There is now overwhelming evidence that in most turbulent flows there exist regions moving with the flow where the velocity and vorticity have a characteristic structure. These regions are called coherent structures because within them the large-scale distributions of velocity and/or vorticity remain coherent even as these structures move through the flow and interact with other structures. Since the flow enters and leaves the bounding surfaces of these structures, a useful definition for coherent structures is that they are open volumes with distinctive large-scale vorticity distributions. Possible fruitful directions for the study of the dynamics of coherent structures are suggested. Most coherent structures research to data was concentrated on measurement and kinematical analysis; there is now a welcome move to examine the dynamics of coherent structures, by a variety of different methods. A few of them will be described

Hunt, J. C. R.

English

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Center for Turbulence Research
Proceedings of the Summer Program 1987
N88-23118 !
315
Coherent Structures-Comments on Mechanisms
By J. C. It. HUNT 1
1. Introductory remarks
There is now overwhelming evidence that in most turbulent flows, despite the flow field be-
ing random, there exist regions moving with the flow where the velocity and vorticity have a
characteristic "structure." These regions are called "coherent structures" because within them the
large-scale distributions of velocity and/or vorticity remain coherent even as these structures move
through the flow and interact with other structures. In most flows the sizes of these structures
vary but the distributions of velocity/vorticity remain similar. There is also evidence that there
is a significant degree of similarity between these distributions in different flows (Hussain 1986).
Since flow enters and leaves the bounding surfaces of these structures, a useful definition, fol-
lowing Hussain (1986), for coherent structures is that they are "open volumes with distinctive
large-scale vorticity distributions."
The following schematic remarks are a personal statement about possible fruitful directions for
the study of the dynamics of coherent structures (hereafter CS). Most CS research to-date has
been concentrated on the measurement and kinematical analysis of CS; there is now a welcome
move to examine the dynamics of CS, by a variety of different methods. A few of them will be
described here.
2. The origins of coherent structures
Coherent structures arise by two main types of mechanism.
(i) Instability (primary or secondary)
When non-uniform flows are generated at high Reynolds number they are usually unstable, so
that some small disturbances to the velocity field amplify. The most unstable disturbances may be
small in amplitude but their length scale is usually of the same order as that of the original mean
flow. Even as the amplitude of the disturbances grow to the same order as that of the velocity of
the original flow, the length scale and flow structure remain similar to that of the original small
disturbance, as in the free shear layer (Fig. 1).
Consequently linear stability theory remains of considerable value in analyzing CS and in esti-
mating their consequences, e.g., their effects on noise generation (e.g., Gaster et al. 1985).
Once the primary instabilities have grown secondary instabilities develop, often with vorticity
in directions perpendicular to that of the primary instability, for example the "braids" that form
in free shear layers with vorticity parallel to the mean flow and that link the primary "roller"
vorticities (e.g., Bernal & Roshko 1986).
A general feature of the growth of the primary and secondary instabilities is that they lead to
a concentration of vorticity in localized regions in the flow as shown by recent non-linear studies
(e.g., Smith 1987). Well-known examples are the rollers and braids in free shear layers, and the
spanwise vortices and "horseshoe" vortices in boundary-layer flows. These regions of concentrated
vorticity, which occur at Reynolds numbers (based on global or local scales) just great enough
for turbulence to persist, are usually found to have characteristics similar to those in coherent
structures at high Reynolds numbers.
(ii) Selective amplification of disturbances by the mean field
Consider an initially isotropic homogeneous turbulent velocity field introduced into a shear flow.
(This is certainly possible on the computer, e.g., Rogallo 1981, and approximately possible in a
wind tunnel, e.g., Champagne et al. 1970; it may broadly approximate the turbulence generated
at the wall or at the outer interface in the boundary layer, e.g., Townsend 1970.)
Computations, both non-linear by Rogallo (1981), and linear by Lee et al. (1987), show that the
shear stretches and rotates the vorticity in such a way that the eddies become elongated and develop
1 University of Cambridge
https://ntrs.nasa.gov/search.jsp?R=19880013734 2020-03-20T06:10:17+00:00Z
816 J. C. R. Hunt
SECONDARY
"CENTRI FUGAL"
INSTABILITIES
I PRIMARY ROLLSSHEAR INSTABILITY
"BRAID"
SUB-STRUCTURES
COHERENT STRUCTURES
FIGURE 1. Coherent structures generated from instabilities: Primary and secondary instabili-
ties in a free shear layer leading to two kinds of coherent structure -- "rolls" and "braids." The
latter are also known as "sub-structures."
sharper boundaries (Fig. 2). This is the explanation of the evolution of the velocity spectrum in
Rogallo's computations from an exponential decay (indicating very smooth velocity distributions
in the eddies) to a k-2 decay (indicating discontinuities on the large scale). Linear calculations
using Rapid Distortion Theory (RDT) also show these effects; similar coherent structures are found
as in the non-linear simulations. (The analysis of the RDT spectra needs to be done.)
3. Interactions between a coherent structure and its surrounding flow
After the formation of a CS, its subsequent existence depends on how it interacts with the
surrounding flow. This varies considerably between flows.
(i) u.ifo,.,. /tow,
Although CS are not generated in uniform flows, they are often moved by the flow or propa-
gate themselves into regions of approximately uniform velocity; CS in wakes far from the body
approximate to this case. The simplest type of CS to consider in this situation is the vortex ring
or the vortex pair (Maxworthy 1977). These CS preserve their structure, but they always lose
some vorticity as they propagate through the flow (Fig. 3a).
Coherent Structures- Comments 317
LOW REYNOLDS NUMBER
HOMOGENEOUS ISOTROPIC TURBULENCE log E(k)
log k
AFTER STRONG SHEAR
_/
dUt >>1
/3 = dy
log E(k)
SHARP GRADIENTS
log k
FIGURE 2. Coherent structures generated by deforming a random velocity field. Distortion of
a typical eddy or CS as shear is applied to homogeneous turbulence. Note the change in spectra
(Rogallo, 1981; Lee et al., 1987)
(ii) Neighboring structure8
In free shear layers, wakes, and jets the CS are formed in thin layers so that the main effect of
the surrounding flow on the CS must be caused by adjacent or nearly adjacent CS. This is quite
different to boundary layers where each CS is surrounded by other CS (Fig. 3b).
One could characterize these interactions as 1 + 1 + 1 + .... These kinds of interactions can
be idealized by considering the interactions between pairs of vortices. Such studies have yielded
useful conservation conditions (Aref 1983), and insights into whether the vortices go round each
other or merge or pair (e.g., Kiya et al. 1986).
(iii) Many structures or a mean fieldf
In turbulent boundary layers each CS is surrounded by others, and also by small-scale incoherent
motion. In such cases it is no longer profitable to examine each interaction between the structures;
it is more practical to represent the whole velocity field as that of the CS, U, plus a mean velocity
field, _, induced by the vorticity within all the CS, and a random component, at I to represent on
the large scale the random locations of the CS, and on the small scale the incoherent motions (Fig.
3c). In other words, 1 + cc + _l, is equivalent to U + u + u I.
At present we understand little about the mechanics of the interaction between a finite volume
of fluid containing vorticity and the surrounding flow. Most research on the dynamics of turbulence
has been focused on the interaction between small perturbations and various kinds of mean flow,
e.g., Townsend (1976) or Landahl & Mollo-Christiansen (1988). This is perhaps surprising since
Prandtl's (1925) mixing length theory is based on a qualitative model of how "lumps of fluid" --
FKissigkeit ballen -- interact with a shear flow.
318 J. C. R. tfunt
(a) l
(b)
PASSING O
_ PAIRING
TEARING
FIGURE 3. (a) A ring vortex in a uniform flow showing slow shedding of vorticity downstream.
(b) Interaction between a Coherent Structure (CS) and neighboring CS (1+1 ---).
Some general questions about this problem were set out by Hunt (1987):
i) What is the applied force and velocity of a volume V in a non-uniform unsteady flow with velocity
u? For inviscid flow (a good idealization for a short time), analytical expressions have recently
been obtained for spherical or cylindrical volumes moving in shearing or uniformly straining flows.
The initial movements of a lump-or CS- when displaced in a shear flow are largely controlled by
inertial forces, in particular the added mass and lift forces. One finds that transverse displacements
generate streamwise velocities of the lumps and thence Reynolds stresses-the CS analogy of the
small perturbation Reynolds stress found in RDT or second-order calculations (Fig. 3d).
(ii) How does the volume deform as it is accelerated when displaced across a shear flow? Flow
visualization of the eddies or CS in a turbulent boundary layer show that as they are ejected from
the wall form into "mushroom" shapes. These are very similar to the shapes of vortex sheets
surrounding volumes of fluid which are suddenly introduced into jets (Coelho g_ Hunt 1987).
In both cases the vortex sheets roll up into concentrated vortices characteristic of the mushroom
Coherent Structures-Comments 319
+
(d)
i U,cs o y
l o (y) =/
I _/o,,:
/ VELOCITY OF THE CS
FIGURE 3. (c) Interaction between a CS and surrounding CS -- equivalent to the interaction
with a mean flow u and turbulence u e. (d) Movement of a volume of fluid across a shear flow
leading to the generation of Reynolds stress, -u_v _ -- an idealization of the mechanics of a CS in
a shear flow (Hunt 1987).
320 J. C. R. Hunt
structures. These vortices appear to have forms that persist-which of course is one of the necessary
features of CS.
The other approach to analyzing the dynamics of CS is to take a typical form of the vorticity
distribution in a CS and to compute its development in an appropriate shear flow. That has
certainly led to some interesting insights (Moin et al. 1986). But it will be necessary to conduct
many such computations to obtain general concepts from them.
4. Effects of coherent structures on the whole flow
Since coherent structures are regions of vorticity they induce a velocity field both within them
and outside them. Consequently, they affect the mean and the fluctuating components of the ve-
locity field. Because they contain the largest and most energetic scales of motion the CS contribute
a significant proportion of the total energy of the turbulence, and the Reynolds stress. The actual
proportion depends on the precise definition of the CS (cf. Adrian 1987).
To compute the velocity fields affected by the CS it is necessary to account for all the key
dynamical processes that affect the movement and the strength of the CS during its "life." Recent
experiments, and detailed analysis of CS found in direct simulations of turbulent shear flows, are
beginning to make this possible (Hussain et al. 1987).
Three examples of dynamical processes that need to be considered are: (i) amplification of the
small-scale vorticity of the surrounding flow at the stagnation and straining regions around the CS,
caused by the relative motion between the CS and the flow. This is similar to the amplification
around bluff bodies and cross jets in turbulent flows (Hussain 1986; Hunt 1973) (Fig. 4); (ii)
distortion of the vorticity and amplification of the "incoherent" turbulence within the CS, by the
internal straining motions and the shear layers on the boundary of the CS. These are particularly
important for determining the growth of the CS, i.e., controlling the spreading of the vorticity of
the CS. They also largely control (iii) the detrainment of vorticity into the "wake" of the CS. This
process has been studied in detail for the simplest type of CS, namely the vortex ring, and the
somewhat more complex jet in a cross flow (Maxworthy,1987; Coelho & Hunt 1987). For those
CS it was fond that at high enough Reynolds number the shed vorticity produces an unstable
velocity distribution similar to that of a bluff body. This is another mechanism whereby one
CS can produce another CS and small-scale turbulence in the surrounding flow. This essentially
depends on a diffusion process initially, whereas in the free shear layer the secondary structures
were produced by inviscid processes.
5. Concluding Remarks
These suggestions for topics of investigation into the dynamics of CS in fact follow many of
the lines of study that are being followed at the University of Houston by Prof. Hussain and
at the Center for Turbulence Research at Stanford/NASA Ames. There is perhaps one area of
significant difference in that at this stage it is not so clear to me how the detailed mechanics of
the CS will emerge from interpreting conditionally sampled velocity measurements in terms of
conditional joint probability density distributions. This approach may lead to the prediction of
certain useful statistics of the flow, but probably not to the dynamics of individual structures.
However, the actual conditionally sampled measurements of velocity will help in elucidating the
velocity/vorticity distributions in the CS, which is necessary for the dynamical studies outlined
here.
I am grateful to Fazle Hussain for organizing the seminar on coherent structures at the CTR
summer school, and for many interesting discussions on coherent structures, which this paper
reflects. I am also grateful to the organizers of the summer school, in particular Parviz Moin and
John Kim, who gave me some interesting other perspectives on coherent structures and how to
analyze them.
REFERENCES
ADRIAN, R. J. 1987 Lecture at CTR Summer School.
AREF, H. 1983 Ann. Rev. Fluid Mech. 15, 345- 389.
INCIDENT
TURBULENCE
Coherent Structures-Comments
(I) STRETCHING
OF VORTIClTY
OF INCIDENT
TURBULENCE
321
(II) SHED
VORTIClTY
FROM CS
FIGURE 4. Effect of a CS on the surrounding flow in high Reynolds number turbulence. (I)
Amplification of turbulence in the surrounding flow. (II) Shedding vorticity which leads to further
turbulence in the surroundings.
BERNAL, L. P. & ROSHKO, A. 1986 J. Fluid Mech. 170, 499-525.
CHAMPAGNE, F H., HARRIS, V. G., _ CORRSIN, S. 1970 J. Fluid Mech. 41, 81-139.
COELHO, S. L. V., & HUNT, J. C. R. 1987 On the initial deformation of strong jets in
cross flows.. J. Fluid Mech.. (Submitted).
GASTER, M., KIT, E., _ WYGNANSKI, I. 1985 J. FLuid Mech. 150, 23-39.
HUNT, J. C. R. 1973 J. FLuid Mech. 61, 625- 706.
HUNT, J. C. R. 1987 Trans. Can. Soc. Mech. Eng. 11, 21-35.
HUSSAIN, A. M. F. 1986 J. Fluid Mech. 173, 303-356.
HUSSAIN, A. M. F., JEONG, J., _ KIM, J. 1987 Proc., CTR Summer School.
KIYA, M., OHYAMA, M., & HUNT, J. C. R. 1986 J. Fluid Mech. 172, 1-15.
LANDAHL, M., &= MOLLO-CHRISTIANSEN, E. 1986 Turbulence and Random Processes
in Fluid Mechanics. Cambridge Univ. Press.
MAXWORTHY, T. 1977 J. Fluid Mech. 81, 465-495.
LEE, M. J., KIM, J., _: MOIN, P. 1987 Proc., 6th Turbulent Shear Flows Conf., Toulouse,
pp. 22-6-1 - 22- 6-6.
MOIN, P., LEONARD, A. & KIM, J. 1986 Phys. Fluids. 29. 955-963.
PRANDTL, L. 1925 . it Z.A.M.M. 5, 136-139.
ROGALLO, R. 1981 NASA T-M 81315.
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307-337.
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SMITH, F. T. 1987 Advances in Turbulence, (Ed., Comte-Bellot & MMthieu), Springer Verlag,
pp. 27-36.
TOWNSEND_ A. A. 1970 J. Fluid Mech. 41, 13-46.
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