VON_KARMAN

Calculates the limiting Darcy friction factor for fully rough flow from relative roughness using the von Karman relation. This represents the asymptotic infinite-Reynolds-number rough-pipe limit where friction depends only on \epsilon/D and no longer varies with Reynolds number.

The limiting relation is:

\frac{1}{\sqrt{f_d}} = -2\log_{10}\left(\frac{\epsilon/D}{3.7}\right)

Although real flows always have finite Reynolds number, this expression is useful as a fully rough reference curve and limiting case.

Excel Usage

=VON_KARMAN(eD)
  • eD (float, required): Relative roughness, [-]

Returns (float): Limiting Darcy friction factor for fully rough flow [-]

Example 1: Typical commercial pipe roughness

Inputs:

eD
0.0001

Excel formula:

=VON_KARMAN(0.0001)

Expected output:

0.0119798

Example 2: Moderate roughness pipe

Inputs:

eD
0.001

Excel formula:

=VON_KARMAN(0.001)

Expected output:

0.0196355

Example 3: Very smooth pipe (small eD)

Inputs:

eD
0.00001

Excel formula:

=VON_KARMAN(0.00001)

Expected output:

0.00806325

Example 4: Very rough pipe (large eD)

Inputs:

eD
0.05

Excel formula:

=VON_KARMAN(0.05)

Expected output:

0.0715507

Python Code

Show Code 
from fluids.friction import von_Karman as fluids_von_karman

def von_karman(eD):
    """
    Calculate Darcy friction factor for rough pipes at infinite Reynolds number from the von Karman equation.

    See: https://fluids.readthedocs.io/fluids.friction.html#fluids.friction.von_Karman

    This example function is provided as-is without any representation of accuracy.

    Args:
        eD (float): Relative roughness, [-]

    Returns:
        float: Limiting Darcy friction factor for fully rough flow [-]
    """
    try:
      eD = float(eD)

      if eD <= 0:
        return "Error: eD must be positive."

      result = fluids_von_karman(eD=eD)
      if result != result:
        return "Error: Result is NaN."
      if result == float('inf') or result == float('-inf'):
        return "Error: Result is not finite."
      return float(result)
    except Exception as e:
      return f"Error: {str(e)}"

Online Calculator

Relative roughness, [-]