2.1. Exact formulation

Here we consider the equations governing the motion of fluid induced by a source of volume and specific momentum flux $(Q_{0},M_{0})$ located at $z=0$ . As discussed in detail in Part 1, we focus on the evolution of integral quantities of the flow, whose governing equations are obtained by integrating pointwise three-dimensional conservation equations over a horizontal disk centred on the axis of the jet. The radius of the disk is $r_{d}(z,t)$ , which is defined according to $\overline{w}(r_{d},z,t)={\it\epsilon}\overline{w}(0,z,t)$ , where ${\it\epsilon}\ll 1$ and $\overline{w}$ is the ensemble-average longitudinal velocity. Hence $r_{d}$ encompasses most of the longitudinal motion comprising the jet and we neglect longitudinal fluxes of momentum and energy entering the jet from the ambient. Integration over the horizontal disk of pointwise conservation equations for volume, momentum and the energy associated with the mean flow results in the following system (Part 1):

Here, $A_{m}\equiv Q_{m}^{2}/M_{m}$ is the characteristic area of the jet, and the dependent variables are the mean volume flux $Q_{m}$ and the mean specific momentum flux $M_{m}$ , hereafter referred to as the volume flux and the momentum flux, respectively:

Given the form of (2.2) and (2.3) and the definitions (2.4a,b ), it will occasionally prove useful to regard $Q_{m}$ and $M_{m}$ not as fluxes, but as the integrals of (specific) momentum and (specific) energy of the mean flow at a given height. The parameter ${\it\alpha}$ is the classical entrainment coefficient (see e.g. Morton, Taylor & Turner Reference Morton, Taylor and Turner1956; Turner Reference Turner1986). Length and velocity scales for the flow are defined as $r_{m}\equiv Q_{m}/M_{m}^{1/2}$ and $w_{m}\equiv M_{m}/Q_{m}$ , respectively. Equations (2.1)–(2.3) comprise an extension to unsteady problems of the integral equations posed by Priestley & Ball (Reference Priestley and Ball1955) and later by Kaminski, Tait & Carazzo (Reference Kaminski, Tait and Carazzo2005), though in the absence of buoyancy. In particular, the only assumptions made in our derivation are that the Reynolds number $\mathit{Re}\gg 1$ and that longitudinal fluxes entering the jet from the ambient at $r=r_{d}$ are negligible.

In (2.1)–(2.3), we use Greek letters to denote dimensionless parameters, corresponding to unknown fluxes and turbulence production terms. Specifically, we define the dimensionless gross flux of momentum

which includes longitudinal turbulent transport ${\it\beta}_{f}$ and the pressure integral ${\it\beta}_{p}$ . Here $w^{\prime }$ represents turbulent fluctuations in the longitudinal velocity field. The pressure $\overline{p}$ and its boundary value $p_{d}\equiv \overline{p}(r_{d},z,t)$ are both relative to a hydrostatic pressure and hence $p_{d}\approx 0$ in the cases considered here. It should be noted that due to the longitudinal development of turbulent transport in the near field, the source momentum flux $M_{0}$ is typically higher than the mean momentum flux $M_{m}$ . Similarly, the dimensionless gross energy flux is defined according to

where ${\it\gamma}_{m}$ corresponds to transport by the mean flow, ${\it\gamma}_{f}$ to turbulent transport and ${\it\gamma}_{p}$ to pressure work. Finally, we define the dimensionless turbulence production and pressure redistribution terms according to

which account for turbulence production due to shear, longitudinal fluctuations and pressure redistribution, respectively. The dimensionless parameters defined in (2.5)–(2.7) allow one to describe the flow in one dimension, without making an assumption about the magnitude or shape of the unknown radial profiles, in terms of the primary unknowns $Q_{m}$ and $M_{m}$ . Table 1 displays the value of these parameters obtained from the DNS of two steady-state turbulent jets, compared with the experimental observations of Panchapakesan & Lumley (Reference Panchapakesan and Lumley1993).

Although the momentum–energy approach of Priestley & Ball (Reference Priestley and Ball1955) predates the volume–momentum formulation of Morton et al. (Reference Morton, Taylor and Turner1956), the latter emerged as the standard model for plumes (see e.g. Hunt & van den Bremer Reference Hunt and van den Bremer2011). Only recently, when Kaminski et al. (Reference Kaminski, Tait and Carazzo2005) showed that a momentum–energy formulation allows one to establish an expression for the entrainment coefficient in terms of turbulence production and several profile constants has there been a revival of interest in the momentum–energy approach. In the intervening years, Fox (Reference Fox1970) also recognised that consideration of the mean flow energetics allows one to understand properties of turbulent entrainment that are not accessible at the level of volume conservation alone. In the analysis of unsteady jets, and by implication unsteady plumes, employing a mean flow energy equation (2.3) brings further advantages.

Pertaining to both type I and type II dispersion, the dimensionless energy flux

has a dominant role in both steady and unsteady jets. Notable is the fact that when $\overline{w}$ is assumed to have a top-hat form, then ${\it\gamma}_{m}=1$ , yet when $\overline{w}$ is assumed to have a Gaussian form ${\it\gamma}_{m}=4/3$ . The difference of $1/3$ between these values greatly exceeds the contributions ${\it\beta}_{f},{\it\beta}_{p}$ and ${\it\gamma}_{f}$ , due to turbulent transport (see table 1 and Part 1). Perhaps more importantly, ${\it\gamma}_{m}$ provides a means of accounting for both type I and type II dispersion in terms of the mean flow: type I dispersion is determined by the steady-state value of ${\it\gamma}_{m}$ , whilst type II dispersion is determined by the response of ${\it\gamma}_{m}$ to local changes in $Q_{m}$ , $M_{m}$ and/or their derivatives. Therefore, to develop a simple model for unsteady jets that accounts for dispersion, we focus on the role of ${\it\gamma}_{m}$ and set ${\it\gamma}_{g}={\it\gamma}_{m},{\it\beta}_{g}=1,{\it\delta}_{g}={\it\delta}_{m}$ in (2.1)–(2.3):

Equations (2.9)–(2.11) focus on the transport of momentum and energy by the mean flow rather than turbulence. The implication of (2.10) and (2.11) is that there is no longitudinal mixing of momentum $Q_{m}$ . In addition, the propagation speed of disturbances in the jet depends only on $M_{m}$ , $Q_{m}$ and the local value of ${\it\gamma}_{m}$ , which in turn is determined by an assumed mean velocity profile. In reality, the presence of turbulent transport modifies the propagation speed and, if there is a local change in the relative intensity of turbulence ${\it\beta}_{f}$ , can result in a discernible redistribution of momentum. In spite of these simplifications we will see that turbulence continues to play a notable role via ${\it\delta}_{m}$ in the final term of (2.11), which represents the conversion of energy in the mean flow to turbulence kinetic energy.

The system (2.9)–(2.11) appears to be over-determined, because in the original three-dimensional equations the mean energy equation is obtained from a combination of the momentum equation and the continuity equation. Thus, in (2.9)–(2.11) there are two independent equations and a third, whose consistency with the others depends on the closure that is employed for ${\it\alpha}$ . In this way it is possible to obtain the expression

which ensures consistency among the three equations. In (2.12) ${\it\alpha}_{prod}$ accounts for the ratio of turbulence production and mean energy flux; ${\it\alpha}_{disp1}$ accounts for type I dispersion and is equal to zero for top-hat jets, for which ${\it\gamma}_{m}=1$ ; and ${\it\alpha}_{disp2}$ accounts for type II dispersion and is zero when the jet is everywhere self-similar. The decomposition (2.12) is similar to that employed by Kaminski et al. (Reference Kaminski, Tait and Carazzo2005) to quantify the effect that buoyancy in plumes has on entrainment in a steady state. For further details the reader is referred to Part 1.

2.2. Existing unsteady jet models

In this section we compare the formulation (2.9)–(2.11), which makes no assumption regarding the radial profile of longitudinal velocity, to the unsteady plume models of Delichatsios (Reference Delichatsios1979), Yu (Reference Yu1990) and Scase et al. (Reference Scase, Caulfield, Dalziel and Hunt2006). For the purposes of comparison, we will assume that ${\it\gamma}_{m}$ is constant, which is equivalent to assuming that the velocity profiles remain self-similar and that type II dispersion is equal to zero. By adopting the traditional perspective of area and momentum conservation, in the absence of buoyancy the models are captured by

Here, ${\it\alpha}_{0}=-{\it\delta}_{m}/(2{\it\gamma}_{m})$ , for constant ${\it\delta}_{m}$ , is the steady-state entrainment coefficient and we have introduced $c$ to account for an inconsistency between a model’s equations and the exact area equation (2.9). Thus, the values $c=0$ or $c=1$ indicate whether the model is inconsistent or consistent, respectively, when considering mean energy and momentum conservation.

The models of Delichatsios (Reference Delichatsios1979) and Yu (Reference Yu1990) are given by $({\it\gamma}_{m},c)=(2,0)$ and $({\it\gamma}_{m},c)=(1,1)$ . Both claim to assume Gaussian profiles for the longitudinal velocity. However, the comparison shows that in neither of these models does one find the exact theoretical value ${\it\gamma}_{m}=4/3$ , which can be obtained by assuming a Gaussian distribution in (2.8). Instead, in Delichatsios (Reference Delichatsios1979), whose equations are ostensibly based on the Lagrangian formulation employed by Middleton (Reference Middleton1975) but are not derived explicitly, one finds ${\it\gamma}_{m}=2$ . A testament to the ambiguity and consequent difficulty surrounding the unsteady area equation is that Yu (Reference Yu1990), also assuming a Gaussian profile, find ${\it\gamma}_{m}=1$ . However, as pointed out by Scase & Hewitt (Reference Scase and Hewitt2012), the derivation of Yu (Reference Yu1990) is questionable in the light of the divergent integral appearing in their mass conservation equation. With $({\it\gamma}_{m},c)=(1,1)$ , the model of Yu (Reference Yu1990) is consistent with (2.13); the model of Delichatsios (Reference Delichatsios1979), on the other hand, is inconsistent in this respect. In conclusion, it appears that neither of these models can be referred to as genuine Gaussian models. In particular, the model of Yu (Reference Yu1990) has the characteristics of a top-hat model, which explains why Scase & Hewitt (Reference Scase and Hewitt2012) identify it as having a similar dynamic response to the model of Scase et al. (Reference Scase, Caulfield, Dalziel and Hunt2006). Indeed, the model of Scase et al. (Reference Scase, Caulfield, Dalziel and Hunt2006), which is derived explicitly using top-hat profiles, is fully consistent with (2.13) and (2.14). Noteworthy, however, is the fact that whilst the equations of Scase et al. (Reference Scase, Caulfield, Dalziel and Hunt2006) are only valid for uniform (top-hat) profiles, the system (2.9)–(2.11) is valid for arbitrary velocity profiles.

Not included in table 2, which shows a comparison of the unsteady plume models, is the model of Vul’fson & Borodin (Reference Vul’fson and Borodin2001), which is unique in stipulating that the plume remains conical, or straight-sided, independently of time-dependent changes that occur at its source; although Scase & Hewitt (Reference Scase and Hewitt2012) remark that the appropriateness of such an assumption is an open question. In this regard we have discovered that Gaussian jets, even if their momentum flux is subjected to a sudden change, have the remarkable property of remaining straight-sided, provided that their source area is fixed. This feature of the jet equations will be discussed in greater detail in § 5, and, more generally, in § 6 we illustrate the effect that any underlying velocity profile has on both the local area and the downstream growth rate of perturbations. Therefore for jets, compared to the models listed in table 2, the model of Vul’fson & Borodin (Reference Vul’fson and Borodin2001) provides the only system that is fully consistent with the assumption of a Gaussian profile.

With the exception of the refinement suggested by Scase & Hewitt (Reference Scase and Hewitt2012), which will be discussed in the following section, none of the models above accounts for the possibility of longitudinal mixing, i.e. for type II dispersion or turbulent transport. However, clear from the results presented in Part 1 is the fact that the longitudinal spreading of fronts is significant and should therefore be included.

2.3. The regularisation of ill-posed models

In Scase & Hewitt (Reference Scase and Hewitt2012) it was pointed out that the unsteady plume models of Delichatsios (Reference Delichatsios1979), Yu (Reference Yu1990) and Scase et al. (Reference Scase, Caulfield, Dalziel and Hunt2006) are ill-posed in their admission of the unbounded growth of short-wave modes. It is not a coincidence, given the discussion of the previous section, that the only model identified as being well-posed was that of Vul’fson & Borodin (Reference Vul’fson and Borodin2001). To render the models well-posed it was recommended (Scase & Hewitt Reference Scase and Hewitt2012) that a diffusion term be introduced in the momentum equation. In this section we identify several problems with the diffusion term that was proposed, and suggest that a more appropriate closure involves the mixing of energy in the mean flow via shear-flow dispersion.

In the absence of buoyancy, the area and momentum equations proposed in Scase & Hewitt (Reference Scase and Hewitt2012) are

where ${\it\beta}_{1}$ is a dimensionless constant. Unlike the original system of equations, the system (2.15) and (2.16) does not admit steady-state solutions of the form $Q_{m}=2{\it\alpha}_{0}M_{m}^{1/2}z$ , i.e. solutions in which the jet radius $r_{m}\equiv Q_{m}/M_{m}^{1/2}=2{\it\alpha}_{0}z$ . This behaviour is in contrast to that of plumes, for which the model was derived, in which the classical steady-state radius $r_{m}=(6/5){\it\alpha}_{0}z$ is unaffected by the additional mixing term. To understand the cause of the problem, it is useful to rearrange (2.16) as

The closure can now be understood as a combination of a modified flux of momentum on the left-hand side and a source term on the right-hand side. Thus, due to the source term, a steady jet does not conserve momentum using the closure proposed by Scase & Hewitt (Reference Scase and Hewitt2012). In fact, a steady-state power-law solution to the system (2.15) and (2.16) can be found:

where $z_{v}$ is the location of a virtual source of zero momentum flux and zero volume flux, $z_{v}\equiv -({\it\Phi}/2+1)Q_{0}/(2{\it\alpha}_{0}M_{0}^{1/2})$ , and $M_{0}$ and $Q_{0}$ are positive quantities denoting the mean momentum flux and volume flux at $z=0$ , respectively. The exponent in (2.18) and (2.19) is found to be

In contrast to plumes, in jets the diffusion term has the undesirable effect of modifying the classical power-law dependences, because jets do not contain a forcing term on the right-hand side of their momentum equation. This illustrates the fact that modification of the plume equations is best tackled at the simpler level of the jet equations, which place a greater restriction on the possible forms that can be introduced.

Of further interest is the fact that (2.15) and (2.16) imply a mean energy balance

which illustrates that a modification of the momentum and/or area equation manifests itself in the mean energy equation. We argue that it is more beneficial to account for longitudinal mixing in the mean energy equation directly, in a manner that is physically realistic.

3.1. Type I dispersion

One of the main advantages of employing the mean energy equation is that it contains nonlinear terms whose value depends on the correlation of mean flow properties over the radius. To appreciate the leading-order effect that this correlation has on the flux $F_{m}$ , we will consider a self-similar, steady state in which the function $g=g_{0}({\it\eta})$ and $f=f_{0}({\it\eta})$ . Indeed, $f_{0}$ and $g_{0}$ can be defined such that their integrals are equal to unity:

When $f_{0}$ and $g_{0}$ differ from uniform distributions of the same width, their correlation might prevent us from equating the top-hat product $w_{m}c_{m}r_{m}^{2}$ with $F_{m}$ :

in general. Thus spatial averaging introduces additional unknown correlations in the same way that they are introduced at a fixed location by a temporal or ensemble average. If the scalar profile $g_{0}$ is narrower than a non-uniform monotonically decreasing distribution $f_{0}$ , then $F_{m}>w_{m}c_{m}r_{m}^{2}$ , due to the concentration of the scalar in regions where $\overline{w}$ is relatively large. To see this, assume that $f_{0}({\it\eta})=2\exp (-2{\it\eta}^{2})$ and

Here ${\it\mu}r_{m}$ is the lateral length scale associated with the radial dependence of $\overline{c}$ . Using (3.3), the scalar flux can be determined exactly:

where, for the particular case of Gaussian profiles $f_{0}$ and $g_{0}$ ,

The parameter ${\it\theta}_{0}$ appeared in the unsteady plume model of Delichatsios (Reference Delichatsios1979) to account for the difference in lateral spread between radial profiles of buoyancy and momentum. In Part 1 we identified those profile effects that are associated with the steady state as giving rise to type I dispersion, and showed that they were responsible for a difference in the rates at which energy and momentum are transported in a jet. A Gaussian approximation to the steady energy distribution in a jet shows that $f_{0}^{2}({\it\eta})=4\exp (-4{\it\eta}^{2})$ . Hence ${\it\mu}^{2}=1/2$ and ${\it\theta}_{0}$ , which in the case of energy we denote as ${\it\gamma}_{0}$ , is equal to $4/3$ .

3.2. Type II dispersion

A difficulty associated with type II dispersion, unlike type I, is that it cannot be investigated without perturbing a jet from a steady self-similar state. In general the longitudinal divergence of the scalar flux $F_{m}\equiv {\it\theta}_{m}w_{m}c_{m}r_{m}^{2}$ is

Type II dispersion occurs when ${\it\theta}_{m}$ is not constant, and therefore results in the second term on the right-hand side of (3.6) being non-zero.

The classical view of dispersion originates from analysis undertaken by Taylor (Reference Taylor1953) of the dispersion of soluble matter in a pipe flow. Taylor demonstrated that in the vicinity of a step reduction in the mean concentration of the solute there emerges a region in which the otherwise uniform solute concentration is modified by a concentration that varies in the radial direction. Responsible for the deformation in the mean concentration profile is the radial variation of longitudinal velocity, because material located on the centreline of the pipe is transported faster than material located near the walls.

Taylor reasoned that for relatively large times the local deformation in the concentration would satisfy a balance between longitudinal advection and radial diffusion:

where $c_{m}(z,t)$ is the mean cross-section concentration and $\tilde{c}(r,z,t)$ is the deformation in the concentration profile, such that the total mean concentration $\overline{c}=c_{m}+\tilde{c}$ . The velocity $\tilde{w}$ is relative to a frame of reference moving with the cross-section mean velocity $w_{m}$ . Equation (3.7) implies that the deformation $\tilde{c}$ has the same radial dependence for all $z$ and $t$ , and has an amplitude that is proportional to ${\it\nu}_{T}^{-1}\partial _{z}c_{m}$ , for a constant eddy viscosity ${\it\nu}_{T}$ .

Using (3.7) to determine $\tilde{c}$ , the scalar flux $F_{m}$ can be obtained as the sum of a steady-state component and a component arising from the correlation of $\tilde{c}$ with $\tilde{w}$ :

For pipe flow, whose boundaries provide a physical radial scale, we will regard $r_{m}$ as the pipe radius. In contrast, jets are unbounded in the radial direction and it is necessary to appeal to moments of the velocity profile in order to determine a suitable radial scale, according to $r_{m}\equiv Q_{m}/M_{m}^{1/2}$ . Indeed, in order for (3.8) to be applicable to jets, we integrate to infinity, but recognise that for pipe flow the integral has an upper limit of $r_{m}$ . In addition, for pipe flow, the self-similar steady-state scalar profile is uniform and ${\it\theta}_{0}=1$ . The main observation to be made from (3.8) is that the second term on the right-hand side will, in accordance with (3.7), depend on $\partial _{z}c_{m}$ . Therefore, the flux $F_{m}$ will vary according to the longitudinal gradient of the mean concentration.

Fundamental to both dispersion in pipe flow and dispersion in jets is the idea that changes in the mean scalar concentration in the longitudinal direction can cause a local departure from self-similarity over the lateral dimension. The aim here is to apply Taylor’s treatment of dispersion in pipes to type II dispersion in jets. Our emphasis is on obtaining a simple and intuitive analogue to Taylor dispersion that focuses on the dominant physics. We do not intend to obtain a complicated model whose relation to classical Taylor dispersion is lost. Hence, we follow a heuristic approach and recast the balance (3.7) in a form that can be applied to jets.

The fundamental difference between dispersion in pipe flow and dispersion in jets is that in the latter the mean steady-state concentration $c_{m0}(z)$ varies in the longitudinal direction. To recast the mean scalar concentration in a jet in a form that is comparable to the mean scalar concentration in a pipe, it is convenient to examine the ratio

which is constant for all $z$ in the case of a steady flow, regardless of the way in which $c_{m0}$ depends on $z$ . Note that for pipe flow $c_{m0}$ is constant and $c_{m}$ can be replaced with $\mathscr{C}_{m}$ in (3.7) without approximation.

In addition, a steady-state scalar distribution in a jet will have a non-uniform radial dependence, which will not necessarily have compact support. Therefore, to describe the steady-state concentration we use the similarity function $g_{0}({\it\eta})$ , introduced in § 3.1. It is also convenient to introduce the function $g_{1}({\it\eta})$ to describe departures from the basic radial dependence of $g_{0}$ . We can therefore express the dimensionless concentration field to leading order according to

where $L_{1}$ determines the relative amplitude of the departures from the steady-state profile $g_{0}$ , and is therefore assumed to be independent of ${\it\eta}$ . In general it is convenient to define $\tilde{w}$ relative to a coordinate moving with the steady velocity ${\it\theta}_{0}w_{m}$ , which means that $\tilde{w}\equiv w_{m}(\,f_{0}-{\it\theta}_{0})$ , where $f_{0}({\it\eta})$ is the self-similar velocity profile introduced in § 3.1.

We will assume that the eddy viscosity ${\it\nu}_{T}$ is constant and equal to ${\it\alpha}_{0}w_{m}r_{m}/3$ . For jets, ${\it\alpha}_{0}w_{m}r_{m}/3$ corresponds to an integral estimation of ${\it\nu}_{T}$ in which ${\it\alpha}_{0}$ is the steady-state entrainment coefficient (Part 1). Consequently, we can express Taylor’s dispersion balance (3.7) in dimensionless form according to

A noteworthy property of the right-hand side of (3.11) is that it is not affected by a linear rescaling of the steady-state concentration: $c_{m0}\mapsto ac_{m0}$ . This property is consistent with the view that the physics of the problem should not depend on the particular steady-state scalar concentration that is used in (3.9) to non-dimensionalise  $c_{m}$ .

Because $L_{1}$ is responsible for all amplitude variations in $z$ and $t$ , it follows that, to within a multiplicative constant,

In this way the form of the leading-order departure from similarity $g_{1}$ is determined by the geometric properties of the steady-state velocity and concentration profiles, and is independent of $z$ and $t$ . However, since we are concerned with integral quantities, it is not necessary to know the precise form of $g_{1}$ . Indeed, due to their conical geometry, we suspect that for jets $g_{1}$ satisfies a slightly different ordinary differential equation in ${\it\eta}$ to that which is implied by (3.11). Nevertheless, from an integral perspective it is the dependence of $L_{1}$ on the dimensionless longitudinal gradient of the mean concentration $\mathscr{C}_{m}$ , expressed in (3.12), that is of particular interest.

In analogy with ${\it\theta}_{0}$ , we can now define another geometric parameter, ${\it\theta}_{1}$ , to characterise the scalar flux arising from the departure from the basic steady-state similarity:

The correlation of the perturbation profile $g_{1}$ with $f_{0}$ determines how the total dimensionless scalar flux depends on gradients in the longitudinal direction:

The dimensionless flux ${\it\theta}_{m}$ consists of the sum of a steady component ${\it\theta}_{0}$ and an unsteady component $L_{1}{\it\theta}_{1}$ . Evident from (3.12) is that the contribution from the latter depends on the extent to which longitudinal gradients $\partial _{z}c_{m}$ deviate from their steady-state value. The constant geometric parameter ${\it\theta}_{1}$ , which depends on $f_{0}$ and $g_{1}$ , determines the sensitivity of the dimensionless flux ${\it\theta}_{m}$ to the departure from similarity that is quantified by $L_{1}$ .

3.3. Closure of ${\it\gamma}_{m}$

Equation (3.14) comprises the dispersion closure advocated in this paper. This closure will now be applied to the dimensionless energy flux ${\it\gamma}_{m}$ . We assume that, like the scalar flux ${\it\theta}_{m}$ , the dimensionless energy flux ${\it\gamma}_{m}$ in a jet can be decomposed into a steady component ${\it\gamma}_{0}$ and an unsteady component $L_{1}{\it\gamma}_{1}$ :

where, like ${\it\theta}_{1}$ , the constant geometric parameter ${\it\gamma}_{1}$ can in principle be determined from the steady self-similar velocity profile $f_{0}$ and the deformation profile $f_{1}$ , which correspond to $g_{0}$ and $g_{1}$ in § 3.2.

To determine $L_{1}$ we focus on the ratio $w_{m}/w_{m0}$ in place of $\mathscr{C}_{m}\equiv c_{m}/c_{m0}$ in (3.12). In addition, we define the local steady-state mean velocity $w_{m0}$ so that $\partial _{z}w_{m0}=-2{\it\alpha}w_{m0}/r_{m}$ and (3.12) becomes

In terms of integral quantities, the dimensionless energy flux (3.15) can therefore be expressed as

The parameters ${\it\gamma}_{0}$ and ${\it\gamma}_{1}$ require calibration before the model can be used. Although it is possible to obtain values of ${\it\gamma}_{0}$ and ${\it\gamma}_{1}$ based on the geometry of the steady-state velocity profile alone, a more convenient and robust approach is to derive their values directly from observation. In Part 1 unsteady jets were produced by imposing an instantaneous step change up and down in the source momentum flux $M_{0}$ . In the steady state we found that ${\it\gamma}_{m}=1.3$ . For the unsteady jets, we estimated a dimensionless dispersion coefficient, $D_{e}/M_{m}^{\ast 1/2}=0.22$ , by comparing self-similar profiles of $M_{m}(z,t)$ to a solution of an advection–dispersion equation for the mean energy. In § 5.2 of this paper we will show that under certain assumptions $D_{e}/M_{m}^{\ast 1/2}=3{\it\gamma}_{1}/(2{\it\alpha}_{0})$ , which, when ${\it\alpha}_{0}=0.07$ , provides a reasonable first estimation ${\it\gamma}_{1}=0.01$ . For further detail regarding the relationship between ${\it\gamma}_{1}$ and $D_{e}$ the reader is referred to § 5.2 of the present work. In addition, we note that ${\it\gamma}_{0}$ is equal to the steady-state value of ${\it\gamma}_{m}$ and therefore set

5.1. A straight-sided equation for momentum and energy (type I dispersion only)

The unsteady equation for area (2.9), neglecting type II dispersion (i.e. ${\it\gamma}_{m}={\it\gamma}_{0}$ ) and using (2.12), can be expressed as

Substitution of the straight-sided solution $Q_{m}=2{\it\alpha}_{0}M_{m}^{1/2}z$ into (5.1) reveals that

Equation (5.2) expresses the fact that when ${\it\gamma}_{0}=4/3$ the jet will remain straight-sided for all time. Further, (5.2) suggests that when ${\it\gamma}_{0}<4/3$ or ${\it\gamma}_{0}>4/3$ the area will increase or decrease, respectively, following a step increase in $M_{m}$ . Indeed, the unsteady similarity solutions obtained by Scase et al. (Reference Scase, Caulfield, Dalziel and Hunt2006) can be rederived for arbitrary ${\it\gamma}_{0}$ (see appendix A) and reveal that when ${\it\gamma}_{0}=4/3$ , the spreading angle of the jet is identical to the steady-state case: $r_{m}=2{\it\alpha}_{0}z$ . In addition, the similarity solutions show that when ${\it\gamma}_{0}<4/3$ or ${\it\gamma}_{0}>4/3$ the spreading angle of the jet is less than or greater than the steady-state spreading angle, respectively. When ${\it\gamma}_{0}=1$ , one recovers the similarity solution for top-hat profiles obtained by Scase et al. (Reference Scase, Caulfield, Dalziel and Hunt2006) and the behaviour $r_{m}=2{\it\alpha}_{0}z/3$ . Contrary to the top-hat plume theory in which there is no means for faster fluid to overtake slower fluid (see e.g. Scase et al. Reference Scase, Aspden and Caulfield2009), we find that specification of a realistic, albeit constant, value of ${\it\gamma}_{0}$ facilitates such overtaking in the form of type I dispersion.

In the light of (5.2), the Gaussian velocity profile can be usefully understood as providing momentum and energy fluxes whose ratio is such that the steady-state area of the jet can be preserved. To visualise this one can consider a control volume located at the front of a starting jet. Energy, $M_{m}$ , is transported into the control volume with dimensionless flux ${\it\gamma}_{0}$ , and it is removed from the control volume owing to turbulence production at a dimensionless rate ${\it\delta}_{m}$ . If ${\it\gamma}_{0}<4/3$ then energy is added to the control volume at a reduced rate, which results in a relative surplus of momentum, $Q_{m}$ , and hence the accumulation of area $Q_{m}^{2}/M_{m}$ . Conversely, ${\it\gamma}_{0}>4/3$ results in the reduction or dispersion of area.

The use of the straight-sided approximation is highly desirable from a modelling perspective, as it provides an algebraic coupling between $Q_{m}$ and $M_{m}$ and therefore reduces the dimension of the system by one. For straight-sided jets, using the relations $Q_{m}=2{\it\alpha}_{0}M_{m}^{1/2}z$ and ${\it\delta}_{m}=-8{\it\alpha}_{0}/3$ , which both follow from ${\it\gamma}_{0}=4/3$ , the momentum equation and mean energy equations (2.10) and (2.11) become

Both (5.3) and (5.4) admit discontinuities at $z=z^{\ast }(t)$ that propagate according to

The nonlinearity of (5.3) and (5.4) ensures that a positive step change ( $\partial _{z}M_{m}<0$ ) in $M_{m}$ gives rise to a shock, whose propagation velocity must be determined from a Rankine–Hugoniot condition (see e.g. Toro Reference Toro1997). In order to determine a unique propagation velocity one must choose whether to impose momentum conservation or energy conservation over the shock. Likening the progression of a positive step change in a jet’s momentum flux to the evolution of a tidal bore, we impose momentum conservation (5.3) over the discontinuity. Substituting $Q_{m}=2{\it\alpha}_{0}M_{m}^{1/2}z$ in (5.3), we find

Here, we use the superscripts $B$ and $A$ to denote the value of a variable or function before/ahead of and after/behind a front, respectively. The characteristic momentum flux (5.6) provided the motivation for the choice of scaling in figure 10 of Part 1. It is also of interest that for a starting jet, for which $M_{m}^{B}=0$ , (5.6) is consistent with the result obtained by Ruban & Vonatsos (Reference Ruban and Vonatsos2008). There, the Bernoulli equation was used along the axis of a planar jet to deduce that the propagation velocity is equal to half the maximum velocity in the jet immediately behind the shock. We note, however, that in other respects their laminar boundary layer solution is very different to the radially integrated turbulent flow considered here. In the present work, if $M_{m}^{B}=0$ in (5.6) then $\text{d}z^{\ast }/\text{d}t=w_{m}^{A}$ , where $w_{m}^{A}$ is the top-hat velocity immediately behind the shock. Noting that the top-hat velocity $w_{m}^{A}$ is equal to half the maximum velocity of a Gaussian profile confirms the consistency between (5.6) and the results of Ruban & Vonatsos (Reference Ruban and Vonatsos2008). In turn, these deductions each agree closely with the observations reported by Turner (Reference Turner1962) for a starting plume.

In (5.4) the inclusion of type I dispersion alone, and not the mixing provided by type II dispersion, has produced a simple first approximation of an unsteady jet. In spite of the absence of longitudinal mixing, (5.4) does not admit the non-physical growth of short-wave modes that were found in the predictions of several unsteady plume models (Scase & Hewitt Reference Scase and Hewitt2012). Indeed, we show in § 6.1 that the evolution of short-wave modes is determined by type I dispersion, and that their growth can therefore be prevented by adopting a physically realistic assumption regarding the jet’s velocity profile.

Although it is clear that the mixing in jets prevents the formation of discontinuities, like other hyperbolic systems, the value of (5.3) or (5.4) is that it provides a description of their limiting behaviour, without the further complications that are introduced when additional mixing terms are present.

5.2. A straight-sided equation for energy (type I and type II dispersion, approximate momentum conservation)

In (5.3) and (5.4) we established a description of a straight-sided unsteady jet that satisfies transport equations for both momentum and mean energy. In general, if the straight-sided relation $Q_{m}=2{\it\alpha}_{0}M_{m}^{1/2}z$ is imposed strictly then one cannot expect to satisfy equations for momentum and mean energy simultaneously unless ${\it\gamma}_{m}=4/3$ . Indeed, the solution for the two dependent variables of the problem is defined by two governing equations. The Gaussian profile, for which ${\it\gamma}_{m}=4/3$ , is the distinguished case for which the three equations (momentum, mean energy and straight-sidedness) are consistent.

As an approximate treatment it is useful to relax the momentum equation and consider a solution that satisfies both straight-sidedness and the mean energy equation exactly. The extent to which the momentum equation will be violated in this approximation depends on the value of ${\it\gamma}_{0}$ and ${\it\gamma}_{1}$ . In particular, if we let ${\it\gamma}_{0}=4/3$ , then the level of approximation in the momentum equation depends on the amount of longitudinal mixing, and therefore on ${\it\gamma}_{1}$ , whose value is small in comparison to $4/3$ . Approximating the momentum balance in this way allows us to obtain a straight-sided version of the full unsteady jet model developed in § 3, which reduces to a simple advection–dispersion equation. As such, it can be readily employed to aid a first estimation of a dispersion coefficient based on experimental or numerical observations.

With the straight-sided solution $Q_{m}=2{\it\alpha}_{0}M_{m}^{1/2}z$ , our model for energy transport (4.2) simplifies to a nonlinear advection–dispersion equation:

To understand the leading-order processes described by (5.7) we consider dimensionless perturbations from $M_{m}=M_{m}^{\ast }$ :

Here, we have employed the characteristic momentum flux $M_{m}^{\ast }$ , rather than $M_{m}^{B}$ for example, because it is $M_{m}^{\ast }$ that determines the propagation velocity of the front. In doing so, we find that the linearised system of equations obtained below can provide a good approximation to the fully nonlinear problem, in which the step-change forcing is relatively large. Substituting this expansion into (5.7) and retaining terms to $O({\it\epsilon})$ yields

which is a linear advection–diffusion equation for a spatially varying velocity field, and has exactly the same form as the equation that was used in Part 1 to estimate the mixing coefficient $D_{e}$ . Indeed, comparison of (5.9) above with (5.9) in Part 1 shows that $D_{e}/M_{m}^{\ast 1/2}=3{\it\gamma}_{1}/(2{\it\alpha}_{0})$ . Assuming similarity of the process in terms of the variable  ${\it\lambda}$ :

where the dimensionless front velocity ${\it\lambda}(z^{\ast })={\it\lambda}^{\ast }=2$ , (5.9) can be expressed as

where the dispersive Péclet number

is consistent with the definition provided in appendix B of Part 1. Indeed, in Part 1 the theoretical justification for using the linear advection–dispersion equation was not provided, yet it was found to give a reasonable agreement with the data. As described in this section, it is the tendency of the jet to remain straight-sided that results in the surprising effectiveness of (5.9). The Péclet number defined in (5.12) has a meaning that is equivalent to the ratio of advection and dispersion parameters ( $K_{a}/K_{d}$ ) used in the recent study of a quasi-two-dimensional jet by Landel et al. (Reference Landel, Caulfield and Woods2012).

In figure 3 we display self-similar, far-field predictions from the general nonlinear model GJM, the straight-sided model GJM-S (5.7), and the linear straight-sided model GJM-SL (5.9), in addition to the collapsed DNS data. In GJM-S and GJM-SL we set ${\it\gamma}_{0}=4/3$ , which is consistent with the value of ${\it\gamma}_{m}$ required for straight-sidedness. We do not include predictions for TJM, as it proved infeasible to provide the number of points that were required for a far-field solution in that case. Notably, having been obtained from the constraint $r_{m}=2{\it\alpha}_{0}z$ , both the linear and nonlinear straight-sided solutions are monotonic, and are therefore not able to reproduce the extrema evident in both the DNS results and the full model. As one might expect, in the step-up (step-down) case the nonlinear model GJM-S predicts a steeper (less steep) front than the linear approximation GJM-SL. The linear, GJM-SL, and nonlinear, GJM-S, straight-sided predictions exhibit a good agreement, which demonstrates the effectiveness of the former when $M_{m}^{\ast }$ is chosen correctly, i.e. as the solution of the idealised nonlinear shock problem. It is also evident in figure 3 that the full model GJM is not able to exactly reproduce the values $M_{m}/M_{m}^{B}>1$ at relatively large values of $z/z^{\ast }$ in the step-up case. An attempt to correct this by increasing ${\it\gamma}_{1}$ would mean that ${\it\gamma}_{1}$ no longer faithfully represents those features of the flow on which its definition in § 3.2 was originally based.

Figure 3. (a) Self-similarity of the dimensionless momentum flux $M_{m}/M_{m}^{B}$ from DNS results compared to predictions using the full model GJM (4.1) and (4.2). (b) Predictions obtained using the nonlinear, straight-sided model GJM-S (5.7), and linear straight-sided model GJM-SL (5.9) are compared to GJM. The constant $z_{v}$ is the value of $z$ at an asymptotic virtual source. In each plot, the left-hand side refers to the step-down case, and the right-hand side to the step-up case.

6.1. The growth rate of source perturbations

In order to determine the cause of the difficulties associated with several unsteady plume models, Scase & Hewitt (Reference Scase and Hewitt2012) analysed their response to infinitesimal perturbations of the source conditions. Here we repeat the analysis for unsteady jets, focusing on a single harmonic of frequency ${\it\sigma}$ . It is convenient to work in terms of the dimensionless variables

where $M_{m0}$ is a constant steady-state momentum flux. We will assume that the dimensionless energy flux ${\it\gamma}_{m}$ is constant. In terms of the dimensionless variables (6.1a,b ), linearised equations enforcing the conservation of momentum and energy, in terms of the dimensionless volume flux $\mathscr{Q}_{1}$ and momentum flux $\mathscr{M}_{1}$ , can be expressed as

We decompose the problem into Fourier modes and, focusing on the harmonic of frequency ${\it\sigma}$ , assume a solution of the form

It follows from (6.2) that

which is a confluent hypergeometric equation, whose general solution is

where M and U are independent solutions of Kummer’s equation (Abramowitz & Stegun Reference Abramowitz and Stegun1972, p. 504), $c_{1}$ and $c_{2}$ are arbitrary constants, and

The limiting form of (6.5) for large $z$ is given by Abramowitz & Stegun (Reference Abramowitz and Stegun1972, p. 508):

The asymptotic solution (6.7) reveals the growth rate of the perturbations in ${\it\zeta}$ , with exponent

which is displayed in figure 4. When ${\it\gamma}_{m}>4/3$ the exponent is less than one and the perturbations decay. However, when ${\it\gamma}_{m}<4/3$ the perturbations grow without bound. The Gaussian profile, or straight-sided jet, for which ${\it\gamma}_{m}=4/3$ , is the special case for which the amplitude of the perturbations remains constant. It is tempting to speculate that it is in order to yield constant-amplitude perturbations that the flow chooses to be Gaussian in form. In the limit ${\it\gamma}_{m}\rightarrow 1$ the perturbations grow exponentially, and the limiting form of the solution is

whilst in the straight-sided limit ${\it\gamma}_{m}\rightarrow 4/3$ the solution degenerates into a single harmonic of constant amplitude:

for constants $c_{3}$ and $c_{4}$ .

Figure 4. Dependence of power-law growth of source perturbations applied to a jet on the dimensionless energy flux ${\it\gamma}_{m}$ . The dashed line indicates the value $-1/8$ of $a({\it\phi})$ as ${\it\phi}\rightarrow \infty$ .

6.2. The response of the jet area

Under the assumption that ${\it\gamma}_{m}$ remains constant, characteristic curves of the system (2.10) and (2.11) are paths along which the total derivatives of several quasi-invariant quantities are decoupled. Using the eigenvectors of the system the invariants are defined in terms of the total derivatives (Part 1):

where the integrating factor $Q_{m}^{-{\it\gamma}_{m}/{\it\lambda}_{n}^{\ast }}$ has been introduced. Here ${\it\lambda}_{n}^{\ast }$ is an eigenvalue of the original system of equations (see Part 1, equation (5.4)):

where $n=1$ and $n=2$ correspond to the slower and faster of the two characteristic curves, respectively. Consequently, the quasi-invariants are

If we consider a case in which the turbulence production is zero, then the original governing equations (2.10) and (2.11) become a homogeneous system and $\text{d}Y_{m}^{n}/\text{d}t=0$ along characteristic curves. Steady-state solutions to this idealised case correspond to a jet that does not entrain and therefore has a constant volume flux and a constant momentum flux. If a discontinuity is imposed at $(z,t)=(0,0)$ then $Q_{m}$ and $M_{m}$ will be constant in a region $S$ , bounded by fast and slow characteristic curves that emanate from $(0,0)$ . Therefore, in the homogeneous case, the value of the invariants, and hence the values $Q_{m}^{S}$ and $M_{m}^{S}$ , of $Q_{m}$ and $M_{m}$ in $S$ , respectively, can be determined by tracing fast and slow characteristic curves to points outside $S$ at which the solution is known:

The solution of these equations is

where ${\it\phi}$ was defined in (6.6) and

Alternatively, (6.15a,b ) can be manipulated to give the velocity $w_{m}^{S}$ and radius $r_{m}^{S}$ in the region $S$ ,

where

If it is assumed that the area is held constant at the source, then

If ${\it\gamma}_{m}=4/3$ , then ${\it\phi}=1/2$ and $r_{m}^{S}=r_{m}^{A}$ , indicating that the jet is straight-sided as expected. Equations (6.19a,b ) are useful because they can provide the change in radius, $r_{m}$ , and change in velocity, $w_{m}^{S}$ , between characteristic curves a priori. Although this analysis is based on the homogeneous system, knowledge of $w_{m}^{S}$ allows one to infer whether each characteristic curve represents a compression or rarefaction wave. This information is summarised in figure 5, which shows that although the nature of the leading wave is determined by the sign of the step change in $M_{m}$ , it is the value of ${\it\gamma}_{m}$ that will determine the nature of the slower characteristic. Specifically, figures 6(a) and 6(b) provide contours of $w_{m}^{S}/w_{m}^{A}$ and $r_{m}^{S}/r_{m}^{A}$ , respectively. The value of each of these variables in the region $S$ is determined by the relative change in source velocity $w_{m}^{A}/w_{m}^{B}$ , and the profile parameter ${\it\phi}({\it\gamma}_{m})$ defined in (6.6). Whether the jet velocity or radius increases or decreases in the region $S$ is therefore indicated by the quadrant of figure 6(a,b) to which the pair $({\it\phi},w_{m}^{A}/w_{m}^{B})$ belongs. The analysis performed in this section supports the inference that was made using the area equation (5.1) in § 5. Namely, when ${\it\gamma}_{m}<4/3$ the radius of the jet increases or decreases in response to an increase or decrease in the source momentum flux, respectively, whilst for ${\it\gamma}_{m}>4/3$ the behaviour is reversed. Real jets, for which ${\it\gamma}_{m}\approx 4/3$ , appear to occupy the special position between these states, in which the area is unaffected by changes in $M_{m}$ .

Figure 5. Response of system characteristic curves following a negative (step-down, a) and positive (step-up, b) change in the source momentum flux. A single bold line denotes a compression wave, three thin lines denote a rarefaction wave, and the symbols $+,-,0$ denote a positive, negative and zero area change, respectively.

Figure 6. Response of relative jet velocity $w_{m}^{S}/w_{m}^{A}$ in (a) and relative radius $r_{m}^{S}/r_{m}^{A}$ in (b), in the region between characteristic curves for the homogeneous hyperbolic problem corresponding to (2.10) and (2.11) with constant ${\it\gamma}_{m}$ and ${\it\delta}_{m}=0$ .