Active drag, part 4: residual thrust and the hydrodynamic reaction

Parts 2 and 3 covered methods that either replace the water beneath the hands with fixed pads or compare two free swims. This post explores two very different ideas. The first tethers the swimmer to a force sensor and works from the balance between propulsion, drag and tether force: drag is inferred from a series of measurements under different conditions with a model of how thrust and resistance change, and the flume and pool versions make different assumptions about that change. The second abandons cycle averages altogether and tries to write the balance of forces on the swimmer’s body for each instant. The post closes with the question this second idea forces into the open: when two laboratories say “active drag”, are they naming the same physical force?

Residual thrust in a flume

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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.

Takagi et al. (1999), after a Japanese version in 1998, developed the flume approach: the swimmer, tethered in a flow, swims at several stroke rates set in advance while the flow speed is raised in steps, and drag at each flow speed is taken as the difference between the fully tethered thrust and the residual thrust. In the refined protocol of Narita, Nakashima, and Takagi (2018), which the authors call measured values of residual thrust (MRT; see also Narita et al., 2017; Narita, Ogita, et al., 2018), the swimmer is tethered to load cells in front and behind, swimming at a stroke rate set by a metronome, and the flow is stepped through eight speeds at 0.05 m/s intervals around the swimmer’s free speed, a window of about ± 0.2 m/s. In slower flow the swimmer pulls ahead against the rear tether; in faster flow the current pushes them back against the front one. Springs keep both lines from going slack (Narita et al., 2017), so the reading is the difference between the two tensions. As the flow speeds up, the residual thrust falls; the flow speed at which it crosses zero is the speed at which the swimmer would swim free, and the fitted curve gives drag at that speed. In the model of Narita et al. (2017), propulsion is allowed to change with flow speed $U$, $F_{\mathrm{p}}=k_{\mathrm{P}}\cdot {(V_{\mathrm{P}}-U)}^2$, where $V_{\mathrm{P}}$ is a notional speed of the propelling limbs and $k_{\mathrm{P}}$ a constant, while resistance is $k_{\mathrm{D}}\cdot U^2$ with $k_{\mathrm{D}}$ a constant; residual thrust is fitted as a quadratic in $U$, and drag at the zero crossing follows from the fitted $k_{\mathrm{D}}$.

In his original work Takagi decomposed the result into passive drag plus a “kinetic drag” that grows linearly with speed, fitting its coefficient $C_{\mathrm{dk}}$ to the Froude number $\mathrm{Fr}$ (speed over the square root of gravity’s acceleration times body length) as $C_{\mathrm{dk}}=0.86\cdot \exp (-3.9\cdot \mathrm{Fr})$ (Takagi et al., 1999), and in doing so questioned the $v^2$ assumption that the perturbation method rests on.

MRT values sit well above passive drag. Testing seven swimmers in full-stroke front crawl, Narita, Nakashima, and Takagi (2018) found 44.4 ± 7.0 N at 1.1 m/s and 77.3 ± 9.3 N at 1.3 m/s, against passive values of 28.2 and 40.5 N, “1.6–1.9 times larger than that in passive conditions” (p. 197); arms only gave 44.0 and 66.1 N. In a separate study, already mentioned in part 2, six swimmers swam arms only both on the MAD system and in the flume (Narita, Ogita, et al., 2018), and fitting $D=k\cdot v^n$, drag $D$ against speed $v$ with constants $k$ and $n$, gave $k=34.5\pm 2.9$ and $n=2.53\pm 0.40$ in the flume against $k=29.1\pm 1.5$ and $n=1.90\pm 0.13$ on the pads. Flume drag was significantly higher than MAD drag, by 19 per cent at 1.0 m/s, 33 per cent at 1.2 m/s and 47 per cent at 1.4 m/s.

The limitations are partly practical and partly conceptual. Practically, it needs a flume, a metronome and a belt; moreover, the flow in a flume varies by 4 to 5 per cent across its section, and wave reflections and visual cues differ from those of a pool (Narita, Nakashima, & Takagi, 2018). Conceptually, the model lets propulsion change with flow, but kP, kD and VP are assumed to stay the same across the speeds tested as long as the stroke pattern is preserved. A set stroke rate helps standardise the task, but it does not show that those parameters stay constant, and the authors themselves name the assumption as a possible source of bias (Narita, Nakashima, & Takagi, 2018). Notably, this is not the assumption of the pool version below, in which the fully tethered thrust is simply taken to equal thrust while moving: the two share the idea of residual thrust, but not their mathematical premises. Test-retest data for flume MRT are limited to two swimmers, whose active drag varied by 6.5 and 3.0 per cent between repeats (Narita et al., 2017).

Residual thrust in a pool

Cortesi et al. (2024) brought the idea into an ordinary pool, for sprinters without a flume. Fourteen national-level sprinters (about 93 per cent of the 50 m world record) first did a 15 s fully tethered test to find their maximal thrust $F_{\mathrm{T}}$, then a series of semi-tethered swims on an isokinetic device at 35 to 85 per cent of $F_{\mathrm{T}}$, with speed and line force recorded. Drag at each speed was taken as $F_{\mathrm{T}}$ minus the semi-tethered force. Expressed as $k=D/v^2$, drag $D$ divided by the square of speed $v$, active drag came out at 38 kg/m against 25 kg/m for passive drag, a ratio of about 1.5. Two cross-checks supported the result: the planimetric estimate, which by construction is 1.5 times passive drag (Gatta et al., 2015), agreed within a Bland-Altman bias of just −2.4 N, and the fitted model predicted a free speed of 2.19 ± 0.08 m/s against a measured 2.22 ± 0.07. At the heaviest load, 85 per cent of $F_{\mathrm{T}}$, however, the apparent $k$ dropped by about 30 per cent; the authors left the cause open (“whatever the reason for this difference”) and urged caution when the applied load is “too high” (Cortesi et al., 2024, p. 22). Leaving that load out, $k$ averaged over loads up to 75 per cent of $F_{\mathrm{T}}$ agrees much better with the fitted model. In perturbation terms, their loads correspond to speed losses of 23 to 55 per cent.

The same group then tested 16 other male sprinters three ways at once (Carmigniani et al., 2025): pool residual thrust as above (a fully tethered test plus semi-tethered swims at 25, 50 and 75 per cent of it); planimetry; and a standing-start sprint. In the standing start the swimmer lay still at the surface for 5 s and then, without breathing and without pushing off the wall, sprinted for about ten cycles with an inertial sensor on the sacrum. Cycle-averaged speed $v$ rose with time $t$ as $v=v_{\max }\cdot \tanh (t/\tau )$, where $v_{\max }$ is the top speed and $\tau$ the time constant of the acceleration; this is the analytical solution of the equation of motion $F_{\mathrm{p}}=k\cdot v^2+(m+M_{\mathrm{A}})\cdot \mathrm{d}v/\mathrm{d}t$ at constant thrust $F_{\mathrm{p}}$, where $k$ is the drag factor, $m$ the body mass, $M_{\mathrm{A}}$ the added mass of water met earlier and $\mathrm{d}v/\mathrm{d}t$ the acceleration; the fit gave $v_{\max }=1.73\pm 0.11$ m/s and $\tau =1.14\pm 0.11$ s. A single curve, however, fixes only two of the three numbers ($F_{\mathrm{p}}$, $k$ and $M_{\mathrm{A}}$). Fixing the added mass at 25 per cent of body mass, as measured in gliding swimmers (Caspersen et al., 2010, as cited in Carmigniani et al., 2025), gives $k=47.7\pm 4.5$ kg/m, close to the 49.2 ± 6.3 kg/m from residual thrust; conversely, taking $k$ and thrust from the tethered tests gives the added mass of 28.7 ± 15.2 per cent of body mass mentioned in part 1, 6 to 53 kg across swimmers. In effect this is one result seen from two sides, the more so because the residual-thrust values are not fully independent of the start: its $v_{\max }$ served as their zero-load point. Planimetry, with its built-in 1.5 times passive drag, gave less: 39.9 ± 5.0 kg/m (more in part 5).

Haskins et al. (2023) tried to take the same subtraction inside the individual cycle. Seven men swam 25 s fully tethered to a load cell and, separately, sprinted from 5 to 25 m against the minimal load of a robotic device, 9.81 N. Cycles were cut at tension minima and normalised to 101 points in time, and drag at each per cent of the cycle was taken as tethered force minus moving force. Cycle means came out at 76 to 140 N at 1.73 m/s, with twin peaks of 175 to 300 N near 20 and 70 per cent of the cycle and near-zero values between them (Haskins et al., 2023, pp. 7–10). The whole calculation rests on two assumptions: the moving swimmer’s force balance has no mass-times-acceleration term, and thrust is taken to be the same at each instant tethered and moving (p. 7). Written out in full, the subtraction returns drag plus body mass times instantaneous acceleration plus the difference in thrust between the two conditions; the second term cancels over a whole cycle but not within it, and the third is not measured. The authors’ own check with sensors on one hand speaks against the assumption: the force impulse lasted 39.1 per cent of the cycle tethered against 26.6 per cent free (Haskins et al., 2023, Supplementary Information, p. 5). And because the moving trial registered only 10 to 25 N, the “drag profile” is essentially the tethered thrust curve shifted down by that amount, so it is no surprise that its peaks fall exactly where thrust peaks. The 88.3 N given as a repeatability figure is not a spread of drag estimates but the largest stroke-to-stroke standard deviation of one swimmer’s tethered force (pp. 11–12).

In a conference poster, Mezêncio et al. (2013) reported an even cruder version of the same subtraction: fully tethered force minus the force while moving at just 0.60 m/s, giving 153.7 ± 9.8 N. Expressed as $k=D/v^2$, drag divided by the square of speed, that is about 427 kg/m, an order of magnitude above any other source; whatever quantity the procedure isolates at so low a speed, it is not drag in the sense the other methods give it.

Related indirect estimates belong here too. Barbosa et al. (2013) showed that intra-cycle speed fluctuation correlates with drag from the perturbation method (r up to 0.87 after controlling for speed, though not all correlations were significant). Morais et al. (2023) used pressure sensors on the hands and, testing 18 juniors both ways, found a drag coefficient that agrees with the perturbation method’s with a between-method coefficient of variation of just 6 per cent (ICC 0.93), the closest agreement between two different methods among the studies reviewed here, with the caveat that the sensor method itself assumes the hands’ thrust alone equals the whole body’s drag, and that the hand forces it measured were about 10 per cent above the perturbation value (68.8 against 62.8 N).

The hydrodynamic reaction method

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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.

The review by Sacilotto et al. (2023) classes this approach as a variant of the perturbation method and calls it the hydrodynamic reaction method (HRM); Kolmogorov (2008) himself calls it the biohydrodynamic method. In his paper it is a method of its own, unlike anything above: it tries to write the swimmer’s equation of motion for each instant,

$$F_{\mathrm{p}(\mathrm{e})}(t)-F_{\mathrm{r}(\mathrm{f}\mathrm{d})}(t)=(m_0+\mathrm{\Delta }m)\frac{\mathrm{d}u}{\mathrm{d}t},$$

where $F_{\mathrm{p}(\mathrm{e})}$ is the effective propulsive force, $F_{\mathrm{r}(\mathrm{f}\mathrm{d})}$ the frontal component of active drag, $m_0$ body mass, $\mathrm{\Delta }m$ an added mass of water of inertial origin ($\mathrm{\Delta }m=m_0\cdot k_{11}$, with $k_{11}$ found from in-situ tests to lie between 0.097 and 0.139, close to the theoretical 0.09 of an elongated spheroid moving along its axis), and $\mathrm{d}u/\mathrm{d}t$ the instantaneous acceleration of the centre of mass.

The hard term is the instantaneous frontal component $F_{\mathrm{r}(\mathrm{f}\mathrm{d})}(t)$. It is computed step by step through the whole cycle (about every 0.06 s, 16 to 25 values per cycle, as described by Kolmogorov et al., 2021) as $C_{x(n)}(t)\cdot (\rho /2)\cdot S_{\mathrm{bs}}(t)\cdot u_{\mathrm{pb}}{(t)}^2$, where $\rho$ is the density of water and the other three terms are described next. The speed $u_{\mathrm{pb}}$ of a reference point on the trunk comes from an acoustic Doppler device, and synchronised video above and below the water, analysed in three dimensions with segment masses and inertias by Troup’s procedure, converts it to the instantaneous speed and acceleration of the centre of mass. The characteristic area $S_{\mathrm{bs}}(t)$ is the immersed volume at that instant, excluding whichever segments are propelling (moving backwards relative to the centre of mass), raised to the power 2/3. Finally, the hydrodynamic force coefficient $C_{x(n)}(t)$ is the sum of a steady-flow coefficient for the body’s posture at that instant, measured in a flume on the swimmer in 6 to 12 typical positions or, where propelling segments are excluded, on a 1.85 m model with detachable segments tested up to 2.2 m/s at Reynolds numbers of 1 to 5 million, and an unsteady increment found by towing a 1.5 m body of revolution under linear, exponential and sinusoidal accelerations (positive acceleration increases viscous resistance, negative decreases it).

Three features of the method matter for this series. First, nothing in it compares two swims, so the equal-power assumption is absent altogether. Second, what the same paper (Kolmogorov, 2008) calls total external mechanical power is taken from the small perturbation method: $P_{\mathrm{uo}}=F_{\mathrm{r}(\mathrm{f}\mathrm{d})}\cdot u_0$ is “useful” power and $P_{\mathrm{to}}=F_{\mathrm{r}(\mathrm{a}\mathrm{d})}\cdot u_0$ total power, $u_0$ being the mean swimming speed, and their ratio is the propelling efficiency, which ranged from 0.626 to 0.837 in nine world top-ten swimmers (two women and seven men) tested before the 1992 and 1996 Olympic Games and the 1994 World Championships. Third, I found no test-retest data for the HRM in the sources reviewed, nor any independent comparison with another method, an absence Sacilotto et al. (2023) also note.

The numbers for the front crawl swimmer at 2.03 m/s (Kolmogorov, 2008) are worth giving in full: passive drag 110 N (coefficient 0.274 on an area of about 0.196 m², computed for the whole body); HRM frontal component of active drag 75 N (coefficient 0.271 on 0.152 m²); active drag by the perturbation method 99 N (coefficient 0.305); propelling efficiency 0.76. For every swimmer in the 2008 table the frontal component is 16 to 37 per cent below the perturbation value. Yet Sacilotto et al. (2023) describe an “apparent cascade of active drag inflation: from VPM and NABA to HRM, passing successively through the MAD-System, ATM and MRT” (p. 1583), placing HRM at the top with “~80 to 90 N for front crawl Olympic champions” (pp. 1582–1583). The 2008 table does not support that place: the high absolute values in it come from the perturbation method applied to sprinters swimming at 2 m/s, and the HRM’s own values are lower.

Which force is being named?

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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.

Kolmogorov (2008) makes explicit a framework whose roots in his work go back to the early 1990s, when the cycle-mean $F_{\mathrm{r}(\mathrm{a}\mathrm{d})}$ was already equated with the mean propulsive force (Kolmogorov et al., 1992). Total propulsive force over a cycle is written as the sum of three terms, $F_{\mathrm{p}}(t)=F_{\mathrm{p}(\mathrm{e})}+F_1+F_2$: an effective part that balances the body’s frontal resistance, a part spent giving backward momentum to the water, and a part lost to eddies and the induced drag of the propelling limbs themselves. “The equality of both mean values of total propelling force $F_{\mathrm{p}}(t)$ and active drag force $F_{\mathrm{r}(\mathrm{a}\mathrm{d})}$ in this cycle is also evident and repeatedly confirmed by experiments” (Kolmogorov, 2008, p. 60), and “this clarifies a physical difference between $F_{\mathrm{r}(\mathrm{a}\mathrm{d})}$ and $F_{\mathrm{r}(\mathrm{f}\mathrm{d})}$, which should be defined simultaneously by different experimental methods” (p. 61). In other words, in this school the perturbation method’s $F_{\mathrm{r}(\mathrm{a}\mathrm{d})}$ is the total propulsive force, while the body’s own resistance is the frontal component $F_{\mathrm{r}(\mathrm{f}\mathrm{d})}$, whose share of that total is the propelling efficiency.

Two implications follow. First, leaving out the propelling limbs helps explain why the HRM’s frontal component can fall below the whole body’s passive drag: it is computed on a smaller reference area (0.152 against about 0.196 m², computed for this swimmer’s whole body) with nearly the same coefficient (0.271 against 0.274). That alone, however, does not explain the earlier, celebrated “paradox” of perturbation-method drag below passive drag, because the perturbation method estimates the other quantity; for this very swimmer the perturbation value (98.9 N) also stays below passive drag (110.5 N), and there Kolmogorov names a further cause, the unsteadiness of the body’s motion itself (Kolmogorov, 2008). Second, under this reading the published comparisons of the perturbation method with the MAD system, towing or the flume are comparisons of total propulsive force with various other operational definitions, none of which isolates the frontal component.

Whether one accepts the framework or not, it brings into sharp relief the point made in part 1: “active drag” currently names at least two different terms of the force balance, and the “cascade of active drag inflation” (Sacilotto et al., 2023, p. 1583) across methods is partly a cascade of different definitions. Part 5, the last, puts all the methods side by side and asks which practical conclusions survive.

The series

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

References

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2. Carmigniani, R., Coloretti, V., Brunel, T., Bosetto, P., Fantozzi, S., Cortesi, M., & Zamparo, P. (2025). A novel method for assessing added mass in front crawl swimming. Journal of Biomechanics, 189, Article 112816. https://doi.org/10.1016/j.jbiomech.2025.112816
3. Cortesi, M., Gatta, G., Carmigniani, R., & Zamparo, P. (2024). Estimating active drag based on full and semi-tethered swimming tests. Journal of Sports Science and Medicine, 23(1), 17–24. https://doi.org/10.52082/jssm.2024.17
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9. Mezêncio, B., Claudino, J. G., Amadio, A. C., Serrão, J. C., Szmuchrowski, L. A., & Huebner, R. (2013). The evaluation of active drag: A new proposal [Poster presentation]. XXIV Congress of the International Society of Biomechanics, Natal, Brazil. https://media.isbweb.org/images/conferences/isb-congresses/2013/poster/ps1-03h.pdf
10. Morais, J. E., Barbosa, T. M., Garrido, N. D., Cirilo-Sousa, M. S., Silva, A. J., & Marinho, D. A. (2023). Agreement between different methods to measure the active drag coefficient in front-crawl swimming. Journal of Human Kinetics, 86, 41–49. https://doi.org/10.5114/jhk/159605
11. Narita, K., Nakashima, M., & Takagi, H. (2017). Developing a methodology for estimating the drag in front-crawl swimming at various velocities. Journal of Biomechanics, 54, 123–128. https://doi.org/10.1016/j.jbiomech.2017.01.037
12. Narita, K., Nakashima, M., & Takagi, H. (2018). Effect of leg kick on active drag in front-crawl swimming: Comparison of whole stroke and arms-only stroke during front-crawl and the streamlined position. Journal of Biomechanics, 76, 197–203. https://doi.org/10.1016/j.jbiomech.2018.05.027
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14. Sacilotto, G., Sanders, R., Gonjo, T., Marinho, D., Mason, B., Naemi, R., Vilas-Boas, J. P., & Papic, C. (2023). “Selecting the right tool for the job” a narrative overview of experimental methods used to measure or estimate active and passive drag in competitive swimming. Sports Biomechanics, 22(12), 1572–1589. https://doi.org/10.1080/14763141.2023.2197858
15. Takagi, H., Shimizu, Y., & Kodan, N. (1999). A hydrodynamic study of active drag in swimming. JSME International Journal Series B: Fluids and Thermal Engineering, 42(2), 171–177. https://doi.org/10.1299/jsmeb.42.171