A hybrid finite element/finite difference algorithm for compressible/incompressible viscoelastic liquids

Authors

  • Amr Guaily,

    Corresponding author
    1. Department of Mechanical and Manufacturing Engineering, University of Calgary, Calgary, AB, Canada T2N 1N4
    • Department of Mechanical and Manufacturing Engineering, University of Calgary, Calgary, AB, Canada T2N 1N4.
    Search for more papers by this author
  • Marcelo Epstein

    1. Department of Mechanical and Manufacturing Engineering, University of Calgary, Calgary, AB, Canada T2N 1N4
    Search for more papers by this author

Abstract

A unified formulation is proposed for modelling compressible/incompressible viscoelastic liquids. The pure hyperbolic nature of the model overcomes some of the drawbacks of available models. The most important of these drawbacks is the mixed nature of the resulting systems of equations, with the subsequent consequence of having no general numerical algorithm for the solution. A new non-dimensionalisation procedure is adopted. A hybrid least-squares finite element/finite difference scheme coupled with a Newton-Raphson's algorithm is used to solve the resulting system of equations. The method is used to predict the velocity and stress fields for different Weissenberg numbers for two benchmark problems.

Abstract

Une formule unifiée est proposée pour la modélisation de liquides viscoélastiques compressibles/incompressibles. La nature hyperbolique pure du modèle compense pour certains des inconvénients des modèles disponibles. L'inconvénient le plus important est la nature mixte des systèmes d'équations résultants, avec comme conséquence de n'avoir aucun algorithme numérique général pour la solution. Une nouvelle procédure de non-dimensionalisation a été adoptée. Un procédé d'élément fini et de différence finie de moindres carrés hybride ajouté à un algorithme de Newton-Raphson est utilisé pour régler le système d'équations résultant. La méthode est utilisée pour prédire les champs de vitesse et de tension pour les différents nombres de Weissenberg pour deux problèmes d'évaluation.

LITERATURE REVIEW

Viscoelastic Flow Solvers

A viscoelastic liquid is a fluid that exhibits a physical behaviour intermediate between that of a viscous liquid and an elastic solid. For this reason, both the mathematical formulation and the experimental techniques used to describe the response of viscoelastic liquids are substantially different from their viscous liquid counterparts. In particular, the numerical implementation of the governing system of equations contains important qualitative differences, such as the character of the equations, the choice of independent variables and the enforcing of boundary conditions. A comprehensive treatment of the associated problematic is presented in Joseph (1990). It is well known that the flow of Newtonian fluids presents a host of mathematical difficulties. It turns out that all of these difficulties exist also for the flow of viscoelastic liquids especially at high Weissenberg number besides other sources of numerical difficulties. One of the sources of difficulties in viscoelastic liquids is the presence of stress boundary layers and singularities. Another source of difficulty stems from the convective behaviour and in the treatment of the stress field as a primary unknown, we refer to Keunings (1990) for more details about challenges in viscoelastic fluids simulations. To the best of our knowledge, this study is the first attempt to formulate the governing equations of viscoelastic liquids as a purely hyperbolic model describing the compressible and the incompressible flow regimes in a unified way. Being a totally hyperbolic system is of great importance for many reasons, the most important of which are: (i) the boundary conditions can be determined without ambiguity, which may not be the situation for other types of systems. (ii) A major source of difficulty in numerical simulation for viscoelastic liquids is the change of type of the system of equations. Phelan et al. (1989) implemented a hyperbolic model for the incompressible flow regime only. Their results appear to be very sensitive to the grid discretisation and their solutions are limited to very coarse spatial discretisations. The mathematical and physical consequences of the elasticity of liquids as well as the asymptotic theories of constitutive models are well explained in Joseph (1990). In polymer processing operations, such as injection moluding and high-speed extrusion, pressure and flow rate may be large. Hence, compressibility effects within the viscoelastic regime may become important and influence resulting flow phenomena. Comprehensive literature reviews for viscoelastic incompressible flow may be found in Baaijens (1998). A variety of formulations have been applied over the last decade or so, such as finite volume (FV) methods (Oliveira and Pinho, 1998; Phillips and Williams, 1999), finite element (FE) method (Marchal and Crochet, 1986; Baloch et al., 1995; Guenette and Fortin, 1995; Matallah et al., 1998), spectral collocation methods (Fietier and Deville, 2003), and hybrid FE/FV methods (Wapperom and Webster, 1998; Aboubacar and Webster, 2001). These would include time-marching and steady-state approaches, leading to various options for the state-of-the-art (e.g., DEVSS as shown by Guenette and Fortin, 1995; Discontinuous-Galerkin and Galerkin-Least-Squares as shown by Baaijens, 1998, and others). The compressible viscoelastic domain is relatively unsheltered in the literature. Georgiou (2003) has addressed non-Newtonian inelastic (Carreau) fluid modelling for compressible flows, with particular interest in slip effects at the wall, relevant to time-dependent Poiseuille flow and extrudate-swell. Brujan (1999) has derived an equation of motion for bubble dynamics, incorporating the effects of compressibility and viscoelastic properties for an Oldroyd-B model fluid. The objective there has been to analyse the physics of cavitation. In Barrett and Gotts (2002), the equations of motion have been transformed into the Laplace domain to analyse a compressible dynamic viscoelastic hollow sphere problem. Hao and Pan (2007) proposed a finite element/operator-splitting method for simulating viscoelastic flow at high Weissenberg numbers. This scheme is stable when simulating lid-driven cavity Stokes flow at high-Weissenberg numbers. Guaily and Epstein (2010) proposed a unified purely hyperbolic model for Maxwell fluids. Misoulis and Hatzikiriakos (2009) re-examined the rheological constitutive equation proposed by Patil et al. (2006) to model the flow of polytetrafluoroethylene (PTFE) paste taking into account the significant compressibility of the paste and implementing the slip law based on the consistent normal-to-the-surface unit vector. The proposed model by Patil et al. (2006) was solved for different cases of extrusion rate and die geometry, with the emphasis being on the relative effects of compressibility and slip. Both of them found to be important and alter significantly the results for velocity, pressure, and other parameters, and of course on the flow rate.

Least-Squares Finite Element Method (LSFEM)

Least-squares schemes for approximating the solution to differential equations were proposed some time ago as a particular variant of the method of weighted residuals. The basic idea is quite straightforward. Given a trial solution expansion with unknown coefficients and satisfying the boundary conditions, construct the corresponding residual for the differential equations. Next, minimise the integral mean square residual to generate an algebraic system. Finally, solve this algebraic system to determine the coefficients and hence the approximation. This approach is appealing because the resulting algebraic system is symmetric and positive definite for a first-order system of differential equations. A theoretical analysis of a class of least-squares methods for the approximate solution of elliptic differential equations was discussed by Varga (1971). Baker (1973) obtained error estimates for the least-squares finite element approximation of the Dirichlet problem for the Laplacian. When compared with the Galerkin finite element method, least-squares finite element methods generally lead to more stringent continuity requirements for the trial functions. Lynn and Arya (1973); and Zienkiewicz et al. (1974) demonstrated that it is possible to reduce the order of continuity requirements at the expense of introducing more unknowns, by first transforming the original differential equation into an equivalent system of first-order differential equations. Subsequently, least-squares finite element methods were applied to boundary-layer flow problems as shown by Lynn and Alani (1976). Fix and Gunzburger (1978) also studied a least-squares method for systems of first-order equations and applied it to the Tricomi equation. A least-squares finite element formulation for the Euler equations governing inviscid compressible flow was proposed by Fletcher (1979). An important feature of his formulation is the designation of groups of variables rather than single variables. Jiang and Chai (1980) applied the least-squares finite element method to a first-order quasi-linear system for compressible potential flow. A least-squares collocation finite element method was investigated by Kwok et al. (1977); and Carey et al. (1980). High-order elements were used for the second-order problem rather than a lower order system being introduced. More recently, least-squares finite element methods have received considerable attention in relation to transonic full potential flow calculations and numerical solution of the Navier–Stokes equations for incompressible viscous flow (Bristeau et al., 1979, 1985). Carey and Jiang (1987) developed a systematic procedure for constructing a least-squares finite element method for partial differential equations. The problem is first recast as a first-order system and the least-squares residual used to form a variational statement. Carey et al. (1998) described a least-squares mixed finite element method and supporting error estimates and briefly summarised some computational results for linear elliptic (steady diffusion) problems. The extension to the stationary Navier–Stokes problems for Newtonian, generalised Newtonian and viscoelastic fluids is then considered. Pontaza et al. (2004) used the approach described by Carey and Jiang (1987) to solve both the Euler and Navier–Stokes equations for the compressible regime. Bolton and Thatcher (2006) used the LSFEM to solve the Navier–Stokes equations in the form of stress and stream functions. Pontaza and Reddy (2006) have presented least-squares based finite element formulations, as an alternate approach to the well-known weak form Galerkin finite element formulations. Gerritsma et al. (2008) described a least-squares spectral element formulation in which time stepping was used to reach steady-state solutions. The time integration method provides artificial diffusion, which suppresses the oscillations in the vicinity of discontinuities. Ahmadi et al. (2009) presented a framework and derivations of 2D higher order global differentiability approximations for 2D distorted quadrilateral elements widely used by LSFEM. Guaily and Megahed (2010) used the LSFEM coupled with a directionally adaptive grid to capture shock waves in compressible inviscid flows.

The present article is organised as follows: the governing equations are presented in second section. In third section, the disadvantage of the standard formulation is explained, followed by our proposed remedy for this disadvantage. In fourth section, we formulate the governing equations in a matrix form. Then, in fifth section, we classify the resulting system of equations. In sixth section, the time integration procedure is explained then the LSFEM formulation is presented in seventh section followed by a description of the method used to describe the boundary conditions in eighth section. In nineth section, numerical results are presented for the sake of validation. tenth section consists of a summary and conclusions.

GOVERNING EQUATIONS

Balance Laws

Conservation of mass yields the scalar equation:

equation image(1)

where ρ denotes the spatial density, V is the velocity field, t denotes time, and equation image is the material time-derivative.

Conservation of momentum yields the vector equation:

equation image(2)

where τ is the extra stress tensor and pL is the isotropic pressure. The separation of the total stress σ into two parts according to:

equation image(3)

is introduced for later convenience in the expression of the constitutive law. Notice that the extra stress tensor is not necessarily traceless for viscoelastic liquids.

Constitutive Equations

The conservation laws (1) and (2) are not sufficient to determine the unknowns corresponding to the flow. Constitutive equations are needed to relate the extra stress tensor to the velocity gradient.

Upper convected Maxwell (UCM) model

The extra stress tensor is linked to the velocity gradient by:

equation image(4)

with

equation image(5)

where µ is the elastic viscosity and the quantity λ has the dimension of time and is known as the relaxation time. It is, roughly speaking, a measure of the time for which the fluid remembers the flow history (Renardy, 2000). And D is the rate of strain tensor. There are many reasons for analysing this particular constitutive equation: elastic effects in a UCM fluid are modelled-albeit in a very simple way. The model only depends on two parameters, λ and µ. Other models can be obtained by simple extensions: adding a quadratic lower order term (αλ/µ)τ2(1 > α > 0), to the left-hand side yields a Giesekus model; letting λ and µ depend on the velocity gradient equation image yields a White–Metzner model, and so on. There is also a link to molecular theories, for example, the linear dumbbell model. What is important for our analysis is to note that many of the modified constitutive equations differ essentially only in lower order terms, that is, in terms containing no derivative. Also the UCM is known to be the most challenging constitutive equation for numerical analysis as it overestimates stresses at higher shear rates as explained by Renardy (2000).

The Compressibility Equation

The compressibility effect is accounted for by considering the modified Tait equation (Thompson, 1972):

equation image(6)

where B is a weak function of the entropy (in practice usually taken as a constant), ρ0 is the liquid density extrapolated to zero pressure, that is, very nearly the density at 1 atm. Different values for B, ρ0, and γ are given in Thompson (1972).

To transform Equation (6) into a first-order partial differential equation, we may define the pressure as:

equation image(7)

with no loss of generality since the pressure term always appears as a spatial derivative. Taking the time derivative for both sides of Equation (6) and using the continuity equation, we get the following non-dimensional equation:

equation image(8)

which is surprisingly the same, in shape, as the energy equation for a perfect gas in an isothermal flow.

FAILURE OF THE STANDARD FORMULATION IN THE INCOMPRESSIBLE LIMIT

From the mathematical point of view, the difficulty of numerical treatment in the unification of compressible and incompressible flows stems from two sources:

  • (i)The property of the equation of mass conservation, whereby it behaves as a hyperbolic equation in the compressible case, but as a constraint equation for the velocity field in the incompressible case. To overcome this difficulty we propose to use the modified Tait equation with equation image in the incompressible limit. Which in practice defines an incompressible fluid while keeping a real set of characteristics.
  • (ii)Another numerical difficulty arises from the standard nondimensionlisation procedure. Indeed, in the standard formulation, the free stream velocity is used to nondimensionalise the variables (Moussaoui, 2003), as equation image, and equation image, where (∞) refers to free-stream values. With this scaling we see that:
    equation image
    where C is the free-stream speed of sound and M is the Mach number, For the incompressible limit M goes to zero, which means that u* tends to unity while p* tends to infinity. In other words, the pressure term in the momentum equation is of order infinity while the advection term is of order unity, which leads to a numerical failure due to round-off error in the momentum equation. This problem is more complicated for viscoelastic liquids, as the speed of sound for air at room temperature is around 330 m/s. Therefore, at say M = 0.3, the speed of the fluid will be approximately 100 m/s. Nevertheless, the speed of sound for compressible liquids is much larger than the speed of sound in air (say five times). In applications such as polymer processing, velocity levels are generally low (of the order of unity as shown by Keshtiban et al., 2005). This is why computation of compressible liquid flows is generally associated with much more severe conditions than those for gas flows.To remedy this problem, in the present study we use the free-stream speed of sound instead of the free-stream velocity as a nondimensionlisation velocity. The nondimensionlised variables become:
    equation image
    where L is a characteristic length. Henceforth, we will omit the asterisk (*) for clarity. The proposed formulation is validated for the compressible and the incompressible flow regimes in Driven Cavity Flow Section.

FORMULATION OF THE GOVERNING EQUATIONS IN MATRIX FORM

In the present study, we consider Equations (1), (2), (4), and (8). Other models can be obtained by just adding lower order terms to Equation (6), a modification that does not affect the type of the equations.

In two-dimensional flow, the velocity vector has the form V = (u,v), and the extra stress tensor:

equation image(9)

The system of equations can be written in non-dimensional vector form as follows:

equation image(10)

We have two options regarding the vector of unknowns q; The first option (pressure-density based system of equations) is to consider the vector of unknowns to be equation image. The matrix At is the identity matrix and

equation image
equation image
equation image

The second option (pressure-based system of equations) is to adopt:

equation image
equation image

and

equation image
equation image

The Reynolds number (based on the free stream speed of sound) is defined as equation image and the Weissenberg number as equation image, which can be related easily to the regular ones using the Mach number as scaling factor.

CLASSIFICATION OF THE SYSTEM OF EQUATIONS

Equation (10) represents a quasi-linear system of first-order partial differential equations. In order to treat such a system numerically, we must first classify it mathematically. The system of Equation (10) can be classified according to the eigenvalues of the matrix equation image, where equation image and equation image are arbitrary scalars as explained in Courant and Hilbert (1962). The rigorous condition for hyperbolicity requires the matrix equation image to have real eigenvalues for every equation image and equation image. Equivalently, according to Brian and Antony (1990), the criterion for hyperbolicity for Equation (10) reduces to the requirement that the matrix Ax or Ay has a full set of real eigenvalues. The condition that the matrix Ax has a complete set of real eigenvalues (for arbitrary velocity and stress tensor fields) is both necessary and sufficient for the system of Equation (10) to be fully hyperbolic. For our system of equations (pressure based system of equations), the eigenvalues for Ax are:

equation image

One can easily show that all of the above eigenvalues are always real, and consequently, that the system of Equation (10) is always purely hyperbolic under the conditions: equation image. These inequalities are easy to satisfy in practice, especially for viscoelastic flows since we have low Reynolds number in practical flows.

It is worth noting that being a totally hyperbolic system is of great importance for many reasons, the most important of which are:

  • The boundary conditions can be determined without ambiguity for all variables including the stresses, which may not be the situation for other types of systems as discussed by Godlewski and Raviart (1996). Marchal and Crochet (1987) proposed a mixed type model, so they have to choose the geometry in such a way that the flow is forced to be fully developed in entry and exit sections which is not consistent with the physics of the problem. While in our model, we do not have this drawback. Since the boundary conditions is determined from the theory of characteristics.

  • A major source of difficulty, which doesn't exist for our proposed model, in numerical simulation for viscoelastic liquids is the change of type of the system of equations which needs a special numerical treatment to have a stable algorithm as explained by Joseph (1990). Several investigators (Brown et al., 1986; Song and Yoo, 1987) have linked the occurrence of a change of type in their numerical schemes to a subsequent loss of convergence. To overcome this issue, Marchal and Crochet (1987) divide the element into several bilinear sub-elements for the stresses, while streamline-upwinding is used for discretising the constitutive equation. By this approach, the computational complexity of the descretised problem is increased significantly.

  • In our proposed model, the stress field is treated as a primary unknown without the need for a special treatment which is not the case with the mixed systems (Marchal and Crochet, 1987).

TIME INTEGRATION

One of the most important parameters in numerical analysis is the size of the time step. This section aims to clarify the significance of using a fully implicit backward Euler scheme in iterating toward the steady state solution. To iterate toward a steady state solution, the unsteady term is descritised using finite difference fully implicit backward Euler scheme:

equation image

Steady state is declared when Δq ≤ tol, for a given tolerance tol. To see the effect of time step and to have a closer look at this choice, let's apply it to one of our governing equations, say the compressibility Equation (8) as it also represents the continuity equation by replacing the pressure by the density and for γ = 1.

Recalling the compressibility Equation (8) in its non-conservative form

equation image(11)

After descritising the unsteady term using Euler backward formula and doing some algebraic work using Taylor series expansion, the modified equation is:

equation image(12)

Since the numerical scheme implicitly deals with Equation (12) rather than Equation (11), it is important to see the modifications occurred to the original equation due to this choice of time integration scheme. By inspection of Equation (12), one can conclude the following:

  • (1)The original advection speeds are changed.
  • (2)A certain amount of dissipation is introduced.
  • (3)The steady state solution is affected by the time step.

In conclusion, we have to choose the time step equation imagein a balanced way to suppress the oscillations and in the same time to minimise both the amount of dissipation added to the original equation and the effect of the time step on the steady-state solution.

LEAST-SQUARES FINITE ELEMENT FORMULATION

Viscoelastic flows remain a demanding class of problems for approximate analysis, particularly at increasing Weissenberg numbers. Part of the difficulty stems from the convective behaviour and in the treatment of the stress field as a primary unknown. This latter aspect has led to the use of higher order piecewise approximations for the stress approximation spaces in recent finite element research (Guenette and Fortin, 1995; Baaijens, 1998). The computational complexity of the descretised problem is increased significantly by this approach. On the other hand the LSFEM appears to be the most viable technique for solving these problems. In addition to the relative ease of its implementation, the LSFEM is a minimisation technique, thus is not subject to the LBB (Ladyshenskaya–Babuska–Brezzi) condition and allow us to use equal-order interpolation functions of all variables, which greatly simplify the discretisation process.

The spatial part of the system given in Equation (10) may be solved iteratively using Newton–Raphson's method, by setting: qn+1 = qn + Δq, neglecting the higher order terms; Equation (10) can be rewritten as:

equation image(13)

where:

equation image

For the pressure-density based system of equations:

equation image
equation image

while for the pressure based system of equations:

equation image
equation image

Defining the residual vector, for more details about the LSFEM (Carey and Jiang, 1987), as:

equation image(14)

the least-squares functional is given by:

equation image(15)

We can now introduce the finite element approximation:

equation image(16)

where ne is the number of nodes per element and Nj (j = 1, …, ne) are the element shape functions. It is worth noting here that the LSFEM allows us to use the same shape functions Nj for all variables, which greatly simplify the discretisation process. While for other finite element techniques, we have to use different shape functions to have a stable scheme. Minimising the least-squares functional Equation (15) yields the following weak form:

equation image(17)

Introducing Equation (16) into Equation (17) results in the linear algebraic system of equations:

equation image(18)

where

equation image

is the coefficient matrix which is symmetric and positive definite, as expected for a system of first-order partial differential equations. For higher order systems, this property is totally lost due to the integration by parts. So, for a higher order system, one should break down the system of equations into the equivalent first order one as suggested by Carey and Jiang (1987). Being symmetric and positive definite allows the use of simple and more efficient iterative solvers like the Jacobe-Conjugate gradient:

equation image(19)

which is the right-hand side for local elements. The integrations are evaluated using Gauss–Legendre quadrature.

BOUNDARY CONDITIONS

For a hyperbolic system of equations, considerations on characteristics show that one must be cautious about prescribing the solution on the boundary. In some particular cases, the boundary conditions can be found by physical considerations (such as a solid wall), but their derivation in general case is not obvious. The problem of finding the “correct” boundary conditions, that is, which lead to a well-posed problem, is difficult in general from both the theoretical and practical points of view (proof of well-posedness, choice of the physical variables that can be prescribed). The implementation of these boundary conditions is crucial in practice; however, it depends very much on the problem as shown in Godlewski and Raviart (1996). The number of boundary conditions to be imposed on a boundary is determined by the theory of characteristics according to the incoming/outgoing characteristics. In the finite element method, the boundary conditions are easily imposed, a fact that can be considered as one of the most important features of the finite element method.

It is worth noting that, as the Weissenberg number approaches zero, the Newtonian term dominates in the constitutive Equation (4) and one can consider it as an algebraic equation (neglecting the upper convected derivative term). In this case, we are left with the Navier–Stokes equations. In other words, we do not have to specify boundary conditions for the stress since in the Newtonian case the flow is completely determined by specifying the velocities and the pressure. On the other hand, as the Weissenberg number increases, the upper convected part in the constitutive law becomes the dominant term and, consequently, the hyperbolicity of the constitutive law becomes stronger so we have to specify boundary conditions for the stress to be consistent with the physics of the problem.

NUMERICAL RESULTS

To validate the present algorithm for both compressible and incompressible flow regimes. In this section, we analysed two problems; the first one is the very well-known test case, lid driven cavity, for the incompressible flows. For the compressible flows, we analysed the channel flow with a bump problem since it tests the ability of the proposed scheme to capture high stress gradients.

Driven Cavity Flow

Geometry

The algorithm described above was used to calculate the flow inside a square cavity. The geometry, the grid, and the boundary conditions are shown in Figure 1a. In order to capture the high gradients, we used a clustered grid as shown in Figure 1a. The clustering equation is taken form Hoffmann and Chiang (1993). Another important feature of using the finite element method for the spatial part of the equations is that we donot need to transform the system of equations into generalised coordinates to be able to use a clustered grid, as would be the case in a pure finite difference scheme.

Figure 1.

Physical domain and the Newtonian solution.

As far the Boundary conditions are concerned, for the upper boundary, we use a constant velocity, u = 0.001, v = 0.0, while the no-slip boundary condition is imposed over the rest of the domain and the pressure is set to p = 1/γ at the lower left corner. There are no other boundary conditions to be imposed, to be consistent with the physics of the problem.

Newtonian limit

To represent the incompressible, Newtonian limit, calculations were performed with Δt = 0.05, Re = 0.001, We = 0.001 on a 20 × 20 (bi-linear elements) clustered grid. These calculations were undertaken to test the accuracy of the proposed scheme by comparing the results with other Newtonian calculations that are available in the literature. The solution was considered to reach the steady state solution when the maximum error, defined as the Euclidean norm equation image, is of order 1E−4 or less for all the variables. We reached this tolerance using about 300 for this case. Note that all the variables plotted in the following figures are nondimensionalised as equation image but the solution was carried out using the free stream speed of sound as the nondimensionalisation velocity as explained in Failure of the Standard Formulation in the Incompressible Limit Section. The stream-trace lines for this limiting case are shown in Figure 1b. The calculations show a vortex located at the point (x = 0.5, y = 0.76). This is in very good agreement with the results of Phelan et al. (1989) who calculated a vortex centre at (x = 0.5, y = 0.75) who solved the steady problem in the lid driven cavity geometry using a split coefficient matrix (SCM) finite difference method. The pressure field is shown in Figure 1c, which is consistent with what is known from the Newtonian literature; the domain is divided into two parts, a lower pressure zone (left) and a higher pressure zone (right). Figure 1d shows the shear stress distribution, which is in a good agreement with Phelan et al. (1989). A mesh convergence test was carried out for different mesh sizes. Figure 2 shows the axial velocity at the centreline of the cavity for different meshes, which shows a mesh-independent solution as well as a very good agreement with the experimental work of Pakdel et al. (1997). Notice that the finer the mesh, the closer the solution is to the experimental data. For quantitative comparison between our algorithm and mixed finite element techniques, Figure 2 shows the axial velocity for Grillet et al. (1999). One has to mention that Grillet et al. (1999) uses a non-physical trick to obtain the solution as they introduce leakage to relieve the corner singularities. To implement this leakage, small rounded channels are included at the corners where fluid can leak through and, by the use of periodic boundary conditions, reenter the cavity on the upstream side. While in the current work we donot need a special treatment for the corner singularities. Also one has to mention that their algorithm is subject to LBB condition so they have to use different shape functions to satisfy the LBB condition. By this approach, the computational complexity of the descretised problem is increased significantly. Finally, we should mention that their mesh consists of 6,312 elements while ours is only 400 elements which means a huge saving in the required memory and faster processing.

Figure 2.

Axial velocity comparison.

Viscoelastic flow

Viscoelastic flow computations were performed with the same flow conditions and for different Weissenberg numbers (We = 0.3, 1, 5, 20) to show the viscoelasticity effects on a 20 × 20 clustered grid. It is worth noting here that we didn't encounter a limit for the Weissenberg number but we stopped the calculations at We = 20 since most of the practical applications involve much lower values. Figure 3 shows the stream-trace lines for different Weissenberg numbers. From this figure, one can see the viscoelastic effects as a distortion to the symmetry of the vortex structure compared to the Newtonian case as reported by Boger and Walter (1993). As well as for very high Weissenberg numbers (more energy and elasticity effects), a secondary vortex is formed as shown in Figure 3d.

Figure 3.

Stream-trace lines for non-Newtonian flow.

Figure 4 shows the resulting shear stress isocontours. It is clear that, the higher the Weissenberg number, the higher the shear stress gradient in the boundary layer, which is a well-known fact about viscoelastic flows. Figure 5 shows the first principal stress difference defined as equation image for different Weissenberg numbers. The effect of high-Weissenberg numbers is evident as the stress gradient becomes more severe for higher Weissenberg numbers.

Figure 4.

Shear stress isocontours for non-Newtonian flow.

Figure 5.

First principal stress difference isocontours for non-Newtonian flow.

Channel Flow With a bump

A challenging problem for this algorithm is the analysis of compressible flow past a parabolic arc bump of height-to-length ratio α = 10%. This problem creates high stress gradients, especially in the vicinity of the bump. The main reason for considering this problem is to show the ability of the proposed algorithm to capture the stress gradient especially in the vicinity of the bump. So the current work could be extended to include axisymmetric effects to be able to simulate viscoelastic flows in a multi-pass rheometer (Mackley and Spittler, 1996, which suffers from high stress gradients). The curved lower wall for the circular arc is given by:

equation image

where b =(α2 − 0.25)/2α and equation image.

Figure 6 shows the physical domain as well as the grid used for the solution.

Figure 6.

Geometry and grid.

Viscoelastic flow computations were performed with (equation image, γ = 7.15, U = 0.2, Re = 1.0, We = 0.1, L1 = 1.5, L2 = 1.5). We are simulating a relatively high-Mach number flow compared to practical values just to show the capabilities of the proposed scheme to capture severe stress gradients. The grid consists of 43 × 15 uniformly distributed bi-linear rectangular elements with 15 elements in the y-direction, 15 elements on the bump, 13 elements in the entrance section, and 15 for the exit section. The pressure-density system of equation is used in the current test case. The boundary conditions at the inlet are:1

equation image

At the exit, the pressure is specified as p = 1/γ and the shear stress as equation image while the no-slip boundary condition is imposed on the upper and lower boundaries.

Results for We = 0.1 are shown in the following figures. Figure 7 shows the axial velocity isocontours, while Figure 8 shows the isocontours of the normal velocity. One easily notices the symmetry about x = 0. Figure 9 shows the first principal stress difference, this figure shows the high-stress-gradient zone to be in the vicinity of the bump. Figure 10 shows the first principal stress difference (Np) and the first stress difference (N1 = S − T) and the shear stress (Q) distributions over the lower boundary, while Figure 11 shows the flow variables distribution at the exit section. It is worth remarking that we obtained the Poiseuille flow distributions for the stresses, thus encouraging us not to put any boundary conditions for the stresses as if we had a Newtonian flow (low-Weissenberg number). We obtained the same results for low-Weissenberg numbers as shown in Figure 12 (where the subscript b stands for the results with specified boundary conditions for the stress while nb stands for the results with no boundary conditions specified for the stresses). This example shows the capability of our numerical scheme to be consistent with the problem physics unlike the mixed type model proposed by Marchal and Crochet (1987) in which they have to choose the geometry in such a way that the flow is forced to be fully developed in entry and exit sections which is not consistent with the physics of the problem. The same calculations were carried out for a finer grid (65 × 20) to check for grid convergence. To save space, we show only the stress distribution over the lower boundary in Figure 13, which assures the scheme convergence.

Figure 7.

The axial velocity isocontours (We = 0.1).

Figure 8.

The normal velocity isocontours (We = 0.1).

Figure 9.

First principal stress difference (We = 0.1).

Figure 10.

Stress distribution over the lower boundary (We = 0.1).

Figure 11.

Flow variables distribution at the exit section (We = 0.1).

Figure 12.

Stress distribution at the inlet section (We = 0.1).

Figure 13.

Flow variables distributions at the exit (finer grid).

Another viscoelastic flow computations were performed with (Δt = 0.1, γ = 7.15, U = 0.1, Re = 1.0, L1 = 1.0, L2 = 1.5) and for different Weissenberg numbers. The pressure system of equation is used in the current test case. The grid consists of 40 × 10 uniformly distributed bi-linear rectangular elements with 10 elements in the y-direction, 10 elements on the bump, 10 elements in the entrance section, and 20 for the exit section. To show the ability of the proposed scheme to capture higher stress gradients, we increased the Weissenberg from be 0.01 to 0.7. The solution was considered to reach the steady state solution when the maximum error, defined as the Euclidean norm equation image, is of order 1E−4 or less for all the variables. We reached this tolerance using about 1,200 for this case. Figure 14 shows the first principal stress difference for different Weissenberg numbers, from the figure, one can notice that the maximum nondimensional first principal stress difference obtained for We = 0.01 is 10.0 compared to 45.0 for We = 0.7. Figure 15 shows the first stress difference N1 for different Weissenberg numbers, it is worth recalling that N1 should be zero for Newtonian flows (We = 0.), which is consistent with the isocontours of Figure 15 (top). Again the higher the Weissenberg number, the higher stress gradients. To show the importance of the compressibility, Figure 16 shows the density isocontours for different Weissenberg numbers. By inspection of this figure, we have about 20% variation in the liquid density for these conditions, which cannot be neglected. Figure 17 shows the corresponding pressure isocontours. Finally, Figure 18 shows the convergence history for this problem for We = 0.3.

Figure 14.

First principal stress difference for different We.

Figure 15.

First stress difference for different We.

Figure 16.

The density isocontours for different We.

Figure 17.

The pressure isocontours for different We.

Figure 18.

Convergence history for We = 0.3.

SUMMARY AND CONCLUSION

A hybrid least-squares finite element/finite difference based technique was used to simulate the flow of viscoelastic fluids. In which the finite difference method was used to advance in time while the LSFEM was used to treat the spatial part of the system of equations. For the first time; the theoretical formulation permits the governing equations to be cast in the form of a totally hyperbolic system of first-order PDEs for both the compressible and the incompressible regimes. Being totally hyperbolic avoids many problems exist in the traditional systems of mixed types. The quality of the numerical results indicates the remarkable performance of the proposed model, as is quite evident from the final results even when using a coarse mesh. The robustness of the technique allows one to use large time steps (Δt = 0.15) to reach the steady state solution in relatively few iterations.

  • 1

    With no loss of generality, we used Poiseuille flow at the inlet to specify the boundary conditions for the stresses.

Ancillary