Friction factor power law with equivalent log law, of a turbulent fully developed flow, in a fully smooth pipe

Abstract

Over the past century, Blasius (Forschungsheft 131:1–41, 1913) gave empirical power law friction factors $\lambda = 0.3164\;\text {Re}^{-1/4}$ for a turbulent pipe flow and later work published other empirical values of power law indexes. The present work deals with open Reynolds momentum equations by matched asymptotic expansions for large Reynolds number. In the overlap region, a rational dual solutions have power law and log law velocities and friction factor. If outer layer flow is neglected, the power-law friction factor becomes $\lambda = m \text {Re}^{-n}$ in a pipe flow, with power law index $n(\text {Re})$ and prefactor $m(\text {Re})$. Further, tangent at a point on power law envelop gives log law at that point. Thus for each value of power index n with prefactor m, the power law theory holds at a point. As an engineering practice, power law at one point is often used in a limited domain of Reynolds number, which compares in that range with experimental and DNS data reported in the literature. The friction factor log law to higher order has been proposed and the second-order effect compares well for lower Reynolds numbers in an entire range of Reynolds numbers for a turbulent pipe flow.

Appendix 1: Reynolds mean momentum equation for turbulent flow in a pipe

The mean momentum equation for an steady turbulent pipe flow, in cylindrical coordinates $(r, \theta , x)$ in streamwise x-direction, can be described as:

$$\begin{aligned} 0 = - \frac{{d}p}{\rho {d}x} + \frac{1}{r} \frac{\partial }{\partial r}\left[ r \left( \nu \frac{\partial U_x}{\partial r} - <u'_x u'_r>\right) \right] \end{aligned}$$

(A1)

Here $U_x$ is the axial mean velocity in pipe, p is the static pressure distribution, $\delta$ is the radius of pipe, $\nu = \mu /\rho$ is the kinematic viscosity of fluid, $\rho$ is the fluid density, $u_\tau = (\tau _w/\rho )$ is the friction velocity and $\tau _w$ is the shear stress on the wall. The first term on the right-hand side is the pressure gradient dp/dx, the second term represents the viscous force $\mu dU_x/dr$ and the third term represents turbulent Reynolds shear stress $- \rho <u'_x u'_r>$ as inertia term. The integration on mean momentum equation (A1) over pipe cross section gives a balance of pressure gradient and wall shear stress as

$$\begin{aligned} \frac{{d}p}{\rho {d}x} = - \frac{2}{\delta } \frac{\tau _w}{\rho } \end{aligned}$$

(A2)

Thus, the mean momentum equation (A1) with balance equation (A2) becomes

$$\begin{aligned} \nu \frac{\partial U_x}{\partial r} - <u'_x u'_r> = - \frac{r}{\delta } \frac{\tau _w}{ \rho } \end{aligned}$$

(A3)

Introducing the coordinate normal from pipe wall $y=\delta - r$, then $du/dr = - du/dy$, $u_r = - u_y$ and $U_x=u$, the mean momentum equation (A1) becomes

$$\begin{aligned} \nu \frac{\partial u}{\partial y} + \tau = u_\tau ^2 \; \left( 1 - \frac{y}{\delta } \right) \end{aligned}$$

(A4)

where $\tau = - <u'_x u'_y>$ is the Reynolds shear stress. The boundary conditions on pipe wall $y=0$ and centerline $y=\delta$ become

$$\begin{aligned} y = 0, \quad u = \tau =0; \quad y = \delta , \quad u - U_c = \tau = 0 \end{aligned}$$

(A5)