scholarly journals On the dynamics of Navier–Stokes equations for a shallow water model

2011 ◽  
Vol 250 (6) ◽  
pp. 2687-2714 ◽  
Author(s):  
Qin Duan
Keyword(s):  
Shallow Water ◽  
Stokes Equations ◽  
Water Model ◽  
Navier Stokes ◽  
Shallow Water Model ◽  
Navier Stokes Equations

scholarly journals Shallow water model for pollutant distribution in the Ypacarai Lake

2021 ◽  
Vol 26 (2) ◽  
pp. 54-76
Author(s):  
Diego Bareiro ◽  
Enrique O’Durnin ◽  
Laura Oporto ◽  
Christian Schaerer
Keyword(s):  
Shallow Water ◽  
Stokes Equations ◽  
Water Model ◽  
Navier Stokes ◽  
Shallow Water Model ◽  
Navier Stokes Equations ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Wind Velocities ◽  
The Impact

In this paper, we analyze the distribution of a non-reactive contaminant in Ypacarai Lake. We propose a shallow-water model that considers wind-induced currents, inflow and outflow conditions in the tributaries, and bottom effects due to the lakebed. The hydrodynamic is based on the depth-averaged Navier-Stokes equations considering wind stresses as force terms which are functions of the wind velocity. Bed (bottom) stress is based on Manning's equation, the lakebed characteristics, and wind velocities. The contaminant transportation is modeled by a 2D convection-diffusion equation taking into consideration water level. Comparisons between the simulation of the model, analytical solutions, and laboratory results confirm that the model captures the complex dynamic phenomenology of the lake. In the simulations, one can see the regions with the highest risk of accumulation of contaminants. It is observed the effect of each term and how it can be used them to mitigate the impact of the pollutants.    

A stationary circular hydraulic jump, the limits of its existence and its gasdynamic analogue

2008 ◽  
Vol 601 ◽  
pp. 189-198 ◽  
Author(s):  
ASLAN R. KASIMOV
Keyword(s):  
Surface Tension ◽  
Hydraulic Jump ◽  
Stokes Equations ◽  
Critical Value ◽  
Water Model ◽  
Navier Stokes ◽  
Jump Conditions ◽  
Shallow Water Model ◽  
Finite Radius ◽  
Navier Stokes Equations

We propose a theory of a steady circular hydraulic jump based on the shallow-water model obtained from the depth-averaged Navier–Stokes equations. The flow structure both upstream and downstream of the jump is determined by considering the flow over a plate of finite radius. The radius of the jump is found using the far-field conditions together with the jump conditions that include the effects of surface tension. We show that a steady circular hydraulic jump does not exist if the surface tension is above a certain critical value. The solution of the problem provides a basis for the hydrodynamic stability analysis of the hydraulic jump. An analogy between the hydraulic jump and a detonation wave is pointed out.

Geometrically intrinsic modeling of shallow water flows

2020 ◽  
Vol 54 (6) ◽  
pp. 2125-2157 ◽  
Author(s):  
Elena Bachini ◽  
Mario Putti
Keyword(s):  
Shallow Water ◽  
Stokes Equations ◽  
Order Approximation ◽  
Bottom Topography ◽  
Water Model ◽  
Navier Stokes ◽  
Bottom Surface ◽  
Finite Volume Scheme ◽  
Navier Stokes Equations ◽  
Mountain Landscape

Shallow water models of geophysical flows must be adapted to geometric characteristics in the presence of a general bottom topography with non-negligible slopes and curvatures, such as a mountain landscape. In this paper we derive an intrinsic shallow water model from the Navier–Stokes equations defined on a local reference frame anchored on the bottom surface. The equations resulting are characterized by non-autonomous flux functions and source terms embodying only the geometric information. We show that the proposed model is rotational invariant, admits a conserved energy, is well-balanced, and it is formally a second order approximation of the Navier–Stokes equations with respect to a geometry-based order parameter. We then derive a numerical discretization by means of a first order upwind Godunov finite volume scheme intrinsically defined on the bottom surface. We study convergence properties of the resulting scheme both theoretically and numerically. Simulations on several synthetic test cases are used to validate the theoretical results as well as more experimental properties of the solver. The results show the importance of taking into full consideration the bottom geometry even for relatively mild and slowly varying curvatures.

scholarly journals CALCULATIONS OF WAVES FORMED FROM SURFACE CAVITIES

1976 ◽  
Vol 1 (15) ◽  
pp. 63 ◽  
Author(s):  
Charles L. Mader
Keyword(s):  
Water Waves ◽  
Wave Motion ◽  
Stokes Equations ◽  
Water Model ◽  
Navier Stokes ◽  
Critical Depth ◽  
Shallow Water Model ◽  
Navier Stokes Equations ◽  
Tsunami Waves ◽  
Long Wave

The wave motion resulting from cavities in the ocean surface was investigated using both the long wave, shallow water model and the incompressible Navier-Stokes equations. The fluid flow resulting from the calculated collapse of the cavities is significantly different for the two models. The experimentally observed flow resulting from explosively formed cavities is in better agreement with the flow calculated using the incompressible Navier-Stokes model. The resulting wave motions decay rapidly to deep water waves. Large cavities located under the surface of the ocean will be more likely to result in Tsunami waves than cavities on the surface. This is contrary to what has been suggested by the upper critical depth phenomenon.

On the Flow Fields of Inertial Impactors

1974 ◽  
Vol 96 (4) ◽  
pp. 394-400 ◽  
Author(s):  
V. A. Marple ◽  
B. Y. H. Liu ◽  
K. T. Whitby
Keyword(s):  
Flow Field ◽  
Stokes Equations ◽  
Relaxation Method ◽  
Water Model ◽  
Navier Stokes ◽  
Visualization Technique ◽  
Navier Stokes Equations ◽  
Theoretical Results ◽  
Plate Distance ◽  
Good Agreement

The flow field in an inertial impactor was studied experimentally with a water model by means of a flow visualization technique. The influence of such parameters as Reynolds number and jet-to-plate distance on the flow field was determined. The Navier-Stokes equations describing the laminar flow field in the impactor were solved numerically by means of a finite difference relaxation method. The theoretical results were found to be in good agreement with the empirical observations made with the water model.

scholarly journals Finite-Volume Solvers for a Multilayer Saint-Venant System

2007 ◽  
Vol 17 (3) ◽  
pp. 311-320 ◽  
Author(s):  
Emmanuel Audusse ◽  
Marie-Odile Bristeau
Keyword(s):  
Shallow Water ◽  
Numerical Investigation ◽  
Finite Volume ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Relevant Alternative ◽  
Navier Stokes Equations ◽  
Numerical Tests ◽  
Shallow Water Models ◽  
Water Models

Finite-Volume Solvers for a Multilayer Saint-Venant SystemWe consider the numerical investigation of two hyperbolic shallow water models. We focus on the treatment of the hyperbolic part. We first recall some efficient finite volume solvers for the classical Saint-Venant system. Then we study their extensions to a new multilayer Saint-Venant system. Finally, we use a kinetic solver to perform some numerical tests which prove that the 2D multilayer Saint-Venant system is a relevant alternative to 3D hydrostatic Navier-Stokes equations.

scholarly journals Congested shallow water model: roof modeling in free surface flow

2018 ◽  
Vol 52 (5) ◽  
pp. 1679-1707 ◽  
Author(s):  
Edwige Godlewski ◽  
Martin Parisot ◽  
Jacques Sainte-Marie ◽  
Fabien Wahl
Keyword(s):  
Shallow Water ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Hyperbolic Equations ◽  
Mechanical Energy ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Water Model ◽  
Shallow Water Model ◽  
Time Step

We are interested in the modeling and the numerical approximation of flows in the presence of a roof, for example flows in sewers or under an ice floe. A shallow water model with a supplementary congestion constraint describing the roof is derived from the Navier-Stokes equations. The congestion constraint is a challenging problem for the numerical resolution of hyperbolic equations. To overcome this difficulty, we follow a pseudo-compressibility relaxation approach. Eventually, a numerical scheme based on a finite volume method is proposed. The well-balanced property and the dissipation of the mechanical energy, acting as a mathematical entropy, are ensured under a non-restrictive condition on the time step in spite of the large celerity of the potential waves in the congested areas. Simulations in one dimension for transcritical steady flow are carried out and numerical solutions are compared to several analytical (stationary and non-stationary) solutions for validation.

scholarly journals The shape of submarine levees: exponential or power law?

2009 ◽  
Vol 619 ◽  
pp. 367-376 ◽  
Author(s):  
V. K. BIRMAN ◽  
E. MEIBURG ◽  
B. KNELLER
Keyword(s):  
Steady State ◽  
Shallow Water ◽  
Power Law ◽  
Sediment Supply ◽  
Water Model ◽  
Navier Stokes ◽  
Entrainment Rate ◽  
Quasi Steady State ◽  
Shallow Water Model ◽  
Power Law Decay

Field observations indicate that the height of submarine levees decays with distance from the channel either exponentially or according to a power law. This investigation clarifies the flow conditions that lead to these respective shapes, via a shallow water model for the overflow currents that govern the levee formation. The model is based on a steady state balance of sediment supply by the turbidity current, and sediment deposition onto the levee, with the settling velocity and the entrainment rate appearing as parameters. It demonstrates that entrainment of ambient fluid is the determining factor for the levee shape. For negligible entrainment rates, levee shapes tend to exhibit exponential profiles, while constant rates of entrainment or detrainment result in power law shapes. Interestingly, whether a levee has an exponential or a power law shape is determined by kinematic considerations only, viz. the balance laws for sediment mass and fluid volume. We find that the respective coefficients governing the exponential or power law decay depend on the settling speeds of the sediment grains, which in turn is a function of the grain size. Two-dimensional, unsteady Navier–Stokes simulations confirm the emergence of a quasi-steady state. The depositional behaviour of this quasi-steady state is consistent with the predictions of the shallow water model, thus validating the assumptions underlying the model, and demonstrating its predictive abilities.

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