27 Internal Fluid Flows

Introduction

Aerospace engineers are more often than not concerned with external fluid flows, e.g., the flow as it develops over the outside surfaces of a body, such as an airfoil, wing, or a complete flight vehicle. However, internal fluid flows are also crucial in many aerospace engineering applications. For example, such flows are encountered inside engines and combustion chambers, engine intake systems, pneumatic or hydraulic systems, fuel systems, heat exchangers and cooling systems, fire suppression systems, and wind tunnels. An internal flow can be defined as a flow in which its downstream development is confined or altered by the boundary surfaces inside which it flows. Internal flows may comprise the flow of gases (such as air) and liquids, called hydraulics, and is a field of fluid dynamics widely studied by civil engineers.

An interesting aerospace example of a combined external/internal flow problem was encountered with the design of the X-43A, which was a hypersonic lifting-body research vehicle, a CFD solution for the flow being shown in the figure below. At hypersonic Mach numbers, the shock waves bend back steeply. Hence, the design used the lower side of the forward fuselage (an external flow) as part of the engine intake to create the necessary sequence of shock waves to slow down the air and increase the pressure before the flow entered the internal flow and the combustion stage of the engine.

The X-43A was developed to test supersonic-combustion ramjet, or “scramjet” engine. The external flow about the forebody was an integral part of preconditioning the air intake to the engine, an example of an combined external and internal flow interaction.
Objectives of this Lesson
  • Appreciate the differences between external flows and internal flows, as well as understand why the action of viscous effects give pressure losses for internal flows.
  • Distinguish between the different effects caused by laminar and turbulent flows through pipes and ducts.
  • Be able to calculate the frictional losses and pressure drops associated with the flows through pipes and ducts using a Moody Chart.
  • Estimate the pumping power requirements to create certain mass flows and/or flow velocities through pipes and ducts, e.g., to produce certain flow velocities in wind tunnels.

Differences Between External & Internal Flows

An interesting consequence of all types of internal flows is that the associated surface boundary layers cannot develop freely. In this regard, the boundary layer thickness and other properties will not be the same as if the boundary layer developed over an external surface, as illustrated in the figure below. This outcome is because, for external flows, the boundaries or surfaces (often just called walls) introduce viscous effects into the flow that progressively diminish away from the wall. However, for internal flows, all parts of the walls of a duct, tube, or pipe constrain the flow development, so the entire cross-section of the internal flow is affected by the walls.

With an external flow the effects diminish away from the all but with an internal flow all of the walls act to constrain the flow developments.

The forgoing behavior is one reason external and internal flows must be carefully distinguished, in that all of the boundaries in an internal flow will affect each other. Because viscous losses (i.e., skin friction) are significant along all flow surfaces, for an internal flow these effects manifest as significant pressure drops over the length of the interior surfaces in which the fluid flows. Evaluating these pressure drops is an integral part of understanding internal flows. The friction and the associated energy drop then appears as heat.

Internal Flow Problems in Aerospace Applications

Internal fluid flow problems occur in many aerospace systems, including fuel, hydraulic, and pneumatic systems. A rocket motor, for example, as shown in the schematic below, has many internal flows that require very high mass flow rates of fuel and oxidizer, which must pass through a series of pumps, pipes, and valves, and so significant losses are incurred. Fuel is also pre-circulated in channels around the exhaust nozzle to keep it cool, over which considerable pressure drops occur. Delivering the fuel and oxidizer to the combustion stage of the rocket motor, which usually involves the flow of cryogenic liquids at extremely high mass flow rates, uses turbopumps.

Schematic showing the internal flow paths of fuel and oxidizer inside a rocket motor, which has to flow through pipes at high volumetric rates.

For aircraft, their hydraulic systems operate at high pressures, so they require relatively high flow rates to operate the large actuators that power the landing gear, flaps, slats, flight control systems, and several other systems. The hydraulic lines may stretch through the structure for considerable distances and usually have multiple systems for redundancy in the event of failure. Airliners, for example, typically have several parallel hydraulic systems, with multiple engine-driven pumps, electrical pumps, air-driven pumps, and hydraulic pumps.

In another example of an internal flow, consider a fuel delivery system on an aircraft where fuel needs to be pumped from one or more fuel tanks (e.g., in the wings) to the engine(s), as shown in the schematic below. The distribution of the fuel lines also involves considerable lengths, often with many twists and turns through the aircraft structure. Therefore, pressure drops, pumping power, and flow rates must be carefully considered so that the engines receive the required fuel volumes at the needed pressures. Redundancy, in this case, is achieved with the use of a mechanical pump as well as an electrical pump.

The delivery of fuel. even for a relatively simple aircraft, requires the consideration of pressure drops, pumping power, and flow rates so that the needed fuel volume is delivered to the engines at the required pressure.

In general, the types of internal flow problems that often arise for aerospace engineers to solve usually fall into these categories:

1. Determining the pressure drop (or the so-called head loss) when flows move through pipes of a given length and diameter for a required volume or mass flow rate.

2. Determining the volume or mass flow rate when the pipe length and diameter are known for a specified (or allowable) pressure drop, which may be constrained for several possible reasons.

3. Finding the pumping power needed from a mechanical pump to obtain a certain exit pressure and/or flow rate for a pipe of a specific diameter and over a certain length.

4. Determining the pipe diameter when the pipe length and flow or mass rate are known for a particular pressure drop. The pipe size needed will also affect its weight and cost, both being significant considerations for aerospace applications.

5. Finding heating effects associated with pumping gases through ducts and pipes, especially at higher flow speeds. Heat is generated because of frictional losses and/or compression of a gas, and heat dissipation may need to be considered in some applications.

Not all of these problems can be solved directly, even if most of the relevant information is known. Design purposes sometimes require iterative solutions starting from some initial estimates. However, some fundamental problems illustrating the process of finding pressure drops and determining pumping power through pipes can be introduced.

Pumps

Many internal flow problems involve using or designing various pumps to move fluid from one place to another, including hydraulic pumps, piston pumps, centrifugal pumps, gear-type pumps, vane pumps, axial flow pumps, fans, etc. For aerospace applications, pumps have to be lightweight (i.e., high power-to-weight ratio), very reliable, and must often contend with harsh operating environments, including large temperature swings and vibration. As a result, pumps are often specified in terms of their volumetric transfer capacity per unit time, e.g., volume/minute.

Consider the example shown in the figure below: a CFD of the internal flow through a diaphragm pump or membrane pump. In this type of pump, which is common in fuel delivery systems, the diaphragm is flexed up and down (by a motor) such that when the volume of the chamber is increased then, the pressure decreases, and a fluid is drawn from one end of the pipe into the chamber. When the chamber pressure increases from the decreased volume, the fluid previously drawn in is forced out through the other pipe. Notice that a pair of butterfly valves prevent reverse fluid flow from the pump.

A good example of an internal flow is this diaphragm pump, which draws fluid from pipe into a chamber then forces it out through the other pipe.

Internal Pipe Flows

Internal flows are encountered in many shapes and sizes of ducts and pipes, the flow through a circular pipe often being used as an exemplar, as shown in the figure below. The effects of viscosity cause the fluid in adjacent layers to slow down gradually and progressively develop a velocity gradient in the pipe, i.e., an axisymmetric form of the boundary layer. To make up for this velocity reduction, the fluid’s velocity at the centerline of the pipe has to increase to balance the net mass flow rate through the pipe. As a result, a significant velocity gradient eventually develops across the entire cross-section of the pipe. The primary manifestation of the viscous friction on the interior walls is a pressure drop along the length of the pipe, which depends on the velocity profile, the volumetric or mass flow rate, and the type of fluid.

An internal flow through a pipe will develop a velocity profile, a manifestation of the viscous friction on the walls being a pressure drop along the length of the pipe.

As previously mentioned, determining the velocity profile and the corresponding mass (or volume) flow rate is one problem encountered in analyzing internal flows. For example, the average velocity V_{\rm avg} at some downstream cross-section of a pipe or duct flow can be determined from the velocity profile by using the principle of the conservation of mass. The mass flow rate \dot{m} is given by

(1)   \begin{equation*} \dot{m} = \rho V_{\rm avg} A_c = \int_{A_c} \rho u(r) dA_c \end{equation*}

where A_c is the cross-sectional area of the pipe and u represents the velocity profile in the pipe as a function of the distance r from the centerline; notice that the profile is assumed to be axisymmetric.

The idea of an average flow velocity in a pipe flow, which is obtained by integrating over the cross section to find the mass (or volume) flow rate and dividing by the area of the pipe.

For the incompressible flow in a circular pipe of radius R = D/2 then

(2)   \begin{equation*} V_{\rm avg} = \frac{\displaystyle{\int_{A_c} \rho u(r) \, dA_c}}{\rho A_c} = \frac{\displaystyle{\int_{A_c} \rho u(r) \, 2 \pi r dr }}{\rho \pi R^2 } = \frac{2}{R^2} \int_0^R u(r) r dr \end{equation*}

Therefore, if the velocity profile u(r) is known (or the mass flow rate or volume flow rate for an incompressible fluid), then the average velocity in the pipe can be determined.

Entrance Length

At the beginning of any internal flow, there is a transition period as the flow organizes itself to become a fully-developed internal flow. As shown in the schematic diagram below, the entrance region/length in such a flow, L_e, is the length needed to transition to a fully developed internal flow. In the entrance region, the velocity profile changes in the flow direction as it adjusts from some initial profile at the inlet to a fully developed profile where the effects of the walls are felt across the cross-sectional area of the entire pipe, duct, or channel.

The flow into a pipe or duct takes time to establish a steady-state condition, but the viscous effects produced by the walls play an important part in determining the final velocity profile. Notice the pressure drop or “loss” as the flow develops along the pipe.

Therefore, a fully developed internal flow is defined as one where the velocity profile does not change along the flow direction, i.e., in the axial downstream direction. At successive axial locations, sections will have the same velocity profile when the duct has the same cross-section. If the pipe changes in cross-section or turns a corner, it will take some distance for the flow to again become fully developed.

As might be expected, the entrance length depends on whether the flow is laminar or turbulent and whether the internal surface is smooth or rough. For example, for the laminar flow in a smooth straight cylindrical pipe of diameter D, then the entrance length is known to be

(3)   \begin{equation*} \frac{L_e}{D} \approx 0.05 Re_D \end{equation*}

where Re_D is the Reynolds number based on the diameter of the pipe, i.e.,

(4)   \begin{equation*} Re_D = \frac{\rho V_{\rm avg} D}{\mu} = \frac{V_{\rm avg} D}{\nu} \end{equation*}

For a fully developed turbulent flow, the entrance length is

(5)   \begin{equation*} \frac{L_e}{D} \approx 4.4 Re_D^{1/6} \end{equation*}

this latter result showing that the flow mixing caused by turbulence greatly speeds up the process of reaching a fully developed internal flow.

Under most practical conditions, the flow in a circular pipe is laminar for Re_{D} \le 2,300 and turbulent for Re_{D} > 4,000 and will be transitional in between, i.e., not entirely one state or another. In practice, the lengths of pipes used are generally many times the entrance length, so for most analyses, the assumption that the entire pipe is turbulent is usually reasonable. Of course, these preceding results will change somewhat for different geometries or if the pipe has a bend, turn, or obstruction, such as a pump, flow meter, etc.

Hydraulic Diameter

For flows through non-circular pipes (for which many types are encountered in engineering practice), the Reynolds number is redefined based on the hydraulic diameter D_h, i.e.,

(6)   \begin{equation*} D_h = \frac{4 A_c}{p} \end{equation*}

where A_c is the cross-sectional area of the pipe, and p is its wetted perimeter. The hydraulic diameter is sometimes referred to as the hydraulic mean diameter. The basic idea is illustrated in the figure below. Naturally, the hydraulic diameter is defined in such a way that it reduces to the physical diameter for circular pipes, i.e., D_h = D.

The principle of hydraulic diameter is to find an equivalent circular pipe based on the wetted perimeter of the non-circular pipe.

For example, for a pipe with a rectangular cross section of width a and height b the area of the duct is A_c = ab and the perimeter is p = 2a + 2b so the equivalent hydraulic diameter for this cross-section is

(7)   \begin{equation*} D_h = \frac{4 A_c}{p} = \frac{4 ab}{2(a + b)} = \frac{2ab}{a+b} \end{equation*}

Notice that in the special case of a square duct where b = a, then D_h = a.

The corresponding Reynolds number will now be calculated by using the hydraulic diameter. However, the concept of a hydraulic diameter works well only for fully developed turbulent flows, i.e., for Re_D greater than about 3,000, where the effects of the details of the pipe geometry on the flow developments is less critical.

Laminar Flow in Pipes

Consider analyzing a steady laminar flow of an incompressible fluid with constant properties in the fully developed region of a smooth straight pipe of circular cross-section. The following solution, a so-called Couette type of flow, is one of the few for viscous internal flows. The approach is exceptionally instructive and is a prerequisite to understanding turbulent flows through pipes, which is a more semi-empirically based analysis.

The flow moves at a constant axial velocity in the fully developed laminar flow. The velocity profile u(r) remains unchanged, i.e., self-similar. It can also be assumed that there is no flow in the radial direction, making this a one-dimensional axisymmetric flow problem. In many cases, the assumption that the flow is steady is a good assumption, but that must be justified on a problem-by-problem basis.

Consider an area (annular disk) of the flow at a given pipe cross-section. The forces acting on the annular disk arise from pressure forces and viscous (shear) forces, as shown in the figure below. The area of the annular disk is 2 \pi r dr its perimeter is 2\pi r.

Diagram for an annular element in the fully developed laminar flow in a circular pipe used to develop the exact laminar flow solution.

Force equilibrium of the annulus requires a balance of pressure and shear forces such that

(8)   \begin{equation*} p (2\pi r dr) - \left( p + \frac{dp}{dx} \right) dx (2\pi r dr) + \tau (2 \pi r dx) - \left( \tau + \frac{d\tau}{dr} \right) (2 \pi (r + dr) ) dx = 0 \end{equation*}

which leads to

(9)   \begin{equation*} r\frac{dp}{dx} + \frac{d\tau}{dr} \, r = 0 \end{equation*}

The shear stresses will be given by Newton’s formula such that

(10)   \begin{equation*} \tau = -\mu \left( \frac{du}{dr} \right) \end{equation*}

the minus sign in this case reflecting that the flow velocity decreases with increasing r away from the centerline, i.e., the gradient is negative. Therefore, the governing equation for the flow becomes

(11)   \begin{equation*} \frac{\mu}{r} \frac{d}{dr} \left( r \frac{du}{dr} \right) = \frac{dp}{dx} \end{equation*}

It can be seen that the left-hand side of the equation is a function of the radial coordinate r and the right hand side is a function of the downstream distance x.The equality in Eq. 11 must hold for any value of r and x, i.e., f (r) = g(x) can be satisfied only if both f(r) and g(x) are equal to the same constant, therefore, dp/dx = constant.

Equation 11 can be solved by rearranging and integrating it twice to give

(12)   \begin{equation*} u(r) = \frac{1}{4 \mu} \left( \frac{dp}{dx} \right) + C_1 \ln r + C_2 \end{equation*}

The velocity profile u(r) is obtained by applying the boundary conditions that du/dr = 0 at the centerline where r = 0 and also that u = 0 at r = R, the latter being the classic no-slip condition. These boundary conditions give the result that

(13)   \begin{equation*} u(r) = -\frac{R^2}{4 \mu} \left( \frac{dp}{dx} \right) \left( 1 - \frac{r^2}{R^2} \right) \end{equation*}

which is a parabolic velocity profile. Also, the axial velocity u is always positive so Eq. 13 shows axial pressure gradient dp/dx must be negative, i.e., pressure in the pipe must decrease in the downstream flow direction, which arises because of viscous losses.

For the foregoing profile, the average velocity can be determined using

(14)   \begin{equation*} V_{\rm avg} = \frac{2}{R^2} \int_0^R u(r) r dr \end{equation*}

which on substitution gives

(15)   \begin{equation*} V_{\rm avg} = -\frac{2}{R^2} \int_0^R \frac{R^2}{4 \mu} \left( \frac{dp}{dx} \right) \left( 1 - \frac{r^2}{R^2} \right) r dr = \frac{R^2}{8 \mu} \left( \frac{dp}{dx} \right) \end{equation*}

Therefore, in terms of V_{\rm avg} then

(16)   \begin{equation*} u(r) = 2 V_{\rm avg} \left( 1 - \frac{r^2}{R^2} \right) \end{equation*}

noticing also that the centerline velocity V_{\rm max} = 2 V_{\rm avg}, i.e., the average velocity is half the maximum centerline velocity, but remembering that this result holds only in this special case.

Quantifying the Pressure Drop

A quantity of interest in the analysis of internal flows is the pressure drop \Delta p, mainly because this drop is directly related to the power requirements of the  pump to maintain a certain flow rate. Recall that the pressure gradient dp/dx is constant for a steady flow through a pipe with constant area, so integrating from location 1 where the pressure is p_1 to another point at a distance L downstream where the pressure is p_2 gives

(17)   \begin{equation*} \frac{dp}{dx} = \frac{p_2 - p_1}{L} \end{equation*}

Because V_{\rm avg} is given by

(18)   \begin{equation*} V_{\rm avg} = \frac{R^2}{8 \mu} \left( \frac{dp}{dx} \right) \end{equation*}

then

(19)   \begin{equation*} \Delta p = p_1 - p_2 = \frac{8 \mu L V_{\rm avg}}{R^2} = \frac{32 \mu L V_{\rm avg}}{D^2} \end{equation*}

Of course, this pressure drop is a direct consequence of the action of viscous effects, so the pressure drop is irrecoverable.

Notice that for a given length of pipe, the pressure drop is proportional to the fluid’s viscosity and flow speed, so the faster a given fluid moves through a pipe or duct, the more significant the pressure drop will be. The pressure drop can also be reduced by using a larger diameter pipe. However, weight and cost are often a significant concern in aerospace systems, where there will be a trade been the weight and cost of a larger pipe (to reduce the losses) versus a bigger pump (to overcome the losses). In practice, this pressure drop will also be different for different pipe cross-sections, different pipe surfaces (smooth or rough), etc.

The pressure loss for a laminar flow can be written as

(20)   \begin{equation*} \Delta p_L = \frac{1}{2} \rho V_{\rm avg}^2 f \left( \frac{L}{D} \right) \end{equation*}

where f is called the friction factor. For a fully developed laminar flow in a smooth circular pipe, the friction factor is

(21)   \begin{equation*} f = \frac{64 \mu}{\rho D V_{\rm avg}} = \frac{64}{Re_D} \end{equation*}

which depends on the Reynolds number only. The friction factors for fully developed laminar flow in pipes of other cross sections such as oval pipes, triangular pipes, etc., are published in various sources, but these latter shapes are rarely encountered in aerospace applications.

The head loss h_L is frequently used in hydraulics to represent the magnitude of the frictional pressure losses, which can be denoted as \Delta p_L. This quantity is equivalent to the additional hydrostatic height that a fluid needs to be raised to overcome the frictional losses. Therefore,

(22)   \begin{equation*} h_L = \frac{\Delta p_L}{\rho g} = f \left( \frac{L}{D} \right) \frac{V_{\rm avg}^2}{2 g} \end{equation*}

After the pressure or head loss is known, the required power P to overcome the pressure loss can be determined from

(23)   \begin{equation*} P = \dot{Q} \Delta p_L = \dot{Q} \rho g h_L = \dot{m} g h_L \end{equation*}

where \dot{Q} is the volume flow rate and h_L is referred to the head loss height, i.e., the barometric height corresponding to the pressure drop \Delta p.

The average velocity for laminar flow in a horizontal pipe (i.e., no gravitational hydrostatic pressure contributions) is

(24)   \begin{equation*} V_{\rm avg} = \frac{\Delta p \, R^2}{8 \mu L} = \frac{\Delta p \, D^2}{32 \mu L} \end{equation*}

so the volume flow rate for laminar flow through the pipe becomes

(25)   \begin{equation*} \dot{Q} = V_{\rm avg} A_c = \frac{\Delta p \, \pi D^4}{128 \mu L} \end{equation*}

this latter equation being known as Poiseuille’s law.

This result means that for a specified flow rate, the pressure drop (and so the required pumping power to overcome the losses) is proportional to the pipe’s length and the fluid’s viscosity but inversely proportional to the fourth power of the diameter. The consequence of this result is that even relatively modest pipe diameter increases can significantly reduce the pumping power. However, this approach may not always prove practical (or even possible) for other reasons, including higher weight and/or costs.

Example #1 – Pressure Drop for Laminar Flow in a Pipe

An engineer is tasked with the design of the fuel delivery system. The system requires a flow through a 200 m length of smooth pipe of 15 mm diameter. The required fuel flow rate is 125 kg hr-1. The fluid properties of the fuel are given as: \rho = 800 kg m-3 and \mu = 0.00164 kg m-1 s-1. All entrance effects can be disregarded.
  1. What is the pressure drop along the length of the pipe?
  2. What pressure capability (in terms of head) is required of the pump?
  3. What are the pumping power requirements?

The cross-sectional area of the pipe is

    \[ A_c = \frac{\pi D^2}{4} = \frac{\pi \times 0.015^2}{4} = 0.0001767 \mbox{ m$^2$} \]

The mass flow rate \dot{m} is given as 125 kg hr^{-1} = 0.0347 kg s^{-1}, i.e.,

    \[ \dot{m} = \rho \, A_c V_{\rm avg} = 0.0347 \mbox{ kg s$^{-1}$} \]

so the average flow velocity in the pipe is

    \[ V_{\rm avg} = \frac{\dot{m}}{\rho \, A_c} = \frac{0.0347}{800.0 \times 0.0001767} = 0.246 \mbox{m s$^{-1}$} \]

The Reynolds number based on pipe diameter is

    \[ Re = \frac{\rho V_{\rm avg} \, D}{\mu} = \frac{ 800.0 \times 0.246 \times 0.015}{0.00164} = 1,797 \]

Notice that this Reynolds number is the laminar regime (so-called Couette flow) so the friction factor f is given by

    \[ f =  \frac{64}{Re} = \frac{64}{1,797} = 0.0356 \]

The pressure drop \Delta p is given by

    \[ \Delta p = \frac{1}{2} \rho V_{\rm avg}^2 f \left( \frac{L}{D} \right) \]

and inserting the values gives

    \[ \Delta p = \frac{1}{2} \times 800.0 \times 0.246^2 \times 0.0356 \times \frac{20}{0.015} = 11,457.4 \mbox{ Pa} = 11.46 \mbox{ k Pa} \]

The equivalent head loss h will be

    \[ h = \frac{\Delta p}{\rho \, g} = \frac{11,457.4}{800.0 \times 9.81} = 1.46 \mbox{ m} \]

Finally, the pumping power P can be determined from

    \[ P = \frac{\dot{m} \, \Delta p}{\rho} = \frac{0.0347 \times 11457.4}{800.0} = 0.497 \mbox{ W} \]

Turbulent Flows in Pipes

Unlike laminar flows, the expressions for the losses in pipes or ducts that contain a turbulent flow are based on both analysis and measurements, which provide semi-empirical relationships. In fact, because of the importance of many industrial applications, many experiments have been performed with pipe flows to measure pressure losses for different flow rates, Reynolds numbers, and surface roughness values.

Representative relative velocity profiles for fully developed laminar and turbulent flows are shown in the figure below. Remember that the fully-laminar velocity profile is parabolic in laminar flow. However, the profile is much “fuller” in turbulent flow, with a more uniform centerline velocity and a sharper drop in flow velocity near the pipe wall, much like a turbulent boundary layer profile develops over an external surface.

Differences in the fully established velocity profiles for fully-laminar and fully- turbulent pipe flows.

Any irregularity or “roughness” on the surface of the pipe, as shown in the figure below, will disturb the development of the boundary layer and so create higher wall stresses. Therefore, the friction factor in a pipe flow with turbulence will depend on the internal surface roughness, and pipes with greater roughness will lead to more significant pressure drops.

It should be kept in mind that “roughness” is a relative concept; in practice, the roughness only starts to become significant when the height of the roughness elements reaches the thickness of the laminar sublayer in the turbulent boundary layer, which is still relatively small. Many extruded plastic pipes are generally considered to be hydrodynamically smooth. However, most other surfaces are rough to some degree, so it will cause the boundary layer becomes more turbulent more quickly, i.e., after a shorter downstream development distance.

Therefore, the friction factor in fully developed turbulent pipe flows (sometimes called Darcy-Welsbach friction factor or just the Darcy friction factor) depends on the Reynolds number and the relative roughness height e/D, which is the ratio of the mean height of roughness of the pipe \epsilon to the pipe diameter D, i.e., for the friction factor then

(26)   \begin{equation*} f = f \left( Re_D, \frac{\epsilon}{D} \right) \end{equation*}

which can also be found from the process of dimensional analysis.

However, the functional form of this dependence in Eq. 26 cannot be obtained from theoretical analysis. Instead, all available results for turbulent flows through pipes are obtained from experiments using artificially roughened surfaces. This is done by using particle grains of a known size bonded to the pipes’ inner surfaces. Ludwig Prandtl’s students conducted the first such experiments in the 1930s, but engineers have done many other experiments. The resulting friction factor f is then calculated from the flow rate measurements and the pressure drop.

The results for pipe friction factors are presented in tabular, graphical, or functional forms, the latter being obtained by curve-fitting the measured data. Commercially available pipes are assessed with equivalent roughness values so that engineers can make appropriate calculations for design and installations. However, the relative roughness of the pipe may increase with everyday use because of corrosion or abrasion (depending on the fluid nature), so friction factors may change over time. Good engineering practice would always make allowances for such effects so that the needed pressure at the end of the pipe can be maintained for its expected service life.

Moody Chart

Most of the available results for common types of pipes have been summarized in the form of a Moody Chart, which engineers widely use to calculate the corresponding Darcy friction factor for an internal flow of interest. The friction factor is then used to calculate the pressure (or head) loss, amongst other things.

On the Moody Chart, which is shown below, the friction factor is shown on the left-hand side ordinate, the Reynolds number based on diameter (or hydraulic diameter) and the average flow velocity is shown on the abscissa, and the relative roughness \epsilon / D is shown on the right-hand side ordinate. If any two of these three parameters are known, the third parameter may be determined using this chart. However, in practice, the Reynolds number and the relative roughness are usually known, so the friction factor is the desired output from reading the chart. Although the Moody Chart was developed for flows through circular pipes, it can also be used for non-circular pipes by replacing the diameter with the equivalent hydraulic diameter, as previously discussed.

Moody chart for the estimation of the Darcy-Weisbach friction factor in circular pipes. It is important to recognize that careful interpolation on the chart will be required to find the needed values. Online calculator versions of the chart are available.

The following points should be understood when using and interpreting the curves on the Moody chart:

1. The laminar flow line represents a theoretical lower bound of pipe friction. There are few practical situations where this is a valid assumption other than for very short, smooth ducts. Nevertheless, the result shows that the friction factor decreases with increasing Reynolds number, and there is no dependency on surface roughness. The laminar result sets the lowest bound on the friction factor for any given Reynolds number.

2. There is a transition region indicated by the shaded area, meaning the flow in this region could be laminar or turbulent. Empirical data here are few and far between and often unreliable, but the zone forms a natural bridge between the theoretical laminar results and the empirical results for turbulent flows. It is uncommon to use this part of the chart as most practical internal flows are fully turbulent.

3. The friction factors increase with roughness for any given Reynolds number at higher Reynolds numbers, another obvious expectation. However, at higher Reynolds numbers, the friction factors become nearly independent of Reynolds number because protruding roughness elements penetrate well into the boundary layer, and the flow will always be turbulent.

4. Notice that even entirely smooth pipes, such as plastic and glass, will still have a friction factor and a pressure drop along their length.

Finally, some words of caution are appropriate when using the Moody Chart. First, like all engineering calculations, it is essential to ensure that the correct units are consistently used when calculating non-dimensional quantities, such as the Reynolds number and relative roughness. Catastrophic mistakes can be made if units are not used carefully and consistently. Second, because of the numerous lines on the chart, it is essential to interpolate carefully. The human eye/brain combination is a suitable interpolator, but greater accuracy may be required. To this end, digital versions of the Moody diagram have been made available, which can be used instead of the chart. More accurate values can be read off the chart using any suitable digitization software.

Example #2 –  Pressure Drop of Turbulent Flow in a Pipe Using the Moody Chart

Oil with a density \rho_{\rm oil}=900 kg/m^3 and kinematic viscosity \nu_{\rm oil}=10^{-5}m^2/s, flows at 0.2 m^3/s through a 500 m length of 300 mm diameter cast iron pipe. The average roughness of the pipe’s surface is 0.26 mm. Calculate:
  1. The average flow velocity in the pipe.
  2. The Reynolds number of the flow. Is the flow in the pipe laminar or turbulent?
  3. The pressure drop and head loss along the pipe.
  4. The minimum power of the pump needed to move the oil.
  1. The average flow velocity is calculated from the volume flow rate, i.e.,

        \[V_{\rm av} = \frac{\dot{Q}}{A }=  \frac{4\dot{Q}}{\pi d^2} = \frac{4 \times 0.2}{\pi \times 0.3^2} = 2.83 \mbox{ m/s} \]

  2. The Reynolds number of the flow in the pipe is

        \[ Re_d = \frac{\rho_{\rm oil} V_{\rm av} d}{\mu_{\rm oil} } =\frac{V_{\rm av} d}{\nu_{\rm oil}} = \frac{2.83\times 0.3 }{10^{-5}} = 84,900 \]

    so flow will be turbulent because the Reynolds number is greater than 2,000 and the Moody chart will be needed to find the friction factor f.

  3. The pressure drop and head loss along the pipe requires the friction factor from the Moody chart. But use the Moody chart the relative surface roughness is needed, which is

        \[ \frac{e}{d} = \frac{0.26}{300} = 0.00087 \]

    From the Moody chart for a Reynolds number of 84,900 and a relative roughness of 0.00087 (using interpolation) then f \approx 0.022. Therefore, the pressure loss over the length of pipe is

        \[ \Delta p  = \frac{1}{2} \rho_{\rm oil} V_{\rm av}^2 f \left( \frac{L}{d}\right) = 0.5\times 900\times 2.83^2 \times 0.022\times \frac{500}{0.3} = 132.15~\mbox{ kPa} \]

  4. The corresponding head loss over this pipe is

        \[ h  = \frac{\Delta p}{\rho_{\rm oil} g} = \frac{132.15\times 10^3}{900\times 9.81} = 14.97~\mbox{m} \]

    The pumping power required will be

        \[ P_{\rm req} = \dot{Q} \, \Delta p_{AB} = 0.2 \times 132.15 \times 10^3 = 26.34\mbox{kW} \]

Design of a Wind Tunnel

Consider now an engineering design problem for the internal flow in a wind tunnel. Wind tunnels are a vital part of the aerospace engineer’s technical repertoire, and tunnel testing is inevitable when designing a new aircraft. Not everything can be predicted before its first flight, and wind tunnels can be used to help establish high confidence in the aircraft’s design, i.e., that the aircraft, when built, will behave as intended.

Wind tunnels come in all shapes and sizes and can have different flow speeds. However, the fundamental purpose of a wind tunnel is to allow measurements to be made under very controlled flow conditions. As a result, measurements are often made on true-to-scale aircraft, wings, and other components; geometric scaling may be required to establish dynamic flow similarity. However, obtaining the correct flight Reynolds number and Mach number in the wind tunnel environment is often challenging.

The design of a modern wind tunnel is a complex affair because it is usually customized to meet a set of unique testing requirements, including the test articles themselves and the types of measurements to be made. A schematic of the new ERAU wind tunnel is shown below. This tunnel is a state-of-the-art, closed-return wind tunnel with a 4 ft by 6 ft rectangular test section 12 ft long with tapered corner fillets. This tunnel allows flow speeds of up to 420 ft/s to be obtained in the test section with exceptional flow quality and low turbulence levels. In addition, the test section has about 70% of its surface area made with optical grade glass, allowing flow measurements using particle image velocimetry (called “PIV”).

The key features of the ERAU wind tunnel.

One of the challenges in wind tunnel design is obtaining uniform flow properties in the test section, i.e., uniform velocities in both magnitude and direction with minimal flow angularity (typically less than 0.1 of a degree) its entire length. Not only does this process need keen attention to the internal flow quality around the entire wind tunnel circuit, including the flow through the fan, but special attention must be made to the contraction before the test section.

By using CFD and taking into account the thickness of the boundary layer and turbulence, it was possible to contour the shapes of the walls of the contraction of ERAU’s wind tunnel to ensure that the best flow uniformity is obtained at the entrance to the test section and also along its entire length. Appropriately shaped and tapered corner fillets from the contraction and along the test section length are also part of the design solution.

The ability to design a wind tunnel with uniform flow properties and low flow angularity along its entire test section is one key to the success of the tunnel as an aerodynamic testing resource.

Pressure Losses in Wind Tunnels

While designing a wind tunnel is a specialist activity, one of the design challenges for a new wind tunnel is determining the motor/fan’s power to create a needed flow velocity (or dynamic pressure) in the test section. To this end, for design, it is necessary to estimate (by calculation) the pressure losses as the flow moves around the tunnel circuit.

As previously discussed, wind tunnels are large ducts with different shapes and cross-sections, transition pieces (adapters), etc. As a result, there will be various pressure losses as the flow moves through these different duct elements at different speeds and Reynolds numbers. In addition, corner vanes are used to help turn the flow and are a cascade of thin airfoils of circular arc plates, and there will be significant frictional pressure losses as the flow passes through the four sets of corner vanes.

In the conventional approach to wind tunnel design, the frictional losses can be estimated for the fan and initial sizing of the motor by breaking the tunnel circuit into its primary parts:

  1. Cylindrical sections (even if just transition pieces).
  2. Corners.
  3. Expanding sections, i.e., diffusers.
  4. Contracting sections, i.e., nozzles.
  5. Turbulence screens.
  6. Heat exchangers.
  7. Other miscellaneous parts.

In each of these sections (and there may be more than one of each), a loss of energy occurs in the form of a loss of static pressure, \Delta p. This can be written as a coefficient of loss, i.e., K = \Delta p/q where q is the local dynamic pressure. This loss is then referenced to the test section values (subscript 0) using

(27)   \begin{equation*} K_0 = \frac{\Delta p}{q} \frac{q}{q_0} = K \frac{q}{q_0} \end{equation*}

Because the pressure or head loss depends on the fourth-power of the tunnel diameter or equivalent hydraulic diameter for non-circular cross-sections, according to Poiseuille’s law then

(28)   \begin{equation*} K_0 = K \left(\frac{D_0}{D}\right)^4 \end{equation*}

where D is the local hydraulic diameter of the tunnel section and D_0 is the hydraulic diameter of the test section. The next step is to refer the energy loss in the flow \Delta E to the test section which will be

(29)   \begin{equation*} \Delta E = K \left(\frac{1}{2} \rho A_0 V_0^3 \right) \left(\frac{D_0}{D}\right)^4 \end{equation*}

where A is the local area and A_0 is the area of the test section. Therefore, the energy loss is

(30)   \begin{equation*} \Delta E = K_0 \left(\frac{1}{2} \rho A_0 V_0^3 \right) \end{equation*}

and so the so-called Energy ratio ER_t can be determined, which is

(31)   \begin{equation*} \mbox{Energy~ratio} = \frac{\mbox{Energy at test section}}{\Sigma~ \mbox{Circuit losses}} = ER_t \end{equation*}

the \Sigma denoting “the sum of.” Therefore,

(32)   \begin{equation*} ER_t = \frac{\frac{1}{2} \rho A_0 V_0^3}{\Sigma K_0 \, \frac{1}{2} \rho A_0 V_0^3} = \frac{1}{\Sigma K_0} \end{equation*}

The resulting energy ratio depends on the inverse sum of the equivalent energy losses for each part of the tunnel circuit, i.e., it is, in effect, the reciprocal of the losses. For a closed-return tunnel, the values of ER_t typically range from 4 to 7. This outcome means that the lower the losses, the higher the energy efficiency, so the lower the power required to be delivered to the air by the fan/motor. Corner vanes and diffuser sections typically contribute to the highest source of losses in a wind tunnel, and as such, they must be carefully designed. Clearly, the overall minimization of the tunnel circuit’s losses is key to reducing the fan’s size and the motor’s power required to drive the flow.

Determining the values of K_0 for each part of the circuit is a straightforward but often lengthy process. As previously discussed, it involves applying the fundamental aerodynamic relationships for turbulent flows through pipes and ducts. In addition, other results are required, such as losses through the corner vanes, turbulence screens, etc. If the losses of the motor and the fan/motor stage were included in the energy ratio, however, it would shed little light on the efficiency of the tunnel design itself. For this reason, it is normally excluded from the pressure loss calculation.

An example, consider the determination of the fan power required to generate a given flow velocity in the test section of a wind tunnel, an effect often called the pumping power. This approach requires that all the various pressure losses in the tunnel circuit be determined, including frictional losses and pressure drop over the walls, turning vanes, screens, etc. Unfortunately, not all of these effects may be known other than being estimated until the wind tunnel is actually built and tested. Therefore, the wind tunnel design may require significant power margins to ensure the specifications are fully met.

Example #3 – Calculating the Power Required for a Wind Tunnel

Estimate the minimum motor power required for a wind tunnel where the maximum flow speed in the test section is to be 230 mph. The test section area is 22.5 ft^2, and the energy ratio of ER_t for the tunnel is 5.2. The fan efficiency is 0.74%, and the motor efficiency is 90%.

The cumulative losses in the tunnel are given by the specification of the energy ratio, so the first step is to find the energy of the flow in the test section, i.e.,

    \[ \mbox{Energy} = \frac{1}{2} \rho A_0 V_0^3 \]

The value of A_0 is 22.5 ft^2 and V_0 = 230 mph = 337.33 ft/s. Assume MSL ISA for the air density, so \rho = 0.002378 slugs/ft^{3}. This gives

    \[ \mbox{Energy} = \frac{1}{2} \rho A_0 V_0^3 = 0.5 (0.002378) (22.5) (337.33)^3 = 1.0269 \times 10^6 \]

Therefore, the power required to be delivered to the air at the fan/motor stage would be

    \[ P_{\rm air~power} = \frac{\mbox{Energy (at test section)} }{ER_t} = \frac{1.0269 \times 10^6 }{5.2} = 360 \mbox{ hp} \]

Taking into account the fan and motor efficiency then the minimum power required would be

    \[ P_{\rm req} = \frac{P_{\rm air~power}}{\eta_p \, \eta_m} = \frac{360}{(0.74)(0.90)} = 539\mbox{ hp} \]

This latter result would only be valid for an empty test section, and to get the same flow speed with an article in the test section, more power would be needed to overcome the drag and “blockage'” of the article. This value is generally not known a priori. However, it is usually considered reasonable to add a margin of power to overcome a winged test article with a wing span of 0.8 of the tunnel diameter (or width), an aspect ratio of 5, and a C_D of 1.0. Furthermore, to account for the potential diversity of test articles (not all will be streamlined shapes), a margin of 50% more power may be needed. The final estimated rated motor power for the tunnel would then be about 900 to 1,000 hp.

Summary & Closure

Internal flows are encountered in various aerospace applications, such as engine air intakes, fuel systems, hydraulic systems, air conditioning systems, wind tunnels, and other applications where fluids flow through pipes and ducts. A significant consideration for internal flows is that frictional effects that arise from the action of viscosity produce significant pressure drops. These pressure drops require a source of power to pump the fluid along the pipe or duct, and the power required is related to the flow velocity and internal surface finish of the pipe or duct. Because such flows are inevitably turbulent, a Moody chart can be used to determine the friction factors and hence estimate the resulting pressure drops for design.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

  • For a hydraulic system on an aircraft, discuss some of the potential design trades in operating the hydraulic system with smaller pipes and higher pressure versus larger pipes and lower pressure.
  • Consider the fuel and oxidizer system for a rocket motor, which requires high mass flow rates of cryogenic liquids. What are the internal flow issues to consider in this case in terms of the delivery of the fuel and oxidizer to the combustion chamber?
  • The corner or turning vanes in a wind tunnel are a large source of losses. Think about the engineering steps that you might take to calculate and also minimize such losses.

Other Useful Online Resources

To learn more about internal flows take a look at some of these online resources: