Solving the Conservation Equations

In this section, we continue the quest started in Section 4.3 for special forms of the governing equations that can be solved using standard mathematical tools. We found that a special form of the momentum equation could be solved by simple integration leading to the powerful Bernoulli Equation. We continue our examination of the governing equations for the case of incompressible steady flow.

Laplace’s Equation

Return now to the continuity equation, Eq. 3.52, which is one of the defining equations for a two-dimensional incompressible-flow problem. The continuity equation may be written in terms of a single dependent variable—either the velocity potential or the stream function—provided that the flow is irrotational. To show this in both

Cartesian and polar coordinates, write the continuity equation in vector form, as in Eq. 3.52—namely, V-V = 0. Now, for an irrotational flow, V = Уф, and substituting this into the continuity equation yields:

Подпись: (4.24)V – (Уф) = У[13] 2ф = 0.

Eq. 4.24 is Laplace’s Equation, an equation that appears in many fields including heat transfer and electrodynamics. Eq. 4.24 in Cartesian coordinates is:

Подпись: (4.25)дх2 ду2

1 дф+1 дФ.

r Э0[14] r dr

Подпись: у2ф=дА + ^^ + ^ї. = о. y dr2 -2 3rt2 ~ 0- Подпись: (4.26)

and in polar coordinates, Laplace’s Equation is:

These two equations each represent a single equation in terms of a single dependent variable, which suggests that a solution is possible.

Regarding the stream function, recall that for irrotational flow, it follows from

Eq. 4.3 that — = — and, from the definition of the stream function, Eq. 4.11, that:

ду Эх

дш, дш

и = —^ and v = -—!-.

ду дх

Substituting the second relation into the first yields:

їш + ЇШ ш = о, (4.27)

дх2 ду2

which is the Laplace’s Equation in Cartesian coordinates.

A parallel development for the stream function shows that it satisfies the Laplace’s Equation in cylindrical coordinates—namely:

Подпись: (4.28)д2 ш+1 д! ш+1 эу=0

дг2 г2 д02 r дг ■

Notice that the original set of three simultaneous partial-differential equations describing a two-dimensional, incompressible, inviscid flow was replaced by the Laplace’s Equation and a scalar equation, the Bernoulli Equation. This represents a considerable mathematical advantage because it is not necessary to deal with any vector variables. A strategy for predicting the pressure distribution in a steady two­dimensional inviscid, irrotational flow then is as follows:

3. Knowing the velocity components, determine the magnitude of the local flow velocity.

4. Use the Bernoulli Equation to solve for the static pressure at any point in the flow field. The stagnation pressure is known or can be measured for a given flow.