# 2d potential flows

For most of the flows, the following quantities are provided: the complex velocity and potential, the scalar potential and stream function and the velocity in polar and Cartesian coordinates. These elementary flows are essential for the implementation and validation of two-dimensional vortex methods. The solutions can be used to validate two-dimensional panel codes. General results from 2D potential flow theory are presented in Sect. Conformal maps are introduced in Sect. Skip to main content.

Advertisement Hide. Elementary Two-Dimensional Potential Flows. Chapter First Online: 07 April This process is experimental and the keywords may be updated as the learning algorithm improves. This is a preview of subscription content, log in to check access.

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### Potential flow

Kerwin, Lecture notes hydrofoil and propellers. Technical report, M. T Google Scholar. Krasny, Desingularization of periodic vortex sheet roll-up. McIntyre, Ib lecture notes on fluid dynamics. Technical report, University of Cambridge Google Scholar. Wald, The aerodynamics of propellers. Zahm, Flow and drag formulas for simple quadrics.A fluid flow that is isentropic and that, if incompressible, can be mathematically described by Laplace's equation.

For an ideal fluid, or a flow in which viscous effects are ignored, vorticity defined as the curl of the velocity cannot be produced, and any initial vorticity existing in the flow simply moves unchanged with the fluid.

Ideal fluids, of course, do not exist since any actual fluid has some viscosity, and the effects of this viscosity will be important near a solid wall, in the region known as the boundary layer.

Nevertheless, the study of potential flow is important in hydrodynamics, where the fluid is considered incompressible, and even in aerodynamics, where the fluid is considered compressible, as long as shock waves are not present. See Boundary-layer flowCompressible flowIsentropic flow. In the absence of viscous effects, a flow starting from rest will be irrotational for all subsequent time. See Potentials. The linearity of the Laplace equation, which also governs other important physical phenomena such as electricity and magnetism, makes it possible to use the principle of superposition to combine elementary solutions in solving more complex problems. See Fluid flow. A potential flow occurs under certain conditions only for an ideal, frictionless fluid. These conditions occur, for example, when the motion begins from a state of rest and a body immersed in an incompressible fluid begins to move, or when a body strikes the surface of the fluid.

In nonideal liquids and gases, a potential flow occurs in those regions where the forces of viscosity are negligible in comparison with those of pressure and no vortices are present.

The study of potential flow is greatly simplified in that it involves the determination of only one function of the coordinates and time, called the potential function. Potential flow A fluid flow that is isentropic and that, if incompressible, can be mathematically described by Laplace's equation.

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See Boundary-layer flowCompressible flowIsentropic flow In the absence of viscous effects, a flow starting from rest will be irrotational for all subsequent time. Mentioned in? Encyclopedia browser?

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Select web site.The simplest element of potential flow, the uniform stream represents a uniform velocity field. A uniform stream is defined by constant velocity components U,V in 2D in cartesian coordinates:. This flow singularity represents a constant fluid flux from a point that results in a radial flow centered around the singularity.

## Potential Flow

A point source is defined by its constant magnitude of flux m, which has units of area per time in 2D and volume per time in 3D. The constant m is positive for a point source and negative for a point sink. The central point is a singularity where conservation of mass is violated and velocity goes to infinity. This flow singularity has a rotating velocity field around its central point. There is no vorticity in the flow outside of the central point despite the rotating particle paths, so the velocity field induced by this singularity is also known as an irrotational vortex.

This flow singularity occurs as the limit of the combination of a source and sink proof here. The central point is a singularity with infinite velocity. In 3D spherical coordinates it is similarly represented by a constant magnitude m units of length 4 per time. This flow element mimics the flow around a sharp corner. In the case of an exterior corner, the center of the flow is a singularity where the velocity goes to infinity. Potential Flow Simulator.It appears that any physical flow is generally three-dimensional.

But these are difficult to calculate and call for as much simplification as possible. This is achieved by ignoring changes to flow in any of the directions, thus reducing the complexity. It may be possible to reduce a three-dimensional problem to a two-dimensional one, even an one-dimensional one at times. Consider flow through a circular pipe. This flow is complex at the position where the flow enters the pipe. But as we proceed downstream the flow simplifies considerably and attains the state of a fully developed flow. A characteristic of this flow is that the velocity becomes invariant in the flow direction as shown in Fig. Velocity for this flow is given by 3. Now consider a flow through a diverging duct as shown in Fig.

Velocity at any location depends not only upon the radial distance but also on the x-distance. This is therefore a two-dimensional flow. Concept of a uniform flow is very handy in analysing fluid flows. A uniform flow is one where the velocity and other properties are constant independent of directions.

Figure 3.Fluid motion can be said to be a two-dimensional flow when the flow velocity at every point is parallel to a fixed plane. The velocity at any point on a given normal to that fixed plane should be constant. A line source is a line from which fluid appears and flows away on planes perpendicular to the line. When we consider 2-D flows on the perpendicular plane, a line source appears as a point source.

By symmetry, we can assume that the fluid flows radially outward from the source. Similar to a line source, a line sink is a line which absorbs fluid flowing towards it, from planes perpendicular to it. When we consider 2-D flows on the perpendicular plane, it appears as a point sink. By symmetry, we assume the fluid flows radially inwards towards the source.

A radially symmetrical flow field directed outwards from a common point is called a source flow. The central common point is the line source described above. As the fluid flows outward, the area of flow increases. As a result, to satisfy continuity equationthe velocity decreases and the streamlines spread out. The velocity at all points at a given distance from the source is the same.

We can derive the relation between flow rate and velocity of the flow. Consider a cylinder of unit height, coaxial with the source.

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The rate at which the source emits fluid should be equal to the rate at which fluid flows out of the surface of the cylinder. The stream function associated with source flow is —. The steady flow from a point source is irrotational, and can be derived from velocity potential.

The velocity potential is given by —. Sink flow is the opposite of source flow. The streamlines are radial, directed inwards to the line source.

As we get closer to the sink, area of flow decreases. In order to satisfy the continuity equationthe streamlines get bunched closer and the velocity increases as we get closer to the source.

As with source flow, the velocity at all points equidistant from the sink is equal. The stream function associated with sink flow is —. The flow around a line sink is irrotational and can be derived from velocity potential. The velocity potential around a sink can be given by —. A vortex is a region where the fluid flows around an imaginary axis.

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For an irrotational vortex, the flow at every point is such that a small particle placed there undergoes pure translation and does not rotate. Velocity varies inversely with radius in this case. The velocity is mathematically expressed as —. The stream function for irrotational vortices is given by —.

A doublet can be thought of as a combination of a source and a sink of equal strengths kept at an infinitesimally small distance apart.In fluid dynamicspotential 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 fieldwhich 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 equationand 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 aerofoilswater waveselectroosmotic flowand groundwater flow. For flows or parts thereof with strong vorticity effects, the potential flow approximation is not applicable.

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: .

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 layerspotential flow theory is not able to provide reasonable predictions of the flow.

For instance in: flow around aircraftgroundwater flowacousticswater wavesand electroosmotic flow. In case of an incompressible flow — for instance of a liquidor a gas at low Mach numbers ; but not for sound waves — the velocity v has zero divergence : .

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. In two dimensions, potential flow reduces to a very simple system that is analyzed using complex analysis see below. Potential flow theory can also be used to model irrotational compressible flow.

The full potential equationdescribing a steady flowis given by: . The full potential equation is valid for sub-trans- and supersonic flow at arbitrary angle of attackas long as the assumption of irrotationality is applicable.

So: . In that case, the linearized small-perturbation potential equation — an approximation to the full potential equation — can be used: .

This linear equation is much easier to solve than the full potential equation: it may be recast into Laplace's equation by a simple coordinate stretching in the x -direction. Multiplying and summing the momentum equation with v iand using the mass equation to eliminate the density gradient gives:.

Note that until this stage, no assumptions have been made regarding the flow besides that it is a steady flow.

Written out in components, the form given at the beginning of this section is obtained. Subsequently, together with adequate boundary conditions, the full potential equation can be solved most often through the use of a computational fluid dynamics code. The full potential equationdescribing a unsteady flow, is given by: .

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Consider the first term. Using Bernoulli's principle we way write. Small-amplitude sound waves can be approximated with the following potential-flow model: .

Potential flow does not include all the characteristics of flows that are encountered in the real world.

Mod-01 Lec-28 Two-dimensional potential flow

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