Elementary irrotational plane flows

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In the following, few examples of elementary irrotational plane flows are shown. Due to the linearity of the Laplace equation, superposition of these simple flow allows for the solution of more complex flow

In all the examples, equipotential and streamlines are shown in yellow and magenta, respectively. These two families of curves always intersect at right angles

 

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THE UNIFORM FLOW

 

 

The first and simplest example is that of a uniform flow with velocity U directed along the x axis

 

 

 

In this case the complex potential is

and the streamlines are all parallel to the velocity direction (which is the x axis). Equipotential lines are obviously parallel to the y axis

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THE SOURCE OR SINK

 

 

The second example is that of a source (or sink), the complex potential of which is

This is a pure radial flow, in which all the streamlines converge at the origin, where there is a singularity due to the fact that continuity can not be satisfied: at the origin there is an input (source, m > 0) or output (sink, m

 

 

The thick magenta line on the left is related to the fact that the complex potential is, in this case, a multi-valued function of space. At any fixed point, the potential is known up to a constant, the so-called cyclic constant, that in this case has the value of . Since, in general, the potential is defined up to a constant, the fact that it is a multi-valued function of space does not create any problem in the determination of the flow field, which is uniquely determined upon deriving the complex potential W with respect to z

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THE VORTEX

 

 

In the case of a vortex, the flow field is purely tangential

 

 

The picture is similar to that of a source but streamlines and equipotential lines are reversed

The complex potential is

There is again a singularity at the origin, this time associated to the fact that the circulation along any closed curve including the origin is nonzero and equal to . If the closed curve does not include the origin, the circulation will be zero

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THE DIPOLE

 

 

We now analyze the case of the so-called hydrodynamic dipole, that results from the superposition of a source and a sink of equal intensity placed symmetrically with respect to the origin. The analogy with electromagnetism is evident. The magnetic field induced by a wire in which a current flows satisfies equations that are similar to those governing irrotational plane flows

The complex potential of a dipole is

if the source and the sink are positioned in (-a,0) and (a,0) respectively

 

 

Streamlines are circles, the center of which lie on the y-axis and they converge obviously at the source and at the sink. Equipotential lines are circles, the center of which lie on the x-axis

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THE DOUBLET

 

A particular case of dipole is the so-called doublet, in which the quantity a tends to zero so that the source and sink both move towards the origin

 

 

 

 

The complex potential of a doublet

is obtained making the limit of the dipole potential for vanishing a with the constraint that the intensity of the source and the sink must correspondingly tend to infinity as a approaches zero, the quantity

being constant (if we just superimpose a source and sink at the origin the resulting potential would be W=0)

Hint: Develop and in a Taylor series in the neighborhood of the origin, assuming small a

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FLOW AROUND A CYLINDER

 

 

The superposition of a doublet and a uniform flow gives the complex potential

that is represented here in terms of streamlines and equipotential lines

 

Note that one of the streamlines is closed and surrounds the origin at a constant distance equal to

Recalling the fact that, by definition, a streamline cannot be crossed by the fluid, this complex potential represents the irrotational flow around a cylinder of radius R approached by a uniform flow with velocity U

Moving away from the body, the effect of the doublet decreases so that far from the cylinder we find, as expected, the undisturbed uniform flow

In the two intersections of the x-axis with the cylinder, the velocity is found to be zero. These two points are thus called stagnation points

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FLOW AROUND A CYLINDER WITH NONZERO CIRCULATION

 

 

 

 

In the last example, we superimpose to the complex potential that gives the flow around a cylinder a vortex of intensity positioned at the center of the cylinder. The resulting potential is

The presence of the vortex does not alter the streamline describing the cylinder, while the two stagnation points below the x-axis

 

 

The streamlines are closer to each other on the upper part of the cylinder and more distant on the lower part. This indicates that the flow is accelerated on the upper face of the cylinder and decelerated on the lower part, with respect to the zero circulation case

The resulting flow field corresponds to the case of a rotating cylinder, which accelerates (with respect to the case of no circulation) fluid particles on part of the cylinder and decelerates them on the remainder of the cylinder

Note the presence of a discontinuity in the potential function (thick yellow line on the left) that is related to the fact that the vortex potential (as mentioned in a previous section) has a nonzero cyclic constant