Kelvin's Theorem

From Euler's equation

This is called Kelvin's Theorem. It relates the rate of change of the total circulation to the line integral of the (pressure change divided by density ) around a closed contour. The integral on the r.h.s. vanishes if either of the following holds:

a) density is constant: r = constant. Note that the condition is that density is constant. If the flow Mach number is below 0.3, we can neglect density changes due to velocity changes: this is the assumption of "incompressible flow". However, if there are temperature changes due to other reasons (e.g., hot air rising), there can be density differences within the closed contour of integration.

b) p = f(r) This is called "barotropic". It means that pressure is a function only of density, i.e, there is a unique relation between pressure and density everywhere inside the closed contour. One counter-examples to this is a case where there is a shock. The isentropic cases is a special case of barotropic, where p = r^(g/(g-1)).

Using Stokes' Theorem,

Thus,

implies:

This means that if a packet of fluid starts out in an incompressible or barotropic flow, with a set amount of circulation, and does not encounter shocks or heat transfer, then its circulation will not change.

Kelvin's theorem for incompressible, barotropic flow:

The time rate of change of circulation around a closed contour consisting of the same fluid elements is zero.

i.e.,

Example: Airfoil starts moving at time t = 0.

Derivation of the Steady Bernoulli Equation from the Euler equation, integrated along a streamline:

Steady:

Multiply the u-component equation by dx:

A Streamline is defined as a locus of points tangent to the flow direction:

Along a streamline,

Using these relations, the u-component equation above becomes:

, so that

Along a streamline in steady incompressible flow.

Pressure Coefficient

where

Using the Bernoulli equation, this can also be written as

Characteristics of the figure above:

1. Leading edge stagnation point: pressure coefficient is +1.0
2. Near the leading edge, and downstream of the stagnation point, the flow acccelerates beyond the freestream velocity, so that the pressure coefficient becomes negative on both the upper and lower surfaces.
3. The point of minimum pressure is upstream of 1/4-chord for most airfoils.
4. The pressure coefficient increases again, and becomes positive (flow slower than freestream) over the aft portion of the airfoil, both surfaces.
5. At the trailing edge, there is again stagnation. If we neglect all viscous losses, the pressure coefficient at the trailing edge should this reach +1.0 again. However, due to viscous losses, some of the stagnation pressure is lost, so that the pressure coefficient at the trailing edge stagnation is less than +1.0
6. Note that the pressure distributions on the upper and lower surfaces are quite similar. There is a small difference between them, either because of the angle of attack or the camber, or due to both. This difference is what gives the lift.

Kutta-Jowkowski Theorem for steady flow

Reference: Eskinazi, S., "Vector Mechanics of Fluids and Magnetofluids". Academic Press, NY 1967, p.284.

Apply Newton's 2nd law of motion to steady flow across a control surface:

From the Bernoulli equation for steady flow,

Since p₀ is constant,

Use the divergence theorem:

Note that

. Substituting,

Note that: is the Dilatation.

is the Vorticity.

If the flow is constant-density, the dilatation is zero. Note: even if it is incompressible flow (i.e., Mach number is so low that density changes due to velocity changes are negligible), there can be density changes due to heating or species gradients.

Thus the Kutta-Jowkwski theorem in steady constant-density flow is:

or,

This acts perpendicular to both the velocity vector and to , the vorticity vector.

Notes:

1. The Lift, which is the force perpendicular to the freestream velocity vector, results from the combination of vorticity and a freestream.

2. Force can come from vorticity and dilatation. The resultant force in compressible flow may not be perpendicular to the freestream velocity vector.

3. If we can somehow get the correct value of vorticity, we can calculate the lift without explicity considering viscosity.

POTENTIAL FLOW

Objective:

Get a method to describe flow velocity fields and relate them to surface shapes consistently. Once the velocity field is known, the pressures and hence the loads can be calculated.

Strategy:

Describe the flowfield as the effect of variations in one scalar quantity.

Consider the vector identity:

, i.e., The curl of the gradient of a scalar function is zero.

Thus, if we have a vector relation of the form

it should be possible to express the vector u as the gradient of the scalar function f.

We defined "vorticity" as

. This is a measure of rotation in the flow; in fact the vorticity is half the angular velocity.

Thus, implies "irrotational flow".

Thus if we have irrotational flow, the vector u can describe the gradient of some scalar function f. This function is called the "velocity potential. Thus,

or, ; ;