Kyle A.Theakerand Robert A.Van Gorder

1Department of Mathematics,University of Central Florida,Orlando,FL 32816-1364,USA

2Mathematical Institute,University of Oxford,Andrew Wiles Building,Radcliffe Observatory Quarter,Woodstock Road,Oxford,OX2 6GG,United Kingdom

1 Introduction

The Lin–Reissner–Tsien equation in dimensional units reads

This equation is used to study transonic gas flows under the transonic approximation.[1−3]Hereˆuis the dimensional velocity potential,ˆx,ˆy＞0 are dimensional spatial coordinates,andˆt＞0 is the temporal coordinate.In Glazatov,[4]existence and uniqueness results of a certain class of solutions to the Lin–Reissner–Tsien equation are proven,subject to specific periodic boundary conditions.Recently,Ref.[5]considered exact and analytical solutions for the Lin–Reissner–Tsien equation(1).In particular,both steady and non-steady similarity solutions were considered.For some parameter values,exact solutions were obtained,while for more general parameter regimes,analytical solutions were found via Taylor series.Under some simplifications,those solutions recovered other more specific exact solutions of Refs.[6–8].Numerical solutions were also employed in Ref.[5]in order to verify the accuracy of the analytical approximations.

For some applications,the equation

In the present paper,we shall consider the Lin–Reissner–Tsien equation on various length scales.We first non-dimensionalize the equation in Sec.2,and are able to show that all spatial length scales enter through a single composite parameter multiplying the nonlinear term.This is useful,as depending on the length scales of interest,the parameter may be large or small.In Sec.3,we seek the generalize some of the results from Ref.[5].We first obtain a slightly more general similarity solution.Then,we turn our attention to mixed wave-similarity solutions,which were not previously considered.Such solutions propagate as a wave in one spatial coordinate,while still exhibiting a global self-similarity.Next,in Sec.4,we consider generalized Lin–Reissner–Tsien equations with forcing terms.Equations similar to(2)have been shown to admit similarity solutions which are relevant in the study of transonic gas flows(i.e.Ref.[11],and references therein).Therefore,in the context of applications,it makes sense to consider forced Lin–Reissner–Tsien equations.We obtain traveling wave solutions,and are able to show that there are non-trivial(in contrast to the fairly simple traveling wave solutions which exist for(1)).We then showthat forced Lin–Reissner–Tsien equations can still admit similarity solutions.We determine the precise class of forcing terms which allow for similarity solutions,before obtaining solutions numerically.We also show that a more restricted class of forcing functions allows for the construction of exact self-similar solutions.Finally,we give concluding remarks in Sec.5.

2 Non-Dimensionalization and Scaling Limits

Let us non-dimensionalize the Lin–Reissner–Tsien equation(1)by the change of variables

HereX,Y,T,Uare constants holding the relative scales of each variable.As we are concerned with spatial scales,while temporal scales are less essential,we can pick the temporal scale to simplify the resulting non-dimensional equation,by takingT=Y2X−1.We find then that

where the composite parameterϵdepends on the remaining length scales like

Therefore,the Lin–Reissner–Tsien equation is reduced to a single equation on non-dimensional scales which depends only on a single scaling group.The spatial length scales are then all encoded in the single parameterϵ,and it is sufficient to study(4)in order to study(1)under any length scales.

Ifϵ≪1,then eitherX≪1 orY≫1,and we have the small-xor large-yscale limit.In such a limit,takingϵ→0 yields

Introducing the new variablesx=ξ−ζ,t=ξ+ζ,we have

which is a two-dimensional wave equation,plus a small perturbation.As such,we expect solutions in the small-ϵregime to behave like solutions to a 2D wave equation.

Ifϵ≫1,then eitherX≫1 orY≪1,and we have the large-xor small-yscale limit.In this case,the equation reduces to

This implies that

in the large-ϵregime.

3 Similarity and Wave-Similarity Solutions for the Lin–Reissner–Tsien Equation

We now turn our attention to obtaining solutions to the scaled the Lin–Reissner–Tsien equation(4).

3.1 Similarity Transformation

Let us take the similarity transformation used in Ref.[5],

We obtain from Eq.(4)the similarity ODE

where prime denotes differentiation with respect to the similarity variable,η.The equation becomes singular for 1−4η2+ϵϕ′(η)=0,or,in theϵ→0 limit,η=1/2.In this limit,(11)reduces to

Then,at such a singular pointη=1/2,the natural boundary condition would read

Hence,the boundary condition

shall be taken atη=1/2.If we solve the linearized equation(12),we obtain

whereAis a free parameter. Yet,for this solution,ϕ(0)=A.Hence,it makes sense to consider the additional boundary condition We shall this be interested in solutions to the boundary value problem consisting of(11)subject to(14)and(16)for small positiveϵ.

3.2 Exact Solutions to the Similarity Problem

Here we obtain exact solutions to the similarity ODE(11).First,assume that

Then,placing this assumption into(11),we obtain the algebraic equation

from which we get the system

LeavingAfixed yet arbitrary,we can findBandCas such:

Thus,

Returning to physical coordinates,we have

The exact solution(23)is the solution given in Eq.(37)of Ref.[5],with two modifications.First,the solution(23)is more general,in that it involves an arbitrary parameter

A∈.Second,the solution(23)also depends on the arbitrary scaling parameterϵ＞0.Therefore,we conclude that this solution is more general than that of Ref.[5].

3.3 Numerical Solutions to Eqs.(11),(14),(16)

With one class of exact solutions obtained,we now turn our attention to numerical solutions of the boundary value problem given by the ODE(11)subject to boundary conditions of the form(14)and(16).Since we are primarily interested in the influence of the scaling parameter,ϵ,on the solutions,we shall takeϕ(0)=1 for these simulations.We plot solutions in Fig.1.

Fig.1 Plot of the numerical solutions of the boundary value problem given by the ODE(11)subject to boundary conditions of the form(14)and(16),given that we fix the parameter A=1.We plot the solutions for various values of ϵ, finding that for sufficiently large ϵ the solutions take on a linear appearance.This makes sense,as the linear solution is the only solution in the limit ϵ→∞.

3.4 Mixed Wave-Similarity Transforms

Let us consider a wave variable along thexdirection;that is to say,z=x−ct.Such solutions were not considered in Ref.[5]or elsewhere.Then(4)becomes

Consider a solution of the formu(y,z)=yaf(zyb),σ=zyb.Then,

Placing these into Eq.(24)gives us

For this equation,we must have 2+2b=0 anda+2+3b=0,which implies thata=1 andb=−1.With these similarity parameters,we obtain

It is clear from the form of(27)that there will always be a constant solution,f(σ)=C.Then,this solution gives the physical solutionu(y,z)=Cy.One may easily verify that this is indeed a solution to(4).

To find a second solution,let us consider the transformationg(σ)=f′(σ),which puts(27)into the form

In the limit whereϵ→0,we simply obtaing(σ)=C1(σ2−c)for arbitrary constantC1.Integrating,we recover

Returning to physical variables,we have

In the more interesting case whereϵis not negligible,and to make this case more tractable we make the change of variable

which gives us the ODE

This ODE permits an exact solution of the form

whereWdenotes the LambertWfunction(which satisfies the implicit functional equationQ=W(Q)exp(W(Q)))andλ/=0 is a constant.This then gives

We then integrate this equation overσto recoverf(σ),

The arbitrary constantσ0must be picked so that a branch of the LambertWfunctionWactually exists,i.e.σ20＞c−e−1.Finally,transitioning back into physical coordinates,we obtain the exact solution to(4),which takes the form

Interestingly,the solution(36)does not always exist on(x,y)∈2.Indeed,we must have that Therefore,the solution exists only forc＜e−1.Ifc≥e−1,then there will exist some region of the plane2for which the solution fails to exist.Physically,this means that solutions of the type(36)have a maximum possible wave speedc=c∗=e−1≈0.367 879.At and beyond this critical value,the solutions will breakdown in finite time if the wave moves too fast(c＞c∗)in the positivexdirection.

To conclude this section,we give numerical plots of the solutions to Eq.‘(27)in Fig.2.

Fig.2 Plot of the numerical solutions to(27)for(a)various values of ϵ for fixed wave speed c=0.2 and(b)various values of the wave speed c for fixed ϵ=0.5.The boundary conditions are fixed as f(0)=0.1 and f′(0)=1 for all plots.

4 Lin–Reissner–Tsien Equation with Forcing Terms

Next we consider the forced Lin–Reissner–Tsien equation

whereFis a forcing term.Such equations are useful in the study of gas dynamics.[11]

4.1 F=F(u)

LetF=F(u).Consider a wave solution

Then,we obtain the ODE

where prime denotes differentiation with respect toz.Unlike the pure traveling wave case discussed in Ref.[5],the inclusion of the forcing function can lead to more complicated dynamics,in contrast to the case of no forcing,for which the pure traveling wave solutions are trivial.

If we multiply Eq.(40)byρ′and integrate,we obtain

Fig.3 Plot of the numerical solutions to(40)given F(u)=αunfor various values of the power-law parameter n.The other parameters are fixed as b=c=1,ϵ=1,and α =1,while boundary conditions are taken as ρ(0)=1, ρ′(0)=0.In order to obtain periodic solutions,we consider only odd n.The solutions do not vary strongly with ϵ,and the role of b2+c is to modify the period of the solutions.The structure of the solutions is most influenced by n.As n increases,the traveling wave solutions become more sharp and the period of oscillation decreases,although the amplitude remains the same.

For various values of the parameters,we may plot the phase portraits in order to understand the behavior of solutions to this equation.On the other hand,we may directly solve the ODE(40)numerically.We do so in Fig.3.

In the special case werec=−b2,so that the wave variable isz=x+by−b2t,we have

which gives us

Suppose that the force scales with a power ofu,sayF(u)=αunfor some positive integernand constant parameterα.Then,we obtain the implicit relation

4.2 F=F(ux,uy,ut)

Consider now the case where the forcing function depends on the derivatives ofu,sayF=F(ux,uy,ut).Under the assumption of a traveling wave solution(39),we find that

where byH(ρ′)we denote Separating variables in Eq.(45)and integrating,we obtain an implicit relation for the functionρ′:

Consider the case where the force scales as a power of the first derivatives ofu,so that we obtainH(ρ′)=β(ρ′)nfor some positive integernand constant parameterβ.We then have three cases:

forn=1,

forn=2,and

forn≥3.If we are able to invert these relations,we may then obtainρ′as a function ofz.Integrating that result would then permit us to recoverρ(z).This may also be done numerically,and we provide plots of the numerical solutions for variousnandϵin Fig.4.

Fig.4 Plot of the numerical solutions to(45)given H(ρ′)= β(ρ′)nfor various values of the power-law parameter n.The other parameters are fixed as b=c=1 and β =1,while boundary conditions are taken as ρ(0)=0, ρ′(0)=1.In(a)we fix ϵ=1 and plot the solutions for various n.As n increases,the solutions uniformly increase in magnitude.In(b)we fix n=2 and plot the solutions for various ϵ.For 0 ＜ϵ＜ 2,the solutions uniformly decrease in magnitude as ϵ increases.At ϵ=2,the problem becomes singular,and for ϵ＞ 2 we then obtain a new type of solution branch.The curve starts out steep,and gradually decreases in slope as ϵ increases toward infinity.

4.3 Forms of F which Permit Similarity Solutions

As discussed in Ref.[11],it is possible to have self similar solutions to equations arising in gas dynamics,even when there is a forcing term present within the governing equation.We seek to find a general form ofF=F(x,y,t)which still allows for a similarity solution.

Due to the similarity transform(10),we should consider

wherea,b,c,γare constant parameters that would be selected based on the physical problem to be studied.Then,under the assumption(10),we find that(38)reduces to

The right hand side of(52)should take the form of a power ofη,the similarity variable.Noting thatηk=xktky−2k,we should havea=k,b−2=−2k,c+3=k.Then,

In other words,the permitted form of the forceFis a power of the similarity variable,η,multiplied by a factory2/t3.Under such an assumption,we have that

We numerically solve(54)for variousk,in order to determine the influence ofϵfor each of these cases.In Fig.5,we plot numerical solutions to(54)in order to determine the influence of the strength of the forcing function on the solutions.

Fig.5 Plot of the numerical solutions to(54)given F(x,y,t)= γηk(y2/t3)for various values of the power law parameter k. The other parameters are fixed as ϵ=1 and γ=1,while boundary conditions are taken as ρ(0)=1, ρ′(0)=0.As we increase k,the solutions uniformly decrease in value,more rapidly tending toward negative infinity as η becomes large.

In addition to numerical solutions,note that it is also possible to obtain exact solutions for the similarity problem(54).Along the lines of the earlier exact solution(17),we assume a polynomial solution

Here,theAj’s are constants to be determined.Ifγ=0,then the solution(55)will reduce to the exact solution(17),withm=2 and bothA0andA1determined as functions of the free parameterA2.

On the other hand,ifγ/=0,then the existence of an exact polynomial solution will depend on the power law parameterk.If a polynomial solution(55)does indeed exist,then the order of the left hand side of(54)with the proposed exact solution plugged in must match the order of the right hand side(which is simplyk).Ifm=0,1,2,3,then the linear terms in(54)will dominate.However,ifm＞3,then the nonlinear termϕ′ϕ′′will have order 2m−3,which is greater thanmform＞3.So,ifk=0,1,we pickm=2,while ifk=3,we pickm=3.It is less clear what to do whenk=2,sincem=2 results in no quadratic terms remaining on the left hand side of(54).While we omit a lengthy argument here,whenk=2,one may show that a polynomial solution would only exist for either complex-valuedϵor complex-valuedγ.However,ifk＞3,then we must be more careful.Ifk=4,observe that there is no integermsuch that 2m−3=4(m=1/2 in this case).Indeed,fork＞4,an exact polynomial solution(55)exists only when 2m−3=khas a positive integer rootm=m∗(k)=(k+3)/2,i.e.kmust be odd.The first few permitted values ofkarek=5(for whichm=4),k=7(for whichm=5),and so on.For other integer values ofk＞0,there are no exact polynomial solutions.Therefore,there are possible polynomial solutions provided the forcing function satisfiesk=0 orka positive odd integer.For other values,numerical simulations can be used,but exact solutions in terms of polynomials are not forthcoming.

We explicitly calculate the first few exact solutions,fi nding that fork=0 we have

fork=1 we have

and fork=3 we have

where

Fork＞3,although solutions are theoretically possible due to order balances discussed above,when calculating the actual solutions we find that the equations for the coefficients in(55)will be over determined.This will result in complex coefficients or parameters,and hence such solutions should be neglected as they are non-physical.Therefore,the exact solutions above are the only polynomial solutions,and exact polynomial solutions fail to exist fork＞3.Meanwhile,note that we see something related in those exact solutions we can obtain.Whenk=0 ork=1,the system of equations for the coefficients is under determined,meaning we always have a free parameter(for us,this isA2).This is exactly why we had the free parameterAin the exact solution(17).Whenk=3,the coeffi-cients of the solution were uniquely determined,which is why the solution fork=3 does not have a free parameter,but rather will only depend on system parametersϵandγ.Still,owing to the nonlinearity,the solution fork=3 is not unique,with two solutions existing(depending on the±root in the definition ofg±(γ,ϵ)).

5 Conclusions

We have extended the results of Ref.[5]in several ways.First,we have found additional solutions to the Lin–Reissner–Tsien equation,including a somewhat more general similarity solution and new mixed wave-similarity solutions.We have also extended the Lin–Reissner–Tsien equation by considering a forcing term.Such forced equations are useful in the study of gas dynamics.[11]For the forced equation,we are able to study a variety of forcing functions,which permit either new wave or similarity solutions.Unlike for the standard Lin–Reissner–Tsien equation,the forced equation permits non-trivial wave solutions.It is interesting that,despite the added complexity due to the forcing term,the forced equation still permits similarity solutions,and for some cases can even still be solved exactly.We are able to determine precisely for which forcing functions exact polynomial solutions will exist.These results suggest that,while complicated,forced Lin–Reissner–Tsien equations can still be solved exactly under some circumstances.For all of the various solutions obtained,numerical simulations verify the behaviors observed in exact or perturbation solutions.

Many of the solutions only exist for certain parameters or parameter regimes.Therefore,some of the parameter values correspond to physically relevant solutions,while parameters for which there are no solution would correspond to a loss of validity of the transonic approximation,or more fundamentally,a breakdown of the transonic gas flow.In such a case,more complicated dynamics,such as turbulence,may arise,which is beyond the scope of the LRT equation.So,when there is a solution,this means that the physical parameters permit a “nice” solution to the transonic gas flow problem.The solutions in Subsec.3.4 further depend on a wave speed,c.We find that left-moving waves(c＜0)are permitted at any velocity,while right-moving waves can propagate only with a velocity bounded like 0＜c＜e−1.For right-moving waves with higher velocity,the wave would likely become unstable and break apart,resulting in turbulence.Note that the break-up is local in nature,in the case ofc＞e−1.This suggests that,give a specific wave speed,we can determine where in space the break-up of the wave solution under the transonic approximation may occur in time,given specific spatial coordinates.

The LRT equation with forcing term was also considered.While the precise form of forcing can be determined by the particular experiment at hand,we provide some examples to illustrate that solutions to forced LRT equations can exist.The form of the forcing term will strongly influence the dynamics of the LRT solutions.If the forcing function scales as a power of the unknown function,then we can expect periodic waves,with the frequency of the waves decreasing as the power of the function increases.Therefore,we have bounded,periodic transonic wave solutions for the gas in this regime.On the other hand,when the forcing function depends on one or more first derivatives of the unknown function,the transonic gas solutions are monotone increasing if we have traveling wave solutions.Therefore,the structure of the forcing term will strongly influence the behavior of traveling wave solutions.

Forced LRT equations also have solutions under a similarity transformation,assuming appropriate forcing term.In such a case,the solutions are highly sensitive to the strength of the nonlinearity in the forcing term.In this case,we also show that certain forcing functions,while theoretically possible,do not give closed-form similarity solutions.This again has to do with the fact that such poorly behaved forcing functions would likely cause breakdown of a solution over time,resulting in a transition to the turbulent regime.

The closed form solutions presented here cast light on when solutions to the LRT equations are possible.In other situations,solutions are not possible(or,not found),and this can indicate other behaviors,such as turbulence,which cannot be captured by the LRT model.Since the solutions have been non-dimensionalized,this means what solutions may be possible at some scales,while at other scales the solutions under the LRT transonic gas model will break down,giving way to turbulent gas dynamics.In particular,solutions always exist whenϵ=0,and are found for smallϵ,as well.In terms of the space and time scales,ϵ=UX3/Y2=UX2/T.Then,ϵ≪1 whenT≫UX2,hence solutions tend to always exist for large timescales(relative to the spatial scales).In contrast,the mixed wave-similarity solutions of Subsec.3.4 are valid either forϵ=0 orϵ＞0,with very different solutions obtained for each case.The former solution can be viewed as the “large-time scale”solution,while the latter can be viewed as the “short-time scale” solution.Therefore,even when solutions are possible at all scales,there are often qualitative differences in the behaviors of the obtained solutions at disparate scales.All of these results will therefore inform us of how solutions should behave at different space or time scales.When coupled with the results for the forced LRT equation,these solutions may then serve as motivation for certain experiments on transonic gas dynamics under specific forcing terms.

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