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Order of Magnitude Analysis and Data-Based Physics-Informed Symbolic Regression for Turbulent Pipe Flow

Yunus Emre Ünal, Özgür Ertunç, Ismail Ari, Ivan Otić

Abstract

Friction losses in rough pipes are often predicted using semi-empirical correlations, such as the Colebrook-White equation (Colebrook,1939), which do not fully replicate Nikuradse's rough-pipe experiments (1950). This study derives scaling relations for the viscous and turbulent contributions to the streamwise pressure drop through an order-of-magnitude analysis of the Reynolds-averaged Navier-Stokes equations and the kinetic-energy transport equations. These relations impose constraints on the local sensitivity of the pressure drop to factors such as mean velocity, roughness, viscosity, and density through exponent envelopes and serve as a physical prior for symbolic regression. By combining Nikuradse's rough-pipe and smooth-pipe data of Zagarola and Smits (1998), we aim to derive compact correlations for the friction factor that fit experimental data while adhering to the derived constraints. A modified genetic programming engine (GPTIPS2) optimizes model structure and evaluates it based on fitness, complexity, and constraint violation. This method yields interpretable expressions that accurately reproduce friction factors across various roughness levels and Reynolds numbers, validated up to $Re \sim 10^7$.

Order of Magnitude Analysis and Data-Based Physics-Informed Symbolic Regression for Turbulent Pipe Flow

Abstract

Friction losses in rough pipes are often predicted using semi-empirical correlations, such as the Colebrook-White equation (Colebrook,1939), which do not fully replicate Nikuradse's rough-pipe experiments (1950). This study derives scaling relations for the viscous and turbulent contributions to the streamwise pressure drop through an order-of-magnitude analysis of the Reynolds-averaged Navier-Stokes equations and the kinetic-energy transport equations. These relations impose constraints on the local sensitivity of the pressure drop to factors such as mean velocity, roughness, viscosity, and density through exponent envelopes and serve as a physical prior for symbolic regression. By combining Nikuradse's rough-pipe and smooth-pipe data of Zagarola and Smits (1998), we aim to derive compact correlations for the friction factor that fit experimental data while adhering to the derived constraints. A modified genetic programming engine (GPTIPS2) optimizes model structure and evaluates it based on fitness, complexity, and constraint violation. This method yields interpretable expressions that accurately reproduce friction factors across various roughness levels and Reynolds numbers, validated up to .
Paper Structure (22 sections, 71 equations, 16 figures, 2 tables)

This paper contains 22 sections, 71 equations, 16 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Derivation of Governing Equations
  4. Order of Magnitude Analysis (OMA)
  5. Constraints for Regression
  6. Constraint C1: Velocity Sensitivity Exponent
  7. Constraint C2: Sensitivity to Roughness at Fixed Reynolds Number
  8. Constraint C3: Viscous Consistency at Fixed Velocity
  9. Constraint C4: Density Sensitivity at Fixed Velocity
  10. Symbolic Regression with Constraints
  11. Results
  12. Pareto Trade-off
  13. Physical Interpretation of Candidate 1
  14. Term $T_1$: Fully Rough Wall Contribution
  15. Term $T_2$: Roughness Onset Gate
  16. ...and 7 more sections

Figures (16)

  • Figure 1: Control volume (CV, red dashed region) for an incompressible fully developed turbulent statistically stationary (a) two-dimensional channel and (b) pipe flow.
  • Figure 2: OMA-based nonlinear regression to Eq. \ref{['eq:order_of_magnitude_pressure_pipe_final3']}.
  • Figure 3: Exponent of surface average mean velocity of OMA-based model and Haaland's model.
  • Figure 4: Exponent of roughness of OMA-based model and Haaland's model.
  • Figure 5: Exponent of viscosity of OMA-based model and Haaland's model.
  • ...and 11 more figures