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 $Re \sim 10^7$.
Predictive correlations for frictional pressure loss in pipe flows are central to hydraulic design, flow metering, and energy systems. Classical treatments express the Darcy-Weisbach friction factor f as a function of the Reynolds number Re and the relative roughness ε/D, and summarize decades of experiments in compact formulas such as the Colebrook-White equation and its explicit approximations such as Haaland equation, together with the Moody diagram that engineers use in practice [1][2][3][4]. The underlying structure is inherently dimensionless: once the flow is fully developed, the pressure gradient can depend only on the ratios of inertial to viscous forces and on the relative wall-roughness.
Dimensional analysis provides a natural starting point for such problems. Buckingham-Π theorem guarantees * ozgur.ertunc@ozyegin.edu.tr that any physically admissible relation among ∆P , U m , ρ, µ, D, L, and ε may be written in terms of a reduced set of dimensionless groups [5,6]. For internal flows, this leads to the familiar choice of the Reynolds number and the relative roughness, so that f = f (Re, ε/D). However, the theorem is silent on which dimensionless groups are most predictive and on how they should be combined. Historically, this ranking has been supplied by physical reasoning and experiments: Blasius used smooth-pipe data to infer a nearly power-law dependence of the friction factor on Re [7], while Nikuradse’s sand-grain roughness experiments mapped out the dependence on ε/D and extended the data into the fully rough regime [8]. At the same time, it has long been recognized that roughness morphology matters: compared with Nikuradse’s uniform sand grains, commercial pipes with broader distributions of protuberance sizes can exhibit a more gradual departure from the smooth-pipe law. In that setting, ε is most consistently interpreted as an effective (equivalent sand-grain) roughness referenced to Nikuradse-style behaviour, enabling different surface types to be compared on a common roughness scale [9]. In a similar spirit, earlier resistance formulations in hydraulics also emphasized relative roughness measures and reported weak power dependence of roughness coefficients on an absolute roughness scale [10]. Semi-empirical correlations such as Colebrook-White interpolate between these regimes but remain implicit and non-trivial to invert.
Recent years have seen growing interest in using datadriven methods to discover dimensionless groups and closure relations directly from experimental or numerical data. The “Virtual Nikuradse” model, for example, constructs a high-fidelity friction-factor surface from Nikuradse-style data via spline and polynomial fits [11]. More broadly, symbolic regression and related techniques seek analytical expressions that reproduce observed data while remaining compact and interpretable [12][13][14][15]. The GPTIPS 2 framework, in particular, provides a multigene genetic-programming implementation for symbolic regression in MATLAB [16], and tools such as PySR offer efficient modern solvers in Python/Julia [15]. Yet most of these applications only enforce dimensional homogeneity and perhaps simple bounds or penalties; they typically do not encode the detailed asymptotics and monotonicity properties that are known for pipe friction. As a result, purely data-driven symbolic models may match frictionfactor data over the training range but exhibit unphysical behaviour when extrapolated: exponents in ∆P versus velocity may drift outside plausible ranges, pressure drop may decrease with increasing roughness in some corner of parameter space, or the dependence on viscosity and density may contradict basic momentum-balance considerations. For engineering use, such pathologies are unacceptable even if they occur in regions with sparse data.
In this work, we combine a problem-specific orderof-magnitude analysis (OMA) with physics-constrained symbolic regression to discover new correlations for the friction factor in turbulent pipe flows. Starting from a stream-wise momentum balance and energy equations in the framework of Reynolds decomposition and averaging, we derive an OMA model for the pressure drop that has separate viscous and roughness-dominated turbulent contributions. The two contributions are blended with a logistic function so that the resulting equation can reflect the transition from viscous effects-dominated smooth turbulent pipe flow to inertial turbulence-dominated rough turbulent pipe flow. Written in dimensionless form, this OMA model implies specific, testable trends for how the pressure drop behaves when one physical parameter is varied while the others are held fixed. We characterize these trends through four logarithmic sensitivities: (i) the effective velocity exponent, (ii) the roughness sensitivity, (iii) the viscous sensitivity, and (iv) the density sensitivity. Evaluated over the Nikuradse and Superpipe parameter ranges, the OMA model yields tight envelopes
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