Download Synthetic schlieren measurements of internal waves

Synthetic schlieren measurements of internal waves
generated by oscillating a square cylinder
Stuart B. Dalziel
Department of Applied Mathematics and Theoretical Physics,
University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
[email protected]
1. Abstract
This paper presents new results showing the similarities and differences in the internal wave field produced by oscillating circular and square cylinders in a linear stratification. For a square cylinder, each of
the four wave beams has the same structure, but this structure differs qualitatively from that from a circular cylinder. As with the waves from a circular cylinder, the wave beams are unimodal in the far field, but
show more than one peak close to the cylinder. In each beam, the portion of the wave beam not crossing
the plain normal to the direction of oscillation has an almost identical structure (quantitatively) to that for
a circular cylinder, whereas the portion of the beam crossing this plane has a very different structure resulting from separation at the corners of the square cylinder. The experimental measurements presented in
this paper are obtained using the recently developed ‘synthetic schlieren’, and an outline the basic principles is given.
2. Introduction
Internal gravity waves are an important dynamical feature of most density-stratified flows and play particularly crucial rôles in the atmosphere and oceans. As a consequence, there have been many theoretical,
numerical and experimental studies of the waves, their generation by flow over obstacles, by the collapse
of turbulence, and excitation through other fluid dynamical and mechanical means.
The classical experiments of Mowbray & Rarity (1967), where a small circular cylinder was oscillated
either vertically or horizontally in a linear stratification, demonstrate the basic dispersive properties of the
waves. Linear theory predicts the motion should be aligned along beams of infinitesimal width for inviscid fluids (Makarov, Neklyudov & Chashechkin 1990), but that viscosity would broaden this, reducing
the shear to finite values.
For a finite size cylinder, the beams are of a width comparable with the diameter of the cylinder (Appleby
& Chrighton 1986, 1987; Voisin 1991; Hurley & Keady 1997). If the diameter of the cylinder is larger
than the viscous scale (gν)1/3/N then there is a gradual transition from a bimodal distribution near the cylinder to a unimodal distribution in the far field. Here g is the acceleration due to gravity, ν is the kinematic viscosity,
N= −
g ∂ρ
ρ 0 ∂z
(1)
is the buoyancy frequency, and ρ is the density. This transition is described by the theory of Hurley &
Keady (1997), and the validity of which has been demonstrated by the experiments of Sutherland et al.
(1999).
This paper provides new results comparing the structure of the wave field produced by a circular cylinder
with that produced by a square cylinder of comparable size. As with the work by Sutherland et al. (1999),
the density perturbations resulting from the waves are measured using the recently developed ‘synthetic
schlieren’ technique. However, whereas Sutherland et al. (1999) used a pattern of lines to obtain measurements of the vertical density gradient, the present paper uses a pattern of random dots to extract both
components of the in-plane density gradient. The synthetic schlieren variant referred to as ‘pattern matching refractometry’ in Dalziel et al. (2000) is employed.
Section 3 outlines the basic optical principle for synthetic schlieren and gives some details of the processing algorithms. Results of the wave fields produced by a circular cylinder and by a square cylinder are
presented in section 4, with the conclusions in section 5.
3. Synthetic schlieren
The basic principle involved is that there is a one to one relationship between density ρ′ and refractive
index n' fluctuations within the fluid, and that these fluctuations cause the features visible on the mask
located on the far side of the tank to appear to move horizontally and/or vertically. For the optical arrangement shown in figure 1, these apparent movements are given by
∆ξ = 12 W (W + 2 B )
1 ∂n ′
,
n 0 ∂x
(2a)
1 ∂n ′
,
(2b)
n 0 ∂z
where W is the width of the tank and B the distance from the tank to the mask. The displacements ∆ξ (in
the x direction along the tank) and ∆ζ (in the vertical z direction) are relative to the world coordinates of
the mask. The nominal refractive index is n0 and the light rays are nominally parallel to the cross-tank
coordinate y.
∆ζ = 12 W (W + 2 B )
∂ρ′∂↑=>=0
Video camera
∂ρ′∂↑=<=0
Mask
Diffuse light source
Figure 1: Sketch of typical optical arrangement for synthetic schlieren.
For salt water the relationship between salinity, density and refractive index are approximately linear, allowing us to write
n
dn
(3)
∇ρ = β 0 ∇ρ ,
dρ
ρ0
where β = (ρ0/n0)(dn/dρ) ≈ 0.184. For the purposes of this paper we shall ignore the deflection of light
rays due to the refractive index contrasts associated with entering and exiting the experimental apparatus.
∇n =
3.1 QUALITATIVE MODE
In its simplest form, synthetic schlieren simply requires images of the mask placed on the far side of the
flow. The absolute value of the difference between an image with a density perturbation and one without
a density perturbation can be shown to be proportional to the apparent movement of the mask, at least
while the apparent movement is small. Figure 2 plots the magnitude of this absolute difference,
|Pij(t) − Pij;0|, as a function of the apparent shift for a mask of randomly located dots. Here, Pij;0 is the pixel
intensities of the unperturbed mask, and Pij(t) the pixel intensities of the perturbed mask.
Displacement (mm)
Relative intensity of absolute difference image
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
Displacement (pixels)
Figure 2: Variations in the mean (+) and root mean square (×) value of |Pij(t) − Pij;0| as the mask is displaced by a
milling machine traverse.
Figure 3 shows a typical example of the internal wave patterns visualised using this mode of synthetic
schlieren. Note the waves emanating from the tangents to the 40mm diameter circular cylinder, and the
gradual reduction in the intensity of the pattern of the pattern away from the cylinder. At the same time,
there is a transition within each of the four wave beams from a bimodal distribution aligned with the tangents, to a unimodal distribution centred on the beam. The use here of |Pij(t) − Pij;0| discards the sign of
the perturbation, resulting in both positive and negative perturbations giving a dark colour in this figure.
Figure 3: Qualitative synthetic schlieren using a mask of two-dimensional objects to visualise the internal wave field
from a circular cylinder oscillating at a frequency ω /N ~ 2−1/2.
3.2 QUANTITATIVE MODE
Quantitative mode synthetic schlieren attempts to measure the apparent displacement of the features
found on the mask. Three distinct methods for achieving this have been described by Dalziel et al. (2000):
line refractometry, dot tracking refractometry and pattern matching refractometry. The first (and simplest)
of these was employed by Sutherland et al. (1999) in their experiments of the structure of internal waves
from a circular cylinder, while this paper adopts the third which arguably represents the most powerful of
the variations.
For those familiar with the group of techniques often referred to as PIV or Particle Image Velocimetry,
the ideas behind pattern matching refractometry will need little explanation. In passing, we note that the
name PIV is itself misleading, as common usage applies this title to what would be more appropriately
described as ‘pattern matching velocimetry’ as there is no need for the patterns to be from particles and
there are other techniques (such as particle tracking velocimetry) which fall under this description.
The basic idea with pattern matching refractometry is to look to see how the patterns present on the mask
appear to move with the changes in the density field. The random dot pattern selected here was chosen to
ensure there was an approximately uniform density of dots at the same time as providing a pattern that is
locally unique. The procedure for identifying the apparent shift is simply one of optimising some measure
of the difference between a region of the unperturbed image and a corresponded shifted region in the perturbed image. In particular, we choose here to minimise
(
)
f abs ∆ξ , ∆ζ ; x i , z j = P (x + ∆ξ , z + ∆ζ ; t ) − P0 (x, z ) ,
(4)
although we note that a range of other functions (such as the cross-correlation function normally used in
PIV) may be used instead. The search algorithm for the apparent displacements ∆ξ, ∆ζ uses a combination of biquadratic fitting of fabs and bilinear interpolation of P to obtain the required subpixel accuracy.
This procedure is embedded in a multigrid framework for improved efficiency. Further details of this
process may be found in Dalziel et al. (2000).
4 Results
Using the pattern matching refractometry mode of synthetic schlieren to obtain ∇ρ′ is straightforward and
only takes a few minutes to set up and calibrate. Figure 4 shows typical images of the internal wave field
measured in this way for a circular (figure 4a) and square (figure 4b) cylinder. In both cases the frequency
of oscillation is close to ω /N = 2−1/2. To assist comparison with the results of Sutherland et al. (1999),
only the vertical component of the density field is shown. The field of view extends over most of the
depth of the tank, allowing the reflections from the tank floor and the free surface to be observed. While a
knowledge of the horizontal component allows us to invert ∇ρ′ and thus obtain ρ′ to within a single arbitrary constant, the relatively small size of the tank used here (605×100×400mm) leads to multiple reflections of the internal waves and a complex structure for ρ′ that is beyond the scope of this paper. The
measurements of ∇ρ′ are less sensitive to these reflections (smaller ρ′ and larger length scales combine to
make ∇ρ′ from the reflected waves much weaker than that from the directly radiated waves).
While the structure of the waves is broadly similar in the two cases, there are a number of important differences. For the circular cylinder, each of the four wave beams are clearly split into two bands close to
the cylinder, emanating tangentially from the cylinder, with viscous diffusion leading to a gradual increase in the perturbation between these bands prior to reflection from the tank floor or free surface.
In contrast, the four wave beams emanating from the circular cylinder appears to have an additional,
broader band originating at the corner of the square, approximately half way between the two tangential
bands. The positioning and form of this broader band suggests that it originates from the separation of the
boundary layer on the vertical faces of the square cylinder, the resulting disturbance to the flow being
confined to the region immediately above and below the cylinder (for the present vertical oscillation).
Additionally, there is some suggestion of horizontal banding from the top and bottom faces of the cylinders. More detailed experiments suggest that this is the result of mixed fluid being generated primarily by
the initial transients of the cylinder, but also from the on-going boundary layer separation and the vanishing of the diffusive flux on the top and bottom faces of the cylinder.
(a)
(b)
Figure 4: Quantitative mode synthetic schlieren measurements of the vertical gradient of the density perturbation for
(a) a circular cylinder and (b) a square cylinder.
Figure 5 shows the cross-beam structure of the internal wave field for the lower left-hand beam at a distance of 8 radii from the cylinder as the cylinder reaches the bottom of its stroke. This figure presents the
perturbation in the vertical density gradient normalised on the oscillation amplitude (in s-2/cm) as a function of the cross-beam position (in cm). Curves are shown for the circular cylinder (solid line), the square
cylinder (dashed line), and the viscous theory of Hurley & Keady (1997; dot-dash line).
0.200
0.150
Key
Perturbation in density gradient
Circular
Square
Theory
0.100
0.050
0.000
-0.050
-0.100
-0.150
-0.200
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
4.0
5.0
6.0
Cross-wave position
Figure 5: Cross-beam structure of the internal wave field 8 radii from the cylinder.
As found by Sutherland et al. (1999), there is good agreement between the theoretical predictions of the
wave field and experimental measurements for a circular cylinder. The level of agreement found in figure
5 is comparable with the earlier results, and is maintained across all phases of the waves.
The wave field from the square cylinder is broadly similar for the upper band of this beam (i.e. the band
that starts at the upper corner of the cylinder and crosses the mid-plane normal to the direction of cylinder
oscillation), a feature that is again observed for all phases of the waves. The somewhat larger amplitude
found in the square cylinder is due in part to the absence of any normalisation based on the size of the
cylinder (the square cylinder was sized approximately midway between the inscribed and circumscribed
square for the circular cylinder). Identical results are found for the comparable band in the other beams.
The lower band of this beam (i.e. the band that does not cross the mid-plane) shows two peaks rather than
the single peak found for the circular cylinder. Each of these peaks is slightly more than half the amplitude of that found for the circular cylinder, and is broader in width, suggesting a higher energy flux is carried away from the cylinder, and we may therefore expect the square cylinder to have a larger drag than
its circular counterpart. Again, identical behaviour is found for the corresponding bands in the other
beams, and similar differences are found for other phases of the oscillation, and for oscillations at different frequencies.
5 Conclusions
The internal gravity wave field produced by a square cylinder oscillating in a linear stratification shows
similarities in the near field with that generated by the oscillation of a circular cylinder. In both cases the
motion is confined to four beams forming a Saint Andrew’s cross centred on the cylinder and bounded by
the tangents to the cylinder at an angle cosθ = ω /N to the vertical. Within each beam, the structure of
parts of the wave field crossing the mid-plane of the oscillation are very close to that of a circular cylinder, whereas the parts of the wave beam that emanate from the horizontal faces (for a vertical oscillation)
show significant differences, creating what is essentially a trimodal distribution across each beam.
These differences in the near field are believed to be primarily due to the separation of the viscous boundary layer found on the vertical faces of the vertically oscillating cylinder, and the subsequent disturbance
to the flow caused by the development of a weak separated wake immediately above and below the cylinder.
The near-field differences will, in turn, lead to a modified structure over the whole of the beam further
from the cylinder, as viscous effects kill off the higher wavenumber components, but ultimately details
will be lost with the development of a unimodal distribution. However, these preliminary results suggest
that the energy flux carried by the waves will be greater from a square cylinder, and that we may expect
the cylinder to experience a larger drag.
The author wishes to thank Mr. Franck Rodriguez for his assistance in running some of the experiments.
References
Appleby, J.C. & Crighton, D.G. (1986) Non-Boussinesq effects in the diffraction of internal waves from
an oscillating cylinder. Q. J. Mech. Appl. Maths 39, 209-231.
Appleby, J.C. & Crighton, D.G. (1987) Internal gravity waves generated by oscillations of a sphere. J.
Fluid Mech. 183, 439-450.
Dalziel, S.B., Hughes, G.O. & Sutherland, B.R. (2000) Whole-field density measurements by ‘synthetic
schlieren’. Exp. in Fluids 28, 322-335.
Hurley, D.G. & Keady, G. (1997) The generation of internal waves by vibrating cylinders, Part 2: Approximate viscous solution for a circular cylinder. J. Fluid Mech. 351, 119-138.
Kistovich, A.V., Neklyudov, V.I. & Chashechkin, Y.D. (1990) Spatial structure of two-dimensional and
monochromatic internal-wave beams in an exponentially stratified liquid. Izv. Atmos. Oceanic Phys.
26, 548-554.
Mowbray, D.E. & Rarity, B.S.H. (1967) A theoretical and experimental investigation of the phase configuration of internal waves of small amplitude in a density stratified liquid, J. Fluid Mech. 28, 1-16.
Sutherland, B.R., Dalziel, S.B., Hughes, G.O. & Linden, P.F. (1999) Visualisation and measurement of
internal waves by “synthetic schlieren”. Part 1: Vertically oscillating cylinder; to J. Fluid Mech. 390,
93-126.
Voisin, B. (1991) Internal wave generation in uniformly stratified fluids. Part 1. Green’s function and
point sources. J. Fluid Mech. 231, 439-480.