Constants of the von Kármán rotating-disk flow

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Numbers

component

$F'(0)$, $G'(0)$, $-H_\infty$

$F'(0)$:

0.510232618867284884402737355913

$G'(0)$:

-0.615922014399353966346933669943

$-H_\infty$:

0.884474110209602423446339215935

Definition

The von Kármán rotating-disk flow is the similarity solution for an infinite disk rotating in an otherwise quiescent incompressible Newtonian fluid [5] [1]. This table gives its three classical characteristic constants.

Parameters

component

— component of the similarity solution (one of radial-shear, azimuthal-shear and axial-inflow)

Formulas

(1)

With $\eta=z\sqrt{\Omega/\nu}$, the dimensional velocities are $u_r=r\Omega F(\eta)$, $u_\theta=r\Omega G(\eta)$ and $u_z=\sqrt{\nu\Omega}\,H(\eta)$.

(2)

The similarity functions satisfy $2F+H'=0$, $F^2-G^2+HF'=F''$ and $2FG+HG'=G''$, with $F(0)=H(0)=0$, $G(0)=1$ and $F(\infty)=G(\infty)=0$.

(3)

$H_\infty=\lim_{\eta\to\infty}H(\eta)$.

Comments

(4)

The axial velocity is negative far from the disk in this convention. The axial-inflow row stores the positive constant $-H_\infty$, so that the dimensional far-field velocity is $-\sqrt{\nu\Omega}\,(-H_\infty)$.

(5)

The three rows are parts of one similarity solution: two wall derivatives and one far-field constant. They are kept together because the classical numerical tables quote them as one set of characteristic parameters [3] [4].

Programs

(P1)

Python

from mpmath import mp
mp.dps = 50
# The attached generate.py solves the two shooting equations for F'(0)
# and G'(0), then reads H at the same finite endpoint.

References

[1]

T. von Kármán, Über laminare und turbulente Reibung, Zeitschrift für angewandte Mathematik und Mechanik 1 (1921), 233-252.

[2]

W. G. Cochran, The flow due to a rotating disc, Mathematical Proceedings of the Cambridge Philosophical Society 30 (1934), 365-375. (doi)

[3]

R. E. White, C. M. Mohr Jr. and J. Newman, The fluid motion due to a rotating disk, Journal of the Electrochemical Society 123 (1976), 383-385. (doi)

[4]

M. Miklavčič and C. Y. Wang, The flow due to a rough rotating disk, Zeitschrift für angewandte Mathematik und Physik 55 (2004), 235-246. (doi)

Links

[5]

Wikipedia: Von Kármán swirling flow

Data properties

Entries are of type: real number

How well the digits are known: heuristic (agreement-checked)

Table is complete: true

How they were obtained:

The values are computed by shooting on $F'(0)$ and $G'(0)$ in the finite-interval boundary-value problem.

The generator solves the same shooting equations at two working precisions, requires the 30 stored digits to agree, checks the first derivative of the integrated $H$ against $H'=-2F$, and checks the rounded values against the constants printed by White, Mohr and Newman [3] and the no-slip case of Miklavčič and Wang [4].