An Exact Integral of Complete Spectral Equations For Unsteady One Dimensional Flow

7/30/2019 An Exact Integral of Complete Spectral Equations For Unsteady One Dimensional Flow

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An exact integral of complete spectral equations forunsteady one-dimensional flow

By GEORGE W. PLATZMAN, The Univereity of Chicago2(Manuscript received June 15, 1964)

ABSTRACTFinite-amplitude, one-dimensional, sothermal flow of a perfect gas is considered. Whenthe flow is organized initially as a linear sound wave, it is equivalent to the solutionof an advection equation. The flow that evolves in this way from an initial state isanalyzed by means of a Fourier series in space, with time-dependent coefficients. Thesecoefficients can be determined exactly when the initial velocity is a pure sinusoid. Theresult is such tha t the Fourier series in space is a Kapteyn series in time. The series isvalid up to the time of shock formation. Throughout this interval the lowest harmonicsinferred from severely truncated spectral equations are in good agreement with theexact solution. However, the spectral representation becomes very inefficient as thediscontinuous state is approached.

1. IntroductionNumerical integration of th e hydrodynamic

equations in th e spectral domain has been car-ried o ut in numerous investigations. Exact solu-tions of severely truncated spectral equationsoften can be obtained, bu t th e errors of spectra1trunc ation are difficult to assess. The purpose ofthe present note is to make a contribution tothis problem and to give a result of intrinsicmathematical interest., namely a n exact solutionof the complete system of nonlinear spectralequations in a special case. This solution isclosed, analy tic a nd time-dependent.

2. One-dimensional isothermal flowThe momentum and mass equations for one-

dimensional isothermal flow of a perfect g as are,respectivelyaa

= - ax : ( 2 . 1a )Presented at the International Symposium onNumerical Weather Forecasting, Oslo, March 1963.This investigation was supported through fundsprovided by the National Science Foundation (GrantNSF-GP-471, Technical Report 16, March 1964). Itwas completed a t the National Center for Atmo-spheric Research, Boulder, Colorado, where the

author was in residence during 1963-64.

au ( 2 . 1 b )

Here (I=cIn(e/eJ, where c =(RT)*s the soundspeed ( =constant) an d el, is a standard density( =constant). I f the second equation is multi-pIied by &1 and added to the first, the result is

The general solution of ( 2 . 2 ) su + a = f ( z ( u + c ) t )u -a =g(z - ( u - c ) t ) ,

( 2 . 3 a )( 2 . 3 b )

where f and g are functions that may be re-garded as fixed by initial conditions:

f ( 4 =u(2,O)+dz, )g(4=u(z, ) -4% )

( 2 . 4 )

We shall assume tha t u(z,O) and o(z,O) can beassigned arbitrarily .

As a special case, consider a disturbanceorganized initially as a linear sound wavemoving along +z with speed +c . Thena(z, )=u(z,0), so ( 2 . 4 )makes f(z)=2 u ( 2 , 0 ) andg(z) =O . Hence a(z,t) u(z , t ) from ( 2 . 3 b ) , and( 2 . 3 ~ )ives

Tellus XVI (1964), 4


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INTEGRAL OF COMPLETE SPECTRAL EQUATIONS FOR UNSTEADY ONE-DIMENSIONAL FLOW 423u =P ( x- u+c ) t ) , (2 .5 )

where P( x) -u(x, 0). The particular integral ( 2 .5 )was obtained first by Poisson in 1808. The dis-tortion implied by (2 .5 ) is the subject of afamous paper by Stokes in 1848 On a difficultyin the theory of sound.3. Advection equation

I n the flow under consideration the soundspeed c is uniform and constant. If the nonlinearterms on the left in ( 2 .1 ) were absent, (2 .5 )would be u =F(x- t); thus, the disturbanceu(z , ) =a(z, ) would be sta tionary in a referenceframe moving with speed c . The distortionproduced by nonlinear effects therefore is bestexamined relative to this frame. I f x -c t isreplaced by z, equations (2 . la ) and ( 2 . l b )become identical to

( +u u=0, (3 .1 )since d =u. he general solution of (3 .1 ) is

u =P(z- t), (3 .2)as is evident also from ( 2 .5 ) .We shall carry outall further discussion of our problem on thebasis of (3 .1 ) and (3 .2 ) .It is clear from (3 .1 ) that du /d t = O alongcharacteristic curves dx/dt =u n the x, t-plane.Therefore, the characteristics are straight lines,and the motion corresponds to t ha t of simplewaves (COURANT nd FRIEDRICHS,948). More-over, (3 . l ) has only one family of characteristics,so the motion is analogous to tha t of t hekinematic waves discussed by LIGHTHILLndWITHAM 1955). It should be noted that in thefactor u + c on the right side of (2 .5 ) , the term uexpresses kinematical effects upon the wave,whereas c expresses dynamical effects. Thelatter can be eliminated, formally, as in (3 .2) ,because c is uniform in the special problembeing considered.A discussion of ( 3 .1 ) would not be completewithout reference t o th e equation

the simplest hydrodynamical-type equation tha tcontains the mechanism of counteraction be-Tellus XVI (1964), 4

tween diffusion and nonlinear convection.BATEMAN1915) pointed out that ( 3 .3 ) has th eexact solution

u = C - Uth[U(z -ct)/Bv],where c is an arbitrary, uniform translationspeed and U an arbitrary, uniform velocity scale.This gives a smooth shock of width v/U pro-pagating along x with speed c . COLE ( 1 9 5 1 )noted that ( 3 .3 ) is invariant under the trans-formationx c t 4%;+t; u - +u; ence, withoutloss of generality we can write c =0 in the pre-ceding solution:

u =- U t h ( U ~ 1 2 ~ ) . ( 3 .4 )This is a steady solution of (3 .3 ) , in which dissi-pation in the shock is exactly balanced by con-vection of kinetic energy from infinity (COLE,1951).The following remark is pertinent in connec-tion with (3 .4 ) . Let u=-Uthkx be an initialcondition for (3 .1 ) ,so that t he solution of (3 .1 ) is

u =-Uthk(x -ut),in accordance with ( 3 .2 ) . This is an unsteadysolution, in which the shock width (initially oforder k -1 ) steadily decreases until a discontinuityis formed. On the other hand, if we regardu=-Uthkx as an initial condition for (3 .3) ,rather than for ( 3 .1 ) , then there is a limitingsteady state-given uniquely by (3.4)-whichis independent of k. The preceding statement,which is plausible intuitively, was proved for-mally by COLE (1951) (in the case k - +m ) .

The most remarkable aspect of ( 3 .3 ) is theexistence of a general, exact solution. This solu-tion was discovered independently by HOPF(1950) and COLE ( 1 9 5 1 ) , and will be given atthe end of the next section. LIGHTHILL (1956)has discussed the matter further, and in parti-cular has shown exactly the circumstances underwhich ( 3 .3 ) is a valid approximation for soundwaves of finite amplitude.Another work which bears a close relation tothe present one is th e well-known investigationof mathematical models of turbulent flow byBURGERS1939, 1948), who studied equationssimilar to (3 .3 ) . Statistical properties of thisequation have been examined by BURGERS(1952), REID (1957), and OGURA1957).It is evident that ( 3 .1 ) is a special case of


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424 GEORGE W. PLATZMANu t

1 I-&U7 +ii -7 0 + aX X

FIG.1. Left panel: profiles of u as a function of x forselected 1. Broken line in triple-valued region showssingle-valuedcomposite function u* represented byFourier series (5.6). Right panel: x, -diagramshowing characteristics u =0 and u = 1. Heavycurve with cusp at t =1 is envelopeof characteristics.(3.3). The significance of equation (3.1) in thepresent investigation is simply that the non-linear term uaulax contains the most rudimen-ta ry form of the mechanism whereby the spec-trum of motion is modified through inertialexchange between spectral components of dif -ferent scales. From the point of view of numeri-cal integration, this equation has been studiedby PLATZMAN1954),OBUKHOV1957), BENTON(1958), and by PHILLIPS1960), who suggestedthe name advection equation.4. Sinusoidal solution

The special case u(x,O)=- Usinkx will beconsidered throught the sequel. The generalsolution (3.2) therefore gives

u =-Usink(z -ut).We adopt the constants U and k- as units ofvelocity and length; thus

u =- in(x -ut), (4.1)on the understanding that u, x, now correspondto what previously were u /U , kx, kUt.

The right panel of Fig. 1 is an x, t-diagram

which shows a few typical characteristics of(4.1).Also shown (by heavy lines) is th e envelopeof the cusp region, inside which the solutionis triple valued ( three characteristics througheach poin t). The left panel of Fig. 1 shows thex, u-configuration of the solution a t selected.times.

Since each value of u is propagated along xwith speed u, i t follows that the wave crest(u >0) is propagated toward x-positive and thetrough (u Iu2 , then u1 is propagated more rapidlythan u2,so the slope S =&/ax of the wave pro-file becomes steeper where S s negative ini-tially, and flatter where S is positive initially.This process may be examined quantita tively bycomputation of th e change of slope along acharacteristic:

by (3.1 .). Integration yieldss=( t+s0-y1, (4.2)

where So s the initial value of S . This equationshows that for each value of u where So1 the solution (4.1) is triple-valued inan expanding zone between the two pointswhere the slope is infinite-one forward-movingand one backward-moving. To ob tain th e coor-dinates of these points, write (4.1) parametri-cally:

u = -sin8 (4.3n)x = 8 -tsin8. (4.3b)

From this we find S =( t - sec8)- for the slopea t any point. Hence, where th e slope is infinite,

x = 8 - an8.t =sec8, (4.4)

which gives th e envelope of characteristics(shown by the two heavy curves in the rightpanel of Fig. 1) . On the envelope, u =(1 - t - * ,



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INTEGRAL OF COMPLETE SPECTRAL EQUATIONS FOR UNSTEADY ONE-DIMENSIONAL FLOW 425

xFIG. . Solutions (4.8) of (3.3) at t =1 and t =2, forReynolds number R =1 (broken curves) and R =100(solid curves). Dotted curve shows initial condition.

so the points where S is infinite graduallyapproach the crest and trough of the wave, andthe regions where S


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426 GEORGE W. PLATZMAN

0 2 3 0

FIQ.3. Fourier coefficients u,(t) evaluated from ( 5 . 5 ) . Left panel: n =1(1)6; ight paneI: n =7(1)10(5)30(10)40. (Note that ordinate scalesof left and right panels differ by a factor of 10).m the result is a modified form of Bessels integral:-u 2, t )= 2 u, ( t ) sin nz, ( 5 .1 )n = l

J , (2) = cos (no- sin 0) cos 0 do .the transform of which is n7c Su ( z , ) sin nxdz. ( 5 . 2 )

(Note that u, is defined as the spectrum ofminus u.)t is remarkable, and one of the mainpoints of this note, tha t in spite of the implicitnature of the solution (4 .1) , the preceding inte-gral for u,(t) can be evaluated explicitly.

By partial integrationun( t )= - - cos nxdu, (5 .3 )

and after substitution of the parametric equa-tions (4 .3 ) ,

cos n (0 - sin e) cos 0 do . (5 .4 )Bessels integral for J , ( z ) is

J , (2)==- cos (n0 sin 0) d0 .If this is inserted on the right side of the re-currence formula

Comparison with (5.4) eads at once toun(t)=BJ,(nt)/n*t. ( 5 . 5 )

Equation ( 5 . 5 ) is a simple, closed formula forthe Fourier coefficients of -u(x,t).

In view of ( 5 . 5 ) he Fourier expansion ( 5 .1 ) isexplicitly

Regarded as a series of Bessel functions, th is isa Kapteyn series, the distinctive feature ofwhich is presence of the order n as a factor inthe argument of J, . It is a curious fact th at (5.6)is completely equivalent to Bessels solution ofKeplers problem concerning the anomalies of aplanetary orbit-the problem which led Besselto define the functions that now bear his name(see WATSON, 944 , 17 .2 ) . The parameters ofKeplers problem and their formal relations tothose of (4 .3 ) are the mean anomaly M Ez ,the eccentric anomaly E S O , and the orbitaleccentricity ~ = t ,n terms of which ( 4 .3 b ) isM =E -&sinE . Keplers problem is inversionof the la tter relation: to express E explicitly interms of M . Writing this as E =M +&sin andintroducing (5.6) for sin E ( =-u y ( 4 .3 a ) ) , wefind



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INTEGRAL OF COMPLETE SPECTRAL EQUATIONS FO R UNSTEADY ONE-DIMENSIONAL FLOW 427c

FIG. . Schematic profile of u &s a function of x fort z 1. Broken curve: original, triple-valued u ; solidcurve: single-valued composite u* represented byseries (6.6).

mE = M + 2 X n - J n ( n e ) innM,n = iwhich is Bessels solution of Keplers problem.

Fourier coefficientsu,(t) were computed from(5.5)for 0 < t 4 . The left panel of Fig. 3 showsresults for n =1(1)6, the right panel (on an ex-panded scale) for n =7 ( 1)10(5)30( 0)40. Allu,(t) are zero initially with exception of u,(O)=1.As t increases from zero, ul( t ) radually declinesas energy cascades through the spectrum. Fig. 3exhibits this process in a striking manner. Ineach harmonic the energy increases monotoni-cally throughout the physical range 0 < c 1and reaches a maximum very soon after t =1.(If t, denotes the time of incidence of this firstmaximum, then t , + + in the limit n+o.)Afterthe first maximum un(t)s oscillatory and decaysto zero (as . ultimately).

When t >1 the function u(x,t) determined by(4.1) is multiple-valued, as shown in Fig. 1 .However, the series ( 5 . 6 ) is convergent andsingle-valued for all t . The meaning of this seriestherefore must be reconsidered for t >1. It ispossible to sum (5 .6 ) as a Kapteyn series in th emanner described by WATSON1944, $ 17.7),but a simpler route is to examine the integral(5.3) from which u,( t ) was obtained. In order tomake the transition from (5.2) to (5.3), it isnecessary to regard (5.2) aa a line integral, asindicated by arrows in Fig. 4. Now, on theinterval of the x-axis in which u s triple-valued,d x is positive along BCD and FGH, but nega-tive along DE F. Therefore, the effect of trans-formation from (5.2) to (5.3) s such tha t on thisinterval u has the composite value u* =u- n++ u , where u u on BCD, u u on DEF,and u=u on FGH. This composite valueis indicated by the solid curve B EH in Fig. 4Tellus XVI (1964), 429- 42899

(and by the broken curves in the left panelof Fig. 1).

The series (5 .6 ) for t >1 therefore representsthe single-valued function indicated by thesolid curve ABEHI in Fig. 4. Since the com-posite function u* n the interval BEH does notsatisfy the basic differential equation (3.1), th epreceding analysis must not be construed ashaving any physical meaning.

At t =1 , the limit of the physical range ofvalidity of ( 5 . 5 ) , the spectral distribution of u,for large n may be found from the asymptoticformula (WATSON, 944, 8.2)

J, (n)mI(l)/(48)nn*. (5.7)According to (5.5), the correspondingu, ~ n - ~ .The energy spectrum at t =1 therefore has thedistribution n-.. This shows an attenuationstronger than that for statistical equilibrium(n-I) by a factor n- l .

The rate of transfer of energy across the spec-trum at wave number n is

Therefore, for large n we have

With ( 5 . 5 ) for u, t ) , this gives at t = 1:de,/dnw -8n-J,(n)[JI,(n)-n-J,(n)]. (5.9)According to WATSON1944, $ 8.2), for large n,

Hence, in view of (5.7), the first term in thebracket of (5.9) dominates the second, and wehave

de,/dn w - 8/n1/3)n-,with the aid of I?(#)r3) =2n/1/3. Integrationwith respect to n yields

en -(8/n1/3)n-1. (5.10)The dependence of en upon n exhibits the degreeto which the spectrum is nonstationary. Theresult given in (5.10) was confirmed by directnumerical evaluation from (5 .8 ) , using (5.5) foru&). This gave ne, =1.364 at n =45 and 1.371


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428 GEORGE W. LATZMANat n =49,which shows n E n to be approximatelyindependent of n. The numerical value of thefactor in (5.10) s 1.470.6. Invariants

After multiplication by mum-', equation (3.1)may be written

a m a= - -at m + l axbut (4.1) eeps u = O a t x =- n and at x = 0 , sointegration with respect to x gives! I o umdx=O,at -nfor any m > - . Since the integral in (6.1) sinvariant, it may be evaluated at t =0, whereu = -sinx . Therefore, after integration withrespect to t ,

where r denotes the gamma-function.Equation (6.2)gives a class of (normalized)integral invariants of the original differentialequation (3.1).Two in particular are of interesthere: namely,

(6.3~)

M , = 2 I o u'dz- 1. (6.3 )n -nThe first of these is proportional to the integra-ted momentum (over a half wave length), thesecond to the kinetic energy.

Spectral forms of (6.3) re obtained by sub-stituting th e series (5.6) nto th e integrals:

mM,=3 2 Jn(nt)/n2t=l (6.4~)M , = 4 2 [Jn(nt) /nt]*=. (6.4b)

n =I, 3.5, ...W

n = I ,2,3, ..The identity (6.4~~)s a special case of a Kapteynseries known as Meissel's expansion (WATSON,1944, 3 17.23),while (6.4b) s a special case of

what WATSON(1944, 3 17.6) refers to as aKapteyn series of t he second kind. (The wholeof th e foregoing analysis can be regarded as adevious but elementary proof of these identi-ties.)It was noted previously that when t >1 theseries (5.6) represents the composite functionu* illustrated by the solid curve in Fig. 4. I nthe interval BEH this function does not satisfythe basic differential equation (3.1).Terefore,when t >1 the identities (6.4) re no longer validand the series have no physical significance.Nevertheless, it is worth recording that M I fort >1 can be evaluated in an elementary manner,as follows.

With reference to Fig. 4,

Mr=# / : nu*dx=#( / A B C +/ F G) u d xThe integrated part s are zero because u =0 a t Aand x = O a t C, E and G . I n xdu we substitutex = 8 +ut an d u = -sine, from (4.3);henM:=&(1 +1 ) (8 cos Ode-tudu).A B C E FGAfter performing the indicated quadratures, we

can write th e result in the following parametricform:'

MI* =cos+ +$+ sin+ (6.8)t =+/sin4,

where +-u,t( = -8, =I,).The points D and F eventually will move out

of th e interval - n


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INTEGRAL OF COMPLETE SPECTRAL EQUATIONS FOR UNSTEADY ONE-DIMENSIONAL FLOW 4291o

0.5

001,FIG. 5. Linear and quadratic integral invariants.Solid curve: M , =1 for 0 < t G 1 and M , =M : from(6.6)fort 3 1. Dots show values of M,computed from24 terms of the series in (6.4). roken line shows M ,computed in the same manner.

It follows that the Kapteyn series M , in(6.4a) s equal to the M : of (6.5) or 1 G t G4.60.The solid curve in Fig, 5 shows M I in the range0 t


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4301.o

0.5

GEORQE W . PLATZMAN1.5

1o

0.5-

u1=sech i tus thJ t .

(Thus, the initial configuration u =-sinx isconverted to u = -sin2x in the limit t-tco.)

The solid curves in Fig. 0 show the exactcoefficients

u1=2J , ( t ) / tug =2Jp(2t)/2tuI =2J8(3t)/3t

in the physical interval 0 < t


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INTEGRAL OF COMPLETE SPECTRAL EQUATIONS FOR UNSTEADY ONE-DIMENSIONAL FLOW 431COLE, JULIAN., 1961, On a quesi-linear parabolicequation occurring in aerodynamics. Quarterly ofApplied Mathematica, 9, pp. 226-236.COURANT,., and K. 0. FRIEDBICHS,948, Super-sonic flow and shock wavm. New York, Interscience

464 pp.HOPF, EBERHARD,960, The partial differentialequation ut+uuI=pZr.ommunicationa on Pureand Applied Mathematica, 8 , pp. 201-230.LIGHTHILL, . J., 1966, Viscosity effects in soundwaves of finite amplitude. In Surveys in Mechanics(G. K. Batchelor and R. M. Davies, Editors), pp.260-361. Cambridge University Press, 476 pp.LIQHTHILL, . J., and G. B. WITHAM,1966, Onkinematic waves. Part I. Proceedings, RoyalSociety of London (A), 229, pp. 281-316.OBUKHOV,. M., 1967, 0 tochnosti predvychislenisadvektionykh izmenenii polei pri chislennomprognoze pogody. Izvmtiu Akademiu NaukS.S.S.R., Seria Ueofizicheakaya,9, pp. 1133-1141.

OGWRA,., 967, A note on the energy transfer inBurgers' model of turbulence. 76th AnniversaryVolume of the Journal of the Meteorobqiml Societyof Japan, pp. 92-94. Meteorological Society ofJapan, 402 pp.PHILLIPS,ORMAN ., 1960, Numerical weatherprediction. Advancea in Computers (Franz L. Alt,Editor), 1,pp. 43-86. New York Academic Press,

316 pp.PLATZMAN,EORQEW., 1964, The computationalstability of boundary conditions in numericalintegration of the vorticity equation. Archiv farMeteorologie, Geophysik, und Bioklimatologie (A).REID, W. H., 1967, On the transfer of energy inBurgers' model of turbulence. Applied Scient$cReeearch (A), 6 , pp. 85-91.WATSON,. N., 1944, A treatise on the theory of Beaeelfunctions, second edition. Cambridge UniversityPress, 804 pp.

7, pp. 29-40.


An Exact Integral of Complete Spectral Equations For Unsteady One Dimensional Flow

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