Pressure distribution potential flow

Previous workers have also made use of potential flow pressure distributions about spheroids, but no allowance was made for lack of fore-and-aft symmetry, while the constant pressure condition was satisfied only near the front stagnation point (SI) or at the equator and poles (H6, Mil). 

Actual shapes of fluid particles deviate from the idealized shape which leads to Eqs. (8-15) and (8-16). Surface pressure distributions derived from observed shapes (W2) are shown in Fig. 8.3 for spherical-cap bubbles at high Re. It is seen that the pressure variation is well described by Eq. (8-15) for 0 < 0 < while the potential flow pressure distribution, Eq. (1-32), gives good agreement up to about 30° from the nose. 

This quantity was plotted in Fig. 10-3.21 Also shown is the measured pressure distribution for Re in the range 30 to 40. It is evident that the measured and predicted distributions are in close agreement over the front portion of the cylinder. Thus it is not surprising that boundary-layer theory, based on the potential-flow pressure distribution, should be quite accurate up to the vicinity of the separation point. It is this fact that explains the ability of boundary-layer theory to provide a reasonable estimate of the onset point for separation, as we shall demonstrate shortly. 

In other words, the pressure distribution in the boundary-layer is completely determined at this level of approximation by the limiting form of the pressure distribution impressed at its outer edge by the potential flow. It is convenient to express this distribution in terms of the potential-flow velocity distribution. In particular, let us define the tangential velocity function ue(x) as... 

For potential flow, ie incompressible, irrotational flow, the velocity field can be found by solving Laplace s equation for the velocity potential then differentiating the potential to find the velocity components. Use of Bernoulli s equation then allows the pressure distribution to be determined. It should be noted that the no-slip boundary condition cannot be imposed for potential flow. 

Fig. 5.28 Distribution of dimensionless modified pressure at surface of spheres at Re = 100, compared with potential flow distribution. (A) Potential flow (p, — Px)/ipU = 1 — 2.25sin fl (B) Rigid sphere (L5) (C) Water drop in air k = 55, y = 790 (L9) (D) Gas bubble k — y 0 (H6).

Consider a bubble rising in a fluidized bed. It is assumed that the bubble is solids-free, is spherical, and has a constant internal pressure. Moreover, the emulsion phase is assumed to be a pseudocontinuum, incompressible, and inviscid single fluid with an apparent density of pp(l — amf) + pamf. It should be noted that the assumption of incompressibility of the mixture is not strictly valid as voidage in the vicinity of the bubble is higher than that in the emulsion phase [Jackson, 1963 Yates et al., 1994]. With these assumptions, the velocity and pressure distributions of the fluid in a uniform potential flow field around a bubble, as portrayed by Fig. 9.10, can be given as [Davidson and Harrison, 1963]... 

These are known as the boundary-layer equations. There is one further simplification that we can always introduce. In particular, if we integrate (10 35), we see that the pressure distribution in the boundary layer is a function of x only. Thus the pressure gradient in the boundary layer (dp/dx) is also independent of Y and must have the same form as the pressure gradient in the outer potential flow, evaluated in the limit as we approach the body surface, namely,... 

Re < 40, and the predicted pressure distribution from potential-flow theory. The ability of the boundary-layer theory to predict separation is probably its most important characteristic. 

Experimental observations of the flow past a circular cylinder show that separation does indeed occur, with a separation point at 0S — 110 . It should be noted, however, that steady recirculating wakes can be achieved, even with artificial stabilization,24 only up to Re 200, and it is not clear that the separation angle has yet achieved an asymptotic (Re —> oo) value at this large, but finite, Reynolds number. In any case, we should not expect the separation point to be predicted too accurately because it is based on the pressure distribution for an unseparated potential flow, and this becomes increasingly inaccurate as the separation point is approached. The important fact is that the boundary-layer analysis does provide a method to predict whether separation should be expected for a body of specified shape. This is a major accomplishment, as has already been pointed out. 

An example of the pressure distribution predicted by potential-flow theory and the experimentally measured distribution for a case in which the boundary-layer separates was shown in Fig. 10-3. 

To illustrate the idea of potential flow and how to use it to calculate forces, let us calculate the pressure distribution on the surface of a cylinder which is immersed in a flow perpendicular to it. If this is a very long cylinder, then there will be negligible change in the flow in the direction of the cylinder s axis, and so the flow will be practically two-dimensional. To find the flow field, we must make a judicious combination of a steady flow, a source, and a sink. Consider first a source and a sink with equal flow rates located some distance A apart on the X axis See Fig. 10.16. The flow between them is given by... 

The fluid pressure distribution at the gas-liquid interface in the potential flow regime is... 

The influence of an external electric field and/or of an external pressure is studied in Sec. Ill, The equations that govern the perturbations of the ion distribution, the electric potential, the pressure, and the velocity are derived. They can be further simplified and linearized when the external electric field is small with respect to the local field induced by the zeta potentials and when the ionic distribution is not disturbed by the flow [24,25], The linearized equations are made dimensionless by an adequate choice of basic units. 

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