Publications
12-21-2016
Wave Motion Induced By Turbulent Shear Flows Over Growing Wave Motion Induced By Turbulent Shear Flows Over Growing
Stokes Waves Stokes Waves
Shahrdad Sajjadi
Embry-Riddle Aeronautical University
, sajja8b5@erau.edu
Serena Robertson
Embry-Riddle Aeronautical University
Rebecca Harvey
Embry-Riddle Aeronautical University
Mary Brown
Embry-Riddle Aeronautical University
Follow this and additional works at: https://commons.erau.edu/publication
Part of the Fluid Dynamics Commons, and the Ocean Engineering Commons
Scholarly Commons Citation Scholarly Commons Citation
Sajjadi, S., Robertson, S., Harvey, R., & Brown, M. (2016). Wave Motion Induced By Turbulent Shear Flows
Over Growing Stokes Waves.
Journal of Ocean Engineering and Marine Energy, 3
(2). https://doi.org/
10.1007/s40722-016-0073-3
This Article is brought to you for free and open access by Scholarly Commons. It has been accepted for inclusion in
Publications by an authorized administrator of Scholarly Commons. For more information, please contact
commons@erau.edu.
Publications
12-21-2016
Wave Motion Induced By Turbulent Shear Flows
Over Growing Stokes Waves
Shahrdad Sajjadi
Embry-Riddle Aeronautical University, sajja8b5@erau.edu
Serena Robertson
Embry-Riddle Aeronautical University
Rebecca Harvey
Embry-Riddle Aeronautical University
Mary Brown
Embry-Riddle Aeronautical University
Follow this and additional works at:
h=ps://commons.erau.edu/publication
Part of the Fluid Dynamics Commons, and the Ocean Engineering Commons
<is Article is brought to you for free and open access by Scholarly Commons. It has been accepted for inclusion in Publications by an authorized
administrator of Scholarly Commons. For more information, please contact commons@erau.edu.
Scholarly Commons Citation
Sajjadi, S., Robertson, S., Harvey, R., & Brown, M. (2016). Wave Motion Induced By Turbulent Shear Flows Over Growing Stokes
Waves. Journal of Ocean Engineering and Marine Energy, 3(2).
h=ps://doi.org/10.1007/s40722-016-0073-3
Journal of Ocean Engineering and Marine Energy manuscript No.
(will be inserted by the editor)
Wave Motion Induced By Turbulent Shear Flows Over
Growing Stokes Waves
Shahrdad G. Sajjadi, Sarena Robertson,
Rebecca Harvey, Mary Brown
Received: date / Accepted: date
Abstract The recent analytical of multi-layer analyses proposed by Sajjadi,
Hunt and Drullion (2014) (SHD14 ther ei n ) is solved numerically for atmo-
spheric turbulent s he ar flows blowing over growing (or unsteady) Stokes (bi-
modal) water waves, of low to moderate ste ep ne ss. For unsteady surface waves
the ampl i t ud e a(t) ∝ e
kc
i
t
, where kc
i
is the wave growth factor, k is the
wavenumber, and c
i
is the complex part of the wave phase speed, and thus the
waves begin to grow as more energy is transferred to them by the wind. This
will then display the critical height to a point where the thickness of the inner
layer kℓ
i
become comparable to the critical height kz
c
, where the mean wi nd
shear velocity U(z) equals the real part of the wave speed c
r
. It is demon-
strated that as the wave steepens further the inner layer exceeds the critical
layer and beneath the cat’s-eye there is a strong reverse flow which will then
affect the surface drag, but at the surface the flow adjusts itself to the orbital
velocity of the wave. We show that in the li mi t as c
r
/U
∗
is very small, namely
slow moving waves (i.e. for waves traveling with a speed c
r
which is much less
than the friction velocity U
∗
), the energy-transfer rate to th e waves, β (being
proportional to momentum flux from wind to waves), computed here using
an eddy-viscosity model, agrees with the asymptotic steady state analysis of
Belcher and Hunt (1993) and the earlier model of Townsend (1980). The non-
separated sheltering flow determi ne s the drag and the ener gy -t r ans fe r and not
the weak critical shear layer within the inner sh ear layer. Computations for
the cases when the waves are trave l i ng faster (i.e. when c
r
> U
∗
) and growing
S.G. Sajjadi
Dept. of Mathematics., E mbry-Riddle A eron autic al Univ., FL 32114, USA,
and Trinity College, Uni versity of Cambridge, UK
E-mail: sajja8b5@erau.edu
S. Robertson, R. Harvey and M. Brown
Dept. of Mathematics., E mbry-Riddle A eron autic al Univ., FL 32114, USA
arXiv:1704.01963v1 [physics.flu-dyn] 5 Apr 2017
2 Shahrdad G. Sajjadi, Sarena Robertson, Rebecca Harvey, Mary Brown
significantly (i.e. when 0 < c
i
/U
∗
) show a critical shear layer forms outside
the inner surface shear layer for steeper waves. Analysis, following Miles (1957.
1993) and SHD14, sh ows that the critical layer produces a significant but not
the dominant effect; a weak lee-side jet is formed by the inertial dynamics
in the critical layer, which adds to the drag produced by the sheltering ef-
fect. The latter begins to decrease when c
r
is significantly exceeds U
∗
, as has
been verified experimentally. Over peaked waves, the inner layer flow on the
lee-side tends to slow and separ at e, which over a growing fast wave deflects
the streamlines and the critical layer upwards on the lee side. This also tends
to increase the drag and the magni tu d e of energy-transfer rate β (from wind
to waves). These complex results, comput ed with relatively simple turbulence
closure model agree broadly with DNS simulations of Sullivan et al. (2000).
Hence, it is proposed, using an earlier study SHD14, that the mechanisms
identified here for wave-induced motion contributes to a larger net growth of
wind driven water waves w he n the waves are non-linear (e.g. bimodal waves)
compared with growth rates for monochromatic waves . This is because in non-
linear waves individual harmonics have stronger positive and weaker negative
growth rates.
1 Introdu cti on
The aim of this paper is to study the flow structure above unsteady Stokes
waves (where the complex wave speed c
i
6= 0) at variou s steep ne ss and for
the range of wave age (c
r
/U
∗
, where c
r
is the wave spee d, U
∗
=
p
τ
w
/ρ
a
is
the air friction velocity τ
w
is the surface shear st r es s, and ρ
a
is the air den-
sity) through numerical integration. We shall investigate the structure of the
wave-induced motion in the vicinity of the critical layer under the influence of
turbulent stresses. We will demonstrate the wave-induced motion, influenced
by the orbital wave velocity, will yield to a phase change around the criti-
cal layer and closed streamlines (cat’s-eye) appear there. Moreover, we shall
demonstrate, as was suggested by in the recent study by SHD14, that when
the wave steepness increases the cat’s-eye elevates from inner layer to the outer
layer. However, SHD14 did not demonstrate the above point explicitly through
neither analytical nor numerical calculations. Hence, the objective her e is to
present numerical results which confirms the SHD14 theory. Moreover, we shall
present results for energy-transfer rate as a function of wave steepness for un-
steady waves, i.e. those waves whose amplitude a vary with time and grow
according to
a(t) = a
0
e
kc
i
t
where a
0
is the initial wave amplitude, kc
i
is the wave growth rate, k is the
wavenumber, and following SHD14, the value of wave complex wave speed is
kept fixed such that |c
i
|/c
r
= 0.1. In this paper, we shall present results for
growing Stokes waves whose steepnesses are in the range 0.01 ≤ ak ≤ 0.1 and
for small to sli ghtly moderate wave ages, namely c
r
/U
∗
≤ 11.5. Also presented
Turbulent Shear Flows Over Growing Stokes Waves 3
here are results for energy-transfer rate from wind to waves as a function of
wave steepness ak and wave age c
r
/U
∗
. We will also investigat e the location of
critical height kz
c
compared with the inner layer height kℓ
i
for various wave
steepnesses at a fix value of the wave age in order to be able to determine
the structure of cat’s-eye formation above the sur f ace as a function of wave
steepness and age.
At the first sight there seems to be some inconsistency between Miles’
(1993) model who constructe d a similar theory to that presented here. How-
ever, we emphasize that the r e are subtle differences between the two models.
Miles’ (1993) theory is briefly reviewed in the next subsection below, but here
we emphasize that Miles considered a steady wave, he neglected the diffu-
sion term in his vorticity-transport equation (so that his formulation becomes
amenable to analytical analysis), and used Charnock similarity argument for
calculating the roughness length. It is well known t h at the evaluation of rough-
ness length z
0
using Charno ck (1955) theory tends to become infinite in the
limit as c
r
/U
∗
↓ 0. Thus, in this study we fix our dimensionless roughness
length to be kz
0
= 10
−4
. We also consider waves witch have finite complex
wave speed, thereby avoiding the singulari ty in the perturbation velocity, both
in phase and out of phase wit h the wave, at the the critical point, wher e
U(z) = c
r
. Hence, we avoid the unphysical singular flow structure resulting
from Miles’ (1957) inviscid theory (see Belcher and Hunt 1998 and SHD14).
These modifications avoids any inconsistency between the present model an d
that of Mile s (1993). Interestingly enough the present theory agre es relatively
well with th at of Belcher and Hunt (1993) in so much as the prediction of the
energy-transfer rate when c
r
/U
∗
is small.
As is well known by oceanographers, the accurate knowledge of air-sea
momentum flux is crucial in dynamic s of ocean-atmosphere modeling. At the
air-water interface the total momentum flux from wind to waves affects vis-
cous stress and wave-induced stress. The wave-induced str e ss is mainly due to
the pressure force actin g on a sloping interface and t r ans fe rs momentum into
surface waves (this is the main focus of the present contribution). The momen-
tum fl ux from wind to the each Fourier component of the wave is determined
by the wave growth rate. This is usually defined as the momentum-transfer
rate from wind to waves per unit wave momentum, and is given by
σ ≡ (kc
E)
−1
(∂E/∂t) = sβ(U
1
/c)
2
where E is the wave energy per unit area, the overbar signifies an average over
the horizontal coordinate, x, s = ρ
a
/ρ
w
≪ 1 is the air-water density ratio,
U
1
= U
∗
/κ is a reference velocity, and κ = 0.41 in von K´arm´an constant. In
the above expression β is the energy-transfer rate from wind to wave which
comprises of a component due the critical layer, β
c
(given by equation 6 be low),
at the elevation z = z
c
above the surface wave where U(z) = c
r
(see figure 1),
and a component, β
T
due to the turb ul ent shear flow blowing over the wave.
Here we shall assume the flow over the surface wave is the positive x- di r ec t ion .