FLIP: A LOW-DISSIPATION, PARTICLE-IN-CELL METHOD FOR FLUID FLOW
←
→
Page content transcription
If your browser does not render page correctly, please read the page content below
Computer Physics Communications 48 (1988) 25—38 25
North-Holland, Amsterdam
FLIP: A LOW-DISSIPATION, PARTICLE-IN-CELL METhOD FOR FLUID FLOW
J.U. BRACKBILL, D.B. KOTHE and H.M. RUPPEL
Los A/amos National Laboratory, Los A/amos, NM 87545, USA
The FLIP (Fluid-Implicit-Particle) method uses fully Lagrangian particles to eliminate convective transport, the largest
source of computational diffusion in calculations of fluid flow. FLIP is an adaptation to fluids of the implicit moment method
for simulating plasmas, in which particles carry everything necessary to describe the fluid. Using the particle data, Lagrangian
moment equations are solved on a grid. The solutions are then used to advance the particle variables from time step to time
step. An adaptive grid and implicit time differencing extend the method to multiple time and space scale flows. Aspects of
FLIP’s properties are illustrated by modeling of a confined eddy, a Rayleigh-Taylor, an unstable subsonic stream, and a
supersonic jet. The results demonstrate FLIP’s instability applicability to hydrodynamic stability problems where low
dissipation is crucial to correct modeling.
1. Introduction mass weighted transport of momentum and en-
ergy, which is usually used, is diffusive. The as-
The particle-in-cell method (PlC) was once the sigmnent of grid quantities to the particles and
only method able to describe highly distorted flow back causes diffusion in momentum and energy.
in two or three dimensions, for which it received “Full-particle” methods like FLIP [8] and PAL
considerable attention [1]. With time, its faults [7] preserve the ability of “classical” PlC to re-
became evident. PlC is noisy, and it has more solve contact discontmuities, but eliminate the
numerical viscosity and heat conduction than are major soujce of numerical diffusion by using a full
acceptable by today’s standards [2]. Were it not Lagrangian representation of the fluid. FLIP also
for its usefulness in simulating plasmas [3], PlC incorporates a variable particle size, adaptive mesh,
would be mainly of historical interest, and implicit solution algorithm, to extend its ap-
The problems with PlC are clearly defined. To plicability to problems with multiple time and
make PlC as accurate as competing methods, its space scales.
viscosity and heat, conduction must be reduced. The properties of FLIP when modeling fluid
The solutions follow one of two different strate- flows in two dimensions are examined here. A
gies for reducing numerical diffusion, improved comparison of FLIP with “classical” PlC, a devel-
“classical” formulations [2,4] and “full-particle” opment of the correspondence between grid and
PlC [5—8].The “classical” PlC method is a par- particle solutions, and an analysis of the ringing
tially Lagrangian description of the fluid. To each instability are reviewed. Several examples, includ-
particle is attributed a mass and a position. The ing calculations of a confined eddy, a
“full-particle” PlC is a fully Lagrangian descrip- Rayleigh—Taylor instability, a subsonic stream, a
tion of the fluid. To each particle is attributed all supersonic jet, and an ion beam target are given.
of the properties of the fluid, including momen-
tum and energy.
“classical” ~icis able to resolve contact dis- 2. The particle-in-cell method
continuities because of its Lagrangian representa-
tion of the mass. As the particles stream from cell In PlC methods for fluid flows, the
to cell during a calculation, they transport mass Navier—Stokes equations in Lagrangian form are
from cell to cell without diffusion. However, the solved on a grid, where they are approximated by
OOlO-4655/88/$03.50 © Elsevier Science Publishers B.V.
(North-Holland Physics Publishing Division)26 J. U. Brackbil/ et al. / FLIP: PlC methodfor fluidflow
finite differences and solved. The equations are arily describe the fluid to model convection. Each
mass, cycle, all of the particle data, except mass and
position, is replaced by data from the grid solu-
dp/dt + PV U— ~ (1) tion of the Lagrangian flow equations. The par-
momentum, tide velocity in Harlow’s PlC [1], for example, is
given by
p dU/dt+ vp— c7(A+2~)v. U+ v• T=O,
(2) ~ (6)
71=~(au1/ax1+aLyax1} V
where U~is the fluid velocity at grid vertex v, and
and energy, S,, is the nearest-grid-point (NGP) assignment
2+ function. The particles are then displaced using an
p dI/dt +pV• U+ A(c’. U) 1.iT• T= 0, (3) interpolated velocity from the grid,
where p, U, I and p are the density, velocity, dx
specific internal energy and pressure, and ~uand A ~Tj7~= U0S1(x~ xv),
~ — (7)
are the shear and bulk viscosities.
In both “classical” and “full-particle” PlC, where S1 is a linear assignment function. The grid
convection is modeled by particles moving through data are then regenerated by projection using eq.
the grid. The data on the grid are regenerated (4) and similar equations for the other fluid varia-
from the particle data when there is relative mo- bles.
tion between the fluid and the grid. In practice, In “full-particle” PlC, the particle data de-
the regeneration requires projecting the particle scribe the fluid completely from cycle to cycle, not
data on to the grid points by accumulating a just during convection. Each cycle, the particle
weighted sum of particle contributions at each data are updated from the grid solution, not re-
grid point. The grid density, for example, is calcu- placed. In FLIP, for example, the particle acceler-
lated from ation is interpolated from the grid data,
~ ~ du~ dU
(4) ã=~~Si(xvxp).
V (8)
p
This difference between “classical” and “full-par-
The particle mass, ~ is shared among the cells, tide” PlC is equivalent to the difference between
c, with centroids, x~,and volumes, J’~,according a partially and a fully Lagrangian description of
to the value of an assignment function, S,~,,which the fluid. The partially Lagrangian description
depends upon the distance between the particle given by the “classical” PlC particles results in
and the cell center. The assignment function has significant diffusion of all non-Lagrangian varia-
bounded support so that it is zero for distances bles.
greater than some value h (usually the mesh spac-
ing). It is a monotone decreasing function of its
argument, and it is normalized to one for any ,~, 3. Diffusion in “classical” PlC
~ S(x~ x~)= ~S(x~
— — x~)= 1, (5) In “classical” PlC, grid variables, such as en-
C V ergy and momentum, are transported from cell to
where S represents any one of the often used cell by the particles as they move through the grid.
nearest-grid-point, linear, or quadratic assignment To transport energy, for example, each particle is
functions [22], which are described in refs. assigned a fraction of the cell energy in proportion
[3,9,10,22]. to its mass before it is moved,
“Classical” and “full-particle” PlC differ in
whether the particle data is preserved from cycle e = m ~
E
—~-s(x0 x— )
to cycle. In “classical” PlC, the particles tempor- p p M~ P C (9)J. U. Brackbill et aL / FLIP: PlC methodforfluid flow 27
where the cell mass, M~,is equal to ~J’~ and the stationary particles cause no change in the solu-
cell internal energy, E~,is equal to ~ (The tion on the grid just as in the original PlC with
superscript 0 denotes data at the beginning of the NGP [1], but with linear or quadratic interpola-
time step.) The particles are then displaced to xi,, tion as well. With their methods and higher order
where the superscript 1 denotes data at the end of interpolation, energy and momentum transport is
the time step, and a new cell energy is computed, a continuous function of particle displacement
E~= e’,, s( xp — x~) and
level the noise of thetooriginal
comparable PlCfinite
ordinary is reduced to a
difference
p methods.
/ i \ ,~ ~ In a sense, EPIC [12] is yet another way to
= ~ >Jm~,Si~x~,
x~’)SI~x~xe).
— — (10) improve “classical” PlC. In EPIC the diffusion,
C C ~° which can be calculated from eq. (10), is canceled
Even when the particles are not displaced, the by an anti-diffusion step (cf. Eastwood, these pro-
new cell energy will not equal the old [2]. Consider ceedings).
the simple example of evenly spaced, equal mass
particles on an evenly spaced grid, x,~.1 = x, + Lix,
in one dimension.
0yields,Fourier transforming eq. (10) 4. “Full particle” PlC
with
E1 — x~,
2 0
= x~ The success of plasma simulation, which is a
k— Sk Ek. ~ ~ “full particle” PlC, has influenced Marder (GAP)
With the commonly used spline interpolation [5], Tajima et al. [6], Morse et al. (PAL) [7], and
functions [22], for example, the Fourier transform Brackbil and Ruppel (FLIP) [8] to try similar
is given by [3], methods for fluid flows. The advantage of the
“full-particle” representation is the elimination of
sin( k i~X/2) “~ 1 numerical viscosity and thermal conduction. By
S~(k)= k i~x/2 (12) making momentum and energy Lagrangian varia-
bles, diffusion of these variables is eliminated just
(Nearest-grid-point (NGP) interpolation cone- as mass diffusion is eliminated in “classical” PlC
sponds to n equal to 0, linear interpolation to n by making mass a Lagrangian variable.
equal to 1, and quadratic interpolation to n equal “Full particle” PlC obviously costs more, be-
to 2.) For k equal to zero, S~(k)is equal to one cause for each particle a position, velocity, mass
and energy is conserved. For all k greater than and energy must be stored. (The extra cost is not
zero, S~ ( k) is positive but less than one. Conse- burdensome now, because increased computer
quently, interpolating energy from the grid to the memory and hardware-indirect-addressing are now
particles and back is diffusive. In fact, using higher available.) More importantly, since each particle
order splines actually increases the diffusion. has its own velocity, interpenetration across
Harlow avoids the zeroth order error in “classi- material interfaces or multistreaming can occur
cal” PlC by using NGP interpolation [1]. With just as in the smoothed particle hydrodynamics
NGP, only those particles residing in cell c con- (SPH) method [9].
tribute to the energy in cell c, and there is no To reduce multistreaming, Tajima et al. [6]
spatial averaging. However, energy is transported calculate the particle displacement from the par-
discontinuously as particles cross cell boundaries, tide velocity, but introduce a Krook collison op-
contributing to noise and diffusion in the results. erator to diminish the relative velocity between
Nishiguchi and Yabe [2], and Clark [4] avoid neighboring particles. This affects the results simi-
the zeroth order error in “classical” PlC in a larly to a viscosity. To eliminate multistreaming,
different way. They use higher order interpolation PAL and FLIP calculate the particle displace-
only for the changes in momentum and energy ment, as in “classical” PlC, by interpolating from
due to transport. Their formulations require that the grid velocity. This eliminates multistreaming28 J. U. Brackbill eta!. / FLIP: PlC methodforfluidflow
completely. (PAL introduces some dissipation to backward Euler time differencing, even angular
suppress the ringing instability [10], which FLIP momentum is conserved by the difference equa-
does not [8].) tions [11].The details are given in ref. [8], to which
some improvements have since been made. Im-
plicit equations are formulated similarly to the
5. The FLIP algorithm method of Casulli and Greenspan [13] using ICCG
(incomplete Cholesky decomposition, conjugate
The FLIP (Fluid-Implicit Particle) algorithm gradient method), with substantial improvements
uses Lagrangian particles to represent the fluid, in speed and reliability.
and a rezonable Lagrangian grid of quadrilateral
zones to calculate the interactions among the par- 5.2. Consistent particle and grid solutions
tides. Variables are assigned to the grid using the
standard von Neumann, Richtmyer staggering (see FLIP is formulated to given consistency be-
Crowley, these proceedings). In its mixture of a tween the solution of the Lagrangian equations of
particle representation of the fluid and a grid for motion on the grid, and the equations of motion
the solution of implicit moment equations, it is of the particles. Consistency is achieved, in part,
similar to the implicit moment method in plasma because the normalization of the assignment func-
simulation [23]. tion given by eq. (5) is dependent on the grid
placement. Furthermore, the particle equations of
5.1. The Lagrangian calculation motion, eqs. (7) and (8), are consistent with the
Lagrangian solution on the grid, because the assign-
Using the interpolation functions discussed ment function, S. is a Lagrangian invariant,
above, the specific internal energy, vertex mass dS dt 0 — 16
and center of mass velocity at the grid points are —
given by, To achieve this, S is defined in terms of natural
coordinates, (~,~) that are calculated by mapping
= ~e~S2(x~ — x~)/p~J’~, (13) each of FLIP’s quadrilateral cells on to a unit
square. With this choice, the grid defines the par-
MV = >m~S1(x~ x0), — (14) tide size. As the grid and particles move, the value
p of the assignment function at a point moving with
/ i ~ the fluid is constant.
U=J~muS1I~x—x)/M.
p p p v V ~15) At each vertex, the natural coordinates assume
integer values, (i, f) which are obviously constant
Linear interpolation is used for vertex-centered as the mesh moves. Elsewhere, the mapping be-
variables, such as position, velocity and mass, to tween physical and natural coordinates is given by
be consistent with the quadrilateral cells of the bilinear interpolation, which can be written,
computation grid. Quadratic interpolation is used —
for cell-centered variables, such as density and Xp — ~ — 1, 7J —J
energy, to give smoother solutions and to suppress
the ringing instability, which is discussed in sec- — S ~ — ~ S 1— J 17
tion 6.
data forBy
theevaluating eqs.Lagrangian
solution of (4) and (13)—(15),
equations the
of — 1~ ~1 i’~
motion on the grid are assembled. The natural coordinates of a particle, whose physi-
The solution of the Lagrangian equations of cal coordinates are xi,, are defined by inverting
motion on the grid, eqs. (1)—(3), can be approxi- this mapping. Because S is a Lagrangian in-
mated in many ways. In FLIP, the equations are variant, the particle’s natural coordinates are con-
formulated using a finite volume method, which stant during the Lagrangian phase, and the par-
conserves mass, momentum and energy. With tide displacement, eq. (7), and acceleration, eq.J. U. Brackbill etat / FLIP: PlC methodforfluidflow 29
(8), are calculated by interpolating from the grid variation among the particles is poorly repre-
values. The correspondence between particle and sented by a linear interpolation function. In this
grid solutions through the Lagrangian phase is sense, it is a measure of how accurately the grid
perfect for linear functions of the dependent vari- resolves the flow gradients. It is this term that is
ables. described as a kinetic pressure in ref. [8]. There it
From eq. (5) and eqs. (13)—(17), one can show was noted that it is large where large changes in
that the total mass, momentum, angular momen- the flow direction or velocity occur.
turn and internal energy for the particles and the The variance of the particle velocity about the
grid are equal, mean, the first term on the right side of eq. (23),
— does not change because of accelerations calcu-
— ~MV, (18) lated on the grid, which have no effect on the
P V relative particle velocity,
= ~MVUV, (19) dv du dU
p V aEV’!~v =0 ~‘25
dt dt ‘—‘dt V5~ ‘
x ui,) = ~ x lx,), (20) V
P V Solving the equations of motion on the grid affects
= ~ ~ (21) only those length scales that the grid resolves. The
p C sub-grid scale information that the particles carry
is unaffected, and can be retrieved by refining the
However, the kinetic energy is not the same for
zomng.
the particles and the gnd. If one defines the . . .
As is required for nonlinear stabihty, the FLIP
particle velocity relative to the mean velocity on . . . .
the grt‘d by formulation
velocity at a is
vertex
dissipative.
during aWhere
time step
the ischange the
in ~
VP = ui,, — ~ IJVS,,J,, S~ = S(x~ xe),
— (22) change in the particle kinetic energy is always
V different from the change in the grid kinetic en-
ergy by the amount
one can show that the difference between particle
and grid kinetic energy is given by
= — ~ T,~~ (26)
~[~mpup2_~MpV2I VV’ .
P V This error is negative defimte, particle by particle,
and order ~at2. When the loss in kinetic energy is
= ~ {m~v~ — ~ u. i~p~u~} , (23) added to the particle energy for each particle,
V,V’ energy is conserved exactly while marntaimng
positive particle internal energy.
where the transfer rnatnx is defined by,
= ~ ~ — (24) 5.3. Adaptive zoning
The transfer matrix is symmetric and diagonally The final step in the computation cycle requires
dominant and, therefore, has positive eigenvalues defining the grid for the next cycle. During this
[8]. Thus, both terms on the right side of eq. (24) step, the particles are stationary in physical space,
are positive definite and the difference in the but their natural coordinates must be recomputed.
kinetic energy of the particles and the grid can The grid may be completely redefined from one
have either sign. However, if one estimates the cycle to the next. It is only required that the grid
relative size of the two terms, the second term is map on to a logical rectangle. In FLIP, the num-
the result of the averaging one does to calculate ber of grid points is fixed, but the grid points may
the grid velocities and is never larger. The first be moved about as needed. Presently, one may
term will be relatively large when the velocity choose to use an Eulerian, fixed grid, a Lagrangian30 J. U Brackbill et at / FLIP: PlC methodfor fluidflow
grid (for a Lagrangian calculation, the particles To minimize I, one must solve Euler equations. In
are superfluous), or an adaptive grid. The adaptive the solution, the weight function, w, controls the
grid is generated by solving a variational problem, mesh spacing. Where w is larger, the grid points
in which one minimizes a functional. The func- are closer together. If w is an appropriate function
tional may be complex allowing one to control of the data, the grid will adapt to provide in-
grid smoothness, orthogonality, spacing and orien- creased resolution where it is needed by moving
tation [24,25]; or it may be simple, giving control grid points. For example, if w is proportional to
only over spacing, such as the functional, the current density in a calculation of magnetohy-
F drodynamic flow, the grid will cluster to resolve
I = f
v
2
~“E+
W
~
21
dv. (27) steep gradients in the magnetic field intensity [27].
Adaptive gridding can be useful in PlC calcula-
4 , . , - . .
Fig. 1. The adaptive grid plot is superimposed on the particles representing a confined eddy. The eddy, which is rotating
counterclockwise, advances one-half rotation period each frame. By the lower-right frame, it has rotated twice. A Kelvin—Helmholtz
instability causes secondary eddies to form, which move and coalesce. In these, the higher kinetic pressure causes the grid points to
cluster.J.U. Brackbill et al. / FLIP: PlC methodfor fluidflow 31
tions. For example, when w is proportional to the more particles per zone, and the adaptive mesh
number of particles per zone, where the particles cannot give the large gains in accuracy observed
are sparse the zones will be larger. This extends with finite difference methods [24].
the range of densities that can be represented by Adaptive gridding has been useful in resolving
making the particle size larger in regions of low material interfaces and in responding to changes
density. When w is proportional to the kinetic in scale that may occur when shocks develop.
pressure, eq. (23), the resolution is automatically Examples of adaptively zoned FLIP calculations
increased where the grid does not resolve the flow are given in refs. [8,14].
gradients. The results of this prescription are
shown in fig. 1, where the particles in a confined,
low-speed eddy are shown superimposed on the 6. The ringing instability
grid at intervals of one-half eddy rotation. The
calculation is performed on a 24 x 24 zone grid With four to sixteen particles per cell in a
with thirty-six particles per zone. This eddy prob- typical PlC calculation, there are many more de-
lem, which is described in more detail in ref. [11], grees of freedom for the particles than there are
is subject to a Kelvin—Helmholtz instability. The for the grid. Thus, the solution of the particle
instability causes secondary eddies to form. equations of motion is underdetermined by the
The resolution required to resolve the flow solution of finite difference equations on the grid.
changes in time, and, consequently, the grid The many different particle solutions which look
changes from frame to frame. The grid is initially the same when projected on to the grid are called
clustered in the boundary layer between the eddy aliases. The aliases interact nonlinearly to drive
and the background fluid. Later, the grid is clus- the well-known finite grid instability in plasma
tered in the secondary eddies, where the gradients simulation [3] and the ringing instability in PlC
in the velocity become steep and the kinetic pres- fluid calculations [10].
sure becomes relatively large. (The kinetic pres- While the assignment function for the vertex
sure is less than one per cent of the fluid pressure.) centered quantities, such as position and velocity,
The corresponding results on an Eulerian grid is determined by the shape of the computation
with the same number of zones are different, but cells, the choice of assignment function for the
the results with the same number of particles on a cell-centered quantities, such as density and en-
mesh with four times as many cells (48 x 48 zones) ergy, is unconstrained. Thus, one may use higher
differ only in detail from those in fig. 1. The order assignment functions to reduce the growth
instability growth rate and resulting large scale rate for the ringing instability [10]. Quadratic in-
structures are similar. The principal difference is terpolation is currently used in FLIP for cell-
that the cost of the 48 x 48 zone calculation is centered variables. There is a disadvantage, how-
twice that for the adaptively zoned calculation. ever, because the ability to resolve gradients is
(The time step for the calculation is the same, as diminished using higher order interpolation [9].
allowed by the implicit formulation, but the 48 X Dispersion analysis of the ringing instability in
48 zone calculation costs twice as much with four PlC for small time steps, uniform grid spacing,
times as many zones, since the grid solutions ~x and uniform flow, U0, yields the dispersion
require half the total computation time.) In three relation,
dimensions, where an adaptive grid would require 2
a tenth as many zones as a fixed grid calculation, 1 — Si(kq)S2(kq)k~a =0,
~ (28)
the savings would be even greater.
There are constraints on adaptivity in pi~ ~ — kqUo]2
—
calculations. In low-speed-flow problems, the where a is the sound speed, ~ the frequency, k
number of particles per zone must be sufficient to the wave number, and kq the alias wave number,
give smoothly varying pressures and densities, even
in the smallest zones. As a result, one must use kq = k + q(2.rr/i~x). (29)32 J. U Brackbill et at / FLIP: PlC methodforfluidflow
The imaginary frequency, labelled gamma in fig. yet for FLIP, but large time steps are observed to
2, corresponds to exponential growth for Mach be very effective in suppressing the instability if
numbers, M equal to U0/a, less than 0.4. The other time step constraints allow them.
minimum growth time is about ten oscillation
periods (With nearest grid point interpolation re-
placing S1 and S2 in eq. (28), the maximum 7. Test problems
unstable M is one, and the minimum growth time
little more than one oscillation period.)
When the total energy is a constant of the With a new method like FLIP, confidence is
motion, the ringing instability is bounded nonlin- accumulated over time by applying it to a wide
early in PlC fluid calculations to relatively low range of problems. From the results, one begins to
amplitudes. The only evidence of the instability is understand its properties, its strengths and its
a disordering of the ordinarily regular patterns of weaknesses, in ways that complement analysis.
particles, with a consequent spatial modulation of FLIP has been applied to many problems, some of
the density or energy on the grid. In fig. 3, the which are: (The reference number before the slash
particles representing the dense fluid in a calcula- is a reference for FLIP results, the reference after
tion of the Rayleigh—Taylor instability [14] are the slash for a description of the experiment, or
plotted. With M equal to 0.001, the particles other computed results.)
clump into patterns that are characteristic of the 1. Interaction of a shock and a thin foil [8]
ringing instability. With quadratic assignment and 2. High speed flow over a step [8/15]
implicit differencing, the effect of the instability is 3. Interaction of a shock in air and a helium
usually limited to unimportant effects like the one bubble [10/16]
shown. Including the effect of a very large time 4. Plane shock (Sod’s problem) [14/17]
step in the dispersion analysis has not been done 5. Noh’s problem [14/18]
6. Rayleigh—Taylor instability [14/19]
7. Imploding sphere and ellipse (unpubl.)
8. Confined eddy [il/Monaghan, these pro-
ceedings]
9. Astrophysical jet [unpubl./20]
One can make several observations about the
performance of FLIP from the results of these
tests. FlIP gives the correct jumps in mass,
momentum and energy across the shock, but the
jumps are several cells thick and not as sharp as
_____ can be for
method, achieved
example with
[15].the
Because
piecewise
FLIP parabolic
does not
have numerical diffusion, it is important to choose
the correct artificial viscosity. The grid Reynolds
/ ~ number, Reg = I u ~x/X, should be equal to one
/ in the shock front to give the minimum shock
N u thickness
The real
without
strength
overshoot.
of FLIP is its ability Both
to model
hydrodynamically unstable interfaces. the
~ no conservation of angular momentum and the ab-
sence of dissipation appear to be essential compo-
Fig. 2. The ringing instability in FLIP occurs for flow Mach nents of this capability. An illustration of the way
numbers less than 0.4. A normalized growth rate of 0.1 corre-
sponds to a growth time of 10 oscillations at the normal FLIP treats shear layers is given by a model
frequency. problem.J. U. Brackbill eta!. / FLIP: PlC methodforfluidflow 33
(A) (B) (C)
Fig. 3. Particles are plotted depicting the dense fluid in a Rayleigh_Taylor instability. In (c), the effect of the ringing instability is
apparent.
___ ___ ~~~~1 ~
(A) (B)
~ fl P (C)
T~1i~ (D)
Fig. 4. A subsonic stream is modeled. The velocity is depicted in (a), the initial particle placement in the background fluid in (b), the
same particles plus injected particles at t = 5 s/a on a 10 x 30 zone mesh in (c), and on a 20 X 60 zone mesh in (d).34 J. U Brackbil! et at / FLIP: PlC methodforfluid flow
7.1. Subsonic stream be seen in the relative displacement of the par-
ticles, which reflects the velocity profile. (Because
As shown in fig. 4a, a stream of fluid enters an the inflow velocity is constant across the aperture
aperture at the bottom of the domain, flows be- at the bottom boundary and varying linearly across
tween two flat plates, and exits at the top. The the stream just inside the boundary, the beam is
stream and the background fluid through which it perturbed.) When the mesh spacing is halved, fig.
flows have the same density and pressure. The 4d, the boundary layer thickness is halved as well,
distance between the plates is five times the stream and the increased gradients in the velocity drive
thickness, s, and the length of the stream is 15 s. the stream unstable.
The Mach number for the flow is 1/Viö~,and the The subsequent development of the instability
artificial viscosity is zero. The particles in the is displayed in fig. 5. With 36 particles per cell and
upper portion of the domain at the initial time are a 10 x 30 mesh, the stream is gently meandering
shown in fig. 4b. The same particles, as well as at t = 10 s/a in fig. 5a. At the same time with a
those that have entered the bottom meanwhile, are 20 x 60 mesh and 9 particles per cell so that the
shown at t = 5 s/a in fig. 4c. The dynamics of total number of particles is the same, fig. Sb, the
these particles is calculated on a mesh with 10 X 30 stream has become turbulent. Evidently, the insta-
zones, L~x= ~ y = s/ 2, and 36 particles per zone. bility can manifest itself only when the gradients
With this resolution, there are just two zones of the flow are resolved by the mesh. Doubling the
across the stream. Because the velocity varies un- grid spacing again (40 x 120 zones) while keeping
early in space, the boundary layer includes the the number of particles per zone constant at 9
entire stream and one zone on either side as can yields the results shown in fig. 5c. The larger
r
HI 4
hI ...
I jiil)1D)~ .?‘~ I’
I Hi... ~
1~~~
‘ ~ I ~
~
‘~1 ‘ 4. ____
(A) (B) (C)
Fig. 5. Background and injected particles as in fig. 4 are plotted at t 10 s/a for a lOx 30 zone mesh, (a), a 20 X 60 zone mesh, (b),
and a 40 x 120 zone mesh, (c). (a) and (b) have the same number of particles. (b) and (c) have 9 particles per cell.J. U Brackbi!l et a!. / FLIP: PlC methodfor fluidflow 35
features of the flow are similar to those in fig. Sb, background density. The jet speed is 30 times the
but more rotation and mixing has Occurred. Be- ambient sound speed, a.
cause there is no viscosity there is no physical In fig. 6, the injected particles are plotted at
length scale, and the results on the scale of the intervals of 2R/a. The interface is very unstable,
grid will not converge as the mesh is refined. On and produces eddies on all scales resolved by the
longer scales, the results should converge, mesh, as is confirmed by velocity vector plots, not
shown. The size of the eddies increases with dis-
tance from the head of the jet, as does the width
7.2. Supersonic jet of the turbulent flow region. The flow of the jet is
unsteady. This is especially evident in figs. 6c and
The unstable stream is a gentle breeze corn- d, where disruptions in the jet itself can be seen
pared with the supersonic jet shown in fig. 6. (The The large density discontinuity between the jet
jet shown is one of many studied by Winider and and the background gas presents some computa-
collaborators [20] in modeling astrophysical jets.) tional difficulties. For example, it is difficult to
The jet is injected through an aperture in the have neither too little nor too much viscosity in
bottom boundary and allowed to flow freely mixed cells. To deal with this problem, the layer
through the right and top boundaries. The left of mixed cells is treated as an internal boundary.
boundary is an axis of symmetry. The domain has In mixed cells, where particles of two or more
dimensions 6R by 16R, where R is the radius of “colors” reside, the particle contributions are given
the jet, and is resolved by a 30 X 80 zone mesh, only to surrounding cells that are mixed, or con-
with 5 zones in the jet. tam particles of the same color. This procedure
The jet is in pressure equilibrium with the yields, typically, a band of mixed cells only one
background gas, but with only one percent of the cell thick separating two materials.
(A) (B) (C) (D)
Fig. 6. Particles in a supersonic jet are plotted at intervals of 2 R /a. The particles are injected at the left-bottom. The left boundary is
an axis of symmetry.36 J. U. Brackbi// et a!. / FLIP: PlC method for fluid f/ow
I •1
I
Fig. 7. Results from a calculation of the response of a target to a heavy ion beam are plotted. Green, blue and
red particles represent lead, lithium and deuterium. respectively. The target is shown at 10. 17, 20. 22, 26 and 29
ns in frames (a) through (1).J. U. Brackbill et at / FLIP: PlC methodfor fluidflow 37
7.3. Ion-beam driven hydrodynamics 8. Current research
FLIP can be applied to more complex prob- Space does not permit more than a brief discus-
lems as well as to the simple fluid flow problems sion of the adaptive zoning capability of FLIP.
described above. For example, one of us (DBK) is Adaptive zoning extends the dynamic range of
presently using FLIP to study ion-beam-driven FLIP, not only in spatial length scales but also in
targets [14]. In these, bismuth ions that have been density variation. It also improves the resolution
accelerated to 10 GeV deposit 4.25 Mi of energy of shocks and other discoiitinuities. Current re-
in 30 ns deep within a target. There, the ion search emphasizes understanding the strategies one
energy is converted to hydrodynamic motion that should use with FLIP to adapt the grid.
compresses and heats a portion of the target FLIP is presently being extended to multi-
material to the densities and temperatures re- material flow with slip interfaces and material
quired for fusion. properties, such as strength, and to magnetohy-
To model such targets, which are very far from drodynamics. Many of the ideas in FLIP are being
thermodynamic equilibrium, the physical model in used in developing a new, adaptively zoned, plasma
FLIP must be extended beyond the Navier—Stokes simulation code in collaboration with D.W. Fors-
equations. Some extensions require cell-based lund. A version of FLIP for three-dimensional
calculations, and are simple to add. For example, flows is an attractive, and apparently feasible,
to model material properties realistically one uses project for the future.
complex equations of state like SESAME [21].
One also must calculate the equilibration of tern- References
peratures between electrons and ions, and the
ionization state of the material. Other extensions [1] F.H. Harlow, Meth. Comput. Phys. 3 (1963) 319.
require calculating gradients, like ray-tracing to [2] A. Nishiguchi and T. Yabe, J. Comput. Phys. 52 (1983)
model ion beam propagation, and energy trans- 390.
port equations to model electron thermal conduc- [3) C.K. Birdsall and A.B. Langdon, Plasma Physics via Com-
puter Simulation (McGraw-Hill, New York, 1985).
tion and convection. Luckily, the underlying grid- [4] R.A. Clark, Los Alamos National Laboratory Report No.
based data structure in FLIP resembles that of
. LA-UR 79-1947, 1979 (unpublished).
many complex physics codes, making it easy to [5] B.M. Marder, Math. of Comput. 29 (1975) 473.
borrow and steal the needed physical models. [6] T. Tajima, J.N. Lebouef and J.M. Dawson, J. Comput.
The results of a calculation of the dynamics of Phys. 38 (1980) 237.
[7) R.L. McCrory, R.L. Morse and K.A. Taggart, Nucl. Sci.
a planar, HIBALL (Heavy Ion Beam And Liquid Eng. 64(1977)163.
Lithium) target are shown in fig. 7. The colors [8] J.U. Brackbill and H.M. Ruppel, J. Comput. Phys. 65
label the target materials, with green for lead, blue (1986) 314.
for lithium, and red for deuterium. The ions, [9] JJ. Monaghan, Comput. Phys. Rep. 3 (1985) 71.
which enter the problem from above, are focused [10] J.U. Brackbill, The Ringing Instability in Particle-in-Cell
Calculations of Low-Speed Flow, J. Comput. Phys. (to be
at infinity but deposit their energy in the lithium. publ.).
The fluence of ions varies sinusoidally by five [11] J.U. Brackbill, Comput. Phys. Commun. 47 (1987) 1.
percent across the target, with several wavelengths [12] J.W. Eastwood, Comput. Phys. Commun. 44 (1987) 73.
included in the domain of the calculation. Where [13) V. Casulli and D. Greenspan, Intern. J. Numer. Meth.
the intensity is highest, the lithium is pushed [14) Fluids 4 (1984)
D.B. Kothe, 1001.
Hydrodynamic Stability and Symmetry of
aside. There, the beam penetrates more deeply and Ion-Beam-Driven Planar Targets, Purdue University the-
causes a highly corrugated shock to form. The sis (1987) unpubl.
highly distorted flow that results is easily modeled [15] P. Woodward and P. Colella, J. Comput. Phys. 54 (1984)
by the particles. 115.
[16] J.-F. Haas and B. Sturtevant, Interaction of Weak Shock
Waves with Cylindrical and Spherical Gas Inhomogenei-
ties, Graduate Aeronautiedl Laboratory, California In-
stitute of Technology, 1986 (unpubL).38 J. U. Brackbill et at / FLIP: PlC methodforfluidflow
[17] GA. Sod, J. Comput. Phys. 27 (1978) 1. [23] R.J. Mason, in: Multiple Time Scales, eds. J.U. Brackbill
[18] W.M. Noh, J. Comput. Phys. 72 (1987) 78. and B.I. Cohen (Academic Press, Orlando, 1985) pp.
[19] BJ. Daly, Phys. Fluids 10 (1967) 297. 233—270.
W.P. Crowley, An Empirical Theory for Large Amplitude J.U. Brackbill, ibid. pp. 271—3 10.
Rayleigh—Taylor Instability, UCRL-72650, Lawrence [24] J.U. Brackbill and J.S. Saltzman, J. Comput. Phys. 46
Livermore National Laboratory (1970). (1982) 342.
[20] M.L. Norman, L. Smarr, K-H.A. Winkler and M.D. Smith, [25] A.E. Giannakopoulos and AJ. Engel, Directional Control
Astron. Astrophys. 113 (1982) 285. in Grid Generation, J. Comput. Phys., to be published.
[21] K.F. Holian, T-4 Handbook of Material Properties Data [26] J.U. Brackbill, Solution Adaptive Grids for Time-Depen-
Bases, LA-1060-MS, Los Alamos National Laboratory, dent Problems in Two and Three Dimensions, J. Comput.
Los Alamos, NM (1984). Phys., submitted.
[221 R.W. Hockney and J.W. Eastwood, Computer Simulation [27] R.D. Milroy and J.U. Brackbill, Phys. Fluids 25 (1982)
Using Particles (McGraw-Hill, New York, 1981). 775.You can also read
