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Overview of CFD Solution Methodologies. Outline. Ingredients Overview of Solution Methodologies Finite Difference Finite Volume (i.e., Control Volume) Finite Element Strengths and Weaknesses. domain discretization (grid). solution of algebraic equations. equation discretization.

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outline
Outline
  • Ingredients
  • Overview of Solution Methodologies
    • Finite Difference
    • Finite Volume (i.e., Control Volume)
    • Finite Element
  • Strengths and Weaknesses
ingredients

domain discretization (grid)

solution of algebraic equations

equation discretization

treatment of convection and source terms

Ingredients

CFD solution method

survey of methods
Survey of Methods
  • Many CFD techniques exist
  • The most common are:
    • Finite Difference
    • Finite Volume or Control Volume
    • Finite Element
  • The focus of this talk is to introduce these three
  • There are certainly many other approaches, including:
    • control volume/finite element
    • spectral
    • spectral element
    • boundary element
    • lattice gas
    • and more!
finite difference method fdm
Finite Difference Method (FDM)
  • Historically, the oldest of the three
  • Techniques published as early as 1910 by L. F. Richardson
  • Seminal paper by Courant, Fredrichson and Lewy (1928) derived stability criteria for explicit time stepping
  • First ever numerical solution: flow over a circular cylinder by Thom (1933)
  • Scientific American article by Harlow and Fromm (1965) clearly and publicly the idea of “computer experiments” for the first time — CFD is born!!
finite volume method fvm
Finite Volume Method (FVM)
  • Has its roots in the Finite Difference Method
  • First well-documented use was by Evans and Harlow (1957) at Los Alamos and Gentry, Martin and Daley (1966)
  • Was attractive because:
    • while variables may not be continuously differentiable across shocks and other discontinuities,
    • mass, momentum and energy would always be conserved
  • Late 70’s, early 80’s saw development of body-fitted grids
  • By early 90’s, unstructured grid methods had appeared

flow

compressible flow over a wedge

contours of density

finite element method fem
Finite Element Method (FEM)
  • Earliest use was by Courant (1943) for solving St. Venant torsion problem
  • Clough (1960) gave the method its name
  • Method was refined greatly in the 60’s and 70’s, mostly for analyzing structural mechanics problem
  • FEM analysis of fluid flow was developed in the mid- to late 70’s

coextrusion

metal insert

contours of velocity magnitude

finite difference basic methodology
Finite Difference: Basic Methodology
  • The domain is discretized into a series of grid points
    • a “structured” (ijk) mesh is required
  • The governing equations are discretized (converted to algebraic form)
    • first and second derivatives are approximated by truncated Taylor series expansions
  • The resulting set of linear algebraic equations is solved iteratively or simultaneously

i

j

j

i

finite difference pro s and con s
Finite Difference: Pro’s and Con’s
  • Advantages:
    • simple derivation, implementation
  • Disadvantages:
    • relatively simple grids
    • mass, momentum, energy not conserved on coarse grids
finite volume basic methodology
Finite Volume: Basic Methodology
  • Divide the domain into control volumes (c.v.’s)
  • Integrate the differential equation over the control volume and apply the divergence theorem.
  • To evaluate derivative terms, values at the control volume faces are needed: have to make an assumption about how the value varies.
  • Result is a set of linear algebraic equations; one for each c.v.
  • Solve iteratively or simultaneously.

finite volume pro s and con s
Finite Volume: Pro’s and Con’s
  • Advantages:
    • basic FV control volume balance does not limit cell shape
    • mass, momentum, energy conserved even on coarse grids
    • efficient, iterative solvers well developed
  • Disadvantages:
    • Simplest implementation uses 1-D assumptions during differencing of convection/diffusion terms — leads to false diffusion (multi-dimensional approaches now available)
finite element basic methodology
Finite Element: Basic Methodology
  • Domain is divided into elements.
  • Most FEM methods use some variant of the Method of Weighted Residuals.
    • we seek an approximate solution to the governing equations
    • therefore, we seek to minimize the residual (or error) in some weighted sense over the domain
  • Choose a shape function which is used to interpolate values between node points.
  • Multiply the governing equations by a weight function and integrate to obtain the “weak” formulation (contains first derivatives, not second).
  • Solve algebraic equations iteratively or simultaneously.
finite element pro s and con s
Finite Element: Pro’s and Con’s
  • Advantages:
    • multi-dimensional shape functions give geometric flexibility and limit false diffusion
    • “natural” imposition of flux boundary conditions
    • mass is conserved locally and globally, momentum is conserved globally and nearly locally
    • for same solution accuracy, generally requires fewest grid points
  • Disadvantages:
    • traditionally used simultaneously solution of all equations (memory and CPU intensive) — iterative solves now available
    • perhaps more complicated to develop, debug and maintain
summary
Summary
  • Finite volume method and finite element method are now the most popular and successful methods
  • Each has its own advantages:
    • FVM seems to enjoy an advantage in memory use and speed for very large problems, higher speed flows and source term dominated flows (like combustion)
    • FEM solutions can be very accurate using generally smaller grids
    • Being a node based scheme with natural imposition of traction boundary conditions, FEM seems better suited for deforming mesh free surface calculations