Projection method (fluid dynamics): Difference between revisions

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{{Short description|Method for numerically solving time-dependent incompressible fluid-flow problems}}

{{expert}}

TheIn [[computational fluid dynamics]], the '''projection method''', also called '''Chorin's projection method''', is an effective means of [[Numerical analysis|numerically]] solving time-dependent [[incompressible flow|incompressible fluid-flow]] problems. It was originally introduced by [[Alexandre Chorin]] andin independently by [[Roger Temam]]1967<ref>

{{Harvard reference

{{Citation

| Surname1 = Temam

| Given1surname1 = R.Chorin

| Given1given1 = A. J.

| Title = Une m&eacute;thode d'approximation des solutions des &eacute;quations Navier-Stokes,

| title = The numerical solution of the Navier-Stokes equations for an incompressible fluid

| Journaljournal = Bull. SocAm. Math. FranceSoc.

| Volume = 98

| Yearvolume = 196873

| Pagesyear = 115&ndash;1521967

| urlissue = 6

| Pagespages = 745928&ndash;762931

}}</ref> as an efficient means of solving the incompressible [[Navier-Stokes equation]]s. The key advantage of the projection method is that the computations of the velocity and the pressure fields are decoupled.

| doi = 10.1090/S0002-9904-1967-11853-6

| url =http://math.berkeley.edu/~chorin/chorin67.pdf

}}</ref><ref>

{{Citation

| Surname1surname1 = Chorin

| given1 = A. J.

| Titletitle = Numerical Solution of the Navier-Stokes Equations

| Journaljournal = Math. Comp.

| Volumevolume = 9822

| Yearyear = 1968

| urlissue = 104

| pages = 745&ndash;762

| url =

| doi=10.1090/s0025-5718-1968-0242392-2

| doi-access = free

}}</ref>

}}</ref> as an efficient means of solving the incompressible [[Navier-Stokes equation]]s. The key advantage of the projection method is that the computations of the [[Flow velocity|velocity]] and the pressure fields are decoupled.

==The algorithm==

The algorithm of the projection method is based on the [[Helmholtz decomposition]] (sometimes called Helmholtz-Hodge decomposition) of any vector field into a [[solenoidal field|solenoidal]] part and an [[irrotational field|irrotational]] part. Typically, the algorithm consists of two stages. In the first stage, an intermediate velocity that does not satisfy the incompressibility constraint is computed at each time step. In the second, the pressure is used to project the intermediate velocity onto a space of divergence-free velocity field to get the next update of velocity and pressure.

==Helmholtz-&ndash;Hodge decomposition==

The theoretical background of projection type method is the decomposition theorem of [[Olga Aleksandrovna Ladyzhenskaya|Ladyzhenskaya]] sometimes referred to as Helmholtz-&ndash;Hodge Decomposition or simply as Hodge Decompositiondecomposition. It states that the vector field <math>\mathbf{u}</math> defined on a [[simply connected space|simply connected]] ___domain can be uniquely decomposed into a divergence-free ([[Solenoidal vector field|solenoidal]]) part <math>\mathbf{u}_{\text{sol}}</math> and an [[Conservative vector field#Irrotational vector fields|irrotational]] part <math>\mathbf{u}_{\text{irrot}}</math>.
<ref>{{cite book | title = A Mathematical Introduction to Fluid Mechanics | author1 = Chorin, A. J. | author2 = J. E. Marsden | edition = 3^rd3rd | publisher = [[Springer Science+Business Media|Springer-Verlag]] | year = 20001993 | isbn = 03879791820-387-97918-2}}</ref> Thus,

Thus,

:<math>

\mathbf{u} = \mathbf{u}_{\text{sol}} + \mathbf{u}_{\text{irrot}} = \mathbf{u}_{\text{sol}} + \nabla \phi

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</math>

This is a [[Poisson equation]] for the scalar function <math>\,\phi</math>. If the vector field <math>\mathbf{u}</math> is known, the above equation can be solved for the scalar function <math>\,\phi</math> and the divergence-free part of <math>\mathbf{u}</math> can be extracted using the relation

:<math>

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This is the essence of solenoidal projection method for solving incompressible

Navier-&ndash;Stokes equations.

==Chorin's projection method==

The icompressibleincompressible Navier-Stokes equation (differential form of momentum equation) may be written as

:<math>

\frac {\partial \mathbf{u}} {\partial t} + (\mathbf{u}\cdot\nabla)\mathbf{u} = - \frac {1}{\rho} \nabla p + \nu \nabla^2 \mathbf{u}

</math>

In [[Alexandre Chorin|Chorin]]'s original version of the projection method, one first computes an intermediate velocity, <math>\mathbf{u}^*</math>, explicitly using the momentum equation by ignoring the pressure gradient term:

<ref>

{{Harvard reference

| Surname1 = Chorin

| Given1 = A. J.

| Title = Numerical Solution of the Navier-Stokes Equations

| Journal = Math. Comp.

| Volume = 22

| Year = 1968

| Pages = 745&ndash;762

| url =

}}</ref>, the intermediate velocity, <math>\mathbf{u}^*</math>, is explicitly computed using the momentum equation ignoring the pressure gradient term:

:<math>

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</math>

where <math>\mathbf{u}^n</math> is the velocutyvelocity at <math>\,n</math>^th time levelstep. In the nextsecond (half of the algorithm, the ''projection)'' step, we havecorrect the intermediate velocity to obtain the final solution of the time step <math>\mathbf{u}^{n+1}</math>:

:<math>

\fracquad (2) \qquad {\mathbf{u}^{n+1} -= \mathbf{u}^*} {\Delta t} = - \frac {1\Delta t}{\rho} \, \nabla p ^{n+1}

</math>

Re-writingOne thecan rewrite abovethis equation forin the velocityform atof <math>\,(n+1)</math>a level,time westep haveas

:<math>

\quad (2) \qquadfrac {\mathbf{u}^{n+1} =- \mathbf{u}^*} {\Delta t} = - \frac {\Delta t1}{\rho} \, \nabla p ^{n+1}

</math>

to make clear that the algorithm is really just an [[operator splitting]] approach in which one considers the viscous forces (in the first half step) and the pressure forces (in the second half step) separately.

Computing the right-hand side of the above equation requires a knowledge of the pressure, <math>\,p</math>, at <math>\,(n+1)</math> level. This is obtained by taking the [[divergence]] and requiring that <math>\nabla\cdot \mathbf{u}^{n+1} = 0</math>, thereby deriving the following Poisson equation for <math>\,p^{n+1}</math>,

Computing the right-hand side of the abovesecond equationhalf requiresstep arequires knowledge of the pressure, <math>\,p</math>, at the<math>\,(n+1)</math> time level. This is obtained by taking the [[divergence]] and requiring that <math>\nabla\cdot \mathbf{u}^{n+1} = 0</math>, which is the divergence (continuity) condition, thereby deriving the following Poisson equation for <math>\,p^{n+1}</math>,

:<math>

\nabla ^2 p^{n+1} = \frac {\rho} {\Delta t} \, \nabla\cdot \mathbf{u}^*

</math>

It is instructive to note that, the equation written in the following wayas

:<math>

\mathbf{u}^* = \mathbf{u}^{n+1} + \frac {\Delta t}{\rho} \, \nabla p ^{n+1}

</math>

is the standard Hodge decomposition if boundary condition for <math>\,p</math> on the ___domain boundary, <math>\partial \Omega</math> isare <math>\nabla p^{n+1}\cdot \mathbf{n} = 0</math>. In practice, this condition is responsible for the errors this method shows close to the boundary of the ___domain since the real pressure (i.e., the pressure in the exact solution of the Navier-Stokes equations) does not satisfy such boundary conditions.

For the explicit method, the boundary condition for <math>\mathbf{u}^*</math> in equation (1) is natural. If <math>\mathbf{u}\cdot \mathbf{n} = 0</math> on <math>\partial \Omega</math>, is prescribed, then the space of divergence-free vector fieldfields will be orthogonal to the space of irrotational vector fields, and from equation (2) one has

:<math>

\frac {\partial p^{n+1}} {\partial n} = 0 \qquad \text{on} \quad \partial \Omega

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# First the system is progressed in time to a mid-time-step position, solving the above transport equations for mass and momentum using a suitable advection method. This is denoted the ''predictor'' step.

# At this point an initial projection maybemay be implemented such that the mid-time-step velocity field is enforced as divergence free.

# The ''corrector'' part of the algorithm is themthen progressed. These use the time-centred estimates of the velocity, density, etc. to form final time-step state.

# A final projection is then applied to enforce the divergence restraint on the velocity field. The system has now been fully updated to the new time.

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[[Category:Computational fluid dynamics]]

[[Category:AppliedMathematical mathematicsphysics]]