Interpretation of color fringes in flowing soap films

Experimental Thermal and Fluid Science 25 (2001) 141±149

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Interpretation of color fringes in ¯owing soap ®lms T.-S. Yang a, C.-Y. Wen a b

b,*

, C.-Y. Lin

b

Department of Mechanical Engineering, National Cheng-Kung University, 70101 Taiwan, ROC Department of Mechanical Engineering, Da-Yeh University, Chang-Hwa, 51505 Taiwan, ROC Received 7 January 2001; accepted 23 June 2001

Abstract Soap ®lms were introduced to carry out classical hydrodynamics experiments on two-dimensional (2-D) ¯ows by Couder [J. Phys. Lett. 42 (1981) 429±431] nearly two decades ago. The thickness of the ®lm, ranging from 1 to 10 lm, varies slightly with the velocity ®eld. This small thickness variation, however, results in fascinating color fringes, thus providing an excellent means for ¯ow visualization. Here, a theoretical investigation is conducted to interpret the physical meaning of the color fringes that appear in soap ®lm ¯ows. It is shown that the color fringes resemble streamlines in the case of steady ¯ow, under some domains of the parameter space. Also, photographs of ¯ows over a circular cylinder and a backward-facing step taken in a horizontal soap ®lm tunnel and numerical simulations under the same conditions of the experiments are presented. The color fringes shown in the photographs are in good agreement with numerically computed streamlines. Ó 2001 Elsevier Science Inc. All rights reserved. Keywords: Soap ®lms; Color fringes; Streamlines

1. Introduction Recently, a new type of experimental technique using soap ®lms to set up two-dimensional (2-D) ¯ows has been introduced by Couder [1], Gharib and Derango [2], and Kellay et al. [3]. Couder [1] stretched the soap ®lm on a large frame and used it as a 2-D towing tank. Gharib and Derango [2] designed and built the ®rst continuously running soap ®lm tunnel, in which a suspended horizontal soap ®lm was set into motion in a horizontal long frame using a planar water jet as a pulling mechanism. Kellay et al. [3] proposed and tested another type of continuously running soap ®lm tunnel. In their setup, a ¯owing down soap ®lm driven by the force of gravity is bounded at its edges by two nylon ®shing lines diverging from a soap reservoir. These ®lms, with thickness ranging from 0.1 to 10 lm, provide the closest physical approximation to the concept of a truly 2-D ¯uid. The ratio of the characteristic length of the ¯ow structures to the ®lm thickness routinely exceeds 104 . Recent experiments performed in these soap ®lm devices have shown certain features of turbulent ¯ow * Corresponding author. Tel.: +886-4-852-8469x2111; fax: +886-4852-6301. E-mail address: [email protected] (C.-Y. Wen).

that resemble those anticipated for a true 2-D system (e.g., [1,4±10]). One fascinating feature of soap ®lms is that, when observed in white light, they show colorful interference patterns. This feature provides an instantaneous and global visualization of the ¯ow structure. The interference patterns result from the small thickness variation with the velocity ®eld. Light of wavelength k is strongly re¯ected when the ®lm thickness h is an integral multiple of k=2. Wu et al. [7] have argued that under conditions of rapid ®lm ¯ow, the thickness obeys the equation of a 2-D passive scalar, such as a dye injected into a ¯ow ¯uid. Rivera et al. [9] used particle image velocimetry (PIV) to measure turbulent velocities and vorticity ®elds produced by a grid inserted in a ¯owing soap ®lm. In the mean time, they obtained semi-quantitative measurements of ®lm thickness from the intensity of the light scattered by the seeding particles and showed that the thickness ®eld was behaving both as a passive scalar and as visualization of the vorticity. Chomaz and Cathalau [11] and Chomaz and Costa [12] took heuristic approaches and derived the dynamical equations for the in-plane velocity, ®lm thickness, and soap concentrations using the idea of soap ®lm particle. They then simulated various regimes on a computer and observed the similar behavior of the thickness variation with the velocity ®eld as Rivera et al. [9]. But as yet, no proper

0894-1777/01/$ - see front matter Ó 2001 Elsevier Science Inc. All rights reserved. PII: S 0 8 9 4 - 1 7 7 7 ( 0 1 ) 0 0 0 8 7 - 5

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Nomenclature C C d f H h h1 ; h2 L p Re S St t U U1

capillary number (lU =r) C=e3 cylinder diameter (m) shedding frequency (1/s) mean thickness of a soap ®lm (m) total thickness of a soap ®lm (m) ®lm surface elevation (m) characteristic length of a soap ®lm ¯ow (m) 2 pressure (N=m ) Reynolds number (UL=m) height of the backward-facing step (m) Strouhal number (fd=U1 ) time (s) characteristic ¯ow velocity (m/s) free-stream velocity (m/s)

demonstrations have validated the above experimental and numerical observations. The objective of this study is to investigate how the thickness depends upon the motion. A theoretical analysis of the three-dimensional (3-D) soap ®lm dynamics is proposed by assuming simply that the thickness of the ®lm is small compared to the characteristic length scale of the in-plane ¯ow. Numerical simulations and photographs of ¯ows over a circular cylinder and a backward-facing step taken in a horizontal soap ®lm tunnel are also presented to support the analytical results. 2. Formulation and scale analysis 2.1. Formulation Consider a soap ®lm stretched on a horizontal frame, as shown in Fig. 1. For a dilute soap solution, it is appropriate to assume that the solution is incompressible with a constant density q and a kinematic viscosity m ˆ l=q, l being the dynamic viscosity of the soap solution. Let the frame be on the x±y plane, and the z-axis perpendicular to the frame. The two (deformed) ®lm

Fig. 1. Schematic of the horizontal soap ®lm and the coordinate system.

XR u; v; w x; y z

length of the cylinder wake bubble; reattachment length of the backward-facing step (m) velocity components in x; y; z directions (m/s) streamwise and spanwise coordinates (m) coordinate perpendicular to the ®lm (m)

Greek symbols e dimensionless thickness l dynamic viscosity (kg/m s) m kinematic viscosity (m2 =s) q Density (kg=m3 ) r surface tension coecient (N/m) Superscripts * dimensionless quantities ± averaged quantities surfaces are described by z ˆ h1 …x; y; t† and z ˆ h2 …x; y; t†, respectively. The ®lm is assumed to be ¯at on average. The Navier±Stokes equations then read ux ‡ vy ‡ wz ˆ 0;

…1†

1 px ‡ m…uxx ‡ uyy ‡ uzz †; …2† q 1 py ‡ m…vxx ‡ vyy ‡ vzz †; vt ‡ uvx ‡ vvy ‡ wvz ˆ …3† q 1 pz ‡ m…wxx ‡ wyy ‡ wzz †; wt ‡ uwx ‡ vwy ‡ wwz ˆ q …4†

ut ‡ uux ‡ vuy ‡ wuz ˆ

where u; v; and w are velocity components in the x, y and z directions, respectively; and p is pressure inside the ®lm. The subscripts t, x and y in the above equations denote partial derivatives with respect to t, x and y, respectively. Here gravity and disjoining pressure are ignored as a ®rst approximation without loss of generality. These forces may be taken into account by simply adding potential body-force terms on the right-hand side of the momentum equations, Eqs. (2)±(4). The direct interaction forces between surfaces represented by a disjoining pressure are composed of repulsion terms (overlapping electrical double layers, forces of entropic origin) and of attraction terms (van der Waals forces) [13]. Because running soap ®lms in experiments are relatively thick (a few micros), the direct interaction between surfaces is weak, it is therefore realistic to neglect the disjoining pressure [2]. The system of governing equations is supplemented by boundary conditions at both free surfaces. The kinematic boundary conditions at the interfaces z ˆ h1 …x; y; t† and z ˆ h2 …x; y; t† may be written as: w ˆ h1t ‡ uh1x ‡ vh1y wˆ

…z ˆ h1 †;

…h2t ‡ uh2x ‡ vh2y †

…z ˆ

…5† h2 †:

…6†

T.-S. Yang et al. / Experimental Thermal and Fluid Science 25 (2001) 141±149

Assuming constant atmospheric pressure and small freesurface deformation, the pressure jumps across the free surfaces are given by the Laplace equation [14]. The normal and tangential force balances at the interfaces z ˆ h1 …x; y; t† and z ˆ h2 …x; y; t† then give the dynamic boundary conditions: p

2lwz ˆ

r…h1xx ‡ h1yy †;

l…uz ‡ wx † ˆ rx ; p

2lwz ˆ

l…vz ‡ wy † ˆ ry ;

r…h2xx ‡ h2yy †;

l…vz ‡ wy † ˆ

ry ;

…z ˆ

…z ˆ h1 †;

l…uz ‡ wx † ˆ h2 †;

rx ;

…7†

…8†

where r is the surface tension coecient. Note that under the assumption of small free-surface deformations, only linear terms are retained in the dynamic boundary conditions. 2.2. Scale analysis First, the equations of motion are normalized by the horizontal length scale L, the mean thickness of the soap ®lm H, the characteristic horizontal velocity U, and the convective timescale L=U . The choice of proper scales for the vertical velocity componentw and the pressure variations in the soap ®lm requires more careful considerations, however. Speci®cally, as one might argue that mass should be conserved in any case, it is tempting to scale w with eU (where e ˆ H =L) so that all terms in the continuity equation, Eq. (1), would formally have the same order of magnitude. However, assuming that the ¯uid pressure is of the same order of magnitude as the viscous stresses, the dynamic boundary conditions, Eqs. (7) and (8), then imply that r…hxx ‡ hyy †= …p 2lwz † ˆ O…e=C†, where C ˆ lU =r is the capillary number. In typical experiments, L is of the order of 1 cm and H ranges from 1 to 10 lm. The characteristic horizontal velocity U is of the order of 1 m/s. The kinematic viscosity m of the soap solution is of the order of 10 5 m2 =s, while the surface tension coecient is of the order of 0.01 N/m. These experimental conditions render e  1 (10 3 ±10 4 ) and C ˆ O…1†. Hence, the choice of the vertical velocity scale w  eU implies that surface tension e€ects are subdominant compared with internal ¯uid pressure and viscous stresses. As a matter of fact, in the asymptotic work of Chomaz [15] on soap ®lm ¯ows, w is scaled with eU and accordingly surface tension e€ects do not enter the `leading-order' problem. It should be emphasized, however, that the soap ®lm owes its very existence to the surface tension e€ects in the ®rst place; the `straightforward' choice of the vertical velocity scale therefore does not make much physical sense. To ensure that surface tension e€ects are retained in the dynamic boundary conditions, here we choose w  e2 r=l and p  re=L instead (see Eqs. (7) and (8)).

143

Accordingly, the Navier±Stokes equations (1)±(4) are normalized to be e …9† ux ‡ vy  ‡ wz ˆ 0; C   e e2 Re ut ‡ u ux ‡ v uy  ‡ w uz C px ˆ …10† ‡ e2 …ux x ‡ uy  y  † ‡ uz z ; C   e e2 Re vt ‡ u vx ‡ v vy  ‡ w vz C py ˆ …11† ‡ e2 …vx x ‡ vy  y  † ‡ vz z ; C  e2 Re   e wt ‡ u wx ‡ v wy  ‡ w wz C C  2 pz e 1 …12† ‡ …w  ‡ wy  y  † ‡ wz z ; ˆ C C xx C where the Reynolds number Re ˆ UL=m and C ˆ C=e3  1 (109 ±1012 ). The asterisks in the above equations denote dimensionless (normalized) quantities. Note also that in the above equations e2 Re  1 (10 4 ±10 6 ). One immediately identi®es several small (or large) parameters in the normalized equations of motion, Eqs. (9)±(12). It is therefore possible to exploit these extreme parameters and employ standard perturbation methods to carry out an asymptotic analysis systematically, as in Chomaz's work [15]. However, that is a major task due to the presence of many parameters of various orders of magnitude, and thus is beyond the scope of this paper. Instead, here we wish to use simpler scale analysis to bring out the dominant balances in the equations of motions and attempt to deduce useful information therefrom. In order to do that e€ectively and correctly, one has to scale the physical variables more carefully, and that is exactly why in the above scaling arguments we insist on retaining the surface tension e€ects in the dominant balances. Speci®cally, the dominant balances in Eqs. (9)±(11) are, returning to dimensional variables, ux ‡ vy ˆ 0;

…13†

0 ˆ luzz ;

…14†

0 ˆ lvzz :

…15†

Note also that all terms in the z-momentum equation, Eq. (12), are much smaller than those retained in the above dominant balances, and therefore Eq. (12) is completely dropped as a ®rst approximation. This may be understood on physical grounds: Since the soap ®lm thickness typically is extremely small compared with the planar dimensions of the frame, ¯uid motion in the ®lmthickness direction therefore is negligible. Similarly, it can be deduced that pressure variation across the ®lm thickness generally is negligible compared with the planar pressure variations, i.e., jpz j  jpx j; jpy j.

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By the same token, the dynamic boundary conditions (Eqs. (7) and (8)) are reduced to p

2lwz ˆ lvz ˆ ry

p

2lwz ˆ lvz ˆ

ry

r…h1xx ‡ h1yy †;

luz ˆ rx ;

…z ˆ h1 †; r…h2xx ‡ h2yy †; …z ˆ

luz ˆ

h2 †:

rx ;

…16† …17†

Furthermore, when the surface tension gradient is moderate so that eDr=lU  1 (which is often satis®ed), the two free surfaces are essentially `shear-free': uz  v z  0

…z ˆ h1 ; h2 †:

…18†

It is worth reiterating here that we have chosen the vertical velocity scale to ensure that surface tension e€ects are retained in the dynamic boundary conditions, and this makes sense because the soap ®lm owes its very existence to the surface tension e€ects. As a result, the reduced continuity equation, Eq. (13), indicates that to leading order the 2-D planar ¯ow appears to be `incompressible' by itself, and this is supported by the experimental evidence to be discussed later. The Laplace equations in Eqs. (7) and (8) also imply that, when gravitational e€ects are subdominant and the surface tension gradient is negligible, the soap ®lm deforms symmetrically so that h1 ˆ h2 . This symmetric deformation has also been described by Lucassen et al. [13]. Integrating then Eqs. (14) and (15) from z to h1 , imposing the shear-free boundary conditions Eq. (18), it transpires that the variations of u and v across the ®lm thickness are also negligible as a ®rst approximation, i.e., uz  v z  0

… h2 < z < h1 †:

proximately constant) following the ®lm element. This explains why the thickness ®eld behaved as a passive scalar in the experimental observations. The color fringes appearing in soap ®lm ¯ows resemble streamlines of steady 2-D ¯ows. It is also worth noting that we have not really ``solved'' the equations of motion to obtain the velocity ®eld. What we intend to do above is integrating the equations of motion across the ®lm thickness, and deducing the conditions on ¯ow parameters ensuring that the variations of u and v across the ®lm thickness are negligible, Eq. (19). And that, in turn, is sucient for the ®lm thickness `seen' by a ¯uid particle to be (approximately) constant. To solve the ¯ow, one would need to specify conditions on the horizontal boundaries as well, which is beyond the scope of the present scaling discussion.

3. Experimental set-up 3.1. Representative ¯ow ®elds To support the above analysis, soap ®lm ¯ows over a circular cylinder and a backward-facing step were chosen as prototype ¯ows. The representative ¯ow ®elds are shown in Fig. 2. For the case of soap ®lm ¯ow over a circular cylinder, a smooth circular cylinder of 4 mm diameter was manufactured from stainless steel. The cylinder was set normal to the main ¯ow, spanning the central part of the working section. The tunnel-wall blockage was 8%. As to

…19†

Moreover, the continuity equation (1) may be integrated from z ˆ h2 …x; y; t† to h1 …x; y; t†, using the kinematic free-surface conditions Eqs. (5) and (6), to yield ht ‡ …hu†x ‡ …hv†y ˆ 0;

…20†

where h…x; y; t† ˆ h1 …x; y; t† ‡ h2 …x; y; t† is the total ®lm thickness and Z Z 1 h1 1 h1 u…x; y; t† ˆ u dz; v…x; y; t† ˆ v dz h h2 h h2 are the depth-averaged velocity components. Note that Eq. (20) is an exact result. In view of Eqs. (13) and (19), …u; v†  …u; v† and ux ‡ vy  0. Accordingly, using Eq. (20), the rate of change of total ®lm thickness following a ¯uid particle is given by Dh ˆ ht ‡ uhx ‡ vhy  ht ‡ uhx ‡ vhy Dt  h…ux ‡ vy †  0:

…21†

This implies that, under the conditions discussed above, the ®lm thickness is constant (or, more precisely, ap-

Fig. 2. Schematic of the representative ¯ow ®elds: (a) 2-D ¯ow over a circular cylinder; (b) 2-D ¯ow over a backward-facing step. The dimensions shown in the ®gures were those used in the experimental test sections and the computational domains.

T.-S. Yang et al. / Experimental Thermal and Fluid Science 25 (2001) 141±149

145

another case of soap ®lm ¯ow over a backward-facing step, an expansion ratio of 4:5 was provided. The section length downstream of the step was 16 times the step height S.

eter of the cylinder) up to about 180. The vortex shedding process can then be used as a practical method to estimate ®lm viscosity. Roshko's famous equation [16] showed that the Strouhal number and the Reynolds number are related by

3.2. Soap ®lm tunnel and instrumentation

St ˆ fd=U1 ˆ 0:212

Experiments were performed in a horizontal soap ®lm tunnel. As shown in Fig. 3, the device consists of a frame (5 cm wide and 12 cm long), with one end positioned in a diluted soap solution reservoir while the other end subjected to a ®lm-pulling mechanism. The main portion of the frame is a ¯at section consisting of two parallel copper rods supported by two legs, one at each end. The size of the frame is limited by the ®lm's tendency to bow at the ¯at section of the tunnel. The contact of the soap ®lm with a clear 2-D water jet provides the pulling mechanism. The tunnel allows uniform free-stream velocities up to 0.9 m/s and longitudinal free-stream turbulence intensities less than 1%. Since the ®lm is continuously fed, evaporation does not pose a serious problem. Experiments were performed with soap ®lm solution of 1.5% commercial liquid detergent (Ivory, Dishwashing Liquid). In order to determine the ¯ow Reynolds number, one needs to estimate the 2-D ®lm viscosity. Gharib and Derango [2] and Wen and Lin [10] have shown that the 2-D vortex shedding process of a circular cylinder at a macroscopic level has a strong similarity to its 3-D counterpart for Reynolds number (based on the diam-

By measuring the velocity and shedding frequency of a known size cylinder in the ®lm and then using St and Roshko's equation to obtain Re, the ®lm viscosity is determined consequently. The ®lm viscosity for 1.5% soap ®lm solution was estimated to be 4:56  10 5 m2 =s. Free-stream velocities and velocity ¯uctuations in the cylinder wake were measured by a commercial onecomponent laser Doppler velocimetry (LDV) system (TSI IFA-750 assembly). The soap ®lm ¯ow is observed by re¯ection, using a white light source (Samson ETSTROBO S-403) and a traditional camera (Nikon FM2). The pulse width of the light source is about 0.5 ms. Schematic of the ¯ow visualization system is shown in Fig. 4.

4:5=Re:

…22†

3.3. Experimental uncertainty The free-stream velocity and turbulence intensity are uniform to within 0:007U1 and 0:001U1 , respectively [2]. Detailed investigations of the uncertainty in the measurements of the ®lm viscosity were given by Gharib

Fig. 3. Schematic of the horizontal soap ®lm tunnel.

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T.-S. Yang et al. / Experimental Thermal and Fluid Science 25 (2001) 141±149

Fig. 5. Comparison of the experimental photograph and the numerical simulation of soap ®lm ¯ow over a circular cylinder, Re ˆ 28: (a) color visualization of wake ¯ow patterns behind a circular cylinder; (b) superimposition of numerical simulated streamlines of the corresponding ¯ow ®eld at the same conditions as the experiment on the photograph. Fig. 4. Schematic of the ¯ow visualization system.

and Derango [2] and Wen and Lin [10]. The estimated uncertainty of the ®lm viscosity is less than 5%. These lead to an estimation of the accuracy of Reynolds numbers of 6%. 4. Numerical method Streamlines of the representative ¯ow ®elds in the geometries equivalent to the two experimental prototype ¯ows were obtained by numerically solving the 2-D incompressible Naiver-Stokes equations. The governing equations were solved by using a ®nite element numerical scheme, embodied in the commercial code ANSYS, FLOTRAN. The dimensions of the calculation domains for both cases are shown in Fig. 2. The FLUID 141, FLOTRAN was used to generate the elements. In the case of ¯ow over a circular cylinder, a total number of 24 939 elements and 25 375 nodes were used, while in the case of ¯ow over a backward-facing step, a total number of 27 794 elements and 28 023 nodes were used. A larger number of elements were placed in the areas where steep variations in velocities were expected, for example, near wall regions. For both cases, a uniform velocity pro®le was applied at the inlet with U1 ˆ measured and V1 ˆ 0, and a zero relative pressure was applied at the outlet. No-slip conditions were applied all along the walls (including the cylinder surface). Convergence of

Fig. 6. Comparison of the experimental photograph and the numerical simulation of soap ®lm ¯ow over a circular cylinder, Re ˆ 35: (a) color visualization of wake ¯ow patterns behind a circular cylinder; (b) superimposition of numerical simulated streamlines of the corresponding ¯ow ®eld at the same conditions as the experiment on the photograph.

T.-S. Yang et al. / Experimental Thermal and Fluid Science 25 (2001) 141±149

Fig. 7. Comparison of the experimental photograph and the numerical simulation of soap ®lm ¯ow over a backward-facing step, Re ˆ 25 (based on the step height, S): (a) color visualization of attached recirculating patterns behind a backward-facing step; (b) superimposition of numerical simulated streamlines of the corresponding ¯ow ®eld at the same conditions as the experiment on the photograph.

the solution was considered satisfactory when the normalized nodal residuals of each calculated variables were smaller than 10 4 .

5. Results and discussion Figs. 5(a) and 6(a) show color visualizations of soap ®lm wake ¯ow patterns behind a circular cylinder. The measured Re based on the diameter of the cylinder are 28 and 35, respectively. The steady pair of eddies is seen in each wake. This ¯ow feature is similar to the 2-D cross-section of a wake in experiments with a ®nite span at the same Re. Corresponding simulated streamlines at the same conditions as the experiments are superimposed on the photograph and are shown in Figs. 5(b) and 6(b). As may be seen, the experimental wake interference patterns are faithfully reproduced by the simulated streamlines. The validity of the numerical simulation was veri®ed by comparing the simulated length of the wake bubble XR with the published result of Fornberg [17]. The current simulated XR normalized by the diameter of the circular cylinder XR =D ˆ 1:67 and 1.94 (see Figs. 5(b) and 6(b)), while the interpolated result from Fornberg [17] XR =D ˆ 1:66 and 1.97, at Re ˆ 28 and 35, respectively. They are in good agreement.

147

Fig. 8. Comparison of the experimental photograph and the numerical simulation of soap ®lm ¯ow over a backward-facing step, Re ˆ 34: (a) color visualization of attached recirculating patterns behind a backward-facing step; (b) superimposition of numerical simulated streamlines of the corresponding ¯ow ®eld at the same conditions as the experiment on the photograph.

To demonstrate other examples, Figs. 7 and 8 show similar exercises for soap ®lm ¯ows over a backwardfacing step. The measured Re based on the step height are 25 and 34 for Figs. 7 and 8, respectively. Again, the comparison shows good agreement. Except for a slight di€erence in the vicinity of the reattachment point in each ®gure, the experimental interference fringes are virtually congruent with the numerical simulated streamlines (see Figs. 7(b) and 8(b)). The possible reason for the deviations of the simulated streamlines and the color fringes in the vicinity of the reattachment point is the di€erence of boundary layer thickness before the step in the numerical and experimental situations. In the experiment, the boundary layer grows from the two legs of the frame above the soap solution. The ¯ow then experiences a smooth 90° bend and goes in the ¯at section (see Fig. 3). On the other hand, in the numerical simulation, the boundary layer grows from the entrance of the ¯at section. Hence, the experimental boundary layer thickness right in front of the step will be thicker than that of numerical simulation. These lead to the positions of experimental reattachment points slightly further than those of numerical simulations. The validity of the numerical simulation was veri®ed by comparing the simulated reattachment length of the backward-

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T.-S. Yang et al. / Experimental Thermal and Fluid Science 25 (2001) 141±149

facing step XR with the published result of Armaly et al. [18]. The current simulated XR normalized by the step height XR =S ˆ 1:81 and 2.14 (see Figs. 7(b) and 8(b)), while the extrapolated results from Armaly et al. [18] XR =D ˆ 1:97 and 2.22, at Re ˆ 25 and 34, respectively. The agreement is satisfactory. From the comparison of the experimental photographs and the numerical simulations of the two representative ¯ow ®elds, clearly, color fringes appearing in soap ®lm ¯ows resemble streamlines of steady 2-D ¯ows as the leading order analysis predicts. In the preceding analysis (see Section 2.2), the thickness ®eld will behave as a passive scalar in the experimental observations and the color fringes appearing in soap ®lm ¯ows will represent streamlines of steady 2-D ¯ows under the conditions that e  1, e2 Re  1, C  1, and eDr=lU  1. The normal experimental conditions make the ®rst three conditions easily met. The thing worth noting is the eDr=lU term. For most parts of the ¯ow ®led, eDr=lU  1. However, the interactions of the soap ®lm with model boundaries involving a meniscus and a contact angle may cause large variations of surface tension in boundary layers and make Dr much higher. Therefore boundaries may be responsible for occasionally peculiar behavior observed in the soap ®lm experiments. This explains what has been observed by Chomaz and Costa [12], ``In my opinion this boundary e€ect constitutes the most dangerous potential source of experimental artifacts.'' 6. Practical signi®cance/usefulness The theoretical and experimental studies of 2-D turbulence, transition mechanisms in shear ¯ows, and the fundamentals of 2-D vortex dynamics have great importance in the oceanic and atmospheric problems. Soap ®lms have been introduced as new experimental tools to study 2-D hydrodynamics, recently. The present research work was carried out to add to the existing knowledge of soap ®lm ¯ows and to elucidate the relationship between the interference pattern shown in soap ®lms and their motion. It is hoped that this study will make soap ®lms more useful for future researches of 2-D hydrodynamics. 7. Conclusions Experimental results suggested that color fringes (constant thickness lines) seen in steadily ¯owing soap ®lm resemble streamlines of 2-D ¯ows. Here we derive the conditions under which the analogy can be drawn. It is shown that in the domains of the parameter space that e  1, e2 Re  1, C  1, and eDr=lU  1, the ®lm thickness is constant (or, more precisely, approximately constant) following the ®lm element and the color fringes will represent streamlines of steady 2-D ¯ows in the experimental observations. The ®rst three conditions are

easily satis®ed in the normal experimental conditions. As to the last condition (eDr=lU  1), it is easily met for most parts of the ¯ow ®eld. However, the interactions of the soap ®lm with model boundaries may make Dr much higher and may be the most dangerous potential source of experimental artifacts. Photographs of ¯ows over a circular cylinder and a backward-facing step taken in a horizontal soap ®lm tunnel and numerical simulations under the same conditions as the experiments are also presented to support the theoretical results. From the comparison of the experimental color fringes and the numerically computed streamlines of the two representative ¯ow ®elds, the analytical results are justi®ed.

Acknowledgements The authors would like to thank Prof. M. Gharib for valuable suggestions in designing the horizontal soap ®lm tunnel and for his comments that have added signi®cantly to this paper. This work was supported by the National Science Council of the Republic of China under Grant NSC 88-2212-E-212-007. References [1] Y. Couder, The observation of a shear ¯ow instability in a rotating system with a soap membrane, J. Phys. Lett. 42 (1981) 429±431. [2] M. Gharib, P. Derango, A liquid ®lm (soap ®lm) tunnel to study two-dimensional laminar and turbulent shear ¯ows, Physica D 37 (1989) 406±416. [3] H. Kellay, X.L. Wu, W. Goldburg, Experiments with turbulent soap ®lms, Phys. Rev. Lett. 74 (1995) 3975±3978. [4] Y. Couder, Two-dimensional grid turbulence in a thin liquid ®lm, J. Phys. Lett. 45 (1984) 353±360. [5] Y. Couder, J.M. Chomaz, M. Rabaud, On the hydrodynamics of soap ®lms, Physica D 37 (1989) 384±405. [6] M. Beizaie, M. Gharib, Fundamentals of a liquid (soap) ®lm tunnel, Exp. Fluids 23 (1997) 130±140. [7] X. Wu, B.K. Martin, H. Kellay, W. Goldburg, Hydrodynamic convection in a two-dimensional Couette cell, Phys. Rev. Lett. 75 (1995) 236±239. [8] M.A. Rutgers, X.L. Wu, R. Bhagavatula, A.A. Petersen, W.I. Goldburg, Two-dimensional velocity pro®les and laminar boundary layers in ¯owing soap ®lms, Phys. Fluids 8 (1996) 2847±2854. [9] M. Rivera, P. Vorobie€, R.E. Ecke, Turbulence in ¯owing soap ®lms: velocity, vorticity and thickness ®elds, Phys. Rev. Lett. 81 (1998) 1417±1420. [10] C.-Y. Wen, C.-Y. Lin, Two-dimensional vortex shedding of a circular cylinder, Phys. Fluids 13 (3) (2001) 557±560. [11] J.M. Chomaz, B. Cathalau, Soap ®lms as two-dimensional classical ¯uids, Phys. Rev. A 41 (1990) 2243±2245. [12] J.M. Chomaz, M. Costa, Thin ®lms dynamics, in: H.C. Kuhlmann, H.J. Rath (Eds.), Free surface ¯ows, CISM courses and Lectures No. 391, 1998, pp. 44±99. [13] J. Lucassen, M. Van Den Tempel, A. Vrij, F. Hesselink, Waves in thin liquid ®lms, Proc. K. Ned. Akad. Wetensch. B 73 (1970) 109±124.

T.-S. Yang et al. / Experimental Thermal and Fluid Science 25 (2001) 141±149 [14] L.D. Landau, Fluid Mechanics, second ed., Butterworth, Heinemann, 1995. [15] J.M. Chomaz, 2D or not 2D, the soap ®lm dilemma, J. Fluid Mech., submitted. [16] A. Roshko, On the development of turbulent wakes from vortex streets, NACA Report 1191, 1954.

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[17] B. Fornberg, A numerical study of steady viscous ¯ow past a circular cylinder, J. Fluid Mech. 98 (1980) 819±855. [18] B.F. Armaly, F. Durst, J.C.F. Pereira, B. Sch onung, Experimental and theoretical investigation of backward-facing step ¯ow, J. Fluid Mech. 127 (1983) 473±496.