The effective Reynolds exponent of turbulent pipe heat transfer and the rigidity of the Dittus-Boelter correlation

The Lyon integral for fully developed turbulent pipe flow of a constant-property fluid is evaluated with a closure containing no constant fitted to heat-transfer data: the Musker eddy viscosityconstrained by the velocity log law, the Reichardt outer factor, and the Kays--Crawford turbulent Prandtl number. The bulk temperature takes the three-term form $\theta_b^+ = (\mathrm{Pr}_{t\infty}/\kappa)\ln\mathrm{Re}_\tau + \beta(\mathrm{Pr}) + C_0$, so that$\mathrm{Nu} = 2\,\mathrm{Re}_\tau\,\mathrm{Pr}/\theta_b^+$, with the thermal log-law intercept $\beta(\mathrm{Pr})$ and the core constant $C_0$ returned by the quadrature. The intercept agrees to within $4\,\%$ with the recent DNS-based expression of Pirozzoli over $0.5 \leq \mathrm{Pr} \leq 16$ and independently supports the small coefficient of $\ln\mathrm{Pr}$ against the classical Kader--Yaglom value. The Nusselt number lies within $\pm 3.6\,\%$ of isoflux pipe simulations at $\mathrm{Pr} = 0.71$ and within $-7$ to $+2\,\%$ of isothermal-wall DNS across the Prandtl range. The Reynolds dependence is a power law modulated by two logarithms, that of the momentum log layer in the friction factor and that of the thermal log layer in the resistance, and the effective exponent $n_{\mathrm{eff}} = \partial\ln\mathrm{Nu}/\partial\ln\mathrm{Re}$ increases from $0.72$ to $0.89$ over $10^4 \leq \mathrm{Re} \leq 10^6$ and $0.7 \leq \mathrm{Pr} \leq 14$, between the asymptotes $1 - 2/\ln\mathrm{Re}$ and $1 - 1/\ln\mathrm{Re}$; exponents from $\mathrm{Pr} = 1$ DNS follow the predicted increase to within $0.03$. ... mehrA power law fitted to this surface has one degree of freedom: prefactor and exponent are related by $\mathrm{d}\ln C/\mathrm{d}n = -\ln\mathrm{Re}_0$ at the calibration Reynolds number $\mathrm{Re}_0$. The Dittus--Boelter pair $(C,\,n) = (0.023,\,0.8)$ of $\mathrm{Nu} = C\,\mathrm{Re}^{n}\mathrm{Pr}^{m}$ is the tangent to the surface at the classical calibration conditions, and the coefficient pairs of two later correlations of the same form, calibrated on supercritical-water experiments by Bishop and by Swenson, lie on the predicted locus at separations that recover their calibration Reynolds numbers, $\mathrm{Re}_0 \approx 1.7\times10^5$ and $4.9\times10^5$. The prefactor is the structural compensation for the exponent: a power-law correlation cannot follow a flow whose bulk Reynolds number varies along the pipe, nor be made to by adjusting the exponent.