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ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 2020, VOL. 14, NO. 1, 1373–1384 https://doi.org/10.1080/19942060.2020.1830857 Heat transfer through converging-diverging channels using Adomian decomposition method a b c d e,f Hayette Saifi , Mohamed Rafik Sari , Mohamed Kezzar , Mahyar Ghazvini , Mohsen Sharifpur and Milad Sadeghzadeh a b Laboratory of Inorganic Materials Chemistry, University BadjiMokhtar of Annaba, Annaba, Algeria; Laboratory of Industrial Mechanics, c d University BadjiMokhtar of Annaba, Annaba, Algeria; Mechanical Engineering Department, University of Skikda, Skikda, Algeria; Department of Ocean and Mechanical Engineering, Florida Atlantic University, Boca Raton, FL, USA; Institute of Research and Development, Duy Tan University, Da Nang 550000, Vietnam; Department of Mechanical and Aeronautical Engineering, University of Pretoria, Pretoria, South Africa; Department of Renewable Energy and Environmental Engineering, University of Tehran, Tehran, Iran ABSTRACT ARTICLE HISTORY Received 23 May 2020 This study consists of proposing a new mathematical method to develop a new model for evalu- Accepted 21 September 2020 ating thermal distributions throughout convergent-divergent channels between non-parallel plane walls in Jeffery Hamel flow. Subsequently, dimensionless equations that govern temperature fields KEYWORDS and velocity are numerically tackled via the Runge–Kutta-Fehlberg approach based on the shoot- Fluid ﬂow; heat transfer; ing method. Additionally, an analytical study is performed by applying an effective computation inclined walls; Adomian technique named Adomian Decomposition Method. Determining the effect of Reynolds and Prandtl Decomposition method; numerical solution numbers on the heat transfer and fluid velocity inside converging/diverging channels can be men- tioned as the fundamental purpose of this research. Based on the results obtained for dimensionless velocity and thermal distributions, a supreme match can be observed between numerical and analytical results indicating the adopted ADM method is valid, applicable, and has great precision. Nomenclature T Temperature, Kelvin Symbol Den fi ition T Ambient temperature, Kelvin V Radial velocity, m/s a Constant V Maximal velocity, m/s max b Constant V Aziumuthal velocity, m/s dThermaldiffusivity V Axial velocity, m/s A Adomian polynomials Greek Symbols B Adomian polynomials η Dimensionless angle Cp Specificheat of studied u fl ids, J/Kg.°K α Channel half-angle, ° f Function φ Viscous dissipation F Dimensionless velocity ρ Fluid density, Kg/m F Solution terms for velocity ν Kinematic viscosity, m /s gFunction µ Dynamic viscosity, Pa.s G Dimensionless temperature ε , ε , ε Strain tensor components rr θθ rθ G Solution terms for temperature Subscripts K Thermal conductivity, W/m.°K r Radial coordinate, m Nu Nonlinear velocity θ Angular coordinate, m Ng Nonlinear temperature z Axial coordinate, m P Fluid pressure,N/m Pr Prandtl number Operators Q Volume u fl x ∂ Derivative operator r Radial coordinate, m L Linear operator Re Reynolds number N Nonlinear operator CONTACT Mohsen Sharifpur [email protected], [email protected]; Milad Sadeghzadeh [email protected] This article has been republished with minor changes. These changes do not impact the academic content of the article. © 2020 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1374 H. SAIFI ET AL. 1. Introduction uid fl between stretchable convergent-divergent channels. Fluid flow throughout a non-parallel channel can be enu- Obtained nonlinear ODEs have been solved numeri- merated as a member of practicable casesusedin vari- cally and analytically with Runge–Kutta scheme and ous applications including chemical, mechanical, biome- Homotopy analysis method, respectively. Energy trans- chanical, civil, and environmental engineering (Gho- fer of Jeffery-Hamel nanouid fl flow between inclined lami et al., 2015; Ramezanizadeh et al., 2019;Xuetal., walls employing Maxwell-Garnetts and Brinkman mod- els was treated analytically by Li et al. (2018)via Galerkin 2012). Thus, understanding the flow in this kind of method. Recently, the eeff ct of magnetic efi ld on u fl id channel to solve engineering problems is extremely cru- flow and heat transfer characteristics has gained much cial (Gao et al., 2018;Goldbergetal., 2010). In this attention and studied extensively by many researchers. way, in the past few years, different numerical and On the other hand, Mahmood et al. (2019) also analyt- experimental studies have been carried out by many ically investigated the thermal performance of a steady researchers to obtain knowledge regarding the flow in two-dimensional incompressible viscous uid fl through- channels and cavities (Baghban et al., 2019;Zaji& out convergent-divergent channels by Spectral Homo- Bonakdari, 2015). Initially, Jeeff ry ( 1915)developed the topy Analysis method (S-HAM) influenced by a trans- celebrated radial 2D flow of an incompressible viscous u fl id through convergent or divergent channels. The verselymagneticfield. renowned Jeffery-Hamel flow is of paramount impor- Over the past few decades, several semi-analytical tance since this is greatly regarded as a member of the methods (Adomian & Adomian, 1994;He, 2003;He& unique exact solution of the Navier-Stokes equation. Due Wu, 2007;S.Liao, 2003;S.J.Liao&Cheung, 2003;Tatari to their considerable importance for many engineering &Dehghan, 2007; Wazwaz, 2000)weredevelopedandex- applications, flows through convergent-divergent chan- tensively used for understanding an extensive range of nels havegained much attention and studied extensively nonlinear initial or boundary-value problems. Among by several researchers. In fact, thermal distributions these methods, Adomian decomposition approach (Ado- between non-parallel plane walls using n fi ite difference mian & Adomian, 1994) introduced by Georges Adomian is highly considered as a member of rigorous methods for method are given by Millsaps and Pohlhausen (Mill- solvingmanyproblemswhichisaquickconvergentseries saps & Pohlhausen, 1953). Eagles (1966)investigated with favorably computable terms without linearization or the stability associated with Jeeff ry-Hamel solutions for discretization. In fact, the ADM approach has attracted divergent channel flow with the use ofresolving the well- researchers’ community to use it for solving many uid fl known Orr-Sommerfeld problem. The temporal stability dynamics problems (Abbasbandy, 2007; Alizadeh et al., of Jeffery-Hamel flow was investigated (Hamadiche et al., 2009; Alizadeh et al., 2009;Gheriebet al., 2020; Kezzar 1994). In this investigation, the critical Reynolds num- &Sari, 2017; RamReddy et al., 2017;Reddy,etal., 2017; bers are computed with respect to the axial velocity and ShakeriAskietal., 2014). volume flux. Uribe et al. ( 1997)analyzedthe tempo- This research aims at modeling and simulating ral and linear stability contributed to several flows for small-width channels via Galerkin approach. Zaturska thermal distributions for different uids fl through Jef- and Banks (2003)presentednewflowsresulting from fery–Hamel flows between non-parallel walls. As a rfi st vortex stretching and mainly created by the renowned step, a new heat transfer model is developed in Jef- Jeffery-Hamel flows. Additionally, this investigation con- fery–Hamel flow. Thereafter, dimensionless velocity and sideredtheinufl encesofconnfi ingside-wallsonanespe- thermal equations arising from mathematical modeling cial Jeeff ry-Hamel flow. Moradi et al. ( 2013)studied are solved analytically and numerically utilizing the Ado- analytically and numerically via differential transforma- mian decomposition method and Runge–Kutta-Fehlberg tion method (DTM) and Runge–Kutta scheme (RK4) based on the shooting approach, respectively. respectively the nonlinear problem of Jeeff ry-Hamel in After establishing and validating the computing code, a nanofluid using several types of solid nanoparticles. it is used to study the velocity and temperature distri- Turkyilmazoglu (2014) extended the classical Jeffery- butions for different working u fl ids such as steam, liq- Hamel flow in convergent/divergent channels in which uid metal, air, and water flowing through convergent or thestationarywallscan stretchorshrink.Theobtained divergent channels. This study mainly shows the eeff ct results reveal which the stretching/shrinking walls can of Reynolds and Prandtl numbers on the studied Jef- significantly aect ff the traditional flow and heat transfer. fery–Hamel flow. Also, acomparison between ADM and Khan et al. (2017)interestedintheSoretand Dufour numerical outcomes is provided for the aim of testing the eecti ff veness of the ADM technique. influences on the Jeeff ry-Hamel flow of second-grade ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 1375 Noted that any constant section, r = cte,istraveledby 2. Governing equations the same quantity Q. 2.1. Hydrodynamical problem For the studied Jeffery–Hamel flow, uids fl can radi- ally move outwards (i.e. in the diverging channel, Consider the steady two-dimensional Jeffery-Hamel flow Q > 0) or inwards (i.e. in converging channel, Q < 0). between non-parallel plane walls. Figure 1 shows a Also, it is well established that the maximal u fl id velocity schematic of the geometry studied in this research. In fact, uniform flow in the z-direction and entirely is undoubtedly provided at θ = 0. In fact, we obtain: radial motion are assumed. It can be written: (V = r.V = f (0) (8) max V(r, θ); V = V = 0). θ z In vector form, the continuity and Navier-Stokes equa- Consequently, f (0) ≥ f (θ ) in the range −α ≤ tions for Jeffery-Hamel flow are expressed as: θ ≤+α. Now introducing the dimensionless parameters ∇.V =0(1) r.V f (θ ) F(η) = = ,(9) ρ[(V∇)V] =−∇P + μ. V (2) r.V f max max The Navier-Stokes equations can be provided as fol- where: η = θ/α with: −1 ≤ η ≤+1 lows for cylindrical coordinates (r,θ,z): Considering the kinematic uid fl viscosity, ν,the dimensionless quantities of the net volume ux fl (Equation ρ ∂ . (rV ) =0(3) r (7)) may be expressed as follows: r ∂r ∂V 1 ∂P r Q V . =− . + ν. = R . F(η)dη (10) ∂r ρ ∂r −1 2 2 ∂ V 1 ∂V 1 ∂ V V r r r r Where the Reynolds number can be introduced like: × + . + . − 2 2 2 2 ∂r r ∂r r ∂θ r rV α f .α (4) max max R = = ν ν 1 ∂P 2.ν ∂V − . + . =0(5) V > 0, α> 0Divergentchannel ρ.r ∂θ r ∂θ max V < 0, α< 0Convergentchannel max where: V :radialvelocity; ρ:density; ν: kinematic viscos- (11) ity; P: u fl id pressure. In accordance with Equation (3), the quantities (r.V ) f expresses the velocity at the channel’s centerline, max which are mainly dependant on θ can be provided below: and, α indicates the channel half-angle. It is also worth noting that (α.r) measures the width of the channel. r.V = f (θ ) (6) By removing the pressure terms in Equations (4) and (5), it can be obtained: Itiswell knownthatthenetvolumefluxinto thechannel from thesourceattheorigin isgivenas: F + 2R αFF + 4α F = 0 (12) +α Q = r.V dθ (7) The boundary conditions associated with the Jeffery- −α Hamel flow in terms of F(η) can be stated like: F(η) = 1, F (η) = 0, (13) atthecenterlineofchannel F(±η) = 0, (14) at the body of channelAccording to Batchelor [40], the Reynolds number based on the center-plane u fl id velocity V , gives a direct measure of the flow intensity max compared to the Reynolds number based on the ux: fl Q 1 Q 1 R = = = R F(η)dη (15) 2ν 2 ν 2 Figure 1. Geometry of Jeﬀery-Hamel ﬂow. −1 1376 H. SAIFI ET AL. The X term in Equation (22) can be determined 2.2. Heat transferin Jeffery–Hamel flow below: In the previous section, the governing Equations (3)–(5) 2 2 R .ν mainly serve to achieve the hydrodynamical solution of e X = (23) α .C the investigated flow inside convergent-divergent chan- nels. For the aim of obtaining the thermal distribution The function G(η) represents the dimensionless thermal in Jeffery-Hamel flow, we introduce the solution of the distribution. hydrodynamical part into the energy equation. In fact, In terms of G(η), the boundary conditions of heat the energy equationis given as: transfer problem are given as: ρ.c (V∇)T = φ + K.∇ T (16) p G (0) = 0, (24) atthecenterlineofchannel where K and c indicate thermal conductivity and the specicfi heat at constant pressure, respectively. G(±η) = 0, (25) Additionally, φ expresses the viscous dissipation term at the body of channel which is provided below: 2 2 2 φ = μ · [2ε + 2ε + ε ] (17) rr θθ rθ 3. Application of Adomian decomposition method to the Jeffery–Hamel problem The components of the strain tensor can be determined by: In this study, the set of nonlinear dieff rential Equations ∂V (12) and (22) under the defined boundary conditions ε = rr ⎪ ∂r (13), (14), (24) and (25) were solved analytically utilizing ∂V V 1 θ r (18) ε = . + θθ the Adomian decomposition method. r ∂θ r 1 ∂V ∂V V It isworthnotingthatfor solvingtheenergyequation r θ θ ε = . + − rθ r ∂θ ∂r r which governs the heat transfer in Jeffery-Hamel flow, it By taking into account the following transformations: is necessary as a rfi st step to solve the hydrodynamical part of the studied flow. G(θ ) T − T = (19) 3.1. Hydrodynamical problem In which T indicates the ambient temperature. The AccordingtotheAdomianalgorithm,Equation(12) can partial dieff rential Equation (16) describing the thermal be written as: distribution in Jeffery–Hamel flow is lowered to an ODE after simplification: LF =−2R αFF − 4α F (26) 2f (θ ) ν In whichL is the differential operator that can be deter- G (θ ) + 4G(θ ) + G(θ ) =− (4f (θ ) + f (θ )) 3 3 d ac mined: L = (d /dη ). (20) −1 The inverse of the operator L is indicatedby L .Itis Where: d = (K/ρc ) is the thermal diffusivity and, upon provided below: introducing the transformation (9) with the following η η η −1 quantities: L = (•)dηdηdη (27) 0 0 0 ⎪ = By considering Equations (27), (26), and the boundary G (θ ) = G (η) (21) conditions (13) and (14), it can be obtained: G (θ ) = G (η) α 2 −1 F(η) = F(0) + F (0)η + F (0) + L (Nu) (28) further reduction gives the dimensionless thermal distribution: where: 2 2 G (η) + 4α G(η) + 2R αP FG(η) Nu =−2R αFF − 4α F (29) e r e 2 2 2 Now, using boundary conditions (13), (14), and + P .X (4α F + F ) = 0 (22) r r F (0) = a, the following can be obtained: In which Re and Pr expressthe Re number given −1 by (11) and Prandtl number (P = (ν/d) = (μ.C /K)), r p F(η) = F = F + L (Nu) (30) n 0 respectively. n=0 ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 1377 where: 3.2. Heat transfer problem WithusingADMontheheattransferproblem,weobtain: F = 1 + a (31) 2 2 2 2 Thefirstfew termsofAdomianpolynomialscan LG + P .X (4α F + F ) =−4α G − 2R αP FG r r e r beobtained with the use of the Adomian decomposition (39) 2 2 algorithm (Adomian & Adomian, 1994). In fact, we get: In whichL can be provided as L = (d /dη ). Additionally, the inverse of L can be provided like: 2 2 3 A =−2aReαη − 4aα η − a Reαη (32) η η −1 2 8 8 L = .dηdη(40) 2 2 3 3 3 4 3 A = aRe α η + aReα η + aα η 1 0 0 3 3 3 3 6 1 The application of Equation (40) on Equation (39) 2 2 2 5 2 3 5 3 2 2 7 + a Re α η + a Reα η + a Re α η yields: 5 5 15 (33) −1 G(η) = G(0) + L (Ng) (41) 1 2 4 where: 3 3 5 2 4 5 5 5 A = aRe α η + aRe α η + aReα η 15 5 5 2 Ng =−4α G − 2R αP FG (42) e r 8 11 6 5 2 3 3 7 + aα η + a Re α η In contrast, if the boundary conditions (24)–(25), and 15 630 G(0) = b are applied, the following could be achieved: 22 22 2 2 4 7 2 5 7 + a Re α η + a Reα η 315 315 −1 3 3 3 9 G(η) = G = G + L (Ng) (43) n 0 11a Re α η 11 3 2 4 9 + + a Re α η n=0 1890 945 4 3 3 11 where a Re α η + (34) 3600 1 2 2 2 2 2 G = b − F Pr Xr η − 2F Pr Xrα η (44) On the other hand, by using Adomian decomposi- 2 tion algorithm (Adomian & Adomian, 1994),thefirstfew Forheattransferproblem,withusingADM(Ado- components of the solution are: mian & Adomian, 1994), the terms of solution and the 1 1 1 Adomian polynomials can be provided: 4 2 4 2 6 F =− aReαη − aα η − a Reαη (35) 12 6 120 2 2 2 2 2 B =−2bFPrReα − 4bα − 4cα η + F FPr ReXrαη 1 1 1 2 2 2 2 2 6 3 6 4 6 F =− aRe α η − aReα η − aα η + 2F PrXrα η 180 45 45 3 2 3 2 2 4 2 3 2 2 10 + 4F Pr ReXrα η + 8F PrXrα η (45) 1 1 a Re α η 2 2 2 8 2 3 8 − a Re α η − a Reα η − 560 280 10800 (36) 2 2 2 2 2 4 G =−bFPrReαη − 2bα η + F FPr ReXrαη 3 3 8 aRe α η 1 1 1 1 2 4 8 5 8 2 2 4 2 2 3 4 F =− − aRe α η − aReα η 3 + F PrXrα η + FF Pr ReXrα η 5040 840 420 6 3 2 3 3 10 2 2 4 10 1 11a Re α η 11a Re α η 2 6 8 2 4 4 − aα η − − + F Pr Xrα η (46) 630 453600 113400 2 5 10 3 3 3 12 3 2 4 12 11a Reα η a Re α η a Re α η − − − 2 2 2 2 2 3 2 4 2 B = 2bF Pr Re α η + 8bFPrReα η + 8bα η 113400 226800 113400 1 4 3 3 14 a Re α η 1 2 3 2 2 4 − (37) − F F Pr Re Xrα η 2 2 Ultimately, the estimated answer for the hydrodynam- 2 2 2 3 4 4 4 − F FPr ReXrα η − F PrXrα η ical problem can be provided: 3 3 2 8 4 3 2 4 4 3 2 5 4 F(η) = F + F + F + F + ... + F (38) − F Pr Re Xrα η − F Pr ReXrα η 0 1 2 3 n 3 3 With using Equations (38) and (14), the constant a 2 6 4 − F PrXrα η (47) could be determined. 3 1378 H. SAIFI ET AL. 1 2 2 1 2 2 2 2 4 3 4 4 4 4 5 4 4 10 G = bF Pr Re α η + bFPrReα η + bα η − F F Pr Re Xrα η 6 3 3 226800 1 1 2 2 2 3 2 2 6 3 4 3 5 10 − F F Pr Re Xrα η − F F Pr Re Xrα η 180 28350 1 1 1 2 2 2 2 3 6 4 6 2 3 2 6 10 − F FPr ReXrα η − F PrXrα η − F F Pr Re Xrα η 45 45 9450 1 4 1 4 3 2 4 6 3 2 5 6 6 5 4 6 10 − F Pr Re Xrα η − F Pr ReXrα η − F Pr Re Xrα η 45 45 56700 4 2 2 6 6 2 7 10 F F − PrXrα η (48) − FPr ReXrα η 45 14175 5 4 3 7 10 − F Pr Re Xrα η 3 3 3 3 4 2 2 2 4 4 B =− bF Pr Re α η − 2bF Pr Re α η 2 1 8 10 − F PrXrα η 5 4 6 4 − 4bFPrReα η − bα η 4 3 2 8 10 − F Pr Re Xrα η 1 1 2 2 3 4 3 3 6 2 3 2 + F F Pr Re Xrα η + F F Pr Re 3 2 9 10 90 15 − F Pr ReXrα η 4 6 2 5 6 × Xrα η + F FPr ReXrα η 2 10 10 − F PrXrα η (51) 2 4 5 4 3 5 6 6 6 + F Pr Re Xrα η + F PrXrα η 45 45 1 1 4 8 4 4 4 4 8 3 3 3 5 8 4 3 2 6 6 3 2 7 6 G = bF Pr Re α η + bF Pr Re α η + F Pr Re Xrα η + F Pr ReXrα η 2520 315 15 15 1 4 16 2 2 2 6 8 7 8 2 8 6 + bF Pr Re α η + bFPrReα η + F PrXrα η (49) 105 315 2 1 8 8 4 5 4 4 10 + bα η − F F Pr Re Xrα η 315 226800 1 1 3 3 3 3 6 2 2 2 4 6 G =− bF Pr Re α η − bF Pr Re α η 3 3 4 3 5 10 − F F Pr Re Xrα η 90 15 2 4 5 6 6 6 − bFPrReα η − bα η 2 3 2 6 10 − F F Pr Re Xrα η 15 45 3 4 3 3 8 + F F Pr Re Xrα η 6 5 4 6 10 − F Pr Re Xrα η 2 3 2 4 8 + F F Pr Re Xrα η 2 7 10 − F FPr ReXrα η 2 5 8 + F FPr ReXrα η 5 4 3 7 10 − F Pr Re Xrα η 1 1 5 4 3 5 8 6 8 + F Pr Re Xrα η + F PrXrα η 8 10 − F PrXrα η 1260 630 4 3 2 6 8 2 8 + F Pr Re Xrα η 4 3 2 8 10 3 2 9 10 − F Pr Re Xrα η − F Pr ReXrα η 4725 14175 1 2 3 2 7 8 2 8 8 + F Pr ReXrα η + F PrXrα η (50) 2 10 10 − F PrXrα η (52) 105 315 Ultimately, the estimated answer of the heat transfer 1 1 4 4 4 4 8 3 3 3 5 8 model in Jeffery-Hamel flow can be provided as: B = bF Pr Re α η + bF Pr Re α η 2520 315 2 2 2 6 8 G(η) = G + G + G + G + ... + G (53) 0 1 2 3 n + bF Pr Re α η Equations (53) and (25) can be employed to calculate 4 2 7 8 8 8 + bFPrReα η + bα η the constant of b. 315 315 ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 1379 Table 1. Thermophysical properties of the studied ﬂuids. Properties LiquidMetal Air Steam Water 2 −11 −5 −5 −6 Kinematic viscosity, ν (m /s) 38758.753 × 10 5.850 × 10 0.372 × 10 1.007 × 10 Prandtl number, Pr 0.01174 0.6843 1.11 7 Heat capacity, C (J/Kg.°K) 913 1063 2305 4178 4. Results and Discussions A parametric investigation has been performed for show- ing the effect of Reynolds and Prandtl numbers on the behavior of heat transfer and uid fl velocity in Jeffery- Hamel flow between non-parallel plane walls. It is worth stating that, for the heat transfer problem, four ranges of uids fl flow including steam, liquid metal, air, and water have been considered. Table 1 states the thermophysical properties of used u fl ids in this study. In this research,bothanalyticalandnumericalsolu- tions were computed. In fact, an analytical solution is gained using the Adomian Decomposition Method Figure 3. Eﬀects of Reynolds number on ﬂuid velocity inside (ADM); however, the numerical solution is achieved divergent channel. by using Runge–Kutta-Fehlberg based on the shooting approach. Figures 2–12 show thermal distributions and the velocity profiles in convergent-divergent channels asso- ciated with the obtained analytical and numerical values for the objective of highlighting the significance of the studied flow. Figure 2 illustrates the influences of Re number on the fluid velocity of the convergent flow. In fact, a flatter profile at the center of the channel with great gradients close to the walls can be obtained by augmenting Re number. As a consequence, the thick- nesses of the boundary layer decreases. For convergent flowcases,itiswellclear thatthebackflowisentirely precluded. Figure 3 illustrates the influence of Reynolds Figure 4. Eﬀect of channel-half angle α on ﬂuid velocity inside number on divergent flow which is to concentrate the convergent channel. Figure 5. Eﬀect of channel-half angle α on ﬂuid velocity inside Figure 2. Eﬀects of Reynolds number on ﬂuid velocity proﬁles divergent channel. inside convergent channel. 1380 H. SAIFI ET AL. Figure 9. Thermal proﬁles under the eﬀect of Reynolds number in diverging channel for steam ﬂow. Figure 6. Thermal proﬁles under the eﬀect of Reynolds number in converging channel for steam ﬂow. Figure 10. Thermal proﬁles under the eﬀect of Reynolds number in diverging channel for air ﬂow. Figure 7. Thermal proﬁles under the eﬀect of Reynolds number in converging channel for air ﬂow. Figure 11. Thermal proﬁles under the eﬀect of Reynolds number in diverging channel for metal liquid ﬂow. Figure 8. Thermal proﬁles under the eﬀect of Reynolds number thecaseofconvergentflow,weobserve thattheback- in converging channel for metal liquid ﬂow. flow phenomenon is precluded, but this phenomenon is highly observed inside divergent channels as depicted in volume ux fl at the center of channels with smaller gradi- Figure 5. ents near the walls. For purely divergent channels, results Heat transfer behavior in convergent-divergentchannels obtained reveal that the flow reversal is highly favored. is displayed in Figures 6–12.Intheconvergentchannel, Figures 4 and 5 illustrate the impact of the channel-half for the case of steam and Air flows as shown in Fig- angle α onthefluidvelocity. Here,thevelocity behav- ures 6 and 7, we notice that the minimum temperature ior is expected to be identical, which occurred in the is observed through the channel axis, while the maxi- case of Re number influence. As shown in Figure 4 for mum temperature occurs in the vicinity of the plates. ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 1381 Figure 12. Thermal proﬁles under the eﬀect of Reynolds number in diverging channel for water ﬂow. Figure 13. CPU time of ADM vs approximation order in converg- ing/diverging channels. Furthermore, as presented in Figure 8,for liquidmetal flow in convergent channels, the thermal profiles show used asaguiderevealsthatthe outcomes areidenti- an identical behavior on the entire channel. cal to each other, which justify and conrm fi that both As displayed in Figures 9–11,the characteristic behav- the Adomian Decomposition method and numerical ior of uid fl temperature in the diverging channel is fairly Runge–Kutta–Fehlberg are valid, applicable, and have various in which the oscillations are worthy. The pres- great precision. ence of oscillations depends on the nature of the studied As depicted in Table 2, for velocity distribution u fl ids. Here, it is highly noted that the apparition of oscil- through convergent/divergent channels when Re = 43 lations is mainly related to the Prandtl number. In fact, and α = 3°, the error between ADM and numerical data we notice the total absence of oscillations for liquid metal is introduced as: flow (Figure 11), while their presence is clearly noticed in the case of Air and Steam flows (Figures 9 and 10). According to the obtained results for both convergent- |F − F | Numerical ADM divergent channels, it can be concluded which the steam (Pr = 1.11) and air (Pr = 0.6843) have vicious behavior. The numerical data of F”(0) for different values of Infact,withincreasingReynoldsnumberReitappears Re and α =±5° are expressed in Table 3.Fromresults that the heat dissipation is very higher near the plates obtained, as drawn in Tables 2 and 3,anappropriate than that observed along the channel axis. Also, it is agreement is monitored between ADM analytical solu- clearly noticed that the heat dissipation is higher for air tion, numerical RK4 solution, and available data in ref- flowwhencomparedtothatoccurredinregardsto the erences (Abbasbandy & Shivanian, 2012;Kezzaretal., steam flow. In contrast, the liquid metal (Pr = 0.01174) 2018). is considered as a conductor fluid for both convergent- On the other hand, Tables 4 and 6 illustrate the numer- divergent channels. In such a case, the heat dissipation ical data of thermal distributions in convergent-divergent for liquid metal flow is lower when compared to that channels (case of liquid metal, Air and Steam) once occurred for the other ranges of uids. fl Reynolds number is equal to 50 and channel-half angle As presented in Figure 12 in the case of the diver- α =3°.Inthesetables,theerrorisintroducedasTable 5: gent channel, it is clearly shown for higher Prandtl value (Pr =7; waterflow case)thatthethermalbehavior G(η) − G(η) Num ADM becomes quite different. In fact, thermal profiles contain Error = G(η) a large number of oscillations with several minima and NUM maxima. Figure 13 shows the behavior of CPU time versus According to the results obtained, it should be stressed channel-half angle (α) and the order of approximation. whichagreatmatch canbeobservedinbothnumerical In fact, obtained results reveal that the CPU time is very and analytical data. short (i.e. few seconds), thus justifying the fast conver- Finally, the values of constant G(0) = b which repre- gence of the adopted ADM algorithm. sents the temperature at the level of channel’s center are For all simulations cases, as displayed in Figures 2–12, gathered in Table 7. These values are calculated for all comparison betweenADM resultsand numericalone temperature curves (Figures 6–12). 1382 H. SAIFI ET AL. Table 2. Comparison between Numerical and ADM solutions for velocity distribution through convergent/divergent channels when Re = 43. Divergingchannel (α =+ 3°) Convergingchannel(α = -3°) η F F |F -F | F F |F -F | ADM ADM ADM ADM Numerical Numerical Numerical Numerical 0.00 1.000000000000 1.000000000000 0.00000000 1.00000000000 1.00000000000 0.00000000 −9 −9 0.25 0.9176760582677312 0.9176760639989867 5.73×10 0.952601226585409 0.952601233559707 6.97×10 −9 −9 0.50 0.6916137868663034 0.6916137895560733 2.68×10 0.7974632865663513 0.7974632807505999 5.81×10 −9 −8 0.75 0.370363266835419 0.37036327154516463 4.70×10 0.497128632405185 0.4971286000590072 3.23×10 1.00 0.000000000000 0.000000000000 0.000000000 0.00000000000 0.0000000000 0.0000000 Table 3. ADM analytical results for F”(0). Convergingchannel (α =−5°) Divergingchannel (α =+5°) F”(0)(Abbasbandy & Shivanian, F”(0) (Kezzar F”(0) F”(0) (Abbasbandy F”(0) (Kezzar F”(0) Re 2012) et al., 2018) [Presentstudy] & Shivanian, 2012) et al., 2018) [Presentstudy] 10 −1.7845468 −1.7845469 −1.7845467711404606 −2.2519486 −2.2519485 −2.251948586722248 20 −1.5881535 −1.5881533 −1.5881534850176322 −2.5271922 −2.5271921 −2.527192251461816 30 −1.4136920 −1.4136921 −1.4136920839885079 −2.8326293 −2.8326295 −2.832629313353397 40 −1.2589939 −1.2589937 −1.2589939169568094 −3.1697121 −3.1697120 −3.169712202009959 50 −1.1219890 −1.1219891 −1.121989146674565 −3.5394156 −3.5394155 −3.539415629020588 Table 4. Comparison between ADM and Numerical results in convergent-divergent channels when Re = 50 and α = 3° (Thermal distribution in the case of liquid metal ﬂow). Converging Diverging −10 −10 −10 −10 η Numerical × 10 ADM × 10 Error Numerical × 10 ADM x10 Error 0 5,884051 5,884084 0.00000560838 7,726855 7,726817 0.00000491791 0,2 5,880418 5,880451 0.00000561185 7,691383 7,691345 0.00000494059 0,4 5,798519 5,798552 0.00000569111 7,369621 7,369586 0.00000474923 0,6 5,366264 5,366292 0.00000521778 6,266984 6,266959 0.00000398916 0,8 3,911092 3,911106 0.00000357956 3,905974 3,905967 0.00000179213 Table 5. Comparison between ADM and Numerical results in convergent-divergent channels when Re = 50 and α = 3° (Thermal distribution in the case of Air ﬂow). Converging Diverging −8 −8 −7 −7 η Numerical × 10 ADM × 10 Error Numerical × 10 ADM × 10 Error 0 1,267192 1,2672 0.0000063131 −2,251551 −2,251531 0.00000888277 0,2 1,352753 1,352762 0.0000066531 −2,094514 −2,094496 0.00000859388 0,4 1,574144 1,574154 0.00000635266 −1,673251 −1,673237 0.00000836695 0,6 1,780953 1,780962 0.00000505347 −1,107112 −1,107102 0.00000903251 0,8 1,561094 1,561098 0.00000256231 −0,5215149 −0,5215095 0.00000103545 Table 6. Comparison between ADM and Numerical results in convergent-divergent channels when Re = 50 and α = 3° (Thermal distribution in the case of Steam ﬂow). η Converging Diverging −10 −10 −9 −9 η Numerical × 10 ADM × 10 Error Numerical × 10 ADM × 10 Error 0 4,509989 4,510017 0.00000620844 −3,053678 −3,053661 0.00000556706 0,2 5,010631 5,010664 0.000006586 −2,715585 −2,71557 0.00000552367 0,4 6,36858 6,368621 0.00000643786 −1,874873 −1,874862 0.00000586706 0,6 7,88654 7,886584 0.00000557913 −0,9252627 −0,9252555 0.00000778157 0,8 7,431525 7,431546 0.0000028258 −0,2338741 −0,2338685 0.00000239445 Table 7. Values of dimensionless temperature at the channel centerline:G(0) = b. Re Convergingchannel(α =− 3°) Divergingchannel(α =± 3°) Liquid Metal Air Steam Liquid Metal Air Steam Water −12 −7 −10 −12 −6 −9 −10 100 1.96×10 4.33×10 7.28×10 3.42×10 −3.586×10 −6.26×10 4.26×10 −12 −7 −10 −11 −8 −9 200 7.04×10 5.98×10 7.83×10 2.20×10 −0.00001049 −2.3×10 2.45×10 −11 −7 −10 −11 −8 −9 300 1.47×10 5.93×10 6.4×10 7.67×10 −0.00002583 −6.14×10 6.63×10 −11 −7 −10 −10 −7 −9 400 2.46×10 5.34×10 5.07×10 1.89×10 −0.00005122 −1.21×10 9.31×10 −11 −7 −10 −10 −7 −8 500 3.63×10 4.65×10 4.07×10 3.78×10 −0.00008621 −2.01×10 1.41×10 ENGINEERING APPLICATIONS OF COMPUTATIONAL FLUID MECHANICS 1383 5. Concluding Remarks References Abbasbandy, S. (2007). A numerical solution of Blasius In this investigation, the steady 2D flows between non- equation by Adomian’s decomposition method and compar- parallel plane walls have been considered. The arising ison with homotopy perturbation method. 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Engineering Applications of Computational Fluid Mechanics – Taylor & Francis
Published: Jan 1, 2020
Keywords: Fluid flow; heat transfer; inclined walls; Adomian Decomposition method; numerical solution
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