Direct Numerical Simulation of Eddy Diffusivity in a Plane Jet
with a second-order reaction
Tomoaki Watanabe 1, Yasuhiko Sakai 1, Kouji Nagata 1, Hiroki Suzuki 2,Yasumasa Ito 1, Osamu Terashima 1, and Toshiyuki Hayase 3
1

Department of Mechanical Science and Engineering, Nagoya University, Japan
Department of Engineering Physics, Nagoya Institute of Technology, Japan
3
Institute of Fluid Science, Tohoku University, Japan
2

1

Introduction

Turbulent flows with chemical reactions can be seen invarious fields. Therefore, it is important to establish numericalmethods to predict turbulent reactive flows.
Reynolds-averaged approach is often used for predictions ofreactive flows. In this approach, Reynolds-averaged scalartransport equations are solved by using models for a turbulent
mass flux <ui> and an averaged production rate by chemical
reactions. Here, ui and  are fluctuation components of i
direction velocity (Ui) and concentration of species  (),respectively, and <*> denotes an ensemble average of variable *.
One of the simplest models for <ui> is a gradient diffusion
model [1], in which <ui> is given byui     Dt ,i ΓxiFig. 1 Sketch of the flow.

(1)

Here, Dt,i is the eddy diffusivity of species  corresponding tothe i direction. Dt,i has a significant effect on the prediction of<ui>. Bilger et al. [2] and Komori et al. [3] haveexperimentally shown that the eddy diffusivity of reactive
species is affected by chemical reactions. However, the effects
of chemical reactions on the eddy diffusivity have not been
investigated sufficiently.
In this study, a planar jet with a second-order chemical
reaction is simulated by DNS, and the eddy diffusivity DtA,x ofreactant species A corresponding to the x direction and theDamkhöler number effects on DtA,x are investigated in detail.
2Direct Numerical Simulation

Figure 1 shows a sketch of the flow. A planar jet containing
reactant A is ejected into a main flow containing the otherreactant B through a slit with width of d. Product R is formed bya second-order chemical reaction which can be written
as A  B  R . We assume that this second-order reaction isisothermal and concentrations of reactive species behave aspassive scalars. The computational domain is resolved by475 425 65 grid points. The grid is equidistant in the x and zdirections. In the y direction, the grid is stretched near the lateralboundaries. The spatial resolution is the same order as theKolmogorov scale.
The governing equations for this problem are theincompressible Navier  Stokes equations, the continuity
equation, and the passive scalar transport equations in theirinstantaneous, local form:
U i
0,
xi

(2)

U i
U i
U i ,p
U j

t
x jxi
x j x j(3)ΓΓΓU j D S .
t
x j
x j x j(4)Here,  is the kinetic viscosity, D is the molecular diffusivity ofspecies  which is assumed to be constant for species A, B, andR, and S is the production rate of  by the chemical reaction
and given by SR =  SA =  SB = kAB, where k is the reaction
rate constant. Equation (4) is solved for concentration of productR and mixture fraction  which is defined by  (A  B  BA  B), where  is the initialconcentration of species . Because SA = SB, the production rateof  by the chemical reaction is zero. The concentrations ofreactants A and B are calculated from R and  by using massconservation law written as follows;Γ A  Γ A0  Γ R , Γ B  Γ B0 1     Γ R .

(5)

The fractional step method is used to solve the governing
equations which are spatially discretized by using central
differences, and the Poisson equation is solved by conjugate
gradient method. The variables are stored on a staggered gridarrangement. The third-order Runge  Kutta method and theCrank  Nicolson method are used for temporal discretization.
The traction-free condition and the convection outflow
condition are applied at the lateral boundaries and the outlet
boundary, respectively. Periodic boundary conditions areapplied in the z direction. Velocity data at the inflow boundaryis generated by superimposing random fluctuations on the mean
velocity profile. The velocity profile at the jet exit is a top-hat
profile with smooth edges which is measured at the exit of theplanar jet in our experimental apparatus [4]. The mean
streamwise velocity UM of the main flow at the inflow boundary
is 0.056UJ, where UJ is the streamwise velocity averaged in thejet exit. The Reynolds number defined by Re = UJd/ is set to2,000. The concentrations of reactive species are imposed at theinflow and lateral boundaries to satisfy that A = A0 and B =
R = 0 in the jet exit, andB = B0 and A = R = 0 in the mainflow. Therefore,  = 1 in the jet exit and  = 0 in the main flow.
Schmidt number (Sc = /D) is set to 1
The initial concentrations are set to A0 = 0.4 [mol/m3] andB0 = 0.2 [mol/m3]. The Damkhöler number Da is the ratio of atime scale of the flow and a time scale of the chemical reaction,and in this study, is defined by Da = k(A0A0)d/UJ.
Simulations are carried out at the different Damkhöler numbers
Da = 0.1, 1, and 10.
31<A>/A0

0.6
0.4
0.2
00Results and discussion

102030x/dFigure 2 shows the axial profile of mean concentration <A>of species A on the jet centreline for Da = 0.1, 1, and 10 and forthe non reactive case. In Fig. 2, <A> is normalized by initialconcentration A0. <> monotonically decreases in x direction.
In Fig. 2, it is also shown that as Da increases, <A> becomes
small and the chemical reaction rapidly proceeds.
Figure 3 shows the axial profiles of streamwise turbulent
mass flux <uA> of species A on the jet centreline for Da = 0.1,1, and 10 and for the non reactive case. <uA> is normalized by
UJ and A0. Figure 3 shows that the chemical reaction makes<uA> large in the upstream region x  22 ~ 30 , but makes small
in the downstream region. <uA> shows the dependency on theDamkhöler number.
Figure 4 shows the eddy diffusivity DtA,x of species A
corresponding to the x direction. In Fig. 4, DtA,x is normalized by
UJ and d. DtA,x is calculated from Eq. (1). Figure 4 shows thatDtA,x becomes small due to the chemical reaction, and the effectof the chemical reaction on DtA,x for Da = 10 is smaller than thatfor Da = 0.1 and 1. Thus, it is found that DtA,x is largely affectedby the chemical reaction, and the change of DtA,x by thechemical reaction depends on the Damkhöler number.
4no reaction
Da = 0.1
Da = 1
Da = 100.8

Fig. 2 Axial profiles of mean concentration of reactant A

0.04<uA>/UJA0no reaction
Da = 0.1
Da = 1
Da = 10

0.03
0.02
0.01
00102030

x/dFig. 3 Axial profiles of streamwise turbulent mass flux ofreactant A

1.5
no reaction
Da = 0.1
Da = 1
Da = 10ConclusionA planar jet with a second-order reaction ( A  B  R ) isinvestigated for the different Damkhöler numbers (Da = 0.1, 1,and 10) by DNS. The eddy diffusivity DtA,x for reactant A
corresponding to the streamwise direction is calculated on thejet centreline from the mean concentration <A> and thestreamwise turbulent mass flux <uA>. The results show that
eddy diffusivity is largely affected by the chemical reaction. For
reactant A, the chemical reaction makes DtA,x small, and theeffect of the chemical reaction on DtA,x for Da = 10 is smaller
than that for Da = 0.1 and 1. The effect of the chemical reaction
on DtA,x depends on the Damkhöler number. These results imply
that the gradient diffusion models may cause a significant errorof the turbulent mass flux estimation when they are applied toreactive flows without considerations of the effect of reactions
on the eddy diffusivity.

DtA, x10.501020x/d30

Fig. 4 Axial profiles of eddy diffusivity of reactant A
corresponding to the streamwise direction
References

Acknowledgements
Part of the work was carried out under the Collaborative
Research Project of the Institute of Fluid Science, Tohoku
University. Part of this work was supported by Grants-in-Aid for(Nos. 22360076, 22360077, 23656133, and 23656134) from theJapanese Ministry of Education, Culture, Sports, Science andTechnology.

[1]

[2]

[3]

[4]

Combest, D. P., Ramachandran, P. A. and Dudukovic, M. P., On
the Gradient Diffusion Hypothesis and Passive Scalar Transport in
Turbulent Flows, Industrial Engineering Chemistry Research, Vol.
50, pp. 8817-8823, 2011
Bilger, R. W., Saetran, L. R. and Krishnamoorthy, L. V., Reactionin a Scalar Mixing Layer, Journal of Fluid Mechanics, Vol. 233,pp. 211-242, 1991
Komori, S., Nagata, K., Kanzaki, T. and Murakami, Y.,
Measurements of Mass Flux in a Turbulent Liquid Flow with a
Chemical Reaction, AIChE Journal, Vol. 39, pp. 1611-1620, 1993
Watanabe, T., Sakai, Y., Yasuhara, H., Nagata, K., Terashima, O.
and Kubo, T., Simultaneous Measurements of Velocity andReactive Scalar in Plane Liquid Jet with the Second-Order
Chemical Reaction, Trans. JSME 78-788 B, pp. 769-782, 2011