Resistance: Flow Laminar and Turbulent Flows In Pipes & Channels
By M.S. Ghidaoui, Spring 2002 I highly recommend you watch the video cassette entitled ``turbulence'' which is available in the HKUST library. The call number is TA 357 F578 1990 V.26. Introduction:
Laminar Flow Velocity Profile: velocity distribution (Laminar Flow) v
t Velocity at a point in a steady laminar flow Turbulent Flow Velocity Profile: v instantaneous velocity distribution (Turbulent flow) v
t Velocity at a point in a turbulent flow---varies with time although the flow boundary conditions are fixed!
Time average velocity distribution
The time averaging is performed over a time T long enough compared to the time scale of turbulence. Characteristics of Laminar Flow: 1. Poor mixing capability (only molecular diffusion). 2. Orderly motion. The fluid moves in layers (laminae). 3. Vorticity is regularly distributed in either 2 or 3 dimensions in space. 4. Near true steady conditions can be achieved. Poor mixing properties (only molecular). 5. Energy dissipation is proportional to viscosity and velocity gradient for du Newtonian fluids (i.e., = ). Laminar flows are usually characterized by dy relatively small energy dissipation (i.e., small head loss). y y
du dy
Shear stress distribution in a pipe. Examples of laminar flows: Often groundwater flows are laminar; blood flows etc. The study of turbulent flows is important for some civil engineering applications. Examples include: Groundwater modeling. Water quality modeling in groundwater. Infiltration and slope stability. Seepage in dams and foundation. Settling of particles in water treatment plants. Characteristics of Turbulent Flow: 1. Efficient mixing of mass, momentum and energy (turbulent diffusion is much higher than molecular diffusion). 2. Disordered motion. 3. Vorticity is irregularly and continuously distributed in 3 space dimensions. 4. Turbulent mixing generally results in high energy dissipation (i.e., large head losses). 5. Instantaneously, the flow is never steady and always 3 dimensional! However, if many velocity samples are collected at a point and averaged in time, the averaged velocity may approach steady condition.
6. Turbulent flow is never reproducible in details. Only time-averaged quantities are reproducible. Examples of Turbulent Flows: pipe flows for water supply; open channel flows (natural and human made); winds and typhoons; ocean flows; flow inside a pump etc. The study of turbulent flows is important for many civil engineering applications. Examples include: water quality modeling; sediment transport; energy dissipation in pipes, sewers and natural open channels; fluid structure interaction (wind loading and structural vibration); dv dv near the pipe wall for turbulent flow is much larger than for dy dy laminar the wall shear in turbulent flow is much larger than the wall shear in laminar flow. Note:
Transition From Laminar to Turbulent Flow & Reynolds Number:
Osborne Reynolds (1883) is the first to conduct a systematic study of the transition from laminar to turbulent flows in a tube. He concluded that whether the flow is laminar or turbulent depends: 1. The characteristic flow velocity (V) (property of the flow). 2. The characteristic length scale: e.g., diameter of pipe; hydraulic radius in channel flow; grain size in porous media flow; particle diameter in settling tank etc. (geometrical property) . 3. The fluid's kinematic viscosity = / (property of the fluid). Reynolds combined these three quantities into the following dimensionless parameter, which we call Reynolds number: VD VD Re = = Reynolds found that there is a critical Reynolds number Rc 2000 at which the transition from laminar to turbulent flow occurs. This transition is believed to be due to flow instability. To explain, in laminar flow, fluid particles (small elements) move in an orderly fashion along their respective streamlines (see the appended photos). That is, fluid layers or laminae appear to slide over each other with the only interaction being at the molecular level. In practice, the fluid layers are subjected to random perturbations due (i) the inevitable existence of bumps in the walls of pipes, channels and other solid objects due to the fact that surfaces of solid bodies are never perfectly smooth, (ii) external disturbances such as traffic and building constructions etc. (iii) low disturbances induced at, say, the entrance of the flow from a reservoir into a pipe or a channel and (iv) other natural and human made random excitations.
It is the growth of these perturbations that leads into flow instability and thus the transition from laminar to turbulent flows. To explain, consider the fluid element that is moving along a perturbed (wavy) streamlines (see figure below). This fluid element is subjected to a centrifugal force whose magnitude is proportional to V2/r, where V is the flow velocity and r is the radius of curvature of a streamline. The centrifugal force tends to cause the fluid element to leave its streamline and move transversally to other streamlines (i.e., mixes with flow). However, the viscous forces tend to inhibit the fluid element from leaving its streamline and move transversally. As a result, the fluid element leaves its streamline and moves in the transverse direction only if the centrifugal force (inertia force) is large in comparison to the viscous force. Hence, it is plausible to think that the state of the flow must depend on a ratio of inertia forces to viscous forces; thus, the Reynolds number. Indeed, both theory and experiments reveal that: Higher magnitude of random perturbations reduces the radius of curvature and increases the centrifugal forces; thus promotes the transition to turbulence. Higher fluid velocity increases the centrifugal forces; thus promotes the transition to turbulence. Higher fluid viscosity increases the shear stress; thus delays the transition to turbulence. Other factors that influence the transition to turbulence are: Fluid acceleration damps turbulence and fluid deceleration promotes turbulence; stable stratification (i.e., light fluid on top and dense fluid on bottom) tends to inhibit turbulence. Stable stratification in the atmosphere is referred to as `inversion'. When inversion occurs, it inhibits vertical turbulent mixing of the pollutants released from traffic, industries etc. and thus leads to increased level of pollution at the ground level. Indeed, the highest pollution levels in Hong Kong often occur during days when the atmosphere is in stable stratification.
V2 /r
The above discussion clearly indicates that the transition from laminar to turbulent is governed by the ratio of inertia Forces to viscous Forces. If L is a characteristic length; V is the characteristic velocity; and T is the characteristic time then the Reynolds number is in general defined as follows:
Re = ma ALV / T L( L / T )V LV LV = = A AV / L V
Note that L/T=V and V/T=a were used in the above derivation. It needs to be emphasized that there is some degree of arbitrariness in the definition of Reynolds number. For example, in pipe flow, either radius or diameter may be
selected as characteristic length scale and either the maximum or the average velocity may be selected as a characteristic velocity scale. What matters is that once you make your decision as to which characteristic variables you use, you need to stick to them. Most Hydraulic Engineers use the pipe diameter and the cross sectional average velocity in their definition of Reynolds number. We will adopt this definition of Reynolds whenever we are referring to pipe flow. The experiments of Reynolds and those of many after him led to the following:
1. 2. 3.
Flow is laminar when Reynolds number is below Rc . Flow transition occurs around Rc (turbulent spots (patches) begin to appear) Flow becomes fully turbulent when Reynolds number is well in excess of Rc .
For pipe flow, Rc 2000 and for groundwater flow Rc 1 . Any ideas why?
Velocity profile In Steady Turbulent Pipe Flow:
Figure 1 The relationship between u and u is: 1 T u = u (r , t )dt T 0 T must be long compared to the time scale of turbulence. If u is found not to depend on time, the time mean flow is steady. It must be noted that for a time scale less than T , all turbulent flows are unsteady! Instantaneously, u (r , t ) = u + u ' u ' velocity fluctuations due to the turbulent motion (sometimes u ' is referred to as a random component of the velocity). Note:
udt = u( r, t )dt - u' dt uT = u(r, t )dt - u' dt uT = uT - u' dt u' dt = 0
0 0 T 0 0 0 0 0
T
T
T
T
T
T
u (r , t ) = u + u ' u ( r, t ) = u + u ' +2u ' u
2 2 2 2 = u T + T u' 2 + 0
T
0
u ( r, t )dt = (u + u' 2 +2u' u )dt =
2 2 0
T
1 T
T
0
u 2 dt = u + u ' 2
2
Figure 2 v= 1 T
T
0
v(r , t )dt = 0 Since the mean flow is steady and one-dimensional.
Referring to the fluid element given in figure 2, the instantaneous net momentum flux due to turbulent fluctuation is as follows: u2rru ] x + x - u2rru ] x + v2rxu ] r + r - v2rxu ] r The time mean of the above expression is as follows:
u 2 2rr
] x + x - u2 2rr ] x + uv2rx ] r + r - uv2rx ] r
For a prismatic pipe of a fully developed turbulent flow, u 2 varies little with x (i.e., the turbulence in the pipe is assumed to be homogeneous with respect to x). Therefore, u2 2rr ] x + x - u 2 2rr ] x 0 As a result, the net time mean of momentum flux due to turbulent fluctuation becomes as follows: u v2rx ] r + r - u v2rx ] r These terms represent the momentum transferred from the flow to the turbulent fluctuations. That is, these terms represent the loss of momentum from the flow due to turbulence. Hence, these turbulent momentum flux terms behave like shear stresses and are usually referred to as Reynolds stresses or Reynolds turbulent stresses. Denoting the Reynolds turbulent stress by , then
= - uv
To determine the velocity distribution, an approximate model for u v is required. The model used here is based on Prandtl's mixing length theory. Prandtl (1925) argued that, in a turbulent flow, flow parcels (lumps or eddies) form and move in all directions inside the pipe in a disordered manner. That is, the lumps of fluids move in both the radial and the axial directions. Let us now study the consequences of this disordered motion. Say that a fluid parcel who is currently located at r + r , where its time mean speed is u (r + r ) , moves along the radial direction to a new position at r (see figure 3). Prandtl stated l that = r , where he called l a mixing length. Prandtl hypothesized that as the lump of fluid moves from its initial position to its
new position it maintains its original momentum and, thus, its original speed u(r+ r). That is, while the parcel of fluid arrives at its new position with a speed u(r+r), the fluid that already exists at position r has velocity u(r). Since u(r+r) < u(r) (see figure below), this difference in velocity will result in fluctuation in velocity at r whose magnitude is as follows:
u =u ( r ) - ( r + r ) r u
du du = l dr dr
r r+r New Position Initial Position
Figure 3 To estimate v', take two lumps of fluid (i.e., lump A and Lump B). Lump (eddy) A fluctuates with u'A and lump (eddy) B fluctuates with u'B. If the two lumps approach one another as they fluctuate, the fluid between them will be pushed outward in the radial direction; inducing an outward radial velocity v' (see figure 4). On the other hand, if the two lumps of fluid move away from each other as they fluctuate, the surrounding fluid will move inward in the radial direction to fill the gap produced by the two fluid lumps moving apart from one another and; thus, induces an inward radial velocity v' (see figure 5).
Figure 4
Figure 5 How large is v' will depend on how fast the two lumps approach (move away) from each other. That is, | v' | | u | + u ' B| u ' | | |
A
du du v' r | |= l | | dr dr or; v ' = const1 r
const 2
du ; const1 0 u' v ' = -const 2 u ' v ' ; dr correlation coefficient
2 2
0 const 2 1
u u = -l12 r r Where l1 needs to be approximated using Prandtl's mixing length theory. Therefore, 2 du du 2 du = - u ' v ' = l1 = l12 dr dr dr Remark: In a steady pipe flow (see figure 6) du du u <0 =- dr dr r 2 du du Therefore, = - l1 dr dr u ' v' = - const 2 const1 r 2
Figure 6 By analogy with laminar flow (recall that for laminar flow = - l12 du = eddy viscosity dr du ), dr
du is a property dr of the flow. That is, varies from flow to flow and from location to location within a particular flow.
2 However, you should note, while is a property of the fluid, = l1
A change of variable y = R - r leads to:
=l
2 1
du du du = l12 dy dy dy
2
This is a nonlinear ordinary differential equation for u. To solve this equation, both and need to be approximated. Here, we use Prandtl's approximations .
1
Prandtl's Approximation of l1 : Prandtl assumed that the mixing length 1 varies linearly with y. That is, l1 = ky ; where k is determined from experiments. So how good is Prandl's approximation? The experiments of Nikuradse (1933) (see figure 7) show that this approximation is valid close to the wall. In addition, these experiments show that, near the wall, the slope k 0.4 for both smooth and rough pipes. It is noted from figure 7 that Prandtl's linear approximation is becomes questionable far away from the pipe wall.
Prandtl's Approximation of : Prandtl assumed that the shear stress remains constant with y. That is, (y)=(y=0)= o. This assumption is acceptable near the wall (i.e., y small). Prandtl defined a friction velocity u o* and related it to shear as follows:
= o = uo* 2 or uo* =
0 (i.e., the shear strees is assumed constant by Prandtl)
With the above approximation of the mixing length and the shear stress, we have: du 0 2 = = l1 (du /dy)2 = uo = k y dy Solving for u gives: * u0 u= ln y + C1 k
Smooth versus Rough pipes: The value of the constant C1 integration depends on whether the pipe is rough or smooth. If the pipe roughness e is smaller than the thickness of the laminar layer L, the flow is treated as smooth. To explain, when e< L, the pipe roughness is ``buried'' under the laminar sub-layer and has little or no little influence on the structure of the inertial (turbulence) layer. However, if e > L, the pipe roughness extends beyond the laminar sub-layer and into the inertial layer and; thus, influences both the turbulence structure, turbulence statistics and, thus, the flow resistance.
Pipe Centerline
Turbulent Region
Viscous (laminar) Sub-layer Buffer Zone (region) Magnified Inside Surface of Pipe Wall In order to Show the Roughness
Smooth Pipe
Pipe Centerline
Turbulent Region
Viscous (laminar) Sub-layer Buffer Zone (region) Magnified Inside Surface of Pipe Wall In order to Show the Roughness
Rough Pipe
Value of C1 and velocity for Smooth Pipes: The constant of integration can be found by setting u = u s at y = y0 which leads to: us 1 = ln y 0 + C1 * u0 k Therefore, us u 1 1 = ln y + * - ln y 0 * u0 k u0 k =
* 1 yu 0 1 ln + k ln u * y k 0 o
=
us 1 ln y / y 0 + * k u0
=
* 1 y u0 u s ln + * * k u0 y 0 u0
us + * u 0
The experiments of Nikuradse show that:
1 us ln * + * 5.5 for smooth pipes. k u0 yo u0 k=0.4 1/k =1/0.4 = 2.5
Therefore, the velocity profile for a turbulent flow in a smooth pipe is: yu * u( y) = 2.5 ln 0 + 5.5 * u0 Recall that Prandtl's assumption are quite accurate close to the wall but not necessarily far away from the wall. Yet, engineers and scientists often accepted that Prandtl's approximation remain valid away from the wall! Therefore, the above velocity profile is assumed to be valid both close to the wall and way from the wall. It is pleasantly surprising, that when compared with experiments, the above velocity profile gives accurate results both close and away from the wall (see figure 8)!
It must be noted that a very short distance from the wall (i.e., y< < R), the velocity is near zero because of the no slip condition. Hence, in the region y< < R the inertial forces are small compared to the viscous forces (i.e., Reynolds number with this region is small). Therefore, the flow in region y< < R is laminar. The region y< < R is often referred to as the laminar sub-layer or viscous sub-layer (see figure 9). Within the laminar sub-layer, the velocity profile for smooth pipe is: * yu 0 u = * v u0 In summary, for a smooth pipe flow: yu * u = 2.5 ln 0 + 5.5 is in good Velocity profile in the turbulent region: * v u0
* agreement with data for yu 0 / >70 (see Figure 10).
Velocity profile within the laminar sub-layer:
* agreement with data for yu 0 / <5 (see Figure 10).
yu * u = 0 * v u0
is in excellent
yu0 Note, the region 5 < < 70 is called the buffer zone (see figure 9). As seen from figure 10, the experimental velocity profile in the buffer zone differs slightly from the both the velocity profile in the laminar sub-layer and the velocity profile in the turbulent region. However, the deviation is generally considered very small and the velocity profile in buffer zone is often determined by matching the velocity profile in the laminar sub-layer and the velocity profile in the turbulent (also called inertial) layer (see figure 10). The point of intersection of the two profiles is:
* * * yu 0 yu 0 yu 0 u = 2.5 ln + 5.5 = 11.5 * v v v u0
Figure 9
Figure 10 (Solid lines are plotted from the equations and dashed curve is plotted by fitting the experimental data. The plot is a semi-log plot. Figure taken from Tritton's book). Value of C1 and velocity for Rough Pipes: The experiments of Nikuradse show that the velocity profile for a rough pope has is similar shape to that of a smooth pipe. However, the constant of integration becomes y u* o 8.5 not 5.5 and the argument of the logarithmic function is y/e and not . That is,
eu 0 u ( y) y = 2.5 ln + 8.5 for > 70 * e u0 For rough pipes, the laminar sub-layer is contained within the grooves of the pipe wall and has, practically, no influence on frictional resistance and head losses on these
pipes. Comparison with experimental data show that the accuracy of the velocity profile for rough pipes is excellent (see Figure 11):
Frictional velocity u* : o Thus far, we have introduced and used u* but we did not mention how we relate this o velocity to the cross sectional average velocity in the pipe V . For convenience, we will drop the over-bar and denote this cross sectional average velocity by V (see Figure 3). According to Darcy-Weisbach: fV2 o= 8 Hence;
o = * = V uo e
f 8
V
u (r)
Figure 12 Remark: Other empirical formulas such as Manning equation and Chezy equation can be used to estimate u* . In fact, in open channel, either the Manning equation or o the Chezy equation is often used to relate u* to V. o Power Law Profiles: The logarithmic velocity profiles compare well with experimental data for a wide range of Reynolds number. However, in practice, it is desirable to have simple form of velocity profiles. The power law profiles are simple to deal with and also fit experiments very well (see Figure 2). Their general relationship is: y = * u0 R u n=6 n=6.6 n=7 n=8.8 n=10 n=10
1/ n
R-r = R
1/ n
r = 1 - R
1/ n
Re=4*10^3 Re=2.3*10^4 Re=1.1*10^5 Re=1.1*10^6 Re=2*10^6 Re=3.2*10^6
Figure 13
Turbulent Velocity Profiles In Other Civil Engineering Applications:
The logarithmic velocity profile and the power law profile are valid in a variety of other turbulent flow applications in civil engineering such as open channel flow and wind engineering. Turbulent Open Channel Flow: The logarithmic velocity profile is: u=
* u0 ln y + C1 k
Let u = u s=0 at y = y0 leads to: us u* y 1 1 u = 0 ln = 0 = ln y 0 + C1 C1 = - ln y 0 * k y0 k k u0 Open channel flow experiments show that k 0.4 1/k =1/0.4 = 2.5 (i.e., remains the same as in pipe flow)! u y * = 2.5 ln y u0 0 Experiments also show that for turbulent flow in smooth open channels we have u ( y) y = 2.5 ln + 5.3 * e u0 yu * u = 0 * v u0 for
eu 0 > 70
* for yu 0 / <5 .
where y= distance measured from channel bottom. Due to its simplicity and relative accuracy, the power law is also often used in open channels: 1/ n u y = u max H where H= Total height of water in a channel. Turbulent Wind Flow:
Studies have shown that steady wind velocity is well represented by the logarithmic velocity profile and that k 0.4 1/k =1/0.4 = 2.5 (i.e., remains the same as in pipe flow and open channel flow)! That is, u z = 2.5 ln * z0 u0 Note that wind engineers often denote the height above ground by z and not y. The height z = z0 is the roughness height. In Australia, the ground is classified into four categories: 1. City Terrain z0 2 m 2. Suburban Terrain z0 0.2 m 3. Airport Terrain z0 0.02 m 4. Ocean Terrain z0 0.002 m Currently Hong Kong engineers use the power law profile and not the logarithmic profile for wind engineering applications. y = u max H where H= reference height. u
1/ n