^{1,a)}and Robert D. Moser

^{1,2,b)}

### Abstract

When assessing the veracity of mathematical models, it is important to consider the uncertainties in the data used for the assessment. In this paper, we study the impact of data uncertainties on the analysis of overlap layer models for the mean velocity in wall-bounded turbulent flows. Specifically, the tools of Bayesian statistics are used to calibrate and compare six competing models of the mean velocity profile, including multiple logarithmic and power law forms, using velocity profile measurements from a zero-pressure-gradient turbulent boundary layer and fully developed turbulent pipe flow. The calibration problem is formulated as a Bayesian update of the joint probability density function for the calibration parameters, which are treated as random variables to characterize incomplete knowledge about their values. This probabilistic formulation provides a natural treatment of uncertainty and gives insight into the quality of the fit, features that are not easily obtained in deterministic calibration procedures. The model comparison also relies on a Bayesian update. In particular, the relative probabilities of the competing models are updated using the calibration data. The resulting posterior probabilities quantify the relative plausibility of the competing models given the data. For the boundary layer, results are shown for five subsets of the turbulent boundary layer data due to Österlund, including different Reynolds number and wall distance ranges, and multiple assumptions regarding the magnitude of the uncertainty in the velocity measurements. For most choices, multiple models have relatively high posterior probability, indicating that it is difficult to distinguish between the models. For the most inclusive data sets—i.e., the largest ranges of Reynolds number and wall distance—the first-order logarithmic law due to Buschmann and Gad-el-Hak is significantly more probable, given the data, than the other models evaluated. For the pipe flow, data from the Princeton Superpipe is analyzed for the region where McKeon *et al.* find a logarithmic layer (600 ⩽ *y* ^{+} ⩽ 0.12δ^{+}). As in the boundary layer case, the first-order logarithmic law by Buschmann and Gad-el-Hak is most probable. However, the parameter values required to fit the data are different from those necessary for the boundary layer. Thus, the present analysis confirms the differences between the boundary layer and pipe flow results observed elsewhere in the literature, casting serious doubt on the universality of overlap layer model parameters.

This work is based on work supported by the Department of Energy [National Nuclear Security Administration] under Award No. DE-FC52-08NA28615. The authors would also like to thank Ernesto E. Prudencio and Sai Hung Cheung, for their work on the QUESO library, and Onkar Sahni and Rhys Ulerich, for their comments on an earlier draft of this paper. Helpful discussions with Professor Lex Smits and Professor Ivan Marusic regarding uncertainties in mean velocity measurements and with Professor Beverley McKeon regarding roughness effects in the Superpipe are gratefully acknowledged.

I. INTRODUCTION

II. BASICS OF BAYESIAN STATISTICAL ANALYSIS

A. Calibration

B. Model comparison

C. Statistical algorithms

III. CALIBRATION PROBLEM FORMULATION

A. Overlap layer mean velocity profile models

B. Calibration data

1. Boundary layer data

2. Pipe flow data

C. The likelihood

1. General concepts

2. Experimental uncertainty

3. Scenario parameter uncertainty

4. Modeluncertainty

5. The complete PDF

D. The prior

IV. ZPG BOUNDARY LAYER RESULTS

A. Model parameter posterior PDFs

1. Classic logarithmic law

2. Buschmann(0) and Oberlack logarithmic law

3. Buschmann(1) logarithmic law

4. Universal power law

5. Afzal power law

B. Model posterior probability

V. PIPE FLOW RESULTS

VI. CONCLUSIONS

### Key Topics

- Calibration
- 58.0
- Error analysis
- 33.0
- Probability theory
- 33.0
- Velocity measurement
- 24.0
- Reynolds stress modeling
- 22.0

## Figures

The VL50 subset of the Österlund^{18} zero-pressure-gradient turbulent boundary layer mean velocity measurements. There are 70 velocity profiles containing a total of 676 measurement points.

The VL50 subset of the Österlund^{18} zero-pressure-gradient turbulent boundary layer mean velocity measurements. There are 70 velocity profiles containing a total of 676 measurement points.

The subset of the McKeon *et al.* ^{12} data from the Princeton Superpipe used here. There are 16 velocity profiles containing a total of 344 measurement points.

The subset of the McKeon *et al.* ^{12} data from the Princeton Superpipe used here. There are 16 velocity profiles containing a total of 344 measurement points.

Marginal prior PDFs for paramters in the logarithmic overlap models. These include the parameters in the classic, Buschmann(0), Buschmann(1), and Oberlack logarithmic laws.

Marginal prior PDFs for paramters in the logarithmic overlap models. These include the parameters in the classic, Buschmann(0), Buschmann(1), and Oberlack logarithmic laws.

Marginal prior PDFs for parameters in the universal and Afzal power laws.

Marginal prior PDFs for parameters in the universal and Afzal power laws.

Marginal prior PDFs for the Colebrook (*C* _{ CR }, left) and Nikuradse (*C* _{ NR }, right) roughness model parameters.

Marginal prior PDFs for the Colebrook (*C* _{ CR }, left) and Nikuradse (*C* _{ NR }, right) roughness model parameters.

Posterior and prior PDFs for κ and *C* in the classic logarithmic law for σ_{γ} = 0.005. The diagonal shows the marginal PDFs while the off-diagonal shows the samples in the κ-*C* plane.

Posterior and prior PDFs for κ and *C* in the classic logarithmic law for σ_{γ} = 0.005. The diagonal shows the marginal PDFs while the off-diagonal shows the samples in the κ-*C* plane.

Posterior and prior PDFs for κ and *C* in the classic logarithmic law for σ_{γ} = 0.0025 and σ_{γ} = 0.01. The diagonal shows the marginal PDFs while the off-diagonal shows the samples in the κ-*C* plane.

Posterior and prior PDFs for κ and *C* in the classic logarithmic law for σ_{γ} = 0.0025 and σ_{γ} = 0.01. The diagonal shows the marginal PDFs while the off-diagonal shows the samples in the κ-*C* plane.

Posterior and prior PDFs for κ, *C*, and *E* in the Buschmann(0) logarithmic law for σ_{γ} = 0.005. The diagonal entries show the marginal PDFs while the off-diagonal entries show the samples projected onto two-dimensional planes in parameter space.

Posterior and prior PDFs for κ, *C*, and *E* in the Buschmann(0) logarithmic law for σ_{γ} = 0.005. The diagonal entries show the marginal PDFs while the off-diagonal entries show the samples projected onto two-dimensional planes in parameter space.

Posterior and prior PDFs for κ, *C*, *E*, κ_{1}, *C* _{1}, *B* _{1}, and *E* _{1} in the Buschmann(1) logarithmic law for σ_{γ} = 0.005. The diagonal entries show the marginal PDFs while the off-diagonal entries show the samples projected onto two-dimensional planes in parameter space.

Posterior and prior PDFs for κ, *C*, *E*, κ_{1}, *C* _{1}, *B* _{1}, and *E* _{1} in the Buschmann(1) logarithmic law for σ_{γ} = 0.005. The diagonal entries show the marginal PDFs while the off-diagonal entries show the samples projected onto two-dimensional planes in parameter space.

Posterior and prior PDFs for *A* and α in the universal power law for σ_{γ} = 0.005. The diagonal entries show the marginal PDFs while the off-diagonal entries show the samples projected onto two-dimensional planes in parameter space.

Posterior and prior PDFs for *A* and α in the universal power law for σ_{γ} = 0.005. The diagonal entries show the marginal PDFs while the off-diagonal entries show the samples projected onto two-dimensional planes in parameter space.

Posterior and prior PDFs for γ and κ and *C* in the Afzal power law for σ_{γ} = 0.005. The diagonal entries show the marginal PDFs while the off-diagonal entries show the samples projected onto two-dimensional planes in parameter space.

Posterior and prior PDFs for γ and κ and *C* in the Afzal power law for σ_{γ} = 0.005. The diagonal entries show the marginal PDFs while the off-diagonal entries show the samples projected onto two-dimensional planes in parameter space.

Posterior and prior PDFs for κ and *C* in the classic logarithmic law obtained using McKeon's Superpipe data.^{12} The diagonal shows the marginal PDFs while the off-diagonal shows the samples in the κ-*C* plane.

Posterior and prior PDFs for κ and *C* in the classic logarithmic law obtained using McKeon's Superpipe data.^{12} The diagonal shows the marginal PDFs while the off-diagonal shows the samples in the κ-*C* plane.

Posterior and prior PDFs for κ, *C*, and *C* _{ CR } in the classic logarithmic law with Colebrook roughness obtained using McKeon's Superpipe data.^{12} The diagonal entries show the marginal PDFs while the off-diagonal entries show the samples projected onto two-dimensional planes in parameters space.

Posterior and prior PDFs for κ, *C*, and *C* _{ CR } in the classic logarithmic law with Colebrook roughness obtained using McKeon's Superpipe data.^{12} The diagonal entries show the marginal PDFs while the off-diagonal entries show the samples projected onto two-dimensional planes in parameters space.

Roughness corrections: Computed from the data (Observed), calibrated Colebrook function (Colebrook (Cal)), and nominal Colebrook function (Colebrook (Nom)).

Roughness corrections: Computed from the data (Observed), calibrated Colebrook function (Colebrook (Cal)), and nominal Colebrook function (Colebrook (Nom)).

Posterior and prior PDFs for κ, *C*, *E*, κ_{1}, *C* _{1}, *B* _{1}, and *E* _{1} in the Buschmann(1) logarithmic law obtained using McKeon's Superpipe data.^{12} The diagonal entries show the marginal PDFs while the off-diagonal entries show the samples projected onto two-dimensional planes in parameter space.

Posterior and prior PDFs for κ, *C*, *E*, κ_{1}, *C* _{1}, *B* _{1}, and *E* _{1} in the Buschmann(1) logarithmic law obtained using McKeon's Superpipe data.^{12} The diagonal entries show the marginal PDFs while the off-diagonal entries show the samples projected onto two-dimensional planes in parameter space.

Posterior and prior PDFs for κ, *C*, *E*, κ_{1}, *C* _{1}, *B* _{1}, *E* _{1}, and *C* _{ CR } in the Buschmann(1) logarithmic law with Colebrook roughness obtained using McKeon's Superpipe data.^{12} The diagonal entries show the marginal PDFs while the off-diagonal entries show the samples projected onto two-dimensional planes in parameter space.

Posterior and prior PDFs for κ, *C*, *E*, κ_{1}, *C* _{1}, *B* _{1}, *E* _{1}, and *C* _{ CR } in the Buschmann(1) logarithmic law with Colebrook roughness obtained using McKeon's Superpipe data.^{12} The diagonal entries show the marginal PDFs while the off-diagonal entries show the samples projected onto two-dimensional planes in parameter space.

## Tables

Summary of overlap layer models, indicating the defining equations and the calibration parameters.

Summary of overlap layer models, indicating the defining equations and the calibration parameters.

Subsets of the Österlund data set that are used to calibrate and compare the overlap layer models.

Subsets of the Österlund data set that are used to calibrate and compare the overlap layer models.

Model posterior probability for σ_{γ} = 0.005. The bold entries indicate the models with the highest posterior plausibility for each data set.

Model posterior probability for σ_{γ} = 0.005. The bold entries indicate the models with the highest posterior plausibility for each data set.

Model posterior probability for σ_{γ} = 0.0025. The bold entries indicate the models with the highest posterior probability for each data set.

Model posterior probability for σ_{γ} = 0.0025. The bold entries indicate the models with the highest posterior probability for each data set.

Model posterior probability for σ_{γ} = 0.01. The bold entries indicate the models with the highest posterior probability for each data set.

Model posterior probability for σ_{γ} = 0.01. The bold entries indicate the models with the highest posterior probability for each data set.

Model log-evidence log (π_{evid}) computed using Princeton Superpipe data. The bold entry indicates the model with the highest evidence.

Model log-evidence log (π_{evid}) computed using Princeton Superpipe data. The bold entry indicates the model with the highest evidence.

Posterior model probability computed using Princeton Superpipe data. The bold entry indicates the model with the highest posterior probability.

Posterior model probability computed using Princeton Superpipe data. The bold entry indicates the model with the highest posterior probability.

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