Computation of steady three-dimensional flow in a model of the basilar artery

J Biomechona Vol. 25, No I?, pp 1451 1465. 1992. Pnnted I” Great Brrtam

CMX-9290192 %5X0+ .oO

Pergamon Press Ltd

COMPUTATION OF STEADY THREE-DIMENSIONAL FLOW IN A MODEL OF THE BASILAR ARTERY

J. K. B. KRIJGER,* R. M. HEETHAAR,~ B. HILLEN,* H. W. HOOGSTRATEN~ and J. RAVENSBERGEN*

Departments of *Functional Anatomy and TMedical Physics University of Utrecht, Catharijnesingel 59, 3511 GG Utrecht. The Netherlands and IDepartment of Mathematics, University of Groningen,

Groningen, The Netherlands

Abstract-The flow in the basilar artery arises from the merging of the flows from the two vertebral arteries. To study the flow phenomena in the basilar artery, computations have been performed using a finite element (FE) method. We consider steady flow in a geometrically symmetric confluence. For simplicity, channels with a rectangular cross-section have been used. Both symmetric and asymmetric flow cases have been considered. The results show that for the Reynolds number of interest the flow downstream of the junction is highly three-dimensional, and that the flow at the end of the basilar artery, where it splits again, will not be fully developed. The computed phenomena have been confirmed by laser Doppler velocity measurements.

NOMENCLATURE

(All quantities are non-dimensional unless indicated otherwise.)

vector containing boundary conditions in FE model downstream half-channel width (2.5 mm) inlet length divergence matrix in FE model pressure matrix in FE model convection matrix in FE model tolerance parameter (%) in the definition of L pressure pressure averaged over cross-section vector containing all pressure unknowns in FE model Reynolds number (= pOl/p) Reynolds number based on hydraulic diameter of downstream channel diffusion matrix in FE model x-, y-. z-component of velocity vector containing all velocity unknowns in FE model mean axial velocity component (m s- ‘) coordinates distance downstream of apex (mm) asymmetry parameter. penalty parameter in FE model dynamic fluid viscosity (kg m s- ‘) fluid density (kg m _ ‘) velocity potential stream function z-component of vorticity in three-dimensional model

INTRODC’CTION

Blood flow in arterial bifurcations has attracted much attention in the literature during the last decade. Al- though flow phenomena in bifurcations are interesting in themselves, most authors study these flows to understand why atherosclerotic plaques often develop

Received in ,jinal form 2 April 1992.

in regions of vessel bifurcations. The location of these plaques is believed to be related to fluid-dynamical phenomena. For example, the flow in the carotid artery bifurcation has been studied intensively. The geometry of this bifurcation is asymmetric. Steady flow in a rigid glass model has been studied experimentally by Bharadvaj et al. (1982a, b), making use of flow visualisation and laser Doppler velocity measurements. Flow in the same configuration has been studied both experimentally and numerically by Rindt et al. (1987) for two-dimensional steady and pulsatile flow. Results of measurements and computations of the complex steady three-dimensional flow in this asymmetric bifurcation can be found in Rindt (1989). Recently, Reuderink (1991) presented computational results for the pulsatile flow in a three- dimensional distensible model of the carotid artery bifurcation. Steady two-dimensional flow in a sym- metrical bifurcation is computed by Bramley and Den- nis (1984) and Bramley and Sloan (1987). Rieu et al. (1989) performed measurements of steady and pulsatile flow in symmetric bifurcation models with vari- ous branching angles. For experimental reasons they used channels with a rectangular cross-section instead of circular tubes. They showed that the main features of the flow are similar to those in circular tubes.

Comparing the three-dimensional and two-dimensional results presented in the literature, it becomes clear that the flow in a bifurcation is highly three- dimensional and that two-dimensional models do not predict satisfactorily what will happen in a real three- dimensional geometry. Nevertheless, two-dimensional models can be useful as a first step in the development of a three-dimensional model and the estimates of general parameters such as inlet length can be given.

Far less is known about the reverse problem of (blood) flow in a confluence. Schroter and Sudlow (1969) studied the flow in a symmetric confluence

1451

1452 J. K. B. KRUGER et al.

experimentally, modelling flow in the human respirat- ory system. They measured velocity profiles and vis- ualised flow patterns both for flow from one main tube into two smaller ones (inspiration) and for the reverse problem (expiration). Pedley et al. (1971) also paid some attention to the problem. The reason that the problem of two arterial flows coming together has attracted little attention is the fact that almost every- where in the human body arteries divide and there exists only one location were two large arteries come together: the basilar artery. This artery arises from the union of the two vertebral arteries; it has a diameter of 3-5 mm and a length of 30-50 mm. At its downstream end the artery bifurcates into both posterior cerebral arteries, which are part of the circle of Wills, an arrangement of arteries that distributes the blood to the brain.

It is important to gain an insight in the flow patterns in the basilar artery for the following reasons. First, the basilar artery is often affected by atherosclerosis, just like the carotid bifurcation. It is interesting to test whether the flow phenomena that are commonly thought to enhance atherogenesis in the carotid bifurcation are also encountered in the basilar artery.

The second point of interest is the question whether the flow in the circle of Willis can be influenced by the flow in the basilar artery. Since the basilar artery is relatively short, the flow will not be fully developed at its end, as can be concluded from our previous research (Krijger et al., 1989). An asymmetry in the flow at the entrance of the basilar artery, caused by unequal inflow from the vertebral arteries, can still be present at the end of the basilar artery. Under such circumstances, the flow in the circle of Willis, which is known to be very sensitive to small disturbances (Hillen et al., 1988), is probably influenced.

To study these issues, we have computed the steady three-dimensional flow of a Newtonian fluid in a geometrically symmetric confluence. For simplicity the cross-section of the channels was chosen to be rectangular. In this way the junction geometry is fully char- acterised by the branching angle, the height and width of the channels and the radius of curvature of the channel outer walls at the junction. This model has the computational advantage that a finite-element mesh is much more easily generated near the junction than in the case of the junction of circular cylinders. Furthermore, the manufacturing of an experimental model is less complicated and the measurements with the laser Doppler technique are not compromised by refraction problems at the wall of the model. Rieu et al. (1989) have demonstrated that bifurcating flow in a model with rectangular channels is to a large extent similar to bifurcating flow in circular cylinders. We think that the same agreement still exists in our case when the flow direction is reversed. We have prelimi- nary experimental evidence that supports this view.

Near the junction the flow has been computed by means of a finite element program, solving the full Navier-Stokes equations. Since this method is com-

putationally very expensive, we used sufficiently far downstream of the junction a simplified version of the governing Navier-Stokes equations. Solving these parabolised Navier-Stokes equations by means of a finite difference method is much cheaper in terms of computer time. A comparison of the results with the solution of the full Navier-Stokes equations shows that the difference is negligible. At the transition of the two parts of the model, the numerical results of the finite element program have been used as an inflow boundary condition for the finite difference program. In our study we do not take into account the fact that, after a finite length the basilar artery bifurcates; small branches of the basilar artery are neglected as well.

First, we computed symmetric flow in our model of the basilar artery for Reynolds numbers (based on the downstream mean flow velocity and downstream half-channel width) between 50 and 250. It was found that for higher Reynolds numbers (2 150) secondary flow effects become very important and they deform the axial velocity profile considerably. Next, to answer the question what will happen when one of the vertebral arteries supplies more blood than the other, asymmetric flow was considered too.

MATHEMATICAL FORMULATION

Two identical channels with rectangular cross- section meet symmetrically under an angle of 60” to form a single rectangular downstream channel. In Fig. 1 we present the geometrical details of this confluence. The curved part of the outer walls in the yz-plane is a circular arc with radius 2.5 mm. We consider steady flow of an incompressible, Newtonian viscous fluid from the two smaller branches, merging into a single flow in the downstream channel. Note that the total cross-sectional area upstream (24 mm2) is larger than the downstream area (20 mm’). This leads to a flow acceleration at the transition of two channels into one, so recirculation regions are not very likely to occur in this flow geometry. This reduction of the cross-section is in accordance with the geometry found in the human vascular system.

Non-dimensional spatial variables (x, y,z), velocity components (u, v, w) and pressure, p, are introduced by scaling their dimensional counterparts with, respectively, the length scale 1=2.5 mm, the (known) downstream mean axial velocity ti and the characteristic pressure plir2, where p denotes the fluid density. Defining the Reynolds number Re = pSl/p, where p is the dynamic viscosity coefficient of the fluid, the governing Navier-Stokes momentum equations can be written as

uu,+vuy+wu,= --P,+$(u,,+v,+u,),

uv,+vuy+wu.= -_p,+~(uxx+v,+v,,),

uw, + vwy + ww, = -_pz+~(wxx+w,,+w*z), (1)

A model of the basilar artery 1453

43 d- lmm -

Cross-section 1 y,z - plane

3

(b)

-4:m - 1

L _ _ _I

Cross-section 2 x,z - plane

Fig. 1. Definition sketches, (a) and (b), and finite element At the channel walls we have the boundary condi- discretisation (c). tions

and the continuity equation reads

u,+u,+w,=o. (2)

At the channel walls the solution should satisfy the no-slip condition II = v = w = 0. At both inflow boundaries we prescribe fully developed flow with a given flux in a rectangular channel, which can be expressed analytically in the form of a Fourier series (see Appen- dix). The outflow boundary condition will be discussed in the next section.

The numerical solution of this boundary value problem will be obtained by use of a fully three- dimensional finite element technique (see next section). The flow in the region consisting of the two inflow channels and a certain finite downstream part of the common channel has been computed by this technique. The flow in the remaining part of the downstream channel has been computed by solving a simplified set of equations, the ‘parabolized Navier-Stokes (PNS) equations, using the computed finite element solution as an entrance boundary condition. The PNS equations are much cheaper to solve than the full Navier-Stokes equations (see Briley and McDonald, 1984).

The PNS equations arise from deletion of the streamwise diffusion terms u,,, u,, and w,, from equations (1) and assuming that the axial pressure gradient

pZ can be replaced by dfi/dz, where i(z) denotes the pressure averaged over the channel cross-section. In general, this simplification is justified when the streamwise evolution of the flow has become relatively slow. The validity of the PNS approximation has been checked by computing the flow field for the symmetric case from 8.0 to 25.0 mm downstream from the apex by both the finite element and the PNS method. The results agree within 1%.

The PNS equations can be written in a convenient form by applying a transformation introduced by Rubin et al. (1977). Defining the ‘velocity potential’, 4, the ‘stream function’, $, and the z-component, R, of the vorticity by

u=&+l(ly, o=&-l(l,, R=ll-0,.

we obtain four equations for 4, $, w and R:

&x+&y= -wz* 1(1,X+$yy=Q>

uR,+vQy+wR,-w,R+w,u,

-w,o,=~(n,,+R,,), (3)

dj 1 uw,+uwg+ww,=dl+~(w”“+wYY).

The pressure gradient dlj/dz is found by adjusting it at each z-location in such a way that the total flux sj w dx dy is constant for all z. This can easily be done since dfi/dz occurs only linearly in the equations.

a4 u=u=w=+--&=o a

-= normal derivative an >

.

Prescribing $ =&$/an = 0 at the wall ensures that the normal velocity is zero. To satisfy the no-slip boundary condition for the tangential velocity, we have to specify an appropriate wall value of R; see Fletcher (1988, p. 288). The entrance boundary conditions for the PNS equations will be discussed in the next section. Since the PNS equations are parabolic, no outflow boundary condition is required.

Briley and McDonald (1984) have shown that the PNS equations are valid when the contribution from 4 to the transverse velocities (u, u) is small compared to the contribution from 1/1. In this so-called small velocity potential approach the secondary velocities can be of the same order of magnitude as the streamwise velocities. An extension of this method has been used to compute the flow in a rectangular channel with a 90” bend (Govindan et al., 1991). The results compare well with a solution of the full Navier-Stokes equations and experimental data.

Finally, we mention that for all computations the plane x = 0 is a symmetry plane. In the case of symmetric inflow, i.e. if the fluxes from the two incoming channels are equal, there is another plane of symmetry: y =O. Prescribing symmetry for the solution can be risky since for non-linear equations like the Navier-Stokes equations the solution of a symmetric

problem is not necessarily symmetric. However, we The element used is a 27-noded isoparametric hexa- have experimental evidence that these phenomena do hedral element with 81 unknowns for the velocity and not play a role for the range of Reynolds numbers we four for the pressure. The velocity components are consider. For higher Reynolds numbers (Re>400) approximated quadratically, the pressure varies lin- symmetry-breaking instabilities, resulting in a period- early per element. The four degrees of freedom in the ically oscillating flow downstream of the apex, have pressure consist of the pressure and three spatial pres- been observed in a Aow visualisation experiment (un- sure derivatives defined at the centre of the element. published results). The resulting sets of linearised equations are solved by

a direct method: asymmetric LU factorisation with-

NUMERICAL METHODS out pivoting. Since a small value of the penalty parameter, E, results in a poor conditioning of the system

Thefinite element method (FEM) matrix, iterative methods are difficult to apply to solve

For a theoretical description of the application of the sparse linear systems of equations.

FEM to fluid flow problems see Cuvelier et al. (1986). The method we use is implemented in the FEM-

Boundary conditions

package SEPRAN. It has also been applied by van de At the inflow boundaries a fully developed flow in

Vosse et al. (1989) to compute the flow in a tube with a rectangular duct with a given volume flux is pre-

a 90” bend for Re between 100 and 500. Due to the scribed (see Appendix). Since the junction has a cer-

strong curvature of the tube, the flow is highly three- tain upstream influence, the inflow boundaries have

dimensional. A comparison with the flow velocity to be located sufficiently far upstream. Numerical

measurements showed a good agreement between experiments have shown that the upstream influence

computation and experiment, although a relatively is not very large; so the length of the inflow channels

coarse computational mesh of only 220 elements with as shown in Fig. 1 is sufficient. For the symmetric flow

2205 nodes was used. case the inflow flux has been chosen such that the

The method makes use of a standard Galerkin mean velocity in the downstream channel is equal to

approach to obtain a finite element discretisation of unity. In the asymmetric flow case the flux in the left

the Navier-Stokes equations. The discretised equa- duct has been multiplied by 1 + a and in the right duct

tions have the form of a set of non-linear algebraic by 1 -a such that the ratio between the left and right

equations: fluxes is equal to the prescribed asymmetry ratio, y. At the fixed walls all velocity components are pre-

{S+N(u)}u+L’p=b (momentum equation), scribed as zero. At the symmetry planes symmetry

Lu=O (continuity equation), boundary conditions have been used. At the outflow boundary we have to prescribe an outflow boundary

where u denotes the vector of velocity unknowns, condition. We use the so-called stress-free outflow p the vector of pressure unknowns, S the diffusion condition; see van de Vosse et al. (1989). This is an matrix, N(u) the convection matrix, L the divergence artificial boundary condition; so near the outflow matrix, b the vector containing the boundary condi- boundary the solution differs from the solution in an tions. For details we refer to van de Vosse et al. (1989) infinitely long duct. Numerical experiments with dif- and Cuvelier et al. (1986). These equations can be ferent locations of the outflow boundary have shown solved iteratively by solving at each step a linearised that the solution is affected only over a distance of one set of equations (Newton-Raphson method). Direct element upstream by the imposition of this outflow solution of the linearised set is very time-consuming boundary condition. since p does not occur in the continuity equation, which results in zero components on the main diag- Finite element mesh and iteration method

onal of the system matrix. This makes a partial pivot- The division of the geometry in finite elements is ing procedure necessary. To circumvent this problem, shown in Fig. l(c). Near the junction the elements the penalty function method is widely used. In this have been concentrated, since in that region the most approach the continuity equation is replaced by complicated flow phenomena are to be expected. It

u,+v,+w,= --Ep, also proved to be necessary to make the elements smaller near the wall parallel to the yz-plane, in order

where E is a small penalty parameter, ~4 1. The dis- to prevent the solution to develop wiggles. The depic- cretised version of this equation is ted mesh was used to compute the asymmetric flow

L.u=&M,p case. It contains 400 elements and 4023 nodes. For the symmetric flow computations only the left half of the

with pressure matrix M,. Elimination of p leads to the mesh has been used (200 elements, 2079 nodes). The single equation Newton-Raphson iterative method we applied to

{S+N(u)}u+iLTM;’ Lu=b solve the non-linear set of equations has the advantage of a fast (quadratic) convergence. However, to ensure convergence a good starting value is needed.

for the vector of velocity unknowns, u. For the iteration process the starting values were

1454 J. K. B. KWGER et al.


provided by the Stokes flow solution (Re=O). Next the solution has been computed for Re = 50, 150 and 250, using the preceding solution as a starting value. For symmetric flow a total number of 14 iterations was needed using 1912 cpu seconds on a Convex ~230 mini super computer. The asymmetric case, with twice as many elements, required 16 iterations, using 7631 cpu seconds; so doubling the number of elements requires four times as much computer time.

PNS equations

The PNS equations are discretised with central differences on a computational grid that is uniform in both the x- and the y-direction. The x-interval is divided into 12 subintervals, the y-interval into 15 subintervals for the symmetric flow case and 30 subintervals for the asymmetric flow case (y=2). In the z-direction the steps are gradually increasing downstream since for larger z-values the flow development becomes more and more slow. A total number of 22 steps has been taken in the z-direction.

As inflow condition the solution computed by the FEM program is used at the location zdim = 10.5 mm, which is one element upstream of the outflow boundary of the FEM domain. The values of (u, u, w) are interpolated on the PNS grid and the required value of R is computed by numerical differentiation. Next the PNS solution is computed in one downstream sweep, solving the coupled set of equations (3) by the same method as used by us previously; see Krijger et al. (1989). The computational cost per grid point is cheaper by a factor 500 compared to the FEM solution method.

RESULTS

Numerical results are presented for Re = 50 and 250 (based on the downstream half-channel width I- 2.5 mm), for both symmetric and asymmetric flow. If the hydraulic diameter of the downstream channel (four times the cross-sectional area divided by the circumference) is chosen as the characteristic length (4.44 mm), we have the Reynolds numbers Rehd = 88.9 and Re,,=444.4, respectively. Since the flow phenomena for the higher Reynolds number Re = 250 are the most interesting ones, the majority of the plots shown apply to that case.

In Fig. 2(a) and (b) axial velocity profiles are depicted for symmetric flow at Re = 250. On the left the dimensional distance from the apex (zdim) is indicated. The level zdim= 2.3 mm is the first downstream level for which the channel width in the y-direction is 5 mm (see Fig. 1). Fig. 2(a) shows that the initial double-hump velocity profile visible in cross-section 1 (the yz-plane) evolves very rapidly into a profile with a single central maximum. Looking at cross-section 2 (the xz-plane), which is perpendicular to cross-section 1, different phenomena can be observed [Fig. 2(b)]. The velocity profile, which initially possesses one maximum, transforms downstream into

a profile with two maxima, after which it changes slowly again into a profile with one maximum as the flow reaches the fully developed state. Near the walls in cross-section 2 the velocity profiles are very steep. For that reason the computational elements have been concentrated there, as indicated in the preceding section.

The cause of these curious phenomena can be seen in Fig. 2(c): it is the structure of the secondary velocities developing in the flow because of the change of direction the flow has to make near the junction. The figure shows the secondary velocities at the first five locations indicated on the scale on the left of Fig. 2(a). The secondary velocities show a four-vortex structure as has already been noted by Schroter and Sudlow (1969). In the downstream direction their magnitude decreases, and the centres of the upper and lower vortices come closer together. Numerical results (not presented here) show that the four-vortex structure persists very far downstream, decaying slowly in magnitude. The phenomena observed from Fig. 2(a) and (b) can be explained from the presence of the four- vortex structure of the secondary velocities. First the two maxima in the initial profiles of Fig. 2(a) are convected towards each other, next the fluid ac- cumulating at the centre of the channel is transported in the negative and positive x-direction [upward and downward in Fig. 2(c)], resulting in double- hump profiles in cross-section 2 downstream of zdim = 14.9 mm.

Axial velocity profiles for asymmetric flow at Re=250 are presented in Fig. 3(a) and (b). The ratio y of the fluxes from the left and the right entrance channel is equal to 2. In Fig. 3(a) a velocity profile with two unequal maxima is present at Zdim = 2.3 mm, the larger maximum being at the left. As the flow evolves downstream, this maximum crosses the channel and at the location zdim = 26.5 mm the situation is completely reversed: the larger maximum is now located at the right. Further downstream the asymmetry disappears slowly, but at the last location depicted the velocity profile is still asymmetric. Fig. 3(b) shows the development of the velocity profiles in cross-section 2. Roughly the same phenomena can be observed as in Fig. 2(b): the initial profile with one central maximum transforms into a double-peaked velocity profile [with a much deeper valley between the peaks than in Fig. 2(b)], and further downstream a profile with a central maximum is developing slowly. The corresponding secondary velocities for the first five z-locations of Fig. 3(a) and (b) are depicted in Fig. 3(c). As in the symmetric case, there is again a four-vortex structure, the two vortices at the left (resulting from the channel with the largest inflow) being stronger than the right-hand-side vortices. Downstream the two stronger vortices gradually tend to dominate the weaker ones. At zdim= 10.5 mm the weaker vortices have almost disappeared. This process continues downstream until only two slowly decaying vortices remain.


cross-section 1 cross-section 2

94.4

75.9

61.1

49.3

39.9

32.4

26.5

21.7

17.9

14.9 12.5

1K . 6.4

;::

(a) (b)

Fig. 2. (a), (b) Velocity profiles of primary velocity w in cross-sections 1 and 2 for symmetric inflow (y = 1) and Re= 250. The scale on the left indicates the downstream distance (z& from the apex, expressed in mm. (c) Vector plots of the corresponding secondary velocity (u, u) at the first five locations of Fig. 2(a) and (b).. The plots (c) have been arranged downstreamwise (upwards from the bottom picture) and their velocity

scale is not the same as in Fig. 2(a) and (b).

A model of the basilar artery

cross- sect ion 1 cross-section 2

94.4

75.9

61.1

49.3

39.9

32.4

26.5

21.7

17.9

14.9

t?S .

K

;:i

(a) (b)

. _ . ..-. -II.-.. .

. ---.-. . .

*.*-- _,), _...

; : ;;,-\\ 1 ’ - : ; : \ \\\ \ I-?-y;:,,, . \\\\4 . .\ -=zdf - 2;) f \ .a,,,

..4

\ . ..-c.. , . __-_. .

. .\_I\ .-. . .

___ccccHcI_. .- . .

* H/P\, ,_ . . *

Fig. 3. Primary velocity profiles and secondary velocity vector plots analogous to Fig. 2 for the asymmetric case (y = 2) and Re = 250. The velocity scalings are the same as in Fig. 2.

In Figs 4 and 5 the axial velocity profiles are depic- Fig. 4(b) the corresponding asymmetric case (with ted in a different way to get a more complete picture of inflow ratio y=2). For this relatively small Re value the flow development in the downstream channel. the flow development is quite fast: at the last location Figure 4(a) shows the symmetric case at Re = 50, and shown (zdim= 10.5 mm) the flow has almost fully de-

1458

Lb)

J. K. B. KRIJGER et al.

(al (b)

Fig. 4. Plots of the primary velocity w at the first five locations of Fig. 2(a) and (b) (z,,,=2.3, 4.4, 6.4, 8.4 and 10.5 mm) for (a) symmetric inflow (y = 1) and (b) asymmetric inflow (y = 2), both for Re = 50. The plots have been arranged

downstreamwise (upwards from the bottom picture).

veloped in both cases. In the asymmetric case [Fig. 4(b)] the initial left-hand-side maximum has just crossed the channel centre line. For higher Reynolds numbers the flow phenomena are more complex as can be seen for Re= 250 in Fig. 5(a) (symmetric case) and S(b) (asymmetric case, r=2). It is clearly visible how the axial velocity profiles are deformed by the presence of secondary velocities. Since in this case the distance for the flow to become fully developed is much longer than in the low Re case, the velocity profiles are shown at 10 locations extending over a larger distance.

Vector plots for the (u, w) velocity components are given in Fig. 6. A plot for symmetric two-dimensional flow (independent of x) at Re= 250, to test the validity of a two-dimensional approach, is given in Fig. 6(a). There is a small upstream influence near the junction since the fluid tends to move towards the outer walls. Downstream of the confluence the flow is almost

Fig. 5. Results analogous to th ose of Figure 4, but now for Re = 250, given at ten locations (z,,,=2.3, 4.4, 6.4, 8.4, 10.5, 14.9,21.7, 32.4,49.3,75.9mm) ‘I he first five locations are the

same as in Fig. 4.


f i= C -

C

-

t

i= -

-

C

C

i=

f z= C

C

C

z=

_

c z ,’ C C C

z c C

c C

c C

c C c

C

c C c

C

c C

C

l= t C

L C

c C c C

C C


___ - _ _ _ _ - - c c c _ - _ - - - -

~cccccccI-~


parallel to the channel walls, the secondary velocity component, u, being present just downstream of the apex, disappearing very fast. At the downstream end of the figure the axial velocity still has two clear maxima.

Figure 6(b) shows the corresponding velocity vectors (u, w) for the three-dimensional case with symmetric inflow at Re= 250 in the plane x=0 (cross-section 1). Since x = 0 is a symmetry plane for the flow, the secondary velocity u is zero here. The flow phenomena downstream of the junction are seen to be quite different from the two-dimensional case. The velocity vectors are inclined towards the centre line; so the fluid is accelerated there and at the downstream end of the figure a velocity profile with one central maximum is formed. The upstream effect of the junction is slight: the velocity vectors remain parallel to the inflow direction until just upstream of the apex. At that point the flow starts to move towards the outer wall. This upstream effect has also been observed by van de Vosse et al. (1989) for flow in a circular duct with a 90” bend. In Fig. 6(c) the velocity vectors (0, w) are drawn for a plane parallel to the symmetry plane x =O, with a distance 1.83 mm from the plane x=0 and 0.17 mm from the fixed channel wall. Since the velocities are smaller near a fixed wall, the vectors in Fig. 6(c) have been enlarged by a factor 2 compared to Fig. 6(b). Downstream of the apex the velocity vectors are directed outwards from the centre line. This hap- pens because in this part of the channel the secondary vortices are carrying fluid outwards from the xz-plane (cross-section 2). The magnitude of the velocity vectors is seen to be increasing in the downstream direction. This effect can also be seen in Fig. 2(b), where large velocities occur near the walls in the first part of the downstream channel. Further downstream in Fig. 2(b) the velocities near the wall become smaller again. The upstream effect of the junction is similar to that in Fig. 6(b); however, the effect is more significant in Fig. 6(c).

Analogous to Fig. 6, we present in Fig. 7 the velocity vector plots for asymmetric flow (inflow ratio 7 = 2) at Re= 250. The two-dimensional pattern is depicted in Fig. 7(a), which shows roughly the same phenomena as the corresponding Fig. 6(a). The velocity in the downstream channel is largely parallel to the outer walls; as a consequence, the region with the largest velocities is at the same side as the inflow channel with the largest flux. Figure 7(b) and (c)for the

three-dimensional case again show flow phenomena which are completely different from the two-dimensional case. In Fig. 7(b) the fluid with the highest velocity is seen to cross the channel centre line; so sufficiently far downstream the highest velocities now occur on the side of the channel opposite to the side with the largest inflow. In Fig. 7(c) the velocity vectors have been enlarged by a factor 2, as in Fig. 6(c). In this figure the secondary velocities in the downstream channel are directed in the opposite direction compared to Fig. 7(b).

To give an impression of the rate of decay of the secondary velocities in the downstream direction, we computed the mean secondary velocity as the cross- sectional average of (u2 + u’)~‘* at a fixed z-value for Zdim 2 2.3 mm. For these values of zdim the channel is straight and it is possible to define unambiguously one primary (z) and a two secondary (x, y) directions. In Fig. 8 the mean secondary velocity is represented as a percentage of the mean axial velocity in the downstream channel for Re= 50 and Re= 250, both for symmetric and asymmetric inflow (y= 2). Two important phenomena can be observed. First, the secondary effects for the asymmetric case are larger than for the symmetric case. Second, the mean secondary velocity is larger and decays slower for the higher Reynolds number.

Finally, we present some computed values of the inlet length; see Table 1. The inlet length L measures the distance downstream from the apex for the flow to become approximately fully developed. It is given implicitly by

MAXlw(x, y, L)-w(x, y. co)1 =& w(O,O. rcl x. Y

(n= 1, 2 or 3). (4)

This definition is analogous to the definition used by us for developing flow in a circular cylinder (see Krij- ger et al., 1989). We recall that L has been non- dimensionalised by the length scale 1= 2.5 mm. The fully developed velocity w(x, y, cc) can be computed by the formula given in the Appendix.

It is interesting to compare the inlet lengths computed with equation (4) with the ones given in Krijger et al. (1989). The conclusions stated in that paper can be summarised as follows: (1) L is proportional to Re;

(2) L is much larger for asymmetric flow than for symmetric flow; and (3) L exceeds the length of the

Table 1. Inlet lenghts for the symmetric (7 = 1) and asymmetric problem (y = 2) for tolerances 1. 2 and 3% and Reynolds numbers 50, 150 and 250

1% 2% 3%

Re=SO

9.9 8.1 7.0

y=l

Re= 150

33.2 28.3 25.6

Re = 250 Re=50

57.9 18.3 50.8 14.9 46.2 12.9

j, = 2

Re= 150

56.4 47.1 41.9

Re = 250

95.8 79.8 70.5

GM 25:12-G

1462 J. K. B. KRIJGER et al.

mean secondary velocity

I I I I I I J 30 a50 5.00 6.50 a00 9.50 11.00 125ll

distance from apex (mm)

Fig. 8. Mean secondary velocity in the downstream channel (expressed as a percentage of the mean axial velocity) as a function of the distance from the apex (zdim) in mm for both Re= 50 and Re=250 in the

symmetric (y = 1) and asymmetric (y = 2) case.

basilar artery (15-25 times its radius), for realistic values of the Reynolds number (Re > 150). Looking at our newly computed inlet lengths we see that conclusion 1 does not hold anymore. Conclusions 2 and 3 remain valid for the flow case considered here.

Comparing our newly computed inlet lengths with the previous ones for Re = 250, we see that the numerical values in the present paper are somewhat larger. This can probably be attributed to the fact that the secondary velocities, which are very important in the case at hand, decay slowly and deform the axial velocity profile over a considerable distance. However, one should keep in mind that a fair comparison of flow in a circular cylinder with flow in a rectangular duct cannot be made.

Experimental validation

To test whether the computed phenomena are present in a real flow, an experimental set-up was made. A physical model having a slightly different geometry from that depicted in Fig. 1 was manufactured from Perspex. The inflow and outflow channels are much longer than those shown in Fig. 1. Measurements were made using the laser Doppler technique, with water as fluid. For a full account of the experimental method we refer to a paper in preparation by Ravens- bergen et al. (1991). In Fig. 9 the results are depicted for the symmetric flow case (y = 1) and Re= 276. The dots denote the measured values, the lines the corres-

ponding computations. It can be seen that there is a good agreement between the measured and the computed values.

DISCUSSION

An important conclusion that emerges from the presented results is that the flow in the basilar artery is highly three-dimensional. The two-dimensional calculations that were made for comparison predict quite dissimilar phenomena. This conclusion is different from the one stated in our previous paper, Krijger et al. (1989). In that paper two-dimensional flow in a straight channel and three-dimensional flow in a circular cylinder has been computed with the double- hump velocity profile being prescribed as an inflow condition. From this earlier research it was concluded that the three-dimensional results do not differ much from the two-dimensional ones. This is caused by the fact that at the entrance of the channel and cylinder zero secondary velocity was prescribed; so these results are valid only for Aow in a confluence with a very small branching angle. For a realistic branching angle (60”), as has been used in the present study, secondary flow can certainly not be ignored. However, the earlier two-dimensional model in which the branching angle was incorporated by prescribing transverse velocities at the entrance of the channel gives results that agree to some extent with the newly computed results. In

89.’

14.’

59.’

44.’

34.’

24.

19.

14.

9.

4.’

3 ( .(


cross-section 1 cross-section 2

1463

Fig. 9. Comparison between computed and measured velocity profiles of the primary velocity w in cross-sections 1 and 2 for symmetric inflow (7’ 1) and Re=276. The scale on the left indicates the

downstream distance (zdim) from the apex expressed in mm.


both cases the initial double-hump velocity profile transforms very fast into a profile with a single central maximum.

In another paper (Krijger et al., 1990) we studied flow in a two-dimensional confluence with a geometry comparable to the one in Fig. 1 (a). Then it was found that the flow phenomena downstream of the junction do not depend on the branching angle. We expect that this result does not hold in the three-dimensional case, since a larger branching angle will give a stronger secondary flow, which is found to have a strong influence on the flow development downstream of the junction.

In our model we made four important assumptions: (i) the channel walls are rigid; (ii) blood behaves as a Newtonian fluid, (iii) the channel has a rectangular cross-section; (iv) the flow is steady. Since the vessel wall movement is small and blood behaves approximately as a Newtonian fluid in vessels of the size of the basilar artery, the first two assumptions are accept- able. Assumption (iii) is, of course, not realistic and we intend to study in the near future flow in a confluence of tubes with a circular cross-section. Anatomical data will be needed to define a standard geometry for the vertebral basilar junction. Assumption (iv) is more reasonable than it seems at first sight. A comparison of two-dimensional pulsatile results (Krijger et af., 1991) with two-dimensional steady results (Krijger et al., 1989) shows that the most important flow phenomena are the same in both cases. This is due to the fact that the dimensionless number characterising the influence of pulsatility, the Womersley parameter, is relatively low (about 2.1). We assume that the influence of unsteady effects will be small in the three- dimensional case too. This assumption will be tested in future calculations and experiments.

Thus, although there are some objections, we be- lieve that the main features of the flow in the real basilar artery are represented in our model. So we will try now to give an answer to the two questions put forward in the introduction. It is commonly thought that atherosclerosis develops in regions of low wall shear stress. Our results show that low velocities near the walls (and, consequently, low wall shear stresses) occur at the apex and at the walls of the channel in cross-section 1 downstream of the junction. A relatively high wall shear stress is present at the walls in cross-section 2 in the first part of the downstream channel. These results have to be compared with anatomical data about the location of atherosclerotic plaques in the basilar artery.

Regarding the second question, the influence on the flow in the circle of Willis, it is found that when the inflow in the basilar artery is asymmetric, the flow is still asymmetric at the location where the artery divides into both posterior cerebral arteries. Since the circle of Willis consists of vessels with a low resistance, this will probable give a larger inflow into one of the halves of the circle of Willis, and this asymmetric inflow will be compensated by a flow in the anterior communicating artery.

Finally, we note that one vertebral artery is often found to have a larger diameter than the other one. Since a larger artery has a lower resistance, the velocity of the blood flowing from this vertebral artery into the basilar artery will be higher than the velocity of the blood supplied by the other vertebral artery. So at the entrance of the basilar artery a velocity profile will be present comparable to the asymmetric velocity profiles studied in this paper. Our model predicts that when fluid coming from the left side is faster than the fluid coming from the right side, the higher velocity peak will cross the channel, which leads to high wall shears at the right wall and low wall shear stress at the left wall. In this respect it is interesting to note that in the literature a relation between the growth of the vessel wall and the shear stress has been reported (La Barbera, 1990). From anatomical studies (Raven- sbergen, 1991; Vawda, 1989; Haverling, 1974) it is clear that quite often an asymmetry of the vertebral arteries is related to a curvature of the basilar artery with the convexity towards the smaller of the two vertebral arteries. Thus, it seems that the asymmetry of the flow can cause the basilar artery to grow asym- metrically and become curved.

REFERENCES

Bharadvaj, B. K., Mabon, R. F. and Giddens, D. P. (1982a) Steady flow in a model of the human carotid bifurcation. Part I-Flow visualisation. J. Biomechonics 15, 349-362.

Bharadvaj, B. K., Mabon, R. F. and Giddens, D. P. (1982b) Steady flow in a model of the human carotid bifurcation. Part II-Laser Doppler anemometer measurements. J. Biomechanics 15, 363-378.

Bramley, J. S. and Dennis, S. C. R. (1984) The numerical solution of two-dimensional flow in a branching channel. Comput. Fluids 12, 339-355.

Bramley, J. S. and Sloan, J. S. (1987) Numerical solution for two-dimensional flow in a branching channel using boundary fitted coordinates. Comput. Fluids 15, 297-311.

Briley, W. R. and McDonald, H. (1984) Three-dimensional viscous flows with large secondary velocity. J. Fluid Mech. 144,47-77.

Cuvelier, C., Segal, A. and van Steenhoven, A. A. (1986) Finite Element Methods and Navier-Stokes Equations. D. Reidel, Dordrecht.

Fletcher. C. A. J. (1988) Snecific techniaues for different flow \ ,. categories. In Computational Techniquesfor Fluid Dynam- ics, Vol. 2. Springer, Berlin.

Govindan, T. R., Briley, W. R. and McDonald, H. (1991) General three-dimensional viscous primary/secondary flow analysis. AIAA J. 29, 361-370.

Haverling. M. (1974) The tortuous basilar artery. Acta Radio/. Diag. 15, 241-249.

Hillen, B., Drinkenburg, A. A. H., Hoogstraten, H. W. and Post, L. (1988) Analysis of flow and vascular resistance of the circle of Willis. J. Biomechanics 21. 807-814.

Krijger, J. K. B., Hillen, B. and Hoogstraten, H. W. (1989) Mathematical models of the flow in the basilar artery. J. Biomechanics 22, 1193-1202.

Krijger, J. K. B., Hillen, B., Hoogstraten, H. W. and van den Raadt, M. P. M. G. (1990) Steady two-dimensional merging flow from two channels into a single channel. Appl. Sci. Res. 47, 233-246.

Krijger, J. K. B., Hillen, B. and Hoogstraten, H. W. (1991) A two-dimensional model of pulsating flow in the basilar artery. J. Appl. Math. Phys. (ZAMP) 42, 649-662.

A model of the basilor artery 1465

La Barbera, M. (1990) Principles of fluid transport systems in zoology. Science 249,992-1000.

Pedley, T. J., Schroter, R. C. and Sudlow, M. F. (1971) Flow and pressure drop in systems of repeatedly branching channels. J. Fluid Me& 46, 365-383.

Ravensbergen, J., Hillen, B., Krijger, J. K. B., Hoogstraten, H. W., Groeneveld, M. and Heethaar, R. M. (1991) The flow in the basilar artery. Eur. J. Morphol. 29, 119.

Reuderink, P. J. (1991) Analysis of the flow in a 3D distensible model of the carotid artery bifurcation. Ph.D. thesis, University of Eindhoven, The Netherlands.

Rieu, R., Pelissier, R. and,Farahifar, D. (1989) An experimental investigation of flow characteristics in bifurcation models. Eur. J. Mech. 8(l), 73-101.

Rindt, C. C. M. (1989) Analysis of the three-dimensional flow field in the carotid artery bifurcation. Ph.D. thesis. Univer-

APPENDIX

FULLY DEVELOPED FLOW IN A RECTANGULAR

CHANNEL

For fully developed flow of a Newtonian fluid with dynamic viscosity p in a rectangular channel with walls given by x =O, x = a, y = 0 and y = b, the velocity w in the axial z-direction is given by

cosh[mx( y- b/2)/a]

>

mm

cosh(mnb/2a) sm --.

cl

sity of Eindhoven, The’Netherlands. Rindt, C. C. M., van de Vosse, F. N., van Steenhoven, A. A., where dp/dz denotes the constant axial pressure gradient. In

Janssen, J. D. and Reneman, R. S. (1987) A numerical and fact, this expression represents the separation-of-variables

experimental analysis of the flow field in a two-dimen- solution of the two-dimensional Poisson equation

sional model of the human carotid bifurcation. J. Biomechanics 20,499-509.

Rubin, R. T., Khosla, P. K. and Saari, S. (1977) Laminar flow in rectangular channels. Comput. Fluids 5, 151-173.

1 dp w,,+wyy=- -

p dz

Schroter, R. C. and Sudlow, M. F. (1969) Flow patterns in models of the human bronchial airways. Res. Physiol. 7, with boundary condition w = 0 on all four channel walls. The

341-355. pressure gradient and the volume flux Q are related in the

van de Vosse, F. N., Van Steenhoven, A. A., Segal, A. and following way:

Janssen, J. D. (1989) A finite element analysis of the steady laminar entrance flow in a 90” curved tube. Int. J. Num. Meth. Fluids 9, 275-287.

Q=j;j;w(x,y)dxdy= -;z $ ( ) L

Vawda. G. H. M. (1989) Lateral bowing of the human basilar artery, a haemodynamic stress phenomenon. S. A$-. J. Sci.

tanh(mnb/‘a)

85, 465. m5 I

Computation of steady three-dimensional flow in a model of the basilar artery

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