12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

FOURIER-AVERAGED NAVIER-STOKES ANALYSIS OF PERIODICWAKES: A NEW TECHNIQUE

Benjamin R.S. Freeman1 , Robert J. Martinuzzi2 , Arman Hemmati3
1,3 Dept. of Mechanical Engineering, University of Alberta, Edmonton, AB, Canada T6G 2R3
2 Dept. of Mechanical and Manuf. Engineering, University of Calgary, Calgary, AB, Canada T2N 1N4
1 brfreema@ualberta.ca, 2 rmartinu@ucalgary.ca, 3 arman.hemmati@ualberta.ca

ABSTRACT

from different flow structures over multiple timescales, particularly regarding interaction between these structures. Several flow assessment techniques have been introduced over the
past decade, which connect mathematical observations withphysical processes in such flows. Two notable methods are theSpectral Proper Orthogonal Decomposition (SPOD) and Bispectral Mode Decomposition (BMD). SPOD was coined by
Picard & Delville (2000) to refer to a specific method of Lumley (Lumley, 1970). This method involves the determinationof modes that vary in both space and time, granting it greater
flexibility than classic POD (Towne et al., 2018). Recently
Towne et al. (2018) formally established a connection between
SPOD and resolvent analysis, in that the two will be identicalfor white-noise (i.e. uncorrelated in space and time) forcing of
the linearized Navier-Stokes equations used in resolvent analysis. This establishes a strong connection between the findingsof SPOD and the linearized equations, which in turns suggeststhat SPOD modes have physical correspondence to important
flow mechanisms. However, the connection relies on a whitenoise assumption and the use of linearized equations in resolvent analysis, which clouds the overall scope of application
when investigating interactions between these structures using
SPOD.
The BMD suggested by Schmidt (2020) uses statisticalmethods to find optimal representations of Fourier modes. In
this case, ‘optimal’ modes (q̂( f )) from a quantity of interest(q(t)) are those that maximize triadic interactions in the flow,
as defined through

Periodic and quasi-periodic fluid motion is observed inengineering and flow applications, which involve a range oftime and length scales. These coherent and regularly recurring structures profoundly affect momentum transport and turbulence energy budget. Hence, understanding the underlying
mechanisms and flow dynamics associated with these structures are important for characterizing the shear flow phenomena. In particular, characterizing these periodic processes isimportant for guiding the design and control of fluid systems.
Here, a new mathematical technique is proposed to assess andquantify the interactions of periodically occurring structures inan unsteady turbulent flow, hereon referred to as the “Fourieraveraged Navier-Stokes” or FANS. In this method, Fourier decomposition in time domain is applied to the Navier-Stokes
equations and particular attention is given to the resulting nonlinear momentum flux terms. Examining the contributions ofeach decomposed momentum component provided physicalinsights into the interactions between structures across multiple time-scales. A general overview of the technique is provided with a summary derivation of the method. Illustrative
case studies highlight the application and benefits of FANS toperiodic flows.

INTRODUCTION
Several techniques have been developed with the aim todeepen our understanding of unsteady turbulent wakes exhibiting strong temporal periodicity. These flows are important tostudy due to their prevalence and complicated physics. Methods such as reduced-order models (ROM) or stability analysis have been widely and successfully used to investigate suchflows. The conventional phase-averaging procedure (Eriksen& Krogstad, 2017) is limited to a single dominant frequency,
and thus it is not suitable, by construction, to analyze complex interactions of different modes of motion. The many
variants of Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD) aim to find optimal reconstructions of the flow kinematics. Thus, they are useful in
examining the flow in a model-free context (Chen et al., 2012;
Noack et al., 2016). However, POD can fail to produce coherent results for flows with multiple important frequencies.
These methods also do not provide a simple approach to compare important interactions between modes. Meanwhile, linear stability analysis tackles the onset and continuation of perturbations through a linearized analysis of the Navier-Stokes
equations, which prevents it from describing nonlinear processes (Beneddine, 2017). Hence, none of these analyses
are capable of providing a description of the forces arisingb( f1 , f2 ) =ZVq̂( f1 )q̂( f2 )q̂( f1 + f2 )dV.

Here, BMD attempts to find correlations between the “input”
of a triadic interaction, that is q( f1 )q( f2 ), and the resulting“output”, q( f1 + f2 ). As a result, BMD may be effective atfinding modes and regions of the flow that are highly involvedin nonlinear interactions, but ignores the specific transfers ofmomentum that are involved in these processes.
To address this gap in analyzing nonlinear interactions
using existing methods, a new formulation based on Fourier
analysis of the momentum fluxes in a flow is proposed– theFourier-Averaged Navier-Stokes, or FANS equations. Applying a Fourier decomposition in time to the momentum equations themselves, and not just the flow field, unveils propertiesof the momentum fluxes at different timescales. Thus, criticalinformation about the flow dynamics can be obtained. This
leads to valuable insights into the periodic features of the unsteady flow. Unlike other frequency-domain methods, FANS
specifically attempts to provide a rigorous link between the1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Note that all summations are indexed by m ∈ (−∞, ∞), omitted
here for cleanliness. Of the terms in the summation, only thekth term survives the integration due to orthogonality. Thus:

flow physics and the governing equations. This methodology
has several similarities to the spatially-spectral methods thathave enjoyed great success in turbulence and instability analysis (e.g. Barkley & Henderson (1996), chapter 10 of Durbin& Pettersson (2011)), however here the specific focus is on atemporal decomposition. The primary hypothesis for FANS isthat many unsteady wakes contain at least one dominant frequency. Examples of such flows are the wakes characterized
by Khalid et al. (2020), Bai & Alam (2018), and Beneddine(2017). The FANS formulations are likewise capable of handling flows with multiple dominant frequencies, where complicated dynamics may result due to interaction between structures. By virtue of their periodic characteristics, the dynamics
of these structures and their resulting effect on the momentum balance of the entire flow would be naturally described bylooking at the momentum budget at that frequency.
To introduce FANS and its application, a brief derivationand two case studies are considered. First, FANS formulations
are introduced, along with methods to carry out calculations of
the key quantities. Then, two case studies on simple periodic
flows are considered. By analysing the wake behind a squarecylinder and an oscillating foil, connections between FANScalculated quantities and physical phenomena are established.
These highlight how FANS can elucidate mechanisms behindcomplicated flow physics in certain flow regimes.

Z T
0

Z T
0

∞

Z T
0

m=−∞

∑ p̂m e jωmt ,

Z T

e
0

e

(
0
dt =
T

k ̸= m
k=m

0

=

(1)

0

∑ ûm · ∇ûn e− jω(n+m−k)t dt

n

(6)

Ĉk

−∇ p̂k + ν∇2 ûk − ∑ ûn · ∇ûk−n| {z } | {z } n̸=
0,k
D̂k
|
{z
}
P̂k

(2)

(7)

χ̂ k

Here, terms corresponding to convection (Cm , χ m ) and pressure (Pm ) are the most important to analyze, as they are representative of the triadic interactions that are highly importantin turbulent flows, e.g., Schmidt (2020). However, this derivation is valid for a continuous, periodic flow field. Since almost
all fluid dynamics research is built on analyzing snapshots of
discrete data, either through simulations or experiment, it isimportant to translate these results into a discrete domain. Supposing that there is a collection of N evenly-spaced snapshots{un } of discrete flow field data, the discrete Fourier transformof this data would be given by

(3)

N−1

ûk = ∑ um e− j2πkm/N

(8)

m=0


−∇ ∑ p̂m e jω(m−k)t + ν∇2 ∑ ûm e jω(m−k)t dt
Z Tn

e− jωkt u · ∇udt = ∑ ûk−n · ∇ûn

T̂k



e− jωkt −∇ ∑ p̂m e jωmt + ν∇2 ∑ ûm e jωmt dt

=

m

Z T

j2πk f ûk + U · ∇ûk + ûk · ∇U ={z}
| {z } |Z T0e− jωkt ∑ ûm e jωmt ·∇ ∑ ûn e jωnt dt

where terms corresponding to n = 0 and k are extracted tohighlight the extreme importance of the mean flow U, and tobring the FANS notation in line with other methods, such as
Reynolds-Averaged-Navier-Stokes (RANS) and linear stability analysis. Collecting these terms together returns a momentum balance corresponding to a mode ûm in terms of itself andthe other Fourier modes:

0

=

0

n̸=0,kThe process is applied term by term. For the pressure anddiffusion fluxes, this process yields:
Z Te− jωkt −∇p + ν∇2 u dt
Z T

Z T

= U · ∇ûk + ûk · ∇U + ∑ ûk−n · ∇ûn ,In order to isolate the momentum balance at a particular
timescale, Eq. 2 is subjected to a Fourier decomposition by
applying an inner product with a test function e− jωkt , exploiting the orthogonality property:− jωkt jωmt

∑ jωmûm e jω(m−k)t dt = jωkûkTerms in the double summation survive only when m = k − n,reducing it to a single summation,

where û, p̂ are the Fourier coefficients corresponding to velocity and pressure, respectively. Assuming incompressible Newtonian flow, the flow field obeys the Navier-Stokes equations,∂t u + u · ∇u = −∇p + ν∇ u.

0

0 m,n

m=−∞

2

e− jωkt u·∇udt =

=

∞

p=

Z T

(5)
The convective term is somewhat more complicated due to its
nonlinear dependence on u.

In order to arrive at an expression representing the balanceof momentum at each timescale, a periodic flow field (consisting of velocity u, and pressure p) with period T = 2π/ω isassumed. This allows us to write the flow field as a Fourier
series in time:Z T

e− jωkt ∂t ∑ ûm e jωmt dt =0

∑ ûm e jωmt ,(4)

The process is similar for the acceleration term:

FORMULATIONu=e− jωkt −∇p + ν∇2 u dt = −∇ p̂k + ν∇2 ûk

Since vectors of dynamics am = e− j2πkm/N are orthogonal under the inner product of am · an = δmn , this process may be repeated for the discrete analog of the Navier-Stokes equations:∑(−∇ p̂m + ν∇2 ûm )e jω(m−k)t dt
2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022∆un
+C[un , un ] = −G[pn ] + νL[un ]∆t(9)Wake splits downstream2j2πk kû +C[U, ûk ] +C[ûk , U] =N
− G[ p̂k ] + νL[ûk ] − ∑ C[ûk−n , ûn ]yusing discrete convection (C), gradient (G) and Laplacian (L)operators, as calculated through whichever method is most appropriate to the data (finite differences, volumes, elements,etc.). Fortuitously, orthogonality of the vectors am may be used
to find the Fourier modes. This results in a discrete formulation that is similar to the continuous FANS equations:0−2

Vortex production
0

5

10

2.500
1.875
1.250
0.625
0.000
−0.625−1.250−1.875−2.500x

Figure 1. Contours of vorticity trailing a square cylinder.
Vortices are produced and shed immediately behind the cylinder, after which time they depart from the centreline at a constant speed.

(10)n̸=0,kFANS formulations are attractive for analyzing unsteady (especially periodic) flows for multiple reasons. For instance,since the regular Navier-Stokes equations may be reconstructed from the decomposed equations (Eq. 7) through a
Fourier series, it can be said that all physical interactions represented in the flow are contained in the FANS equations. Another attractive aspect is ease of implementation. FANS-based
analysis depends on a discrete Fourier transform of snapshots(usually an FFT) and discrete derivative calculations, both ofwhich are widely available in software packages commonlyused for fluid dynamics research. The following sections will
provide an example of this analysis on familiar periodic flows
to highlight the application of the method.

flow. Second order accurate spatial and temporal discretization methods are used for solving the convective, diffusive andadvective terms. The Pressure-Implicit with Splitting of Operators (PISO) method is used for coupling velocity and pressure
fields. Specific details of each simulation are detailed separately in the coming sections.
Square Cylinder: cylinders at low Re are common case studies for investigating periodic flows due to their regularity andsimplicity, e.g., Schmidt (2020). For this investigation, asquare cylinder oriented normal to the streamwise direction at
Re = 100 was selected since the resulting flow exhibits purely
periodic vortex shedding in a street extending from behind thecylinder, and it is below the threshold for 2D-3D transitions.
The incompressible, laminar solver icoFOAM is selected forthe simulation. A simple structured mesh with local refinement
around the cylinder is used, following the mesh description in
Bai & Alam (2018).
Oscillating Foil: the second case study is a teardrop foil undergoing combined sinusoidal pitching and heaving motion.
The reduced frequency of the motion ( f ∗ = f c/U∞ ) is 0.4,
while the pitching amplitude is 8◦ and heaving amplitude is
h/c = 0.25. At Re = 1000, this flow is characterized by regular
shedding of both leading and trailing edge vortices due to the
combined motion. An Overset Grid Assembly in OpenFOAM
was used for the simulations. More computational details aredescribed in Verma et al. (2022). Five cycles are considered
for data analyses after reaching statistical convergence.

Existing methods in literature
The above formulation (FANS) bears similarities to existing methods based on a temporal Fourier decomposition thathave been discussed in literature. For example, similar formulations have been used as a simulation tool in the past, namely
the Spectral Time Discretization (STD) by Carte et al. (1995),
harmonic balance (HB) technique by Hall et al. (2002), andSelf-Consistent method (SCM) by Mantič-Lugo et al. (2015).
The HB and STD are specifically high-accuracy, fast simulation methods for periodic flows, however notably the HB
is used for compressible flows with externally forced periodicity, primarily turbomachinery. Meanwhile, SCM can beviewed as an extension of linear stability analysis used to calculate saturation dynamics and sensitivity maps, albeit at significant effort. In this way, the FANS formulations may not
constitute a new mathematical approach, however, their application as a tool to obtain physics-based insights using existing data is novel. The previous applications strictly focus
on solving for an unknown flow field, whereas the current
work contends that the resulting formulations are also usefulin post-processing flow field data. A key advantage of using
the formulations for postprocessing opens up insights from alarger variety of flows and data acquisition methods. In particular, simulating flows with a large number of harmonics
or broadband spectrum may be difficult or infeasible with atime-spectral method. However, investigating physics involving periodically-occuring structures may still be worthwhile.
Using the formulations for postprocessing allows these investigations to occur.

ANALYSIS
Square Cylinder: Results of the FANS decomposition of thewake behind a square cylinder are shown in Figures 2 - 4.
Fourier decomposition shows the presence of two strong frequencies, corresponding to the primary vortex shedding frequency (Mode 1) and its harmonic (Mode 2). Figure 2a shows
the spatial distribution of Mode 1, from which the vortex street
can be clearly identified as it splits and moves downstream.
Figure 2b shows the corresponding harmonic structure that isstrongest in the near-wake, where there are interactions between two vortex streets. Using the momentum budgets in thewake region, it is shown that the main frequency represents a
self-sustaining structure, while the harmonic arises naturally
due to the convection induced by the velocity fluctuations.
Effect of the nonlinear forcing term (χu1 ) is not criticalin the flow as suggested by the results in Figure 4a, where
the wake is largely dominated by the pressure, convective, andtemporal terms. Since χu1 is the only connection to other fluctuating modes, and the other terms are linear in û1 , this indicates that the primary frequency exists as a self-sustaining

CASE STUDIESThe main case studies completed include a twodimensional direct numerical simulation (DNS) of an oscillating foil with combined pitching and heaving motion at Re =1000 and periodic vortex shedding behind a square cylinder
at Re = 100. OpenFOAM is used as the main computational
platform, although different methods are used for solving the3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

0
−2
0

5

10
x+

2
y+

y+

2

0.24
0.18
0.12
0.06
0.00
−0.06
−0.12
−0.18
−0.24

0
−2
0

5

10
x+

(a)

(b)

Contours of real parts of ûm for streamwise velocity, u, for (a) Mode 1 and (b) Mode 2.

Figure 2.
magnitude, shifted in space by half a vortex length.

Imaginary parts are of similar

is the only other significant mode along with the mean flow:

Pv1

0.27800.1853

2
y+

0.064
0.048
0.032
0.016
0.000
−0.016−0.032−0.048−0.064

χu2 =0.0927∑ ûnk ∂k û2−n ≈ û1k ∂k û1(11)n̸=0,20.00000−0.0927−2This indicates how FANS analyses highlight the origin of triadic interactions of the fundamental frequency, which leads
to the generation of harmonics. This finding corroborates
those of other analyses on cylinder wakes, e.g., BMD and selfconsistent method (Schmidt, 2020; Mantič-Lugo et al., 2015).
These conclusions come directly out of the analysis with little
computational effort.

−0.18530510x−0.2780+

Figure 3. Pressure flux Pv1 corresponding to mode v̂1 , showing strong pressure fluctuations in the region of vortex production.

Oscillating Foil: In order to deepen the discussion of FANS
and its representation of unsteady wakes, a periodically pitching and heaving foil is analyzed. This oscillatory wake is characterized by pairs of vortices shed during each oscillation cycle from the leading and trailing edges on both top and bottomof the foil, as depicted in Figure 5. The contours of vorticity show strong shearing of the flow between the leading andtrailing edge vortices as they develop. These connections linkthese structures strongly in this region. To isolate the fluid motion around the foil for FANS analyses, a reference frame fixed
on the foil is considered. Note that the original formulations(Eqs. 7, 10) will then contain added terms due to the Coriolis and centripetal forces. The contribution of these terms at aharmonic k is denoted as τ k .
The first mode of the streamwise velocity, corresponding
to the foil oscillation frequency, is shown in Figure 6. Effectof the alternated shedding of leading and trailing edge vortices
from the foil edges can be seen in their respective locations.
Note the meandering of the wake due to the change in reference frame.
The momentum flux terms (P̂1 , Ĉ1 ) dominate the momentum transport phenomenon at the foil oscillatory frequency in
Figures 7 and 8. This is due to flow stagnation in the narrow nose region at the leading edge, the location of whichmoves during the cycle by the foil rotation. This oscillation
contributes to shedding of the leading edge vortices.
The motion of leading edge vortices along the foil body is
an important factor leading to changes in the thrust generated
by an oscillating foil (Verma et al., 2022). Thus, it would beinteresting to see if such a motion can be isolated by flux termsin FANS formulations.
As seen in Figure 9, the effect of this motion is more apparent at higher harmonics due to the interaction between the
foil and leading edge vortices. This vortex motion drives theinter-harmonic triadic coupling along the foil boundary. In

process. This observation about the vortex street is in agreement with the observations of linear stability analysis reported
for example by Noack et al. (2003). The pressure flux contourfor v̂1 , suggests the origin of the split wake. This separationof vortices away from the centreline can be attributed to strongpressure fluctuations in the near-wake, as shown in Figure 3.
Pressure fluctuations are generated by the combined effectof distortions of the mean flow and primary velocity fluctuations in the vortex formation region. Meanwhile, the effect ofthese pressure fluctuations on the streamwise velocity is highlighted by Figure 4a. Note that the streamwise fluid acceleration (Tu1 ) and pressure gradient terms (Pu1 ) at the fundamental
frequency are in phase as the vortices convect downstream,
driven by the mean flow convection (Cu1 ). This indicates thatthe fluctuations of the pressure deficit region serve to dampen
the streamwise fluctuations, which slows and distorts vortices.
This corresponds to the intuitive idea that pressure deficit behind the square cylinder entrains freestream fluid into the wakeregion, slowing structures that depart in the streamwise direction, reducing the convection rate of the shed vortices. Here,FANS intuitively represents this physical flow process.
FANS analyses highlight the origins of the harmonic as
well. Figure 4b shows the importance of the forcing term χu2
in perpetuating the harmonic. The magnitude of this momentum flux is comparable to the mean-flow convective (Cu2 ), pressure (Pu2 ), and temporal (Tu2 ) terms, indicating that the ‘forcing’ generated by χu2 continues to create a harmonic even after
the wake has split and vortices are no longer interacting. Basedon an order-of-magnitude analysis, χu2 is generated entirely by
the mechanism associated with the primary frequency, since it

412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

0.10
0.1

Tu1
1
Du
1
Pu
1
Cu
1
χu

0.0
−0.1
0

5

10
x

Tu2

0.05

2
Du

Pu2

0.00

2
Cu

χ2u

−0.05

15

0

5

+

x

(a)
Figure 4.

10

15

+

(b)

Real parts of momentum budgets of ûm behind the cylinder along y+ = 1 for (a) Mode 1 and (b) Mode 2.

Vortices fromprevious cycle

Shed TEV
Shed LEV
Developing LEVDeveloping TEV
Figure 5.

Vortex shedding pattern of foil undergoing simultaneous heaving and pitchingĈ, û1

û1

0.034

0.05

1.65580.05

0.8279y

0.017

0.00000.00-0.0000.00-0.8279-0.017
−0.05

−0.05

-0.034
0.00

0.05

0.10

0.00

0.15

0.05

0.10

0.15

-1.6558

x

Figure 8.
1

Figure 6. Real part of the first mode of the streamwise velocity fluctuations

P̂ , ûturn, this convective coupling generates strong pressure gradients on the foil (Figure 10), which have a pronounced effecton thrust generations. The appearance of these structures in
FANS analyses shows how it may be connected to important
performance characteristics, such as thrust generation of an oscillating foil and the utilized power. Finally, Figure 9 shows asignificant degree of triadic interaction just past the trailingedge through χ̂ 3 ≈ ∇ · (û1 û2 ). This can be inferred due to thehigh saturation of contours in the wake behind the foil. As thecolor scheme is referenced to the maximum momentum flux inthis region, dark contours are indicative of locations where theinter-harmonic coupling is dominant. These convective couplings suggestive of the shearing between the shed trailing andleading edge vortices, earlier seen in the developing vortex pat-11.65580.05

0.82790.00000.00-0.8279−0.05
0.00

Figure 7.
mode 1

0.05

0.10

0.15

Streamwise convective flux corresponding to mode

-1.6558

Streamwise pressure gradient corresponding to5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

χ̂, û3

0.0919utilized in a fluid dynamics context. The simplicity and connection to physical phenomena presented by FANS formulations make it an attractive technique for analyzing periodic andquasi-periodic flows.

0.0000REFERENCES

0.18390.05

0.00

Bai, H. & Alam, Md. M. 2018 Dependence of square cylinder
wake on reynolds number. Physics of Fluids 30 (1), 015102.
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Fluid Mech. 322, 215–241.
Beneddine, S. 2017 Characterization of unsteady flow behavior by linear stability analysis. Theses, Univ. Paris-Saclay.
Carte, G., Dušek, J. & Fraunié, P. 1995 A spectral time discretization for flows with dominant periodicity. Journal ofComputational Physics 120 (2), 171–183.
Chen, K. K., Tu, J. H. & Rowley, C. W. 2012 Variants ofdynamic mode decomposition: Boundary condition, koopman, and fourier analyses. Journal of Nonlinear Science 22,887–915.
Durbin, P. A. & Pettersson, R. B. A. 2011 Statistical theory
and modeling for turbulent flows, 2nd edn. Wiley.
Eriksen, Pål Egil & Krogstad, Per-Åge 2017 An experimental study of the wake of a model wind turbine using phaseaveraging. International Journal of Heat and Fluid Flow67, 52–62, symposium on Experiments and Simulations in
Fluid Dynamics Research.
Hall, K. C., Thomas, J. P. & Clark, W. S. 2002 Computationof unsteady nonlinear flows in cascades using a harmonicbalance technique. AIAA J. 40 (5), 879–886.
Khalid, M. S. U., Wang, J., Akhtar, I., Dong, H. & Liu, M.
2020 Modal decompositions of the kinematics of crevalle
jack and the fluid–caudal fin interaction. Bioinspiration andBiomimetics 16 (1), 016018.
Lumley, J.L. 1970 Stochastic Tools in Turbulence, 1st edn.
Mantič-Lugo, V., Arratia, C. & Gallaire, F. 2015 A selfconsistent model for the saturation dynamics of the vortex shedding around the mean flow in the unstable cylinder
wake. Physics of Fluids 27 (7), 074103.
Noack, Bernd R., Afanasiev, Konstantin, Morzynksi, Marek,
Tadmor, Gilead & Thiele, Frank 2003 A hierarchy of lowdimensional models for the transient and post-transientcylinder wake. Journal of Fluid Mechanics 497, 335–363.
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2016 Recursive dynamic mode decomposition of transientand post-transient wake flows. Journal of Fluid Mechanics
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subsonic round jet. International Journal of Heat and Fluid
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Schmidt, O. T. 2020 Bispectral mode decomposition of nonlinear flows. Nonlinear Dynamics 102 (4), 2479–2501.
Towne, A., Schmidt, O. T. & Colonius, T. 2018 Spectral
proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis. Journalof Fluid Mechanics 847, 821–867.
Verma, S., Freeman, B. R. S. & Hemmati, A. 2022 Effects ofreynolds number and average angle of attack on the laminar
scaling of oscillating foils. Phys. Fluids 34 (3), 031905.

-0.0919
−0.05
0.00

0.05

0.10

0.15

-0.1839

Figure 9. Inter-harmonic streamwise momentum flux corresponding to the 3rd harmonic of an oscillating foilP̂ , û30.18390.05

0.09190.00000.00-0.0919
−0.05
0.00

0.05

0.10

0.15

-0.1839

Figure 10. Streamwise pressure gradient corresponding to
the 3rd harmonic of an oscillating foil

tern in Figure 5. As this connection between vortices occurs
only during select portions of the cycle, it is natural that the resulting coupling would be found at a timescale corresponding
to a higher harmonic. This suggests that FANS can highlightcomplicated interactions between vortex structures, as well ascomplementing other forms of wake analyses.

Conclusion
The Fourier-Averaged Navier-Stokes equations represent
a novel method based on Fourier decomposition of a flow togain insights into the dynamics of periodic phenomena. In
FANS operations, the momentum budgets corresponding to individual frequencies are extracted. Here, the FANS formulations are presented and a basic analysis is performed by applying them to the periodic wake of a square cylinder and oscillating foil. These results show that the primary frequency, corresponding to the vortices traveling downstream, are largely unaffected by higher frequency content in the wake. This agrees
with linear stability analysis. Meanwhile, FANS builds uponthese findings by showing that the flow harmonic is generated naturally by a convective process of the large-scale structures, which is an important source of secondary velocity fluctuations in the wake. Likewise, the role of convective- andpressure-based interactions around and behind an oscillating
foil were shown in connection to the motion of the leading andtrailing edge vortices. In this way, FANS-based analysis is aconvenient method to analyze nonlinear interactions by leveraging information from the governing equations. Each of theabove findings indicates that FANS is capable of recreating
the results of other methods, using techniques that are widely6