Abstract
Liquid jet impingement is used in industries for cleaning or cooling the surfaces, since this process is characterized by high heat or mass transport rates. The impinging jet spreads radially outwards and creates a wall film flow, which is bounded by a hydraulic jump. The existing models describing the extent of the radial flow zone and the position of hydraulic jump are only applicable for small nozzle-to-target distances and low flow rates. In this work, the model is extended to include the effect of splattering liquid, which may reduce the extent of the radial flow zone considerably. The splattering in combination with the hydraulic jump position is investigated experimentally for a liquid jet impinging horizontally onto a vertical wall. In addition, the high-speed images of the jet and of the impingement region provide further insight into the splattering mechanisms. It is found that for large nozzle-to-target distances the splattered mass fraction is determined only by the jet Weber number. The hydraulic jump position can be predicted using the extended model with deviations of less than 20% in this region.
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Abbreviations
- \(a\) :
-
Image height, m
- \(b\) :
-
Image width, m
- \(d\) :
-
Jet diameter, can differ from \(d_{\text{N}} ,\;\; {\text{m}}\)
- \(d_{\text{N}}\) :
-
Nozzle diameter (orifice), m
- \({\text{fr}}\) :
-
Frame rate (high-speed camera), \(/ {\text{s}}\)
- \(g\) :
-
Gravitational constant \(9.81 \;\;{\text{m /s}}^{2} , {\text{kg/s}}^{2}\)
- h :
-
Film thickness, function of \(r\) and \(\varphi\), kg
- \(I\) :
-
Momentum of wall jet, function of \(r\) and \(\varphi\), \({\text{kg}}\;{\text{m/s}}\)
- \(L\) :
-
Nozzle distance, distance between nozzle and target, m
- \(l_{\text{N}}\) :
-
Nozzle length, m
- \(\dot{M},\dot{M}_{\text{r}} ,\dot{M}_{\text{s}}\) :
-
Mass flow, draining mass flow, splattered mass flow, \({\text{kg/s}}\)
- \(M\) , \(M_{\text{r}}, M_{\text{s}}\) :
-
Mass, drained mass, splattered mass, \({\text{kg}}\)
- \(r\) :
-
Radial coordinate measured from point of impingement, m
- \(r_{\text{s}}\) :
-
Radial position of splattering at which droplets leave the wall jet, m
- \(R\) :
-
Radial position of the hydraulic jump, function of \(\varphi , \;\;{\text{m}}\)
- \(R_{\text{q}}^{\text{u}} , R_{\text{q}}^{\text{l}} , R_{\text{q}}\) :
-
RMS roughness of the upper boundary, or lower boundary computed from \(y^{\text{u}} , y^{\text{l}}\). Mean value of the former two
- \(t,t_{\text{s}}\) :
-
Image time, shutter time (high-speed camera), s
- \(u_{\text{drop}}\) :
-
Drop velocity at impact, \({\text{m/s}}\)
- \(u_{\text{jet}}\) :
-
Free liquid jet velocity, \({\text{m/s}}\)
- \(\bar{u}_{\text{jet}}\) :
-
Mean free liquid jet velocity \(\frac{4}{{\pi d_{\text{N}}^{ 2} }}\frac{{\dot{M}}}{\rho }\), \({\text{m/s}}\)
- \(\bar{u}\) :
-
Mean velocity of the wall jet, function of \(r\) and \(\varphi\), \({\text{m/s}}\)
- \(\bar{u}_{\gamma }\) :
-
Surface tension speed \(\gamma \frac{5r\pi }{{3\dot{M}}}\), \({\text{m/s}}\)
- \(U\) :
-
Dimensionless drop speed \(Ca \lambda_{\text{drop}}^{{\frac{3}{4}}}\), -
- \(X\), \(X_{\text{num}}\) :
-
Measured and predicted horizontal position of the hydraulic jump, m
- \(Y,Y_{\text{num}}\) :
-
Measured and predicted vertical position of the hydraulic jump, m
- \(y^{\text{u}} , y^{\text{l}}\) :
-
Vertical pixel position of upper and lower jet boundaries in the stitched images, m
- \(z\) :
-
Wall film coordinate measured perpendicularly from the wall, m
- \(\beta\) :
-
Contact angle (water wall), \({\text{rad}}\)
- \(\gamma\) :
-
Surface tension (water–air), \({\text{N/m}}\)
- \(\lambda\) :
-
Wave length, m
- \(\lambda_{\text{drop}}\) :
-
Dimensionless distance between drops \(\left( {v/f} \right)^{{\left( {1/2} \right)}} \sigma /\left( {\rho v^{2} } \right),\)-
- \(\mu\) :
-
Dynamic viscosity (water), \({\text{kg/m}}\; {\text{s}}\)
- \(\nu\) :
-
Kinematic viscosity (water), \(/ {\text{m}}^{2} \; {\text{s}}\)
- \(\rho\) :
-
Density (water), \({\text{kg/m}}^{3}\)
- \(\rho_{g}\) :
-
Density of surrounding gas, \({\text{kg/m}}^{3}\)
- \(\xi\) :
-
Splattered mass fraction, -
- \(\tau\) :
-
Wall shear stress, \({\text{kg/m}}\; {\text{s}}^{2}\)
- \(\varphi\) :
-
Azimuthal angle measured from the vertical in the plane of the wall, rad
- \(Ca\) :
-
Capillary number of a drop \(We_{{d_{\text{N}} }} /Re_{{d_{\text{N}} }}\)
- \(Re_{{d_{\text{N}} }}\) :
-
Reynolds number of free liquid jet \(\frac{{4\dot{M}}}{{\pi d_{\text{N}} \eta }}\)
- \(We_{{d_{\text{N}} }}\) :
-
Weber number \(\frac{{\rho \bar{u}_{\text{jet }}^{ 2} d_{\text{N}} }}{\gamma }\)
- \(We_{\text{g}}\) :
-
Gas Weber number \(\frac{{\rho_{g} u_{\text{jet}}^{ 2} d}}{\gamma }\)
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Appendix
Appendix
1.1 Image processing for characterization of jet surface roughness
Illustration of image processing operations: a raw cropped image, b image after background subtraction, c detected edges, d processed image
Illustration of image stitching: a processed image at \(t\) with a predefined detail to find in (box), b processed image at \(t + \Delta t\) with a predefined area to search in (box), c result of 2D cross correlation (pink and green areas show deviations between a and b, and d ongoing stitched image with the inserted cropped detail
Stitched image of a water jet originating from a 4 mm convergent nozzle with two 90° deflections upstream the nozzle entering (convU) at \(L/d_{\text{N}} = 100\) and \(Re_{{d_{\text{N}} }} =\) 59,600. The imperfect stitching of the formed droplets due to their lower velocity is clearly visible
Stitched image of a water jet originating from a 4 mm convergent nozzle with two 90° deflections upstream the nozzle entering (convU) at \(L/d_{\text{N}} = 167\) and \(Re_{{d_{\text{N}} }} =\) 59,600
To create the stitched images, the recorded images were stored in 5 stacks of equal size, to reduce the size of the stitched images and thereby increase the speed of the roughness calculation. The images were processed using the following steps, some of them illustrated in Fig. 16:
-
Cropping the relevant region of the image
-
Subtraction of background image
-
Conversion of 8bit grayscale image into Boolean image with edge detection, using the “Canny method” (Canny 1986)
-
Adding a “true” line to the rear and back of the image
-
Filling of closed geometries
-
Eliminating droplets one order of magnitude smaller than the largest object (jet).
After processing, the images were correlated and stitched as illustrated in Fig. 17. Parts (a) and (b) of this figure show processed images at the time \(t\), and \(t + \Delta t\) respectively. The region within the box of Fig. 17a was correlated with the region in the box of Fig. 17b, which was predefined by the expectancy of the jet speed and a tolerance. Note, that the left side of the box was set to the position, at which the nozzle showed at the set up (compare Sect. 2.5). In Fig. 17c, the result of the correlation is shown. The pink areas at the border of the jet highlight regions where a jet is present in the box of Fig. 17a, but not in the found position in Fig. 17b, vice versa for the green area. These minor deviations are due to the change of the jet surface during the period \(\Delta t = 1/{\text{fr}}\). The region of Fig. 17a was then cropped at the right to reach a width of the travelled distance and stitched to the ongoing stitched image of Fig. 17d. In this way, images counting up to 100 kPi (equivalent to a length of ca. 4.5 m) in length were constructed.
Clips of the derived binary stitched images are shown in Fig. 7 for jets originating from all 4 mm nozzle configurations with \(Re_{{d_{\text{N}} }} =\) 59,600, while non-binary images are shown in the following Figs. 18 and 19 for a jet originating from a 4 mm convergent nozzle with two 90° deflections upstream the nozzle entering for illustration purposes.
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Wassenberg, J.R., Stephan, P. & Gambaryan-Roisman, T. The influence of splattering on the development of the wall film after horizontal jet impingement onto a vertical wall. Exp Fluids 60, 173 (2019). https://doi.org/10.1007/s00348-019-2810-6
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DOI: https://doi.org/10.1007/s00348-019-2810-6