Prediction of Fully Developed Turbulent Convection with Minimal Empiricism

For a circular tube, let a be the internal radius. We may define the nondimensional tube radius as a^+=\rho au_{\tau}/\mu, where the shear velocity is defined as u_{\tau}=\sqrt{\tau_w/\rho}. Churchill expressed the time-averaged conservation equation for fully developed flow of a constant property fluid in a round tube as

\frac {\mathrm du^+}{\mathrm dy^+}=\left(1-\frac {y^+}{a^+}\right)\left[1-\left(\overline{u'v'}\right)^{++}\right] \tag{1}

Alternative Analogy

For the near-wall turbulent core region, defined as 30<y^+<0.1a^+, the general semi-logarithmic expression derived by Millikan (1938) is known to agree well with experimental data per Zagarola (1996). In this region, we may approximate 1-y^+/a^+\cong1 and (\overline{u'v'})^{++}\cong1-\frac 1{0.436y^+}. For much of the derivation, we neglect \gamma by assuming it to be zero and then re-apply it at the end. From the generalized nondimensional temperature distribution,

\frac {\mathrm dT^+}{\mathrm dy^+}=\frac 1{1+\frac {\mathrm{Pr}}{\mathrm{Pr}_t}(0.436y^+-1)} \tag{2}

Indefinitely integrating and assuming \mathrm{Pr}_t is independent of y^+ gives the nondimensional temperature distribution as

T^+=\frac 1{0.436}\left(\frac {\mathrm{Pr}_t}{\mathrm{Pr}}\right)\ln\left[y^+-\frac 1{0.436}\left(1-\frac {\mathrm{Pr}_t}{\mathrm{Pr}}\right)\right]+\Psi\left(\frac {\mathrm{Pr}_t}{\mathrm{Pr}}\right) \tag{3}

Hence, the nondimensional temperature at the centerline is

T_c^+=\frac 1{0.436}\left(\frac {\mathrm{Pr}_t}{\mathrm{Pr}}\right)\ln\left[a^+-\frac 1{0.436}\left(1-\frac {\mathrm{Pr}_t}{\mathrm{Pr}}\right)\right]+\Psi\left(\frac {\mathrm{Pr}_t}{\mathrm{Pr}}\right) \tag{4}

Using Zagarola’s approximation for the time-mean velocity at the centerline

u_c^+=7.64+\frac {\ln a^+}{0.436} \tag{5}

Then Eq. 4 becomes

T_c^+=\frac {\mathrm{Pr}_t}{\mathrm{Pr}}(u_c^+-7.64)+\Psi\left(\frac {\mathrm{Pr}_t}{\mathrm{Pr}}\right)

From the definition of the Nusselt number, then

\mathrm{Nu}=\frac {2a^+}{T_m^+}=\frac {2a^+}{T_c^+}\frac {T_c^+}{T_m^+}=\frac {2a^+}{\frac {\mathrm{Pr}_t}{\mathrm{Pr}}(u_c^+-7.64)+\Psi\left(\frac {\mathrm{Pr}_t}{\mathrm{Pr}}\right)}\frac {T_c^+}{T_m^+} \tag{6}

The integration constant \Psi can be determined by observing that Eq. 6 must reduce down to \mathrm{Nu}_1 for \gamma=0 and \mathrm{Nu}_{\infty} for infinitely high \mathrm{Pr} only if

\Psi\left(\frac {\mathrm{Pr}_t}{\mathrm{Pr}}\right)=\frac {\mathrm{Pr}_t}{\mathrm{Pr}}\left[13.62\left(\frac {\mathrm{Pr}_t}{\mathrm{Pr}}\right)^{2/3}-5.98\right] \tag{7}

Substituting Eq. 7 into Eq. 6 gives

\frac 1{\mathrm{Nu}}=\left(\frac {\mathrm{Pr}_t}{\mathrm{Pr}}\right)\frac 1{\mathrm{Nu}_1}+\left[1-\left(\frac {\mathrm{Pr}_t}{\mathrm{Pr}}\right)^{2/3}\right]\frac 1{\mathrm{Nu}_{\infty}} \tag{8}

where the formulations for \mathrm{Nu}_{\infty} and \mathrm{Nu}_1 are re-stated for convenience.

\begin{aligned} \mathrm{Nu}_1 & =\frac {u_m^+}{u_c^+}\left(\frac {T_c^+}{T_m^+}\right)_1\frac {\mathrm{Re_D}f}2 \\ \mathrm{Nu}_{\infty} & =0.07343\left(\frac {\mathrm{Pr}}{\mathrm{Pr}_t}\right)^{1/3}\mathrm{Re_D}\left(\frac {C_f}2\right)^{1/2} \end{aligned}