as long wave length supposition. From the above mentioned literature it is seen that Power law fluid in channel and cavity are discussed separately in various physical configurations and constraints. But the thread to this communication is to introduce a newly innovated combined physical configuration containing channel and cavity known as channel driven cavity. This manuscript is fabricated after getting motivation form the experimental study conducted by Kwon and Arceneaux [37–40]. This newly proposed sketch is highly utilized in pressure vessel and hydraulic industry. One of the most impressing features of channel driven cavity is that it covers various benchmark problems in it like flow in cavity, flow in a channel, forward and backward facing step processes, contraction and expansion phenomenon. So the development and fabrication of this configuration will serve as benchmark study for all type of problems and validate the previously computed results.
3
Fig. 1 – Schematic diagram of problem.
Fig. 2 – A P2 − P1 finite element pair-Location of degrees of freedom.
2. Mathematical formulation and viscosity model
at u = v = 0 (no slip boundaries by setting). A parabolic inlet velocity governed by relation with Umax = 0.3 is injected to the channel and a do nothing boundary condition at the outlet is chosen. A very fine hybrid grid is generated to improve the accuracy of the solution, and excessive number of elements around the obstacle is gathered for better findings. Power law fluid is considered representing the properties of shear thinning, shear thickening and Newtonian fluids. Square cylinder with centres positioned at (0.9, 1.5), (1.5, 1.5) and (2.1, 1.5) are considered. The Reynold number chosen here are Re = 20 and Re = 50.
The conservation law for steady, incompressible fluid flow are expressed as under
4.
∂ui = 0, ∂xi
(1)
∂ 1 ∂p 1 ∂ (u u ) + = ∂xj i j ∂xi ∂xj
() ˙
∂uj ∂ui + ∂xi ∂xj
, i, j = 1...3,
(2)
Relation representing viscosity variation in power law fluid with shear rate is represented as below () ˙ = m() ˙
n−1
,
(3)
where m is the fluid consistency coefficient; n is the power law index; ˙ is the shear rate.
3.
Physical configuration
Consider a channel driven cavity shown schematically in Fig. 1. The cavity of unit dimension is adjusted on lower side of channel. The upper and lower walls of the channel are set
Methodology
Mathematical modelling of most of engineering problems is highly complexed in nature so extraction of solution is also tough task. In this regards various computational approaches are available in literature [38–41] but each method has restrictions. Among these methods the most renowned finite element method is employed to narrate the flowing features of fluid in channel driven cavity. In most of nonstructural flow domains finite element method is capitalized because a large system is firstly divided in to finite number of smaller and simpler parts which are known as finite elements. In this direction the conforming element pair P2 − P1 is selected for the velocity and pressure approximations. The location of degrees of freedom for this finite element pair is shown in Fig. 2. This pair has 15 local degrees of freedom for problem at hand. This element is a stable element pair satisfying inf–sup condition. After the discretization of domain into smaller elements algebraic system of equations is constructed and Newton’s method is applied to solve discrete non-linear algebraic systems and the inner linear sub prob-
Please cite this article in press as: Mahmood R, et al. CFD analysis for characterization of non-linear power law material in a channel driven cavity with a square cylinder by measuring variation in drag and lift forces. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.02.010
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Table 1 – Number of degree of freedom at different refinement levels. Refinement level
# of elements
# of Dof
1 2 3 4 5 6 7 8 9
769 1167 1786 2962 3590 6707 16,139 39,724 52,844
4032 6130 9245 15,171 21,648 33,213 78,925 191,180 250,024
lems are solved using a direct solver. The convergence criteria for the nonlinear iteration is set as
n+1 − n −6 n+1 < 10 ,
where represents the general solution component (Table 1). The computational grids at coarse level for three different setting of obstacles are shown in Fig. 3 named as G1, G2, G3 . The process of mesh refinement is a key step in validating any finite element model and to raise the trust of reader on attained physical outcomes regarding the work. The main theme of finite element scheme is to divide the domain into subdomains called elements and to solve the governing equations locally in each element. In view of the importance of discretization of fluid domain into finite elements meshing at multiple levels of refinements is performed but here only coarse level is shown.
5.
Results and discussion
Fig. 4(a–c) illuminates momentum distribution by varying power law index (n) from 0.5 to 1.5 and by restricting Re = 20. Here power law flow shows shear thinning behavior for n = 0.5, Newtonian fluid for n = 1 and shear thickening fluids for n = 1.3, 1.5. Since parabolic velocity is induced at inlet and other boundaries are kept at no slip conditions so variation in velocity near obstacles and other portions of channel driven cavity is observed. It is concluded that similar parabolic velocity profile is exhibited at outlet and additionally circulating flow in cavity is generated and formation of vortices takes place. Fig. 5(a–c) is plotted to measure change in pressure throughout the physical domain and especially in vicinity of obstacle placed at three different locations i.e. for grids G1, G2 and G3 by varying (n) within the range of (0.5 ≤ n ≤ 1.5) and by restricting Re = 20 and Re = 50. It is seen from the portrays that pressure shows non-linear behavior near the obstacle and becomes linear along the downstream as expected in channel flow. Here observation about optimum pressure is made that it has maximum value at face of obstacle which is interacting with fluid. Fig. 6(a–c) represents line graphs to express the variation in velocity at different portions of physical configuration. From the above sketches it is seen that at inlet perfect parabolic plateau is achieved because viscosity has no active role on injected velocity. Whereas, the fluid flow with in the channel changes its behavior near obstacle and above the cavity. In
Fig. 3 – The computational grids at coarse level.
addition it is also explicated that velocity of fluid decreases by increasing power law (n) because the viscosity of fluid enriches and more resistance is offered to fluid. Tables 2a–2c interprets variation in pressure across the obstacle located at three different positions in channel like at centre of G1 , G2 , G3 for increasing magnitude of power law index (n) and by restricting Re = 20 and Re = 50. From the attained numerical data it is enumerated that by increasing (n) pressure drop increments. As power law fluid behaves as shear thinning fluid for (n < 1) and become Newtonian fluid at (n = 1) and elucidates the properties of shear thickening fluid so by increasing the magnitude of (n) the viscosity of power law
Please cite this article in press as: Mahmood R, et al. CFD analysis for characterization of non-linear power law material in a channel driven cavity with a square cylinder by measuring variation in drag and lift forces. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.02.010
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Fig. 4 – a): velocity profile for G1, at Re = 20 for various n. b): velocity profile for G2, at Re = 20 for various n. c): velocity profile for G3 at Re = 20 for various n.
fluid intensifies and strikes the obstacle with more strength and thus pressure drop mounts. Variation in pressure gradient is also evidenced by the numeric data attained that for shear thinning case i.e. n varying from (0.5 to 0.9) the pressure gradient approaches to 0.0923169010721 at obstacle location G1 and for shear thickening case i.e. n varying form (1.1 to
1.5) the pressure gradient approaches to 0.1066536252091 at obstacle location (0.9, 1.5) for Re = 20. It is also observed that by changing the position of obstacles and by varying power law index (n) the pressure gradient varies. The manipulated variation is proved form tabular values that at n = 0.5 i.e. shear thinning case for restriction Re = 20 and at location (0.9, 1.5) the
Please cite this article in press as: Mahmood R, et al. CFD analysis for characterization of non-linear power law material in a channel driven cavity with a square cylinder by measuring variation in drag and lift forces. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.02.010
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Fig. 5 – a): Pressure profile for G1 at Re = 20 for various n. b): Pressure profile for G2 at Re = 20 for various n. c): Pressure profile for G3 at Re = 20 for various n.
Table 2a – Pressure gradient at centre of G1 for various n. n 0.5 0.7 0.9 1 1.1 1.3 1.5
Re = 20 ıp = pB − pA
Re = 50 ıp = pB − pA
0.0828604860287 0.0875478576104 0.0923169010721 0.0946981050108 0.0970734765920 0.1018252633472 0.1066536252091
0.0667172348720 0.0681923622698 0.0712711286253 0.0729454316371 0.0746737862482 0.0782369401251 0.0818976603145
Table 2b – Pressure gradient at centre of G2 for various n. n
0.5 0.7 0.9 1 1.1 1.3 1.5
Re = 20 ıp = pB − pA
Re = 50 ıp = pB − pA
0.0781110642119 0.0835134542417 0.0884731804764 0.0907965522813 0.0930166504224 0.0971431850531 0.1008965588128
0.060473759292558 0.062975223522683 0.066025509148481 0.067540127154250 0.069033824796297 0.071935602822738 0.074701729700852
Please cite this article in press as: Mahmood R, et al. CFD analysis for characterization of non-linear power law material in a channel driven cavity with a square cylinder by measuring variation in drag and lift forces. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.02.010
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Table 2c – Pressure gradient for centre of G3 for various n. n
0.5 0.7 0.9 1 1.1 1.3 1.5
Re = 20 ıp = pB − pA
Re = 50 ıp = pB − pA
0.078875015959165 0.084948175601868 0.090939257630179 0.093886809719276 0.096799896069806 0.102543398288502 0.108245801052848
0.07497012329253 0.07546257422997 0.07857883957756 0.08031802033540 0.08209801458882 0.08566981906243 0.08918381056459
pressure drop is 0.0828604860287 whereas in similar physical parameter restriction but at location G2 the pressure change is at 0.0781110642119 and at obstacle positon G3 variance is 0.0781110642119. These depicted values shows that near the inlet pressure is maximum and above the cavity pressure drops with less intensity and then against lift up after the crossing over of fluid from cavity. Tables 3a–3c represents fluctuation in benchmark hydrodynamic quantities like drag and lift coefficients on outer surface of square obstacle placed at G1 , G2 , G3 . It is found that by increasing power law index drag and lift coefficients upsurge whereas reverse behavior in both these hydro dynamical forces against uplift in Reynold number (Re). Negative values of lift coefficient ( CL ) shows that lift forces in upward direction are dominant. The reason behind negative values of lift coefficient is that the obstacles are placed in such a way that fluid fell into the cavity and generates upward thrust on the obstacle thus it generates such numerical pattern. From the calculated data about the drag coefficient at G1 , G2 and G3 . It is deduced that drag has maximum magnitude of 3.7728996641145 at G1 and by fixing n = 0.5 and Re = 20 where as it reduces ne ar G2 in the situation when obstacle is placed over the cavity having magnitude of 3.42145328808411 and again increases at G3 attaining values to 3.7720752163646. Whereas contrasting behavior for lift coefficient is addressed in such a way that lift forces at G1 has lowest magnitude of 0.0595685094797372 but at obstacle location G2 upraise in the values of lift coefficient scaled up to 0.131684889330373 is found and exceeds to 0.148163843109242 at G3 . Here negative coefficients are measured when obstacle is placed near entrance and over cavity location because upward lift forces are dominant and positive coefficients are attained at G3 because downward lift forces are active.
6.
Fig. 6 – a): Cutline graph for G1 at Re = 20 for various n. b): Cutline graph for G2 at Re = 20 for various n. c): Cutline graph for G3 at Re = 20 for various n.
Conclusions
Present analysis addresses flow features of generalized nonNewtonian fluid model in a newly proposed channel driven cavity with square obstacle placed at different locations. Physical problem is converted into mathematical formulation in terms of two dimensional Navier stokes equation along with boundary constraints. A well reputed computational method known as finite element method is obliged to report solution
Please cite this article in press as: Mahmood R, et al. CFD analysis for characterization of non-linear power law material in a channel driven cavity with a square cylinder by measuring variation in drag and lift forces. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.02.010
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Table 3a – Drag and Lift coefficient for various n at centre of G1 . Re = 20
n
0.5 0.7 0.9 1 1.1 1.3 1.5
Re = 50
CD
CL
CD
CL
3.7728996641145 4.39734366354035 5.06524900959966 5.41036840362092 5.76189873503173 6.48436659610317 7.23835554996017
−0.0595685094797372 −0.0982655193149808 −0.126331547935937 −0.139383783328286 −0.152376171231296 −0.179346101202571 −0.208594154852365
3.13559638538286 3.27601270221956 3.40831687113724 3.60938018927641 3.8242188137267 4.28052787519498 4.76085165782894
0.080429028303184 0.0199484099864073 −0.0213180374297233 −0.0381682604247344 −0.0522645659953361 −0.0752052654851581 −0.0946279098319215
Table 3b – Drag and Lift coefficient for various n at centre of G2 . Re = 20
n
0.5 0.7 0.9 1 1.1 1.3 1.5
Re = 50
CD
CL
CD
CL
3.42145328808411 3.94249969109768 4.47115320125086 4.73294084643189 4.99109332511408 5.49367373603658 5.97625443204032
−0.131684889330373 −0.188402284884305 −0.244004274207543 −0.27296816440631 −0.303086293886686 −0.367692822712802 −0.438927745734126
2.73584762852872 2.80285467590667 3.10838039593571 3.27997812518368 3.45764447194587 3.81969512462804 4.18057223905749
0.0707420862086769 −0.0059892212984909 −0.0554138185168249 −0.0788924001693645 −0.100483678955644 −0.140253781064674 −0.178795536650709
Table 3c – Drag and Lift coefficient for various n at centre of G3 . Re = 20
n
0.5 0.7 0.9 1 1.1 1.3 1.5
Re = 50
CD
CL
CD
CL
3.7720752163646 4.43394249262315 5.1421025531268 5.50943775652902 5.88434297328834 6.65537502041482 7.45799797881548
0.148163843109242 0.18283312337039 0.212373573471148 0.227081667591816 0.241935828009557 0.272146175182898 0.303438489023832
2.90913902124884 3.03515112042785 3.42710065642627 3.65010100634513 3.8842833042712 4.37485986732887 4.88522236953334
−0.0876392972574064 0.00456305137305655 0.0497091049331152 0.0670300669690129 0.0816464735626635 0.106574494980338 0.12886105155848
and physical happening. The obtained outcomes are offered via graphical trends. Velocity and pressure plots at various locations within the configuration are manifested. Bar lines representing the behavior of momentum distribution of power law fluid at inlet, obstacles and near outlet regions are illuminated. Tabular data regarding the pressure variation and drag and lift coefficients on the outer surface of obstacles is enumerated. The key outcomes are listed below. i) Pressure drops with more intensity as the magnitude of power law index uplifts. ii) Placement of obstacle effects the pressure drop by first increasing at G1 , decreases at G2 and eventually boost at G3 . iii) Reynold number has decrementing attribute on pressure constraint difference in vicinity of square obstacle. iv) Drag and lift coefficients have low magnitude for shear thinning case of Power law fluid in comparison to shear thickening version of the fluid. v) Negative values of lift coefficient are attained when upward forces are dominant whereas positive coefficients are accomplished when downward forces exceeds.
vi) By increasing Power law index velocity profile delineates and evidenced by drawing cut lines at various location in configurations. vii) Pressure has stagnant magnitude at obstacle side where fluid in interacting with it and at corners of cavity. viii) It is found that by replacing n < 1 power law model presents class of shear thinning materials and for n > 1 the shear thickening model representation is achieved. For n = 1 the model reduces to Newtonian fluid with m as the constant viscosity.
Conflict of interest The authors declare no conflicts of interest.
Appendix A. Supplementary data Supplementary material related to this article can be found, in the online version, at doi:https://doi.org/10.1016/j. jmrt.2020.02.010.
Please cite this article in press as: Mahmood R, et al. CFD analysis for characterization of non-linear power law material in a channel driven cavity with a square cylinder by measuring variation in drag and lift forces. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.02.010
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references
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[22] Schäfer M, Turek S. Benchmark computations of laminar flow around a cylinder. Flow Simulat High Perform Comput 1996;2:547–66. [23] Coelho PM, Pinho FT. Vortex shedding in cylinder flow of shear-thinning fluids II, Flow characteristics. J Non-Newt Fluid Mech 2003;11:177–93. [24] Chhabra RP, Soares AA, Ferreira JM. Steady non-Newtonian flow past a circular cylinder: A numerical study. Acta Mech 2004;172:1–16. [25] Sivakumar P, Bharti RP, Chhabra RP. Steady flow of power-law fluids across an unconfined elliptical cylinder. Chem Eng Sci 2007;62:1682–702. [26] Patnana VK, Bharti RP, Chhabra RP. Two dimensional unsteady flow of power-law fluid over a cylinder. Chem Eng Sci 2009;64:2978–99. [27] Soares AA, Ferreira JM, Caramelo L, Anacleto J. Effect of temperature-dependent viscosity on forced convection heat transfer from a cylinder in cross-flow of power-law fluids. Int J Heat Mass Transf 2010;53:4728–40. [28] Chandra A, Chhabra RP. Influence of power-law index on transitional Reynolds numbers for flow over a semi-circular cylinder Appl. Math Model 2011;35:5766–85. [29] Sanyal A, Dhiman A. Wake interactions in a fluid flow past a pair of side-by- side square cylinders in presence of mixed convection. J Phys Fluid 2017;29:315–24. [30] Yang K, Zhao X, Kawamura R. From bead to rod: Comparison of theories by measuring translational drag coefficients of micron-sized magnetic bead-chains in Stokes flow. PLoS ONE 2017;17, http://dx.doi.org/10.1371/journal.pone.0188015. [31] Noferesti S, Ghassemi H, Nowruzi H. Numerical investigation on the effects of obstruction and side ratio on non-Newtonian fluid flow behaviour around a rectangular barrier. J Appl Math Comput Mech 2019;18:53–67. [32] Paliwal B, Sharma A, Chhabra RP, Eswaran V. Power-law fluid flow past a square cylinder: momentum and heat transfer characteristics. Chem Eng Sci 2003;58:5315–29. [33] Shahzadi I, Ahsan N, Nadeem S, Issakhov A. Analysis of bifurcation dynamics of streamlines topologies for pseudoplastic shear thinning fluid: biomechanics application. Physc A Stat Mech Appl 2020;540, http://dx.doi.org/10.1016/j.physa.2019.122502. [34] Shahzadi I, Nadeem S. Impact of curvature on the mixed convective peristaltic flow of shear thinning fluid with nanoparticles. Canad J Phys 2016;94, http://dx.doi.org/10.1139/cjp-2015-0508. [35] Nadeem S, Shahzadi I. Inspiration of induced magnetic field on nano hyperbolic tangent fluid in a curved channel. AIP Adv 2016;6, http://dx.doi.org/10.1063/1.4940757. [36] Nadeem S, Shahzadi I. Mathematical analysis for peristaltic flow of two phase nanofluid in a curved channel. Commun Theor Phys 2015;64, http://dx.doi.org/10.1088/0253-6102/64/5/547. [37] Kwon YW, Arceneaux SM. Experimental study of channel driven cavity flow For fluid structure interaction. J Pressure Vessel Tech 2017;139:034502. [38] Armaly BF, Durst F, Pereira JCF. Experimental and theoretical investigation of backward-facing step flow. J Fluid Mech 1983;127:473–96. [39] Ghia U, Ghia K, Shin C. High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. J Comput Phys 1982;48:387–411. [40] Botella B, Peyret R. Benchmark spectral results on the lid driven cavity flow. Comput Fluids 1998;27:421–33.
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Original Article
CFD analysis for characterization of non-linear power law material in a channel driven cavity with a square cylinder by measuring variation in drag and lift forces Rashid Mahmood a , S. Bilal a , Afraz Hussain Majeed a , Ilyas Khan b,∗ , Kottakkaran Sooppy Nisar c a
Department of Mathematics, Air University Sector E-9 AIR Complex, P. O. 44000, Islamabad, Pakistan Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, 72915, Vietnam c Department of Mathematics, College of Arts and Sciences, Prince Sattam bin Abdulaziz University, Wadi Aldawaser, 11991, Saudi Arabia b
a r t i c l e
i n f o
a b s t r a c t
Article history:
Current communication is manifested to investigate flowing features of power law mate-
Received 11 December 2019
rial in a newly proposed physical configuration namely the channel driven cavity. Since
Accepted 3 February 2020
power law fluid discloses the dynamical features of shear thinning, shear thickening and
Available online xxx
Newtonian materials so in present communication it is considered for depiction of flow
Keywords:
below the channel. The flow is induced with parabolic inlet velocity and a Neumann condi-
attributes. To achieve the desired outcomes from the work, a unit length cavity is placed Finite element method
tion is applied at the outlet, while no slip condition is set at all other boundaries. A square
Power law material
cylinder is placed in the channel with varying positions giving rise to three computational
Drag and lift coefficients
grids named as G1 , G2 and G3. Mathematical modelling is constructed by obliging funda-
Channel-driven cavity
mental conservation and rheological laws for power law fluid. Since the representative equations are complex in nature so an efficient computational procedure based on finite element method (FEM) is executed. A hybrid computational grid is generated at coarse level and then further refinement is done to improve the accuracy of the solution. The solution is approximated by adopting P2 − P1 element based on second and first order polynomial shape functions. Graphical trends against involved parametric variables are adorned. In addition for more physical insight of problem velocity and pressure plots and line graphs are added. Furthermore, the hydrodynamical benchmark quantities like pressure drop, drag and lift coefficients are evaluated in tabular form around the outer surface of obstacle. © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
∗
Corresponding author. E-mail: [email protected] (I. Khan). https://doi.org/10.1016/j.jmrt.2020.02.010 2238-7854/© 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: Mahmood R, et al. CFD analysis for characterization of non-linear power law material in a channel driven cavity with a square cylinder by measuring variation in drag and lift forces. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.02.010
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Nomenclature # EL # DOF p ıp pA pB ui
Number of Elements Number of degrees of freedom Dimensionless pressure Pressure drop Pressure at point A of obstacle Pressure at point B of obstacle Density of the fluid Dimensionless velocity components
Shear rate Mean inflow velocity Maximum inflow velocity Power law index Consistency index Drag coefficient Lift coefficient
•
Umean Umax n m CD CL
1.
Introduction
The flow in a space generated between two parallel surfaces with relative tangential velocity is called channel flow. Research concerning with dynamics of channel flow has attained superb attention of researchers due to its extraordinary applications in multiple disciplines. Such types of flows are utilized in hydraulics, aviation, chemical engineering, process engineering [1–6]. Flow in a channel is generated either by generating pressure gradient or by drag force. The fluid executed by effective pressure difference in a channel is termed as Poiseuille flow and was identified by Jean Poiseuille in 1938. Comprehensive work on Poiseuille flow was by various researchers which can be accessed through [7–11]. Later on researchers tried to work on other factors that create the fluid flow in a channel. A comprehensive literature review regarding fluid flow in a channel under various physical constraints is enclosed in references [12–16]. Interpretation about fluid flow in confined enclosures has intended fluid dynamists towards itself due to its technoscientific utility in real life processes. Aidun et al. [17] pointed the direct relevancy of cavity flows to coaters and melting spinning processes which are used to manufacture microcrystalline materials. Zumbrunnen et al. [18] disclosed the importance of eddies structures formation in cavity in drag reducing riblets and synthetization of fine polymer composites. Initially fluid experts thought that flow generation inside cavity is produced by moving the walls with prescribed velocity or making some thermal change through boundaries. For the sake of interest of readers few recent investigations regarding fluid flow in cavities are engrossed in Refs. [19–21]. Interaction of fluid with solid structures has received an overwhelming attraction during last few decades due to pervasive implications in heat exchanger tubes, cooling systems for nuclear power plants, transmission cables, bridges, high rise buildings, and electronic equipment. After getting view about these procedural applications experts are taking keen interest due to the fact that these are capitalized to mount the lift and to demise the drag. The square shaped obstacle
among these different structures of cylinders serve as fundamental component of design and structures. These structures often interact with fluids and experiences flow induced forces which lead to the failure under certain unfavourable circumstances. In this regard premier work on obstacle flow was done by Schaefer and Turek [22] in which they explored benchmark problem for incompressible flow around a cylinder and constructed comparison between findings made with different solutions approaches. Initially viscous flow around obstacles was considered but due to technological changes across the frontiers of industrial work, non-Newtonian fluids have gained considerable attention. Such fluids are defined on the basis of their rheological pattern. Fluid experts interprets that the models disobeying Newton law of viscosity and expressing non-linear relation between deformation rate and shear stress are called as non-Newtonian fluids. For better physical interpretation they characterize them into fundamental three classes are i) shear thinning (Pseudoplastic) ii) shear thickening (Dilatant) iii) thixotropic fluids. On the basis of this diversification various fluid models have been proposed. Among these Ostwald-de Waele model commonly known as power law fluid is one of the generalized model that expresses the properties of shear thinning and thickening fluids. Shear thinning and thickening materials in channel and cavities are interrogated abundantly due to their potential applications like multiphase mixtures, i.e., foams, suspensions, emulsions, etc. and high molecular weight polymeric systems, i.e., solutions, melts, blends, etc. exhibit shear-thinning(pseudoplastic fluids, n < 1) and/or shear-thickening (dilatant fluids, n > 1). Some recent available literature is discussed here. Computational finite element analysis emphasizing the roles of Reynolds number and power law index on the global and detailed flow characteristics of shear thinning fluid demarcated in flow regime across obstacle was elucidated by Coelho et al. [23]. Chhabra et al. [24] investigated steady and incompressible flow of power law fluid past a circular cylinder by implementing finite difference scheme for different magnitudes of Reynold number 1, 20 and 40. Sivakumar et al. [25] studied the flowing features of power-law fluid past an elliptic cylinder and examined the behaviour of shear-thinning (n < 1) and for shear-thickening (n > 1). Patnana et al. [26] investigated uplift in drag coefficient against fixed magnitude of Reynold number and power law index in incompressible generalized Newtonian fluid across an unconfined circular cylinder. Power law fluid across circular cylinder to check the dependence of individual drag coefficient on fluid model parameter was adumbrated by Soares et al. [27]. Chandra and Chhabra [28] examined the influence on flow separation around surface of cylinder immersed in power law fluid across a semi-cylinder and under transitional Reynolds numbers for the flow separation over the surface of cylinder. Sanyal and Amit [29] measured variation in drag and lift coefficients over the outer surface of cylinder at low Reynold number within range of 0 ≤ Re ≤ 50. The measurement about depending relationship of translational velocity with drag coefficient in case of power law fluid was studied by Yang et al. [30]. Noferesti et al [31] performed analysis on characteristics of generalized Newtonian fluid over a quadrilateral cylinder and found increment in drag coefficient by raising aspect ratio of cylinder. In recent years Paliwal et al. [32] have scrutinized flow of power law
Please cite this article in press as: Mahmood R, et al. CFD analysis for characterization of non-linear power law material in a channel driven cavity with a square cylinder by measuring variation in drag and lift forces. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.02.010
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liquids past a square cylinder confined between slip boundaries within the range of5 ≤ Re ≤ 40. Analysis on bifurcation of stream lines by considering shear thinning fluid model inside channel as well as through axisymmetric tube was executed by Shahzadi et al. [33]. They utilized methodology of dynamical systems to inspect the local bifurcation and changes in the topologies. Discussion about mixed convection peristaltic flow of shear thinning tangent hyperbolic fluid under the inclusion of nanoparticles in a curved channel was manifested by Shahzadi and Nadeem [34]. Nadeem and Shahzadi [35] investigated the impact of thermophoresis and Brownian movement on peristaltic motion of shear thinning hyperbolic fluid in a curved channel. Nadeem and Shahzadi [36] conducted theoretical analysis for peristaltic motion of water base nanofluid containing distinct types of the nanoparticles like Cu, TiO2 , and Al2 O3 and by modelling the equations under low Reynolds number as well as long wave length supposition. From the above mentioned literature it is seen that Power law fluid in channel and cavity are discussed separately in various physical configurations and constraints. But the thread to this communication is to introduce a newly innovated combined physical configuration containing channel and cavity known as channel driven cavity. This manuscript is fabricated after getting motivation form the experimental study conducted by Kwon and Arceneaux [37–40]. This newly proposed sketch is highly utilized in pressure vessel and hydraulic industry. One of the most impressing features of channel driven cavity is that it covers various benchmark problems in it like flow in cavity, flow in a channel, forward and backward facing step processes, contraction and expansion phenomenon. So the development and fabrication of this configuration will serve as benchmark study for all type of problems and validate the previously computed results.
3
Fig. 1 – Schematic diagram of problem.
Fig. 2 – A P2 − P1 finite element pair-Location of degrees of freedom.
2. Mathematical formulation and viscosity model
at u = v = 0 (no slip boundaries by setting). A parabolic inlet velocity governed by relation with Umax = 0.3 is injected to the channel and a do nothing boundary condition at the outlet is chosen. A very fine hybrid grid is generated to improve the accuracy of the solution, and excessive number of elements around the obstacle is gathered for better findings. Power law fluid is considered representing the properties of shear thinning, shear thickening and Newtonian fluids. Square cylinder with centres positioned at (0.9, 1.5), (1.5, 1.5) and (2.1, 1.5) are considered. The Reynold number chosen here are Re = 20 and Re = 50.
The conservation law for steady, incompressible fluid flow are expressed as under
4.
∂ui = 0, ∂xi
(1)
∂ 1 ∂p 1 ∂ (u u ) + = ∂xj i j ∂xi ∂xj
() ˙
∂uj ∂ui + ∂xi ∂xj
, i, j = 1...3,
(2)
Relation representing viscosity variation in power law fluid with shear rate is represented as below () ˙ = m() ˙
n−1
,
(3)
where m is the fluid consistency coefficient; n is the power law index; ˙ is the shear rate.
3.
Physical configuration
Consider a channel driven cavity shown schematically in Fig. 1. The cavity of unit dimension is adjusted on lower side of channel. The upper and lower walls of the channel are set
Methodology
Mathematical modelling of most of engineering problems is highly complexed in nature so extraction of solution is also tough task. In this regards various computational approaches are available in literature [38–41] but each method has restrictions. Among these methods the most renowned finite element method is employed to narrate the flowing features of fluid in channel driven cavity. In most of nonstructural flow domains finite element method is capitalized because a large system is firstly divided in to finite number of smaller and simpler parts which are known as finite elements. In this direction the conforming element pair P2 − P1 is selected for the velocity and pressure approximations. The location of degrees of freedom for this finite element pair is shown in Fig. 2. This pair has 15 local degrees of freedom for problem at hand. This element is a stable element pair satisfying inf–sup condition. After the discretization of domain into smaller elements algebraic system of equations is constructed and Newton’s method is applied to solve discrete non-linear algebraic systems and the inner linear sub prob-
Please cite this article in press as: Mahmood R, et al. CFD analysis for characterization of non-linear power law material in a channel driven cavity with a square cylinder by measuring variation in drag and lift forces. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.02.010
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Table 1 – Number of degree of freedom at different refinement levels. Refinement level
# of elements
# of Dof
1 2 3 4 5 6 7 8 9
769 1167 1786 2962 3590 6707 16,139 39,724 52,844
4032 6130 9245 15,171 21,648 33,213 78,925 191,180 250,024
lems are solved using a direct solver. The convergence criteria for the nonlinear iteration is set as
n+1 − n −6 n+1 < 10 ,
where represents the general solution component (Table 1). The computational grids at coarse level for three different setting of obstacles are shown in Fig. 3 named as G1, G2, G3 . The process of mesh refinement is a key step in validating any finite element model and to raise the trust of reader on attained physical outcomes regarding the work. The main theme of finite element scheme is to divide the domain into subdomains called elements and to solve the governing equations locally in each element. In view of the importance of discretization of fluid domain into finite elements meshing at multiple levels of refinements is performed but here only coarse level is shown.
5.
Results and discussion
Fig. 4(a–c) illuminates momentum distribution by varying power law index (n) from 0.5 to 1.5 and by restricting Re = 20. Here power law flow shows shear thinning behavior for n = 0.5, Newtonian fluid for n = 1 and shear thickening fluids for n = 1.3, 1.5. Since parabolic velocity is induced at inlet and other boundaries are kept at no slip conditions so variation in velocity near obstacles and other portions of channel driven cavity is observed. It is concluded that similar parabolic velocity profile is exhibited at outlet and additionally circulating flow in cavity is generated and formation of vortices takes place. Fig. 5(a–c) is plotted to measure change in pressure throughout the physical domain and especially in vicinity of obstacle placed at three different locations i.e. for grids G1, G2 and G3 by varying (n) within the range of (0.5 ≤ n ≤ 1.5) and by restricting Re = 20 and Re = 50. It is seen from the portrays that pressure shows non-linear behavior near the obstacle and becomes linear along the downstream as expected in channel flow. Here observation about optimum pressure is made that it has maximum value at face of obstacle which is interacting with fluid. Fig. 6(a–c) represents line graphs to express the variation in velocity at different portions of physical configuration. From the above sketches it is seen that at inlet perfect parabolic plateau is achieved because viscosity has no active role on injected velocity. Whereas, the fluid flow with in the channel changes its behavior near obstacle and above the cavity. In
Fig. 3 – The computational grids at coarse level.
addition it is also explicated that velocity of fluid decreases by increasing power law (n) because the viscosity of fluid enriches and more resistance is offered to fluid. Tables 2a–2c interprets variation in pressure across the obstacle located at three different positions in channel like at centre of G1 , G2 , G3 for increasing magnitude of power law index (n) and by restricting Re = 20 and Re = 50. From the attained numerical data it is enumerated that by increasing (n) pressure drop increments. As power law fluid behaves as shear thinning fluid for (n < 1) and become Newtonian fluid at (n = 1) and elucidates the properties of shear thickening fluid so by increasing the magnitude of (n) the viscosity of power law
Please cite this article in press as: Mahmood R, et al. CFD analysis for characterization of non-linear power law material in a channel driven cavity with a square cylinder by measuring variation in drag and lift forces. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.02.010
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Fig. 4 – a): velocity profile for G1, at Re = 20 for various n. b): velocity profile for G2, at Re = 20 for various n. c): velocity profile for G3 at Re = 20 for various n.
fluid intensifies and strikes the obstacle with more strength and thus pressure drop mounts. Variation in pressure gradient is also evidenced by the numeric data attained that for shear thinning case i.e. n varying from (0.5 to 0.9) the pressure gradient approaches to 0.0923169010721 at obstacle location G1 and for shear thickening case i.e. n varying form (1.1 to
1.5) the pressure gradient approaches to 0.1066536252091 at obstacle location (0.9, 1.5) for Re = 20. It is also observed that by changing the position of obstacles and by varying power law index (n) the pressure gradient varies. The manipulated variation is proved form tabular values that at n = 0.5 i.e. shear thinning case for restriction Re = 20 and at location (0.9, 1.5) the
Please cite this article in press as: Mahmood R, et al. CFD analysis for characterization of non-linear power law material in a channel driven cavity with a square cylinder by measuring variation in drag and lift forces. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.02.010
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Fig. 5 – a): Pressure profile for G1 at Re = 20 for various n. b): Pressure profile for G2 at Re = 20 for various n. c): Pressure profile for G3 at Re = 20 for various n.
Table 2a – Pressure gradient at centre of G1 for various n. n 0.5 0.7 0.9 1 1.1 1.3 1.5
Re = 20 ıp = pB − pA
Re = 50 ıp = pB − pA
0.0828604860287 0.0875478576104 0.0923169010721 0.0946981050108 0.0970734765920 0.1018252633472 0.1066536252091
0.0667172348720 0.0681923622698 0.0712711286253 0.0729454316371 0.0746737862482 0.0782369401251 0.0818976603145
Table 2b – Pressure gradient at centre of G2 for various n. n
0.5 0.7 0.9 1 1.1 1.3 1.5
Re = 20 ıp = pB − pA
Re = 50 ıp = pB − pA
0.0781110642119 0.0835134542417 0.0884731804764 0.0907965522813 0.0930166504224 0.0971431850531 0.1008965588128
0.060473759292558 0.062975223522683 0.066025509148481 0.067540127154250 0.069033824796297 0.071935602822738 0.074701729700852
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Table 2c – Pressure gradient for centre of G3 for various n. n
0.5 0.7 0.9 1 1.1 1.3 1.5
Re = 20 ıp = pB − pA
Re = 50 ıp = pB − pA
0.078875015959165 0.084948175601868 0.090939257630179 0.093886809719276 0.096799896069806 0.102543398288502 0.108245801052848
0.07497012329253 0.07546257422997 0.07857883957756 0.08031802033540 0.08209801458882 0.08566981906243 0.08918381056459
pressure drop is 0.0828604860287 whereas in similar physical parameter restriction but at location G2 the pressure change is at 0.0781110642119 and at obstacle positon G3 variance is 0.0781110642119. These depicted values shows that near the inlet pressure is maximum and above the cavity pressure drops with less intensity and then against lift up after the crossing over of fluid from cavity. Tables 3a–3c represents fluctuation in benchmark hydrodynamic quantities like drag and lift coefficients on outer surface of square obstacle placed at G1 , G2 , G3 . It is found that by increasing power law index drag and lift coefficients upsurge whereas reverse behavior in both these hydro dynamical forces against uplift in Reynold number (Re). Negative values of lift coefficient ( CL ) shows that lift forces in upward direction are dominant. The reason behind negative values of lift coefficient is that the obstacles are placed in such a way that fluid fell into the cavity and generates upward thrust on the obstacle thus it generates such numerical pattern. From the calculated data about the drag coefficient at G1 , G2 and G3 . It is deduced that drag has maximum magnitude of 3.7728996641145 at G1 and by fixing n = 0.5 and Re = 20 where as it reduces ne ar G2 in the situation when obstacle is placed over the cavity having magnitude of 3.42145328808411 and again increases at G3 attaining values to 3.7720752163646. Whereas contrasting behavior for lift coefficient is addressed in such a way that lift forces at G1 has lowest magnitude of 0.0595685094797372 but at obstacle location G2 upraise in the values of lift coefficient scaled up to 0.131684889330373 is found and exceeds to 0.148163843109242 at G3 . Here negative coefficients are measured when obstacle is placed near entrance and over cavity location because upward lift forces are dominant and positive coefficients are attained at G3 because downward lift forces are active.
6.
Fig. 6 – a): Cutline graph for G1 at Re = 20 for various n. b): Cutline graph for G2 at Re = 20 for various n. c): Cutline graph for G3 at Re = 20 for various n.
Conclusions
Present analysis addresses flow features of generalized nonNewtonian fluid model in a newly proposed channel driven cavity with square obstacle placed at different locations. Physical problem is converted into mathematical formulation in terms of two dimensional Navier stokes equation along with boundary constraints. A well reputed computational method known as finite element method is obliged to report solution
Please cite this article in press as: Mahmood R, et al. CFD analysis for characterization of non-linear power law material in a channel driven cavity with a square cylinder by measuring variation in drag and lift forces. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.02.010
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Table 3a – Drag and Lift coefficient for various n at centre of G1 . Re = 20
n
0.5 0.7 0.9 1 1.1 1.3 1.5
Re = 50
CD
CL
CD
CL
3.7728996641145 4.39734366354035 5.06524900959966 5.41036840362092 5.76189873503173 6.48436659610317 7.23835554996017
−0.0595685094797372 −0.0982655193149808 −0.126331547935937 −0.139383783328286 −0.152376171231296 −0.179346101202571 −0.208594154852365
3.13559638538286 3.27601270221956 3.40831687113724 3.60938018927641 3.8242188137267 4.28052787519498 4.76085165782894
0.080429028303184 0.0199484099864073 −0.0213180374297233 −0.0381682604247344 −0.0522645659953361 −0.0752052654851581 −0.0946279098319215
Table 3b – Drag and Lift coefficient for various n at centre of G2 . Re = 20
n
0.5 0.7 0.9 1 1.1 1.3 1.5
Re = 50
CD
CL
CD
CL
3.42145328808411 3.94249969109768 4.47115320125086 4.73294084643189 4.99109332511408 5.49367373603658 5.97625443204032
−0.131684889330373 −0.188402284884305 −0.244004274207543 −0.27296816440631 −0.303086293886686 −0.367692822712802 −0.438927745734126
2.73584762852872 2.80285467590667 3.10838039593571 3.27997812518368 3.45764447194587 3.81969512462804 4.18057223905749
0.0707420862086769 −0.0059892212984909 −0.0554138185168249 −0.0788924001693645 −0.100483678955644 −0.140253781064674 −0.178795536650709
Table 3c – Drag and Lift coefficient for various n at centre of G3 . Re = 20
n
0.5 0.7 0.9 1 1.1 1.3 1.5
Re = 50
CD
CL
CD
CL
3.7720752163646 4.43394249262315 5.1421025531268 5.50943775652902 5.88434297328834 6.65537502041482 7.45799797881548
0.148163843109242 0.18283312337039 0.212373573471148 0.227081667591816 0.241935828009557 0.272146175182898 0.303438489023832
2.90913902124884 3.03515112042785 3.42710065642627 3.65010100634513 3.8842833042712 4.37485986732887 4.88522236953334
−0.0876392972574064 0.00456305137305655 0.0497091049331152 0.0670300669690129 0.0816464735626635 0.106574494980338 0.12886105155848
and physical happening. The obtained outcomes are offered via graphical trends. Velocity and pressure plots at various locations within the configuration are manifested. Bar lines representing the behavior of momentum distribution of power law fluid at inlet, obstacles and near outlet regions are illuminated. Tabular data regarding the pressure variation and drag and lift coefficients on the outer surface of obstacles is enumerated. The key outcomes are listed below. i) Pressure drops with more intensity as the magnitude of power law index uplifts. ii) Placement of obstacle effects the pressure drop by first increasing at G1 , decreases at G2 and eventually boost at G3 . iii) Reynold number has decrementing attribute on pressure constraint difference in vicinity of square obstacle. iv) Drag and lift coefficients have low magnitude for shear thinning case of Power law fluid in comparison to shear thickening version of the fluid. v) Negative values of lift coefficient are attained when upward forces are dominant whereas positive coefficients are accomplished when downward forces exceeds.
vi) By increasing Power law index velocity profile delineates and evidenced by drawing cut lines at various location in configurations. vii) Pressure has stagnant magnitude at obstacle side where fluid in interacting with it and at corners of cavity. viii) It is found that by replacing n < 1 power law model presents class of shear thinning materials and for n > 1 the shear thickening model representation is achieved. For n = 1 the model reduces to Newtonian fluid with m as the constant viscosity.
Conflict of interest The authors declare no conflicts of interest.
Appendix A. Supplementary data Supplementary material related to this article can be found, in the online version, at doi:https://doi.org/10.1016/j. jmrt.2020.02.010.
Please cite this article in press as: Mahmood R, et al. CFD analysis for characterization of non-linear power law material in a channel driven cavity with a square cylinder by measuring variation in drag and lift forces. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.02.010
JMRTEC-1375; No. of Pages 9
ARTICLE IN PRESS j m a t e r r e s t e c h n o l . 2 0 2 0;x x x(x x):xxx–xxx
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Please cite this article in press as: Mahmood R, et al. CFD analysis for characterization of non-linear power law material in a channel driven cavity with a square cylinder by measuring variation in drag and lift forces. J Mater Res Technol. 2020. https://doi.org/10.1016/j.jmrt.2020.02.010