A numerical method for solving steady 2D and axisymmetrical viscoelastic flow problems with an application to inertia effects in contraction flows

Transcription

1 A numerical method for solving steady 2D and axisymmetrical viscoelastic flow problems with an application to inertia effects in contraction flows Martien A. Hulsen, Delft University of Technology, Laboratory for Aero and Hydrodynamics, Rotterdamseweg 145, 2628 AL Delft, The Netherlands. Faculty of Mech. Eng. and Marine Tech., MEMT report no. 11, ISBN September 26, 1990

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3 Summary This report contains the description of a numerical method for simulating incompressible, isothermal, steady viscoelastic fluid flows in 2D and axisymmetrical geometries. The method has been applied to the flow through a four-to-one contraction to study the influence of inertia on the vortex growth. After a general introduction the basic equations and the constitutive equations are given. In Chapter 3 a numerical method is described for solving these equations in rather general geometries. The momentum equation and the continuity equation are discretized by the finite-element method. The constitutive equations are discretized by a characteristics-based method. The characteristics (stream lines for steady flow) are computed by a method based on quadratic interpolation of the stream function. Stability and integration aspects of the discretized constitutive equation are fully discussed. The equations are solved by a Picard-type iteration scheme. The method is capable of solving viscoelastic fluid flows at high Deborah numbers for many popular fluid models. In Chapter 4the method is applied to the flow through a four-to-one contraction. The influence of the elongational properties of the fluid and of the inertia forces are studied in particular. It appears that the vortex growth regime and the divergent flow regime, which are observed in experiments for some fluids, can be found for a choice of the material parameters where both the elongational stresses and the inertia forces are large for the flow rate considered. After studying the type and the vorticity it is concluded that the appearance of a divergent flow regime is a critical phenomenon (i.e. a change of type for a critical velocity). iii

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5 Contents 1 Introduction 1 2 Basic equations Momentum equation and continuity equation Constitutive models Numerical methods Introduction Discretization of the momentum and the continuity equation Discretization of the constitutive equation General procedure Computation of streamlines Stability and integration aspects Iteration scheme Computing times Conclusions and future developments Application: Influence of inertia and elongational properties on the flow through a four-to-one contraction Introduction Problem description Results Streamlines Centerline velocities Axial velocity profiles Some remarks on the convergence of the iteration scheme Does the appearance of the divergent flow regime correspond to a critical phenomenon? Conclusions References 50 A Visco-elastic fluid models 54 B Linearization of the constitutive equations and eigenvalue computation for stability 57 v

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7 Chapter 1 Introduction The availability of fast computers with large memory has enormously stimulated the development of new computational methods for fluid flows. In particular the methods for solving viscoelastic fluid flows are now at such a state of development that simulations at industrial process conditions are possible. Since this is only a very recent development, numerical research in viscoelastic fluid flow has mainly been directed towards the goal of finding algorithms that work at all ( the high-weissenberg-number problem ). As such algorithms are available now, future research will be more directed towards applications to real flows for the study of complicated flow phenomena such as vortex enhancement in contraction flows (see, for example, Hulsen & van der Zanden 1990), unsteady flows, inertia/elasticity interaction (see Chapter 4). The application of computer codes to complicated flows is very important for industrial polymer processing. However, it is also important for the fundamental aspects of rheology. It opens up the possibility to test the behaviour of viscoelastic material models under much more severe deformation conditions than is possible in simple flows such as steady simple shear and steady elongation. It may also be feasible in the future to fit the parameters of a model not only to the behaviour in simple shear/elongation but also to the behaviour in complex flows. This is particular interesting for fluids where the measurement of basic material functions such as the elongational viscosity is very difficult in practice (e.g. polymer solutions). Complicated flow phenomena in even relatively simple geometries are mostly not tractable by analytical methods and can be predicted by numerical methods only. In this report a numerical method for solving viscoelastic fluid flows in complex geometries is described. After a review of the basic equations, the method will be described in detail. The computer codes that have been developed will subsequently be applied to the flow through a four-to-one contraction. In particular the influence of inertia on the size and shape of the corner vortex will be studied. 1

8 Chapter 2 Basic equations The basic equations describing the flow of viscoelastic fluids consist of the basic laws of continuum mechanics and the constitutive equation describing a particular fluid. To simplify the basic equations, we assume that the flow is incompressible and isothermal. The last assumption means that we do not have to consider the energy equation. 2.1 Momentum equation and continuity equation The balance of linear momentum and the conservation of mass in a fixed bounded space Ω become ρ v + grad p = ρf + div t, (2.1) div v = 0, (2.2) where ρ is the density, v the velocity, f a body force per unit mass, p the pressure and t is the extra-stress tensor. The tensor t is symmetrical and vanishes in equilibrium. The extra-tensor t is determined by the deformation history of a material particle and has to be specified by the constitutive equation (model) of the particular fluid. Many models have been proposed in the literature. For an overview of various models we refer to the books of Tanner (1985), Bird et al. (1987) and Larson (1988). 2.2 Constitutive models All the models we will consider in the following have a viscous term given by t =2η s d + τ, (2.3) where d = 1 2 (h + ht ) with h T = grad v is Euler s rate-of-deformation tensor and η s is the extra-viscosity, which may be zero. The stress tensor τ is either given by differential equations (differential model) or by an integral equation (integral model). Although the algorithm given in this report has also been adapted for integral models (see Hoitinga 1990), we will not consider such models here. The differential models are given by superpostition of substresses t k K τ = t k, (2.4) k=1 2

9 where K is the number of modes. The substresses t k,k=1,...,k satisfy the following differential equations λ k t k + γ k (d : t k )(t k + η k I)+f λ k (t k )=2η k d, (2.5) k where λ k and η k are the relaxation time and viscosity parameter of mode k respectively and ( ) denotes the Gordon-Schowalter derivative of a tensor ( ) = ( ) (h ξd) () () (h ξd) T, (2.6) with 0 ξ 2. The term with factor γ k in equation (2.5) has been included for the Larson differential model (Larson 1988). The function f k is an isotropic tensor function f k (a) =f 1k (I 1,I 2,I 3 )I + f 2k (I 1,I 2,I 3 )a + f 3k (I 1,I 2,I 3 )a 2, (2.7) where f ik, i =1, 2, 3 are functions of the invariants I 1,I 2,I 3 of a, given by I 1 =tra, I 2 = 1 2 [I2 1 tr(a 2 )], I 3 = det a. (2.8) Examples of differential models are (Larson 1988) Johnson-Segalman f 1k = f 3k =0,f 2k =1,γ k =0, Larson ξ =0,f 1k = f 3k =0,f 2k =1,γ k 0, Giesekus ξ =0,f 1k =0,f 2k =1,f 3k = constant, γ k =0, Phan-Thien/Tanner f 1k = f 3k =0,f 2k = Y (I 1 ), γ k =0. The Leonov model also belongs to the class of models given above. In Appendix A we have described the models in more detail. The equations (2.1) (2.5) form a system of partial differential equations in v, p and t k, k =1, 2,...,K, which has to be supplemented by proper initial and boundary conditions. For a discussion on the type of the equations and whether the system is a properly posed initial value problem we refer to van der Zanden et al. (1985), Joseph et al. (1985), Hulsen (1986), van der Zanden & Hulsen (1988), Hulsen (1988b) and the recent book of Joseph (1990) (for some results related to this subject, see also Hulsen 1988c, Hulsen 1990) In the following we will only consider steady flows. In that case the system is of a mixed elliptic-hyperbolic type with particle trajectories (stream lines) being the real characteristics. This means that the stresses t k, k =1, 2,...,K must be specified at the inflow boundaries to represent fluid memory. We note that for models with η s = 0 the latter condition is theoretically not correct because not all stress components can be prescribed (Hulsen 1986, van der Zanden & Hulsen 1988, Renardy 1988), but in problems with long entry sections this appears to be not a problem for practical computations. However, models with η s = 0 are more difficult to compute than models with η s 0 if only a small number of modes are present. Hence, we recommend to use models having η s 0 in that case. 3

10 Chapter 3 Numerical methods 3.1 Introduction Choosing proper numerical methods for the system (2.1)-(2.5) requires special care. First we have to take into account the mixed elliptic-hyperbolic type (for the steady flow case). This requires an upwind-type of method for the hyperbolic part. Marchal & Crochet (1987) apply the finite-element method to the complete system, but treat the convective term of the constitutive equations with a nonconsistent upwind scheme. This leads to a first-order accurate scheme. In a finite-difference context these upwind schemes are known to be very stable but not very accurate (Leonard 1979). A higher-order accurate upwind scheme known as the Lesaint-Raviart method or discontinuous finite-element method (see Johnson 1987) has been used by Fortin & Fortin (1989). This method looks promising, although it requires a special ordering of the elements. We have used a streamlineintegration scheme for the integration of the constitutive equation similar to the scheme described by Shen (1984). This method, which resembles the methods for integral models (see, for example, Dupont et al. 1985, Luo & Tanner 1986) involves computation of the stream lines and integration with an explicit time integration scheme along the stream lines. A second problem with the constitutive equation (2.5) and (2.6) is the handling of a reasonable number of modes (for example, Hulsen & van der Zanden 1990 have used eight modes). Simultaneously solving for v, p, t k, k =1,...,K with for example, the coupled mixed finite element technique (Marchal & Crochet 1987), would require excessive computer resources. Therefore we prefer to use a decoupled method, which involves solving alternatively for ( v, p) and t k, k =1,...,K. We note that unlike the mixed method, solving for v, p, t k simultaneously is also not very practical for the Lesaint-Raviart scheme, the streamlineintegration methods and the methods for integral models. A disadvantage of a decoupled scheme is that usually (much) more iterations are required to obtain convergence. In the following sections we will discuss the numerical methods in more detail. 4

11 3.2 Discretization of the momentum and the continuity equation For steady flows equations (2.1) and (2.2) become ρ v grad v + grad p div t = ρf, (3.1) div v = 0. (3.2) We will discretize these equations by standard Galerkin finite-element techniques. Let W and Q be suitable spaces on Ω for the velocity field and the pressure field respectively. By using Gauss theorem and substituting equation (2.3), the equations (3.1) and (3.2) can be put into the following weak form [ρ v grad v, w]+ 2η s d, grad w ( p, div w )+ τ, grad w = [ σ, w ] Γ +[ρf, w] w W, (3.3) (div v, q) = 0 q Q, (3.4) where w and q are so-called test functions, σ = n ( pi + t) is the traction on the boundary Γ with outward normal n and (.,.),.,. and [.,. ] Γ are defined by [ p, q ]= p qdω, ( p, q )= pq dω, Ω Ω p, q = p : q dω, [ p, q ] Γ = p qdγ. Ω The discrete (approximate) formulation of (3.3) and (3.4) is found by the Galerkin method, which means restricting the spaces W and Q to finite-dimensional subspaces W h and Q h. In the finite element method the basis functions of W h and Q h consist of piecewise polynomials. Applying the Galerkin method we find a system of equations of the following form Ñ(ṽ)+η s S ṽ + L T + (τ ) =, (3.5) p Q F L ṽ = 0, (3.6) where ṽ and are the unknown velocity and pressure (column) vectors respectively, Ñ(ṽ) is p a vector due to the inertia term, and are constant matrices, (τ ) is a vector of internal forces due to the stress S tensor L τ and is an external Q F force due to the surface traction σ and the body force ρf. Note that in equation (3.5) the stress tensor τ has not yet been discretized. We will not give expressions for the vectors and matrices in (3.5) and (3.6) because they can be found in any textbook on finite-element methods for fluid mechanics (see, for example, Cuvelier, Segal & van Steenhoven 1986). To eliminate the pressure vector p, we use the penalty method. This method consists of using the following perturbed equation instead of (3.4) (Cuvelier et al. 1986) (ε p p + div v, q) =0, (3.7) 5 Γ

12 where ε p is a small parameter. In discretized form we get p = 1 ε p D 1 L ṽ, (3.8) where is a symmetric matrix. Substitution of (3.8) into (3.5) yields D Ñ(ṽ)+(η s S + C )ṽ + Q (τ )=F, (3.9) where = C L T D 1 L /ε p is the penalty matrix. In our computations we have used the modified P 2 + P 1 Crouzeix-Raviart triangular element (Cuvelier et al. 1986). The original element has extended quadratic velocity basis functions and piecewise discontinuous linear pressure basis functions. This leads to an element having seventeen degrees of freedom: fourteen velocity components, one centroid pressure and two pressure gradient components. The two velocity components of the central node and the pressure gradient components can be eliminated on element level. Hence, we have effectively an element with twelve velocity components (of six boundary nodes) and one centroid pressure. The latter is eliminated by the penalty method, also on element level. 3.3 Discretization of the constitutive equation General procedure When we denote a particle trajectory (stream line) by x = x(τ), where τ is a curve coordinate, the constitutive equation (2.5) can be written as a set of ordinary differential equations for t k : dt k dτ = F k( d x dτ, v, h, t k), k =1, 2,...,K. (3.10) Assuming that x(τ), v( x), h( x) are known, this equation can be integrated provided a starting condition has been given, for example t k (τ = τ 0 )=t 0 k, k =1, 2,...,K. (3.11) We only integrate (3.10) for the nodal points of the elements. We find the stresses in other points by interpolation. For the P + 2 P 1 element we compute t k at the six boundary nodal points and interpolate quadratically. For the computation of t k in some point x p, we have to find the upstream part of the stream line starting form x p. Since the element we use obeys (element-wise) mass-conservation, we can uniquely compute the stream function values in the nodal points up to an integration constant by computing the volume flux through the element sides. Six nodal point values of the stream function on an element leads to piecewise quadratic interpolation. Streamlines are contour lines of the 6

13 stream function, which consist in this case of piecewise quadratic curves (ellips, hyperbola) with possibly degenerated forms (parabola, straight line, point). See the next subsection for more details. Before we integrate equation (3.10) we have to specify the initial condition in (3.11). Here we have the following three possibilities i. The stream line crosses an inflow boundary. The value of t 0 k is taken from the specified boundary value, for example according to a developed profile. ii. The stream line crosses an element(side) where the stresses have already been computed. The value of t 0 k is found by quadratic interpolation. iii. The particle travel time from the current point on the stream line to the point x p is longer than a specified time T k. The tracking is stopped and the value of t 0 k is set to an interpolated value from the previous iteration. In our computations we have taken T k =3λ k, where λ k is the relaxation time of mode k. We note that as long as we have a convergent iteration scheme, the interpolation in iii. is allowed. Furthermore, we have taken T k several relaxation times so that the influence of t 0 k on the resulting stresses is limited in most cases. We found that T k =3λ k is a good choice. Increasing T k beyond this value increases the computation times but does not change the results. The possibility iii. has been introduced to avoid excessive computation times due to many small integration steps for modes with small relaxation times and to deal with recirculation zones and points at rest. In recirculation zones it is not always possible to fulfill ii., in particular when only a few stress points in the zone have been computed yet. The particle travel time in iii. does not have to be very accurate. Therefore it is estimated by integration of dt/dτ = d x/dτ / v with a composite mid-point rule having one subinterval per element (Ralston & Rabinowitz 1978). Points at rest (for example a center of recirculation, points on a fixed wall) are special because the stream line reduces to a single point, which means that particles stay there forever and condition iii. is always fulfilled. The integration of (3.10) is performed by taking τ to be the real time coordinate of a fluid particle. In that case the dependence of the right-hand side of (3.10) on d x/dτ = v =0 disappears Computation of streamlines In Hulsen (1988b) the stream lines are approximated by dividing an element into four sub-cells. In each cell the stream function is interpolated linearly. This leads to a stream line segment consisting of a straight line. With the same information on the stream function (stream function values in six nodal points per element), it is also possible to interpolate quadratically. This leads to quadratic stream lines, which will be described in the following. We expect that a higher order 7

14 Figure 3.1: A stream line starting from P 0. interpolation leads to a more accurate scheme. Remember that a piecewise linear stream function leads to a piecewise constant velocity only. We have not actually compared the accuracy of the two schemes, but it may be worthwhile to do this, because the quadratic scheme turns out to be rather complicated. Streamlines for a quadratic stream function We consider planar steady incompressible flow with a constant velocity gradient h = (grad v) T, with tr h = 0, which follows from equation (2.2). We introduce the stream function ψ as follows v = ψ y e x ψ x e y = u e x + v e y, (3.12) where ( e x, e y, e z ) is an orthonormal vector base in IR 3. It is also useful to introduce a vector v that is normal to the stream lines having a length equal to v v = grad ψ = v e x + u e y. (3.13) We want to find the stream line starting from a point P 0 where the velocity is v 0. The position vector x of a point P is defined relative to P 0. The arbitrary constant of ψ is chosen such that the curve ψ( x) = 0 is the stream line on which P 0 is lying. In Figure 3.1 the various quantities that have been introduced are illustrated. 8

15 For a constant velocity gradient we have the following expressions for v, v and ψ where the symmetric tensor k has been defined by v = h x + v 0, (3.14) v = k x + v 0, (3.15) ψ = 1 x k x + 2 v 0 x, (3.16) k = h yx e x e x + h xy e y e y + h xx ( e x e y + e y e x ). (3.17) Based on the sign of the principal values of k, we define the following three cases for the quadratic form ψ = 1 2 x k x + v 0 x = 0 (stream line through P 0) I. det k = 0, one or two principal values of k are equal to zero. II. det k > 0, principal values of k have equal sign. III. det k < 0, principal values of k have opposite sign. For case I we may have a straight line, a parabola or the complete (x, y)-plane as a solution for ψ = 0. For case II we have an ellips or a point and for case III a hyperbola or two straight lines. We will give more detailed subdivisions in the following. Note that det h = det k so that the division I III can also be made with det h. To find an expression for the stream line it is more convenient to write x as a function of a time co-ordinate τ: x(τ), with x(0) = 0. For τ > 0we are in downstream part and for τ<0in the upstream part of the stream line. Substitution of d x/dτ = v into (3.14) gives d x dτ = h x + v 0, x(0) = 0. (3.18) Differentiating (3.18) leads to d 2 x dτ = h d x 2 dτ = h h x + h v 0. (3.19) Using h h =( det h)i + (tr h)h =( det h)i and the definition w 0 = h v 0, we find that d 2 x dτ + (det h) x = w d x 0, x(0) = 0, 2 dτ (0) = v 0. (3.20) According to the division into I III based on det k = det h we find the following solution of (3.20) 9

16 I. det h =0: x(τ) = 1 2 w 0τ 2 + v 0 τ, (3.21) II. det h > 0; γ 2 = det h: III. det h < 0; γ 2 = det h: x(τ) = w 0 γ (1 cos(γτ)) + v 0 sin(γτ), (3.22) 2 γ x(τ) = w 0 γ (cosh(γτ) 1) + v 0 sinh(γτ). (3.23) 2 γ Note that if we take lim γ 0 in II and III we find I. Remember that the equations in (3.21) are valid both for τ>0 (downstream) and τ<0 (upstream). It is useful for understanding the type of flow to consider the eigenvalues and eigenvectors of h given by Multiplying with h we find that h y = λ y, y 0. (3.24) h h y =( det h) y = λh y = λ 2 y, (3.25) so that λ 2 = det h. For the three classes I III we find I. det h = 0. Two identical eigenvalues λ = 0. The case h = 0 is trivial, any vector is an eigenvector. For h 0 we find a one-dimensional eigenvector space (line) which is the kernel (nullspace) of the mapping h, i.e. all vectors x such that h x = 0. The range of the mapping h is equal to the kernel because for any z = h y we have h z = h h y =( det h) y = 0. The eigenvectors point in the direction of shearing. II. det h > 0. No real eigenvalues and eigenvectors. III. deth < 0. Two real eigenvalues and eigenvectors. The eigenvalues are ± det h Looking at the equations in (3.21) we see that if v 0 is an eigenvector of h the vector w 0 = h v 0 is in the direction of v 0 and the particle trajectory is a straight line. Checking whether v 0 is an eigenvector can be done by checking whether v 0 0 and v 0 w 0 = 0. Based on these observations we make the following subdivisions a. v 0 = 0, 10

17 b. v 0 0, v 0 w 0 = 0, i.e. v 0 is an eigenvector of h, c. v 0 0, v 0 w 0 0, i.e. v 0 is not an eigenvector of h. This leads to the following nine cases Ia. No movement; x(τ) = 0. Ib. Straight line in the direction of v 0. Note that w 0 = 0. Ic. Parabola. IIa. No movement; x(τ) = 0; center of recirculation. IIb. Not possible. IIc. Ellips. IIIa. No movement; x(τ) = 0; Stagnation/separation point. IIIb. Straight line, on asymptotic line of hyperbola. IIIc. Hyperbola. We make the following remarks concerning the division given above. We could further subdivide Ia. and Ib. according to 1. h = 0 2. h 0 This leads to the following possibilities Ia1: No flow, ψ = 0 for all x. Ia2: On shear line with zero velocity. Shear line is also given by ψ =0. Ib1: Rigid body translation with velocity v 0 ; ψ = 0 for all x. Ib2: On shear line with non-zero velocity; v 0 is in the direction of the shear line. For case Ic there is no point with zero velocity because from (3.14) we find that v = 0 h x = v 0, which has no solution for x since v 0 is not in the range of the mapping h. For IIa the point is a local extremum for ψ and for IIIa the point is a saddle point for ψ. Points with zero velocity are given by h x + v 0 = 0. (3.26) 11

18 For IIc and IIIc we find a single point x c given by { w0 /γ x c = w 0 /(det h) = 2 for IIc (center) w 0 /γ 2 for IIIc (stagnation/separation) (3.27) which follows from multiplication of (3.26) by h. For the practical implementation we also need the inverse of (3.21). For that purpose we introduce the vector w 0, which is defined similarly to v 0, i.e. w 0 = w 0y e x + w 0x e y if w 0 is given by w 0 = w 0x e x + w 0y e y. It is easy to verify that w 0 = v 0 h. We will treat the relevant cases b and c separately. Ib. Straight line. Ic. Parabola τ = v 0 x v 0 v 0. (3.28) 2 v 0 x τ = v 0 v 0 + ( v 0 v 0 ) 2 +2( v 0 w 0 )( v 0 x). (3.29) IIc. Ellips p = cos(γτ) =1 γ 2 v 0 x v 0 w 0 q = sin(γτ) =γ w 0 x v 0 w 0 τ = 1 γ arctan(p ). (3.30) q IIIb. Asymptotic line to hyperbola. v 0 is eigenvector of h: w 0 = α v 0, α 2 = γ 2 = det h =( w 0 v 0 )( v 0 v 0 ). We find that x(τ) = v 0 (e ατ 1)/α so that τ = 1 α ln(1 + α x v 0 v 0 v 0 ) (3.31) Note that if x x c (= w 0 /γ 2 ) the time variable τ. IIIc. Hyperbola sinh(γτ) =γ w 0 x v 0 w 0 = p τ = 1 γ ln(p + p 2 +1). (3.32) Extension to axisymmetrical co-ordinates is straightforward. In that case we have rv r = ψ/ z and rv z = ψ/ r. Assuming that ψ is a quadratic function in r and z, we can use exactly the same equations if we replace (x, y) by(r, z). The vector v defined in (3.12) is not the velocity but r times the velocity, and the tensor h (which is derived by differentiating ψ twice) is not the velocity gradient. Note that the curve co-ordinate τ in (3.18) does not have a dimension of time. For our purpose this is of no importance, since we are only interested in the geometric shape of the curves (τ is only a curve co-ordinate). 12

19 Practical implementation For the six-node element we also have available six nodal values of the stream function ψ. With quadratic interpolation it is easy to compute a linear velocity field in each element by equation (3.12) and a constant velocity gradient h by differentiating v. We note that these are not the velocity field and velocity gradient field given by the finite element interpolation. The basic procedure to find a stream line is as follows. We start at some point x s. and compute the value of the stream function ψ s = ψ( x s ). We compute the stream line (either upstream or downstream) by computing the intersection points with the element sides, i.e. the points where ψ = ψ s. Hence the stream line segment in an element connects the intersections of the element sides. An exception is the first segment of the streamline if the starting point x s is a vertex of an element. In that case the segment goes from the vertex to an opposite element side of one of the elements connected to that nodal point. It is never checked whether the stream line happens to cross a nodal point during tracking. This means that if a stream line crosses a vertex of an element exactly, the length of some stream line segments may be actually zero. The stream line in an element is given by the equations in (3.21) and the residence time in an element by (3.28), with x being the point where the stream line leaves the element. We note that the time coordinate τ should be considered as a curve coordinate as described in section and not the real time t. The latter is found by integration of dt/dτ = d x(τ)/dτ / v, where x(τ) is the curve given by (3.21) and v is the finite element velocity field. In its basic form the procedure given above can be rather complicated because the number of points on an element boundary where ψ = ψ s may vary from element to element and it is difficult to choose the right one if there is more than one possible exit point. For example if a center of recirculation is inside an element there may be six such points. In order to avoid such complications we adopt a simplified procedure with modified mid-side nodal values of ψ. These values of ψ are only modified if there is a local extremum within an element side. In that case the value of the mid-side node of ψ is taken to be the nearest value such that the extremum is shifted to the end points of the interval. This may be written as follows if ψ 2 >ψ 2max then ψ 2 = ψ 2max where if ψ 2 <ψ 2min then ψ 2 = ψ 2min ψ 2max = 1 4 max(ψ 1 +3ψ 3, 3ψ 1 + ψ 3 ) ψ 2min = 1 4 min(ψ 1 +3ψ 3, 3ψ 1 + ψ 3 ) with ψ 2 being the mid-side nodal value and ψ 1,ψ 3 the end values of the element side. The procedure has been depicted in Figure 3.2. This procedure has some 13

20 Figure 3.2: Modification of the midside nodal value of ψ. important consequences There is both a unique entry and exit point if a stream line intersects an element. The boundary intersection points may easily be found by solving a quadratic equation for ψ. Intersection of an element side can easily be checked by ψ [ψ 1,ψ 3 ]. The direction of the stream line (upstream or downstream) can easily be found from the sign of ψ 1 ψ 3. This makes life a lot easier and in practice the modified stream function has only slightly different stream lines than the original, except for the exact position of centers of recirculation and stagnation points on coarse meshes. This means for example that recirculations of the size of an element and the high curvature of stream lines near stagnation points are not properly resolved. We must point out one special simplification of the algorithm. First, all points with velocity v = 0 are found by considering the value of v from the finite element interpolations. No stream line is needed here. If a point does have v 0 but still belongs to the cases Ia, IIa and IIIa, this point is at rest 14

21 in the approximate velocity field v, given by (3.12) with quadratic ψ, only and not in the real velocity field, given by the finite element interpolations. We do not consider these points to be at rest and connect the entry and exit points of an element (which always exist!) simply by a straight line. Examples of such points are points on the center line in axisymmetrical problems and points in an element having two sides on fixed walls. The latter example is a special case of an element with identical ψ-values on the vertices but with different ones on the midside nodal points. These are changed by the scheme depicted in Figure 3.2 and made equal to the values at the vertices, resulting in a constant ψ. Centers of recirculation are found by a local extremum of the stream function (at a vertex nodal point), so that no stream line can be found in that case and a point at rest ( v = 0) is assumed. In recirculation zones it is sufficient to track a stream line only during one cycle. Therefore it is checked whether an element side has been crossed before and tracking is stopped then, because the stream line is completely known by then. For an actual implementation the following items need some further attention. Checking whether det h =0, v = 0 and v 0 w 0 = 0 needs to be replaced by det h <ɛ 1, v < ɛ 2 and v 0 w 0 <ɛ 3, where ɛ 1, ɛ 2 and ɛ 3 are small. This may lead to unexpected problems. For example the equation for the travel time of a parabola (equation (3.29)) is based on det h being exactly zero. If we decide that the particle is on a parabolic curve segment based on det h <ɛ 1 it is possible that the square root becomes negative in some cases (which we have encountered in practice). To avoid these problems we have deleted the parabolic case altogether and have replaced it by a straight line segment. We note that checking whether v 0 w 0 = 0 has actually been implemented as v 0 w 0 <ɛ 3 γ v 0 2 to avoid points close to stagnation points (i.e. v 0 is small) being falsely identified as hyperbolic asymptotic lines. On boundaries which are stream lines (free boundaries, symmetry lines) the stream function is constant. In practice however, a small deviation is found due to round-off errors and a slight compressibility due to the penalty method. Therefore it is possible that a stream line leaves the domain on such boundaries. To present this, it is checked whether the element side is on a boundary and the value of ψ s is close to the stream function values at the boundary. If this is true, a small shift is made to ψ s to pull the stream line back into the domain. The exit points of the stream line is computed with the new ψ s -value. In the element where the correction occurs, the entry and exit point are connected by a straight line because these belong to different values of ψ s and the quadratic stream line approach may fail due to inconsistencies (e.g. negative square root arguments etc.). We ran into trouble with the above procedure at the boundary for elements 15

22 where all ψ-values are nearly equal. Therefore we have decided to consider this as a special case (actually case Ia1) and assumed the point to be at rest ( v = 0). The exit point of an element may be a point with v = 0. In that case tracking is stopped and it is assumed that the stream line ends in a points at rest just before the exit point to avoid v being zero on a regular stream line segment. A similar situation arises if the exit point is a stagnation point of case IIIb. The curve co-ordinate τ would become infinite. This is handled in the same way, i.e. the stream line is stopped just before this point and a point at rest is assumed for the rest of the streamline. If we find a local minimum or maximum for ψ in the starting point (if it is a vertex nodal point) we assumed a center of recirculation. However, this is not true for a node on a boundary. If we have encountered a point where no stream line can be found, it is first checked whether it is inside the domain or not. If it is inside the domain a center of recirculation is assumed ( v = 0). If not, the value of ψ s may be close to the boundary stream function value as described above, and ψ s is shifted towards the domain in that case. If this is not true, the stream line is assumed to leave the domain (which corresponds to an inflow or outflow boundary). Recirculations are found by checking a repeated crossing of the same element side, which is easy to apply and fast. However, it is possible in a rare situation that the same element side is not crossed and the element side in a connected element is crossed instead although the intersection points are identical (which is a common vertex nodal point in this case). The reason is the different treatment of the starting point, which may be a vertex nodal point, compared to the treatment of the element side intersection points. If the situation happens to occur, which may result in exceeding the maximum number crossing points (set to some default value), another check on recirculation is made by comparing co-ordinates directly and appropriate action is undertaken Stability and integration aspects The integration of (3.10) is performed with a standard explicit fourth-order Runge-Kutta method (see, for example, Ralston & Rabinowitz 1978). The step size is taken such that τ f lim τ lim where τ lim is the linear stability limit and f lim 1 is a factor usually chosen to be 0.75 to avoid small oscillations due to non-linearities. The stability limit τ lim is based on the linearized form of (3.10) given by dε k dτ = E k : ε k, (3.33) 16

23 where the fourth-order tensor E k is given by E k = F k, (3.34) t k and ε k is a perturbation of the exact solution due to a perturbation of the initial value given by (3.11) (see also van der Zanden & Hulsen 1988, Hulsen 1988b). In matrix form we may write (3.33) as follows dε k dτ = Ē k : ε k, (3.35) where ε k is a column vector with all the components of interest of the tensor ε k, for example (ε xx,ε xy,ε yy ) T for planar flow and (ε rr,ε rz,ε zz,ε θθ ) T for axisymmetrical flow. The matrix Ē k is the matrix version of the fourth-order tensor E k with respect to the chosen components of ε k. The stability limit τ lim is defined as the maximum value of τ such that τe i belongs to the stability region of the integration method for all eigenvalues E i of Ē k. If the real part of E i is positive, which can occur in some parts of the flow, we use E i instead of E i (see also Hulsen 1988b). We note that in this case stability of the integration method does not make much sense. Stability is fully determined by the condition of the problem, i.e. the growth of the solution due to the positive real part of the eigenvalue. The condition of the problem is dependent on the visco-elastic fluid model that is used and the actual parameters that are chosen. In particular the quasi-linear models (Johnson-Segalman) behave badly in this respect. In Appendix B we have given expressions for finding the fourth-order tensor E k for the models given by (2.5) (2.6). For some models it is possible to find analytical expressions for the eigenvalues E i (Johnson-Segalman, Giesekus, Leonov 2D) and for other models these have to be computed numerically (Larson, Phan-Thien/Tanner, (extended) Leonov). Numerical computations of eigenvalues leads to larger CPU-times. In practice we have found an increase of a factor of two. In Appendix B we have discussed some methods for finding approximate eigenvalues. In our implementation we have the possibility of limiting the stepsize further according the following criteria Step limited by the relative change of the magnitude of the velocity in a step τ d x dτ = s f 1 v / d v, (3.36) ds where f 1 is a factor, e.g. 0.1, s is the arc-length coordinate and v 2 d v /ds = v h v, evaluated at the start of the interval. Step limited by the radius of curvature R of the streamline τ d x dτ = s f 2R, (3.37) 17

24 where f 2 is a factor, e.g. 0.1, and v 3 R 1 = v h v, again evaluated at the start of the interval. Step limited by the relative change of the r-co-ordinate in axisymmetrical flow τ d x dτ = s f r 3 v v r, (3.38) where f 3 is a factor, e.g. 0.75, and v r is the radial velocity evaluated at the start of the interval. Step limited by the deformation in the step τ d x dτ f 4 v 1 max i,j τ = t f 4 1 max i,j h ij h ij for a streamline, (3.39) for a point at rest, (3.40) where f 4 is a factor, e.g. 0.5 and h ij are the components of the tensor h evaluated at the start of the interval. The first three criteria are only applicable to real stream lines and not to points at rest. The fourth criterium is a generalization of the first three involving all velocity gradient components and all points including points at rest. There are no general rules for giving values for the factors f 1,...,f 4. The values given above are just examples that have been used in practice. The fourth criterium is very useful and for some more difficult models (Larson, Phan-Thien/Tanner) even necessary to compute high Deborah number flows with initial vectors of in the iteration process (see next section). A costly alternative would be computing 0 many intermediate solutions. Step size limiting by deformation (fourth criterium) is only necessary at the start-up of the iteration process for most problems. After a few iterations the factor f 4 may be set large ( ). In our implementation a step τ cannot span more than one element so that τ is also limited by the size of the elements. There is no step size limit based on an accuracy criterion. This can lead to large approximation errors for the stresses in some isolated points in regions with very high gradients, e.g. near sharp corners. In particular the tensor b k (see Appendix A) can become indefinite for some points, although theoretically this cannot occur (Hulsen 1990). Unfortunately, negative values of b k may blow up the quadratic term in equation (2.5) in the integration process (Hulsen 1988c). To avoid these disastrous nonlinear instabilities, we have built in a limiter for b k which consists of adding a hydrostatic term to the configuration tensor b k when it becomes indefinite due to numerical errors: b k = b k + βi, (3.41) 18

25 where b k is the computed (indefinite) tensor and b k the corrected tensor. The value of β is chosen to be the minimum value such that all the principal values of b k are greater than or equal to zero. If b k is positive definite then β = 0 and the tensor is not changed. An alternative to this procedure would be a step size strategy based on local truncation errors (Ralston & Rabinowitz 1978), but we have not done that. For points on a fixed wall we have a simple shear type of deformation. However due to numerical approximation errors of the velocity gradient, small stretching components can be present and lead to large spurious elongational stresses on the wall for high Deborah numbers. These stresses can develop due to the long (infinite) residence time of these particles. They can be avoided by imposing an exact simple shear flow gradient h sh on the fixed wall. We have used the following expression for h sh, which scales Euler s rate-of-deformation tensor d h sh = ω ɛ d + ω, (3.42) where d = d 1(tr d)i, ω = h d is the vorticity tensor, ω = 1 curl v and 2 2 ɛ is the maximum principle value of d. Note that equation (3.42) is co-ordinate system invariant and that for a shear flow ɛ = ω and thus no scaling takes place. There are other possibilities to define h sh, for example scaling ω by ɛ/ω, but we have not considered these because (3.42) works satisfactory. 3.4 Iteration scheme The system to solve is (3.9) together with the stream line integration of (2.5), which may be written as follows (ṽ,τ ) = 0, (3.43) R = T (ṽ), (3.44) τ where τ is the vector of discretized τ values, T (ṽ) represents the stream line integration procedure and the residual vector R (ṽ,τ ) is given by R (ṽ,τ ) =Ñ(ṽ)+(η s S + C )ṽ + Q (τ ) F. (3.45) For solving this system we use a simple Picard iteration scheme with a viscous iteration matrix: ( ) (ṽ M i + (ṽ )+ ηs C i+1 ṽ i ) = (ṽ R i,τ i ), (3.46) i+1 τ = T (ṽ i+1 ), (3.47) where the matrix M (ṽ i ) is due to the linearization of the inertia term Ñ(ṽ), η is the iteration viscosity and ṽ 0 =0, τ 0 =0. The best value of η depends on the model parameters, but η = η 0, with η 0 the zero-shear-rate viscosity, is 19

26 often a good choice. For high Deborah numbers and for models with severe elongational thickening some relaxation is often necessary and η should be chosen somewhat larger than η 0. For the matrix (ṽ M i ) we may choose any of the matrices that is used for solving the Navier-Stokes equations (Picard or Newton). For creeping flows we have and thus the iteration matrix in equation (3.46) does not depend on the M iterand =0 ṽ i in that case. The matrix has to be factored only once at the start of the iteration process and the iterands can be found by backsubstitution only. This saves a lot of computing time. For factoring at each iteration is necessary. We note that the Picard iteration scheme M 0 (for creeping flows, i.e. =0 ) is similar to the scheme already used by Viriyayuthakorn & Caswell (1980) M for integral models and used by others more recently (for example Luo & Tanner 1986, Hulsen 1988b). For detecting convergence we use the following criteria ɛ v = max vk i+1 vk i k max k v i+1 k ɛ v, ɛ 2 r = R T R ˆR T ˆR ɛ 2 r, (3.48) where ˆR denotes the residual vector of the system including the equations of the imposed velocities on the boundary ( reaction forces ) and ɛ v and ɛ r are small parameters, for example We note that a convergence criterium of 10 3 is not uncommon for slowly converging Picard schemes such as (3.46) (3.47). For example, Luo & Tanner (1986) reached a value of To reach values for ɛ v that are usually obtained in a Newton-Raphson scheme (say ) would lead to much more iterations, probably not worth the effort. However, faster iteration schemes are needed to reduce the already large number of iterations (O(300)) for obtaining 10 3 at high Deborah and Reynolds numbers. We have the experience that the scheme given above does not converge very well in particular for higher Deborah numbers. These problems can however be resolved if we substitute in the constitutive equation (3.10) not the discontinuous velocity gradients h following directly from the finite-element discretization but smoothed velocity gradients that are made continuous by pre-averaging. This has been done by defining nodal velocity gradient values found by averaging element values and interpolating the nodal values quadratically over elements. Smoothing the velocity gradients is not uncommon in numerical methods for integral models (Luo & Mitsoulis 1990) and in the boundary element method for differential models (Tran-Cong & Phan-Thien 1988). The averaging of the velocity gradients has an unfortunate side-effect, it can lead to velocity modes having an effective viscosity that is near zero, in particular for regular meshes. The wave length of these modes is of the order of the mesh size. This may be illustrated with a one-dimensional regular mesh x i = ih, with linearly interpolated velocity v h.ifv h (x i+1 )= v h (x i ), the velocity gradients in the two elements connected to a mesh point are of opposite value. Averaging leads to a zero velocity gradient value in the mesh points, and thus also to zero 20

27 Table 3.1: CPU timings in seconds. iteration number 1 >1 system building system solving stress computation per mode 7.9 (viscous) stresses. The convergence factor of these modes in the iteration scheme is near 1, which determines the final convergence rate if the long wavelength modes have converged. We can avoid this by increasing the value of η s with a value of η s but at the same time cancelling it by subtracting averaged viscous stresses. The residual vector R in the equations (3.43) and (3.46) now becomes R (ṽ,τ ) =((η s + η s )S + C )ṽ + Q (τ 2 η s d ) F, (3.49) where d denotes the vector of Euler s rate-of-deformation tensors based on the vector h, which is the is the vector of averaged velocity gradients. The procedure in (3.49) only increases the viscosity of short wavelength velocity modes. In our computations we have chosen η s = K k=1 η k, which means that for low Deborah numbers we have Q 0. Although somewhat masked by the formulation of the boundary element method, a similar procedure has been used by Tran-Cong & Phan-Thien (1988). 3.5 Computing times For our computations we have used a HP s mini computer which is rated about fourteen times as fast as a VAX11/780 for floating point computations. To give an idea of the computing times involved, we give some approximate timings in Table 3.1 for mesh4 (669 elements, 1440 nodal points) used by Hulsen & van der Zanden (1990) with a Giesekus model, no inertia terms and for a low Deborah number of 0.4. The decreased system building/solving time after the first iteration is due to the constant iteration matrix in the Picard scheme. Approximate timings for other conditions can be found from the following (approximate) observations The system building time is proportional to the number of elements. The system solving time is proportional to the square of the number of nodal points. The stress computation time is proportional to the number of nodal points and the number of modes. 21

28 The stress computation increases for higher Deborah number because of an increased number of time steps. In our case up to 2.4times for the highest Deborah number (De = 256). If inertia terms are taken into account the building/solving step takes significantly more time ( 2 times as much) and must be fully performed at each iteration. The eigenvalues for the Giesekus model (see Appendix A) can be computed analytically. For models where this is not possible the stress computation times is approximately two times larger. To give two examples, computations with mesh2 (397 elements 876 nodal points) for one mode took about 1 minute and 40 seconds of CPU time (16 iterations) for a small Deborah number and for mesh4with an eight-mode model and a high Deborah number about 13 hours (296 iterations). We note that the stress computation takes up 98% of the CPU time in the latter example. The code for the stress computation is not vectorizable, which means that no substantial decrease in CPU time is expected on traditional super computers. On a Convex 240 vector machine we only found a factor of 1.5 decrease in CPU time where we normally find a factor of 4to 5 for Newtonian flow problems (compared to the HP s). However, the stress computation would be ideal for computers with many parallel processors. 3.6 Conclusions and future developments We have described a numerical method for simulating viscoelastic fluid flows in complex 2D and axisymmetrical geometries. The method is based on a combined finite element/characteristics method. With this method we are able to compute viscoelastic fluid flows up to very high Deborah numbers for various constitutive fluid models. The computing times at present are acceptable for research purposes, but are still rather long for industrial design applications. Improvement of the convergence rate of the iteration scheme is strongly needed. The method is currently restricted to isothermal, steady and 2D or axisymmetrical flows. Many applications require extensions in one or more of these areas and future developments will be directed towards non-isothermal, 3D and timedependent flows. Some applications will include deformation-induced anisotropic heat conduction and three-dimensional flow through non-circular pipe bends. 22

29 Chapter 4 Application: Influence of inertia and elongational properties on the flow through a four-to-one contraction 4.1 Introduction In recent years there has been much interest in contraction flows. In part this is due to their practical importance for various industrial polymer manufacturing processes. However, advances in numerical methods for simulation of viscoelastic flows have also stimulated both experimental and theoretical research in contraction flows. For recent reviews on this subject we refer to Boger (1987) and White, Gotsis & Baird (1987). Earlier research on contraction flows was mainly directed towards the instabilities at higher flow rates ( melt fracture ), see, for example, den Otter (1970). Later also the stable flow patterns and possible mechanisms for the observed large vortices for some fluids were studied (see, for example, Ballenger & White 1971, Cogswell 1972, White 1973, White & Kondo 1977). It was indeed intriguing that some melts (for example LDPE) showed large vortices and other melts (for example HDPE) did not. Cogswell (1972) was the first to suggest that the elongation viscosity is of dominant importance whether vortex growth is seen or not. This idea was also adopted by White & Kondo (1977) and they introduced the term stress-relief-mechanism. They interpret the development of vortices in terms of the development of large extensional stresses in the contraction. They argued that LDPE exhibits such stresses and HDPE does not. In a series of papers on this subject White & Baird (1986,1988a,1988b) showed the importance of elongational properties in planar entry flows by linking the amount of stress growth with the appearance of large vortices. They also did numerical simulations, but these were much less convincing because of convergence problems for higher shear rates. By assuming viscous power-law models for shear and elongational properties Binding (1988) presented an approximate analysis for contraction flows. Although we believe that the transient stress growth behaviour on both shear and elongation are more important than the steady viscosity functions, Binding s results support the conclusion that the elongational properties relative to the shear properties are important and not the elongational properties alone. This important result was also found, but not explained, by White & Baird (1988a). 23

30 The application of numerical methods for a realistic prediction of vortices that are observed in LDPE melts has only been undertaken very recently. White & Baird (1988b) use a Phan-Thien/Tanner model with one relaxation time leading to a rather crude fit of material functions. However, multi-relaxation time K- BKZ integral models have been used by most other authors (Dupont & Crochet 1988, Luo & Mitsoulis 1990, Hoitinga 1990). The reason is that a good fit of LDPE requires a broad relaxation spectrum and mainly multi-relaxation time integral models have been used for that purpose (see, for example, Laun 1978). However, Bird, Armstrong & Hassager (1987) show that a good fit can be made too with a multi-mode Giesekus model. This multi-relaxation-time differential model has been used by Hulsen & van der Zanden (1990) to show that such models are capable of giving realistic results in simulations of contraction flows for LDPE melts up to high flow rates and that in this respect (multi-mode) integral models are not necessarily superior to (multi-mode) differential models. They also study the stress behaviour in the contraction in detail to support the stress-relief-mechanism and the importance of elongational properties relative to shear properties for vortex growth. Some polymer solutions have even larger elongational/shear ratios than LDPE melts and show an even more pronounced vortex-growth behaviour and instabilities at higher flow rates (Boger 1987). The absolute values of the viscosities of polymer solutions are usually orders of magnitude smaller than for polymer melts. This means that for these fluids the Reynolds number may become of the order of unity or higher and inertia may play a significant role in some flows. An effect in contraction flows that is believed to be due to inertia is the divergence of streamlines before entering the contraction (Cable & Boger 1978, Yoganathan & Yarlagadda 1984, Wunderlich, Brunn & Durst 1988, Bol 1989a). Off-center velocity maxima in the contraction region are reported. This so-called divergent flow regime, which is a steady flow, is found only if the flow rate is approached from below. If the flow-rate is decreased from the unsteady flow at high flow rates, the flow remains unsteady in the range where the divergent flow regime was found. See the cited papers for a discussion on the vortex growth regime in which the vortex growths and no divergence of stream lines occurs and the divergent flow regime in which the vortex diminishes and divergence of the stream lines occurs. In this chapter we will investigate the influence of both the inertia and the elongational properties to find a vortex growth regime and a divergent flow regime. We will also try to find an answer to the question whether the divergence of streamlines may be considered as a critical phenomenon. 4.2 Problem description We consider the flow through an axisymmetrical contraction. The geometry has been depicted in Figure 4.1. The contraction ratio 4:1 has been chosen because 24

31 Figure 4.1: Geometry of the 4:1 axisymmetrical contraction. it is rather widely studied in the literature (Boger 1987) and because it has been defined as a benchmark geometry for the workshop on numerical simulation of viscoelastic flow. In order to be able to prescribe developed flow conditions for the velocity at the inflow and outflow and the extra-stress components at the inflow, the length of the entry and exit sections have been taken 20 and 60 times the downstream radius respectively. These values have also been used by Dupont and Crochet (1988) and turn out to be sufficient for our computations. We note that for the higher Deborah numbers the exit length is not long enough to obtain developed stress exit conditions, but this is no necessity as far as properly posed boundary conditions are concerned. We assume no-slip conditions on the fixed wall and symmetry conditions on the center line. Hence, we have the following types of proper boundary conditions according to the mixed hyperbolic-elliptic type of the equations Dirichlet conditions for v on the complete boundary (except for the center line). Developed flow at the inflow and outflow boundary and v = 0 onthe fixed wall. v r =0,t rz = 0 on the center line. Prescribed t k, k =1, 2,...,K at the inflow boundary according to developed flow conditions. For a description of the computation of developed flow conditions we refer to Hulsen (1988a). 25

32 To characterize the elasticity in the flow we will introduce a Deborah number De as follows De = λ n v z d, (4.1) D d where v z d is the mean velocity according to the flow rate in the down stream channel, D d is the diameter of the down stream channel and λ n is the natural time defined by N 1 ( γ) λ n = lim γ 0 2τ( γ) γ =(K η k λ k )/η 0. (4.2) k=1 Here N 1 ( γ) and τ( γ) are the first normal stress difference and shear stress in a simple shear flow respectively, γ is the shear rate and the zero-shear-rate viscosity η 0 is given by η 0 = η s + K k=1 η k. For high Deborah numbers, De is not really a measure for the amount of elasticity in the flow and only serves as a dimensionless linear scale for the flow rate (Hulsen & van der Zanden 1990). The size of the inertia forces in the flow is characterized by a Reynolds number, which we define here as follows Re = ρ v z d D d η 0. (4.3) The Reynolds number Re is the ratio of the inertia stresses ρ v z 2 d and the linear viscous stresses η 0 v z d /D d. We note that Cable & Boger (1978) use a generalized Reynolds number based on the actual (shear) stresses in the flow instead of the linear viscous stresses. The values of such a Reynolds number are usually much larger than obtained from (4.3) due to shear-thinning of the fluid. We have not used such a number because it has only been properly defined for power-law fluids and only gives credit for the shear stresses in the flow and not the much larger elongational stresses. For our simulations we have used a one-mode Phan-Thien/Tanner model (see Appendix A) having η s /η 0 = 1 and an upper-convected derivative (ξ = 0). We 6 have used two different values for ɛ, the parameter in the model that controls the non-linear response. One value leading to a relatively small elongational viscosity (ɛ = 0.25) and the other one leading to a large elongational viscosity (ɛ =0.02). These two values for ɛ have been recommended by R.I. Tanner for the 7th Workshop on Numerical Computations in Viscoelastic Flows to be held in The shear properties of the model have been depicted in Figure 4.2 and the elongational properties in Figure 4.3. The natural time for this model is given by λ n = 5 λ, where λ is the relaxation time of the single mode we have 6 used. Figure 4.4 shows (a part of) the mesh that we have used for our computations. In Figure 4.4 we have also given the number of elements and nodal points. The size of the elements at the sharp corner is approximately 6% of the down stream radius R. 26

33 Figure 4.2: Viscosity η and first normal stress coefficient Ψ 1 for the one-mode Phan-Thien Tanner model with η s = 1 6 η 0 and two different values of ɛ: ɛ =0.25 and ɛ =

34 Figure 4.3: Elongational viscosity η E for the one-mode Phan-Thien Tanner model with η s = 1 6 η 0 and two different values of ɛ: ɛ =0.25 and ɛ =0.02. Figure 4.4: Part of the finite element mesh. 28

35 Table 4.1: Corresponding Deborah numbers De and shear rates γ w used in the computations. De λ γ w (ɛ =0.25) λ γ w (ɛ =0.02) Results We have computed flows for the Deborah numbers given in Table 4.1. For each of the Deborah numbers we have given the corresponding values of λ γ w, where γ w is the shear rate at the wall in the down stream channel Streamlines No inertia terms (Re 1) In the Figures 4.5 and 4.6 we have depicted the streamlines for the various Deborah numbers and for ɛ =0.25 and ɛ =0.02 respectively. We see that vortex growth is small for ɛ =0.25. For ɛ =0.02, where the ratio of elongation/shear properties is much larger, the vortex growth is also much larger, although there is hardly any vortex growth between De = 8 and De = 16. These findings correspond well with the results of Hulsen & van der Zanden (1990), who discuss the importance of elongational properties versus shear properties for vortex growth. Including inertia terms In Figure 4.7 we have depicted the stream lines for ɛ = 0.25 for the various Deborah numbers but including the inertia terms with Re = 10 for De = 16. Comparing this with Figure 4.5 we see that due to the inertia terms the vortex decreases and virtually disappears for increasing flow rate. This behaviour is typically found in the experiments of Raiford et al. (1989) for the polymer solution they used. The results for ɛ =0.02 are given in Figure 4.8. We see that the picture is completely different and that we find indeed a vortex growth regime for De =1, 2 and 4and a divergent flow regime for De = 8 and 16. Note that both the decrease of the vortex size and the divergence of the stream lines in the divergent flow regime corresponds well qualitatively to the experimental results of Cable & Boger (1978). The results in Figures 4.7 and 4.8 show that in order to find a divergent flow regime both the inertia and elasticity effects must be large (Re > 1 and De > 1), but that also the elongational/shear ratio must be large enough such that the vortex can grow for the lower flow rates (vortex growth regime). For the higher flow rates inertia terms start to dominate and the divergent flow regime appears. 29

36 Figure 4.5: Streamlines for the model with ɛ =0.25 and Re =0. 30

37 Figure 4.6: Streamlines for the model with ɛ =0.02 and Re =0. 31

38 Figure 4.7: Streamlines for the model with ɛ =

39 Figure 4.8: Streamlines for the model with ɛ =

40 Figure 4.9: Center-line velocities relative to the mean velocity in the down stream tube Centerline velocities In Figure 4.9 we have depicted the center-line velocity at De = 16 for the various parameters shown. We see that for ɛ = 0.25 there is hardly any velocity overshoot. The effect of inertia terms is only a shift downstream and a smoother transition at the contraction for this case. The shift is also found in the experiments of Raiford et al. (1989). For ɛ =0.02 the overshoot at the contraction is much larger. The effect of the inertia terms is again a shift downstream, but also a still larger overshoot and an undershoot before entering the contraction region. The large overshoot has also been found experimentally by Bol (1989b) for a polymer solution. The undershoot in the divergent flow regime has also been observed experimentally by Cable & Boger (1978), Bol (1989b) and Wunderlich et al. (1988), although the latter results are for planar flow. The undershoot is a result of the mass conservation of the flow at the point where the stream lines diverge Axial velocity profiles In Figure 4.10 we have depicted the axial velocity profiles in the flow at various positions for ɛ =0.25 and De = 16. The profiles should be thought of as being 34

41 Figure 4.10: Axial velocity profiles for the model with ɛ =0.25 and De =

42 Figure 4.11: Axial velocity profiles for the model with ɛ =0.02 and De = 16. positioned at the cross sections where the profiles intersect the fixed wall. The separating streamline of the corner vortex has been plotted by a dashed line. For Re = 10 we note that there is a small off-center maximum in the velocity profile at the contraction, although this can hardly be seen in the picture. In Figure 4.11 the velocity profiles for ɛ =0.02 at De = 16 are given. For Re = 10 we see off-center axial velocity maxima in the contraction region, which are not present for Re = 0. These maxima have also been found experimentally in the divergent flow regime by Cable & Boger (1978), Yoganathan & Yarlagadda (1984), Wunderlich, Brunn & Durst (1988) and Bol (1989a). Here we should make a note that although for De =8,Re = 5 the flow is clearly in the divergent flow regime (see Figure 4.8) there are not yet off-center velocity maxima in this case. However, the undershoot in the center-line velocity is found in this case. 36

43 4.4 Some remarks on the convergence of the iteration scheme In the iteration scheme (3.46), (3.47) we used Newton linearization for the matrix M. For ɛ =0.25 we have used η =2η 0. We found convergence ( ɛ v =10 3, ɛ r =10 3 ) within iterations for any of the cases described above. For ɛ = 0.02, which corresponds to a model with large elongational versus shear properties, obtaining convergence is much more difficult. For the case Re = 0 we used η =4η 0 for De = 1 and 2 and η =7η 0 for De = 4, 8 and 16. For the latter case convergence ( ɛ v =10 3, ɛ r =10 3 ) was obtained after 82 iterations. We also tried higher Deborah numbers (De = 24) but we were not able to find a converged solution. Increasing η did not help. Inspecting the intermediate velocity iterands we found that the stream lines had a divergent flow character similar to the divergent flow regime for non-zero Reynolds number (but no decrease in vortex size). However the streamlines were not steady. It is unclear whether unsteadiness is a result of the iteration process or that no steady solution exists at all. We note that real flows also become unsteady and even turn to large chaotic motion flows at high flow rates, but these are all three-dimensional flows! For ɛ = 0.02 and non-zero Reynolds numbers convergence is even more difficult than for zero Reynolds numbers. For De = 1 and 2 we used η =7η 0, for De =4: η = 13η 0 and for De = 8 and 16 we used η = 25η 0 to obtain convergence ( ɛ v =5 10 4, ɛ r =10 4 ). For the latter case we used 250 iterations! The reason for the difficult convergence may be the larger extensional velocity gradients leading to larger extensional stresses (see Figure 4.9) but also the formation of shock-like structures in the vorticity, as discussed in section 4.5 may corrupt the scheme. We also tried higher Reynolds numbers, but we were not able to find any converged solutions. In this respect we note that in real flows the divergent flow regime is only one of the solutions that exist at the corresponding flow rate. Hence, difficulties in the numerical scheme are also not unlikely. 4.5 Does the appearance of the divergent flow regime correspond to a critical phenomenon? Joseph (1990) discusses several observed phenomena that may be attributed to a change of type of the system of equations. Examples are the Nusselt number and drag becoming independent of the velocity and regions of stagnation for a critical velocity, all for flow over circular cylinders. A change of type occurs when the local velocity becomes equal to the local wave speed c of the so-called elastic shear waves. This is the critical velocity. We can introduce a Mach number Ma = U/c, where U is the magnitude of the velocity. If Ma < 1 the flow is subcritical, ifma > 1 the flow is supercritical and for Ma = 1 the flow is called (trans)critical. The quantity that is directly involved in the change of type is the vorticity ω = curl v. In the domains where Ma < 1 the vorticity equation is 37

44 elliptic, and it is hyperbolic for Ma > 1. This is all very similar to the situation in gas dynamics, although the quantities involved here are derived quantities (vorticity) and not primitive variables such as pressure and velocity. Change of type for a critical velocity is, strictly speaking, only possible for models of Maxwell type, i.e. models with η s = 0. For Jeffreys-type models (η s 0) no change of type can occur. However if η s is small, the characteristics of the system with η s = 0 substituted, may be considered as subcharacteristics determining the flow field in large parts of the domain, except where gradients are large. This situation may be compared with the convection-diffusion equation if convection dominates. In that case, in large parts of the domain the solution is determined by convection only. In our Maxwell-type system shocks in the vorticity may appear. Including a term with small η s smoothes the shocks (Joseph 1990). For the Maxwell-type models having an upper-convected derivative it is possible to show that the vorticity ω fulfills the following second-order equation (Joseph et al. 1985, Hulsen 1986, Joseph 1990) 2 ω div(e grad ω) = l.o.t., (4.4) t2 where l.o.t. stands for lower-order terms and the tensor e is given by K η k e =( )I + τ ρ v v = λ k k=1 K k=1 η k λ k b k ρ v v, (4.5) with τ according (2.4) and b k = I +(λ k /η k )τ k (see Appendix A). For steady flows we have div(e grad ω) = l.o.t. (4.6) This equation for the vorticity ω is elliptic whenever e is a (negative or positive) definite tensor (all three eigenvalues having equal sign) and hyperbolic whenever e is indefinite (eigenvalues having different signs). We note that for the models we use, the tensors b k are all positive definite (see Hulsen 1990) so that the tensor e is either positive definite or indefinite. For planar and axisymmetrical flows we can find the elliptic/hyperbolic regions by considering the sign of = e xx e yy e 2 xy or = e rr e zz e 2 rz respectively as follows > 0: elliptic < 0: hyperbolic. For = 0 the flow is transcritical, but this is usually a region of lower dimension (curve) separating regions of ellipticity and hyperbolicity. The region > 0 corresponds to the Ma < 1 region and < 0 to the Ma > 1 region. Now returning to our contraction flow we may ask ourselves: is the appearance of a divergent flow regime a critical phenomenon? We have a Jeffreys-type model 38

45 with η s = 1 η 6 0. Thus η s is smaller than η 0 but not several orders of magnitude smaller. However, considering the elongational viscosities η E (see Figure 4.3), which are of the order of O(6η 0 ) for ɛ =0.25 and O(80η 0 ) for ɛ =0.02, the ratio is η s /η E is somewhat smaller. Strictly speaking there is no change of type and, to be sure, we should have run our computations again for a model having η s = 0. We have not done that because for a one-mode model convergence is much more difficult and much more iterations are necessary at high Deborah and Reynolds numbers, possibly due to the increased shear-thinning. We have also some doubts on the correct treatment of hyperbolicity in our scheme for the Maxwell-type models with only a limited relaxation spectrum (small number of modes). For models with a large spectrum there is no problem because there is always enough diffusion (viscosity) of the small relaxation times (see Hulsen & van der Zanden 1990). We also refer to the book of Joseph (1990, p. 199), where he notes that, with respect to critical phenomena, the question whether polymer solutions should be modelled by Maxwell or Jeffrey models, is still unresolved. However, considering the remark on subcharacteristics given above, we have computed the value of to get some indication of a change of type. In the Figures 4.12 and 4.13 we have depicted the > 0 and < 0 regions in the flow for the models with ɛ = 0.25 and ɛ = 0.02 respectively. Comparing Figure 4.13 with Figure 4.8 we see that the divergent flow regime starts to appear when the < 0 domain starts spreading in the domain upstream of the contraction. This supports the critical phenomenon hypothesis. However, for ɛ = 0.25 the region where < 0 is even larger (due to the smaller values of τ zz in equation 4.5). This means that it is not the only explanation. Typical of supercritical flows is the appearance of upstream regions of silence where ω 0 (Joseph 1990). The only non-zero component of ω has been denoted by ω. The large production of vorticity due to the contraction is not felt in these regions because the liquid velocity is larger than the speed of the waves that should transport this information upstream. In the Figures 4.14 and 4.15 we have given contour plots of the vorticity for the models with ɛ =0.25 and ɛ =0.02 respectively. The lines with ω = 0 have been indicated with a 0 (the axis of symmetry also has ω = 0). We see that an upstream region where ω 0 appears and that the separation between the regions with ω 0 and ω 0 becomes sharper for higher flow rates. This can be seen more clearly in the 3D pictures given in Figure 4.16 and Figure In the last picture of Figure 4.17 there even seems to appear a shock-like structure in ω, although no real shock can occur because η s 0. We should note that the vertical axis (vorticity) in each of the figures is scaled relative to the maximum value, which is a non-linear function of De (or Re). The vorticity on the fixed wall in the down stream channel scales approximately linear with De and may be used as a reference to compare the figures. We have seen that the appearance of the divergent streamlines corresponds to the flow becoming supercritical, but only for ɛ =0.02 and not for ɛ =0.25. We 39

46 Figure 4.12: Regions where > 0 and < 0 for the model with ɛ =

47 Figure 4.13: Regions where > 0 and < 0 for the model with ɛ =

48 Figure 4.14: Vorticity contours of the model with ɛ = Contours of zero vorticity are indicated with a 0. 42

49 Figure 4.15: Vorticity contours of the model with ɛ = Contours of zero vorticity are indicated with a 0. 43

50 Figure 4.16: Vorticity ( ω) for the model with ɛ =

51 Figure 4.17: Vorticity ( ω) for the model with ɛ =

52 Figure 4.18: Vorticity contours of the model with ɛ =0.02, De = 16, Re = 0. Contours of zero vorticity are indicated with a 0. believe that this is due to the larger elongational stresses leading to pronounced vortex growth in the Re = 0 case. There are two competing effects: the increase in the vortex due to elongational stresses and the decrease due to the inertia. These must balance in some way to maintain a vortex. If the flow becomes supercritical the region of ω 0( potential flow) interacts with the vortex increasing mechanism, which is a vorticity producing mechanism. The latter effect can be seen if we compute the vorticity for the case ɛ =0.02, De = 16, Re = 0 (last picture in Figure 4.6). In Figure 4.18 we have given a contour plot and in Figure 4.19 a 3D-plot. The incompatibility of the two vorticity distributions leads to the strange divergent streamline pattern. For ɛ = 0.25 the vortex enhancement mechanism is too weak to maintain the vortex at the point where the flow becomes supercritical. The flow pattern is already close to the flow field found in the Newtonian case, which is given in Figure The contour plot and the 3D plot of the vorticity are given in the Figures 4.21 and These figures together with the Figures 4.7, 4.14 and 4.16 show that in the upstream flow a region having ω 0 can develop without having much effect on the stream line pattern, because ω is already close to zero even for the Newtonian case. We may conclude that for the contraction flow we have studied, the supercritical flow may lead to unexpected effects only when the vorticity-free region is incompatible with the flow field effects that are due to the viscoelasticity. Hence, when studying supercritical flow, the viscoelastic constitutive equation should be chosen carefully because critical phenomena may be a result of the interaction between viscoelastic effects (such as vortex enhancement) and inertia effects. We have shown that the divergent flow regime is probably a critical phenomenon, but we have to stress that in our case we have a Jeffreys-type model and that no real change of type occurs, only the subcharacteristics change. Furthermore, we have to stress that our algorithm is based on the type that is found 46

53 Figure 4.19: Vorticity ( ω) for the model with ɛ =0.02, De = 16, Re =0. Figure 4.20: Streamlines for a Newtonian model with Re =

54 Figure 4.21: Vorticity contours for a Newtonian model with Re = 10. Figure 4.22: Vorticity ( ω) for a Newtonian model with Re =0. 48