Thomson Theorem about Circulation Constancy

Let's allocate closed contour (see fig. 2.17) within frictionless liquid, which movement can be considered as barotropic. The contour consists of the same liquid particles. Such liquid contour moves together with the liquid and changes its shape together with the liquid.

Let's take arch of finite size on the contour. Velocity circulation over arch can be represented as (2.48)

. (3.62)

Let's determine . Differentiating equation (3.62), we find

.Carrying out the differentiation of the subintegral expression and taking into account that the contour is liquid, we shall have

;

;

.Let's substitute the found values with derivatives under the integral:

.

Expression in the last brackets represents itself . Considering that body forces have potential, and the liquid is barotropic, we shall write down the Eulers equations (3.5) in the following form:

;

; (3.63)

.

Replacing the projections of the total acceleration , , with their values from the differential equations of motion (3.63), we shall have

or

,from here

. (3.64)

As the values in brackets, i.e. , and , are single-valued functions, then for closed contour, i.e. when points and coincide, the expression

or .Thus, the velocity circulation will not depend on time. This result received by Thomson can be formulated in the form of the following theorem: the velocity circulation over the closed contour, which passes through the same particles of frictionless liquid does not change in time, with presence of the body forces possessing the single-valued potential and barotropy.It follows from the Thomson theorem, that if frictionless liquid starts moving from quiesant state and is continuous, then the velocity circulation over arbitrary closed contour within flow will be equal to zero because at the initial moment of time it was equal to zero. Such flow, as follows from the Stokes theorem, will be vortex-free, and the field of velocities will be potential.

Hence, the velocity circulation can appear in a flow of frictionless liquid under condition of the potentiality of the body forces and presence of barotropy, as it follows from equality (3.64) only when either function of pressure , or flow velocity has a jump on some surfaces, and due to this reason the difference (3.64) differs from zero.

Let's consider an arbitrary wing airfoil as an example. At the initial moment of movement the liquid is motionless and the velocity circulation at this moment on any closed contour drawn within liquid, is equal to zero, including the liquid contour covering the airfoil (fig. 3.11,).

With the increasing of the movement the liquid flows, passing around the wing, start to come down from its trailing edge. If the velocity of the flow coming down on this edge will have finite size and not equal to zero, then liquid contour will start to stretch and drift into the flow by one part, as it is shown in fig. 3.11,b.

abFig. 3.11. Occurrence of the circulation around the wing airfoil:a the liquid is motionless along the airfoil; b the liquid moves along the airfoil

Let's cut the stretched contour at the airfoil trailing edge into two parts: contour , covering the airfoil, and contour , came down from the airfoil. According to the theorem of velocity circulation change over the closed liquid contour proved earlier we shall have during all time of movement

,

where is the velocity circulation over contour ; is the velocity circulation over contour .It follows from here, that at any moment of time

.

Thus, the velocity circulation over contour , i.e. over the airfoil contour will always be equal to the velocity circulation taken with the opposite sign over contour , came down from the airfoil to the flow after the beginning of the movement.

Let's notice, that the velocity circulation over contour will differ from zero only in case when tangent velocities have a jump on this contour when transitioning one contour side to opposite one along the normal. Such jump of tangent velocities behind the airfoil placed in frictionless liquid can be created if the airfoil trailing edge is sharp. Then, with increasing of the movement, the flows moving from below of the wing will approach to the aft edge with the smaller velocity, rather than the flows moving from above, this is the reason of jump of tangent velocities created at the junction of flows.

Increasing of circulation on the airfoil will continue with the increasing movement of the flow incoming the airfoil. The line of the jumps of velocities created during this time is curling up into a spiral behind the airfoil. This vortex spiral is carried away by the flow far from the airfoil, after that the motion near the airfoil becomes steady.

When movement of the flow near the airfoil became steady, the lift will be defined using circulation according to the Zhukovsky theorem. Let's notice in addition, that formation of vortexes behind the airfoil occurs not only with appearance of the circulation on the airfoil , but also with its disappearance, but in the last case the vortex of an opposite sign will be created behind the airfoil.