D4 Computational Fluid Dynamics - CFD

Topic

Computational fluid dynamics is a technique that numerically solves the equations governing fluid motion around a swimming fish; in doing so, it yields complete fields of velocity, pressure, vorticity, and stress throughout the fluid domain surrounding the fish—information that is impossible to obtain experimentally with comparable resolution and coverage. In ichthyology, this technique is used to study propulsion hydrodynamics, the drag on bodies of various shapes, fin interactions, and the hydrodynamics of entire schools of fish, at scales exceeding the capabilities of the particle image velocimetry technique described earlier in this series.

The equations that computational fluid dynamics solves numerically to describe the incompressible flow of a Newtonian fluid are essentially the mathematical expression of Newton's second law applied to a continuous fluid: at any given point in the fluid, the sum of pressure forces, viscous forces, and the inertial force associated with the fluid's own motion equals the acceleration experienced by that portion of the fluid.

For a real fish, the flow around its body is typically turbulent because the swimming Reynolds number far exceeds ten thousand—the order of magnitude at which flow ceases to be orderly. This turbulence necessitates the inclusion of a turbulence model within the numerical solution process. These models follow two distinct approaches: some represent only the average effect of turbulent fluctuations that the simulation does not resolve directly, while others explicitly resolve the larger scales of turbulence and average only the smaller scales. In the first approach, the average effect of turbulent fluctuations is incorporated into the equations via an additional, fictitious viscosity—added to the fluid's actual viscosity—that increases or decreases depending on the intensity of the local turbulence.

The geometry of the swimming fish does not remain static within this simulation; it is represented as a boundary that moves and deforms with each swimming cycle. This deformation is either transferred—step-by-step over time—to the computational mesh discretizing the fluid domain, or handled by keeping the mesh fixed and treating the fish's body as an additional condition imposed upon that fixed mesh (an alternative method known as the immersed boundary method).

Based on this simulation of the flow around the moving fish geometry, parameters such as the drag coefficient, pressure distribution over the body, thrust force, and propulsive efficiency are obtained by integrating the pressure distribution and viscous stress across the fish's entire surface.

Validation of the computational fluid dynamics model is performed by comparing these simulated results with experimental data—obtained via particle image velocimetry applied to the fish's actual wake—and with direct force measurements on real fish. Once properly validated, the computational fluid dynamics model reproduces the same vorticity patterns observed in the actual wake, and its force values ​​fall within a 5% to 15% margin of the experimentally measured values; this confirms that the comprehensive velocity, pressure, vorticity, and stress fields yielded by the technique—which are impossible to measure directly, as noted at the start of this model—accurately describe the actual fluid surrounding the fish.

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