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Why can't potential flow describe the boundary layer? What is the physical truth behind it?
In fluid dynamics, the concept of potential flow plays an integral role in many areas of engineering and science. Potential flow usually describes the flow of a fluid with no curl, which assumes that the flow is incompressible and there are no vortices when the fluid has a small viscosity. If we analyze under these conditions, we can use velocity potential and Laplace's equation to characterize the flow. However, potential flow cannot effectively describe boundary layer phenomena, which becomes a major challenge in fluid mechanics.
The reason why the characteristics of potential flow cannot describe the boundary layer is fundamentally due to the existence of curl and the particularity of the velocity field.
In the definition of potential flow, the velocity field is considered to be the gradient of a scalar function, which makes the curl of the velocity field always zero. In such a flow, there is no rotation or vortex generation in the fluid. Therefore, potential flow can effectively explain the behavior of flows in a large range, especially in the flow field outside the aircraft, ground water flow, acoustics, and water waves. However, the assumption of potential flow breaks down when we consider the boundary layer - the layer of flow close to the surface of a solid object.
The boundary layer is a fluid layer formed due to the friction on the surface of a solid object and its influence on the flow velocity field. In this layer, the irregular motion of the fluid leads to the generation of curl, and the flow velocity varies with the distance from the solid object. These situations cannot be reasonably described in the potential flow theory. For example, on an aircraft wing, when a fluid contacts the wing surface, vortices are generated near the wing surface due to friction, and the appearance of these vortices limits the application of potential flow.
The change of the curl and velocity field of the fluid in the boundary layer is an important physical reason why the potential flow cannot be resolved.
In addition, the non-uniqueness of the potential flow makes it impossible to describe the flow behavior of the boundary layer. The velocity potential in the underlying flow is not unique, which means that when applied to the boundary layer, choosing different initial conditions may lead to different solutions that do not reflect the actual flow situation. In the boundary layer, the dynamic behavior of the fluid is often strongly affected by the boundary conditions, which once again challenges the validity of the potential flow theory.
In the boundary layer, the Navier-Stokes equations of fluid dynamics are a more appropriate description of the change in flow velocity. This set of equations takes into account fluid viscosity and vortex effects and is more accurate than potential flow theory in describing flows near contacting solid surfaces. The flow behavior of the fluid in the boundary layer becomes complex and involves various interactions, such as the rate of change of flow velocity, friction, and even abnormal changes in pressure.
It can be seen that the limitation of potential flow is that it does not consider the viscosity and curl effects in the flow.
As for the practical application of potential flow, even though it is still very effective in some large-scale flows, when dealing with complex boundary layer problems, scientists and engineers usually rely on more advanced mathematical models to capture These details. Boundary layer theory in fluid dynamics provides effective tools to analyze these phenomena and is the key to understanding and designing fluid dynamic systems.
With the advancement of technology, the emergence of computational fluid dynamics (CFD) has made flow simulation more accurate. These methods can include rotational effects and boundary conditions, giving us a deeper understanding of flow. However, in the analysis of various fluid models, the understanding and learning of the underlying flow model is still the basis.
The boundary between bubble spectrum and potential flow shows the challenges and opportunities in future fluid dynamics research.
Ultimately, we can't help but ask, in such complex fluid dynamics, are there still unexplored potential flow applications?
