Stabilization techniques for pressure recovery applied to - - PowerPoint PPT Presentation

Stabilization techniques for pressure recovery applied to POD-Galerkin methods for the incompressible Navier-Stokes equations

• G. Stabile, G. Rozza

SISSA, International School for Advanced Studies, MathLab, Mathematics Area, Trieste, Italy

Quantification of Uncertainty Improving Efficiency and Technology QUIET 2017 – Trieste 18-21/07/2017

Framework and motivations

• In order to efficiently apply Uncertainty Quantification in computational fluid

dynamics problems one needs inexpensive computational models to solve the forward problem. In this direction the development of efficient and reliable reduced order models (ROMs) would be a great advantage.

• It is well known that Galerkin based ROMs of the incompressible Navier-Stokes

equations suffer from stability issues for what concern the pressure term. The considered system of PDEs consists in the unsteady parametrized incompressible Navier Stokes Equations.

            

∂u ∂t + (u · ∇)u − ∇ · ν∇u = −∇p

in Ω ∇ · u = 0 in Ω u = u(µ)

• n ∂Ω,in

u = 0

• n ∂Ω,0

(µ∇u − pI)n = 0

• n ∂Ω,out

(1) The offline stage is performed using a Finite Volume Method (OpenFOAM) while the projection and online stage are based on the in-house package ITHACA-FV.

1/ 3

• G. Stabile

Stabilization techniques applied to POD-Galerkin methods for the Navier–Stokes equations

Methods Reduced Order Modelling

Most of the problems require high dimensional parametrized simulations.

POD Reduced Basis Approximation of the fields Galerkin projection of the governing equation onto the reduced basis spaces Full order model: high dimensional system of PDEs: Solved with a Finite Volume technique U=[u(t 1),u(t 2),...,u(tn)] P=[p(t1), p(t2),..., p(t n)] Generation of the reduced Basis spaces V=[ϕ1,ϕ2,...,ϕNu] Q=[χ1,χ 2,...,χ N p] u≈ur=∑i=1

Nu ai ϕi

p≈ pr=∑i=1

N p bi χi

Poisson equation for pressure Enrichment of the velocity space to ensure the Inf-Sup condition Lower Dimensional System Of ODEs M ˙ a=aB+b K

T+a' C a

K a=0

OFFLINE STAGE ONLINE STAGE

• STAB. METHODS

Stability issues

2/ 3

• G. Stabile

Stabilization techniques applied to POD-Galerkin methods for the Navier–Stokes equations

Conclusions References

[1] G. Stabile, S. Hijazi, A. Mola, S. Lorenzi, and G. Rozza. POD-Galerkin reduced order methods for CFD using Finite Volume Discretisation: vortex shedding around a circular cylinder. In Press, 2017 [2] G. Stabile and G. Rozza, Stabilized Reduced order POD-Galerkin techniques for finite volume approximation of the parametrized Navier–Stokes equations, submitted, 2017. [3] F. Ballarin, A. Manzoni, A. Quarteroni, and G. Rozza, Supremizer stabilization of POD-Galerkin approximation of parametrized steady incompressible Navier–Stokes equations, International Journal for Numerical Methods in Engineering, vol. 102, no. 5, pp. 1136–1161, 2015.

3/ 3

• G. Stabile

Stabilization techniques applied to POD-Galerkin methods for the Navier–Stokes equations