Gamma Function in Submarine Pipelines Free Span Analysis

We all know how to calculate the factorial of any positive integer (ex. n! = n × (n - 1) × (n - 2) ... × 1), but have we considered what if we want to calculate the factorial of fraction number like "3.2". That is where Gamma Function rise from ashes.

From Wikipedia,

In mathematics, the gamma function (represented by Γ). It can be defined via a convergent improper integral for complex numbers with positive real part:

The gamma function is like a magic tool that helps us find a smooth curve that connects the points of the factorial sequence. Imagine you have a sequence of points where each point is represented by ((n, n!)), where (n) is a positive integer and (n!) (n factorial) is the product of all positive integers up to (n). For example, (3! = 1 * 2 * 3 = 6).

The simple formula for the factorial, (x! = 1 * 2 *....* x), only works when (x) is a positive integer. Unfortunately, no basic function can do this for non-integer values. That's where the gamma function comes in! The gamma function, denoted as (Γ(x+1)), provides a way to extend the concept of factorials to non-integer values, creating a smooth curve that fits perfectly with the factorial points. So, in simpler terms, the gamma function is a clever way to connect the dots of the factorial sequence with a smooth curve, even for non-integer values of (x).

We use it in Subsea Pipeline design all the time even without noticing it. The gamma function is often used in various fields such as combinatorics, probability, and statistics. It helps extend the concept of factorials beyond whole numbers, making it useful in complex calculations and analyses.

In Free Span Analysis for Submarine Pipelines as an example. When fluid flows past a pipeline free span, vortices are formed on either side of the free span. These vortices are periodically shed, breaking away from the pipe. If the frequency of vortex shedding coincides with the natural frequency of the pipeline free span, potentially damaging vibrations (resonant) may be set up in the free span. These vibrations may either be in-line with the fluid flow (in-line vibrations) or perpendicular to the fluid flow (cross flow vibrations). Inline and crossflow vibrations can be analysed by a response model, which is an empirical model providing maximum steady state VIV amplitude response as a function of basic hydrodynamic and structural parameters.

A 3-parameter Weibull distribution is often appropriate for modelling of the long-term statistics for the current velocity Uc or significant wave height, Hs. In probability theory and statistics, the Weibull distribution is a continuous probability distribution. It models a broad range of random variables, largely in the nature of a time to failure or time between events.

The probability density function of a Weibull random variable is

If we examine the function closely, we'll find that it is simply a gamma function with (1+1/k) parameter.

Gamma function helps in modeling the probability of failure due to fatigue, which is a critical aspect of ensuring the structural integrity and safety of submarine pipelines.

References:

1- https://en.wikipedia.org/wiki/Gamma_function

2- https://en.wikipedia.org/wiki/Weibull_distribution

3- https://en.wikipedia.org/wiki/Probability_density_function

4- DNV-RP-F105_09_2021 - Free Span