**4. Simulation**

To design a forced flow apparatus to suppress the light deflection effect successfully, the described conflicting interest between the reduction of the light deflection and the forced cooling of the component needs to be addressed. To this end, the interaction of a forced air flow with the thermal field around an object was simulated to find a compromise. The simulations were conducted using ANSYS Discovery AIM. While the overall aim was to measure the geometry of crank shafts, the simulated object was simplified to a cylinder. Its main axis was concentric to the *y*-axis of world coordinate system (see Figure 1). It was placed in a rectangular, friction-less flow channel of dimensions *dx* = 200 mm × *dz* = 400 mm. The gravitational acceleration of *g* = 9.81 m s<sup>−</sup><sup>1</sup> was assumed to point in negative *z*-direction. The simulation mode itself was transient and halted after *ts* = 5 s. At that stage, a stationary state was reached.

To estimate the flow velocities induced by the density gradient around the hot object, the temperature of the cylinder was assumed to be *ϑc* = 1000 °C, while the temperature of the ambient air was set to *ϑ*∞ = 20 °C. To simulate a simplified interaction between the nozzle flow and the heat-induced convection, a forced uniform flow was introduced into the simulation. According to Section 3, the direction of the forced flow was in the negative *z*-direction. The forced flow itself was modeled to be homogeneously distributed in the entire area of the flow channel used. Different forced flow velocities *vf f* were combined with different cylinder diameters *dc* to investigate its influence.

The light deflection effect was evaluated in relation to the thickness of air *dl* with a temperature *ϑair* > 110 °C. The thickness of the air layer was extracted from the simulation by measuring in a straight line up from the top center point of the cylinder (see Figure 2b). The cooling of the component was also simulated through the temperature of the topmost point on the cylinder, i.e., *<sup>ϑ</sup>c*(*zc* = *max*).

**Figure 1.** Comparison of the simulated convective flow over a cylinder with a diameter *dc* = 30 mm and a temperature *ϑc* = 1000 °C. The ambient temperature is *ϑ*∞ = 20 °C Left side: Results for the simulated temperature. Right side: Resulting air velocity in the *z*-direction.

**Figure 2.** Simulation of the effect of different forced air flow velocities *vf f* on the development of a heat-induced temperature field around a hot cylinder: (**a**) *vf f* = 0.0 m s<sup>−</sup>1; (**b**) *vf f* = 0.2 m s<sup>−</sup>1; (**c**) *vf f* = 0.55 m s<sup>−</sup>1; (**d**) *vf f* = 1.0 m s<sup>−</sup>1. The fixed boundary conditions are the following: diameter of cylinder *dc* = 37.5 mm; initial temperature of cylinder *ϑc* = 1000 °C, temperature of air *ϑ*∞ = 25 °C. The procedure for the calculation of the air layer thickness *dl*is sketched in (**b**).

#### *4.1. Simulation Results and Discussion*

The developed temperature and velocity fields in the simulation without an additional forced air flow are shown in Figure 1. The size of both fields in *y* was not investigated here, since only a 2D cylindrical object was simulated. Overall, the shape of the convective flow over the cylindrical object was qualitatively similar to the one measured by, e.g., Beermann [14]. The simulation of the free convection field was therefore considered to be valid.

The effect of different forced air flow velocities on the development of the heat-induced temperature field is shown in Figure 2 in combination with Figure 3. Figure 2 shows the sectional view through the simulated temperature field at different forced flow velocities *vf f* . There was an equilibrium-like state between *vf f* and *vc f* for *vf f* = 0.2 m s<sup>−</sup><sup>1</sup> (see Figure 2b). The simulation results for *vf f* > 1.0 m s<sup>−</sup><sup>1</sup> were conducted (see Figure 2), but are not shown here, since the differences with respect to Figure 2d were considered to be marginal.

Figure 3 shows the relation between the thickness of the air layer *dl* with *ϑair* > 110 °C and the cooling rate Δ *Tf f* as a function of the forced air flow velocity *vf f* . We estimated a function to fit into the air layer data (red dashed line), corresponding to the general style of:

$$d\_l = \frac{a}{bv\_{ff}^3 + c}.\tag{1}$$

In this equation, *a*, *b*, and *c* are arbitrary values not corresponding to any physical attributes. The equation itself is similar to an inverse relation between *dl* and *vf f* , as the thickness can be considered to be infinite for *dl*(*vf f* → 0) → inf and zero for larger values *dl*(*vf f* → inf) → 0, which corresponded to the development of the field in Figure 2. The displayed line was considered

a helpful tool for the visualization of the development of the thickness of the hot air layer; therefore, a description of the optimized parameters was omitted.

Prior to the simulation, the thickness of the air layer was expected to also be a function of the diameter of the cylinder *dl* = *f*(*dc*). This seemed to be the case up to *vf f* ≤ 0.3 m s<sup>−</sup>1, while being less distinct for *vf f* < 0.3 m s<sup>−</sup>1. The only small differences between the free convection velocities *vf c*(*dc*) hinted at a marginal influence of the cylinder diameter on the free convection.

Considering the cooling effect of the forced flow onto the cylinder (blue lines in Figure 3), the results showed an increasing cooling effect with increasing flow velocities. The cooling effect was assumed to be inversely proportional to the radius of the cylinder, which was considered to be due to the decreasing volume-surface quotient.

**Figure 3.** The thickness of the hot air layer *dl* and the cooling of the component as functions of the cylinder diameter *dc* and the forced air flow velocities *vf f* . The initial velocity of the free convective flow *vf c* varies and is shown in Table 1.

**Table 1.** Initial velocity of the free convective flow *vfc* in relation to the cylinder diameter *dc*.


## *4.2. Simulation Conclusions*

The simulations showed a direct connection between forced flow velocity *vf f* to the layer thickness *dl* and the cooling rate Δ*Tf f* . While a minimum layer thickness was desired, the reduction of the layer thickness for flow velocities *vf f* > 1ms−<sup>1</sup> was considered marginal compared to the increase of the cooling rate Δ*Tf f* through the forced air flow. Therefore, the flow velocities for the design of the actuator were chosen to be 1ms−<sup>1</sup> < *vf f* < 4ms−<sup>1</sup> across the whole cylinder surface, assuming a linear increase in cooling Δ*Tf f* with further increasing *vf f* .

To transfer the findings from the simulation to the design of a laboratory setup, a setup with three nozzles was implemented. The main aim was to achieve a mostly homogeneous velocity distribution over the area of a cylinder (diameter *dc* = 50 mm; length *lc* = 250 mm) at a distance of 500 mm. The comparison between the simulated velocities and the measured velocities of the nozzle setup is shown in the following section 5. When considering the interaction between the light from the

FPS and the forced air flow, it was concluded that the measurement light would most be affected by the warmer areas under the hot object (light blue areas in Figure 2c,d). Therefore, the combined setup should include nozzles and FPS viewing the object from roughly the same direction.
