# Time-Series-Data Interpolation Applied to Boundary-Layer Profiles Measured on Different Flights

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Flight-Test Bed, Measurement System and Equipment, and Test Conditions

#### 2.1. Flight-Test Bed and Measurement System Onboard

#### 2.2. Paint-Based Riblet

#### 2.3. Test Conditions

^{+}(Equation (1)) and $R{e}_{u\infty}$ (or Re

_{flight}) (Equation (3)) is presented in Figure 3a for assumed on-design conditions at semi-cruise conditions described in Section 2.1 and in Figure 3b for all time-series data, including off-design conditions from take-off to landing. Note that Figure 3a corresponds to the conditions and analysis presented in a previous study [32] and Figure 3b shows extended data for all time-series data obtained during one flight.

^{+}has a linear relationship with $R{e}_{u\infty}$ with R

^{2}value of 0.99 for these variables. Figure 3b also shows strong linearity between the two variables, although slight discrepancies can be seen. This kind of discrepancy and outliers from the linear fit correspond to near-take-off and -landing conditions. Therefore, while the flight condition becomes stable regardless of whether it is in ascent/descent or cruising, these two variables are linearly related. For this reason, s

^{+}is used to define flight conditions, such as the flight Mach numbers and altitudes, which were obtained by the onboard ADS system, to obtain a specific value of s

^{+}in the range of 5 to 40. In conjunction with the physics behind the s

^{+}and the Reynolds number ($R{e}_{u\infty}$), $R{e}_{u\infty}$ was varied by the changes in Mach number and flight altitude in order to examine a wide range of s

^{+}conditions.

^{7}(as presented in Figure 3b) and Mach numbers of 0 (take-off) to 0.82. It should be noted that in all the data ranges, Mach numbers below 0.4 were eliminated, since datasets in this low range were mainly involved around take-off and landing (i.e., outliers in Figure 3b), when riblets would not be expected to have any effect. Note that this low Mach-number range is also important to evaluate the effect of riblets on reduction in skin friction drag for off-design conditions. Since the current riblet was not expected to have an effect in this low Mach number range, the datasets were eliminated, as mentioned above. This data elimination also helped to reduce data scattering, which is discussed in the Results and Discussion sections.

^{+}, as mentioned above, and several targeted conditions were made in a single flight. Therefore, the flight profile had several acceleration and deceleration scenarios, as shown in Figure 4a. Note that the figure shows the slight difference in flight-sequence scenarios between the first flight (used as reference case in a later section) and the second flight (used as compared case in a later section), such as the varying duration of maneuvers, ensuring that total flight durations were different. Despite the differences in time, a wide range of flight conditions was applied in each case.

## 3. Data-Reduction Methods

#### 3.1. Response Surface Methodology and Governing Parameters for RSM-Based Interpolation

_{∞}, Re

_{∞}, α, etc.) may be possible. For instance, a sufficiently large number of datasets would increase the accuracy of the fitting curve in response surface model, using as many variables as possible. However, available data points would generally be limited, and some additional consideration was given to selecting the governing parameters.

_{p}. Considering the measurement uncertainties for these parameters, this isentropic assumption can contain approximately 2.4% of uncertainty. Note that the application of these two variables to the RSM successfully interpolated the boundary layer profile with an uncertainty range of 4% [32]. A detailed discussion of the uncertainty is given in Section 4.1.

#### 3.2. Interpolation of Boundary-Layer Profile

_{p}distribution through the interpolation technique using RSM as a function of the Mach number and total pressure with good accuracy. The interpolation interval was 0.001 for C

_{p}distribution, considering the significant digits for the ADS system. Similar to the previous method [32], the restriction was added to the interpolation to ensure that the generated response surface did not involve any extrapolations and to increase the accuracy. For example, if the C

_{p}distribution made by the RSM method exceeds the boundary values determined by the Mach number and the total pressure, it is considered as an outlier and is limited to the boundary values of the constrained area [32]. This reconstruction is implemented at every C

_{p}value of 0.001 that exceeds boundary values. This process strengthens the robustness of the interpolation as it ensures that only interpolation is used and prevents potential errors due to extrapolation.

#### 3.3. Residual-Norm Calculation (Residual-Norm-Based Interpolation)

_{∞}, P

_{0}

_{∞}, T

_{0}

_{∞}, s

^{+}, α, and V

_{∞}are considered.

## 4. Results and Discussion

#### 4.1. Errors Associated with RSM-Based Interpolation

_{∞}= 0.75, α = 0.66°, and s

^{+}= 18.5 and “Com” denotes flight conditions with M

_{∞}= 0.70, α = 1.23°, s

^{+}= 17.9). These flight conditions and data were found by using the residual-norm evaluation. The “original” and “interpolated” profiles denote the raw profile and interpolated profile, respectively. For example, in Figure 6, the label Ref (Original) represents the original boundary-layer-profile data for the reference case; and the data labeled Ref (Interpolated) represents the interpolated data for the reference case. The same rule is applied to the other dataset. The third-order response surface model with two variables, x

_{1}= P

_{0}

_{∞}and x

_{2}= M

_{∞}, was employed here for the RSM-based interpolation [32]. It should be noted that the RSM-based interpolation aim to eliminate the differences in flight conditions. Therefore, if the premised conditions between two different flights are exactly the same, it is not necessary to implement the RSM-based interpolation.

#### 4.2. Influence of Parameter Selection on Case Identification

^{+}or α for the norm calculation. The errors appearing around 40% were found to be cases in which the variables associated with flight velocity or Mach number were not used. It was found that the inclusion of the flight velocity or Mach number reduced the error levels to 10%. Figure 7b shows much lower levels of errors. Thus, the RSM-based interpolation is beneficial in reducing the overall error level for any combination of variables. The minimum value in the maximum error can be seen at the combination numbers of 22, 28, and 34 in Figure 7b. Since the use of more variables offers higher accuracy, combination No. 34, which gives a combination of four variables (M

_{∞}, P

_{0}

_{∞}, s

^{+}, and V

_{∞}), was chosen.

^{+}. The solid lines correspond to the reference case (Flight 1) and the dotted lines (Flight 2) correspond to the compared case. Each parameter was normalized by its maximum value. The main reason for showing this plot is to show that the residual-norm approach can find the flight condition that is most closely matched to the reference data at each timepoint. According to Figure 4a, which shows original datasets for both cases, it is obvious that the total duration of the flight time for each case is different and, thus, each flight procedure is different. Accounting for this difference in flight procedure, the horizontal axis on each plot indicates the time starting from a timepoint at which the flight Mach number exceeded 0.4 in the ascent phase. This makes the comparison between these two different flights appropriate. It should be noted that the flight procedures for these two flights were not identical, but the difference was small because these two flights were intended to have the same flight conditions and procedures. The resulting datasets (dotted lines), which had the minimum norm values of the variables, showed good agreement with the reference data (solid lines). This fact also indicates that the differences in the testing duration had no influence on the residual-norm calculation and that the corresponding comparison data could be found throughout the testing scenario.

#### 4.3. Errors Associated with the Residual-Norm Calculation

_{∞}< 0.4 were eliminated before calculating the residual-norm procedure.

#### 4.4. Application of the Interpolation Method to Evaluate u^{+} Gain by Riblets

^{+}(representing the skin friction drag to the quantitative skin-friction coefficient (c

_{f})). However, this study does not contain any data such as calibration curves; hence, a quantitative evaluation is not possible within the scope of this study. Therefore, instead of evaluating the effect of riblets on the reduction in the skin friction drag in a quantitative manner, this section discusses the applicability of the proposed residual-norm-based interpolation method to evaluate the skin-friction-drag reduction in a qualitative manner.

^{+}or u

^{+}gain when using riblets was calculated first. In order to calculate the Δu

^{+}with the riblets, the focus was on the boundary-layer profile in the log-law region, where it usually appeared around y

^{+}of 100–1000. To convert the boundary-layer profile in a log scale, the following y

^{+}- u

^{+}plot was employed. The Δu

^{+}is the reduction in skin friction drag. The Y

^{+}and u

^{+}were calculated from Equations (6)–(8).

^{+}-y

^{+}plots for cases s

^{+}of 17 and 31. The smooth-wall case and the riblet-wall case are compared. The error bar is 5.7% for u

^{+}, which was obtained from Table 1. The positive riblet effect is shown in Figure 10a at s

^{+}= 17, where Δu

^{+}is expected to be the maximum, as shown through the u

^{+}gain in the region of y

^{+}< 2000. The s

^{+}= 31, which is shown in Figure 10b, is where Δu

^{+}is expected to be negative or exert no positive effect, indicating that the riblet would reduce the skin-friction-drag reduction [30,44].

^{+}of 100 to 1000, Δu

^{+}at an average of y

^{+}= 200–500 considering some data-scattering features was compared between the reference and compared cases.

^{+}plotted against the s

^{+}. The Δu

^{+}obtained without using RSM-based interpolation (red) was compared with those with RSM-based interpolation (black). The error bar was 5.7%, as discussed in Section 4.3. The RSM-based interpolated datasets showed slightly less data scattering, although almost the same Δu

^{+}was obtained throughout the s

^{+}range.

^{+}based on the datasets with RSM-based interpolation, shown in Figure 11, for the Δu

^{+}range of −2 to 2, and also shows the bin-averaged Δu

^{+}calculated with an s

^{+}bin-range of 2.5s

^{+}. A quadratic curve fitted to the bin-averaged dataset is also superimposed in the figure. The designed riblets were expected to show their maximum performance in reducing skin friction drag around the s

^{+}of 17 [30,44]. The curve fitting clearly indicates the expected performance trend, which suggests that the positive effect of skin friction reduction appears to peak in the s

^{+}range of around 10 to 20. Although some data scattering remains, the positive values of Δu

^{+}in the s

^{+}range of 10–30 predominate over those in the negative values. In the region where the s

^{+}ranges below 10 and above 30, a negative expected performance of the riblets is displayed, in that the skin friction drag increases. Thus, qualitatively, the findings here are significant in that the RSM-based interpolation reduces the error level, which makes the trend of the Δu

^{+}clearer.

## 5. Conclusions

^{+}(Δu

^{+}) when using the riblets. The results showed that the Δu

^{+}peaked in the nondimensional riblet width (s

^{+}) range of 10–20, which was around the designed values; and at the other s

^{+}ranges, the effectiveness of the riblets in reducing skin friction drag was reduced, as expected.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

c_{f} | local skin-friction coefficient |

C_{f} | total skin-friction coefficient |

C_{p} | pressure coefficient |

M | Mach number |

q | uncertainty |

r | Residual-norm value |

R | gas constant |

Re | Reynolds number |

s | width of riblet, μm |

s^{+} | non-dimensional width of riblet |

T | temperature, K or °C |

U | streamwise velocity component (U-velocity), m/s |

u_{τ} | friction velocity |

x | coordinate in streamwise direction |

α | angle of attack, deg |

γ | specific heat ratio |

μ | dynamic viscosity |

ν | kinematic viscosity, m^{2}/s |

θ | boundary layer thickness, mm |

ρ | density, kg/m^{3} |

σ | standard deviation |

Subscripts | |

0 | standard condition |

com | compared case |

flight | flight condition |

i, j | index number |

interpolated | interpolated data |

pitot | pitot-rake data |

ref | reference case |

s | static condition |

total | total value |

var | variable |

∞ | freestream condition |

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**Figure 1.**The JAXA flight-test bed and paint-based riblet surface attached to the aircraft: (

**a**) Flight-test bed; (

**b**) mounted pitot rakes.

**Figure 3.**Relationship between s

^{+}and unit Reynolds number for: (

**a**) on-design (target) cruise conditions; (

**b**) all conditions from time-series data over Mach 0.4.

**Figure 4.**Representative flight history showing Mach number: (

**a**) Original flight histories for two flights; (

**b**) original and 1-s-averaged flight histories for the smooth wall.

**Figure 5.**Data interpolation using the response surface model reconstructed per Mach number and total pressure (original experimental data are superimposed as black dots for comparison).

**Figure 6.**Example case (M

_{∞}= 0.73, α = 0.78°, P

_{∞}= 42.0 kPa, T

_{∞}= 221.6 K) of the original velocity profiles superimposed with the interpolated profiles.

**Figure 7.**Average of maximum percentage errors presented for outer pitot probes (nos. 44 and 45) for all measured flight cases (

**a**) without RSM-based interpolation and (

**b**) with RSM-based interpolation.

**Figure 8.**Time-series datasets comparing each reference and compared case found by residual-norm calculations with four variables: (

**a**) M

_{∞}; (

**b**) P

_{0∞}; (

**c**) V

_{∞}; and (

**d**) s

^{+}. Each variable was normalized by its maximum value.

**Figure 9.**Maximum percentage uncertainty for all pitot probes for all measured flight cases caused by interpolation.

**Figure 10.**Comparison of boundary-layer profiles between smooth-wall and riblet-wall cases at (

**a**) a near design point (s

^{+}= 17) and (

**b**) an off-design point (s

^{+}= 31).

**Figure 12.**Enlarged Δu

^{+}obtained from RSM-based interpolated data, bin-averaged Δu

^{+}, and their curve fits.

Uncertainty Source | Symbol | Value, % | Note |
---|---|---|---|

Total pressure | q_{1} | 0.025 | Atmospheric air pressure measured by the onboard air-data-sensing system. |

Mach number | q_{2} | 1.01 | Obtained by the onboard air-data-sensing system. |

Pitot measurement | q_{3} | 3.5 | Max. percentage error obtained as 1σ over an averaged value for 1-min duration for all probes. |

Response surface methodology (RSM) (interpolation) | q_{4} | 1.9 | When RSM-based interpolation is used. |

Residual-norm calculation | q_{5} | 3.9 | Errors obtained from Figure 9, including pitot-pressure-measurement errors. |

Total error | q_{total} | 5.7 | With RSM-based interpolation. |

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## Share and Cite

**MDPI and ACS Style**

Takahashi, H.; Kurita, M.; Iijima, H.; Koga, S.
Time-Series-Data Interpolation Applied to Boundary-Layer Profiles Measured on Different Flights. *Aerospace* **2023**, *10*, 322.
https://doi.org/10.3390/aerospace10040322

**AMA Style**

Takahashi H, Kurita M, Iijima H, Koga S.
Time-Series-Data Interpolation Applied to Boundary-Layer Profiles Measured on Different Flights. *Aerospace*. 2023; 10(4):322.
https://doi.org/10.3390/aerospace10040322

**Chicago/Turabian Style**

Takahashi, Hidemi, Mitsuru Kurita, Hidetoshi Iijima, and Seigo Koga.
2023. "Time-Series-Data Interpolation Applied to Boundary-Layer Profiles Measured on Different Flights" *Aerospace* 10, no. 4: 322.
https://doi.org/10.3390/aerospace10040322