# A Comparative Study on Numerical Flow Simulations of a Centrifugal Electronic Cooling Fan Using Four Different Turbulence Models

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## Abstract

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## 1. Introduction

- (a)
- a comparative study on the results obtained for different turbulence models,
- (b)
- detailed results of the outlet flow field.

## 2. Fan Data and Nomenclature

#### 2.1. Fan Geometry

#### 2.2. Coordinate Systems, Evaluation Lines, Surfaces and Sections

#### 2.3. Fan Performance Data

## 3. Simulation Setup and Meshing

#### 3.1. Computational Domain

#### 3.2. Boundary Conditions and Rotation Modeling

#### 3.3. Grid Generation

#### 3.4. Numerical Settings and Turbulence Models

- k-$\u03f5$ realizeable (abbreviated as rke),
- Shear Stress Transport k-$\omega $ (abbreviated as SST),
- Reynolds Stress Omega (abbreviated as RSM) and
- Scale-Adaptive Simulation (abbreviated as SAS),

#### 3.4.1. k-ϵ Realizeable Turbulence Model

- convective term discretization: second order upwind
- pressure discretization: second order
- near-wall treatment: Enhanced Wall Treatment [53]
- temporal treatment:
- (a)
- steady, pressure-velocity coupling: SIMPLE
- (b)
- pseudo-transient, automatic time step, pseudo time scale factor 0.5, pressure-velocity coupling: Coupled
- (c)
- transient, time step: 1 × 10
^{−5}s, pressure-velocity coupling: Coupled

#### 3.4.2. SST k-ω Turbulence Model

- convective term discretization: second order upwind
- pressure discretization: second order
- temporal treatment: pseudo-transient, automatic time step, pseudo time scale factor: 0.05
- pressure-velocity coupling: Coupled
- transition modeling: one-equation model by Menter and Smirnov [54]

#### 3.4.3. RSM Turbulence Model

- Reynolds Stress Model: Stress-Omega
- convective term discretization: bounded central differencing
- pressure discretization: second order
- temporal treatment: transient, time step: 1 × 10
^{−4}s - pressure-velocity coupling: Coupled

#### 3.4.4. SAS Turbulence Model

- convective term discretization: bounded central differencing
- pressure discretization: second order
- temporal treatment: transient, time step: 5 × 10
^{−6}s - pressure-velocity coupling: Coupled
- transition modeling: one-equation model by Menter and Smirnov [54]

## 4. Numerical Tests

#### 4.1. URANS Grid Independence Study

^{−5}s were conducted for all meshes and the simulations were run until the monitors demonstrated that a steady state had been reached. Uncertainty estimations were performed according to the method described in [55]. Figure 6 shows the convergence behavior of the four different meshes for two selected integral flow quantities. The massflow-weighted total pressure difference between fan inlet and outlet $\Delta {p}_{\mathrm{total}}$ as well as the impeller outlet volumetric flow rate ${Q}_{i}$ were chosen for this purpose. These quantities are plotted over the mean cell size ratio $h/{h}_{\mathrm{very}\mathrm{fine}}$. The mean cell sizes h were calculated by evaluating the average cell volume and the values correspond well with the average sizes in the jet domain given in Table 4. The solid blue line in each graphic represents the flow quantity result for the individual meshes. The black dashed line represents the power series expansion calculated from these results and the blue vertical dashed lines represent an estimate for the uncertainty which contains the exact solution with a probability of 95%. The results show mostly monotonic grid convergence with a continuously decreasing uncertainty. The maximum deviation in the total pressure difference and the volumetric flow rate amount to 4.5% and 2.2%, respectively.

- For $-0.5\le x/D<-0.25$ all the velocity magnitude profiles show little deviation except for the near-wall region where the improved near-wall resolution of the two finer meshes leads to a steeper gradient.
- For $-0.25<x/D<0.05$ the two coarser meshes overestimate the velocity since the upper lobe of the C-shape experiences slightly stronger diffusion. It is also notable that the curves of the two finer meshes are almost identical in that region.
- For $0.2<x/D\le 0.5$ the results exhibit the strongest differences, which mainly result from the crossflow in the outlet plane. The two coarser meshes significantly underestimate the x-velocity in the upper lobe of the C-shape and consequently the velocity profiles in the vicinity of the wall do not exhibit a discernible local velocity maximum. It is also notable that this maximum is more pronounced for the fine grid in comparison to the very fine grid, which might be influenced by slight differences in the flow through the final two impeller blade channels ($\varphi =337.5\xb0\u2013360\xb0$).

#### 4.2. Steady vs. Transient Solvers

- realizable k-$\u03f5$, steady (abbreviated as rke-s);
- realizable k-$\u03f5$, pseudo-transient, global pseudo time step and time scale factor 0.5 (abbreviated as rke-pt);
- realizable k-$\u03f5$, unsteady, time step size 1 × 10
^{−5}s (abbreviated as rke-t).

#### 4.3. Estimation of Turbulent Scales

## 5. Numerical Simulations and Results

- k-$\u03f5$ realizeableThe results were evaluated from the transient simulation mentioned in Section 4.2. The scaled residuals for the continuity equation dropped by eight orders of magnitude.
- SST k-$\omega $Steady convergence was unsatisfactory. The results presented were obtained by a pseudo-transient simulation with a time scale factor of 0.05. The scaled residuals for the continuity equation dropped by five orders of magnitude.
- Reynolds-Stress OmegaSteady convergence was impossible to attain. Transient simulation with a time step size of 1 × 10
^{−4}s yielded stable convergence, where the scaled residuals for the continuity equation dropped by three orders of magnitude. - SASA time step size of 5 × 10
^{−6}s was necessary to ensure $\mathrm{CFL}\le 1$ in the blade channels. Scaled residuals for the continuity equation dropped by five orders of magnitude.

#### 5.1. Overall Performance Results

#### 5.1.1. Fan Performance Results

#### 5.1.2. Impeller Performance Results

#### 5.1.3. Efficiencies

#### 5.2. Results at the Impeller In- and Outlet

#### 5.2.1. Velocity Triangles

#### 5.2.2. Velocity Distributions at the Impeller Outlet

- $\varphi =30\xb0\phantom{\rule{0.166667em}{0ex}}\dots \phantom{\rule{0.166667em}{0ex}}180\xb0$: Velocity magnitudes tend to increase continuously up to $\varphi =120\xb0$ and subsequently decrease slightly. Differences over the impeller height remain below 5 $\frac{\mathrm{m}}{\mathrm{s}}$.
- $\varphi =180\xb0\phantom{\rule{0.166667em}{0ex}}\dots \phantom{\rule{0.166667em}{0ex}}300\xb0$: Velocity magnitudes rise steadily until they reach the maximum at approximately $\varphi =300\xb0$. With the exception of the realizable k-$\u03f5$ model results, the differences over the impeller height increase as well.
- $\varphi =300\xb0\phantom{\rule{0.166667em}{0ex}}\dots \phantom{\rule{0.166667em}{0ex}}30\xb0$: Velocity magnitudes decrease strongly and reach their minima in the vicinity of the nose. The deviations over the evaluation lines become very high, since the drop in velocity is more pronounced for the upper evaluation lines. The minimum values are followed by a steep increase until $\varphi $ reaches 30°.

#### 5.3. Volute Flow Results

#### 5.4. Fan Flow Field Overview and Blade Channel Details

#### 5.5. Outlet Flow Field

- The z-velocity and the velocity magnitude contours exhibit a C-shape.
- The x-velocity field shows negative values in the center, i.e., a flow from right to left, and positive values in the upper and lower lobe of the C-shape.
- The y-velocity of the vertical part of the C-shape is zero at $y/D\approx 0$ and exhibits positive values as the upper left corner of the fan outlet is approached and likewise, negative values as the lower left corner is approached.
- At the end of the lower lobe (lower right corner of the fan outlet) the x-velocity is noticeably positive. Conversely, the end of the upper lobe does not show a significant downward movement.

- The C-shape is the widest for the realizable k-$\u03f5$ model. Consequently, the velocity magnitude or z-velocity values are the lowest.
- The x-velocities in the upper and lower lobes are also lowest for the realizable k-$\u03f5$ model.
- The y-velocities values in the upper left corner are highest for the realizable k-$\u03f5$ model. The contours are significantly smoother than those of the other models in the entire outlet plane.
- The SST model shows the highest local velocities for the x- and z-components as well as for the magnitude.
- The SST model is the only one which produces local backflow. Negative z-velocities occur in the vicinity of the lower right corner.
- The flow structures in-between the lobes have experienced the lowest diffusion for the SST model.
- The lower lobe is significantly shorter in the SST and RSM results, and it exhibits a notable upwards movement at its end. This is a direct consequence of the flow development in the vicinity of the nose, as previously discussed in the context of Figure 15e. In the SAS results, the lower lobe extends towards the lower right corner, since, the subvertex identified in evaluation surface S33 was considerably weaker. As previously mentioned, these differences might be influenced by the MRF modeling of the impeller rotation.
- The SAS model shows the thinnest vertical part of the C-shape. The upper and lower lobe extend over the entire width of the fan outlet. As a consequence, in a thin layer at the right border of the outlet, the y-velocities show pronounced values in up- and downward direction, respectively.
- The z-velocities in-between the upper and lower lobe are the lowest for the SAS model.

#### 5.6. Comparison of Turbulence Quantities

#### 5.6.1. Turbulent Kinetic Energy

- Inlet area upstream of the impeller blades:The turbulent kinetic energy is essentially zero for both the SST and SAS model. For the realizable k-$\u03f5$ model, k is already produced when the air accelerates while moving through the fan inlet port. In the RSM case, the inlet turbulence field exhibits transient behavior leading to patches of increased turbulent kinetic energy which change over time, while in other areas k is almost zero.
- Blade channels:For the realizable k-$\u03f5$ model the flow around the blades is modeled as fully turbulent due to the turbulent inflow and the enhanced wall treatment model applied at the blade boundary layers. This leads to production of k especially between the suction side of the main blade and the pressure side of the splitter blade, as well as at the suction side of the splitter blade. Essentially, in those areas where the other turbulence models show significant flow separation the realizable k-$\u03f5$ model responds with a high production of k, which prevents massive flow separation. All $\omega $-based models show high values of $\omega $ in the wall-adjacent cells, which leads to turbulent viscosity ratios below one in the RSM case. The intermittency transmission model applied in the SST and SAS cases predicts intermittency values below 0.3 over the entire blade boundary layers, which leads to even lower turbulent viscosity ratios in these areas. This consequently facilitates the development of flow separations.
- Outlets of the blade channels 1–27:In the case of the realizable k-$\u03f5$ model, high production of k takes place in the shear-layers where the small jets emanating from the blade channels merge with the volute flow. This leads to elevated turbulent kinetic energy over the entire circumference of the impeller outlet. In addition to the low flow separation in the blade channels, these high levels of turbulence generated at the impeller outlet lead to highly blurred velocity contours and a lack of resolution of flow details, as was discussed in Section 5.3. The high degree of flow separation predicted in the SST and RSM cases leads to thin jets entering the volute, where they get deflected in circumferential direction. Distinct areas of high turbulent kinetic energy develop in the shear layers. In general, the production of k in the SST model and the RSM case are relatively similar, which explains the observation that the resulting mean velocity fields in the volute only exhibit small differences, as presented in the context of Figure 14 and Figure 15. The levels of modeled k in the SAS case reach values above 1.5 in distinct spots downstream of the impeller outlet. Further discussion regarding the ratio of modeled and resolved k will follow later in this section.
- Outlets of the blade channels 28–32 towards the fan outlet:The impeller outlet area downstream of the final blade channels is characterized by a high degree of unsteady fluctuations and turbulence production, since the main volute flow follows the outer wall towards the fan outlet. A high degree of turbulence production is predicted by all models. However, the differences are quite apparent. The realizable k-$\u03f5$ model shows a confined area with high levels of k, which strongly decreases towards the outlet. The SST model only produces similar levels of turbulent kinetic energy in distinct regions, where jets exit the blade channels. Particularly in this area, the pseudo-transient approach leads to a certain degree of fluctuations in the turbulence field, so that both size and location of the regions of higher turbulent kinetic energy change over the course of the simulation. In the RSM case, the turbulence field is highly unsteady. Only small regions of elevated levels of k related to jets emanating from the blade channels appear. Unsteady patches of higher k can occur towards the outlet, but generally, turbulence levels decrease strongly. In the SAS case confined regions of high levels of k can be detected at the impeller outlet, depending on the unsteady local flow situation. In these cases large turbulent eddies are locally not fully resolved.

#### 5.6.2. Reynolds Stresses

#### 5.6.3. Grid Resolution Dependency of SAS Results

#### 5.7. Comparison with CTA and PIV Measurements

#### 5.7.1. Simulated vs. Experimental Time-Averaged Outlet Flow Field Data

- The measurement and all simulated contours agree concerning the width of the upper lobe, but only the SAS results agree with the measurements in indicating a stagnation point in the upper right corner.
- All simulation models overpredict the width of the vertical part of the C-shape on the left side of the contours. In this regard, the SAS model demonstrates the highest level of agreement.
- In the measurement the lower lobe of the jet ends at approximately $x/D=0.2$. Here, the best fit is obtained by the RSM model. Moreover, the SST model exhibits a nearly similar behavior.
- The measurements exhibit a region of increased velocity outside of the C-shape, starting above the right corner of the outlet at $x/D=0.5$ and $y/D=-0.2$ and ending at $x/D=0.1$ and $y/D=0.2$. This detail is not predicted by any of the simulation models. The SST model is the only model which shows elevated velocities in the proximity of this area, however, the shape of the velocity contours do not match the experiments. A plausible reason for these differences might be related to the MRF modeling, which does not accurately represent certain dynamic effects, especially regarding the outflow of the final three blade channels and the flow around the nose.
- Although the SAS model overestimates the width of the lower lobe of the C-shape, in Figure 26a the location of the velocity maximum aligns relatively well with the measurements.

#### 5.7.2. Simulated vs. Experimental RSME Data at the Outlet

#### 5.7.3. Comparison of the Near-Field Jet Development

## 6. Conclusions

- The realizable k-$\u03f5$ model shows steady convergence. Pseudo-transient and transient simulations do not lead to significant differences in the results. The main reason is very high turbulent kinetic energy production at the outlet of the blade channels, which, due to high turbulent diffusion, does not allow for the development of distinct and unsteady vortex structures.
- For the SST k-$\omega $ model steady convergence could not be obtained since the turbulent kinetic energy production at the blade outlet channel was similar to that of the more sophisticated models, which allowed for the resolution of a high degree of flow details and vortex structures. Only the SST model produced inflow in a small area close to the lower right corner of the outlet port. Reasonable convergence could be obtained using the pseudo-transient solver. However, the results showed significant fluctuations, particularly in the highly unsteady flow region towards the outlet. One main advantage of using a pseudo-transient solver in this situation is the reduced solving time and the greatest disadvantage is the lack of information regarding the temporal evolution of the results.
- In general, the results produced by the RSM-$\omega $ model are satisfactory. Several aspects of the outlet flow field could be predicted with reasonably accuracy, particularly the length of the lower lobe of the C-shape. The complexity of the RSM case required an unsteady simulation, however, a relatively high time-step was permissible, which lead to overall lower computational costs compared to the SAS model.
- The SAS model was able to predict the development of the left part and the upper lobe of the C-shape very accurately. However, discrepancies from the measurements were observed for the lower lobe. The model could adapt well to the relatively abrupt jumps in the mesh discretization without producing discernible errors in the pressure or velocity field. The requirement to satisfy the CFL condition lead to a very small time step and long simulation times to produce robust mean values.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**CT scan of the investigated centrifugal fan, which shows good agreement with the CAD model.

**Figure 2.**Sections through the fan and rotor, showing the definition of the coordinate systems and evaluation surfaces.

**Figure 3.**Visualization of the nineteen evaluation lines located at the outer surface of the impeller.

**Figure 5.**Details of the generated fine mesh; the region colored in cyan in the lower two images indicates the rotating reference frame. (

**a**) Fan wall mesh; looking into the fan via the outlet port. (

**b**) Mesh detail for a cut-plane at $y/D=0.2$, including outlet and volute nose. (

**c**) Mesh detail for blade channel 27 for a cut-plane at $y/D=0.2$.

**Figure 6.**Convergence trend of the massflow weighted total pressure difference between fan inlet and the trend of the VFR at the impeller outlet; solid blue lines represent the simulation results, dashed black lines the power series expansion, and vertical dashed lines the 95% uncertainty estimates. (

**a**) Total pressure difference between fan in- and outlet. (

**b**) VFR at the impeller outlet.

**Figure 7.**Mesh convergence study: Comparison of the normalized velocity magnitude results on a horizontal evaluation line over the fan outlet at $y/D=0.25$ (black dashed line) and visualization of the velocity contours of the fine mesh. (

**a**) Evaluation line results. (

**b**) Normalized velocity magnitude contours for the fine mesh.

**Figure 8.**Height-averaged normalized velocity magnitude results at the rotor outlet of the steady, pseudo-transient and transient realizable k-$\u03f5$ simulations and the deviations with respect to the transient results. (

**a**) Normalized velocity magnitude result. (

**b**) Differences of the normalized velocity magnitude.

**Figure 9.**Fraction of cell width $\Delta {x}_{c}$ to integral length scale L and the Kolmogorov length scale ${\eta}_{K}$, respectively, obtained from URANS simulations. A cut-plane at $y/D=0.25$ is used to evaluate the results for the fine and very fine mesh, including the detailed situation between selected blade channels. Turbulent length scale estimation for the (

**a**) fine mesh: fan cross section, (

**b**) very fine mesh: fan cross-section, (

**c**) fine mesh: blade channel detail, (

**d**) very fine mesh: blade channel detail.

**Figure 10.**Velocity triangles at the impeller inlet and outlet calculated from time- and spatially-averaged SAS results. Common nomenclature for velocity triangles is used here, $\overrightarrow{C}\phantom{\rule{0.166667em}{0ex}}\dots $ absolute velocity, $\overrightarrow{W}\phantom{\rule{0.166667em}{0ex}}\dots $ relative velocity, $U\phantom{\rule{0.166667em}{0ex}}\dots $ impeller tangential velocity. Indices 1 and 2 refer to the rotor inlet and outlet, respectively. (

**a**) Velocity triangle at the impeller inlet. (

**b**) Velocity triangle at the impeller outlet.

**Figure 11.**Normalized time-averaged velocity magnitude results of the realizable k-$\u03f5$ simulation at the impeller outlet surface. The aspect ratio of the plot (impeller outlet height vs. outlet circumference) is increased by a factor of 8. Note: No moving average filter was applied.

**Figure 12.**Normalized time-averaged velocity magnitude results for the evaluation lines at the impeller outlet surface for the different turbulence models; bottom (blue, L1), mid (green, L10), top (red, L19), others (grey).

**Figure 13.**Time-averaged normalized tangential, axial and radial velocity averaged over all 19 evaluation lines at the impeller outlet as well as the blade channel volume flow rates for the different turbulence models.

**Figure 14.**Time-averaged normalized velocity magnitude results for six different evaluation surfaces of the fan volute and the different turbulence models. (

**a**) includes a sketch of the cross-section of the fan for orientation purposes: ➀ …upper impeller ring, ➁ …blade channel, ➂ …impeller base plate, ➃ …lower impeller gap, ➄ …lower volute extension. Velocity magnitude for evaluation surface (

**a**) S5, (

**b**) S15, (

**c**) S20, (

**d**) S30, (

**e**) S33, (

**f**) S35.

**Figure 15.**Time-averaged normalized axial and radial velocity results for three different evaluation surfaces of the fan volute and the different turbulence models. (

**a**) Axial velocity for evaluation surface S20. (

**b**) Radial velocity for evaluation surface S20. (

**c**) Axial velocity for evaluation surface S30. (

**d**) Radial velocity for evaluation surface S30. (

**e**) Axial velocity for evaluation surface S33. (

**f**) Radial velocity for evaluation surface S33.

**Figure 16.**Normalized relative velocity magnitude and normalized static pressure results for a selected time step at an evaluation surface at $y/D=0.2$, slightly above the middle height of the impeller outlet. Note: The white dashed line indicates the MRF border. Overlaid LIC lines: (

**a**) shows LIC lines calculated from the relative velocity field, (

**b**) from the absolute velocity field.

**Figure 17.**Normalized relative velocity magnitude results for selected blade channels at an evaluation surface inside the fan at $y/D=0.2$, slightly above the middle height of the impeller outlet.

**Figure 18.**Time-averaged normalized velocity magnitude, x-, y- and z- velocity results at the fan outlet (S37) for the different turbulence models.

**Figure 19.**Normalized turbulent kinetic energy of all turbulence models for selected time steps at an evaluation surface at $y/D=0.2$, slightly above the middle height of the impeller outlet.

**Figure 20.**Normalized turbulent kinetic energy at the evaluation surfaces S33 and S37 (fan outlet) for the different turbulence models.

**Figure 21.**Normalized time-averaged resolved turbulent kinetic energy results of the SAS model at an evaluation surface at $y/D=0.2$ and at the evaluation surfaces S33 and S37 (fan outlet).

**Figure 22.**Comparison of normalized Reynolds stress results of the SST k-$\omega $ and RSM simulation for a selected time step at an evaluation surface at $y/D=0.2$.

**Figure 23.**Comparison of normalized $\overline{{u}^{\prime}{w}^{\prime}}$ Reynolds stress results of the SST k-$\omega $ and RSM simulation for a selected time step at an evaluation surface at $y/D=0.2$.

**Figure 24.**Cell-value contours of SAS simulation results for velocity, pressure and turbulent kinetic energy; details in the vicinity of blade channels 24 and 25; black lines indicate borders of regions with uniform mesh resolution, the white line indicates the boundary of the rotating reference frame. (

**a**) normalized time-averaged velocity magnitude. (

**b**) normalized time-averaged static pressure. (

**c**) normalized time-averaged resolved turbulent kinetic energy. (

**d**) normalized modeled turbulent kinetic energy.

**Figure 25.**Cell-value contours of SAS simulation results for different turbulence quantities; details in the vicinity of blade channels 24 and 25; black lines indicate borders of regions with uniform mesh resolution, the white line indicates the boundary of the rotating reference frame. (

**a**) normalized modeled specific dissipation rate. (

**b**) normalized time-averaged turbulent viscosity ratio. (

**c**) normalized time-averaged resolved $\overline{{u}^{\prime}{w}^{\prime}}$-stress. (

**d**) normalized time-averaged modeled $\overline{{u}^{\prime}{w}^{\prime}}$-stress.

**Figure 26.**Time-averaged normalized $\sqrt{{V}_{x}^{2}+{V}_{z}^{2}}/{\overline{V}}_{z,f}$ and $\sqrt{{V}_{y}^{2}+{V}_{z}^{2}}/{\overline{V}}_{z,f}$ velocity contours at the fan outlet slightly downstream of evaluation surface S37 at $z/D=0.05$ for the CTA measurements and the different turbulence models. Note: The white lines indicate the fan outlet edges.

**Figure 27.**RMSE values of $\sqrt{{V}_{x}^{2}+{V}_{z}^{2}}$ and $\sqrt{{V}_{y}^{2}+{V}_{z}^{2}}$ at the fan outlet slightly downstream of evaluation surface S37 at $z/D=0.05$ for the CTA measurements and selected turbulence models. Note: The white lines indicate the fan outlet edges.

**Figure 28.**Time-averaged normalized ${V}_{z}/{\overline{V}}_{z,f}$ velocity contours of various $xy$-planes showing the development of the jet’s cross section with increasing outlet distance. Note: Red lines indicate FWHM.

Fan Casing | ||
---|---|---|

outer dimensions | 65.3 mm × 64.5 mm × 23.5 mm | |

outlet dimension | $D\times D$ | 20 mm × 20 mm |

inlet diameter | ${D}_{\mathrm{in}}$ | 31 mm |

characteristic length | D | 20 mm |

fan outlet area | ${A}_{f}$ |
400 mm^{2} |

Impeller | ||

main blade outer diameter | $2\phantom{\rule{0.166667em}{0ex}}{r}_{2}$ | 45.0 mm |

main blade inner diameter | $2\phantom{\rule{0.166667em}{0ex}}{r}_{1}$ | 35.9 mm |

splitter blade inner diameter | $2\phantom{\rule{0.166667em}{0ex}}{r}_{\mathrm{s}}$ | 38.5 mm |

impeller height | ${h}_{\mathrm{i}}$ | 9.0 mm |

number of blade pairs | ${N}_{\mathrm{B}}$ | 32 |

impeller outlet area | ${A}_{i}$ |
1417 mm^{2} |

blade sweep | forward-curved | |

Nominal Fan Parameters | ||

nominal supply voltage | ${V}_{n}$ | 12 V |

nominal supply current | ${I}_{n}$ | 135 mA |

nominal impeller speed | ${n}_{n}$ | $5000\text{}\mathrm{rpm}$ |

nominal VFR | ${Q}_{n}$ | $10\text{}\frac{{\mathrm{m}}^{3}}{\mathrm{h}}$ |

nominal total fan pressure | ${p}_{n,\mathrm{total}}$ | $75\mathrm{Pa}$ |

nominal overall fan efficiency | ${\eta}_{n}$ | 12% |

measured supply voltage | ${V}_{m}$ | 13.5 V |

measured supply current | ${I}_{m}$ | 187.2 m |

measured impeller speed at $13.5$ $\mathrm{V}$ | ${n}_{m}$ | 5033 rpm |

measured VFR | ${Q}_{m}$ | $13.54\text{}\frac{{\mathrm{m}}^{3}}{\mathrm{h}}$ |

measured static fan pressure | ${p}_{m}$ | 0 Pa |

Notation | Variables | Dimension | Normalized Dimension |
---|---|---|---|

outer domain | $a\times b\times c$ | $800\times 900\times 500$ ${\mathrm{mm}}^{3}$ | $40D\times 45D\times 25D$ |

jet domain | $d\times e\times f$ | $220\times 600\times 110$ ${\mathrm{mm}}^{3}$ | $11D\times 30D\times 5.5D$ |

fan domain | $g\times h\times i$ | $75\times 100\times 75$ ${\mathrm{mm}}^{3}$ | $3.75D\times 5D\times 3.75D$ |

Variable | Notation | Result |
---|---|---|

Density | $\rho $ | $1.225\text{}\frac{\mathrm{k}\mathrm{g}}{{\mathrm{m}}^{3}}$ |

Dynamic viscosity | $\mu $ | 1.7894 × 10^{−5} $\frac{\mathrm{k}\mathrm{g}}{\mathrm{m}\text{}\mathrm{s}}$ |

Notation | Very Rough | Rough | Fine | Very Fine |
---|---|---|---|---|

Number of cells | 5 × 10^{6} | 9 × 10^{6} | 21 × 10^{6} | 36 × 10^{6} |

Cell size outer domain | 15 mm | 12 mm | 8 mm | 6 mm |

Cell size jet dom. | 3.75 mm | 3.0 mm | 2.0 mm | 1.5 mm |

Cell size impeller dom. | 0.938 mm | 0.75 mm | 0.5 mm | 0.375 mm |

Cell size blade section | 0.469 mm | 0.188 mm | 0.125 mm | 0.094 mm |

average fan ${y}^{+}$ | 3.1 | 2.5 | 1.9 | 1.4 |

Turbulence Model | $\Delta {\mathit{p}}_{\mathbf{total}}$ (Pa) | ${\mathit{Q}}_{\mathit{f}}$ ($\frac{{\mathbf{m}}^{3}}{\mathbf{h}}$) | ${\overline{\mathit{V}}}_{\mathit{z},\mathit{f}}$ ($\frac{\mathbf{m}}{\mathbf{s}}$) | ${\mathit{R}\mathit{e}}_{\mathit{f}}$ |
---|---|---|---|---|

rke-t | 58.0 | 12.4 | 8.69 | $1.19\times {10}^{4}$ |

SST-pt | 60.1 | 11.6 | 8.17 | $1.12\times {10}^{4}$ |

RSM-t | 52.8 | 11.8 | 8.30 | $1.14\times {10}^{4}$ |

SAS | 56.3 | 12.2 | 8.59 | $1.18\times {10}^{4}$ |

**Table 6.**Comparison of dimensionless fan performance parameters obtained with the four different turbulence models.

Turbulence Model | $\mathit{\phi}$ | $\mathit{\psi}$ | $\mathit{\sigma}$ | $\mathit{\delta}$ |
---|---|---|---|---|

rke-t | 0.184 | 2.142 | 0.242 | 2.824 |

SST-pt | 0.173 | 2.220 | 0.229 | 2.935 |

RSM-t | 0.175 | 1.950 | 0.254 | 2.821 |

SAS | 0.181 | 2.082 | 0.246 | 2.820 |

**Table 7.**Comparison of impeller performance results obtained with the four different turbulence models.

Turbulence Model | $\Delta {\mathit{p}}_{\mathit{i},\mathbf{total}}$ (Pa) | $\Delta {\mathit{p}}_{\mathit{E},\mathbf{total}}$ (Pa) | $\Delta {\mathit{p}}_{\mathit{c}}$ (Pa) | $\Delta {\mathit{p}}_{\mathit{d}}$ (Pa) | $\Delta {\mathit{p}}_{\mathit{r}}$ (Pa) | ${\mathit{Q}}_{\mathit{i}}$ ($\frac{{\mathbf{m}}^{3}}{\mathbf{h}}$) | ${\overline{\mathit{V}}}_{\mathit{r},\mathit{i}}$ ($\frac{\mathbf{m}}{\mathbf{s}}$) |
---|---|---|---|---|---|---|---|

rke-t | 128.7 | 136.1 | 30.9 | 74.5 | 30.7 | 13.3 | 2.61 |

SST-pt | 137.8 | 120.5 | 30.9 | 61.2 | 28.4 | 12.7 | 2.48 |

RSM-t | 125.0 | 124.7 | 30.9 | 64.6 | 29.2 | 12.9 | 2.52 |

SAS | 126.5 | 125.9 | 30.9 | 66.5 | 28.5 | 13.3 | 2.61 |

Turbulence Model | $\mathit{\eta}$ (%) | ${\mathit{\eta}}_{\mathit{m}\mathit{e}}$ (%) | ${\mathit{\eta}}_{\mathit{i}}$ (%) | ${\mathit{\eta}}_{\mathit{b}}$(%) | ${\mathit{\eta}}_{\mathit{v}}$ (%) |
---|---|---|---|---|---|

rke-t | 7.9 | 29.5 | 63.7 | 45.0 | 93.1 |

SST-pt | 7.6 | 26.9 | 71.2 | 43.0 | 92.0 |

RSM-t | 6.9 | 27.2 | 65.1 | 42.2 | 92.0 |

SAS | 7.6 | 29.0 | 63.8 | 44.6 | 92.0 |

**Table 9.**Comparison of the velocities and angles of the velocity triangles at the impeller inlet for the different turbulence models.

Turbulence Model | ${\mathit{\alpha}}_{1}$ (°) | ${\mathit{\beta}}_{1}$ (°) | ${\mathit{U}}_{1}$ ($\frac{\mathbf{m}}{\mathbf{s}}$) | ${\mathit{W}}_{1}$ ($\frac{\mathbf{m}}{\mathbf{s}}$) | ${\mathit{C}}_{1}$ ($\frac{\mathbf{m}}{\mathbf{s}}$) |
---|---|---|---|---|---|

rke-t | 53.3 | 32.9 | 9.40 | 7.55 | 5.12 |

SST-pt | 50.3 | 31.8 | 9.40 | 7.30 | 5.00 |

RSM-t | 51.2 | 32.2 | 9.40 | 7.37 | 5.05 |

SAS | 50.8 | 33.9 | 9.40 | 7.31 | 5.27 |

**Table 10.**Comparison of the velocities and angles of the velocity triangles at the impeller outlet for the different turbulence models.

Turbulence Model | ${\mathit{\alpha}}_{2}$ (°) | ${\mathit{\beta}}_{2}$ (°) | ${\mathit{U}}_{2}$ ($\frac{\mathbf{m}}{\mathbf{s}}$) | ${\mathit{W}}_{2}$ ($\frac{\mathbf{m}}{\mathbf{s}}$) | ${\mathit{C}}_{2}$ ($\frac{\mathbf{m}}{\mathbf{s}}$) |
---|---|---|---|---|---|

rke-t | 12.4 | 88.0 | 11.78 | 2.61 | 12.15 |

SST-pt | 12.8 | 70.4 | 11.78 | 2.63 | 11.18 |

RSM-t | 12.7 | 76.0 | 11.78 | 2.60 | 11.44 |

SAS | 12.9 | 81.3 | 11.78 | 2.64 | 11.68 |

**Table 11.**Comparison of the volume flow rates derived from CTA measurements and the conducted simulations.

Source | ${\mathit{Q}}_{\sqrt{{\mathit{V}}_{\mathit{x}}^{2}+{\mathit{V}}_{\mathit{z}}^{2}}}$ ($\frac{{\mathbf{m}}^{3}}{\mathbf{s}}$) | ${\mathit{Q}}_{\sqrt{{\mathit{V}}_{\mathit{y}}^{2}+{\mathit{V}}_{\mathit{z}}^{2}}}$ ($\frac{{\mathbf{m}}^{3}}{\mathbf{s}}$) | ${\mathit{Q}}_{{\mathit{V}}_{\mathit{z}}}$ ($\frac{{\mathbf{m}}^{3}}{\mathbf{s}}$) |
---|---|---|---|

rke-t | 13.2 | 13.0 | 12.9 |

SST-pt | 12.3 | 12.1 | 11.9 |

RSM-t | 12.4 | 12.2 | 12.1 |

SAS | 13.0 | 12.6 | 12.5 |

Measurements (CTA) | 13.4 | 12.9 | - |

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## Share and Cite

**MDPI and ACS Style**

Kirchhofer, M.; Krieger, M.; Hofer, D.
A Comparative Study on Numerical Flow Simulations of a Centrifugal Electronic Cooling Fan Using Four Different Turbulence Models. *Energies* **2023**, *16*, 7864.
https://doi.org/10.3390/en16237864

**AMA Style**

Kirchhofer M, Krieger M, Hofer D.
A Comparative Study on Numerical Flow Simulations of a Centrifugal Electronic Cooling Fan Using Four Different Turbulence Models. *Energies*. 2023; 16(23):7864.
https://doi.org/10.3390/en16237864

**Chicago/Turabian Style**

Kirchhofer, Martin, Michael Krieger, and Dominik Hofer.
2023. "A Comparative Study on Numerical Flow Simulations of a Centrifugal Electronic Cooling Fan Using Four Different Turbulence Models" *Energies* 16, no. 23: 7864.
https://doi.org/10.3390/en16237864