An incompressible, time developing 3-D mixing layer with idealized initial conditions was simulated numerically. Consistent with the suggestions from experimental measurements, the braid region between the dominant spanwise vortices or rolls develops longitudinal vortices or ribs, which are aligned upstream and downstream of a roll and produce spanwise distortion of the rolls. The process by which this distortion occurs is explained by studying a variety of quantities of dynamic importance (e.g., production of enstrophy, vortex stretching). Other quantities of interest (dissipation, helicity density) are also computed and discussed. The currently available simulation only allows the study of the early evolution (before pairing) of the mixing layer. New simulations in progress will relieve this restriction

Colonius, T.

Hussain, Fazle

Moin, P.

Moser, R. D.

Rogers, M. M.

English

NASA Technical Reports Server

N89- 24543 
Center for Turbulence Rereowh 
Proceedingr of the Summer Program 1988 
49 
Dynamics of coherent structures 
in a plane mixing layer 
By F. Hussainl, R. MoserZ, T. Colonius3, P. M0in~1~ AND M. M. Rogers2 
An incompressible, time-developing three-dimensional mixing layer with idealized 
initial conditions has been simulated numerically. Consistent with the suggestions 
from experimental measurements, the braid region between the dominant spanwise 
vortices or rolls develops longitudinal vortices or ribs, which are aligned upstream 
and downstream of a roll and produce spanwise distortion of the rolls. The process 
by which this distortion occurs has been explained by studying a variety of quantities 
of dynamic importance (e.g. production of enstrophy, vortex stretching). Other 
quantities of interest (dissipation, helicity density) have also been computed and 
are discussed. The currently available simulation only allows the study of the early 
evolution (before pairing) of the mixing layer. New simulations in progress will 
relieve this restriction. 
Introduction 
While there is no doubt about the occurrence of large-scale coherent structures 
in turbulent shear flows, there is doubt about their role and dynamical significance. 
Unfortunately, a mathematical definition of coherent structures or a theory of tur- 
bulence based on coherent structures has not yet been developed, nor is there any 
in sight. Until the development of a theoretical framework , or even to formu- 
late one, we must continue to improve our understanding of coherent structures. 
Studies of coherent structures in shear flows, in particular the mixing layer, have 
focused on the geometric form of the structures (morphology) and the relationship 
of the structures to topological features of the flow (e.g. the stagnation line and 
saddle between rolls in the mixing layer); also of interest are quantitative measures 
of dynamical quantities such as coherent Reynolds stress and coherent production. 
Experimental observations of coherent structures in the mixing layer have relied 
on flow visualization techniques (which suffer from the indirect relationship of flow 
markers to the hydrodynamics) and quantitative point-wise measurements (which 
are unable to measure the full three-dimensional flow field). Such measurements 
strongly suggest the complex morphology of the coherent structures in mixing lay- 
ers, in particular the counter-rotating longitudinal vortices (ribs) in the braid region 
between the rolls (Bernal & Roshko, 1986, Hussain, 1983). However, the details of 
these structures are not currently accessible from experimental data. Fortunately, 
1 University of Houston 
2 NASA Ames Research Center 
3 Stanford University PRECEDING PAGE BLANK NOT FILMED 
https://ntrs.nasa.gov/search.jsp?R=19890015172 2020-03-20T03:01:56+00:00Z
50 F. Hwsain, R.  D. Moser, T. Coloniw, P.  Moin and M.  M. Rogers 
direct numerical simulation of turbulent flows can provide the 3D flow field with 
adequate resolution, albeit at low Re (Metcalf e t  aJ., 1987, Hussain 1986). 
Several numerical simulations were performed to study the details of the co- 
herent structures observed experimentally in mixing layers. A three-dimensional 
time-developing mixing layer with prescribed initial conditions was simulated. The 
initial conditions were chosen to produce structures similar to those observed ex- 
perimentally. These simulations are different from the experiments they are meant 
to mimic in several important ways. First, the experimental flows are spatially 
developing, whereas the simulations are time developing. The time-developing sim- 
ulation approximates the evolution of a structure in the spatially-developing flow 
as it travels downstream. Second the initial conditions do not necessarily reflect 
the conditions present in an experiment. In the simulations the initial conditions 
are very smooth and simple, there is no small-scale random motion, thus the re- 
sulting flow fields are not turbulent. These simple conditions are used so that the 
coherent structures can be studied in their simplest form. The question of how 
the observed structures evolve from conditions actually present in experiments, and 
how they respond to small-scale random disturbances is left to future work. Finally, 
the Reynolds number in the simulations is quite low compared to experiments. 
The simulations were performed using a numerical method similar to that dis- 
cussed by Cain, Ferziger & Reynolds (1984). In this method periodic boundary con- 
ditions are applied in the streamwise (2) and spanwise ( z )  directions, and Fourier 
series are used in these directions. An infinite domain in the cross-stream direction 
(y) is treated by using a coordinate transformation which maps the domain into a 
finite intend.  Fourier series are then used in the finite interval. In the particular 
simulation to be discussed below, 32 Fourier modes were used in the x and z direc- 
tions and 64 modes were used in the y direction. The length of the computational 
domain in the x direction was 4.48~6, and in the z direction it was 2 . 7 ~ 6  where 6 
is the initial vorticity thickness of the layer. The initial velocity field consisted of 
an error function velocity profile and a pair of three-dimensional disturbances. The 
first disturbance leads to the two-dimensional Kelvin-Helmholtz roll-up of the mix- 
ing layer and ultimately pairing, the initial spanwise vorticity in this disturbance is 
niven by 
~ 
X X 
W ,  = O . ~ ~ I ( ~ ) C O S ( - )  + O . O ~ ~ ~ ( Y ) C O S ( - ) .  
1.126 2.246 
where the functions fi and fi are Rayleigh eigenfunctions and the wave-lengths are 
chosen to be the most unstable and its subharmonic. The second disturbance leads 
to the longitudinal vortices in the braid region between the Kelvin-Helmholtz rolls. 
This disturbance initially consists of an array of streamwise vortices described by 
~ 
Z z 
w:. = 0.0591(y)sin( - ) + 0.02592(y)~in(-) 0.6756 1.356 
I where the functions 91 and 92 were chosen to represent streamwise vortices and the 
wave-lengths were chosen to be the most unstable and its subharmonic. A Reynolds 
number of 1000 based on initial vorticity thickness and velocity difference was used 
in this simulation. 
Coherent structures in a plane mizing layer 5 1  
I I I I I I I t I I 1 I 1 I I I 
I 
I 1 I I I 1 I I I I I I I I 1 I 
FIGURE 1. Vorticity contours in plane 1; ( a ) ,  wz,  (b), wy and (c), w,, and plane 2; 
(d), w, at time I. Contour increment is O.lU/&. Negative contours are dashed in 
(a)and (a), positive contours are dashed in (c)and (d). Streamwise (2) direction is 
horizontal; cross stream (y) direction is vertical. 
Results and Discussion 
For the current discussion we have selected two instants in the simulation de- 
scribed above: time I ( t  = 26.46/AU) is when the initial Kelvin-Helmholtz roll-up 
is saturating and the two rolls are nearly identical, and time I1 ( t  = 356/AU) is 
a short time later when the pairing interaction has started. To study the pairing 
process, a much later time is required; however, the current simulation was stopped 
at t = 17.5 because of lack of resolution. Higher resolution simulations are being 
performed to address questions of pairing. In the discussion to follow, two dimen- 
sional contour plots of various quantities will be presented in two planes in the flow 
field. The first plane (plane 1) is an x-y plane (plane of constant z )  which passes 
through the center of the longitudinal vortices (location of maximum streamwise 
vorticity). The second plane (plane 2) is also an x-y plane, and it passes halfway 
between the longitudinal vortices (streamwise vorticity is zero in this plane). 
Contours of the three components of vorticity in both planes at time I are shown 
in figure 1 (wz and wy are identically zero in plane 2). The role-up of the shear- 
layer resulting in the concentration of spanwise vorticity in two large rolls is clearly 
evident. Note that there is substantial spanwise vorticity in the braid region between 
the rolls (as much as -0.55) compared to the initial maximum spanwise vorticity 
(-2) As the roll-up continues, spanwise vorticity continues to be swept from the 
braid region; at time I1 (figure 2) the spanwise vorticity in the braid region has been 
reduced to -0.25. The streamwise and cross-stream vorticity (wz and wy in figures 
l ( a , b )  are concentrated in the braid region as expected; these are the longitudinal 
52 F. Hussain, R .  D. Moser, T .  Colonius, P.  Moin and M .  M.  Rogers 
1 1 
1 1 
FIGURE 2. Vorticity contours in plane 1; (a), w,, (a), w,, and ( c ) ,  w,, and 
plane 2; (4, w, at time 11. Contour increment is (a), O.l5U/6, (b,c,d), O.lU/6. 
Negative contours are dashed in (a)and (b), positive contours are dashed in (c)and 
(d). Streamwise (z) direction is horizontal; cross stream (3) direction is vertical. 
(or rib) vortices. They include both wz and w,, because they are inclined. There 
are, of course, an array of counter-rotating rib vortices, the ones in this plane have 
positive vorticity but in other planes the vorticity would be negative. The rib 
vortices lie along the diverging separatrix of the stagnation point between the two 
main rolls. Therefore they are subjected to a plane strain which stretches them 
along the separatrix. The result is an increase in the vorticity magnitude with time 
(wz = 0.65 at time I, and 0.98 at time 11, see figures 1 and 2). 
In the rolls there is a somewhat weaker region of streamwise and cross-stream 
vorticity of opposite sign to that in the ribs (wa = -0.35 at time I). This is ap- 
parently a result of the three-dimensional distortion of the rolls by the ribs. This 
distortion is most apparent at time I1 where there is a marked difference in the 
spanwise vorticity contours in planes 1 and 2 (figures 2(c,d)). In plane 2, the span- 
wise vorticity is concentrated near the bottom of the rolls and is very strong there 
(-3.05) compared to the initial maximum spanwise vorticity (-2.0). In plane 1, 
the vorticity is more evenly distributed through the roll and has a maximum value 
of -1.85. The mechanism by which the ribs produce three-dimensionality in the 
rolls can be understood by examining the rate of production of enstrophy (w iS i jw j ,  
where Si, = ;(aui/azj + auj /az i ) )  at time I1 (figure 3). In plane 2 there is a 
region of strong enstrophy production coinciding with the concentrated region of 
spanwise vorticity. Above it there is also a region of negative enstrophy production. 
The strong production is a consequence of the stretching of the spanwise vorticity. 
This stretching occurs in the region between the counter-rotating rib vortices where 
Coherent dtructures in a plane mizing layer 53 
,.----. 
:' .-2 
FIGURE 3. Rate of enstrophy production in (a), plane 1 and (b), plane 2 at 
time 11. Contour increment is 0.2U3/&i3. Negative contours are dashed. Streamwise 
(z) direction is horizontal; cross stream (y) direction is vertical. 
1 1 
FIGURE 4.
Contour increment is 0.0005U3/6. 
stream (y) direction is vertical. 
Dissipation of kinetic energy in (a) plane 1 and (b) plane 2 at time 11. 
Streamwise (z) direction is horizontal; cross 
the strain a w / a z  is large and negative. Between rib vortices where a w / a z  is pos- 
itive compression occurs resulting in the negative enstrophy production mentioned 
above. In plane 1, which is through the center of the rib vortices (8w/8z  = 0), the 
enstrophy production in the rolls is small. There is however enstrophy production 
in the braid region, corresponding to the stretching of the rib vortices. 
Another quantity of interest is the dissipation of kinetic energy, which is shown 
at time I1 in figure 4. The dissipation is rather weak in plane 1 (0.003) compared to 
plane 2 (0.009). In particular, the dissipation in plane 2 is concentrated in the region 
of large w, and large enstrophy production discussed in the previous paragraph. 
Thus the three dimensional distortion of the rolls results in significant dissipation of 
kinetic energy. The work of Moffatt (1985) on inviscid flows suggests that dissipation 
and helicity density (uiwi) should be spatially exclusive; however, Hussain (1986) 
suggests that this may not be the case. Helicity density and dissipation contours 
in plane 1 at time I are shown in figure 5. Helicity density is identically zero in 
plane 2. The helicity density is concentrated in the region where the ribs meet the 
rolls and is zero near the stagnation point between the rolls; this is as suggested 
by Hussain (1986). We note that there is also a concentration of dissipation in the 
braid region, thus at this time the dissipation and helicity density are not spatially 
exclusive. This may be a consequence of low Reynolds number or the early stage of 
54 
1 1 
F. Hubrain, R. D .  Morer, T. Colonius, P. Moin and M. M. Rogers 
FIGURE 5. Dissipation of kinetic energy (a )  and helicity density (b) in plane 1 at 
time I. Contour increment is (a),  0.00025U3/6 and (a), 0.1Uz/6. Negative contours 
are dashed. Streamwise (2) direction is horizontal; cross stream (y) direction is 
vertical. 
1 
FIGURE 6. 
Streamwise (2) direction is horizontal; cross stream (y) direction is vertical. 
Helicity density in plane 1 at time 11. Contour increment is 0.1U2/6. 
development of the flow. At time I1 the dissipation is not significantly concentrated 
in the braid region (see figure 4), whereas the helicity density is nearly the same as 
the earlier time (figure 6). Thus at time I1 the peak of dissipation occurs in plane 2, 
where the helicity density is zero, and the peak of the helicity density occurs in 
plane 1 where the dissipation is minimum. 
Summary 
A simulation of a time-developing mixing layer with idealized initial conditions 
has produced flow structures which are at least qualitatively similar to those ob- 
served in experimental mixing layers. With the availability of the three-dimensional 
flow field, it is possible to investigate a variety of structural and dynamical ques- 
tions by computing any number of flow quantities (e .9.  vorticity, production of 
enstrophy, dissipation etc.). For example, in this brief study we were able to ob- 
serve and explain the three-dimensional distortion of the Kelvin-Helmholtz rolls by 
the rib vortices, and we were able to study the relationship of helicity density and 
dissipation. A more complete study of higher-resolution simulations should provide 
great insight into the dynamics and topology of coherent structures in the mixing 
layer. This is the focus of our ongoing research. 
Coherent atructurea in a plane mizing layer 55 
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BERNAL, L. P. & ROSHKO, A. 1986 Streamwise vortex structure in plane mixing 
layers. J.  Fluid Mech. 170, 499-525. 
CAIN, A. B., FERZIGER, J .  H. & REYNOLDS, W. C. 1984 Discrete orthogonal 
function for non-uniform grids using the fast Fourier transform. J .  Comp. Phys. 
HUSSAIN, A. K.  M. F. 1983 Turbulence and Chaotic Phenomena in Fluids (ed. T. 
HUSSAIN, A. K.  M. F. 1986 Coherent structures and turbulence. J.  Fluid Mech. 
MOFFATT, H. K.  1985 Magnetostatic equilibria and analogous Euler flows of arbi- 
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METCALF, R. W., ORSZAG, S. A., BRACHET, M. E., MENON, S. & RILEY, J .  J .  
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Dynamics of coherent structures in a plane mixing layer

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