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Fluid Mechanics

8.1 Buckingham Pi Theorem

3 min readLast Updated on July 19, 2024

The Buckingham Pi Theorem is a game-changer in fluid mechanics. It simplifies complex problems by reducing variables to dimensionless groups called pi terms. This powerful tool helps identify key parameters and apply dynamic similarity across different systems.

By following specific steps, you can use this theorem to create dimensionless parameters like Reynolds number and Froude number. These numbers provide crucial insights into fluid flow behavior, making it easier to compare and analyze various fluid mechanics scenarios.

Buckingham Pi Theorem

Principles of Buckingham Pi Theorem

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  • Powerful tool in dimensional analysis reduces number of variables by combining them into dimensionless groups called pi terms (π\pi terms)
  • States that a physically meaningful equation with nn variables can be rewritten using p=nkp = n - k dimensionless parameters, where kk is the number of independent physical dimensions
  • Particularly useful in fluid mechanics to simplify complex problems, identify important parameters, and apply dynamic similarity
    • Dynamic similarity allows comparison of fluid flow behavior between different systems (scaled models and real-world applications)
  • Based on the principle of dimensional homogeneity, which requires the dimensions of terms on both sides of an equation to be the same

Application for dimensionless parameters

  • To apply the Buckingham Pi Theorem, follow these steps:
    1. List all variables involved in the problem, including dependent and independent variables
    2. Determine the number of independent dimensions (kk) among the variables (mass, length, time)
    3. Choose kk repeating variables that include all independent dimensions, usually density (ρ\rho), velocity (VV), and length (LL)
    4. Form p=nkp = n - k dimensionless pi terms by combining each remaining variable with the repeating variables to ensure the resulting term is dimensionless
  • When forming pi terms, follow these rules:
    • Each pi term must be dimensionless
    • Each pi term should include one non-repeating variable
    • Repeating variables can be raised to different powers in each pi term to achieve dimensionless groups
  • The resulting dimensionless pi terms will be the key parameters governing the fluid flow problem

Significance of dimensionless parameters

  • Dimensionless parameters obtained from the Buckingham Pi Theorem often have physical interpretations that provide insights into fluid flow behavior
  • Common dimensionless numbers in fluid mechanics include:
    • Reynolds number (Re): Ratio of inertial forces to viscous forces, indicates relative importance of these forces in a flow, characterizes laminar and turbulent flows
    • Froude number (Fr): Ratio of inertial forces to gravitational forces, used in open-channel flows and flows with a free surface
    • Mach number (Ma): Ratio of flow velocity to the speed of sound in the fluid, characterizes compressible flows
    • Strouhal number (St): Relates frequency of vortex shedding to flow velocity and a characteristic length, often used in unsteady flows and vortex dynamics
  • These dimensionless numbers enable comparison of fluid flow behavior across different scales, geometries, and fluid properties by applying dynamic similarity principles

Variable reduction with Pi Theorem

  • Consider fluid flow through a pipe, where pressure drop (ΔP\Delta P) depends on pipe diameter (DD), pipe length (LL), fluid velocity (VV), fluid density (ρ\rho), and fluid dynamic viscosity (μ\mu)
    • Variables involved: ΔP\Delta P, DD, LL, VV, ρ\rho, and μ\mu
    • Three independent dimensions: mass (M), length (L), and time (T)
  • Choose ρ\rho, VV, and DD as repeating variables, as they include all independent dimensions
  • Form three pi terms (π1\pi_1, π2\pi_2, π3\pi_3) using the remaining variables:
    • π1=ΔPρV2\pi_1 = \frac{\Delta P}{\rho V^2} (pressure coefficient)
    • π2=LD\pi_2 = \frac{L}{D} (aspect ratio)
    • π3=ρVDμ\pi_3 = \frac{\rho VD}{\mu} (Reynolds number)
  • Original problem with six variables can now be expressed as a relationship between three dimensionless pi terms:
    • π1=f(π2,π3)\pi_1 = f(\pi_2, \pi_3) or ΔPρV2=f(LD,ρVDμ)\frac{\Delta P}{\rho V^2} = f(\frac{L}{D}, \frac{\rho VD}{\mu})
  • Reduction in variables simplifies the analysis and helps identify key parameters governing fluid flow behavior in the pipe
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© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.

© 2025 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.