G Papadopoulos

F Theoleyre

RAW

The wireless medium presents significant specific challenges to achieve properties similar to those of wired deterministic networks. At the same time, a number of use cases cannot be solved with wires and justify the extra effort of going wireless. This document presents wireless use cases demanding reliable and available behavior.

Introduction

Based on time, resource reservation, and policy enforcement by distributed shapers, Deterministic Networking provides the capability to carry specified unicast or multicast data streams for real-time applications with extremely low data loss rates and bounded latency, so as to support time-sensitive and mission-critical applications on a converged enterprise infrastructure.

Deterministic Networking in the IP world is an attempt to eliminate packet loss for a committed bandwidth while ensuring a worst case end-to-end latency, regardless of the network conditions and across technologies. It can be seen as a set of new Quality of Service (QoS) guarantees of worst-case delivery. IP networks become more deterministic when the effects of statistical multiplexing (jitter and collision loss) are mostly eliminated. This requires a tight control of the physical resources to maintain the amount of traffic within the physical capabilities of the underlying technology, e.g., by the use of time-shared resources (bandwidth and buffers) per circuit, and/or by shaping and/or scheduling the packets at every hop. Available is even more challenging than it is with wires, due to the numerous causes of loss in transmission that add up to the congestion losses and the delays caused by overbooked shared resources.

Key attributes of

The wireless and wired media are fundamentally different at the physical level, and while the generic Problem Statement [START_REF] Finn | Deterministic Networking Problem Statement[END_REF] for DetNet applies to the wired as well as the wireless medium, the methods to achieve RAW necessarily differ from those used to support Time-Sensitive Networking over wires. This draft extends the "Deterministic Networking Use Cases" document [START_REF] Grossman | Deterministic Networking Use Cases[END_REF] and describes a number of additional use cases which require "reliable/predictable and available" flows over wireless links and possibly complex multi-hop paths called Tracks. This is covered mainly by the "Wireless for Industrial Applications" use case, as the "Cellular Radio" is mostly dedicated to the (wired) transport part of a Radio Access Network (RAN). Whereas the "Wireless for Industrial Applications" use case certainly covers an area of interest for RAW, it is limited to 6TiSCH, and thus its scope is narrower than the use cases described next in this document. --------------------> Airport <----------------------> o Interoperability: All 192 members of the international Civil Aviation Organization (ICAO) must be able to use these solutions.

# # # # # # # # # # # # # # # # # # # # # Satellite-based # # # Communications # # # SatCOM (#) # # # # Aircraft # # % % # # % % # # % Air-Air % # # % Communications % # # % LDACS A/A (%) % # # % % # Aircraft % % % % % % % % % % Aircraft # | Air-Ground | # | Communications | # | LDACS A/G (|) | # Communications in | | # and around airports | | # AeroMACS (-) | | # | | # Aircraft-------------+ | | # | | | # | | | # Ground network | | Ground network | SatCOM <-

The Need for Wireless

In a high mobility environment such as aviation, the envisioned solutions to provide worldwide coverage of data connections with inflight aircraft require a multi-system, multi-link, multi-hop approach. Thus air, ground and space based datalink providing technologies will have to operate seamlessly together to cope with the increasing needs of data exchange between aircraft, air traffic controller, airport infrastructure, airlines, air network service providers (ANSPs) and so forth. Thus making use of wireless technologies is a must in tackling this enormous need for a worldwide digital aeronautical datalink infrastructure.

Requirements for RAW

Different safety levels need to be supported, from extremely safety critical ones requiring low latency, such as a WAKE warning -a warning that two aircraft come dangerously close to each other -and high resiliency, to less safety critical ones requiring low-medium latency for services such as WXGRAPH -graphical weather data.

Overhead needs to be kept at a minimum since aeronautical data links provide comparatively small data rates in the order of kbit/s.

Policy needs to be supported when selecting data links. The focus of RAW here should be on the selectors, responsible for the routing path a packet takes to reach its end destination. This would minimize the amount of routing information that has to travel inside the network because of precomputed routing tables with the selector being responsible for choosing the most appropriate option according to policy and safety.

Amusement Parks

Use Case Description

The digitalization of Amusement Parks is expected to decrease significantly the cost for maintaining the attractions. By monitoring in real-time the machines, predictive maintenance will help to reduce the repairing cost as well as the downtime. Besides, the attractions may use wireless transmissions to interconnect Attractions may rely on a large set of sensors and actuators, which react in real time. Typical applications comprise:

o Emergency: safety has to be preserved, and must stop the attraction when a failure is detected.

o Video: augmented and virtual realities are integrated in the attraction. Wearable devices (e.g., glasses, virtual reality headset) need to offload one part of the processing tasks.

o Real-time interactions: visitors may interact with an attraction, like in a real-time video game. The visitors may virtually interact with their environment, triggering actions in the real world (through actuators) [robots].

o Geolocation: visitors are tracked with a personal wireless tag so that their user experience is improved.

o Predictive maintenance: statistics are collected to predict the future failures, or to compute later more complex statistics about the attraction's usage, the downtime, its popularity, etc.

Specifics

Amusement parks comprise a variable number of attractions, mostly outdoor, over a large geographical area. Wearable devices are extensively used for a user experience personalization. They typically need to support wireless transmissions. Personal tags may help to reduce the operating costs [disney-VIP] and to increase the number of charged services provided to the audience (VIP tickets, interactivity, etc.) Some applications rely on more sophisticated wearable devices such as digital glasses or Virtual Reality (VR) headsets for an immersive experience.

Requirements for RAW

The network infrastructure has to support heterogeneous traffic, with very different critical requirements. Thus, flow isolation has to be provided.

We have to schedule appropriately the transmissions, even in presence of mobile devices. Nowadays, long-range wireless transmissions are used for best-effort traffic, and [IEEE802.1TSN] is used for critical flows using Ethernet devices. However, we need an IP enabled technology to interconnect large areas, independent of the PHY and MAC layer to maximize the compliancy.

We expect to deploy several different technologies (long vs. short range) which have to cohabit in the same area. Thus, we need to schedule appropriately the transmissions to limit the co-technology interference, so that an end-to-end delay across multiple technologies can be guaranteed. It is needed to understand which technologies RAW will cover and how they can be used cohabitating in the same area. Motion control that monitors dynamic activities may require even faster rates in the order of a few ms. Finally, some industries exhibit hybrid behaviors, like canned soup that will start as a process industry while mixing the food and then operate as a discrete manufacturing when putting the final product in cans and shipping them.

In all those cases, a packet must flow reliably between the sensor and the PLC, be processed by the PLC, and sent to the actuator within the control loop period. In some particular use cases that inherit from analog operations, jitter might also alter the operation of the control loop. A rare packet loss is usually admissible, but typically 4 losses in a row will cause an emergency halt of the production and incur a high cost for the manufacturer.

Unmeasured Data

A secondary use case deals with monitoring and diagnostics. This socalled unmeasured data is essential to improve the performances of a production line, e.g., by optimizing real-time processing or maintenance windows using Machine Learning predictions. For the lack of wireless technologies, some specific industries such as Oil and Gas have been using serial cables, literally by the millions, to perform their process optimization over the previous decades. But few industries would afford the associated cost and the Holy Grail of the Industrial Internet of Things is to provide the same benefits to all industries, including SmartGrid, Transportation, Building, Inside the factory, wires may already be available to operate the Control Network. But unmeasured data are not welcome in that network for a number of reasons. On the one hand it is rich and asynchronous, meaning that using they may influence the deterministic nature of the control operations and impact the production. On the other hand, this information must be reported to the carpeted floor over IP, which means the potential for a security breach via the interconnection of the Operational Technology (OT) network with the Internet technology (IT) network and possibly enable a rogue access.

The Need for Wireless

Ethernet cables used on a robot arm are prone to breakage after a few thousands flexions, a lot faster than a power cable that is wider inn diameter, and more resilient. In general, wired networking and mobile parts are not a good match, mostly in the case of fast and recurrent activities, as well as rotation.

When refurbishing older premises that were built before the Internet age, power is usually available everywhere, but data is not. It is often impractical, time consuming and expensive to deploy an Ethernet fabric across walls and between buildings. Deploying a wire may take months and cost tens of thousands of US Dollars.

Even when wiring exists, e.g., in an existing control network, asynchronous IP packets such as diagnostics may not be welcome for operational and security reasons (see Section 4.2.1). An alternate network that can scale with the many sensors and actuators that equip every robot, every valve and fan that are deployed on the factory floor and may help detect and prevent a failure that could impact the

Specifics

Uninterrupted Stream Playback

Considering the uninterrupted audio or video stream, a potential packet losses during the transmission of audio or video flows cannot be tackled by re-trying the transmission, as it is done with file transfer, because by the time the packet lost has been identified it is too late to proceed with packet re-transmission. Buffering might be employed to provide a certain delay which will allow for one or more re-transmissions, however such approach is not efficient in application where delays are not acceptable.

Synchronized Stream Playback

In the context of ProAV, latency is the time between the transmitted signal over a stream and its reception. Thus, for sound to remain synchronized to the movement in the video, the latency of both the audio and video streams must be bounded and consistent. The devices need the wireless communication to support video streaming via 802.11 wireless LAN for instance.

During the public address, the deployed announcement speakers, for instance along the platforms of the train stations, need the wireless communication to forward the audio traffic in real time.

Requirements for RAW

The network infrastructure needs to support heterogeneous types of traffic (including QoS).

Content delivery with bounded (lowest possible) latency.

The deployed network topology should allow for multipath. This will enable for multiple streams to have different (and multiple) paths through the network to support redundancy.

6. Wireless Gaming

Use Case Description

The gaming industry includes [IEEE80211RTA] real-time mobile gaming, wireless console gaming and cloud gaming. For RAW, wireless console gaming is the most relevant one. We next summarize the three:

o Real-time Mobile Gaming: Different from traditional games, real time mobile gaming is very sensitive to network latency and stability. The mobile game can connect multiple players together in a single game session and exchange data messages between game server and connected players. Real-time means the feedback should present on screen as users operate in game. For good game experience, the end to end latency plus game servers processing time should not be noticed by users as they play the game.

o Wireless Console Gaming: Playing online on a console has 2 types of internet connectivity, which is either wired or Wi-Fi. Most of the gaming consoles today support Wi-Fi 5. But Wi-Fi has an especially bad reputation among the gaming community.

Specifics

Some of the use cases/tasks involving drones require coordination among drones. Others involve complex compute tasks that might not be performed using the limited computing resources that a drone typically has. These two aspects require continuous connectivity with the control center and among drones.

Remote maneuvering of a drone might be performed over a cellular network in some cased, however, there are situations that need very low latencies and deterministic behavior of the connectivity.

Examples involve platooning of drones or share of computing resources among drones (e.g., a drone offload some function to a neighboring drone). UAVs cannot be connected through any type of wired media, so it is obvious that wireless is needed.

Requirements for RAW

The network infrastructure is actually composed by the UAVs themselves, requiring self-configuration capabilities.

Heterogeneous types of traffic need to be supported, from extremely critical ones requiring ultra low latency and high resiliency, to traffic requiring low-medium latency.

When a given service is decomposed into functions --hosted at different drones --chained, each link connecting two given functions would have a well-defined set of requirements (latency, bandwidth and jitter) that have to be met.

Edge Robotics control

Use Case Description

The Edge Robotics scenario consists of several robots, deployed in a given area (for example a shopping mall), inter-connected via an access network to a network's edge device or a data center. The robots are connected to the edge so complex computational activities are not executed locally at the robots, but offloaded to the edge. This brings additional flexibility in the type of tasks that the robots do, as well as reducing the costs of robot manufacturing (due to their lower complexity), and enabling complex tasks involving coordination among robots (that can be more easily performed if robots are centrally controlled).

A simple example of the use of multiples robots is cleaning, delivering of goods from warehouses to shops or video surveillance. Multiple robots are simultaneously instructed to perform individual tasks by moving the robotic intelligence from the robots to the network's edge (e.g., data center). That enables easy synchronization, scalable solution and on-demand option to create flexible fleet of robots.

Robots would have various forms of wireless connectivity: o IEEE 802.11: for connection to the edge and also inter-robot communications (e.g., for coordinated actions). o Cellular: as an additional communication link to the edge, though primarily as backup, since ultra low latencies are needed.

Specifics

Some of the use cases/tasks involving robots might benefit from decomposition of a service in small functions that are distributed and chained among robots and the edge. These require continuous connectivity with the control center and among drones.

Robot control is an activity requiring very low latencies between the robot and the location where the control intelligence resides (which might be the edge or another robot).

The Need for Wireless

Deploying robots in scenarios such as shopping malls for the aforementioned applications cannot be done via wired connectivity.

Requirements for RAW

The network infrastructure needs to support heterogeneous types of traffic, from robot control to video streaming.

When a given service is decomposed into functions --hosted at different robots --chained, each link connecting two given functions would have a well-defined set of requirements (latency, bandwidth and jitter) that have to be met.

Acknowledgments

The work of Carlos J. Bernardos in this draft has been partially supported by the H2020 5Growth (Grant 856709) and 5G-DIVE projects (Grant 859881).

12. Informative References

  Deterministic Networking include: o time synchronization on all the nodes, o centralized computation of network-wide deterministic paths, o multi-technology path with co-channel interference minimization, o frame preemption and guard time mechanisms to ensure a worst-case delay, and o new traffic shapers within and at the edge to protect the network. Wireless operates on a shared medium, and transmissions cannot be fully deterministic due to uncontrolled interferences, including self-induced multipath fading. RAW (Reliable and Available Wireless) is an effort to provide Deterministic Networking on across a path that include a wireless physical layer. Making Wireless Reliable and Papadopoulos, et al. Expires September 8

Papadopoulos

  Process Control designates continuous processing operations, e.g., heating Oil in a refinery or mixing drinking soda. Control loops in the Process Control industry operate at a very low rate, typically 4 times per second. Factory Automation, on the other hand, deal with discrete goods such as individual automobile parts, and requires faster loops, in the order of 10ms.

	Internet-Draft	RAW use cases scenarios	March 2020
	4. Wireless for Industrial Applications
	4.1. Use Case Description
	A major use case for networking in Industrial environments is the
	control networks where periodic control loops operate between a
	sensor that measures a physical property such as the temperature of a
	fluid, a Programmable Logic Controller (PLC) that decides an action
	such as warm up the mix, and an actuator that performs the required
	action, e.g., inject power in a resistor.
	4.2. Specifics
	4.2.1. Control Loops
	Papadopoulos, et al.	Expires September 8, 2020	[Page 9]

  Over the course of the recent years, major Industrial Protocols, e.g.,[ODVA] with EtherNet/IP[EIP] and[Profinet], have been migrating towards Ethernet and IP. In order to unleash the full power of the IP hourglass model, it should be possible to deploy any application over any network that has the physical capacity to transport the industrial flow, regardless of the MAC/PHY technology, wired or wireless, and across technologies. RAW mechanisms should be able to setup a Track over a wireless access segment such as TSCH and a backbone segment such as Ethernet or WI-Fi, to report a sensor data or a critical monitoring within a bounded latency. Some of these applications require low latency capability. For instance, when the application provides interactive play, or when the audio takes plays in real time (i.e. live) for public addresses in train stations or in theme parks.

	Internet-Draft	RAW use cases scenarios	March 2020
	5. Pro Audio and Video
	5.1. Use Case Description
	Many devices support audio and video streaming by employing 802.11
	wireless LAN.
	production. IEEE Std. 802.15.4 Time-Slotted Channel Hopping (TSCH)
	[RFC7554] is a promising technology for that purpose, mostly if the
	scheduled operations enable to use the same network by asynchronous
	and deterministic flows in parallel.
	4.4. Requirements for RAW
	As stated by the "Deterministic Networking Problem Statement"
	[RFC8557], a Deterministic Network is backwards compatible with
	(capable of transporting) statistically multiplexed traffic while
	preserving the properties of the accepted deterministic flows. While
	the [I-D.ietf-6tisch-architecture] serves that requirement, the work
	at 6TiSCH was focused on best-effort IPv6 packet flows. RAW should
	be able to lock so-called hard cells for use by a centralized
	scheduler, and program so-called end-to-end Tracks over those cells.
	Papadopoulos, et al.	Expires September 8, 2020	[Page 11]

The professional audio and video industry ("ProAV") includes: o Virtual Reality / Augmented Reality (VR/AR) o Public address, media and emergency systems at large venues (airports, train stations, stadiums, theme parks).

Internet-Drafts. The list of current Internet-Drafts is at https://datatracker.ietf.org/drafts/current/.

Internet-Drafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time.


Adrien Toutant
email: adrien.toutant@univ-perp.fr

Numerical simulations of unsteady viscous incompressible flows using general pressure equation

published or not. The documents may come

Numerical simulations of unsteady viscous incompressible flows using general pressure equation

Introduction

Incompressible Navier-Stokes (INS) equations correspond to a mixture of hyperbolic and elliptic partial differential equations [1]. The pressure in [START_REF] Chorin | A numerical method for solving incompressible viscous flow problems[END_REF] is not an independent thermodynamic variable. It can be seen as a Lagrangian multiplier of the incompressibility constraint. It is determined by the Laplace or Poisson equation:

∂ α ∂ α P = -(∂ β u α )(∂ α u β ) (1)
The physical meaning of (1) is that in a system with infinitely fast sound propagation, any pressure disturbance induced by the flow is instantaneously propagated into the whole domain. The absence of time evolution equation for pressure is a well known difficulty. Indeed, solving the Poisson equation is often the most costly step in simulation. This difficulty has motivated the search of alternative numerical approaches to determine pressure without solving the Poisson equation. Six different ways have been found:

1. The so-called artificial compressibility method (ACM) where a pressure evolution equation is postulated [START_REF] Chorin | A numerical method for solving incompressible viscous flow problems[END_REF]. Many modifications of the Chorin method were suggested in order to compute accurate solutions. For example, Ohwada and Asinari [START_REF] Ohwada | Artificial compressibility method revisited: Asymptotic numerical method for incompressible Navier-Stokes equations[END_REF] proposed to introduce a dissipation term in order to improve the quality of numerical solution obtained with artificial compressibility method. The addition of the dissipation term of the artificial continuity equation is useful for surpressing checkerboard instability when a collocated grid system is employed. This revisited ACM is noted SCI for surpressing checkerboard instability. In this paper, a staggered grid system is used. It is shown by numerical results that checkerboard instability is avoided thanks to the staggered grid system and without a dissipation term. The dissipation term proposed in this paper allows to obtain the exact pressure evolution in the case of Taylor-Green vortices and to damp accoustic waves. 2. The Lattice Boltzmann method (LBM) which uses a velocity-space truncation of the Boltzmann equation from the kinetic theory of gases [START_REF] Benzi | The lattice boltzmann equation: theory and applications[END_REF]. LBM and ACM have similarities [START_REF] Ohwada | Artificial compressibility method and lattice boltzmann method: Similarities and differences[END_REF]. A hybrid method between ACM and LBM has been proposed [START_REF] Asinari | Link-wise artificial compressibility method[END_REF]. 3. An inverse kinetic theory which permits the identification of the (Navier-Stokes) dynamical system and of the corresponding evolution operator which advances in time the kinetic distribution function and the related fluid fields [START_REF] Tessarotto | Exact pressure evolution equation for incompressible fluids[END_REF]. The pressure evolution equation obtained by this method is non-asymptotic. The full validity of INS equations is preserved. 4. The kinetically reduced local Navier-Stokes (KRLNS) wich establishes a simplified equation for the grand potential [START_REF] Ansumali | Thermodynamic theory of incompressible hydrodynamics[END_REF]. 5. The entropically damped form of artificial compressibility (EDAC) wich employs a thermodynamic constraint to damp pressure oscillations [START_REF] Clausen | Entropically damped form of artificial compressibility for explicit simulation of incompressible flow[END_REF]. 6. The general pressure equation (GPE) wich derives the exact pressure evolution from compressible Navier-Stokes equation [START_REF] Toutant | General and exact pressure evolution equation[END_REF]. GPE can be seen as a physical justification of ACM. GPE contains an additional dissipation term. KRLNS and EDAC contain exactly the same diffusion term.

For KRLNS, the diffusion term is applied to grand potential instead of pressure. In fact, the KRLNS is exactly the same as GPE if one replaces the negative of the grand potential by pressure. The difference between EDAC and GPE comes from pressure advection. In SCI, the dissipation term is chosen proportional to pressure and not proportional to the Laplacian of pressure. Similarities and differences between GPE, ACM, EDAC, KRLNS and SCI are detailled in section 2.

KRLNS [START_REF] Karlin | Kinetically reduced local Navier-Stokes equations: An alternative approach to hydrodynamics[END_REF][START_REF] Borok | Kinetically reduced local Navier-Stokes equations for simulation of incompressible viscous flows[END_REF][START_REF] Hashimoto | Simulation of doubly periodic shear layers using Kinetically Reduced Local Navier-Stokes equations on a GPU[END_REF][START_REF] Hashimoto | Higher order numerical simulation of unsteady viscous incompressible flows using kinetically reduced local Navier-Stokes equations on a GPU[END_REF] and EDAC [START_REF] Clausen | Entropically damped form of artificial compressibility for explicit simulation of incompressible flow[END_REF] have been applied for the simulation of different viscous incompressible flows: three-dimensional Taylor-Green vortex flow, steady-state cavity flow, two-dimensional Taylor-Green vortex flow and doubly periodic shear layers. This paper realizes a similar work with the GPE approach. It proposes a discretization of GPE in a staggered grid system with second order centered schemes in space and a third order Runge-Kutta scheme in time. In this paper, GPE [START_REF] Toutant | General and exact pressure evolution equation[END_REF] is applied for the simulation of different unsteady viscous incompressible flows: two-dimensional Taylor-Green vortex flow, the traveling wave and the doubly periodic shear layers. The results obtained with GPE are compared with those obtained by ACM, EDAC and a classical finite volume method with a Poisson equation. They show that the proposed general pressure equation (GPE) allows to solve general, time-accurate, incompressible Navier-Stokes flows.

The outline of the paper is as follows. The computational approach is described in section 2. Sections 3 presents the results obtained for unsteady analytical solutions of viscous incompressible flows: the two-dimensional Taylor-Green vortex flow and the traveling wave. The more complex case of doubly periodic shear layers is studied in section 4.

Computational approach

The classical incompressible Navier-Stokes (INS) equations consist of the equation for the momentum and the continuity equation:

∂ t u α + ∂ β (u β u α ) + ∂ α P = 1 Re ∂ β ∂ β u α ∂ β u β = 0 (2)
With the general pressure equation (GPE), the continuity equation is replaced by:

∂ t P + 1 M a 2 ∂ β u β = γ ReP r ∂ β ∂ β P (3)
With artificial compressibility methods (ACM), the previous equation is approximated by:

∂ t P + 1 M a 2 ∂ β u β = 0 (4)
With SCI, a dissipation term is added to the equation of ACM

∂ t P + γP + 1 M a 2 ∂ β u β = 0 ( 5
)
where γ is a positive function of space and time [START_REF] Ohwada | Artificial compressibility method revisited: Asymptotic numerical method for incompressible Navier-Stokes equations[END_REF]. It is worth noting that Ohwada and Asinari choose a discretization that introduces an other dissipation term than γP . This other dissipation term is proportional to the pressure Laplacian. However, it cannot be compared with the pressure Laplacian used in GPE, EDAC or KRLNS. Indeed, in SCI, the pressure Laplacian is multiplied by the cell size: its contribution is negligible when grid independence is reached. With EDAC, pressure equation is written:

∂ t P + u β ∂ β P + 1 M a 2 ∂ β u β = γ ReP r ∂ β ∂ β P (6)
The KRLNS method is based on a subtitution,

G = P -E k ( 7
)
where G is stated to be the negative of the grand potential and E k is the kinetic energy. The time-evolution of G is given by:

∂ t G + 1 M a 2 ∂ β u β = γ ReP r ∂ β ∂ β G (8)
ACM (eq. ( 4)) corresponds to the limit of GPE (eq. ( 3)) with P r = +∞. The diffusive term in GPE, KRLNS and EDAC can be surprising. GPE is derived from a rigorous asymptotic analysis of the compressible Navier-Stokes equations. The diffusive term in GPE comes from the conductive heat transfer of the enthalpy equation. Indeed, the enthalpy equation is written in terms of pressure. The conductive heat transfer is written in terms of pressure using the following equation:

∂ β ∂ β T = ∂ 2 T ∂P 2 ρ (∂ β P )(∂ β P ) + ∂ 2 T ∂ρ 2 P (∂ β ρ)(∂ β ρ) + ∂T ∂P ρ ∂ β ∂ β P + ∂T ∂ρ P ∂ β ∂ β ρ (9)
For sufficiently large time-space scales (see the discussion in [START_REF] Ansumali | Thermodynamic theory of incompressible hydrodynamics[END_REF]), one can neglect density variation (ρ, χ T and α are supposed constant and evaluated at equilibrium) and temperature variation (around a globally uniform equilibrium temperature) becomes a function of pressure:

∂ β ∂ β T ≈ ∂T ∂P ρ ∂ β ∂ β P = χ T α ρ ∂ β ∂ β P (10)
where χ T is the isothermal compressibility coefficient

χ T = 1 ρ ∂ρ ∂P T
and where

α ρ is the isobaric thermal expansion coefficient α ρ = -1 ρ ∂ρ ∂T P
. For more details on the establishment of GPE equation, the reader will refer to [START_REF] Toutant | General and exact pressure evolution equation[END_REF]. The diffusive term of GPE is the difference from artificial compressibility models. The difference between GPE and EDAC resides essentially in the advective term of pressure that is neglected in GPE and not in EDAC. As it is shown by the numerical results (see section 3), the pressure advection of EDAC conducts to errors for pressure estimation (compared to pressure obtained with INS). However, the pressure advection term is usually neglected in EDAC. In this case, there is therefore no difference between EDAC and GPE. Furthermore, it is worth noting that these errors will disappear with the physical Mach number that is supposed to be very small. However, in these methods, the Mach number becomes an artificial parameter, which may be larger than the physical Mach number. The other difference between GPE and EDAC comes from the physical interpretation of the diffusive term. For GPE, the diffusive term is physical, it comes from the conductive heat transfer of the enthalpy equation. For EDAC, the diffusive term is an artificially way to minimize density fluctuations. In GPE, the Prandtl number like the Mach number can become an artificial parameter, which may be smaller than the physical Prandtl number. As it is shown by the numerical results (see section 4), a small Prandtl number allows to damp acoustic waves. KRLNS is exactly the same as GPE if one replaces the negative of the grand potential by pressure. In order to clarify the differences between SCI, EDAC, KRLNS and GPE, one can subtract the different pressure equations. One obtains:

GP E -SCI = 1 ReP r ∂ β ∂ β P + γP (11a) GP E -KRLN S = -u β ∂ β P + ∂ β (u β E k ) + 1 Re ∂ β u α ∂ β u α (11b) GP E -EDAC = -u β ∂ β P (11c)
It is shown in [START_REF] Clausen | Entropically damped form of artificial compressibility for explicit simulation of incompressible flow[END_REF] that

• the convection of kinetic energy ∂ β (u β E k ) is likely negligible based on the difference in acoustic and convective speeds.

• the kinetic energy dissipation 1 Re ∂ β u α ∂ β u α is important in obtaining a qualitatively correct pressure evolution.

The present numerical results show that pressure advection in EDAC conducts to errors in pressure evolution. However, the pressure advection term is usually neglected in EDAC. Concerning the Prandtl number, simulations realized with EDAC and KRLNS generally assume P r = γ. In this paper, the effect of the ratio P r γ is studied. Because the Prandtl number and the heat capacity ratio only appear in the coefficient of pressure diffusion γ ReP r , it is useless to study separately their effect. Consequently, it is assumed without loss of generality that γ = 1 and the effect of the Prandtl number is studied. It is shown that small Prandtl number in GPE allows accoustic wave damping.

The equation for the momentum, GPE, ACM and EDAC are discretized with the same scheme without subiteration. The temporal integration is performed using the case 7 of the third order low storage Runge-Kutta schemes proposed by Williamson [START_REF] Williamson | Low storage Runge-Kutta schemes[END_REF]. Due to acoustic, the time scale of INS is related to that of GPE, ACM, EDAC and KRLNS equations; [START_REF] Karlin | Kinetically reduced local Navier-Stokes equations: An alternative approach to hydrodynamics[END_REF]. This means that the time step used in GPE, ACM, EDAC and KRLNS is M a times smaller than that of INS. This justifies the use of a third Runge-Kutta scheme instead of a fourth Runge-Kutta scheme like in the work of Hashimoto et al. [START_REF] Hashimoto | Simulation of doubly periodic shear layers using Kinetically Reduced Local Navier-Stokes equations on a GPU[END_REF][START_REF] Hashimoto | Higher order numerical simulation of unsteady viscous incompressible flows using kinetically reduced local Navier-Stokes equations on a GPU[END_REF]. It is worth noting that very low Mach numbers do not imply accuracy problems. They allow very small velocity divergence but they imply very small time-step. One objective of the paper consists in realizing a parametric study on the Mach number to estimate how big the Mach number can be in order to increase the time-step without modifying the results (see section 4.3). Concerning the spatial discretization, we will restrict ourselves to two spatial dimensions to limit the length of the presentation; the extension to 3D is straightforward. The chosen spatial discretization corresponds to a staggered uniform grid system. The velocity components are distributed around the pressure points. The continuity is centered at pressure points. The momentum equation corresponding to each velocity component is centered at the respective velocity point. Staggered grid compared to collocated grid has the advantage that pressure gradient is naturally calculated at velocity points and velocity divergence at pressure points. Furthermore, it allows an easier comparison of the results with INS softwares that use mainly staggered grid system. Finally, staggered grid are generally better for avoiding checkerboard instability. The proposed discretization allows simulations with classical ACM. As it is shown in section 4.2, classical ACM is the most interesting for computational cost because only velocity divergence has to be calculated. The setting is illustrated in fig. 1. All used schemes are second order centered schemes:

t GP E = t ACM = t EDAC = t KRLN S = M a t IN S
(∇ • U ) ij ≡ u i+1j -u ij ∆ x + v ij+1 -v ij ∆ y (12a) (∇ • (uU )) ij ≡ 0.25 (u i+1j + u ij ) 2 -(u i-1j + u ij ) 2 ∆ x + (12b) (u ij+1 + u ij )(v i-1j+1 + v ij+1 ) -(u ij-1 + u ij )(v i-1j + v ij ) ∆ y (∂ x P ) ij ≡ P ij -P i-1j ∆ x (12c) (U • ∇P ) ij ≡ 0.5 u i+1j P i+1j -P ij ∆ x + u ij P ij -P i-1j ∆ x + (12d) v ij+1 P ij+1 -P ij ∆ y + v ij P ij -P ij-1 ∆ y (∆u) ij ≡ u i+1j -2u ij + u i-1j ∆ 2 x + u ij+1 -2u ij + u ij-1 ∆ 2 y (12e)
The convective term (∇ • (uU )) ij is discretized with the standard divergence form in a staggered grid system [START_REF] Harlow | Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with Free Surface[END_REF]. It is conservative a priori for momentum equation and it is conservative for kinetic energy if (∇•U ) ij = 0 [START_REF] Morinishi | Fully conservative higher order finite difference schemes for incompressible flow[END_REF]. Like LBM, the proposed discretization is local: the pressure and the velocity in one specified grid point only depends of its immediate neighbors of the previous time step. It will greatly facilitate the parallelization of the method and it is anticipated that this method will be very efficient for massively parallel simulation. In particular, using graphics processing unit (GPU) is useful. Hashimoto et al [START_REF] Hashimoto | Simulation of doubly periodic shear layers using Kinetically Reduced Local Navier-Stokes equations on a GPU[END_REF][START_REF] Hashimoto | Higher order numerical simulation of unsteady viscous incompressible flows using kinetically reduced local Navier-Stokes equations on a GPU[END_REF] realized parallel computations of KRLNS on GPU by using the CUDA library provided by the NVIDIA. They found that the computational time of KRLNS approach using 4th order approximations is 2.7 times faster than that of pseudo spectral method. The speedup of parallel computation is at least 5. Because KRLNS and GPE use the same equation (KRLNS for grand potential and GPE for pressure), the GPE gain of computational time due to parallel computation on GPU is expected to be the same than for KRLNS.

Unsteady analytical solutions of the Navier-Stokes equation

The analytical solutions of the Navier-Stokes equation simulated in this section are the two-dimensional Taylor-Green vortices and the two-dimensional traveling wave problem. They correspond to unsteady viscous incompressible flows. For GPE and ACM, they are solved using a 2 n x2 n computational grid for n = 4, 5, 6 and 7. The Reynolds, Prandtl and Mach numbers for each run are respectively Re = 100, P r = 1 and M a = 0.02 (note that the Prandtl number is not a parameter for ACM). The time step for each run is ∆t = 10 -4 .

Decaying vortices

The flow considered in this section corresponds to two-dimensional Taylor-Green vortices [START_REF] Taylor | Mechanism of the production of small eddies from large ones[END_REF] in a doubly-periodic unit square:

u(x, y, t) = cos(2πx)sin(2πy)e -8π 2 Re t (13a) v(x, y, t) = -sin(2πx)cos(2πy)e -8π 2 Re t (13b) p(x, y, t) = - 1 4 (cos(4πx) + cos(4πy))e -16π 2 Re t (13c)
It is worth noting that the previous velocity and pressure fields are exact solutions of INS eq. ( 2) and GPE eq. ( 3) with P r = 1. They are not solution of ACM eq. ( 4) or KRLNS or EDAC. Indeed, one gets [START_REF] Le | An improvement of fractional-step methods for the incompressible Navier-Stokes equations[END_REF]:

∂ β (u β u α ) + ∂ α P = 0 (14a) ∂ t u α = 1 Re ∂ β ∂ β u α (14b) ∂ α u α = 0 ( 14c
)
∂ t P = 1 ReP r ∂ α ∂ α P. ( 14d
)
For each case, the L 2 norms of the error in the u-component of the velocity and in the pressure were computed at time t = 0.1 (for symmetry reasons, the v-component of the velocity has exactly the same norm of error than the u-component). The results are shown in fig. 2. The convergence rates quoted in the caption of the figure are computed by taking the logarithm to base 2 of the error ratio from 64x64 and 128x128 computations. Because Taylor-Green vortices are exact solution of GPE eq. ( 3) but are not solution of ACM eq. ( 4) and EDAC eq. ( 6), the convergence rates of GPE are better than those of ACM and EDAC. The convergence rate of GPE for the velocity is exactly equal to 2 (the order of the spatial schemes). For the 128x128 mesh, the L2 norm of GPE error is smaller than for ACM and EDAC.

Traveling wave solution of the Navier-Stokes equations

The flow considered in this section corresponds to the traveling wave solution of the Navier-Stokes equations in a doubly-periodic unit square:

u(x, y, t) = 1 3 + 2 3 cos(2π(x -t))sin(2π(y -t))e -8π 2 Re t (15a) v(x, y, t) = 1 3 - 2 3 sin(2π(x -t))cos(2π(y -t))e -8π 2 Re t (15b) p(x, y, t) = - 1 9 (cos(4π(x -t)) + cos(4π(y -t)))e -16π 2 Re t (15c)
It is worth noting that the previous velocity and pressure fields are exact solution of INS eq. ( 2). They are not solution of GPE eq. ( 3) or ACM eq. ( 4) or EDAC eq. ( 6). For each case, the L 2 norms of the error in the u-component of the velocity and in the pressure were computed at time t = 0.

Doubly periodic shear layers

The case of doubly periodic shear layers is from [START_REF] Brown | Performance of under-resolved two-dimensional incompressible flow simulations[END_REF][START_REF] Minion | Performance of Under-resolved Two-Dimensional Incompressible Flow Simulations, II[END_REF]. The initial conditions on the periodic unit square are given by:

u = tanh(ρ(y -0.25)), if y < 0.5. tanh(ρ(0.75 -y)), otherwise. (16a) v = δsin(2π(x + 0.25)) ( 16b
)
where ρ is the shear layer width parameter and δ the strength of the initial perturbation. Here, ρ = 80, and δ = 0.05 are used. These parameters correspond to those used in [START_REF] Minion | Performance of Under-resolved Two-Dimensional Incompressible Flow Simulations, II[END_REF][START_REF] Hashimoto | Simulation of doubly periodic shear layers using Kinetically Reduced Local Navier-Stokes equations on a GPU[END_REF][START_REF] Hashimoto | Higher order numerical simulation of unsteady viscous incompressible flows using kinetically reduced local Navier-Stokes equations on a GPU[END_REF]. The strength of the initial perturbation is the same in [START_REF] Brown | Performance of under-resolved two-dimensional incompressible flow simulations[END_REF][START_REF] Minion | Performance of Under-resolved Two-Dimensional Incompressible Flow Simulations, II[END_REF]. The shear layer with parameter of 80 correspond to the thin layer of [START_REF] Minion | Performance of Under-resolved Two-Dimensional Incompressible Flow Simulations, II[END_REF] (it is an intermediate value between the thick layer (ρ = 30) and the thin layer (ρ = 100) from [START_REF] Brown | Performance of under-resolved two-dimensional incompressible flow simulations[END_REF]). The Reynolds, Prandtl and Mach numbers for each run are respectively Re = 10 000, P r = 1 and M a = 0.02 (note that the Prandtl number is not a parameter for ACM). The Prandtl and Mach numbers are modified for the parametric study in section 4.3. The time step for each run is ∆t = 10 -5 (except for the simulations with a Mach number M a = 0.002 whose the time step is ∆t = 10 -6 ).

Mesh independence

The shear layer problem was computed with 64x64, 128x128, 256x256, 384x384, 512x512, 768x768 and 1024x1024 grids. Mesh independence is studied with velocity, pressure, velocity divergence, kinetic energy, vorticity and enstrophy. Kinetic energy E k , vorticity ζ and enstrophy E are defined by

E k = 1 2 (u 2 + v 2 ) (17a) ζ = ∂v ∂x - ∂u ∂y (17b) E = S ζ 2 dS (17c)
Kinetic energy can not be defined uniquely on a staggered grid. In this work, kinetic energy is descretized at pressure points and vorticity is discretized at nodes (see fig. 1):

(E k ) ij = 1 4 (u 2 ij + u 2 i+1j + v 2 ij + v 2 ij+1 ) (18a) ζ ij = v ij -v i-1j ∆ x - u ij -u ij-1 ∆ y (18b)
Figure 4 shows the vorticity field at t = 1 for the different grids. The results from the 64x64 and 128x128 grids show the beginning of unstable oscillations and additional spurious vortices. The additional spurious vortices have disappeared in the 256x256 and 384x384 grids but small oscillations remain. The results from the 512x512 and 768x768 grids appear identical (the 1024x1024 grid -not shown here-gives the same results). It indicates that the 512x512 grid allows computing a reasonably accurate solution. Figure 5 shows the velocity divergence field at t = 1 for different grids. This figure confirms that the 512x512 grid is sufficient. One can distinguish different frequencies corresponding to the different initial velocity scales. It is worth noting that the average of the velocity divergence is equal to zero. Figures 6 and7 display the computed kinetic energy and enstrophy of the solutions for the different grids. It is clear that these plots are not diagnostic to determine mesh independance (since 256x256 grid seems sufficient with these plots). The time evolution of the root mean square velocity divergence (eq. ( 19)) is given in fig. 8 for the different grids. One notes that the maximum value does not depend on the grid and that again the 512x512 grid is accurate enough. Figures 9 to 13 show the spatial evolution in function of x at y=0.25 and t=1 of x-velocity, y-velocity, kinetic energy, pressure and vorticity respectively. Because the velocity divergence evolves faster in the ydirection (see figure 5), figure 14 represents the spatial evolution in function of y at x=0.25. Note that the point (x,y)=(0.25,0.25) corresponds to the vortex centre. All these figures confirm that the 512x512 grid is accurate enough. This grid is also the one used by [START_REF] Hashimoto | Simulation of doubly periodic shear layers using Kinetically Reduced Local Navier-Stokes equations on a GPU[END_REF][START_REF] Hashimoto | Higher order numerical simulation of unsteady viscous incompressible flows using kinetically reduced local Navier-Stokes equations on a GPU[END_REF]. Consequently, for the comparison with classical finite volume method with a Poisson equation and for the parametric study, the grid is 512x512.

Comparison with incompressible simulations

In this section, the results obtained with GPE, ACM, EDAC and INS are compared. The INS simulations are realized with the TrioCFD software developped by CEA (French atomic agency) [START_REF] Calvin | An object-oriented approach to the design of fluid mechanics software[END_REF]. The selected spatial and temporal schemes in TrioCFD are the same as those implemented for GPE. The INS time step is ∆t = 10 -3 for the 256x256 grid and ∆t = 5 10 -4 for the 512x512 grid. The pressure is solved with the classical pressure-Poisson formulation [START_REF] Chorin | Numerical solution of the Navier-Stokes equations[END_REF][START_REF] Minion | Performance of Under-resolved Two-Dimensional Incompressible Flow Simulations, II[END_REF]. The Poisson equation is solved with the conjugate gradient method and the Symmetric Successive Over-Relaxation preconditioning technique. With a Mach number of 0.02 for GPE (ACM or EDAC) and a residual norm tolerance of 10 -12 for the Poisson equation (INS), it is found that computational time of GPE, ACM and EDAC is respectively 1.3, 1.5 and 1.2 times smaller than that of INS. ACM is the most interesting in terms of computational time because only velocity divergence has to be calculated. EDAC is the least interesting because pressure advection and diffusion have also to be computed. Only the pressure and the vorticity fields (fig. 15 and16 respectively) are presented because the results with velocity and kinetic energy are the same. Figures 15 and16 indicate that the results obtained with GPE, ACM and INS are in very good agreement. The differences between GPE, ACM and EDAC are not distinguishable. The differences with INS are smaller than those due to the mesh. These results demonstrate that the proposed discretization of GPE, ACM and EDAC allows describing the time evolution of velocity and pressure without subiteration. They also show that the additional term in the EDAC equation compared to the GPE equation is negligible. It confirms the asymptotic expansion realized in [START_REF] Toutant | General and exact pressure evolution equation[END_REF] that conducts to neglect pressure advection (u β ∂ β P ) for small Mach number. The test of different time steps for GPE, ACM and EDAC shows that ∆t = 2.5 10 -5 is stable and that ∆t = 5 10 -5 is not stable for the 512x512 grid. Consequently, it is found that the proposed discretization conducts to the same stability time step for GPE, ACM and EDAC.

Prandtl and Mach numbers effect

GPE depends on three dimensionless numbers: the Reynolds, Prandtl and Mach numbers. In this section, a parametric study on the Prandtl and Mach numbers is realized. The effect of the Reynolds number is not studied because it is considered here as a physical parameter while the Prandtl and Mach numbers are considered as numerical parameters. The objective is to estimate

• how small the Prandtl number can be in order to damp acoustic waves without modifying the results and,

• how big the Mach number can be in order to increase the stability time step without modifying the results.

The tested Prandtl numbers are 0.001, 0.01, 0.1, 1, 10 and +∞. The tested Mach numbers are 0.2, 0.1, 0.05, 0.025, 0.02 and 0.002. Figures 17,[START_REF] Taylor | Mechanism of the production of small eddies from large ones[END_REF]19 and 20 show the results obtained for all the pairs (M a, P r) with M a ∈ {0.2, 0.02 and 0.002} and P r ∈ {0.1, 1 and 10}. Figures 17 and18 show that a Mach number M a = 0.2 is too big and modifies integral quantities such as kinetic energy and enstrophy. In contrast, the Prandtl number seems to have no effect on these integral quantities. The spatial evolution of pressure and vorticity presented in figures 19 and 20 confirms that M a = 0.2 modifies the results and that the Prandtl number has almost no effect.

The same profiles with more values of the Mach number and P r = 1 are plotted in figures 21 and 22. Very low Mach numbers do not imply accuracy problems but they imply very small time-step. Indeed, due to acoustic, the time scale of GPE, t GP E , is related to that of INS equations, t IN S ; t GP E = M at IN S . A Mach number equal to 0.05 -value typically used in lattice Boltzmann simulations-offers a good compromize between the accuracy of the results and the stability time step. This value is coherent with the order of magnitude proposed for KRLNS by [START_REF] Karlin | Kinetically reduced local Navier-Stokes equations: An alternative approach to hydrodynamics[END_REF]. For GPE, this value has to be confirm in other configurations.

The rms velocity divergence is defined by

(∂ α u α ) rms = V (∂ α u α ) 2 dV (19)
It is worth noting that the mean velocity divergence V (∂ α u α )dV is exactly equal to zero due to periodic conditions and the used staggered grid system. Figure 23 shows that the maximum of the rms velocity divergence is proportional to the Mach number. The proportionality constant seems to be equal to 0.3. This linear scaling and this proportionality constant have also been found for KRLNS [START_REF] Karlin | Kinetically reduced local Navier-Stokes equations: An alternative approach to hydrodynamics[END_REF]. A Fourier transform of the temporal signal gives that the nondimensionalized period is also proportional to the Mach number (T = 7.10 -3 for M a = 0.02 and T = 7.10 -4 for M a = 0.0021 ). The period corresponds to the plane wave associated to the thickness of the shear layer. Indeed, noting δ sl the nondimensionalized shear layer thickness, one gets δ sl ≈ T 2πM a = 0.056. Figures 24 and 25 seem to show that the Prandtl number has almost no effect on velocity and pressure. However, figure 26 indicates that the Prandtl number impacts significantly the velocity divergence. Indeed, according to the Prandtl number, we can distinguish the acoustic waves or/and the shear layer vortex in the velocity divergence field. For a Prandtl number smaller or equal to 0.001, acoustic waves are completely damped at t = 1 and only shear layer vortex are observable. For a Prandtl number between 0.01 and 0.1, both acoustic waves and shear layer vortex are present. For a Prandtl number greater or equal to one, only acoustic waves are visible. Furthermore, it is worth noting that, even if a small Prandtl number conducts to acoustic wave damping, a smaller Prandtl number does not imply a smaller velocity divergence. These observations are coherent with the acoustic plane wave solution of GPE equations. In one dimension with the acoustic approximation, GPE equations are The acoustic plane wave solution is given by

∂ t u + ∂ x P = 1 Re ∂ x ∂ x u (20a)
∂ t P + 1 M a 2 ∂ x u = 1 ReP r ∂ x ∂ x P (20b)
u = 1 -P r 2ReP r ksin(ωt -kx) + ω k cos(ωt -kx) M a 2 e -at (21a)
P = cos(ωt -kx)e -at (21b)

ω 2 = k 2 1 M a 2 - 1 4Re 2 1 - 1 P r 2 k 2 (21c) a = 1 + P r 2ReP r k 2 (21d)
For P r = 1, this solution gives the classical relation k 2 = ω 2 c 2 (in dimensional variables and with c the speed of sound). The expression of the damping coefficient is coherent with the fact that both viscosity and thermal conductivity are involved in acoustic wave attenuation [1]. The smaller the Prandtl number, the higher the damping coefficient ( lim P r→0 a = +∞). One can compare the effect of the Mach and Prandtl numbers on the velocity divergence in figures 26 and 27. A small Mach number allows the decrease of both acoustic waves and shear layer vortex in the velocity divergence field. A small Prandtl number allows the damping of acoustic waves but not of the shear layer vortex. A smaller Mach number implies a smaller velocity divergence. It is not the case for the Prandtl number.

Conclusion

The proposed general pressure equation (GPE) allows to solve general, timeaccurate, incompressible Navier-Stokes flows. GPE should be considered as an analog of the Lattice Boltzmann Method. The velocity divergence is not equal to zero but it can be limited by an arbitrary maximum. General pressure equation (GPE), artificial compressibility method (ACM) and entropically damped form of artificial compressibility (EDAC) have been applied for simulations of Taylor-Green vortices, traveling wave and doubly periodic shear layers in order to investigate their accuracy, efficiency and the capability to capture the correct transient behavior. The proposed discretization uses a staggered grid system, second order centered schemes in space and a third order Runge-Kutta scheme in time without subiteration. The solutions obtained by GPE, ACM and EDAC are in excellent agreement with analytical solutions or with that of a classical finite volume method using a Poisson equation. It is found that the GPE, ACM and EDAC approaches can capture the correct transient behavior without subiteration, and keep the divergence fluctuation at smaller level by giving an appropriately low Mach number. GPE L2 norm of errors are smaller than those of ACM and EDAC. For Taylor-Green vortices, the better convergence rate of GPE is explained by the fact that this flow is analytical solutions of GPE but not of ACM and EDAC. Consequently, the additional diffusion term of GPE compared to ACM seems physically crucial and the additional pressure advection term of EDAC compared to GPE seems to be useless. The onedimensional acoustic plane wave analytical solution of GPE gives the acoustic damping coefficient. This coefficient is inversely proportional to the Prandtl number (for small Prandtl number). Numerically, it is shown that a small Prandtl number allows to damp acoustic waves.

Figure 1 :

 1 Figure 1: Staggered grid system

Figure 4 :

 4 Figure 4: Vorticity at t=1

Figure 5 :

 5 Figure 5: Velocity divergence at t=1

Figure 8 :Figure 9 :- 1 Figure 10 :Figure 11 :- 3 Figure 12 :Figure 13 :-Figure 14 :

 89110113121314 Figure 8: Time evolution of root mean square velocity divergence (top: whole time of the simulation, bottom: zoom on one representative peak)

Figure 15 :

 15 Figure 15: Spatial evolution of pressure in function of x at y=0.25 and t=1 (top: whole profile, bottom: zoom around the peak x∈ [0.2, 0.3]). Comparison between incompressible, ACM, EDAC and GPE simulations for two different meshes (256x256 and 512x512).

Figure 16 :

 16 Figure 16: Spatial evolution of vorticity in function of x at y=0.25 and t=1 (top: whole profile, bottom: zoom around the peak x∈ [0.2, 0.3]). Comparison between incompressible, ACM, EDAC and GPE simulation for two different meshes (256x256 and 512x512).

Figure 17 :

 17 Figure 17: Kinetic energy decay for different Mach and Prandtl numbers.

Figure 18 :

 18 Figure 18: Enstrophy decay for different Mach and Prandtl numbers.

Figure 23 :

 23 Figure 23: Time evolution of root mean square velocity divergence (top: whole time of the simulation, bottom: zoom on the last 0.05s). Effect of the Mach number.

Figure 24 :

 24 Figure 24: Spatial evolution of pressure in function of x at y=0.25 and t=1. Effect of the Prandtl number.

Figure 25 :

 25 Figure 25: Spatial evolution of vorticity in function of x at y=0.25 and t=1. Effect of the Prandtl number.

Figure 26 :

 26 Figure 26: Velocity divergence at t=1 for Ma=0.02 and for different Prandtl numbers.

  [START_REF] Tessarotto | Exact pressure evolution equation for incompressible fluids[END_REF]. The results are shown in fig.3. The convergence rates quoted in the caption of the figure are computed by taking the logarithm to base 2 of the error ratio from 64x64 and 128x128 computations. As for the Taylor-Green vortices, GPE L2 norm of error is smaller than for ACM and EDAC.

		0.01
		0.001
	L2 norm of error	0.0001
		1e-05
		GPE velocity error
		ACM velocity error
		EDAC velocity error
		GPE pressure error
		ACM pressure error
		EDAC pressure error
		1e-06
		10	100
		Number of grid points

Figure

2

: Convergence results for the Taylor-Green vortices. The computed convergence rates are GPE velocity, 2.00; ACM velocity, 1.42; EDAC velocity, 1.2; GPE pressure, 1.87; ACM pressure, 0.57; EDAC pressure, 0.1.

  Figure 19: Spatial evolution of pressure in function of x at y=0.25 and t=1 (top: whole profile, bottom: zoom around the peak x∈ [0.2, 0.3]). Effects of the Prandtl and Mach numbers.

			1	Ma=0.2 Pr=1
					Ma=0.02 Pr=1
					Ma=0.002 Pr=1
			0.1
	Pressure RMS velocity divergence	0.0001 0.001 0.01
														Ma=0.2 Pr=0.1
														Ma=0.2 Pr=1
														Ma=0.2 Pr=10
		1e-05										Ma=0.02 Pr=0.1 Ma=0.02 Pr=1
														Ma=0.02 Pr=10
													Ma=0.002 Pr=0.1
														Ma=0.002 Pr=1
													Ma=0.002 Pr=10
		1e-06
				0	1	0.1	0.2	0.2	0.3 0.3	0.4 0.4	0.5 0.5	0.6 0.6	0.7 0.7	0.8 0.8	0.9 0.9	1 1
											x Time
		-2.4	0.1
		-2.6
		0.01
		-2.8
	Pressure RMS velocity divergence	-3.2 -3 0.001
		-3.4												Ma=0.2 Pr=0.1
		0.0001										Ma=0.2 Pr=1
														Ma=0.2 Pr=10
														Ma=0.02 Pr=0.1
		-3.6												Ma=0.02 Pr=1 Ma=0.02 Pr=10
													Ma=0.002 Pr=0.1 Ma=0.2 Pr=1
														Ma=0.002 Pr=1 Ma=0.02 Pr=1
		-3.8 1e-05									Ma=0.002 Pr=10 Ma=0.002 Pr=1
			0.2	0.95	0.21	0.22 0.96	0.23	0.24 0.97	0.25	0.26 0.98	0.27	0.28 0.99	0.29	0.3 1
											x Time

For Ma=0.2, the signal is not enough periodic to isolate one frequency.