COMPUTATION OF PRESSURES

Solution of Eq. (13.12) will provide the velocity potential and the velocity components. Once the flow field is determined the resulting pressures can be computed by the Bernoulli equation. In the inertial frame of reference this equation is (note that [УФ]2 = УФ • УФ = [и2 + v2 + w2])

(in X, Y, Z coordinates)

The magnitude of the velocity УФ is independent of the frame of reference (only the resolution of the velocity vector into its components is affected) and therefore the form of the first term in this equation remains unchanged. The

time derivative of the velocity potential, however, is affected by the frame of reference and must be evaluated by using Eq. (13.11); therefore, the pressure difference p*. — p will have the form:131

(in x, y, z coordinates)

In the case of three-dimensional panel methods it is often simpler to use the instantaneous Bernoulli equation, in its original form (Eq. (2.35)):

Prct ~ P = Q2 vjt | ЗФ p 2 2 dt

Here Q and p are the local fluid velocity and pressure values and pref is the far-field reference pressure and uref is the magnitude of the kinematic velocity as given in Eq. (13.8):

vref= -[Vo + JlXr] (13.25)

It is often convenient to express this kinematic velocity in the direction of the moving x, y, z frame as [£/(t), V(t), W(f)], which can be obtained by a simple transformation fx (which is a function of the momentary rotation angles ф, в, rp, resembling Eq. (13.7o))

U j "ref,

v І =МФ, в, VO vrefy w) urefz

The total velocity at an arbitrary point on the body (or collocation point к in the case of a numerical solution) is the sum of the local kinematic velocity (e. g., the reference velocity in Eq. (13.25)) plus the perturbation velocity

Qa (Vref/> "ref,,,> "ref,)*: (ф» Яті Яп)к (13.27)

C — P Pref 1 ‘*■’p 12 2P"ref

where (l, m, n)k are the local tangential and normal directions (see Fig. 9.10) and the components of vref in these directions are obtained by a transformation similar to Eq. (13.26). The local perturbation velocity is (q,, qm, qn) = (ЗФ/dl, ЭФ! Эт, дФ/дп) and of course the normal velocity component on the solid body is zero. The pressure coefficient can now be computed for each panel as

or if we use Eq. (13.23) then the pressure coefficient becomes:

Note that in situations such as the forward flight of a helicopter rotor uref can be selected as the forward flight speed or the local blade speed at each section on the rotor blade. Consequently different values of the pressure coefficient will be obtained—and this matter is usually left to be settled by the programmer. The contribution of an element with an area of ASk to the aerodynamic loads AF* is

AF* = – CPt(pv2ref)k AS* n* (13.29)

(Note that ure{ here has a subscript k, which means that it depends on the body coordinates. This is usually not recommended, but may be used in cases such as the forward flight of a helicopter rotor).

Once the potential field and the velocity field are obtained, the corresponding pressure field is calculated using Eq. (13.28a) and additional information such as forces, moments, surface velocity surveys, etc., can be obtained.