Gerardo Hernandez-Duenas

A Hybrid Algorithm for the Baer-Nunziato Model Using the Riemann Invariants

The paper considers the Baer-Nunziato model for two-phase flow in porous media, with discontinuous porosity. Computing solutions of the Riemann problem rests on capturing the jump in the solution across the porosity jump. A recent study (Lowe in J. Comput. Phys. 204:598–632, 2005) showed that numerical discretizations may fail to correctly capture the jump conditions across the so-called compaction wave, and yield incorrect solutions. We have formulated the Baer-Nunziato system using the Riemann invariants across the porosity jump, and propose a hybrid algorithm that uses the Riemann invariants formulation across the compaction wave, and the conservative formulation away from the compaction wave. The paper motivates and describes the hybrid scheme and present numerical results.

Shallow Water Flows in Channels

We consider the shallow water equations for flows through channels with arbitrary cross section. The system forms a hyperbolic set of balance laws. Exact steady-state solutions are available and are controlled by the relation between the bottom topography and the channel geometry. We use a Roe-type upwind scheme for the system. Considerations of conservation, near steady-state accuracy, velocity regularization and positivity near dry states are discussed. Numerical solutions are presented illustrating the merits of the scheme for a variety of flows and demonstrating the effect of the interplay between the topography and the geometry on the solution.

Minimal Models for Precipitating Turbulent Convection

Simulations of precipitating convection would typically use a non-Boussinesq dynamical core such as the anelastic equations, and would incorporate water substance in all of its phases: vapor, liquid and ice. Furthermore, the liquid water phase would be separated into cloud water (small droplets suspended in air), and rain water (larger droplets that fall). Depending on environmental conditions, the moist convection may organize itself on multiple length and time scales. Here we investigate the question, “What is the minimal representation of water substance and dynamics that still reproduces the basic regimes of turbulent convective organization?” The simplified models investigated here use a Boussinesq atmosphere with bulk cloud physics involving equations for water vapor and rain water only. As a first test of the minimal models, we investigate organization or lack thereof on relatively small length scales of approximately 100 km and time scales of a few days. It is demonstrated that the minimal models produce either unorganized (“scattered”) or organized convection in appropriate environmental conditions, depending on the environmetal wind shear. For the case of organized convection, the models qualitatively capture features of propagating squall lines that are observed in nature and in more comprehensive cloud resolving models, such as tilted rain-water profiles, low-altitude cold pools, and propagation speed corresponding to the maximum of the horizontally averaged, horizontal velocity.

Stability and Instability Criteria for Idealized Precipitating Hydrodynamics

AbstractA linear stability analysis is presented for fluid dynamics with water vapor and precipitation, where the precipitation falls relative to the fluid at speed VT. The aim is to bridge two extreme cases by considering the full range of VT values: (i) VT = 0, (ii) finite VT, and (iii) infinitely fast VT. In each case, a saturated precipitating atmosphere is considered, and the sufficient conditions for stability and instability are identified. Furthermore, each condition is linked to a thermodynamic variable: either a variable θs, which denotes the saturated potential temperature, or the equivalent potential temperature θe. When VT is finite, separate sufficient conditions are identified for stability versus instability: dθe/dz > 0 versus dθs/dz < 0, respectively. When VT = 0, the criterion dθs/dz = 0 is the single boundary that separates the stable and unstable conditions; and when VT is infinitely fast, the criterion dθe/dz = 0 is the single boundary. Asymptotics are used to analytically characteriz...

A Scheme for Shallow Water Flow with Area Variation

We consider the shallow water equations for flows through channels with variable area. The system is obtained by depth/width averaging of the Euler equations and forms a hyperbolic set of balance laws. Exact steady‐state solutions are available and are controlled by the relative positions of the bottom crest and channel throat. We present a Roe‐type upwind scheme for the system. Considerations of conservation, near steady‐state accuracy, velocity regularization and positivity near dry states are discussed. Numerical solutions are presented illustrating the merits of the scheme for a variety of flows including sub‐, trans‐ and supercritical flows and drainage problems, with emphasis on effects of the interplay between topography and geometry on the solution.

A Hybrid Method to Solve Shallow Water Flows with Horizontal Density Gradients

In this work, a model for shallow water flows that accounts for the effects of horizontal density fluctuations is presented and derived. While the density is advected by the flow, a two-way feedback between the density gradients and the time evolution of the fluid is ensured through the pressure and source terms in the momentum equations. The model can be derived by vertically averaging the Euler equations while still allowing for density fluctuations in horizontal directions. The approach differs from multi-layer shallow water flows where two or more layers are considered, each of them having their own depth, velocity and constant density. A Roe-type upwind scheme is developed and the Roe matrices are computed systematically by going from the conservative to the quasi-linear form at a discrete level. Properties of the model are analyzed. The system is hyperbolic with two shock-wave families and a contact discontinuity associated to interfaces of regions with density jumps. This new field is degenerate with pressure and velocity as the corresponding Riemann invariants. We show that in some parameter regimes numerically recognizing such invariants across contact discontinuities is important to correctly compute the flow near those interfaces. We present a numerical algorithm that correctly captures all waves with a hybrid strategy. The method integrates the Riemann invariants near contact discontinuities and switches back to the conserved variables away from it to properly resolve shock waves. This strategy can be applied to any numerical scheme. Numerical solutions for a variety of tests in one and two dimensions are shown to illustrate the advantages of the strategy and the merits of the scheme.