Nonlinear development of subsonic modes on

compressible mixing layers: a unified strongly

nonlinear critical-layer theory

Clifford A. Sparks and Xuesong Wu ∗

Department of Mathematics, Imperial College London, SW7 2AZ, UK

This paper is concerned with the nonlinear instability of compressible mixing lay-ers in the regime of small to moderate values of Mach number M , in which subsonicmodes play a dominant role. At high Reynolds numbers of practical interest, previousstudies have shown that nonlinear evolution of instability modes is controlled primar-ily by nonlinear, viscous and non-equilibrium effects in the critical layer (i.e. a thinregion surrounding the level at which the mean-flow velocity equals the phase speedof the instability wave under consideration). In the incompressible limit (M = 0), thecritical-layer dynamics is strongly nonlinear, with nonlinearity being associated withthe logarithmic singularity of the velocity fluctuation (Goldstein & Leib 1988). Incontrast, in the fully compressible regime (M = O(1)), nonlinearity is associated with asimple-pole singularity in the temperature fluctuation, and enters in a weakly nonlinearfashion (Golstein & Leib 1989). In this paper, we first consider a weakly compressibleregime, corresponding to the distinguished scaling M = O(ε1/4), for which the stronglynonlinear structure persists but is affected by compressibility at leading order (whereε � 1 measures the magnitude of the instability mode). A strongly nonlinear systemgoverning the development of the vorticity and temperature perturbation is derived.It is further noted that the strength of the pole singularity is controlled by T ′c, themean temperature gradient at the critical level, and for typical base-flow profiles, T ′cis small even when M = O(1). By treating T ′c as an independent parameter of O(ε

1/2),we construct a composite strongly nonlinear theory, from which the weakly nonlinearresult for M = O(1) can be derived as an appropriate limiting case. Thus the stronglynonlinear formulation is uniformly valid for moderate O(1) Mach numbers. Numericalsolutions show that this theory captures vortex roll-up process, which remains the mostprominent characteristics of compressible mixing-layer transition. The theory offers aneffective tool for investigating the nonlinear instability of mixing layers at high Reynoldsnumbers.

I. Introduction

Mixing layers are an important prototype of shear flows for studying a fundamental instability mecha-nism: inviscid Kelvin-Helmholtz (or Rayleigh) instability. The interest in its instability and transition toturbulence is primarily motivated by their relevance to many practical applications such as combustionand other chemical reactions, where mixing process is of serious technological concern. Recent researcheson supersonic mixing layers in particular have been driven by the plan to develop scramjet engines topropel high-speed aircrafts (Curran 1996). The projected engine uses air-breathing technique and com-bustion takes place in non-premixed mode and at supersonic speeds. Because of extremely short residencetime of reactants in the combustor, effective mixing is crucial for operation. A compressible mixing layerprovides the simplest conceptual model in which relevant mixing processes could be investigated.In typical applications, the relevant Reynolds numbers are high so that the flow is slowly evolving

in the streamwise direction and local parallel-flow stability theory is deemed applicable. In the incom-pressible regime, linear stability analysis shows that a mixing layer in the absence of reverse flow isconvectively unstable and thus behaves like a noise amplifier (Ho & Huerre 1984, Huerre & Monkewitz1990). Dominant unstable waves, usually excited by ambient unsteady perturbations, roll up under non-linear effects to form concentrated spanwise vortices, or ‘rollers’, which then undergo repeated pairing.At the location of pairing, the shear-layer thickness increases abruptly (Winant & Browand 1974, Ho &

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Huerre 1984). These rollers eventually break down to small-scale motions. A most striking feature isthat predominantly two-dimensional large-scale coherent roller-like structures persist in a fully turbulentstate (Brown & Roshko 1974).Large-scale rollers as well as the associated small-scale motions directly influence mixing process. The

presence of the former implies that the conventional concept of gradient diffusion for turbulent mixingmay not be applicable. For instance, Broadwell & Breidenthal (1982) demonstrated that a model takinginto account large-scale structures would give significantly different predictions from diffusion models. Inorder to characterise mixing, one has to describe first the formation and structure of rolled-up vortices.That in turn requires investigating the nonlinear development of instability waves.The classical weakly nonlinear theory (Stuart 1960), formulated for exactly parallel flows at finite

Reynolds numbers, has been adapted to spatially developing mixing layers, and a certain degree of successhas been achieved; see the review of Liu (1989). However, such adoptions reply on ad-hoc treatmentsof nonlinear and non-parallel-flow effects. In order to treat systematically the delicate balance andinteraction of various physical effects, it is necessary to take the Reynolds number to be asymptoticallylarge. This has led to a self-consistent approach, known as non-equilibrium critical-layer theory; forreviews see Goldstein (1994), Cowley & Wu (1995).Non-equilibrium critical-layer theory for a planar mode on an incompressible mixing layer was de-

veloped by Goldstein & Leib (1988). A key observation is that as an initially linear mode propagatesdownstream, its growth rate diminishes due to the thickening shear-layer width, and there emerges acritical layer, i.e. a region surrounding the level at which the mean velocity UB equals the phase speedof the wave, c say. Outside this region, the perturbation remains linear and inviscid, but nonlineareffects become significant within the critical layer because the effective mean flow (UB − c), about whichlinearization is made, is vanishingly small locally. In addition, the non-equilibrium effect associated withthe slow modulation of the wave envelope, may also appear at leading order. Goldstein & Leib (1988)showed that a planar mode on an incompressible mixing layer evolves into a nonlinear stage, wherethe critical layer is non-equilibrium, and strongly nonlinear in the sense that nonlinearity enters as a‘coefficient’ of the operator governing the dynamics rather than as an inhomogeneous term of a linearsystem in the weakly nonlinear theory (cf. Stuart 1960). They derived an evolution system consistingof an amplitude equation coupled with a transport equation for the vorticity within the critical layer.Due to the strongly nonlinear nature, the disturbance in the critical-layer rolls up to form concentratedvortices, or “rollers”. In the inviscid limit considered, pathologically small-scale motions develop withinthe critical layer. Goldstein & Hultgren (1988) introduced a small amount of viscosity into analysis, sothat both diffusion and nonlinear terms appear at leading order. The viscous effect was found to causethe vorticity to diffuse into a regular pattern.Linear instability theory for compressible shear flows was first formulated by Lees & Lin (1946). The

effect of compressibility is characterised by a Mach number, defined as

M = U∗/a∗ (1)

where U∗ and a∗ are the reference flow velocity and sound speed respectively. Lees & Lin showed thata necessary condition for a shear flow to be inviscidly unstable is for the base-flow profile to have ageneralised inflection point, i.e. (

U ′/T)′= 0, (2)

where a prime denotes differentiation with respect to the transverse coordinate. A significant conse-quence of including compressibility is that a flow that is stable in the incompressible limit may beunstable. Moreover, there may exist a multitude of instability modes at high Mach numbers. Extensivecomputational work has been carried out to identify these modes. For boundary layers, the major resultsare documented in the report by Mack (1984).For a compressible mixing layer forming between two streams with velocities U1 and U2 and tem-

peratures T1 and T2, in addition to the (fast) free-stream Mach number M = U1/a1, the instabilityproblem involves extra parameters, i.e. the free-stream velocity and temperature ratios, βU = U2/U1 andβT = T2/T1. It has been suggested (Bogdanoff 1983, Papamoschou & Roshko 1988) that the combinedeffect of compressibility, βU and βT may be characterised by a single parameter, a convective Machnumber, which for a mixing layer of a single species is defined as

Mc = (U1 − U2)/(a1 + a2) = (1− βU )M/(1 + β1/2T ),

where a1 and a2 denote the sound speeds in the fast and slow streams respectively. Note that Mc maybe less than unity even if M > 1.

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Instability modes may be classified according to their propagation speeds relative to the free-streamvelocity. If the relative velocity is smaller/larger than the free-stream sound speed, the mode is called asubsonic/supersonic mode. The eigenfunction of a subsonic mode decays exponentially in the free stream.Lees & Lin (1946) showed that subsonic neutral modes must have a phase speed given by c = U(yc),where yc is the point at which (2) is satisfied. Thus for a subsonic neutral mode, the critical level isregular. Supersonic modes exist at sufficiently high Mach numbers. The eigenfunction of a growingsupersonic mode still attenuates exponentially but in an oscillatory manner for large y. However, aneutral supersonic mode does not vanish but rather remains bounded in the free stream, and its criticallayer does not coincide with an inflection point and hence is singular. These modes are of significancebecause they radiate highly directional Mach waves to the far field (Tam & Burton 1984, Wu 2005).There have been extensive studies of linear instability of compressible mixing layers. Temporal

analysis has been carried out by Lessen, Fox & Zien (1965, 1966), Blumen (1970), Blumen et al. (1975)and Sandham & Reynolds (1991). Spatial analysis was performed by Jackson & Grosch (1989, 1991)among others. The main findings of these linear instability calculations may be summarised as follows.

1. The overall instability is weakened as compressibility, i.e. the Mach number, increases.

2. For small and moderate Mach numbers, the dominant instability corresponds to two-dimensionalsubsonic modes.

3. When the Mach number exceeds a critical value, the fastest growing disturbances are obliquemodes, which may be subsonic or supersonic with respect to one of the free streams.

Nonlinear instability of a fully compressible mixing layer (M = O(1)) was considered by Goldstein& Leib (1989). They studied the evolution of a single oblique wave propagating at an arbitrary angle θto the streamwise direction. The instability wave was taken to be a subsonic mode, and the dominantnonlinearity is associated with a pole singularity in the leading-order temperature fluctuation, whichmakes nonlinearity appear sooner than in the incompressible case. As a result, nonlinearity operates in aweakly nonlinear fashion, that is, nonlinear effects appear successively as inhomogeneous terms at higherorders (cf. Stuart 1960). By analyzing the weakly nonlinear interaction within the critical layer, anamplitude equation containing cubic nonlinearity was derived as expected. However, unlike the classicalLandau equation, the nonlinear term is non-local because of the non-equilibrium effect being includedat leading order in the critical-layer equations. The amplitude develops a singularity at a finite distancedownstream for sufficiently small viscosity, but evolves into an equilibrium state when viscosity exceedsa critical value. The analysis of Goldstein & Leib (1989) was subsequently extended by Leib (1991) toinclude supersonic modes.The two existing nonlinear instability theories, a strongly nonlinear critical-layer theory in the in-

compressible regime (M = 0) and a weakly nonlinear one in the compressible regime (M = O(1)), haveno overlapping. From the mathematical viewpoint, it is certainly desirable to unify them by consideringan intermediate weakly compressible regime.Experiments indicate that in the moderate to fully compressible regime with M = O(1) (including

low supersonic speeds M > 1), mixing layers continue to behave in more-or-less the same manner as inthe incompressible regime. In particular, vortex roll-up remains the most significant feature during itstransition to turbulence. Dominant coherent structures are spanwise-oriented rollers of Brown-Roshko-type up to Mc ≈ 0.6 (Clemens & Mungal 1995, Elliott, Samimy & Arnette 1995, Urban & Mungal 2001,Olsen & Dutton 2003). However, the phenomenon of vortex roll-up cannot be captured by the weaklynonlinear theory for M = O(1).The aim of this paper is to develop a nonlinear theory capable of describing vortex roll-up. This

is to be achieved via two steps. In the first, we consider a distinguished weakly compressible regime,corresponding toM = O(ε1/4), for which the critical-layer dynamics is strongly nonlinear but is substan-tially modified by compressibility. The next step extends the analysis to the fully compressible regimeby constructing a composite theory that is uniformly valid for M = O(1). The extension is based onthe observation that T ′c, the temperature gradient at the critical layer, is numerically small for practicalprofiles (e.g. the Lock profile) even whenM = O(1). We are thus prompted to treat T ′c as an independentparameter of O(ε1/2) so as to include the compressible nonlinear effect (associated with the singularityin the temperature perturbation) in the strongly nonlinear structure.The proposed theoretical development is important since vortex roll-up plays a crucial role in the

entrainment and mixing of the two streams, and appropriate characterization of large-scale roller struc-tures is a prerequisite for investigating their subsequent breakdown into small-scale motions. The presenttheoretical work is pertinent to high Reynolds numbers and therefore complements direct numerical sim-ulations, which can be conducted only at very low Reynolds numbers (e.g. Sandham & Reynods 1991,Ragab & Sheen 1991).

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II. Formulation of the problem

Consider a mixing layer which forms between two parallel streams. Their velocities are denoted byU1, U2 (U1 > U2), and their temperatures by T1, T2 respectively. The fluid is assumed to be a perfectgas. The flow will be described in terms of Cartesian coordinates (x, y) with x and y being paralleland normal to the streamiwse direction respectively and normalized by the share-layer thickness l at thelocation of interest, which is to be taken as the origin of the coordinate system. The time variable t isnormalized by l/U1. The non-dimensional velocity field (u, v), the pressure p, density ρ and temperatureT are introduced by

(u, v) = (u∗, v∗)/U1, ρ = ρ∗/ρ1, p =

γM2

ρ1U21

p∗, T = T ∗/T1,

where * signifies dimensional quantities, ρ1 the density of the fast stream, γ is the ratio of specific heatsγ = cp/cv, and M is the Mach number

M = U1/a1 (3)

with a1 being the speed of sound in the fast stream. The Reynolds number R and Prandtl number Prare defined as

R = ρ1U1l/μ1, P r = cpμ1/k. (4)

The viscosity μ is assumed to be a function of the temperature T only, i.e. μ = μ(T ).The unperturbed steady flow in the mixing layer is governed by the classical boundary-layer equations,

the solution of which gives rise to the (compressible) Lock (1951) profile. However, as is common inshear-layer instability studies, most of our numerical calculations will be formed for the tanh profile,

U0 =1

2

(1 + βU + (1− βU ) tanh(η + s)

)(5)

where η is now defined as

η =

∫ y

0

ρ0(y)dy, (6)

with ρ0 = 1/T0 being the mean density, and s being a constant (chosen such that the critical level islocated at η = 0).Under the assumption of unity Prandtl number, the mean temperature profile is related to the velocity

profile via the Crocco relation

T0 = 1−(1− T0

) (1− U0)(1− βU )

−1

2

(γ − 1

)M2[(U0 − 1)(U0 − βU )

]. (7)

In order to study the stability, we now perturb the basic flow. The perturbed flow can be written as

(u, v, p, T, μ

)=(UB , VB , PB , TB , μB

)+(u, v, γM2p, T , μ

)(8)

where the perturbation is assumed to be a small-amplitude instability wave with p, T , u, v � 1. Asthe mode propagates downstream, its linear growth rate gradually reduces as the shear layer thickens,and eventually becomes nearly neutral it approaches its neutral point xn. Without loss of generality,the origin of the coordinate is taken to be the neutral position. Thus close to the origin, a critical layeremerges, and nonlinear effects may then become important in this layer, whilst the unsteady flow outsidethe critical layer remains linear. The analysis must therefore be performed for two distinct regions – alinear inviscid outer layer and a nonlinear viscous inner region, i.e. critical layer.The perturbation is of the travelling wave form. In the main part of the shear layer, it can be

expressed as

q = ε[(A(X , T )q0(y)e

iζ + c.c.)+Δq1(X , T , y, ζ) +O(Δ

2, ε) + ∙ ∙ ∙]

(9)

where q represents any of u, v, p or T and ε� 1. The variable ζ is defined as

ζ = αx− ωt (10)

which represents a moving coordinate. Both the wavenumber α and frequency ω(= αc) are real in thevicinity of xn, where the disturbance evolves nonlinearly. The nonlinear development can be describedby introducing an amplitude function A(X , T ), which depends on the slow streamwise and temporalvariables

X = Δx, T = Δt. (11)

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Figure 1. Sketch of the flow structure. An instability wave with an amplitude of O(ε) enters the nonlinear criticallayer regime at a distance of O(ε1/2R) upstream of the neutral position xn. The strongly nonlinear evolution occurson the scale of Δ−1 ∼ ε−1/2.

The length/time scale on which the amplitude evolves, Δ−1, is to be determined later.At leading-order, the pressure of the perturbation satisfies the compressible Rayleigh equation. For

subsonic modes of interest, the solution for the pressure, and velocities are all regular, but close to thecritical level yc = 0 the temperature fluctuation T0 behaves as

T0 ∼T ′cU ′c

1

y,

where T ′c and U′c represent the the mean temperature and velocity gradients at yc respectively. Thus T0

exhibits a pole singularity if T ′c 6= 0.The second term in expansion (9) can be expressed as

q1 =

∞∑

m=−∞

q(m)1 (X , T , y)e

imζ(with q

(−m)1 = q

(m)∗1

)(12)

where * represents complex conjugate. Here all harmonics are generated simultaneously due to the non-linear interaction within the critical layer. The pressure of the fundamental component, p(1), satisfies aninhomogeneous equation, which has a solution satisfying the boundary conditions only when a solvabilitycondition is imposed, namely,

−3

U ′c(d+2 − d

−2 ) = 2i

(cJ1∂A

∂X+ J2

∂A

∂T

), (13)

where J1, J2 are constants which can be expressed in terms of the leading-order eigenfunction for thepressure.Close to the critical level yc = 0, the second-order streamwise velocity exhibits a logarithmic singu-

larity,

u(1)1 ∼

i

αU ′c

(U ′′′cU ′c−T ′′cTc

)(AT + cAX ) ln |y|+ . . . . (14)

The left-hand side of (13) represents a streamwise velocity jump,

−3Tcα2U ′c

(d+2 − d−2 ) = u

(1)1 (y = 0+)− u

(1)1 (y = 0−). (15)

For a linear perturbation, the velocity jump is the well-known ’−π’ phase shift of the logarithm acrossthe critical level. However, when the amplitude of the disturbance reaches a threshold magnitude, thejump is influenced by nonlinear effects within the critical layer.

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III. Strongly nonlinear critical-layer dynamics

The singular nature of the outer solution at yc implies that either or both of viscous and non-equilibrium effects, which are negligible in the main part of the flow, have to be included within thecritical layer in order to smooth out the singularity. More importantly, nonlinearity within the criticallayer controls the overall development of the disturbance.In the weakly compressible regime M � 1, it is anticipated that the strongly nonlinear structure (see

figure 1) should remain essentially intact so that the critical layer has a width of O(ε1/2) (Goldstein &Leib 1988). It follows that the appropriate inner variable must be defined as

Y = y/ε1/2 = O(1). (16)

Non-equilibrium and viscous effects both appear at leading order if (Goldstein & Hultgren 1989)

Δ = ε1/2, R−1 = Λε3/2, (17)

where Λ is an O(1) parameter (Haberman 1972).The distinguished scaling is the one such that the nonlinearity effects of both the velocity and tem-

perature perturbations are comparable. An estimate of contribution from the nonlinear interactionassociated with the temperature fluctuation indicates that this occurs when T ′c = O(ε

1/2). It followsfrom the Crocco relation (7) that we require M ∼ ε1/4, and temperature difference across the shear layer(1− βT ) ∼M2. Thus we write

M =Mε1/4, 1− βT = βM2, (18)

whereM and β are O(1) parameters.However, for certain profiles (e.g. the tanh profile), T ′c = 0 when βT = 1 and M = O(1). In this case,

an scaling analysis shows that the critical-layer dynamics is also strongly nonlinear. Moreover, numericalsolutions indicate that for the Lock profile, when βT is close to 1, T

′c is numerically quite small even

for M = O(1). This implies that weakly nonlinear theory based on the scaling T ′c = O(1) (Goldstein &Leib 1989, Leib 1991) is unlikely to be accurate. To overcome this shortcoming, we tacitly treat T ′c as anindependent parameter with T ′c = O(ε

1/2), whilst still treating M = O(1), so that a strongly nonlinearcritical-layer theory may be formulated.In all these three cases, the solution for the perturbation in the critical layer expands as

q = εq0 + ε3/2(q1 + (ln ε

1/2)q1L) + ε2(q2 + (ln ε

1/2)q2L) + ∙ ∙ ∙ , (19)

where q represents any of p, u, v, or T , and qi = qi(ζ,X , T , Y ). This expansion is suggested by writingthe outer asymptotic solution in terms of Y .Near to yc, the basic temperature and velocity profiles are approximated as

T0 = Tc +M2{ε1/2Y T ′

cM +1

2εY 2T ′′

cM +1

6ε3/2Y 3T ′′′

cM + ∙ ∙ ∙}, (20)

U0 = c+ ε1/2Y U ′c +

1

2M2εY 2U ′′

cM +1

6ε3/2Y 3U ′′′c + ∙ ∙ ∙ . (21)

Inserting these expansions into the compressible Navier-Stokes equations, we can find the solution atO(ε) and O(ε3/2) which matches to the outer solution. At O(ε2), we arrive at the evolution equations

LT = iα(Aeiζ − c.c.

)TY + iαT

′cMM

2(Aeiζ − c.c.

)−T ′′cU ′c

{(AT + cAX

)eiζ + c.c.

}, (22)

LQ = iα(Aeiζ − c.c.

)QY +

(T ′′cTc−U ′′′cU ′c

){(AT + cAX

)eiζ + c.c.

}

−iαU ′cTc

(Aeiζ − c.c.

)TY − ΛU

′c

(μc − Tcμ

′(Tc))TY Y , (23)

where Q and T represent the vorticity and temperature within the critical layer, and

L ≡ αU ′cY∂

∂ζ+∂

∂T+ c∂

∂X− ΛTcμc

∂2

∂Y 2. (24)

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Equating the streamwise velocity jump in the inner solution to that in the outer solution, we obtain from(13) and (15) the amplitude equation

∫ ∞

−∞

∫ 2π

0

Qe−iζdζdY = 4πiTc

α2

(cJ1∂A

∂X+ J2

∂A

∂T

). (25)

In the following, we shall consider the simple case where a single mode is excited upstream. Theamplitude A takes on the form

A(X , T ) = eiΩT A(X ), (26)

where Ω = (ωn − ω)/ε1/2 represents a (scaled) deviation of the forcing frequency from the neutralfrequency. The ensuing disturbance initially consists primarily of a fundamental wave only, so that theappropriate initial condition is

A(X )→ aeκX as X → −∞, (27)

where <{κ} is the linear growth rate, and a is a (complex) amplitude parameter. However, as thedisturbance evolves into the nonlinear regime all harmonics are generated at the same order, and so Tand Q take the form

Q =

∞∑

m=−∞

Qm(X , Y )eim(ζ+ΩT ), T =

∞∑

m=−∞

Tm(X , Y )eim(ζ+ΩT ), (28)

where Y = αU ′cY +Ω.To remove the parameters a and Ω from equations (22)–(27), we now introduce the re-scaled variables

(cf. Goldstein & Hultgren 1988)

X = (αΩ/2)X − x0, Y = (2/αcΩ)Y , ζ = ζ + arga+ κix0, (29)

A = (8U ′c/c2Ω2)A,

(Qm, Tm

)= −4(αU ′2c /cΩ

2U ′′′c )(Qm, Tm

)(30)

M2 = −(αU ′2c |T′cM|/ΩTcU

′′′c )M

2, Λ = 8Tcμc(U′2c /αc

3Ω3)Λ, κ = (2/αΩ)κ, (31)

where κ = κr + iκi, and x0 = κ−1r ln

((c2Ω2)/(8U ′c|a|)

). Then Tm and Qm satisfy the re-scaled equations

( ∂∂X+ imY

)Tm +

i

2

∂

∂Y

(A∗Tm+1 − ATm−1

)− Λ∂2Tm

∂Y 2= δm,1

(iM2A− σ(

αc

2

d

dX+ i)A

), (32)

( ∂∂X+ imY

)Qm +

i

2

∂

∂Y

(A∗(Qm+1 − Tm+1)− A(Qm−1 − Tm−1)

)

+ Λ∂2

∂Y 2

[(1− Tc

μ′(Tc)

μc

)Tm − Qm

]= δm,1(1 + σ)

(αc2

dA

dX+ iA

), (33)

∫ ∞

−∞Q1(X , Y )dY = −2i

TcU′2c

αU ′′′c

(αc2J1d

dX+ iJ2

)A, (34)

where σ = −U ′cT′′c /(TcU

′′′c ). The initial condition (27) becomes

A(X )→ eκX as X → −∞. (35)

IV. Weakly nonlinear limit: amplitude equation

In this section, we shall show that for M � 1, the weakly nonlinear theory can be recovered fromthe system (32)–(34). To facilitate the derivation, we re-introduce Ω as follows

T † = T , X † = (1/Ω)X − x†0, Y† = ΩY , ζ† = ζ + κ†ix

†0, (36)

A† = Ω2A, Q†m = Ω2Qm, T

†m = Ω

2Tm, (37)

M†2 = Ω∣∣M2

∣∣, Λ† = Ω3Λ, κ† = Ωκ, x†0 =

1

κ†rln( 1Ω2

). (38)

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For large M, we may ignore the linear inhomogeneous term multiplied by σ in (32). This together withthe rescaling (36)–(38) changes equations (32)–(34) to

( ∂∂X †

+ imY †)T †m +

i

2

∂

∂Y †

(A†∗T †m+1 − A

†T †m−1

)− Λ†

∂2T †m∂Y †2

= iδm,1sgn(T′c)M

†2A†, (39)

( ∂∂X †

+ imY †)Q†m +

i

2

∂

∂Y †

(A†∗(Q†m+1 − T

†m+1)− A

†(Q†m−1 − T†m−1)

)

+Λ†∂2

∂Y †2

[(1− Tc

μ′(Tc)

μc

)T †m − Q

†m

]= δm,1(1 + σ)

(αc2

dA†

dX †+ iΩA†

), (40)

∫ ∞

−∞Q†1(X

†, Y †)dY † = −2iTcU

′2c

αU ′′′c

(αc2J1d

dX †+ iΩJ2

)A†. (41)

The initial condition (35) is now

A†(X †)→ eκ†X † as X † → −∞. (42)

In order to scale M out of (39)–(41), we perform the substitutions

X = M†(2/5)X †, Y = M†(−2/5)Y †, A = A†, (43)

Λ = M†(−6/5)Λ†, κ = M†(−2/5)κ†, Ω = M†(−2/5)Ω, (44)

and expand T †m and Q†m as

T †m = M†(8/5){T (0)m + M†(−4/5)T (1)m + M†(−8/5)T (2)m + ∙ ∙ ∙

}, (45)

Q†m = M†(8/5){Q(0)m + M

†(−4/5)Q(1)m + M†(−8/5)Q(2)m + ∙ ∙ ∙

}. (46)

Inserting these into equations (39)–(41), we obtain for n ≥ 0,

( ∂

∂X+ iY

)T(0)1 − Λ

∂2T(0)1

∂Y 2= isgn(T ′c)A, (47)

∂T(1)0

∂X+i

2

∂

∂Y

(A∗T

(0)1 − AT (0)−1

)− Λ∂2T

(1)0

∂Y 2= 0, (48)

( ∂

∂X+ 2iY

)T(1)2 +

i

2

∂

∂Y(−AT (0)1 )− Λ

∂2T(1)2

∂Y 2= 0, (49)

( ∂

∂X+ iY

)T(2)1 +

i

2

∂

∂Y

(A∗T

(1)2 − AT (1)0

)− Λ∂2T

(2)1

∂Y 2= 0, (50)

( ∂

∂X+ iY

)Q(0)1 + Λ

∂2

∂Y 2

((1− Tc

μ′(Tc)

μc)T(0)1 − Q(0)1

)= 0, (51)

∂Q(1)0

∂X+i

2

∂

∂Y

(A∗(Q

(0)1 − T

(0)1 )− A(Q

(0)−1 − T

(0)−1 )

)

+Λ∂2

∂Y 2

((1− Tc

μ′(Tc)

μc)T(1)0 − Q(1)0

)= 0, (52)

( ∂

∂X+ 2iY

)Q(1)2 +

i

2

∂

∂Y

(−A(Q(0)1 − T

(0)1 )

)+ Λ

∂2

∂Y 2

((1− Tc

μ′(Tc)

μc)T(1)2 − Q(1)2

)= 0, (53)

( ∂

∂X+ iY

)Q(2)1 +

i

2

∂

∂Y

(A∗(Q

(1)2 − T

(1)2 )− A(Q

(1)0 − T

(1)0 )

)

+Λ∂2

∂Y 2

((1− Tc

μ′(Tc)

μc)T(2)1 − Q(2)1

)= (1 + σ)

(αc2

dA

dX+ iΩA

), (54)

where we have used the fact that T(n)−m = T

(n)∗m and Q

(n)−m = Q

(n)∗m .

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Solving these equations in sequence using the Fourier transform, we can derive the amplitude equation

dA

dX= κA+ sgn(T ′c)C

∫ ∞

0

∫ ∞

0

ℵ(λ, μ; Λ)A(X − λ)A(X − λ− μ)A∗(X − 2λ− μ)dμdλ, (55)

where

C =παU ′′′c κ

παU ′′′c (1 + σ) + 2iTcU′2c J2, (56)

ℵ(λ, μ; Λ) = λ2[1−1

6Λ(1−Tc

μcμ′(Tc)

)λ2(2λ+ 3μ

)]e−

13 Λ(2λ+3μ)λ

2

. (57)

Thus we have shown that the weakly nonlinear theory of Goldstein & Leib (1989) for M = O(1) can berecovered as a limiting case of the current strongly nonlinear formulation. Hence we can conclude thatthe latter theory is valid for all 0 ≤M ≤ O(1).

V. Numerical solutions

The strongly nonlinear system (32)-(35) has been solved numerically. The results will be presented toshow the development of the amplitude and contours of the normalised total vorticity and temperaturewithin the critical layer; the latter are defined respectively as

Ω1 =[Y −

2

αc

(1 + M2

)]2+ 2(1 + σ

)<{Aeiζ

}−2

αc

∞∑

m=−∞

Qmeimζ , (58)

τ1 = σ[Y −

2

αc

(1−M2

σ

)]2+ 2σ<

{Aeiζ

}+2

αc

∞∑

m=−∞

Tmeimζ , (59)

where ζ = ζ +ΩT .We take γ = 7/5, and use Chapman viscosity law μ = T and the tanh profile. The velocity ratio is

taken to be a representative value βU = 1/2. In addition to Mach number M , another parameter, whichcharacterises the viscous effect (or Reynolds number), is

Λ =αc3

8TcμcU ′2cΛ =

Λ

Ω3=1

ω3R(60)

whereω = (ωn − ω). (61)

Note also that

M2 ∝T ′cMM

2

Ω=T ′cω. (62)

VI. Results

VI.A. Case I: βT = 1

The first case corresponds to two streams of equal temperature (βT = 1) and a relatively low Machnumber (M = 1/2). Figure 2 shows amplitude development and instantaneous growth rate for a range

of Λ. For small Λ, the amplitude exhibits a sinusoidal-like behaviour with an oscillating growth rate.Increasing Λ suppresses and eventually eliminates the oscillation, accompanied by an increase in theamplitude. For all Λ 6= 0, the amplitude eventually grows at a rate that ultimately tends to zero. Thesimilarity with the results of Goldstein & Hultgren (1988) suggests that the final asymptotic state is closeto |A| ∼ (ΛX )2/3. The effect of the Mach number on the final asymptotic state will be shown below.Vorticity roll-up is the most prominent feature of mixing-layer transition. Figure 3 shows vorticity

contours for Λ = 0.001, which corresponds to a high Reynolds number R = 6.4 × 107 for ω = 1/40).There is a great degree of roll-up, with mixing occurring mostly along the “braids”, where the contoursare closely clustered. Figure 4 shows the corresponding temperature contours. Temperature roll-upappears almost identical to vorticity roll-up. This may be explained by noting that since M = 0 and|Tm| � |Qm|, the temperature and vorticity equations, (32) and (33), are essentially the same.

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Figure 2. Amplitude ln|A| (top) and instantaneous growth rate <{A′/A} (bottom) vs. X for βT = 1, M = 1/2 and

Λ = 0.001, 0.1, 1, 10, 100.

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Figure 3. Vorticity contours in the (ζ, Y )-plane for βT = 1, M = 1/2, Λ = 0.001: (a) X = −0.986; (b) X = 0.986; (c)X = 2.958; (d) X = 4.930. (Contour levels shown are 0, 1, 2, . . . , 9 and 10.)

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Figure 4. Temperature contours in the (ζ, Y )-plane for βT = 1, M = 1/2, Λ = 0.001: (a) X = −0.986; (b) X = 0.986;(c) X = 2.958; (d) X = 4.930. (Contour levels shown are 0, −0.003, −0.006, . . . , −0.027 and −0.030.)

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Figure 5. Vorticity contours in the (ζ, Y )-plane for βT = 1, M = 1/2, Λ = 0.1: (a) X = −0.986; (b) X = 0.986; (c)X = 2.958; (d) X = 4.930. (Contour levels shown are 0, 1, 2, . . . , 9 and 10.)

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Figure 6. Development of higher harmonics, as measured by Hm (63) vs. X , for βT = 1, M = 1/2 and Λ = 0.001:m = 2 (solid), m = 3 (long-dashed), m = 5 (short-dashed) and m = 10 (dash-dotted).

Physically, roll-up is associated with the development of higher harmonics, which are simultaneouslygenerated by nonlinear interactions. Figure 6 shows some of the harmonics, measured by

Hm=

∫ ∞

−∞QmdY , (63)

for the case of Λ = 0.001. As is illustrated, the first harmonic m = 2 has the most significant magnitude,but the first few low harmonics (e.g. m ≤ 5) also retain appreciate amplitudes. Higher harmonicsgradually become smaller.For moderate viscous value Λ = 0.1, or Reynolds number (R = 6.4 × 105 for ω = 1/40), contours

feature a regular cat’s eye structure of Kelvin, and there is almost no roll-up (figure 5).Overall, the nonlinear instability characteristics are almost identical to the incompressible case of

Goldstein & Hultgren (1988). This is not surprising, since T ′c = 0 (for the tanh profile) and M2 is quite

small so that the governing equations are very close to those for the incompressible case.In order for the effect of Mach number to be reflected more clearly, we replace A by α2A and κX by

κX + ln(α2), which removes the dependence of the coordinates on α, introduced via the normalisation(29)–(31).

Figure 7 shows the amplitude development for βT = 1 and various M for three different Λ. The valueM = 3.96 is chosen to be 99% of Mmax, the maximum Mach number for subsonic modes. Increasing theMach number has a stabilising effect for Λ = 0.1, 10, and for these two values of Λ the solution is moreoscillatory at larger Mach numbers. Recall that in the incompressible case, the amplitude evolves intoan asymptotic state |A| ∼ (ΛX )2/3 (Goldstein & Hultgren 1988). In order to see if a similar algebraicgrowth occurs, calculations for the Λ = 10 have been advanced far downstream, and the results aredisplayed in figure 8. Straight lines of slope 2/3 clearly are approached at large distances, indicating thatthe algebraic growth is unchanged. The ultimate asymptotic state does not depend on Mach number forβT = 1, except for perhaps a constant multiplying (ΛX )2/3.Figures 9 and 10 show the vorticity and temperature contours for the case M = 3 and Λ = 0.001.

In this fully compressible regime, roll-up remains a prominent feature, consistent with experimentalobservations. Indeed, both contour plots look similar to those for the low-Mach-number case M = 1/2

(cf. figures 3 and 4). As in the incompressible limit, increasing Λ reduces roll-up; see further calculationsin Sparks (2006).

VI.B. Case II: βT 6= 1

Next we consider the case where the two streams have different the temperatures, i.e. βT 6= 1. In thelow-Mach-number regime M = 1/2, the development of the amplitude is shown in figure 11 for βT = 1/2

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Figure 7. Amplitude ln|α2A| vs. κrX + ln(α2) for βT = 1, and M = 0.1 (solid), 3 (dashed), 3.96 (dash-dotted). Top:

Λ = 0.001; middle: Λ = 0.1; bottom: Λ = 10.

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Figure 8. Amplitude ln|α2A| vs. ln|(κrX + ln(α2))Λ| for βT = 1, Λ = 10, and M = 0, 3, 3.96. Dashed lines are

approximate asymptotes, with slope 2/3.

and a range of Λ. In this case |T ′c| 6= 0, and consequently the results contrast greatly with those forβT = 1.For small Λ (e.g. 0.001 and 0.1), the amplitude exhibits a quite chaotic-like behaviour with growth

rate oscillating violently about zero. The initial behaviour is somewhat similar to that found in Goldstein& Wundrow (1990). A main difference from their results is that the growth rates initially decrease beforeincreasing. They proposed that the increase was due to the Bjerknes force, although this is unlikely tobe the cause of the (decrease or) increase in the present case. For small Λ, we were unable to advancethe solutions arbitrarily farther downstream, due to pathological development of small-scale structures,which are hard to resolve numerically.For moderate values of Λ (Λ = 1, 10), solutions could be advanced far downstream, and the amplitude

appears to attenuate or decay slowly. However, farther downstream the amplitude begins to grow again(Sparks 2006). For the largest value of Λ (Λ = 100) considered, the amplitude ultimately equilibrates.

Figure 12 shows vorticity contours for βT = 1/2 and Λ = 0.1. Vorticity roll-up is much stronger thanfor βT = 1. Appearance of closed “islands” similar to those found in Goldstein & Wundrow (1990) caninitially be seen. Further downstream irregular small-scale development causes contours to break up andform tiny islands or excessive “wiggles”. As vorticity rolls up, the temperature contours (figure 13) alsowind up to form tight spirals, indicating a significant degree of mixing. Interestingly, the temperaturecontours appear rather “regular”, despite the fact that temperature roll-up appears also as a “cause” ofthe vortex roll-up through coupling.The effect of βT on the amplitude is rather complicated, depending on Λ, as is shown in figure 14.

For a moderate or relatively large Λ (Λ ≥ 1), increasing βT above unity has a stabilising effect. Inparticular, for Λ = 10 the amplitude equilibrates (rather than undergoing (quasi-)algebraic growth as in

the case of βT = 1). For Λ = 1, the amplitude decays, but a further examination (Sparks 2006) showsit remains at an almost zero level for a prolonged period. The disturbance re-emerges again, and thiscycle of extinction and resurrection repeats itself in quasi-periodic fashion. Decreasing βT below unityhas a relatively moderate effect on the amplitude for relatively large Λ. For small Λ (e.g. Λ ≤ 0.1), bothincreasing βT above unity or decreasing βT below unity causes the amplitude to undergo increasinglycomplex oscillation; see the bottom plot of figure 14.

VI.C. Comparisons between strongly and weakly nonlinear predictions

The weakly nonlinear amplitude equation (55) was formally derived from the strongly nonlinear systemin the limit M � 1. The calculations presented in the previous sections focussed on M ≡ T ′c/ω = O(1),for which (55) is not expected to give a good approximation. The validity of (55) is now examined bycomparing its predictions with that of the strongly nonlinear theory for different ω (and hence M).

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Figure 9. Vorticity contours in the (ζ, Y )-plane for βT = 1, M = 3, Λ = 0.001: (a) κrX + ln(α2) = −1.325; (b)

κrX + ln(α2) = −0.101; (c) κrX + ln(α

2) = 1.123; (d) κrX + ln(α2) = 2.347. (Contour levels shown are 0, 3.75, 7.5, . . . ,

30 and 33.75.)

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Figure 10. Temperature contours in the (ζ, Y )-plane for βT = 1, M = 3, Λ = 0.001: (a) κrX + ln(α2) = −1.325; (b)

κrX + ln(α2) = −0.101; (c) κrX + ln(α

2) = 1.123; (d) κrX + ln(α2) = 2.347. (Contour levels shown are −3.6, −3.2, −2.8,

. . . , −0.4 and 0.)

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Figure 11. Amplitude ln|A| vs. X for βT = 1/2, M = 1/2: (a) Λ = 0.001; (b) Λ = 0.1; (c) Λ = 1; (d) Λ = 10; (e)

Λ = 100.

As an example, we consider the case where βT = 1/2, M = 1/2. Figure 15 shows comparisons forω = 1/40, 1/250, 1/4000. For ω = 1/40, which is the value used in previous calculations, the results fromthe two theories overlap in the linear regime only (figure 15), but deviate immediately once the nonlineareffect becomes important. For ω = 1/250, there is an appreciable overlap in the nonlinear regime, with

the extend of the overlapping also depending on Λ. For small to moderate Λ, the weakly nonlinearsolutions deviate eventually from the strongly nonlinear ones to develop a singularity, whereas for largeΛ, the two solutions are almost identical for all X . For extremely small ω = 1/4000, the two theories giveessentially the same results. The agreement severs a useful check on our numerical codes. Sparks (2006)monitored the amplitudes of harmonics Hm for m up to 5, and found that their magnitudes decreaserapidly with m, thereby satisfying a pre-request for weakly nonlinear theory to be valid. Meanwhile,vorticity exhibits very little roll-up as a result.

VII. Discussion and conclusions

In this paper, we carried out a theoretical investigation of the nonlinear instability of compressiblemixing layers. Attention is focussed on the regime of the low to moderate (but including supersonic)Mach numbers, where subsonic modes play a dominant role in causing transition. By considering thedistinguished weakly compressible regime, we presented a theory which ‘bridges’ the strongly and weaklynonlinear critical-layer theories formulated respectively for the incompressible limit and the compressibleregime with O(1) Mach numbers. The analysis is then extended to construct a composite theory uni-formly valid for Mach numbers 0 ≤ M ∼ O(1). The amplitude development may exhibit a rich varietyof behaviours, including explosive growth, equilibration, algebraic growth, violent “chaotic” oscillationand even the near-extinction followed by resurrection. A significant success of the theory is that itis able to describe one of the most remarkable phenomenon in mixing-layer transition, vortex roll-up.The compressible weakly nonlinear theory, though formally valid for M = O(1), is found to be a poorapproximation for the majority of parameters of interest.As an asymptotic theory valid for high Reynolds numbers, it complements the DNS approach, which

could currently be performed only at fairly moderate Reynolds numbers. The composite theory providesan efficient tool to study the nonlinear instability and transition of mixing layers, and should prove usefulwhen an extensive parametric study is required for design purpose.

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Figure 12. Vorticity contours in the (ζ, Y )-plane for βT = 1/2, M = 1/2, Λ = 0.1: (a) X = 0; (b) X = 1.475; (c)X = 2.950; (d) X = 4.425. Contour levels shown are −30, −15, 0, . . . , 60 and 75.)

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Figure 13. Temperature contours in the (ζ, Y )-plane for βT = 1/2, M = 1/2, Λ = 0.1: (a) X = 0; (b) X = 1.475; (c)X = 2.950; (d) X = 4.425.(Contour levels shown are 370, 380, 390, . . . , 500 and 510.)

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Figure 14. Amplitude development: ln|A| vs. κrX for βT = 1/2, 7/8, 1, 3/2, 2. Λ = 10 (top), 1 (middle), 0.1 (bottom).

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Figure 15. Amplitude ln|A| vs. X for βT = 1/2, M = 1/2 and Λ = 0, 100. Dashed lines show equivalent weaklynonlinear results. ω = 1/40 (top), 1/250 (middle) 1/4000 (bottom).

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