Turbulent Blood Flow in a Ventricular Assist Device

Application Challenge AC7-03 © copyright ERCOFTAC 2021

Evaluation

Experimental Validation of URANS and LES

For simulations in turbopumps, hydraulic characteristics such as the head $H$ are among the most important result variables. Also in the field of CFD applications in VADs, it is common to use the head to validate the numerical calculation. In fact, the measurement of the head is the only experimental validation of the simulated flow field in a large number of literature studies (e.g., in [4], [5], [28], [29]). Since the pressure in the pump is coupled via the governing equations to the flow field, the comparison of the heads can be used as the first stage of flow field validation in VADs.

Fig. 5.1. Validation of the numerically calculated pressure heads

H

with experimental data. The figure shows the head curve for

n=7'900~1/min

.

The experimental and numerical results are given in Figure 5.1. For the operating point at $Q=4.5~l/min$ , good agreement between numerical and experimental results can be observed with a deviation of $\Delta H/H_{exp}=-0.12\%$ for LES and $\Delta H/H_{exp}=-4.82\%$ for URANS. For the smaller flow rate $Q=2.5~l/min$ the deviations are slightly larger with $\Delta H/H_{exp}=-4.84\%$ for LES and $\Delta H/H_{exp}=-5.66\%$ for URANS.

In summary, the discrepancy between numerically and experimentally determined head is still within an acceptable range for both LES and URANS. From this, it can be concluded that both numerical models are valid for reproducing the real pressure buildup of the VAD.

Fluid Mechanical & Hemodynamical Evaluation of LES and URANS

Inner Efficiencies $\eta _{i}$

The inner efficiencies according to Eq. (2) of the impeller (index: $i$ ) and the whole pump (index: $p$ ) are given in Tab. 5.1. The deviation between the URANS and the reference LES case are minor for both operation points with a maximum deviation of $+3.2~\%$ . It can be concluded that the turbulence-modelling URANS method can reflect the efficiencies, and hence the global losses, as accurately as the turbulence-resolving LES method.

Table 5.1 Inner efficiencies of the impeller and the whole pump.
Flow Rate	Parameter	LES	URANS	Relative deviation to LES [%]
$Q=4.5~l/min$	$\eta _{i,i}~[\%]$	$49.1$	$48.8$	$-0.6$
$Q=2.5~l/min$	$\eta _{i,i}~[\%]$	$46.8$	$48.3$	$+3.2$
$Q=4.5~l/min$	$\eta _{i,p}~[\%]$	$34.2$	$33.1$	$-3.3$
$Q=2.5~l/min$	$\eta _{i,p}~[\%]$	$29.7$	$29.5$	$-0.7$

Equivalent Shear Stresses $\tau _{eff}$ in the time-averaged flow field

The computed effective stresses $\tau _{eff}$ (Eq. (9) in section Description) are plotted for both operating points in Fig. 5.2. The stresses from the reference LES are compared to the URANS computations (The stresses in the instanteous flow field can also be seen in Fig. 1.3 in Description). As can be seen from the LES results, relevant stresses above 9 Pa (threshold for vWF degradation) and 50 Pa (platelet activation) are present within the flow channel of the rotor and the outlet guide vane. In general, the stresses are underpredicted with URANS. Nevertheless, similar hot-spots for significant stresses $\geq 9~Pa$ are observable for the partial operation point ( $Q=2.5~l/min$ ). This is in contrast to the nominal operation point ( $Q=4.5~l/min$ ), where greater deviations in computed stresses are noticable between LES and URANS. Despite the fact that the URANS can reflect the high stresses in the gap vortex and the trailing edge flow regions (red areas marked with (A) and (B)), other relevant regions in the blade channels of the impeller (C) or in the area between the impeller and the outlet guide vane (D) cannot be adequately reflected for the nominal operation point. In these areas of the pump, complex interactions occur between secondary flows are present (explained in Refs. [2] and [27]), which are directly resolved by the LES. In contrast, the URANS turbulence model cannot adequately model the impact of these complex turbulent flow interactions on the effective stress field.

Fig. 5.2. Effective stresses

\tau _{eff}

of LES and URANS. The figure displays a cylindical cut (explained in Fig. 1.3. in Description) through the rotor and outlet guide vane at a radius of 80% of the outer radius

R_{2}

. Top row: partial operating point at

Q=2.5~l/min

. Bottom row: nominal operating point at

Q=4.5~l/min

. Note 1: The "hard" cut in stresses directly between impeller and outlet guide vane domain is present due to averaging procedures in different frames of references. Note 2: The blades (I,II and 1-3) and the blade channels are mirrored to get an overview of the entire blade channels.

Hemodynamical Evaluation: Hemolysis Index ${MIH}$ and Volumetric Threshold Analysis $I_{\tau _{eff}}$

The computed MIH indices (defined by Eq. (11) in section Description) are shown in Table 5.2. As already displayed in the stress fields in Fig. 5.2., the LES computes highest hemolysis values in both operating points. This is partly due to the coarser spatial and temporal resolution of the stresses in URANS (also recognizable from the grid convergence study in Fig. 3.3 in section CFD Simulations). But also the already mentioned insufficient resolution of the secondary structures in the blade channels leads to the larger deviations at at $Q=4.5~l/min$ .

Table 5.2 Modified index of Hemolysis ${MIH}~[-]$ . The constants $(C,\alpha ,\beta )$ of Zhang et al. [30] were used in Eq. (11). The percentage value in brackets indicate the relative deviation between URANS and LES.
$MIH$ by Eq. (11)	partial load $Q=2.5~l/min$	nominal load $Q=4.5~l/min$
LES - reference	$83.4$	$59.1$
URANS	$79.8~(-4.3\%)$	$50.3~(-12.2\%)$

Tables 5.3. and 5.4. show the percentage of the entire VAD volume in which certain stress thresholds for van-Willebrand-degradation (vWF; $>9Pa$ ), platelet activation ( $>50Pa$ ) and hemolysis ( $>150Pa$ ) are exceeded. Again, the computed results of URANS are lower than those from LES. Relative deviations of maximal $-15.0\%$ ( $Q=4.5~l/min$ ) are observable for the stress thresholds of $9~Pa$ and $50~Pa$ . Only for the stress threshold above $150~Pa$ , larger deviations are present for all URANS cases, which is due to the coarser near-wall grid density, where the highest stresses are present. These near-wall stresses affect the numerical hemolysis prediction greatly (see Ref. [29] or [31]).

As can be seen from these hemodynamic results, the discrepancies between URANS and LES in the evaluation of blood damage are many times greater than those for the previously evaluated fluid mechanical parameters such as head or efficiency.

Table 5.3 Analysis of volumes $I_{\tau _{eff}}$ , in which the thresholds for vWF degradation (>9 Pa), platelet activation (>50 Pa) and hemolysis (>150 Pa) are exceeded, at the partial load. The percentage value in brackets indicate the relative deviation between URANS and LES.
$Q=2.5~l/min$	$I_{\tau _{eff}>9~Pa}~[\%]$	$I_{\tau _{eff}>50~Pa}~[\%]$	$I_{\tau _{eff}>150~Pa}~[\%]$
LES - reference	$8.41$	$0.87$	$0.12$
URANS	$7.68~(-8.7\%)$	$0.81~(-6.9\%)$	$0.10~(-16.7\%)$

Table 5.4 Analysis of volumes $I_{\tau _{eff}}$ , in which the thresholds for vWF degradation (>9 Pa), platelet activation (>50 Pa) and hemolysis (>150 Pa) are exceeded, at the nominal load. The percentage value in brackets indicate the relative deviation between URANS and LES.
$Q=4.5~l/min$	$I_{\tau _{eff}>9~Pa}~[\%]$	$I_{\tau _{eff}>50~Pa}~[\%]$	$I_{\tau _{eff}>150~Pa}~[\%]$
LES - reference	$11.34$	$0.97$	$0.13$
URANS	$9.64~(-15.0\%)$	$0.84~(-13.4\%)$	$0.10~(-23.1\%)$

Influence of Modelled Turbulent Stresses on the Hemodynamical Evaluation with URANS

The effective stress from Eq. (9) is defined to include the contribution of the whole flow field in the stress definition. As Konnigk et al. [6] were able to show, this stress definition can be used to achieve a better consistency in the computed stresses between different simulation methods (RANS, LES, DNS). For RANS methods, the contribution of turbulence to the stresses is accounted for including the modeled turbulent dissipation $\langle \varepsilon _{mod}\rangle$ to Eq. (9). However, in previous analyses of stresses in VAD simulations, this contribution from the turbulence model was often not considered [5,32-34]. For this reason, the influence of this modeled term on the hemodynamic evaluation with URANS will be examined in this subsection.

Fig. 5.3. Effective stresses

\tau _{eff}

of URANS using different stress definitions at partial load. The figure displays a cylindical cut (explained in Fig. 1.3. in Description) through the rotor and outlet guide vane at a radius of 80% of the outer radius

R_{2}

.

The stress fields of the URANS simulation at the partial operating point are shown in Fig. 5.3. In this figure, the effective stresses according to Eq. (9) (left sub-figure) are compared with the effective stresses where the contribution of the modeled turbulence is omitted (right sub-figure). A huge portion of relevant stresses ( $>9Pa$ ) cannot be reflected, when the contribution from the turbulence model is not considered. In particular, the turbulent stresses in the blade channel and in the outlet guide vane are significantly underpredicted, which is considered critical since these areas will also not be considered in the hemodynamic optimization of the VAD.

The influence of this underprediction on the blood damage prediction can be seen in Table 5.5. where the volumetric analysis was performed with the two stress definitions. Especially for the threshold of $9~Pa$ , approximately half of the volume will not be computed, when $\varepsilon _{mod}$ is not considered.

Table 5.5 Analysis of volumes $I_{\tau _{eff}}$ , in which the thresholds for vWF degradation (>9 Pa), platelet activation (>50 Pa) and hemolysis (>150 Pa) are exceeded, at partial load. The percentage value in brackets indicate the relative deviation between the different stress definitions.
$Q=2.5~l/min$	$I_{\tau _{eff}>9~Pa}~[\%]$	$I_{\tau _{eff}>50~Pa}~[\%]$	$I_{\tau _{eff}>150~Pa}~[\%]$
URANS - $\tau _{eff}$ according to Eq. (9)	$7.68$	$0.81$	$0.10$
URANS - $\tau _{eff}$ according to Eq. (9) without $\langle \varepsilon _{mod}\rangle$	$4.22~(-45.1\%)$	$0.65~(-19.8\%)$	$0.09~(-10.0\%)$