Periodic Reporting for period 1 - StratifiedGRANULAR (Modelling of rheologically stratified granular flows by a multi-layer depth-averaged approach)
Berichtszeitraum: 2019-10-01 bis 2021-09-30
Geophysical granular flows are a major hazard to men and infrastructures. This MSCA project is intended to develop a multi-layer model for describing highly-sheared granular flows characterized by a rheological stratification. The project aims at: (1) overcoming the mathematical issues related to the loss of hyperbolicity; (2) describing the essential physics of granular avalanches, whose rheology is strongly depending on the solid volume fraction.
The MSCA researcher developed a well-posed depth-averaged model, where various layers are advected in a dynamically coupled way. The stress and shear-rate tensors are related through the well-established µ(I) rheology. The evolution of the volume fraction is captured by an advection-diffusion transport equation that incorporates a dilatancy law depending on the inertial number. To avoid short-wave instabilities of Hadamard type, a physically-based viscous regularization, using an approximation of the in-plane stress gradients, is proposed. The model has been numerically integrated by a finite volume scheme with a high-resolution lateralized Harten–Lax–van Leer (LHLL) solver. For the comparison with the velocity and volume fraction profiles in steady state, an experimental dataset from the Environmental and Maritime Hydraulics Laboratory (LIDAM) of the University of Salerno (Italy) has been used, where the sidewall velocity profiles have been obtained by granular particle image velocimetry, g-PIV (Sarno et al., Adv. Powder Technol., 2018), and the near-wall volume fraction profiles have been estimated by the stochastic-optical method, SOM (Sarno et al., Granul. Matter, 2016). Additional experimental datasets from the literature have been used to validate the model in the transient state.
The objectives of the project have been:
(1) development of a multi-layer model for granular flows, where a suitable constitutive law is implemented and the evolution of the volume fraction is consistently considered;
(2) taking into account the effects due to irregular topographies;
(3) numerical integration of the model;
(4) validation of the model through laboratory experiments;
(5) broadening the skills of the MSCA researcher in mathematical modelling and computational fluid dynamics;
(6) helping the MSCA researcher to develop scientific independence.
The MSCA researcher developed a well-posed depth-averaged model, where various layers are advected in a dynamically coupled way. The stress and shear-rate tensors are related through the well-established µ(I) rheology. The evolution of the volume fraction is captured by an advection-diffusion transport equation that incorporates a dilatancy law depending on the inertial number. To avoid short-wave instabilities of Hadamard type, a physically-based viscous regularization, using an approximation of the in-plane stress gradients, is proposed. The model has been numerically integrated by a finite volume scheme with a high-resolution lateralized Harten–Lax–van Leer (LHLL) solver. For the comparison with the velocity and volume fraction profiles in steady state, an experimental dataset from the Environmental and Maritime Hydraulics Laboratory (LIDAM) of the University of Salerno (Italy) has been used, where the sidewall velocity profiles have been obtained by granular particle image velocimetry, g-PIV (Sarno et al., Adv. Powder Technol., 2018), and the near-wall volume fraction profiles have been estimated by the stochastic-optical method, SOM (Sarno et al., Granul. Matter, 2016). Additional experimental datasets from the literature have been used to validate the model in the transient state.
The objectives of the project have been:
(1) development of a multi-layer model for granular flows, where a suitable constitutive law is implemented and the evolution of the volume fraction is consistently considered;
(2) taking into account the effects due to irregular topographies;
(3) numerical integration of the model;
(4) validation of the model through laboratory experiments;
(5) broadening the skills of the MSCA researcher in mathematical modelling and computational fluid dynamics;
(6) helping the MSCA researcher to develop scientific independence.
The project activities have been declined via 4 Work Packages (WPs):
Work Package 1:
The goal of WP1 was to develop a well-posed multi-layer model for granular flows in the dense-collisional regime. A first activity was to study the simple case of two layers, in order to overcome the issues due to the hyperbolicity loss and, possibly, extend the method of the extra momentum exchanges also to the multilayer case. A theoretical study on the eigenstructure of the two-layer model has been performed by asymptotic analysis. This study increased the insight into the mathematical structure of two- and multi-layer models and revealed a new family of stability criteria, valid for any flow condition.
The second part of WP1 was devoted to the development of the multi-layer model. The viscoplastic µ(I) rheology has been employed as a constitutive law, while the evolution of the volume fraction has been considered through an advection-diffusion transport equation, implementing a dilatancy law, depending on the inertial number, and also designed to account for possible non-local effects. To regularize the ill-posed first-order multi-layer model, the structure of the µ(I) rheology has been successfully exploited, by considering an approximation of the in-plane stress gradients.
Work Package 2:
In WP2 the multi-layer model has been extended to the case of irregular basal topographies. A fixed Cartesian frame of reference with one axis parallel to the mean bed slope is considered, while the topography irregularities are treated by introducing a superimposed bed function. For the closure of the multi-layer model, suitable discretized expressions, consistent with the chosen constitutive and dilatancy laws, have been developed. As well, different closures for the mass exchanges among the layers have been investigated.
Work Package 3:
In WP3 the model has been numerically integrated. A multi-stage time-splitting approach, with the advection part of the partial differential equation system being solved through a lateralized LHLL approximate Riemann solver, has been used. High-resolution in space has been obtained by a MUSCL predictor-corrector treatment. For the numerical treatment of the diffusion-like term an explicit scheme has been implemented. To guarantee the numerical stability of this stage of the numerical scheme, a multi-step time-marching approach has been considered. An advantage of the proposed multi-layer model, over fully 3D models, is represented by its higher computational efficiency. Indeed, the computational costs are found to increase at most linearly with the number of the layers. Moreover, thanks to its well-posedness for any value of the inertial number, the multi-layer model can safely be used on irregular topographies with arbitrary slopes. Various numerical simulations have been performed in order to investigate the main properties of the model.
Work Package 4:
In WP4 the MSCA researcher compared the multi-layer model with experimental data. A laboratory dataset, previously gathered at the LIDAM, has been employed for comparisons of both velocity and volume fraction profiles in the steady state. Additionally, two experimental datasets from the literature have been used for the comparisons in the unsteady state. For a reliable comparison of the multi-layer model with experimental data, a crucial task was to consider the effects of the sidewall resistances that are typically non-negligible in the majority of experimental setups. To this end, a rate-independent Coulomb-friction law has been introduced at the sidewall surfaces of each layer. The comparisons with experimental data confirmed the good capabilities of the multi-layer model in capturing the essential physics of granular flows in both steady and transient states. A relatively worse agreement with the experimental volume fraction was observed in the upper more dilute flow region, which suggests that a more sophisticated dilatancy law would be advisable in a future development of the model.
Work Package 1:
The goal of WP1 was to develop a well-posed multi-layer model for granular flows in the dense-collisional regime. A first activity was to study the simple case of two layers, in order to overcome the issues due to the hyperbolicity loss and, possibly, extend the method of the extra momentum exchanges also to the multilayer case. A theoretical study on the eigenstructure of the two-layer model has been performed by asymptotic analysis. This study increased the insight into the mathematical structure of two- and multi-layer models and revealed a new family of stability criteria, valid for any flow condition.
The second part of WP1 was devoted to the development of the multi-layer model. The viscoplastic µ(I) rheology has been employed as a constitutive law, while the evolution of the volume fraction has been considered through an advection-diffusion transport equation, implementing a dilatancy law, depending on the inertial number, and also designed to account for possible non-local effects. To regularize the ill-posed first-order multi-layer model, the structure of the µ(I) rheology has been successfully exploited, by considering an approximation of the in-plane stress gradients.
Work Package 2:
In WP2 the multi-layer model has been extended to the case of irregular basal topographies. A fixed Cartesian frame of reference with one axis parallel to the mean bed slope is considered, while the topography irregularities are treated by introducing a superimposed bed function. For the closure of the multi-layer model, suitable discretized expressions, consistent with the chosen constitutive and dilatancy laws, have been developed. As well, different closures for the mass exchanges among the layers have been investigated.
Work Package 3:
In WP3 the model has been numerically integrated. A multi-stage time-splitting approach, with the advection part of the partial differential equation system being solved through a lateralized LHLL approximate Riemann solver, has been used. High-resolution in space has been obtained by a MUSCL predictor-corrector treatment. For the numerical treatment of the diffusion-like term an explicit scheme has been implemented. To guarantee the numerical stability of this stage of the numerical scheme, a multi-step time-marching approach has been considered. An advantage of the proposed multi-layer model, over fully 3D models, is represented by its higher computational efficiency. Indeed, the computational costs are found to increase at most linearly with the number of the layers. Moreover, thanks to its well-posedness for any value of the inertial number, the multi-layer model can safely be used on irregular topographies with arbitrary slopes. Various numerical simulations have been performed in order to investigate the main properties of the model.
Work Package 4:
In WP4 the MSCA researcher compared the multi-layer model with experimental data. A laboratory dataset, previously gathered at the LIDAM, has been employed for comparisons of both velocity and volume fraction profiles in the steady state. Additionally, two experimental datasets from the literature have been used for the comparisons in the unsteady state. For a reliable comparison of the multi-layer model with experimental data, a crucial task was to consider the effects of the sidewall resistances that are typically non-negligible in the majority of experimental setups. To this end, a rate-independent Coulomb-friction law has been introduced at the sidewall surfaces of each layer. The comparisons with experimental data confirmed the good capabilities of the multi-layer model in capturing the essential physics of granular flows in both steady and transient states. A relatively worse agreement with the experimental volume fraction was observed in the upper more dilute flow region, which suggests that a more sophisticated dilatancy law would be advisable in a future development of the model.
The impact of the MSCA action, foreseen in the project proposal, is fully achieved, as documented by the published papers.
It is worth remarking that the proposed multi-layer model is mathematically well-posed for all inertial numbers, thanks to the terms accounting for the in-plane stress gradients. This nice property makes the model an excellent candidate for the description of geophysical granular flows on irregular topographies and under different flow conditions. Hence, the model represents a significant breakthrough advancement over pre-existing models based on the same µ(I) rheology that are only conditionally well-posed, and it is also less computationally expensive than fully 3D models. Moreover, to the best of our knowledge, this is the first multi-layer model for granular flows, where the varying volume fraction field is consistently described.
The proposed multi-layer model is expected to be of great interest to geophysicists and hydraulic engineers, and has the potential to become a computationally affordable tool to simulate field-scale events for the mitigation of natural hazards of this type.
It is worth remarking that the proposed multi-layer model is mathematically well-posed for all inertial numbers, thanks to the terms accounting for the in-plane stress gradients. This nice property makes the model an excellent candidate for the description of geophysical granular flows on irregular topographies and under different flow conditions. Hence, the model represents a significant breakthrough advancement over pre-existing models based on the same µ(I) rheology that are only conditionally well-posed, and it is also less computationally expensive than fully 3D models. Moreover, to the best of our knowledge, this is the first multi-layer model for granular flows, where the varying volume fraction field is consistently described.
The proposed multi-layer model is expected to be of great interest to geophysicists and hydraulic engineers, and has the potential to become a computationally affordable tool to simulate field-scale events for the mitigation of natural hazards of this type.