# The influence of splattering on the development of the wall film after horizontal jet impingement onto a vertical wall

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### 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.

### Graphic abstract

## List of symbols

## Latin characters

- \(a\)
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

## Greek

- \(\beta\)
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

## Dimensionless numbers

- \(Ca\)
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 \(\frac{{\rho_{g} u_{\text{jet}}^{ 2} d}}{\gamma }\)

## Notes

## References

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