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Wave interaction in a nonequilibrium gas flow
International Journal of Non-Linear Mechanics 40 (2005) 1031 – 1040 www.elsevier.com/locate/nlm
Wave interaction in a nonequilibrium gas flow V.D. Sharma∗ , Gopala Krishna Srinivasan Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India Received 28 April 2004; received in revised form 20 December 2004
Abstract By employing the method of multiple time scales, we derive here the transport equations for the primary amplitudes of resonantly interacting high-frequency waves propagating into a non-equilibrium gas flow. Evolutionary behavior of nonresonant wave modes culminating into shocks or no shocks, together with their asymptotic decay behavior, is studied. Effects of non-linearity, which are noticeable over times of order O(−1 ), are examined, and the model evolution equations for resonantly interacting multi-wave modes are derived. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Wave interaction; Nonequilibrium gas flows; Shock waves
1. Introduction A number of problems relating to the effects of non-linearity and rate processes on wave propagation in gases with internal relaxation have been studied previously; in this context the contributions due to Blythe [1], Ockendon and Spence [2], Chu [3], Parker [4], Becker [5], Clarke and McChesney [6] and Clarke [7] are worth mentioning. Asymptotic methods for the analysis of weakly non-linear hyperbolic waves have received great attention in the past. ChoquetBruhat [8], while dealing with small-amplitude waves, proposed a method similar to that of Varley and ∗ Corresponding author. Tel.: +91 22 576 7482; fax: +91 22 572 3480. E-mail addresses: [email protected] (V.D. Sharma), [email protected] (G.K. Srinivasan).
0020-7462/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2005.02.003
Cumberbatch [9] to discuss shockless solutions of hyperbolic systems which depend upon a single phase function. A general discussion of single-phase progressive waves has been given by Germain [10]; using the progressive wave approach and the related procedures, Sharma et al. [11,12] and Radha and Sharma [13] analyzed the wave propagation problems in different material media. Based on a formal version of the weakly non-linear geometrical optics, Hunter and Keller [14] established a general non-resonant multiwave mode theory which has led to several interesting generalizations by Majda and Rosales [15], and Hunter et al. [16] to include resonantly interacting multi-wave mode features. In this paper, using the resonantly interacting multi-wave theory, we examine small-amplitude highfrequency asymptotic waves for one-dimensional unsteady planar and non-planar flows of a general
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inviscid relaxing gas, where at the leading order, many waves coexist and interact with one another resonantly, and obtain evolution equations which describe the resonant wave interactions inherent in the system. It is assumed that the rate of change of energy in the lagging mode depends only on the local values of pressure, entropy and internal energy, and the particular nonequilibrium phenomenon of interest is the vibrational relaxation; the rotational and translational modes are assumed to be in local thermodynamical equilibrium throughout.
2. Preliminaries and the basic equations It is well known that at high temperatures (∼ 1000 K), the internal energy of the gas molecules consists of translational, rotational and vibrational components. When the gas is in equilibrium, each internal energy mode is characterized by the same temperature. A rate of transfer of energy from one mode to another mode can be observed by inducing a small change in any of these temperatures and observing the rate of return to equilibrium. For instance, when a gas is compressed by the mechanical action of a piston or by the passage of a shock front, the whole energy goes initially to increase the translational energy, and it is followed by a relaxation from translational mode to rotational mode and also from translational mode to vibrational mode until the equilibrium between these modes is re-established. This is called a relaxation process and is governed by an equation of the form (1d) below, with q replaced by Ti , a measure of the energy in the molecular vibrational state, and = (T − Ti )/ where T is the translational temperature and the relaxation time; this corresponds to a gas model with single relaxing internal degree of freedom in which the internal energy is characterized by T and Ti . Under actual flow conditions, the internal energy is always tending toward the equilibrium for the new conditions; this time-lag, known as the relaxation time, is short for translation and rotation, and one may consider that these modes are adjusted instantaneously in all flow problems. However, the relaxation time of vibrational modes is much longer, as a result the adjustment of translational and rotational non-equilibrium in a gas flow requires relatively few collisions as compared
with that of vibrational non-equilibrium. When gradients in the flow field are large, as for example in boundary layers, translational non-equilibrium results in the phenomena known as viscous stress and heat conduction; similarly, rotational non-equilibrium gives rise to bulk viscosity, and thus, the molecular studies of situations involving translation and rotation in a gas require the effects of viscosity and heat conduction to be considered. Here, we shall assume that the flow is everywhere in instantaneous translational and rotational equilibrium, or, in other words, that all effects of viscosity (both bulk and shear), heat conduction and diffusion are negligible. For simplicity, we shall allow for only one non-equilibrium process, that is vibrational non-equilibrium, and so introduce in addition to the pressure and entropy, a third nonequilibrium variable q to specify the vibrational state of the gas. The introduction of an additional variable requires an additional equation (1d) known as the rate equation for the non-equilibrium process. To keep the treatment within desired length, we presume that the reader is acquainted with the governing equations as presented in the standard texts (see for example, Refs. [3,6,17]). Thus the basic equations governing the one-dimensional unsteady planar and non-planar flows of a relaxing gas with general thermodynamic properties in the absence of viscosity, heat conduction and body forces can be written in the form mu (1a) + af2 = 0, pt + up x + af2 ux + x px = 0, (1b) ut + uux + St + uS x = − qt + uq x = ,
, T
(1c) (1d)
where t is the time, x the spatial coordinates being either axial in flow with planar (m = 0) geometry or radial in cylindrically (m = 1) and spherically (m = 2) symmetric flows. The state variable u denotes the gas velocity, p the pressure, S the entropy, q the energy in the molecular vibrational state, af the frozen sound speed given by af2 = (jp/j)S,q , and the rate of change of energy in the vibrational mode, which is assumed to be a known function of p, S and q. The symbols , T and denoting respectively, the density, temperature and the affinity of internal transformation
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characterized by the variable q, are regarded as functions of p, S and q and are given by (invoking Gibbs’ relation T dS = dh − dp/ − dq)
jh 1 = , jp
jh = T, jS
jh = , jq
where h is the specific enthalpy related to p, S and q through the canonical equation of state h = h(p, S, q), and the entity is given by =−(/T )j/jS+j/jq. When the internal transformation attains a state of equilibrium, we have
(p, S, q) = 0 = (p, S, q) ⇒ q = q(p, ¯ S).
In this limit, the perturbations caused by the wave are of size O(), and they depend significantly on ¯ (x, t)/, where the fast characteristic variable ¯ = ¯ = (1 , 2 , 3 , 4 ) and ¯ = ( , , , ); indeed 1 2 3 4 i , 1 i 4, is the phase of the ith wave associated with the characteristic speed i . We denote the left and right eigenvectors of A0 associated with the eigenvalue i by L(i) and R (i) , respectively, satisfying the normalization conditions L(i) R (j ) = ij , 1 i, j 4, where ij denotes the Kronecker symbol. In fact, we have 2 af0 0 ∓af0 (1,4) L = , , 0, 0 , 2 2
Using matrix notation, Eqs. (1) may be cast in the following form:
L(2) = (0, 0, af−2 , 0), 0
jU jU +A + F = 0, jt jx
R (2) = (0, 0, af20 , 0),
(2)
where U and F are column vectors defined as U = (p, u, S, q) and F =(af2 +maf2 x −1 , 0, /T , −). A is a 4 × 4 matrix having components Aij , the non-zero ones are as follows: A11 = A22 = A33 = A44 = u, 1 A12 = af2 , A21 = . It follows immediately that system (2) admits four families of characteristics, two of which represent waves propagating in the ±x directions with the modified frozen sound speed u ± af , and the remaining two form a set of double characteristics representing entropy waves or particle paths. We consider waves, which propagate through a constant background state U0 = (p0 , 0, S0 , q0 ). The characteristic speeds at U = U0 are given by 1 = −af0 , 2 = 3 = 0, 4 = af0 . The subscript 0 refers to evaluation at U = U0 , and is synonymous with a state of equilibrium.
3. High-frequency limit Here we use the method of multiple scales to obtain small-amplitude high-frequency asymptotic solutions of (2), when the time scale 0 = −(j/jq)−1 0 , defined by the relaxation or rate phenomenon, is large compared with the signal period ˜ , that is = ˜ /0 >1.
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L(3) = (0, 0, 0, 1),
R (1,4) = (0 af20 , ∓af0 , 0, 0),
R (3) = (0, 0, 0, 1).
4. Weakly non-linear resonant waves We seek asymptotic solutions of (2) as → 0 of the form U ∼ U0 + U1 (x, t, ¯ ) + 2 U2 (x, t, ¯ ) + O(3 ), (3) where U1 is a smooth bounded function of its arguments and U2 is bounded in (x, t) in a certain bounded region of interest having at most sub-linear growth in as → ±∞. Now we use (3) in (2), expand A and F in a Taylor series in powers of about U = U0 , replace the partial derivatives j/jX (X being either x or t) by j/jX + −1 4i=1 (j i /jX)j/ji , and equate to zero the coefficients of 0 and 1 in the resulting expansions, to get 4 j i j i jU1 I = 0, + A0 O( ): jt jx ji i=1 4 j j jU2 O(1 ): I i + A0 i jt jx ji 0
(4)
i=1
jU1 jU 1 − A0 − (U1 · ∇F )0 jt jx 4 j i jU 1 (U1 · ∇A) , − jx ji
= −
i=1
(5)
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where I is the 4 × 4 unit matrix, and ∇ is the gradient operator with respect to the dependent variable U . Since each of the phases i satisfies the eikonal equation
j j Det I i + A0 i = 0, (6) jt jx we choose the simplest phase function solutions of this equation, namely
i (x, t) = x − i t,
1 i 4.
(7)
It follows from (4) that for each phase i , jU1 /ji is parallel to the right eigenvector R (i) , and thus U1 =
4
i (x, t, i )R (i) ,
(8)
i=1
where i = (L(i) · U1 ) is a scalar function called the wave amplitude, that depends on the ith phase variable i ; the dependence of i on i describes the wave form whether it is an oscillatory wave-train or a pulse. We suppose that i (x, t, i ) has zero mean value with respect to the phase variable i , that is, T 1 lim i (x, t, ) d = 0. (9) T →∞ 2T −T We then use (8) in (5) and solve for U2 . To begin with we write U2 =
4
mj R (j ) ,
j =1
which we substitute in (5) and premultiply the resulting expression by L(i) to yield the following system of decoupled inhomogeneous first-order PDEs: 4 j =1
(i − j )
jm i jj
j i j i − i − L(i) (U1 · ∇F )0 jt jx 4 jU1 − L(i) (U1 · ∇A) , 1 i 4. jj
=−
j =1
Suppressing the (x, t) dependence in k , and denoting by d i /ds the ray derivative j i /jt + i j i /jx,
the equation for mi assumes the form 4
(i − j )
j =1
=−
jm i jj
d i i j k d i − ai i − iii i − j k j ds d i jk j,k
:= Hi (x, t, 1 , . . . , 4 ),
1 i 4.
(10)
The characteristic ODEs for the ith equation in (10) are given by
˙ j = i − j
for j = i, ˙ i = 0, m ˙ i = Hi .
We asymptotically average (10) along the characteristics and appeal to the sub-linearity of U2 in , which ensures that expression (3) does not contain secular terms. The constancy of i along the characteristics and the vanishing asymptotic mean value of m ˙ i along the characteristics implies that the wave amplitudes i , 1 i 4, satisfy the following coupled system of integro-differential equations:
j i j i j i + i + ai i + iii i jt jx ji T 1 + ij k lim j (i + (i − j )s) T →∞ 2T −T i=j =k × k (i + (i
− k )s) ds = 0,
(11)
where k = j k /jk and the coefficients ai and ij k are given by ai = L(i) (R (i) · ∇F )0 ,
ij k = L(i) (R (i) · ∇A)0 R (k) . (12)
Here it may be noticed that for plane (m = 0) waves, coefficients ai vanish in the absence of internal relaxation as (2) is free from the source term (i.e. F = 0), and (11) reduces to the equations obtained in [18]. Thus, the coefficients ai in (11) describe the growth or decay behavior of the wave amplitude i on account of non-planar wave form and the relaxation phenomenon present in the medium. The interaction coefficients ij k , which are asymmetric (in j, k), measure the strength of coupling between the j th and kth wave modes (j = k) that can generate an ith wave (i = j = k). The coefficients iii account for the nonlinear self-interaction, and are non-zero for genuinely non-linear waves; these coefficients are indeed zero
V.D. Sharma, G.K. Srinivasan / International Journal of Non-Linear Mechanics 40 (2005) 1031 – 1040
for linearly degenerate waves. If all the coupling coefficients ij k (i = j = k) are zero or the integral averages in (11) vanish, the waves do not resonate and (11) reduces to a system of uncoupled Burgers’ equations. The coefficients ai , iii and ij k , given by (12), provide a qualitative picture of the non-linear interaction process present in the system under consideration, and can be easily determined in the following form; the non-zero ones being: 1 1 ma f0 b− , a3 = , 2 x 0 1 ma f0 a4 = b+ , 2 x j(af ) 4 1 2 44 = − 11 = af0 = , jp S,q 0 2 a j ( a ) f f 124 = − 421 = 0 = 1 , 0 jS p,q 0 1 j(a f ) 1 4 34 = − 31 = = 2 , 0 jq p,S a1 =
(13)
0
where b = (1/0 )(af20 /ae20 − 1) is positive and serves as the amplitude attenuation rate on account of relaxation, 0 = −(j/jq)−1 0 is the relaxation time of the 1/2 medium, and ae0 = (jp/j)S=S0 ,q=q0 is the equilibrium sound speed. It then follows that Eq. (11) reduces after some simplifications to
j 1 j 1 1 j 1 ma f0 1 − 1 − af0 + b− jt jx 2 x j1 T 1 1 + − lim K x, t, T →∞ 2T −T 2 (14)1 × 4 (x, t, ) d = 0, j 2 = 0, jt
(14)2
j 3 3 =− , jt 0
(14)3
ma f0 j 4 j 4 1 j 4 b+ 4 + 4 + af0 + jt jx 2 x j4 T 1 4 + + lim K x, t, T →∞ 2T −T 2 (14)4 × 1 (x, t, ) d = 0,
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where the coefficients b, and 0 are the same as in (13) and the kernel K is defined as + + K x, t, = 1 2 x, t, 2 2 + + 2 3 x, t, , (15) 2 with 1 and 2 being the same as in (13). For an inert gas, which needs only a pair of thermodynamic variables such as (p, S) or (, S) for the state of the medium to be completely specified, we obtain 2 j(a) , = 0, b = 0, = a0 2 jp S 0 20 a03 j(a −2 jT /j) a02 j(a) 1 = = , 0 jS p 2 j S 0 0 + + K x, t, = 1 2 x, t, , 2 2 and thus, in the absence of internal relaxation, Eqs. (14) reduce exactly to those derived in [15] for the corresponding plane (m = 0), inviscid ordinary gas dynamic case. The integral average term in (14)1 (respectively (14)4 ) represents contribution to the wave amplitude 1 (respectively, 4 ) on account of the non-linear interactions of the wave fields 2 and 3 with the acoustic wave field 4 (respectively, 1 ); this, indeed, plays an important role in explaining how the energy carried by the non-decaying wave ( 2 ) and the energy released on account of internal relaxation get partitioned among the acoustic modes. The non-linear terms proportional to 1 1 and 4 4 in (14)1 and (14)4 account for non-linear self-interactions which generate higher harmonics leading to the distortions of the wave profile and consequent shock formation; indeed, the two acoustic wave fields 1 and 4 exhibit a strong effect from the non-linearity present in system (1). The absence of self-interaction terms (14)1,4 show that the wave amplitudes 2 and 3 , corresponding to the repeated eigen mode, are linearly degenerate; thus we can solve (14)2,3 for 2 and 3 and then use the expressions in K (given by (15)) that appears in (14)1,4 . It may be observed that the wave fields 2 and 3 , associated with the repeated eigenvalue do not interact with each other; however they do interact with an acoustic wave field to produce resonant contributions
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only toward the other acoustic wave field. Similarly the acoustic wave fields 1 and 4 may or may not interact with each other but in either case their net contribution, which is directed toward the entropy field only, is always zero. If the initial data U10 (x, ) are specified at t = 0, these determine the initial data for i that is functions 0i (x, ) so that at i |t=0 = 0i (x, i ). Thus, 2 (x, t, )= 02 (x, ), 3 (x, t, )= 03 (x, )e−t/0 , and subsequently system (14) reduces to a pair of equations for the acoustic wave fields 1 and 4 , coupled through a linear integral operator involving the kernel
K(x, t, ) = 1 02 (x, ) + 2 e−t/0 03 (x, ).
(16)
The pair of resonant asymptotic equations is much closer to the original system as it includes (to leading order) a model version of the non-linear resonantly interacting wave dynamics inherent in the full Euler equations (1). If the initial data 0i (x, ) are 2 periodic functions of the phase variable , only even harmonics in 02 (x, ) and 03 (x, ) contribute in (14), and the pair of resonant asymptotic equations take the following simplified form: j 1 j 1 1 j 1 1 ma f0 1 − 1 − − af0 + b− jt jx 2 x j1 2 1 + × K x, t, 4 (x, t, ) d = 0, (17) 2 − j 4 j 4 1 j 4 1 ma f0 4 + 1 + + af0 + b+ jt jx 2 x j4 2 4 + × K x, t, 1 (x, t, ) d = 0, (18) 2 − where K is given by (16).
inviscid Burgers’ equation (14) with vanishing integral average terms, is uniformly valid to the leading order until shock waves have formed in the solution. In terms of the characteristic field associated with r (r = 1, 4), and defined by the equations
r dr , = dx af0
dt er , = dx af0
(20)
where er = −1 or +1 according as r = 1 or 4, respectively, the decoupled equation (14)1 can be written as d r m ber − r , (21) = dx 2af0 2x along any characteristic curve belonging to this field, and yields on integration
r = 0r (sr , r )(x/sr )−m/2 exp(−bt/2),
(22)
along the rays sr = x − er af0 t = const, where the function 0r is obtained from the initial condition (19), and the fast variable r parametrizes the set of characteristic curves (20)1 . We thus obtain from (20)
r = r − er 0r (sr , r )Jr(m) (t),
(23)
t (m) where Jr (t)= 0 (1+(af0 er /sr )t )−m/2 exp(−bt /2) dt . Eq. (22) modify the usual results of gas dynamics concerning the evolution of cross-sectional area of a ray tube; in fact, the wave amplitude decays exponentially in contrast to the corresponding classical gas dynamics case, where it decays like x −m/2 (see [14]). Thus, the solution of (1), satisfying (19), with U10 (x, x/t) having compact support, is given by p = p0 + 0 af20 x −m/2 e−bt/2 ((x − af0 t)m/2 × 04 (x − af0 t, 4 ) + (x + af0 t)m/2
5. Non-linear geometrical acoustics solution The asymptotic approximate solution (3) of the hyperbolic system in (1) or (2), satisfying smallamplitude oscillating initial data U (x, 0) = U0 + U10 (x, x/) + O(2 ),
(19)
is non-resonant if the functions U10 (x, x/) are smooth with a compact support [18]; indeed expansion (3) with U1 given by (8), where the wave amplitude i , 1 i 4, solve a reduced system of decoupled
× 01 (x + af0 t, 1 )) + O(2 ), u = af0 x −m/2 e−bt/2 ((x − af0 t)m/2 04 (x − af0 t, 4 ) − (x + af0 t)m/2 01 (x + af0 t, 1 )) + O(2 ), S = S0 + af20 02 (x, x/), q = q0 + 03 (x, x/), where the fast variables r , given by (23), are chosen such that at t = 0, r = x/, and the initial data for i (1 i 4) are determined from the data U10
V.D. Sharma, G.K. Srinivasan / International Journal of Non-Linear Mechanics 40 (2005) 1031 – 1040
specified at t = 0 as 1 0 , 1 (x, 1 ) = 20 af20 (p10 (x, 1 ) − 0 af0 U10 (x, 1 )),
02 (x, x/) = S10 (x, x/), 03 (x, x/) = q10 (x, x/), 1 0 (p10 (x, 4 ) + 0 af0 U10 (x, 4 )). 4 (x, 4 ) = 20 af20 This completes the solution of (1) and (19); any multivalued overlap in this solution has to be resolved by introducing shocks into the solution. It may be observed from (23) that for planar flows (m = 0), there will be an overlap in the region t > 0, formed by those characteristics for which j 01 /j1 > 0 (respectively, j 04 /j4 < 0), assuming that > 0. Thus, in the wave field 1 (respectively, 4 ), a shock can occur only if 01 > b/2 > 0 (re spectively, 04 < − b/2 < 0) at time t = t1 (respectively, t = t4 ) given by t1 = −(2/b) ln(1 − b/ 1 ) (respectively, t4 = −(2/b) ln(1 + b/ 4 )), where 0r ≡ j 0r /jr and r = Sup{ 0r (r )}. 5.1. Cylindrical and spherical waves The cases of converging/diverging waves belong to a particular class of piston motion and are indeed of special interest as they permit to establish, in particular, the trend of the influence of relaxation on the formation and propagation of shock waves. If we consider the situation in which the initial data occurring on the surface with radial coordinates x = x1 propagate into the gas-filled region 0 x x1 , it will give rise to converging waves. It may be noted from (22) that for m=1, 2 the rays focus and the wave amplitude 1 → ∞ as t → t ∗ = /af0 where = inf{s1 } indicating the formation of a caustic at t = t ∗ , where the geometrical optics approximation is rendered invalid; the reader is referred to the work of Hunter–Keller [19], who investigate the singular behavior near the caustic. (1) Since the integral J1 in (23) converges to a finite positive limit (say K1 ) as t → t ∗ , it follows that a shock can occur in the wave field 1 at time tc < t ∗ (1) given by the solution of J1 =1/1 in the wave field 1 , only when the initial value 01 exceeds a critical value 1/K1 .
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However unlike the cylindrical case, as the integral (2) J1 becomes unbounded in a finite time, a shock is always formed before the caustic in the wave field 1 , no matter how small the initial value 01 associated with a contracting piston motion with a spherical symmetry. Further, if we consider the situation when the initial data, occurring on the surface with radial coordinate x = x1 , propagate into the gas-filled region x > x1 , it will give rise to diverging waves. It may be (m) noticed that the integral J4 (t) in (23) converges to a finite limit lm (m = 1, 2) as t → ∞, implying thereby that a shock can appear in the wave field 4 in a finite (m) time t =ts given by the solution of J4 (ts )=−1/4 . Thus, except for the wave field 1 associated with a spherically symmetric flow, if the magnitude of the initial data 0r is less than a critical value the solution to the leading order of approximation remains shock free. In fact, when >1, or the fraction of thermal energy proportional to b that exists in the relaxing mode is sufficiently large, or the initial profile 0r is such that r → 0, or, the derivative 0r (r ) is small, the initial data do not give rise to shocks. 5.2. Shocks As discussed in the preceding sub-section, a shock wave may be initiated in the flow region, and once it is formed, it will propagate by separating the portions of the continuous region. At shocks, the correct generalized solution satisfies the Rankine–Hugoniot jump conditions for the shock location sr (t) in the t–r plane; thus if sr (t) be the shock location in the r wave field, it can be shown following [14] that 1 dsr (+) = rrr ( (−) r + r ), dt 2
r = 1, 4,
(24) (+)
which is the shock speed in the r –t plane. Here r (−) and r are the limiting values of r from the upstream and downstream sides of the shock, respectively. For the front shock, which has an undisturbed (+) flow ahead of it, r = 0. Dropping the superscripts (−) s on r and r , and using (22) we have dr
= er 0r (sr , r )(x/sr )−m/2 exp(−bt/2). dt 2
(25)
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Eq. (25), together with another relation for the shock speed obtained from (23), yields the following relation which relates r to t on the shock: r 2er (m) 0r (t ) dt . (26) Jr (t) = − ( 0r )2 0 Eq. (26) together with
r = r − 2( 0r )−1
r 0
0r (t ) dt ,
(27)
which is obtained from (23) for the front shock determines the shock path parametrically. If 0r (0) = 0, the shock starts immediately at the origin, and the relation between r and t is given by r ∼ −( /2)er ∗r Jr (t); ∗r = 0r |r =0 , so that r ∼ (m) ( /2)er ∗r Jr (t), implying that the shock starts with velocity 21 er ∗r (x/sr )−m/2 exp −bt/2, which is quite consistent with (25). In case of a source producing only outgoing waves, if the compressive phase (0) is followed by a rarefaction so that r (r ) → 0 as ∗ r → r , the wavelets in the neighborhood of the wavelet r = ∗r will never reach the shock, and then the shock is asymptotic to r ∼ ∗r + Ker 2 Jr(m) , 1/2 ∗r 0r (t ) dt , K = −er 0
and the amplitude in (22) behaves like t er af0 t −m/2 r ∼ 2/ Ke−bt/2 1 + e−bt /2 sr 0 −m/2 −1/2 er af0 t × 1+ dt . (28) sr Relation (28) shows that the shock decays exponentially; indeed, the shock amplitude depends very strongly on the internal energy (proportional to b) that exists in the relaxing mode and depends on the wave geometry through the parameter m as well as the body (piston) shape through the factor K. It is evident from (28) that for an inert gas (b = 0), plane (m = 0), cylindrical (m = 1) and spherical (m = 2) shock waves decay like t −1/2 , t −3/4 and t −1 (log t)−1/2 , respectively; these results are in agreement with earlier results [17].
5.3. Analysis of the integro-differential system We close this section with a few comments on the analysis of the integro-differential system (17)–(18), that parallels the treatment in [20]. Introducing the ray derivative, Eqs. (17) and (18) for the planar case (m = 0) assume the following forms: j 1 1 + 1 b 1 d 1 − K x, t, + − 1 ds 2 j1 2 − 2 × 4 (x, t, ) d = 0, (29) j 4 4 + 1 b 4 d 4 + K x, t, + + 1 ds 2 j4 2 − 2 (30) × 1 (x, t, ) d = 0, The system has linear part that can be cast in the form b Ys + Y + LY = 0, 2
(31)
where Y = [ 1 , 4 ]T and L is the operator given by 1 1 + K x, t, 4 (x, t, ) d , LY = − 2 − 2 4 + 1 K x, t, 2 − 2
(32) × 1 (x, t, ) d . We assume that the kernel K(x, t, ) is summable on the interval [−, ] for the variable . The operator norm of L as an operator on the spaces L∞ (R) and BV (R), is given by KL1 [−,] . On multiplying (31) by Y T and integrating with respect to we see that the L2 norm Y (·, t) of solutions of (31) decays exponentially and we have the estimates LU L∞ KL1 U L∞ , Var(LU ) KL1 Var(U ),
(33)
where we have denoted by Var(U ) the BV-norm of U . We now assume for simplicity that 1 in (16) vanishes and 03 is a 2-periodic function of alone, which is even as it was observed in Section 4 that only even harmonics contribute to the integro-differential terms. Multiplying (31) by ebs/2 , integrating and using the Gronwall lemma, we get the decay estimate
Y (·, t)L∞ exp(−bt/2 + 2 0 03 L1 × (1 − e−t/0 ))Y (·, 0)L∞ ,
(34)
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with a similar estimate for the BV norm. So it follows immediately that due to relaxation effects we have a uniform L∞ and BV estimates for the solutions of (31). If, instead, the coefficient 1 in (16) is different from zero, the exponential term in (34) gets modified to
exp(−bt/2 + 1 t 02 L1
+ 2 0 03 L1 (1 − e−t/0 )).
In any case if the attenuation rate b due to the relaxation is strong enough, we still have a uniform estimate on the L∞ and BV norms of solutions of (31). For entropy solutions of the pure inviscid Burgers’ equation ws + ( /2)ww = 0,
(35)
the L∞ and BV norms are monotone decreasing in time. The method employed in [20] is to alternately interlace approximate solutions of (31) and (35), obtain in the limit a solution to the full non-linear system (29)–(30) and estimates (34) hold for the solutions of the full non-linear problem. In conclusion, we would like to add that the case considered in [20] is conservative admitting reductions to local systems with traveling wave solutions. With relaxation effects, the systems governing nonequilibrium flows do not generate traveling wave solutions but the solutions enjoy better time decay. 6. Results and conclusion We use the method of multiple scales to obtain small-amplitude high-frequency asymptotic solutions to the basic equations governing one-dimensional unsteady planar and radially symmetric flows of a relaxing gas, and then study wave motion influenced by the effects of non-linearity, internal relaxation and geometric spreading. The theory of weakly non-linear geometrical acoustics is used to analyze situations where many waves coexist and interact with one another resonantly. The basic idea underlying the procedure is to separate the rapidly varying part of the solution from the slowly varying part; this is accomplished by averaging the solution with respect to the fast variables. Transport equations for the wave amplitudes are derived along the rays of the governing system; these constitute a system of inviscid Burgers’ equations with quadratic non-linearity coupled through a
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linear integral operator with a known kernel. The coefficients appearing in the transport equations provide a measure of coupling between the various modes and set a qualitative picture of the interaction process involved therein. In the absence of internal relaxation, the transport equations reduce exactly to those derived in [15,18] for the corresponding inviscid gas-dynamic planar (m = 0) case. It is observed that the wave fields associated with the particle paths do not interact with each other; however they do interact with an acoustic wave field to produce resonant contribution toward the other acoustic field. The acoustic wave fields may or may not interact, but in either case their net contribution, which is directed toward the entropy field, is always zero. Indeed, the governing system of Euler equations, thus, reduces to a pair of resonant asymptotic equations for the acoustic wave fields, which is closer to the original system as it includes to the leading order a model version of the non-linear resonantly interacting wave dynamics inherent in the full Euler equations. For a non-resonant multi-wave mode case, proposed by Hunter and Keller [14], the reduced system of transport equations gets decoupled with vanishing integral average terms, and the occurrences of shocks and caustics in the acoustic wave fields are analyzed. It is found that in a contracting piston motion having spherical symmetry, a shock is always formed before the formation of a focus no matter how small be the initial wave amplitude; this is in contrast with the corresponding cylindrical situation where a shock forms before the focus only if the initial amplitude exceeds a critical value. It is found that unlike the wave field associated with a contracting piston motion having spherical symmetry, which always culminates into a shock before the caustic, the solution to the leading order approximation can be shock free, provided the fraction of the thermal energy that exists in the relaxing mode is sufficiently large. In case of a source producing only outgoing waves, asymptotic forms of the shock amplitude and trajectory are obtained which show that the exponentially decaying shock amplitude depends very strongly on the initial energy that exists in the relaxing mode. We point out certain analogies and distinctions between the integro-differential systems arising in the case of a relaxing gas vis a vis the ordinary gas dynamic cast considered in [20]. In addition to the monotone decrease in L∞ andBV norms of entropy solutions, the effect of relaxation is
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to cause an exponential attenuation of these norms, as shown in (34). In particular, there are no traveling wave solutions or time recurrent solutions that have been observed in the ordinary gas dynamic case. Also the iterative procedure for solving the coupled integrodifferential system outlined in [20] may be employed in the relaxing case, however the estimates are not merely local but uniform in time due to the apriori exponential time decay. Acknowledgements The research supported by the CSIR research Grant 25(0122)/02/EMR-II, Human Resource Development Group, New Delhi, is gratefully acknowledged. References [1] P.A. Blythe, Nonlinear wave propagation in a relaxing gas, J. Fluid Mech. 37 (1969) 31–50. [2] H. Ockendon, D.A. Spence, Nonlinear wave propagation in a relaxing gas, J. Fluid Mech. 39 (1969) 329–345. [3] B.T. Chu, Weak nonlinear waves in nonequilibrium flow, in: P.P. Wegener (Ed.), Nonequilibrium Flows, vol. I, part II, Marcel Dekker, New York, 1970. [4] D.F. Parker, Propagation of damped pulses through a relaxing gas, Phys. Fluids 15 (1972) 256–262. [5] E. Becker, Chemically reacting flows, Ann. Rev. Fluid Mech. 4 (1972) 155–194. [6] J.F. Clarke, A. McChesney, Dynamics of Relaxing Gases, Butterworth, UK, 1976. [7] J.F. Clarke, Lectures on plane waves in reacting gases, Ann. Phys. Fr. 9 (1984) 211–306.
[8] V. Choquet-Bruhat, Ondes asymptotique et approchees pour systemes d’equations aux derivees partielles nonlineaires, J. Math. Pures Appl. 48 (1969) 119–158. [9] E. Varley, E. Cumberbatch, Nonlinear high frequency sound wave, J. Inst. Math. Appl. 2 (1966) 133–143. [10] P. Germain, Progressive waves, 14th Prandtl Memorial Lecture, Jarbuch, der DGLR, 1971, pp. 11–30. [11] V.D. Sharma, L.P. Singh, R. Ram, The progressive wave approach analyzing the decay of a saw-tooth profile in magnetogasdynamics, Phys. Fluids 30 (1987) 1572–1574. [12] V.D. Sharma, R.R. Sharma, B.D. Pandey, N. Gupta, Nonlinear analysis of a traffic flow, Z. Angew. Math. Phys. 40 (1989) 828–837. [13] Ch. Radha, V.D. Sharma, Propagation and interaction of waves in a relaxing gas, Philos. Trans. R. Soc. Lond. 352 A (1995) 169–195. [14] J.K. Hunter, J. Keller, Weakly nonlinear high frequency waves, Comm. Pure Appl. Math. 36 (1983) 547–569. [15] A. Majda, R. Rosales, Resonantly interacting weakly nonlinear hyperbolic waves, Stud. Appl. Math. 71 (1984) 149–179. [16] J.K. Hunter, A. Majda, R. Rosales, Resonantly interacting, weakly nonlinear hyperbolic waves II. Several space variables, Stud. Appl. Math. 75 (1986) 187–226. [17] G.B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974. [18] A. Majda, Nonlinear geometric optics for hyperbolic systems of conservation laws, in: C. Dafermos et al. (Ed.), Oscillation Theory, Computation and Methods of Compensated Compactness, IMA volumes in Mathematics and its Applications, vol. 2, Springer, New York, 1986. [19] J.K. Hunter, J. Keller, Caustics in nonlinear waves, Wave Motion 9 (1987) 429–443. [20] A. Majda, R. Rosales, M. Schonbeck, A canonical system of integro-differential equations arising in resonant nonlinear acoustics, Stud. Appl. Math. 79 (1988) 205–262.
Wave interaction in a nonequilibrium gas flow V.D. Sharma∗ , Gopala Krishna Srinivasan Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India Received 28 April 2004; received in revised form 20 December 2004
Abstract By employing the method of multiple time scales, we derive here the transport equations for the primary amplitudes of resonantly interacting high-frequency waves propagating into a non-equilibrium gas flow. Evolutionary behavior of nonresonant wave modes culminating into shocks or no shocks, together with their asymptotic decay behavior, is studied. Effects of non-linearity, which are noticeable over times of order O(−1 ), are examined, and the model evolution equations for resonantly interacting multi-wave modes are derived. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Wave interaction; Nonequilibrium gas flows; Shock waves
1. Introduction A number of problems relating to the effects of non-linearity and rate processes on wave propagation in gases with internal relaxation have been studied previously; in this context the contributions due to Blythe [1], Ockendon and Spence [2], Chu [3], Parker [4], Becker [5], Clarke and McChesney [6] and Clarke [7] are worth mentioning. Asymptotic methods for the analysis of weakly non-linear hyperbolic waves have received great attention in the past. ChoquetBruhat [8], while dealing with small-amplitude waves, proposed a method similar to that of Varley and ∗ Corresponding author. Tel.: +91 22 576 7482; fax: +91 22 572 3480. E-mail addresses: [email protected] (V.D. Sharma), [email protected] (G.K. Srinivasan).
0020-7462/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2005.02.003
Cumberbatch [9] to discuss shockless solutions of hyperbolic systems which depend upon a single phase function. A general discussion of single-phase progressive waves has been given by Germain [10]; using the progressive wave approach and the related procedures, Sharma et al. [11,12] and Radha and Sharma [13] analyzed the wave propagation problems in different material media. Based on a formal version of the weakly non-linear geometrical optics, Hunter and Keller [14] established a general non-resonant multiwave mode theory which has led to several interesting generalizations by Majda and Rosales [15], and Hunter et al. [16] to include resonantly interacting multi-wave mode features. In this paper, using the resonantly interacting multi-wave theory, we examine small-amplitude highfrequency asymptotic waves for one-dimensional unsteady planar and non-planar flows of a general
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inviscid relaxing gas, where at the leading order, many waves coexist and interact with one another resonantly, and obtain evolution equations which describe the resonant wave interactions inherent in the system. It is assumed that the rate of change of energy in the lagging mode depends only on the local values of pressure, entropy and internal energy, and the particular nonequilibrium phenomenon of interest is the vibrational relaxation; the rotational and translational modes are assumed to be in local thermodynamical equilibrium throughout.
2. Preliminaries and the basic equations It is well known that at high temperatures (∼ 1000 K), the internal energy of the gas molecules consists of translational, rotational and vibrational components. When the gas is in equilibrium, each internal energy mode is characterized by the same temperature. A rate of transfer of energy from one mode to another mode can be observed by inducing a small change in any of these temperatures and observing the rate of return to equilibrium. For instance, when a gas is compressed by the mechanical action of a piston or by the passage of a shock front, the whole energy goes initially to increase the translational energy, and it is followed by a relaxation from translational mode to rotational mode and also from translational mode to vibrational mode until the equilibrium between these modes is re-established. This is called a relaxation process and is governed by an equation of the form (1d) below, with q replaced by Ti , a measure of the energy in the molecular vibrational state, and = (T − Ti )/ where T is the translational temperature and the relaxation time; this corresponds to a gas model with single relaxing internal degree of freedom in which the internal energy is characterized by T and Ti . Under actual flow conditions, the internal energy is always tending toward the equilibrium for the new conditions; this time-lag, known as the relaxation time, is short for translation and rotation, and one may consider that these modes are adjusted instantaneously in all flow problems. However, the relaxation time of vibrational modes is much longer, as a result the adjustment of translational and rotational non-equilibrium in a gas flow requires relatively few collisions as compared
with that of vibrational non-equilibrium. When gradients in the flow field are large, as for example in boundary layers, translational non-equilibrium results in the phenomena known as viscous stress and heat conduction; similarly, rotational non-equilibrium gives rise to bulk viscosity, and thus, the molecular studies of situations involving translation and rotation in a gas require the effects of viscosity and heat conduction to be considered. Here, we shall assume that the flow is everywhere in instantaneous translational and rotational equilibrium, or, in other words, that all effects of viscosity (both bulk and shear), heat conduction and diffusion are negligible. For simplicity, we shall allow for only one non-equilibrium process, that is vibrational non-equilibrium, and so introduce in addition to the pressure and entropy, a third nonequilibrium variable q to specify the vibrational state of the gas. The introduction of an additional variable requires an additional equation (1d) known as the rate equation for the non-equilibrium process. To keep the treatment within desired length, we presume that the reader is acquainted with the governing equations as presented in the standard texts (see for example, Refs. [3,6,17]). Thus the basic equations governing the one-dimensional unsteady planar and non-planar flows of a relaxing gas with general thermodynamic properties in the absence of viscosity, heat conduction and body forces can be written in the form mu (1a) + af2 = 0, pt + up x + af2 ux + x px = 0, (1b) ut + uux + St + uS x = − qt + uq x = ,
, T
(1c) (1d)
where t is the time, x the spatial coordinates being either axial in flow with planar (m = 0) geometry or radial in cylindrically (m = 1) and spherically (m = 2) symmetric flows. The state variable u denotes the gas velocity, p the pressure, S the entropy, q the energy in the molecular vibrational state, af the frozen sound speed given by af2 = (jp/j)S,q , and the rate of change of energy in the vibrational mode, which is assumed to be a known function of p, S and q. The symbols , T and denoting respectively, the density, temperature and the affinity of internal transformation
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characterized by the variable q, are regarded as functions of p, S and q and are given by (invoking Gibbs’ relation T dS = dh − dp/ − dq)
jh 1 = , jp
jh = T, jS
jh = , jq
where h is the specific enthalpy related to p, S and q through the canonical equation of state h = h(p, S, q), and the entity is given by =−(/T )j/jS+j/jq. When the internal transformation attains a state of equilibrium, we have
(p, S, q) = 0 = (p, S, q) ⇒ q = q(p, ¯ S).
In this limit, the perturbations caused by the wave are of size O(), and they depend significantly on ¯ (x, t)/, where the fast characteristic variable ¯ = ¯ = (1 , 2 , 3 , 4 ) and ¯ = ( , , , ); indeed 1 2 3 4 i , 1 i 4, is the phase of the ith wave associated with the characteristic speed i . We denote the left and right eigenvectors of A0 associated with the eigenvalue i by L(i) and R (i) , respectively, satisfying the normalization conditions L(i) R (j ) = ij , 1 i, j 4, where ij denotes the Kronecker symbol. In fact, we have 2 af0 0 ∓af0 (1,4) L = , , 0, 0 , 2 2
Using matrix notation, Eqs. (1) may be cast in the following form:
L(2) = (0, 0, af−2 , 0), 0
jU jU +A + F = 0, jt jx
R (2) = (0, 0, af20 , 0),
(2)
where U and F are column vectors defined as U = (p, u, S, q) and F =(af2 +maf2 x −1 , 0, /T , −). A is a 4 × 4 matrix having components Aij , the non-zero ones are as follows: A11 = A22 = A33 = A44 = u, 1 A12 = af2 , A21 = . It follows immediately that system (2) admits four families of characteristics, two of which represent waves propagating in the ±x directions with the modified frozen sound speed u ± af , and the remaining two form a set of double characteristics representing entropy waves or particle paths. We consider waves, which propagate through a constant background state U0 = (p0 , 0, S0 , q0 ). The characteristic speeds at U = U0 are given by 1 = −af0 , 2 = 3 = 0, 4 = af0 . The subscript 0 refers to evaluation at U = U0 , and is synonymous with a state of equilibrium.
3. High-frequency limit Here we use the method of multiple scales to obtain small-amplitude high-frequency asymptotic solutions of (2), when the time scale 0 = −(j/jq)−1 0 , defined by the relaxation or rate phenomenon, is large compared with the signal period ˜ , that is = ˜ /0 >1.
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L(3) = (0, 0, 0, 1),
R (1,4) = (0 af20 , ∓af0 , 0, 0),
R (3) = (0, 0, 0, 1).
4. Weakly non-linear resonant waves We seek asymptotic solutions of (2) as → 0 of the form U ∼ U0 + U1 (x, t, ¯ ) + 2 U2 (x, t, ¯ ) + O(3 ), (3) where U1 is a smooth bounded function of its arguments and U2 is bounded in (x, t) in a certain bounded region of interest having at most sub-linear growth in as → ±∞. Now we use (3) in (2), expand A and F in a Taylor series in powers of about U = U0 , replace the partial derivatives j/jX (X being either x or t) by j/jX + −1 4i=1 (j i /jX)j/ji , and equate to zero the coefficients of 0 and 1 in the resulting expansions, to get 4 j i j i jU1 I = 0, + A0 O( ): jt jx ji i=1 4 j j jU2 O(1 ): I i + A0 i jt jx ji 0
(4)
i=1
jU1 jU 1 − A0 − (U1 · ∇F )0 jt jx 4 j i jU 1 (U1 · ∇A) , − jx ji
= −
i=1
(5)
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where I is the 4 × 4 unit matrix, and ∇ is the gradient operator with respect to the dependent variable U . Since each of the phases i satisfies the eikonal equation
j j Det I i + A0 i = 0, (6) jt jx we choose the simplest phase function solutions of this equation, namely
i (x, t) = x − i t,
1 i 4.
(7)
It follows from (4) that for each phase i , jU1 /ji is parallel to the right eigenvector R (i) , and thus U1 =
4
i (x, t, i )R (i) ,
(8)
i=1
where i = (L(i) · U1 ) is a scalar function called the wave amplitude, that depends on the ith phase variable i ; the dependence of i on i describes the wave form whether it is an oscillatory wave-train or a pulse. We suppose that i (x, t, i ) has zero mean value with respect to the phase variable i , that is, T 1 lim i (x, t, ) d = 0. (9) T →∞ 2T −T We then use (8) in (5) and solve for U2 . To begin with we write U2 =
4
mj R (j ) ,
j =1
which we substitute in (5) and premultiply the resulting expression by L(i) to yield the following system of decoupled inhomogeneous first-order PDEs: 4 j =1
(i − j )
jm i jj
j i j i − i − L(i) (U1 · ∇F )0 jt jx 4 jU1 − L(i) (U1 · ∇A) , 1 i 4. jj
=−
j =1
Suppressing the (x, t) dependence in k , and denoting by d i /ds the ray derivative j i /jt + i j i /jx,
the equation for mi assumes the form 4
(i − j )
j =1
=−
jm i jj
d i i j k d i − ai i − iii i − j k j ds d i jk j,k
:= Hi (x, t, 1 , . . . , 4 ),
1 i 4.
(10)
The characteristic ODEs for the ith equation in (10) are given by
˙ j = i − j
for j = i, ˙ i = 0, m ˙ i = Hi .
We asymptotically average (10) along the characteristics and appeal to the sub-linearity of U2 in , which ensures that expression (3) does not contain secular terms. The constancy of i along the characteristics and the vanishing asymptotic mean value of m ˙ i along the characteristics implies that the wave amplitudes i , 1 i 4, satisfy the following coupled system of integro-differential equations:
j i j i j i + i + ai i + iii i jt jx ji T 1 + ij k lim j (i + (i − j )s) T →∞ 2T −T i=j =k × k (i + (i
− k )s) ds = 0,
(11)
where k = j k /jk and the coefficients ai and ij k are given by ai = L(i) (R (i) · ∇F )0 ,
ij k = L(i) (R (i) · ∇A)0 R (k) . (12)
Here it may be noticed that for plane (m = 0) waves, coefficients ai vanish in the absence of internal relaxation as (2) is free from the source term (i.e. F = 0), and (11) reduces to the equations obtained in [18]. Thus, the coefficients ai in (11) describe the growth or decay behavior of the wave amplitude i on account of non-planar wave form and the relaxation phenomenon present in the medium. The interaction coefficients ij k , which are asymmetric (in j, k), measure the strength of coupling between the j th and kth wave modes (j = k) that can generate an ith wave (i = j = k). The coefficients iii account for the nonlinear self-interaction, and are non-zero for genuinely non-linear waves; these coefficients are indeed zero
V.D. Sharma, G.K. Srinivasan / International Journal of Non-Linear Mechanics 40 (2005) 1031 – 1040
for linearly degenerate waves. If all the coupling coefficients ij k (i = j = k) are zero or the integral averages in (11) vanish, the waves do not resonate and (11) reduces to a system of uncoupled Burgers’ equations. The coefficients ai , iii and ij k , given by (12), provide a qualitative picture of the non-linear interaction process present in the system under consideration, and can be easily determined in the following form; the non-zero ones being: 1 1 ma f0 b− , a3 = , 2 x 0 1 ma f0 a4 = b+ , 2 x j(af ) 4 1 2 44 = − 11 = af0 = , jp S,q 0 2 a j ( a ) f f 124 = − 421 = 0 = 1 , 0 jS p,q 0 1 j(a f ) 1 4 34 = − 31 = = 2 , 0 jq p,S a1 =
(13)
0
where b = (1/0 )(af20 /ae20 − 1) is positive and serves as the amplitude attenuation rate on account of relaxation, 0 = −(j/jq)−1 0 is the relaxation time of the 1/2 medium, and ae0 = (jp/j)S=S0 ,q=q0 is the equilibrium sound speed. It then follows that Eq. (11) reduces after some simplifications to
j 1 j 1 1 j 1 ma f0 1 − 1 − af0 + b− jt jx 2 x j1 T 1 1 + − lim K x, t, T →∞ 2T −T 2 (14)1 × 4 (x, t, ) d = 0, j 2 = 0, jt
(14)2
j 3 3 =− , jt 0
(14)3
ma f0 j 4 j 4 1 j 4 b+ 4 + 4 + af0 + jt jx 2 x j4 T 1 4 + + lim K x, t, T →∞ 2T −T 2 (14)4 × 1 (x, t, ) d = 0,
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where the coefficients b, and 0 are the same as in (13) and the kernel K is defined as + + K x, t, = 1 2 x, t, 2 2 + + 2 3 x, t, , (15) 2 with 1 and 2 being the same as in (13). For an inert gas, which needs only a pair of thermodynamic variables such as (p, S) or (, S) for the state of the medium to be completely specified, we obtain 2 j(a) , = 0, b = 0, = a0 2 jp S 0 20 a03 j(a −2 jT /j) a02 j(a) 1 = = , 0 jS p 2 j S 0 0 + + K x, t, = 1 2 x, t, , 2 2 and thus, in the absence of internal relaxation, Eqs. (14) reduce exactly to those derived in [15] for the corresponding plane (m = 0), inviscid ordinary gas dynamic case. The integral average term in (14)1 (respectively (14)4 ) represents contribution to the wave amplitude 1 (respectively, 4 ) on account of the non-linear interactions of the wave fields 2 and 3 with the acoustic wave field 4 (respectively, 1 ); this, indeed, plays an important role in explaining how the energy carried by the non-decaying wave ( 2 ) and the energy released on account of internal relaxation get partitioned among the acoustic modes. The non-linear terms proportional to 1 1 and 4 4 in (14)1 and (14)4 account for non-linear self-interactions which generate higher harmonics leading to the distortions of the wave profile and consequent shock formation; indeed, the two acoustic wave fields 1 and 4 exhibit a strong effect from the non-linearity present in system (1). The absence of self-interaction terms (14)1,4 show that the wave amplitudes 2 and 3 , corresponding to the repeated eigen mode, are linearly degenerate; thus we can solve (14)2,3 for 2 and 3 and then use the expressions in K (given by (15)) that appears in (14)1,4 . It may be observed that the wave fields 2 and 3 , associated with the repeated eigenvalue do not interact with each other; however they do interact with an acoustic wave field to produce resonant contributions
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only toward the other acoustic wave field. Similarly the acoustic wave fields 1 and 4 may or may not interact with each other but in either case their net contribution, which is directed toward the entropy field only, is always zero. If the initial data U10 (x, ) are specified at t = 0, these determine the initial data for i that is functions 0i (x, ) so that at i |t=0 = 0i (x, i ). Thus, 2 (x, t, )= 02 (x, ), 3 (x, t, )= 03 (x, )e−t/0 , and subsequently system (14) reduces to a pair of equations for the acoustic wave fields 1 and 4 , coupled through a linear integral operator involving the kernel
K(x, t, ) = 1 02 (x, ) + 2 e−t/0 03 (x, ).
(16)
The pair of resonant asymptotic equations is much closer to the original system as it includes (to leading order) a model version of the non-linear resonantly interacting wave dynamics inherent in the full Euler equations (1). If the initial data 0i (x, ) are 2 periodic functions of the phase variable , only even harmonics in 02 (x, ) and 03 (x, ) contribute in (14), and the pair of resonant asymptotic equations take the following simplified form: j 1 j 1 1 j 1 1 ma f0 1 − 1 − − af0 + b− jt jx 2 x j1 2 1 + × K x, t, 4 (x, t, ) d = 0, (17) 2 − j 4 j 4 1 j 4 1 ma f0 4 + 1 + + af0 + b+ jt jx 2 x j4 2 4 + × K x, t, 1 (x, t, ) d = 0, (18) 2 − where K is given by (16).
inviscid Burgers’ equation (14) with vanishing integral average terms, is uniformly valid to the leading order until shock waves have formed in the solution. In terms of the characteristic field associated with r (r = 1, 4), and defined by the equations
r dr , = dx af0
dt er , = dx af0
(20)
where er = −1 or +1 according as r = 1 or 4, respectively, the decoupled equation (14)1 can be written as d r m ber − r , (21) = dx 2af0 2x along any characteristic curve belonging to this field, and yields on integration
r = 0r (sr , r )(x/sr )−m/2 exp(−bt/2),
(22)
along the rays sr = x − er af0 t = const, where the function 0r is obtained from the initial condition (19), and the fast variable r parametrizes the set of characteristic curves (20)1 . We thus obtain from (20)
r = r − er 0r (sr , r )Jr(m) (t),
(23)
t (m) where Jr (t)= 0 (1+(af0 er /sr )t )−m/2 exp(−bt /2) dt . Eq. (22) modify the usual results of gas dynamics concerning the evolution of cross-sectional area of a ray tube; in fact, the wave amplitude decays exponentially in contrast to the corresponding classical gas dynamics case, where it decays like x −m/2 (see [14]). Thus, the solution of (1), satisfying (19), with U10 (x, x/t) having compact support, is given by p = p0 + 0 af20 x −m/2 e−bt/2 ((x − af0 t)m/2 × 04 (x − af0 t, 4 ) + (x + af0 t)m/2
5. Non-linear geometrical acoustics solution The asymptotic approximate solution (3) of the hyperbolic system in (1) or (2), satisfying smallamplitude oscillating initial data U (x, 0) = U0 + U10 (x, x/) + O(2 ),
(19)
is non-resonant if the functions U10 (x, x/) are smooth with a compact support [18]; indeed expansion (3) with U1 given by (8), where the wave amplitude i , 1 i 4, solve a reduced system of decoupled
× 01 (x + af0 t, 1 )) + O(2 ), u = af0 x −m/2 e−bt/2 ((x − af0 t)m/2 04 (x − af0 t, 4 ) − (x + af0 t)m/2 01 (x + af0 t, 1 )) + O(2 ), S = S0 + af20 02 (x, x/), q = q0 + 03 (x, x/), where the fast variables r , given by (23), are chosen such that at t = 0, r = x/, and the initial data for i (1 i 4) are determined from the data U10
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specified at t = 0 as 1 0 , 1 (x, 1 ) = 20 af20 (p10 (x, 1 ) − 0 af0 U10 (x, 1 )),
02 (x, x/) = S10 (x, x/), 03 (x, x/) = q10 (x, x/), 1 0 (p10 (x, 4 ) + 0 af0 U10 (x, 4 )). 4 (x, 4 ) = 20 af20 This completes the solution of (1) and (19); any multivalued overlap in this solution has to be resolved by introducing shocks into the solution. It may be observed from (23) that for planar flows (m = 0), there will be an overlap in the region t > 0, formed by those characteristics for which j 01 /j1 > 0 (respectively, j 04 /j4 < 0), assuming that > 0. Thus, in the wave field 1 (respectively, 4 ), a shock can occur only if 01 > b/2 > 0 (re spectively, 04 < − b/2 < 0) at time t = t1 (respectively, t = t4 ) given by t1 = −(2/b) ln(1 − b/ 1 ) (respectively, t4 = −(2/b) ln(1 + b/ 4 )), where 0r ≡ j 0r /jr and r = Sup{ 0r (r )}. 5.1. Cylindrical and spherical waves The cases of converging/diverging waves belong to a particular class of piston motion and are indeed of special interest as they permit to establish, in particular, the trend of the influence of relaxation on the formation and propagation of shock waves. If we consider the situation in which the initial data occurring on the surface with radial coordinates x = x1 propagate into the gas-filled region 0 x x1 , it will give rise to converging waves. It may be noted from (22) that for m=1, 2 the rays focus and the wave amplitude 1 → ∞ as t → t ∗ = /af0 where = inf{s1 } indicating the formation of a caustic at t = t ∗ , where the geometrical optics approximation is rendered invalid; the reader is referred to the work of Hunter–Keller [19], who investigate the singular behavior near the caustic. (1) Since the integral J1 in (23) converges to a finite positive limit (say K1 ) as t → t ∗ , it follows that a shock can occur in the wave field 1 at time tc < t ∗ (1) given by the solution of J1 =1/1 in the wave field 1 , only when the initial value 01 exceeds a critical value 1/K1 .
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However unlike the cylindrical case, as the integral (2) J1 becomes unbounded in a finite time, a shock is always formed before the caustic in the wave field 1 , no matter how small the initial value 01 associated with a contracting piston motion with a spherical symmetry. Further, if we consider the situation when the initial data, occurring on the surface with radial coordinate x = x1 , propagate into the gas-filled region x > x1 , it will give rise to diverging waves. It may be (m) noticed that the integral J4 (t) in (23) converges to a finite limit lm (m = 1, 2) as t → ∞, implying thereby that a shock can appear in the wave field 4 in a finite (m) time t =ts given by the solution of J4 (ts )=−1/4 . Thus, except for the wave field 1 associated with a spherically symmetric flow, if the magnitude of the initial data 0r is less than a critical value the solution to the leading order of approximation remains shock free. In fact, when >1, or the fraction of thermal energy proportional to b that exists in the relaxing mode is sufficiently large, or the initial profile 0r is such that r → 0, or, the derivative 0r (r ) is small, the initial data do not give rise to shocks. 5.2. Shocks As discussed in the preceding sub-section, a shock wave may be initiated in the flow region, and once it is formed, it will propagate by separating the portions of the continuous region. At shocks, the correct generalized solution satisfies the Rankine–Hugoniot jump conditions for the shock location sr (t) in the t–r plane; thus if sr (t) be the shock location in the r wave field, it can be shown following [14] that 1 dsr (+) = rrr ( (−) r + r ), dt 2
r = 1, 4,
(24) (+)
which is the shock speed in the r –t plane. Here r (−) and r are the limiting values of r from the upstream and downstream sides of the shock, respectively. For the front shock, which has an undisturbed (+) flow ahead of it, r = 0. Dropping the superscripts (−) s on r and r , and using (22) we have dr
= er 0r (sr , r )(x/sr )−m/2 exp(−bt/2). dt 2
(25)
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Eq. (25), together with another relation for the shock speed obtained from (23), yields the following relation which relates r to t on the shock: r 2er (m) 0r (t ) dt . (26) Jr (t) = − ( 0r )2 0 Eq. (26) together with
r = r − 2( 0r )−1
r 0
0r (t ) dt ,
(27)
which is obtained from (23) for the front shock determines the shock path parametrically. If 0r (0) = 0, the shock starts immediately at the origin, and the relation between r and t is given by r ∼ −( /2)er ∗r Jr (t); ∗r = 0r |r =0 , so that r ∼ (m) ( /2)er ∗r Jr (t), implying that the shock starts with velocity 21 er ∗r (x/sr )−m/2 exp −bt/2, which is quite consistent with (25). In case of a source producing only outgoing waves, if the compressive phase (0) is followed by a rarefaction so that r (r ) → 0 as ∗ r → r , the wavelets in the neighborhood of the wavelet r = ∗r will never reach the shock, and then the shock is asymptotic to r ∼ ∗r + Ker 2 Jr(m) , 1/2 ∗r 0r (t ) dt , K = −er 0
and the amplitude in (22) behaves like t er af0 t −m/2 r ∼ 2/ Ke−bt/2 1 + e−bt /2 sr 0 −m/2 −1/2 er af0 t × 1+ dt . (28) sr Relation (28) shows that the shock decays exponentially; indeed, the shock amplitude depends very strongly on the internal energy (proportional to b) that exists in the relaxing mode and depends on the wave geometry through the parameter m as well as the body (piston) shape through the factor K. It is evident from (28) that for an inert gas (b = 0), plane (m = 0), cylindrical (m = 1) and spherical (m = 2) shock waves decay like t −1/2 , t −3/4 and t −1 (log t)−1/2 , respectively; these results are in agreement with earlier results [17].
5.3. Analysis of the integro-differential system We close this section with a few comments on the analysis of the integro-differential system (17)–(18), that parallels the treatment in [20]. Introducing the ray derivative, Eqs. (17) and (18) for the planar case (m = 0) assume the following forms: j 1 1 + 1 b 1 d 1 − K x, t, + − 1 ds 2 j1 2 − 2 × 4 (x, t, ) d = 0, (29) j 4 4 + 1 b 4 d 4 + K x, t, + + 1 ds 2 j4 2 − 2 (30) × 1 (x, t, ) d = 0, The system has linear part that can be cast in the form b Ys + Y + LY = 0, 2
(31)
where Y = [ 1 , 4 ]T and L is the operator given by 1 1 + K x, t, 4 (x, t, ) d , LY = − 2 − 2 4 + 1 K x, t, 2 − 2
(32) × 1 (x, t, ) d . We assume that the kernel K(x, t, ) is summable on the interval [−, ] for the variable . The operator norm of L as an operator on the spaces L∞ (R) and BV (R), is given by KL1 [−,] . On multiplying (31) by Y T and integrating with respect to we see that the L2 norm Y (·, t) of solutions of (31) decays exponentially and we have the estimates LU L∞ KL1 U L∞ , Var(LU ) KL1 Var(U ),
(33)
where we have denoted by Var(U ) the BV-norm of U . We now assume for simplicity that 1 in (16) vanishes and 03 is a 2-periodic function of alone, which is even as it was observed in Section 4 that only even harmonics contribute to the integro-differential terms. Multiplying (31) by ebs/2 , integrating and using the Gronwall lemma, we get the decay estimate
Y (·, t)L∞ exp(−bt/2 + 2 0 03 L1 × (1 − e−t/0 ))Y (·, 0)L∞ ,
(34)
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with a similar estimate for the BV norm. So it follows immediately that due to relaxation effects we have a uniform L∞ and BV estimates for the solutions of (31). If, instead, the coefficient 1 in (16) is different from zero, the exponential term in (34) gets modified to
exp(−bt/2 + 1 t 02 L1
+ 2 0 03 L1 (1 − e−t/0 )).
In any case if the attenuation rate b due to the relaxation is strong enough, we still have a uniform estimate on the L∞ and BV norms of solutions of (31). For entropy solutions of the pure inviscid Burgers’ equation ws + ( /2)ww = 0,
(35)
the L∞ and BV norms are monotone decreasing in time. The method employed in [20] is to alternately interlace approximate solutions of (31) and (35), obtain in the limit a solution to the full non-linear system (29)–(30) and estimates (34) hold for the solutions of the full non-linear problem. In conclusion, we would like to add that the case considered in [20] is conservative admitting reductions to local systems with traveling wave solutions. With relaxation effects, the systems governing nonequilibrium flows do not generate traveling wave solutions but the solutions enjoy better time decay. 6. Results and conclusion We use the method of multiple scales to obtain small-amplitude high-frequency asymptotic solutions to the basic equations governing one-dimensional unsteady planar and radially symmetric flows of a relaxing gas, and then study wave motion influenced by the effects of non-linearity, internal relaxation and geometric spreading. The theory of weakly non-linear geometrical acoustics is used to analyze situations where many waves coexist and interact with one another resonantly. The basic idea underlying the procedure is to separate the rapidly varying part of the solution from the slowly varying part; this is accomplished by averaging the solution with respect to the fast variables. Transport equations for the wave amplitudes are derived along the rays of the governing system; these constitute a system of inviscid Burgers’ equations with quadratic non-linearity coupled through a
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linear integral operator with a known kernel. The coefficients appearing in the transport equations provide a measure of coupling between the various modes and set a qualitative picture of the interaction process involved therein. In the absence of internal relaxation, the transport equations reduce exactly to those derived in [15,18] for the corresponding inviscid gas-dynamic planar (m = 0) case. It is observed that the wave fields associated with the particle paths do not interact with each other; however they do interact with an acoustic wave field to produce resonant contribution toward the other acoustic field. The acoustic wave fields may or may not interact, but in either case their net contribution, which is directed toward the entropy field, is always zero. Indeed, the governing system of Euler equations, thus, reduces to a pair of resonant asymptotic equations for the acoustic wave fields, which is closer to the original system as it includes to the leading order a model version of the non-linear resonantly interacting wave dynamics inherent in the full Euler equations. For a non-resonant multi-wave mode case, proposed by Hunter and Keller [14], the reduced system of transport equations gets decoupled with vanishing integral average terms, and the occurrences of shocks and caustics in the acoustic wave fields are analyzed. It is found that in a contracting piston motion having spherical symmetry, a shock is always formed before the formation of a focus no matter how small be the initial wave amplitude; this is in contrast with the corresponding cylindrical situation where a shock forms before the focus only if the initial amplitude exceeds a critical value. It is found that unlike the wave field associated with a contracting piston motion having spherical symmetry, which always culminates into a shock before the caustic, the solution to the leading order approximation can be shock free, provided the fraction of the thermal energy that exists in the relaxing mode is sufficiently large. In case of a source producing only outgoing waves, asymptotic forms of the shock amplitude and trajectory are obtained which show that the exponentially decaying shock amplitude depends very strongly on the initial energy that exists in the relaxing mode. We point out certain analogies and distinctions between the integro-differential systems arising in the case of a relaxing gas vis a vis the ordinary gas dynamic cast considered in [20]. In addition to the monotone decrease in L∞ andBV norms of entropy solutions, the effect of relaxation is
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to cause an exponential attenuation of these norms, as shown in (34). In particular, there are no traveling wave solutions or time recurrent solutions that have been observed in the ordinary gas dynamic case. Also the iterative procedure for solving the coupled integrodifferential system outlined in [20] may be employed in the relaxing case, however the estimates are not merely local but uniform in time due to the apriori exponential time decay. Acknowledgements The research supported by the CSIR research Grant 25(0122)/02/EMR-II, Human Resource Development Group, New Delhi, is gratefully acknowledged. References [1] P.A. Blythe, Nonlinear wave propagation in a relaxing gas, J. Fluid Mech. 37 (1969) 31–50. [2] H. Ockendon, D.A. Spence, Nonlinear wave propagation in a relaxing gas, J. Fluid Mech. 39 (1969) 329–345. [3] B.T. Chu, Weak nonlinear waves in nonequilibrium flow, in: P.P. Wegener (Ed.), Nonequilibrium Flows, vol. I, part II, Marcel Dekker, New York, 1970. [4] D.F. Parker, Propagation of damped pulses through a relaxing gas, Phys. Fluids 15 (1972) 256–262. [5] E. Becker, Chemically reacting flows, Ann. Rev. Fluid Mech. 4 (1972) 155–194. [6] J.F. Clarke, A. McChesney, Dynamics of Relaxing Gases, Butterworth, UK, 1976. [7] J.F. Clarke, Lectures on plane waves in reacting gases, Ann. Phys. Fr. 9 (1984) 211–306.
[8] V. Choquet-Bruhat, Ondes asymptotique et approchees pour systemes d’equations aux derivees partielles nonlineaires, J. Math. Pures Appl. 48 (1969) 119–158. [9] E. Varley, E. Cumberbatch, Nonlinear high frequency sound wave, J. Inst. Math. Appl. 2 (1966) 133–143. [10] P. Germain, Progressive waves, 14th Prandtl Memorial Lecture, Jarbuch, der DGLR, 1971, pp. 11–30. [11] V.D. Sharma, L.P. Singh, R. Ram, The progressive wave approach analyzing the decay of a saw-tooth profile in magnetogasdynamics, Phys. Fluids 30 (1987) 1572–1574. [12] V.D. Sharma, R.R. Sharma, B.D. Pandey, N. Gupta, Nonlinear analysis of a traffic flow, Z. Angew. Math. Phys. 40 (1989) 828–837. [13] Ch. Radha, V.D. Sharma, Propagation and interaction of waves in a relaxing gas, Philos. Trans. R. Soc. Lond. 352 A (1995) 169–195. [14] J.K. Hunter, J. Keller, Weakly nonlinear high frequency waves, Comm. Pure Appl. Math. 36 (1983) 547–569. [15] A. Majda, R. Rosales, Resonantly interacting weakly nonlinear hyperbolic waves, Stud. Appl. Math. 71 (1984) 149–179. [16] J.K. Hunter, A. Majda, R. Rosales, Resonantly interacting, weakly nonlinear hyperbolic waves II. Several space variables, Stud. Appl. Math. 75 (1986) 187–226. [17] G.B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974. [18] A. Majda, Nonlinear geometric optics for hyperbolic systems of conservation laws, in: C. Dafermos et al. (Ed.), Oscillation Theory, Computation and Methods of Compensated Compactness, IMA volumes in Mathematics and its Applications, vol. 2, Springer, New York, 1986. [19] J.K. Hunter, J. Keller, Caustics in nonlinear waves, Wave Motion 9 (1987) 429–443. [20] A. Majda, R. Rosales, M. Schonbeck, A canonical system of integro-differential equations arising in resonant nonlinear acoustics, Stud. Appl. Math. 79 (1988) 205–262.