IMPLICITIZATION OF THE MULTIFLUID SOLVER AND EMBEDDED FLUID

STRUCTURE SOLVER

Charbel Farhat, Arthur Rallu, Alex Main and Kevin Wang

Department of Aeronautics and AstronauticsDepartment of Mechanical Engineering

Institute for Computational and Mathematical EngineeringStanford UniversityStanford, CA 94305

OUTLINE

Implementation of implicit time-stepping for fluid-fluid interaction

Numerical results and timing for the fluid-fluid solverShock tube problemTurner Implosion

Numerical results and timing for the embedded

fluid-structure solver2D Imp mode 45

Implementation of implicit time-stepping for fluid-fluid interaction

Finite volume method with MUSCL (Roe’s solver)

12

12

Fj,j+1 = Fj+1/2 (nj,j+1) = (Fj + Fj+1 )- | F’ |j+1/2 (Wj+1 – Wj)

= Roe (Wj, Wj+1, gs, ps) (stiffened gas)

j j + 1

j + 1/2

Interface capturing via the level-set equation

COMPUTATIONAL FRAMEWORK

+ = 0@t

(rf)@

@x

@(ruf)(conservation form)

FVM with exact local Riemann solver for multi-phase flows

j j + 1j - 1 j + 1/2j - 1/2

Wjn

- Fj,j+1 = Roe (Wjn, W*

n, EOSj)

W*n

Fj+1,j = Roe (Wj+1n, W*

n, EOSj+1)

W*n

Wj+1n

FVM-ERS

C. Farhat, A. Rallu and S. Shankaran, "A Higher-Order Generalized Ghost Fluid Method for the Poor for the Three-Dimensional Two-Phase Flow Computation of Underwater Implosions", Journal of Computational Physics, Vol. 227, pp. 7674-7700 (2008)

- W*n and W*

n determined from the exact solution of local two-phase Riemann problems

Wave structure and Riemann problem

x

t rarefactioncontact discontinuity

shock

watergas

rL uL pL

rIL,pI,uI ,rIR

j j + 1j + 1/2

rR uR pRWnpj+1 Wn

pj

RL(pI; pL,rL) + RR(pI; pR,rR) + uR – uL = 0

uI = (uL + uR) + (RR(pI; pR,rR) -RL(pI; pL,rL))2 21 1

Exact solution of the analytical problem (Tait’s EOS)

- Newton’s method pI, rIL, rIR, uI

LOCAL RIEMANN SOLVER

Dt- Wjn+1 = Wj

n - (Fj,j+1 - Fj,j-1) (forward Euler)

Dx

GFMP with exact local Riemann solver

~

- Unpack Wn+1 using fn and solve the level-set equation to get fn+1~

- Pack Wpn+1 using fn+1 to get the updated solution Wn+1

j j + 1j - 1

j + 1/2j - 1/2

- If fjn fj+1

n > 0 thenFj,j+1 = Fj+1,j = Roe (Wj

n, Wj+1n, EOSj = EOSj+1)

If fjn fj+1

n < 0 thenFj,j+1 = Roe (Wj

n, WjRn(rIL, pI, uI), EOSj)

Fj+1,j = Roe (Wj+1n, W(j+1)R

n(rIR, pI, uI), EOSj+1)

FVM-ERS (EXPLICIT)

Dt- Wjn+1 = Wj

n - (Fj,j+1 - Fj,j-1) (backward Euler)

Dx

Implicit Extension of FVM-ERS method

~

- Unpack Wn+1 using fn and solve the level-set equation to get fn+1~

- Pack Wpn+1 using fn+1 to get the updated solution Wn+1

j j + 1j - 1

j + 1/2j - 1/2

- If fjn fj+1

n > 0 thenFj,j+1 = Fj+1,j = Roe (Wj

n+1, Wj+1n+1,EOSj = EOSj+1)

If fjn fj+1

n < 0 thenFj,j+1 = Roe (Wj

n+1, WjRn+1,EOSj)

Fj+1,j = Roe (Wj+1n+1, W(j+1)R

n+1, EOSj+1)

FVM-ERS (IMPLICIT)

IMPLICIT FLUID-FLUID

Backward Euler advancement requires the solution of a nonlinear equationUse Newton’s method, which requires Jacobians of the flux functions

dFj,j+1 dFj,j+1 dWjn dFj,j+1 dW*

n

dpj dWjn dpj dW*

n dpj

+=

dFj,j+1 dFj,j+1 dW*n

dpj+1 dW*n dpj+1

=

Need Jacobians of two-phase Riemann problems

STIFFENED GAS

Local two phase Riemann solver for stiffened gas (SG)- stiffened gas requires the solution of the equation

uL + FL(rL, pL; pI) = uIL

uIR = uR + FR(rR, pR; pI)

=

dFL dFL dFL

Taking the total differential yields derivatives of pI , uI

drL dpL dpI

duL +

drL dpL dpI

+ +

dFR dFR dFR drR dpR dpI

= duR +

drR dpR dpI

+ +

Derivatives of rIR , rIL then come from the Riemann invariants

Jacobians for Tait-Tait, SG-Tait follow the same derivation

OTHER EOS

Perfect Gas (PG) is a subset of SG (with = 0p )

Also support Tait EOS for compressible liquids

p = Arb + B

JWL EOS

Jones-Wilkins-Lee (JWL) equation of state for modeling explosive products of combustion (and in particular Trinitrotoluene — a.k.a. TNT)

where A, B, R1, R2, w and r0 are material constants

p = A(1 - )e-R1 + B(1 - )e-R2 + wrewr

R1r0

wr

R2r0

r0r

r0r

- Highly nonlinear function p(r,e)- Presence of exponentials

Solution of exact Riemann problem involves a system of two nonlinear equations

uL + FL(rL, pL; rIL) = uIL

uIR = uR + FR(rR, pR; rIR)

GL(rL, pL; rIL) = pIL

pIR = GR(rR, pR; rIR)=

=

- FL and GL depend on the nature of the interaction in the

phase modeled by the JWL EO shock algebraic equation rarefaction differential equation

JWL EOS

(1)

(2)

Rarefaction wave in a JWL medium

- Algebraic entropy (s) formula for the JWL EOS- No obvious algebraic Riemann invariants for the

JWL EOS- No analytical Jacobians of the invariants

either

x

t rarefaction

rR,uR ,pR

rIR,uIR ,pIR- The isentropic evolution in the rarefaction fan between two constant states is given by

complex Riemann problem

r

c(r,p)

+_dudr

=r +1w

p - Ae-R1 + Be-R2

r0r

r0r

= s

(k)

SG-JWL RIEMANN SOLVER

(1) (2)

JWL EOS

Riemann invariants are tabulated for the explicit time stepping scheme

For implicit time-stepping, where Jacobians are required, they are not tabulated; rather they are computed on-line by solving an ODE

Relatively cheap compared to other aspects of the simulation

Support both SG-JWL and JWL-JWL

TIME INTEGRATORS

We support two different time integratorsBackward EulerThree Point Backward Difference (3BDF)

Backward Euler estimates the time derivative at time

n+1 at node i byDt

Win+1 - Wi

ndWi

dt= (1)

The integration of the fluid equations at time step n+1

assumes that node i is of the same phase; thus there is no problem

~

3BDF

3BDF approximates the derivative at time step n+1 as

Dt

a0Win+1 -a1Wi

n + a2Win-

1

dWi

dt= (2)

But node i at time n-1 may be of a different phase

Because density can be discontinuous across a fluid

interface, Win-1 and Wi

n are not necessarily related in this case

~

3BDF

When node i has changed phase between time step n and n+1, replace Wi

n-1 with W*n-1

Where W*n-1 is the exact solution of the two phase

Riemann problem on the upstream side of the interface at node i at time step n-1

i-1 ii-2

n-1 n

i+1

W*n-1

LEVEL SET 3BDF

A similar issue arises when we use the 3BDF integrator on the level set

3BDF requires fn+1, fn , and fn-1

After reinitialization fn-1 no longer exists

Solution is to use a special integrator

Dt

2fin+1 - 2fi

nd fi

dt= -

1 dfin

2 dt

The final term can be estimated from the spatial

fluxes at time step n

LIMITATIONS

The fluid interface may cross no more than one cell per time step

AERO-F automatically ensures this is not violated by

reducing the time step as necessary

- Required to handle phase change

SHOCK TUBE PROBLEM

1D Shock tube with air to the left, water to the right.

Air modeled as a perfect gas ( = 1.4g ); water modeled as a stiffened gas ( = 4.4, = 6.0 g p x 108)

Simulation to t=1e-5 s in 3D AERO-F code

r = 50 (kg/m3)u = 0.0 (m/s)p = 105 (Pa)

r = 1000.0 (kg/m3)u = 0.0 (m/s)p = 109 (Pa)

Air Water

SHOCK TUBE RESULTS

SHOCK TUBE RESULTS

TURNER IMPLOSION

Implosion of a spherical air bubble

Air modeled as a perfect gas ( = 1.4g ); water modeled as a stiffened gas ( = 7.15, = g p 2.89 x 108 Pa)

780,000 grid points

Simulation to t=0.5 ms Airp=0.1 MPa Water

p=7 MPa

VALIDATION

Turner (2007): implosion of a glass sphere (D = 0.0762 m)

Air (P = 105 Pa)

Water (P = 6.996 MPa)

x

z

(0.5m, 0.5m)

(0.5m, -0.5m)

(0, 0) Sensor

TURNER RESULTS

Explicit (FE), CFL=0.5

Implicit (3BDF), CFL=100

TURNER RESULTS

TURNER RESULTS

TURNER TIMING

Method CPU time

Explicit (FE) 17867 s

Implicit (BE) 4130 s

Simulation performed on a Linux cluster using 168 processors

Speedup of 4.33

EMBEDDED FLUID-STRUCTURE

For embedded fluid structure, fluid-fluid Riemann

problem is replaced by a fluid structure Riemann problem

* could also be a shock

x

t rarefaction*

contact discontinuity

fluid 2fluid 1

i jMij

rR uR pR Wnj

x = x(t)

not involved

pI, rIR us

t+ = 0

w (w

)

F

x

w( ,0) x = W , if x ≥ 0

jn

u(x(t), t) = u (Mij) ∙ nG(Mij)

s

us = uR + R2(pI(2); pR , rR)

- Closed form Jacobians exist as well

(Fluid 2, shell) problem

x

t rarefaction*

contact discontinuity

fluid 2fluid 1

i jMij

rR uR pR Wnj

x = x(t)

not involved

pI, rIR us

ONE-SIDED RIEMANN PROBLEM

- Closed form algebraic solution of the problem exists

The flux across the face at Mij is then given by

FLUX COMPUTATION

Fji = Roe (us, pI(2), Wn

j , EOS(2), uji ) Fij = Roe (us, pI

(1), Wni ,

EOS(1), uij )

i j

G

Mijfluid 1 fluid 2

Dt- Wjn+1 = Wj

n - (Fn+1j,j+1

- Fn+1j,j-1) (backward Euler)

Dx

Implicit Extension of Embedded FSI method

~

EMBEDDED FSI (IMPLICIT)

- Update uncovered nodes to compute Wjn+1

Solve for Wjn+1 using Newton’s method

Requires Jacobians of fluid-structure Riemann problem

- Closed form solution exists for stiffened gas

3BDF FOR FSI

In this case, use W*n-1 as Wi

n-1

W*n-1 is the solution of the exact two phase

Riemann problem on the upstream side of the structure boundary at node i at time step n-1

The same difficulty exist when using 3BDF for embedded fluid-structure When node i has been uncovered, Wi

n-1 does not exist

i-1 ii-2

n-1 n

i+1

W*n-1

structure

2D Imp45

2D Implosion problem

air ( p = 14.5 psi )

water( p = 1500 psi)

Simplified IMP45 using a thin slice of the aluminum tube

Explicit simulation uses dt = 0.75 x 10-8 Implicit simulation uses dt = 3.0 x 10-6

IMP45 RESULTS

Pressure at a sensing node

IMP45 RESULTS

Pressure fields at t=0.4 ms

Clockwise from left: Explicit (RK2), Implicit (BDF), Implicit (BE)

IMP45 TIMING

Method CPU time

Explicit (FE) 12153 s

Implicit (BE) 882 s

Implicit (3BDF) 1115 s

Simulation performed on a Linux cluster using 64 processors

Speedup of 13.8, 10.9

Implicitization of fluid-fluid interaction in AEROF

SUMMARY

Equipment of the FSI solver in AERO-F with an implicit integrator

- Validation on shock tube and implosion problems

- Development of new scheme for three point backward difference integration

- Validation on 2D implosion problem

- Speedups of ~ 4-5

- Speedups of ~12