# A Comprehensive Presentation of the Turbulent Plane Jet Theory with Passive Scalar

Authors: D. Violeau
Journal: Mathematical Problems in Engineering
 (2017)

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## 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.


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## Body

## 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 coordinateζ and a function of the dimensionless transverse coordinate ξ 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 k-ϵ governing equations, giving their dependency to ζ. 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 widthR (see Figure 1). We work in the x,z plane, with z being the coordinate along the axis of the jet, ν the fluid molecular kinematic viscosity, and K the scalar molecular diffusion coefficient. The spatial distribution of (Reynolds-averaged) longitudinal (or axial) fluid velocity w¯ and scalar concentration c¯ will be searched as function of the fluid discharge velocity and concentration U and C.Figure 1
Notation. The curve shows a typical axial velocity profile.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 vectorA=w¯,c¯T of conserved quantities and, assuming Prandlt’s boundary layer assumptions to be valid, write the governing equations as(1)∂u¯∂x+∂w¯∂z=0,(2)∂w¯A∂z=-∂∂xu¯A+u′A′¯,where primes refer to turbulent fluctuations and overbars to Reynolds-averaging. The symmetry condition through the z-axis implies(3)u¯0,z=0∂w¯∂x0,z=0∂c¯∂x0,z=0.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:(4)limx→±∞w¯x,z=0limx→±∞u′w′¯x,z=0limx→±∞c¯x,z=0.limx→±∞u′c′¯x,z=0Experiments (e.g., Bradbury, 1965; see the review by Pope, 2000) show that the plane jet half-widthx∗z is a linear function of z. As in the classical theory of the turbulent plane jet without a scalar, we thus define(5)ξ=xx∗zζ=z-z0R,where z0 is a virtual origin. x/(z-z0) is thus a linear function of ξ and assumes autosimilarity of the jet. We denote by S≑dx∗/dz the slope of the half-width.Then, it is known that integrating the momentum and scalar governing equations along the cross-stream direction gives(6)w¯U=αwζ1/2fwξc¯C=αcζ1/2fcξ.For the two unknown functionsfi, we may impose fi0=1 for each i∈w,c, since the dimensionless constants αi are unknown. The above dependencies in ζ-1/2 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 fieldA in the following generic form (in most cases the power βA will be set to 1/2 later on, according to the previous subsection):(7)Ax,z=αAA0ζβAfAξ,where A0 is a dimensional constant (U or C 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(8)∂A∂x=αAA0SRζβA+1fA′ξ(9)∂A∂z=-αAA0RζβA+1βAfAξ+ξfA′ξ(here and in the sequel, the prime denotes differentiation with respect to ξ, except in the turbulent fluctuations u′, etc.). Using (1) and applying (9) to A=w¯ with βw=1/2 we get(10)∂u¯∂x=-∂w¯∂z(11)=αwURζ3/212fwξ+ξfw′ξ.We now define the antiderivative offw as(12)Fwξ=∫0ξfwξ′dξ′with Fw0=0. Thus, (11) gives the transverse dimensionless jet velocity as(13)u¯U=Sαwζ1/2ξfwξ-12Fwξ.From that, we can compute the advection of the arbitrary fieldA:(14)∂u¯A∂x+∂w¯A∂z=-αwαAUA0RζβA+3/2βAfwξfAξ+12FwξfA′ξ(15)=-αwαAUA0RζβA+3/2fA1/2ξFwβAfA1/2′ξFwβA-1ξ.For the two components of the vectorA defined above (axial velocity and concentration), since the powers are equal to 1/2 we find(16)∂u¯Ai∂x+∂w¯Ai∂z=-αwαiUAi02Rζ2Fwfi′ξwith Aw0=U and Ac0=C.We now address the diffusion ofA. We model the turbulent flux and stress with the eddy viscosity assumption:(17)u′w′¯=-νT∂w¯∂x(18)u′c′¯=-νTσc∂c¯∂x,where σc is the turbulent Schmidt number. We can put these two formulae together by defining σi,i∈w,c where σw=1. Now, from (6) the diffusion terms in (2) read(19)-∂u′Ai′¯∂x=∂∂xνTσi∂Ai∂x=αiS2σiAi0R21ζ5/2∂∂ξνTξ,ζfi′ξ.According to (3) and (4), the dimensionless unknown functions introduced so far are subject to the following boundary conditions:(20)limξ→±∞Fw′ξ=0limξ→±∞Fw′′ξ=0Fw0=0Fw′′0=0,(21)limξ→±∞fcξ=0limξ→±∞fc′ξ=0fc′0=0with the arbitrary additional conditions Fw′0=1 and fc0=1. Now, the momentum and concentration equations (2) can be rewritten using (16) and (19) and integrating once with respect to ξ:(22)Fwξfiξ=-2S2σiαwURνTξ,ζζ1/2fi′ξ,where i∈w,c.


### 2.3. Turbulent Closure

Equation (22) tells us that the eddy viscosity should depend on ζ as ζ1/2 in the turbulent plane jet, as already known [1]. This is in agreement with the generic form (7) with βνT=-1/2 and AνT=UR:(23)νTUR=ανTζ-1/2fνTξ.Thus, (22) gives(24)Fwξfiξ+2ανTσiαwfνTξfi′ξ=0(i∈w,c). We may first try to compute the eddy viscosity from the standard k-ϵ model; that is, νT=Cμ(k2/ϵ). Under the steady boundary layer assumptions, the TKE and dissipation rate governing equations read(25)∂u¯k∂x+∂w¯k∂z︸-Ck=Cμk2ϵ∂w¯∂x2︸Pk+Cμσk∂∂xk2ϵ∂k∂x︸Dk-ϵ(26)∂u¯ϵ∂x+∂w¯ϵ∂z=CμCϵ1k∂w¯∂x2+Cμσϵ∂∂xk2ϵ∂ϵ∂x-Cϵ2ϵ2kwith Cμ=0.09,σk=1.0,σϵ=1.3,Cϵ1=1.44, and Cϵ2=1.92 [14]. Contrary to the momentum and concentration, we cannot write conserved fluxes of k and ϵ because of source terms in the governing equations (25) and (26). However, as for the other unknowns, we may seek for k and ϵ in the following form:(27)kU2=αkζβkfkξϵU3/R=αϵζβϵfϵξ,where αk and αϵ are two unknown dimensionless constants. Using the general relations (16) and (19), the substitution of these models into (25) and (26) gives(28)-αwαkζβk+3/2fk1/2Fwβkfk1/2′Fwβk-1=Cμαw2αk2S2αϵζ2βk-βϵ+3fw′2fk2fϵ+Cμαk3S2σkαϵζ3βk-βϵ+2fk2fk′fϵ′-αϵζβϵfϵ-αwαϵζβϵ+3/2fϵ1/2Fwβϵfϵ1/2′Fwβϵ-1=CμCϵ1αw2αkS2ζβk+3fw′2fk+Cμαk2S2σϵζ2βk+2fk2fϵ′fϵ′-Cϵ2αϵ2αkζ2βϵ-βkfϵ2fk(we dropped the explicit dependency of all functions on ξ for the sake of clarity). The dependency on ζ should cancel in both equations, which is possible if and only if βk=1 and βϵ=5/2. Hence, the TKE in the jet is inversely proportional to the dimensionless distance from the jet virtual origin ζ, in agreement with Gutmark and Wygnanski [10]. With the relation giving νT as a function of k and ϵ, this gives us(29)νTUR=Cμαk2αϵζ1/2fk2ξfϵξin agreement with (23) with fνT=fk2/fϵ. Note that the dependency of k and ϵ on ξ gives the size of the large turbulent eddies (or integral scale) Lt=Cμ3/4k3/2/ϵ as(30)LtR=Cμ3/4αk3/2αϵζfk3/2ξfϵξ.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 set(31)γi≑4Cμαk2S2σiαwαϵfor i∈w,c,k,ϵ. Then, the momentum and concentration equations read(32)Fwfi+γi2fk2fi′fϵ=0for i∈w,c, while the k-ϵ equations become(33)fwfk+12Fwfk′+CμαS2fw′2fk2fϵ+γk4fk2fk′fϵ′-1αfϵ=0(34)52fwfϵ+12Fwfϵ′+CμCϵ1αS2fw′2fk+γϵ4fk2fϵ′fϵ′-Cϵ2αfϵ2fk=0,where(35)α≑αwαkαϵ.Equations (33) and (34) should be provided with boundary conditions identical to (21). Since the dimensionless constants αk and αϵ have not yet been specified, we may choose fk0=fϵ0=1.


## 2.1. State of the Art and Notation

We consider a steady incompressible plane turbulent jet determined by an infinite orifice (or slot) of widthR (see Figure 1). We work in the x,z plane, with z being the coordinate along the axis of the jet, ν the fluid molecular kinematic viscosity, and K the scalar molecular diffusion coefficient. The spatial distribution of (Reynolds-averaged) longitudinal (or axial) fluid velocity w¯ and scalar concentration c¯ will be searched as function of the fluid discharge velocity and concentration U and C.Figure 1
Notation. The curve shows a typical axial velocity profile.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 vectorA=w¯,c¯T of conserved quantities and, assuming Prandlt’s boundary layer assumptions to be valid, write the governing equations as(1)∂u¯∂x+∂w¯∂z=0,(2)∂w¯A∂z=-∂∂xu¯A+u′A′¯,where primes refer to turbulent fluctuations and overbars to Reynolds-averaging. The symmetry condition through the z-axis implies(3)u¯0,z=0∂w¯∂x0,z=0∂c¯∂x0,z=0.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:(4)limx→±∞w¯x,z=0limx→±∞u′w′¯x,z=0limx→±∞c¯x,z=0.limx→±∞u′c′¯x,z=0Experiments (e.g., Bradbury, 1965; see the review by Pope, 2000) show that the plane jet half-widthx∗z is a linear function of z. As in the classical theory of the turbulent plane jet without a scalar, we thus define(5)ξ=xx∗zζ=z-z0R,where z0 is a virtual origin. x/(z-z0) is thus a linear function of ξ and assumes autosimilarity of the jet. We denote by S≑dx∗/dz the slope of the half-width.Then, it is known that integrating the momentum and scalar governing equations along the cross-stream direction gives(6)w¯U=αwζ1/2fwξc¯C=αcζ1/2fcξ.For the two unknown functionsfi, we may impose fi0=1 for each i∈w,c, since the dimensionless constants αi are unknown. The above dependencies in ζ-1/2 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 fieldA in the following generic form (in most cases the power βA will be set to 1/2 later on, according to the previous subsection):(7)Ax,z=αAA0ζβAfAξ,where A0 is a dimensional constant (U or C 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(8)∂A∂x=αAA0SRζβA+1fA′ξ(9)∂A∂z=-αAA0RζβA+1βAfAξ+ξfA′ξ(here and in the sequel, the prime denotes differentiation with respect to ξ, except in the turbulent fluctuations u′, etc.). Using (1) and applying (9) to A=w¯ with βw=1/2 we get(10)∂u¯∂x=-∂w¯∂z(11)=αwURζ3/212fwξ+ξfw′ξ.We now define the antiderivative offw as(12)Fwξ=∫0ξfwξ′dξ′with Fw0=0. Thus, (11) gives the transverse dimensionless jet velocity as(13)u¯U=Sαwζ1/2ξfwξ-12Fwξ.From that, we can compute the advection of the arbitrary fieldA:(14)∂u¯A∂x+∂w¯A∂z=-αwαAUA0RζβA+3/2βAfwξfAξ+12FwξfA′ξ(15)=-αwαAUA0RζβA+3/2fA1/2ξFwβAfA1/2′ξFwβA-1ξ.For the two components of the vectorA defined above (axial velocity and concentration), since the powers are equal to 1/2 we find(16)∂u¯Ai∂x+∂w¯Ai∂z=-αwαiUAi02Rζ2Fwfi′ξwith Aw0=U and Ac0=C.We now address the diffusion ofA. We model the turbulent flux and stress with the eddy viscosity assumption:(17)u′w′¯=-νT∂w¯∂x(18)u′c′¯=-νTσc∂c¯∂x,where σc is the turbulent Schmidt number. We can put these two formulae together by defining σi,i∈w,c where σw=1. Now, from (6) the diffusion terms in (2) read(19)-∂u′Ai′¯∂x=∂∂xνTσi∂Ai∂x=αiS2σiAi0R21ζ5/2∂∂ξνTξ,ζfi′ξ.According to (3) and (4), the dimensionless unknown functions introduced so far are subject to the following boundary conditions:(20)limξ→±∞Fw′ξ=0limξ→±∞Fw′′ξ=0Fw0=0Fw′′0=0,(21)limξ→±∞fcξ=0limξ→±∞fc′ξ=0fc′0=0with the arbitrary additional conditions Fw′0=1 and fc0=1. Now, the momentum and concentration equations (2) can be rewritten using (16) and (19) and integrating once with respect to ξ:(22)Fwξfiξ=-2S2σiαwURνTξ,ζζ1/2fi′ξ,where i∈w,c.


## 2.3. Turbulent Closure

Equation (22) tells us that the eddy viscosity should depend on ζ as ζ1/2 in the turbulent plane jet, as already known [1]. This is in agreement with the generic form (7) with βνT=-1/2 and AνT=UR:(23)νTUR=ανTζ-1/2fνTξ.Thus, (22) gives(24)Fwξfiξ+2ανTσiαwfνTξfi′ξ=0(i∈w,c). We may first try to compute the eddy viscosity from the standard k-ϵ model; that is, νT=Cμ(k2/ϵ). Under the steady boundary layer assumptions, the TKE and dissipation rate governing equations read(25)∂u¯k∂x+∂w¯k∂z︸-Ck=Cμk2ϵ∂w¯∂x2︸Pk+Cμσk∂∂xk2ϵ∂k∂x︸Dk-ϵ(26)∂u¯ϵ∂x+∂w¯ϵ∂z=CμCϵ1k∂w¯∂x2+Cμσϵ∂∂xk2ϵ∂ϵ∂x-Cϵ2ϵ2kwith Cμ=0.09,σk=1.0,σϵ=1.3,Cϵ1=1.44, and Cϵ2=1.92 [14]. Contrary to the momentum and concentration, we cannot write conserved fluxes of k and ϵ because of source terms in the governing equations (25) and (26). However, as for the other unknowns, we may seek for k and ϵ in the following form:(27)kU2=αkζβkfkξϵU3/R=αϵζβϵfϵξ,where αk and αϵ are two unknown dimensionless constants. Using the general relations (16) and (19), the substitution of these models into (25) and (26) gives(28)-αwαkζβk+3/2fk1/2Fwβkfk1/2′Fwβk-1=Cμαw2αk2S2αϵζ2βk-βϵ+3fw′2fk2fϵ+Cμαk3S2σkαϵζ3βk-βϵ+2fk2fk′fϵ′-αϵζβϵfϵ-αwαϵζβϵ+3/2fϵ1/2Fwβϵfϵ1/2′Fwβϵ-1=CμCϵ1αw2αkS2ζβk+3fw′2fk+Cμαk2S2σϵζ2βk+2fk2fϵ′fϵ′-Cϵ2αϵ2αkζ2βϵ-βkfϵ2fk(we dropped the explicit dependency of all functions on ξ for the sake of clarity). The dependency on ζ should cancel in both equations, which is possible if and only if βk=1 and βϵ=5/2. Hence, the TKE in the jet is inversely proportional to the dimensionless distance from the jet virtual origin ζ, in agreement with Gutmark and Wygnanski [10]. With the relation giving νT as a function of k and ϵ, this gives us(29)νTUR=Cμαk2αϵζ1/2fk2ξfϵξin agreement with (23) with fνT=fk2/fϵ. Note that the dependency of k and ϵ on ξ gives the size of the large turbulent eddies (or integral scale) Lt=Cμ3/4k3/2/ϵ as(30)LtR=Cμ3/4αk3/2αϵζfk3/2ξfϵξ.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 set(31)γi≑4Cμαk2S2σiαwαϵfor i∈w,c,k,ϵ. Then, the momentum and concentration equations read(32)Fwfi+γi2fk2fi′fϵ=0for i∈w,c, while the k-ϵ equations become(33)fwfk+12Fwfk′+CμαS2fw′2fk2fϵ+γk4fk2fk′fϵ′-1αfϵ=0(34)52fwfϵ+12Fwfϵ′+CμCϵ1αS2fw′2fk+γϵ4fk2fϵ′fϵ′-Cϵ2αfϵ2fk=0,where(35)α≑αwαkαϵ.Equations (33) and (34) should be provided with boundary conditions identical to (21). Since the dimensionless constants αk and αϵ have not yet been specified, we may choose fk0=fϵ0=1.


## 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 ξ [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 fk2ξ=fϵξ, 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(36)FwFw′+γw2Fw′′=0(37)Fwfc+γc2fc′=0.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]):(38)Fwξ=δtanh⁡ξδfwξ=Fw′ξ=sech2ξδwith δ≑γw. Then, (37) can be readily solved with its boundary conditions and using the definition (31) to give(39)fcξ=sech2σcξδ.Finally, the distribution of axial velocity and concentration can be synthetized in the following formula:(40)Aiξ,ζAi0=αiζ1/2sech2σiξδ(recall Aw=u¯ and Ac=c¯, by definition of the vector A, with Aw0=U and Ac0=C). In a more comprehensive way, the final formulae for axial velocity and concentration are(41)w¯ξ,ζU=αwζsech2ξδ(42)c¯ξ,ζC=αcζsech2σcξδ.The constants can be deduced by integrating the fluxesw¯A over the cross-stream directions and using dx=x∗zdξ=SRζdξ, which gives(43)αwαiJi=Qi∗S,where(44)Ji≑∫-∞+∞fwξfiξdξ(i=w,c). The dimensionless fluxes of momentum and scalar Qi∗ 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 Qw∗≈Qc∗=1. Then we compute the integrals Ji:(45)Ji=∫-∞+∞sech21+σiξδdξ=2πδ1+2σiΓ1+σiΓ1/2+σi,where Γ denotes Euler’s function of first kind. We finally get(46)αw=123Sδ(47)αc=1+2σc3πSδΓ1/2+σcΓ1+σc.Note that (31) and (46) give(48)αk2αϵ=964Cμαw3=3Sδ3/28Cμ.We can see thatS and δ 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 ξ/δ, as all subsequent functions will do. In the sequel, we will thus use the following notations:(49)ϕ≑Sδ(50)η≑ξδ.We now determine the model constants. By definition of the jet half-width we havew¯1,ζ=(1/2)w¯0,ζ; therefore, (41) gives us(51)δ=1ln1+2≈1.135.Figure2 shows the present model for the velocity and scalar concentration profiles (41) and (42) for S=0.0970, which is in agreement with experimental data (e.g., [1, 2, 9, 10]). With that, (49) yields ϕ≈0.1101. For the concentration profile, σc=0.63 was chosen in order to fit the data, while other authors recommend σc=0.71 ([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 σc=0.59 for the theory of the plane jet, but his concentration profile is in a modest agreement with experimental data.Figure 2
Profiles of axial velocity (blue) and concentration (green). Present models#1 (see (41) and (42), solid lines) and #2 (see (52) and (54), dashed lines) versus measurements (symbols) from Fischer et al. [13].With the present values, (46) and (47) yield αw≈2.610 and αc≈2.325. 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:(52)fwη=e-η2.With the above Gaussian profile, the definition of the jet half-width gives(53)δ=1ln⁡2≈1.201in place of (51). With the abovementioned value of S, we now have ϕ≈0.1165. 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 calculateFw and solving (37) for the concentration give another theoretical profile:(54)fcη=exp⁡-σce-η2+πηerf⁡η-1.The integralsJi also have new values, and (43) gives(55)αw=2πϕ21/4≈2.617(56)αc=π2ϕ21/4∫-∞+∞e-η2exp⁡-σce-η2+πηerf⁡η1dη-1≈2.350.The integral (56) giving αc was obtained numerically using σc=0.70. 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 σc, 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, Pk=ϵ, the corresponding terms in (33) (3rd and last terms) give(57)fϵ=-CμαSfw′fk.With the momentum equation (22) (for i=w), relation (29), and definitions (31) and (35), we get(58)αkfk=Sαw22CμFwfwαϵfϵ=-αw32Fwfwfw′(note that fw′≤0). Only the product αkfk and αϵfϵ can be predicted, since we arbitrarily set fk0=fϵ0=1. With (29) again and (41), relation (46), and definition (50), we get(59)αkfkη=38Cμsinh⁡ηcosh3⁡ηαϵfϵη=183ϕ3/2sinh2⁡ηcosh6⁡η.From (29) and (30) we now obtain(60)νTζUR=98ϕ33/2ζ(61)Ltη,ζR=8ϕ83/2ζcosh3⁡ηsinh⁡η.From model (17), we further get(62)u′w′¯η,ζU2=-98ϕ33/2αwS1ζfw′=381ζsinh⁡ηcosh3⁡η.In a similar way, we obtain the scalar flux as(63)u′c′¯η,ζUC=-98ϕ33/2αcSσc1ζfw′=1+2σc4πΓ1/2+σcΓ1+σc1ζsinh⁡ηcosh1+2σc⁡η.Equations (59) also give the Taylor microscale λ≑10νk/ϵ1/2 [1, 2] as(64)λη,ζνz-z0/U=10ϕ3ζ3Cμ1/2cosh3⁡ηsinh⁡η.Note that (64), as well as (61), diverges for η=0. 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 Ck and Dk in the form of the simplified equation (33) as(65)Ckη,ζU3/R=3323Cμϕ1ζ5/23-2sinh2⁡ηsinh⁡ηcosh5ηDkη,ζU3/R=916σk3Cμϕ1ζ5/2sinh2⁡η-2sinh⁡ηcosh5⁡η.Recall that, with the present approximation, the (dominant) production termPk is assumed equal to the dissipation ϵ.In order to make all dimensionless quantities independent of the axial dimensionless coordinateζ, it is relevant to use the centerline velocity and scalar distributions, obtained from (41) and (42):(66)W0ζ≑w¯0,ζ=αwUζC0ζ≑c¯0,ζ=αcCζ.Table1 presents a summary of our analytical formulae where W0ζ,C0ζ, and x∗z are used to make all quantities nondimensional. It should be noted that using z-z0 in place of x∗z would avoid keeping S and δ 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 η (defined by (50)) instead of ξ, we used the following notation for an arbitrary function fη:(67)f˙η≑dfdη=δf′ξF~η≑∫0ηfη′dη′=1δFξ.Table 1
It presents theoretical profiles of the quantities of interest in a self-similar turbulent jet. The functionsfw, fc, and so on depend on η, although not explicitly specified everywhere. Three auxiliary functions are used for clarity (defined below the table).
Terms
General
WithP=ϵ
Withfk2=fϵ in (32)
Withfwη=e-η2
w
¯
η
,
ζ
W
0
ζ
f
w
f
w
sech
2
⁡
η
e
-
η
2
c¯η,ζC0ζ
f
c
f
c
sech
2
σ
c
⁡
η
Ψ
1
η
a
F~wη
—
—
tanh
⁡
η
π
2
erf
⁡
η
-f˙wη
—
—
2
sinh
⁡
η
cosh
3
⁡
η
2
η
e
-
η
2
-f˙cη
—
—
2
σ
c
sinh
⁡
η
cosh
1
+
2
σ
c
⁡
η
π
σ
c
erf
⁡
η
Ψ
1
η
a
kη,ζW02ζ
α
k
α
w
2
f
k
ϕ
2
C
μ
F
~
w
f
w
ϕ
2
C
μ
sinh
⁡
η
cosh
3
⁡
η
ϕ
4
π
C
μ
e
-
η
2
erf
⁡
η
ϵ=Pkη,ζW03ζ/x∗z
S
α
ϵ
α
w
3
f
ϵ
S
1
2
F
~
w
f
w
-
f
˙
w
S
sinh
2
⁡
η
cosh
6
⁡
η
S
π
2
η
e
-
2
η
2
erf
⁡
η
νTζW0ζx∗z
1
S
C
μ
α
k
2
α
w
α
ϵ
f
k
2
f
ϵ
1
S
ϕ
2
2
F
~
w
f
w
-
f
˙
w
1
S
ϕ
2
4
≈
0.0312
1
S
π
ϕ
2
8
erf
⁡
η
η
Ltη,ζx∗z
1
S
C
μ
3
/
4
α
k
3
/
2
α
ϵ
f
k
3
/
2
f
ϵ
1
S
ϕ
3
/
2
2
F
~
w
f
w
-
f
˙
w
1
S
8
ϕ
8
3
/
2
cosh
3
⁡
η
sinh
⁡
η
1
S
π
1
/
4
ϕ
3
/
2
4
1
η
erf
⁡
η
e
-
η
2
u′w′¯η,ζW02ζ
-
1
ϕ
C
μ
α
k
2
α
w
α
ϵ
f
k
2
f
˙
w
f
ϵ
ϕ
2
F
~
w
f
w
ϕ
2
sinh
⁡
η
cosh
3
⁡
η
π
ϕ
4
e
-
η
2
erf
⁡
η
u′c′¯η,ζW0ζC0ζ
-
1
ϕ
C
μ
σ
c
α
k
2
α
w
α
ϵ
f
k
2
f
˙
c
f
ϵ
ϕ
2
σ
c
F
~
w
f
w
f
˙
c
f
˙
w
ϕ
2
sinh
⁡
η
c
o
s
h
1
+
2
σ
c
η
π
ϕ
8
erf
2
⁡
η
Ψ
1
η
a
η
λη,ζνx∗z/W0ζ
10
S
α
w
α
k
α
ϵ
f
k
f
ϵ
δ
10
C
μ
1
/
2
1
-
f
˙
w
δ
5
C
μ
1
/
2
cosh
3
⁡
η
sinh
⁡
η
δ
5
C
μ
1
/
2
e
η
2
η
Ckη,ζW03ζ/x∗z
S
α
k
α
w
2
f
w
f
k
+
1
2
F
~
w
f
˙
k
S
ϕ
4
C
μ
F
~
w
3
f
w
2
+
F
~
w
f
˙
w
S
ϕ
4
C
μ
3
-
2
sinh
2
⁡
η
sinh
⁡
η
cosh
5
⁡
η
S
π
C
μ
ϕ
8
Ψ
2
η
b
Dkη,ζW03ζ/x∗z
S
1
ϕ
2
C
μ
α
k
3
σ
k
α
w
3
α
ϵ
d
d
η
f
k
2
f
˙
k
f
ϵ
S
ϕ
4
C
μ
σ
k
d
d
η
F
~
w
f
w
f
w
2
+
F
~
w
f
˙
w
-
f
˙
w
S
ϕ
2
C
μ
σ
k
s
i
n
h
2
η
-
2
cosh
5
⁡
η
sinh
⁡
η
S
ϕ
8
C
μ
σ
k
Ψ
3
η
c
-
Ψ
2
η
b
a
Ψ
1
η
≑
exp
⁡
-
σ
c
e
-
η
2
+
π
η
erf
⁡
η
-
1
.
b
Ψ
2
η
≑
e
-
η
2
erf
⁡
η
3
e
-
η
2
-
π
η
erf
⁡
η
.
c
Ψ
3
η
≑
e
-
2
η
2
e
-
η
2
/
η
-
π
1
+
1
/
2
η
2
erf
⁡
η
.Table1 presents the theoretical profiles obtained with(i)
the most general assumptions in the present theory (i.e., similarity, column#2),(ii)
the same assumptions with production-to-dissipation equilibrium assumption (column#3),(iii)
the previous assumptions with the classical velocity profile (41) (i.e., using the crude assumption fk2=fϵ in (32), column #4, hereafter referred to as model variant #1),(iv)
the same theory but using the empirical velocity profile (52) (column #5, hereafter referred to as model variant #2).In the case of model variant#2, the present formulae for Pk, νT, and u′w′¯ are in agreement with Agrawal and Prasad [8].Figure3 shows the distributions of dimensionless k, u′w′¯, and λ as a function of ξ. We prefer using the latter on the horizontal axis since η is defined from δ, 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 λ, 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.Figure 3
Profiles ofk, u′w′¯, and λ. Present models #1 (from (41), solid lines) and #2 (from (52), dashed lines) versus measurements (symbols) from Bradbury [2]. The analytical formulae of these three quantities are given in Table 1.
(a)
TKEk
(b)
Transverse Reynolds stressu′w′¯
(c)
Taylor microscaleλThe 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,Pk, Ck, Dk, and ϵ, Figure 4 shows their distributions versus ξ for the two present model variants (recall that our model assumes Pk=ϵ). 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 Pk) in the center part of the jet (i.e., ξ>0.5), 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 Pk=0), 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 Ck and Dk around ξ=1.2.Figure 4
Profiles of TKE budget: productionPk (blue), dissipation -ϵ (green), convection Ck and diffusion Dk. Present models #1 (from (41), solid lines) and #2 (from (52), dashed lines) versus measurements (symbols) from Bradbury [2]. The analytical formulae of these three quantities are given in Table 1.Table2 gives the model constants for the last two particular cases. They both provide similar values. The integrals of Pk, Ck, and Dk 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 #2, based on the empirical velocity profile (52), predicts an exactly vanishing integral of the energy diffusion.Table 2
Model constants and useful figures of the present models. The experimental values are from the references below the table.
Constants
Withfk2=fϵ in (32)
Withfwη=e-η2
Experimental
δ
1
ln
⁡
1
+
2
≈
1.135
1
ln
⁡
2
≈
1.201
—
ϕ
≈
0.1101
≈
0.1165
—
σ
c
≈
0.63
≈
0.70
0.59–0.75b
α
w
1
2
3
ϕ
≈
2.610
2
π
ϕ
2
1
/
4
≈
2.617
—
α
c
1
+
2
σ
c
3
π
ϕ
Γ
1
/
2
+
σ
c
Γ
1
+
σ
c
≈
2.325
π
2
ϕ
2
1
/
4
∫
-
∞
+
∞
e
-
η
2
Ψ
1
η
a
d
η
-
1
≈
2.350
—
α
k
2
ϵ
3
ϕ
3
/
2
8
C
μ
≈
0.0879
2
π
1
/
4
ϕ
3
/
2
4
C
μ
≈
0.0987
—
∫
0
+
∞
P
k
ξ
,
ζ
d
ξ
W
0
3
ζ
/
x
∗
z
2
15
ϕ
≈
0.0147
1
8
π
3
ϕ
≈
0.0149
0.016
9
c
∫
0
+
∞
C
k
ξ
,
ζ
d
ξ
W
0
3
ζ
/
x
∗
z
1
sqrt
⁡
C
μ
ϕ
2
16
≈
0.00253
2
C
μ
t
a
n
-
1
2
2
ϕ
2
16
≈
0.00246
0.003
8
c
∫
0
+
∞
D
k
ξ
,
ζ
d
ξ
W
0
3
ζ
/
x
∗
z
-
ϕ
2
8
σ
k
C
μ
≈
-
0.00505
0
0
c
aSee note (a) of Table 1.
bTurner [16], Mellor and Yamada [18], Sukoriansky et al. [17], Law [12], and Violeau (2009).
cBradbury [2].Finally, one should mention that Bradbury’s [2] experiments yield almost constant turbulent turnover timescale; that is, k∗/ϵ∗≈5.0, where k∗ and ϵ∗ are the dimensionless versions of k and ϵ 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.


## 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 ξ [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 fk2ξ=fϵξ, 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(36)FwFw′+γw2Fw′′=0(37)Fwfc+γc2fc′=0.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]):(38)Fwξ=δtanh⁡ξδfwξ=Fw′ξ=sech2ξδwith δ≑γw. Then, (37) can be readily solved with its boundary conditions and using the definition (31) to give(39)fcξ=sech2σcξδ.Finally, the distribution of axial velocity and concentration can be synthetized in the following formula:(40)Aiξ,ζAi0=αiζ1/2sech2σiξδ(recall Aw=u¯ and Ac=c¯, by definition of the vector A, with Aw0=U and Ac0=C). In a more comprehensive way, the final formulae for axial velocity and concentration are(41)w¯ξ,ζU=αwζsech2ξδ(42)c¯ξ,ζC=αcζsech2σcξδ.The constants can be deduced by integrating the fluxesw¯A over the cross-stream directions and using dx=x∗zdξ=SRζdξ, which gives(43)αwαiJi=Qi∗S,where(44)Ji≑∫-∞+∞fwξfiξdξ(i=w,c). The dimensionless fluxes of momentum and scalar Qi∗ 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 Qw∗≈Qc∗=1. Then we compute the integrals Ji:(45)Ji=∫-∞+∞sech21+σiξδdξ=2πδ1+2σiΓ1+σiΓ1/2+σi,where Γ denotes Euler’s function of first kind. We finally get(46)αw=123Sδ(47)αc=1+2σc3πSδΓ1/2+σcΓ1+σc.Note that (31) and (46) give(48)αk2αϵ=964Cμαw3=3Sδ3/28Cμ.We can see thatS and δ 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 ξ/δ, as all subsequent functions will do. In the sequel, we will thus use the following notations:(49)ϕ≑Sδ(50)η≑ξδ.We now determine the model constants. By definition of the jet half-width we havew¯1,ζ=(1/2)w¯0,ζ; therefore, (41) gives us(51)δ=1ln1+2≈1.135.Figure2 shows the present model for the velocity and scalar concentration profiles (41) and (42) for S=0.0970, which is in agreement with experimental data (e.g., [1, 2, 9, 10]). With that, (49) yields ϕ≈0.1101. For the concentration profile, σc=0.63 was chosen in order to fit the data, while other authors recommend σc=0.71 ([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 σc=0.59 for the theory of the plane jet, but his concentration profile is in a modest agreement with experimental data.Figure 2
Profiles of axial velocity (blue) and concentration (green). Present models#1 (see (41) and (42), solid lines) and #2 (see (52) and (54), dashed lines) versus measurements (symbols) from Fischer et al. [13].With the present values, (46) and (47) yield αw≈2.610 and αc≈2.325. 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:(52)fwη=e-η2.With the above Gaussian profile, the definition of the jet half-width gives(53)δ=1ln⁡2≈1.201in place of (51). With the abovementioned value of S, we now have ϕ≈0.1165. 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 calculateFw and solving (37) for the concentration give another theoretical profile:(54)fcη=exp⁡-σce-η2+πηerf⁡η-1.The integralsJi also have new values, and (43) gives(55)αw=2πϕ21/4≈2.617(56)αc=π2ϕ21/4∫-∞+∞e-η2exp⁡-σce-η2+πηerf⁡η1dη-1≈2.350.The integral (56) giving αc was obtained numerically using σc=0.70. 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 σc, 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, Pk=ϵ, the corresponding terms in (33) (3rd and last terms) give(57)fϵ=-CμαSfw′fk.With the momentum equation (22) (for i=w), relation (29), and definitions (31) and (35), we get(58)αkfk=Sαw22CμFwfwαϵfϵ=-αw32Fwfwfw′(note that fw′≤0). Only the product αkfk and αϵfϵ can be predicted, since we arbitrarily set fk0=fϵ0=1. With (29) again and (41), relation (46), and definition (50), we get(59)αkfkη=38Cμsinh⁡ηcosh3⁡ηαϵfϵη=183ϕ3/2sinh2⁡ηcosh6⁡η.From (29) and (30) we now obtain(60)νTζUR=98ϕ33/2ζ(61)Ltη,ζR=8ϕ83/2ζcosh3⁡ηsinh⁡η.From model (17), we further get(62)u′w′¯η,ζU2=-98ϕ33/2αwS1ζfw′=381ζsinh⁡ηcosh3⁡η.In a similar way, we obtain the scalar flux as(63)u′c′¯η,ζUC=-98ϕ33/2αcSσc1ζfw′=1+2σc4πΓ1/2+σcΓ1+σc1ζsinh⁡ηcosh1+2σc⁡η.Equations (59) also give the Taylor microscale λ≑10νk/ϵ1/2 [1, 2] as(64)λη,ζνz-z0/U=10ϕ3ζ3Cμ1/2cosh3⁡ηsinh⁡η.Note that (64), as well as (61), diverges for η=0. 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 Ck and Dk in the form of the simplified equation (33) as(65)Ckη,ζU3/R=3323Cμϕ1ζ5/23-2sinh2⁡ηsinh⁡ηcosh5ηDkη,ζU3/R=916σk3Cμϕ1ζ5/2sinh2⁡η-2sinh⁡ηcosh5⁡η.Recall that, with the present approximation, the (dominant) production termPk is assumed equal to the dissipation ϵ.In order to make all dimensionless quantities independent of the axial dimensionless coordinateζ, it is relevant to use the centerline velocity and scalar distributions, obtained from (41) and (42):(66)W0ζ≑w¯0,ζ=αwUζC0ζ≑c¯0,ζ=αcCζ.Table1 presents a summary of our analytical formulae where W0ζ,C0ζ, and x∗z are used to make all quantities nondimensional. It should be noted that using z-z0 in place of x∗z would avoid keeping S and δ 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 η (defined by (50)) instead of ξ, we used the following notation for an arbitrary function fη:(67)f˙η≑dfdη=δf′ξF~η≑∫0ηfη′dη′=1δFξ.Table 1
It presents theoretical profiles of the quantities of interest in a self-similar turbulent jet. The functionsfw, fc, and so on depend on η, although not explicitly specified everywhere. Three auxiliary functions are used for clarity (defined below the table).
Terms
General
WithP=ϵ
Withfk2=fϵ in (32)
Withfwη=e-η2
w
¯
η
,
ζ
W
0
ζ
f
w
f
w
sech
2
⁡
η
e
-
η
2
c¯η,ζC0ζ
f
c
f
c
sech
2
σ
c
⁡
η
Ψ
1
η
a
F~wη
—
—
tanh
⁡
η
π
2
erf
⁡
η
-f˙wη
—
—
2
sinh
⁡
η
cosh
3
⁡
η
2
η
e
-
η
2
-f˙cη
—
—
2
σ
c
sinh
⁡
η
cosh
1
+
2
σ
c
⁡
η
π
σ
c
erf
⁡
η
Ψ
1
η
a
kη,ζW02ζ
α
k
α
w
2
f
k
ϕ
2
C
μ
F
~
w
f
w
ϕ
2
C
μ
sinh
⁡
η
cosh
3
⁡
η
ϕ
4
π
C
μ
e
-
η
2
erf
⁡
η
ϵ=Pkη,ζW03ζ/x∗z
S
α
ϵ
α
w
3
f
ϵ
S
1
2
F
~
w
f
w
-
f
˙
w
S
sinh
2
⁡
η
cosh
6
⁡
η
S
π
2
η
e
-
2
η
2
erf
⁡
η
νTζW0ζx∗z
1
S
C
μ
α
k
2
α
w
α
ϵ
f
k
2
f
ϵ
1
S
ϕ
2
2
F
~
w
f
w
-
f
˙
w
1
S
ϕ
2
4
≈
0.0312
1
S
π
ϕ
2
8
erf
⁡
η
η
Ltη,ζx∗z
1
S
C
μ
3
/
4
α
k
3
/
2
α
ϵ
f
k
3
/
2
f
ϵ
1
S
ϕ
3
/
2
2
F
~
w
f
w
-
f
˙
w
1
S
8
ϕ
8
3
/
2
cosh
3
⁡
η
sinh
⁡
η
1
S
π
1
/
4
ϕ
3
/
2
4
1
η
erf
⁡
η
e
-
η
2
u′w′¯η,ζW02ζ
-
1
ϕ
C
μ
α
k
2
α
w
α
ϵ
f
k
2
f
˙
w
f
ϵ
ϕ
2
F
~
w
f
w
ϕ
2
sinh
⁡
η
cosh
3
⁡
η
π
ϕ
4
e
-
η
2
erf
⁡
η
u′c′¯η,ζW0ζC0ζ
-
1
ϕ
C
μ
σ
c
α
k
2
α
w
α
ϵ
f
k
2
f
˙
c
f
ϵ
ϕ
2
σ
c
F
~
w
f
w
f
˙
c
f
˙
w
ϕ
2
sinh
⁡
η
c
o
s
h
1
+
2
σ
c
η
π
ϕ
8
erf
2
⁡
η
Ψ
1
η
a
η
λη,ζνx∗z/W0ζ
10
S
α
w
α
k
α
ϵ
f
k
f
ϵ
δ
10
C
μ
1
/
2
1
-
f
˙
w
δ
5
C
μ
1
/
2
cosh
3
⁡
η
sinh
⁡
η
δ
5
C
μ
1
/
2
e
η
2
η
Ckη,ζW03ζ/x∗z
S
α
k
α
w
2
f
w
f
k
+
1
2
F
~
w
f
˙
k
S
ϕ
4
C
μ
F
~
w
3
f
w
2
+
F
~
w
f
˙
w
S
ϕ
4
C
μ
3
-
2
sinh
2
⁡
η
sinh
⁡
η
cosh
5
⁡
η
S
π
C
μ
ϕ
8
Ψ
2
η
b
Dkη,ζW03ζ/x∗z
S
1
ϕ
2
C
μ
α
k
3
σ
k
α
w
3
α
ϵ
d
d
η
f
k
2
f
˙
k
f
ϵ
S
ϕ
4
C
μ
σ
k
d
d
η
F
~
w
f
w
f
w
2
+
F
~
w
f
˙
w
-
f
˙
w
S
ϕ
2
C
μ
σ
k
s
i
n
h
2
η
-
2
cosh
5
⁡
η
sinh
⁡
η
S
ϕ
8
C
μ
σ
k
Ψ
3
η
c
-
Ψ
2
η
b
a
Ψ
1
η
≑
exp
⁡
-
σ
c
e
-
η
2
+
π
η
erf
⁡
η
-
1
.
b
Ψ
2
η
≑
e
-
η
2
erf
⁡
η
3
e
-
η
2
-
π
η
erf
⁡
η
.
c
Ψ
3
η
≑
e
-
2
η
2
e
-
η
2
/
η
-
π
1
+
1
/
2
η
2
erf
⁡
η
.Table1 presents the theoretical profiles obtained with(i)
the most general assumptions in the present theory (i.e., similarity, column#2),(ii)
the same assumptions with production-to-dissipation equilibrium assumption (column#3),(iii)
the previous assumptions with the classical velocity profile (41) (i.e., using the crude assumption fk2=fϵ in (32), column #4, hereafter referred to as model variant #1),(iv)
the same theory but using the empirical velocity profile (52) (column #5, hereafter referred to as model variant #2).In the case of model variant#2, the present formulae for Pk, νT, and u′w′¯ are in agreement with Agrawal and Prasad [8].Figure3 shows the distributions of dimensionless k, u′w′¯, and λ as a function of ξ. We prefer using the latter on the horizontal axis since η is defined from δ, 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 λ, 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.Figure 3
Profiles ofk, u′w′¯, and λ. Present models #1 (from (41), solid lines) and #2 (from (52), dashed lines) versus measurements (symbols) from Bradbury [2]. The analytical formulae of these three quantities are given in Table 1.
(a)
TKEk
(b)
Transverse Reynolds stressu′w′¯
(c)
Taylor microscaleλThe 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,Pk, Ck, Dk, and ϵ, Figure 4 shows their distributions versus ξ for the two present model variants (recall that our model assumes Pk=ϵ). 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 Pk) in the center part of the jet (i.e., ξ>0.5), 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 Pk=0), 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 Ck and Dk around ξ=1.2.Figure 4
Profiles of TKE budget: productionPk (blue), dissipation -ϵ (green), convection Ck and diffusion Dk. Present models #1 (from (41), solid lines) and #2 (from (52), dashed lines) versus measurements (symbols) from Bradbury [2]. The analytical formulae of these three quantities are given in Table 1.Table2 gives the model constants for the last two particular cases. They both provide similar values. The integrals of Pk, Ck, and Dk 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 #2, based on the empirical velocity profile (52), predicts an exactly vanishing integral of the energy diffusion.Table 2
Model constants and useful figures of the present models. The experimental values are from the references below the table.
Constants
Withfk2=fϵ in (32)
Withfwη=e-η2
Experimental
δ
1
ln
⁡
1
+
2
≈
1.135
1
ln
⁡
2
≈
1.201
—
ϕ
≈
0.1101
≈
0.1165
—
σ
c
≈
0.63
≈
0.70
0.59–0.75b
α
w
1
2
3
ϕ
≈
2.610
2
π
ϕ
2
1
/
4
≈
2.617
—
α
c
1
+
2
σ
c
3
π
ϕ
Γ
1
/
2
+
σ
c
Γ
1
+
σ
c
≈
2.325
π
2
ϕ
2
1
/
4
∫
-
∞
+
∞
e
-
η
2
Ψ
1
η
a
d
η
-
1
≈
2.350
—
α
k
2
ϵ
3
ϕ
3
/
2
8
C
μ
≈
0.0879
2
π
1
/
4
ϕ
3
/
2
4
C
μ
≈
0.0987
—
∫
0
+
∞
P
k
ξ
,
ζ
d
ξ
W
0
3
ζ
/
x
∗
z
2
15
ϕ
≈
0.0147
1
8
π
3
ϕ
≈
0.0149
0.016
9
c
∫
0
+
∞
C
k
ξ
,
ζ
d
ξ
W
0
3
ζ
/
x
∗
z
1
sqrt
⁡
C
μ
ϕ
2
16
≈
0.00253
2
C
μ
t
a
n
-
1
2
2
ϕ
2
16
≈
0.00246
0.003
8
c
∫
0
+
∞
D
k
ξ
,
ζ
d
ξ
W
0
3
ζ
/
x
∗
z
-
ϕ
2
8
σ
k
C
μ
≈
-
0.00505
0
0
c
aSee note (a) of Table 1.
bTurner [16], Mellor and Yamada [18], Sukoriansky et al. [17], Law [12], and Violeau (2009).
cBradbury [2].Finally, one should mention that Bradbury’s [2] experiments yield almost constant turbulent turnover timescale; that is, k∗/ϵ∗≈5.0, where k∗ and ϵ∗ are the dimensionless versions of k and ϵ 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.



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*Source: 4369895-2017-05-25.xml*