A pulsatile 3D flow relevant to thoracic hemodynamics: CFD - 4D Flow MRI comparison

Application Challenge AC7-04 © copyright ERCOFTAC 2021

Best Practice Advice

Key Fluid Physics

The flow considered in the CFD-4D Flow MRI comparison has to reflect some properties found in the cardiovascular system. Thereby it is necessary to consider a pulsatile flow, with intermediate values of the Reynolds number within the laminar-turbulent range. Furthermore the geometry also has to be representative to study typical flow patterns. The present geometry allows to study typical flow in aneurysm, arch, flow split and flow merge.

Application Uncertainties

There are several sources of uncertainties which can explain the differences between measurements and simulation.

A first reason could come from the experimental inlet conditions which are used in the simulation. The assumption is made that the flow is fully-developed at the position where the 2D cine PC-MRI image is made thanks to the flow straightener set upstream. In order to increase the signal-to-noise ratio, the 2D acquisition was made on a 6 mm slice, which implies that the velocity field used as the inlet is a spatial average over this slice width. Furthermore, the process to apply this experimental condition on the inlet includes a spatio-temporal interpolation.

Another small difference between experiment and CFD is the geometry of the domain, where there is a 60 μm geometric tolerance in the 3D-printing process of the phantom. The wall position is another source of discrepancy. Indeed there are partial volume artefacts in MR images for voxels straddling the surface of the phantom, which lead to a poor definition of the wall boundaries.

Computational Domain and Boundary Conditions

The phantom was designed in order to remove some of the uncertainties inherent to the cardiovascular system. Thereby all the walls in the computational domain were assumed rigid, consistent with the material selected for manufacturing the phantom, and a no-slip boundary condition was prescribed. For the outlet condition, the choice was made to have a single outlet in the experimental phantom in order to safely apply a zero-pressure condition in the simulation. As shown in Fig. 1, the set-up included a rigid tube and a flow straightener upstream the inlet insuring the developed flow profile expected.

According to the MRI acquisition process, it seems unrealistic to be able to capture all flow features occurring within the phantom with this imaging method, especially as cycle-to-cycle fluctuations may appear for such high Reynolds number unsteady flows. Thus phase-averaging the CFD simulation appears to be more relevant than instantaneous CFD velocity fields for comparison with the MRI fields. Finally, the downsampling process to interpolate these phase-averaged CFD fields onto the MRI grid is also required to mimic the PC MRI resolution.

Discretisation and Grid Resolution

To resolve most of the turbulent fluctuations of the flow in LES frameworks, high order schemes are preferred as they reduce the numerical diffusion and dispersion. The spatial discretization scheme used in the present AC provides a 4th-order accurate approximation of the finite-volume integration on regular grids. As shown in section on numerical accuracy, a tetrahedral mesh with a characteristic length of 0.7 mm was found fine enough to resolve the main flow features.

Physical Modelling

To model turbulence, the LES strategy is chosen rather than RANS. In LES, the largest turbulent scales are explicitly resolved as a solution of the low-pass filtered Navier-Stokes equations while the subgrid scales are modelled, whereas in RANS all the scales are averaged and the entire turbulence spectrum is modelled. According to Kolmogorov’s theory, the statistics of the smallest structures are universal and only depend on the rate of kinetic energy dissipation and the viscosity of the fluid. On the contrary, large scales fluctuations are geometry-dependant and their shape is less generic. As it generally harbours many large scale fluctuations induced by the complex geometries of the cardiovascular system (Chnafa et al., 2016 [19]), LES strategy seems then more adapted to solve this type of flows. This is even more true because the flow regime is more transitional than turbulent (peak Reynolds number of order 2000).

Recommendations for Future Work

Some recommendations for future work are attempts to achieve more physiological models, both for the experimental and the simulation sides. A pulsatile sinusoidal flow rate was used as inlet boundary condition in this AC. Instead one could use a flow rate closer to the one observed clinically in the aorta. The geometry has been assumed rigid. In reality, blood vessel walls deform and are usually modelled as elastic or hyperelastic. Another uncertainty in the modelling are the inlet boundary conditions. PIV measurements could be conducted to validate numerical predictions. Finally, using an hybrid mesh with prismatic elements near the wall boundaries for better prediction of this region could be investigated.

References

[1] T. Puiseux, Simulations numériques pour l’Imagerie par Résonance Magnétique à contraste de phase. PhD thesis, Universit de Montpellier, 2019.

[2] T. Puiseux, A. Sewonu, O. Meyrignac, H. Rousseau, F. Nicoud, S. Mendez, and R. Moreno, “Reconciling PC-MRI and CFD: an in-vitro study,” NMR in Biomedicine, vol. 32, no. 5, p. e4063, 2019.

[3] T. Puiseux, A. Sewonu, R. Moreno, S. Mendez, and F. Nicoud, “Numerical simulation of time-resolved 3d phase-contrast magnetic resonance imaging,” PLoS ONE, vol. 16, no. 3, p. e0248816, 2021.

[4] V. Moureau and G. Lartigue, “YALES2.” https://www.coria-cfd.fr/index.php/YALES2, 2021. Accessed: 2021-06-25.

[5] V. Moureau, P. Domingo, and L. Vervisch, “Design of a massively parallel CFD code for complex geometries,” Comptes Rendus Mecanique, vol. 339, no. 2, p. 141–148, 2011.

[6] V. Moureau, P. Domingo, and L. Vervisch, “From large-eddy simulation to direct numerical simulation of a lean premixed swirl flame: Filtered laminar flame-pdf modeling,” Combustion and Flame, vol. 158, p. 1340–1357, 2011.

[7] A. Chorin, “Numerical solution of the Navier-Stokes equations,” Mathematics of Computation, vol. 22, p. 745–762, 1968.

[8] C. Chnafa, S. Mendez, and F. Nicoud, “Image-based large-eddy simulation in a realistic left heart,” Computers & Fluids, vol. 94, p. 173–187, 2014.

[9] M. Malandain, N. Maheu, and V. Moureau, “Optimization of the deflated conjugate gradient algorithm for the solving of elliptic equations on massively parallel machines,” Journal of Computational Physics, vol. 238, no. Supplement C, pp. 32–47, 2013.

[10] S. Mendez and F. Nicoud, “YALES2BIO.” https://imag.umontpellier.fr/~yales2bio/, 2021. Accessed: 2021-06-25.

[11] J. Kim and P. Moin, “Application of a fractional-step method to incompressible Navier-Stokes equations,” Journal of Computational Physics, vol. 59, no. 2, pp. 308–323, 1985.

[12] F. Nicoud, H. Toda, O. Cabrit, S. Bose, and J. Lee, “Using singular values to build a subgrid-scale model for large eddy simulations,” Physics of Fluids, vol. 23, no. 8, p. 085106, 2011.

[13] F. Nicoud, C. Chnafa, J. Sigüenza, V. Zmijanovic, and S. Mendez, Large-Eddy Simulation of Turbulence in Cardiovascular Flows, pp. 147–167. Cham: Springer International Publishing, 2018.

[14] H. Baya Toda, O. Cabrit, K. Truffin, G. Bruneaux, and F. Nicoud, “Assessment of subgrid-scale models with an les-dedicated experimental database: the pulsatile impinging jet in turbulent cross- flow,” Physics of Fluids, vol. 26, no. 7, p. 075108, 2014.

[15] J. Sigüenza, S. Mendez, D. Ambard, F. Dubois, F. Jourdan, R. Mozul, and F. Nicoud, “Validation of an immersed thick boundary method for simulating fluid-structure interactions of deformable membranes,” Journal of Computational Physics, vol. 322, pp. 723– 746, 2016.

[16] S. Pope, Turbulent Flows. Cambridge University Press, 2000.

[17] A. Yoshizawa and K. Horiuti, “A statistically-derived subgrid-scale kinetic energy model for the large-eddy simulation of turbulent flows,“ Journal of the Physical Society of Japan, vol. 54, no. 8, pp. 2834–2839, 1985.

[18] D. Steinman, C. Ethier, and B. Rutt, “Combined analysis of spatial and velocity displacement artifacts in phase contrast measurements of complex flows,” Journal of Magnetic Resonance, vol. 7, no. 2, pp. 339–346, 1997.

[19] C. Chnafa, S. Mendez, and F. Nicoud, “Image-based simulations show important flow fluctuations in a normal left ventricle: What could be the implications?,” Annals of Biomedical Engineering, vol. 44, no. 11, p. 3346–3358, 2016.

Contributed by: Morgane Garreau — University of Montpellier, France