Fluid Mechanics Chapter 10 Laminar and Turbulent Flows

Fluid Mechanics Chapter 10Laminar and Turbulent FlowsFOSTEMINTI International University

Steady and Uniform Laminar flow in Circular PipesIncompressible fluid - density of the fluid is assumed to be constantSteady flow - conditions such as velocity, depth, cross sections do not change with timeUniform flow - conditions do not change with space

Laminar flow: Re < 2000 (viscosity effect is significant)

Basic principles used are:application of momentum equationapplication of the shear stress - velocity gradient relationshipknowledge of the flow condition at the pipe wallInfluid dynamics,laminar flow(orstreamlineflow) occurs when a fluid flows in parallel layers, with no disruption between the layers.[1]At low velocities, the fluid tends to flow without lateral mixing, and adjacent layers slide past one another like playing cards. There are no cross-currents perpendicular to the direction of flow, noreddiesor swirls of fluids.[2]In laminar flow, the motion of the particles of the fluid is very orderly with all particles moving in straight lines parallel to the pipe walls.[3]Laminar flow is a flow regime characterized by highmomentum diffusionand low momentumconvection.3Steady and Uniform Laminar flow in Circular PipesFig. 10.4: Forces acting on an annular element in a laminar pipe flow

Applying momentum equation to the fluid element in the flow direction,Steady and Uniform Laminar flow in Circular PipesVelocity u at a radius r,(10.16)

Equation (10.16) shows the variation of local fluid velocity u across the pipe.This velocity profile may be seen to be parabolic. The negative sign is present due to the fact that the pressure gradient will be negative in the flow direction.

The maximum velocity will occur on the pipe centreline, at (r = 0),

Maximum velocity, (10.17)

Steady and Uniform Laminar flow in Circular PipesThe incremental flow Q through an annular element of radial width r at radius r across the flow from r = 0 to r = R will be, Q = u2rrVolume flow rate, Q = R u 2rdr (10.18)

In terms of a pressure drop, p over a length l of pipe of diameter d,

Volume flow rate,(10.19)

Steady and Uniform Laminar flow in Circular PipesVelocity u at a radius r,(10.16)

Maximum velocity, (10.17)

Volume flow rate, (10.19)

Mean velocity of flow,(10.20)

Fig. 10.5: Velocity distribution in laminar flow in a circular pipe

Steady and Uniform Laminar flow in Circular PipesHagen-Poiseuile equation for pressure loss p (N/m2) in a pipeline, of length l (m) and diameter d (m),

(10.21)where = dynamic viscosity (Ns/m2) and Q = discharge (m3/s) Since discharge, Q = Au = (d2/4)u,

(10.22)

Example 10.2

Steady and Uniform Turbulent flow in Bounded Conduits Fig. 10.6: Turbulent flow in a bounded conduit

Applying momentum equation to the fluid element in the flow direction yields,p1A p2A olP + W sin = 0

Steady and Uniform Turbulent flow in Bounded ConduitsChezy formula:

Mean velocity,(10.29)where C = Chezys roughness coefficient, m = hydraulic radius, i = energy slope

Darcy-Weisbach equation:Head loss due to friction,(10.30)

where f = friction factor which can be obtained from a Moody Chart. v = mean velocity (m/s), l = length of pipe (m), and d = diameter of pipe (m)

Steady and Uniform Turbulent flow in Circular PipesThe head loss in turbulent flow in a pipe is given by the Darcy equation (10.30),

where f = friction factor which can be obtained from a Moody Chart. v = mean velocity (m/s), l = length of pipe (m), and d = diameter of pipe (m)hf l;hf v2;hf l/d;hf depends on the surface roughness of the pipe walls;hf depends on fluid density and viscosity;hf is independent of pressure.

Head Loss or Friction head or Resistance head is due to the frictional forces acting against a fluid's motion by the container.Moody Chart (for friction factor f)

Fig. 10.7: Variation of friction factor f with Reynolds number and pipe wall roughnessThe Moody Chart Common reference for calculation of losses in turbulent pipe flowLogarithmic plot of (f versus Re) for a range of (k/d) values where k = roughness of surface of the pipe, d = diameter of the pipe1.The straight line is labelled laminar flow. f = 16/Re

2. For values of (k/d < 0.001) the curves approach the Blasius curve due to the presence of the laminar sublayer. f = 0.079/Re1/4

3.At high Reynolds numbers, or pipes having a high (k/d) values, all the roughness particles are exposed to the flow above the laminar sublayer. This condition is represented on the Moody Chart by portions of the (f versus Re) curves which are parallel to the Re axis.

Laminar & Turbulent flows

Example 10.3

Steady and uniform Turbulent flow in Open ChannelsChezy Formula (m/s)(10.29)

where v = mean velocity (m/s)C = Chezy's coefficient (L1/2T-1) m = hydraulic radius = Area/Wetted perimeter (m) i = slope of energy lineFor steady uniform flow, the slope of the energy line (i) is equal to the bed slope (S) where v = mean velocity (m/s) C = Chezy's coefficient (L1/2T-1)R = m = hydraulic radius = Area/Wetted perimeter (m)S = bed slope

Discharge (m3 /s)(10.36)

Example 10.4A rectangular open channel has a width of 4.5 m and a slope of 1 vertical to 800 horizontal. Find the mean velocity of flow and the discharge when the depth of water is 1.2 m and if C in the Chezy formula is 49.

Chezy formula: mean velocity, v = C(mi) Di = S =1/800 and m = A/PBA = BD = 4.5x1.2 = 5.4 m2 P = 2D + B = 2x1.2 + 4.5 = 6.9 mm = A/P = 5.4/6.9 = 0.783 m

Mean velocity V = 49(0.783/800) = 1.53 m/s

Discharge Q = AV = 5.4x1.53 = 8.27 m3 /sLosses of energy in pipelinesLosses of energy in a pipeline are:(a) Frictional resistance to flow

using Darcy equation (m)(b) Separation losses due to disturbance of the normal flow at pipe fitting such as valve, bend, junction or sudden changes of section including pipe entry and exit.

These losses are conveniently expressed as energy loss in m (N-m/N), that is, the head loss in terms of the fluid in the pipe, and related to the velocity head (v2/2g) as, (m)where K is the fitting loss coefficient.

Separation losses in Pipe flow

Separation losses in Pipe flow

Losses of energy in pipelines

Pipe fittings (valves)

Losses in pipe fittings, bends and at pipe entry Table 10.2: Head loss coefficients for a range of pipe fittings

Head loss(m)

The EndLaminar Flow (Re < 2000)Turbulent Flow (Re > 4000)

1. Equation for local velocity u

v = uavg = umaxwhere u = local velocity R = radius of the pipe r = any distance from the center of the pipe P = pressure drop l = length of the pipe

2. Hagen-Poiseuille equation for pressure drop

3. Darcy-Weisbach equation for head loss due to pipe friction

Where hf = head loss v = mean velocity f = friction factor l = length of the pipe d = diameter of the pipe

1. Darcy-Weisbach equation for head loss due to pipe friction

Where hf = head loss v = mean velocity f = friction factor l = length of the pipe d = diameter of the pipe

f can be obtained from

Moody Chart

or

Blasius equation

2. Pressure head loss

Power required to maintain the flow = gQhf