S. H. Kim Research Associate.
K. B. Chandran Professor of Biomedical and Mechanical Engineering. Fellow ASME
C. J. Chen Professor and Chairman of Mechanical Engineering. Fellow ASME University of Iowa, Iowa City, Iowa
Numerical Simulation of Steady Flow in a Two-Dimensional Total Artificial Heart Model In this paper, a numerical simulation of steady laminar and turbulent flow in a twodimensional model for the total artificial heart is presented. A trileaflet polyurethane valve was simulated at the outflow orifice while the inflow orifice had a trileaflet or a flap valve. The finite analytic numerical method was employed to obtain solutions to the governing equations in the Cartesian coordinates. The closure for turbulence model was achieved by employing the k-e-E model. The SIMPLER algorithm was used to solve the problem in primitive variables. The numerical solutions of the simulated model show that regions of relative stasis and trapped vortices were smaller within the ventricular chamber with the flap valve at the inflow orifice than that with the trileaflet valve. The predicted Reynolds stresses distal to the inflow valve within the ventricular chamber were also found to be smaller with the flap valve than with the trileaflet valve. These results also suggest a correlation between high turbulent stresses and the presence of thrombus in the vicinity of the valves in the total artificial hearts. The computed velocity vectors and turbulent stresses were comparable with previously reported in vitro measurements in artificial heart chambers. Analysis of the numerical solutions suggests that geometries similar to the flap valve (or a tilting disk valve) results in a better flow dynamics within the total artificial heart chamber compared to a trileaflet valve.
Introduction One of the critical problems with the total artificial heart (TAH) in clinical use is the thromboembolic complications. Most recipients of the TAH implanted as a total replacement have exhibited physiological complications due to stroke resulting from emboli occluding the blood vessels in the brain. The thromboembolic complications have been linked to abnormal fluid dynamic stresses in the flow chamber. Elevated turbulent stresses have been shown to result in thrombus formation (Stein and Sabbah, 1974), and damage to platelets and erythrocytes (Sutera and Mehrjardi, 1975; Hung et al., 1976; Sutera, 1977; Sallam and Huang, 1984). Platelet adhesion or chemical reactions with thrombus formation have been correlated with increasing rates of shear (Turitto and Baumgartner, 1975; Voisin et al., 1976). More recently, Hung et al. (1991) demonstrated marked increases in relative blood viscosity, erythrocyte rigidity, fibrinogen concentration, and platelet aggregation in patients supported by TAH and ventricular assist device. Jarvis et al. (1991) demonstrated, through an in vitro study of blood flow in the Penn state artificial heart, that bulk turbulent stresses may play an important role in mediating blood damage. The deposition of thrombi appear to be concentrated in the vicinity of inflow and outflow valves of the TAH (Levinson et al., 1986). In order to understand Contributed by the Bioengineering Division for publication in the JOURNAL OF BIOMECHANICAL ENGINEERINO. Manuscript received by the Bioengineering Division July 24, 1991; revised manuscript received January 10, 1992. Associate Technical Editor: J. M. Tarbell.
the problems of sublethal and/or lethal damage to formed elements in blood due to the flow dynamics within the chamber, several studies have been reported on the measurement of turbulent shear stresses within a TAH. Philips et al. (1979) estimated magnitudes of turbulent stresses ranging 30 to 570 Pa past a tilting disc valve in a TAH using laser Doppler technique. Tarbell et al. (1986) employed a pulsed ultrasonic Doppler technique and reported maximum magnitudes of 2.5 Pa and 21 Pa for the wall shear and Reynolds stress respectively in a ventricular assist device. However, due to a practical limitation of measurements within a TAH especially in the vicinity of the inflow and outflow valves, data obtained from experimental measurements are limited. Computational fluid dynamics provides an alternative technique in which more detailed analysis of fluid dynamic effects of flow within a TAH can be evaluated. Extensive numerical analysis on natural heart flow dynamics has been reported by Peskin et al. (1989). They initially solved the Navier-Stokes equations for a two dimensional model of the left ventricle and mitral valve by a finite difference scheme. The heart was considered to be submerged in an infinite pool of blood, with the mitral valve and ventricular walls moving at the local fluid velocity and acting as regions in space where extra forces are applied to the fluid. They reported atrial and ventricular pressure and flow patterns which showed reasonable agreement with the experimental data for the case of the natural mitral valve. They extended the studies to perform three dimensional numerical calculations of blood flow in the human heart. RogNOVEMBER 1992, Vol. 114/497
Journal of Biomechanical Engineering
Copyright © 1992 by ASME Downloaded From: http://biomechanical.asmedigitalcollection.asme.org/ on 05/20/2015 Terms of Use: http://asme.org/terms
V
J L Trileaflet valve
Trileaflet valve
f j Flap valve /
Fig. 2 Fig. 1
Geometry of the total artificial heart models
ers et al. (1989) presented numerical solution of incompressible Navier-Stokes equations for steady-state and time-dependent laminar flow through a model of a Penn state TAH. The analysis included the three-dimensional geometries and the moving boundaries and the solution were obtained for a Reynolds number of 100 based on unit length and velocity. Flow distal to prosthetic mechanical and polyurethane valves has been demonstrated to induce turbulent stresses (Yoganathan et al., 1986; Chandran et al., 1989a). As an initial attempt to analyze the effect of turbulent flow past valve prostheses in a TAH, the present study was restricted to steady flow in a simplified two-dimensional model of a TAH employing Cartesian coordinates. Two-dimensional representation of polyurethane flap and trileaflet valves were employed at the inflow and outflow orifices in order to compare the results of the numerical simulations with in vitro results (Chandran et al. 1989b, 1991). Results are presented for the flow field and computed mechanical stresses within the model TAH chamber. Possible correlation between the resulting mechanical stresses and regions of thrombus deposition in the TAH chambers are discussed.
Model for the T A H In this initial attempt to describe the turbulent flow within a TAH chamber, a two-dimensional simplified geometry was employed as shown in Fig. 1. A trileaflet polyurethane valve was simulated in the outflow orifice of the two models represented in Fig. 1. A trileaflet valve was also simulated in the inflow orifice of Model I while a polyurethane flap valve was simulated in the inflow orifice of Model II. The geometry of the trileaflet and flap valves can be approximately represented in the two-dimensional Cartesian coordinates. At present, TAH's are used only as a bridge to transplant for a period of several weeks and the mechanical prostheses at the inflow and outflow orifices constitute a substantial portion of the total cost for the device. In order to reduce the cost, several prototype polyurethane valves are under development. In vitro measurements of pressure drop and turbulent stresses distal to the polyurethane valves in a TAH are available (Chandran et al., 1989b, 1991) and hence were also modelled in this study. In the case of polyurethane valve in Model I three different dimensions of valve leaflet were considered in order to investigate the effects of the valve leaflet flow area as well as valve leaflet height on the fluid dynamic characteristics of the flow. The dimensions of the chamber and the valves, considered in this study, are representative of prototype TAH and valves as shown in Table 1. The extended trileaflet valve height and the extended gap between the leaflet and the channel, correspond to a 25 percent increase from the prototype. Such a parametric study was undertaken to analyze the effect of various dimensions of the inflow valve on the flow dynamics within the TAH. The grid distributions for the models are shown on Fig. 2. The same number of grid nodes are used in the outflow channel
Grid distribution of the total artificial heart models
Table 1 The dimensions of the model and each valve Height = 45 Chamber Width = 74 Diameter = 1 9 Inflow channel Length = 13 Diameter = 19 Outflow channel Length = 32 Standard: W, = 2.6, H, = 8 Trileaflet valve Type I: W, = 3.3, H, = 9 Type II: W, = 2.6,H,= 10 Flap valve All dimensions are in mm.
i r Table 2 Model I Model II Model III
Number of grid nodes of each model
Inflow channel 27x9 27x9 31x9
Outflow channel 27x23 27x23 27x23
Chamber 77x23 77x35 81x37
in both models and different number of grids are used in the chamber and inflow channel due to the different chamber and valve geometry, which are presented in Table 2. More grids are located in the vicinity of both walls and valve leaflets where the velocity gradients are higher. Same grid distributions are used for both laminar flow and turbulent flow calculations. The study was restricted to steady flow and hence both the inflow and outflow valves were in the fully open position as shown in Fig. 1.
Numerical Solution Technique The problem considered herein is the two-dimensional steady flow of a homogeneous, incompressible, Newtonian fluid through a TAH. The governing equations solved are NavierStokes equations for laminar flow or the ensemble averaged Navier-Stokes equations along with the k-e turbulence model (Chen and Chang, 1987; Chen, 1988) for turbulent flow. The governing equations and the equations for the k-e turbulence model are included in the Appendix. Nondimensionalization is performed on the governing equations for the laminar and turbulent flows using uniform inlet velocity, V0, the diameter of the inflow channel, D and the density, p (constant). Transformed governing equations are also included in the Appendix. Uniform velocity profile was specified at the inlet for both laminar and turbulent flow simulation. The inlet boundary conditions of k and e are difficult to specify since experimental data are not available at this time. Assumed values of k = 0.001 and e = 0.0001 were used for the turbulence parameters at the inlet. Solutions were also obtained by arbitrarily varyings of k and e. The flow patterns and turbulent stresses within the
498 / Vol. 114, NOVEMBER 1992 Downloaded From: http://biomechanical.asmedigitalcollection.asme.org/ on 05/20/2015 Terms of Use: http://asme.org/terms
Transactions of the ASME
Re - 600
Fig. 3
Re
_
6 0 0 0
Flow patterns with trileaflet inflow valve
chamber were found to be independent of the assumed turbulence parameters at the inlet. Large velocity gradients are present as the flow passes past the leaflets of the valves at the inflow orifice. The turbulence generated by the velocity gradients dominates the flow behavior within the chamber rather than the turbulence parameters upstream to the inflow orifice. At the solid wall boundaries the no slip conditions (U = 0, V = 0) are specified for the laminar flow calculations. For the turbulent flow calculations, due to the steep gradients of velocities and turbulent variables, a very fine grid is needed in the wall region which demands excessive computational time. In order to overcome this difficulty, the wall function approach was employed which is described in the Appendix. The values of dependent variables at the outlet boundary are usually unknown before computation and should be computed as a part of the solution. In this study a relatively long channel is chosen for the outflow channel where the flow is expected to be approximately fully developed and the outlet boundary conditions of k and e are specified such that there is little change in the k, e and the velocity component along the flow direction at the outlet. The finite analytic (FA) numerical method described by Chen and Chen (1984) is used to obtain the numerical solution of governing equations for two-dimensional convective transport equation in the Cartesian coordinates system. In the FA formulation, the discretized algebraic equation for each element is obtained from the local analytic solution of the partial differential equations governing the physical system. The finite analytic solution provides an algebraic expression for the value of an interior node as a function of the surrounding boundary nodes. The system of finite analytic algebraic equations is then obtained for all elements in the flow regime. In order to solve for pressure, the velocity variables in the continuity equation are replaced by the pressure variable. The solution of these algebraic equations with the given proper boundary and initial conditions provides the numerical solution of the problem. The description of this method is included in Appendix. A staggered numerical grid is employed, in which the pressure and turbulence parameters, turbulent kinetic energy (k) and its rate of dissipation (e), are calculated directly on the computed grid node, while the U velocity is computed east of it, halfway to the next node, and Kis computed to the north. An algorithm is needed for determining the pressure field and the corresponding velocity field which satisfies the continuity equation because the correct pressure field is usually unknown until the problem is actually solved. In this study, Patankar's SIMPLER (1980) method was used for obtaining the solution. Solutions for wall-driven cavity flow were initially obtained in order to verify the accuracy of the computer programs developed in this study. After the numerical results for the cavity flow was verified with previously reported numerical and experimental results, and the grid independence was established for the model, the numerical solutions for flow in a TAH were obtained. Further details of the solution procedure and numerical verifications are included in Kim (1991). For the laminar flow calculations, solutions are obtained at Journal of Biomechanical Engineering
Fig. 4 Profiles of V/V0 and UVIV02 where Vis the vertical velocity component and l/and Vare the horizontal and vertical fluctuation velocities, respectively
a Reynolds number of 600, based on the diameter of the inflow channel and the mean velocity in the inlet. It was the maximum Reynolds number in the laminar flow calculation at which convergence was obtained. For turbulent flow, solutions are obtained at a Reynolds number of 6000. The flow rate through the inflow valve is 15 1/min corresponding to a typical peak flow rate expected during the diastolic phase of the pulsatile flow in a TAH. It is equivalent to the mean velocity of 88 cm/ s at the inlet. The fluid simulated in the calculations is Newtonian with viscosity and specific gravity representing that of blood, which are 3.0 cp and 1.07, respectively. Numerical Results The velocity field past the standard polyurethane trileaflet valve within the ventricular chamber for laminar and turbulent flows are displayed in Fig. 3(a) and 3(b), respectively. As can be observed, a jet-like flow emanates past the leaflets at the inflow orifice and a vortex is formed in the center of the chamber. In turbulent flow, the vortex within the chamber is larger than that for the laminar flow. Zones of relative stasis are also observed behind the leaflets both in laminar and turbulent flow cases similar to those observed in in vitro experiments (Chandran et al., 1989b). Thrombus deposition may be enhanced by such stagnation and low velocity reverse flow regions in the total artificial heart. Trapped vortices were observed at the bottom left and right corners. These vortices in turbulent flow are considerably smaller than that in laminar flow due to the fact that in turbulent flow, stronger mixing is generated by the turbulent eddy motion. The pressure drop across the inflow valve is about 1.3 KPa at a Reynolds number of 6000. The transvalvular pressure drop is calculated as the pressure difference between the inlet and 10 mm downstream to the mitral valve in the middle of the chamber. The predicted mean pressure drop across the trileaflet outflow valve between the top of chamber and 20 mm downstream from the valve is around 0.27 KPa. Figure 4(a) shows the F-velocity (vertical velocity) profile with the standard polyurethane trileaflet valve normalized with respect to the inlet mean velocity (V0) at various positions of the model. A high velocity jet with a maximum magnitude of 1.67 is observed distal to the inflow valve. After the flow passes through the inflow valve it rapidly decelerates toward the bottom wall. The maximum magnitude of outflow jet is also 1.67. The outlet flow eventually recovers to fully developed flow. The velocity profile at the center of the chamber shows a tendency for a large vortex which was demonstrated earlier in Fig. 3. The profiles of uv, normalized by V\, at different locations are shown in Fig. 4(b). High values of stresses are predicted at the orifice near the valve leaflets as well as on the left wall. A maximum turbulent stress of 470 Pa was computed distal to the inflow valve near the edge of the jet like flow. In the central region the shear stresses are almost constant due to the vortical flow with a small relative motion. The distribution of wall shear stress along the inner wall of the leaflets are shown on Fig. 5. Each leaflet is labeled NOVEMBER 1992, Vol. 114 / 499
Downloaded From: http://biomechanical.asmedigitalcollection.asme.org/ on 05/20/2015 Terms of Use: http://asme.org/terms
B A
n r
Nondimensional distance
Nondimenaional dlstan'
Fig. 7 Profiles of V/V„ and UV/V
K. B. Chandran Professor of Biomedical and Mechanical Engineering. Fellow ASME
C. J. Chen Professor and Chairman of Mechanical Engineering. Fellow ASME University of Iowa, Iowa City, Iowa
Numerical Simulation of Steady Flow in a Two-Dimensional Total Artificial Heart Model In this paper, a numerical simulation of steady laminar and turbulent flow in a twodimensional model for the total artificial heart is presented. A trileaflet polyurethane valve was simulated at the outflow orifice while the inflow orifice had a trileaflet or a flap valve. The finite analytic numerical method was employed to obtain solutions to the governing equations in the Cartesian coordinates. The closure for turbulence model was achieved by employing the k-e-E model. The SIMPLER algorithm was used to solve the problem in primitive variables. The numerical solutions of the simulated model show that regions of relative stasis and trapped vortices were smaller within the ventricular chamber with the flap valve at the inflow orifice than that with the trileaflet valve. The predicted Reynolds stresses distal to the inflow valve within the ventricular chamber were also found to be smaller with the flap valve than with the trileaflet valve. These results also suggest a correlation between high turbulent stresses and the presence of thrombus in the vicinity of the valves in the total artificial hearts. The computed velocity vectors and turbulent stresses were comparable with previously reported in vitro measurements in artificial heart chambers. Analysis of the numerical solutions suggests that geometries similar to the flap valve (or a tilting disk valve) results in a better flow dynamics within the total artificial heart chamber compared to a trileaflet valve.
Introduction One of the critical problems with the total artificial heart (TAH) in clinical use is the thromboembolic complications. Most recipients of the TAH implanted as a total replacement have exhibited physiological complications due to stroke resulting from emboli occluding the blood vessels in the brain. The thromboembolic complications have been linked to abnormal fluid dynamic stresses in the flow chamber. Elevated turbulent stresses have been shown to result in thrombus formation (Stein and Sabbah, 1974), and damage to platelets and erythrocytes (Sutera and Mehrjardi, 1975; Hung et al., 1976; Sutera, 1977; Sallam and Huang, 1984). Platelet adhesion or chemical reactions with thrombus formation have been correlated with increasing rates of shear (Turitto and Baumgartner, 1975; Voisin et al., 1976). More recently, Hung et al. (1991) demonstrated marked increases in relative blood viscosity, erythrocyte rigidity, fibrinogen concentration, and platelet aggregation in patients supported by TAH and ventricular assist device. Jarvis et al. (1991) demonstrated, through an in vitro study of blood flow in the Penn state artificial heart, that bulk turbulent stresses may play an important role in mediating blood damage. The deposition of thrombi appear to be concentrated in the vicinity of inflow and outflow valves of the TAH (Levinson et al., 1986). In order to understand Contributed by the Bioengineering Division for publication in the JOURNAL OF BIOMECHANICAL ENGINEERINO. Manuscript received by the Bioengineering Division July 24, 1991; revised manuscript received January 10, 1992. Associate Technical Editor: J. M. Tarbell.
the problems of sublethal and/or lethal damage to formed elements in blood due to the flow dynamics within the chamber, several studies have been reported on the measurement of turbulent shear stresses within a TAH. Philips et al. (1979) estimated magnitudes of turbulent stresses ranging 30 to 570 Pa past a tilting disc valve in a TAH using laser Doppler technique. Tarbell et al. (1986) employed a pulsed ultrasonic Doppler technique and reported maximum magnitudes of 2.5 Pa and 21 Pa for the wall shear and Reynolds stress respectively in a ventricular assist device. However, due to a practical limitation of measurements within a TAH especially in the vicinity of the inflow and outflow valves, data obtained from experimental measurements are limited. Computational fluid dynamics provides an alternative technique in which more detailed analysis of fluid dynamic effects of flow within a TAH can be evaluated. Extensive numerical analysis on natural heart flow dynamics has been reported by Peskin et al. (1989). They initially solved the Navier-Stokes equations for a two dimensional model of the left ventricle and mitral valve by a finite difference scheme. The heart was considered to be submerged in an infinite pool of blood, with the mitral valve and ventricular walls moving at the local fluid velocity and acting as regions in space where extra forces are applied to the fluid. They reported atrial and ventricular pressure and flow patterns which showed reasonable agreement with the experimental data for the case of the natural mitral valve. They extended the studies to perform three dimensional numerical calculations of blood flow in the human heart. RogNOVEMBER 1992, Vol. 114/497
Journal of Biomechanical Engineering
Copyright © 1992 by ASME Downloaded From: http://biomechanical.asmedigitalcollection.asme.org/ on 05/20/2015 Terms of Use: http://asme.org/terms
V
J L Trileaflet valve
Trileaflet valve
f j Flap valve /
Fig. 2 Fig. 1
Geometry of the total artificial heart models
ers et al. (1989) presented numerical solution of incompressible Navier-Stokes equations for steady-state and time-dependent laminar flow through a model of a Penn state TAH. The analysis included the three-dimensional geometries and the moving boundaries and the solution were obtained for a Reynolds number of 100 based on unit length and velocity. Flow distal to prosthetic mechanical and polyurethane valves has been demonstrated to induce turbulent stresses (Yoganathan et al., 1986; Chandran et al., 1989a). As an initial attempt to analyze the effect of turbulent flow past valve prostheses in a TAH, the present study was restricted to steady flow in a simplified two-dimensional model of a TAH employing Cartesian coordinates. Two-dimensional representation of polyurethane flap and trileaflet valves were employed at the inflow and outflow orifices in order to compare the results of the numerical simulations with in vitro results (Chandran et al. 1989b, 1991). Results are presented for the flow field and computed mechanical stresses within the model TAH chamber. Possible correlation between the resulting mechanical stresses and regions of thrombus deposition in the TAH chambers are discussed.
Model for the T A H In this initial attempt to describe the turbulent flow within a TAH chamber, a two-dimensional simplified geometry was employed as shown in Fig. 1. A trileaflet polyurethane valve was simulated in the outflow orifice of the two models represented in Fig. 1. A trileaflet valve was also simulated in the inflow orifice of Model I while a polyurethane flap valve was simulated in the inflow orifice of Model II. The geometry of the trileaflet and flap valves can be approximately represented in the two-dimensional Cartesian coordinates. At present, TAH's are used only as a bridge to transplant for a period of several weeks and the mechanical prostheses at the inflow and outflow orifices constitute a substantial portion of the total cost for the device. In order to reduce the cost, several prototype polyurethane valves are under development. In vitro measurements of pressure drop and turbulent stresses distal to the polyurethane valves in a TAH are available (Chandran et al., 1989b, 1991) and hence were also modelled in this study. In the case of polyurethane valve in Model I three different dimensions of valve leaflet were considered in order to investigate the effects of the valve leaflet flow area as well as valve leaflet height on the fluid dynamic characteristics of the flow. The dimensions of the chamber and the valves, considered in this study, are representative of prototype TAH and valves as shown in Table 1. The extended trileaflet valve height and the extended gap between the leaflet and the channel, correspond to a 25 percent increase from the prototype. Such a parametric study was undertaken to analyze the effect of various dimensions of the inflow valve on the flow dynamics within the TAH. The grid distributions for the models are shown on Fig. 2. The same number of grid nodes are used in the outflow channel
Grid distribution of the total artificial heart models
Table 1 The dimensions of the model and each valve Height = 45 Chamber Width = 74 Diameter = 1 9 Inflow channel Length = 13 Diameter = 19 Outflow channel Length = 32 Standard: W, = 2.6, H, = 8 Trileaflet valve Type I: W, = 3.3, H, = 9 Type II: W, = 2.6,H,= 10 Flap valve All dimensions are in mm.
i r Table 2 Model I Model II Model III
Number of grid nodes of each model
Inflow channel 27x9 27x9 31x9
Outflow channel 27x23 27x23 27x23
Chamber 77x23 77x35 81x37
in both models and different number of grids are used in the chamber and inflow channel due to the different chamber and valve geometry, which are presented in Table 2. More grids are located in the vicinity of both walls and valve leaflets where the velocity gradients are higher. Same grid distributions are used for both laminar flow and turbulent flow calculations. The study was restricted to steady flow and hence both the inflow and outflow valves were in the fully open position as shown in Fig. 1.
Numerical Solution Technique The problem considered herein is the two-dimensional steady flow of a homogeneous, incompressible, Newtonian fluid through a TAH. The governing equations solved are NavierStokes equations for laminar flow or the ensemble averaged Navier-Stokes equations along with the k-e turbulence model (Chen and Chang, 1987; Chen, 1988) for turbulent flow. The governing equations and the equations for the k-e turbulence model are included in the Appendix. Nondimensionalization is performed on the governing equations for the laminar and turbulent flows using uniform inlet velocity, V0, the diameter of the inflow channel, D and the density, p (constant). Transformed governing equations are also included in the Appendix. Uniform velocity profile was specified at the inlet for both laminar and turbulent flow simulation. The inlet boundary conditions of k and e are difficult to specify since experimental data are not available at this time. Assumed values of k = 0.001 and e = 0.0001 were used for the turbulence parameters at the inlet. Solutions were also obtained by arbitrarily varyings of k and e. The flow patterns and turbulent stresses within the
498 / Vol. 114, NOVEMBER 1992 Downloaded From: http://biomechanical.asmedigitalcollection.asme.org/ on 05/20/2015 Terms of Use: http://asme.org/terms
Transactions of the ASME
Re - 600
Fig. 3
Re
_
6 0 0 0
Flow patterns with trileaflet inflow valve
chamber were found to be independent of the assumed turbulence parameters at the inlet. Large velocity gradients are present as the flow passes past the leaflets of the valves at the inflow orifice. The turbulence generated by the velocity gradients dominates the flow behavior within the chamber rather than the turbulence parameters upstream to the inflow orifice. At the solid wall boundaries the no slip conditions (U = 0, V = 0) are specified for the laminar flow calculations. For the turbulent flow calculations, due to the steep gradients of velocities and turbulent variables, a very fine grid is needed in the wall region which demands excessive computational time. In order to overcome this difficulty, the wall function approach was employed which is described in the Appendix. The values of dependent variables at the outlet boundary are usually unknown before computation and should be computed as a part of the solution. In this study a relatively long channel is chosen for the outflow channel where the flow is expected to be approximately fully developed and the outlet boundary conditions of k and e are specified such that there is little change in the k, e and the velocity component along the flow direction at the outlet. The finite analytic (FA) numerical method described by Chen and Chen (1984) is used to obtain the numerical solution of governing equations for two-dimensional convective transport equation in the Cartesian coordinates system. In the FA formulation, the discretized algebraic equation for each element is obtained from the local analytic solution of the partial differential equations governing the physical system. The finite analytic solution provides an algebraic expression for the value of an interior node as a function of the surrounding boundary nodes. The system of finite analytic algebraic equations is then obtained for all elements in the flow regime. In order to solve for pressure, the velocity variables in the continuity equation are replaced by the pressure variable. The solution of these algebraic equations with the given proper boundary and initial conditions provides the numerical solution of the problem. The description of this method is included in Appendix. A staggered numerical grid is employed, in which the pressure and turbulence parameters, turbulent kinetic energy (k) and its rate of dissipation (e), are calculated directly on the computed grid node, while the U velocity is computed east of it, halfway to the next node, and Kis computed to the north. An algorithm is needed for determining the pressure field and the corresponding velocity field which satisfies the continuity equation because the correct pressure field is usually unknown until the problem is actually solved. In this study, Patankar's SIMPLER (1980) method was used for obtaining the solution. Solutions for wall-driven cavity flow were initially obtained in order to verify the accuracy of the computer programs developed in this study. After the numerical results for the cavity flow was verified with previously reported numerical and experimental results, and the grid independence was established for the model, the numerical solutions for flow in a TAH were obtained. Further details of the solution procedure and numerical verifications are included in Kim (1991). For the laminar flow calculations, solutions are obtained at Journal of Biomechanical Engineering
Fig. 4 Profiles of V/V0 and UVIV02 where Vis the vertical velocity component and l/and Vare the horizontal and vertical fluctuation velocities, respectively
a Reynolds number of 600, based on the diameter of the inflow channel and the mean velocity in the inlet. It was the maximum Reynolds number in the laminar flow calculation at which convergence was obtained. For turbulent flow, solutions are obtained at a Reynolds number of 6000. The flow rate through the inflow valve is 15 1/min corresponding to a typical peak flow rate expected during the diastolic phase of the pulsatile flow in a TAH. It is equivalent to the mean velocity of 88 cm/ s at the inlet. The fluid simulated in the calculations is Newtonian with viscosity and specific gravity representing that of blood, which are 3.0 cp and 1.07, respectively. Numerical Results The velocity field past the standard polyurethane trileaflet valve within the ventricular chamber for laminar and turbulent flows are displayed in Fig. 3(a) and 3(b), respectively. As can be observed, a jet-like flow emanates past the leaflets at the inflow orifice and a vortex is formed in the center of the chamber. In turbulent flow, the vortex within the chamber is larger than that for the laminar flow. Zones of relative stasis are also observed behind the leaflets both in laminar and turbulent flow cases similar to those observed in in vitro experiments (Chandran et al., 1989b). Thrombus deposition may be enhanced by such stagnation and low velocity reverse flow regions in the total artificial heart. Trapped vortices were observed at the bottom left and right corners. These vortices in turbulent flow are considerably smaller than that in laminar flow due to the fact that in turbulent flow, stronger mixing is generated by the turbulent eddy motion. The pressure drop across the inflow valve is about 1.3 KPa at a Reynolds number of 6000. The transvalvular pressure drop is calculated as the pressure difference between the inlet and 10 mm downstream to the mitral valve in the middle of the chamber. The predicted mean pressure drop across the trileaflet outflow valve between the top of chamber and 20 mm downstream from the valve is around 0.27 KPa. Figure 4(a) shows the F-velocity (vertical velocity) profile with the standard polyurethane trileaflet valve normalized with respect to the inlet mean velocity (V0) at various positions of the model. A high velocity jet with a maximum magnitude of 1.67 is observed distal to the inflow valve. After the flow passes through the inflow valve it rapidly decelerates toward the bottom wall. The maximum magnitude of outflow jet is also 1.67. The outlet flow eventually recovers to fully developed flow. The velocity profile at the center of the chamber shows a tendency for a large vortex which was demonstrated earlier in Fig. 3. The profiles of uv, normalized by V\, at different locations are shown in Fig. 4(b). High values of stresses are predicted at the orifice near the valve leaflets as well as on the left wall. A maximum turbulent stress of 470 Pa was computed distal to the inflow valve near the edge of the jet like flow. In the central region the shear stresses are almost constant due to the vortical flow with a small relative motion. The distribution of wall shear stress along the inner wall of the leaflets are shown on Fig. 5. Each leaflet is labeled NOVEMBER 1992, Vol. 114 / 499
Downloaded From: http://biomechanical.asmedigitalcollection.asme.org/ on 05/20/2015 Terms of Use: http://asme.org/terms
B A
n r
Nondimensional distance
Nondimenaional dlstan'
Fig. 7 Profiles of V/V„ and UV/V
