An energy-consistent depth-averaged Euler system: derivation and properties

In this paper, we present an original derivation process of a non-hydrostatic shallow water-type model which aims at approximating the incompressible Euler and Navier-Stokes systems with free surface. The closure relations are obtained by aminimal energy constraint instead of an asymptotic expansion. The model slightly differs from thewell-known Green-Naghdi model and is confronted with stationary andanalytical solutions of the Euler system corresponding to rotationalflows. At the end of the paper, we givetime-dependent analytical solutions for the Euler system that are alsoanalytical solutions for the proposed model but that are not solutionsof the Green-Naghdi model. We also give and compare analytical solutions of thetwo non-hydrostatic shallow water models

A multilayer Saint-Venant system with mass exchanges for Shallow Water flows. Derivation and numerical validation

The standard multilayer Saint-Venant system consists in introducing fluid layers that are advected by the interfacial velocities. As a consequence there is no mass exchanges between these layers and each layer is described by its height and its average velocity. Here we introduce another multilayer system with mass exchanges between the neighborhing layers where the unknowns are a total height of water and an average velocity per layer. We derive it from Navier-Stokes system with an hydrostatic pressure and prove energy and hyperbolicity properties of the model. We also give a kinetic interpretation leading to effective numerical schemes with positivity and energy properties. Numerical tests show the versatility of the approach and its ability to compute recirculation cases with wind forcing.Comment: Submitted to M2A

A 2D model for hydrodynamics and biology coupling applied to algae growth simulations

Cultivating oleaginous microalgae in specific culturing devices such as raceways is seen as a future way to produce biofuel. The complexity of this process coupling non linear biological activity to hydrodynamics makes the optimization problem very delicate. The large amount of parameters to be taken into account paves the way for a useful mathematical modeling. Due to the heterogeneity of raceways along the depth dimension regarding temperature, light intensity or nutrients availability, we adopt a multilayer approach for hydrodynamics and biology. For free surface hydrodynamics, we use a multilayer Saint-Venant model that allows mass exchanges, forced by a simplified representation of the paddlewheel. Then, starting from an improved Droop model that includes light effect on algae growth, we derive a similar multilayer system for the biological part. A kinetic interpretation of the whole system results in an efficient numerical scheme. We show through numerical simulations in two dimensions that our approach is capable of discriminating between situations of mixed water or calm and heterogeneous pond. Moreover, we exhibit that a posteriori treatment of our velocity fields can provide lagrangian trajectories which are of great interest to assess the actual light pattern perceived by the algal cells and therefore understand its impact on the photosynthesis process.Comment: 27 pages, 11 figure

Layer-averaged Euler and Navier-Stokes equations

In this paper we propose a strategy to approximate incompressible hydrostatic free surface Euler and Navier-Stokes models. The main advantage of the proposed models is that the water depth is a dynamical variable of the system and hence the model is formulated over a fixed domain.The proposed strategy extends previous works approximating the Euler and Navier-Stokes systems using a multilayer description. Here, the needed closure relations are obtained using an energy-based optimality criterion instead of an asymptotic expansion. Moreover, the layer-averaged description is successfully applied to the Navier-Stokes system with a general form of the Cauchy stress tensor

Information capacity of the Hopfield model

The information capacity of general forms of memory is formalized. The number of bits of information that can be stored in the Hopfield model of associative memory is estimated. It is found that the asymptotic information capacity of a Hopfield network of N neurons is of the order N^3b. The number of arbitrary state vectors that can be made stable in a Hopfield network of N neurons is proved to be bounded above by N

Smart Market for Woodchips

This paper presents the outline for a smart market geared to Quebec s woodchip industry. It aims to explain how such a market operates, and to show how it fits into or modifies the existing trading system. The proposed smart market has two main objectives. The first is to enhance the efficiency of the woodchip market. The electronic market is a favoured means of enhancing market flexibility and thus controlling the accumulation of unsold inventory. Such a mechanism maximizes the industry s overall profits by ensuring that each firm can identify all business opportunities within its reach and realize the additional profits associated with them. The second main objective is continuity of trade practices. Long-term contracts between sawmills and paper mills are an important market reality, and they will be an integral part of the smart market. The keystone of our proposal is the Optimized Periodic Market (OPM). Somewhat along the lines of the Encan électronique du porc (EÉP, or electronic pork auction), set up by the Fédération des producteurs de porcs du Québec (Quebec s pork producers federation), we are proposing to establish an electronic woodchip exchange which would open up periodically (monthly, for instance) to allocate Quebec s entire woodchip output. At the close of an electronic market session, exchanges for the month will be determined, and everyone will know who he is to receive his woodchips from, or who he is to deliver them to, along with the price of each transaction and the merchandise shipment schedule. The OPM fulfils three main functions: (1) it co-ordinates players actions with a view to reducing inventory surpluses and meeting paper mills needs as closely as possible; (2) it keeps transportation costs down by determining the least costly exchanges between sawmills and paper mills with respect to supply; and (3) it offers an effective tool for renegotiating long-term contracts. An OPM session is in three main stages: pre-sale, sale and after-sale. At the pre-sale stage, the system gathers the information it will use to determine optimum exchanges. This information concerns two key variables in the woodchip market: quality requirements, and transportation costs. Buyers first express their technical requirements: on a BBS, they indicate the quality of the woodchips they wish to purchase and the penalties that will be applied should delivery not be in compliance with the specified technical requirements. The sellers are then invited to provide a production cost indicator for their woodchips that meets each paper mill s requirements. Sellers must also indicate the unit transportation costs (UTC) which they will have to pay if they deliver to each of the buyers. An auction mechanism for transportation costs, where buyers and third parties will be able to try to lower transportation costs, may also be set up. The sale stage involves an auction which takes place in successive rounds. Buyers bids are demand schedules, and sellers bids are supply schedules. At the start of the opening round, buyers and sellers enter their bids through an electronic interface linked to a server. Sellers must also provide a list of the types of woodchips they wish to market, and this list will be valid for the duration of the sale. Depending on the classification of the woodchips they put up for sale and their ability to sort species, they will have the choice of offering low density chips, high density chips, and/or jack pine chips. Buyers take part in a single market grouping together the three types of woodchips. At the start of the opening round, they have to specify the minimum and maximum proportions of each type of woodchips they wish to acquire. When all the bids have been tallied, the system generates the quantities traded and the equilibrium price paid or received by each participant. In light of this information, players can raise or lower their bids in line with their assessment of the market. Buyers can also revise the proportions of each type of woodchips they are asking for. Another round is then initiated, ending with the re-optimization of the system and disclosure of the new allocations. If no one wishes to alter his bid after the prices and quantities are announced at the end of a round, the efficient bilateral exchanges become official, and the delivery schedule for the period is established. Bid changes from one round to the next are governed by an eligibility rule whose function is to give impetus to the market, to prompt participants to be active in the market, and to reveal their real needs as early as possible. The OPM is a medium-term market in that it lies between a long-term compensatory market and a short-term spot market. Sawmills and paper mills wishing to guarantee their supply by signing long-term agreements will be able to go the compensatory contract route. A long-term compensatory contract between a sawmill and a paper mill specifies a price and quantity of woodchips for each OPM period (each month). In this way, the paper mill is guaranteed a certain quantity of woodchips at a certain price, and the sawmill is guaranteed a price for a given portion of its output. Data from these compensatory contracts are then taken into consideration by the OPM. Somewhat like term markets, compensatory contracts are agreements on financial flow alone. It is the OPM which determines the actual exchanges of woodchips. The proposed market mechanism is designed to ensure that participation in the OPM is to everyone s benefit. With flexibility, transparency and the implementation of an open competitive mechanism, beneficial participation for all is one of the guiding principles of the smart market for woodchips. A participant bound by a compensatory contract is guaranteed to find himself, after an OPM session, in a situation at least as beneficial as that provided for in the contract. The spirit of the compensation mechanism is that if one of the partners receives an unfavourable price on the OPM, he will be compensated by his partner. In this way, the OPM compensation rule and renegotiation mechanism offer a procedure guaranteeing: (1) each participant the equivalent of what is negotiated in the long-term contract; and (2) all participants the possibility of discovering and realizing additional profits that are accessible but have not yet been realized. The proposed smart market is rounded out by a very short-term spot market on which sawmills will be able to liquidate their surpluses and paper mills will be able to procure additional quantities of woodchips, thus dealing with surpluses and unfilled requirements not anticipated in the OPM session. The spot market may be presented in the form of a BBS through which each facility will be able to initiate an auction.

An experimental study of boundary-layer transition induced vibrations on a hydrofoil

This paper aims at characterizing experimentally laminar to turbulent transition induced vibrations. Here, the transition is known to be triggered by a Laminar Separation Bubble that results from a laminar separation of the boundary-layer flow on a hydrofoil. In this study we consider two NACA66312 (Mod) laminar hydrofoils at low angles of incidence (mostly 2° and 4°) and Reynolds numbers ranging from Re=450 000 to 1 200 000, in order to get transitional regimes. The first hydrofoil, made of steel (E=2.1×1011 Pa), is referred to as the rigid hydrofoil, although it is seen to vibrate under the action of the LSB. To better understand the possible interaction between the flow and the foil vibrations, vibration measurements are repeated using a flexible hydrofoil (E=3×109 Pa) of same geometry (under zero loading) and in close configurations. The experiments are carried out at the French Naval Academy Research Institute (IRENav, France). Wall pressure and flow velocity measurements enable a characterization of the laminar separation bubble and the identification of a vortex shedding at a given frequency. It is hence shown that the boundary-layer transition induces important foil vibrations, whose characteristics in terms of frequency and amplitude depend on the vortex shedding frequency, and can be coupled with natural frequencies of the hydrofoils