3.4: Kelvin’s circulation theorem

3.4: Kelvin’s circulation theorem For any flow governed by Euler’s equation , circulation round a closed chain of fluid particles is conserved. This is • Remarks : • A closed chain of fluid particles : • a loop which consists continuously of the same fluid particles; i.e. each element dl is moving with the fluid. • b) : has been extended slightly from its application at a single point to indicate that each point of the loop is moving with the fluid.

Prove the theorem : Using Euler’s eq. for the 1st term on RHS: Where, ( using the fact that the line integral about a closed loop of a perfect differential is zero.) 2nd term on RHS Since is a position vector,

Application of the Kelvin’s Circulation Theorem --------the generation of lift on an aerofoil The measured pressure shows that pressures above the wing are less than those below. From the Bernoulli Theorem for irrotational flow,

If pup<plow, u up>ulow must be satisfied. The Kutta-Joukowski Lift Theorem: The lift force perpendicular to the streamline is Where u is flow speed at infinity. So, and Why u up>ulow

KCT explains the formation of negative circulation Ґ At Consider a loop ‘abcda’ which is large enough to be away thin boundary layer on the aerofoil and away a thin wake

At t=t1, portition of curve ‘aecda’ has positive vorticity from votex and from Stokes’s theorem. And, from KCT, Therefore, from K-J Lift Theorem

3.5: The persistence of irrotational flow from Stoke’s theorem Hence, the vorticity flux through a close chain of fluid particles is conserved in inviscid flow. ----------- Cauchy---Lagrange theorem For a 2-D flow, vorticity eq. C-L theorem is obvious. For a 3-D flow , C-L theorem is not as obvious as in 2-D flow. (see Lec. 6) If at t=0, at is one solution for (2). If a portion of the fluid is initially in irrotational motion, that portion will always be in irrotational motion. C-L theorem

Physical explanation of C-L theorem : In Euler’s eq. (i.e. inviscid) , only stresses are , which act normally to the particle surface and cannot apply a couple (can’t generate torque as pressure act through the center of mass of an element) to the particle to bring it into rotation since i. HENCE, the study of inviscid motion may be reduced to the study of irrotational motion. For irrotational flow , velocity field can be expressed by velocity potential . And for incompressible fluid . ( ) (only when vorticity=0, can velocity write in velocity potential form) Stream function : for 2-D flow

is known as the stream function because it is constant along a streamline. On a streamline (i.e. definition). dy v Or dx u Streamline give a ‘snapshot’ of the velocity field at any instant (steady flow) . For 2_D irrotationality flow. We have Cauchy-Riemann conditions

Equipotential lines (on which is constant) and streamlines are orthogonal, as This demonstration fails at stagnation points where the velocity is zero.

3.4: Kelvin’s circulation theorem