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...h-order solution of viscoelastic fluids. The solid mathematical theory behind the DG method as well as its compatibility with the nature of fluid flow problems has made the DG a great candidate to seek the high-order solution of viscoelastic problems. Current study, as one of the first attempts, tries to address if DG method can be utilized to obtain high-order solution of viscoelastic problems. The DG method has been utilized to solve the pure elliptic and conservative hyperbolic problems in the last three decades. A complete review of DG solutions for elliptic problems can be found in Refs. [19] and [20] and solution of hyperbolic problems in Ref. [21]. The DG solution of incompressible flows including Stokes and Navier–Stokes problems is more recently introduced. The solutions of Stokes problem 1Corresponding author. Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received January 20, 2014; final manuscript received October 4, 2014; published online November 20, 2014. Assoc. Editor: D. Keith Walters. Journal of Fluids Engineering MARCH 2015, Vol. 137 / 031205-1Copyright VC 2015 by ASME Downloaded From: https://f...

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...tical theory behind the DG method as well as its compatibility with the nature of fluid flow problems has made the DG a great candidate to seek the high-order solution of viscoelastic problems. Current study, as one of the first attempts, tries to address if DG method can be utilized to obtain high-order solution of viscoelastic problems. The DG method has been utilized to solve the pure elliptic and conservative hyperbolic problems in the last three decades. A complete review of DG solutions for elliptic problems can be found in Refs. [19] and [20] and solution of hyperbolic problems in Ref. [21]. The DG solution of incompressible flows including Stokes and Navier–Stokes problems is more recently introduced. The solutions of Stokes problem 1Corresponding author. Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received January 20, 2014; final manuscript received October 4, 2014; published online November 20, 2014. Assoc. Editor: D. Keith Walters. Journal of Fluids Engineering MARCH 2015, Vol. 137 / 031205-1Copyright VC 2015 by ASME Downloaded From: https://fluidsengineering.asmedigitalcollection.asme.org on 06/29/2...

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...olution of viscoelastic fluids. The solid mathematical theory behind the DG method as well as its compatibility with the nature of fluid flow problems has made the DG a great candidate to seek the high-order solution of viscoelastic problems. Current study, as one of the first attempts, tries to address if DG method can be utilized to obtain high-order solution of viscoelastic problems. The DG method has been utilized to solve the pure elliptic and conservative hyperbolic problems in the last three decades. A complete review of DG solutions for elliptic problems can be found in Refs. [19] and [20] and solution of hyperbolic problems in Ref. [21]. The DG solution of incompressible flows including Stokes and Navier–Stokes problems is more recently introduced. The solutions of Stokes problem 1Corresponding author. Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received January 20, 2014; final manuscript received October 4, 2014; published online November 20, 2014. Assoc. Editor: D. Keith Walters. Journal of Fluids Engineering MARCH 2015, Vol. 137 / 031205-1Copyright VC 2015 by ASME Downloaded From: https://fluidsengi...

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...t in stresses near the contraction region recommending using finer grids. Alves et al. [13] obtained convergent solution up to We 3 for the creeping flow of an Oldroyd-B fluid using fully implicit control volume method. Recently, Zhang et al. [14] used a semi-implicit characteristic based split mesh-free algorithm for viscoelastic flow simulation. They obtained stable and convergent solution of 4:1 planar contraction up to We 4. In another approach, Malaspinas et al. [15] were able to achieve high Weissenberg solution in different problems using the lattice Boltzmann method. Also, DG method [16] has been used to study the viscoelasticity problem but with only low-order accuracy. Among the extensive efforts that have been carried out to obtain the numerical solution of viscoelastic fluid flow, there have been only few studies [17,18] performing the high-order flow analysis. Implementing higher-order methods enables us to simulate the fluid flow more accurately with less numerical diffusion especially for problems with sharp gradients and local discontinuities especially on coarser grids. However, convergence and stability issues of higher-order methods still need to be improved. In th...

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...Dt /h;Tij Dk 1 We /h;Tij Dk þ /h;uk @Tij @xk Dk 1 2 /h; n:ukTij Dk þ /h; @ui @xk Tkj Dk 1 2 /h; n: uk Tkj Dk /h; @uj @xk Tik Dk þ 1 2 /h; n: uk Tik Dk þ 1 bð Þ We /h; @ui @xj Dk 1 bð Þ 2We /h; n:uið ÞDkþ 1 bð Þ We /h; @uj @xi Dk 1 bð Þ 2We /h; n:uj Dk (38) By computing the viscoelastic stresses using Eq. (38) the flow variables updating is completed at step nþ 1. 4 Results and Discussion First, in order to verify the high-order accuracy of the implemented DG method, a Newtonian fluid flow is simulated. The Kovasznay flow [42], which is the flow behind a grid, is chosen as the benchmark problem for the Newtonian flow. The computational domain and types of the boundary conditions for the Kovasznay flow are shown in Fig. 1. The exact solution of this problem is [42]: kK 1 2Re ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4Re2 þ 4p2 r u 1 ekK x cos 2pyð Þ v kK 2p ekK x sin 2pyð Þ p 1 2 1 e2kKx (39) Since in this paper the viscoelastic problem is simulated for the creeping flow and for low Reynolds problem of Re 1, the Kovasznay flow is also simulated here for the case of Re ...

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...es [5]. Early studies included non-Newtonian fluid models designed to predict the shear thinning and shear thickening behavior in which the viscosity is a function of rate of strain tensor. Those kind of fluid flows were simulated with less difficulty using finite difference and finite element methods developed earlier to simulate the Newtonian fluid flow [6,7]. However, those models were not capable to describe the elastic behavior of some of the non-Newtonian fluid flows. Early studies [6,7] of the viscoelastic fluids were limited to slow moving fluids with low levels of elasticity We 1ð Þ [8] with less applicability for real world problems. More sophisticated models utilized to explain complex viscoelastic behavior of the fluid flows including the Oldroyd-type models [8,9]. However, the existing numerical methods at that time could not be easily applied to Oldroyd-type fluid flow model. In the early 1980s, one of the main problems of the Rheology community was to solve the problems with high Weissenberg number [6,8]. The main issue was achieving a stable and convergent solution for problem of viscoelastic fluids with We 1 in geometries with abrupt changes such as planar or axisy...

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...erical dissipation and diffusion are necessary [31,32]. If the solution is smooth enough, using high-order polynomial basis functions can provide high accuracy results even on a coarse mesh. The spatial discretization used in this paper leads to two elliptic equations that needed to be solved in order to simulate the flow field. There exist various methods for elliptic type problems [19,20,33] treating the numerical fluxes at the interfaces of the neighboring elements differently. The interior penalty (IP) method uses a penalization technique to ensure the smoothness of the numerical solution [34]. The characteristic parameter in the IP methods is a tuning a penalty parameter, which affects the solution accuracy as well. The temporal discretization of the equations is performed using a fractional-step method [34–37]. In the current study, the nodal discontinuous formulation [38] is used to simulate the viscoelastic Oldroyd-B flow in a 4:1 planar contraction. The viscoelastic stresses are decomposed into two separate terms, stresses due to the Newtonian solvent and stresses due to the viscoelastic polymers. A separate system of equations is needed to be solved for the polymer stresses. ...

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...gineers were capable to use the computational methods for simulation of non-Newtonian fluid flows, which are used widely in industries such as mineral processing industries [1], micronano fluidic [2], polymer solution [3,4], and food industries [5]. Early studies included non-Newtonian fluid models designed to predict the shear thinning and shear thickening behavior in which the viscosity is a function of rate of strain tensor. Those kind of fluid flows were simulated with less difficulty using finite difference and finite element methods developed earlier to simulate the Newtonian fluid flow [6,7]. However, those models were not capable to describe the elastic behavior of some of the non-Newtonian fluid flows. Early studies [6,7] of the viscoelastic fluids were limited to slow moving fluids with low levels of elasticity We 1ð Þ [8] with less applicability for real world problems. More sophisticated models utilized to explain complex viscoelastic behavior of the fluid flows including the Oldroyd-type models [8,9]. However, the existing numerical methods at that time could not be easily applied to Oldroyd-type fluid flow model. In the early 1980s, one of the main problems of the Rheolog...

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...y 1980s, one of the main problems of the Rheology community was to solve the problems with high Weissenberg number [6,8]. The main issue was achieving a stable and convergent solution for problem of viscoelastic fluids with We 1 in geometries with abrupt changes such as planar or axisymmetric contractions. The simulation of the flow with large We number is still an issue and many methods have been developed to solve this problem. The clear reason of the problem is still somewhat an open ended question, and there is no specific explanation that is agreed by everyone. Lately, Aboubacar et al. [10] investigated the solution of the Oldroyd-B creeping flow in the 4:1 planar contraction problem. They solved the momentum equations using the finite element method and solved the stress equations using the finite volume method. They were able to obtain a stable solution for We 4:4, which was a great improvement in the field. They explained the stability issue for higher We due to sudden changes in the stresses in the contraction region and cited that refining the meshes would overcome the stability issue. Phillips and Williams [11,12] used a semi-Lagrangian finite volume method using particl...

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...anation that is agreed by everyone. Lately, Aboubacar et al. [10] investigated the solution of the Oldroyd-B creeping flow in the 4:1 planar contraction problem. They solved the momentum equations using the finite element method and solved the stress equations using the finite volume method. They were able to obtain a stable solution for We 4:4, which was a great improvement in the field. They explained the stability issue for higher We due to sudden changes in the stresses in the contraction region and cited that refining the meshes would overcome the stability issue. Phillips and Williams [11,12] used a semi-Lagrangian finite volume method using particle tracking for both creeping and inertial cases in planar contraction of 4:1 for Oldroyd-B fluid. They were able to obtain steady and stable solution up to We 2.5. They explained the loss of convergence due to the steep gradient in stresses near the contraction region recommending using finer grids. Alves et al. [13] obtained convergent solution up to We 3 for the creeping flow of an Oldroyd-B fluid using fully implicit control volume method. Recently, Zhang et al. [14] used a semi-implicit characteristic based split mesh-free algorit...

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...Then using the IP method for the Poisson equation, the spatial discretization of Eq. (20) appears as r/h;rphð ÞDk 1 2 n:r/h; n:phð Þ@Dk /h; n:frphgð Þ@Dk þ IP/h; n:phð Þ@Dk n:r/h; ph @Dk \@XO /h; 2n:rph @Dk \@XIWþ IP/h; p h @Dk \@XO /h; Re Dt r:~u Dk n:r/h; poð Þ@Dk \@XO þ /h; @pnh @n @Dk \@XIW þ IP/h; poð Þ@Dk \@XO (35) where again /h is the test function. The IP in Eq. (35) is IP coefficient, which was introduced earlier IP C N þ 1ð Þ 2 h ; C 1 (36) where h is the length scale of each element; a proper range of C can be found in Ref. [41]. In Eq. (35), the superscript “O” denotes the outflow boundary, “I” the inflow boundary, and “W” indicates wall boundaries. By solving Eq. (35), the pressure field is computed and ~~u is then calculated using Eq. (19). In the next step, Eq. (24) needs to be discretized in order to find the velocity field. The Helmholtz equation (24) is discretized employing the IP method similar to Eq. (20). The discretized equations for x- and y-velocity fields are similar and only the one for x-velocity field is presented here r/h;ruhð ÞDkþ Re Dt /h; uhð ÞDk 1 2 n:r/h; n:uhð Þ@Dk /h; n:fruhgð Þ@Dkþ ...

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... al. [14] used a semi-implicit characteristic based split mesh-free algorithm for viscoelastic flow simulation. They obtained stable and convergent solution of 4:1 planar contraction up to We 4. In another approach, Malaspinas et al. [15] were able to achieve high Weissenberg solution in different problems using the lattice Boltzmann method. Also, DG method [16] has been used to study the viscoelasticity problem but with only low-order accuracy. Among the extensive efforts that have been carried out to obtain the numerical solution of viscoelastic fluid flow, there have been only few studies [17,18] performing the high-order flow analysis. Implementing higher-order methods enables us to simulate the fluid flow more accurately with less numerical diffusion especially for problems with sharp gradients and local discontinuities especially on coarser grids. However, convergence and stability issues of higher-order methods still need to be improved. In the current research, we intend to extend the implementation of the DG method to compute the high-order solution of viscoelastic fluids. The solid mathematical theory behind the DG method as well as its compatibility with the nature of fluid fl...

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...anation that is agreed by everyone. Lately, Aboubacar et al. [10] investigated the solution of the Oldroyd-B creeping flow in the 4:1 planar contraction problem. They solved the momentum equations using the finite element method and solved the stress equations using the finite volume method. They were able to obtain a stable solution for We 4:4, which was a great improvement in the field. They explained the stability issue for higher We due to sudden changes in the stresses in the contraction region and cited that refining the meshes would overcome the stability issue. Phillips and Williams [11,12] used a semi-Lagrangian finite volume method using particle tracking for both creeping and inertial cases in planar contraction of 4:1 for Oldroyd-B fluid. They were able to obtain steady and stable solution up to We 2.5. They explained the loss of convergence due to the steep gradient in stresses near the contraction region recommending using finer grids. Alves et al. [13] obtained convergent solution up to We 3 for the creeping flow of an Oldroyd-B fluid using fully implicit control volume method. Recently, Zhang et al. [14] used a semi-implicit characteristic based split mesh-free algorit...

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...solved in order to simulate the flow field. There exist various methods for elliptic type problems [19,20,33] treating the numerical fluxes at the interfaces of the neighboring elements differently. The interior penalty (IP) method uses a penalization technique to ensure the smoothness of the numerical solution [34]. The characteristic parameter in the IP methods is a tuning a penalty parameter, which affects the solution accuracy as well. The temporal discretization of the equations is performed using a fractional-step method [34–37]. In the current study, the nodal discontinuous formulation [38] is used to simulate the viscoelastic Oldroyd-B flow in a 4:1 planar contraction. The viscoelastic stresses are decomposed into two separate terms, stresses due to the Newtonian solvent and stresses due to the viscoelastic polymers. A separate system of equations is needed to be solved for the polymer stresses. The temporal discretization of the momentum equations results in a decoupled system of equations for the pressure and velocity in form of Helmholtz and Poisson equations which are elliptic problems. The DG–IP formulation is used to solve those equations. In the first step, the solution ...

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...tensor. Those kind of fluid flows were simulated with less difficulty using finite difference and finite element methods developed earlier to simulate the Newtonian fluid flow [6,7]. However, those models were not capable to describe the elastic behavior of some of the non-Newtonian fluid flows. Early studies [6,7] of the viscoelastic fluids were limited to slow moving fluids with low levels of elasticity We 1ð Þ [8] with less applicability for real world problems. More sophisticated models utilized to explain complex viscoelastic behavior of the fluid flows including the Oldroyd-type models [8,9]. However, the existing numerical methods at that time could not be easily applied to Oldroyd-type fluid flow model. In the early 1980s, one of the main problems of the Rheology community was to solve the problems with high Weissenberg number [6,8]. The main issue was achieving a stable and convergent solution for problem of viscoelastic fluids with We 1 in geometries with abrupt changes such as planar or axisymmetric contractions. The simulation of the flow with large We number is still an issue and many methods have been developed to solve this problem. The clear reason of the problem is s...

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... were able to obtain steady and stable solution up to We 2.5. They explained the loss of convergence due to the steep gradient in stresses near the contraction region recommending using finer grids. Alves et al. [13] obtained convergent solution up to We 3 for the creeping flow of an Oldroyd-B fluid using fully implicit control volume method. Recently, Zhang et al. [14] used a semi-implicit characteristic based split mesh-free algorithm for viscoelastic flow simulation. They obtained stable and convergent solution of 4:1 planar contraction up to We 4. In another approach, Malaspinas et al. [15] were able to achieve high Weissenberg solution in different problems using the lattice Boltzmann method. Also, DG method [16] has been used to study the viscoelasticity problem but with only low-order accuracy. Among the extensive efforts that have been carried out to obtain the numerical solution of viscoelastic fluid flow, there have been only few studies [17,18] performing the high-order flow analysis. Implementing higher-order methods enables us to simulate the fluid flow more accurately with less numerical diffusion especially for problems with sharp gradients and local discontinuities e...

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...he bottom of the channel and salient corner (XrÞ obtained in this study, is compared with those of the literature in Fig. 8. The intensity of the corner vortex [43] has also been used as a verification measure. Figure 9 compares the corner vortex intensity of current study with those found in Refs. [13,39, and 43]. Considering Figs. 8 and 9, it can be seen that the results presented in the current paper are in good agreement with other studies for both length and intensity of the corner vortex. Both of those parameters decrease as the We number increases similar to prediction of other studies [13,39,43]. Using the high-order DG method even on such a coarse mesh is capable of providing reliable results for relatively high We numbers. The high-order DG solution of Oldroyd-B fluid flow in this study remains stable for We numbers up to 4.5 but further elasticity increasing leads to loss of convergence. While the authors have not reached a definitive explanation for the occurred instability for such We numbers, it is expected that employing more suitable numerical fluxes in Eq. (38) would enhance the stability of solution even for higher We numbers. The benchmark problem is considered again now f...

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... al. [14] used a semi-implicit characteristic based split mesh-free algorithm for viscoelastic flow simulation. They obtained stable and convergent solution of 4:1 planar contraction up to We 4. In another approach, Malaspinas et al. [15] were able to achieve high Weissenberg solution in different problems using the lattice Boltzmann method. Also, DG method [16] has been used to study the viscoelasticity problem but with only low-order accuracy. Among the extensive efforts that have been carried out to obtain the numerical solution of viscoelastic fluid flow, there have been only few studies [17,18] performing the high-order flow analysis. Implementing higher-order methods enables us to simulate the fluid flow more accurately with less numerical diffusion especially for problems with sharp gradients and local discontinuities especially on coarser grids. However, convergence and stability issues of higher-order methods still need to be improved. In the current research, we intend to extend the implementation of the DG method to compute the high-order solution of viscoelastic fluids. The solid mathematical theory behind the DG method as well as its compatibility with the nature of fluid fl...

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...in the field. They explained the stability issue for higher We due to sudden changes in the stresses in the contraction region and cited that refining the meshes would overcome the stability issue. Phillips and Williams [11,12] used a semi-Lagrangian finite volume method using particle tracking for both creeping and inertial cases in planar contraction of 4:1 for Oldroyd-B fluid. They were able to obtain steady and stable solution up to We 2.5. They explained the loss of convergence due to the steep gradient in stresses near the contraction region recommending using finer grids. Alves et al. [13] obtained convergent solution up to We 3 for the creeping flow of an Oldroyd-B fluid using fully implicit control volume method. Recently, Zhang et al. [14] used a semi-implicit characteristic based split mesh-free algorithm for viscoelastic flow simulation. They obtained stable and convergent solution of 4:1 planar contraction up to We 4. In another approach, Malaspinas et al. [15] were able to achieve high Weissenberg solution in different problems using the lattice Boltzmann method. Also, DG method [16] has been used to study the viscoelasticity problem but with only low-order accuracy. A...

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...shes would overcome the stability issue. Phillips and Williams [11,12] used a semi-Lagrangian finite volume method using particle tracking for both creeping and inertial cases in planar contraction of 4:1 for Oldroyd-B fluid. They were able to obtain steady and stable solution up to We 2.5. They explained the loss of convergence due to the steep gradient in stresses near the contraction region recommending using finer grids. Alves et al. [13] obtained convergent solution up to We 3 for the creeping flow of an Oldroyd-B fluid using fully implicit control volume method. Recently, Zhang et al. [14] used a semi-implicit characteristic based split mesh-free algorithm for viscoelastic flow simulation. They obtained stable and convergent solution of 4:1 planar contraction up to We 4. In another approach, Malaspinas et al. [15] were able to achieve high Weissenberg solution in different problems using the lattice Boltzmann method. Also, DG method [16] has been used to study the viscoelasticity problem but with only low-order accuracy. Among the extensive efforts that have been carried out to obtain the numerical solution of viscoelastic fluid flow, there have been only few studies [17,18] p...

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...creeping flow Fig. 10 Variation of corner vortex length with We number for the inertial flow Journal of Fluids Engineering MARCH 2015, Vol. 137 / 031205-7 Downloaded From: https://fluidsengineering.asmedigitalcollection.asme.org on 06/29/2019 Terms of Use: http://www.asme.org/about-asme/terms-of-use that the size of the vortices in the inertial flow is smaller than the creeping flow in general. Moreover, the size of the vortex in the creeping flow is more sensitive to changes of We number than the inertial flow. The length of the corner vortex of the current study with those of the literature [11,14,44,45] is compared in Fig. 11. Figure 11 shows that the length of the corner vortex obtained in this research is in good agreement with majority of other studies especially for We numbers less and equal to We 2. In the numerical simulation, for 2, elements with polynomials of order eight was used. However, when the We number was raised to 2.5 and beyond the convergence issue appeared for elements with polynomial order of eight and, the order of elements was reduced to seven to obtain a converged and stable solution. The length of the corner vortex is reduced (as expected) with increase in We numbe...