3.2 Inviscid and viscous flows

Fluid dynamics explores how liquids and gases move. This section dives into two key flow types: inviscid (ignoring friction) and viscous (considering friction). Understanding these helps us grasp real-world fluid behavior in everything from blood flow to jet streams.

Inviscid flows simplify calculations for high-speed scenarios, while viscous flows account for friction effects. We'll look at governing equations, boundary layers, and flow transitions. This knowledge is crucial for engineering applications and understanding natural phenomena.

Inviscid vs Viscous Flows

Characteristics and Applications

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• Inviscid flows neglect viscosity effects, while viscous flows account for internal fluid friction
• Reynolds number determines the relative importance of inertial forces to viscous forces in fluid flow
  • High Reynolds number flows approximate inviscid conditions (jet streams)
  • Low Reynolds number flows exhibit significant viscous effects (blood flow in capillaries)
• Inviscid flow models apply to high Reynolds number scenarios
  • Used in aerodynamics (flow around aircraft wings)
  • Applicable to large-scale atmospheric flows (global wind patterns)
• Viscous flows feature boundary layers where fluid velocity changes rapidly near solid surfaces
  • Boundary layer thickness varies with flow conditions and surface geometry
  • Affects heat transfer and drag in engineering applications (heat exchangers, vehicle aerodynamics)

Governing Equations and Flow Behavior

• No-slip condition fundamental principle in viscous flows
  • Fluid particles adhere to solid boundaries, creating velocity gradients
  • Leads to development of boundary layers and viscous drag
• Inviscid flow models use simplified equations
  • Euler equations describe inviscid, incompressible flow
  • Neglect viscous terms, reducing computational complexity
• Viscous flows governed by more complex Navier-Stokes equations
  • Include viscous stress terms and heat conduction effects
  • Require advanced numerical methods for solution (finite element analysis, computational fluid dynamics)
• Laminar to turbulent flow transition occurs in viscous fluids
  • Influenced by flow velocity, fluid properties, and surface roughness
  • Critical Reynolds number marks onset of turbulence (pipe flow transitions around Re ≈ 2300)
  • Turbulent flows exhibit chaotic motion and enhanced mixing (atmospheric turbulence, oceanic currents)

Potential Flow and Irrotational Flow

Fundamental Concepts and Equations

• Potential flow represents inviscid, incompressible flow where velocity field expressed as gradient of scalar potential function
  • Velocity potential ϕ defined as $\mathbf{v} = \nabla \phi$
  • Simplifies flow analysis and allows use of powerful mathematical techniques
• Irrotational flow characterized by zero vorticity
  • Vorticity ω defined as curl of velocity field: $\omega = \nabla \times \mathbf{v}$
  • Fluid particles do not rotate as they move through flow field
• Continuity equation for incompressible flow simplifies to Laplace equation in terms of velocity potential
  • $\nabla^2 \phi = 0$
  • Allows use of harmonic function properties in flow analysis
• Stream functions describe two-dimensional incompressible flows
  • Related to velocity potential through Cauchy-Riemann equations
  • Streamlines represent paths of fluid particles in steady flow

Analysis Techniques and Applications

• Complex potential theory combines velocity potential and stream function
  • Utilizes complex analysis techniques to solve two-dimensional potential flow problems
  • Complex potential $F(z) = \phi + i\psi$, where z is complex coordinate
• Conformal mapping transforms complex flow geometries into simpler ones
  • Facilitates solution of potential flow problems around complex shapes
  • Preserves angles and streamline patterns during transformation
• Superposition of elementary flows constructs solutions for complex potential flow problems
  • Elementary flows include uniform flow, source/sink flow, vortex flow
  • Example: flow around circular cylinder obtained by superimposing uniform flow and doublet
• Applications of potential flow theory
  • Aerodynamics (lift calculation for airfoils)
  • Hydrodynamics (wave propagation, ship hull design)
  • Groundwater flow (well hydraulics, contaminant transport)

Viscosity and Energy Dissipation

Viscous Flow Characteristics

• Viscosity measures fluid's resistance to deformation
  • Responsible for energy dissipation in fluid flows
  • Dynamic viscosity μ relates shear stress to velocity gradient: $\tau = \mu \frac{du}{dy}$
• Navier-Stokes equations incorporate viscous effects
  • Govern viscous fluid motion in most practical applications
  • Include terms for convection, diffusion, pressure gradients, and body forces
• Viscous flows exhibit shear stress between adjacent fluid layers
  • Leads to velocity gradients and energy dissipation
  • Results in development of boundary layers near solid surfaces

Energy Dissipation and Boundary Layer Theory

• Viscous dissipation function quantifies rate of mechanical energy conversion to heat
  • Due to internal friction in fluid
  • Important in high-speed flows and lubrication theory
• Boundary layer theory developed by Ludwig Prandtl
  • Describes thin region near solid surfaces where viscous effects dominate
  • Divides flow into viscous boundary layer and inviscid outer flow
• Blasius solution provides analytical description of laminar boundary layer velocity profile over flat plate
  • Self-similar solution to boundary layer equations
  • Predicts boundary layer thickness growth as $\delta \sim \sqrt{x}$, where x is distance along plate
• Viscosity influences drag forces on objects moving through fluids
  • Skin friction drag results from viscous shear stresses
  • Form drag caused by pressure differences due to flow separation
  • Total drag coefficient varies with Reynolds number and object geometry

Vorticity in Fluid Flow

Vorticity Fundamentals and Transport

• Vorticity measures local rotation in fluid flow
  • Defined as curl of velocity field: $\omega = \nabla \times \mathbf{v}$
  • Vector quantity indicating axis of rotation and magnitude
• Vorticity transport equation describes evolution of vorticity in fluid flow
  • Includes terms for vortex stretching, tilting, and diffusion
  • $\frac{D\omega}{Dt} = (\omega \cdot \nabla)\mathbf{v} + \nu\nabla^2\omega$
• Kelvin's circulation theorem states circulation conservation in inviscid, barotropic flows
  • Provides insights into vortex dynamics and stability
  • Circulation $\Gamma = \oint \mathbf{v} \cdot d\mathbf{l} = \int_S \omega \cdot d\mathbf{S}$

Vortex Dynamics and Applications

• Vortex lines and vortex tubes fundamental concepts in rotational fluid flows
  • Vortex lines tangent to vorticity vector at every point
  • Vortex tubes formed by collection of vortex lines
• Biot-Savart law relates induced velocity field to vorticity distribution
  • $\mathbf{v}(\mathbf{r}) = \frac{1}{4\pi}\int_V \frac{\omega(\mathbf{r}') \times (\mathbf{r} - \mathbf{r}')}{|\mathbf{r} - \mathbf{r}'|^3} d\mathbf{r}'$
  • Used in analysis of vortex-induced flows (wing tip vortices, tornado dynamics)
• Helmholtz's vortex theorems describe behavior of vortex filaments in inviscid fluids
  • Include continuity and conservation properties
  • Provide foundation for understanding vortex dynamics in ideal fluids
• Vortex shedding occurs when fluid flows past bluff bodies
  • Leads to formation of von Kármán vortex street
  • Causes flow-induced vibrations in structures (bridges, tall buildings)
  • Frequency of vortex shedding characterized by Strouhal number: $St = \frac{fD}{U}$