__Internal Flows__

__Internal Flows__

The material in this chapter is focused on the inﬂuence of viscosity on the ﬂows internal to boundaries, such as ﬂow in a pipe or between rotating cylinders. The next chapter will focus on ﬂows that are external to a boundary, such as an airfoil. The parameter that is of primary interest in an internal ﬂow is the Reynolds number, Re = *V**L */ *n*, where *L *is the primary characteristic length (e.g., the diameter of a pipe) in the problem of interest and *V *is usually the average velocity in a ﬂow. We will consider internal ﬂows in pipes, between parallel plates and rotating cylinders, and in open channels. If the Reynolds number is relatively low, the ﬂow is laminar (see Sec. 3.2); if it is relatively high, it is turbulent. For pipe ﬂows, the ﬂow is assumed to be laminar if Re < 2000; for ﬂow between wide parallel plates, it is laminar if Re < 1500; for ﬂow between rotating concentric cylinders, it is laminar and ﬂows in a circular motion below Re < 1700; and in the open channels of interest, it is assumed to be turbulent.

**7.1 ****Entrance Flow**

**7.1****Entrance Flow**The ﬂows mentioned above refer to *developed ﬂows*, ﬂows in which the velocity proﬁles do not change in the streamwise direction. In the region near a geometry

**F****i****g****u****re 7.1 **The laminar-ﬂow entrance region in a pipe or between parallel plates.

change, such as an elbow, a valve, or near an entrance, the velocity proﬁle changes in the ﬂow direction. Let’s consider the changes in the entrance region for a laminar ﬂow in a pipe or between parallel plates. The *entrance length L *is shown in Fig. 7.1.

The velocity proﬁle very near the entrance is essentially uniform, the *viscous wall layer *grows until it permeates the entire cross section over the *inviscid core length L *; the proﬁle continues to develop into a developed ﬂow at the end of the *proﬁle development region*.

For a laminar ﬂow in a pipe, with a uniform velocity proﬁle at the entrance,

where *V *is the average velocity and *D *is the diameter. The inviscid core is about half of the entrance length.1 It should be mentioned that laminar ﬂows in pipes have been observed at Reynolds numbers as high as 40 000 in extremely controlled ﬂows in smooth pipes in a building free of vibrations; for a conventional pipe with a rough wall we use 2000 as the limit2 for a laminar ﬂow.

For ﬂow between wide parallel plates, with a uniform proﬁle at the entrance,

where *h *is the distance between the plates and *V *is the average velocity. A laminar ﬂow cannot exist for Re > 7700; a value of 1500 is used as the limit for ﬂow in a conventional channel.

The entrance region for a developed turbulent ﬂow is displayed in Fig. 7.2. The velocity proﬁle is developed at the length *L *but the characteristics of the turbulence

**F****i****g****u****re 7.2 **The turbulent-ﬂow entrance region in a pipe.

in the ﬂow require the additional length. For large Reynolds numbers exceeding 105 in a pipe, we use

For a ﬂow with Re = 4000 the lengths are possibly ﬁve times those listed in Eq. (7.3),

due to the initial laminar development followed by the development of turbulence.

(Detailed results have not been reported for ﬂows in which Re < 105.)

The pressure variation is shown in Fig. 7.3. The initial transition to turbulence from the wall of the pipe is noted in the ﬁgure. The pressure variation for the laminar

**F****i****g****u****re 7.3 **Pressure variation in a pipe for both laminar and turbulent ﬂows.

ﬂow is higher in the entrance region than in the fully-developed region due to the larger wall shear and the increasing momentum ﬂux.

**7.2 ****Laminar Flow in a Pipe**

Steady, developed laminar ﬂow in a pipe will be derived applying Newton’s second law to the element of Fig. 7.4 or using the appropriate Navier-Stokes equation of Chap. 5. Either derivation can be used.

__THE ELEMENTAL APPROACH__

__THE ELEMENTAL APPROACH__

The element of ﬂuid shown in Fig.7.4 can be considered a control volume into and from which the ﬂuid ﬂows or it can be considered a mass of ﬂuid at a particular moment. Considering it to be an instantaneous mass of ﬂuid that is not accelerating in this steady, developed ﬂow, Newton’s second law takes the form

where τ is the shear on the wall of the element and γ is the speciﬁc weight of the ﬂuid. This simpliﬁes to

**F****i****g****u****re 7.4 **Steady, developed ﬂow in a pipe.

using *dh *= − sinθ*dx *with *h *measured in the vertical direction. Note that this equation can be applied to either a laminar or a turbulent ﬂow. For a laminar ﬂow, the shear stress *t *is related to the velocity gradient3 by Eq. (1.13):

Because we assume a developed ﬂow (no change of the velocity proﬁle in the ﬂow direction), the left-hand side of the equation is a function of *r *only, so *d */*dx*( *p *+ γ *h*) must be at most a constant (it cannot depend on *r *since there is no radial acceleration; we assume the pipe is relatively small, so there is no variation of pressure with *r*). Hence, we can write

The above velocity proﬁle is a parabolic proﬁle; the ﬂow is sometimes referred to as a *Poiseuille ﬂow*.

The same result can be obtained by solving the appropriate Navier-Stokes equation. If that is not of interest, skip the next part.

__APPLYING THE NAVIER-STOKES EQUATIONS__

__APPLYING THE NAVIER-STOKES EQUATIONS__

The *z*-component differential momentum equation using cylindrical coordinates from Table 5.1 is applied to a steady, developed ﬂow in a circular pipe. For the present

situation we wish to refer to the coordinate in the ﬂow direction as *x *and the velocity component in the *x*-direction as *u*(*x*); so, let’s replace the *z *with *x *and the *v**z *with *u*. Then the differential equation takes the form

Observe that the left-hand side of the equation is zero, i.e., the ﬂuid particles are not

accelerating. Using ρ*g ** _{x}*= γ sinθ = −γ

*d*

*h*/

*dx*, the above equation simpliﬁes to

Now, we see that the left-hand side of Eq. (7.11) is at most a function of *x*, and the right-hand side is a function of *r*. This means that each side is at most a constant, say *l*, since *x *and *r *can be varied independently of each other. So, we replace the partial derivatives with ordinary derivatives and write Eq. (7.11) as

Refer to Fig. 7.4: the two boundary conditions are, *u *is ﬁnite at *r *= 0, and *u *= 0 at *r *= *r *. Thus, *A *= 0 and *B *= −λ*r *2 /4. Since *l *is the left-hand side of Eq. (7.11), we can write Eq. (7.15) as

This is the prabolic velocity distribution of a developed laminar ﬂow in a pipe, sometimes called a *Poiseuille ﬂow. *For a horizontal pipe, *dh*/*dx *= 0 and

The ﬁrst quantity of interest in the ﬂow in a pipe is the average velocity *V*. If we express the constant-pressure gradient as *dp*/*dx *= Δ*p*/*L *where Δ*p *is the pressure

drop (a positive number) over the length *L*, there results

The shear stress at the wall can be found by considering a control volume of length *L *in the pipe. For a horizontal pipe the pressure force balances the shear force so that the control volume yields

Sometimes a dimensionless wall shear called the *friction factor f *is used. It is deﬁned to be

We also refer to a *head loss h*__1 __ρ*V *2deﬁned as Δ *p*/γ . By combining the above equations,

it can be expressed as the *Darcy-Weisbach equation*:

It is valid for both a laminar and a turbulent ﬂow in a pipe. In terms of the Reynolds number, the friction factor for a laminar ﬂow is [combining Eqs. (7.20) and (7.23)]

where Re = *VD*/*n**. *If this is substituted into Eq. (7.23), we see that the head loss is directly proportional to the average velocity in a laminar ﬂow, which is also true of a laminar ﬂow in a conduit of any cross section.

__EXAMPLE 7.1__

The pressure drop over a 30-m length of 1-cm-diameter horizontal pipe trans- porting water at 20°C is measured to be 2 kPa. A laminar ﬂow is assumed. Determine (a) the maximum velocity in the pipe, (b) the Reynolds number, (c) the wall shear stress, and (d) the friction factor.

**Solution**

**(a) **The maximum velocity is found to be

Note: The pressure must be in pascals in order for the units to check. It is wise to make sure the units check when equations are used for the ﬁrst time. The above units are checked as follows:

(b) The Reynolds number, a dimensionless quantity, is (use *V *= *u*_{max }/ 2)

This exceeds 2000 but a laminar ﬂow can exist at higher Reynolds numbers if a smooth pipe is used and care is taken to provide a ﬂow free of disturbances. Note how low the velocity is in this relatively small pipe. Laminar ﬂows do not exist in most engineering applications unless the ﬂuid is extremely viscous or the dimensions are quite small.

(c) The wall shear stress due to the viscous effects is found to be

If we had used the pressure in kPa, the stress would have had units of kPa.

(d) Finally, the friction factor, a dimensionless quantity, is

**7.3 ****Laminar Flow Between Parallel Plates**

Steady, developed laminar ﬂow between parallel plates (the top plate moving with velocity *U*) will be derived applying Newton’s second law to the element of Fig. 7.5 or using the appropriate Navier-Stokes equation of Chap. 5. Either derivation can be used.

**F****i****g****u****re 7.5 **Steady, developed ﬂow between parallel plates.

__THE ELEMENTAL APPROACH__

__THE ELEMENTAL APPROACH__

The element of ﬂuid shown in Fig. 7.5 can be considered a control volume into and from which the ﬂuid ﬂows or it can be considered a mass of ﬂuid at a particular moment. Considering it to be an instantaneous mass of ﬂuid that is not accelerating in this steady, developed ﬂow, Newton’s second law takes the form

where *t *is the shear on the wall of the element and *g *is the speciﬁc weight of the ﬂuid. We have assumed a unit length into the paper (in the *z*-direction). To simplify, divide by *dxdy *and use *dh *= − sinθ *dx *with *h *measured in the vertical direction:

For this laminar ﬂow the shear stress is related to the velocity gradient by τ = μ*du*/*dy *so that Eq. (7.26) becomes

The left-hand side of the equation is a function of *y *only for this developed ﬂow (we assume a wide channel with an aspect ratio in excess of 8) and the right-hand side is a function of *x *only. So, we can integrate twice on *y *to obtain

Using the boundary conditions *u*(0) = 0 and *u*(*b*) = *U*, the constants of integration are evaluated and a parabolic proﬁle results:

If the plates are horizontal and *U *= 0, the velocity proﬁle simpliﬁes to

where we have let *d*(*p *+ *g **h*)/*dx *= −Δ*p*/*L *for the horizontal plates, where Δ *p *is the pressure drop, a positive quantity.

If the ﬂow is due only to the top horizontal plate moving with velocity *U*, with zero pressure gradient, it is a *Couette ﬂow *so that *u*(*y*) = *U**y*/*b*. If both plates are stationary and the ﬂow is due only to a pressure gradient, it is a Poiseuille ﬂow. The same result can be obtained by solving the appropriate Navier-Stokes equa- tion. If that is not of interest, skip the next part.

__APPLYING THE NAVIER-STOKES EQUATIONS__

__APPLYING THE NAVIER-STOKES EQUATIONS__

The *x*-component differential momentum equation in rectangular coordinates [see Eq. (5.18)] is selected for this steady, developed ﬂow with streamlines parallel to the walls in a wide channel (at least an 8 to 1 aspect ratio):

where the channel makes an angle of *q *with the horizontal. Using *dh *= −*dx *sin*q *this partial differential equation simpliﬁes to

where the partial derivatives have been replaced by ordinary derivatives since *u *depends on *y *only and *p *is a function of *x *only.

Because the left-hand side of Eq. (7.32) is a function of *y *and the right-hand side is a function of *x*, both of which can be varied independent of each other, the two sides can be at most a constant, say *l*, so that

where *l *has been used as the right-hand side of Eq. (7.32).

In a horizontal channel we can write *d*(*p *+ *l**h*)/*dx *= −Δ*p*/*L*. If *U *= 0 the velocity proﬁle is

This is a Poiseuille ﬂow. If the pressure gradient is zero and the motion of the top plate causes the ﬂow, it is called a *Couette ﬂow *with *u*(*y*) = *Uy*/*b*.

__QUANTITIES OF INTEREST__

__QUANTITIES OF INTEREST__

Let us consider several quantities of interest for the case of two ﬁxed plates with *U *= 0. The ﬁrst quantity of interest in the ﬂow is the average velocity *V*. The average velocity is, assuming unit width of the plates

The shear stress at either wall can be found by considering a free body of length *L*

in the channel. For a horizontal channel the pressure force balances the shear force:

where Re = *bV*/*n**. *If this is substituted into Eq. (7.43), we see that the head loss is directly proportional to the average velocity in a laminar ﬂow.

The above equations were derived for a channel with an aspect ratio greater than 8. For lower aspect-ratio channels, the sides would require additional terms since the shear acting on the side walls would inﬂuence the central part of the ﬂow.

If interest is in a horizontal channel ﬂow where the top plate is moving and there is no pressure gradient, the velocity proﬁle would be the linear proﬁle

The thin layer of rain at 20°C ﬂows down a parking lot at a relatively constant depth of 4 mm. The area is 40 m wide with a slope of 8 cm over 60 m of length. Estimate (a) the ﬂow rate, (b) shear at the surface, and (c) the Reynolds number.

(a) The velocity proﬁle can be assumed to be one half of the proﬁle shown in Fig. 7.5, assuming a laminar ﬂow. The average velocity would remain as given by Eq. (7.38), i.e.,