Laminar and Turbulent Flow

Flow phenomena can be divided into three main types:

• Laminar (streamline)

• Transitional

• Turbulent

In laminar flow there are no disturbances, and therefore all flow particles move in the same direction. Transitional flow is the flow regime that takes place dur­ing the change from streamline to turbulent flow. In the case of turbulent flow the particles move in a given flow direction, but the flow is erratic and random.

Laminar Tube Flow

When the Reynolds number is under 2000, it is shown empirically that the flow in a smooth tube is laminar. This flow has a parabolic velocity pro­file, as shown in Fig. 4.3.

Now consider a cylindrical volume element in a flow stream. The radius of the element is r and its length is L. The force produced by the flow in this volume is due to the viscosity, which is

ItttLt ~ — lirrLri-^1, r ~ ~^~dr

The pressure difference (drop) between the ends of the element produces a force Apirr2, and considering the force balance,

Apirr2 = -2 TrrLv-^.

R ‘ ar

Simplifying this gives

Laminar and Turbulent Flow

Denoting w as ml at r = 0 and noting that vm = 0 at r = R, the integration gives

Laminar and Turbulent Flow

W = m i =

(4.40)

T

T

подпись: tP-&P

Laminar and Turbulent Flow

Laminar and Turbulent Flow

L

A p = 8r>V™L = 32rlVmL R2 D2 ‘

(4.44)

Equation (4.44) is the Hagen-Poiseuille law, which shows that the pressure loss during laminar flow is linearly proportional to the flow velocity.

The following equations show the relationship between the pressure loss and the friction factor:

O2irrV"’ dr = 2rl

=

For a parabolic velocity profile the velocity expression is

Vm

^(R2-r2).

4tjL

Laminar and Turbulent Flow

(4.41;

 

Volume flow is calculated by integrating the expression for the velocity over the surface:

 

Laminar and Turbulent Flow

2 4

 

Hence

 

_ ApirR 8tjL

 

(4.42)

 

The mean velocity is

 

(4.43)

 

The pressure difference is

 

Laminar and Turbulent Flow

Ati = g L pvm 32rivniL ap 2 d2

? = 64—ZL_ = 64 * pDvm Re

This connection is valid for laminar flow, as Re < 2000.

 

(4.45)

(4.46)

 

Turbulent Flow

Laminar flow after transition usually turns into turbulent flow when

Re > 2000. It has been shown that the pressure loss of a turbulent flow is

Caused by a friction factor with the magnitude of

F= 0.079Re"1/4, when Re < 105. (4.47)

This equation is the Blasius equation.

The shearing stress r0 on the surface of the flow duct and the pressure loss can now be solved from Eq. (4.48), given below:

A 2. A mac l/4n-l/4 7/4

To = fjPVm = 0.0395pi’ D vm

Ap = 2/^pv2 = 0.158Lpv1/4D-s/4v™ . (4.48)

Thus Ap ~ vli75.

 

When Re > 105, the following equation, derived by means of the logarith­mic velocity distribution by Prandtl and the empirical research results of Ni — kuradse, is valid:

= 4.01 In (Re7/) — 0.4 . (4.49)

Surface Roughness

In the previous section it was assumed that the surface of the flow duct was smooth. In reality duct surfaces are rough to varying degrees, which has an effect on the magnitude of friction. Thus Eqs. (4.47) and (4.49) rep­resent the lowest possible levels of f; in other words, the effect of roughness is zero.

To allow for the effect of roughness one can use the results of empirical tests in ducts that have been artificially roughened with particles glued on the surface. This approach allows roughness levels to be determined as a function of the particle diameter k. The following friction factor equation has been de­rived for large Reynolds numbers:

J_ = 4.oin^ + 3.48. /4.50)

If k

This is an ultimate case, when the friction factor is no longer a function of the Reynolds number and is a function of roughness; the pressure loss is now Ap ~ w~, where w is the fluid velocity in the duct. The surface roughness of typical manufactured ductworks varies between the values of a theoretically fully smooth duct and an artificially roughened one. Accordingly the pressure loss varies between Ap ~ w]JS — w~ and Ј = f(Re, roughness).

With most forms of duct, the roughness given by the following Colebrook and White equation can be used (Eq. (4.51)). This equation has been deter­mined by calculating an equivalent roughness, corresponding to the sand par­ticle tests results and taking into account that with large Reynolds numbers the friction factor’s dependency on the Re value is minimal.

5.4

(4.51;

подпись: (4.51;+

Jf

подпись: jfRe//- 3.7 Id

This equation represents the change-over section between a smooth tube and a fully developed rough flow.

In practice the friction factors are calculated either by integration of Eq, (4.51) or by reference to a Moody chart. This is based on Eq. (4.51) by using equivalent roughness values representing the sand particle roughness (see Table 4.3).

Figure 4.4 shows the Moody chart for tubes when k = 0.03 mm, which is the case for steel tubes. Friction factors for other values of k can be attained by using the following ratio:

D

подпись: dCase

And determining the corresponding diameter from the Moody chart, which is derived from this equation.

TABLE 4.3 Equivalent Roughness Values for Various Materials

Material k,,quiv

TOC o "1-5" h z Commercial or follower steel 0.046

Asphalted cast iron 0.120

Cialvanized steel 0.150

Cast iron 0.26

Wooden surface 0.18-9

Concrete 0.3-3

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