• A better nondimensionalization scheme for slender laminar flows: The Laplacian operator scaling method

Considering the Cartesian velocity field $\mathit{u}=\left(u,v,w\right)$, the equation governing low-inertia, weakly time-dependent, and predominantly $z$-directional flows is a Poisson equation derived from the $z$-component momentum equation:

$$k=\Delta w=\left(\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}+\frac{{\partial }^2}{\partial z^2}\right)w,$$

(1)

subject to boundary conditions. As is often helpful, assuming acceptable scales for $x$, $y$, $z$ and $k$ can be identified for the problem, scale analysis may be performed on Eq. (1) to determine the velocity scale for $w$ with which Eq. (1) can in turn be nondimensionalized. In situations where the “cross-flow coordinates” $x$ and $y$ can be parametrized and scaled in terms of $z$ and $t$, it is proposed that such spatially and temporally dependent length scales $x\sim {\overline{x}}_s$ and $y\sim {\overline{y}}_s$ with $z\sim z_s=L$ may be employed to compute a spatially and temporally dependent velocity scale $w\sim {\overline{w}}_s$. Provided the flow is slender, ${\left({\overline{x}}_s/L\right)}^2⪡1$, treating $\Delta$ as a single scalable quantity such that $\Delta \sim {\overline{\Delta }}_s$, “scale analysis” on Eq. (1) is performed to determine

$$w\sim {\overline{w}}_s=\frac{k}{{\overline{\Delta }}_s}\equiv \frac{k}{\frac{1}{{\overline{x}}_s^2}+\frac{1}{{\overline{y}}_s^2}}\equiv \frac{k{\overline{x}}_s^2}{1+{\overline{T}}_{sxy}^2},$$

(2)

where ${\overline{T}}_{sxy}={\overline{x}}_s/{\overline{y}}_s$. The overbar for ${\overline{x}}_s$ and ${\overline{y}}_s$ denotes local, potentially $z$- and/or $t$-dependent $x$- and $y$-coordinate length scales such that, in general ${\overline{x}}_s={\overline{x}}_s\left(z,t\right)$, ${\overline{y}}_s={\overline{y}}_s\left(z,t\right)$, $L=L\left(t\right)$, $k=k\left(z,t\right)$, and thus ${\overline{w}}_s={\overline{w}}_s\left(z,t\right)$. The method introduces a notation ${\overline{\Delta }}_s$ for the local $z$- and $t$-dependent Laplacian operator scale which is treated as its own term—a minor twist on the more common and intuitive term-by-term scaling method.¹1. W. B. Krantz, Scaling Analysis in Modeling Transport and Reaction Processes (Wiley, Hoboken, NJ, 2007). The “Laplacian scale” ${\overline{\Delta }}_s$ obeys commutative laws (e.g., ${\overline{\Delta }}_s{\overline{w}}_s={\overline{w}}_s{\overline{\Delta }}_s$)—it is not an operator but the scale of an operator. Using velocity scale (2) and spatially dependent length scales, Eq. (1) when nondimensionalized becomes

$$1=\frac{1}{\left(1+{\overline{T}}_{sxy}^2\right)}\frac{{\partial }^2{\overline{w}}^{\ast }}{\partial {\overline{x}}^{\ast 2}}+\frac{{\overline{T}}_{sxy}^2}{\left(1+{\overline{T}}_{sxy}^2\right)}\frac{{\partial }^2{\overline{w}}^{\ast }}{\partial {\overline{y}}^{\ast 2}},$$

(3)

where ${\overline{x}}^{\ast }=x/{\overline{x}}_s$, ${\overline{y}}^{\ast }=y/{\overline{y}}_s$, and ${\overline{w}}^{\ast }=w{\overline{\Delta }}_s/k=w\left(1+{\overline{T}}_{sxy}^2\right)/k{\overline{x}}_s^2$. The result of Eq. (3) is forwarded as a modified two-dimensional (2D) Poisson equation that, despite having $z$- and $t$-dependent variables and coefficients, can lead to narrower bounds for numerical coefficients for the area-averaged velocity,

$$⟨{\overline{w}}^{\ast }⟩=\frac{{\overline{x}}_s{\overline{y}}_s}{A}\int \int {\overline{w}}^{\ast }d{\overline{x}}^{\ast }d{\overline{y}}^{\ast }\equiv F_i,$$

(4)

where $A$ is the dimensional section area. We do not make any significant effort to examine (or prove) the generality of this claim. Instead we demonstrate the usefulness of the approach with several basic example problems where ${\overline{\Delta }}_s=\text{const}$, ${\overline{\Delta }}_s={\overline{\Delta }}_s\left(z\right)$, and ${\overline{\Delta }}_s={\overline{\Delta }}_s\left(z,t\right)$.

As briefly mentioned in Sec. I A, the Laplacian scaling method characterizes the geometric dependence of viscous diffusion with greater accuracy than the hydraulic diameter scaling. It does so by characterizing the effective viscous length with greater accuracy. If one exploits these benefits to compute Fanning friction factors $f$ for such flows (Fig. 11) one might define a Laplacian diameter,

$$D_{\Delta }\equiv {\left(\frac{F_i}{{\overline{\Delta }}_s}\right)}^{1/2}={\left(\frac{F_n{A}_s}{3\left({\overline{T}}_{sxy}+{\overline{T}}_{sxy}^{-1}\right)}\right)}^{1/2},$$

(28)

and a Laplacian Reynolds number, ${\text{Re}}_{\Delta }=\rho U{D}_{\Delta }/\mu$, to find that

$$2.67\dots \le {\text{Po}}_{\Delta }\equiv f{\text{Re}}_{\Delta }\le 2\sqrt{3}.$$

(29)

This result implies that the Laplacian Poiseuille number, ${\text{Po}}_D\approx 3\left(\pm 10\%\right)$, is a weak $O\left(1\right)$ function of the section geometry and may be treated approximately as a constant for laminar flows. As a consequence, $f\approx 3/{\text{Re}}_D$ for all applicable cross sections. This compares to $14.2\dots \le \text{Po}\equiv f\text{Re}\le 24$ for traditional laminar flows, where Re is scaled on $D_{\text{hyd}}$.

In summary, the Laplacian scaling method amounts to nothing more than treating the operator as its own geometric scale quantity, ${\overline{\Delta }}_s$. It appears to be a quick and effective method for certain predominately 2D fields with possibilities for weak three-dimensional as well as temporal dependence. The subsequent process of nondimensionalization yields a modified 2D Poisson equation with $z$- and $t$-dependent variables and coefficients [Eq. (3)], which when solved for the dimensionless area-averaged velocity $⟨{\overline{w}}^{\ast }⟩\equiv F_i$ [Eq. (4)] possesses a weak dependence on numerical data—a dependence that is further reduced when $F_n$ from Eq. (26) is employed. In such situations exact or approximate average quantities are readily determined as are the subsequent equations that depend on them.