## Abstract

Studies are presented to elucidate the role of steady streamwise vortex structures, initiated upstream from weak Görtler vortices in the absence of explicit vortex generators, and their excited nonlinear wavy instabilities in the intensification of scalar mixing in a spatially developing mixing region. While steady streamwise vortex flow gives rise to significant mixing enhancement, the excited nonlinear wavy instabilities, which in turn modify the basic three-dimensional streamwise vortices, give rise to further mixing intensification which is quantitatively assessed by a mixedness parameter. Possibility of similarity between the dimensionless streamwise momentum and scalar transport problems leading to an extended Reynolds analogy is sought. This similarity is shown earlier to hold for the steady streamwise vortex flow in the absence of nonlinear wavy instabilities (Liu & Sabry 1991 *Proc. R. Soc. A* **432**, 1–12). In this paper, the momentum conservation equations for the nonlinear wavy or secondary instabilities together with the advected fluctuation scalar problems are examined in detail. The presence of the streamwise fluctuation pressure gradient, which prevents the similarity, is estimated in terms of the fluctuation dynamical pressure and its relative importance to advective transport. It is found from scaling that the fluctuating streamwise pressure gradient, though not completely negligible, is sufficiently unimportant so as to render similarity between fluctuation streamwise velocity and fluctuation temperature and concentration a distinct possibility. The scalar fluctuations are then inferable from the fluctuation streamwise velocity and that the Reynolds stresses of the nonlinear fluctuations and the scalar fluxes are also similar. The nonlinear instability-modified mean streamwise momentum and the modified mean heat and mass transport problems are also similar, thus providing a complete ‘Reynolds analogy’, rendering possible the interpretation of the scalar mixedness for a gaseous medium for which the Prandtl and Schmidt numbers are near unity. It is found that the nonlinearity of the wavy instability, which induces scalar fluxes modifying the mean scalar transport, further intensifies scalar mixedness over a significant streamwise region which is well above that achieved by the steady, unmodified streamwise vortices alone for the numerical example corresponding to the most amplified wavy-sinuous mode.

## 1. Introduction

Mixing enhancement and control are of great technological interest in a wide spectrum of industrial processes. In exhaust nozzles of aeroengines, enhanced mixing is directed towards noise reduction and scalar pollutant reductions. In combustors, enhanced mixing leads to control of the sequence of reactions. In chemical process industries, the control of mixing again leads to control of the desired sequence of reactions. We address issues in the use of streamwise vortices in mixing enhancement, which is more akin to continuous flow mixers and reactors rather than stationary stirred-tank operations (Bałdyga & Bourne 1999). The generation of streamwise vortices usually involves external intrusions into the flow. For instance, tabs in jet exhaust nozzles (Bradbury & Khadem 1975; Zaman *et al*. 1994) were used for reduction of jet noise and winglet-type vortex generators (Fiebig 1995, 1996; Carletti *et al*. 1995) were inserted into the flow field in heat exchanger applications and in ejectors for mixing enhancement. Deformation of jet exhaust nozzle trailing edge into lobed wall geometry is also used to generate streamwise vortices (Crouch *et al*. 1977; Presz *et al*. 1986; Tillman *et al*. 1991; Eckerle *et al*. 1992; McCormick & Bennett 1994; Yu *et al*. 1995; Tsui & Wu 1996; Waitz *et al*. 1997), which are directed at alleviating noise and emission problems in aeroengines and suppression of engine observables (Wang & Li 2006). Lobed injectors and secondary flows in combustion systems are also found useful (Schadow *et al*. 1989; Swithenbank *et al*. 1989). The control of the sequence of mixing and combustion in order to lower pollutant levels from air-breathing engines has been steadily advanced by Smith *et al*. (1997), Strictland *et al*. (1998), Majamaki *et al*. (2003) and Mitchell *et al*. (2004). In certain vortex generation techniques, it is possible to estimate the relatively large penalty due to drag of the vortex generators relative to the benefit attained: a doubling of heat transfer enhancement ratio could incur a quadrupling of the drag penalty ratio (e.g. Fiebig 1995, 1996). In addition, considerable surface area, and hence skin friction drag, is incurred in lobed nozzles.

Our interest in the present paper is to address the possibilities of passive scalar mixing intensification. To this end, the understanding gained from the intensification of momentum mixing (Girgis & Liu 2002), which provides the crucial advective effects in the scalar transport problem, is thoroughly made use of. The minimally intrusive mixers, such as the wavy-vortex momentum mixer discussed by Girgis & Liu (2002) in the absence of explicit vortex generators, appear to be novel from the point of view of minimizing energy waste compared to, say, use of winglet vortex generators. They studied the fundamental incompressible momentum-mixing enhancement with streamwise vortices in the mixing region. In place of generation via explicit vortex generators placed in the flow, their streamwise vortices are the nonlinear consequence of upstream wall-bounded weak Görtler (1940) vortices of varying strength, depending on the streamwise length-scale allowed for their development upstream. For reviews of Görtler vortices, which form the basis of the fundamental steady streamwise vortices, one is referred to Floryan (1991) and Saric (1994). The wall-bounded momentum problem including further intensification of momentum transport by Reynolds stresses of the excited wavy instability is studied by Girgis & Liu (2006). They showed that the skin friction could well overshoot that of the local turbulent value, even in the absence of turbulent flow, under the influence of wavy instability of steady streamwise vortices. The nonlinear modification of the steady streamwise vortex problem by the Reynolds stresses of the wavy disturbance is responsible for the modification of the mean flow, which in turn causes intensification of skin friction beyond that attainable without such instabilities. In the studies of the scalar transport, though uncoupled from the momentum problem in the incompressible approximation, integration of the heat transport and the species transport equations would, in general, be necessary.

Liu & Sabry (1991) and Liu & Lee (1995) studied the advected scalar problem in the absence of wavy instabilities for wall-bounded flows. Owing to scaling in the steady-flow problem, the streamwise pressure gradient in the streamwise momentum equation is absent (e.g. Floryan & Saric 1982). It is observed that the steady advected scalar problems for temperature and mass concentration in an inert binary mixture are similar to the streamwise momentum problem in their respective dimensionless forms provided that the upstream initial and boundary conditions are also similar and that the Prandtl and Schmidt numbers are unity. This then allows aspects of scalar transport to be inferable from the streamwise momentum problem. The smoke observations in an air wind tunnel of Ito (1980, 1985, 1988) showed that the downstream nonlinear development of the steady Görtler vortex organized the smoke or iso-concentration lines into mushroom shapes. Through the similarity arguments given (Liu & Sabry 1991), it is then possible to interpret the amazing similarity to the iso-total streamwise velocity lines obtained by hot-wire measurements (Swearingen & Blackwelder 1987). For Prandtl and Schmidt numbers away from unity, Liu & Lee (1995) integrated the scalar transport equation with parameters corresponding to the dominant spanwise mode. Owing to limited diffusion for Prandtl number (or Schmidt number) much larger than unity, such as for a liquid, the iso-temperature scalar mushroom structure is much curtailed when compared with the full mushroom structure of the iso-total streamwise velocity lines. For fluorescene in water, the Schmidt number is of the order of a thousand. The advected scalars in stream tubes, starting from upstream conditions and which develop downstream spiralling around the counter-rotating vortices, have very limited diffusion spread across these spiral concentration ‘jets’ and hence, cumulatively, splatter into a very narrow cross-sectional structures in the cross-sectional planes downstream as is observed by Peerhossaini & Wesfreid (1988).

The possibility of similarity between momentum and scalar transport, found possible for the steady streamwise vortex problem (Liu & Sabry 1991), is here extended to the nonlinear wavy instability problem. In the presence of wavy instabilities accompanied by the fluctuation velocity-induced scalar fluxes, the search for similarity between momentum and scalar transport is much more intricate than for the steady streamwise vortex flow problem. This similarity is found possible, although approximate. Through this, the physical and quantitative interpretation of scalar transport and mixedness intensification in a mixing region owing to fluctuation transport are the main contributions of the present paper for fluids with Prandtl and Schmidt numbers not far from unity.

The formulation of the scalar transport problem under streamwise vortices is given in §2. This brings into the picture the three-dimensional advecting velocity field of the streamwise vortices and their wavy instability-induced scalar fluxes. The nonlinear momentum problem for the mean flow and the coupled fluctuations are briefly reviewed in §3 for completeness. The plan of attack of the momentum problem necessitates discussion of the resemblance, at least in physical description if not complete mathematical congruity, between the present class of problems and those in the body of Stuart's work on nonlinear hydrodynamic stability; this is discussed in §4. The possibility for similarity between the streamwise momentum and scalar transport including the nonlinear wavy instabilities, Reynolds stresses and the vector heat and concentration fluxes are discussed in §5. This similarity would allow the scalar transport problem to be inferred from the results of the momentum problem in the case of Prandtl and Schmidt numbers near unity. In §6, profiles associated with scalar transport are given to illustrate the modification of the steady iso-temperature (or iso-concentration) profile, whose variance contributes to the mixedness parameter, and the modified fluctuation temperature (or concentration) profile in the mixing region. The intensification of scalar mixing for fluids of Prandtl and Schmidt numbers near unity is discussed in §7 in terms of a mixedness parameter; although steady streamwise vortices enhance mixedness compared with the case in their absence, the nonlinear wavy instability of the steady streamwise vortices modifies the steady flow to such an extent so as to provide significant further intensification of mixedness. The overall oscillating structures of the iso-concentration and iso-thermal lines, as would be observed in light sheet observations over downstream cross-sectional planes, are described in §8. Brief concluding remarks are given in §9. Parameters of the physical problem, which are the basis of a numerical illustration, are stated and discussed in appendix A.

## 2. The scalar transport problem

As in the momentum problem described by Girgis & Liu (2002), an incompressible fluid is considered under the assumption of low Mach numbers and small temperature loading. The thermodynamic and the inert mass transport problems are uncoupled from each other and from the momentum problem. As such, the nonlinearities are associated with the momentum problem. Scaling from the large Reynolds number flow momentum problem (e.g. Girgis & Liu 2002) is imposed on the advected, passive scalar transport problem where the advection velocities are known from the momentum problem. We refer to Girgis & Liu (2002, 2006) for the discussion of the momentum problem. To accommodate the nonlinear wavy instabilities, a Reynolds splitting is used, as in problems of nonlinear hydrodynamic stability (Stuart 1956*a*,*b*). The steady Reynolds-averaged mean flow problem is then augmented by the presence of the Reynolds stresses of the unsteady wavy instabilities, which must be solved simultaneously with the wavy-fluctuation momentum problem.

A schematic of both the wall-bounded flow upstream and the mixing region is shown in figure 1. The slightly curved wall generates the streamwise vortices. For simplicity, a one-stream mixing region in steady flow is considered. Dimensionless coordinates *x*, *y*, *z* are defined in (2.1). Along the vortex generating wall, the streamwise direction *x* is measured along the wall from the leading edge, *y* is the normal to the wall and *z* is the spanwise direction parallel to the trailing edge. In the mixing region description, *x* is re-measured from the trailing edge, whereas *y* and *z* play a similar role in the cross-sectional plane perpendicular to the streamwise direction. The upper free stream velocity is *U*_{0}, while the lower stream has zero velocity. Cross-sectional plane ‘streamlines’ are obtained by connecting vectors of cross-sectional velocities *v* and *w* in the *y* and *z* plane at a given streamwise location. These depict a pair of counter-rotating vortices within one spanwise wavelength *λ*_{z}. The mushroom structures are lines of iso-*u* and in the steady case, in the absence of wavy instabilities (Liu & Sabry 1991), they also represent the dimensionless temperature and normalized concentration, iso-*θ* and iso-*z*_{i} lines (the dimensionless quantities *u*, *θ* and *z*_{i} are defined in equation (2.1)) for Prandtl and Schmidt numbers unity. The downwash region is where the cross-sectional streamlines are pointed downwards (*v*<0) and the upwash region is where *v*>0 in figure 1.

For description of the effect of wavy instabilities, it is essential that the already scaled mean flow problem and the nonlinear instability problem be cast into similar dimensionless independent variables. In this case, the rescaled wavy instability normalizations (Girgis & Liu 2002, 2006) are favoured. The dimensionless total flow quantities and coordinates, represented in lower case, are written as the sum of the dimensional quantities represented by the respective capital symbols. The flow and scalar quantities are written in the Reynolds splitting form, where the steady flow is without prime and fluctuation quantities are indicated by primes,(2.1)where the scalar quantities are defined in terms of the temperature *T* and mass fraction *K*_{i} of the *i*th species in a binary mixture; *u*, *v* and *w* are the corresponding velocity components of the streamwise, normal and spanwise coordinates *x*, *y* and *z*, respectively; *p* is the dimensionless pressure; *t* is the time; *θ* is the dimensionless temperature; and *z*_{i} is the normalized concentration. On the solid surface upstream of the mixing region, *T*_{w} is the constant wall temperature and *T*_{e} is the temperature at the upper edge of the boundary layer. The upper stream of the one-stream mixing layer has properties *U*_{0}, *T*_{e} and *K*_{e} and the lower stream is at zero velocity and at temperature *T*_{w} and concentration *K*_{w}.

We start from the incompressible form of the energy equation in terms of the static enthalpy with constant transport properties and heat capacity. The Reynolds splitting *θ*+*θ*′, as in equation (2.1), is substituted into the energy equation. Upon Reynolds averaging, the mean heat transport equation is obtained as(2.2)where ** u** is the vector velocity of the three-dimensional steady counter-rotating streamwise vortex flow;

**′ is the three-dimensional wavy-fluctuation velocity written in vector form; is the parabolized Laplacian in the**

*u**yz*cross-sectional plane which resulted from the scaling imposed from the momentum problem; the over bar is the time average over at least the largest period of the fluctuation; is the heat flux vector, which modifies the mean transport of the temperature

*θ*, similar to the Reynolds stresses modifying the mean velocity in the nonlinear hydrodynamic stability (and turbulent shear flow) problem;

*Re*=

*U*

_{0}

*δ*

_{0}/

*ν*is the Reynolds number, where

*δ*

_{0}is a viscous length-scale defined by

*δ*

_{0}=

*νX*

_{0}/

*U*

_{0},

*ν*is the kinematic viscosity and

*X*

_{0}is a reference length chosen to be the distance from the leading edge on the plate where the streamwise vortices were imposed upstream (the physical problem is independent of

*X*

_{0}as discussed by Lee & Liu 1992 and Benmalek & Saric 1994). The Prandtl number is

*Pr*=

*ν*/

*α*, where

*α*is the thermal diffusivity and

*ν*is the kinematic viscosity. The fluctuation energy equation is obtained by subtracting the mean energy equation from the overall energy equation,(2.3)In the above equation, partial differentiation is indicated by subscripts. Again, the scale analysis from the fluctuation momentum problem imposes its scale and structure on the fluctuation energy equation. The mean flow advection is provided only by the

*x*-component velocity of the streamwise vortices, which dominates over that of the counter-rotating velocities in the

*yz*plane. Owing to the strong gradients of the mean flow in the cross-sectional plane and not in the streamwise direction, fluctuation advection of the mean temperature is confined to the

*yz*plane. The fluctuation transports of mean temperature, the third and fourth terms on the left-hand side of equation (2.3), are the sources of the fluctuation temperature. This is similar to the sources of the streamwise velocity fluctuation in hydrodynamic stability. Since the scale of the fluctuations is very nearly isotropic, the Laplacian on the right-hand side of equation (2.3) is tentatively three dimensional prior to distinguishing streamwise wavy characteristics from the streamwise wave envelop. The last term in equation (2.3) is the divergence of excess fluctuation heat flux vector, which is responsible for harmonics in the temperature fluctuation. The modification of the mean temperature distribution

*θ*is effected directly by the Reynolds-averaged heat fluxes , indirectly through the fluctuation scalar advection by the Reynolds stress-modified mean flow

*u*as well as by the fluctuation velocities in equation (2.3), and the mechanisms of diffusion and harmonic generation on the right-hand side of equation (2.3).

Referring to the schematic in figure 1 and equation (2.1), the location of the trailing edge in wall coordinates is denoted by subscript *x*_{G}, which is measured from the leading edge of the plate. Subsequently, in mixing region coordinates, the distance measured from the trailing edge is simply denoted by *x*. The conditions for the mean temperature are as follows: in the mixing region,in the upper region, representing the single stream uniform flow, and in the lower regionPeriodic boundary conditions are imposed in the spanwise directionwhere *λ*_{z} is also used as the dimensionless spanwise wavelength normalized by *δ*_{0}. The upstream initial condition for the mixing region is the wall-bounded solution (Liu & Lee 1995) at the trailing edge

The scalar problem is indeed a passive one in that the advecting momentum problem imposes the structure and scales. The temperature fluctuation analysis follows that of the momentum fluctuation problem (§3), and the fluctuation temperature is expressed as(2.4)where c.c. denotes the complex conjugate and is the complex amplitude of the fluctuation temperature and is a function of spatial variables (*x*, *y*, *z*). The streamwise wavenumber *α* and frequency *ω* are imposed by the momentum problem, which are discussed in more detail in §3 and in appendix A.

The concentration transport for an inert mixture is briefly discussed and is similarly described by(2.5)for the mean concentration and by(2.6)for the fluctuation concentration. The scaling and the advective transport mechanisms are similar to the temperature problem, except that the Schmidt number *Sc*=*ν*/*D*_{12} replaces the Prandtl number, where *D*_{12} is the binary mass diffusion coefficient. Discussions are confined to the situation where boundary and initial conditions are similar to that for the temperature, with *z*_{i} replacing *θ*.

For the same geometrical flow field and at the same Reynolds number, there is complete similarity between the dimensionless temperature and normalized concentration, i.e. *θ*=*z*_{i}, and between the dimensionless vector scalar transport , provided that the dimensionless upstream initial and boundary conditions for *θ*, *θ*′ and *z*_{i}, are similar and if *Sc*=*Pr* or that the Lewis number *Le*=*Pr*/*Sc* is unity.

## 3. Description of the nonlinear momentum problem

Advection by the fluid velocities plays an essential role in the present class of problems and, for completeness, is briefly discussed in the following. Debonis (1992) simulated the flow field of a two-dimensional mixer ejector nozzle. The temperature profiles obtained showed the development of mushroom-like structures in terms of the iso-temperature contours. This strongly resembles the iso-streamwise velocity contours in Görtler vortices (Sabry & Liu 1991; Lee & Liu 1992), which is expected from similarity considerations at least for an incompressible steady streamwise vortex flow (Liu & Sabry 1991). Grosch *et al*. (1997) numerically simulated the presence of small tabs in a hot supersonic two-dimensional jet. An empirical relation was used to model the presence of tabs by introducing a pair of incompressible, counter-rotating vortices as upstream condition. As expected, these simulations, which are for steady flow, showed that mixing was enhanced. Such steady flows are susceptible to secondary instabilities giving rise to observed oscillations of the streamwise vortices (Ito 1980, 1985, 1988; Swearingen & Blackwelder 1987; Peerhossaini & Wesfreid 1988). It will be shown that the averaged stresses and scalar fluxes from the wavy instabilities will provide further and significant intensification of transport and mixing.

In the incompressible approximation, the scalar transport problems are linear in that it is presumed that the advective effects are known from the nonlinear momentum transport problem (Girgis & Liu 2002, 2006). The full momentum problem is briefly discussed here. The steady-flow scaling follows from the pioneering work of Floryan & Saric (1982) and Hall (1983, 1988) in which the streamwise history and the parabolicity of the problem were brought out. Extensive applications appear in papers such as those of Sabry & Liu (1988, 1991), Lee & Liu (1992) and Benmalek & Saric (1994) for the parabolic steady streamwise counter-rotating vortex flow. They pointed out the importance of upstream initial conditions, which must be consistent with the local linear Görtler vortex problem when the initial disturbances are small.

The mean continuity equation for an incompressible fluid is(3.1)The streamwise mean momentum equation is(3.2)The advection is carried out by the full three-dimensional velocity components of the mean flow.

The normal, or *y*-component, mean momentum equation is(3.3)where *r*_{c}=*R*/*δ*_{0} is the dimensionless upstream wall radius. It is taken to be *r*_{c}≫1 in the mean flow scaling in addition to *Re*≫1. The Görtler centrifugal mechanism *Nu*^{2}/*r*_{c} is used to generate streamwise vortices on the wall upstream of the mixing region so that *N*=1 on the wall. The mixing region is considered to be tangent to the trailing edge of the wall so that *N*=0. For an incompressible fluid, the density is absorbed as the denominator of the pressure. The spanwise mean momentum equation is(3.4)The parabolic nature of the steady mean flow supports dominant viscous diffusion and pressure gradients only in the *yz* cross-sectional plane, with the streamwise pressure gradient notably absent. It is clear that mean momentum transport in equations (3.2)–(3.4) is similar to that, for instance, in Floryan & Saric (1982) in the absence of the wavy fluctuations; here, the mean flow transport is augmented by the Reynolds stresses, as would be in the case of nonlinear instabilities (Stuart 1956*a*,*b*). The boundary conditions for the mixing region arein the upper region, representing the ‘core region’ in a nozzle flow, andin the lower region. Periodic boundary conditions are imposed in the spanwise directionwhere *λ*_{z} is also used as the dimensionless spanwise wavelength normalized by *δ*_{0}. The upstream initial condition for the mixing region is the wall-bounded solution at the trailing edge (from Lee & Liu (1992), which is recomputed in Girgis & Liu (2002)),

The coupled fluctuation problem is obtained by subtracting the mean flow equations from that of the total flow,(3.5)(3.6)(3.7)(3.8)

In order to proceed, the unsteady wavy fluctuations are represented in normal mode form, reflecting the observations (Swearingen & Blackwelder 1987), with the fundamental wave characteristics given by the initial local linear theory,(3.9)where c.c. denotes the complex conjugate and , , and are the fluctuation complex amplitudes and are functions of (*x*, *y*, *z*). Girgis & Liu (2002) gave extensive discussions in favour using the characteristics from the linear theory (e.g. Yu & Liu 1991, 1994; see also Hall & Horseman 1991; Li & Malik 1995; Park & Huerre 1995) at upstream conditions, which remain rather robust as the wavy instability develops downstream according to observations (Swearingen & Blackwelder 1987). This procedure then replaces the formal parabolized stability theory, known as PSE (Herbert 1997) as is practised elsewhere (Li & Malik 1995). Spatial growth of the disturbances will be considered; therefore, *α* is a complex number, *α*=*α*_{r}+i*α*_{i}, that combines the spatial wavenumber *α*_{r}=2*π*/*λ*_{x} and the spatial amplification rate *α*_{i}, where *λ*_{x} is the fluctuation wavelength in the *x*-direction and *ω* is the dimensionless fluctuation frequency. Equation (3.9) represents the fluctuations as rapidly oscillating waves exp i(*αx*−*ωt*) propagating in the streamwise direction, which are embedded within slowly varying wave amplitudes or wave envelops . Thus, the parabolization process here amounts to attributing the dominant streamwise changes of flow quantities to the rapidly varying wavy part compared to the slow variations of the wave envelop. Upon substitution of equations (3.9) into (3.5)–(3.8), equating coefficients of the same order, the parabolic secondary instability equations in spectral form are obtained. In the absence of harmonics, the fundamental perturbation complex amplitudes are obtained with the absence of the explicit nonlinear effects of the excess Reynolds stress terms on the right-hand sides of equations (3.6)–(3.8). Nonlinearities enter through the modified mean flow *u* on the left-hand sides of equations (3.6)–(3.8), which are nonlinearly coupled to the mean flow equations (3.1)–(3.4) through the Reynolds stresses on their right-hand sides. For simplicity, the spectral form of the equations is not presented here.

The streamwise initial conditions for the fluctuation problem in the mixing region are imposed at the trailing edge of the wall, *X*_{g} (or its dimensionless form, *x*_{g}). It is a wall parameter measured from the leading edge of the upstream plate and indicates the extent in which the upstream mean streamwise vortex is allowed to develop to the desired stages of nonlinearity. Girgis & Liu (2002) suggested an *X*_{g} that produced the most vigorous mixedness parameter development. The upstream initial conditions for the fluctuation problem in the mixing region are imposed at the trailing edge of the wall, using the linear stability profile (Yu & Liu 1991, 1994). It is a temporal theory and is recomputed by Girgis & Liu (2002) for convenience. The linear secondary instability equations are a special case of equations (3.5)–(3.8) in a locally parallel flow or ‘equilibrium’ form devoid of streamwise advection over large distances. The source mechanism is identical to the nonlinear theory. Only the most amplified mode, which is the sinuous mode, is considered and the sinuous-wave amplitudes are represented as (Yu & Liu 1991)(3.10)where *nz* is the number of points in the spanwise direction implying spanwise periodic boundary conditions and the subscript ‘1’ represents the first sinuous frequency mode in Yu & Liu's (1991) notation. Specific values of parameters associated with the wavy instability for a specific example are given in appendix A.

## 4. Relation to classical works in nonlinear hydrodynamic stability

The importance of the nonlinear momentum problem (Girgis & Liu 2002) cannot be underestimated. The scalar transport problem, which is linear, is intricately related to the momentum problem through advective effects of the three-dimensional streamwise vortex velocities, including the effect of the scalar flux vectors. It is thus helpful to briefly discuss the scheme of approach of the momentum problem that impresses or imposes its scales and solution ideas on the scalar transport problem. The solution of the nonlinear momentum problem, which in this case is the nonlinear instability of developing counter-rotating streamwise vortices, is strongly guided by the spirit of earlier works by Meksyn & Stuart (1951) and Stuart (1956*a*,*b*, 1958, 1960, 1962*a*,*b*, 1963, 1965). In the computation of the mean field steady-flow momentum equations, the Reynolds stresses are first calculated from upstream initial linear secondary instability equations (Yu & Liu 1991, 1994) and are incorporated into the mean field numerical integration to advance one streamwise increment. The modified mean flow at the end of this increment is then used to provide the mean flow advection velocities and velocity gradients in the nonlinear fluctuation equation so that the fluctuations are advanced one streamwise increment and so on. Thus, each pair of advancing leapfrog streamwise step is indeed a Meksyn–Stuart increment in the computation of streamwise development. In this case, the Reynolds stresses of the wavy fluctuations play a dominant role (Stuart 1956*a*,*b*), as will be the scalar fluxes in the scalar transport problem. In the present class of problems, the subsequent development of the nonlinear instabilities appears to be fixed by the frequency and wavelength of the upstream linearized fluctuations even from observations. Hence, the use of a single fundamental mode (Stuart 1958) with its wavy characteristics furnished by the upstream linear theory and nonlinear wave envelop described by the nonlinear differential equations is justified in this case and parallels the single-mode shape assumption discussed by Stuart (1958). In the momentum problem computations of Girgis & Liu (2002), the mean flow is considerably modified by the Reynolds stress effects of the wavy fluctuations (3.2)–(3.4), and this is reflected by the Stuart constant *k*_{1} (Stuart 1960); in turn, the fundamental fluctuation component is modified (the Stuart constant *k*_{3}). Harmonic generation (Stuart constant *k*_{2}), though not accounted in the detailed computations of Girgis & Liu (2002), would be included in the general formulation through the presence of excess Reynolds stresses in the fluctuation equations (3.6)–(3.8). The discussion of energy transfer mechanism between odd and even modes (Stuart 1962*a*) is interpreted as fundamental–harmonic interactions. Equally, it could also be interpreted in terms of even–odd mode interactions reflecting the fundamental–subharmonic interactions (Kelly 1967; Liu 1981; Nikitopoulos & Liu 2001); these would be reflected in the general formulation also. In the present problem, of course, three-dimensional nonlinear effects abound (Stuart 1962*b*), including the stretching and tilting of spanwise vorticity by streamwise counter-rotating vorticity elements studied kinematically by Stuart (1965) and which is illustrated in the steady Görtler flow problem (Sabry & Liu 1991) using the dynamical equations. In the nonlinear instability of wavy fluctuations, these three-dimensional effects depicted earlier by Stuart (1965) are manifested through the normal fluctuation advection of spanwise vorticity of the mean three-dimensional streamwise vortex flow and the spanwise fluctuation advection of the normal vorticity of the mean flow. The wavy instabilities in the present open flow problem owe their origins from the earlier studies of the instabilities of Taylor vortices (Davey *et al*. 1968), as much as the modern studies of the basic Görtler (1940) vortices to studies of Taylor (1923) vortices.

There is no attempt here to provide as complete a literature survey (Saric 1994) and commentary as is done in Girgis & Liu (2002, 2006) for the wavy instability and Sabry & Liu (1991) and Lee & Liu (1992) for the steady streamwise vortices. However, it is emphasized here that the dominant nonlinear physical features in the present class of three-dimensional problems involving wavy instabilities of developing streamwise vortices are anticipated much earlier in the nonlinear hydrodynamic stability studies of Stuart; these nonlinear features transcend and reach far beyond the restrictions of the original assumptions of nearly equilibrium flow in the classical works on nonlinear hydrodynamic stability.

## 5. The search for Reynolds analogy

The application of the simplified but coupled nonlinear momentum system, (3.1)–(3.4) and (3.5)–(3.8), to the mixing region problem is discussed in Girgis & Liu (2002), with applications to finding and assessing the mixedness of the overall streamwise velocity *u*. In the absence of nonlinear wavy fluctuations, the Reynolds stresses and the heat and mass flux vectors, and , respectively, are absent from the mean flow equations. It is shown by Liu & Sabry (1991) that the total *x*-component velocity *u* of the streamwise vortices, the temperature *θ* and the concentration *z*_{i} are similar provided that the Prandtl and Schmidt numbers are unity, *Sc*=*Pr*=1, and that the initial and boundary conditions are also similar in their dimensionless form.

The system is somewhat more intricately involved in the presence of nonlinear wavy fluctuations. An examination of the mean streamwise momentum equation (3.2) for *u* and the uncoupled mean scalar transport equations (2.2) and (2.5) for *θ* and *z*_{i}, respectively, shows that owing to the Reynolds stresses and the scalar fluxes, the similarity between the fluctuations becomes an issue. From previous discussions, it is already possible to have for Lewis number unity from observations of the fluctuation scalar transport equations (2.3) and (2.6). Equation (3.6) for *u*′ indicates that the streamwise fluctuation pressure gradient, , which was retained in the computational solutions discussed by Girgis & Liu (2002), stands out as an impediment for similarity between the streamwise fluctuation velocity *u*′ and the scalar quantities *θ*′ and .

We now estimate the order of magnitude of the fluctuation streamwise pressure gradient in equation (3.6) relative to other effects. The relative magnitudes of the fluctuation velocity components, estimated from the fluctuation continuity equation (3.5), are in dimensional form *U*′/*λ*_{x}, *V*′/*δ*_{0} and *W*′/*λ*_{z}, where *λ*_{x} is the streamwise wavelength of the wavy instability, *δ*_{0} is the viscous length-scale (appendix A) measuring the region in which the *Y* derivative is of importance and *λ*_{z} is the spanwise wavelength set by the steady streamwise vortices. The dynamical pressure of the fluctuations estimates the fluctuation pressure and hence its streamwise gradient is estimated asIn terms of *U*′ alone, with continuity estimates of *V*′ and *W*′, thenThe dominant effect is given bysince results of the momentum problem indicate that *δ*_{0}/*λ*_{x}≪1, *λ*_{z}/*λ*_{x}<1. The mean flow advective effect, which is the second term on the left-hand side of equation (3.6), is estimated in dimensional form as *ρU*_{0}*U*′/*λ*_{x}. The fluctuation transports of mean momentum, given by the third and fourth terms on the left-hand side of equation (3.6), are the sources of the wavy instability and are estimated in dimensional form asThe parabolized viscous diffusion effect, which balances the advective and source mechanisms, is of the same order. The fluctuation streamwise pressure gradient relative to advective and instability source mechanism is then of the order *U*′/*U*_{0}≪1. In this case, the dimensionless streamwise momentum equation (3.6) is approximately represented as(5.1)

Accepting this approximation as far as similarity is concerned, we now compare (5.1) with the fluctuation scalar transport equations (2.3) and (2.6). It is essential to emphasize that, for similar dimensionless upstream initial conditions and boundary conditions, and for the same *Re* and if *Pr*=1, *Sc*=1 we would have(5.2)and in addition, the dimensionless scalar flux vectors are similar to those Reynolds stresses that involve the transport of streamwise fluctuation momentum *u*′,(5.3)

Simultaneously, the dimensionless mean flow streamwise momentum equation (3.2) is similar to that of the mean scalar transport equations (2.2) and (2.4), so that we arrive at an analogy for the wavy instability-modified mean flow(5.4)The three-dimensional steady mean flow field of the streamwise vortices, ** u**, described by equations (3.2)–(3.4), is solved simultaneously with the fluctuation velocities (3.5)–(3.8) in the incompressible problem (e.g. Girgis & Liu 2002), independently of the scalar transport of heat ((2.2) and (2.3)) and mass ((2.5) and (2.6)). Thus, equations (5.2)–(5.4) are viewed as extended Reynolds analogies (Reynolds 1874; Squire 1953) for the present class of problems that include the extended analogy involving the structure of the flow quantities prior to averaging. The streamwise momentum problem can now be used to study the scalar mixing intensification for fluids with Prandtl and Schmidt numbers in the vicinity of unity.

## 6. Mean and fluctuation scalar structure

The wavy instability-modified mean flow profile *u*=*θ*=*z*_{i} is shown in figure 2 for the downstream location *X*=30 cm from the trailing edge for the parameter values discussed in appendix A. It is at this location where the mixedness parameter (figure 4; §7) for the modified mean flow is twice that of the case in the absence of wavy instabilities. The iso-*θ* and iso-*z*_{i} contour lines are modified by the Reynolds-averaged heat and concentration fluxes induced by the wavy-fluctuation velocities, and , respectively. The significant distortion is due to the local transfer from the mean temperature-squared *θ*^{2}/2 to that of the fluctuation via the gradient–flux conversion mechanisms (Tennekes & Lumley 1972)which is similar to the instability mechanismfor kinetic energy transfer from the mean motion to the *x*-component contribution to the kinetic energy . For the present most amplified sinuous mode, the dominant effect is associated with the spanwise mean flow gradient (Yu & Liu 1994). Similarly, the distortion of the *θ* contours of figure 2 is attributable to the dominant mechanism associated with the velocity-sinuous mode manifested through the production of by the spanwise advection of heat against the spanwise mean temperature gradient . The contours in figure 2 also indicate that the mean flow has spread considerably into the lower region (*y*<0) of the mixing layer. The discussion of the mechanisms leading to the temperature contours also applies to the concentration contours. The depleted structure on the shoulders of the otherwise smooth mushroom-like contour indicates that the outer contours of higher values of *θ*, or of *z*_{i}, are being transported towards the interior due to the nonlinear effects of the wavy instability, thus promoting mixing in the mean.

For the same downstream location as that for the mean flow iso-*u* contours of figure 2, figure 3 shows the contours of iso- and iso- lines, where r.m.s. denotes the root mean square. For *Pr*=*Sc*=1, the contours are inferred from iso- of Girgis & Liu (2002) in view of equation (5.2). The scalar contours possess symmetrical, double regions of high intensity. These correspond to regions where the structure of the temperature gradient ∂*θ*/∂*z* dominates, typical for the sinuous mode (fig. 12 of Girgis & Liu 2002). Since the strong mean flow gradients are located only in the upper regions, the fluctuations are therefore lacking in the lower regions of the mixing layer. The strong fluctuations in the upper region coincide with the depleted regions of the strongly modified mean temperature distribution of figure 2. In the absence of wavy instabilities, the mushroom structures of iso-*θ* contours are otherwise smooth as depicted in figure 1.

## 7. Intensification of mixing for fluids of Prandtl and Schmidt numbers near unity

Mixedness is assessable in different ways (e.g. Roshko 1976). The mixedness parameter is defined here in terms of the variance *σ*_{u} of an advected time-averaged scalar quantity (Tsui & Wu 1996). In the momentum problem (Girgis & Liu 2002), the advected scalar used in the definition is the overall streamwise component of mean velocity *u*,wherewhere is the integration of the time-averaged quantity *u* carried out over the finite cross-sectional region *A* in the *yz* plane. The mixedness parameter is then defined aswhere *σ*_{u}(0) is the variance at the trailing edge. Such a parameter was used, for instance, as an indication of mixedness downstream of multilobe mixers (e.g. Tsui & Wu 1996). In view of the similarity arguments leading to the extended Reynolds analogy equation (5.4), the mixedness parameters for the scalars *θ*, *z*_{i}, respectively, can be directly obtained from that of *u*,(7.1)wherewith the corresponding definitionsand that

Figure 4 shows the development of as a function of the downstream distance from the trailing edge. The solid line, denoted by ref, is the situation in the absence of streamwise vortices where the mixedness development is accomplished by molecular transport alone. The dashed line represents the case of steady streamwise vortices in the absence of wavy instabilities. The dashed-dotted line is the result including the effect of wavy instability-induced fluxes on the mixedness parameter. At approximately *X*≈30 cm (*x*=*X*/*δ*_{0}≅230) from the trailing edge, its value is approximately four times that in the absence of streamwise vortices and more than twice that of the steady case. At the same streamwise location, the centreline values of the modified mean flow are accelerated to ( from fig. 4 of Girgis & Liu 2002) and are 50% larger than the corresponding quantities in an unmodified streamwise vortex flow. This accelerated development is another indication of mixing intensification by the averaged effect of the wavy instabilities.

## 8. Visual depiction of oscillating iso-concentration and iso-thermal structures

It is shown in §5 that for the same *Re* and *Pr*=1, *Sc*=1 as well as similar dimensionless upstream initial and boundary conditions, and *θ*=*z*_{i}=*u*. In this case, the overall instantaneous structures are also similar,Thus, the instantaneous profiles *u*+*u*′ (fig. 18 of Girgis & Liu 2002) could be directly used to infer information about the instantaneous isothermal lines *θ*+*θ*′ and the iso-concentration lines *z*_{i}+*z*′. Experimental observations of downstream developing, oscillating mushroom structures are reported, for example, by Aihara *et al*. (1985) and Ito (1985) for smoke seeding in air that resembles a gaseous mixture for which the Schmidt number would indeed be close to unity; recent observations in an air wind tunnel are discussed by Momayez *et al*. (2004). In gaseous flows, the scalar structures are similar to the streamwise velocity structure. Observations in a water tunnel using light-sensitive fluorescene are reported in the classical experiments of Peerhossaini & Wesfreid (1988). In this case, the Schmidt number is of the order of 10^{3} and the theoretical description would require separate numerical computation for quantitative interpretation. However, for steady nonlinear streamwise vortices, it is shown (Liu & Lee 1995) that the scalar mushroom structure for large *Pr*≫1, *Sc*≫1 is very much curtailed relative to the streamwise velocity structure. The relative curtailment is due to the smallness of thermal and mass diffusivities relative to momentum diffusivity; thus, scalar diffusion across spiralling streamtubes about the counter-rotating vortices is relatively limited. The light sheet in the experimental observations, which cuts across a *yz* cross-sectional plane at a given downstream location, reveals the effect of the propagating wavy instabilities manifested in the predominant spanwise side-to-side oscillations of the mushroom structure. Depicting such oscillations is the nonlinear fluctuating scalar sinuous mode, superimposed upon the nonlinear structure of the scalar flux-modified mean scalar structure and is shown in figure 5. The progression in time through one period of oscillation is indicated in figure 5*a–f*, which are, respectively, 0, 5/24, 9/24, 13/24, 17/24 and 21/24 of a period for a physical frequency of 130 Hz corresponding to the most vigorous sinuous wavy instability mode. If the varicose mode were dominant, the observations would give a predominantly symmetrical oscillation about the axis of the mushroom structure as the wavy instability propagates through a *yz* cross-sectional plane at a given streamwise station.

In the present problem, computations are carried out for a fixed spanwise wavelength of the basic streamwise vortex flow. It should be pointed out that this fixed spanwise wavelength was found in Swearingen & Blackwelder (1987) wind tunnel experiment in a particular spanwise region where detailed measurements are carried out. However, the spanwise wavelength has different but similar values in different spanwise regions in the same experiment (Peerhossaini & Wesfreid 1988; Swearingen & Blackwelder 1987). In the former observations, multiple spanwise structures are at instances, strongly interacting with one another. This feature could possibly be captured by starting computations with a multiple of spanwise structures distributed in the spanwise region.

## 9. Concluding remarks

The possibility of generating streamwise vortices without using explicit vortex generators or lobed trailing edges is intriguing. The present class of streamwise vortices, even in its steady state without wavy instabilities, is amplifying downstream. The stimulation of most amplified wavy instabilities of the steady steamwise vortices further and significantly intensifies mixedness in the mixing region. In the mixedness development of figure 4, the possibility of exciting and embedding wavy instabilities of higher frequency could further intensify the mixedness over an earlier region in the streamwise development. This is because the higher frequency modes develop and peak earlier in the streamwise development, such as that found in nonlinear mode interaction problem of two-dimensional wavy instabilities in a two-dimensional mixing region (Nikitopoulos & Liu 2001). In this case, it was found that the higher frequency modes, which are harmonics of and thus coupled to the prevailing fundamental mode, would also reinforce the fundamental mode as the higher frequency modes decay in the midst of amplification and development of the fundamental mode. For the present three-dimensional system, these ideas need to be tried out and confirmed. It would be of interest to test these ideas experimentally. V. Kottke (2001, private communication) experimented, in an unpublished work, with the exit flow of a U-shaped channel exiting into still air, with the exit end cut to various lengths to promote mixing via the exiting streamwise vortices of various strengths. The quantification of results from experiments of this type, much in the same way the present contribution has been done by the way of theory, is of great interest. Clearly, there is no dearth of fundamental problems that remain in the choice of intensification of mixing that minimizes the impact of the mixer, such as the present wavy mixer in the absence of explicit vortex generators, including the integration of the parabolized scalar transport problem for a liquid system at large Prandtl and Schmidt numbers.

Tani (1962) noted that Görtler vortices can persist well into the turbulent boundary layer region. As such, these large-scale streamwise vortices are but another class of large-scale coherent structures in free turbulent shear flows and would be worthy of study for the intensification of mixing in a stimulated turbulent shear flow such as that described for predominantly spanwise-oriented structures (Liu 1988) in free turbulent shear flows.

## Acknowledgments

Some aspects of the present work were presented at Eurotherm 79, Workshop on Mixing and Heat Transfer in Chemical Reaction Processes, Cargèse, Corsica, from 31 July to 5 August 2006. I express thanks and appreciation to the anonymous referees for their helpful comments and to L. Le Bas, F. Pring and M. Williams for rendering publication in PRS-A a most pleasant experience.

## Footnotes

- Received February 17, 2007.
- Accepted March 30, 2007.

- © 2007 The Royal Society