Benjamin Connell

18.325

4/19/00

G. I. Taylor, 1952

Though an extensive study into the body movements associated with fish locomotion had been previously been undertaken by James Gray, it is explained that, until the writing of this paper, "attempts to analyse swimming from the point of view of hydrodynamics have failed." The difficulties in solving the problem were two-fold, stemming from technological limitations. First was the difficulty in mechanically recreating a swimming body. Second was in solving for the equations of equilibrium of an unsteady body in a turbulent flow field. As will be described, Taylor simplified the problem to a level at which it could be solved with the resources available. The theoretical solutions were compared to data acquired in the study of swimming aquatic animals performed by Gray.

The common body motion associated with the swimming of fish (and eels and snakes) is described by Gray as a wave propagated opposite the direction of propulsion, increasing significantly in amplitude from head to tail. The picture below shows such a wave progressing from right to left.

Observations in nature indicate that the above depicted unsteady boundary has a net propulsive effect to the right. Taylor’s goal in Analysis of the Swimming of Long and Narrow Animals was to calculate this propulsive effect for a long, cylindrical swimming body.

The simplifications made by Taylor in his solution to the problem were rather extensive. The assumption upon which his model was based is that for a long cylindrical swimming body, the drag force on each small section of body length is exactly the same as if that section were part of an infinite straight cylinder in a flow. In this sense, the snake was approximated as a series of pieces of infinitely long cylinders inclined at different angles to the flow. A significant advantage to this approach for Taylor, was the availability of data for the drag on an infinite cylinder in a flow. The obvious shortcoming of this method is that the flow at any point on the body is considered unperturbed by the rest of the body, the "narrow" assumption.

A significant deviation from nature in Taylor’s model was the imposition of a constant amplitude for the body wave, eliminating the growth effect displayed in the above picture. The symmetry provided by this simplification of geometry is exploited in the development of the solution. The study is then limited to a single wave-length in the middle of the body, and head and tail effects are not considered, the "long" assumption.

Taylor’s formulation for the swimming of a long and narrow body starts by defining the hydrodynamic forces on a straight infinite cylinder inclined to a flow. The geometry and boundary motion of a flexible waving cylinder are then discussed, permitting application of the straight infinite cylinder theory to sections of the waving cylinder. Equations of equilibrium of the modeled system are then solved, yielding the possible combinations of parameters for the system. The variables for a given cylindrical body are:

l – Wavelength of the body wave

B – Amplitude of the body wave

U – Phase speed of the body wave

V – Swimming speed (opposite direction of U)

With the equations of equilibrium, V can be found for a given combination of l, B, and U.

The forces on an inclined cylinder are considered in two components, the normal component, N, and the longitudinal component, L. Likewise, the flow can also be considered in the same two components Q_N and Q_L. Two non-directional components of drag must also be considered. The first is the form drag due to separation of the flow around a bluff body, weighted by the drag coefficient C_D. The second is frictional drag, as the drag of flow over a flat plate, weighted by the frictional drag coefficient C_F. Taylor asserts that the form drag is due only to the normal component of the flow, as this is the direction in which a cross-section of the cylinder forms the most bluff-body, as shown below.

It can be seen in the above picture how the longitudinal component of the cross-section, C1, serves only to streamline the body. Thus, only the normal component of the flow is taken as significant when considering separation of the flow and the resulting form drag. Frictional drag is considered important in both the normal and longitudinal directions.

The normal component of the drag on a cylinder of diameter d inclined at angle i to the flow is given by

(1)

Using and , (1) becomes

(2)

The longitudinal component of the hydrodynamic force is considered to come entirely from frictional drag. It is found by integrating the viscous stress in the axial direction around the cylinder. Taylor starts with a special solution to the Navier-Stokes Equations. Calling "z" the axial or longitudinal direction, equilibrium in the axial direction is given by

(3)

For an infinite cylinder where we are considering no axial dependence of the flow or pressure, and steady flow in the axial direction, (3) becomes

(4)

The form of this equation opens the door to a solution by analogy. The equation governing heat conduction in a flow is of the same form

(5)

Where T is temperature, K is thermal conductivity, and s is specific heat.

The boundary conditions for the two problems are

Drag Eq. Heat Eq.

On Cylinder: w=0 T=T₀

At Infinity: w=W T=0

The viscous stress is given, by definition, as , and the total longitudinal force is found by integrating this around the cylinder.

(6)

The total heat loss from a cylinder is given by the analogous equation

(7)

From (6) and (7) and the boundary conditions, the relationship is formed

or (8)

This put Taylor in a position to use extensive prior research and experimentation on heat loss from a cylinder in a flow. It is from this prior research that he obtains the relationship dependent on the normal component of the flow

(9)

where A is a constant determined by experimentation. Combining with (8)

with A₂ having absorbed the thermal constants and a into A. Our final equation for the longitudinal force is then

(10)

A₂=5.4

The value of the constant, A₂, was determined by using experimental data available for the drag on a cylinder inclined to a stream.

Taylor next launches into a discussion and speculation of how N and L would be affected by different types of roughness of the cylinder surface. The essential point is that varying the form of the surface will affect the frictional drag coefficient, C_F. It’s noteworthy that, depending on the nature of the roughness, the frictional coefficient can be directional (differing in its affect on the normal vs. longitudinal force). Though not stated directly, this is what Taylor shows in his pictures and force predictions.

One wavelength of the swimming cylindrical body in a body-fixed reference frame will be defined by parameters as depicted.

In the above, U is the phase speed of the propagating wave, and V is the swimming speed (which translates to mean flow velocity in the body-fixed frame). Simplification of the ensuing description is achieved by considering a coordinate system that is phase-locked to the propagating wave (i.e. moving to the left at U) as follows.

The net body motion, which was zero in the body-fixed frame, will now be U to the right. The appropriate boundary condition is satisfied by defining a body motion, q, which we will call "slithering". With the assumption that the body is inextensible, the magnitude of q must be constant. In order for the net body motion to be U, q must cover the arc length of one wavelength in the amount of time that U covers l.

(11)

The next step is to make the connection between the above picture and a system of stationary cylinders inclined to a flow. Consider one section of the body in this context, the upslope at the left side of Fig.4, at angle q to the mean flow.

This must now be transformed to a stationary cylinder in a flow. Knowing from (11) that q>U>U-V, we get

From Fig.6 the values of Q and i to be used in (2) and (10) are found.

(12)

(13)

The value of q=f(U,l,B) is found numerically for each case using (11).

On inspection of Fig.6, we can begin to understand how wave propagation can cause motion in either direction, depending on the nature of the roughness of the cylinder. If we consider the frictional drag in Fig.6 as negligible compared to the form drag, then the normal component of Q and its normal force will dominate and the body will be propelled to the right. However, if the longitudinal frictional force dominates, as it might for some types of very rough surfaces, then the body will be propelled to the left.

Taylor’s initial assumption in the application of equilibrium equations is that the fish length is an integer multiple of the wavelength. This allows application of equilibrium equations over a single wavelength. Consider in Fig.4 that x is the streamwise direction, y the transverse direction, and q the angle of the body at a given point to the x-direction. We have force balance in the y-direction by symmetry.

(14)

If we consider the body to be swimming at a constant speed (no acceleration), then we also have force balance in the x-direction given by

(15)

The system is now determined by (2), (10), (12), (13), (14), and (15). The equations set up the complete relationship between the parameters l, B, U, and V.

At this point Taylor goes about finding solutions to the system, which one might imagine to be a bit of a mess. By example, after defining a and z to the parameters of the system, he was faced with solving the four definite integrals

with and .

Taylor’s Table 3 shows the results of his numerical evaluation of these integrals for 10 values of a. Though it’s a bit unfair, we used our ability to numerically evaluate these integrals ~instantaneously to check his work. Our integrals were split into 1000 sections. Table 1 compares the values, Taylor’s first, and ours in parentheses.

Table 1

None of Taylor’s values appear remarkably far removed from those we calculated. The table above is only one of several that appear in Taylor’s solution to the system, suggesting significantly laborious calculations.

Taylor presents his solutions in the form of "swimming diagrams." These diagrams are plots of B/l vs. V/U, with contours representing constant Re. Diagrams are first generated for smooth cylinder values of N and L. Photographic series obtained from James Gray of both a snake and a leech swimming are interpreted and plotted on the swimming diagram. The location of the snake and leech points on the plot are compared with the predictions of the model for the equivalent B/l and Re. Though the real snake had a faster swimming speed (V) than that predicted by the model, the values for the leech were very nearly the same.

The solutions for rough cylinders (using rough cylinder N and L expressions) are presented with similar swimming diagrams, but with contours of C_D/C_F. Taylor was quite surprised to find that for very low values of C_D/C_F the model predicted negative values of V/U, implying swimming in the direction of wave propagation. This seems reasonable in the context of Figs.4-6, which show how the longitudinal component of the drag force (scaled by C_F) pushes the body in the direction of wave propagation. Taylor shows pictures of natural confirmation of such swimming in a marine worm.

The model is also used to calculate the swimming parameters which maximize efficiency. The power put into the fluid by the body is defined as

(16)

Eqs. (10) and (11) are used to develop predictions for values of l, B, and U which minimize the energy put into the fluid by the body for a given swimming speed. The snake’s swimming parameters were not far off of those predicted for maximum efficiency, while the leech’s parameters were not very close.

Taylor was presented with a difficult problem to solve given the resources available to him in 1952. The approach does at first seem a bit surprising, but creativity can be a worthy substitute for brute force. The most glaring shortcoming of the model used by Taylor is that each section of the body is considered to be a straight infinite cylinder in a constant flow field. In this sense, no part of the body is considered to perturb the flow for any other part of the body. For a vanishingly narrow body it seems that this would be appropriate, but it’s hard to imagine that to be the case for a snake. The body is also approximated by Taylor to be propagating a wave of constant amplitude, while in nature there is a marked increase in wave amplitude over a single wavelength as can be seen in the pictures Taylor presents. Though these considerations seem very significant, Taylor went through the rigorous procedure of generating predictions for equilibrium-satisfying swimming parameters. Amazingly enough, the model’s predicted swimming parameters did not fall far from parameters observed in nature.

Additional advances in the study of the hydrodynamics of fish swimming came not too long after Taylor’s model was published, and the work continues today. In 1960-1961, two important papers on the subject were published. In Swimming of a waving plate, T. Yao-Tsu Wu undertook a two-dimensional potential flow investigation of flow over a waving plate. His description allowed a wave of increasing amplitude and termination of the body in a wake. The study of the nature of the shedding vortices in the wake showed formation of a propulsive jet. M. J. Lighthill in Note on the swimming of slender fish also developed a potential flow model, but for a fish (rather than a plate) using the "slender-body" approximation. Lighthill’s calculations also consider the thrust effects of vortex shedding in the wake. As computational power increases today, so does the ability to accurately model the viscous hydrodynamics of fish swimming, but the problem is still far from solved. Further work is still needed to accurately model the viscous problem at the Reynold’s numbers necessary for practical engineering purposes.