analytic solutions of hydrodynamics equations

Analytic solutions of hydrodynamics equations S. V. Coggeshall Los Alamos National Laboratory, Los Alamos, New Mexico 87545

(Received 8 May 1990; accepted 26 November 1990)

Many similarity solutions have been found for the equations of one-dimensional (1-D) hydrodynamics. These special combinations of variables allow the partial differential equations to be reduced to ordinary differential equations, which must then be solved to determine the physical solutions. Usually, these reduced ordinary differential equations are solved numerically. In some cases it is possible to solve these reduced equations analytically to obtain explicit solutions. In this work a collection of analytic solutions of the 1-D hydrodynamics equations is presented. These can be used for a variety of purposes, including (i) numerical benchmark problems, (ii) as a basis for analytic models, and (iii) to provide insight into more complicated solutions.

I. INTRODUCTION

The technique of similarity solutions for partial differential equations is a valuable method for obtaining special classes of solutions. Similarity variables are formed by certain combinations of the dependent and independent variables that allow the partial differential equations to be reduced to ordinary differential equations. These special solutions describe evolutions pertaining to particular choices of initial/boundary conditions that allow the solutions to proceed. Even though they are only a small class of all possible flows, similarity solutions have been extensively used and provide a great deal of information about more complicated evolutions. In many cases, similarity solutions are attractors of the dynamical evolution, in that the system asymptotically approaches a similarity solution.’ In particular, similarity solutions can be used for (i) exact solutions pertaining to certain initial/boundary conditions, (ii) benchmark solutions for numerical codes, (iii) ideal solutions with special properties, and (iv) insight into more general flow behaviors.

Many publications exist that describe various similarity solutions to various forms of hydrodynamics equations. A partial list can be found in Ref. 2 (also see Refs. 3-7). There are basically two methods for finding similarity solutions to partial differential equations. The first technique, more widely used, is that of dimensional analysis.’ In this method the equations and initial/boundary conditions are analyzed to identify dimensionless combinations of variables, which then provide the similarity variables. This technique is straightforward, quick, and has become a useful tool in many studies of the solutions of partial differential equations.

The second method is Lie group analysis of partial differential equations,’ briefly described in this paper. With the Lie group technique, the partial differential equations are analyzed through the use of a differential operator U, the generator of the group. More details of the application of this technique to the equations considered here can be found in Ref. 2.

Much of the Lie group method of finding similarity variables involves lengthy algebra. (This algebra can be

done using an algebraic symbol-manipulating computer program.) The benefits of this technique over the dimensional analysis are that the Lie group method can uncover types of similarity that cannot be discovered by the first method. Furthermore, the Lie group method identifies variable transformations by which new solutions can be generated from existing ones.

In this paper we take the properties discovered in Ref. 2 and use them lirst to write a set of global transformations by which a given solution can generate others. Next, we solve the reduced ordinary differential equations for some particular solutions, arriving at a collection of closed-form analytic solutions. These particular solutions of the reduced equations are not the most general solutions, so the collection of solutions in this paper is neither complete nor unique. However, they can be quite useful for a wide variety of applications.

In Sec. II we introduce the physical model and the equations whose solutions are given. In Sec. III we describe the group properties of the model equations. The physical transformations for each group parameter are given, along with the composite transfarmation of all parameters. In Sec. IV we list a number of analytic solutions to the hydrodynamics equations. In Sec. V we demonstrate several of these solutions as examples of interesting physical problems.

II. MODEL

The equations considered in this work are those for one-dimensional, one-temperature inviscous hydrodynamic flow. Thermal conduction (nonlinear, in general) is included but can be dropped if desired. A perfect gas equation of state is assumed, so that the pressure P and energy per unit mass E can be written as

P= rp T, d

E=[l?/(y- l)lT,

where l? is the gas constant and y is the adiabatic exponent. With these assumptions, the equations for continuity, mo- mentum and energy transport become

757 Phys. Fluids A 3 (5), May 1991 / 0899-8213/91/050757-13$02.00 0 1991 Ameriqan Institute of Physics 757 Downloaded 27 Jan 2004 to 128.165.156.80. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp

pt + up, + pq + kpu/r=O,

ut + uu, + ( l/p)rTp, + rT,.=O, (1)

--&(T,+uT,)+TTu,+TT;+j F,fy -0. c 1

In these equations, k is a geometry factor set to 0, 1, or 2 for planar, cylindrical, or spherical geometry, respectively.

The heat flux F can be represented through a radiation diffusion approximation as

F= - (ci1/3)Vap. (2)

Here c is the speed of light, a is the radiation constant,and A(p,T) is the radiation mean free path, which can be re- lated to the Rosseland mean opacity K by A = l/~p. We can approximate the function A(p,T) using a power-law lit:

/I (p, T) =/2&T p. (3) Using (2)) we can recover various forms of conductivity K by noting

F= - (4cAaT3/3)VTr - K(p,T)VT,

and choosing the appropriate values of &, oz, and /3 in (3). For radiation diffusion, values of o range from - 1 to - 2, with I</3 < 3. Classical conduction requires a - f and

p-$.

Ill. GROUP PROPERTIES

Using the result of Ref. 2 we can write down the Lie group invariance properties of the system equations ( 1). We allow each system variable in the model to change through the transformation

x--t 2 =e”x, (4) where x is any one of the independent variables r, t, or the dependent variables p, u, or T. We can rewrite Eqs. ( 1) in terms of the new variables Z using the relations (4) along with the chain rule and obtain a set of partial differential equations ( ? ) in terms of these new variables X. When the transformations (4) are invariance transformations, these new equations ( T ) are identical in form to the original set ( 1). That is, the variables X satisfy the same equations as do the variables x. In Ref. 2 we found the group generator U that formed the Lie group invariance transformation for Eqs. (1).

The transformation operator e” is defined through

e”= 1 + U + cr2/2! + U3/3! + * f 0,

where

Unx~Uu(“-l)(Ux), n>l (Vo-1).

The operator e” forms the global transformation, whereas the operator U forms the infinitesimal transformation around the identity transformation. The operator U is called the generator of the multiparameter group of transformations and for the present set of equations ( I), is written

a a 3 3 ci=e,m+E2~+9lrlpi~2~+~3~T. (5)

The coordinate functions cj and nj for the system ( 1) were found in Ref. 2 allowing a general equation of state. For the perfect gas assumption they reduce to

{I =ria2 + a3t + $4) + ad + a7,

&=a33 -t a4t + a5,

rll=p[al- (k-t l)(a2+a3t+hdl,

172=u(a2--3t--h4) fa3rfa6,

q3=T(2a2 - 2a3t - a4),

along with the conditions

(6)

kag==ka,=O,

q[y- (k+3)/(k+ l)l=O,

(71

(8)

This last condition (9) was reported in Ref. 2 to be required only when ;1#0. Actually, it is also not required when V*F=O, so it is required only when V*FfO:

Each parameter ai in (6) creates a separate group generator Vi* written as

Each generator Vi forms a group (in the mathematical sense) of transformations, with ai the group parameter. The single-parameter coordinate functions can be identi- fled from the multiparameter coordinate functions through

Note that the multiparameter group generator, U=BaJJ;, includes the group parameters a, whereas the single- parameter generators do not include the group parameters. Traditionally, in the theory of one-parameter groups, the group parameter is not included in the generator. For multiparameter groups the parameters must be included in the form of U to distinguish the separate transformation groups.

Now that we have the single-parameter generators U, we can ask what is the physical significance associated with each specific transformation. These single-parameter generators transform the variables just as in (4), except now we explicitly add the group parameter:

x--r 5 = eaiuix, for each i. (10)

This relation can be used to calculate the global transformations from the coordinate functions of the infinitesimal transformation, by explicitly exponentiating the operator.

An alternative method is to solve the relations

d7 z=s”l,

dZ dF dE dF -= dai 6-i I

-&=77:, z=v;, -&=17:, I 1

758 Phys. Fluids A, Vol. 3, No. 5, May 1991 S. V. Coggeshall 758

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with the coordinate functions evaluated at the tilded variables. The original, untilded variables appear as the inte- gration constants, and the global transformation equations are obtained.

With the global transformations we can identify the physical properties associated with each particular group parameter. For each group parameter in (6), we list the corresponding generator, its physical meaning and its global transformation relations. In some of the transformations, we have relabeled the group parameters by their exponentials, eir eni.

1. Parameter: al; group generator: U,=p(a/$.~); physical interpretation: scale transformation on p; transformation:

7 =r, z =t, p =e,p,

z =u, T--T.

2. Parameter: a,; group generator: U,=r(W&) c (k + l)~(d/ap) + ~(a/&) + 2T(d/dT); physical in-

terpretation: space scale transformation; transformation:

7 =e2r,

‘ii =e2u,

3. Parameter: -i- t’ (a/at) --

7 =t, jj =e,‘k+l)p,

‘T =e$

group generator: U3=rt(d/&) (2; l)pt(a/ap) + (r - ut) (Wr3u) - 2tT

x (8/6’T); physical interpretation: projective group in the r-t plane; transformation:

7 =r/(l - a$), Z -t/( 1 - a3t>,

~=p(l-a3tp+1, Z==u(l--aa3t) +a3r,

;Ir = T( 1 - a3t)2.

4. Parameter: a4; group generator: U4 = ir( d/Jr) + t(cS’/dt> - +Ck + l)p(a/+) - $4d/du) - T(S’T);

physical interpretation: combined scale transformation on all variables; transformation:

7 =ei’2r, -f =e,t, p =e, (k+ 1)‘2p,

-. u =e<’ ‘12u, T=e,-‘T.

5. Parameter: a5; group generator: Us= (Zl/dt); physical interpretation: translation in time; transformation:

7 =r, z ==t+u5, p =p,

ii =u, T==T.

6. Parameter: as; group generator: U,= t(d/dr) + d/&d; physical interpretation: Galilean boost in velocity; transformation:

7 =r f a6t, T =t, p =p,

is =u + afj, T=T.

7. Parameter: a,; group generator: U, = a/&; physical interpretation: translation in space; transformation:

7 =r+a7, f =t, p =p,

ii =u, T = T.

759 Phys. Fluids A, Vol. 3, No. 5, May 1991

Each of these transformations may be taken individu- ally to obtain a new solution from a known one. Addition- ally, they may be combined in any number and order. One representation of a combined invariance transformation for solutions of ( 1) is

7 = (e2ei’2r + a6e4t)/( 1 - a3e4t) + a7,

T =teJ( 1 - a3e4t) -I- a5,

ij =pe,e2F(k+ ‘)ec (k+ 1)‘2( 1 - a3e4t)k+ l, (11)

ii = ( e2e4- 1’2 u + aa> (1 - a3e4t) + @3(e2ei’2r + aged), F =e:e; ’ T( 1 - a3e4t) ‘.

The parameters aj and ej may take any values. This combined transformation ( 11) can be used to generate complicated, nontrivial solutions from existing ones. That is, using any known solution set p(r,t), u(r,t), T(r,t), the functions p(F,?), i7(7,3, F(Y,:;iF> also form a solution set for any choices of the a/s and efs. Recall, however, that the restrictions (7):(o) apply to parameters a3, a6, and a7. An example of this extension of solutions is given in Appendix A.

The transformations given by (6) form a seven- parameter Lie group of transformations. The classification of this group system is important to identify the complete list of group invariant solutions. This classification is partly done in Appendix Bi but it is not possible to complete it, as a result of the inability to solve the reduced (ordinary) differential equations for their most general explicit solutions. Therefore the following list of analytic solutions does not form a complete basis from which all group invariant solutions may be constructed. Each of the 22 solutions- is, however, a group invariant solution. The physical interpretation of the group transformation that generates each solution can be found in Sec. III.

IV. ANALYTIC SOLUTIONS

The coordinate functions (6) can be used to generate systematically different similarity solutions to Eqs. ( 1) . In Ref. 2 we listed these similarity solutions (solutions IV B I-10). We point out that these are allowed reductions of the partial differential equations to’ ordinary differential equations, and not complete analytic solutions to ( 1). However, closed-form analytic solutions can indeed be obtained through any analytic solution of these reduced ordinary differential equations.

One method of finding analytic solutions to these reduced equations is to assume a simple form of the solutions. The ordinary differential equations are written for the similarity variables f(X), g(X), and h(X). We can look for solutions to the ordinary differential equations by representing f, g, and h as power laws in X,

f=f&ya, g=g&‘, h=h&F,

and solve the resulting algebraic equations for the allowed values of the parameters fo, gc, h,, a, b, and c. This is one method that can be used to obtain analytic solutions of ( 1)

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using the reduced similarity solution equations. With this method the following 22 closed-form analytic solutions of (1) were obtained. For each solution, we. identify the group generator(s) that allow the formation of the reduced (ordinary) differential equations, from which the given solution is obtained.

(1) UlJJ$

p(r,t)=polbt-b-‘kl, u(r,t)==r/t,

T(r,t)=Tor-btb-((y-lI)(k+ 1).

Free parameters: .b, k, po, and To.

(2) &JJ,:

p(rJ) =p0+ 2(b+k+1)/[2+(y--I)(k+ 111

9

2 r u(ryt)=2+ (y- l)(k+ 1) 2

Xy- l)(k+ 1) 12

T(r’t)=I’(b+2)[(y- l)(k+,l) +2]?

Free parameters: b, k, and pe.

(3) ul,u2,u5:

p(r,t) =pofmk- ‘eb’,

u(r,t) = - brh,

T(r,t) =b2?/[v21Y(k - v - l)],

with

y=(k - l)/(k+ 1).

Free parameters: v, 6, k, and po.

(4) u5:

p(r,t) =por-2W(y+ ‘I,

u(r,t) =uor -My-- ‘My+ 1) 9

T(r,t)=[ui(l --)/(2yr)]r-2k(y-1)‘(ySI).

Free parameters: k, uc, and pc. Note: The oniy ~physical (T> 0) solution is for y< 1.

(5) U,:

p(r,t) =porw2, u(r,t> =uot,

T(r,r) =uor/l?,

with

k=2 and y=$.

Free parameters: p. and u,,.

(6) u1,udJ5:

p(r,t) =po9/(72 - r’)(k+ l+ b)‘2,

u(r,t) = - rt/(? - t’),

T(i,t)=7”~+‘[I’(d+2)(i?-?)~],

with

y=(k+ 3)/(k+ 1).

Free parameters: b, k, r, and pc. Note. When b=3 and k=2, y= 3 and this becomes Kidder’s 1974 solution.9 (Some details of this solution are given in Appendix C.)

(7) v,,u,,u,:

Rb/y&k+ l)y-I-b]/(y- Or--k- 1

p(r,t> = ’ (R~-b/Y-RRiZ-b/Y)I/(Y-‘)

x(&)k+1-b’y

x~((i-;)lD)2-b~Y- (~~~~“~]“(~el),

u(r,t) =-- rt/(? - t2),

T(r,t)=aY- lW2 - r(2y-b) (h)“”

j ( (14r3),;)-~y (y-] ,

with

y=(k+ 3)/(k+ 1).

Free parameters: b, k, r, R,, and R, Note: When b=O and k=2, y= 5 and this becomes Kidder’s 1976 hollow shell solution.” (Again, see Appendix C for the details on this solution.)

r-_

(8) Ul,U2, with conduction:

p(rJ> =p0r (k-l)/(B-a+4)t-k-l-(k-1)/(P-cr+4)

9

u(i’,t> =r/t,

T(rt)=Tor(‘-k)/‘~-a+4)t(l-~)(k+1)+(k--1)/t~-a+4) 2

Free parameters: a, P, k, po, and T,.

(9) U2,U4, with conduction:

p (r,f> =por - Cl/a)(2@-+ kt7)

Xt-2[a(k+1)-ZZP-k-7]/~[2+(y-lI)(k+1)1 1

760 Phys. Fluids A, Vol. 3, No. 5, May 1991 S. V. Coggeshall 760 .


2 P u(r,t) = 2+(Y-l)wtl)t’ 2a(y- 1)Ck-k 1) r;?

T(v,t)=f[2+(y-l)(k+1)]2(2a-2~-k-7);”~

Free parameters: a, fl, k, and po.

. - (10) Us, wrth conductton:

-k p(r,t) =por ,

u(rJ)=(4cilea/3)[(Y-- l)/(ry)]kpg:‘~f3,

T(r,t) = To@,

with

a=B+4- l/k, k#O. Free parameters: /3, k, po, and T,.

(11) Ut,U, with conduction:

p(r,t),por’~-~)(~+1)-2t~---((y-ll)(k+1) ,

u(r,t) =r/t,

T(r,t)=ToJ!-(Y-l)(k+l)f--2,~

with

a=-B+4+ (k- 1)/[2- (y- l)(kf l)].

Free parameters: k, pot To, and either a or 8.

< 12) U& with conduction:

p(r,t) =por-2k’(y+ I),

u(,,t)=-;uoi/i(*-y)‘(l+y),

T(r,t)=[u;(l - y)/(2ry)]r2k(‘-r)‘c1+r),

with

a=(P+4)(1--14 + (k- l)(y+ l)i2k, k#O.

Free parameters: k, po, uo, and either a or 0. Note: Need y < 1 for physical ( T > 0) solution.

(13) ut, Uz, with conduction:

p(r,t)=po~(U-P-“)t-2/(a----4)-k-l, _

u(r,t) -r/t,

xt[“(k+~)-----2l/(P+3)-+2(n-l)/[(~+3)(a-~-4)] ,

with

3r To=

a- I+ (B-/-3)(Y-,m 1) 4c@(y - 1) 8+3

ypt-aP+4-a 1'U3+3) -~ \ 2 ) Free parameters: a, /?, k, and pP

(14) Ur, LJ,, LJ,, with conduction: ’

p (r,t> =por - k - 6, u(r,t) =# TT,(k - b)/b,

T( r,t) = TOP,

with

b=(k- l-ak)/[2+a-2(8+4)1

and

To+&-J!yp)’

2a-216@

X[2b+ :;- l)(k+b)lZ 1 - l/(5 + 26)

Free parameters: a, 8, k, and p.

(15) U,,U,, with conduction:

P(r,f)~p0~/(~-~-4)t-k-1-2/(a-~-4,,

u(r,t) =r/t,

T(r,t) = T,r - 2/(c~-j3-4)~-2 9

with

2 k+4-a(k+ 1) -2(@+4) a-p--4= a- l

and

( 3 r

Toz I-a

z&E@-% l)(y- 1) RI

x-2+ [2- (Y- 1)Ck-k l,]

1

l /W + 3)

X(a-B-4)) .

Free parameters: k, po, and either a or p.

(16) U2, U,, with conduction:

p(r,t) =por- k-6, u(r,t) =uo4,

T(r,t)=u;br?‘[IYk - b)],

with

a=1 - I/k, fi=ia - 3, k#O,

and



1 6cAou k

-y-- iy- 11 T(r,t) =o.

PO=

b’5k- 1)/2

The shock location is

‘uo(k- b)(k-1)nr(3k-11)‘2[2b+ (y- l)(k+ b)lke R= - +(y-- l)uot.

Free parameters: k, uo, and pe Free parameters: b, k, and q,.

( 17) U2, U,, with conduction:

pb-J) =pop (Z/3 - 4)/(1 - df(2/3 + 5)/(u - 1)

,

u (r,t) = uor/t, T( r,t> = Tog/?,

with

X-f+5 "=2fi-4+ (1 -a)(k+ 1)’

(a - 1)(2!3+ 5) T"=r[2fi-4+ (1 -a)(k+ l)]’

[9-- (1 -a)ik+ l)] x [W-4+2(1 -ar)] ’

( -2+d2+-(~--1)(k-t1)1 “= 2[a(2D-4)/(1 -a) +2fi+k+7]

X &5 T;~-3)“‘a-1r.

Free parameters: a, /?, and k.

(18) U,, U,, with conduction:

pb,t) =p0r -(l/a)(W+k+7)

x(~_iZ)-(1/2)(k+1)+(1/2a)(~+k+7) 7

u(r,t) = - H/(72 - 2),

a2 vz Tir~t)=r(2a-228-kk7)‘(72’

with

y=ik+3V(k+ 1).

Free parameters: CL, /3, k, r, and po.

(19) U2, LJJ, shock, no conduction: Region 1:

pir,t9 =pol: iy + 1 )/iy - 19 lk + l,

Uiv)=o, 7+-,t) =u& - i)/2r;

region 2:

pb,t) ‘pot (r - uOt)/rlk,

U(r,t) =uoi <09,

(20) U2,Us,U+ shock, no conduction: D Region 1:

pirA=pdiy+ l)/iy- l)lk+t(l -at) -k-

u(r,t) = - ar/( 1 -at),

T(r,t) = [ui(y - 1)/2r] (1 - at) -2;

region 2:

p(r,t)=poCl -at) -k-‘[(r- uot)/rlk,

u(r,t> = (uo - izr)/(l - at), T(r,t) =o.


R=[uo(y- l)/&z][t(l -2czt)/(l -at)].

Free parameters: CI, k, po, and uo.

(21) U,Uh; shock, no conduction: Region 1:

p(r,t) +)F3, u(r,t) =o, T(r,t) =-To?;

region 2:

p(M)==pOr-3, u(r,t)=r/t, T(r,t)=O,

with

k=2, y=5.


R=Z/(rT,t”).

Free parameters: p. and T,.

(22) Us,. shock, no conduction: Region 1:

p(r,t)=plr-2k”y+ ‘),

u(r,t) =ulr - k(y- IMy+ 11 t

T(r,t)=[u;( 1 - y)/2yIJp-2k+ l)‘(Y+ l);

region 2:

u(r,t> =u2r -kCy-- IMy+ 1) 7

T(r,t) = [ui( 1 - y)/2yIyr-2k’Y- IMy+ ‘),

with

762 762 Phys. Fluids A, Vol. 3, No. 5, May 1991 S. V. Coggeshall


FIG. 1. Time dependence of the pressure for Example 1.

[ (1 - YMvl (&1- &2) = hpd(p1- p2> 1 cf.4 - a2 and

UT - u;= [Y(U1- u2)2/cpl - pd’l tp; - p:,. The shock location is

R--t,.-b’- I)/(?‘+ “( p2uz - p1u*)/(p2 - p1) + c(r).

Free parameters: pr, ur, and k. Note: Need y < 1 for- a physical ( T > 0) solution.

V. PHYSICAL EXAMPLES

A. Example 1

For this first example we select solution (1) from Sec. IV. We set y = $ and choose spherical geometry (k=2). For these parameters, this particular analytic solution becomes

The pressure is P=TpT==I’poT,,t - ‘, which is uniform throughout the system (i.e., does not depend on r) . We can consider this flow at some (initial) time ti and thereafter. At time ti, the system has the form

p=piPy u =uir, T= Tir- b, --h-3 where pi = potr , Ui = l/t, and Ti = Tot;- 2. We can

choose the initial velocity profile to. be either positive or negative (incoming or outgoing) by selecting ti to be either positive or negative. We are assured that T and p are real and positive because of the arbitrary choice of the constants p. and To.

Figure 1 shows the system pressure as a function of time. A positive choice of ti will cause material to flow out from the center of the sphere and the pressure will decrease as t - 5. With a negative choice for t, material flows inward and the pressure correspondingly rises (without bound). Note that for all choices of b, the requirement that p and T be real and positive implies that P is also real and positive for ti either positive or negative. This can be seen by Iook- ing at all possible [real) choices for b, determining the


2.0 cn

2 -ii b P

a. -2 3 kl

c, 2

0.a

(a)

2.0

E ._) 4 k 8

ok 2 .? L a,

3 2

0.0

(b)

Radius

Radius

z .A 4 ki 0”

lk -2 3 tii

% r

(c) Kodius

PIG, 2. Initial spatial profiles for Example 1 with (a) b > 0, (b) b=O, and (c) b<O.

requirements on p. and To for that choice of b, and check- ing that P is indeed positive.

Consider first an expanding system (ti> 0). We can choose b to be either positive, negative, or zero. The corresponding material initial conditions are shown qualitatively in Figs. 2 (a) -2 (c) . Let us investigate the properties of the system for ti> 0 and b > 0. We set b = 2.

The initial conditions for b =2, shown in Fig. 2 (a), are

p(r,t[) =pi?( =potfm5rZ), U(r,ti) =r/tf,

T(r,ti)=Ti/rZ(=To/rZ). (12)

We can restrict the study to a finite system, with initial radius r, by requiring the proper pressure boundary condition at that Lagrangian location for all time. We can therefore consider a finite expanding sphere with initial

S. V. Coggeshall 763


0.0 0.0 O-O (a) -2.0 0.0 2.0

Radius Time

FIG. 4. Space-time trajectories for the Row in Example 1.

s) .-r: ::

y < 1. The solution has no explicit time dependence, but it 3 has a nonzero velocity and therefore material flow. For 2 example, choose k=2 and y = f. The solution is

p(r,t) =por- 8’3, u(r,t) =uo313, 0.0

0.0 2.0 (b)

T(r t) = (ui/21’)r4’3. , Radius

. Consider first an infinite region with these profiles, shown in Fig. 5. With positive velocity everywhere, material is always flowing out from the origin, where the density is infinite, and the flow velocity increases as the material moves out. These nontrivial profiles are fixed for all time, and form an interesting, steady-state hydrodynamic flow pattern.

We can also choose u. to be negative, which represents inward flow in this “stationary” system. Furthermore, we can consider a finite boundary radius R(t), obtained by

s 4 5 6 integrating dr/dt= uoF3,

1 & 1

2.0 R(t)= (uot/3 + R;B)3, R(0) =RO, Radius

and consider the flow inside of this finite radius sphere. The

FIG. 3. Evolution of the material profiles for Exampk 1 with 6 = 2. Note pressure boundary condition is

in (c) the temperature dependence does not change in time. Only the boundary of the finite region is changing in (c).

P,(t) IR(ty=I’pT= (po@2)R -4/3.

profiles given by (12) for r<r, with P(R,t)=poTotw5, where R(t) =rit/ti. The solutions for such a system are shown parametrically in Figs.* 3 (a)-3 (c).

The inward flowing ( tj < 0) solution can easily be seen from these pictures by letting t-r - t (U becomes negative). We can draw a combined space-time plot for material trajectories for this solution, shown in Fig. 4. The lines are trajectories for Lagrangian position or material “pack- ets.” At any time, the spatial profiles are shown in Figs. 3(a)-3(c). Note that the temperature profile does not change in time.

6. Example 2 ‘0.0 Radius

2’.0

Solution (4) of Sec.-IV is quite interesting. First note that it requires an unusual value of the adiabatic constant, FIG. 5. Material properties for all time for Example 2.



This finite system can be either expanding or contracting, depending on the choice of the sign of u@ We note that the system is everywhere supersonic; for all space and time, u=2c,

Notice that solutions (lo), (12), (14)) and (16) also do not have explicit time dependence, even though they have nonzero velocities. These solutions all include condu- tion, which precisely balances out the equations to allow the solution to exist.

C. Example 3

The next example we consider is solution (IS). Again we choose k=2 for spherical geometry, which gives y = 3 for this solution. We first choose a = - 1 and /? = 2

for a mean free path corresponding to radiation conduction. With these choices the solution becomes

p(r,t> =piJr’3/(? - w,

u(r,t) = - rt/(? - t2),

T(r,t) =?/[ 15l?(72 - t2)2],

with the conduction mean free path given by .

This is a Kidder-type9 solution with heat conduction. It does have a rather high exponent on the radial coordinate for the density. The effect of this high radial exponent is minimized if we consider the corresponding hollow shell version of this solution. The details of this particular hollow shell solution are not presented here. It relates to solution (18) as solution (7) relates to solution (6).

With a different choice of a and fi, we can construct the identical solution presented in Ref. 9, this time including heat conduction, by requiring k-2 (y= $), and b = 3 in solution (18) to obtain

p(r,t)=porv(? - t2>3, u(r,t)= - rt/($ - ty,

T(r,t)=212/[5I-(72 - t2>‘].

1.0

0.0 -r 0.0 r

Time

FIG. 6. Material trajectories for the Kidder-type solutions. Note the existence of both an expansion and a contraction phase.

D. Example 4

Efficient, high-gain inertial confinement fusion (ICF) requires optimal compression and ignition of deuterium- tritium (DT) fuel. Mainline target designs achieve this with spherical implosions using careful drive pulse shap- ing. The shaped drive pulse achieves the simultaneous formation of a hot ignitor region surrounded by a spherical shell of cold, maximally compressed DT fuel. This main fuel region is formed by compressing an initial hollow shell of cryogenic DT in a nearly Fermi-degenerate state. Start- ing from very low entropy, a near-isentropic compression is required to approach a state of maximum compression. One form of an isentropic compression pulse shape was derived by Kidder,‘.” and is generalized here as solutions (6)) (7)) and (17). This particular form of pulse shape is optimum for a certain choice of initial conditions. For other initial conditions, other pulse shapes are required. Using the nonshock solutions, (l)-( 18), we can write down the pressure pulse required to form an isentropic compression. A laser drive pulse can then be obtained using the relationship between laser intensity and ablation pressure P(t) --I(t)‘.‘.

For example, in solution (2) we find

Now we require a = - 2&‘3 - 3 for the fit d = A.20paTp. In general, we find that we can add heat conduction to solution (6) through the substitution

P(r,t) = I)T

=Poib + 2t - 2(b+k+ 1)/p+ (y- l)(k+ 111-2 3

b= - (2fi + 9)/a.

Any intermediate solution between the shown b= 13 and b = 3 solutions can be chosen.

In Fig. 6 we show the trajectories of fluid elements in this Kidder-type solution. These trajectories are independent of a, p, and b, which determine the radial dependence of the density and temperature. The striking feature of these solutions [which include solutions (6), (7), and (IS)] is the existence of both an expansion and contraction phase. Kidder described only the contraction phase,‘*i’ be- ginning at t=O, where the material velocity is zero everywhere. Reinicke” considered both phases and described the solution as a “birth and death” solution.

where PO is some constant. If we monitor the pressure at some position moving in the material at the material velocity (a Lagrangian position), we must solve for that position as a function of time and eliminate Y from this relation. This yields the pressure felt at some specific material surface, or Lagrangian position, as a function of time. We solve

dr 2 P dt=--= 2+ (y- 1)tk-t 1);

to find

r=Rot2/[2+ (y- l)(k+ 1)l



When this is substituted in the above relation for pressure, we find the pressure on any material surface that results in isentropic compression for this example is

p(t),p,t-2y(k+1)/[2+(y-l)(k+l)l

We now list the required isentropic drive pressures for all the solutions that do not involve shocks.

Solution ( 1) :

p,pot-rW1);

solution (2) :

p=pot-Wk+ 1)/P+ (Y- lI(k+ I)].

solution (3) :

p,p@k- l)Vu. ,

solution (4) :

p=p/ - WC1 -b)

or

p(t) =Pge-‘jTTg(k-ljS for b=l;

solution (15) : p= pofk - w - 7;

solution ( 16) :

p= pot@ - WC 1 - b)

or

P(t)==P,e -‘dk-‘) for b=l;

solution ( 17) :

p,p,t[2~+5-uO(2B--4)I/(~-11)+2(~o--1). >

solution ( 18) : p,pot- W/[&r- 1) + Y+ 11 p,p@ - $1 -(k-f-3)/2.

or Most of these solutions have not been offset in time to describe a drive pulse of duration r. This has been explicitly done in the generation of solutions (6), (7), and (18). P=Poe2*~d(1-~) for k=(p+ I)/(1 -y)* 7

solution (5) : *

P= Pot - 2;

solution (6) :

p=po@ - 3) - (k+ 3)/2;

solution (7) :

p=po(p - 2) - (k+ 3V2;

solution (8) :

p--pot-W- 1). >

solution (9) :

p=pot-VW 1)/P+ (Y- 1)W-t 111. >

solution (10) :

P=P,;

solution ( 11) :

p=p,t-r(k+ 0;

solution (12) :

p,pot-WV+- 1) +Y+ 11

or

P(t) =Poe 2uf’(1-yy) for k=(y+ I)/(1 -y); 0

solution (13) :

p,pop/KP+3bB-411 + wP+3)

with

a=(k+ l)(a2+$-88a:f88-2aP+ 16);

solution (14) :

0.0 0.2 0.4 0.6 0.6 1.0 Time (t/tat11

0.0 0.2 0.4 0.6 0.6 1.0 Time Ct/toul

Time lt/toul

FIG. 7. Drive pulse shapes (time dependence) required for isentropic compression depend on the particular initial conditions, but are all qualitatively identical. [a)-(c) show the required pressure drives on a material surface for spheri%al, cylindrical, and planar geometries, respectively. These curves use y= 5 and an arbitrary scale factor Pn

766 Phys. Fluids A, Vol. 3, No. 5, May i991 S. V. Coggeshall 766


In all other solutions, we can replace t with r - t and understand that t goes from zero to r.

Four of these solutions, numbers (3)-(5) and (12), require a nonphysical value of y (y < 1). Solution (10) is an interesting solution that requires constant drive pressure. Since conduction is included it is not really isentro- pit, and it goes to a trivial solution as conduction is turned off (&,+O). In the solution as written, conduction balances out the compression to allow constant pressure. This solution works well with classical conduction values of 01 and 0.

For the other solutions in this example we can plot the pressure profile required for an isentropic compression. These are shown in Figs. 7(a)-7( c) .for spherical, cylindrical, and planar geometries, using y = 3, a = - 1, fi = 2. The solutions have been scaled with an arbitrary value of Pa and a small vertical offset at t=O. This offset is important for ICF designs because it initiates a small shock that forms the central ignitor that is surrounded by the compressed fuel.” In actual ICF designs there is a relationship between the value of the initial pressure offset and the duration 7.

The same values of PO were used in the k=2 and k=l curves. Some of the k=O curves use larger values of PO in order for the shapes to be seen. Values of PO could be chosen for each geometry that will bring all the curves very close to overlapping. Note that all the solutions for any geometry are qualitatively identical, being everywhere con- cave upward with a very rapid rise just before 7. In prac- tice, one “flattens out” the singularity at t=r by setting

P=P(t), for t<r--6,

P=P(7 - E), for t>7-e.

VI. DISCUSSION

In this paper we have presented 22 analytic solutions to the equations of hydrodynamics ( 1). Several of these solutions are new, and many are extensions and generali- zations of previous ones. In particular, the analytic solutions including heat conduction [(8)-(18)] are generally new, with the exception of solution (17) .I1 Solution (6) is an extension of Kidder’s 1974 solution.’ Here the solution is still isentropic, but now each Lagrangian position can be ona separate adiabat, as is common in inertial fusion tar- gets. All adiabats become identical for the choice b =3. Solution (7) is the corresponding extension to the hollow shell solution.” Solution (18) is a different extension of Kidder’s 1974 solution, where we now include heat conduction.

Heat conduction is not included in solutions (l)-(7). Heat conduction is included in solutions (8)-( 18). Solu- tions (19)-( 22) are propagating shocks with no heat conduction. Some of these solutions~ require unusual choices of the adiabatic exponent y [solutions (3)-(S), and (12)]. Many more solutions are certainly possible. It is likely that all of these can be written for hollow shells, as solutions (7) was for (6).


The solutions listed here are obtained by solving nonre- dundant branches of the reduced ordinary differential equations, listed in Ref. 2. We look for particular forms of solutions of these equations, constructed by simple power laws. Any analytic solutions of these reduced equations can provide analytic solutions of the original equations ( 1). The collection of special solutions presented here is neither unique nor complete. It does not provide a basis for representing all group invariant solutions to the hydrodynamic equations ( 1) (see the comments in Appendix B) . It does provide a number of interesting and nontrivial explicit, closed-form analytic solutions of these equations. These solutions can be used (i) for benchmarking numerical codes, (ii) as a basis for constructing simple physical models, and (iii) to provide insight into more general, complicated flows. In particular, the presentation in Exam- ple 4 of the required external pressure drive profiles for a large class of isentropic implosions in various geometries demonstrates an important result using these solutions.

ACKNOWLEDGMENTS

The author would like to thank J. Meyer-ter-Vehn for his helpful discussions on the subject of global transformations and extensions of existing solutions. The author also thanks Roy Axford for his patient tutelage on the subject of Lie groups. Additionally, the author expresses gratitude to the Max-Planck-Institut fur Quantenoptik in Garching bei Miinchen, where this work was completed.

APPENDIX A: EXAMPLE OF AN EXTENSION OF AN EXISTING SOLUTION

In this appendix we show an example of how the global invariance transformations ( 11) can be used to generate a new solution from a known one. First, we write down a trivial solution of ( 1 ), which can be found by inspection:

p(r,t) ‘PO& u(r,t) =o, (Al)

T(r,t) =Tor- ‘.

We wish to add time dependence to this solution, so we will use the transformation corresponding to as. We also include a, to shift the time axis to a convenient location. We set a,=0 for i#3 or 5, and ei = 1 for all i. With these choices, ( 11) becomes

7; =r/( 1 - ust), 7=t/(l -aa,t) +a5,

p “p( 1 - qtp+ 1,

5 =u(l -ua3t) +a3r, T=T(l.-aBt)2.

We will need t expressed in terms of??

-f L,? + u5 3 t= I -a5

----. 3 1 +as(Z -0s)

This gives 1 (l-a$)=[l+a3(1--Us)]- .

S. V. Coggeshall

(A21

767 Downloaded 27 Jan 2004 to 128.165.156.80. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp

TABLE I. Commutator table.

When we substitute the solution (Al ) into (A2), eliminate the old variables, we then have the (extended) solution in the new variables. For example,

p=po4(1-ag)k+’

=poP(l -a$)b+k+l

=po7b/[1 +a3(7 -a5)]b+k+l.

Similarly, we find

ii=@/[l--a,(: -a5>],

F=To7 -b[l -a,(-7 --u~)]~--.

We must also consider conditions (7)-( 9). We allowed a3 to be nonzero and V-F=O, so the only condition to keep is (8). With this condition, along with the choice ~1~ = - 11 a3, we find that this new solution in the “tilded” variables is identical in form to solution (1) in Sec. IV. The overtil- des are irrelevant, since this solution satisfies the same Eqs. ( 1 ), written in the “overtilded” variables. We therefore see that a nontrivial solution can be generated from a trivial solution by using the global transformations ( 11).

APPENDIX B: CLASSIFICATION OF THE SEVEN-PARAMETER GROUP

Here we attempt the classification of the seven- parameter group given in Eqs. (6). This classification process begins by forming an optimal system of generators, from which all other group invariant solutions can be found.

First, we construct the table of commutators, which displays the structure constants of the Lie group. Next, we investigate the conjugate map of the group structure by forming the table of the adjoint operators of the group generators. The construction of this adjoint table makes

TABLE II. Adjoint table.

use of the previous table of commutators. With this adjoint table we can then identify an optimal system of generators which spans the solution space of the group.

The structure constants cf are determined by calculat- ing the commutators of all generators U, through U,, as

[ Vi> Uj] E UiUj -- UjUi= /iI CiU[y i, j=l,..., 7.

We form a table of all such commutators (see Table I). We note that the one-parameter subgroup G’ ( U,) is a central ideal of the seven-parameter group G7.

The next step in the classification process is to form a similar table of adjoint operators, defined through

Ad[exp(eUi)] UjZUj- ~[UitUj]

This list of adjoint operators is found in Table II. Following the procedure outlined in Olver,’ we find an

optimal system of one-parameter subgroups to be the following eight generators:

u2 + au4 -t bU1, a& -I- Ud + bU1,

aU,+ u,+ U4+bU1, aU~+bU2+U5+cU1~

au,+ U2+bU1, U,-I-~UI, Ue-kaUlt

aU5 -I- bU4 -I- U3 f cult

where a, b, and c are arbitrary parameters. The final step in the classification process is to list the

general analytic solutions of the reduced differential equations for each of these eight basis group generators. From these eight solutions, all other group invariant solutions may be obtained by some combination of group transformations. These-eight general solutions would then form a complete basis, from which all other group invariant solutions may be obtained.

However, it is not possible to solve the reduced equations corresponding to these eight basis generators for the most general explicit solutions. Therefore, the classification of solutions is not completed for this somewhat complicated system of equations. For example, the reduced (ordinary) differential equations corresponding to the basis generator U2 + aU4 + bU,, with no heat conduction, are’

Ad

Ul

u2

u3

rr,

u5

u6

Ul

Cl3

U3

U3 e-W,

u, - 2EU4 f $u,

U3 u, - EUb

u4

u4 u4 -k EU,

u4

u4 -ys u4 +~~ub u, - IEU,

U5’

u5 LJ5 + 2EUq -I tw,

etUS U5

u, -I- EU,

u5

U6 e'U6

ub e’Lu

;;6-e;,

U6

ub

UT eEU,

u, f EU6 et/= u,

tr,

Ul

U?



[b - (k-i- l)c]h -I- A’( f - CX) + hf’ + (khf/X) =o, We can therefore introduce a new dependent variable H that will reduce (C2) to ouadrature. The new variable His

(c- l)f+f(f--CX) + (Igh’/h) + lY=o, a group invariant:

where b and c=a + l/2 are constants. These reduced equations can be solved numerically for a variety of interesting cases. They can also be solved analytically for some special cases.

Even though some specific solutions can be found, these reduced equations cannot be solved explicitly in general. This is especially true when one includes heat conduction, which makes these equations much more complicated. There are also the possible solutions for shocks of various types, e.g., strong, weak, isothermal, conducting, with and without thermal precursors.

APPENDIX c: DETAILS OF SOLUTIONS (6) AND (7); DEMONSTRATION OF LIE GROUP THEORY APPLIED TO ORDINARY DIFFERENTIAL EQUATIONS

In this appendix we show explicitly how solutions (6) and (7) are constructed from the reduced equations. In doing so, we demonstrate the use of Lie group reduction techniques as applied to ordinary differential equations.

The reduced equations for this case are listed in Ref. 2, Sec. IV B 1, with d=z=w=s=v=O. If we further set f-0, only the second ordinary differential equation re- mains. Setting r=Q, this equation becomes

- ?x + rgh’/h + rg’ = 0. (Cl)

If we look for solutions of the form h=aXb, g=cXd for some constants a, b, c, and d, we find

d-l and c=?/r(b + 2),

for T#O. When written in terms of the physical variables, this becomes solution (6).

The power-law assumption does not provide the general solution to (Cl), only a particular form of solution. We can solve this equation more generally with only the assumption of isentropic flow. Here we require

Pap7 3 g(X) =C(X) [h(X)]?- ’

for the similarity temperature g and density h. By allowing the constant C to depend on X, say through C=cXb, we are allowing isentropic flow where each Lagrangian position X has a fixed adiabat, but at any time the entropy is not uniform in space. This is the typical case for inertial fusion capsules.

We look for a solution of (Cl) with g=cXbhy - I, which provides the ordinary differential equation

- ?X/I’c + bXb- ‘hr- ’ + yXbhr- 2h’=-=0. cc21

This rather complicated nonlinear ordinary differential equation can be solved using Lie group techniques.

We tind that this equation is invariant under the group with generator

uH=O j H=hx’b-2’/(?‘-1).

We compute

dH dh “=E=H.rfHhE

b-2 =y--.IX’b-2)/(y-lblh

+ X’b-2)/(Y- “h’.

When this is substituted into (C2), it becomes

xH’=(?/yrc)H2-Y- [(2y-bb)/y(y- 1)1H,

which is separable. The quadrature can be done, giving

fl-1=K~-2+b’yf?(j’- l)/[Cr(2y-b)] (,hY- lxb-2)

For b= 2y, the h solution is

h=X-2{K+ [?(y- l)/cl?y]lnX}‘/(Y-I).

Written in terms of the original variable h, we find the general isentropic solution to Eq. (C2) as

,+{fl-&‘-WY

+ [?(y- i)/cr(2y-b)]x2-b)1'(y-'1).

This solution, when written in terms of the physical variables, becomes solution (7). The value of K is chosen so that p(R,O) = T(R,O) =0, Ri and R, being the inner and outer radii of a hollow shell, respectively.

‘G. I. Barenblatt and Ya. B. Zel’dovich, Annu. Rev. Fluid Mech. 4, 285 (1972).

2S. V. Coggeshall and R. A. Axford, Phys. Fluids 29, 2398 ( 1986). ‘A. V. Voloshinov, Izv. Vyss. Ucheb. Zaved. Mat. 18, 31 (1974). *J. R. Sanmartm and A. Barrero, Phys. Fluids 21, 1957 (1978). ‘S. I. Anisimov and N. A. Inogamov, Zh. Prikl; Mekh. Tekh. Fiz. 4,20

(1980): 6J. Meyer-ter-Vehn and C. Schalk, 2. Naturforsch. Teil A. 37, 955

(1982). 7See, for example, G. Birkhoff, Hydrodynamics, A Study in Logic, Fact,

and Simultude (Princeton U.P., Princeton, NJ, 1960); L. I. Sedov, Similarity and Dimensional Methods in Mechanics (Academic, New York, 1959).

“See the references cited in Ref. 2, along with P. J. Olver, Applications of Lie Groups to Differential Equations (Springer, New York, 1986).

9R. E. Kidder, Nucl. Fusion 14, 53 (1974). ‘OR. E. Kidder, Nucl. Fusion 16, 33 ( 1976). “P Reinicke and J. Meyer-ter-Vehn, Laser Interacyion With Matter

(World Scientific, Singapore, 1989), p. 132. I2 R. E. Kidder, Nucl. Fusion 19, 223 (1979).



analytic solutions of hydrodynamics equations

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