Numerical study of light scattering by aboundary-layer flow

Darek J. Bogucki and Julian A. Domaradzki

Temperature inhomogeneities in free, isotropic turbulence have the effect of scattering light in near-forward angles. We investigate numerically modifications of free turbulence by a rigid wall and its effecton the propagation of light through turbulence. The wall is a 5 cm optical window placed at the leadingedge of an instrument towed with speeds of 0.1 and 1 m�s in free turbulence. The turbulent flow fieldpresents inhomogeneities of an embedded passive scalar (Pr � 7, temperature in water), which aremodified by the boundary layer developing on the window. We find that the developing laminar boundarylayer has a negligible effect on light scattering for the investigated geometry when considered in termsof the volume-scattering function (differential cross section). This indicates that the boundary layer is notan obstacle for optical measurements of turbulence. © 2005 Optical Society of America

OCIS codes: 010.0010, 010.7060.

1. Introduction

The existence of a boundary-layer flow associatedwith rigid walls is a fact of life for any optical in-strument operating in an aquatic environment.Consequently measurements carried out by such in-struments must factor in the optical properties of theboundary-layer flow in water. Here we examine theeffect of a boundary layer for an instrument thatmeasures turbulence optically.

Most oceanic flows are turbulent and are charac-terized by a wide range of coexisting scales of motionranging between a few millimeters and a few meters.The turbulent fluctuations of passive scalars (such astemperature and salinity) in water cause fluctuationsin the density of the fluid, hence in its refractiveindex. These inhomogeneities in the refractive indexare primarily caused by the temperature field.1

The scattering properties of oceanic, isotropic, ho-mogeneous turbulence were recently investigated byBogucki et al.,1 who showed excellent agreement

between the numerical estimates of the volume-scattering function (VSF) (the differential cross sec-tion for light scattering) due to turbulence and theVSF observed in a convection tank. The measure-ments were carried out across a convective tank witha fully turbulent core and a boundary layer at theentrance and at the exit. Despite passing through twoboundary layers, the measured VSF was the same asfor homogeneous and isotropic turbulent flows simu-lated numerically. These results indirectly suggestthat the boundary layer (BL) did not effectively mod-ify the scattered light wavefront for the path lengthemployed in the tank.

Although the effect of the BL on light propagation(and its implications for measurements) has been theobject of frequent speculations, few definitive resultshave been documented. The light propagationthrough the BL has been studied only in the contextof underwater optical holographic interferometry.2,3

These studies qualitatively established that theboundary layer did not affect the holographic images.

Here we report on a numerical study of a lightbeam propagating through a BL flow in a configura-tion mimicking typical optical in situ oceanographicmeasurements. Specifically, we have examined a BLflow where a fully developed, isotropic, homogeneousturbulent water parcel is advected with variousspeeds of the mean flow over a no-slip, varying-lengthboundary.

To accomplish this, first we carried out a directnumerical simulation (DNS) of homogeneous, isotro-pic turbulence with a passive scalar with Pr � 7

D. J. Bogucki (dbogucki@rsmas.miami.edu) is with the Rosen-stiel School of Marine and Atmospheric Science, Division of Ap-plied Marine Physics, University of Miami, 4600 RickenbackerCauseway, Miami, Florida 33149-1098. J. A. Domaradzki is withthe Department of Aerospace and Mechanical Engineering, Uni-versity of Southern California, Los Angeles, California 90089-1191.

Received 8 November 2004; revised manuscript received 16March 2005; accepted 17 March 2005.

0003-6935/05/255286-06$15.00/0© 2005 Optical Society of America

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(water). Next this turbulent field was embedded in aBL flow with two mean flow speeds, 0.1 and 1 m�s,developing over a 5 cm rigid wall. Subsequently, theBL flow with embedded isotropic turbulence hasevolved over time equal to one or two periods corre-sponding to the advection of a mean flow over thewindow. Finally, using a geometrical optics approxi-mation,1 the scattering of a light beam propagatingthrough the boundary layer has been calculated andcompared with scattering on unmodified, isotropicturbulence. The effect of light scattering on the BLflow is expressed by the VSF; hence it is applied to awide array of measurement possibilities.

Below we present the basic underlying theory oflight scattering on turbulence and the numerical cal-culations of the turbulent flow and of the light prop-agation. Finally we discuss the results and theiroceanographic implications.

2. Light Scattering on Turbulence

A. Theory

For short propagation paths L, such as required forour calculations, O�1� m, the geometrical opticsapproximation was found to be accurate for the near-forward-scattering problems.1 In the scalar and near-forward approximation to Maxwell’s equation thefunction � representing the complex amplitude oflight propagating in the z direction over the pathlength L can be expressed as4

�(x, y, z � L) � A0(x, y)exp�i� �0

L

n(x, y, z�)dz��,

(1)

where A0 is the initial value of ��x, y, z� at z � 0, � isthe light wavenumber, � � 2���, where � is thewavelength, and n�x, y, z� is the deviation of the localrefractive index from its mean value.

The value taken by the exponent in Eq. (1) at each�x, y� position describes how the initially plane-parallel wavefront has been distorted by a light-beampassage through the turbulent volume.

To quantify the effect of light scattering we use theVSF or VSF (�). The VSF describes the angular dis-tribution of the scattered radiation as the scatteredradiant intensity dI��� in a direction � per unit scat-tering volume dV divided by the incident irradianceEO:

VSF(�) �dI(�)EOdV . (2)

The smallest-scale temperature inhomogeneitiesthat are responsible for light scattering depend on theparameters of a turbulent flow. The relative contri-bution from different scales of the flow to the lightscattering depends on the three-dimensional temper-ature variance spectrum ET�k�, where k is the wave-number. The physical interpretation of ET�k� is that

the temperature variance for scales between k andk � dk is proportional to ETdk. In general the spec-trum ET�k� depends parametrically on molecularproperties, the kinematic viscosity � and diffusivityD, as well as on the kinetic energy dissipation rate �and the temperature variance dissipation rate �.

The rate of dissipation of turbulent kinetic energy� determines the size of the smallest temperaturestructures through the Batchelor length scale, �B

� �D2��.4 We showed previously,5 for isotropic ho-mogeneous turbulence, that the largest contributionto scattering comes from the scales of flow in thevicinity of the Batchelor length scale �B. This is anattractive and important observation because itmakes light scattering on the turbulent flow a uni-versal process (independent of the flow type, i.e., BLor isotropic flow, as long as the dissipation range isfar removed from the energy containing range) de-pending only on the local parameters of the flow � and�.

B. Numerical Calculations of Light Scattering onTurbulent Flow

Since the general approach to calculating light scat-tering on the simulated turbulent flow is presentedelsewhere1,5; here we simply outline the major steps.To numerically compute the VSF, we consider thesurface of the constant phase at a distance z � L fromthe entrance plane:

�(x, y) ��0

L

n(x, y, z�)dz�. (3)

A vector N normal to the isosurface,

N � ���

�x ,��

�y , 1�, (4)

is parallel to the outgoing ray direction and can becalculated at each mesh point at the exit plane. TheVSF is a properly normalized probability distributionof ray angles. The spatially varying fluctuations inthe refractive index, n�x, y, z� in Eq. (3), are calcu-lated from the fluctuations of a temperature field,obtained in the DNSs of turbulence with the temper-ature as a passive scalar.

3. Numerical Calculations of Free Turbulence Modifiedby the Boundary-Layer Flow

We assume that our imaginary instrument (typical ofin situ measurements) is towed in water; ocean tur-bulence (or free turbulence) is continually enteringthe field of view of the optical window while retainingproperties of the originating isotropic, homogeneousflow. After free turbulence comes into contact withthe rigid optical window, it undergoes stretching,which may result in local sharpening of the scalargradient near the boundary. This gradient sharpen-ing was speculated to alter light-scattering proper-

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ties. In this paper we aim to refute this claim andmore specifically to address the following questions:

(1) What are the light-scattering properties of freeocean turbulence?

(2) How are the scattering properties modified bythe presence of rigid boundaries introduced by theinstrument?

To address the first question we performed directnumerical simulations of isotropic turbulence for theconditions collected in Table 1. To address the secondquestion we performed DNSs of the same isotropicturbulence field, evolving in the mean boundary layerflow, for the conditions corresponding to the first twoentries in Table 2. The DNS calculations were done intwo consecutive stages: First, we generated a fullydeveloped, isotropic, homogeneous turbulent flow. Inthe next stage this flow was numerically advectedover a no-slip, 5 cm long boundary with two speeds ofthe mean flow. In addition we allowed the initialisotropic turbulent flow to freely decay for a periodequal to the advection time for a 0.1 m�s speed overthe 5 cm optical window, corresponding to 0.5 s. Thiswas done to compare the VSF for turbulence with andwithout the influence of the boundary after the sameevolution time.

The ratio of the BL thickness � to the total light-beam path length L is an important parameter de-termining the influence of the boundary layer on thelight scattering. In our simulations the BL thicknessis �15% of the light-beam path length of 2.41 cm.

Since this path length is much shorter than the typ-ical optical path length of optical measurements (tensof centimeters), our investigation represents an up-per bound for the estimated impact of the BL onscattering.

A. Isotropic Turbulence

We assume (following Ref. 5) that most frequently therefractive-index fluctuations of the oceanic water isdetermined by the temperature distribution. We thensimulate isotropic turbulence in water � � 10 6

m2�s� with passive scalar with the value of thePrandtl number, Pr � 7, characteristic of tempera-ture in water. The intensity of turbulence is deter-mined by the rms turbulent velocity u and the kineticenergy dissipation rate ε, which for oceanic conditionsis in the range from 10 4 to 10 10 m2�s3. Our simula-tions used ε of 3.7 � 10 6 m2�s3. This value is repre-sentative of a very energetic ambient water column;hence our results are likely overestimates of the ef-fect of scattering for a typical oceanic measurement.

The smallest relevant scales of turbulent velocityand the temperature field are determined by the Kol-mogoroff and Batchelor length scales and �B, re-spectively:

� � �3

� 14, �B � Pr 1�2�. (5)

The large scales of turbulence are determined bythe integral length scale Lint. The large and smalleddy turnover times, and �, respectively, are de-fined as

� � Lint�u, � � ��vK � (��)12. (6)

The relevant physical parameters of simulated tur-bulence for Pr � 7 are collected in Table 1 (all quan-tities are in SI units).

The simulations were performed by using a spec-tral code with periodic boundary conditions6 and aresolution of 1283 mesh points. The simulation do-main is a cube of a linear size L (see Table 1) in thecomputational space always equal to 2�. The form ofthe initial energy spectrum is

E(k) �1254

2

�exp( 5�k), (7)

where the exponential form is characteristic of thedissipation range of turbulence.7,8 The spectrum isimplemented in the simulation performed with theresolution of N3 modes (mesh points) by specifyingthe maximum resolved wavenumber �kmax � K. For� � 3.7 � 10 6 we took K � 5, guaranteeing anexcellent resolution of the dissipation range. The pas-sive scalar was initialized with the same spectralform as Eq. (7). The simulations were run with forc-ing at low wavenumbers of more than 1.25 large eddy

Table 1. Relevant Parameters, in SI Units, for DNS Simulations ofTurbulent Flow Field

Parameter � 3.7 � 10 6 �m2�s3�

L �m� 0.0241Lint �m� 0.00768� �m� 0.00073�B �m� 0.00028� �m� 0.00576u �m�s� 0.0024532 u2 �m2�s2� 0.92 � 10 5

(s) 3.2� �s� 0.52R� �m� 15

Note: L, size of simulation domain; Lint, integral length scale ofturbulence; , Kolmogoroff length scale; �B, Batchelor length scale;�, Taylor microscale; u, rms turbulent velocity; 3

2 u2, turbulentkinetic energy; , large eddy turnover time; �, Kolmogoroff timescale; R�, microscale Reynolds number.

Table 2. Characteristics of the Boundary-Layer Flow

u0 �m�s� � �m� Re tadv �s�

0.1 3.5 � 10 3 348 0.51 1.1 � 10 3 1100 0.059 3.6 � 10 4 3200 0.0056

Note: u0, free-stream velocity; �, boundary layer thickness at x� 5 cm; Re, Reynolds number based on �; tadv, advection time withvelocity u0 for a distance of 5 cm.

5288 APPLIED OPTICS � Vol. 44, No. 25 � 1 September 2005

turnover times, when the turbulence reached a fullydeveloped, statistically steady state. This fully devel-oped turbulent field served as an initial condition forsubsequent runs, both for boundary layer and freelydecaying, isotropic turbulence.

B. Isotropic Turbulence Evolving in a Boundary-LayerFlow

The simulated BL flow corresponds to an instrumenttowed in water with an optical window with a lineardimension of �5 cm placed at the leading edge. InTable 2 we collect values of the BL thickness � (adistance from the wall where the flow velocity isequal to 0.99u0, where u0 is the mean, free-streamvelocity) and the Reynolds number Re for differenttowing velocities and a location, x � 5 cm, from theleading edge. The boundary layer thickness � is cal-culated from9

� 4.9 �xu01�2

, (8)

and the Reynolds is

Re � u0��. (9)

The critical Reynolds number, indicating the tran-sition to a turbulent flow, is Rec 1800 [or 520 for theReynolds number based on the displacement thick-ness, �1 � 1.7208 �x�u0�1�2].9 Therefore at the twolowest velocities in Table 2 the BL is stable. At highervelocities the BL is unstable, but the Reynolds num-bers are sufficiently low for the flow not to be fullyturbulent but merely in the transition to turbulence.Only two low-velocity cases were considered for sim-ulations. The simulations were performed by using aspectral BL code10,11 with the mean velocity obtainedas a solution of the Blasius BL (the analytical solu-tion for a laminar flat-plate boundary layer with aconstant external flow velocity) for the parameters inTable 2. The simulations were initialized by usingvelocity and a scalar field for Pr � 7 from the isotropicrun, interpolated to the stretched BL mesh. The BLsimulations were then run for one or two advectiontimes from Table 2, corresponding to either one ortwo complete times of propagation over the opticalwindow. The advection time is 0.05 and 0.5 s for 1 and0.1 m�s flow speed, respectively. Additionally, start-ing from the same initial condition, we have simu-lated decaying isotropic turbulence over a time of0.5 s. The simulation parameters for the boundarylayer runs (b1 and b2) as well as for the decayingisotropic turbulence run �i2� are listed in Table 3.

Using the flows calculated above, we assembled theDNS database and propagated a light beam throughit by using Eq. (1). The results are presented for thefollowing flows and times T, measured from the in-stant at which the fully developed turbulence hasbeen established:

(1) initial, fully evolved isotropic, homogeneousturbulent flow �T � 0 s�;

(2) isotropic turbulence decayed from the initialcondition (T � 0.5 s, run i2);

(3) flow advected once over the optical windowwith low speed (u0 � 0.1 m�s, T � 0.5 s, run b1);

(4) flow advected once over the optical windowwith high speed (u0 � 1 m�s, T � 0.05 s, run b2);

(5) flow advected twice over the optical windowwith high speed (u0 � 1 m�s, T � 0.1 s, run b2).

C. Calculations of the Volume-Scattering Function

To calculate the VSF of light scattered on therefractive-index inhomogeneities, we numerically in-tegrated temperature fluctuations over variable z (inthe boundary layer this is the direction normal to thewall) and computed the derivatives of the integratedfield with respect to x and y [Eq. (4)].

The numerical temperature corresponding to theDNS, Tc�x, y, z�, is converted by a suitable transfor-mation to the real-world temperature field, Tr�x, y, z�,with a prescribed temperature variance dissipation �r

and prescribed r. Requiring that the temperaturespectra ET�k� in the dissipation range have a univer-sal form defines the transformation for the tempera-ture field1:

Tr � (�r��c)1�2(c�r)

1�4(r�c)1�4Tc, (10)

where �c and c are results from the DNS calculationsperformed with the kinematic viscosity c and r and�r are the actual dissipation rates in a fluid with theviscosity r for which the Tr field is sought. For sim-ulated turbulence r � 3.7 � 10 6 m2�s3, and thetemperature dissipation rate was chosen to corre-spond to the values used in earlier experiments, �r

� 2 � 10 6 °C2�s. Subsequently the dimensional tem-perature field is converted to fluctuations of the re-fractive index by using the linear relationship12:n�x, y, z� � 1.2 � 10 4 � T�x, y, z� for an ambienttemperature of T0 � 20 °C and the wavelength �� 670 nm.

The final calculations of the VSF are done in astraightforward manner from Eq. (2), as described byBogucki et al.5

4. Results

The results in a form of the scattering angle maps arepresented in Fig. 1. The maximum value of the scat-tered light in our experiment is around 0.1 mrad. The

Table 3. Simulated Databases for the Boundary-Layer Runs(b1 and b2) and the Decaying Turbulence Run (i2)

Parameter Run b1 Run b2 Run i2

u0 �m�s� 0.1 1.0 0c �m2�s� 10 6 10 6 10 6

�t �s� 0.0005 0.00005 0.0005Nt 1000 2000 1000tadv �s� 0.5 0.05 N�A

Note: u0, mean free-stream velocity; c, kinematic viscosity; Nt,number of time steps; tadv, advection time.

1 September 2005 � Vol. 44, No. 25 � APPLIED OPTICS 5289

dark spots correspond to maximum scatter, thelighter shades to less scatter.

The maps in Fig. 1 can be interpreted more intu-itively if we notice that in the geometrical optics ap-proximation the gradient of the scattering angle isrelated to the intensity of the emerging light13; hencethe observed differences correspond to different mapsof light intensity after propagation through theboundary layer.

Note differences when comparing the scatter anglemap for the isotropic flow at the initial time (Fig. 1,top, left, no BL) to an isotropic but decaying flow (Fig.1, top, right, no BL) after T � 0.5 s. The decaying flowbecomes a weaker light scatterer since the moleculardiffusivity erodes the strongest temperature gradientstructures.

Comparing the initial scatter (Fig. 1, top, left, noBL) with the scatter on turbulence advected with alow speed over the optical window (Fig. 1, bottom,left, with BL), the scattering maps show some erosionin the high-scattering regions but much less than fora freely decaying flow (Fig. 1, top, right, no BL). Thisis because turbulence entering the window is activelymaintained by the mean BL flow. Similar behavior isobserved for the high-speed propagation case (Fig. 1,bottom, right, with BL).

In Fig. 2 we compare the VSF for the initial turbu-lent field and the decayed field for run i2. The calcu-

lated VSFs are self-similar down to the smallestnondimensional angle of � � 10 5 rad. At the smallestangles the VSFs become more erratic because therelatively less scattering energy was observed atthose angles contributing to noisiness in the results.

In Fig. 3 we compare the VSF for all BL runs: that

Fig. 1. (Color online) Map of the scattering angle for all runs. Panel A: initial, isotropic homogeneous flow �T � 0 s�; Panel B: isotropic,homogeneous, decaying flow after time T � 0.5 s; Panel C: flow advected once over the optical window with low speed �0.1 m�s�, T� 0.5 s; Panel D: flow advected twice over the optical window with high speed �1 m�s�, T � 0.1 s.

Fig. 2. (Color online) VSF for the isotropic homogeneous turbu-lence forced run �T � 0 s� and the decaying run �T � 0.5 s�.

5290 APPLIED OPTICS � Vol. 44, No. 25 � 1 September 2005

due to homogeneous isotropic turbulence �T � 0 s�,that due to the flow that passed once over the windowat a speed of 0.1 m�s �T � 0.5 s�, that of the flow thathad passed once over the BL at a high speed of1 m�s �T � 0.05 s�, and finally that which passedtwice over the BL �T � 0.1 s�. All VSFs collapse wellto a self-similar shape with the exception of the an-gular range from � � 0 to � � 10 5 rad, which weattribute to the finite resolution of the runs. Figure 3makes a compelling case that the presence of the BLhas no discernible effect on the VSF. This is not sur-prising and confirms previous investigations.1

5. Conclusions

The simulated BL flow over typical optical windowsused in in situ measurements �5 cm� is laminar attypical towing speeds as long as the window is locatedat the leading edge. The developing BL over the win-dow is speed dependent and is thickest at lower tow-ing speeds. For analyzed towing speeds of 0.1 and

1 m�s, and a ratio of the total optical path length andthe BL thickness of the order of 10, we have foundthat the influence of BL flow on light scattering byturbulence advected over the window is negligible.Hence we do not expect the boundary-layer effect toplay a significant role at higher ��1 m�s� speeds.

This work was supported by a grant from Spectra-Scan Corporation.

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Fig. 3. (Color online) Comparison of VSF for all BL flows.

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