Deep-Sea ROV in a Current

What is modelled

A vehicle on a tether has to fight two drag forces when a current flows past it: the drag of its own body and the drag of the long thin cable. Both grow with the square of the water speed, so doubling the current quadruples the force and, because a propeller needs power to the power three halves, more than eight times the power. The thrusters are modelled as ideal actuator discs scaled by a figure of merit. The tether is taken as a straight line from the ship, and half of its drag is assumed to be carried by the vehicle end.

Equations

• Vehicle drag F = ½ ρ Cd A u2; tether cross-flow drag Ft = ½ ρ Ct d L u2 cos3φ, with L = h / cosφ.
• Static thrust of a propeller disc T = (η P)2/3 (2 ρ Ad)1/3, so the power is P = T3/2 / (η √(2 ρ Ad)).
• Holding station needs T ≥ (c1 + c2) u2, giving the strongest current umax = √(T / (c1 + c2)).

Checks run in the page

• The momentum balance and energy flux of one propeller.
• The tether drag added up piece by piece.
• The strongest holdable current found by a search.
• The transit speed from the equation of motion.

Honest simplifications

A straight tether in a uniform current, a fixed cross-flow drag coefficient of 1.2, half of the tether drag on the vehicle, no tether weight, sag or strumming, no thruster interaction with the hull and a constant figure of merit. A real tether bows into a curve and real thrusters lose thrust near the hull, so these numbers show the scaling and are not a dive plan.

• Press Strong current, thick tether and see how the tether can cost more than the vehicle body.
• Press Big thrusters and compare the holding current with the work-class preset.
• Move the tether angle from 0 to 70 degrees at a fixed depth and watch the drag climb as the tether lengthens.