Glossary

11.2 Fluid dynamics and the Navier-Stokes equations

5 min read•august 15, 2024

Fluid dynamics is the study of how liquids and gases move. The are the key to understanding this complex behavior, describing how fluids flow under various forces and conditions.

These equations are crucial in many fields, from weather prediction to designing airplanes. They help us model real-world fluid behavior, but solving them can be tricky due to their complexity.

Derivation of Navier-Stokes Equations

Conservation Laws and Principles

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  • Conservation laws in fluid dynamics encompass mass conservation (), momentum conservation, and energy conservation
  • Navier-Stokes equations derived by applying Newton's second law to fluid motion considering internal and external forces
  • Continuity equation expresses relationship between fluid density and based on mass conservation principle
  • Momentum conservation yields momentum equation accounting for convective acceleration, gradients, and viscous forces
  • Energy conservation principle used to derive energy equation incorporating heat transfer and viscous dissipation
  • Complete set of Navier-Stokes equations consists of continuity equation, momentum equations (one for each spatial dimension), and energy equation
  • Assumptions and simplifications often applied during derivation process (incompressibility, constant ) to obtain more manageable forms

Derivation Process and Components

  • Start with control volume analysis to apply conservation laws to fluid elements
  • Express mass conservation as rate of change of mass within control volume equals net mass flux across boundaries
  • Apply Newton's second law to fluid element to derive momentum equations
  • Account for surface forces (pressure, viscous stresses) and body forces (gravity) in momentum equations
  • Utilize constitutive relations (Newtonian fluid assumption) to relate stress to strain rate
  • Derive energy equation by applying first law of thermodynamics to fluid element
  • Incorporate Fourier's law of heat conduction for thermal energy transport
  • Combine derived equations to form complete set of Navier-Stokes equations

Terms in Navier-Stokes Equations

Velocity and Acceleration Terms

  • Time derivative term (∂u∂t\frac{\partial u}{\partial t}) represents local acceleration of fluid describing velocity changes at fixed point in space over time
  • Convective acceleration term (u⋅∇uu \cdot \nabla u) accounts for velocity change due to fluid motion from one location to another with different velocities
  • Material derivative (DuDt=∂u∂t+u⋅∇u\frac{Du}{Dt} = \frac{\partial u}{\partial t} + u \cdot \nabla u) combines local and convective acceleration terms
  • Velocity field u(x,y,z,t)u(x, y, z, t) describes fluid motion in three-dimensional space and time
  • Acceleration field a(x,y,z,t)a(x, y, z, t) represents rate of change of velocity field

Pressure and Force Terms

  • Pressure gradient term (−∇pρ-\frac{\nabla p}{\rho}) describes force exerted on fluid due to pressure differences causing flow from high to low pressure regions
  • Viscous term (ν∇2u\nu \nabla^2 u) represents diffusion of momentum due to fluid viscosity accounting for internal friction between fluid layers
  • (ff) includes external forces acting on fluid (gravity, electromagnetic forces)
  • τij\tau_{ij} represents internal forces acting on fluid element surfaces
  • ϵij\epsilon_{ij} describes deformation rate of fluid element

Additional Terms and Conditions

  • Incompressibility condition (∇⋅u=0\nabla \cdot u = 0) ensures fluid volume remains constant during flow (applicable to many liquid flows and low-speed gas flows)
  • Relationship between stress and strain rate tensors describes how fluid deforms under applied forces
  • (ω=∇×u\omega = \nabla \times u) measures local rotation of fluid elements
  • (ν=μρ\nu = \frac{\mu}{\rho}) represents ratio of dynamic viscosity to density
  • (α=kρcp\alpha = \frac{k}{\rho c_p}) in energy equation describes rate of heat diffusion in fluid

Solving Navier-Stokes Equations

Simplified Forms and Analytical Solutions

  • simplifies for very slow, viscous flows where inertial forces negligible compared to viscous forces
  • applies to inviscid, irrotational flows allowing analytical solutions using complex analysis techniques (airfoil theory)
  • derived by neglecting viscous terms in Navier-Stokes equations applicable to high flows away from boundaries
  • Parallel flow solutions obtained for steady, unidirectional flows between parallel plates or in pipes (, )
  • approximations simplify equations for high Reynolds number flows near solid surfaces leading to boundary layer equations
  • applies to thin film flows where viscous forces dominate and flow predominantly in one direction (journal bearings)
  • find approximate solutions for slightly perturbed flows from known exact solutions (small amplitude water waves)

Numerical Methods and Computational Approaches

  • Finite difference methods discretize spatial and temporal derivatives on structured grids
  • divide domain into control volumes and apply conservation laws
  • use variational formulation and basis functions for spatial discretization
  • employ global basis functions for high-accuracy solutions in simple geometries
  • resolves all scales of turbulent motion without modeling
  • resolves large-scale motions and models small-scale
  • models time-averaged equations with turbulence closure models

Fluid Behavior with Navier-Stokes

Flow Regimes and Transitions

  • Reynolds number (Re=ρULμRe = \frac{\rho UL}{\mu}) indicates relative importance of inertial forces to viscous forces helping predict flow regimes
  • occurs at low Reynolds numbers characterized by smooth, predictable fluid motion with parallel streamlines (blood flow in small vessels)
  • Turbulent flow occurring at high Reynolds numbers characterized by chaotic, irregular fluid motion with rapid velocity fluctuations (atmospheric flows)
  • Transition from laminar to turbulent flow analyzed using linear stability analysis of Navier-Stokes equations
  • marks onset of turbulence varying with flow geometry and conditions (pipe flow, boundary layer flow)
  • observed in transitional flows with alternating laminar and turbulent regions

Vorticity and Boundary Layer Dynamics

  • Vorticity dynamics described by vorticity equation derived from Navier-Stokes explain formation and evolution of rotational motion in fluids
  • relates vorticity evolution to viscous and baroclinic effects
  • Boundary layer separation predicted by analyzing pressure gradient and velocity profiles near solid surfaces using Navier-Stokes equations
  • forms behind bluff bodies due to periodic vortex shedding
  • develops in rotating fluids due to balance between Coriolis force and viscous force
  • intensifies vorticity in three-dimensional flows leading to energy cascade in turbulence

Non-Newtonian and Complex Fluid Behavior

  • requires modifications to standard Navier-Stokes equations to account for complex rheological properties
  • exhibit decreased viscosity with increasing shear rate (ketchup, paint)
  • show increased viscosity with increasing shear rate (cornstarch suspension)
  • display both viscous and elastic properties (polymer solutions)
  • exhibit yield stress below which they behave as solids (toothpaste)
  • couples Navier-Stokes equations with Maxwell's equations for electrically conducting fluids (plasma, liquid metals)

Key Terms to Review (49)

Aerodynamics: Aerodynamics is the study of the behavior of air as it interacts with solid objects, particularly those in motion. This field focuses on understanding how forces like lift, drag, and thrust affect the performance of objects like aircraft, cars, and even buildings in windy conditions. The principles of aerodynamics are rooted in fluid dynamics and are essential for designing efficient vehicles and structures that can effectively navigate through air.
Bingham Plastics: Bingham plastics are a type of non-Newtonian fluid that exhibit both solid and liquid characteristics, defined by a yield stress below which the material behaves like a solid and above which it flows like a viscous fluid. This unique behavior is important for understanding how certain materials, such as muds, slurries, and pastes, behave under stress, especially in fluid dynamics applications.
Body force term: The body force term refers to forces that act throughout the volume of a fluid rather than at its boundaries. This includes forces such as gravity, electromagnetic forces, or any other distributed force that affects the fluid as a whole. Understanding the body force term is crucial in analyzing fluid motion, especially when applying the Navier-Stokes equations, which describe how velocity fields evolve in response to these forces.
Boundary layer: A boundary layer is a thin region adjacent to a boundary, such as a surface or interface, where the effects of viscosity are significant, leading to velocity and temperature gradients. This concept is crucial in understanding how fluids behave near solid surfaces, influencing phenomena like drag and heat transfer in various applications.
Continuity equation: The continuity equation is a fundamental principle in fluid dynamics that expresses the conservation of mass in a flowing fluid. It states that the rate of mass entering a control volume must equal the rate of mass exiting that volume, which can be mathematically represented as $$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0$$, where $$\rho$$ is the fluid density and $$\mathbf{v}$$ is the flow velocity. This equation plays a crucial role in deriving the Navier-Stokes equations, linking fluid motion and changes in density over time, ensuring that mass is conserved as fluids move and change within a system.
Couette flow: Couette flow refers to the steady, laminar flow of a viscous fluid between two parallel plates, where one plate is stationary and the other is moving. This flow illustrates fundamental principles of fluid dynamics, particularly in the study of shear stress and velocity profiles in response to applied forces, often analyzed through the Navier-Stokes equations.
Critical Reynolds Number: The critical Reynolds number is a dimensionless quantity used in fluid dynamics that indicates the transition between laminar and turbulent flow. It is defined as the point where the flow starts to become unstable and chaotic, leading to a change in the behavior of the fluid. This number is vital in understanding how fluids behave under different conditions, particularly when analyzing the Navier-Stokes equations which describe the motion of viscous fluid substances.
Direct numerical simulation (DNS): Direct numerical simulation (DNS) is a computational approach used to solve the Navier-Stokes equations directly, without any turbulence modeling. It provides detailed and high-fidelity predictions of fluid flow by resolving all relevant scales of motion, which makes it particularly useful in studying complex fluid dynamics phenomena.
Ekman Layer: The Ekman layer is a thin layer of fluid near the surface of the ocean or atmosphere where the flow is influenced by both friction and the Coriolis effect. In this layer, the movement of water or air is affected by the balance between the wind stress and the Earth's rotation, resulting in a spiraling pattern of flow that diverges with depth. This concept is essential for understanding how wind interacts with ocean currents and plays a role in large-scale fluid dynamics.
Euler Equations: Euler equations are a set of hyperbolic partial differential equations that describe the motion of inviscid fluid flow. They are foundational in fluid dynamics, particularly in the study of compressible flows, and are derived from the conservation laws of mass, momentum, and energy. Understanding these equations is crucial for analyzing various fluid behaviors, including shock waves and wave propagation.
Finite Difference Method: The finite difference method is a numerical technique used to approximate solutions to differential equations by discretizing them into finite difference equations. This method transforms continuous derivatives into discrete differences, enabling the analysis of various partial differential equations through grid-based approximations.
Finite Element Methods: Finite Element Methods (FEM) are numerical techniques used to find approximate solutions to complex problems described by partial differential equations. By breaking down a large problem into smaller, simpler parts called finite elements, FEM allows for the analysis of complicated geometries, material properties, and boundary conditions, making it particularly valuable in engineering and applied sciences.
Finite Volume Methods: Finite volume methods are numerical techniques used for solving partial differential equations, especially those arising in fluid dynamics. They work by dividing the domain into a finite number of small volumes and applying the integral form of the conservation laws to these volumes, allowing for the calculation of fluxes across the volume boundaries. This approach is particularly effective for problems governed by the Navier-Stokes equations, which describe the motion of fluid substances.
Incompressible Flow: Incompressible flow refers to the type of fluid motion where the fluid density remains constant throughout the flow field. This simplification is particularly useful in fluid dynamics as it allows for a more straightforward analysis, especially in situations involving low-speed flows where density changes are negligible. Understanding incompressible flow is essential for applying the Navier-Stokes equations, which describe how fluids move under various conditions, including viscosity and pressure variations.
Inlet Boundary Condition: An inlet boundary condition is a specific type of boundary condition used in fluid dynamics to define the behavior of fluid entering a computational domain. It sets the values for parameters such as velocity, pressure, and temperature at the inlet, influencing how fluid flows through the entire system. These conditions are crucial for solving the Navier-Stokes equations, as they provide the necessary information to initiate simulations and analyze fluid behavior in various applications.
Intermittency: Intermittency refers to the irregular and sporadic nature of certain physical phenomena, often observed in fluid dynamics where flow patterns fluctuate between different states, such as laminar and turbulent flow. This behavior can significantly impact the stability and predictability of systems governed by the Navier-Stokes equations, which describe the motion of fluid substances. Understanding intermittency is essential for predicting flow behaviors and designing effective fluid systems.
Kelvin's Circulation Theorem: Kelvin's Circulation Theorem states that the circulation around a closed curve moving with the flow of an inviscid, incompressible fluid remains constant over time. This theorem connects to the fundamental principles of fluid dynamics and is crucial for understanding the behavior of fluid motion, particularly in relation to vorticity and the conservation of momentum.
Kinematic viscosity: Kinematic viscosity is a measure of a fluid's internal resistance to flow under the influence of gravity, defined as the ratio of dynamic viscosity to fluid density. This property is crucial in fluid dynamics as it affects how fluids move and behave in different situations. Kinematic viscosity helps predict flow patterns and is essential for understanding the behavior of fluids in various applications, such as engineering, meteorology, and oceanography.
Laminar flow: Laminar flow is a type of fluid motion characterized by smooth, orderly layers of fluid that flow parallel to each other, typically occurring at lower velocities and with low turbulence. This flow regime is crucial in understanding how fluids behave under various conditions, particularly when it comes to analyzing the forces acting on them as described by the Navier-Stokes equations. The study of laminar flow helps in predicting how fluids interact with surfaces and other fluids, which is fundamental in many applications such as engineering and physics.
Large eddy simulation (LES): Large eddy simulation (LES) is a mathematical modeling technique used in fluid dynamics to simulate the turbulent flows of fluids by resolving the large-scale structures of turbulence while modeling the smaller scales. This approach allows for a more accurate representation of turbulent flows compared to traditional methods, as it captures the dominant energy-carrying eddies in the flow. LES plays a crucial role in understanding complex fluid behavior and helps predict how fluids interact with their environments.
Lubrication Theory: Lubrication theory is a simplified approach to fluid dynamics that focuses on the behavior of thin films of fluid, which are typically used to reduce friction and wear between solid surfaces. This theory derives its significance from applications where the fluid layer's thickness is much smaller than the characteristic length scale of the problem, allowing for approximations that simplify the governing equations, often leading to the Navier-Stokes equations.
Mach number: Mach number is a dimensionless quantity used in fluid dynamics that represents the ratio of the speed of an object moving through a fluid to the speed of sound in that same fluid. It is a crucial concept when analyzing compressible flow, particularly when determining the behavior of fluids at high velocities, and plays a significant role in understanding shock waves and flow regimes.
Magnetohydrodynamics (MHD): Magnetohydrodynamics (MHD) is the study of the behavior of electrically conducting fluids in the presence of magnetic fields. This field combines principles of fluid dynamics and electromagnetism, focusing on how magnetic forces interact with the motion of conductive fluids such as plasmas, liquid metals, and electrolytes. MHD plays a critical role in various applications including astrophysics, engineering, and geophysics, where understanding the flow and behavior of these fluids under magnetic influences is essential.
Method of characteristics: The method of characteristics is a technique used to solve certain types of partial differential equations (PDEs), particularly first-order PDEs, by transforming the PDE into a set of ordinary differential equations along characteristic curves. This approach allows for tracking the evolution of solutions over time, making it especially useful in contexts where shock formation and discontinuities are present.
Navier-Stokes equations: The Navier-Stokes equations are a set of nonlinear partial differential equations that describe the motion of fluid substances, capturing the relationship between the velocity field and the pressure within a fluid. These equations are fundamental in fluid dynamics, allowing us to model various phenomena, from simple flows to complex turbulent behavior, linking closely with classification types and conservation laws.
No-slip condition: The no-slip condition is a fundamental concept in fluid dynamics that states the velocity of a fluid at a solid boundary is equal to the velocity of the boundary itself. This means that at the interface between a fluid and a solid surface, the fluid has zero relative velocity, causing it to stick to the surface. This principle is crucial for accurately modeling fluid flow around objects and plays a significant role in the Navier-Stokes equations.
Non-newtonian fluid behavior: Non-newtonian fluid behavior refers to the characteristics of fluids whose viscosity changes under different conditions of stress or shear rate, unlike Newtonian fluids that have a constant viscosity. This unique property means that non-newtonian fluids can exhibit complex behaviors, such as becoming thinner or thicker when stirred or subjected to force. Understanding non-newtonian fluids is crucial in fluid dynamics and is modeled by the Navier-Stokes equations, which describe how these fluids flow under varying forces.
Perturbation methods: Perturbation methods are mathematical techniques used to find an approximate solution to a problem that cannot be solved exactly. By introducing a small parameter to the system, these methods allow for the analysis of changes in the solution due to slight variations in the parameters or initial conditions. This approach is particularly useful in various fields, as it provides insights into complex systems by simplifying them without losing significant detail.
Poiseuille Flow: Poiseuille flow refers to the laminar flow of a viscous fluid through a cylindrical pipe, characterized by a parabolic velocity profile. This flow regime is governed by the balance of viscous forces and pressure gradients, and it provides key insights into fluid behavior in scenarios involving low Reynolds numbers, where inertial forces are negligible compared to viscous forces.
Potential flow theory: Potential flow theory is a mathematical approach used to describe the flow of incompressible and irrotational fluids. This theory simplifies fluid dynamics by assuming that the fluid has no viscosity and the flow can be represented as a potential function, allowing for the analysis of flow patterns and behaviors in various scenarios.
Pressure: Pressure is the force exerted per unit area on a surface, often described in terms of fluid dynamics. It plays a crucial role in understanding how fluids behave, influencing flow patterns, stability, and the interactions between fluids and solid boundaries. In the study of equations that govern fluid motion, pressure emerges as a central variable, impacting the equations used to describe various physical phenomena.
Reynolds Number: Reynolds Number is a dimensionless quantity used to predict flow patterns in different fluid flow situations. It helps determine whether the flow will be laminar or turbulent, based on the ratio of inertial forces to viscous forces. Understanding this number is crucial for analyzing fluid dynamics and plays a vital role in solving the Navier-Stokes equations, which describe how fluids move.
Reynolds-Averaged Navier-Stokes (RANS): Reynolds-Averaged Navier-Stokes (RANS) refers to a set of equations used to describe the motion of fluid substances, accounting for the effects of turbulence by averaging the Navier-Stokes equations over time. This approach simplifies the complexity of turbulent flow by separating the mean flow from the fluctuating components, making it easier to analyze and predict fluid behavior in various applications, such as engineering and meteorology.
Shear-thickening fluids: Shear-thickening fluids are non-Newtonian fluids that increase in viscosity when subjected to shear stress, meaning they become thicker and more resistant to flow as the applied force increases. This behavior is contrary to shear-thinning fluids, which become less viscous with increased shear. The properties of shear-thickening fluids have significant implications in fluid dynamics, particularly in the context of the Navier-Stokes equations, where their unique behavior can affect flow patterns and stability.
Shear-thinning fluids: Shear-thinning fluids, also known as pseudoplastic fluids, are materials whose viscosity decreases when subjected to shear stress or strain. This behavior is significant in fluid dynamics, as it affects how these fluids flow and interact with forces in different environments, especially when analyzing the motion of fluids governed by the Navier-Stokes equations.
Shock Wave: A shock wave is a type of propagating disturbance that moves faster than the speed of sound in a medium, creating a sharp change in pressure, temperature, and density. These waves are often formed in high-speed flows, such as those seen in supersonic flight or during explosive events, and they result in sudden changes that can drastically affect the surrounding environment. The understanding of shock waves is crucial for analyzing and predicting behaviors in fluid dynamics and related fields.
Spectral methods: Spectral methods are a class of numerical techniques used to solve differential equations by transforming the problem into a spectral space, often utilizing orthogonal functions like Fourier series or polynomials. These methods are highly effective for problems with smooth solutions and can provide very accurate results, particularly when analyzing complex phenomena like shock formation and stability in fluid dynamics.
Stokes Flow Equation: The Stokes Flow Equation describes the motion of viscous fluid under the influence of very low Reynolds numbers, where inertial forces are negligible compared to viscous forces. This equation is significant in fluid dynamics as it simplifies the Navier-Stokes equations by focusing on the balance between pressure and viscous stress, making it applicable in situations like the flow around small particles or bubbles in a fluid.
Strain rate tensor: The strain rate tensor is a mathematical representation that quantifies the rate of deformation of a material when it is subjected to stress. This tensor is essential for describing how fluids deform under various flow conditions and plays a critical role in the formulation of the Navier-Stokes equations, which model fluid dynamics. Understanding the strain rate tensor helps in analyzing the behavior of fluids, particularly in cases of viscous flow where the deformation is continuous over time.
Stress Tensor: The stress tensor is a mathematical representation that describes the internal forces acting within a material body, providing insight into how these forces are distributed and how they impact the material's deformation. It is crucial for understanding fluid dynamics and the behavior of materials under various forces, particularly in the context of the Navier-Stokes equations, which govern the motion of fluid substances.
Thermal diffusivity: Thermal diffusivity is a measure of how quickly heat can spread through a material. It is defined as the ratio of the material's thermal conductivity to its volumetric heat capacity, and it plays a critical role in understanding how temperature changes propagate through materials in fluid dynamics and related equations. A higher thermal diffusivity indicates that a material can reach thermal equilibrium faster, which is essential for processes involving heat transfer in fluids.
Turbulence: Turbulence refers to the chaotic, irregular motion of fluid particles that occurs when a fluid flows at high velocities or through complex geometries. It is characterized by vortices, eddies, and rapid variations in pressure and flow velocity, significantly impacting the behavior of fluids. This phenomenon plays a crucial role in understanding fluid dynamics and is described mathematically by the Navier-Stokes equations, which model the motion of viscous fluid substances.
Velocity Field: A velocity field is a mathematical representation of the velocity of a fluid at different points in space, often depicted as a vector field. This concept is crucial in fluid dynamics, as it provides insight into how fluids flow and interact with their surroundings, allowing for the formulation of equations that describe fluid behavior, including the Navier-Stokes equations. Understanding the velocity field helps in analyzing fluid motion and the forces acting within it.
Viscoelastic fluids: Viscoelastic fluids are materials that exhibit both viscous and elastic properties when deformed. This means they have the ability to flow like a viscous liquid under certain conditions, while also being able to return to their original shape, like an elastic solid, when the deforming stress is removed. Understanding these fluids is crucial in analyzing complex fluid dynamics, especially in systems governed by the Navier-Stokes equations.
Viscosity: Viscosity is a measure of a fluid's resistance to deformation and flow, essentially quantifying how 'thick' or 'sticky' a fluid is. It plays a crucial role in fluid dynamics, affecting how fluids move and interact with surfaces. The concept of viscosity is integral to understanding the behavior of fluids described by the Navier-Stokes equations, as it directly influences the momentum transfer within the fluid and is pivotal in predicting flow patterns and turbulence.
Von Kármán vortex street: A von Kármán vortex street is a repeating pattern of swirling vortices caused by the unsteady separation of flow around an obstacle, such as a cylinder, in a fluid. This phenomenon is critical in fluid dynamics, illustrating how fluid flows can become unstable and lead to organized structures in the wake of objects, which can be analyzed using the Navier-Stokes equations to understand the underlying mechanics.
Vortex stretching mechanism: The vortex stretching mechanism is a phenomenon in fluid dynamics where the distortion of vorticity (the measure of rotation in a fluid) occurs due to the stretching of vortices. This process enhances the intensity of vorticity, leading to the amplification of turbulent motions in fluids, particularly in the context of the Navier-Stokes equations that describe fluid flow. The vortex stretching contributes to the transfer of energy within turbulent flows, making it a crucial aspect for understanding turbulence development and behavior.
Vorticity: Vorticity is a measure of the local rotation of fluid elements in a flow field, quantifying the tendency of particles to spin about an axis. It plays a crucial role in fluid dynamics by helping to describe the motion of fluids, especially in relation to turbulence and the behavior of vortex structures. The concept is integral to understanding complex flow patterns and is directly linked to the Navier-Stokes equations, which govern the movement of fluid substances.
Weather modeling: Weather modeling is the use of mathematical equations and computer simulations to represent and predict atmospheric phenomena. These models help meteorologists understand complex weather systems by utilizing principles from fluid dynamics, thermodynamics, and the Navier-Stokes equations to forecast conditions like temperature, precipitation, and wind patterns over time.
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