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Boundary Conditions
Equation 2.15 as a fundamental equation is solved with the proper boundary conditions. In general the external flow problems will be studied. Therefore, we need to impose the boundary conditions accordingly as follows.
i. At infinity, all disturbances must die out and only free stream conditions prevail.
ii. The time dependent boundary conditions at the body surface must be given as the time dependent motion of the body.
The equation of a surface for a 3-D moving body in Cartesian coordinate system is given as follows
B(x, y, z, t) = 0
Let us take the material derivative of this surface in the flow field q = ui + vj + wk.
DB 0B 0B OB 0B „
= +u + v + w =0
Dt 0t Ox 0y 0z
For the steady flow it simplifies to
0B 0B 0B
u + v + w = 0 0x 0y 0z
The external flows studied here require to find the pressure distribution at the lower and upper surfaces of the body immersed in a free stream. For this purpose,
we need to know the upper and lower surface equations of a body in a free stream in x direction. If we show the direction normal to the flow with z, then the single valued surface equation, with the aid of Eq. 2.16, reads as
B(x, y, z, t) = z – za(x, y, t) = 0 (2.18)
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Now, we can take the material derivative of Eq. 2.18 with the aid of Eq. 2.17
Note that, OB = 1 is used for the convective term in z direction. Here, the explicit expression of vertical velocity component w is named ‘downwash’ in aerodynamics. This downwash at the near wake is the indicative of the lifting force on the body. The direction of the force and the downwash are the same but their senses are opposite. Accordingly, for the downward downwash the force is then upward. In other words, downward velocity component at the wake region creates a clockwise circulation which in turn generates the lifting force together with the free stream.
Equations 2.15 and 2.19 are not linear. In order to solve those equations together, linearization is necessary. Once the equations are linearized we can also employ the superpositioning technique for solving them.
