Lecture+11_Dimensional+analysis

Dimensional Analysis in Engineering Fluid Mechanics

Introduction

  • Course Name: Engineering Fluid Mechanics (CWR 3201)

  • Instructor: Dr. Kibler

Lecture Outline

  1. Dimensions and Units

    • Fundamental concepts surrounding dimensions and units of measurement.

  2. Dimensional Homogeneity

    • The importance of maintaining consistent dimensions in equations.

  3. Dimensional Analysis and Similarity

    • Introduction to the П (pi) theorem.

Dimensions and Units

  • Primary Dimensions:

    • There are 7 primary dimensions used in fluid mechanics.

    • Units can be assigned to these dimensions, defining various physical quantities.

  • Non-Primary Dimensions:

    • All other dimensions can be derived by combining primary dimensions mathematically.

    • Example:

      • Velocity (V) = Length (L) / Time (t)

      • Formula: 𝑉 = 𝐿/𝑡

Dimensional Homogeneity

  • Definition:

    • All equations must be dimensionally homogeneous; each term in the equation must have the same dimensions.

    • Key Principle:

      • A formula that lacks dimensional homogeneity is incorrect, while one that is dimensionally homogeneous is not guaranteed to be correct.

Physical Models and Prototypes

  • Importance of Dimensional Analysis:

    • Required for simulating small-scale models and extrapolating to full-scale prototypes.

  • Examples:

    • They include model buildings and model cars for experimental designs.

Physical River Models

  • Application:

    • Used to predict behaviors in fluvial systems, such as:

      • Consequences of dam removal.

      • Sediment transport dynamics.

      • Changes in river channels.

  • Case Studies:

    • Marmot Dam Removal (Sandy River, Oregon)

    • Glines Canyon Dam Removal (Elwha River, Washington)

      • Noted as the largest dam removal in history.

Similarity in Dimensional Analysis

  • Conditions of Complete Similarity:

    1. Geometric Similarity:

      • Equivalence of length scales.

    2. Kinematic Similarity:

      • Equivalence of time scales (requires geometric similarity).

    3. Dynamic Similarity:

      • Equivalence of force scales (requires kinematic and geometric similarities).

Model Scaling Example: DoD Reefense Project

  • Objective:

    • Develop habitats to support self-sustaining oyster populations and mitigate coastal erosion and flooding.

  • Key Features:

    • Use of multi-material reef structures with environmentally-friendly materials.

    • Engineered micro and macro-scale topographies to enhance oyster recruitment and wave attenuation.

    • Focus on increasing oyster growth and resilience through genomic selection.

Lab Testing of Scaled Models

  • Testing Phases:

    1. Design

    2. Test

    3. Build

    4. Monitor

П (Pi) Theorem

  • Purpose:

    • To derive nondimensional parameters (∏) for application in scaled models.

  • Definition of ∏:

    • A dimensionless parameter, such as the Reynolds number:

    • Formula: [ Re = \frac{V_{avg} L}{ u} = \frac{\rho V_{avg} L}{\mu} ]

К (Pi) Theorem Relationships

  • Expression:

    • ( П_1 = f(П_2, П_3, П_4, …, П_k) )

    • Establishes dependencies of model parameters on known prototype parameters.

  • Conditional Relationships: [ П_{2,m} = П_{2,p} \text{ AND } П_{3,m} = П_{3,p} … \text{ THEN } П_{1,m} = П_{1,p} ]

Steps in the Pi Theorem

  1. Define dependent variable (z) and independent variables (t, g).

  2. Identify primary dimensions and repeating parameters.

  3. Establish nondimensional groups (Πs).

  4. Example:

    • Deriving the height of a falling ball related to time, initial velocity, and gravitational force.

Nondimensional Variables in Open-Channel Hydraulics

  • Froude Number (Fr):

    • Another dimensionless metric assessing hydraulic conditions.

    • Ratio of inertial forces to gravitational forces.

    • Critical Depth: Defined as the flow depth where inertial and gravitational forces balance (Fr = 1).

  • Flow Conditions:

    • Subcritical Flow (Fr < 1): Downstream conditions influence upstream conditions.

    • Supercritical Flow (Fr > 1): Flow disturbances cannot propagate upstream; energy dissipates rapidly during transitions.

Scaling Free Surface Flows

  • Relevance:

    • Important in civil and water resources engineering applications such as flooding, dam design, and sediment transport.

  • Scaling Requirements:

    • Ensure matching Reynolds and Froude numbers between models and prototypes for accurate predictions, especially in turbulent flows: [ \frac{ν_m}{ν_p} = \left( \frac{L_m}{L_p} \right)^{\frac{3}{2}} ]

  • In turbulent conditions, matching Froude number takes priority over Reynolds number.