Abstract
We present a unified vision of the existing theoretical models for the turbulent plane jet, leading to new analytical profiles for scalar concentration and turbulent quantities, including a complete turbulent kinetic energy budget. Integrals of the budget terms are also computed. The present model is split into two variants. Both compare fairly well with referenced experimental data.
1. Introduction
The theory of the turbulent planar jet has already motivated a lot of publications. Pope [1] made a thorough description of the standard theoretical approach based on the boundary layer approximation and self-similarity, leading to a theoretical profile for the velocity field. This method, based on mass and momentum conservation laws, has been extensively discussed (see, e.g., [2–4]) and is in good agreement with experiments except far away from the jet center part where it overestimates the axial jet velocity field [1, 2]. On the other hand, some authors prefer using a purely empirical Gaussian profile [5–8] which provides a better fit to experimental data. Theoretical profiles for the turbulent quantities such as eddy viscosity (as far as this concept is admitted) and the various terms of the turbulent kinetic energy (TKE) budget have also been provided [1, 8] and compared to the experimental data obtained by Bradbury [2], Heskestad [9], Gutmark and Wygnanski [10], Ramapriyan and Chandrasekhara [11], among others. However—to our knowledge—there exists no publication giving a complete and consistent method leading to theoretical approximated profiles for all these quantities. As an example, Agrawal and Prasad [8] give theoretical formulae for Reynolds stresses, TKE production, and eddy viscosity from the Gaussian velocity assumption, but they do not provide any experimental validation. Finally, the distribution of nonbuoyant scalar concentration in a plane jet has been poorly addressed; Kotsovinos and List [5] suggest a Gaussian empirical profile while Law [12] gives a theoretical formula based on scalar flux conservation, but his mathematical expression is cumbersome and requires a function defined in the complex plane.
In this paper, we first build a theoretical formalism that allows treating all transported quantities in a similar way, on the assumptions of boundary layer approximation and self-similarity. In particular, conserved quantities (i.e., momentum and scalar concentration) are treated on the same foot, leading to unified dimensionless governing equations and solutions. The solutions are obtained as the product of two terms: a power law for the axial dimensionless coordinate and a function of the dimensionless transverse coordinate that is determined by solving the governing equations. With this method, the standard velocity profile is recovered while a new analytical profile for scalar concentration is given. Other quantities like TKE and dissipation rate are searched in the same generic form and substituted in the governing equations, giving their dependency to . By assuming a production-to-dissipation equilibrium, analytical formulae are then obtained for all quantities of interest, including the TKE budget terms and Taylor microscale. The latter method is then applied to the Gaussian empirical velocity profile, giving a second variant of this theory. It is found that all present analytical formulae are in good agreement with experimental observations except some of them (dissipation, convection, and diffusion of TKE) in the center part of the jet. The present approach thus provides a unified way to understanding the (approximate) theory of plane turbulent, self-preserving jet.
2. Theory
2.1. State of the Art and Notation
We consider a steady incompressible plane turbulent jet determined by an infinite orifice (or slot) of width (see Figure 1). We work in the plane, with being the coordinate along the axis of the jet, the fluid molecular kinematic viscosity, and the scalar molecular diffusion coefficient. The spatial distribution of (Reynolds-averaged) longitudinal (or axial) fluid velocity and scalar concentration will be searched as function of the fluid discharge velocity and concentration and .
In the sequel, we assume the molecular fluxes and stresses to be negligible with respect to turbulent fluxes and stresses, so that the Reynolds and Schmidt numbers have no effect. We define the vector of conserved quantities and, assuming Prandlt’s boundary layer assumptions to be valid, write the governing equations aswhere primes refer to turbulent fluctuations and overbars to Reynolds-averaging. The symmetry condition through the -axis implies
On the other hand, the boundary conditions far away from the jet are vanishing of axial velocity and concentration, as well as turbulent shear stress and scalar flux:
Experiments (e.g., Bradbury, 1965; see the review by Pope, 2000) show that the plane jet half-width is a linear function of . As in the classical theory of the turbulent plane jet without a scalar, we thus definewhere is a virtual origin. is thus a linear function of and assumes autosimilarity of the jet. We denote by the slope of the half-width.
Then, it is known that integrating the momentum and scalar governing equations along the cross-stream direction gives
For the two unknown functions , we may impose for each , since the dimensionless constants are unknown. The above dependencies in are in agreement with Kotsvinos and List [5] and Pope [1], as well as with Gutmark and Wygnanski’s [10] measurements.
2.2. Advection and Diffusion
We now investigate the advection and diffusion of an arbitrary field in the following generic form (in most cases the power will be set to later on, according to the previous subsection):where is a dimensional constant ( or in the present case). The purpose of the next developments is to write the governing equations (momentum, scalar, and turbulent quantities) in a unified form. We get(here and in the sequel, the prime denotes differentiation with respect to , except in the turbulent fluctuations , etc.). Using (1) and applying (9) to with we get
We now define the antiderivative of aswith . Thus, (11) gives the transverse dimensionless jet velocity as
From that, we can compute the advection of the arbitrary field :
For the two components of the vector defined above (axial velocity and concentration), since the powers are equal to we findwith and .
We now address the diffusion of . We model the turbulent flux and stress with the eddy viscosity assumption:where is the turbulent Schmidt number. We can put these two formulae together by defining where . Now, from (6) the diffusion terms in (2) read
According to (3) and (4), the dimensionless unknown functions introduced so far are subject to the following boundary conditions:with the arbitrary additional conditions and . Now, the momentum and concentration equations (2) can be rewritten using (16) and (19) and integrating once with respect to :where .
2.3. Turbulent Closure
Equation (22) tells us that the eddy viscosity should depend on as in the turbulent plane jet, as already known [1]. This is in agreement with the generic form (7) with and :Thus, (22) gives(). We may first try to compute the eddy viscosity from the standard model; that is, . Under the steady boundary layer assumptions, the TKE and dissipation rate governing equations readwith , and [14]. Contrary to the momentum and concentration, we cannot write conserved fluxes of and because of source terms in the governing equations (25) and (26). However, as for the other unknowns, we may seek for and in the following form:where and are two unknown dimensionless constants. Using the general relations (16) and (19), the substitution of these models into (25) and (26) gives(we dropped the explicit dependency of all functions on for the sake of clarity). The dependency on should cancel in both equations, which is possible if and only if and . Hence, the TKE in the jet is inversely proportional to the dimensionless distance from the jet virtual origin , in agreement with Gutmark and Wygnanski [10]. With the relation giving as a function of and , this gives usin agreement with (23) with . Note that the dependency of and on gives the size of the large turbulent eddies (or integral scale) as
Thus, the large turbulent eddy size on the jet axis is proportional to the distance to the origin of the jet, in agreement with observations [1]. We now setfor . Then, the momentum and concentration equations readfor , while the equations becomewhere
Equations (33) and (34) should be provided with boundary conditions identical to (21). Since the dimensionless constants and have not yet been specified, we may choose .
3. Solutions and Validation
3.1. Velocity and Scalar Profiles
The system given by (32), (33), and (34) is a nonlinear coupled system of four unknown functions (recall (32) contains two equations), and it looks hard to find analytical solutions. Similar systems have been proposed to study round jets and plumes [15] and then solved by numerical integration. In the present case, the theory of plane jet without any scalar assumes that turbulent quantities do not depend on the horizontal dimensionless coordinate [1]. This is obviously a very coarse hypothesis, since the turbulence is mainly confined within the jet. However, having no method to solve the complete abovementioned system, we will keep this assumption as a first approximation. In other words, we assume , in contradiction with Bradbury’s [2] experiments (a more advanced assumption will be presented in Section 3.2). This crude assumption is known to provide a decent velocity profile. It is easy to see that it is also in disagreement with (33) and (34). Thus, in this first approach the latter two equations are dropped and we only need to solve (32), which now reads
With the boundary conditions mentioned above, (36) has the well-known solution for the self-similar distribution of the jet axial velocity (see, e.g., [1]):with . Then, (37) can be readily solved with its boundary conditions and using the definition (31) to give
Finally, the distribution of axial velocity and concentration can be synthetized in the following formula:(recall and , by definition of the vector , with and ). In a more comprehensive way, the final formulae for axial velocity and concentration are
The constants can be deduced by integrating the fluxes over the cross-stream directions and using , which giveswhere(). The dimensionless fluxes of momentum and scalar can be calculated upstream of the orifice. We may assume the Reynolds-averaged velocity to have a log-profile and the scalar concentration to be uniformly distributed along the pipe section, giving . Then we compute the integrals :where denotes Euler’s function of first kind. We finally get
We can see that and appear everywhere through their product, which is a natural consequence of the arbitrary definition of the jet slope. Moreover, the profiles (41) and (42) depend on , as all subsequent functions will do. In the sequel, we will thus use the following notations:
We now determine the model constants. By definition of the jet half-width we have ; therefore, (41) gives us
Figure 2 shows the present model for the velocity and scalar concentration profiles (41) and (42) for , which is in agreement with experimental data (e.g., [1, 2, 9, 10]). With that, (49) yields . For the concentration profile, was chosen in order to fit the data, while other authors recommend ([16], experimental), 0.72 ([17], theoretical), 0.75 ([18], theoretical), or 0.74 (Violeau, 2009, theoretical) for simple shear flows. On the other hand, Law [12] suggests for the theory of the plane jet, but his concentration profile is in a modest agreement with experimental data.
With the present values, (46) and (47) yield and . One can see the model for the concentration distribution is in a very good agreement with Fischer et al.’s [13] data. It should be noted that our solution (39) matches exactly the theoretical profile proposed by Law [12], although the latter was presented in a much more complicated way involving a function of complex variables.
As for the velocity profile, another formula is used by several authors (e.g., [5–8]), which is purely empirical:
With the above Gaussian profile, the definition of the jet half-width givesin place of (51). With the abovementioned value of , we now have . Figure 2 shows that (52) fits the experimental velocity profile better than (41) away from the center part of the jet.
Using this new velocity profile to calculate and solving (37) for the concentration give another theoretical profile:
The integrals also have new values, and (43) gives
The integral (56) giving was obtained numerically using . The latter value was set in order to fit Fischer et al.’s [13] experimental results for the concentration (Figure 2). It can be seen that (39) and (54) are graphically very close to each other (for these values of , that are not the same in the two models).
3.2. Turbulent Quantities
We now want to provide analytical formulae for the turbulent quantities. Assuming a production-to-dissipation equilibrium in the TKE equation (25), that is, , the corresponding terms in (33) ( and last terms) give
With the momentum equation (22) (for ), relation (29), and definitions (31) and (35), we get(note that ). Only the product and can be predicted, since we arbitrarily set . With (29) again and (41), relation (46), and definition (50), we get
From (29) and (30) we now obtain
From model (17), we further get
In a similar way, we obtain the scalar flux as
Equations (59) also give the Taylor microscale [1, 2] as
Note that (64), as well as (61), diverges for . The present assumptions are consequently not valid in the center part of the jet. This will be confirmed later.
Finally, it is also interesting to look at the terms of the TKE budget (25). Equations (59) allow writing the convection and diffusion terms and in the form of the simplified equation (33) as
Recall that, with the present approximation, the (dominant) production term is assumed equal to the dissipation .
In order to make all dimensionless quantities independent of the axial dimensionless coordinate , it is relevant to use the centerline velocity and scalar distributions, obtained from (41) and (42):
Table 1 presents a summary of our analytical formulae where , and are used to make all quantities nondimensional. It should be noted that using in place of would avoid keeping and in the dimensionless turbulent quantities. We made the present choice in order to keep consistent with most publications on the plane turbulent jet. Note that, in order to use only functions of (defined by (50)) instead of , we used the following notation for an arbitrary function :
Table 1 presents the theoretical profiles obtained with(i)the most general assumptions in the present theory (i.e., similarity, column ),(ii)the same assumptions with production-to-dissipation equilibrium assumption (column ),(iii)the previous assumptions with the classical velocity profile (41) (i.e., using the crude assumption in (32), column , hereafter referred to as model variant ),(iv)the same theory but using the empirical velocity profile (52) (column , hereafter referred to as model variant ).
In the case of model variant , the present formulae for , , and are in agreement with Agrawal and Prasad [8].
Figure 3 shows the distributions of dimensionless , , and as a function of . We prefer using the latter on the horizontal axis since is defined from , which is not the same for the present two theoretical model variants. Moreover, this allows comparing the present models with Bradbury’s [2] experiments. We can see that, despite the crude assumptions made, both models give decent predictions except for , where the models only catch the order of magnitude. Moreover, in the jet center part the velocity gradients become very small, breaking the production-to-dissipation equilibrium assumption. The TKE is consequently not well predicted by the present analytical models in this area.
(a) TKE
(b) Transverse Reynolds stress
(c) Taylor microscale
The present method can also be used to give estimations of the root-mean-square of turbulent velocity fluctuations. However, the results (not given here) are not satisfactory, since our assumptions are not in agreement with a marked turbulence anisotropy.
In order to provide a validation of the proposed formulae for the terms in the TKE budget, that is, , , , and , Figure 4 shows their distributions versus for the two present model variants (recall that our model assumes ). It can be seen that they are in fair agreement with Bradbury’s [2] experimental results. However, as previously both models fail in predicting the budget contributions (except ) in the center part of the jet (i.e., ), where the data obviously show that the production-to-dissipation equilibrium is broken. As a consequence of the vanishing velocity gradient on the centerline (which yields ), the (nonzero) dissipation is balanced by diffusion and advection in this part of the jet. Nevertheless, outside of the jet center part the different terms of the budget reproduce the measured data fairly well, including the change of sign of and around .
Table 2 gives the model constants for the last two particular cases. They both provide similar values. The integrals of , , and along the transverse coordinate can be calculated analytically, and their values (displayed in the same table) are consistent with Bradbury’s [2] experiments. Note that the model variant , based on the empirical velocity profile (52), predicts an exactly vanishing integral of the energy diffusion.
Finally, one should mention that Bradbury’s [2] experiments yield almost constant turbulent turnover timescale; that is, , where and are the dimensionless versions of and given by the left column of Table 1. This provides a clue to further simplify (33). However, we could not get any more accurate analytical predictions with this approach.
4. Conclusions
We have built a consistent a complete theory of the turbulent plane jet under several assumptions: the boundary layer simplification and the self-similarity hypothesis for the velocity and concentration profiles and then the production-to-dissipation equilibrium for turbulent quantities. The last assumption can also be applied to a more accurate (but empirical) velocity profile, giving two model variants. All profiles are written in dimensionless form involving powers of the dimensionless (self-similar) transverse coordinate and are compared to experimental data; the agreement is generally good; it is excellent for the scalar concentration but less accurate for the turbulent quantities, especially in the TKE budget, where the production-to-dissipation equilibrium fails. Integrals of the budget terms are also computed. The present work provides a synthetic approach to estimate all quantities of interest in a turbulent plane jet and can easily be extended to the round jet case of other boundary layer flows like turbulent plumes.
Notation
| Scalar concentration scale (—) | |
| Reynolds-averaged scalar concentration (—) | |
| Turbulent fluctuations (—) | |
| Centerline concentration (—) | |
| Turbulent kinetic energy convection () | |
| 0.09, constant of the model (—) | |
| Turbulent kinetic energy diffusion () | |
| Transverse distribution functions (—) | |
| Molecular kinematic diffusion coefficient () | |
| Turbulent kinetic energy () | |
| Integral scale (m) | |
| Turbulent kinetic energy production () | |
| Reynolds number (—) | |
| Size of the jet orifice (m) | |
| Molecular Schmidt number (—) | |
| Jet slope (—) | |
| Velocity scale () | |
| Reynolds-averaged transverse velocity () | |
| Centerline axial velocity () | |
| Reynolds-averaged axial velocity () | |
| Spatial coordinates (m) | |
| Jet half-width (m) | |
| Normalising constants (—) | |
| Auxiliary model constants (—) | |
| Model constants (—) | |
| Turbulent kinetic energy dissipation rate () | |
| Taylor microscale (m) | |
| Molecular kinematic viscosity () | |
| Turbulent kinematic viscosity () | |
| Turbulent Schmidt number (—) | |
| 1, auxiliary notation (—) | |
| Dimensionless coordinates (—). |
Conflicts of Interest
The author declares that he has no conflicts of interest.