This article was downloaded by: [The University of Manchester Library]On: 01 April 2015, At: 07:08Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

Click for updates

Computer Methods in Biomechanics and BiomedicalEngineeringPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/gcmb20

Assessment of turbulence models for pulsatile flowinside a heart pumpMohammed G. Al-Azawyab, A. Turan & A. Revellaa School of Mechanical, Aerospace and Civil Engineering, The University of Manchester,Manchester, UKb Mechanical Engineering Department, College of Engineering, Wasit University, Wasit, IraqPublished online: 27 Mar 2015.

To cite this article: Mohammed G. Al-Azawy, A. Turan & A. Revell (2015): Assessment of turbulence models for pulsatile flowinside a heart pump, Computer Methods in Biomechanics and Biomedical Engineering, DOI: 10.1080/10255842.2015.1015527

To link to this article: http://dx.doi.org/10.1080/10255842.2015.1015527

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Assessment of turbulence models for pulsatile flow inside a heart pump

Mohammed G. Al-Azawyab*, A. Turan and A. Revella

aSchool of Mechanical, Aerospace and Civil Engineering, The University of Manchester, Manchester, UK; bMechanical EngineeringDepartment, College of Engineering, Wasit University, Wasit, Iraq

(Received 24 June 2014; accepted 2 February 2015)

Computational fluid dynamics (CFD) is applied to study the unsteady flow inside a pulsatile pump left ventricular assistdevice, in order to assess the sensitivity to a range of commonly used turbulence models. Levels of strain and wall shearstress are directly relevant to the evaluation of risk from haemolysis and thrombosis, and thus understanding the sensitivityto these turbulence models is important in the assessment of uncertainty in CFD predictions. The study focuses on a positivedisplacement or pulsatile pump, and the CFD model includes valves and moving pusher plate. An unstructured dynamiclayering method was employed to capture this cyclic motion, and valves were simulated in their fully open position to mimicthe natural scenario, with in/outflow triggered at control planes away from the valves. Six turbulence models have been used,comprising three relevant to the low Reynolds number nature of this flow and three more intended to investigate differenttransport effects. In the first group, we consider the shear stress transport (SST) k2 v model in both its standard andtransition-sensitive forms, and the ‘laminar’ model in which no turbulence model is used. In the second group, we comparethe one equation Spalart–Almaras model, the standard two equation k2 1 and the full Reynolds stress model (RSM).Following evaluation of spatial and temporal resolution requirements, results are compared with available experimentaldata. The model was operated at a systolic duration of 40% of the pumping cycle and a pumping rate of 86 BPM (beats perminute). Contrary to reasonable preconception, the ‘transition’ model, calibrated to incorporate additional physicalmodelling specifically for these flow conditions, was not noticeably superior to the standard form of the model. Indeed,observations of turbulent viscosity ratio reveal that the transition model initiates a premature increase of turbulence in thisflow, when compared with both experimental and higher order numerical results previously reported in the literature.Furthermore, the RSM is indicated to provide the most accurate prediction over much of the flow, due to its ability to morecorrectly account for three-dimensional effects. Finally, the clinical relevance of the results is reported along with adiscussion on the impact of such modelling uncertainties.

Keywords: left ventricular assist device; computational fluid dynamics; turbulence modelling; transition modelling;dynamic mesh

1. Introduction

In recent years, artificial heart devices have emerged as a

promising alternative therapy for patients suffering from

heart disease. A recent report from the American Heart

Association stated that the number one cause of mortality

is cardiac disease; this includes heart failure, coronary

heart disease, high blood pressure and stroke. Artificial

hearts are particularly attractive given that the number of

available donor hearts is very small, and in general far

lower than potential demand.

The main pumping chambers of the natural heart are

the ventricles, which have a large mass, and the majority of

pumping is undertaken by the left ventricle. A ventricular

assist device (VAD) is an artificial heart device that is used

to support the heart’s pumping function for either short or

long term. They can be used as a replacement for either

side of the heart, or both at once, though left ventricular

assist devices (LVADs) are the most common. Aside from

the mechanical reliability of the devices, the main causal

risks associated with heart pumps are thrombosis and

haemolysis, both of which are directly related to the flow

field. Nowadays, computational fluid dynamics (CFD)

plays a leading role in the investigation of the flow physics

for medical device design, and can thus be used as a tool to

evaluate and avoid potential for damage caused to the

blood.

There are two basic categories of VAD; the positive

displacement or ‘pulsatile’ pump is also known as a 1st

generation device while 2nd and 3rd generation devices

refer to those which instead employ a centrifugal pump

design, also known as ‘continuous flow’ pumps.

Continuous flow pumps are much smaller in size and

with less moving parts, are less complex than the pulsatile

pump and easier to install in most patients. However, these

pumps do not mimic the natural pulsatile flow of the

cardiovascular system, and so the patient effectively does

not have a discernible pulse. Studies have indicated that

there are detrimental effects on health associated with the

lack of a pulse, such as association with wall flexibility and

plaque build-up, and so pulsatile flow VADs remain a

q 2015 Taylor & Francis

*Corresponding author. Email: mohammed.al-azawy@postgrad.manchester.ac.uk

Computer Methods in Biomechanics and Biomedical Engineering, 2015

http://dx.doi.org/10.1080/10255842.2015.1015527

Dow

nloa

ded

by [

The

Uni

vers

ity o

f M

anch

este

r L

ibra

ry]

at 0

7:08

01

Apr

il 20

15

focus of research; e.g. a study by Sezai et al. (1997)

indicated that continuous blood flow provided by a

centrifugal pump may have detrimental physiological

effect on the renal circulation, though Allen et al. (1997)

reported that these effects may be temporary. Motivated

by this sustained interest in pulsatile flow devices and also

by the need to validate CFD results against readily

available data, this study focuses on the evaluation of first

generation positive-displacement pumps. Some obser-

vations can be expected to be relevant to continuous flow

devices also, since the flow conditions are similar in many

aspects.

1.1 Turbulence modelling inside the cardiovascularsystem

The flow of blood in the human body is predominantly

laminar (Re is usually 300 and sometimes less) (Lee &

Jerry 2007). However, the blood flow can become

turbulent in the case of high velocity rates in descending

arteries. In addition, turbulent flow may also occur in some

pathological cases, such as in stenotic heart valves and in

the expansion flow from the inlet to the chamber (Lee &

Jerry 2007). The blood flow through the natural ventricles,

arteries, heart assist devices or artificial heart pumps is

expected to exhibit a combination of both laminar and

turbulent flow. The pulsatile nature of the flow will give

rise to a cycle of transition and re-laminarisation, forwards

and backwards between the two states. Under peak

conditions, the Reynolds number within the LVAD can be

of the order of 104 (Medvitz 2008), and so turbulence is

guaranteed to be present for at least part of the cycle.

Upstream of the pump, the presence of turbulence is less

definite, though still likely. While the classic transitional

Reynolds number for the flow in a smooth pipe is

approximately 2300, turbulence can be anticipated in a

practical scenario at values as low as 500 where local

instabilities are introduced via wall roughness, tight bends

or protuberances along the pipe. In positive displacement

pumps, the stream of flow entering the chamber passes

directly over a valve, set in the fully open position, which

would act to induce an immediate flow separation, giving

rise to an immediate source of turbulence right from the

start of the cycle (Bluestein et al. 2002). Furthermore, even

in the absence of a valve, previous studies have indicated

that the expansion of the flow from the inlet to the main

chamber could cause transition to turbulence at a Reynolds

number as low as 754 (Konig et al. 1999b).

While the use of CFD is widespread, it remains a

considerable challenge and in complex cases as that

considered here, errors are expected to arise from many

sources. Some examples relevant here include inadequate

spatial/temporal resolution, the choice and implementation

of the numerical discretisation and gradient calculation,

the manner in which the boundaries are permitted to move

and the physical model of the blood. Therefore, validation

with experimental data is crucial, not only to understand

the level of confidence one can have in the results, but also

to assess the relative impact of each factor on the overall

error of prediction. The use of turbulence models in CFD is

well known to introduce complications and prediction

inaccuracies and is often cited as the ‘weak link’ in the

predictive accuracy of CFD. In common turbulence

models, the need for semi-empirical closures arises from

the derived form of the Reynolds averaged Navier–Stokes

(RANS) family of models, in which one considers to

model the time-averaged flow rather than the instan-

taneous flow. The alternative to this are either direct

numerical simulation (DNS), large eddy simulation (LES)

or a combination of RANS-LES (see e.g. Haase et al.

(2009)), though these are considerably more expensive

than RANS when used correctly, since they must be run

for sufficient time so that time-averages can be obtained.

In DNS, it is assumed that all scales of motion are

resolved, and the associated computational requirements

increase drastically with the Reynolds number, as it

switches from the laminar regime to turbulence. This is

due to the very nature of turbulence; coherent patches of

motion or ‘eddies’ which exist and remain at increasingly

small scales as the flow inertia is increased further. While a

purely laminar flow requires no turbulence modelling and

can be solved directly, without such consideration, a

substantial increase in resolution is required once the flow

becomes transitional or fully turbulent.

A study in 2003 by Avrahami (2003) assumed that the

flow inside the ‘Berlin’ pulsatile VAD to be fully laminar,

for a case where the mean and peak Reynolds numbers

were 1350 and 4200, respectively, and employed a

numerical resolution consistent with this assumption.

While the study provided useful insight into the flow, the

limitations of the laminar model were recognised and the

incorporation of turbulence and transitional-flow models

was recommended.

A range of experimental studies have been reported for

the flow of blood within a positive displacement pump,

using techniques such as particle image velocimetry (PIV)

and laser Doppler anemometry (LDA). Konig et al.

(1999a) and Konig and Clark (2001) investigated the flow

inside a VAD using flow visualisation and laser Doppler

velocity measurements. The author conducted tests using

two different Newtonian fluids, one with low viscosity and

the other with high viscosity, and included a plastic

pumping chamber in the experimental set-up. Some

numerical work was also conducted in the same study for

the purpose of comparison, and again the flow was

assumed to be fully laminar, i.e. no turbulence model was

used. Results indicated a reasonably good qualitative

agreement of the flow field but substantially under-

estimated recorded levels of velocity compared with the

M.G. Al-Azawy et al.2

Dow

nloa

ded

by [

The

Uni

vers

ity o

f M

anch

este

r L

ibra

ry]

at 0

7:08

01

Apr

il 20

15

LDA measurements. Of particular note, the predicted

qualitative agreement is worse for the lower viscosity case,

i.e. when the Reynolds number is higher and turbulence is

more prevalent (Konig et al. 1999a).

A series of experimental work at Penn State University

was initiated by Hochareon (2003) and Hochareon et al.

(2004), who undertook experimental work using a 50 cc

artificial heart pump design, employing PIV to obtain flow

field measurements of the flow within a sac-type artificial

heart. The study used a combination of conventional PIV

with particle-tracking velocimetry to achieve accurate wall

shear stress (WSS) estimation. After investigating the wall

shear and velocity measurements, the authors observed that

some areas exhibited low wall shear rate or flow stagnation,

indicating that these areas did not receive enough wall

washing and would thus be associated with higher risk of

thrombosis. Nanna et al. (2011) continued the work by

reporting PIVmeasurements on three new designs, in which

the position and orientation of the outlet port was

investigated to assess the impact on the flow field within

the chamber and specifically on thrombosis. The PIV data

were recorded at six planar positions within the pump and

results indicated that while differences were observed, the

effect of the outlet port position was relatively low.

In parallel to this work a series of companion CFD studies

were performed by Medvitz (2008) and Medvitz et al.

(2007, 2009), making use of the experimentally obtained

data to provide detailed assessment of CFD methods. In the

majority of the work an implicit LES method was

employed, in which again, no turbulence model is used

for the subgridscale modelling; inferring instead that small-

scale turbulence is approximately represented by numerical

dissipation, without the need for physical modelling. This

approach has some practical advantages, but is subject to

strong dependence on mesh resolution and is difficult to

justify from a physical perspective without a significantly

high mesh resolution. The same study also tested the

Spalart–Almaras (SA) model (Spalart & Allmaras 1992), a

popular one-equation RANS model and noted that, while it

was developed primarily for significantly higher Reynolds

number flow, it was broadly able to reproduce the same flow

field.

Based on Bluestein’s study (Bluestein et al. 2002) of the

turbulence induced by mechanical heart valves, the

traditional high Reynolds number based turbulence models,

for example k2 1, are inadequate for the use in such low

Reynolds number physiological flows. Instead, models

which have some natural suitability for such flows are more

appropriate, such as those based on the Wilcox ðk2 vÞ, orthe shear stress transport (SST) model of Menter (1994).

Indeed, similar conclusions are reported in many similar

CFD studies of physiological flow such as those byBluestein

et al. (2000, Bluestein et al. 2002) and Yin et al. (2004).

Even with ever-increasing computational speed, a need

for fast and efficient turbulence models will persist, and

thus the motivation to find a suitable RANS-based

approach is strong. More recently, there has been renewed

interest in the development of RANS models capable of

operating at low Reynolds number; sensitive to the effects

of transition and re-laminarisation relevant to the present

work. This work aims to assess one such recently

developed model for transitional flows (Menter et al. 2006)

versus the standard SST model it is based on (Menter

1994). Transitional flow is extremely complicated, and in

general is difficult to reproduce using single-point closure

modelling, i.e. models in which only local information is

used, since it is by nature highly non-local. As such careful

testing is required before these models can be confidently

employed in flows where one might expect them to be

needed. In what follows, we build on our preliminary study

(Al-Azawy et al. 2015) to compute the flow through the

50cc Penn State LVAD design V2 (Medvitz 2008), and

aim to evaluate the impact of the predictive uncertainties

from these models in addition to a model without a

turbulence model, the so-called ‘laminar’ model and a

suitable Reynolds stress model (RSM) (Launder et al.

1975). Results are also compared with predictions from

two other common models: the one equation SA model

(Spalart & Allmaras 1992) and the standard k-epsilon

(Launder & Spalding 1974).

2. Case description

In this study, a model of a VAD is constructed following

the work described in the previous section by Medvitz

(2008) on a 50cc LVAD test rig. Specifically, the V2

design was selected because this design, according to a

recent study, gave the best desirable flow behaviour

compared with other designs (Nanna et al. 2011). Figure 1

(a) shows the V2 design, which illustrates the position of

Bjork–Shiley valves and the pusher plate. The mitral

valve (23mm) and aortic valve (21mm) were simulated

without supported struts for the sake of simplicity. The

model was investigated under physiological operating

conditions at 86 BPM (beats per minute) and 4.2 LPM

(litres per minute). The details of experiments in a mock

circulatory loop were illustrated by Rosenberg et al.

Outlet port (a)

Aorticvalve

Inlet port

Mitralvalve

Pusherplate

(b)

Figure 1. (a) Model geometry showing fully opened valves and(b) numerical mesh M5.

Computer Methods in Biomechanics and Biomedical Engineering 3

Dow

nloa

ded

by [

The

Uni

vers

ity o

f M

anch

este

r L

ibra

ry]

at 0

7:08

01

Apr

il 20

15

(1981) and Hochareon (2003). The chamber in the in vitro

test was made from acrylic and a non-transparent

polyurethane diaphragm; the working fluid used was a

blood analogue of 50% sodium iodide, 34.47% water,

15.5% glycerine and 0.03% xanthan gum by weight. This

fluid is non-Newtonian and the kinematic viscosity is

4:3 £ 1026m2 s21. According to the movement of the

pusher plate, the resultant flow rates are shown in Figure 2,

with a peak systolic flow rate of 18 LPM and a peak

diastolic flow rate of 12 LPM. Avertical line is included to

denote the onset of systole.

The non-dimensional Reynolds number is given in the

following equation for an arbitrary diastolic ratio as

defined by Deutsch et al. (2006) and Bachmann et al.

(2000):

Re ¼ UL

n¼ 4

pn

� �SV

dinðN=RÞ ; ð1Þ

where SV is the stroke volume of the chamber that is equal

to the stroke length times the piston surface area, R is the

ratio of diastolic time ðt dÞ to cycle time T , n is the kinematic

viscosity, U is a characteristic flow velocity and N is the

beat rate. Bachmann chose the characteristic length scale

ðLÞ as the mitral port diameter ðdinÞ and the time scale was

chosen as diastolic time ðt d ¼ R=NÞ. According to this

equation, the Reynolds number of the present device is

1849. From the present results, the peak mitral Reynolds

number was recorded to be of the order of 3000.

3. Numerical description

In each case, the unsteady Navier–Stokes equations were

solved using a commercially available CFD software

(ANSYS FLUENT V.14) (ANSYS FLUENT Theory

Guide 2011) based on the finite volume method as follows:

›ui›xi

¼ 0;

›ui›t

þ uj›ui›xj

¼ 21

r

›p

›xiþ ›

›xjðnþ ntÞ ›ui

›xj

� �;

ð2Þ

where ui is the velocity gradients, xi is the Cartesian

coordinate in the ith direction, p is the pressure and r is thedensity. In the context of this work in which we have

investigated the use of turbulence models, n is the laminar

kinematic viscosity and nt is the turbulent viscosity,

calculated via additional transport equations representative

of the turbulence. The pressure–velocity coupling is

obtained by using the SIMPLEC algorithm (Van Doormaal

& Raithby 1984; Van Doormaal et al. 1987). Spatial

discretisation is second-order upwind while a first-order

implicit scheme is applied in time and the Green–Gauss

cell based scheme is used for gradient reconstruction.

Following the work of Medvitz et al. (2009) in their

study on the same case, an incompressible Newtonian fluid

is assumed here also. Given the dimensions of the device

considered, this is a reasonable assumption; see e.g.

Amornsamankul et al. (2006). More significantly, it is

anticipated that variation arising from different turbulence

model predictions will be far greater than that for which

different non-Newtonian models would indicate; thus it is

not of primary concern.

In the present simulations the total pressure and static

pressure were set at the inflow and outflow boundaries,

respectively, according to the in vitromeasurements which

indicated a device mean static pressure rise of 80mmHg

(Medvitz 2008). Therefore, the total pressure was set to

zero at the inlet and the static pressure was set to 80mmHg

at the outlet, in order to achieve the 80mmHg average

pressure rise. However, in order to minimise an adverse

impact of boundary conditions on the flow inside the

device, the inflow and outflow pipes were extended so that

the flow was given adequate space to fully develop. The

inlet was located seven inlet diameters upstream of the

mitral valve, and the outlet was located 15 inlet diameters

downstream of the aortic valve, following Medvitz (2008).

3.1 Turbulence modelling

In this study, six different approaches to modelling

turbulence were included. For clarity, the full model

equations are not listed below; the reader is instead

referred to the relevant source in each case. Four different

models were used to approximate nt in Equation (2):

. the standard SA model (Spalart & Allmaras 1992)

. the standard k2 1 model of Launder and Spalding

(1974)

Time (sec)

Vol

ume

flow

rate

(L

PM)

Push

er p

late

dis

plac

emen

t (m

m)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7–20

–15

–10

–5

0

5

10

15

20

0

5

10

15

Pusher–plateInlet portOutlet port

Figure 2. Flow rates at inlet and outlet ports and pusher platemovement.

M.G. Al-Azawy et al.4

Dow

nloa

ded

by [

The

Uni

vers

ity o

f M

anch

este

r L

ibra

ry]

at 0

7:08

01

Apr

il 20

15

. the standard k2 v SST model (Menter 1994)

. the transition-SST (Menter et al. 2006), also known

as the g2 Reu model.

Two further approaches were used where nt is not

needed:

. the standard RSM (Launder et al. 1975) adapted for

low Reynolds number flows via additional of the vequation following (Wilcox 1998). This approach

provides the Reynolds stresses uiuj directly and thus

eliminates the need for an eddy viscosity altogether.. the laminar equations, in which no turbulence model

is used and nt is zero.

Each of these models bring specific advantages as

well as intrinsic limitations, though they are all

offered as possible options via the present (and many

other) commercial CFD solvers. As such it is useful

for prospective users to review each in turn.

However, the flow here is of a low Reynolds nature

and so the standard models of Spalart and Allmaras

(1992) and Launder and Spalding (1974), tuned for

high Reynolds number flows, will be less relevant.

All others offer low Reynolds number features, but

transitional effects are especially challenging to

predict correctly, and thus the difference between

the standard and transitional forms of the SST is of

particular interest.

The turbulent intensity at the inlet was set to 3% in all

simulations, corresponding to best practise in the literature

(see e.g. Rodefeld et al. 2010; Kennington et al. 2011); the

authors also tested intensity levels of 5% and 7% for the

SST model and no notable differences were observed.

In general, a low level of turbulent intensity is desirable,

since high levels with long exposure times can cause lysis

activation (Konig & Clark 2001).

3.2 Dynamic modelling of the valves and pusher plate

In positive displacement pumps (piston-driven), it is

necessary to model the valve closure and the pusher plate

movement to maintain unidirectional flow and to acquire

the proper behaviour for the diastolic and systolic phases.

For modelling of the valve, various candidate methods are

available; either a dynamic mesh, immersed-boundary or a

binary flow model (where the flow is either fully closed or

fully opened). A ‘valve closer’ model was implemented by

Medvitz et al. (2007) using a binary model, along with a

variable viscosity model, as used by Avrahami (2003) and

Stijnen (2004). The valve closing and opening times are

short compared with the duration of diastole and systole

and no significant effect on the flow inside the chamber

was observed (Avrahami et al. 2006).

In this study, the mitral and aortic valves were fixed in

the fully open position during the pump cycle and without

supported struts, for the sake of computational simplicity.

To mimic the closed valve an interface was fixed

immediately above the valve, which was set to be a wall

during one part of the cycle and an open interface during

the next. The procedure of the diastolic and systolic phases

is shown in Figure 3. During diastole the flow enters from

the inlet port as the pusher plate expands, and the interface

above the aortic valve will be set as a wall, whereas during

systole the pusher plate will pump the flow towards the

outlet port and instead, the interface above the mitral valve

will be defined as a wall. The procedure is then repeated.

The dimensions in this study correspond to reference

experimental work, where the pusher plate diameter was

63.5mm, and the maximum chamber thickness, zc was

18.8 mm corresponding to a maximum volume of

approximately 50cc. The thickness of the chamber was

varied cyclically from a minimum of z=zc ¼ 0:218 to a

maximum of z=zc ¼ 1, described using the sine and cosine

series to match the in vitro waveform. The motion of the

Inlet port

Mitralvalve

Mitralvalve

Aorticvalve

Aorticvalve

Interface (wall)

Pusher platemovingdirection(expansion)

Diastolic phase

Y

Z X

Outlet port Interface (wall)

Pusher platemovingdirection(compression)

Systolic phase

Figure 3. Domain configuration during diastole (left) and systole (right).

Computer Methods in Biomechanics and Biomedical Engineering 5

Dow

nloa

ded

by [

The

Uni

vers

ity o

f M

anch

este

r L

ibra

ry]

at 0

7:08

01

Apr

il 20

15

pusher plate was modelled using a dynamic mesh layering

method, and cells were added/removed in increments of

Dz=zc ¼ 0:03. The time of the diastolic phase

(0 # t , 0:43 s) is longer than the systolic phase

(0:43 # t , 0:7 s), with the velocity of the wall introducedas follows:

For the diastolic and systolic phase:

Vdiastolewall ¼ A

2p

Tsin

2p

Tt

� �

Vsystolewall ¼ A

2p

Tcos

2p

Tt

� �;

ð3Þ

where Vwall is the velocity of the moving wall, the pusher

plate is represented as a function of time, t is the flow time

(s), A is the distance between the moving wall and the mid-

stroke position and T ¼ 0:7 s is a one cycle period.

3.3 Spatial and temporal resolution

Five different meshes were constructed to investigate the

spatial mesh resolution requirements for the three-

dimensional (3D) simulations, as shown in Table 1.

In this study, Pointwise CFD mesh generation software

(V16.04 R4) from Pointwise, Inc. (2011) was used to build

the CFD meshes, as shown in Figure 1(b). Figure 4

displays the variation of x-velocity with mesh size for the

SST k2 vmodel. A prism mesh with five layers was used

to resolve the boundary layer of the moving pusher plate,

where near wall resolution is assessed using the non-

dimensional distance to the first near-wall grid point,

yþ ¼ y=mffiffiffiffiffiffiffiffirtw

p, where y is the distance from the first cell

centre to the wall, m is the blood viscosity, r is the density

of blood and tw is the WSS. In all cases, this was set to the

recommended value of yþ , 1 for all locations inside the

chamber, and within yþ , 2:4 for the rest of the device.

The meshes were compared at the end of the diastolic

phase (at time t/T ¼ 0.61) at which point the pusher plate

is fully extended and the valves are fully open. For the two

finest meshes, the prediction of velocities are observed to

be very similar in contrast with the first two meshes which

vary significantly. These differences were a consequence

of the resolution of the boundary layer around the valves

and walls of the chamber. The computational time

increases with increasing mesh size; the mesh M4

(2,313,005 cells at the onset of diastole) is adequate to

capture the properties of the flow within the chamber and

near the valves and is selected for the following

discussions.

For transient simulations, one has to consider a time

step size based on the local flow velocity across each mesh

Table 1. Details of mesh models.

Mesh M1 M2 M3 M4 M5

No. of cells at onset of diastole 755,060 1,375,566 1,965,151 2,313,005 2,785,928No. of cells at onset of systole 840,260 1,533,846 2,193,811 2,541,665 3,014,623

–0.03 –0.02 –0.01 0 0.01 0.02 0.03

–0.3

–0.2

–0.1

0

0.1

0.2

0.3

M1M2M3M4M5

Position(m)

X-v

eloc

ity (

m/s

)

(b)

M1M2M3M4M5

–0.03 –0.02 –0.01 0 0.01 0.02 0.03

–0.2

–0.15

–0.1

–0.05

0

0.05

0.1

0.15

0.2

Position (m)

X-v

eloc

ity (

m/s

)

(a)

Figure 4. Time averaged x-velocity located on the plane z=zc ¼ 0:42 along a horizontal centreline (a) and a vertical centreline (b) at timet=T ¼ 0:614.

M.G. Al-Azawy et al.6

Dow

nloa

ded

by [

The

Uni

vers

ity o

f M

anch

este

r L

ibra

ry]

at 0

7:08

01

Apr

il 20

15

cell, to ensure that each point is able to correctly capture

the flow rate at that point. To achieve this the Courant–

Friedrichs–Lewy number CFL ¼ ðDtDxÞ=U is employed

during the simulation. In general, in the zone of interest,

CFL should of the order of unity for unsteady analysis.

In this study, five different time steps (in seconds) were

tested: f0:0001; 0:0005; 0:001; 0:003; 0:006} with the

same mesh and the same conditions. The time step Dt ¼0:001 s was found to be satisfactory, resulting a maximum

CFL number around 1 inside the chamber.

To obtain a fully converged unsteady solution, the

simulation was allowed to continue until a time periodic

flow was obtained. Figure 5 shows the history of velocity

magnitude at three points in the chamber and the test

performed for five pump cycles of flow with the same

conditions. In this study, the fourth cycle has been chosen

to extract the data of the simulation.

4. Results

4.1 Time evolution of mean flow

The current numerical set-up was first validated by

comparing the time-dependent mean flow field through the

device against the available PIV experimental data. Traces

of time-dependent mean velocity magnitude were

recorded at three extraction points in the chamber located

at 25% of chamber’s radius from the wall on the plane

z=zc ¼ 0:159. In vitro PIV and numerical data published

by Medvitz et al. (2009) were used in the computational

comparisons.

Figure 6 provides the comparison of velocity

magnitude at three points inside the chamber; proximal

to (i) the mitral port, (ii) the bottom of the chamber and

(iii) the aortic port, as shown. The results from the six

models tested are presented in two groups to facilitate

comparison. In the first column, the laminar model is

compared with both the standard and transition versions of

the SST model in order to understand the significance of

laminar, transitional and full turbulence modelling. In the

second column, focus is placed on increasing model

complexity; comparison is between a one equation model

(SA), a two equation model (k-epsilon) and a seven

equation model (RSM).

From the figure, it can be seen that there is little

variation between all models tested at the mitral port,

where the flow is injected into the chamber over the valve.

The exception is the RSM for which a closer agreement

with the experimental and numerical predictions are

reported, most likely associated with an improved ability

to model the swirling flow resulting from the presence of

the valve in its fully open position. The fact that all other

models produce very similar results at this stage supports

the finding that RSM is needed here.

At the bottom of the chamber, the prediction from the

RSM is again superior compared with the others, although

the standard SST model is here returning predictions

similar to those from the RSM. This is with the notable

exception of the peak of the velocity at the location

t=T ¼ 0:2, which is differs for both models. Seemingly the

RSM agrees better with the experimental results, whilst

SST agrees more with the reference numerical results.

While one is unable to say with certainty which is the more

‘correct’, the presence of this lag helps illustrate an

advantage of using RSM since it accounts for more the

realistic transport, or ‘history’, of 3D effects.

The above-mentioned observations help indicate the

limitations of the linear stress–strain relationship, implicit

in all standard eddy viscosity models (EVMs). More

advanced turbulence models such as non-linear EVMs

(NLEVMs) allow for a functional relationship between

stress and strain that can improve the response of the

model to effects such as streamline curvature and/or swirl.

Despite offering some improvements however, NLEVMs

remain unable to fully incorporate the history effects of 3D

turbulence, which would allow it to be more realistically

de-coupled from the local mean strain field. In order to

achieve the latter capabilities, a full RSM is required.

Particularly during diastole, the other models report

very similar results, though some differences are observed

during systole. The strongest variation between these

models is reported at the most downstream location,

proximal to the aortic valve; this is to be expected since it

represents the accumulation of different flow features

arising upstream. Compared with the standard SST model,

the transition model returns a higher value of flow rate

during diastole, and a lower value during systole.

It appears that the higher levels of turbulence in this

region act to ‘smooth’ out the peak variations. Similar

observations are made for the standard k-epsilon model

and to a lesser extent, the SA model. In all three cases, thisFigure 5. History of velocity magnitude for five cycles at threepoints in the chamber.

Computer Methods in Biomechanics and Biomedical Engineering 7

Dow

nloa

ded

by [

The

Uni

vers

ity o

f M

anch

este

r L

ibra

ry]

at 0

7:08

01

Apr

il 20

15

can be explained by the higher levels of turbulent viscosity

resulting from erroneously high turbulence, and the

consequential higher levels of momentum diffusion

according to Equation (2).

4.2 Examination of flow field

Figure 7 displays comparisons of levels of turbulence

viscosity ratio (TVR) for all turbulent models used in this

study.1 Planes are taken at six points during the cycle,

corresponding to early, peak and late instances in first

diastole and then systole. To aid cross-comparison, marks

are provided to indicate the location of points where data

was extracted in Figure 6.

At the start of the cycle, it is observed that levels of

TVR are first increased with the arrival of the incoming

flow over the valve, as is expected. This turbulence is then

convected around the bottom section of the curved

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ExperimentalNumerical[Medvitz]Spalart-Allmarask-epsilonRSM

0.4

0.2

0.6

0.8

1

Vel

ocity

(m

/s)

t/T

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.4

0.2

0.6

0.8

1 ExperimentalNumerical[Medvitz]LaminarSSTK-omegaTransition-SST

Vel

ocity

(m

/s)

t/T

(a)

ExperimentalNumerical[Medvitz]Spalart-Allmarask-epsilonRSM

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

Vel

ocity

(m

/s)

t/T

ExperimentalNumerical[Medvitz]LaminarSSTK-omegaTransition-SST

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

Vel

ocity

(m

/s)

t/T

(b)

1ExperimentalNumerical[Medvitz]Spalart-Allmarask-epsilonRSM

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

Vel

ocity

(m

/s)

t/T

ExperimentalNumerical[Medvitz]LaminarSSTK-omegaTransition-SST

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

Vel

ocity

(m

/s)

t/T

(c)

Figure 6. Cyclic variation of velocity magnitude at (a) mitral port, (b) bottom of the chamber, and (c) aortic port. Reference data(experiment and numerical) from Medvitz et al. (2009).

M.G. Al-Azawy et al.8

Dow

nloa

ded

by [

The

Uni

vers

ity o

f M

anch

este

r L

ibra

ry]

at 0

7:08

01

Apr

il 20

15

chamber and is also transported into the central region.

Compared with the standard SST model, the transition

model indicates peak values in more or less the same

locations, though the levels are significantly higher. The

transition model uses an empirical correlation to respond

to certain trigger points in a laminar boundary layer flow

by increasing levels of turbulence in a way that mimics

natural transition. We recall that it is the turbulent

viscosity which dictates the extent to which momentum is

diffused, or ‘smoothed-out’ in the centre of the chamber,

as noted previously in discussion of Figure 6.

The TVR is also predicted to be high for both the k-

epsilon and the SA models, which in their standard forms

are tuned for much higher Reynolds numbers than the

present flow, and would thus not be expected to respond

correctly. In contrast, values of TVR for both SST andRSM

remain around two orders of magnitude lower. Indeed, the

standard SST model includes a limiter on nt, which may be

responsible for its favourable performance, while the RSM

is instead naturally able to adjust to the correct levels of

turbulence since it does not compute nt directly.Figure 8 illustrates the vorticity magnitude at the

plane z=zc ¼ 0:21, as defined in Equation (4) where vi is

the vorticity vector and 1ijk is the Levi-Civita cyclic

operator.

kvk ¼ffiffiffiffiffiffiffiffiffiffiffiffi2vivi

p; where vi ¼ 1ijk

›uk›uj

: ð4Þ

(b) SST-k omega (c) Transition-SST (e) RSM(a) Spalart-Allmaras (d) k-epsilon

t/T=0.5

t/T=0.643

t/T=0.8

t/T=0.86

t/T=0.3

t/T=0.143

Turbulent Viscosity Ratio 1.0E-05 3.9E-05 1.5E-04 5.8E-04 2.3E-03 8.8E-03 3.4E-02 1.3E-01 5.1E-01 2.0E+00 7.7E+00 3.0E+01

Figure 7. TVR at plane z=zc ¼ 0:21. Figures on far right indicate relative position of displacement pump during cycle. Crosses in firstrow correspond to the location of velocity data sampling points presented in Figure 6.

Computer Methods in Biomechanics and Biomedical Engineering 9

Dow

nloa

ded

by [

The

Uni

vers

ity o

f M

anch

este

r L

ibra

ry]

at 0

7:08

01

Apr

il 20

15

Levels of vorticity observed within the chamber were

observed to reach peak levels during diastole, especially

proximal to the inlet port at around t/T ¼ 0.3. A higher

degree of unsteadiness is observed in the case of the

standard SST model, consistent with the above-mentioned

observations that excessive turbulence from the transition

model (as well as for the k-epsilon model and to a lesser

extent the SA model) act to smooth out peaks in the flow-

rate. It should be noted that such smoothing will reduce the

frequency of instances of high strain rate, i.e. relevant to

assessing the propensity of flow features likely to cause

haemolysis. Results from both the laminar and standard

SST models are very similar during diastole, indicating

that turbulence predicted by the SST model is minimal.

In contrast the flow pattern is slightly different for the

RSM in peak diastole, corresponding to improved results

observed in Figure 6 and justifying the need for a

turbulence modelling closure.

Transition is a particularly challenging feature to

capture using engineering turbulence models because it is,

by nature, induced by small instabilities that may arise

from different parts of the domain (i.e. non-local).

Transition modelling relies on high accuracy numerical

approximations to the governing equations to reduce

numerical noise that may otherwise hide, or indeed

amplify, the small-scale flow features that induce

(c) SST-k omega (d) Transition-SST (f) RSM(b) Spalart-Allmaras (e) k-epsilon

t/T=0.5

t/T=0.643

t/T=0.8

t/T=0.86

t/T=0.3

t/T=0.143

(a) Laminar

0 50 100 150 200 250 300 350 400 450 500

Vorticity (1/s)

Figure 8. Contours of vorticity magnitude at the plane z=zc ¼ 0:21. Figures on far right indicate relative position of displacement pumpduring cycle.

M.G. Al-Azawy et al.10

Dow

nloa

ded

by [

The

Uni

vers

ity o

f M

anch

este

r L

ibra

ry]

at 0

7:08

01

Apr

il 20

15

transition. Despite the sensitivity studies conducted in

Section 3, high orders of spatial and temporal accuracy are

difficult to provide in such complex geometries, and in this

work the round-off errors are likely to be significant.

It may be that the transition model in its present form is

overly sensitive to such errors, and hence indicates a

higher level of turbulence than is otherwise expected.

Given the operational Reynolds number of this case,

improved or specifically tailored transition modelling is an

area of high relevance to the current application, and

indeed is the subject of ongoing further study by several

groups, including that of the authors.

4.3 Clinical relevance of results

To analyse the behaviour of flow inside a typical blood

pump, the shear stress and strain rate should be

investigated in appropriate zones inside the pump as a

function of time, in order to assess the impact on the

prediction of potential blood clot damage models which

use these quantities.

Figure 9 provides comparison of the evolution of the

strain rate invariant kSk predicted by the four more

promising models from the previous section, tested at

along two arcs located at 8% of chamber’s radius from the

wall (wr) and at a distance of z=zc ¼ 0:186 from the front

wall as indicated in the figure. The strain rate invariant is

defined in Equation (5).

kSk ¼ ffiffiffiffiffiffiffiffiffiffiffiffi2SijSij

p; where Sij ¼ 1

2

›ui›xj

þ ›uj›xi

� �: ð5Þ

This parameter provides a scalar measure of local

mean-flow velocity gradients, where kSk is high, there

may be potential for haemolysis, while below a certain

value, there may instead be a risk of platelet activation and

thrombosis.

Throughout diastole, a patch of high strain rate is

predicted to occur at a location along the arc between the

2 o’clock and 3 o’clock locations when no model is used

(Figure 9(a)), while this patch is much reduced with the

turbulence models. The transition model appears to

provide the lowest prediction of strain rate, which is

consistent with previous observations that excessive

turbulence would reduce high gradients. A patch of high

strain rate is found throughout systole to occur at a

location between 10 o’clock and 11 o’clock (Figure 9

(b)) in all cases, corresponding to the outflow.

As expected, the maximum shear rates are found near

the mitral valve in peak diastole and near the aortic

valve in peak systole. It is also important to note the

variation in minimum values, since previous studies in

pulsatile LVADs have shown that thrombus deposition

is correlated to areas of low strain rate, associated to

flow stagnation.

Figure 10 shows contours of WSS, ktwallk, which is

computed from the viscous stress tensor tij and the surfacenormal vector nj as follows:

ktwallk ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitwalli twalli

q; ð6Þ

twalli ¼ tij�njffiffiffiffiffiffiffiffinjnj

p ; ð7Þ

tij ¼ 2ðnþ ntÞSij: ð8ÞIn the figure, contours of WSS are plotted over the

surface of the device in various stages of the cycle. Results

are displayed for the laminar model as well as for standard

SST, transition-SST and RSM. Baldwin et al. (1994) stated

that exposure to shear stresses higher than ~150 N=m2 was

likely to lead to blood damage, indicating that this would not

be the case for the vast majority of the results reported here,

where ktwallk # 10. However, we note that in some cases

values in excess of 150 N=m2 are reported at a small

number of cells close to the narrow gap between the aortic

valve and the wall.2 It is likely that this is not physical, but

rather it is an error associated with limitations of near-wall

modelling or imperfections in the mesh, although this

remains to be elucidated by further study. Either way, values

over the majority of the domain are far lower than those

expected to damage the blood. Compared with the bulk of

the flow, highest levels ofWSS are observed to occur around

both valves, particularly the aortic valve during systole.

While observed differences are small, the RSM predicts the

highest values of the results reported. Differences between

models are relatively low, mostly likely due to the fact that

the same mesh has been used and the wall treatments used

are similar in each case. It is expected that WSS would be

highly sensitive to both different near-wall mesh resolution,

though this remains to be quantified in future work.

While precise error bounds on such thresholds are

difficult to provide in practice, and will depend on many

clinical factors, it has been demonstrated by this work that

the selected method of turbulence modelling is yet another

factor that should be considered. Erroneously high levels of

turbulence act to reduce peak values of flow rate, and thus

reduce the measured instances of strain rate, and could

thereby be of impact in the evaluation of risk of haemolysis.

5. Conclusion

CFD is capable of taking a leading role in the investigation

of the flow physics within the development of LVADs;

enabling the evaluation of potential blood damage where

experimental data are not readily available. In this work,

the unsteady transitional flow through a model of an

LVAD was simulated numerically, in which the motion of

Computer Methods in Biomechanics and Biomedical Engineering 11

Dow

nloa

ded

by [

The

Uni

vers

ity o

f M

anch

este

r L

ibra

ry]

at 0

7:08

01

Apr

il 20

15

the pusher plate was incorporated via a dynamic mesh

layering method. The application of five relevant industrial

turbulence models was reported, as well as the prediction

when no model was used, i.e. assuming a laminar flow.

The comparisons with available experimental and

numerical data indicate that for much of the flow, a closer

agreement with reference results is obtained from the

RSM, while the standard SST k2 vmodel also performed

somewhat better than average. Other models, predomi-

nantly tuned for higher Reynolds number flows, tended to

over predict turbulence and hence diffusion of momentum.

The transition SST model appears to indicate premature

and excessive levels of transition to turbulence, which has

the counter-intuitive effect of leading to greater levels of

turbulence than with the standard SST model. In all cases,

exhibiting an over-prediction of turbulence, levels of strain

rate and WSS are observed to be reduced as a

consequence, which could potentially lead to under-

estimation of clinical risk.

Despite the identified shortfalls of the turbulence

models tested in this work, it seems that with careful

consideration, RANS computations can be used as a tool for

Strain rate (s–1)

RSM

Transition-SST

SST-k omega

3002752502252001751501251007550250

wr

RSM

SST-k omega

Transition-SST

12 o’clock 12 o’clock

12 o’clock 12 o’clock

3 o’clock

3 o’clock12 o’clock

t/T t/T

9 o’clock 12 o’clock

Position

t/Tt/T t/T

t/T

3 o’clockposition

9 o’clock

9 o’clock

Position

Position

(a) Laminar Laminar (b)

t/T t/T

3 o’clock12 o’clock 9 o’clock 12 o’clockPosition Position

position Position

Figure 9. Evolution of strain rate along arcs on the plane z1=zc ¼ 0:186, (a) from 3 to 12 o’clock (proximal to the inlet port), and (b)from 9 to 12 o’clock (proximal to the outlet port).

M.G. Al-Azawy et al.12

Dow

nloa

ded

by [

The

Uni

vers

ity o

f M

anch

este

r L

ibra

ry]

at 0

7:08

01

Apr

il 20

15

design and optimisation of these devices. Compared with

LES they retain significant advantages in terms of reduced

cost, the practical ability to attain mesh converged solutions

and the faster realisation of time-averaged data. Further

analysis of more advanced turbulence models is required to

provide a more accurate prediction of the flow in this

complicated application, and the impact such models may

have on predicted levels of shear rate and WSS. This is

deemed necessary to provide insight into the design and

development of VADs with respect to thrombosis and

haemolysis, and is the focus of ongoing work.

Acknowledgements

The authors acknowledge the assistance given by IT Services andthe use of the Computational Shared Facility at the University ofManchester.

Disclosure statement

No potential conflict of interest was reported by the authors.

Funding

The financial support from the Higher committee for educationdevelopment in Iraq and University of Wasit is greatlyacknowledged.

Notes

1. It is noted that the laminar model does not provide thisquantity, while for RSM this must be computed a posteriorias cmðuiui=2Þ2=1n:

2. In the figure, contours are displayed in the range 0 #ktwallk # 16 for clarity, but the location of the maximaproximal to the aortic valve coincideS with the locationswhere these high values were found.

(a)

t/T=0.143

(b)

(c)

(d)

t/T=0.3

t/T=0.8

t/T=0.86

Wall shear stress(N/m2)

RSMSST k -omega Transition -SST

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Laminar

X

Y

Z

wall-shear

Figure 10. WSS at early/peak diastole and peak/late systole phase.

Computer Methods in Biomechanics and Biomedical Engineering 13

Dow

nloa

ded

by [

The

Uni

vers

ity o

f M

anch

este

r L

ibra

ry]

at 0

7:08

01

Apr

il 20

15

References

Al-Azawy MG, Turan A, Revell A. 2015. Investigating the use ofturbulence models for flow investigations in a positivedisplacement ventricular assist device, in 6th Europeanconference of the International Federation for Medical andBiological Engineering p. 395–398. Vol. 45. doi:10.1007/978-3-319-11128-5_255.

Allen GS, Murray KD, Olsen DB. 1997. The importance ofpulsatile and nonpulsatile flow in the design of blood pumps.Artif Organs. 21(8):922–928. doi:10.1111/j.1525-1594.1997.tb00252.x.

Amornsamankul S, Wiwatanapataphee B, Wu YH, Lenbury Y.2006. Effect of non-Newtonian behaviour of blood onpulsatile flows in stenotic arteries. Int J Biol Med Sci.1(1):42–46.

ANSYS, Inc. ANSYS FLUENT theory guide, 2011. Release14.0. Southpointe, USA.

Avrahami I, Rosenfeld M, Raz S, Einav S. 2006. Numerical modelof flow in a sac-type ventricular assist device. Artif Organs.30(7):529–538. doi:10.1111/j.1525-1594.2006.00255.x.

Avrahami I. 2003. The effect of structure on the hemodynamicsof artificial blood pumps [PhD thesis]. Tel Aviv: Tel-AvivUniversity.

Bachmann C, Hugo G, Rosenberg G, Deutsch S, Fontaine A,Tarbell JM. 2000. Fluid dynamics of a pediatric ventricularassist device. Artif Organs. 24(5):362–372. doi:10.1046/j.1525-1594.2000.06536.x.

Bluestein D, Li YM, Krukenkamp IB. 2002. Free emboliformation in the wake of bi-leaflet mechanical heartvalves and the effects of implantation techniques.J Biomech. 35(12):1533–1540. doi:10.1016/S0021-9290(02)00093-3.

Bluestein D, Rambod E, Gharib M. 2000. Vortex shedding as amechanism for free emboli formation in mechanical heartvalves. J Biomech Eng. 122(2):125–134. doi:10.1115/1.429634.

Baldwin JT, Deutsch S, Geselowitz DB, Tarbell JM. 1994. LDAmeasurements of mean velocity and Reynolds stress fieldswithin an artificial heart ventricle. J Biomech Eng.116(2):190–200. doi:10.1115/1.2895719.

Deutsch S, Tarbell JM, Manning KB, Rosenberg G, FontaineAA. 2006. Experimental fluid mechanics of pulsatileartificial blood pumps. Ann Rev Fluid Mech. 38(1):65–86.doi:10.1146/annurev.fluid.38.050304.092022.

Haase W, Braza M, Revell A. 2009. DESider – a European efforton hybrid RANS-LES modelling: results of the European-Union funded project, 2004–2007. Vol. 103 Springer.doi:10.1007/978-3-540-92773-0.

Hochareon P. 2003. Development of particle image velocimetry(PIV) for wall shear stress estimation within a 50cc PennState artificial heart ventricular chamber [PhD thesis].University Park, PA: The Pennsylvania State University.

Hochareon P, Manning KB, Fontaine AA, Tabell JM, Deutsch S.2004. Wall shear-rate estimation within the 50cc Penn Stateartificial heart using particle image velocimetry. J BiomechEng. 126(4):430–437. doi:10.1115/1.1784477.

Kennington JR, Frankel SH, Chen J, Koenig SC, Sobieski MA,Giridharan GA, Rodefeld MD. 2011. Design optimizationand performance studies of an adult scale viscous impellerpump for powered Fontan in an idealized total cavopulmon-ary connection. Cardiovasc Eng Technol. 2(4):237–243.doi:10.1007/s13239-011-0058-2.

Konig CS, Clark C. 2001. Flow mixing and fluid residence timesin a model of a ventricular assist device. Med Eng Phys.23:99–110.

Konig CS, Clark C, Mokhtarzadeh-Dehghan MR. 1999a.Comparison of flow in numerical and physical models of aventricular assist device using low- and high-viscosity fluids.Proc Inst Mech Eng H J Eng Med. 213:423–432. doi:10.1243/0954411991535031.

Konig CS, Clark C, Mokhtarzadeh-Dehghan MR. 1999b.Investigation of unsteady flow in a model of a ventricularassist device by numerical modelling and comparison withexperiment. Med Eng Phys. 21:53–64.

Launder BE, Reece GJ, Rodi W. 1975. Progress in thedevelopment of a Reynolds-stress turbulence closure.J Fluid Mech. 68(3):537–566. doi:10.1017/S0022112075001814.

Launder BE, Spalding D. 1974. The numerical computation ofturbulent flows. Comput Methods Appl Mech Eng.3(2):269–289. doi:10.1016/0045-7825(74)90029-2.

Lee W, Jerry F. 2007. Applied biofluid mechanics. USA:McGraw Hill. doi:10.1036/0071472177.

Medvitz RB. 2008. Development and validation of a compu-tational fluid dynamic methodology for pulsatile blood pumpdesign and prediction of thrombus potential. [PhD thesis]University Park, PA: Pennsylvania State University.

Medvitz RB, Kreider JW, Manning KB, Fontaine AA,Deutsch S, Paterson EG. 2007. Development andvalidation of a computational fluid dynamics method-ology for simulation of pulsatile left ventricular assistdevices. ASAIO J. 53(2):122–131. doi:10.1097/MAT.0b013e31802f37dd.

Medvitz RB, Reddy V, Deutsch S, Manning KB, Paterson EG.2009. Validation of a CFD methodology for positivedisplacement LVAD analysis using PIV data. J BiomechEng. 131(11):1110091–1110099. doi:10.1115/1.4000116.

Menter FR. 1994. Two-equation eddy-viscosity turbulencemodels for engineering applications. AIAA J.32(8):1598–1605. doi:10.2514/3.12149.

Menter FR, Langtry RB, Likki SR, Suzen YB, Huang PG, VolkerS. 2006. A correlation-based transition model using localvariables – Part I: model formulation. J Turbomach.128(3):413–422. doi:10.1115/1.2184352.

Nanna JC, Wivholm JA, Deutsch S, Manning KB. 2011. Flowfield study comparing design iterations of a 50cc leftventricular assist device. ASAIO J. 57(5):349–357. doi:10.1097/MAT.0b013e318224e20b.

Pointwise, Inc. Pointwise Tutorial Workbook, 2011. Release16.04R4, USA.

Rodefeld MD, Coats B, Fisher T, Giridharan GA, Chen J, BrownJW, Frankel SH. 2010. Cavopulmonary assist for theuniventricular Fontan circulation: von Karman viscousimpeller pump. J Thorac Cardiovasc Surg.140(3):529–536. doi:10.1016/j.jtcvs.2010.04.037.

Rosenberg G, Phillips WM, Landis DL, Pierce WS. 1981. Designand evaluation of the Pennsylvania State University mockcirculatory system. ASAIO. 4(2):41–49.

Sezai A, Shiono M, Orime Y, Nakata K, Hata M, Yamada H, IidaM, Kashiwazaki S, Kinishita J, Nemoto M, et al. 1997. Renalcirculation and cellular metabolism during left ventricularassisted circulation: comparison study of pulsatile andnonpulsatile assists. Artif Organs. 21(7):830–835. doi:10.1111/j.1525-1594.1997.tb03752.x.

Spalart PR, Allmaras SR. 1992. A one-equation turbulencemodel for aerodynamic flows. 30th Aerospace SciencesMeeting and Exhibit. doi:10.2514/6.1992-439.

Stijnen JMA. 2004. Interaction between the mitral and aorticheart valve an experimental and computational study. [PhDthesis] Eindhoven University.

M.G. Al-Azawy et al.14

Dow

nloa

ded

by [

The

Uni

vers

ity o

f M

anch

este

r L

ibra

ry]

at 0

7:08

01

Apr

il 20

15

Van Doormaal JP, Raithby GD. 1984. Enhancements of thesimple method for predicting incompressible fluid flows.Numer Heat Transfer. 7(2):147–163. doi:10.1080/01495728408961817.

Van Doormaal JP, Turan A, Raithby GD. 1987. Evaluation ofnew techniques for the calculation of internationalrecirculating flows, AIAA 25th Aerospace Sciences Meeting,Reno, AIAA-87-0059. doi:10.2514/6.1987-59.

Wilcox DC. 1998. Turbulence modeling for CFD. Vol. 2. LaCanada, CA: DCW Industries.

Yin W, Alemu Y, Affeld K, Jesty J, Bluestein D. 2004. Flow-induced platelet activation in bileaflet and monoleafletmechanical heart valves. Ann Biomed Eng.32(8):1058–1066. doi:10.1114/B:ABME.0000036642.21895.3f.

Computer Methods in Biomechanics and Biomedical Engineering 15

Dow

nloa

ded

by [

The

Uni

vers

ity o

f M

anch

este

r L

ibra

ry]

at 0

7:08

01

Apr

il 20

15