# Data-Driven Calibration of Rough Heat Transfer Prediction Using Bayesian Inversion and Genetic Algorithm

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Test Case Geometry and Setup

#### 2.1. Physical Geometry and Boundary Conditions

#### 2.2. Mesh and Numerical Setup

^{+}below 1 for all the roughness ranges tested. The growth rate normal to the wall is 1.1. A close-up on the mesh near the floor is shown on Figure 2. The main outcomes of the mesh convergence study are detailed in the Appendix A.

## 3. 2PP Thermal Correction Model

_{t}. The model takes two roughness input parameters: the roughness height k (m) and the equivalent roughness k

_{s}(m). Equation (1) allows the computation of ΔPr

_{t}. In Equation (1), Pr is the laminar Prandtl number and d is the distance to the wall.

## 4. PCE Metamodeling

#### 4.1. Design of Experiment (DOE)

_{s}/k are defined to set the ranges for the sampling. Note that the ratio k

_{s}/k is used instead of k

_{s}alone, since it will allow one to directly evaluate the relation between the roughness height and the equivalent roughness. The present work is included in the broader scope of in-flight aircraft icing. Therefore, the typical ranges of variation of k and k

_{s}/k are obtained from the icing literature [33,34]. These ranges are wide enough to ensure that they are suitable for the current study, which is not specifically a simulation in icing conditions. The compilation of the distribution of all the input parameters is given in Table 1. The distributions are chosen as uniform since there is no a priori knowledge of the experimental roughness pattern.

#### 4.2. Metamodels Generation

_{i}is the output of interest, X = (X

_{1}, X

_{2}) is the input parameter vector, M

_{i}is the corresponding PCE metamodel defined by its coefficients ${y}_{\alpha}$ and the multivariate polynomials ${\psi}_{\alpha}$ of the decomposition. α = (α

_{1}, α

_{2}) is the multi-index with two components (since there are two input parameters; see Table 1). The multivariate polynomials ${\psi}_{\alpha}$ are obtained as the tensor product of the two (in the present application) univariate basis polynomials φ (Equation (5)).

_{i}[9].

_{c}is the heat transfer coefficient in W/m

^{2}K.

_{2}, predicts the mean relative error compared to the experimental results of [2]. The mean relative error ε is the average of the relative errors taken on every mesh point (Equation (6)).

_{points}is the number of mesh points in the study zone, and h

_{c,CFDi}and h

_{c,EXPi}are the CFD predicted h

_{c}at point i and the experimental h

_{c}at point i, respectively.

_{3}is a multi-output metamodel evaluating in one estimation N values of h

_{c}along the wall. Setting N to a high value allows one to predict the complete h

_{c}distribution on the entire rough wall. The metamodel M

_{1}is separated from the metamodel M

_{3}: this separation allows one to compare the cases where the starting h

_{c}is calibrated alone (M

_{1}) and where several locations are simultaneously calibrated (M

_{3}).

^{2}coefficient (Equation (7)).

_{CFD}and Y

_{PCE}are the CFD and PCE predictions, respectively. ${\overline{Y}}_{CFD}$ is the mean value of the output of interest (CFD). An R

^{2}coefficient close to 1 ensures a PCE metamodel with a good accuracy, since it predicts outputs close to what the full CFD simulation gives. Once the metamodels are established, the next step is to use them for the sensitivity study and the calibration purpose.

## 5. Sensitivity Study

_{i}denotes the output of interest when the ith input parameter is fixed. The total Sobol index, which is monitored in the present study, for the ith input parameter for a generic three parameters study is given by Equation (10).

- above 80% is very important;
- between 50% and 80% is important;
- between 30% and 50% is unimportant;
- below 30% is negligible.

## 6. Model Calibration

#### 6.1. Bayesian Inversion Calibration

_{i}) of input X

_{i}, based on the assumed prior distributions (see Table 1) π(θ) and on the information provided by the experimental data. Here, θ denotes the distribution parameters and the | symbol denotes the conditional dependence. The posterior distributions are the distributions of the input parameters knowing the information brought by the experimental data.

_{i}|θ) is called the likelihood function and π(X

_{i}) is seen as a normalization constant called the marginal likelihood. The new posterior distributions are generally not uniform anymore and allow identifying the input parameter values that will produce an output that best fits the experimental data. Additionally, a discrepancy between the PCE output and the CFD prediction is given to the solver, along with the experimental observations themselves. For instance, a discrepancy between 0 W/m

^{2}K and 15 W/m

^{2}K means that up to 15 W/m

^{2}K of difference between the PCE and CFD predictions is expected. Numerically speaking, the Bayesian inversion is performed with the UQLab tool [19]. The computation of the posterior distribution is made using a Markov chain Monte Carlo (MCMC) algorithm. The samplers used in the study are the affine invariant ensemble algorithm (AIES) or the Metropolis–Hastings (MH) algorithm. For the purposes of the work, the MCMC solver is tuned to perform 70,000 iterations and generates 15 chains. The Bayesian solver setup for each metamodel is summarized in Table 3. For the metamodel M

_{2}(predicting the mean relative error with experimental data), the experimental observation is 0% of mean relative error. This means that the objective of the calibration is to obtain a mean relative error close to 0%.

#### 6.2. Calibration Using a Genetic Algorithm

_{3}outputs multiple (N) values, the genetic algorithm is executed N times. Each one of the executions gives a set of calibrated parameters and the final retained parameters are obtained by doing an average or a weighted average over the N solutions. Most of the time, the simple averaging produces unsatisfactory results. The weighted average, with appropriate weight(s) on the most relevant output(s), allows one to improve the results, as shown in the next section.

## 7. Results

#### 7.1. CFD Results before Calibration

_{s}= 1.55 mm. This baseline simulation also provides useful information about the convergence of the CFD configuration used. The iterative convergence residual curves are displayed on Figure 3.

#### 7.2. Visualization of the DOE outputs

#### 7.3. Characteristics of the Metamodels and Accuracy

_{c}along the rough zone, one to predict the mean relative error with experimental results and a third one to predict N values of h

_{c}along the wall (see Table 2). For each metamodel, the resulting PCE is described with the polynomial degree p

_{PCE}and the number of terms in the expression (Equation (4)). Additionally, the R

^{2}coefficient is calculated to assess the metamodel accuracy (Equation (5)). The values obtained are gathered in Table 5. For the present study and for the rest of the paper, the metamodel M

_{3}is used with N = 6, meaning it computes six values of h

_{c}regularly spaced along the wall. Note that the number of terms in Equation (4), for a two input parameters study, is equal to $\frac{({p}_{PCE}+2)!}{{p}_{PCE}!2!}$.

^{2}coefficients as low as 94.6%. For graphical visualization, the regression for M

_{2}(with R

^{2}of 99.96%) is plotted on Figure 6 where Y

_{PCE}and Y

_{CFD}are the mean errors in percentage with the literature, as predicted by PCE and CFD, respectively. For synthesis and concision, only the regression for M

_{2}is displayed, since it is the worst among the three (see Table 5).

^{2}regression coefficient. This R

^{2}assessment shows that the metamodels generated are accurate and reliable enough to be used in the present application.

#### 7.4. Sensitivity Study

_{s}/k is the most dominant parameter of influence, with 88% to 97% sensitivity. According to the classification made by [38], k is a negligible parameter in the model sensitivity while k

_{s}/k is a very important parameter. Therefore, the relation between k

_{s}and k is more critical for the model output than their absolute values in millimetres.

#### 7.5. Bayesian Inversion Calibration

_{1}and M

_{2}, the calibrated roughness parameters retained are the mean values of the posterior distribution. For metamodel M

_{3}, the maximum a posteriori (MAP) is retained. The calibrated roughness parameters retained are listed in Table 7, along with the value (mean or MAP) chosen.

_{3}calibration for both k and k

_{s}/k roughness parameters, respectively.

_{3}: k = 1.8 mm and k

_{s}/k = 2.8 (i.e., k

_{s}= 5.0 mm). The observations highlight that the ratio k

_{s}/k (Figure 8b) is finely calibrated with a smaller uncertainty compared to the roughness height k (Figure 7b). The posterior distribution for the ratio k

_{s}/k exhibits a narrower peak, typical of a small variance.

_{1}and M

_{2}, the calibrated roughness parameters from Table 7 are inputted into the CFD solver and the simulation is run to verify the new heat transfer obtained after calibration. Figure 9 shows the heat transfer using the calibrated roughness parameters for M

_{1}and M

_{2}.

_{1}allows one to obtain an average relative error with the experimental data of 4.7%. The relative error is computed as the average for the entire rough zone among all the grid points. The experimental data being available at about only 25 locations, the experimental values are interpolated at the grid points to allow the relative error calculation. The calibration for M

_{2}presents similar errors with the experimental data with 4.8% of relative mean error. Globally, both calibrations present satisfactory results, showing less than 5% of error on average compared to the experimental results. The results for the metamodel M

_{1}are higher than the one for the metamodel M

_{2}, which is due to the higher roughness parameters in the case of M

_{1}. This observation is in accordance with the usual experimental observation, where higher roughness elements lead to an enhanced heat transfer [2].

_{3}. Since M

_{3}outputs h

_{c}values at different locations, it is possible to use the calibrated inputs to plot both PCE metamodel and CFD predictions.

#### 7.6. Genetic Algorithm Calibration

_{3}, the genetic algorithm outputs N calibrated parameters corresponding to each of the N locations. The final retained parameters are computed by average or weighted average (where the initial point at the start of the rough zone has a weight of three).

_{1}is 8.2% and is 5.7% for metamodel M

_{2}.

_{3}. The calibration shows better agreement in the middle and at the end of the wall. This feature has a drawback: the agreement at the beginning is not as good as for the metamodels M

_{1}and M

_{2}. Taking the average roughness parameters gives a large overestimation of h

_{c}at the start of the rough zone. Performing a weighted average, giving a weight of three to the first point, tends to lower the starting h

_{c}value and globally lowers the values everywhere.

#### 7.7. Comparison of Both Calibration Methods

_{1}and M

_{2}(see Figure 11). By purely looking at the values in Table 9, the Bayesian inversion provides more satisfying and consistent results compared to the genetic algorithm. Note that the uncertainty in the experimental results was not considered. Given the oscillations observed on the experimental curve, one can estimate that the uncertainty in the experimental data is between 6% and 10%.

_{1}and M

_{2}present a better agreement at the beginning of the rough zone for the genetic algorithm calibration (Figure 11) compared to the Bayesian inversion (Figure 9). When calibrating the starting value of h

_{c}with the metamodel M

_{1}, the genetic algorithm performs better since the starting value of h

_{c}after calibration agrees better with the experimental data compared to the Bayesian inversion. On the other hand, the Bayesian inversion presents a better agreement between the calibrated results and the experimental data in the middle and at the end of the rough zone. For the metamodel M

_{3}, the trend is the opposite: the Bayesian inversion (Figure 10) performs better at the beginning of the rough zone while the genetic algorithm (Figure 12) exhibits a better agreement at the end of the rough zone.

_{3}), the Bayesian inversion calibrates the model by working simultaneously on all outputs, thus estimating correlated calibrated parameters. The genetic algorithm works on each output independently, and requires a human intervention to establish the correlation between the outputs, like in the case of the weighted-average metamodel M

_{3}. Modeller intervention can have a large impact on the results. Table 9 shows that changing the weight of the first output of the metamodel M

_{3}from one to three in the averaging decreases the error from 10% to 7%. An optimization of the weights in the future could improve the results even further.

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Symbols | |

d | Distance to the wall (m) |

E | Mean value of a dataset |

${e}_{a}^{ij}$ | Relative error between mesh i and j for a scalar quantity |

g | Adimensional factor in the thermal correction model |

h_{c} | Heat transfer coefficient (W/m^{2}K) |

k | Roughness height (m) |

k_{s} | Equivalent roughness (m) |

l | Mesh characteristic length (m) |

M_{i} | Metamodel |

N_{points} | Number of mesh points in the rough wall |

N | Number of outputs for a multi-output metamodel |

$Pr$ | Laminar Prandtl number |

p | Convergence order for mesh study |

p_{PCE} | Degree of polynomial chaos expansion |

R^{2} | Regression coefficient |

$R{e}_{S}$ | Roughness Reynolds number |

S_{i} and S_{i,j} | First and second order Sobol indices |

S_{Ti} | Total Sobol index |

${u}_{\tau}$ | Friction velocity (m/s) |

V | Variance of a dataset |

x | Local abscissa along the channel (m) |

X = (X_{1},X_{2})
| Vector of input variables for a metamodel |

Y_{i} | Output of interest of a metamodel |

y_{α} | Coefficient of the PCE term of index α |

Greek letters | |

α = (α_{1},α_{2}) | Multi-index of the PCE decomposition |

$\Delta P{r}_{t}$ | Turbulent Prandtl number correction |

$\nu $ | Kinematic viscosity of air (m^{2}/s) |

ε | Mean relative error |

${\psi}_{\alpha}$ | Multivariate polynomial of index α |

${\phi}_{{\alpha}_{i}}$ | Univariate polynomial of index α_{i} |

π(θ|X_{i}) | Posterior distribution of input X_{i} |

π(θ) | Prior distribution with parameters θ |

π(X_{i}) | Marginal likelihood |

Subscripts | |

CFD | CFD-predicted value |

EXP | Experimental value |

## Appendix A

_{s}= 1.55 mm. The mesh convergence methodology suggested by [43] is used to compute the convergence order p and the grid convergence index (GCI). The flow quantities monitored are h

_{c}at three locations: x = 0.16 m, x = 0.46 m and x = 0.80 m. The results are gathered in Table A1.

h_{c} (x = 0.16 m) | h_{c} (x = 0.46 m) | h_{c} (x = 0.8 m) | |
---|---|---|---|

p | 0.7 | 1.7 | 2.1 |

GCI^{21} | 0.9% | 0.2% | 0.1% |

GCI^{32} | 1.4% | 0.7% | 0.5% |

_{c}at 0.8 m on the fine mesh, the interpretation of the GCI value is that the uncertainty on the monitored h

_{c}due to the mesh refinement is 0.1%. Applying the same interpretation for each value denotes a satisfactory mesh convergence study. To be conservative and confident about the quality of the CFD results, even with the various roughness patterns planned to be run, the fine mesh is retained for the rest of the process.

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**Figure 2.**Close view on the mesh in the near-wall region: (

**a**) Inlet area; (

**b**) transition straight/curved zone (scale in m).

Parameter | Minimum | Maximum | Distribution |
---|---|---|---|

k (mm) | 0.41 | 4.32 | Uniform |

Ratio k_{s}/k | 0.2 | 6.5 | Uniform |

Metamodel | Output(s) of Interest |
---|---|

M_{1} | h_{c} at the starting point of the rough zone, W/m^{2}K |

M_{2} | Mean relative error with experimental h_{c} (%) |

M_{3} | h_{c} values at N equally spaced locations along the rough zone (multi-output), W/m^{2}K |

Metamodel Calibrated | Objective of the Calibration | Sampler | Discrepancy | Experimental Observation Supplied |
---|---|---|---|---|

M_{1} | Recovering the same starting value of h_{c} | AIES | Uniform [0; 15] W/m^{2}K | 255.1 W/m^{2}K |

M_{2} | Having a mean relative error with experimental h_{c} of 0% | AIES | Uniform [0; 5]% | 0% |

M_{3} | Recovering the same h_{c} values at the N equally spaced locations along the rough zone | MH | Uniform [0; 15] W/m^{2}K | Experimental h_{c} at the N locations (W/m^{2}K) |

Metamodel Calibrated | Objective of the Calibration | Objective Function Used |
---|---|---|

M_{1} | Recovering the same starting value of h_{c} | |M_{1} − 255.1| |

M_{2} | Having a mean relative error with experimental h_{c} of 0% | |M_{2}| |

M_{3} | Recovering the same h_{c} values at the N equally spaced locations along the rough zone | |M_{3}[i] − h_{c}[i]|i = 1…N |

Metamodel | Output of Interest | PCE Degree p _{PCE} | Number of Terms in Equation (4) | R^{2}Coefficient |
---|---|---|---|---|

M_{1} | h_{c} at the starting point of the rough zone, W/m^{2}K | 10 | 66 | 0.99994 |

M_{2} | Mean relative error with experimental h_{c} (%) | 10 | 66 | 0.99962 |

M_{3} | h_{c} values at 6 equally spaced locations along the rough zone (multi-output), W/m^{2}K | Y1:10 | 66 | 0.99994 |

Y2:9 | 55 | 0.99993 | ||

Y3:12 | 91 | 0.99999 | ||

Y4:10 | 66 | 0.99992 | ||

Y5:12 | 91 | 0.99999 | ||

Y6:12 | 91 | 0.99997 |

Metamodel | Total Sobol Indices |
---|---|

M_{1} | k: 0.1445 k _{s}/k: 0.8868 |

M_{2} | k: 0.3061 k _{s}/k: 0.9772 |

M_{3} | k: 0.1167 k _{s}/k: 0.9061 |

Calibrated Metamodel | Values Retained | Final Calibrated Roughness Parameters |
---|---|---|

M_{1} | Mean | k = 2.2 mm k _{s} = 6.4 mm |

M_{2} | Mean | k = 1.6 mm k _{s} = 4.2 mm |

M_{3} | MAP | k = 1.8 mm k _{s} = 5.0 mm |

Calibrated Metamodel | Values Retained | Final Calibrated Roughness Parameters |
---|---|---|

M_{1} | k = 1.9 mm k _{s} = 3.1 mm | |

M_{2} | k = 4.3 mm k _{s} = 8.2 mm | |

M_{3} | Average among the N values | k = 3.0 mm k _{s} = 8.3 mm |

M_{3} | Weighted average among the N values | k = 2.9 mm k _{s} = 7.1 mm |

Calibrated Metamodel | h_{c} Mean Relative Error with Experimental Data | |
---|---|---|

Bayesian Inversion | Genetic Algorithm * | |

M_{1} | 4.7% | 8.2% |

M_{2} | 4.8% | 5.7% |

M_{3} | 5.4% | Avg: 10% W-Avg: 7.0% |

_{3}, Avg: Average among the N outputs; W-Avg: Weighted average.

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**MDPI and ACS Style**

Ignatowicz, K.; Solaï, E.; Morency, F.; Beaugendre, H.
Data-Driven Calibration of Rough Heat Transfer Prediction Using Bayesian Inversion and Genetic Algorithm. *Energies* **2022**, *15*, 3793.
https://doi.org/10.3390/en15103793

**AMA Style**

Ignatowicz K, Solaï E, Morency F, Beaugendre H.
Data-Driven Calibration of Rough Heat Transfer Prediction Using Bayesian Inversion and Genetic Algorithm. *Energies*. 2022; 15(10):3793.
https://doi.org/10.3390/en15103793

**Chicago/Turabian Style**

Ignatowicz, Kevin, Elie Solaï, François Morency, and Héloïse Beaugendre.
2022. "Data-Driven Calibration of Rough Heat Transfer Prediction Using Bayesian Inversion and Genetic Algorithm" *Energies* 15, no. 10: 3793.
https://doi.org/10.3390/en15103793