Universal Scaling Laws for a Generic Swimmer Model

Chapter 1

Two over five. Eight over eight. [physics.bio-ph]

Universal Scaling Laws for a Generic Swimmer Model

We have developed a minimal model of a swimmer without body deformation based on force and torque dipoles which allows accurate three-D Navier-Stokes calculations. Our model can reproduce swimmer propulsion for a large range of Reynolds numbers, and generate wake vortices in the inertial regime, reminiscent of the flow generated by the flapping tails of real fish. We performed a numerical exploration of the model from low to high Reynolds numbers and obtained universal laws using scaling arguments. We collected data from a wide variety of micro-organisms, thereby extending the experimental data presented in. Our theoretical scaling laws compare very well with experimental data across the different regimes, from Stokes to turbulent flows. We believe that this model, due to its relatively simple design, will be very useful for obtaining numerical simulations of collective effects within fish schools composed of hundreds of individuals.

Introduction The wide variety of means employed by living creatures to move in aquatic environments is fascinating. Motion generally involves a complex interplay between the deformation of the body and the surrounding fluid. From the smallest organisms, like bacteria, to colossal blue whales, the differences in length scale L, velocity V and mode of locomotion are so vast that the elaboration of a universal model to describe swimming across these scales might seem impossible.

The importance of inertia with respect to viscous dissipation is quantified by the Reynolds number, which expresses the ratio of stress due to inertia to stress due to viscosity: R e equals rho V L over eta, where rho and eta represent fluid density and viscosity, respectively. The Reynolds numbers associated with aquatic living species span several decades, typically ranging from R e superscript fish approximately ten to the power of three to ten to the power of six for fish to R e superscript micro-organism less than or approximately ten to the power of negative three for micro-organisms like spermatozoa. Consequently, the drag force exerted by the fluid on the swimmer is largely dependent on the species considered, originating from either fully viscous dissipation at the small scale, or turbulent inertia at the large scale.

As a result, each swimming model tends to offer a tailored approach specific to the flow regime considered and the corresponding body deformation. Most models focus on periodic body deformation, coupled with the surrounding fluid, and resolve the full swimming cycle. This provides specific approaches that address swimming at small scales, where viscous forces dominate (e.g. for micro-organisms), differently from the macroscopic strategies of large fish or mammals, which can leverage the inertia of the surrounding fluid to break time reversibility. The highly diverse physical origins of particular swimming patterns represent an obstacle to the exploration of a more comprehensive and universal viewpoint.

In this letter, we propose a different approach. Deformation kinematics, such as undulations, oscillations and pulsations, are ignored, and locomotion is described using force and torque dipoles applied by a solid body of finite size L on a fluid. While a similar description has already been used for micro-swimmers at low R e, to the best of our knowledge, it has never yet been employed for high Re, when inertia starts to dominate viscous forces. In this work, we study the motion described by our model over eight decades of Reynolds numbers ten to the power of negative five less than or approximately R e less than or approximately ten to the power of four, theoretically and numerically, comparing our results with experimental data. Although the direct effect on swimming velocity of the specific deformation of the swimmer's body is acknowledged, particularly as it can reduce the drag force, we would like to emphasize that we are not attempting to provide a detailed and precise analysis of a particular mode of locomotion, but rather a more universal description in terms of the forces applied to the fluid. While our swimmer model is minimal, the motion of the surrounding fluid is accurately captured using the full numerical resolution of the three-D Navier-Stokes equation, and our approach encompasses the different swimming regimes of a wide variety of aquatic species. Our model can also remarkably reproduce the characteristic wake vortices observed behind fish due to the flapping of their tails.

Our approach furthermore exhibits universal scaling laws which link the swimming Reynolds number R e to a new dimensionless group, the thrust number defined below. We identify three different regimes: the Stokes regime R e less than one, a laminar regime one less than R e less than ten to the power of three and a turbulent regime R e greater than ten to the power of three to ten to the power of four, and calculate the theoretical exponents of the scaling laws in the three regimes using simple scaling analyses independently of the space dimension. These compare very well with the numerical simulations produced by our generic swimmer model. We also validate our results with experimental data presented for the laminar and turbulent regimes, and further extend this validation with data collected on micro-swimmers for the Stokes regime.

Swimmer model. The model uses a time-dependent force dipole combined with a torque dipole, both attached to a rigid body B of ellipsoidal shape. An autonomous swimming body creates its own motion, therefore the total sum of forces and torques must cancel out, due to the third law of Newton of the fluid-body system. The model developed by Filella et al. presents some conceptual similarities. It represents each fish as a point-like active particle bearing a dipole in a potential two-D inviscid fluid, which allows consideration of the hydrodynamic interactions between fish in the far-field limit. However, since our aim is to explore a wide spectrum of Reynolds numbers from viscous to inertial regimes, we solved the full incompressible Navier-Stokes equations in three-D and two-D.

The swimmer exerts on the fluid a force dipole negative F, F like that generally used for a microswimmer at a low Reynolds number. We used a pusher-like model that reproduces the force distribution of a fish at high Reynolds numbers. This approach can easily be extended to a more detailed model by using more complex force distributions. As shown in Figure one (a), the force dipole is composed of one force applied in the fluid at the rear of the body X, mimicking a swimming organ, and an opposite force exerted inside the body at the center-of-mass XB. The force is time-dependent with pulsation w and pusher-like: F(t) equals (seven/half) Fo cosine wt with Fo greater than zero XB-Xt. The absolute value in the expression of F(t) enforces the pusher nature of the swimmer. We also consider a torque dipole: T(t) equals To cosine wt, colocated with the force dipole. The torque at the back represents the stroke of the swimming organ and causes the vortex street in the fish's wake at high Reynolds numbers. An opposite torque is applied in the body (Figure one (a)), and represents the counter-reaction of the rest of the body. For practical reasons in the scaling analyses below, we also introduce force and torque densities fo equals Fo divided by L cubed and To equals To divided by L cubed respectively. Averaging these dipoles over one period of time (twenty-seven divided by w) results in a simple static force dipole (negative Fo, Fo) (Figure one (b)), while the average torque cancels out. This time averaging over one beating period is very similar to models of pushers and pullers beating at low Reynolds numbers.

Scaling laws. The hydrodynamic nature of our model allows for simple scaling arguments, inspired by Gazzola et al. for inertial flows but translated to a more generic framework and extended to the non-inertial Stokes regime. In the following, we present three D arguments, but they remain valid two D (see Supplemental Material). We also consider that all the lengths scale as L.

The body of the swimmer is submitted to different dominant drag forces depending on the Reynolds number. To describe this effect across all swimming regimes, we introduce the thrust number Th as the ratio between the applied force density f zero multiplied by inertial forces rho absolute value D v divided by D t is similar to rho v squared L to the negative one, and the square of viscous forces absolute value eta Delta v squared is similar to eta v L to the negative two squared, which gives

(one)

The thrust number appears naturally at all scales of Reynolds numbers, as shown below. It contains the force term at the origin of the motion, which is characterized by the Reynolds number. It therefore provides a convenient method for evaluating velocity as a function of force. Let us consider classic scaling arguments:

· At high Reynolds numbers, the boundary layer around the body is turbulent and pressure drag dominates. The corresponding force scales as f zero L cubed is similar to rho v squared L squared. Since v is approximately equal to eta R e divided by rho L, we obtain R e is similar to rho f zero L cubed divided by eta squared to the one half equals Th to the one half.

· For small but finite Reynolds numbers, the regime is laminar and the viscous force in the boundary layer dominates: f zero L cubed is similar to eta v divided by delta L squared where delta is the thickness of the boundary layer, which obeys the Blasius law delta is similar to L R e to the negative one half. This finally leads to R e is similar to Th to the two thirds.

· At low Reynolds numbers, the Stokes drag force dominates and the force applied on the body compensates the drag: f zero L cubed is similar to eta v L. This gives R e is similar to rho f zero L cubed divided by eta squared equals Th.

From the Stokes to the turbulent regime, we observe that the exponent alpha of the scaling R e is similar to Th to the alpha is always below one and decreases. It suggests a diminishing swimming performance as the Reynolds number of the swimmer increases. To confirm these three successive regimes, we present below numerical simulations with our swimmer model, exploring a large range of values for the thrust and Reynolds numbers.

Numerical simulations. We performed direct numerical simulations of the incompressible Navier-Stokes equations in the presence of our model swimmer, a rigid ellipsoid body with the force and torque dipoles attached. The corresponding fluid momentum balance equation writes:

(two)

where V and P are respectively the velocity and pressure fields, D over D T denotes the material derivative D V over D T equivalent to partial V over partial T plus left V dot nabla right V, E left V right equivalent to left nabla V plus right. left nabla V superscript T right over two is the strain-rate tensor, rho and eta denote the density and viscosity fields, and script F left T right text and script T left T right respectively represent the time-varying force and torque dipoles attached to the swimmer, as introduced above. The rigid body script B of the swimmer is accounted for with a fictitious domain penalty method inspired by; in practice, this can be implemented simply with a spatially-variable viscosity left twenty-nine right: eta equals eta sub F plus left eta sub B minus eta sub F right H sub script B where H sub script B is the indicator or Heaviside function of script B. In practice, a viscosity ratio eta sub B over eta sub F equals ten to the power of three to ten to the power of six is applied between the fluid and the swimmer body to ensure that the rigid motion constraint E left V right equals zero is satisfied within script B. Density rho is defined as constant inside and outside script B, thus making the swimmer neutrally buoyant. Note that this fictitious domain penalty method allows the swimmer to be treated directly as part of the Navier-Stokes equations through the viscosity field, thereby avoiding the need to deal with moving boundary conditions and potential remeshing issues at the body interface in the discrete setting. The incompressible Navier-Stokes equations are solved numerically using an implicit script P two minus script P one finite element method implemented in the parallel FEEL++ library. The position and orientation of the swimmer are updated at each time-step using a first order Euler scheme with the translational and rotational velocities computed from the fluid velocity field in script B. A comprehensive derivation of the numerical model and technical details are provided in SM.

This numerical framework is used to explore a large range of Reynolds numbers left ten to the power of negative five less than R E less than right ten to the power of four and Thrust numbers left ten to the power of negative three less than T H less than ten to the power of six, in order to evaluate the dependence of R E as a function of T H while varying each of the model's different parameters left L, eta sub F, F sub zero, omega and tau sub zero right separately (see SM). As shown in Fig. two, the numerical simulations are in perfect agreement with the scaling laws presented above, displaying the Stokes regime in the range ten to the power of negative five less than or approximately R E less than or approximately ten to the power of two, the laminar regime for ten to the power of two less than or approximately R E less than or approximately ten to the power of three and the turbulent regime for ten to the power of three less than or approximately R E, with fitting exponents that match the predictions. Note that the R E ranges of each regime depend on the aspect ratio of the swimmer a sub R, which is kept constant. We also found that omega and tau sub zero do not play any role in the R E left T H right dependency, which confirms that all the important parameters are embedded in the T H number.

Wake and vortices. Although torque plays no role in the scaling of R E as a function of T H, it is essential to reproduce the wake at the rear of the swimmer in the inertial regime. The torque dipole can be used to account for flagellum, body undulation or tail beating to generate a reverse von Karman vortex street, as observed in the wake of a fish at high R E number.

Figures three left a, b right illustrate the typical wakes obtained with left T sub zero, F sub zero right equals left four thousand, four thousand right, left six hundred, six hundred right in two D simulations. The vortices created at the same frequency are spaced further apart as the Reynolds number left i.e. velocity right increases from R E equals one hundred to R E equals nine hundred sixty. The length over which they dissipate becomes shorter towards the viscous regimes (Fig. three c), vanishing completely at low R E (Fig. three d). Indeed, no vortices are present behind micro-organisms.

Comparison with experimental data. T sub C compare our scaling results with experimental data, forces must be expressed in terms of observable data, such as undulation or tail beating frequency. Although force F sub zero, torque tau sub zero and pulsation omega are independent quantities in the model, physical constraints exist between these quantities in living organisms. We made the reasonable assumption that the size of the swimming organ scales with the size of the body L. Force and torque generated by the swimming organ are such that tau sub zero is approximately equal to F sub zero L.

In inertial regimes, i.e. excluding the Stokes regime that we address separately, the instantaneous force creates a transient acceleration of the fluid, which scales with L omega squared, i.e. F sub zero is approximately equal to rho L omega squared. Introducing the observation-based swimming number S sub W equals rho omega L squared divided by eta, the theoretical force drive can be related to experimentally measurable data as T sub H is approximately equal to S sub W squared. The scaling laws previously derived from our model can thus be reformulated in terms of S sub W: in the laminar regime Re is approximately equal to T sub H to the two-thirds is approximately equal to S sub W to the four-thirds while in the turbulent regime Re is approximately equal to T sub H to the one-half is approximately equal to S sub W, in accordance with the results of Gazzola et al.

In the Stokes regime, the swimming organ is balanced by viscous drag, so that F sub zero L cubed is approximately equal to eta L v is approximately equal to eta L squared omega. Reintroducing again the swimming number S sub W, we obtain T sub H is approximately equal to S sub W.

leading to Re approximately equal to S sub W. Note that this is a natural consequence of the absence of inertia: each stroke creates a net displacement that scales with approximately L, inducing a swimming velocity v approximately equal to L omega.

Figure four shows the experimental data collected from micrometer- to meter-size aquatic organisms along with the corresponding fitted scaling laws. In addition to the results from Gazzola et al., an excellent agreement between experimental data and hydrodynamic scaling laws is also obtained in the Stokes regime.

Note that v approximately equal to L omega in both the Stokes and turbulent regimes, so that the corresponding Strouhal number St equals S sub W divided by Re is constant, as illustrated in Figure five. The transition between St divided by twenty-seven approximately equal to zero point three and St divided by twenty-seven approximately equal to one occurs in the laminar regime where St is approximately equal to Re to the negative one-fourth.

Conclusion. By avoiding direct fluid-structure coupling, our generic swimmer model provides an efficient model for hydrodynamic propulsion, while retaining the salient features of swimming organisms across several decades of Reynolds numbers. High numerical stability and efficiency ensure fully tractable two D and three D simulations, thereby paving the way to large scale simulations with hundreds of agents. It also proposes a methodology for progressive refinement of the hydrodynamic field, by retaining higher moments of forces and torques, resulting in more complex propulsion models. This approach broadens our understanding of the swimming of aquatic organisms by revealing the universal relationship between the velocity of a swimmer and the force exerted by its swimming organ. The sub-linear dependence demonstrated between Re and Th suggests diminishing swimming performance as the swimmer's Reynolds number increases. The scaling laws obtained also match the experimental data obtained from thousands of aquatic animals, ranging from large mammalians to micro-organisms. Our results shed new light on the general mechanisms underlying swimming and provide an efficient and robust numerical framework to investigate the collective behavior of swimmers in complex environments.