Whole-field density measurements by `synthetic schlieren'

S. B. Dalziel, G. O. Hughes, B. R. Sutherland

Abstract This paper outlines novel techniques for pro-ducing qualitative visualisations of density ¯uctuationsand for obtaining quantitative whole-®eld density mea-surements in two-dimensional density-strati®ed ¯ows.These techniques, which utilise image processing tech-nology, are much simpler to set up than the classicalschlieren and interferometry methods, and provide usefulinformation in situations where shadowgraph is of little orno value. Moreover, they may be set-up to analyse muchlarger domains than is feasible with the classicalapproaches, and do not require high quality opticalwindows in the experimental apparatus. Ultimately thegreatest strength of these techniques is the ability toextract accurate, quantitative measurements of thedensity ®eld.

Application of these techniques is illustrated by an in-ternal wave ®eld produced by an oscillating cylinder. Re-cent theoretical advances for this classical problem make itthe ideal test bed. Results are presented for both a circularand a square cylinder oscillating vertically in a linearstrati®cation. Further aspects of the techniques are illus-trated by considering thermal convection from a hand and¯ow over an obstacle towed through a density strati®ed¯uid.

1IntroductionQualitative visualisations of density ¯uctuations have longbeen used to assist with the understanding of ¯ows indensity strati®ed systems. The techniques range from theintroduction of passive tracers, such as dye lines or neu-trally buoyant particles, to shadowgraph, schlieren andinterferometric techniques which rely on refractive indexvariations induced by the stratifying agent. Numerousexamples of these visualisations may be found within theliterature (e.g. van Dyke 1982), as have attempts to extractquantitative information from them (e.g. Irvin and Ross1991; Greenberg et al. 1995).

In this paper we introduce a number of novel techniquesfor obtaining both high quality visualisations and accuratemeasurements of the density ®eld. These techniques arebased on the same optical principles as the classicalschlieren method, but have the advantages of being mucheasier to set-up and of being adaptable to much larger do-mains. The name which has been chosen for the simplest ofthese techniques ± `qualitative synthetic schlieren' ± re¯ectsthe relationship with classical schlieren, although we notethat it is the optics (external to the ¯ow) used to form theschlieren image which are ``synthesised'', not the opticalinhomogeneities of the medium being visualised. The mostsophisticated of the quantitative techniques has beennamed `pattern matching refractometry' to indicate theorigin of the approach in `pattern matching velocimetry'(more frequently called `particle image velocimetry').

In total we shall outline four techniques: `qualitativesynthetic schlieren', `line refractometry', `dot tracking re-fractometry' and `pattern matching refractometry'. The®rst of these has two modes of operation, `line mode' and`dot mode' and these represent the basic qualitative visu-alisation behind the remaining three quantitative tech-niques. For this reason we often refer to all four techniquesas `synthetic schlieren'. These techniques are primarily ofvalue for analysing two-dimensional ¯ows where the cur-vature of the density ®eld is suf®ciently small that classicalshadowgraph techniques are of limited value, althoughthere is an overlap in the range of applicability. They mayalso be used for ¯ows where the density gradients are toosmall for techniques relying on the concentration of apassive dye (e.g. light induced ¯uorescence or dye ab-sorption) to give useful results. While limited informationcan be extracted from ¯ows with three-dimensional vari-ations in the density ®eld, we shall not discuss these inany detail.

Experiments in Fluids 28 (2000) 322±335 Ó Springer-Verlag 2000

Received: 27 October 1998 /Accepted: 24 May 1999

S. B. Dalziel, G. O. Hughes1, B. R. Sutherland2

Department of Applied Mathematics and Theoretical PhysicsThe University of Cambridge, Silver StreetCambridge CB3 9EW, England

Present addresses:1 Research School of Earth Sciences, The Australian NationalUniversity, Canberra, A.C.T. 0200, Australia2 Department of Mathematical Sciences, University of Alberta,Edmonton, Alberta T6G 2G1, Canada

Correspondence to: S. B. Dalziel

SBD wishes to acknowledge the support of AWE, Yorkshire Waterand the Isaac Newton Trust. GOH thanks the Association ofCommonwealth Universities for his time in DAMTP andacknowledges the support of NERC under grant GR3/9356 for theexperiments conducted in the EnFlo towing-tank. BRSacknowledges support from NERC under grant GR3/09399. Theauthors also wish to thank an anonymous reviewer for drawingour attention to early work on the Moire fringe methods in the1940s and 1950s.

322

This paper is intended to complement and expand onthe subset of these techniques presented by Sutherlandet al. (1999) in their analysis of the internal wave ®eld instrati®ed salt water solutions produced by an oscillatingcylinder, and to provide more detail than may be foundin Dalziel et al. (1998). We again employ internal wave®elds as our primary method of illustrating the tech-niques. It is therefore appropriate to review some aspectsof ¯ow in a density-strati®ed ¯uid, although the intentionhere is to illustrate the power of synthetic schlierenrather than present a detailed analysis of the ¯uiddynamics.

If a ¯uid element is perturbed vertically in a densitystrati®ed ¯uid, it experiences a buoyancy force directed torestore it to its original position. This buoyancy force, incombination with the inertia of the ¯uid element, act as asimple harmonic oscillator with a frequency (the buoyancyfrequency)

N � ÿ g

q0

oqoz

� �1=2

; �1�

where q is the ¯uid density with reference density q0, g isgravity and z is oriented vertically upward. Oscillation in adirection at an angle h to the vertical reduces the restoringforce, giving rise to the dispersion relation for small am-plitude waves in a Boussinesq ¯uid,

x � �N cos h ; �2�where x is the frequency of the waves. The ¯uid motionand group velocity are at an angle h to the vertical, whilethe phase velocity is perpendicular to this (the sign of thevertical component of the phase velocity is opposite to thatof the group velocity).

A linearly strati®ed ¯uid �N � const� is therefore able tosupport waves of frequencies 0 � x � N . Mowbray andRarity (1967) demonstrated theoretically and experimen-tally (using classical schlieren) that for waves forced by anoscillating cylinder, the ¯uid motion was con®ned with aregion of ¯uid resembling a ``St. Andrew's Cross''. This is insharp contrast to the ¯ow found when the ¯uid is unstrat-i®ed (e.g. Lin and Rockwell 1997 who looked at the vortexshedding from an oscillating cylinder). Other authors havesubsequently used different experimental techniques toanalyse these waves (e.g. Peters 1985 and Merzkirch andPeters 1992 used interferometry). Recent work by Hurleyand Keady (1997) has provided additional theoretical de-tails of the structure of this wave ®eld for cylinders of a ®nitesize in a viscous ¯uid, The ®rst experimental results con-®rming their prediction are provided by Sutherland et al.(1999). In the experiments presented here a circular cylin-der of radius R � 20 mm is oscillated vertically in a linearstrati®cation with N � 1 sÿ1 and x=N � 2ÿ1=2.

Internal waves are also generated if an obstacle is towedhorizontally through a strati®cation. For a ¯ow which issteady in the frame of reference of the obstacle, the hori-zontal component of the phase velocity is equal to thespeed of towing U, giving rise to waves of frequencyx � 2pk=U , where k is the length scale associated with theobject. The need to transport energy away from theobstacle combined with the dispersion relation give rise to

an asymmetric pattern of waves with the crests tilted up-stream (see Turner 1973; Gill 1982).

After reviewing the basic optics in Sect. 2, we outlinethe experimental set-up used for the classical schlieren andMoire fringe methods in Sect. 3. In Sect. 4 the qualitativemode of synthetic schlieren is introduced and Sects. 5±7discuss the quantitative modes. Finally, Sect. 8 presentsour conclusions.

2Basic opticsAll four techniques introduced in this paper utilise thesame optical phenomenon, namely that a ray of light isbent towards regions of larger refractive index. A detaileddescription of this process is required in order to makequantitative measurements. This description may startfrom a number of points. Sutherland et al. (1999) analyseda ray of light by invoking Snell's law. Here we shall startfrom Fermat's variational principle for the behaviour oflight in an inhomogeneous medium

dZ

n�x; y; z�ds � 0 ; �3�

where s is oriented along the light ray and n�x; y; z� is therefractive index ®eld (the ratio of the speed of light througha vacuum to that through the medium). We select our co-ordinate system �x; y; z� with x along the length of the tank,y across the width (the direction in which variations in the¯ow are negligible) and z vertically upward.

Rather than solving the full variational problem, we re-strict ourselves to rays of light which always have a com-ponent in the y direction so that their paths may bedescribed by x � n�y� and z � f�y�. This restriction issimply a requirement that light is able to cross the tank, afundamental requirement for the methods described in thispaper. The variational principle then gives rise to a pair ofcoupled ordinary differential equations (Weyl 1954) relat-ing the light path to the gradients of n in the x-z plane:

d2ndy2� 1� dn

dy

� �2

� dfdy

� �2" #

1

n

on

ox; �4a�

d2fdy2� 1� dn

dy

� �2

� dfdy

� �2" #

1

n

on

oz: �4b�

For synthetic schlieren we are interested primarily in lightrays which remain approximately parallel to the y direc-tion. Under this restriction the terms �dn=dy�2 and�df=dy�2 may be neglected, effectively decoupling (4a) and(4b). The two-dimensionality of the ¯ow and the weakvariations in density (and hence weak variations in therefractive index) along the ray path allow these expres-sions to be integrated across the width W of the tank toobtain the path of the light ray across the tank:

n � ni � y tan /n ÿ1

2y2 1

n

on

ox; �5a�

f � fi � y tan /f ÿ1

2y2 1

n

on

oz; �5b�

323

where ni, fi describe the incident location and tan /n �dn=dy�y � y0� and tan /f � df=dy�y � y0� describe thehorizontal and vertical components (respectively) of theangle at which the light ray enters the tank (measuredrelative to the y direction).

As shall be seen in the following sections, we are in-terested in how an image of a mask placed some distance Bbehind the tank (the reason for choosing B 6� 0 will begiven in Sect. 4) appears to change as the result of ¯ow-induced refractive index variations relative to the refrac-tive index variations in the absence of the ¯ow. Speci®callywe wish to analyse the changes in the image formed by thecamera as a shift in the origin of the light ray reaching thecamera. By back-tracing the light rays received by thecamera the apparent shift (Dn, Df) in the origin of the lightray is given by

Dn � 1

2W�W � 2B� 1

n0

on0

ox; �6a�

Df � 1

2W�W � 2B� 1

n0

on0

oz: �6b�

Here we have decomposed the refractive index ®eld n inton0 � nbase � n0 where n0 is the nominal refractive index ofthe medium (e.g. n0 � 1:3332 for water), nbase representsspatial variations associated with the ``known'' base state(e.g. the changes introduced by adding a quiescent linearbackground strati®cation) and n0 is the variation causedby the ¯ow under consideration (e.g. the internal wave®eld). In obtaining (6) we have assumed the variationsnbase and n0 in the refractive index ®eld are small com-pared with nominal value n0.

In principle there should be some correction made to(6) for the refractive index contrasts between air, thematerial the tank is made of and the working ¯uid. Detailsof this correction, using Snell's law, are presented inSutherland et al. (1999). In practice, however, provided theincident light rays are approximately normal to the tankwalls (which is the requirement to ignore the �d=dy�2terms in (5)) the additional refraction at these optical in-terfaces need not be considered explicitly.

As stated above, there is normally a constitutive rela-tionship between the density of the ¯uid and the refractiveindex. To a good approximation the relationship betweenrefractive index and density for salt water is linear (Weast1981), allowing us to write

rn � dn

dqrq � b

n0

q0

rq ; �7�

where

b � q0

n0

dn

dq� 0:184 ; �8�

and q0 is the nominal reference density (1000 kg mÿ3).Substitution into (6) then gives the relationship betweendensity ¯uctuations q0 and apparent movement of thesource of a light ray

Dn � 1

2W�W � 2B� b

q0

oq0

ox; �9a�

Df � 1

2W�W � 2B� b

q0

oq0

oz: �9b�

3Schlieren and Moire fringesThe classical schlieren method makes use of the de¯ectionof a ray of light to mask off a greater or lesser portion ofthe light source with the edge of a ``knife''. Figure 1 il-lustrates a typical set-up. The light source in combinationwith the ®rst parabolic mirror produces a ®eld of nomi-nally parallel light rays. As these light rays pass throughthe tank they are bent towards the gradient of the refrac-tive index, and hence density. If the basic state is repre-sented by a linear strati®cation then (5) shows the lightrays will execute a parabolic mirror which projects animage onto a screen via a lens.

The knife edge is positioned near the focal point of thesecond parabolic mirror and lens, and acts as an asym-metric aperture. Typically it is adjusted so that half of thelight passing through the tank with quiescent conditions isobscured by the knife edge. When the ¯ow produces re-fractive index variations n0, the light rays leaving the tankare de¯ected so that the mirror no longer focuses them atthe knife edge. Depending on the direction of this de¯ec-tion and the orientation of the knife edge, the amount oflight striking the knife may be increased (decreased),leading to a reduction (intensi®cation) of the light reach-ing the corresponding position on the screen. The orien-tation of the knife edge in Fig. 1 is chosen so that theintensity of the image increases with positive oq0=o". Thesymbol " is used here to represent the de¯ection directioncorresponding to the normal to the knife edge (e.g. upwardfor the arrangement shown in Fig. 1).

There are a number of practical dif®culties with thisclassical schlieren. First, the setting up of the opticalcomponents, especially the parabolic mirrors, requiresconsiderable care. Second the cost of parabolic mirrors

Parabolic mirror

Light source

Parabolicmirror

Knife edge

Lens

Screen

∂ρ'/∂↑ = 0∂ρ'/∂↑ > 0

∂ρ'/∂↑ < 0

∂ρ'/∂↑ > 0

∂ρ'/∂↑ < 0

Fig. 1. Sketch of optical arrangement for classical schlierenmethod

324

rapidly becomes prohibitive with increasing size. Third,high quality images can only be formed when the appa-ratus itself is optically good (e.g. optical quality glassrather than Perspex which is prone to scratching), addingfurther to the cost. Moreover, if the base state described bynbase is not itself either constant or linear, then the un-perturbed light rays are not parallel, yielding an unwantedstructure in the image formed for this base state. Finally,the need for a point source of illumination limits the in-tensity of illumination available.

Using the knife edge illustrated in Fig. 1 it is possible tovisualise only the component of rq0 normal to the edge.To overcome this a number of techniques have used dif-ferent geometries to mask the focal point in place of aknife edge, as have methods for obtaining quantitativeinformation from the resulting images through accuratedetermination of the intensity of the image (e.g. Irvin andRoss 1991) or determining the colour produced by a col-our schlieren system (e.g. Greenberg et al. 1995).

The Moire fringe method (e.g. Burton 1949; Mortensen1950; Sakai 1990) represents an optically simpler alter-native to the classical schlieren. Two masks of alternatingclear and opaque lines (or dots), with a spacing greaterthan the diffraction limit, are produced with one beingpositioned on either side of the ¯ow. Diffuse lightingilluminates the mask behind the tank. The mask in frontof the tank is positioned and scaled in such a way that,for a ®xed observer and no perturbations to the ¯owthe masks appear to have the same line spacing. Whenthe ¯ow is initiated the density ®eld is perturbed and thelight rays reaching the observer originate from a differentlocation on the mask behind the tank. This locationmay result in either more or less light reaching the ob-server depending on the magnitude and direction of rq0and the phase relationship between the two masks. Sucha situation is illustrated in Fig. 2 where the phasing ofthe lines on the two masks has been selected so as toincrease the mean intensity (averaged over a regionlarger than the line spacing) for positive oq0=o" (here "represents the direction perpendicular to the lines). Ifthere had been no phase difference between the two setsof lines then the image would have got darker for eithersign of oq0=o", while a 180� phase difference would havestarted from a black image and got brighter forjoq0=o"j > 0.

There are a number of experimental dif®culties withthis Moire fringe arrangement. The production of suitablemasks is relatively straight forward using modern laserprinters, provided the geometry of the masks is related by asimple scalar. Their positioning, however, is fairly criticalas accurate alignment of both masks and the observer isnecessary. Achieving the required level of alignment for astrati®ed ¯ow may require the mask nearer the observer tohave a non-uniform line spacing, even when the maskbehind the tank is uniform. Moreover, the precise posi-tioning of the mask and spacing of the lines will depend onthe background strati®cation. A decrease in the buoyancyfrequency of this background strati®cation, for example,will require the mask located in front of the tank to bepositioned higher than that required when the strati®ca-tion is stronger, while a nonlinear strati®cation willrequire a nonlinear relationship between the lines on thetwo masks.

The optical arrangement illustrated in Fig. 2 is not theonly one which may be used. Replacing the observer with apoint light source and the diffuse light source with aprojection screen will yield an equivalent result by tracingin the reverse direction along the light rays. This dualityleads to an alternative method of generating the larger ofthe two masks. The geometry of this larger mask, locatedfurther from the light source, may be determined bysimply projecting the smaller mask through the tank(when it contains the ambient strati®cation) onto the lo-cation of the second (larger) mask. The required geometrymay then be recorded in situ by positioning photographic®lm in place of the mask. After developing, this photo-graphic ®lm may be used as the mask. Any phase differ-ence required between the two masks can be achieved byapplying a suitable translation to the regular geometry ofthe smaller mask prior to exposing the ®lm.

If carefully set-up, the Moire fringe method is capable ofproducing high quality visualisation with excellent spatialresolution at only a small fraction of the cost for con-ventional schlieren. However, as we will see in the nextsection, synthetic schlieren avoids most of the dif®cultieswith setting up the Moire fringe method.

4Qualitative synthetic schlierenThe photographic production of one of the masks for theMoire fringe method outlined in the previous sectionforms the basis of the qualitative synthetic schlierenmethods introduced in this section. The fundamental ideais to record digitally (rather than photographically) theapparent location of one of the masks. The image recordedin this way then forms a virtual mask against which theapparent movements due to refractive index gradients maybe measured. This approach works in both the diffuse lightsource (such as shown in Fig. 3) and point light sourcearrangements. In either case the real mask is placed be-tween the light source and the tank, while the virtual maskis created from the appearance of the real mask in theabsence of perturbations to the quiescent state. The mainrestriction is that the depth of ®eld of the video camera (orlight source if projecting the mask through the tank) islarge enough such that both the ¯ow and the mask are

Diffuselight source

Mask

Mask

Observer

∂ρ'/∂↑ = 0∂ρ'/∂↑ > 0

∂ρ'/∂↑ < 0

∂ρ'/∂↑ > 0

∂ρ'/∂↑ < 0

Fig. 2. Sketch of optical arrangement for Sakai's Moire method

325

reasonably well in focus so that we may identify the lightfalling on a pixel in the video camera with a single raypassing through the tank. This is not a severe restriction asthe experimental arrangement allows for high levels of il-lumination to be used (unlike classical schlieren, there isno need for the illumination to originate from a pointsource), and hence small lens apertures do not cause dif-®culties.

We shall start by considering a mask with lines par-allel to the x direction. With such a set of lines we will beable to visualise oq0=oz through the vertical de¯ection ofthe lines (9b), but the uniformity of the basic mask inthe x direction makes the visualisation insensitive tooq0=ox.

4.1Basic operationFigure 4a sketches the typical appearance of the mask forquiescent conditions and with density perturbations re-sulting in an apparent shift. For the present we shall as-sume the optics are perfect (this assumption will berelaxed in subsequent sections) and an image of the maskis formed on the surface of the image sensor in the videocamera. For a typical video camera operating in frameintegration mode the CCD sensor consists of an array ofalmost touching rectangular pixels, each of which returns asignal Pij�t� proportional to the spatial average of the lightintensity striking the ijth pixel:

Pij�t� � 1

DxDz

Zxi�Dx=2

xiÿDx=2

Zxj�Dz=2

xjÿDz=2

p�x; z; t�dz

264375dx : �10�

Here the intensity striking the CCD sensor is p�x; z; t� andthe pixel images a region of the mask of size Dx� Dz. Theeffect of this averaging or pixelisation is illustrated inFig. 4b.

For simplicity we will start by assuming that sharptransitions in intensity in the mask produce identicallysharp transitions in the intensity p�x; z; t� falling on theCCD sensor. Consider a mask consisting of nominallyhorizontal lines oriented parallel to the x axis. If thelines are perfectly black (p � 0) and perfectly white(p � 1) then

p0�x; z� � 0 0 � z mod g < a1 otherwise

n�11�

represents the intensity produced by the mask underquiescent conditions. Here g is the spacing of the lines onthe mask and the lines consist of alternating black andwhite regions of widths ag and �1ÿ a�g, respectively. Theintensity of the corresponding digitised image will be

Pij;0 �0 1

2 Dz � zj mod g < agÿ 12 Dz

1 ag� 12 Dz � zj mod g < gÿ 1

2 Dz

12�

zjmod gÿagDz agÿ 1

2 Dz � zj mod g < ag� 12 Dz

12ÿ

zjmod gÿgDz gÿ 1

2 Dz � zj mod g < 12 Dz

8>>>>><>>>>>::

�12�An expression similar to (12) holds when the density ®eldis perturbed. Provided the perturbations are small suchthat Df � g, where Df is given by (9b), and variations inDf occur over a length scale large compared with g, thenwe may replace g in (12) with gÿ Df. Qualitative modesynthetic schlieren works by calculating jPij�t� ÿ Pij;0j, thisdifference simply being jDfj=Dz and is proportional tojoq0=ozj for pixels which contain an edge in both the Pij;0

and Pij�t� images. The constant of proportionality dependson the width of the tank, the relative positions of the maskand the screen, giving

Diffuselight source

Mask

Video camera

∂ρ'/∂↑ = 0 ∂ρ'/∂↑ > 0 ∂ρ'/∂↑ < 0

∂ρ'/∂↑ > 0

∂ρ'/∂↑ < 0DigitisedimageP (∂ρ'/∂↑)

❘ P(∂ρ'/∂↑) − P(0) ❘

Fig. 3. Sketch of optical arrangement for line-mode syntheticschlieren

p0

p

p0

p

ba

Fig. 4. Idealised images for qualitativesynthetic schlieren. a Location of lines;b image formed in CCD sensor

326

oq0

oz

���� ����ij

� 2q0

bDz

W�W � 2B�

�Pij�t� ÿ Pij;0

�� �� for pixels containing an edge

unknown elsewhere

8<: :

�13�In practice the equipment used in this study comprised astandard video camera (Cohu 4910 series set to frameintegration mode and no gamma correction) producing ananalogue signal which was subsequently digitised to adifferent horizontal resolution by a frame grabber card(Data Translation DT2862). While this combination makesthe situation somewhat more complex than the ideal casedescribed above, the qualitative effect is much the sameand jPij�t� ÿ Pij;0j is proportional to the vertical (for hor-izontal lines) gradient in the perturbation density.

Figure 5a shows an example of the St. Andrew's crossinternal wave ®eld from an oscillating cylinder in a lineardensity-strati®cation (x=N � 2ÿ1=2, N � 1 sÿ1) obtainedin this manner. For maximum sensitivity and spatial res-olution we would have exactly one line of the mask forevery two rows of pixels on the CCD sensor. In practice,however, the spacing and alignment may not be this pre-cise so that the average line spacing exceeds this ideal. As aresult the images formed from jPij�t� ÿ Pij;0j contain stri-ations where there is no data, corresponding to pixels notcontaining the edge of a line. The sensitivity of this tech-nique is illustrated in Fig. 5b which shows thermal con-vection in air driven by the heat of a hand. The ambienttemperature was 26 �C and the hand surface temperaturewas only marginally above this.

4.2SensitivityFrom (13) we can see that the sensitivity may be controlledby the position of the mask relative to the density ¯uctu-

ations in the ¯ow. Increasing the distance from the objectto the mask B increases the sensitivity, although it isnecessary to account also for the apparent reduction in thesize of the mask (and hence the increase in Dz) as themask is moved further from the camera. If we assumethe unperturbed light rays are straight (thus ignoring theapparent foreshortening due to refractive index changesencountered along the way), we may calculate the relativesensitivity S as the ratio of the apparent de¯ection of amask at some distance B behind the tank to that withB � 0:

S � L

L� B1� 2

B

W

� �� 1� B

W

L�W

L� B: �14�

Here we assume the distance between the camera and frontwall of the tank is ®xed at LÿW , and the camera zoomremains ®xed so that the same region of the tank is vi-sualised for all B. Note that as B is increased, the size of Dzis increased as L=�L� B� compared with B � 0.

The sensitivity to thermal convection illustrated inFig. 5b illustrates one of the dif®culties associated withthese synthetic schlieren methods, namely the sensitivityto thermal noise in the laboratory environment. For a ®xed®eld of view and distance L� B from the mask, (13) showsthat the optical system is most sensitive to thermal noiseclose to the camera, the sensitivity being proportional tothe distance between the thermal anomaly and the mask. Itis thus important to minimise the thermal noise, especiallyclose to the camera. Fortunately thermal noise usuallyevolves on a timescale either much faster or much slowerthan that for the ¯ow being considered, facilitating itsremoval through a temporal ®ltering process, although wewill not go into details here.

By setting up the mask on an accurate traverse (thebed of a milling machine equipped with a digital readoutwith a resolution of approximately �0:010 mm on rela-tive displacements) we have assessed the sensitivity of thetechnique without taking any special precautions against

a b

Fig. 5. Line mode qualitative synthetic schlieren. White represents no difference between Pij(t) and Pij;0. a Internal wave ®eldproduced in a linear salt strati®cation (described by a buoyancy frequency N � 1 s)1) by oscillating a circular cylinder 20 mm inradius at a frequency x � N/Ö2. b Thermal convection from a hand

327

thermal noise. Figure 6 shows how the mean level ofjPij�t� ÿ Pij;0j changes with the displacement of the maskof lines. From this we see that displacements as small as0.01 pixels may be detected with the expected linear re-sponse for displacements up to 1 pixel and only a slightdeparture from linearity for displacements up to 2 pixels.The reduction in jPij�t� ÿ Pij;0j for larger displacementsis the result of aliasing of adjacent lines which have aspacing of a little over 4 pixels. For this test the mask wasaligned as closely as possible with the lines parallel tothe scan of the standard frame integration video cameraused for all the results presented in this paper. The maskwas viewed through a zoom lens set to a focal length of50 mm to image a region 420 mm across. The output ofthis camera was digitised directly at a resolution of512� 512 pixels. We note, in passing, that the thermalenvironment in the workshop where these tests wereundertaken was signi®cantly quieter than in the labora-tory.

4.3Variations on a themeThe above analysis was based on a mask consisting ofhorizontal lines which are sensitive to ¯uctuations in thevertical gradient of density. Clearly this is not the onlycon®guration possible. The use of vertical lines yields ameasure of the ¯uctuations in the horizontal gradient, or apattern of dots yields both components of the gradient.Figure 7a shows the same internal wave ®eld obtainedfrom a regular array of dots, while Fig. 7b illustrates thesame for a random array of dots (produced by introducinga random perturbation to a regular array). Careful in-spection of either of these reveals information about thedirection of the gradient vector in addition to jrq0j.Figure 7c illustrates this directional information with anenlarged view of the image presented in Fig. 7a. Thedouble-crescent shape is oriented with the gradientdirected from one crescent to the other.

The sensitivity study outlined in Sect. 4.2 for the hori-zontal lines was repeated for the two mask geometries usedfor Fig. 7. The corresponding displacement versusjPij�t� ÿ Pij;0j curves are shown in Fig. 8. We again observea linear response for displacements less than 1 pixel. Forthe regular array of dots used for Fig. 8a, we see an aliasingof the (imperfectly aligned) dots for larger displacements,while for the random dots used for Fig. 8b the image

Fig. 6. Variations in the mean (+) and root mean square (´)value of |Pij(t) ) Pij;0| as the mask is displaced by a milling ma-chine traverse for a mask of lines aligned with the scan lines ofthe video camera. The mask was traversed normal to the scanlines of the video camera

Fig. 7. Qualitative synthetic schlieren using a mask of two-di-mensional objects to visualise the internal wave ®eld shown in®gure 5. The mask geometries are a a regular array of dots andb a random array of dots. An enlarged view of the regular arrayimage is shown in c

328

saturates at larger displacements. Tests have shown com-parable results are obtained when traversing the dotsparallel and normal to the scan lines of the video camera.

With a standard video camera the one-dimensionalmask of the line-mode synthetic schlieren is less sensitiveto signal noise than for the two-dimensional masks (i.e.dots) due to the ®xed one-to-one relationship betweenrows of pixels on the CCD sensor and rows of pixels in thedigitised image. If the mask is rotated by 90� so that thelines appear vertical we ®nd the noise component is in-creased by a factor of two, reducing the measurable min-imum displacement to approximately 0.02 pixels. Theincreased noise arises from the electronic jitter in thevideo signal and frame grabber which manifest themselvesas random horizontal displacements of the image of theorder of 0.04 pixels. This jitter was not noticeable when thelines were horizontal as the mask was approximatelyuniform in the direction of the jitter. The same issues arisewhen using the more complex dot mask geometries. Thenet result is that the maximum sensitivity may be achievedonly when the lines on the mask are aligned closely withthe scan lines of the video signal. We note, however, that ifa digital camera and frame grabber were to be used, thenthis problem with signal jitter would be eliminated.

Moreover, modern high resolution digital cameras willenable substantially higher sensitivities to be achieved.

One of the features of the synthetic schlieren techniquesis the ability to scale it up to visualise ¯ows at much largerscales than is feasible with conventional schlieren. Forexample, Fig. 9 shows the ¯ow over a h � 0:1 m high `hill'towed at a velocity U � 0:055 msÿ1 along a tank contain-ing a nominally N � 1:3 sÿ1 strati®cation, giving a Froudenumber Fr � U=Nh � 0:4. The ®eld of view here is 1.3 mlong and 0.6 m high. The ultimate size limit which may bevisualised is given by the trade off between thermal noiseand sensitivity, and the ability to produce a mask con-taining a suitable texture.

4.4Preserving the signBy preserving the sign information contained inPij�t� ÿ Pij;0 we can establish the direction of rq0.Figure 10a is calculated from the same pair of images asFig. 7a and c, but without taking the absolute value of thedifference. Each crescent pair consists of one light crescentand one dark crescent. The gradient vector is directedfrom the darker toward the lighter crescent.

The simplicity of calculating Pij�t� ÿ Pij;0, or sometransformation of this, enables this qualitative measure ofoq0=o" or rq0 to be determined in real-time using rela-tively simple hardware. Indeed the Data TranslationDT2862 frame grabber card used for this work is a designten years old yet allows calculation of f �Pij�t� ÿ Pij;0� at fullframe-rate video using a combination of an on-board ar-ithmetic logic unit and on-board look-up tables. For theimages presented in Figs. 5, 7 and 9 the look up tablefunction was simply the calculation of the absolute valueand the introduction of a scaling factor, while for Fig. 10aan offset is added instead.

A relatively simple extension of the above techniqueallows us to visualise the sign of the density gradientsrelatively easily for line mode. Whereas with the MoireÂfringe method a p=2 phase shift between two sets of lines is

Fig. 8. Variations in the mean (+) and root mean square (´) value of |Pij(t) ) Pij;0| as the mask is displaced by a milling machinetraverse for a a regular array of dots and b a random array of dots. The mask was traversed normal to the scan lines of the videocamera for the regular array of dots, and parallel to the scan lines for the random array of dots

Fig. 9. Flow over a hill towed to the left through a linear saltstrati®cation. The region of the tank visualised is approximately2.5 m long and 1 m deep

329

used, such phase shifting is not practical and unnecessaryfor synthetic schlieren. By utilising a line spacing greaterthan the ``ideal'' two pixels, we may construct

Pij;s � sign�oPij;0=oz� ; �15�and use this, where jopij;0=ozj is suf®ciently large, to obtainthe image

Pij;o � Pij;s�Pij�t� ÿ Pij;0� : �16�Figure 10b demonstrates the result of this additionalprocessing on the images used for Fig. 5a to obtain thesign of the vertical gradient of the perturbation density.The image Pij;o is everywhere proportional to oq0=oz, butthe constant of proportionality may vary spatially as theresult of p0�x; y� taking values other than the 0 and 1 statedin (11). An obvious extension would be to factor out theillumination by normalising Pij;0 and Pij�t� by the intensityof a mask containing no black lines. In the next section weshall introduce a more versatile approach to obtaining anaccurate quantitative measure of rq0 from the image pairPij;0 and Pij�t�.

5Line refractometryTo obtain quantitative information from the apparentmotion of the lines on the mask we must introduce amore general model for the intensity reaching the CCDsensor in the video camera. The need for this modelarises from an inability to form a sharp, uniform imageof the mask on the CCD sensor. Typically the optics leadto some blurring, and the illumination of the mask isnonuniform, rendering the functional relationship be-tween the images Pij;0 and Pij�t�, and the density gradientrq0 more complex than suggested by (13). In this sectionwe analyse the image formed by a mask of horizontallines and show how quantitative information may beextracted.

Suppose the changes in the refractive index gradientgive an apparent vertical displacement of the mask by

some amount Df at time t. We shall assume the curva-ture in q0 is small so that Df varies only over lengthscales large compared to the features contained in themask.

As we have seen, the intensity of a pixel is related tothe (unknown) intensity falling on the CCD sensor by(10). The combination of optical imperfections, noise andimperfections in the mask will ensure that p�x; z; t� is acontinuous function, even when the mask contains dis-crete steps. We may approximate p�x; z; t� using a piece-wise quadratic interpolation in a manner similar to thatemployed for numerical solution of the advection equa-tion in control volume techniques. The idea here is thatthe approximation Pij�t� � p�xi; zj; t� (approximating theintegral in (10) by the so-called mid-point rule for nu-merical integration) has an error O�Dz3� which is of thesame order as the error in a quadratic interpolation ofthe intensity �xi; zj�. More speci®cally, if P̂0;ij�zÿ zj� isthe quadratic interpolation of the unperturbed imagearound �xi; zj�, we look to solve for the valuezÿ zj � Dfij such that P̂0;ij�Dfij� � Pij�t�. Thus the ap-parent displacement (in the z direction) of the mask Dfij

is given by the roots of

P0;0 ÿ P� 12�P0;1 ÿ P0;ÿ1�Df

� 12�P0;1 ÿ 2P0;0 � P0;ÿ1�Df2 � 0 : �17�

Here we have used the shorthand P � Pij�t�, P0;0 � P0;ij,P0;ÿ1 � P0;i;jÿ1 and P0;1 � P0;i;j�1. To avoid ambiguity as towhich root of (17) should be taken, we solve (17) only ifP0;0 is intermediate between P0;ÿ1 and P0;1, and the in-tensity contrast across the three lines is suf®ciently large(i.e. jP0;1 ÿ P0;ÿ1j > DPmin). Further, we select the root of(17) with smallest jDfj, effectively limiting Df to be lessthan the spacing of the lines on the mask.

As an alternative to solving the quadratic expression forDf given by (17), we may utilise a binomial expansion toshow that his process has the same O�Dz2� accuracy asassuming Df is quadratic in P0;ij. This latter approach wasused by Sutherland et al. (1999) and gives

a b

Fig. 10. Preserving the sign of the density gradient with qualitative synthetic schlieren. a As for ®gure 7a but without taking theabsolute value of the difference. b As for ®gure 5a but calculating a Pij;s (Pij(t) ) Pij;0) with light regions representing positive ¶q¢/¶zand dark regions negative ¶q¢/¶z

330

Df � �Pÿ P0;0��Pÿ P0;ÿ1��P0;1 ÿ P0;0��P0;1 ÿ P0;ÿ1��ÿ �Pÿ P0;0��Pÿ P0;1��P0;ÿ1 ÿ P0;0��P0;ÿ1 ÿ P0;1�

�Dz : �18�

As with (17), Df is calculated from (18) only if P0;0 isintermediate between P0;ÿ1 and P0;1, and there is suf®cientintensity contrast across the three lines.

Once Df has been determined from either (17) or (18),it is mapped from pixel space into physical space and (9) isapplied to determine oq0=oz. Points for which Dz could notbe calculated (typically points where oP=oz is too small, asmay occur if a line is centred on a pixel and would lead toan ambiguity in the sign of the displacement) are replacedby interpolated values using a Gaussian weighting func-tion. The ®nal result is scaled and used to construct animage representing oq0=oz such as that shown in Fig. 11afor the same internal wave ®eld as shown in Figs. 5a and 7.By assuming there is no density perturbation on the planepassing through the centre of the cylinder, this densitygradient may be integrated with respect to z to obtain thedensity perturbation shown in Fig. 11b. We note, however,that wave re¯ections in this relative small tank�605� 100� 400 mm� generate density perturbationswhich extend throughout the domain and render the as-sumed `boundary condition' inappropriate. The experi-ments of Sutherland et al. (1999), which were performed ina much larger tank, do not suffer the same problems withre¯ections and demonstrate the full power of this tech-nique, providing more complete and accurate measure-ments of the internal wave ®eld than was previouslypossible.

6Dot tracking refractometryWith dot tracking refractometry, the mask consists of aregular array of dots. The precise position of each dot isdetermined using an intensity-weighted centroid relativeto some local threshold (determined, for example, from

the intensity histogram of a window larger than but ap-proximately centred on the dot). The centroids of the dotsare found ®rst for the base ¯ow, and then for each ¯owcontaining a density perturbation to be analysed. Thedifference in these two centroid locations �Dn;Df�ij for dotij is related to the in-plane density gradient Dq0ij through(9). This two-dimensional density gradient ®eld may thenbe integrated once, using an iterative method, to obtain thedensity perturbation ®eld. The one arbitrary constant (themean perturbation density) introduced by this process cannormally be assumed to vanish.

The apparently simple procedure outlined above ismade somewhat more complicated through the dots in thebase and perturbed images not lying on a regular grid(although the mask is itself regular), and the possibility ofthe dots disappearing or being distorted to such an extentbyrq0 that ambiguities arise as to which dots are paired inthe two images. Failure to consider these issues eithergives rise to spurious displacement vectors (and henceerrors in q0) or limits the apparent displacements to bemuch less than the spacing between the dots.

The spatial resolution of the dot tracking refractometryis simply the spacing between the dots which, for a givenCCD array, will necessarily be greater than the line spacingused for line refractometry as each dot must be resolved byat least two pixels in each direction. Sensitivity tests usingthe same set-up as described in Sect. 4.2, suggest thatapparent displacements as small as 0.02 pixels can bemeasured for line refrectometry and 0.04 pixels for dottracking refractometry. While the horizontal spatial reso-lution of dot tracking is substantially less than the pixelresolution achieved by line refractometry, dot tracking isable to recover both components of the density gradientand cope with larger apparent displacements of themask. As a result it may be used in some situationswhere the line refractometry does not yield the requiredinformation.

Figure 12a shows q0 calculated from dot tracking re-fractometry for the same internal wave ®eld used forFig. 7a. The density perturbation calculated from dot

a b

Fig. 11. Line mode refractometry for the internal wave ®eld shown in Fig. 5a: a vertical perturbation density gradient ¶q¢/¶z andb density perturbation q¢

331

tracking refractometry is somewhat smoother than linerefractometry because a knowledge of oq0=ox removes theaccumulation of errors associated with the uncoupledvertical integrations used for the latter. The superimposedarrows represent the gradient in the perturbation densityrq0 and illustrate the size and direction of the apparentmovement of the mask. Figure 12b shows the completedensity ®eld for the waves over the towed hill presentedqualitatively in Fig. 9.

The dot tracking method may also be used in con-junction with lines with only minor modi®cations to thealgorithm. The results are similar to those obtained withthe interpolation method presented in the previous sec-tion. The main limiting feature of both dot tracking andline interpolation is the trade off between a limited dy-

namic range and spatial resolution. In both cases themaximum useable apparent displacement is comparablewith the dot or line spacing. Larger apparent displace-ments lead to ambiguities. At the same time the spatial

resolution is limited to the spacing of the dots or lines.Hence increasing the dynamic range leads to a reducedspatial resolution. In many circumstances it is possible toachieve a dynamic range of O(100:1) between the largestand smallest values of Dq0 while retaining an adequatespatial resolution. In other ¯ows higher spatial resolutionsmay be required without compromising the dynamicrange and the next section outlines an alternate to ex-tracting the apparent motion of the mask which offers animproved balance between these two considerations.

7Pattern matching refractometryThe ®nal variant we consider here draws on the sameideas as employed for particle image velocimetry (PIV; amore appropriate name would be pattern matching ve-locimetry or PMV). For the results presented in this paperthe mask consists of a randomised array of dots (as usedfor Fig. 7b) so that each window large enough to coverseveral dots will contain a unique pattern of dots. The onlyrequirement for the mask, however, is that it has a strongtexture which does not repeat itself except over lengthscales large compared with the apparent movement of themask. Ideally it should have high contrast and the indi-vidual features of a size just detectable by the videocamera used.

As with the techniques outlined in the previous sec-tions, the refractive index perturbations associated with q0will then appear to shift the texture of the mask. Providedthe curvature terms o2q0=ox2 and o2q0=oz2 do not lead tolarge distortions of the patterns, establishing the apparentdisplacement ®eld is simply a matter of determining theshift that has to be applied to a window containing a sub-region of the textured image to map it back on to theimage of the unperturbed ¯ow.

This process is somewhat cleaner than the comparablePIV problem as we can control the uniformity of the``particle seeding'', and we do not have to worry aboutthree-dimensional effects causing particles to enter orleave the ``light sheet''. Dots, or whatever texture is usedfor the mask, can still disappear if the curvature in thedensity perturbation is too large, but this does not repre-sent a dif®culty in practice except in so far as the gradientswill not be resolved adequately in such regions.

As with PIV, a variety of pattern matching algorithmsare possible. Indeed our ®rst tests of this method utilisedsimple PIV routines written in 1991 based on optimisingone of three objective functions:

In each of these hu�x; z�i represents the average of ucalculated over all pixels falling in the interrogation win-dow described by xi ÿ wx � x � xi � wx andzj ÿ wz � z � zj � wz, where xi; zj describes the location

a

b

Fig. 12. Dot tracking refractometry. a Perturbation density q¢(greyscale) and gradient Ñq¢ (arrows) for oscillating cylinderinternal wave ®eld in Fig. 7a. b Complete density ®eld(showing contours of constant density) for ¯ow over a towed hillshown in Fig. 9

fcross�Dn;Df; xi; zj� � ÿ hP�x� Dn; z� Df; t�P0�x; z�i ÿ hP�x� Dn; z� Df; t�ihP0�x; z�ihP�x� Dn; z� Df; t�2i ÿ hP�x� Dn; z� Df; t�i2� �1=2 hP0�x� z�2i ÿ hP0�x; z�i2

� �1=2�19�

fabs�Dn;Df; xi; zj� � hjP�x� Dn; z� Df; t� ÿ P0�x; z�ji ; �20�fsq�Dn;Df; xi; zj� � h�P�x� Dn; z� Df; t� ÿ P0�x; z��2i : �21�

332

of the centre of the window of width 2wx and height 2wz.The fcross objective function is that typically used by mostinvestigators for PIV (note that here the cross correlationfunction is evaluated directly rather than imposing a pe-riodicity constraint and using Fast Fourier Transforms).The fabs and fsq objective functions are computationallysimpler (and were originally introduced as they could becalculated by the frame hardware an order or magnitudefaster than the CPU could process the images) andtherefore faster to compute. In all cases the objectivefunction is minimised (hence the negative sign in (19)),although there are a number of algorithms for doing so. Inthis paper we consider two such algorithms.

The ®rst algorithm minimises the objective function forinteger values of Dn and Df to obtain a pixel-resolutionmeasure of the apparent displacement. A surface (biqua-dratic in this case) is then ®tted in the neighbourhood ofthis minimum to provide subpixel resolution for the op-timal shift. The performance of the pattern matching re-fractometry using this algorithm, and the relative merits offcross, fabs and fsq, have been assessed by a sensitivity studysimilar to that in Sect. 4.2. The results of this study arepresented in Table 1 in the column headed ``Simple''. Thevalues quoted are the mean and standard deviation of thedisplacements calculated using interrogation windows15� 15 pixels at 625 separate locations. The same pair ofimages was used for each objective function for a giventraverse displacement.

The second algorithm is similar to the ®rst but ratherthan restricting the calculation of fcross; fabs or fsq to in-teger values of �Dn; Df�, we interpolate the image to ob-tain values for P�x� Dn; z� Df; t� (for use in (19), (20) or(21)) with noninteger �Dn;Df��<1� in the neighbourhoodof the minimum identi®ed by the biquadratic ®t. Theobjective functions fcross, fabs or fsq are calculated from theinterpolated image and a biquadratic function is again®tted in the neighbourhood of the minimum to obtain theoptimal �Dn;Df�. This process can be repeated (withP�x� Dn; z� Df; t� calculated over decreasing intervals)if so desired. For the results presented in Table 1 the imageinterpolation was restricted to be bilinear. To improve thecomputational ef®ciency of this process, multigrid meth-ods are employed whereby for successive outer iterationsthe spatial resolution is doubled and the search region forthe pattern matching is reduced.

Comparison of the results presented in Table 1 usingthe two competing algorithms and three different objectivefunctions suggests some improvement may be gained byusing the interpolating algorithm with a reduction in thestandard deviation of the displacements calculated for a

a

b

c

Fig. 13. Pattern matching refractometry. a Horizontal gradient¶q¢/¶z, b vertical gradient ¶q¢/¶z, and c perturbation density q¢for oscillating cylinder internal wave ®eld shown in Fig. 7b

Table 1. Displacements (inmm) calculated using differentmatching algorithms

Description Simple Interpolate three times

Nominal displacementfrom digital readout

0.100 � 0.010 1.000 � 0.010 0.100 � 0.010 1.000 � 0.010

fcross, biquadratic ®t 0.066 � 0.017 0.975 � 0.020 0.086 � 0.012 1.011 � 0.018fabs, biquadratic ®t 0.051 � 0.014 0.939 � 0.019 0.072 � 0.011 0.972 � 0.018fsq, biquadratic ®t 0.067 � 0.015 0.975 � 0.020 0.071 � 0.011 0.989 � 0.018

Average 0.061 � 0.015 0.963 � 0.020 0.076 � 0.011 0.991 � 0.018

333

given image pair and a slight convergence of the differentobjective functions. The difference between the three ob-jective functions with the interpolating algorithm allows usto estimate an upper bound for the error associated withthe algorithm/objective function combination as approxi-mately 0.01 mm (�0:01 pixels). The standard deviationsfor each algorithm, function and displacement are ap-proximately constant and somewhat less than 0.02 mm(0.02 pixels). We believe this standard deviation repre-sents the optical distortion, thermal and signal noise in thepair of images. The ®nal component of the difference be-tween the displacement from the digital readout, and thatcalculated by the pattern matching is due to mechanicalimperfections in the traverse and digital readout and isconsistent with the �0:010 mm quoted by the manufac-turer.

Figure 13 illustrates the performance of patternmatching refractometry (using an 11� 11 pixel interro-gation window) with the same ¯ow as used for previousmethods. Figure 13a and b show the two components ofthe gradient of the density perturbation, while Fig. 13cshows the density perturbation itself. Note that, as with theearlier density ®elds, re¯ections from the boundaries ofthe tank make this wave ®eld much more complex thanexpected and remove some of the normal symmetries.

Figure 14 compares the experimentally measured gra-dient in the perturbation density ®eld with viscous theory.The solid curve shows the experimental measurement ob-tained by pattern matching refractometry along a linenormal to the lower left beam of the internal waves withx=N � 2ÿ1=2 at a distance of 8 radii from the cylinder asthe cylinder reaches the bottom of its stroke. The dashedcurve is the prediction from the theory of Hurley andKeady (1997) for the same conditions. The systematicdifferences are here dominated by re¯ections of the in-ternal waves from the tank boundaries. A similar level ofagreement is found at other phases of oscillation. Suth-erland et al. (1999) have presented a more detailed com-parison and analysis of this ¯ow using line refractometry

and found some systematic differences related to theboundary layer approximations used in Hurley and Kea-dy's theory.

When compared with the dot tracking refractometry inSect. 6 the advantages of pattern matching refractometryare the ability to operate at higher spatial resolutions andwith larger apparent displacements of the mask. Thespatial resolution for pattern matching refractometry isagain controlled by the dot spacing, but here the dots willtypically be smaller and closer together. The use of over-lapping pattern windows optimises the resolution. Thelarger the windows the greater the resolution in rq0, butthe lower the spatial resolution. In dot tracking refract-ometry the maximum apparent displacements were limit-ed by the dot spacing as larger movements lead to aliasingerrors. With pattern matching refractometry the patternscan have substantial apparent displacements as each re-gion of the pattern retains its own unique identity. On thedown side the computational cost per displacement vectoris substantially larger, but with the ever-increasing per-formance of computer hardware, this is unlikely to rep-resent a serious issue.

8ConclusionsThe synthetic schlieren techniques outlined in this paperare able to provide qualitative visualisations of a sensi-tivity and nature comparable with classical schlieren, yetare substantially easier to set up and may be scaled to copewith ¯ows in signi®cantly larger domains without expen-sive parabolic mirrors. Obtaining quantitative measure-ments of the density perturbation is a relatively simpleextension, with a number of distinct methods of obtainingthe required information being described. It is unfortunatethat the set-up for the experiments used here to illustratethe techniques resulted in the visualisations being con-taminated by re¯ected waves, and that there was a highlevel of thermal noise in the laboratory signi®cantly re-ducing the signal to noise ratio.

Fig. 14. Comparison between experiment(solid line) theory (dashed line) for thecross-beam structure of the vertical gra-dient in the perturbation density a dis-tance 8R from the cylinder. The cross-beam coordinate r is centred on the wavebeam with the orientation shown in theinsert

334

Perhaps the biggest drawback compared with tradi-tional schlieren is the reduced spatial resolution. For many¯ows, such as the internal waves used here to illustrate themethods, this is unimportant as the curvature in thedensity ®eld remains small. In some cases, however, it isimportant to resolve both the regions of high curvature,where the apparent displacement of light rays may occurover scales comparable to or smaller than the spacing ofthe mask elements, and regions of low curvature in q0.We are currently investigating a hybrid of shadowgraphyand the synthetic schlieren techniques reported hereby which simultaneous measurements of r2q0 and rq0could be made to improve the resolution of fronts, shocksor layers.

While quantitative information requires the ¯ow to betwo-dimensional, useful measurements and qualitativevisualisations may also be obtained in weakly three-di-mensional and turbulent ¯ows, as illustrated by the imageof thermal convection from a hand shown in Fig. 5b. Themethods described here may also be used to obtain mea-surements of depth or layer depth when a free surface orinternal interface are present, provided the refractive in-dices are constant within the layer (or layers). Furtherdiscussion on these alternative applications is beyond thescope of this paper.

Each of the three quantitative techniques reported herehas its strengths. We believe, however, that ultimatelypattern matching refractometry will prove to be the mostuseful and versatile. It is the easiest to set up as its does notrequire alignment of the mask and video system, and theprecise form of the mask is unimportant. Moreover, itoffers the measurement of both components of rq0 whileachieving both a large dynamic range and good spatialresolution.

ReferencesBurton RA (1949) A modi®ed schlieren apparatus for large areas

of ®eld. J Opt Soc Amer 39: 907±909

Dalziel SB; Hughes GO; Sutherland BR (1998) Synthetic schlie-ren. In: Carlomagno; Grant (ed) Proceedings of the 8th Inter-national Symposium on Flow Visualization, ISBN 0 9533991 09, paper 062

Gill AE (1982) Atmosphere-Ocean Dynamics, Academic Press,London, 662 pp

Greenberg PS; Klimek RB; Buchele DR (1995) Quantitativerainbow schlieren de¯ectometry. Appl Optics 34: 3810±3822

Hurley DG; Keady G (1997) The generation of internal waves byvibrating cylinders, Part 2: Approximate viscous solution for acircular cylinder. J Fluid Mech 351: 119±138

Irvin BR; Ross J (1991) Observations of pattern evolution inthermal convection with high-resolution quantitative schlierenimaging. Phys Fluids A 3: 1699±1710

Lin J-C; Rockwell D (1997) Quantitative interpretation of vorticesfrom a cylinder oscillating in quiescent ¯uid. Expts in Fluids 23:99±104

Merzkirch W; Peters F (1992) Optical visualisation of internalgravity waves in strati®ed ¯uid. Optics and Lasers in Engi-neering Vol. 16 pp. 411±425

Mortensen TA (1950) An improved schlieren apparatus em-ploying multiple slit-gratings. Rev Sci Instrum 21: 3±6

Mowbray DE; Rarity BSH (1967) A theoretical and experimentalinvestigation of the phase con®guration of internal waves ofsmall amplitude in a density strati®ed liquid. J Fluid Mech 28:1±16

Peters F (1985) Schlieren interferometry applied to a gravity wavein a density-strati®ed liquid. Expts in Fluids 3: 261±269

Sakai S (1990) Visualisation of internal gravity waves by MoireÂmethod. Kashika-Joho 10: 65±68

Sutherland BR; Dalziel SB; Hughes GO; Linden PF (1999) Visu-alisation and measurement of internal waves by ``syntheticschlieren''. Part I: Vertically oscillating cylinder; to J FluidMech 390: 93±126

Turner JS (1973) Buoyancy effects in ¯uids, Cambridge Univer-sity Press, 368 pp

Weast RC (1981) Handbook of chemistry and physics C.R.C.Press, 62 edition

Weyl FJ (1954) Analysis of optical methods. In Ladenburg RW(ed) Physical measurements in gas dynamics and combustion.Princeton University Press, Princeton, New Jersey, 3±25

van Dyke M (1982) An album of ¯uid motion, Parabolic press,Stanford, 176 pp

335