capturing material interfaces
TRANSCRIPT
ELSEVIER Comput. Methods Appl. Mech. Engrg. 17C' (1999) 15-25
Computer methods in applied
mechanics and engineering
Capturing material interfaces Farhad Ali
Dr. A.Q. Khan Research Laboratories, p.o. Box 562, Rawalpindi, Pakistan
Received 17 December 1996.
Abstract
Captunng material interfaces using the MacCormack difference scheme with flux corrected transport over both Lagrangian and Eulerian grids, without complicated transport formulations are investigated. Several simple techniques of damping spurious oscillations, resulting from capturing material interfaces using Eulerian grids, are applied locally. Formulat:ons based on actual physical conditions are found to be superior to artificial dissipative mechanisms. © 1999 Elsevier Science S.A. All rights reserved.
I. Introduction
Material interfaces being physical discontinuities are difficull to resolve using conventional capturing schemes, as opposed to shocks which behave like sharp jumps but are, in fact, continuous transitions over a distance of a few Angstrom units [ 1 ]. Most of the sophisticated shock capturing schemes resolve shocks quite accurately but need special tailoring to give acceptable representation of interfaces in Eulerian descriptions. Capturing interfaces in one-dimensional Lagrangian descriptions give satisfactory results but extensions to multi-dimensions degrade this leverage by frequent rezoning [2] and slide lines algorithms, which in turn overweigh the physics in highly distorted flows and flows involving slip surfaces.
Shock fitting/tracking methods [3,4] which build discontinuities into the numerical solution, produce sharp discontinuities and are easy to apply in one-dimensional calculations, but entails high programing complexities in extension to multi-dimensions [5].
On the other hand, in Eulerian calculations materials diffuse across the interfaces at non-physical rates and give rise to spurious oscillations. Both these phenomena being mutually exclusive are difficult to handle and are mainly caused by the first-order advection terms in the transport equations. To avoid some of the disadvantages of both the descriptions, mixed, coupled, hybrid and Arbitrary Lagrangian-Eulerian (ALE) approaches [6-14] have been proposed. Particle-in-cell (PIC) [15] was a first step in this direction.
Moving mesh methods [13,16] and adaptive mesh techniques [17] are also steps in this direction. Off-shoots of this approach are the free-Lagrange methods (SPH, etc.) [18] which do not use struclured grids. Instead, depending on the situation, Lagrange particles (nodes) are regrouped, every time to utilize full advantages of the Lagrangian calculations in an adaptive manner [19].
All these approaches shift the difficulty and labour involved from one form to another for easy handling depending on the speciality and background of the user. To avoid slide lines, rezoning and re-griding fantasies of the Lagrangian methods, attempts have been made to float the interfaces in a fixed Eulerian grid. Marker-and-cell (MAC) has been a very popular technique, of this kind, in free surface problems [20-23]. To avoid complicated logic in handling intersecting surfaces that occur in penetration calculations and tangling of interfaces due to inaccuracies in the Lagrangian velocity of the marker particles, the Volume Of Fluid (VOF) method was devised for pure Eulerian or ALE calculations [24]. Other similar approaches include Simple Line Interface Calculation (SLIC) algorithm [25-27], and the donor-acceptor mass transport technique [27,28].
Due to high sophistication in the shock capturing schemes, the search for simpler techniques to handle
0045-7825/99/$ - see front matter © 1999 Elsevier Science S.A. All lights rese:rved. PII: S0045-7825(98)00186-8
16 F. Ali / Comput. Methods Appl. Mech. Engrg. 170 (1999) 15-25
material interfaces in Eulerian formulation is still hotly pursued. Most of the high resolution shock capturing schemes solve the flow equations ignoring all discontinuities, and then some form of filtration either as a separate step or embedded in the main algorithm is carried out to cancel excessive diffusion where it is clearly not needed, ensuring not to introduce unphysical extrema in the flow field. These schemes resolve shocks very accurately, but at material interfaces spurious oscillatNns spoil the results severely.
In this paper some simple techniques of damping these oscillations are investigated to compare their effectiveness when the MacCormack difference scheme with flux corrected transport (FCT) of Boris and Book [29] is used to solve the compressible fluid flow equations on a one-dimensional Eulerian grid. In the next section, numerical procedure and the oscillation damping techniques are given together with the governing equations. In Section 3, numerical results are given to highlighl performance of different strategies. Section 4 contains some concluding remarks.
2. Equat ions and techniques
The inviscid compressible flow equations for smooth flows, ignoring heat conduction and surface tension, in a one-dimensional conservative form are the Euler equations
OF OG Ot +-~-r + S = 0 (1)
with F = [p, pu, pE] T, G = [pu, pu 2 + p , u(pE +p)]V and S =: (au/r~)[p , pu, (pE +p)]T. Where u denotes _ l 2 with 8 the s~_ecific internal energy of the fluid, ce = 0, 1 velocity, p mass density, p pressure and E - 8 + ~ u
or 2 for rectangular, cylindrical or spherical symmetry, i.e. r --- x, ~/(x 2 + y ~ and ~/(x 2 + y2 + z 2) (with x, y and z the Cartesian coordinates), respectively.
A Lagrangian equivalent could be
OF OG 0--7- + ~ + S = 0 (2)
with F = [V, u, E] T, G = r " [ - u , p, up] T and S = [ 0 , - a p V ] r , 0] T. Here, V = 1/p, the partial time derivative represents material derivative and m is the Lagrange coordinate given by
fr r p r ~ m = dr , 0
where r o = r at time t = O. Particle path is given by
Or
Ot
u the particle velocity. Either of these sets needs to be supplemented by an equation of state of the type,
p = f ( p , e ) , (3)
depending on the characteristics of the medium. Amongst the shock capturing methods, the MacCormack difference scheme is the only member of the S ~
t~
family [30] that has enjoyed wide-ranging applications due to its good accuracy in nonlinear transport phenomena. Denoting the numerical solution by F, its predicted value by F ° and its corrected value by F c, this scheme approximates (1) by
F (t, j ) = F(i, j ) - At - - + S ( j ) Ar ( j )
1 [ ( A _ G P ( j ) ) ] (4) FC(i, j ) = -~ F(i, j ) + F°(i, j ) - At -~-~-~ + SP(j)
where A+ and A_ are the first-order forwardand backward difference operators, respectively.
F. Ali / Comput. Methods Appl. Mech. Engrg. 170 (1999) 15-25 17
F(2, j ) G ( j ) - F(1, j ) F(i, j ) + (i - 1)(4 - i) F(i - 1, j ) F(1, j ) p ( j ) / 2 ,
O~ S ( j ) r ( j ) ~ _ , {G(j) - (i - 1)(3 - i)(4 - i ) p ( j ) [ 2 } , (5)
A r ( j ) = r ( j + l) - r ( j ) ,
GP(j) and SP(j) are evaluated using F p. The index i, (i = l, 2 or 3) stands for the ith component of F and j varies over all the spatial nodes. To improve resolution of the captured discontinuities, that may be present either at the outset or generated during the flow, Flux Corrected Transport (FCT) of Boris and Book [29] supplement the algorithm usually. Here, the simple form of FCT given by Book et al. [31-33], is used to improve resolution. In case of spherical symmetry, multiplication of the diffusive and anti-diffusive fluxes by the weights ( r j + l / 2 / r ) ) ~' is proposed [31] but due to its nominal effects, is not used here. The above algorithm apply to (2) also, without any significant change.
With this algorithm both shocks and contact discontinuities are captured very well in Lagrangian description but Eulerian calculations give high oscillations near an interface. To damp these oscillations, some simple recipies have been investigated as follows:
(1) Nittman [34] has applied the FCT-Shasta algorithm of Boris and Book [29] to the problem of a shock wave hitting a contact discontinuity. A smaller anti-diffusion coefficient (thus heavy diffusion) has been shown to damp the pressure oscillations between the transmitted and reflected waves, slightly suppressing the density profile also. Effect of such a step is investigated first in the next section.
(2) MacCormack and Baldwin [35] have introduced a dissipative term
E(g~kX)4 ~X ~ T ~X 2 (6)
with 0 ~< e ~< 0.5 and x the independent space variable, to the right-hand side of the conservation laws, for smoothing pressure oscillations in shock boundary layer interactions when using the MacCormack difference scheme. We will add its difference form
1 ( A+6 A_Fj~ -~(Arj +Ar j_ , ) \Dj+ , Ar, D j A r , _ , ] (7)
with
lujl + la%l, D j = e i + 2p i + ,
Cj =- j ,
to the right-hand side of the corrector, in the vicinity of the interface. Here, 82 is, the second-order second difference operator and Fj = F ( j ) , etc.
(3) Another similar approach is the application of fourth-order diffusion by Taki and Fujiwara in detonation modelling, when a moving coordinate with Chapman Jouguet detonation velocity is used [32]. After the corrector step of the MacCormack scheme and before FCT diffusion, they apply
F2 ew = F ; - l a , ( F ) _ 2 - 4 F j _ , + 6 ~ - 4 ~ + , + ~ + 2 ) with /z = 1 /48 . (8)
This expression will be used, as it is, to a few neighboring nodes of the interface. Both the preceding dissipative terms may result in damping the oscillations without smearing the interface, since normally artificial viscosity does not contribute to the smearing of a c'ontact discontinuity [136].
(4) In a bubble liquid-system Sugimura et al. [37], use u c = 2Ub, derived from impedance matching, with u c the velocity of the interface and u b the particle velocity of gas behind the shock wave. In the region of diffused interface they use continuity of pressure and temperature at the interface in the form Pa = Pe and L=T .
While temperature need not be continuous in general, the continuity of pressure and the normal
18 F. Ali / Comput. Methods Appl. Mech. Engrg. 170 (1999) 15-25
component of velocity (u here) hold across an interface in the absence of shocks always [7]. In shock capturing schemes, shocks are treated as continuous transitions and hence continuity of p and u can be utilized across an interface. Utilizing the continuity of u we calculate the interface velocity u h (say) by linear interpolation between nodes Mr, and M b + 1 when the position of interface (say) r h [r(Mb), r(M h + 1)], as
h × u (M h + 1 ) + H × u(Mb) ub = h + H ' (9)
w i t h h = r b - r (M b) a n d H = r ( M b + l ) - r b.
Continuity of p is implemented by taking
p(Mb) = p ( M b + 1). (10)
(5) Fry and Book [38], in interaction of shock waves with internal boundaries use Dalton's law of partial pressures for calculation of pressure in mixed cells. The application of this law in the neighbourhood of the interface allow pressure at a node to be a linear combination of pressure at other neighboring nodes. To see its suitability as wiggle suppressor we will take
h × p ( M b + 1) + H X p(Mb) Ph = h + H '
A r ( M b - 1) x Pb + h X p ( M b - - 1) p(M,,) = a t (M, , - 1) + h ' (11)
A r ( M D × Pb + H × p ( M b + 2) p ( M b + 1) - Ar(Mh) + H
(6) Bjorn Sjogreen [39] has introduced several filters to elimirate spurious oscillations, of which the simplest one is given by Let A t =pj+, - p j and A 2 = p j - & - l .
If A,A 2 < 0 and IAI]<IA21
take A = sgn(A,) m i n { 2 ]A2I, JAIl } ,
p j = p j + A and p j _ ] = p j _ ~ - - A ;
otherwise
take A = sgn(A1) m i n { 1 JAil, IA2]},
pj = p j + A and Pj+I =Pj+1 - A . (12)
The effect of this filter, on pressure profile, in the vicinity of an interface will be illustrated next. (7) The artificial compression method (ACM) developed by Harten [40,41], that can be used with standard
finite difference schemes to prevent smearing of shocks and contact discontinuities, use a sort of modified flux so that a contact discontinuity for the original equation is a shock for the modified equation and a shock for the original equation remains a shock for the modified equation. A conservation law is solved in a two-step manner as follows: (i) Approximate the conservation law by a monotone finite difference scheme in conservation form
which smears the discontinuity as it propagates it, say
= Q F 7 .
(ii) Compress the smeared discontinuity while not being prapagated, i.e. it should not involve any motion and not alter the physical time in the solution obtained in step (i). For this Harten [41] chose
n + l Fj = Q ' (F)
F. Ali / Comput. Methods Appl. Mech. Engrg. 170 (1999) 15-25 19
with Q' given by
Fy +' =Fj - At(~ +1/2(~) -- ~ j - , /2 (~)) [ A r , (13)
where
1 ~ + l n = ~-(D/+, - Dj) -IDa+, - DjIss, Dj = SS max{O, min(lAr A + 6 I , ssl Ar A Fj[)},
SS = sgn(A +Fj).
Results will be presented, where ACM is applied in the near vicinity of an interface supplementing the existing algorithm, and when ACM is applied, instead of FCT anti-diffusion, in the whole domain.
Other similar remedies could be the use of law of conservation of energy in a non-conservative form having no convective derivative [28], MacKinnon and Carey's [42] type of special difference formulae utilizing vanishing of flux jumps, differencing based on Vp continuity [4311, and Mao's [44] balancing of extrapolated values on the two sides of the interface, etc. Local grid refinement, especially floating strips of finer grids enclosing the interfaces, could also prove promising. Nearly all of these techniques need substantial supplementary arrangements to give sufficiently smooth results without loss of accuracy.
3. Numerical results
In this section results are presented for a high intensity converging shock impinging on a metallic sphere. Lagrangian calculations are performed first, to ascertain the capturing ability of the numerical algorithm, i.e. MacCormack scheme with FCT, in resolving the main shock, the reflected shock as well as the gas-metal interface. Comparisons of the techniques given in Section 2, are then carried out when the interface floats in a fixed Eulerian grid as a continuous transition region of very small thickness whose position is determined by either (i) the largest local gradient of density; or (ii) a massless marker particle moving with the local fluid velocity. The shock strength is very high compared to the yield strength of the metal and hence both the
> , 6 ̧
ca M E T A L GAS
~ , ,,, r ! 26.0 27.0 28.0 29.0 30.0
0.4'
0.3-
tQ 0.2"
~' o . I .
M E T A L
26'.0 l
27.0 28.0 29'.0 ' 30'.0
-0 .25 ' ro o - 0 . 2 0 .
> -0.15'
r~ _0.I01
- 0 . 0 5 .
~" 0.00"
M E T A L ~AS
26'.0 27 0 28'.0 29'.0 30.0
o . 00s-
o. 006-
"~0. 004" ~ •
0. 002"
Fig. 1. Initial data.
M E T A L
26'.0 27.0 28'.0 29'.0 301.0
20 F. Ali I Comput. Methods Appl. Mech. Engrg. 170 (1999) 15-25
mediums can be treated inviscid and compressible. Initial distributions of density, pressure, particle velocity and internal energy are given in Fig. 1. Equation of state in the gas region, 27 ~< r ~< 30, is the perfect gas law
P = (9' - 1)pe, (14)
and in the metal, 25 ~< r < 27, the polynomial equation
p = A/z + B/z 2 + C/x 3 , (15)
with/x = (P /Po - 1), Po initial density, A = 0.98, B = 3.03 and C = 4.65. The number of grid points (nodes) in the two regions are the same (400 and 500, respectively) in both the descriptions and results are plotted at constant intervals of time (say) tj, t z, t 3 and t4, in Figs. 2 -4 , to ease comparison. The resulting Lagrangian grid sizes are A m , - 11.25 in the gaseous region and Am 2 --~ 21.21 in the metal, while in the Eulerian description Ar, --~ 7.5 X 10 3 and Ar e --~ 2.8 X 10 -3 are the corresponding grid spacings.
Time step At, is calculated using
At = min{Al, A2}
A,<2 ~ = max{[Iu( j ) I + c(j)]/[2 Ar( j ) ]}
with j varying over the gaseous nodes (those in the metal region). Analogous expressions for At are used in Lagrangian calculations. To ensure stability, At is multiplied by a safety factor, 0.75, in both the descriptions. Due to differing grid sizes in the Lagrangian and Eulerian descriptions the respective time steps are different. Similarly, in the Eulerian formulation different smoothing techniques give different particle velocities giving rise to slightly different time steps. Thus, in each of the following filgures results are plotted at four different time intervals, approximately given by t~ = 0.06, the solid line; t 2 -'= 0.4, the small dashes; t 3 ~ 0.8, the longer dashes; and t 4 ~ 1.7, the longest dashes.
Plotted in Fig. 2 are the density, pressure, particle velocity and internal energy, of the Lagrangian calculations, at these time intervals. The transmitted (into the metal region) and reflected (back into the gaseous region) wave,
• lO- r-.. , - ¢ ~ I . ' ~ Jt 0 .6" I ' 'li=
8" ' ! h~I 0 . 5 " >, i~ ® ~ ' ~ 0 , 4 " -,-I IG I
o~O. 3- IZl h
~ 0 . 2 " 4" I1~ . I ! L o . 1-
2 o.o i i i i
26.0 27'.0 28'.0 29'.0 30'.0
,I I% I ",~ I%. I ~: I - , ,-. ~ i , I \ i I I~4. I "t' I,'/
26'.0 27'.0 28'.0 ' 29'.0 i
30.0
-0.20" - , - t
,~ -o.ls- ,,--I
-0.i0"
® -0.05" ,.--I U
-,~ O. O0 }.-i ,~ 0 . 0 5 -
0.i0- 26'.0
I " - L I '
I i " I ! :
I f%
c,j ".,csX \ I \
\ I \ I
27'.0 2 8 . 0
0.012"
O. 010"
0 . 0 0 8 -
0.006"
0.004"
O.O02- H
0.000-
29'.0 30'.0 26'.0 27'.0
Fig. 2. Results of the Lagrangian calculations at four intervals of time.
28'.0 29'.0 r
i
3 0 . 0
>,
- ,4 la
,-%
>,
. ' 4 l a
Cl
>,
. ' 4 Iii
10-
8"
6"
4"
2"
10"
8"
6"
4"
2"
10"
8"
6"
4"
2"
F. Ali I Comput. Methods AppI. Mech. Engrg. 170 (1999) 15-25
k
I~'-- ~'~:
No smoother
0.8"
~ 0 . 6 -
~ 0 . 4 "
0.2"
~ . ~ 0.0-
26'.0 27.0 28'.0 29.0 30.0 26'.0 27 0 28.0 29.0 30'.0
f~
I i I ',
I '
26'.0 27.0 28'.0 29'.0 30'.0
Anti-diff. coef.
~4
t~
= 0.Ii
1 . 4 "
1.2"
1.0-
0.8"
0.6"
0.4"
0.2"
0.0"
i I I i i
, . . I -,.2 dl"k4_
I I : l l l - " ~ " ~
26.0 27.0 28.0
| . . . ~. I'-.~ ~ 11,. 0 '
I ,,~ I , : l • J . [
26'.0 27.0 28'.0 " 29'.0 30'.0
Viscosity
0.8-
~0 .6"
~ 0,4"
0.2-
0.0"
ol l, IN I '
I ! I , J !
' .q i
29'. 0 30'. 0
26'.0 27.0 28'.0 29'.0 30'.0
21
-'4 Ul
10-
8-
6-
I t
4-
2- 26'.0
4th order diffusion
0.8-
II
27 0 28.0 29'.0 30.0
~0.6-
~ 0 . 4 - t,,.
0.2-
O.J-
a|
I ~' . I "
I " <
I J
26.0 27.0
,,',,
28'.0 29'.0 r r-
Fig. 3. Position of maximum density gradient as Int,~rface position in all graphs.
i
30.0
22 F. Ali / Comput. Methods Appl. Mech. Engrg. 170 (1999) 15-25
! ! '{ I It
;I
;I c
• | •
26'.0 27 0 28.0
No smoother 0.8-
0.6-
~ o . 4 ,
0 . 2 '
29'.0 30 ~. 0 2 .o 27.0 28'.0 " 29 ' . 0 3 0 . 0
8-
6"
4"
2- ' ! ' , !
26.0 27 0 28.0 , %
2 9 . 0 3 0 . 0
Pressure cont~ nuity
0.7~
o. ~2 o.s :
~ o . ¢ V l
~ o . 3 . 0 . 2 '
0 . 1
o.o~
K
I
] 26'. 0 27.0 28'.0 " 29'.0
t
30.0
8
~ 6.
4'
8 '
~ 6-
4
I :!I,I
'1 i c ~
2g.o 2 7 . 0 ' 2 i . 0 29'.o
Pressure interpolation
0.5"
~ o . 4 - l.x\,
~o.3 {
o.2. [ ,
o.1- : : I
2 6 . 0 ,, , ,~
30.0 27.0 28'.0
I t , I I t
! i
2g.o ' 27'.0 28'.0" 29'.0 r
Filter
3 0 . 0
0.6"
0.5- Q)
~0.4" t@ ~ o . 3 - ~ 0 . 2 -
0 .1"
E\ , ~ill: t "q :'l ,~, I t % 1 1 " k .
I ,
I t I , i t _~ ; ' ~ ,
26.0 27.0 28.0 r
Fig. 4. Interface position by interpolated u in all the graphs.
29'.0 ~ 30'.0
29'.0 30'.0
F. Ali / Comput, Methods Appl. Mech. Engrg. 170 (1999) 15-25
I
26'.0 27 .0 28'.0 ~ 29 ' .0
ACM near interface
0.7- 0.6-
o.5- 0 . 4 '
o.3- 0.2-
0.1-
! I \ , , , I
I I I : i : J 1
ig m ! ,
30.0 26.0 27.0 28'.0 29.0 30'.0
23
4: 21
; it
26'.0 27'.0 28';0 " 29'.0 r
ACM in the whole domain
0.5- J~ I . l \ , ~]ll',
o.4- I \ I ' , . , ', I ! ~ .
o.3- 0.2-
0.1"
" - n
" 30'.0 26.0 27.0 28.0 29.0 30.0 r
Fig. 4. Continued.
as well as the interface are very accurately resolved. Density and internal energy are discontinuous across the interface which moves linearly (r = U × t), as is apparent from the interface position at times t,, t 2 . . . . . l 4.
Pressure and velocity are continuous across the interface; and the transmitted and reflected waves have nearly equal and opposite velocity, again manifested in the respective positions of these waves. Hence, for an overall picture of the main events, i.e. strength, speed, position and resolution of these discontinuities, only the density and pressure plots seem sufficient for further investigation.
In multi-material flows, position of interface need be known tO employ appropriate equations of state properly. In Lagrangian calculations the associated node travels with the interface and hence distinguishing different material regions never becomes a problem. In contrast, due to the fixed-in-space nature of the grid in Eulerian calculations, certain mechanism is required to keep track of the interface position. A crude way of doing so is to consider interface position equivalent to the position of local maximum of density gradient. This seems appropriate in our case here, due to highly differing densities of the two materials. To this end, position of max{Vp} is determined before each computational cycle which is then followed, if desired, by an oscillation damping/smoothing technique of Section 2.
In Fig. 3 are plotted density and pressure profiles at times t~, t 2 . . . . . t 4 utilizing such formulations. The density profile, in nearly all the graphs, is comparable to that of Fig. 2 but pressure oscillates heavily near the interface though positions as well as amplitudes of both the transmitted and reflected waves match Fig. 2 quite satisfactorily. Results displayed in the first row are obtained when no wiggle smoother of Section 2 is used. The second row represent the case when the anti-diffusion coefficient is reduced. In the third and fourth row, viscosity given by (7) and the 4th-order diffusion (8) are applied, at nodes M b - 3 to M b + 3 only, respectively. All of these graphs portray comparable over- and under-shoots except the second row where, though started later, they exceed the others substantially. Hence, these ad hoc arrangements are of no use in damping the interfacial oscillations.
Results plotted in Fig. 4 are obtained when after each computational cycle the interface is moved as a massless marker particle using the local fluid velocity u h given by (9). Again, resolution, speed and amplitudes
24 F. Ali / Comput. Methods Appl. Mech. Eng,g. 170 (1999) 15-25
of the transmitted and the reflected waves are reasonably good but the spurious oscillations near the interface have polluted the results up to varying degrees.
In the first row are given the plots when no smoothing/damping mechanism is used. Magnitude of the oscillations is reduced compared to that of Fig. 3. Pressure continuity (10) implementation, plotted in row second, damps the oscillations quite reasonably. Pressure interpolation (11) suppress the wiggles initially but spreads them in the whole region behind the transmitted wave, as shown in the third row. In the fourth row are plotted the results when the filter given by (12) has been applied at nodes M~ - 3 to M b + 3. It fits some where between the second and the third rows in suppressing the wiggles.
Application of ACM given by (13) near the interface, at nodes M b - 3 to M b + 3, gives the results plotted in the fifth row. Amplitude of the oscillations is reduced compared to those of rows 1, 3 and 4; and the wave profile is very smooth compared to that of all the preceding rows. ACM application, instead of FCT anti-diffusion, in the whole computational domain, displayed by the plots of row 6, give a diffused reflected wave exhibiting still noticeable undershoots near the interface. Limited improvement by ACM supplementing a TVD scheme and the level set method for interface capturing, compared to interface tracking has been reported by Li et al. [45] in the case of a Rayleigh-Taylor instability.
4. Concluding remarks
The problem investigated in this paper portrays the difficulty in capturing material interfaces. The oscillations near a material interface, in Eulerian description, do not manifest Ihere severity in capturing an ordinary contact discontinuity, like that of the Sod's shock tube problem [36] or Nittmann's single state equation formulation [34]. In view of the computations presented here, suitability of the simple remedies, mentioned in Section 2, for wiggles near an interface in descending order of priority could be: (i) Pressure continuity (10); (ii) ACM (13) in the vicinity of interface; and (iii) Filter (12) in the neighbourhood of the interface; to supplement the numerical algorithm described in that section. All the other choices either give no or insignificant improvement in the existing results.
It is possible that some other similar maneuver, or a suitable combination of those mentioned in Section 2, improve the results further but, it is quite clear from this study that embedding more relevant physics in the numerical algorithm could prove a better remedy than pure mathematical/numerical patching. Such physical recipies may include supplementing the pressure and velocity continuity by the jumps, in density and internal energy; and /o r implementation of any or all of these conditions in some other suitable form.
Acknowledgements
Useful discussions with Dr M. Alam and Dr M. Akram are gratefully acknowledged. Continuous support and encouragement by Dr A.Q. Khan and Dr F.H. Hashmi has greatly contributed to carry out this study.
References
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