MAE 3130: Fluid MechanicsLecture 10: Internal and External Flows

Spring 2003

Dr. Jason RoneyMechanical and Aerospace Engineering

Outline• Overview of Viscous Pipe Flow• Laminar Pipe Flow• Turbulent Pipe Flow• Dimensional Analysis of Pipe Flow• Overview of External Flows• Boundary Layer Characteristics• Pressure Gradients Effects• Lift and Drag• Some Example Problems

Viscous Pipe Flow: Overview

Pipe Flow is important in daily operations and is described in general as flow in a closed conduit (pipes and ducts). It is also known as an internal flow.

Some common examples are oil and water pipelines, flow in blood vessels, and HVAC ducts.

When real world effects such as viscous effects are considered, it is often difficult to use only theoretical methods. Often theoretical, experimental data, and dimensional analysis is used,

Some common pipe flow components are shown:

Viscous Pipe Flow: Overview

Pipe flow versus Open-channel flow:

Pipe Flow:Open-Channel Flow:

•Pipe is completely filled with fluid•Pressure Gradients drive the flow•Gravity can also be important

•Pipe is not full of fluids•Pressure gradient is constant•Gravity is the driving force

i.e., flow down a concrete spill way.

Viscous Pipe Flow: Flow RegimeOsborne Reynolds Experiment to show the three regimes Laminar, Transitional, or Turbulent:

Laminar

Transitional

Turbulent

“Experiment”:

Viscous Pipe Flow: Flow Regime

If we measure the velocity at any given point with respect to time in the pipe:

1. Turbulence is characterized by random fluctuations.2. Transitional flows are relatively steady accompanied by occasional burst.3. Laminar flow is relatively steady.

For laminar flow there is only flow direction:

For turbulent flow, there is a predominate flow direction, but there are random components normal to the flow direction:

Reynolds Number Dependency:

Re > 4000

Re < 2100

2100< Re < 4000

Viscous Pipe Flow: Entrance and Fully DevelopedThe entrance region in a pipe flow is quite complex (1) to (2):

The fluid enters the pipe with nearly uniform flow.The viscous effects create a boundary layer that merges.When they merge the flow is fully developed.

There are estimates for determining the entrance length for pipe flows:

and

Viscous Pipe Flow: Entrance and Fully Developed

For very low Reynolds numbers (Re = 10), the entrance length is short:

For many practical engineering problems:

For large Reynolds number flow the entrance length can be several pipe diameters:

Bends and T’s affect Fully Developed Flow:

Pipe is fully developed until the character of the pipe changes.

It changes in the bend and becomes fully developed again after some length after the bend.

In many flows, the fully developed region is greater than the developing region.

Many disruptions can cause the flow to never be fully developed.

Viscous Pipe Flow: Pressure and Shear Stress

The shear stress in laminar flow is a direct result of momentum transfer along the randomly moving molecules (microscopic).

The shear stress in turbulent flow is due to momentum transfer among the randomly moving, finite-sized bundles of fluid particles (macroscopic).

The physical properties of shear stress are quite different between the two.

Fully Developed Laminar Flow: Overview

Both turbulent and laminar flows become fully developed in long enough straight pipes. However, the details of the two flows are quite different.

Some important quantities that we calculate: velocity profiles, pressure drop, head loss, and flow rate.

Although most flows are turbulent rather than laminar, and many pipes are not long enough to allow the attainment of fully developed flow, a fully understanding of fully developed laminar flow is important.

This study is the basis for more complex analysis, and there are some cases where these assumption are good.

The equations or a description can be obtained in three different ways (1) Momentum applied to a fluid element, (2) Navier-Stokes equations, and (3) dimensional analysis methods.

Fully Developed Laminar Flow: Fluid Element Method

Basic Pipe flow is governed by a balance of viscous and pressure forces.

Consider and cylindrical fluid element within a pipe:

Free-Body Diagram:

Fully Developed Laminar Flow: Fluid Element MethodNow since neither the pressure gradient nor the length depend on r, the R.H.S. must also be independent of r.

Then, Then at r = 0, =0, and at r = D/2, is the wall shear stress.

Now,

A small shear stress can produce a large pressure difference if the pipe is relatively long.

The shear profile is linear.

The shear stress for laminar Newtonian Flow:

Velocity decreases from the center-line.

Fully Developed Laminar Flow: Fluid Element Method

Now, recall

Substitute, the shear stress definition, and rearrange:

Integrate,

Apply the boundary conditions, no-slip, u =0 at r = D/2, and solve for C1:

Vc = centerline velocity

Also, we can write in terms of shear stress:

Fully Developed Laminar Flow: Fluid Element Method

Find the Volumetric Flow Rate:

or,

The average velocity is V.

The average velocity is 1/2Vc

Hagen-Poiseuille Flow

Fully Developed Laminar Flow: Fluid Element Method

1. The flowrate is directly proportional to the pressure drop.2. The flowrate is inversely proportional to the viscosity.3. The flowrate is inversely proportional to the pipe length.4. The flowrate is inversely proportional to the pipe diameter to the 4th power.

Some general remarks:

We could adjust the equations for non-horizontal pipes:

Mean Velocity:

Volumetric Flow:

Lastly, we could develop these flows from Navier-Stokes as in Lecture 8.

Fully Developed Laminar Flow: Dimensional Analysis

Important Variables:

Density is not important because of “fully developed”

Number of dimensionless groups:

Two dimensional groups:

If we state that the pressure drop has to be directly proportional to l:

Or, solving for V, and substituting in Q:

C in this case must be determined by experimental analysis. It is 32 for circular pipes.

Now, return to

and divide by dynamic pressure

Fully Developed Laminar Flow: Dimensional Analysis

Now, simplifying,

Now, rewriting, f is the Darcy Friction Factor (dimensionless).

Fully Developed Turbulent Flow: Overview

Turbulent flow is the least understood of all flow phenomena, yet is more likely to occur than laminar flow, so we address ways of describing the flow.

Transition from Laminar to Turbulent Flow in a Pipe:

Fully Developed Turbulent Flow: Overview

One see fluctuation or randomness on the macroscopic scale.

One of the few ways we can describe turbulent flow is by describing it in terms of time-averaged means and fluctuating parts.

mean fluctuating

Fully Developed Turbulent Flow: Overview

Now consider, the time average of the fluctuating parts:

The fluctuations are equally distributed on either side of the average.

Now, consider the average of the square of the fluctuations:

Turbulence Intensity:

Indication of the “gustiness” of the flow.

in Atmosphere, In “good” wind tunnel

Fully Developed Turbulent Flow: OverviewNow, shear stress:

However, for turbulent flow.

Laminar Flow:

Shear relates to random motion as particles glide smoothly past each other.

Shear comes from eddy motion which have a more random motion and transfer momentum.

For turbulent flow:

“Experiment”:

Is the combination of laminar and turbulent shear. If there are no fluctuations, the result goes back to the laminar case. The turbulent shear stresses are positive, thus turbulent flows have more shear stress.

Turbulent Flow:

Fully Developed Turbulent Flow: Overview

The turbulent shear components are known as Reynolds Stresses.

Shear Stress in Turbulent Flows: Turbulent Velocity Profile:

In the outer layer: tirb > laminar 100 to 1000 time greater.

In viscous sublayer: laminar > turb 100 to 1000 times greater.

The viscous sublayer is extremely small.

Fully Developed Turbulent Flow: Velocity ProfileThe velocity profile for turbulent flow is been obtained through experimental analysis, dimensional analysis, and semiempirical theoretical efforts.

In the viscous sublayer:

is the friction velocity, and

for a smooth wall, “Law of the Wall”

In the overlap region:

From dimensional analysis arguments

Possible outer region approximation:

Some alternative, approach include the Power-Law equation:

n = 7 for many practical flows.

Fully Developed Turbulent Flow: Velocity Profile

The power-law equation is not valid near the wall, since that would give an infinite velocity gradient.

Also, the shear does not go to zero at the center-line.

n, chosen based on the Reynolds number.

Turbulent velocity profiles are relatively flat in a pipe flow.

“Profiles”:

Dimensional Analysis of Pipe Flow: Moody ChartMost turbulent pipe flow data is based on experiments. In turbulent flow, in order to do dimensional analysis we consider the roughness of the pipe, as well as density which relates to momentum.

Variables:

roughness

Roughness is important in the viscous sub-layer in turbulent flows, if it protrudes sufficiently in this layer.

The viscous layer in laminar flow is so large, that small roughness does not play a role.

Then range of roughness for validity of this analysis is for:

Then, the dimensionless groups are the following:

Dimensional Analysis of Pipe Flow: Moody Chart

As for laminar flow, the pressure drop must be proportional to the pipe length:

Recalling the definition of the friction factor:

Then the friction factor is one of our dimensionless groups:

Then using experiments, we can find the above relationship with various manufactured pipe roughness values:

“Moody Chart”

Colebrook Relation for Non-Laminar part of the Moody Chart (curve fit):

Dimensional Analysis of Pipe Flow: Moody Chart

Laminar

Marks Reynolds Number independence

Dimensional Analysis of Pipe Flow: Moody Chart

Energy Equation relation to Pipe Flow:

’s account for non-uniform velocity profiles.

Darcy-Weisbach Equation:

For fully developed pipe flow in a horizontal pipe:

And,

Some Example Problems