The Advection Upstream Splitting Method (AUSM) is a

numerical method In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm. Mathem ...

used to solve the

advection In the fields of physics, engineering, and earth sciences, advection is the transport of a substance or quantity by bulk motion of a fluid. The properties of that substance are carried with it. Generally the majority of the advected substance is a ...

equation in

computational fluid dynamics Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid dynamics, fluid flows. Computers are used to perform the calculations required ...

. It is particularly useful for simulating compressible flows with shocks and discontinuities. The AUSM is developed as a numerical

inviscid Viscosity is a measure of a fluid's rate-dependent resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for example, syrup h ...

flux Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications in physics. For transport phe ...

function for solving a general system of

conservation equation In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of mass-energy, conservation of linear moment ...

s. It is based on the upwind concept and was motivated to provide an alternative approach to other upwind methods, such as the Godunov method, flux difference splitting methods by Roe, and Solomon and Osher, flux vector splitting methods by Van Leer, and Steger and Warming. The AUSM first recognizes that the inviscid flux consist of two physically distinct parts, i.e., convective and pressure fluxes. The former is associated with the flow (

) speed, while the latter with the acoustic speed; or respectively classified as the linear and nonlinear fields. Currently, the convective and pressure fluxes are formulated using the

eigenvalue In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...

s of the flux Jacobian matrices. The method was originally proposed by Liou and Steffen for the typical compressible aerodynamic flows, and later substantially improved in Liou, M.-S., “A Sequel to AUSM: AUSM+” J. Comput. Phys., Vol. 129, 364-382, 1996. to yield a more accurate and robust version. To extend its capabilities, it has been further developed in Liou, M.-S., “A Sequel to AUSM, Part II: AUSM+-up” J. Comput. Phys., Vol. 214, 137- 170, 2006. for all speed-regimes and

multiphase flow In fluid mechanics, multiphase flow is the simultaneous Fluid dynamics, flow of materials with two or more thermodynamic Phase (matter), phases. Virtually all processing technologies from Cavitation, cavitating pumps and turbines to paper-making ...

. Its variants have also been proposed.

Features

The Advection Upstream Splitting Method has many features. The main features are: *accurate capturing of shock and contact discontinuities *

entropy Entropy is a scientific concept, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the micros ...

-satisfying solution *positivity-preserving solution *algorithmic simplicity (not requiring explicit eigen-structure of the flux Jacobian matrices) and straightforward extension to additional conservation laws *free of “carbuncle” phenomena *uniform accuracy and convergence rate for all

Mach number The Mach number (M or Ma), often only Mach, (; ) is a dimensionless quantity in fluid dynamics representing the ratio of flow velocity past a boundary to the local speed of sound. It is named after the Austrian physicist and philosopher Erns ...

s. Since the method does not specifically require

eigenvector In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by ...

s, it is especially attractive for the system whose eigen-structure is not known explicitly, as the case of two-fluid equations for multiphase flow.

Applications

The AUSM has been employed to solve a wide range of problems, low-Mach to

hypersonic In aerodynamics, a hypersonic speed is one that exceeds five times the speed of sound, often stated as starting at speeds of Mach 5 and above. The precise Mach number at which a craft can be said to be flying at hypersonic speed varies, since i ...

aerodynamics Aerodynamics () is the study of the motion of atmosphere of Earth, air, particularly when affected by a solid object, such as an airplane wing. It involves topics covered in the field of fluid dynamics and its subfield of gas dynamics, and is an ...

large eddy simulation Large eddy simulation (LES) is a mathematical model for turbulence used in computational fluid dynamics. It was initially proposed in 1963 by Joseph Smagorinsky to simulate atmospheric air currents, and first explored by Deardorff (1970). LES is ...

and aero-acoustics, direct numerical simulation, multiphase flow, galactic relativistic flowWada, K. and Koda, J., “Instabilities of Spiral Shock – I. Onset of Wiggle Instability and its Mechanism,”

Monthly Notices of the Royal Astronomical Society ''Monthly Notices of the Royal Astronomical Society'' (MNRAS) is a peer-reviewed scientific journal in astronomy, astrophysics and related fields. It publishes original research in two formats: papers (of any length) and letters (limited to ...

, Vol. 349, 270-280 (11), 2004. etc.

References

{{DEFAULTSORT:Advection upstream splitting method Computational fluid dynamics Numerical differential equations

Features

Applications

See also

References