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The Friction Factor for Laminar Flow: Breaking Down the Equation & Calculations

Key Takeaways

  • Drag occurs in laminar flow or turbulent flow, and its strength is quantified using a drag coefficient, also known as the friction factor.

  • The friction factor will depend on the Reynolds number and has some simple expressions for laminar flow.

  • The total drag force depends on a pressure-driven drag coefficient, and the total drag force will change once flow becomes turbulent.

Using a simulation to determine friction factor for laminar flow

CFD simulations can be used to determine the friction factor for laminar flow

Airflow or fluid flow around a moving object drives an important force: drag. Drag is composed of two constituent forces that act between the surrounding fluid and the moving object—friction drag and pressure drag. Either of these forces may dominate depending on the shape of an object, its velocity, the density of the surrounding fluid, and the fluid’s viscosity. When doing an initial survey of a design and estimating the amount of drag force acting on a moving object, it’s helpful to use an equation to estimate the drag for a particular object before running more thorough CFD simulations.

In this article, we’ll look at some of the common expressions used to estimate the friction factor for laminar flow, which will determine the amount of friction drag acting on an object. The friction factor is part of a more comprehensive drag coefficient, which can be used to calculate the total drag based on fluid velocity and the cross-sectional geometry of the moving object. As we’ll see, the friction factor depends on the Reynolds number and the fluid flow regime (laminar vs. turbulent flow). However, the friction factor for laminar flow is a good place to start estimating drag for many systems for several reasons, which we’ll discuss below.

Estimating Drag With the Friction Factor

The first place to start developing an estimate of drag is to look at laminar flow across an object, ideally for incompressible flow. There are several reasons to start at this point:

  • This is one of the simplest calculations that can be done to determine drag.
  • This will be accurate for a range of object velocity values and Reynolds numbers in practical situations.
  • In aerodynamics, it is preferred to be operating in the laminar regime anyways, so this estimate will be needed in general.
  • In aerodynamics, the flow will tend to be in the linear (friction) regime until aircraft/vehicles reach very high speeds.

Since the friction factor is clearly important as a starting point for systems design, it helps to have some simple models to estimate its contribution to total drag. First, we need to break drag into its two respective contributions. Then, we can create a model for the skin friction force acting on an object.

Defining Drag and the Friction Factor

In general, the total drag force acting on an object is defined in terms of an integral:

Drag force equation

Drag force equation

The coefficient Cf in the integral is the drag coefficient. A surface integral is used because the fluid velocity might vary across the integration surface, such as with a spherical object. This means that the fluid velocity, and thus Cf, is not always a constant. Furthermore, the value of Cf is defined in terms of two other coefficients:

Cf = Cf + Cd

The drag coefficient is defined in terms of a friction factor (cf) and pressure drag factor (cd)

The coefficient cf is incorrectly called the Moody friction factor; in fact there are many friction factors attributed to different authors. Both of the coefficients shown above can be functions of Reynolds number, so they will have different forms in the laminar and turbulent flow regimes. At this point, we can use some empirical observations or theoretical models to derive some expressions for the friction factor for various geometries.

Friction Factor for Laminar Flow

The friction factor for laminar flow has a simple expression for a range of systems:

Friction factor for laminar flow

Friction factor for laminar flow equation

For a closed pipe, we have k = 64. Typical values in similar geometries range from k = 24 to 96 for laminar flow in smooth, enclosed systems.

Another way to examine the effects of friction factor on laminar flow is from empirical data in a Moody diagram. An example is shown below. In this diagram, the left half of the graph shows the inversely proportional relationship between Reynolds number and friction factor for laminar flow. As the Reynolds number increases, there is a sudden increase in friction (near Re = 2000) as the system transitions into the turbulent regime. Once the flow is turbulent, the friction factor vs. Reynolds number curve depends on surface roughness, as indicated on the right y-axis in the graph.

Moody diagram

Example Moody diagram. [Source]

Moody diagrams are useful for extracting trends in specific systems, but they have less generalizability to systems with different geometries. However, they can be constructed from measurements and provide very reliable interpolations. When the system design needs to change or when pressure starts to dominate, CFD simulations should be used to determine the relative contributions from pressure and drag.

When Do Pressure Drag and Turbulence Dominate?

Eventually, when the object/fluid velocity becomes large enough, the cross-sectional area where pressure acts becomes too large, or the external pressure associated with the flow is too large (again, at high velocity), then pressure drag will start to dominate. At this point, the drag coefficient must include its pressure-related term. In addition to a pressure term, turbulent flow will eventually begin to dominate as the Reynolds number increases, and drag coefficient formulas will be different at high Reynolds numbers. The expressions shown in the previous section will no longer be valid and alternative models must be used to describe drag.

Perhaps the most common equation describing the friction factor in turbulent flow is the Colebrook–White equation:

Colebrook-White equation

Colebrook-White equation

This is a transcendental equation with the roughness factor (e/D) taken as a parameter. It can be solved numerically with a random search algorithm or it can be solved graphically. A good review of this equation and its uses in convective cool can be reviewed in the literature:

Real engineered systems will have complex geometries and velocity distributions, making the calculation of friction factor for laminar flow difficult. However, these problems become tractable for designers who use the complete set of CFD meshing and simulation tools from Cadence. Modern numerical approaches used in aerodynamics simulations, turbulent and laminar flow simulations, reduced fluid flow models, and much more can be implemented with these tools.

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