Stratified shear flows occur in a wide range of …



Mixing Mechanisms and Resulting Droplet Distributions in a Fuel-Water Stratified Shear Flow

X. Wu and J. Katz

Department of Mechanical Engineering

The Johns Hopkins University,

Department of Mechanical Engineering, Baltimore, Maryland 21218

1. Introduction

Numerous studies have already been conducted on the mean flow and turbulence within a miscible stratified shear flow. It has been found that mixing and the evolution of turbulence depend on the gradient Richardson number, [pic] (detailed definitions follow), the initial value of the Taylor micro-scale Reynolds number, Re(=qλ/ν, and the initial value of the shear number S*=Sq2/(. (Rohr et al., 1988; Piccirrilo & Van Atta, 1997; Jacobitz et al., 1997, Holt et al., 1992; Shih et al., 2000). Several investigations, for example by Shih et al. (2000), show that at high Reynolds numbers the values of S* tend to a constant of about 11, irrespective of the initial conditions. The growth or decay of turbulent kinetic energy within the shear layer depends on Rig. According to the DNS data of Jacobitz et al. (1997), the critical Richardson number, Ricr, separating conditions of growth and decay, can vary between 0.04 and 0.17 depending on S* and Re(. At high Reynolds numbers Ricr depends mostly on the Reynolds number. These results are consistent (with some variations) with the experimental data of Piccirillo & Van Atta (1997) as well as simulations by Holt et al. (1992) and Shih et al. (2000), but are lower than the theoretical stability condition of Ricr~0.25 developed by Miles (1961). An alternative parameter, the turbulent Froude number, Frt, which is based on the local turbulence properties and density gradients, has also been proposed as a dominating parameter to characterize the turbulence and mixing (Ivey and Imberger, 1991; Kaltenbach et al., 1994; Piccirillo & Van Atta, 1997; Shih et al., 2000).

The flow structure, stability and mixing in the near field of miscible stratified shear flows have also been studied extensively. It is already widely accepted that in homogeneous shear flow the mixing is dominated by large-scale coherent structures and related secondary instabilities (e.g. Brown & Roshko, 1974; Winant and Browand, 1974; Corcos & Sherman, 1984). The underlying physics is considerably more complicated in stratified shear flows, as demonstrated in many investigations (e.g. Keulegan, 1949; Ellison & Turner, 1959; Kato & Phillips, 1969; Thorpe, 1968, 1971, 1973; Moore & Long, 1971; Narimousa & Fernando, 1987; Koop and Browand, 1979; Sullivan & List, 1994) and extended reviews by Turner (1986), Fernando (1991), and Staquet & Sommeria (1996). As a general conclusion, the mixing process and the depth of the penetration at the interface depend on the Richardson number that has several definitions. The Reynolds and Peclet numbers are also important if viscosity and molecular diffusion play significant roles, i.e. at low Reynolds numbers and high diffusivity. Fernando (1991) provides an extensive summary on empirical relations for entrainment laws based on the Richardson number. He shows that the entrainment rates vary substantially, but all clearly decreases with increasing Ri.

Sullivan and List (1994) categorize the different regimes of mixing in stratified shear layers. At very low Richardson number the shear layer resembles that of homogeneous flow with rollup of large vortices and subsequent breakdown to turbulence at a wide range of scales. As the Richardson number is increased slightly, the mixing process is dominated by Kelvin-Helmholtz (K-H) instabilities and the effect of small-scale turbulence becomes less evident. With further increase in Richardson number, the rollup of K-H vortices stops and the mixing occurs mostly due to shear-driven breaking of interfacial waves, changing and reducing (but far from eliminating) the extent of mixing. At even higher Richardson number, the interface becomes stable and the convective mixing diminishes. Mixing continues, however, by molecular diffusion for miscible fluids. Close looks at the vortex structures and vortex interactions in miscible stratified shear flows at low Richardson numbers have been taken by Koop and Browand (1979), Schowalter et al. (1994) and Atsavapranee & Gharib (1997). They all observe that as the Richardson number increases the growth of instabilities and vortex interactions are suppressed.

Much less information is available on the mechanics and extent of entrainment stratified in shear layers created at interfaces between immiscible liquids, where entrainment implies formation of droplets. The present paper deals with such a system, i.e. with the mixing process in a stratified shear layer of water and fuel. This problem has practical significance since stratified shear flows containing water-fuel interfaces exist, for example, in ship and large storage fuel tanks, as well during fuel spills. Relevant background for the latter can be found in Milgram et al (1978) and Milgram & Van Houten (1978), including measurements and analysis of the size of oil slicks, rates of oil droplets coalescence, as well as photographs showing the formation of droplets, interfacial waves and eddy rollup at high speeds. In this paper we provide detailed data on the structure and turbulence statistics, enrtrainment mechanisms and droplet statistics in a water-fuel, two-dimensional stratified shear layer. Relevant parameters, including different forms of the Richardson number, distributions of void fraction, mean velocity, turbulence production and dissipation and turbulent kinetic energy are also calculated. Some data on the interfacial structures and droplet statistics have already been presented previously (Wu and Katz, 1999, 2000) and are only introduced briefly here. A relevant parallel study has focused on a water jet impinging on a fuel water interface (Friedman and Katz, 1999, 2000; Friedman et al., 2001).

We start by describing the experimental setup and measurement techniques (Section 2). Characteristic interfacial structures along with sample velocity and vorticity distributions are presented in Section 3.1. The dimensions of the shear layer and “mixture layer” (region with dense fuel-water mixture), distributions of mean velocity, Richardson numbers and turbulence parameters are discussed in Section 3.2. The droplet statistics are introduced in Section 3.3.

2. Experimental Setup and Procedures

Counter flowing layers of water and diesel fuel generate the two-dimensional stratified shear layer. A sketch of the experimental setup is presented in Figure 1. The mean water velocity can be varied between 0.2 and 1.2 m/s. The diesel fuel velocity is very low, 15 cm, the values at 0.8 m/s become significantly larger than those at 1.2 m/s. The trends of [pic] are similar for both flow rates and cannot be used for characterizing the differences in entrainment rates. As discussed in Wu and Katz (2000) and in Section 3.3, at 0.8 m/s the total number of droplets increases rapidly up to x~20 cm and then at a much slower pace up to x~30 cm. Further downstream the total number of droplets does not change significantly. Conversely, at 1.2 m/s, the total number of droplets and those in the mixture layer continue to increase over the entire facility. When these trends are compared to the distributions of [pic] they suggest that droplet production decreases substantially when [pic] and stops when [pic].

Sample distributions of u’, the rms values of axial velocity fluctuations (plotted up to α=0.5) are presented in Figure 8. Estimates of the turbulent kinetic energy based on the two available velocity components and assuming that the third one is an average of the other two, i.e. [pic], are presented in Figure 9. They both show the expansion of the turbulent shear layer and reduction in peak magnitudes with increasing axial distance. Consistent with Figure 5b, the overall widths of the shear layer for the two cases are comparable. The turbulence level is maximum at α=0.5 point and decreases with increasing distance from the interface. At α=0.5, for example at x=11 and 22 cm, the ratio of maximum k’ (at different elevations) is about 0.57, higher than the ratio of U2 (0.44). However, at the same elevation and slightly away from the interface, for example at y=0, the turbulent kinetic energy scales well with the U2. The differences near the α=0.5 point occur since they do not represent similar locations in the shear layer, as Figure 5b indicates. As discussed in the introduction, previous studies of miscible stratified shear flows show the existence of a critical Richardson number Ricr, at which k’ remains almost a constant along the flow direction. In the present results k’ decreases in the streamwise direction, indicating that the flow is above the critical conditions.

To study the characteristics of the turbulence one can calculate the turbulence spectra using the PIV data. One dimension spatial spectra, Eii(k1) (no summation over i, k1 is the wavenumber along the flow direction) are calculated using FFT of the first difference (and appropriate compensation), pre-whitening, a Hanning windowing function (and appropriate scaling), as described in Doron et al. (2001). The results of all 126 data sets at the same location are averaged. Sample spectra for both velocity components, within and below the shear layer, are presented in Figure 10. Since [pic] in the inertial range of isotropic turbulence, where the subscripts 1 and 2 refer to spectra of u and v, respectively (Monin and Yaglom, 1971), the spectra of the vertical velocity are multiplied by 3/4. Each spectrum is also fitted with a –5/3 slope line. Consistent with the similarity in rms values, E11(k1) and 0.75 E22(k1) are very close to each other in the resolved range, indicating the turbulence is nearly isotropic. On the other hand there is difference of more than one decade between the spectra within and outside of the shear layer. Using the fitted –5/3 slope lines one can estimate the energy dissipation rate, ε, from (Tennekes and Lumley, 1972)

[pic]

where ck=1.7. Once ε is known one can estimate the Kolmogorov scale, (, the Taylor microscale, ( and the integral scale, l, using (as in Liu et al., 1994)

[pic]

Figure 11presents the distributions of ε with error bars indicating the range of instantaneous values. Figure 12 shows the resulting estimates of ( and (. Outside of the shear layer ε is 2-3 orders of magnitude weaker than the levels inside the shear layer. The free stream dissipation rates, ~3x10-5 and ~3x10-4 at 0.8 and 1.2 m/s, respectively, represent the turbulence level at the entrance to the test section. With increasing x, as the shear layers expand, the regions with increased levels of dissipation expand. At the same elevation ε decreases with axial distance. At 1.2 m/s ε is about five times higher than that at 0.8 m/s, although this ratio fluctuate in the 3-10 range. The minimum kolmogorov scale in the shear layer increases with axial distance from about 65μm to 100μm at 0.8 m/s, and fluctuates between 55μm to 85μm at 1.2 m/s.

In both cases the values of ( in the shear layer vary between 2-3 mm. The resulting peak values of Re( (Re(= q(/ν where [pic]) at 0.8 m/s fluctuate between 600-800 at x ................
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