# Active drag, part 4: residual thrust and the hydrodynamic reaction (ELI5: in plain English)

> Swimming on a rope in a water treadmill, and filming every moment of a stroke: two more ways to get at a swimmer's drag, and the question of what 'active drag' even means, in plain English.

- Source: https://swimas.co.nz/blog/active-drag-4-residual-thrust/eli5/
- The full version: https://swimas.co.nz/blog/active-drag-4-residual-thrust/
- Author: V. Karpov, swimas (Swimming Art & Science), New Zealand
- Published: 18 Oct 2026
- Topics: Active drag, Biomechanics; strokes: FR

The second and third posts covered methods that either replace the water beneath the hand with fixed pads or compare two free swims. This post explores two quite different ideas. The first tethers the swimmer to a force sensor and measures how hard they pull on the line under varying conditions, calculating drag from the balance between propulsion, drag and tether force. The second abandons stroke-cycle averages altogether, attempting instead to map the forces acting on the swimmer's body at every single instant. We will finish with the fundamental question this second approach brings to a head: when two laboratories say "active drag", do they actually mean the same force?

## Tethered in a flume

A flume is essentially a treadmill for swimmers: the water flows towards you while you swim on the spot. Now, imagine walking on an ordinary treadmill with a belt round your waist and two cords from it, each through a spring balance: one to the wall behind you, the other to the wall in front. If the treadmill belt moves more slowly than your walking pace, you move forward and the rear cord goes taut: its balance shows how hard you are pulling forward. If the treadmill belt moves faster, it carries you backwards and the front cord goes taut instead. At one specific belt speed, both balances read zero: that is your natural walking pace.

Japanese researchers led by Hideki Takagi built the "residual thrust" method around this exact concept. A swimmer in a flume is connected front and back by lines attached to force sensors, with springs keeping the lines from going slack. They swim at a stroke rate dictated by a metronome while the current is adjusted in steps across eight speeds, spaced 0.05 metres per second apart and centred around the swimmer's natural pace (from roughly 0.2 metres per second slower to 0.2 faster). When the current is slower than the swimmer, they pull forward; when it is faster, the current pushes them back. What remains of the swimmer's propulsion once water resistance is subtracted is called residual thrust. As the current speeds up, residual thrust drops until, at a certain flow velocity, it reaches zero: this represents the speed at which the swimmer would travel freely in still water. A curve plotted through the measured data points then reveals the active drag at that speed.

**Figure 1.** Tied in place in a flume while the water speeds up step by step. The rope force falls from forwards to backwards; where it is zero, the push equals the drag. The numbers are made up to explain, from the model.

An animation of residual thrust measured in a flume. The swimmer swims in place against the flow, breathing through a snorkel, wired from a waist belt obliquely up to load cells on beams over the two ends of the flume, a spring in each wire, the stroke rate held at 46 cycles a minute by a metronome. The flow steps through eight speeds, 1.15 to 1.50 metres a second, 0.05 apart. The tethers read the residual thrust as the difference of their tensions: forward while the flow is slower than the swimmer's own speed, the rear tether holding the swimmer back, and backward once it is faster, the front tether holding the swimmer against the flow; springs keep the other wire taut, drawn grey. Each step leaves its mean on a chart of residual thrust against flow speed:

| Flow speed, m/s | Residual thrust, N |
| --: | --: |
| 1.15 | +32.9 |
| 1.20 | +22.0 |
| 1.25 | +11.0 |
| 1.30 | 0.0 |
| 1.35 | -11.1 |
| 1.40 | -22.2 |
| 1.45 | -33.4 |
| 1.50 | -44.7 |

At the end the fit T = α·U² − β·U + γ, where T is the residual thrust, U the flow speed and α, β and γ fitted constants, follows Narita et al.'s model: drag D = kD·U² and a propulsion Fp = kP·(VP − U)² that falls as the flow rises, where kD and kP are constants of drag and propulsion and VP a notional speed of the propelling limbs, all three taken as unchanged while the technique is kept. It gives kD = β²/4γ − α = 45.6 kg/m and crosses zero at U0 = 1.30 m/s, the self-propelled speed, where propulsion equals drag: the active drag at that speed is D = kD·U0² = 77 N. The means are illustrative, computed from the model with kD set for Narita 2018's whole-stroke crawl, 77 N at 1.3 m/s (passive 41 N); kP = 34.2 kg/m and VP = 2.8 m/s are chosen, not measured.


The resulting figures come out well above passive drag. In seven swimmers using a full stroke, active drag measured 44 newtons at 1.1 metres per second and 77 newtons at 1.3 metres per second: 1.6 to 1.9 times their passive drag. When six swimmers were tested using arms only, both in the flume and on the MAD pads from the second post, flume drag came out 19 to 47 per cent higher and climbed more steeply with speed.

The method's limitations are partly practical and partly a matter of principle. Practically, it requires a flume, where water flow is inherently uneven: current velocity varies by 4 to 5 per cent across the channel's cross-section, and neither the wave patterns nor the visual surroundings match those of a standard pool. As a matter of principle, the calculation assumes the swimmer's stroke mechanics remain identical across every current speed. While a metronome keeps stroke rate constant, it does not guarantee that the swimmer pulls the same way at each setting, a caveat the authors themselves highlight as a potential source of error. Repeatability data exist for only two swimmers, whose drag measurements varied by 6.5 and 3 per cent between trials.

## A tether in an ordinary pool

In 2024, one research group adapted this concept for an ordinary pool to test sprinters without a flume. Fourteen national-level sprinters (capable of roughly 93 per cent of world-record pace over 50 metres) first swam all-out for 15 seconds against a stationary, rigid tether to establish their maximal thrust. Next, they swam a series of efforts attached to a specialised device that allowed them to move forward while resisting them with 35 to 85 per cent of that maximal thrust. Active drag at each resulting speed was calculated as maximal stationary thrust minus the tension on the line. This introduces its own core assumption: that a swimmer's propulsive force on a stationary tether equals the force they generate while moving through the water.

Active drag came out at roughly 1.5 times passive drag. Two cross-checks supported these findings: the numbers closely matched those derived from planimetry, familiar from the second post (though planimetry, by its very design, always yields roughly one and a half times passive drag), and the predicted free-swimming speed (2.19 metres per second) landed remarkably close to the actual measured speed (2.22 metres per second). Under the heaviest load, however, calculated drag suddenly plummeted by about 30 per cent, prompting the authors to urge caution when resistance is set "too high".

A year later, the same group tested sixteen more sprinters using three protocols simultaneously. One of the three is particularly ingenious: the swimmer floats motionless on the surface and then, without breathing or pushing off the wall, sprints from a dead stop while a sensor on their lower back tracks how velocity builds. Much like a car accelerating from a standstill, how quickly a swimmer reaches top speed depends on their power, mass and drag. From this velocity build-up curve, counting the added mass of entrained water as a quarter of body mass, the calculated drag matched the tether results almost exactly. The two measurements are not entirely independent, however, as part of the sprint data was also fed into the tether calculation.

In 2023, another group attempted to use tether data to peer inside the stroke cycle itself. They broke down the force recorded on a stationary tether and the force recorded while moving into matching phases of the stroke cycle, subtracting one from the other. The resulting "drag" curve featured two prominent peaks per cycle. While moving, however, the swimmers pulled against the line with only 10 to 25 newtons, meaning the derived curve was essentially just the stationary thrust curve shifted downwards; no surprise, then, that its peaks aligned perfectly with peak thrust. Moreover, performing a point-by-point subtraction within a single stroke ignores intracyclic body accelerations, and hand-mounted sensors revealed that stroke mechanics change between tethered and free swimming: the propulsive phase lasted 39 per cent of the cycle on the tether, compared to just 27 per cent when swimming freely.

Among all the studies reviewed, the closest agreement between two distinct methods came from pressure sensors worn on the hands: across 18 junior swimmers, the resulting drag coefficient differed from that of the small perturbation method by just 6 per cent. Yet this approach, too, rests on a premise of its own: that the thrust generated by the hands alone equals the total drag acting on the entire body.

## The hydrodynamic reaction method: every instant on its own

Every method discussed so far yields a single average figure per stroke cycle. Kolmogorov tried something fundamentally different: calculating the forces acting on the body at each discrete instant, roughly every 0.06 seconds, or 16 to 25 times per cycle. It is akin to stepping through a video frame by frame and computing, for each individual image, exactly how hard the water presses against the front of the body.

Achieving this required several instruments working in concert. An acoustic Doppler device tracks trunk velocity, while above- and underwater video, analysed in three dimensions, yields the velocity and acceleration of the swimmer's centre of mass. The frontal area meeting the oncoming water is derived from the submerged volume of the body, deliberately excluding whichever parts of the arms and legs are actively stroking at that instant. Finally, the drag coefficient is drawn from flume tests, both of the live swimmer held in 6 to 12 typical postures and of a 1.85-metre mannequin with detachable limbs, complete with a mathematical correction for the fact that water resists an accelerating or decelerating body differently than one moving at a constant velocity.

**Figure 2.** Filmed above and below the water, with the trunk's speed measured by sound. The blue part of the body counts towards the drag; the limbs moving backwards relative to the body are left out. The drag is worked out every 0.06 seconds. The lines are drawn to explain; only their averages are from the study.

An animation of the hydrodynamic reaction method, shown at a third of the real speed. The swimmer swims freely at 2.03 metres a second through a section watched by video cameras above and below the water, with an acoustic Doppler device reading the speed of a point on his trunk. The sources give the cameras' mark, 40 m from the start, but not where they or the device stand, so these are drawn schematically. The immersed body is tinted blue: this is the volume behind S(bs). The segments of the arms and legs that are moving backwards relative to the centre of mass at that instant, the propelling ones, are left out and stay as painted. Below, through one stroke cycle of 1.32 s, two records build up: the trunk speed u(pb)(t), swinging between about 1.9 and 2.1 metres a second, and the frontal drag component Fr(fd)(t) = Cx(n)(t)·(ρ/2)·S(bs)(t)·u(pb)(t)², where ρ is the density of water and Cx(n) and S(bs) are explained at the end, a value every 0.06 s, 23 in the cycle, between about 55 and 95 newtons, lower while more of the limbs are left out. Their mean over the cycle is 75 N, the frontal component, against 99 N from the small perturbation method and a passive drag of 110 N for the same swimmer (Kolmogorov 2008). The coefficients Cx(n) come from flume tests of the swimmer in 6 to 12 positions and of a 1.85 m model; S(bs) is the immersed volume, without the propelling segments, to the power two thirds.


This method boasts one major advantage: because it does not compare two separate swims, the equal-power assumption vanishes altogether. Yet the reviewed literature contains no data on its day-to-day repeatability, nor any cross-comparisons with other methods conducted by independent laboratories.

Consider the numbers recorded for a single freestyler travelling at 2.03 metres per second: passive drag was 110 newtons; the hydrodynamic reaction method gave 75 newtons; and the small perturbation method, on that very same swimmer, gave 99 newtons. In fact, for every swimmer in that study, the hydrodynamic reaction method produced values 16 to 37 per cent lower than those of the small perturbation method.

## Which force are we talking about?

Why did one and the same swimmer produce three entirely different numbers? Kolmogorov's answer is simple: because they represent three different forces. He divides the total force a swimmer generates over a stroke cycle into three distinct components. The first balances the frontal resistance of the body itself: this is the useful, effective force. The second goes into accelerating water backwards. The third is lost to spinning up eddies and overcoming the local drag of the stroking arms and legs themselves.

**Figure 3.** Two things called active drag. The total push splits into a useful part, which balances the body's drag, and parts wasted on moving water. Different schools give the name to different parts.

A diagram of the total propulsive force over a cycle split into three parts: the effective propulsion Fp(e), which balances the frontal drag; F1, the force spent accelerating water backwards; and F2, the force lost to eddies and the induced drag of the limbs, so that Fp(t) = Fp(e) + F1 + F2 (Kolmogorov 2008, equation 10). Beside it, what each school calls active drag: for the MAD, ATM, MRT and tethered methods, the mean resistance that propulsion balances; for Kolmogorov's SPM, Fr(ad), read as the total propulsive force itself (set out in 2008); for Kolmogorov's HRM, Fr(fd), the frontal component, the resistance of the body, 63 to 84 per cent of Fr(ad) in the 2008 data. An example from that paper, crawl at 2.03 m/s: passive drag 110 N; Fr(fd) 75 N; Fr(ad) by the SPM 99 N; the propelling efficiency ep = Fr(fd)/Fr(ad) = 0.76. The smaller reference area, 0.152 against about 0.196 square metres, at nearly equal coefficients, helps explain why the frontal component falls below passive drag; the perturbation-method value, a different quantity, is below it too.


Think of it like a salary before and after tax. The word is the same, "salary", but the figures are quite different: one is gross pay (what you earn), and the other is net pay (what actually reaches your pocket). According to Kolmogorov, the small perturbation method measures the swimmer's total gross force, the "earned" figure. The hydrodynamic reaction method, by contrast, measures only the net, useful portion that directly balances body drag. The ratio of this useful force to total force represents the propelling efficiency of the stroke: among nine swimmers ranked in the world's top ten, this efficiency ranged from 0.63 to 0.84.

Two crucial conclusions follow from this framework. First, it becomes clear how active drag measured by the hydrodynamic reaction method can fall below passive drag: it is calculated over a smaller surface area that excludes the actively stroking limbs. Second, when researchers compare the small perturbation method against MAD pads, winch towing or flume tethers, then, if Kolmogorov is right, they are comparing the swimmer's total gross force against entirely different physical quantities.

## What to remember

Whether or not one fully accepts Kolmogorov's framework, it vividly illustrates the central point of the first post: the term "active drag" currently serves as a label for at least two distinct physical forces. A 2023 review described a "cascade of active drag inflation", noting that each successive method tended to produce higher numbers than the last. In part, that inflation is actually a cascade of shifting definitions. The residual thrust method in a flume and the pool tether both yield active drag values of 1.5 to 2 times passive drag, and each rests on its own core assumption: that stroke mechanics across varying currents, or thrust output on a tether, remain identical to free swimming. Meanwhile, the hydrodynamic reaction method dispenses with comparing multiple swims, but it has yet to be independently verified outside its creators' laboratory.

In the fifth and final post, we will place all of these methods side by side to see which practical conclusions survive scrutiny.

## The series

1. [What active drag is and why it has to be inferred](/blog/active-drag-1-what-it-is/eli5/)
2. [Pads and masks: the MAD system and the energetic method](/blog/active-drag-2-pads-and-masks/eli5/)
3. [Two swims, one power: the perturbation family](/blog/active-drag-3-two-swims-one-power/eli5/)
4. Residual thrust and the hydrodynamic reaction (this post)
5. [Why the methods disagree, and what a coach can use](/blog/active-drag-5-why-they-disagree/eli5/)

The sources for this post are listed in [the full version](/blog/active-drag-4-residual-thrust/#references).