IMPLICITIZATION OF THE MULTIFLUID SOLVER AND EMBEDDED FLUID STRUCTURE SOLVER Charbel Farhat, Arthur...
-
Upload
jasmine-thornton -
Category
Documents
-
view
219 -
download
0
Transcript of IMPLICITIZATION OF THE MULTIFLUID SOLVER AND EMBEDDED FLUID STRUCTURE SOLVER Charbel Farhat, Arthur...
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