New definitions of entrainment and mixing based on the passive scalar field in the plane mixing layer are proposed. The definitions distinguish clearly between three fluid states: (1) unmixed fluid, (2) fluid engulfed in the mixing layer, trapped between two scalar contours, and (3) mixed fluid. The difference betwen (2) and (3) is the amount of fluid which has been engulfed during the pairing process, but has not yet mixed. Trends are identified from direct numerical simulations and extensions to high Reynolds number mixing layers are made in terms of the Broadwell-Breidenthal mixing model. In the limit of high Peclet number (Pe = ReSc) it is speculated that engulfed fluid rises in steps associated with pairings, introducing unmixed fluid into the large scale structures, where it is eventually mixed at the Kolmogorov scale. From this viewpoint, pairing is a prerequisite for mixing in the turbulent plane mixing layer

Broadwell, J. E.

Mungal, M. G.

Reynolds, W. C.

Sandham, N. D.

English

NASA Technical Reports Server

Center for Ilfrrbulence Rerearch 
Proceedingr of the Summer Progrum 1988 
Scalar Entrainment in the Mixing Layer 
69 
By N. D. Sandham’, M. G. Mungal’, 
J. E. Broadwel12 and W. C. Reynolds’ 
New definitions of entrainment and mixing based on the passive scalar field in the 
plane mixing layer are proposed. The definitions distinguish clearly between three 
fluid states - (a) unmixed fluid (b) fluid ‘engulfed’ in the mixing layer, trapped 
between two scalar contours, and (c) mixed fluid. The difference between (b) and 
(c) is the amount of fluid which has been engulfed during the pairing process, 
but has not yet mixed. Trends are identified from direct numerical simulations 
and extensions to high Reynolds number mixing layers are made in terms of the 
Broadwell-Breidenthal mixing model. In the limit of high Peclet number (Pe  = 
ReSc) it is speculated that engulfed fluid rises in steps associated with pairings, 
introducing unmixed fluid into the large scale structures, where it is eventually 
mixed at the Kolmogorov scale. From this viewpoint pairing is a prerequisite for 
mixing in the turbulent plane mixing layer. 
1. Introduction 
Existing definitions of entrained fluid are not specific about the state of the fluid 
(whether mixed or unmixed) that is carried by large scale structures. Corcos & 
Sherman [1984] used the instantaneous streamlines in the moving reference frame 
to identify a cat’s-eye structure boundary. Similarly Hernan & Jimenez [1982] in 
their digital analysis of movies of the mixing layer, used a best-fit ellipse to frame 
each structure, and measured the area within the ellipses to estimate entrainment. 
Neither definition distinguishes between entrained fluid that is mixed, entrained 
fluid that is doomed to be mixed, and appear to allow for the possibility of ‘en- 
trained’ fluid subsequently leaving the ellipse. 
In this paper we use insights from direct numerical simulations to provide ideas 
on how the large-scale structure dynamics affect scalar transport in the mixing 
layer. For this purpose a two-dimensional time-developing code was used to solve 
the compressible Navier-Stokes equations (Sandham (1988)). The simulations were 
made at low convective Mach number ( M ,  = 0.4) so that the results can be ap- 
plied to the low-speed mixing layer. The simulations are of course limited by two- 
dimensionality and the low Reynolds and Schmidt numbers that can be handled 
numerically. However some basic trends emerge clearly in the simulations and by 
using the Broadwell-Breidenthal model of the post-mixing-transition layer we are 
able to make some qualitative statements about the influence of pairing on mixing 
in the high Reynolds number mixing layer. 
1 Stanford University 
2 California Institute of Technology 
https://ntrs.nasa.gov/search.jsp?R=19890015174 2020-03-20T03:01:52+00:00Z
70 N. D .  Sandham, M.  G. Mungal, J .  E.  Broadwell and W. C. Reynolds 
I '  
FIGURE 1. Time series of a pairing cycle. 
2. Observations From Direct Simulations 
From flow visualisations it is apparent that the pairing process is the dominant 
mechanism for local adjustment of the mixing layer eddy length scale to allow 
growth in the streamwise direction. To investigate this basic process we consider the 
simple case of the numerically-simulated time-developing mixing layer undergoing a 
pairing, shown in Figure 1. The mixing layer is viewed at approximately equal time 
steps in the cycle. At the first step shown it has already undergone one pairing and 
by the last (5th) time step the structure has filled its periodic box and is unable to 
grow or pair any further. The Reynolds number based on vorticity thickness was 
initially 200 and rose to 2000 by the end of the simulation. The Schmidt number 
was 1, and a 300 x 300 grid was used. 
If the structure is assumed to maintain self-similarity from the first time step 
to the last, then the area must rise by a factor of 4. Two vortices have paired; 
so in the final structure half the fluid came from the original two structures and 
Scalar entrainment in the mizing layer 
 
71 
- - _ _ _ _ _ - - -  
(a) 
I 1 
FIGURE 2. Scalar cut-and-connect process showing the 0.1 and 0.9 scalar contours 
(a) before (b) after. 
the other half was introduced during the pairing cycle. Mixing by diffusion is one 
mechanism for the growth of the structure. A second mechanism can be identified 
from the scalar contour plots as a scalar cut-and-connect, which has the effect of 
trapping nearly-unmixed fluid within the structure. This is shown in more detail 
in Figure 2. Unmixed fluid is wrapped around the structure at time 1 (see Figure 
l ) ,  with mixing only occurring by diffusion. Then, between times 1 and 2 this fluid 
is cut off from the outside and becomes trapped in the structure. Between times 3 
and 5 mixing is occurring in two places - diffusion from the outside, and diffusion 
of the fluid inside the structure which was captured by the pairing. 
This observation suggested new definitions of fluid state based on the scalar con- 
tours. Engulfed fluid ( E )  is that fluid contained between the outermost scalar 
contours, for example fluid between the levels 0.1 and 0.9, where 0 and 1 represent 
the free-stream values. Mized fluid (M) is that fluid throughout the mixing layer 
which is molecularly mixed between two levels (e.g. 0.1 and 0.9). The difference 
E - M is the fluid that has been engulfed, but is yet to mix. 
The effect of Peclet number on the process was investigated by running 150 x 150 
simulations at a Reynolds number of 200 and Schmidt numbers of 0.25 and 1.0. 
The results are shown in Figures 3(a) for Sc = 0.25 and 3(b) for Sc = 1.0. Engulfed 
fluid E was found by integrating around scalar contours, while mixed fluid M was 
calculated by scanning the computational domain for mixed fluid and adding area 
increments. The two measures are equal so long as all the fluid within the newly 
defined structure is mixed. It is seen from Figure 3(a) that at low Schmidt number 
molecular diffusion is very strong, and fluid is essentially diffusion-mixed before it 
is engulfed. The plots are shown for various scalar cutoff levels 0.1 - 0.9 (containing 
72 N .  D.  Sandham, M.  G. Mungal, J .  E .  Broadwell and W.  C. Reynolds 
(a) :I 
8 -  0 
8 -  0 
8-  
-k 
rl 
0 
N 
9 
0 
0- 
0 
Q w 
Ly 
Q 
LEGEND I - 
0 E 0.1-0.9 
O M 0.1-0.9 
A E 0.2-0.8 
_...___..-- -. 
--- 
+ M 0.2-0.0 I 
TIME 
1.0 
0 :  I 1 I 1 1 I I I I 8 I 
TIME 
FIGURE 3. Growth of engulfed and mixed fluid at Sc = (a) 0.25 and (b) 1.0. 
80% of the mixed fluid), 0.2 - 0.8 (60%), and 0.3 - 0.7 (40%). The effect of lowering 
the limits on the cutoff is to exclude some of the diffusion effects and give some 
indication of how a higher Schmidt number flow would behave. At Sc = 1.0 (Figure 
3(b)) and taking the 40% and 60% limits there is a clear trend emerging. The E 
Scalar entrainment in the mizing layer 73 
curve shows a sharp jump at the moment of the scalar cut-and-connect, trapping 
fluid within the structure, which later mixes. By extrapolating these results we 
speculate that in the limit of infinite P e ,  for simulations with Re fixed below the 
mixing transition, the function E would be a step function, while the function M 
would remain essentially zero since the computations have no three-dimensional 
small scales to increase the interfacial area for mixing. 
In the following section we use the Broadwell-Breidenthal picture of the mixing 
layer past the mixing-transition to include the small scales of the vortex core in our 
model pairing event, and to make some predictions about the influence of pairing 
on the mixing process. 
I 
3. Extension of Ideas to High Reynolds Numbers 
A simplified model of mixing in the plane mixing layer was proposed by Broadwell 
& Breidenthal [1982] and has been extended to include chemical reactions of arbi- 
trary rate by Broadwell & Mungal [1988]. In the model, fluid is viewed as mixing 
(1) in laminar strained-flame regions between the two free-stream fluids, and (2) 
inside the structure at the Kolmogorov scale. In the former process the amount of 
product that is formed has Reynolds and Schmidt number dependence, while in the 
latter process it does not. The model correctly predicts many features of the mixing 
layer experiments, including the differences between liquid and gas experiments and 
the variation of product formation with equivalence ratio, that cannot be explained 
with conventional gradient-diffusion models. 
In the limit of high Peclet number, such as occurs in liquid mixing layers and in 
gases at very high Reynolds numbers, mixing in the laminar strained flame regions 
is reduced to zero and we need only consider mixing at the Kolmogorov scale. The 
ideas from the previous section show that the pairing is responsible for introducing 
unmixed fluid into the structures in a discrete manner. Immediately after the scalar 
cut-and-connect process the structure contains one-half old fluid and one-half new, 
unmixed fluid. The vorticity field at this stage consists of the two pairing vortices, 
plus their associated streamwise vortices and small scales. It is assumed that a 
cascade in scales occurrs and that mixing finally occurs at the Kolmogorov scale 
when the interfacial area of fluid has increased dramatically. Schematically the 
proposed process is given in Figure 4. From the Broadwell-Breidenthal model the 
time scale for mixing to occur after engulfment would be given by TM - TE - L l  AU, 
where L is a characteristic length scale at the start of the cascade. 
Some experimental results of Roberts and Roshko [1985] can be interpreted in the 
light of the above ideas. Roberts performed chemical reactions in a liquid mixing 
layer ( P e  + m) at high Reynolds number and found that, when the layer was 
forced, new product formation dropped to zero. We can now postulate that the 
effect of forcing is to lock the mixing layer, stopping pairing and hence preventing 
new fluid being engulfed, which in turn prevents any new product being formed. 
In reaching the above conclusions for Pe + 00 it has been assumed that the 
increase in interfacial area due to the presence of streamwise vortices in the flow 
is not sufficient to outweigh the reduction in diffusion coefficient. At lower Peclet 
74 N. D .  Sandham, M.  G.  Mungal, J .  E .  Broadwell and W.  C.  Reynolds 
5 
I I 
Te Tm 
Time 
0 
FIGURE 4. Prediction of growth in engulfed and mixed fluid for P e  + 00. 
numbers, for example in the gaseous mixing layer, these streamwise vortices will be 
important, and have been invoked by Mungal & Dimotakis [1985] to explain some 
features of ‘ramping’ in the temperature time traces of a reacting mixing layer. 
Finite Peclet numbers will have the effect of reducing the jump in engulfed fluid 
and giving the E and M curves a positive slope between pairings, due to mixing 
in the strained laminar diffusion layers. At the Reynolds number used by Mungal 
& Dimotakis (6.8 x lo4 based on visual thickness) it was estimated that about half 
the mixing occurred in these diffusion layers, and we would expect the jump in E 
to be reduced by approximately a factor of two. 
The method used by Hernan & Jimenez [1982] to estimate entrainment in a 
gaseous mixing layer at high Reynolds number (but finite Pe)  involved framing 
each structure with a best-fit ellipse. Figure 5 shows how this might work for the 
structure at time 5 in the pairing cycle. For times 2 and 3 (see Figure 1) it was 
more difficult to fit any meaningful ellipse to the structures. This method evidently 
includes unmixed fluid that is not contained in our engulfed fluid definition, and 
may exclude some mixed fluid. The amount of extra fluid included is a function of 
the particular structure orientation, so that the area of the ellipse may change even 
if no mixing is taking place. The conclusion of Hernan & Jimenez that entrainment 
mostly occurs between pairings should therefore be viewed with caution, since it 
may be influenced by the method that they used to measure entrainment. Also, we 
should not interpret their measurements of entrainment as giving any information 
about mixing. The model proposed here for finite P e  is that mixed fluid rises 
steadily in between discrete jumps which are associated with the mixing of fluid 
that was engulfed during pairing and has reached the Kolmogorov scale. The size 
of each jump is speculated to be Reynolds number dependent, reducing as Reynolds 
number is reduced. 
Scalar entrainment in the mizing layer 75 
\ 
I \ I 
FIGURE 5. Result of fitting an ellipse to a structure. 
4. Conclusions 
Definitions of entrainment and mixing based on the scalar field of a numerically 
simulated plane mixing layer have been used to study the fundamental effects of 
pairing on the mixing process. It was found that pairing is responsible for the 
process of ‘engulfment’, bringing unmixed fluid into the structure, while actual 
mixing lagged behind. Use of the Broadwell-Breidenthal mixing model allowed 
the results from two-dimensional simulations to be extended to the high Reynolds 
number mixing layer, leading to a model picture of pairing for P e  + 00 as a step 
function doubling in fluid contained in a structure, followed by molecular mixing 
after a time lag for the cascade in scales to reach the Kolmogorov scale. It is 
concluded that for Pe + 00 pairing is necessary for mixing, helping to explain 
some experimental findings of Roberts & Roshko [1985]. 
REFERENCES 
BROADWELL, J .  E. & BREIDENTHAL, R. E. 1982 A simple model of mixing and 
chemical reaction in a turbulent shear layer. J. Fluid Mech. 125, 397-410. 
BROADWELL, J .  E. & MUNGAL, M. G. 1988 Molecular mixing and chemical 
Twenty-Second International SympoJium reactions in turbulent shear layers. 
on Cornbudion, Seattle, August 1988. 
CORCOS, G. M. & SHERMAN, F. S. 1984 The mixing layer : deterministic models 
of a turbulent flow. Part 1. Introduction and the two-dimensional flow. J. Fluid 
Mech. 139, 29-65. 
HERNAN, A. H. & JIMENEZ, J .  1982 Computer analysis of a high-speed film of 
the plane turbulent mixing layer. J. Fluid Mech. 119, 323-345. 
-76 N .  D.  Sandham, M. G .  Mungal, J .  E.  Broadwell and W.  C. Reynolds 
MUNGAL, M. G. & DIMOTAKIS, P. E. 1984 Mixing and combustion with low heat 
ROBERTS, F. A. & ROSHKO, A. 1985 Effects of periodic forcing on mixing in 
SANDHAM, N. D. 1988 Ph. D. Thesis, Stanford University. In preparation. 
release in a turbulent shear layer . J.  Fluid Mech. 148, 349-382. 
turbulent shear layers and wakes . AIAA-85-0570. 


Scalar entrainment in the mixing layer

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server