Potential flow

In fluid dynamics, potential flow describes the velocity field as the gradient of a scalar function: the velocity potential. As a result, a potential flow is characterized by an irrotational velocity field, which is a valid approximation for several applications. The irrotationality of a potential flow is due to the curl of the gradient of a scalar always being equal to zero.while using the summation convention: since j occurs more than once in the term on the left hand side of the momentum equation, j is summed over all its components (which is from 1 to 2 in two-dimensional flow, and from 1 to 3 in three dimensions). Further: In fluid dynamics, potential flow describes the velocity field as the gradient of a scalar function: the velocity potential. As a result, a potential flow is characterized by an irrotational velocity field, which is a valid approximation for several applications. The irrotationality of a potential flow is due to the curl of the gradient of a scalar always being equal to zero. In the case of an incompressible flow the velocity potential satisfies Laplace's equation, and potential theory is applicable. However, potential flows also have been used to describe compressible flows. The potential flow approach occurs in the modeling of both stationary as well as nonstationary flows.Applications of potential flow are for instance: the outer flow field for aerofoils, water waves, electroosmotic flow, and groundwater flow. For flows (or parts thereof) with strong vorticity effects, the potential flow approximation is not applicable. In fluid dynamics, a potential flow is described by means of a velocity potential φ, being a function of space and time. The flow velocity v is a vector field equal to the gradient, ∇, of the velocity potential φ: Sometimes, also the definition v = −∇φ, with a minus sign, is used. But here we will use the definition above, without the minus sign. From vector calculus it is known that the curl of a gradient is equal to zero: and consequently the vorticity, the curl of the velocity field v, is zero: This implies that a potential flow is an irrotational flow. This has direct consequences for the applicability of potential flow. In flow regions where vorticity is known to be important, such as wakes and boundary layers, potential flow theory is not able to provide reasonable predictions of the flow. Fortunately, there are often large regions of a flow where the assumption of irrotationality is valid which is why potential flow is used for various applications. For instance in: flow around aircraft, groundwater flow, acoustics, water waves, and electroosmotic flow. In case of an incompressible flow — for instance of a liquid, or a gas at low Mach numbers; but not for sound waves — the velocity v has zero divergence: with the dot denoting the inner product. As a result, the velocity potential φ has to satisfy Laplace's equation where ∇2 = ∇ ⋅ ∇ is the Laplace operator (sometimes also written Δ). In this case the flow can be determined completely from its kinematics: the assumptions of irrotationality and zero divergence of flow. Dynamics only have to be applied afterwards, if one is interested in computing pressures: for instance for flow around airfoils through the use of Bernoulli's principle.