ESTIMATION OF BOUNDARY LAYER THICKNESS.

the thickness of a boundary layer which has not separated can be easily estimated in the following way. whereas friction forces can be negleted with respect to inertia forces outside the boundary layer, owing to low viscosity, they are of a comparable order of magnitude inside it. the force per unit volume is, as explained in sectio I e is equal to -----. for a plate of lenght l the gradient ---- is proportional to U/l, where U denotes the velocity outside the boudary layer. hence the inertia force is of the order ----. on the other hand the friction force per unit volume is equal to---, which , on the assumption of laminar flow, is equal to ----. the velocity gradient --- in a direction perpendicular to the wall is of the order --- so that the friction force per unit volume is ---. from the condition of equality of the friction and inertia forces the following relation is obtained :

-----

or, solving for the boudary layer thickness -- :

-----

hence for laminar flow in the boundary layer we have :

----

the dimensionless boundary-layer thickness, referred to the length of the plate, l, becomes :

----

where -- denotes the reynolds number related to the lenght of the plate, l. it is seen from eq. 2.1 that the boundary layer thickness is proportional to --- and to --. if l is replaced by the variable distance x from the leading edge of the plate, it is seen that -- increaseas proportinately to ---. on the other hand the relative boundary layer thickness--- decreases with increasing reynolds number as --- so that in the limiting case of frictionless flow, with ----, the boundary layer thickness vanishes.

we are now in a position to estimate the shearing stress --- on the wall, and consequently, the total drag. according to newton's law friction. we have

---

where subscript 0 denotes the value at the wall, ---. with the estimate ---- we obtain --- and, inserting the value of -- from eq.. we have

---

thus, the frictional stress near the wall is proportional to ---

we can now form a dimensionless stress with reference to ---, as explained in chap. i and obtain

-----

this result agrees with the dimensionless analysis in chap. 1, which prdicted that the dimensionless shearing stress could depend on the reynold number only.

the total drag D on the plate is equal to ---, where b denotes the widht of the plate. hence, eith the aid of eq. 2.3 we obtain

----

the laminar frictional drag is thus seen to be proportional to --- and ---. proportionality to -- means that doubling the plate length doesnt double the drag, and this result can be understood by considering that the downstream part of the plate experiences a smaller drag than the leading portion because the boundary layer thicker towards the trailing edge. finally, we can write down an expression for the dimensionless drag coefficient in accordance with eq. 114. hence eq 2.4 gives that :

----

the numerical factor follows from II blassius exact solution, and is ---, so that the drag of a plate in parallel laminar flow becomes

---

DEFENITION OF BOUNDARY LAYER THICKNESS :

the definition of the boundary layer thickness is to a certain extent arbitary because transition from the velocity in the boundary to that outside it takes place asymptotically. this is, howefer, of no practical importance, because the velocity in the boundary layer attains a value which is very close to the external velocity already at a small distance from the wall. it is possible to define the boundary layer thickness as that, distance from the wall where the velocity differs by 1 per cent from the external velocity. with this definition the numerical factor in eq (2.2). has the value 5. instead of the boundary-layer thickness, another quantity. the displacement thickness --, is sometimes used, fig. 2.3. it is defined by the equation :

SEPARATION AND VORTEX FORMATION

the boundary layer near a flat plate in parallel flow and at zero incidence is particulary simple, because the static pressure remains constant in the whole filed of flow. since outside the boundary layer the velocity remains constant in the same applies to the pressure bevause in the frictionless flow bernoulli's equation remains valid. furthermore, the pressure remains sensibly constant over the width of the boundary layer at a given distance x. hence the pressure over the width of the boundary layer has the same applies magnitude as outside the boundary layer at the same distance, and the same applies to cases of arbitrart body shapes when the pressure outside the boundary layer various along the wall with the length of are. this fact is expressed by saying that the external pressure is impressed on the

boundary layer. hence in the case of the motion past a plate the pressure remains constant throughout the boundary layer.

the phenomenon of boundary layer separation mentioned previously is intimately connected with the pressure distribution in the boundary layer. in the boundary layer on a plate no separation takes place as no back flow occurs.

in order to explain the very important phenomenon of boundary layer separation let us consider the flow about a blunt body, e.g. about a circular cylinder, as shown in fig. 2.4 . in frictionless flow, the fluid particles are accelerated on the upstream half from D to E, and decelerated on the downstream half from E to F. hence the pressure decreases from D to E increases from E to F. when the flow is started up the motion in the first instant is very nearly frictionless, and remains so as long as the boundary layer remains thin. outside the boundary layer there is a transformation of pressure into kinetic energy along D E, the reserve taking place along E F, so that a particle arrives at F with the same velocity asit had at D. A fluid particle which moves in the immediate vicinity of the wall in the boundary layer remains under the influence of the same pressure field as that exiting outside, because the external pressure is impressed on the boundary layer. owing to the large friction forces in the thin boundary layer such a particle consumes so much of its kinetic energy on its path from D to E that the remainder is too small to surmount the "pressure hill" from E to F. such a particle cannot move far into rhe region of increasing pressure between E and F and its motions is, eventually, arrested. the external pressure causes it then to move in the opposite directon.

SEPARATION

the boundary layer theory succeeds in this manner, with the aid of the phenomenon of separation. in throwing light on the occurence of pressure or form drag in addtion to viscous drag. the danger of boundary layer separation exists always in region with an adverse pressure gradient and the likehood of its occurence increases in the case of steep pressure curves i.e behind bodies with bluent end.

the streamlines in the boundary layer near separation are shown diagrammatically in fig. 2.12. owing to the reversal of the flow there is a considerable thickening of the boundary layer. and associated with it. there is a flow of boundary layer material into he outside region. at the point of separation one streamline intersect the wall at a definite angle, and the point of separation itself is determined by the condition that the velocity gradient normal to the wall vanishes there :

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c. turbulent flow in a pipe and in a boundary layer

measurements show that the type of motion through a circular pipe which was calculated in section 1d