In fluid flows, fluid-structure interactions and other fluids-related problem, nonlinear dynamics play an important role in determining the development, response or output. Understanding these dynamics is essential for development of analytical models and prediction and control purposes. Higher-order statistical analysis has been shown to be an effective tool that can be applied to identify nonlinear couplings and measure energy transfer rates. These techniques have been applied by our group and others to investigate transition of shear flows, energy cascading in turbulence, oceanographic and geophysical flows, and fluid-structure interactions. The results of these investigations revealed important nonlinear characteristics regarding these problems. In this paper, we review these techniques and explain their usefulness in identification and quantification of nonlinear dynamics.

1.
Beall
J. M.
,
Kim
Y. C.
, and
Powers
E. J.
,
1982
Estimation of Wave Number and Frequency Spectra Using Fixed Probe Pairs
,”
J. Appl. Phys.
, Vol.
53
, pp.
3933
3940
.
2.
Choi
D.
,
Miksad
R., W.
and
Powers
E. J.
,
1985
Application of Digital Cross-Bispectral Analysis Techniques to Model the Nonlinear Response of a Moored Vessel System in Random Seas
,”
J. Sound and Vibration
, Vol.
99
, pp.
309
326
.
3.
Corke, T. C., 1987, “Measurements of resonant phase locking in unstable axisymmetric jets and boundary layers,” Nonlinear Wave Interactions in Fluids R. W. Miksad, T. R. Akylas and T. Herbert, eds., Winter Annual Meeting of ASME/AMD, Vol. 87, pp. 37–65.
4.
Dalzell, J. F., 1975, “The Applicability of the Functional Input-Output Model to Ship Resistence in Waves,” Report SIT-DL-75-1794, Stevens Institute of Technology.
5.
Dalzell, J. F., 1976, “The Application of the Functional Polynomial Mode to the Ship Added Resistance Problem,” Proc. 11th Symposium on Naval Hydrodynamics.
6.
Davila, J. B., 1990, “Wavenumber Resonance in Nonlinear Wave Interactions in the Wake of a Flat Plate,” Ph.D. dissertation, The University of Texas at Austin.
7.
Elgar
S.
, and
Guza
R. T.
,
1985
, “
Observations of bispectra of shoaling surface gravity waves
,”
J. Fluid Mechanics
,
161
, pp.
425
448
.
8.
Elgar
S.
,
Van Atta
C. W.
, and
Gharib
M.
,
1990
, “
Cross-Bispectral Analysis of the Coupling Between a Vibrating Cylinder and its Wake in Low Reynolds Number Flow
,”
J. Fluids and Structures
, Vol.
4
, pp.
59
71
.
9.
Hajj, M. R., 1990. An Experimental Investigation of Nonlinear Interactions Associated With the Subharmonic Growth in a Plane Mixing Layer. Ph.D. dissertation, the University of Texas at Austin.
10.
Hajj
M. R.
,
Miksad
R. W.
, and
Powers
E. J.
,
1992
, “
Subharmonic Growth by Parametric Resonance
,”
J. Fluid Mech.
Vol.
236
, pp.
385
413
.
11.
Hajj
M. R.
,
Miksad
R. W.
, &
Powers
E. J.
,
1993
, “
Fundamental-subharmonic interaction: effect of phase relation
,”
J. Fluid Mech.
Vol.
256
, pp.
403
426
.
12.
Hasselmann, K., Munk, W., and MacDonald, G., 1963, “Bispectra of Ocean Waves,” Times Series Analysis, M. Rosenblatt, ed., Wiley, New York, pp. 125–139.
13.
Hasselman
K.
,
1962
, “
On the Nonlinear Energy Transfer in a Gravity-Wave Spectrum,” Part 1, General Theory
,”
J. Fluid Mech.
,
12
, pp.
481
500
.
14.
Haubrich
R. A.
,
1965
, “
Earth Noises, 5 to 500 Millicycles per Second
,”
J. Geophysical Research
, Vol.
70
, pp.
1415
1427
.
15.
Herbers
T. H. C.
,
Lowe
R. L.
, and
Guza
R. T.
,
1992
, “
Field Observations of Orbital Velocities and Pressure in Weakly Nonlinear Surface Gravity Waves
,”
J. Fluid Mech.
Vol.
245
, pp.
413
435
.
16.
Herring
J. R.
,
1980
, “
Theoretical Calculations on Turbulent Bispectra
,”
J. Fluid Mech.
, Vol.
97
, pp.
205
213
.
17.
Herring
J. R.
, and
Metais
O.
,
1992
, “
Spectral Transfer and Bispectra for Turbulence With Passive Scalars
,”
J. Fluid Mech
, Vol.
235
,
103
122
.
18.
Hinich
M. J.
, &
Clay
C. S.
,
1968
, “
The Application of the Discrete Fourier Transform in the Estimation of Power Spectra, Coherence, and Bispectra of Geophysical Data
,”
Reviews of Geophysics
, Vol.
6
, pp.
347
363
.
19.
Jones, F. L., 1983, “Experimental Study of Nonlinear Wave Interactions and Modulations During Transition of a Symmetric Wake,” Ph.D. dissertation, the University of Texas at Austin.
20.
Jones
F. L.
,
Ritz
Ch. P.
,
Miksad
R. W.
,
Powers
E. J.
, and
Solis
R. S.
,
1988
, “
Measurement of the Local Wavenumber and Frequency Spectrum in a Plane Wake
,”
Exp. Fluids
6
, pp.
365
372
.
21.
Kim
K. I.
, and
Powers
E. J.
,
1988
, “
A digital method of modeling quadratically nonlinear systems with a general random input
,”
IEEE Trans. Acoustics, Speech, Signal Processing
, Vol.
36
, pp.
1758
1769
.
22.
Kim, S. B., Powers, E. J., Miksad, R. W., and Fischer, F. J., 1991, “Quantification of nonlinear energy transfer to sum and difference frequency responses of TLP’s,” Proc. First Intl. Offshore and Polar Engng Conf. Edinburgh, U.K.
23.
Kim
Y. C.
, and
Powers
E. J.
,
1979
, “
Digital Bispectral Analysis and Its Application to Nonlinear Wave Interactions
,”
IEEE Trans. Plasma Sci.
,
PS-7
, pp.
120
131
.
24.
Lii
K. S.
,
Rosenblatt
M.
, and
Van Atta
C.
,
1976
, “
Bispectral Measurements in Turbulence
,”
J. Fluid Mech
, Vol.
77
, pp.
45
62
.
25.
Lumley
J. L.
, and
Takeuchi
K.
,
1976
, “
Application of Central-Limit Theorems to Turbulence and Higher-Order Spectra
,”
J. Fluid Mech.
, Vol.
74
, pp.
433
468
.
26.
Miksad
R. W.
,
Jones
F. L.
,
Powers
E. J.
,
Kim
Y. C.
, and
Khadra
L.
,
1982
, “
Experiments on the Role of Amplitude and Phase Modulations During the Transition to Turbulence
,”
J. Fluid Mech.
, Vol.
123
, pp.
1
29
.
27.
Miksad
R. W.
,
Jones
F. L.
,
Ritz
C. P.
, and
Powers
E. J.
,
1987
, “
The Role of Nonlinear Wave Interactions in Laminar-Turbulent Transition
,”
Arch. Mech
, Vol.
39
,
3
, pp.
177
205
.
28.
Miles
W. H.
,
Pezeshki
C.
, and
Elgar
S.
,
1992
, “
Bispectral analysis of a fluid elastic system; the cantilevered pipe
,”
J. Fluids and Structures
, Vol.
6
, pp.
633
640
.
29.
Monkewitz
P. A.
, and
Huerre
P.
,
1982
, “
The Influence of the Velocity Ratio on the Spatial Stability of Mixing Layers
,”
Phys. Fluids
25
, pp.
1137
1143
.
30.
Nayfeh, A. H., and Balachandra, B., 1995, Applied Nonlinear Dynamics, Wiley, New York.
31.
Nayfeh
A. H.
, and
Bozatli
A. N.
,
1980
, “
Nonlinear Interaction of Two Waves in Boundary-Layer Flows
,”
Phys. Fluids
, Vol.
23
(
3
), pp.
448
458
.
32.
Ritz
Ch. P.
, and
Powers
E. J.
,
1986
, “
Estimation of Nonlinear Transfer Functions for Fully Developed Turbulence
,”
Physica
D
20D
, pp.
320
334
.
33.
Ritz
Ch. P.
,
Powers
E. J.
,
Miksad
R. W.
, and
Solis
R. S.
,
1988
, “
Nonlinear Spectral Dynamics of a Transitioning Flow
,”
Phys. Fluids
, Vol.
31
, pp.
3577
3588
.
34.
Ritz
Ch. P.
,
Powers
E. J.
, and
Bengtson
R. D.
,
1989
, “
Experimental Measurement of Three-Wave Coupling and Energy Cascading
,”
Phys. Fluids
, Vol.
B1
, pp.
153
163
.
35.
Solis, R. S., 1987, “Experiments on the Effects of Mean Flow Unsteadiness on the Transition to turbulence,” Ph.D. dissertation, The University of Texas at Austin.
36.
Townsend, A. A., 1976, The Structure of Turbulent Shear Flow, Cambridge, U.K., Chapter 3.
37.
Whitham, G. B., 1974, Linear and Nonlinear Waves, Wiley, New York.
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