# Feasibility of Optical Flow Field Measurements of the Coolant in a Grinding Machine
## 2 Measurement Approach  
In order to measure the flow velocity field with optical measurement techniques, flow characteristics must be visualised, from which the fluid motion can be derived. As shown in Figure 1, the flow gets torn up into droplets after interaction with the grinding wheel. There, the appearing droplets are unambiguously allocated in the shadowgram imaging, so that their movement can be observed with high-speed measurements and an SIV measurement approach. The 2d displacement field behind the interaction with the grinding wheel is calculated with a cross-correlation of consecutive images captured at the times \(t_{1}\) and \(t_{2}\)[24]. For this purpose, the obtained images are divided into interrogation areas for each and the cross-correlation function  
\[cc(\Delta\tilde{x},\Delta\tilde{y})=\sum_{\tilde{x}}\sum_{\tilde{y}}I_{1}( \tilde{x},\tilde{y})\cdot I_{2}(\tilde{x}+\Delta\tilde{x},\tilde{y}+\Delta \tilde{y}), \tag{1}\]  
is evaluated, with \(I_{1}\) and \(I_{2}\) as the image intensity at \(t_{1}\) and \(t_{2}\), respectively, over the image position \((\tilde{x},\tilde{y})\). The displacement \((\Delta\tilde{x},\Delta\tilde{y})\) of the cross-correlation maximum from the origin represents the displacement of the imaged particles or flow structures between the consecutive images. The velocity field is thus calculated by dividing the displacement from each interrogation area by the time interval \(t_{2}-t_{1}\) between the considered images. In contrast to the MWF behaviour after the wheel interaction, the situation for the MWF supply flow and its interaction with the grinding wheel is different. The MWF exiting the nozzle and hitting the grinding wheel is a continuous liquid free jet that forms structures on the surface due to friction with the stagnant surrounding air [25], which leads to local changes in the free jet surface. The appearing inhomogeneities on the surface represent characteristic flow structures and are visualisable with the shadowgraphy method. When applying SIV, these surface structures are used as flow tracers for a velocity determination (analogue to the image processing in the regions after the interaction with the grinding wheel). The occurrence of the surface structures mainly depends on the outlet shape of the nozzle, the fluid properties of the MWF (with respect to the fluid properties of the surrounding air) and the fluid velocity at the outlet. One important characteristic quantity is the Reynolds number  
\[Re=\frac{v_{\text{sup}}\cdot d_{\text{h}}}{\
u} \tag{2}\]  
with \(v_{\text{sup}}\) as the supply flow velocity, \(d_{\text{h}}=\frac{4\cdot w\cdot h}{2w+2h}\) the hydraulic nozzle outlet diameter, \(w\) and \(h\) the width and height of the rectangular nozzle outlet and \(\
u\) the kinematic viscosity of the MWF. As a theoretical value for the supply flow velocity \(v_{\text{sup}}\) at the nozzle exit, the continuity equation yields the approximation  
\[v_{\text{sup,theo}}\approx\frac{Q}{w\cdot h} \tag{3}\]  
with \(Q\) as the provided volume flow. As classified by [PERSON] and [PERSON] [26], liquid jet properties can be derived from the Reynolds number. For \(1000<Re<3000\), the liquid jet remains laminar after exiting the nozzle for a certain length and is named a semi-turbulent jet. For Reynolds numbers above 3000, the jet is considered fully turbulent with turbulent structures immediately after the nozzle outlet. On the other hand, for pipe flows, the laminar-turbulent transition point is given at the critical Reynolds number around 2300 and the clearly turbulent flow region is above 4000 [27]. Note that the cited Reynolds numbers are for round nozzle exit, so that the findings can be applied to the studied rectangular nozzle orifice only to a certain extent.