incompressible flows sauro succi. incompressible flows
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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,…