Glossary

12.4 Computational fluid dynamics

5 min read•august 16, 2024

Computational fluid dynamics (CFD) is a powerful tool for simulating fluid flow. It uses numerical methods to solve complex equations that govern fluid behavior. CFD helps engineers and scientists understand and predict flow patterns in various applications, from aerodynamics to weather forecasting.

This section dives into the math behind CFD. We'll look at the key equations, discretization techniques, and numerical schemes used to model fluid flow. We'll also explore stability and accuracy issues that arise when solving these equations computationally.

Governing equations for fluid flow

Conservation laws and Navier-Stokes equations

Top images from around the web for Conservation laws and Navier-Stokes equations

  • On the role of the Helmholtz-Leray projector in the space discretization of the Navier-Stokes ... View original

    Is this image relevant?

  • Index notation with Navier-Stokes equations - Physics Stack Exchange View original

    Is this image relevant?

  • On the role of the Helmholtz-Leray projector in the space discretization of the Navier-Stokes ... View original

    Is this image relevant?

1 of 3

Top images from around the web for Conservation laws and Navier-Stokes equations

  • On the role of the Helmholtz-Leray projector in the space discretization of the Navier-Stokes ... View original

    Is this image relevant?

  • Index notation with Navier-Stokes equations - Physics Stack Exchange View original

    Is this image relevant?

  • On the role of the Helmholtz-Leray projector in the space discretization of the Navier-Stokes ... View original

    Is this image relevant?

1 of 3
  • of mass, momentum, and energy form the foundation for deriving fluid flow equations
  • describe the motion of viscous fluid substances incorporating momentum and continuity equations
  • expresses conservation of mass in a fluid system relating fluid density, velocity, and time
    • Example: For incompressible flow, the continuity equation simplifies to ∇⋅v=0\nabla \cdot \mathbf{v} = 0
  • derived from Newton's second law describes the balance of forces acting on a fluid element
    • Example: The momentum equation for a Newtonian fluid takes the form ρDvDt=−∇p+μ∇2v+ρg\rho \frac{D\mathbf{v}}{Dt} = -\nabla p + \mu \nabla^2\mathbf{v} + \rho\mathbf{g}
  • based on the first law of thermodynamics accounts for energy transfer and conversion in fluid systems
    • Example: For a compressible flow, the energy equation can be expressed as ρcpDTDt=DpDt+∇⋅(k∇T)+Φ\rho c_p \frac{DT}{Dt} = \frac{Dp}{Dt} + \nabla \cdot (k\nabla T) + \Phi

Boundary conditions and simplifications

  • Boundary conditions and initial conditions essential components in formulating complete set of governing equations for specific fluid flow problems
    • Example: No-slip condition at solid walls (velocity of fluid at wall equals velocity of wall)
    • Example: Specified inlet velocity profile for flow entering a domain
  • Simplifications and assumptions applied to general equations to derive specialized forms for specific fluid flow scenarios
    • (constant density) simplifies continuity equation
    • eliminate time-dependent terms
  • Dimensionless numbers used to characterize flow regimes and simplify equations
    • (ratio of inertial forces to viscous forces)
    • (ratio of flow velocity to speed of sound)

Discretization of fluid flow equations

Finite difference methods

  • approximate derivatives using Taylor series expansions replacing continuous derivatives with discrete differences
  • Forward, backward, and used for spatial discretization
    • Example: Central difference approximation for second derivative: ∂2u∂x2≈ui+1−2ui+ui−1Δx2\frac{\partial^2 u}{\partial x^2} \approx \frac{u_{i+1} - 2u_i + u_{i-1}}{\Delta x^2}
  • Explicit and for unsteady problems
    • Example: Forward Euler (explicit) scheme: un+1−unΔt=f(un)\frac{u^{n+1} - u^n}{\Delta t} = f(u^n)
  • Advantages include simplicity and ease of implementation
  • Challenges include handling complex geometries and maintaining stability

Finite volume methods

  • Finite volume methods divide domain into control volumes applying conservation laws to each volume to derive discretized equations
  • Integral form of conservation laws used as starting point for discretization
  • Flux calculations at control volume faces crucial for accuracy
    • Example: Upwind scheme for convective flux calculation
  • Suitable for complex geometries and ensuring conservation properties
  • Cell-centered and vertex-centered approaches for variable storage

Finite element methods

  • use variational principles to approximate solutions dividing domain into elements and using for interpolation
  • Weak form of governing equations derived using weighted residual methods (Galerkin method)
  • Element types include triangular and quadrilateral elements in 2D, tetrahedral and hexahedral in 3D
  • Shape functions used for interpolation within elements
    • Example: Linear shape functions for 1D element: N1(x)=1−xL,N2(x)=xLN_1(x) = 1 - \frac{x}{L}, \quad N_2(x) = \frac{x}{L}
  • Well-suited for complex geometries and higher-order approximations
  • Assembly process combines element contributions into global system of equations

Numerical schemes for Navier-Stokes

Pressure-velocity coupling algorithms

  • used to handle pressure term in incompressible flow simulations
  • SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithm
    • Iterative process for solving pressure and velocity fields
    • Pressure correction step to enforce continuity
  • PISO (Pressure Implicit with Splitting of Operators) algorithm
    • Non-iterative approach suitable for transient simulations
    • Multiple pressure correction steps for improved accuracy
  • separate pressure and velocity calculations
    • Example: Projection method for incompressible Navier-Stokes equations

Discretization schemes for convective terms

  • account for flow direction in convective term discretization
    • First-order upwind scheme (robust but diffusive)
    • Higher-order upwind schemes (QUICK, MUSCL) for improved accuracy
  • Central difference schemes provide second-order accuracy but may introduce oscillations
  • combine upwind and central difference approaches
    • Example: Flux limiter methods for shock capturing in compressible flows

Time-marching schemes and solvers

  • (forward Euler, Runge-Kutta) simple but stability-limited
  • (backward Euler, Crank-Nicolson) allow larger time steps
  • accelerate convergence for large-scale problems
    • Geometric multigrid for structured grids
    • Algebraic multigrid for unstructured grids
  • Iterative solvers for large linear systems
    • Gauss-Seidel and Jacobi methods
    • Krylov subspace methods (GMRES, BiCGSTAB)

Stability and accuracy of numerical methods

Stability analysis techniques

  • Courant-Friedrichs-Lewy (CFL) condition necessary for stability in explicit time-marching schemes for hyperbolic PDEs
    • Example: For 1D advection equation, CFL condition: uΔtΔx≤1\frac{u\Delta t}{\Delta x} \leq 1
  • assesses stability of finite difference schemes for linear PDEs
    • Fourier analysis of error propagation
    • Amplification factor determines stability
  • for systems of equations
    • Eigenvalue analysis of amplification matrix

Accuracy assessment and error analysis

  • Consistency, stability, and convergence fundamental properties determining accuracy and reliability of numerical schemes
  • quantifies difference between discrete approximation and exact differential equation
    • Example: Taylor series expansion to determine order of accuracy
  • assess spatial and temporal resolution requirements for accurate solutions
    • Systematic refinement of grid spacing and time step size
    • Richardson extrapolation estimates order of accuracy and improves precision of numerical solutions
  • Validation and verification processes crucial for assessing accuracy and reliability of CFD simulations
    • Comparison with analytical solutions (method of manufactured solutions)
    • Benchmarking against experimental data and high-fidelity simulations

Key Terms to Review (45)

ANSYS Fluent: ANSYS Fluent is a powerful computational fluid dynamics (CFD) software tool used for simulating fluid flow, heat transfer, and chemical reactions in various applications. It allows engineers and researchers to create detailed models of fluid behavior in complex geometries and operating conditions, making it essential for optimizing designs in fields like aerospace, automotive, and energy.
Central difference schemes: Central difference schemes are numerical methods used to approximate derivatives by utilizing values at points symmetrically located around a target point. These schemes are particularly useful in solving partial differential equations and play a significant role in computational fluid dynamics by providing a way to calculate gradients in space and time efficiently.
Conservation Laws: Conservation laws are fundamental principles that state certain physical quantities remain constant within a closed system as long as no external forces are applied. These laws are crucial in understanding the behavior of fluid flows, as they apply to mass, momentum, and energy, which help describe how fluids interact and change over time.
Continuity Equation: The continuity equation is a fundamental principle in fluid dynamics that expresses the conservation of mass within a fluid flow. It states that the mass of fluid entering a system must equal the mass of fluid exiting the system, assuming there are no sources or sinks. This concept is crucial for understanding how fluids behave in various conditions and is foundational in computational fluid dynamics to ensure accurate simulations of fluid motion.
Convergence Criteria: Convergence criteria refer to the specific conditions or rules used to determine when an iterative method has reached a satisfactory solution. These criteria help identify whether the sequence of approximations generated by numerical methods is approaching the true solution within a defined tolerance, ensuring accuracy and stability in calculations.
Courant-Friedrichs-Lewy Condition: The Courant-Friedrichs-Lewy (CFL) condition is a fundamental criterion used in numerical analysis to ensure the stability and convergence of solutions for certain numerical methods applied to partial differential equations (PDEs). It essentially provides a relationship between the time step size and spatial grid size, indicating that information must propagate through the grid within each time step to avoid numerical instability. This concept is crucial when dealing with methods such as finite difference, finite volume, or method of lines when solving PDEs like the heat and wave equations or in computational fluid dynamics.
Dirichlet boundary condition: A Dirichlet boundary condition specifies the values a solution must take on the boundary of the domain, essentially fixing the solution at those boundary points. This type of boundary condition is crucial in numerical methods as it helps in defining well-posed problems where the values at the edges are known and can significantly influence the solution throughout the domain.
Energy equation: The energy equation is a fundamental principle in fluid dynamics that relates the conservation of energy within a fluid system, often expressed in terms of kinetic energy, potential energy, and internal energy. This equation helps in understanding how energy is transferred and transformed within flowing fluids, providing insights into various phenomena such as turbulence, flow separation, and shock waves.
Euler's equations: Euler's equations describe the motion of an inviscid fluid and are foundational in fluid dynamics. They express the conservation of momentum, encapsulating the relationship between pressure, density, and velocity in a flowing fluid. These equations are crucial for understanding various phenomena in computational fluid dynamics, such as turbulence, flow separation, and shock waves.
Explicit time discretization schemes: Explicit time discretization schemes are numerical methods used to solve differential equations by approximating the equations over discrete time intervals. These schemes calculate the state of a system at a new time step solely based on the information from the previous time step, making them easier to implement but potentially unstable for certain problems. Their simplicity and direct approach are often leveraged in computational fluid dynamics to model fluid behavior over time.
Explicit time-marching schemes: Explicit time-marching schemes are numerical methods used to solve time-dependent problems in computational mathematics by advancing the solution through time in a straightforward manner. These schemes are characterized by their direct computation of the solution at the next time step from the known values at the current time step, making them easier to implement and understand. However, they often require careful consideration of stability criteria to ensure accurate results.
Finite difference methods: Finite difference methods are numerical techniques used to approximate solutions to differential equations by discretizing continuous functions. This approach involves replacing derivatives with finite differences, which makes it easier to solve equations that describe dynamic systems, particularly in contexts involving stochastic processes and fluid dynamics. These methods are essential for analyzing various mathematical models where exact solutions are difficult or impossible to obtain.
Finite Element Methods: Finite element methods (FEM) are numerical techniques used to find approximate solutions to boundary value problems for partial differential equations. They break down complex structures or fields into smaller, simpler parts called finite elements, making it easier to analyze physical phenomena. This method is widely applied in engineering and scientific computations, allowing for effective modeling of various systems across multiple disciplines.
Finite volume method: The finite volume method is a numerical technique used for approximating solutions to partial differential equations, particularly in fluid dynamics. This method involves dividing a physical domain into small, discrete control volumes and applying the integral form of the conservation laws to these volumes, ensuring that fluxes across the boundaries are calculated accurately. It's particularly effective in handling complex geometries and conserving quantities like mass, momentum, and energy.
Fractional step methods: Fractional step methods are numerical techniques used to solve partial differential equations (PDEs) by breaking down the solution process into smaller, manageable steps. These methods separate the different physical processes represented in the equations, allowing for more efficient computation and improved accuracy in fluid dynamics problems. By decoupling the various components of the solution, fractional step methods can significantly reduce computational costs while still capturing the essential features of fluid flow.
Grid convergence studies: Grid convergence studies are systematic analyses used to determine the effect of grid refinement on the results obtained from numerical simulations. They are essential in computational modeling, particularly in fluid dynamics, to ensure that the results are not dependent on the grid size but rather converge to a true solution as the grid is made finer. This process helps validate the accuracy of numerical methods and assesses the reliability of simulations.
Grid generation: Grid generation refers to the process of creating a computational mesh or grid that is used in numerical simulations, particularly for solving partial differential equations in fluid dynamics. This grid defines how the physical space is divided into discrete elements where calculations are performed, making it crucial for accurately modeling fluid flow and other phenomena. The quality and structure of the grid can significantly influence the accuracy and efficiency of computational fluid dynamics simulations.
Hybrid Schemes: Hybrid schemes are computational methods that combine different numerical techniques to solve complex problems, particularly in computational fluid dynamics. These schemes leverage the strengths of various approaches, such as finite volume methods and spectral methods, to achieve higher accuracy and efficiency in simulations, making them essential for modeling fluid flow and other dynamic systems.
Implicit time discretization schemes: Implicit time discretization schemes are numerical methods used to solve time-dependent differential equations by approximating the equations at discrete time steps, where the unknowns at the next time step depend on both known and unknown values. These schemes are particularly useful in computational fluid dynamics, as they allow for stability in simulations of fluid flow and can handle stiff problems more effectively than explicit methods.
Implicit time-marching schemes: Implicit time-marching schemes are numerical methods used to solve differential equations, where the solution at the next time step is defined in terms of both the current and future states of the system. These schemes are particularly useful in computational fluid dynamics as they allow for larger time steps while maintaining stability, especially for stiff problems. They involve solving a system of equations at each time step, which can handle complex interactions in fluid flow more effectively than explicit methods.
Incompressible Flow Assumption: The incompressible flow assumption is a simplification used in fluid dynamics, stating that the density of a fluid remains constant regardless of pressure changes within the flow field. This assumption is crucial when dealing with liquids, where density changes are negligible compared to gases. It allows for easier mathematical modeling and computational analysis in computational fluid dynamics, enabling the use of simpler equations and reducing computational costs.
Laminar flow: Laminar flow is a smooth, orderly movement of fluid in parallel layers with minimal disruption between them. This type of flow is characterized by low velocities and low turbulence, allowing the fluid to flow in an even manner, often observed in scenarios involving viscous fluids or under conditions of low Reynolds numbers. In computational fluid dynamics, understanding laminar flow is crucial as it affects various calculations and simulations in modeling fluid behavior.
Large Eddy Simulation: Large Eddy Simulation (LES) is a mathematical modeling technique used in computational fluid dynamics to simulate the turbulent flow of fluids by resolving the larger scales of turbulence while modeling the smaller scales. This approach provides a more accurate representation of turbulent flows compared to traditional methods, as it allows for the investigation of flow structures and energy transfer in a more detailed manner. By focusing on the large eddies that dominate the flow, LES captures essential physical phenomena critical for understanding complex fluid behaviors.
Lattice boltzmann method: The lattice Boltzmann method is a computational technique used for simulating fluid dynamics by modeling the microscopic behavior of particles on a discrete lattice. It provides a mesoscopic approach to fluid flow, enabling the study of complex fluid interactions and boundary conditions in an efficient manner, which makes it particularly useful in computational fluid dynamics.
Mach Number: Mach number is a dimensionless quantity representing the ratio of the speed of an object to the speed of sound in the surrounding medium. It serves as a key indicator in fluid dynamics, particularly in the study of compressible flows, determining whether an object is traveling at subsonic, transonic, or supersonic speeds, which influences flow behavior and shock wave formation.
Matrix stability analysis: Matrix stability analysis is a mathematical technique used to determine the stability of numerical methods employed in solving differential equations, particularly partial differential equations (PDEs). This analysis evaluates how perturbations in the initial conditions or parameters affect the solution's behavior over time. It's essential for understanding how numerical methods can converge to the correct solution and avoid instabilities, which is crucial when dealing with phenomena like heat conduction or fluid dynamics.
Momentum equation: The momentum equation is a fundamental principle in fluid dynamics that describes the conservation of momentum for a fluid flow system. It relates the forces acting on the fluid to the changes in momentum, allowing for the analysis of fluid behavior under various conditions. This equation is essential in computational fluid dynamics, as it helps predict how fluids move and interact with their surroundings.
Multigrid methods: Multigrid methods are a powerful computational technique used to solve large linear systems of equations efficiently, particularly in numerical simulations. They work by operating on multiple levels of discretization to accelerate the convergence of iterative solvers, making them especially effective for boundary value problems, sparse linear systems, and fluid dynamics applications. By utilizing both coarse and fine grids, these methods reduce computational time significantly while maintaining accuracy.
Navier-Stokes Equations: The Navier-Stokes equations are a set of nonlinear partial differential equations that describe the motion of fluid substances. These equations account for various factors such as viscosity, pressure, and external forces, allowing for the modeling of complex fluid behaviors in different scenarios, from simple laminar flows to turbulent conditions.
Neumann boundary condition: The Neumann boundary condition specifies the value of a derivative of a function on a boundary, often representing a flux or gradient at that boundary. This type of condition is crucial in numerical methods, as it helps to define how solutions behave at the edges of the domain, influencing both stability and accuracy in computations.
OpenFOAM: OpenFOAM is an open-source software toolkit for computational fluid dynamics (CFD) that provides a comprehensive range of tools for simulating fluid flow, heat transfer, and chemical reactions. It is widely used in engineering and research to solve complex fluid dynamics problems due to its flexibility and customizable features, enabling users to develop specific applications tailored to their needs.
PISO Algorithm: The PISO algorithm, which stands for 'Pressure Implicit with Splitting of Operators', is a numerical method used to solve incompressible fluid flow problems in computational fluid dynamics. It enhances the performance of traditional pressure-velocity coupling methods by using an iterative approach to ensure mass conservation while efficiently updating pressure and velocity fields. This algorithm is particularly beneficial in handling complex geometries and flows, making it a staple in computational simulations.
Pressure-velocity coupling algorithms: Pressure-velocity coupling algorithms are numerical methods used in computational fluid dynamics (CFD) to solve the Navier-Stokes equations for fluid flow, specifically addressing the interdependence of pressure and velocity fields. These algorithms ensure that the computed pressure and velocity fields remain consistent with each other during the simulation process, which is crucial for obtaining stable and accurate results in fluid flow problems.
Reynolds Number: The Reynolds number is a dimensionless quantity used in fluid mechanics to predict flow patterns in different fluid flow situations. It provides insight into whether the flow is laminar or turbulent, which is crucial for understanding the behavior of fluids in various applications, including those modeled in computational fluid dynamics. The Reynolds number is calculated using the fluid's density, velocity, characteristic length, and dynamic viscosity, allowing engineers and scientists to analyze and design systems involving fluid motion.
Reynolds-Averaged Navier-Stokes: The Reynolds-Averaged Navier-Stokes (RANS) equations are a set of equations used to model the behavior of fluid flows by averaging the effects of turbulence over time. This approach decomposes the fluid properties into mean and fluctuating components, allowing for the simplification of complex turbulent flow problems. RANS is widely utilized in computational fluid dynamics as it provides a balance between accuracy and computational efficiency when simulating turbulent flows.
Shape functions: Shape functions are mathematical functions used in numerical methods, particularly in the context of finite element analysis, to interpolate the solution over an element based on the values at the nodes. They play a critical role in defining how the displacement or other physical quantities vary within an element, helping to approximate complex geometries and behaviors in computational fluid dynamics.
Simple algorithm: A simple algorithm is a straightforward, step-by-step procedure for solving a problem or completing a task, typically characterized by its clarity and efficiency. These algorithms are designed to be easy to understand and implement, making them ideal for introductory programming and computational tasks. In computational fluid dynamics, simple algorithms can help model fluid flow and behavior in a manageable way, often serving as building blocks for more complex simulations.
Stability Analysis: Stability analysis is a method used to determine the behavior of a system in response to small perturbations or changes. It helps assess whether small deviations from an equilibrium state will grow over time, leading to instability, or will decay, returning to the equilibrium. Understanding stability is crucial in various fields, as it informs the reliability and robustness of systems under different conditions.
Steady-state conditions: Steady-state conditions refer to a situation in which the properties of a fluid system, such as velocity, pressure, and density, remain constant over time, despite the flow of fluid through the system. This concept is crucial in analyzing fluid behavior because it simplifies the mathematical modeling and computational simulations used to predict fluid dynamics, allowing for more straightforward calculations and stable solutions.
Time-stepping methods: Time-stepping methods are numerical techniques used to solve time-dependent partial differential equations by breaking down the problem into discrete time intervals or steps. These methods allow for the approximation of the solution at specific time points, making it easier to analyze dynamic systems like fluid flow, heat transfer, and other processes that evolve over time. They are essential in simulations where predicting behavior at each time step is critical for understanding the overall system's dynamics.
Truncation Error Analysis: Truncation error analysis refers to the assessment of errors that arise when a mathematical procedure or algorithm is approximated by a simpler or finite representation. This concept is crucial in understanding how accurate numerical solutions can be when applied in fields like computational fluid dynamics, where continuous equations are represented by discrete methods. By evaluating truncation errors, one can gain insights into the reliability of simulations and models used for predicting fluid behaviors.
Turbulent flow: Turbulent flow is a type of fluid motion characterized by chaotic and irregular fluctuations in velocity and pressure. This complex behavior typically occurs at high velocities or in large fluid systems where inertial forces dominate over viscous forces, leading to mixing and eddies. Understanding turbulent flow is crucial for predicting fluid behavior in various applications, such as aerodynamics, hydrodynamics, and energy transfer.
Upwind schemes: Upwind schemes are numerical methods used to solve hyperbolic partial differential equations, particularly in fluid dynamics. They are designed to handle advection-dominated problems by taking into account the direction of the flow, ensuring stability and accuracy when approximating the solution. These schemes help prevent non-physical oscillations and instabilities that can occur when the numerical grid does not align with the flow direction.
Viscosity: Viscosity is a measure of a fluid's resistance to deformation or flow, which describes how thick or sticky a fluid is. In computational fluid dynamics, viscosity plays a crucial role in determining how fluids behave under various conditions, influencing factors like flow rate and turbulence. Understanding viscosity is essential for accurately simulating fluid motion and interactions in various applications.
Von Neumann stability analysis: Von Neumann stability analysis is a mathematical method used to assess the stability of numerical schemes for solving partial differential equations (PDEs). It focuses on examining how errors propagate through a numerical solution over time, helping to determine whether small perturbations will grow or diminish, which is crucial for ensuring reliable simulations in computational mathematics.
© 2024 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.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide