A series of circular cylinder wake simulations were undertaken in order
to investigate the influence of stable stratification on the turbulence
energy spectrum. The simulations target measurements made by Pao in a
tow tank capable of various levels of background stable stratification
via a salinity gradient in the vertical direction. Pao reported results
for three Froude numbers Fr=∞, Pr=3.8, Fr=0.64.
Here Fr=U/ND, where U
is speed of the cylinder relative to the undisturbed fluid, N is the
buoyancy frequency, and D is the cylinder diameter. The Reynolds numbers
vary slightly from case to case but are all close to Re=UD/ν=4200, where
ν is the kinematic viscosity. Pao computed frequency spectra from a
streamwise velocity time series taken at a position x=20D downstream of the
cylinder and displaced vertically z=1D above the wake centerline.
The mesh used for the simulations is shown below. The first view shows the entire domain, but with only one out of five mesh lines drawn. If the resolution were not reduced in drawing the image, it would appear as a solid object (i.e. the image is not able to resolve individual mesh lines). The second view shows a zoom of the mesh around the cylinder, again with a skip factor of five. The third view shows an increased zoom level with a skip factor of one (thus all mesh lines are shown). The mesh itself is a C-mesh topology containing 3000 points in the streamwise/azimuthal direction, 640 points in the normal direction, and 320 points in the spanwise direction. The mesh points are highly clustered near the cylinder surface in order to resolve the boundary layers. There is also clustering along the wake centerline and along vertical lines near emanating from the aft end of the cylinder. These latter two features are not really desired, but are natural consequences of the orthogonal mesh generation procedure in connection with the mesh point distribution near the cylinder surface. The clustering artifacts do not really cause a problem, except for the fact that the mesh points involved could be better used elsewhere in the domain. The domain extends six cylinder diameters in front of, as well as above and below the cylinder. It extends 25.5 cylinder diameters downstream and three diameters in the spanwise direction.
Each simulation is started with low amplitude random velocity fluctuations but no mean motion. The mean velocity is then ramped up to the value U over a period of D/U time units. This mimics the laboratory procedure where the cylinder is initially at rest and then accelerated up to speed U over a short time interval. The wake thus evolves both in space and time during the initial portion of the simulation. The wake completely fills the streamwise extent of the domain at a time of about Ut/D=30. Data collection for time series begin no earlier than a time of Ut/D=38 and continue for 20 non-dimensional time units. Detailed time series are taken on planes located at x/D=5, 10, 15, 20, and 25.
The mesh used for the simulations is shown below. The first view shows the entire domain, but with only one out of five mesh lines drawn. If the resolution were not reduced in drawing the image, it would appear as a solid object (i.e. the image is not able to resolve individual mesh lines). The second view shows a zoom of the mesh around the cylinder, again with a skip factor of five. The third view shows an increased zoom level with a skip factor of one (thus all mesh lines are shown). The mesh itself is a C-mesh topology containing 3000 points in the streamwise/azimuthal direction, 640 points in the normal direction, and 320 points in the spanwise direction. The mesh points are highly clustered near the cylinder surface in order to resolve the boundary layers. There is also clustering along the wake centerline and along vertical lines near emanating from the aft end of the cylinder. These latter two features are not really desired, but are natural consequences of the orthogonal mesh generation procedure in connection with the mesh point distribution near the cylinder surface. The clustering artifacts do not really cause a problem, except for the fact that the mesh points involved could be better used elsewhere in the domain. The domain extends six cylinder diameters in front of, as well as above and below the cylinder. It extends 25.5 cylinder diameters downstream and three diameters in the spanwise direction.
Each simulation is started with low amplitude random velocity fluctuations but no mean motion. The mean velocity is then ramped up to the value U over a period of D/U time units. This mimics the laboratory procedure where the cylinder is initially at rest and then accelerated up to speed U over a short time interval. The wake thus evolves both in space and time during the initial portion of the simulation. The wake completely fills the streamwise extent of the domain at a time of about Ut/D=30. Data collection for time series begin no earlier than a time of Ut/D=38 and continue for 20 non-dimensional time units. Detailed time series are taken on planes located at x/D=5, 10, 15, 20, and 25.
Overall mesh. Only 1 out of 5 mesh lines are drawn.
Close-up of mesh near the cylinder. Only 1 out of 5 mesh lines are drawn.
Closer zoom of mesh near the cylinder. All mesh lines are drawn.
Case 1: Fr=∞
The Fr=∞ has N=0 and thus is neutrally stratified.
This run serves as a baseline to which the stably stratified cases can be
compared. Pao also took measurements in an unstratified case at Re=4210 and
we reproduced this Reynolds number here.
Animation of the vorticity magnitude field is shown below. Early into the simulation the boundary layers separate from the cylinder and produce a pair of quasi-stationary wake vortices. The flow in these vortices then becomes unstable and turbulence erupts. The color map in the animation is adjusted abruptly once turbulence is present in order to accommodate the increased dynamic range. The wake is initially fairly steady but then evolves into a quasi-periodic vortex shedding mode. The wake spreads considerably with downstream distance and the individual vortex blobs associated with the shedding merge to form a fairly homogeneous turbulent wake by the domain exit.
Animation of the vorticity magnitude field is shown below. Early into the simulation the boundary layers separate from the cylinder and produce a pair of quasi-stationary wake vortices. The flow in these vortices then becomes unstable and turbulence erupts. The color map in the animation is adjusted abruptly once turbulence is present in order to accommodate the increased dynamic range. The wake is initially fairly steady but then evolves into a quasi-periodic vortex shedding mode. The wake spreads considerably with downstream distance and the individual vortex blobs associated with the shedding merge to form a fairly homogeneous turbulent wake by the domain exit.
Vorticity in a view of the entire domain
Vorticity in a view zoomed near the cylinder
Velocity Statistics
Below are a collection of velocity statistics taken from the time series at the streamwise locations x/D=5, 10, 15, 20, and 25.
The statistics show the expected decay of the velocity deficit and spreading of the wake with downstream distance. They also show decay in the velocity fluctuations and shear stresses.
Velocity Spectra
Below are a collection of streamwise velocity frequency spectra taken at the streamwise locations x/D=5, 10, 15, 20, and 25 and the vertical locations z/D=0.0,0.1,0.2...4.0. In order to view the spectra efficiently they are grouped into movies that scan either in the vertical direction at fixed x/D, or scan in the streamwise direction at fixed z/D.
The spectra generally show 1-1.5 decades of inertial range, followed by a gradual roll-off to a dissipation range at a Kolmogorov-scaled wavenumber kη=0.2-0.3. At some stations the roll-off appears to pause at a k-4 slope for about a half decade in wavenumber. Near the wake edge the spectra behave much differently with the inertial and dissipation ranges giving way to a single, steep quasi-power law decay. Interestingly, at isolated locations, the power law is very close to k-4 and may extend for 1-2 decades in wavenumber! Stations where this behavior is seen clearly are (x/D,z/D) = (5,2.2), (10,3.0), and (15,3.7). With reference to the vorticity magnitude plot above, these stations correspond almost exactly to the "feet" of the vorticity magnitude distribution where the steep fall-off near the wake edge is mainly complete. Thus these stations are only intermittently turbulent at best, which is consistent with the absence of an inertial range in the spectra.
Scan in z at fixed downstream location x/D = 5
Scan in z at fixed downstream location x/D = 10
Scan in z at fixed downstream location x/D = 15
Scan in z at fixed downstream location x/D = 20
Scan in z at fixed downstream location x/D = 25
Scan in x at fixed vertical location z/D = 0.0
Scan in x at fixed vertical location z/D = 0.2
Scan in x at fixed vertical location z/D = 0.4
Scan in x at fixed vertical location z/D = 0.6
Scan in x at fixed vertical location z/D = 0.8
Scan in x at fixed vertical location z/D = 1.0
Scan in x at fixed vertical location z/D = 1.4
Scan in x at fixed vertical location z/D = 1.8
Scan in x at fixed vertical location z/D = 2.2
Scan in x at fixed vertical location z/D = 2.6
Scan in x at fixed vertical location z/D = 3.0
Case 2: Fr=3.8
Figure 10 in pao's report shows a spectra displaying a k-4 range.
This case is listed as being or Fr=3.8 and thus we reproduced this Froude
number here. After the simulation was complete and the spectra was compared
with Pao's measurements it became apparent that Pao had mislabeled the data
in Figure 10. Figure 11 in Pao's report shows a spectra for Fr=0.64 but
this data is in very close agreement with the data in Figure 10. One would
not expect such close agreement for two cases where the Froude number differs
by a factor of 6. Furthermore the turbulent energy in either case is almost
two orders of magnitude less than for the unstratified case. These
observations lead us to suspect that both figure 10 and 11 show data for
Fr=0.64, and that spectra for the Fr=3.8 case is not contained in the report.
While Fr=0.64 is simulated in Case 3, we still undertook full
post-processing of the Fr=3.8 case in order to better assess the effects
of stable stratification.
The vorticity magnitude field is shown in the animation directly below. In
this case animation data was not saved during the early part of the simulation
and thus the movie shows stationary fully-turbulent state after all starting
transients have passed. In comparing with Case 1, it is clear the stable
stratification limits the wake vertical growth. It also results in a more
homogeneous turbulent region with fewer quiescent zones mixed with turbulent
ones.
Vorticity in a view of the entire domain
An animation of the density fluctuation is shown in the figure below. Again
the sequence shown excludes the starting transient and is only for the
fully-developed turbulent state. Significant vertical transport results in
an array of alternating sign density fluctuation mushroom-shaped blobs that
do not mix significantly by the domain exit. Although internal waves are
present, they are barely visible using a color scale that highlights the
unmixed density regions. The faint internal waves are seen best near the
domain exit on the lower side of the wake.
Density fluctuation in a view of the entire domain
Velocity Statistics
Below are a collection of velocity statistics taken from the time series at the streamwise locations x/D=5, 10, 15, 20, and 25.
Compared with the unstratified case, the Fr=3.8 case shows a larger velocity deficit, smaller velocity fluctuations (especially the vertical fluctuation), and restricted vertical extent (as evidenced by the turbulent shear stress). These are all expected manifestations of stable stratification.
Velocity Spectra
Below are a collection of streamwise velocity frequency spectra taken at the streamwise locations x/D=5, 10, 15, 20, and 25 and the vertical locations z/D=0.0,0.1,0.2...2.0. In order to view the spectra efficiently they are grouped into movies that scan either in the vertical direction at fixed x/D, or scan in the streamwise direction at fixed z/D.
The spectra for this case are similar to unstratified spectra shown in Case 1. There is generally at least one decade of inertial range followed by a gradual roll-off to a dissipation range at a Kolmogorov-scaled wavenumber kη=0.2-0.3. Perhaps more pronounced than in the unstratified case, the roll-off occasionally appears to pause at a k-4 slope for about a half decade or more in wavenumber. Stations where this is seen are (x/D,z/D) = (5,0.2), (5,1.7), and (10,0.3). As in the unstratified case, more extended regions of k-4 are seen near the wake edges at (x/D,z/D) = (5,2.3), (10,2.7), (15,2.6), (20,2.9). Once again these stations correspond very closely to the "feet" in the vorticity magnitude distribution.
One other additional interesting feature is the spectral "bump" at a scaled frequency of fD/U~2 that appears near the wake edge for the further downstream stations. Since this feature is absent in the unstratified case it is presumably a buoyancy effect, the most likely of which would be the signature of internal waves.
Scan in z at fixed downstream location x/D = 5
Scan in z at fixed downstream location x/D = 10
Scan in z at fixed downstream location x/D = 15
Scan in z at fixed downstream location x/D = 20
Scan in z at fixed downstream location x/D = 25
Scan in x at fixed vertical location z/D = 0.0
Scan in x at fixed vertical location z/D = 0.2
Scan in x at fixed vertical location z/D = 0.4
Scan in x at fixed vertical location z/D = 0.6
Scan in x at fixed vertical location z/D = 0.8
Scan in x at fixed vertical location z/D = 1.0
Scan in x at fixed vertical location z/D = 1.4
Scan in x at fixed vertical location z/D = 1.8
Scan in x at fixed vertical location z/D = 2.2
Scan in x at fixed vertical location z/D = 2.6
Scan in x at fixed vertical location z/D = 3.0
Case 3: Fr=0.64
Based on the discussion in the introduction to Case 2 above, we suspect that
Pao's spectra displaying a k-4 range is for Fr=0.64. We reproduced
this Froude number here and indeed we see a k-4 range for this
case.
The vorticity magnitude field is shown in the animation directly below. In
comparing with both Cases 1 and 2, it is clear the extreme stable
stratification in this case severely limits turbulence in the wake. Vigorous
3D turbulence only exists for a few wake diameters downstream, beyond which
sheet-like structures and internal waves appear to dominate the wake. the
wake vertical extent is also severely limited, never extending much more than
one cylinder diameter.
Vorticity in a view of the entire domain
An animation of the density fluctuation is shown in the figure below. the pattern is significantly different as compared with Case 2. Here the density fluctuations are dominated by internal waves, with the turbulent mixing appearing as a secondary effect. Significant vertical transport results in an array of alternating sign density fluctuation mushroom-shaped blobs that do not mix significantly by the domain exit. Although internal waves are present, they are barely visible using a color scale that highlights the unmixed density regions. The faint internal waves are seen best near the domain exit on the lower side of the wake.
Density fluctuation in a view of the entire domain
Velocity Statistics
Below are a collection of velocity statistics taken from the time series at the streamwise locations x/D=5, 10, 15, 20, and 25.
Compared with the unstratified case, the Fr=0.64 case shows a larger velocity deficit, smaller velocity fluctuations (especially the vertical fluctuation), and restricted vertical extent (as evidenced by the turbulent shear stress). These are all expected manifestations of stable stratification.
Velocity Spectra
Below are a collection of streamwise velocity frequency spectra taken at the streamwise locations x/D=5, 10, 15, 20, and 25 and the vertical locations z/D=0.0,0.1,0.2...2.0. In order to view the spectra efficiently they are grouped into movies that scan either in the vertical direction at fixed x/D, or scan in the streamwise direction at fixed z/D.
The spectra for this case follow a continuous variation from Cases 1 and 2. Within the wake turbulent core there is still generally at least one decade of inertial range followed by a roll-off to a dissipation range at a Kolmogorov-scaled wavenumber kη=0.2-0.3. Even more pronounced than in Case 2, the roll-off often appears to pause at a k-4 slope for about a half decade or more in wavenumber. Stations where this is seen are (x/D,z/D) = (5,0.5-1.0), (10,0.7-1.0), (15,0.2), and (15,0.7-0.9). The "feet" of the vorticity magnitude distribution are more difficult to identify in this case since the vorticity remains elevated for some distance beyond the wake edge (as deduced from the animations). With some uncertainty, the "feet" are probably at about z/D=1 for the x/D=5 and 10 stations, 1.5 for x/D=15 and then closer to 0.75 for x/D=20 and 25. Thus once again it appears that the most extended regions of k-4 are located near the wake edge.
The spectral "bump" near the wake edge is also present in this case but it is shifted to a higher scaled frequency of fD/U~6. This fact leads more support to the idea that this is the signature of internal waves since the buoyancy frequency is higher in this case.
Scan in z at fixed downstream location x/D = 5
Scan in z at fixed downstream location x/D = 10
Scan in z at fixed downstream location x/D = 15
Scan in z at fixed downstream location x/D = 20
Scan in z at fixed downstream location x/D = 25
Scan in x at fixed vertical location z/D = 0.0
Scan in x at fixed vertical location z/D = 0.2
Scan in x at fixed vertical location z/D = 0.4
Scan in x at fixed vertical location z/D = 0.6
Scan in x at fixed vertical location z/D = 0.8
Scan in x at fixed vertical location z/D = 1.0
Scan in x at fixed vertical location z/D = 1.4
Scan in x at fixed vertical location z/D = 1.8
Scan in x at fixed vertical location z/D = 2.2
Scan in x at fixed vertical location z/D = 2.6
Scan in x at fixed vertical location z/D = 3.0