Notes on Foundations of Computational Fluid Dynamics - Module 2

Overview

• Welcome back to the discourse on Computational Fluid Dynamics (CFD).
• Module 2 of the first week.
• Previous class summary: syllabus overview, outcomes from the course, definitions of key terms, and review of equations.

Key Definitions and Concepts

Fluid Characteristics

• Velocity Field: A vector field that represents the velocity of fluid particles in space.
• Vorticity: A measure of the local rotation of the fluid.
• Viscosity: A property of fluid that measures its resistance to flow.
• Fluid Definition Based on Viscosity: Fluids can be classified based on their viscosity characteristics.

Flow Visualization

• Importance: Characterizes and visualizes fluid motion.
• Types of Characteristic Lines:
  • Timeline: Curves formed by fluid particles at a specific instant in time (important for unsteady flow).
  • Path Line: The path followed by individual fluid particles over time (Lagrangian description).
  • Streak Line: The locus of all fluid particles that pass a specific point (visualized by dye injection).
  • Stream Line: Curves tangent to the velocity field, indicating fluid motion direction at any point.

Fluid Dynamics Equations

Basic Equations

• Three Fundamental Equations:
  • Conservation of Mass (Continuity Equation)
  • Conservation of Momentum
  • Conservation of Energy
• Additional equations might be necessary depending on specific problems (e.g., ideal gas equation for compressible flows).

System vs. Control Volume

• System: A fixed identifiable mass that can change boundaries over time.
• Control Volume: A defined volume in space through which fluid can flow, allowing for analysis of mass, momentum, and energy transfer.
• Control Surface: Boundary of the control volume.

Reynolds Transport Theorem

• Relates system and control volume properties.
• Extensive Property (n): Related to intensive properties (e.g., mass, momentum, and energy).
• Mathematical Form:
  n = ∫(n)dm = ∫(n)ρdV

Conservation of Mass

• Integral Form:
  [ \frac{\partial}{\partial t} \int_{V} \rho dV + \int_{S} \rho V \cdot n dA = 0 ]
• Differential Form:
  [ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho V) = 0 ]
• Special cases for steady and incompressible flows simplify these equations.

Conservation of Momentum

• Based on Newton's Second Law (F = ma).
• Equation Structure:
  [ \rho \frac{D V}{D t} = -\nabla P + \mu \nabla^2 V + F_{external} ]
  • Where ( F_{external} ) can represent various forces (e.g., gravity, magnetic).
• Momentum Equation in Scalar Form:
  • Three directional equations: x, y, z components.

Navier-Stokes Equations

• Derived from conservation of mass, momentum, and energy principles.
• Characteristics of Non-linear Terms:
  • Convective acceleration terms have nonlinear behavior, requiring special attention in numerical analysis.

Summary of Important Concepts

• Focus on flow description, conservation equations, and the significance of non-linear convective terms in momentum equations.
• Next class: Further details on momentum equations and conservation of energy.

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Thank you for attending!