A Saint-Venant Model for Overland Flows with Precipitation and Recharge
Abstract
:1. Introduction
2. Navier–Stokes Equations with Infiltration and Recharge
2.1. Geometric Set-Up and the Two-Dimensional Navier—Stokes Equations
2.2. The Wet Boundary
2.3. The Free Surface
2.4. Mixing Friction
3. Saint-Venant System with Recharge via Vertical Averaging
3.1. Dimensionless Navier–Stokes Equations
3.2. Remark Slip vs. No-Slip Boundary Condition
3.3. First-Order Approximation of the Dimensionless Navier–Stokes Equations
3.4. The Saint-Venant System with Recharge
3.5. Example (Lake at Rest and Filling the Lake)
3.6. Why the Mixing Friction?
4. Entropy
4.1. Theorem (Hyperbolicity and Stability)
- (a)
- System (70) is strictly hyperbolic on the set .
- (b)
- If is smooth and , we have the velocity balance equation
- (a)
- The Jacobian of (70)’s flux function is given byFor these eigenvalues to be real and distinct, we require that ; the Jacobian matrix is thus diagonalizable and system (70) is strictly hyperbolic on the set .
- (b)
- We rewrite the conservation of momentum equation in system (70) in terms of the unknowns , with , asApplying the product rule to the first term of (82) and substituting in the conservation of mass equation, we getSubstituting this into (85) and dividing by h throughout, we getMaking the further substitution
4.2. Remark (Friction Effects)
4.3. Theorem (Entropy Production)
4.4. Remark (Entropy–Entropy-Flux Pairs and Entropy Production)
4.5. Remark (Discontinuous Solutions)
5. The Numerical Model
5.1. Well-Balanced Schemes
5.2. Kinetic Function
5.3. Proposition (Macroscopic–Microscopic Relations)
5.4. Kinetic Connection to Saint-Venant
5.5. Remark (Advantages of the Kinetic Formulation)
- (i)
- In contrast to previous work (e.g., [19]), the kinetic Equation (113) contains an extra term accounting for precipitation and infiltration effects. This departure is crucial for the derivation of the fluxes that lead to a well-balanced scheme in the presence of such terms.
- (ii)
- We also note that, even though the Maxellian M is constructed for still water steady states, where , we can still use it here to ensure a well-balanced scheme.
- (iii)
- In general, it is easier to find a numerical scheme to solve Equation (113) for M that has the properties we desire, such as entropy stability, than to solve the full Saint-Venant system for h and u. However, in finding M, we can calculate h and by virtue of the macro-/microscopic relations (Proposition 5.3). In fact, M is never calculated explicitly; rather, the function
- (iv)
- As shown and fixed by Xia et al. [48], Buttinger-Kreuzhuber et al. [49], and Taccone et al. [50], some other well-balanced numerical methods fail to correctly represent the effect of the topography, especially when the water height h is close to zero, while the kinetic approach used herein supports arbitrarily small h.
5.6. Discretization and Kinetic Fluxes
- (i)
- : movement of water with positive velocity () from within cell to cell ;
- (ii)
- : movement of water with negative velocity () from within cell to cell . This term is decomposed a second time into components reflecting whether the water has enough energy to overcome the topography and friction to enter or leave the cell.
6. Numerical Tests
6.1. Comparison with Real-World Data
6.2. Single-Level and Three-Level Cascades
- (1)
- The total length of the rainfall process ,
- (2)
- The topography of the slope onto which the rain falls, for which we consider a constant slope (the single cascade) with and a decreasing slope (the three-level cascade, see Figure 6) with
- (3)
- The rain-induced friction level , for which we take and 5 for both the single cascade and the three-level cascade.
7. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Ersoy, M.; Lakkis, O.; Townsend, P. A Saint-Venant Model for Overland Flows with Precipitation and Recharge. Math. Comput. Appl. 2021, 26, 1. https://doi.org/10.3390/mca26010001
Ersoy M, Lakkis O, Townsend P. A Saint-Venant Model for Overland Flows with Precipitation and Recharge. Mathematical and Computational Applications. 2021; 26(1):1. https://doi.org/10.3390/mca26010001
Chicago/Turabian StyleErsoy, Mehmet, Omar Lakkis, and Philip Townsend. 2021. "A Saint-Venant Model for Overland Flows with Precipitation and Recharge" Mathematical and Computational Applications 26, no. 1: 1. https://doi.org/10.3390/mca26010001
APA StyleErsoy, M., Lakkis, O., & Townsend, P. (2021). A Saint-Venant Model for Overland Flows with Precipitation and Recharge. Mathematical and Computational Applications, 26(1), 1. https://doi.org/10.3390/mca26010001