Pressure-Correction Methods

Pressure-Correction Methods

Pressure-Correction Methods refer to a class of methods used in computational fluid dynamics for solving the Navier-Stokes equations normally for incompressible flows.

Common Properties

The equations solved in this approach arise from the implicit time integration of the incompressible Navier-Stokes equations.

overbrace{ ho Big(underbrace{frac{partial mathbf{v{partial t_{egin{smallmatrix} ext{Unsteady}\ ext{acceleration}end{smallmatrix + underbrace{left(mathbf{v} cdot abla ight) mathbf{v_{egin{smallmatrix} ext{Convective} \ ext{acceleration}end{smallmatrixBig)}^{ ext{Inertia =underbrace{- abla p}_{egin{smallmatrix} ext{Pressure} \ ext{gradient}end{smallmatrix + underbrace{mu abla^2 mathbf{v_{ ext{Viscosity + underbrace{mathbf{f_{egin{smallmatrix} ext{Other} \ ext{forces}end{smallmatrix

Due to the non-linearity of the convective term in the momentum equation that is written above, this problem is solved with a nested-loop approach. While so called "global"or "inner iterations" represent the real time-steps and are used to update the variables mathbf{v} and p, based on a linearized system, and boundary conditions; there is also an "outer loop" for updating the coefficients of the linearized system.
The outer iterations comprise two steps:

  1. solve the momentum equation for a "provisional" velocity based on the velocity and pressure of the previous outer loop.
  2. plug the new newly obtained velocity into the continuity equation to obtain a correction.
The correction for the velocity that is obtained from the second equation one has with incompressible flow, the non-divergence criterion or continuity equation

ext{div}mathbf{v} = 0

is computed by first calculating a residual value dot{m}, resulting from spurious "mass flux", then using this "mass imbalance" to get a new pressure value. The pressure value that is attempted to compute, is such that when plugged into momentum equations a divergence-free velocity field results. The mass imbalance is often also used for control of the outer loop.
The name of this class of methods stems from the fact that the correction of the velocity field is comptued through the pressure-field.

The discretization of this is typically done with either the finite element method or the finite volume method. With the latter, one might also encounter the dual mesh, i.e. the computation grid obtained from connecting the centers of the cells that the initial subdivison into finite elements of the computation domain yielded.

Implicit Split-Update Procedures

Another approach which is typlically used in FEM is the following.

The aim of the correction step is to ensure "conservation of mass". In continuous form for compressible substances mass, conservation of mass is expressed by ablacdotleft( ho(mathbf{x})mathbf{v}(mathbf{x}) ight) = frac{frac{d}{dt}p(mathbf{x})}{c^2} where c^2 is the square of the "speed of sound". For low Mach numbers and incompressible media c is assumed to be infinite, which is the reason for the above continuity equation to reduce to egin{array}{r c l} ext{div}mathbf{v} &=& 0\ ablacdotmathbf{v} &=& 0end{array}The way of obtaining a velocity field satisfying the above, is to compute a pressure which when substituted into the momentum equation leads to the desired correction of a preliminary computed intermediate velocity.

Applying the divergence operator to the copmpressible momentum equation yields
egin{array}{r c l l} ablacdotpartial_t mathbf{v} &=& - ablacdot(mathbf{v}cdot abla)mathbf{v} + ablacdot abla^2mathbf{v} - abla^2 p\ partial_t ablacdotmathbf{v} &=& - ablacdot(mathbf{v}cdot abla)mathbf{v} + abla^2 ablacdotmathbf{v} - abla^2 p\ 0 &=& - ablacdot(mathbf{v}cdot abla)mathbf{v} - abla^2 p\ abla^2 p &=& - ablacdot(mathbf{v}cdot abla)mathbf{v} & (ast)end{array}
(ast) then provides the governing equation for pressure computation.

The idea of pressure-correction also exists in the case of variable density and high Mach numbers, although in this case there is a real physical meaning behind the coupling of dynamic pressure and velocity as arising from the "continuity equation"
egin{array}{r c l} partial_t ho &=& ablacdot( ho mathbf{v})\ partial_t ho &=& frac{1}{c^2}partial_t pend{array}
p is with compressibility, still an additional variable that can be eliminated with algebraic operations, but its variablilty is not a pure artifice as in the compressible case, and the methods for its computation differ significantly from those with ho =constant.


* A. Meister, J. Struckmeier: Hyperbolic Partial Differential Equations, 1st Edition, Vieweg, 2002

External links

* [ ISNaS - incompressible flow solver]
* [ Application of Temperature and/or Pressure Correction Factors in Gas Measurement]

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Cardiac output — (Q or or CO ) is the volume of blood being pumped by the heart, in particular by a left or right ventricle in the time interval of one minute. CO may be measured in many ways, for example dm3/min (1 dm3 equals 1000 cm3 or 1 litre). Q is… …   Wikipedia

  • Partial androgen insensitivity syndrome — Classification and external resources AIS results when the function of the androgen receptor (AR) is impaired. The AR protein (pictured) mediates the effects of androgens in the human body. ICD 10 …   Wikipedia

  • Van Deemter equation — The Van Deemter equation in chromatography relates the variance per unit length of a separation column to the linear mobile phase velocity by considering physical, kinetic, and thermodynamic properties of a separation.[1] These properties include …   Wikipedia

  • Economic Affairs — ▪ 2006 Introduction In 2005 rising U.S. deficits, tight monetary policies, and higher oil prices triggered by hurricane damage in the Gulf of Mexico were moderating influences on the world economy and on U.S. stock markets, but some other… …   Universalium

  • Regression dilution — is a statistical phenomenon also known as attenuation . Consider fitting a straight line for the relationship of an outcome variable y to a predictor variable x, and estimating the gradient (slope) of the line. Statistical variability,… …   Wikipedia

  • eye disease — Introduction  any of the diseases or disorders that affect the human eye (eye, human).       This article briefly describes the more common diseases of the eye and its associated structures, the methods used in examination and diagnosis, and the… …   Universalium

  • QT interval — Schematic representation of normal ECG trace (sinus rhythm), with waves, segments, and intervals labeled. In cardiology, the QT interval is a measure of the time between the start of the Q wave and the end of the T wave in the heart s electrical… …   Wikipedia

  • Comparison of analog and digital recording — This article compares the two ways in which sound is recorded and stored. Actual sound waves consist of continuous variations in air pressure. Representations of these signals can be recorded using either digital or analog techniques. An analog… …   Wikipedia

  • United States — a republic in the N Western Hemisphere comprising 48 conterminous states, the District of Columbia, and Alaska in North America, and Hawaii in the N Pacific. 267,954,767; conterminous United States, 3,022,387 sq. mi. (7,827,982 sq. km); with… …   Universalium

  • Business and Industry Review — ▪ 1999 Introduction Overview        Annual Average Rates of Growth of Manufacturing Output, 1980 97, Table Pattern of Output, 1994 97, Table Index Numbers of Production, Employment, and Productivity in Manufacturing Industries, Table (For Annual… …   Universalium

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.