TY - JOUR
T1 - Ill posedness in modelling two-dimensional morphodynamic problems
T2 - Effects of bed slope and secondary flow
AU - Chavarrías, Víctor
AU - Schielen, Ralph
AU - Ottevanger, Willem
AU - Blom, Astrid
N1 - Accepted Author Manuscript
PY - 2019
Y1 - 2019
N2 - A two-dimensional model describing river morphodynamic processes under mixed-size sediment conditions is analysed with respect to its well posedness. Well posedness guarantees the existence of a unique solution continuously depending on the problem data. When a model becomes ill posed, infinitesimal perturbations to a solution grow infinitely fast. Apart from the fact that this behaviour cannot represent a physical process, numerical simulations of an ill-posed model continue to change as the grid is refined. For this reason, ill-posed models cannot be used as predictive tools. One source of ill posedness is due to the simplified description of the processes related to vertical mixing of sediment. The current analysis reveals the existence of two additional mechanisms that lead to model ill posedness: secondary flow due to the flow curvature and the effect of gravity on the sediment transport direction. When parametrising secondary flow, accounting for diffusion in the transport of secondary flow intensity is a requirement for obtaining a well-posed model. When considering the theoretical amount of diffusion, the model predicts instability of perturbations that are incompatible with the shallow water assumption. The effect of gravity on the sediment transport direction is a necessary mechanism to yield a well-posed model, but not all closure relations to account for this mechanism are valid under mixed-size sediment conditions. Numerical simulations of idealised situations confirm the results of the stability analysis and highlight the consequences of ill posedness.
AB - A two-dimensional model describing river morphodynamic processes under mixed-size sediment conditions is analysed with respect to its well posedness. Well posedness guarantees the existence of a unique solution continuously depending on the problem data. When a model becomes ill posed, infinitesimal perturbations to a solution grow infinitely fast. Apart from the fact that this behaviour cannot represent a physical process, numerical simulations of an ill-posed model continue to change as the grid is refined. For this reason, ill-posed models cannot be used as predictive tools. One source of ill posedness is due to the simplified description of the processes related to vertical mixing of sediment. The current analysis reveals the existence of two additional mechanisms that lead to model ill posedness: secondary flow due to the flow curvature and the effect of gravity on the sediment transport direction. When parametrising secondary flow, accounting for diffusion in the transport of secondary flow intensity is a requirement for obtaining a well-posed model. When considering the theoretical amount of diffusion, the model predicts instability of perturbations that are incompatible with the shallow water assumption. The effect of gravity on the sediment transport direction is a necessary mechanism to yield a well-posed model, but not all closure relations to account for this mechanism are valid under mixed-size sediment conditions. Numerical simulations of idealised situations confirm the results of the stability analysis and highlight the consequences of ill posedness.
KW - nonlinear instability
KW - river dynamics
KW - sediment transport
UR - http://www.scopus.com/inward/record.url?scp=85064980349&partnerID=8YFLogxK
U2 - 10.1017/jfm.2019.166
DO - 10.1017/jfm.2019.166
M3 - Article
AN - SCOPUS:85064980349
SN - 0022-1120
VL - 868
SP - 461
EP - 500
JO - Journal of Fluid Mechanics
JF - Journal of Fluid Mechanics
ER -