Suction is one of the methods of boundary layer control, which have the aim of reducing drag on bodies in an external flow or of reducing losses of energy in channels. This method was suggested by L. Prandtl in 1904 as one of the means of preventing or "delaying" boundary layer separation. To effect suction, the surface should have holes (slots, porous sections, perforations etc.). These holes serve for sucking the portion of the boundary layer which is nearest to the wall and which is travelling at the lowest velocity. As a result, the boundary layer velocity profile becomes more "filled up" (see Boundary Layer) and thus is more stable as far as separation is concerned.
Suction is applied in practice for increasing the efficiency of diffusers with high compression ratio of the working fluid (with large convergence angles) by means of delaying early separation of the boundary layer. Boundary layer suction through slots near the trailing edge is used for increasing lift and decreasing drag of aerofoils operating at large incidence angles.
It has been demonstrated in practice that continuous suction through a porous wall is more effective than suction through slots. For example, for aerofoils, the same increase of lift force can be achieved by sucking a smaller amount of fluid through pores than through slots.
Suction is also an effective means of the boundary layer laminarization, which decreases friction losses (see Boundary Layer). The effect of suction on the laminar boundary layer stability is due to decrease of the boundary layer thickness (a thinner boundary layer is less liable to turbulization) and also due to the changes in the velocity profile (it becomes more filled).
Of practical importance for applying suction is the necessity to determine the minimum suction fluid necessary to keep the boundary layer laminar, because an excess of suction flow rate may result in such a power consumption that this would make insignificant the power economy achieved by decreasing the drag force. It is necessary that the suction rate V_{w} < 0 be small as compared with the external flow velocity U_{e}, namely, (V_{w}/U_{e}) = 0.0001 − 0.01 (Figure 1). In this case the assumptions of the boundary layer theory remain valid for the wall layer.
With uniform suction (V_{w}(x) = const) through a plate which is traversed (in the longitudinal direction) by an incompressible flow with velocity U_{e}, the system of boundary layer equations is written as (see nomenclature on Figure 1)
Under the following boundary conditions
and taking into account ∂U/∂x = 0 the following asymptotic solutions exist
For integral boundary layer thicknesses, namely, displacement thickness δ* and momentum loss thickness δ** (see Boundary Layer) we obtain the following relationships:
where δ*, δ** are independent of distance x along the boundary layer.
The shear stress τ_{w} at the wall and the plate drag C_{f} coefficient are determined from the relationships
and do not depend in the flow viscosity. The boundary layer thickness on the plate leading edge is equal to zero, but it grows asymptotically in the downstream direction (see Figure 1) approaching the values corresponding to Eq. (4). The velocity profile shape corresponding to Eq. (3) is also approached asymptotically along the initial section.
The asymptotic velocity profile corresponding to Eq. (3) is formed at the end of the initial section:
The boundary layer described by Eqs. (3)-(5) is termed an "asymptotic suction layer".
The asymptotic velocity profile (3) is more "filled up" than the Blasius profile characteristic for the case of the flow over a plate without suction, and also than for flow at the initial section near the plate edge when there is suction. An asymptotic solution is also possible for a plate in a gas flow. In this case the velocity profile is described by Eq. (7):
where y_{1} = _{e} dy.
The shear stress is calculated from Eq. (8)
The integral momentum relationship in the case under consideration is obtained in the similar way as that for the boundary layer without suction and is described by the equation
For solution of Eq. (9) (see Boundary Layer). The term V_{w}U_{e} in the right hand side of this equation takes into account the momentum variations caused by suction.
In 1935, from Eq. (9) L. Prandtl obtained an approximate relationship Eq. (10) for the minimum suction velocity V_{w min separ} at which separation is prevented. He assumed that at all points of the surface the velocity profile corresponds to the conditions such that τ_{w} = 0. It followed that
The practically important problem of determining minimum suction flow rate V_{w min lam} for which the laminar flow condition is preserved can be solved using the above asymptotic solution for the boundary-layer near the plate.
K. Bussman and H. Munz in 1942 calculated the laminar boundary layer stability at the plate with an asymptotic velocity profile. As a result they determined the critical value of a Reynolds number when the boundary layer becomes turbulent
This value is 100 times larger than the corresponding Re_{c} for the plate without suction, which is the proof of the efficiency of this process for stabilizing laminar boundary layers.
Using Eq. (4) for δ* and the value of Re_{c} we obtain the condition of the laminar boundary layer stability with suction
Actually, the suction intensity must be somewhat higher than that determined by Eq. (11) because this equation was derived under the assumption of existence of an asymptotic velocity profile (Eq. (3)) starting from the plate leading edge. In the initial section of the boundary layer the Blasius velocity profiles have a smaller stability limit and therefore more intensive suction is required for preserving the laminar boundary layer in this section.
REFERENCES
Prandtl, L. (1904) Über Flussigkeitsbewegung bei sehr Kleiner Reibung: Verhandl, III int, Math, Kongr, Heidelberg.
Prandtl, L. (1935) The mechanics of viscous fluid, Durand W. F. Aerodynamic Theory III.
Bussman, K. and Munz, H. (1942) Die Stabilitat der laminaren Reibungsschicht mit Absangung. Sb. at. Luftfahrtforschung 1, 36-39.
References
- Prandtl, L. (1904) Ãœber Flussigkeitsbewegung bei sehr Kleiner Reibung: Verhandl, III int, Math, Kongr, Heidelberg.
- Prandtl, L. (1935) The mechanics of viscous fluid, Durand W. F. Aerodynamic Theory III.
- Bussman, K. and Munz, H. (1942) Die Stabilitat der laminaren Reibungsschicht mit Absangung. Sb. at. Luftfahrtforschung 1, 36-39.