Propellant Sloshing — Theory & documentation

Overview

The report gives force and moment transfer functions of sloshing propellant for ballistic-missile autopilot analysis. It was the first such analysis for circular tanks. Its key assumptions are:

the liquid is inviscid and incompressible, and its motion is small, so the free-surface condition can be linearised;
the tank is a rigid upright circular cylinder;
the axial acceleration a=(T-D)/m is steady over the interval of interest.

The natural frequencies are proportional to \sqrt a, so a varying thrust acceleration would make the problem non-linear. In practice the variation over the time of interest is small, and the frequencies are evaluated at the current acceleration.

Natural frequencies

The velocity potential is a sum of terms in J_1(\lambda_n r/R)\cos\varphi, multiplied by hyperbolic functions of depth. No flow through the wall requires

J_1^\prime(\lambda_n)=0:\qquad \lambda_1=1.84118,\ \lambda_2=5.33144,\ \lambda_3=8.53632,\ \dots

These are the modern values given in the errata. The linearised free-surface condition then gives the natural frequencies, for a fill depth h=M/(\rho\pi R^2):

\omega_n^2=\frac{a}{R}\,\lambda_n\tanh\frac{\lambda_n h}{R}

The report writes the argument as \lambda_nM/(\pi\rho R^3), which is the same thing.

Two limits are useful:

For a deep fill, \tanh\to1 and \omega_n^2\to a\lambda_n/R, independent of depth.
For a shallow fill, \omega_n\to\lambda_n\sqrt{ah}/R, the shallow-water wave speed.

Equivalent mechanical model

For vehicle dynamics each mode can be replaced by a pendulum that produces the same side force at the same frequency. The report’s side-force transfer function for a sideways velocity \nu_z of the tank (eq. 5-47) is

\frac{F_z}{\nu_z}=-\pi\rho a^2s\left\{\frac{m}{\pi\rho a^2}+\sum_{n=1}^{\infty}\frac{2a\,s^2\tanh(\lambda_nm/\pi\rho a^3)}{\lambda_n(\lambda_n^2-1)\left[s^2+a_t\frac{\lambda_n}{a}\tanh(\lambda_nm/\pi\rho a^3)\right]}\right\}

in the report’s symbols (a the tank radius, m the liquid mass, a_t the thrust acceleration). Each term of the sum is a mass that answers at its own natural frequency. With M=\pi\rho R^2h its size is

\frac{m_n}{M}=\frac{2\tanh(\lambda_nh/R)}{\lambda_n(\lambda_n^2-1)\,h/R},\qquad L_n=\frac{a}{\omega_n^2}=\frac{R}{\lambda_n\tanh(\lambda_nh/R)}

where L_n is the length of a pendulum that swings at \omega_n in the thrust “gravity”. The rest of the liquid moves with the tank.

Because \sum_n 2/(\lambda_n^2-1)=1, almost all of a shallow fill sloshes, about 84 % of it in the first mode. In a deep tank only a small top layer does. The higher modes carry little mass, which is why the first mode dominates stability analyses.

Where the pendulum hangs

The moment transfer function for the same sideways motion (eq. 5-49) has the same poles. Dividing the residue of mode n in the moment by its residue in the force gives the lever arm of the mode’s force, measured from the report’s origin, a distance h_0 above the free surface:

\frac{R}{\lambda_n}\,\frac{(\lambda_nh_0/R)\tanh x-2\,\mathrm{sech}\,x+1}{\tanh x}=h_0+e_n,\qquad e_n=\frac{R}{\lambda_n}\,\frac{1-2\,\mathrm{sech}\,x}{\tanh x},\quad x=\frac{\lambda_nh}{R}

A pendulum passes its side force to the tank through its hinge, so e_n is the depth of the hinge below the still surface (negative: above it). The bob hangs L_n lower, at e_n+L_n=(2R/\lambda_n)\tanh(\lambda_nh/2R), which is always inside the liquid.

Deep fill: the hinge is R/\lambda_1\approx0.54R below the surface and the bob 1.09R below it.
Shallow fill: the bob sits on the bottom and the hinge is far above the surface (a long pendulum): the liquid slides almost bodily.

This location is derived on this page from the report’s two transfer functions; the report does not state it in this form. The rigid-liquid term of eq. (5-49), a^2/4+(h+\tfrac12m/\pi\rho a^2), appears misprinted (it lacks the liquid depth as a factor) and is not used.

Verdict: separation from the pitch frequency

Sloshing liquid pushes the tank sideways at the slosh frequency. If that frequency is close to the rocket’s pitch (weathercock) frequency, or to an autopilot’s bandwidth, the two motions can drive each other. The calculator compares the first slosh frequency with the pitch frequency you enter (the Pitch-Yaw Wave Number calculator gives it) through the separation

S=\max\left(\frac{f_1}{f_p},\ \frac{f_p}{f_1}\right)

SAFE: S\ge r, the separation you want (default 2);
MARGINAL: 1.25\le S<r;
UNSAFE: S<1.25.

These bands are a project choice, not a published limit; the report gives no criterion.

The fixes are solved exactly, on the side of the pitch frequency where the slosh is now:

Tank diameter, same propellant mass: bisection on R, since f_1 falls steadily as the tank widens (a wider tank both raises R and makes the pool shallower).
Pitch frequency: f_p\le f_1/r or f_p\ge r f_1.
Thrust acceleration: at a fixed depth f_1\propto\sqrt a, so a_{req}=a\,(f_{target}/f_1)^2.

As the tank drains, every slosh frequency falls towards zero, so a slosh that starts above the pitch frequency crosses it at the depth

h_\times=\frac{R}{\lambda_1}\,\mathrm{artanh}\frac{(2\pi f_p)^2R}{a\lambda_1}

(at the same acceleration and pitch frequency; none when the argument reaches 1). The drawing shades the liquid levels inside the bands, and the drain plot shows the whole history. Baffles are not modelled: the report notes that they add damping and change the frequencies only by changing the tank geometry, and gives no method for them.

Verification

tests/slosh.test.js checks the following:

the ten Bessel roots against J_1^\prime=0, evaluated from Bessel’s integral;
the frequency formula, including its deep and shallow limits;
the pendulum analogue;
the slosh-mass sum rule and the slosh mass as the residue of eq. (5-47);
the hinge and bob depths in their deep and shallow limits;
that the crossing depth, the diameter fix and the pitch-frequency fix each hit their target frequency.

Notes on the method

Fill depth. The frequencies use \tanh(\lambda_nh/R) with h=M/(\rho\pi R^2), as in the report. The tabulated example puts the density where the propellant mass belongs, which removes the effect of fill depth.
Equivalent mechanical model. The slosh masses, pendulum lengths and hinge depths are read from the poles and residues of the report’s transfer functions (eqs. 5-47 and 5-49). Only their magnitudes are used.
Not modelled. Baffles, tank-bottom shape and damping. Baffles mainly add damping; they change the frequencies only by changing the tank geometry.

References

C. P. Hoult, “An Analysis of the Motion of Liquids in a Partially-Filled Enclosure.” View in library
C. P. Hoult, “Errata for ‘An Analysis of the Motion of Liquids in a Partially-Filled Enclosure’,” RS&T, 27 August 2013. View in library

Propellant Sloshing — theory