Incompressible Flows

Sauro Succi

Incompressible flows

Incompressible constraint

Kinematic Constraint: elliptic (time-consuming)

“Slow” flows: sound speed to infinity (fluid<<sound)

Matrix Formulation

Cruelly non-local: no way!

Many options…

Colocated/StaggeredExplicit/Implicit,Exactly/Quasi Incompressible,……

Colocated; Control Volume

Hourglass in simple geos

No hourglass in complex ones

Staggered: stronger VP coupling

Staggered

Laborious, good for surfint > simple geos

No hourglass, VP coupled

Isotropic Laplacians

Colocated: Hourglass instability

Complex geos

Spherical cows!

Staggered: complicatedColocated: no hourglass

Modern FV: Implicit diffusion with structured colocated FV leads to 9-diag regular matrices,Can be solved efficiently with ADI.

Poisson solver has no hourglass, but still veryExpensive because the coeff’s are inhomogeneous

Handling non-locality

Rapid Poisson Solvers

Artificial compressibility

Predictor-Corrector methods

Explicit/Implicit time marching

Rapid Poisson: Spectral

Fourier transform: f(x) to f(k)

And back : f(k) to f(x)

Differential to algebraic problem

1. FT

2. Solve

3. IFT

2d homog. Inc. turbulence

Spectral: plus and minus

Problems:N^2 complexityPeriodic Geometries

RemediesFFT: N^2 to N*logNPeriodic constraint basically remains

Two basic families

Exactly Incompressible (EI)

Artificial Compressibility (AC)

Exactly incompressible

Strictly incompressible: elliptic

Two hyperbolic+one elliptic, stiff matrix

EI: Explicit

Divfree is enforced in time, but Poisson very CPU intensive ->Rapid Elliptic Solvers (RES)

Solve Poisson for p^0, then advance U^0 to U^1

Artificial Compressibility

Fictitious (pseudo)-time Exact at steady stateHard to soft constraint

Full Time-dependent

Exact at steady-state (only)

Divergence dynamics

Small-amplitude oscillations around epsilon=O(Mach^2)

“Hydrodynamic Charge”Similar to gravity: curvature of u

AC: Chorin

Pseudodyn is stable: small flucts around p0divu>0 p goes down and viceversaDivfree remains O(epsilon) all along, No Poisson, but dt very small

AC: another version ?

Pseudodyn is stable: small flucts around p0divu>0 p goes down and viceversaDivfree remains O(epsilon) all along, No Poisson, but dt very small

AC: Explicit: WRONG!

Divfree is not conserved in time, No Poisson, but p1 not ok: iteration needed:WRONG: if p0 obeys poisson divu frozen = 0!!!

Wrong: divfree frozen to 0

Hard vs Soft Constraints

Electronic structure: Born-Oppenheimer, Car-Parrinello: softOrbital Orthogonality : hard

Biomolecular dynamics: hard

FluidCompressibility: soft

With f hard to invertHard:

Soft: No need to invert f

CFL stability conditions

Diffusion is very-constraining Advection: ExplicitDiffusion: Implicit

EI: Linearly-Implicit

Poisson less of a drag: implicit anyway

Predict-Correct

Predict u*(p=0):

Correct u*:

Require:(Projection)

u^{n+1} isnow div-free

AC: Implicit Diffusion (Linear)

Summary

Exactly Incompressible:Explicit: Divfree is forced via Poisson, but Poisson solver is a dragRemedies: RPS: Rapid Poisson Solver (simple geo’s)Implicit: large dt, Poisson less of a drag, implicit anyway

Artificial Compressibility:Exact only at steady-state.Divfree is only quasi-conserved to O(eps) Can leave with it if steady-state is the only targetLess so for dynamics Implicit: PS no longer a drag, implicit anyway

Nonlinearity

Nonlinearity-Picard iteration

The face of the discrete operators:Finite Differences

MAC staggered grid(FD)

Pressure equation

Staggered grid: X component

Y-component

Explicit/Implicit

Boundary conditions

Spherical cows! ?

Boundary Conditions: Dirichlet

Boundary Conditions: Neumann

One-sided derivatives

End of Lecture

Colocated: Hourglass instability

Colocated

Simple, economic > complex geos

Hourglass, VP uncoupled

Incompressible/Compressible

Viscous/Inviscid

Steady/Unsteady

Navier-Stokes equations

Special features of NSE

Vector 3d Non-Linear

Non-local (incompressible)

Complex geos

Mathematical structure

3 explicit: soft and matrix-free. But … Incompressibility holds only at steady-state, OK if steady-state is the only target

Fully Explicit (AC)

nw ned

swsw

n

ew

s

P E

N

W

S

NE

SE

SW SE

Vertex-centered Colocated

Nonlinearly-Implicit

Nonlinear iterations, k=0,1,…