William W. Schultz

Power Requirements of Swimming: Do New Methods Resolve Old Questions?

Abstract A recurring question in the study of fish biomechanics and energetics is the mechanical power required for tail-swimming at the high speeds seen among aquatic vertebrates. The quest for answers has been driven by conceptual advances in fluid dynamics, starting with ideas on the boundary layer and drag initiated by Prandtl, and in measurement techniques starting with force balances focussing on drag and thrust. Drag (=thrust) from measurements on physical models, carcasses, kinematics as inputs to hydromechanical models, and physiological power sources vary from less than that expected for an equivalent rigid reference to over an order of magnitude greater. Estimates of drag and thrust using recent advances largely made possible by increased computing power have not resolved the discrepancy. Sources of drag and thrust are not separable in axial undulatory self propulsion, are open to interpretation and Froude efficiency is zero. Wakes are not easily interpreted, especially for thrust evaluation. We suggest the best measures of swimming performance are velocity and power consumption for which 2D inviscid simulations can give realistic predictions. Steady swimming power is several times that required for towing an equivalent flat plate at the same speed.

One‐Dimensional Liquid Fibers

Axisymmetric Newtonian viscous fibers are examined under isothermal conditions. An expansion based on lubrication scaling is used to derive systematically the one‐dimensional equations for the fiber. At lowest order these are identical with those obtained by others, e.g., Matovich and Pearson. One‐dimensional theory limitations on jet shape and inertial, gravitational, and surface tension effects are obtained from higher‐order approximations. Similar approaches can be used to analyze fibers having complex rheological behaviors.

Modeling and Parametric Study of Torque in Open Clutch Plates

The relative motion of the friction and separator plates in wet clutches during the disengaged mode causes viscous shear stresses in the fluid passing through the 100 microns gap. This results in a drag torque on both the disks that wastes energy and decreases fuel economy. The objective of the study is to develop an accurate mathematical model for the above problem with verification using FLUENT and experiments. Initially we two consider flat disks. The mathematical model calculates the drag torque on the disks and the 2D axisvmmetnc solver verifies the solution. The surface pressure distribution on the plates is also verified. Then, 3D models of one grooved and one flat disk are tested using CFD, experiments and an approximate 3D mathematical model. The number of grooves depth of groove and clearance between the disks are studied to understand their effect on the torque. The study determines the pressure field that eventually affects aeration incipience (not studied here). The results of the model, computations and experiments corroborate well in the single-phase regime.

Spectral method solution of the Stokes equations on nonstaggered grids

Abstract The Stokes equations are solved using spectral methods with staggered and nonstaggered grids. Numerous ways to avoid the problem of spurious pressure modes are presented, including new techniques using the pseudospectral method and a method solving the weak form of the governing equations (a variation on the “spectral element” method developed by Patera). The pseudospectral methods using nonstaggered grids are simpler to implement and have comparable or better accuracy than the staggered grid formulations. Three test cases are presented: a formulation with an exact solution, a formulation with homogeneous boundary conditions, and the driven cavity problem. The solution accuracy is shown to be greatly improved for the driven cavity problem when the analytical solution of the singular flow behavior in the upper corners is separated from the computational solution.

Chebyshev pseudospectral method of viscous flows with corner singularities

Chebyshev pseudospectral solutions of the biharmonic equation governing two-dimensional Stokes flow within a driven cavity converge poorly in the presence of corner singularities. Subtracting the strongest corner singularity greatly improves the rate of convergence. Compared to the usual stream function/ vorticity formulation, the single fourth-order equation for stream function used here has half the number of coefficients for equivalent spatial resolution and uses a simpler treatment of the boundary conditions. We extend these techniques to small and moderate Reynolds numbers.

An efficient finite element method for treating singularities in Laplace's equation

We present a new finite element method for solving partial differential equations with singularities caused by abrupt changes in boundary conditions or sudden changes in boundary shape. Terms from the local solution supplement the ordinary basis functions in the finite element solution. All singular contributions reduce to boundary integrals after a double application of the divergence theorem to the Galerkin integrals, and the essential boundary conditions are weakly enforced using Lagrange multipliers. The proposed method eliminates the need for high-order integration, improves the overall accuracy, and yields very accurate estimates for the singular coefftcients. It also accelerates the convergence with regular mesh refinement and converges rapidly with the number of singular functions. Although here we solve the Laplace equation in two dimensions, the method is applicable to a more general class of problems.

Moderate and steep Faraday waves: instabilities, modulation and temporal asymmetries

Mild to steep standing waves of the fundamental mode are generated in a narrow rectangular cylinder undergoing vertical oscillation with forcing frequencies of 3.15 Hz to 3.34 Hz. A precise, non-intrusive optical wave profile measurement system is used along with a wave probe to accurately quantify the spatial and temporal surface elevations. These standing waves are also simulated by a two-dimensional spectral Cauchy integral code. Experiments show that contact-line effects increase the viscous natural frequency and alter the neutral stability curves. Hence, as expected, the addition of the wetting agent Photo Flo significantly changes the stability curve and the hysteresis in the response diagram. Experimentally, we find strong modulations in the wave amplitude for some forcing frequencies higher than 3.30 Hz. Reducing contact-line effects by Photo-Flo addition suppresses these modulations. Perturbation analysis predicts that some of this modulation is caused by noise in the forcing signal through ‘sideband resonance’, i.e. the introduction of small sideband forcing can generate large modulations of the Faraday waves. The analysis is verified by our numerical simulations and physical experiments. Finally, we observe experimentally a new form of steep standing wave with a large symmetric double-peaked crest, while simulation of the same forcing condition results in a sharper crest than seen previously. Both standing wave forms appear at a finite wave steepness far smaller than the maximum steepness for the classical standing wave and a surface tension far smaller than that for a Wilton ripple. In both physical and numerical experiments, a stronger second harmonic (in time) and temporal asymmetry in the wave forms suggest a 1:2 resonance due to a non-conventional quartet interaction. Increasing wave steepness leads to a new form of breaking standing waves in physical experiments.

Slender Viscoelastic Fiber Flow

The one‐dimensional equations describing Newtonian fiber flow have been derived previously using lubrication scaling of the axisymmetric equations of motion. Here, we present an extension of that analysis using a generalized convected Maxwell model. We find that the non‐Newtonian behavior of the fluid must be severely limited for the one‐dimensional equations to be determinate. In this sense, the primary effect of viscoelastic behavior is to make the flow more two dimensional. Limiting cases based on previous one‐dimensional models are shown to violate the axisymmetric equations of motion. In addition, questions are raised about the validity of elongational viscosity measurements of highly elastic fluids.

Kinematic and dynamic evolution of deep water breaking waves

Experiments were performed to exploit the dispersive properties of unsteady surface waves and to induce breaking by using a modified chirp pulse technique to focus the wave energy at a specific location in the Naval Research Laboratory deep water wave channel. The experiments have resulted in a highly resolved archive of breaking events ranging from wave steepening and incipient breaking to spilling and to plunging. The potential energy density, the crest front steepness, the horizontal asymmetry, and other geometric properties of an incipient breaker vary only within a moderate band about their mean values over the extent of these experiments. Thus the properties of an incipient unsteady breaker are well defined. The application of the phase-time or Hilbert transform method to the data set provides new insights into the local properties of the unsteady wave breaking. Recently, spectral and piecewise-linear algorithms for two-dimensional potential flow were developed and used by Schultz et al. [1994] to compare the onset of breaking for several methods of energy input to the unsteady wave system. The computations show that steep plunging waves occur when energy input rates are large. The various energy input methods exhibit similar breaking trends in the limit as the energy input rate becomes small in that incipient spilling breakers form when the potential energy is approximately 52 to 54% of the energy for the most energetic Stokes wave, with the formation of a singularity immediately before the crest.

Potential Energy in Steep and Breaking Waves

Abstract : We find that potential energy rather than wave height is a better experimental and analytic criterion for determining when wave breaking will occur. A simple two-dimensional, periodic algorithm is developed and used to compare breaking onset criteria for energy input from (1) converging sidewalls, (2) a submerged disturbance and (3) wave focusing. Wave-breaking criteria (potential energy or the more classical peak-to-peak wave height) are a function of the rate of energy input. Large plunging waves occur for large energy input rates with a smooth transition to smaller spilling waves for lesser energy input rates. The first two kinds of energy input show similar trends in the limit as the energy input rate becomes small. The third case, wave focusing, is the subject of an ongoing investigation. The effects of wave modulation and reflection are also discussed.