# Discharge Coefficients of Ports with Stepped Inlets

## Abstract

**:**

## 1. Introduction

_{d}, is also found to be sensitive to approach flow conditions that may not be well predicted or may be only crudely defined as a boundary condition to the calculation, as discussed by McGuirk and Spencer [3].

_{d}is found. The discharge coefficient of chamfered ports is found to be less sensitive to the flow conditions than radiused ports. Empirical correlation for chamfer effects have been presented in the literature [9] for holes that have a chamfer angle of 45°, but otherwise empirical relationships are not available for chamfering because of the limited data available to calibrate them. Naturally, the increase in discharge coefficient brought about by chamfering is dependent upon the chamfer angle with a maximum found at a chamfer angle of around 30°. Once a given depth of chamfer is achieved, the discharge coefficient is found to be fairly insensitive to chamfer depth [6] and chamfer tooling mis-alignment [10], making them attractive because of the lower manufacturing tolerances required to produce them.

_{d}can then be made.

_{d}correlations (such as length to diameter ratio (L/D), for example). Instead, only two or three values of other main parameters are studied to ensure stepped port behavior is consistent with these existing correlations and exhibits no significant coupling between the step feature geometry and other parameters.

## 2. Materials and Methods

_{d}, quoted in various works to be between 1.8 × 10

^{4}and 3.5 × 10

^{4}[13,14]. Reynolds number variation in C

_{d}has been tested in this work in the range 5 × 10

^{4}to 2 × 10

^{5}, which has agreed with these previous works and all results presented are for jet Reynolds numbers above 5 × 10

^{4}. Results from both rigs agree with previous studies, within experimental uncertainty. Other numerous parameters affecting discharge coefficients have been studied previously and several empirical correlations exist to describe each of their behaviors.

#### 2.1. Water Flow Facility

#### 2.2. Airflow Facility

#### 2.3. Instrumentation

^{3}.

_{d}obtained from the Stolz equation was always below 2.0% (refer to the standard for further details). The pressure drop across each plate was monitored using an inverted water manometer connected to D and D/2 static pressure tappings. Typically, each return mass flow rate could be measured to better than ±2.5% over the flow conditions used for these tests. This has been verified by checking the measured outlet mass flows against the mass flows obtained by integrating the Laser Doppler Anemometer (LDA) determined diametral velocity profiles through the annulus and core of a blank test section (i.e., with no ports).

#### 2.4. Data Reduction

_{d}and α minimised for the current experimental setup. Inputs to this calculation were the inlet axial velocity profiles, the exit orifice plate pressure drops, and the pressure transducer voltage. Calibration coefficients for each of these then allows the respective inlet, exit, and ideal mass flow rates to be calculated in kg/s.

_{d}is defined as the actual to ideal mass flow ratio and assuming constant density locally and using the calibrated venturi meter (C

_{d}

_{,v}= 0.994) as the reference (actual mass flow), the port C

_{d}was calculated using the following:

_{d}is quoted as an average of the tests over the range 6 × 10

^{4}and 3 × 10

^{5}) and uncertainty analysis shows the C

_{d}is reported at better than 3% with 95% certainty.

#### 2.5. Test Configurations

_{d}correlation methods and the stepped port results are listed in Table 2. Stepped port configurations are given in Table 3. The 12-mm diameter stepped ports are able to closely match all of the important geometric ratios found typically on an engine. Four step length, s, (hence four angles, θ) are considered for the 12 mm ports (cases A to D). At around two-thirds scale, it would be difficult to manufacture a test section with accurate eccentricity between the cold skin and tile ports. Approximately full-scale ports (diameter 18 mm) were thus employed to explore the effect of eccentricity in three directions, in which the outer sleeve could be moved relative to the inner ring to create the eccentricity (cases ε1, ε2, and ε3 have the cold skin upstream, downstream, and to one side of the smaller port, respectively). These are benchmarked against case E, which had no eccentricity but otherwise matched all other geometric parameters. To compare the effects of the step inlet to inlet chamfering, some airflow tests were also performed for single ports in axisymmetric flow with a range of step inlet angles comparable to the chamfering angles that had been considered in previous studies [6]. These are reported separately at the end of the results section.

_{p}/V

_{c}. Discharge coefficients are known to be insensitive to this parameter and no correlations feature in C

_{d}as a result. In these tests, the port jet to cross flow velocity ratio was kept in the range 2 < V

_{p}/V

_{c}< 5. Experience on the facility used here has shown no measurable change in C

_{d}in this range. However, exceeding a value of 5 for V

_{p}/V

_{c}can cause problems because the jet impingement becomes very strong and forms a recirculation large enough to give unsteady pressure measurements in the core, and was thus avoided. These flow conditions are summarized in Table 4. Readers interested in further details, such as the approach flow velocity profiles, may see references [3,16].

## 3. Results

#### 3.1. Plain Port

_{d}at high α, which is thought to be because of the formation of an unsteady vortex over a variable number of the ports at this high ingestion fraction with low port to annulus area ratio. Otherwise, the scatter of the measured C

_{d}is generally reasonable and within ±3% as expected from the uncertainty analysis.

_{d}predicted by the correlation of Chin et al. [11] for the plain circular port configurations are also shown in Figure 6 by the solid lines (the configuration they have been calculated for is indicated by unfilled versions of the respective symbol for the experimental data). The correlation has been further modified to also consider the L/D of the ports using the method outlined by McGreehan and Schotsch [8].

_{d}of a port is very sensitive to the port’s length. Significant scatter can be seen in the measured C

_{d}’s of different workers in this range because of the L/D sensitivity (Lichtarowicz [13]). This effect could be further compounded by the crossflow inducing non-axisymmetric flow. Secondly, and more important to consider here, is that from experience, some doubt may be cast on C

_{d}correlations as the annulus height is reduced in comparison to the port diameter. For the plain port geometry, this ratio, H/D, was 1.0, and it has been seen for this geometry that unsteady through-port vortices can be formed at high α (Spencer [16]). The higher levels of scatter in the measurements for α > 0.9 seen in Figure 7 are thought to be caused by these vortices. To avoid this problem in the subsequent stepped port measurements, it was decided to scale the ports so that H/D (and all other geometrical ratios) were higher and more representative of the engine configuration. This was indeed seen to reduce scatter in subsequent data.

#### 3.2. Stepped Ports, with Crossflow

_{d}with alpha is shown in Figure 7 for stepped inlet port configurations A to E. Cases A and E are highlighted with filled symbols, cases B, C, and D use the respective character to indicate measurement points. Each set has a best fit correlation line plotted through it. It can be seen that there is a significant increase in C

_{d}compared with the plain ports of the previous section. Here, the discharge coefficients reach up to 0.83 at high alpha compared with around 0.62 to 0.64 for plain ports. Principally, this significant increase is because the size of the vena contracta has been increased by the presence of the step, which in turn has increased the coefficient of contraction. With no empirical data available for stepped ports, correlations for its effect are not available. Here, the correlation of Chin et al. [11] is fitted to the data, with modifications for L/D as described earlier using McGreehan and Schotsch [8] for plain ports. An additional correction is added by increasing r/D also using the method of the authors of [8] until the correlation best fits the experimental data available here, see Appendix A. In total, five step configurations with varying step angles are considered and this process has been repeated for each. At lower values of alpha, it can be seen that the correlation appears to increasingly overpredict the discharge coefficient. This suggests that the directionality of the flow can be important to the discharge coefficient, as might be expected, but stepped port C

_{d}appears to fall more rapidly compared with expectation from this correlation with reducing α at lower values (α < 0.5).

_{d}. To predict C

_{d}for stepped ports in the range 50 < θ < 90°, a linear distribution maybe acceptable as indicated.

#### 3.3. Stepped Ports, Eccentricity

_{d}with alpha was not well captured by empirical correlations at low α, suggesting a different flow direction sensitivity of stepped ports compared with plain or radiused ports. As a result, it was considered sensible to investigate if eccentricity of the holes in the hot and cold skins had an effect. A single magnitude of eccentricity was considered of ε/D = 0.083, but in three directions, as indicated in the bottom right of Figure 9. The correlation of the work of [11] did not fit this data very well in variation with α, therefore, to compare the change in C

_{d}between the configurations, a best fit quadratic was fitted to the measured data. The percentage difference between these smoothed curves and the datum (concentric) case is then shown in Figure 10.

_{d}, reducing by 7% to 10% across the range of α considered. At high port mass ingestion (α), moving the cold skin downstream has a reduced effect, from −7% (ecc.1) down to −2% (ecc.2) at α = 1.0. At lower α, both eccentricity directions have a similar effect of a 10% reduction. Sideways eccentricity does not appear to be too detrimental to C

_{d}at high α, but C

_{d}is reduced by 5% at α = 0.4. These figures should be seen in the context of a systematic uncertainty in C

_{d}of around 2.5%, relevant when considering ratios to the concentric datum case C

_{d}

_{0}.

#### 3.4. Axisymmetric Stepped Ports, Low Step Angles

_{d}value, which the results here have confirmed. The C

_{d}of chamfered ports is maximised once their depth exceeds ~0.08D, beyond this value, increasing the size of the chamfer has little effect. Thus, in Figure 12, the high C

_{d}values of just above 0.95 at a chamfer angle of 30° are constant with t/D.

_{d}is seen for L/D of both 0.12 and 0.2. The shorter port sees a larger percentage increase than the longer port (~25% cf. ~10%) in discharge coefficient compared with the sharp-edged ports (C

_{d}

_{0}), and the maximum value in this range occurs at 45°. Insensitivity to increased step height of t/D = 0.16 for an L/D of 2.0 is also seen for two points that map closely to the t/D = 0.08 curve. It would appear the C

_{d}is increased using stepped inlets, with the peak value (at θ = 45°) being about half of the increase obtained by chamfering at θ = 30°. However, it does appear from this and the previous sections that there is limited sensitivity of C

_{d}to step height, as is the case for chamfer depth.

_{d}in cases in Figure 12 highlight some coupling between L/D and t/D that exist for short ports (low L/D). Coupling of this nature is compensated for in C

_{d}correlations for inlet radiusing. Radiusing, unlike chamfering and stepped inlets, continues to increase C

_{d}to values very close to 1.0, which is obtained once r/D is ~0.82. Thus, the data in Figure 12 can also be presented in the form of an equivalent r/D value by best fitting a correlation to the data using the best r/D to replicate the measured value. In this way, the t/D to L/D coupling can be deconvolved by the empirical data that has trained correlations, where here, we use McGreehan and Schotsch [8] approach as briefly outlined in Appendix A.

_{d}correlation based on the evidence from this study. Provided t/D > 0.07 and t < 0.75L, the effective r/D in existing correlations can be chosen from Figure 13 for the appropriate value of step angle. This should provide a C

_{d}estimate within typical measurement accuracy.

_{d}increase in the 45–50° region with an equivalent r/D of around 0.15, with step angles below ~25° and above ~80–85°, the step has no discernable effect on C

_{d}.

## 4. Conclusions

_{d}of 25% over the equivalent sharp-edged port was found (L/D = 0.25, t/D = 0.08).

_{d}was reduced by up to 10% for the orientations of eccentricity tested and approach flow directionality behavior was changed compared with concentric inlet features.

_{d}, though further work is required to verify this because this value fell between the two testing configurations.

## Funding

## Conflicts of Interest

## Nomenclature

Symbol | |

A | Area |

C_{d} | Discharge Coefficient |

$\dot{m}$ | Mass flow rate |

p | Static pressure |

q | Dynamic pressure |

V | Velocity |

α | Port ingestion fraction |

β | Port to annulus area ratio |

ρ | Density |

Subscript | |

a | Annulus inflow |

b | Bleed (Annulus outflow) |

c | Core inflow |

o | Core outflow |

p | Port |

v | Venturi |

:l | Length corrected value |

:r | Radius corrected value |

0 | Base value (no inlet step) |

Symbols for the geometrical descriptions of ports are defined in Figure 5 and Table 2, Table 3 and Table 4. |

## Appendix A

_{d}

_{0}) is corrected for inlet radiusing via the following:

_{d:r}, not explained fully here for brevity. These formulae have been used to calculate an effective r/D that best matches the stepped or chamfered port behavior. ‘r/D’ in the f equation has thus been changed to minimize the rms error in fitting the correlation of the literature [11] (taken as the baseline ‘C

_{d}

_{0}’ value) to the data for cross flow cases. For axis-symmetric flow, C

_{d}

_{0}has been taken to be the sharp-edged value (no step/no chamfer case) for the same L/D port. ‘r/D’ has then been changed to obtain the best fit to experimental values of C

_{d}.

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**Figure 7.**Stepped port C

_{d}for cases A to E. Chin et al. [11] correlation best fit by adjusting r/D.

Parameter | Measurement Device & Uncertainty |
---|---|

Port pressure drop | Furness, ±(1 Pa + 0.5%Δp) |

Port mass flow | ${\dot{m}}_{p}={\dot{m}}_{o}-{\dot{m}}_{c}$, ±3.0% |

Velocity: annulus in, core in | LDA, TSI IFA550, ±0.02 m/s |

Mass flow: annulus out, core out | BS1044 Orifice plate, ±2.5% |

Temperature | k-Type thermocouple, ±0.5 K |

Dimension | Symbol | Plain | Plain | Plain | Plain | Plain |
---|---|---|---|---|---|---|

Port Diameter (mm) | D | 20 | 20 | 20 | 20 | 20 |

Number of Holes | n | 6 | 3 | 6 | 3 | 2 |

Pitch to Diameter | Z/D | 2.62 | 5.24 | 2.62 | 5.24 | 7.85 |

Annulus Height/D | H/D | 1.00 | 1.00 | 0.25 | 0.25 | 0.25 |

Port Length/D | L/D | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 |

Port/Annulus Area | β | 0.250 | 0.125 | 1.143 | 0.571 | 0.381 |

Dimension | Symbol | A | B | C | D | E | ε1,2,3 | Typical |
---|---|---|---|---|---|---|---|---|

Port Diameter (mm) | D | 12 | 12 | 12 | 12 | 18 | 18 | 18 |

Number of Holes | n | 10 | 10 | 10 | 10 | 7 | 7 | - |

Pitch to Diameter | Z/D | 2.62 | 2.62 | 2.62 | 2.62 | 2.47 | 2.47 | 2.5 |

Annulus Height/D | H/D | 1.67 | 1.67 | 1.67 | 1.67 | 1.1 | 1.1 | 1.9 |

Port Length/D | L/D | 0.28 | 0.28 | 0.28 | 0.28 | 0.27 | 0.27 | 0.28 |

Step Height/D | t/D | 0.0708 | 0.0708 | 0.0733 | 0.0683 | 0.0601 | 0.0601 | 0.056 |

Step Length/D | s/D | 0.0890 | 0.104 | 0.1708 | 0.25 | 0.0972 | 0.0972 | 0.097 |

Step Angle | θ | 51.5° | 55.6° | 66.8° | 74.7° | 58.3° | 58.3° | 60° |

Eccentricity/D | ε/D | 0 | 0 | 0 | 0 | 0 | 0.083 × 3 | - |

Port/Annulus Area | β | 0.15 | 0.15 | 0.15 | 0.15 | 0.24 | 0.24 | 0.16 |

Property | Symbol | Minimum | Maximum |
---|---|---|---|

Port Reynolds Number | Re_{p} | 5 × 10^{4} | 2 × 10^{5} |

Port/Annulus Mass Flow ratio | α | 0.3 | 1.0 |

Port/Core Velocity ratio | V_{p}/V_{c} | 2.0 | 5.0 |

Dimension | Symbol | Chamfer | Step | Step | Step |
---|---|---|---|---|---|

Port Diameter (mm) | D | 25 | 25 | 25 | 25 |

Port Length/D | L/D | 2.00 | 2.00 | 0.24 | 2.08 |

Step Height/D | t/D | 0.08 | 0.08 | 0.08 | 0.16 |

Step Length/D | s/D | 0, 0.029, 0.037, 0.046, 0.056, 0.067, 0.080 | 0.046, 0.080 | ||

Step Angle θ = tan^{−1}(s/t) | θ | 0°, 20°, 25°, 30°, 35°,40°, 45° | 30°, 45° |

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**MDPI and ACS Style**

Spencer, A.
Discharge Coefficients of Ports with Stepped Inlets. *Aerospace* **2018**, *5*, 97.
https://doi.org/10.3390/aerospace5030097

**AMA Style**

Spencer A.
Discharge Coefficients of Ports with Stepped Inlets. *Aerospace*. 2018; 5(3):97.
https://doi.org/10.3390/aerospace5030097

**Chicago/Turabian Style**

Spencer, Adrian.
2018. "Discharge Coefficients of Ports with Stepped Inlets" *Aerospace* 5, no. 3: 97.
https://doi.org/10.3390/aerospace5030097