The most important parameter that characterizes a gas flow is the Reynolds number

(9.1)

where d is the typical transverse dimension of the channel; U is the velocity of the gas flow averaged over the channel cross-section; and v is the kinematic viscosity.

© Springer Science+Business Media New York 2015 S. P. Melnikov et al., Lasers with Nuclear Pumping, DOI 10.1007/978-3-319-08882-2_9

At Reynolds numbers, Re, lower than the critical value, Rec, a flow is laminar. At Re > Rec, the laminar mode makes the transition to the turbulent mode. Under normal conditions for a flow in a round tube, Rec«2,300. For channels with another transverse cross-section shape, it is close to this same value [2, 3].

During a laminar gas flow in a cylindrical channel, the pressure differential along its length, bL, comes to [4]

др = 8UpA,

r1

while for a channel formed by two parallel planes, the distance between which equals d,

др = (9.2)

During a turbulent flow in a channel with any transverse profile, the pressure differential along the channel’s length is determined by the expression [5—7]

_ 2 b

др = tpU2-L, (9.3)

2de

where £ is the resistance coefficient. The channel’s equivalent diameter, de, is expressed by way of the area of the channel’s transverse cross-section, Sc, and the total perimeter, nc, of the channel surface that the gas washes over,

de = 4Sc/nc-

The resistance coefficient can be determined based on the Blasius law [6, 7]

£ = 0,3164Re~°,25.

Other formulas also exist for determining the £ parameter [3-5], but they all yield a similar result. An estimate based on formula (9.2) for U ~ 10 m/s and bL ~ 1 m yields др < 3 x 10 4 atm. From a similar estimate based on formula (9.3) for U~100 m/s, we get ДР < 0.03 atm. In both cases, the ДP parameter value is negligible as compared to the optimum pressure value of P ~ 1 atm used in NPLs. Consequently, the gas pressure in a laser channel can be roughly regarded as homogeneous.

In this case, the heat balance equation for a laser channel with a stationary gas flow that operates in the steady-state mode takes the forms:

qcScbi = P0U0Sccp(Tb — 7q),

where cp is the specific heat capacity of the gas at a constant pressure; qc is the specific power deposition averaged over the volume; T0, Tb are the gas temperatures
at the laser channel inlet and outlet; U0 is the gas velocity at the channel inlet; and p0 is the gas density at the channel inlet. Thus,

qA

p0cp(Tb — T0)

Because the operating length of the laser channels is bL > 1 m, then at a specific power deposition of q > 10 W/cm3, gas circulation along a channel already requires the use of high gas velocities (U0 ~ 100 m/s) in order to prevent conspicuous active medium overheating (for example, by AT = Tb — T0 < 100°K). In the presence of a channel transverse dimension of d ~ 1 cm, these velocities correspond to Re > 104. These Re values markedly exceed the Rec value at which a laminar flow mode makes the transition to a turbulent mode. Because bL >> d and Re >> Rec, a well — developed turbulent flow is then established in the bulk of the cell volume, which can adversely affect the optical quality of the active medium as a whole.