33. Local disturbances in laminar flow

All the foregoing in this chapter referred to local disturbances in turbulent flow. In laminar flow local disturbances are usually insig­nificant as compared with friction and, secondly, the disturbance laws are much more complicated and considerably less studied than in the case of turbulent flow.

In the latter regime of flow local losses can be assumed proportion­al to the square of the velocity (rate of discharge), the loss coeffici­ents being determined mainly by the local feature causing the dis­turbance and being practically independent of the Reynolds number. In laminar flow, however, the loss of head h_t at a local feature must be taken as the sum

(8.16)

where h, = loss of head due to friction forces (viscosity) in the lo­cal feature and proportional to viscosity and, the first power of the velocity; h_t = loss due to flow separation and turbulence in the local feature or immediately down­stream, which is proportional to the square of the velocity.

Thus, for example, in the case of flow through a calibrated jet (Fig. 78) the loss of head upstream from section 1-1 is due to friction, and downstream, to turbu­lence.

Taking into account the resistance law for laminar flow, Eqs (6.6) and (6.7) with the correction for entrance length, and also Eq. (4.17), the sum can be rewritten as follows

(8.16)

where A and В are dimensionless constants depending on the shape of the local feature.

Dividing the expression (8.16) by the velocity head, we obtain the general expression of local loss coefficient in laminar flow:

(8.17)

The ratio between the first and second terms in Eqs (8.16) and (8.17) depends on the shape of the local feature and the Reynolds number.

When the local feature is a thin tube substantially longer than it is wide, with a rounded inlet and outlet, as shown in Fig. 79a, and small values of Re, the loss of head is determined mainly by friction, and the resistance law is almost linear. In such cases the second term in Eqs (8.16) and (8.17) is zero or very small in compari­son with the first one.

If, on the other hand, friction in the local feature is negligible, as in the case of the sharp edge in Fig. 796, the stream breaks away and eddies form, Re is relatively high and the loss of head is approxi­mately proportional to the second power of the velocity (and rate of discharge).

When the Reynolds number varies over a wide range in a local feature, the local disturbance may be described by a linear law (when Re is low), a quadratic law (when Re is high) and an inter­mediate law for transitional values of Re. A typical plot of С as a function of Re in logarithmic coordinates is presented in Fig. 80 for a broad band of Reynolds numbers through four different orifices. The sloped straight lines correspond to a linear resistance law (when ζ varies inversely as Re), the curved portions represent the transient regions, and the horizontals denote a quadratic law, when ζ does not depend on Re. Such graphs for specific local disturbances are usually plotted empirically.

Sometimes the binomial expression of minor losses is replaced by an exponential monomial of the form

where к = dimensional quantity;

m — exponent varying from 1 to 2 depending on shape of

local feature and Reynolds number.

When the resistance law is nearly linear for a local feature and given Reynolds number minor losses are frequently expressed in terms of equivalent pipe lengths: to the actual length of a pipe is added a length whose resistance is equal to that of the local features. Thus,

(8.18)

and

(8.19)

The equivalent length (referred to the pipe diameter) is usually determined experimentally for different local features.

The formula for the loss of head in an abrupt expansion developed in Sec. 29 for turbulent flow is not valid for laminar flow. In laminar flow the assumptions made in developing the formula, viz., uniform velocity distribution across sections 1-1 and 2-2, uniform pressure across the area S₂ at section 1-1 _y and absence of shearing stress, are no longer tenable.

Recent laboratory experiments show that the loss coefficient in an abrupt enlargement when the Reynolds number is very low (Re<9) hardly depends on the area ratio and is determined mainly by Re, the equation being of the form

This means that no separation takes place and the divergence loss is proportional to the first power of the velocity. At 9 < Re < 3,500 the loss coefficient depends both on the Reynolds number and on the area ratio. At Re > 3,500, the "Borda-Carnot loss" formula [Eq. (8.1)] can be used.

When a liquid flowing through a pipe with a velocity v_x is discharged into a large reservoir (v₂ = 0) all the kinetic energy is dissipated. In steady laminar flow through a circular pipe the kinetic energy is

If the flow is not steady, i. e., if the pipe length I < l_ent the coefficient a should be taken from the diagram in Fig. 47.

Example. Determine the loss coeficient of a jet of diameter d_f = 1 mm and length I = 5 mm installed in a pipe of diameter d = 6 mm (see Fig. 7 8), as a function of Re.

1.2.3.4.

Solution. Regarding the jet as the entrance section of a pipe and assuming that all the kinetic energy is dissipated when the flow diverges, the loss of head in the jet can be represented as the sum (neglecting the loss due to contraction):

Going over from the velocity in the jet vj to the velocity in the pipe v, the loss coefficient in the jet is:

Assigning several values of Re in the pipe, find the values of Re in the jet from the relationship

Making use of the graph in Fig. 47 to find the coefficients к and a, the neces­sary computations give the table:

Table 3

Re	10	100	,-200	300	400	500
	9,500	3,140	2,500	2,200	2,030	1,885