3rd Year CFD Flashcards
Why is CFD used? What is it good for?
Low cost, Fast, Simulates real conditions
What type of equations are used for distributed properties?
Partial Differential Equations, often non-linear
What are “analytical” solutions?
exact solutions which are only possible for a limited class of problems – typically artificial and idealised
What are “numerical” solutions?
approximate solutions
What is the difference between ODE and PDE?
PDE has more than one independent variable
What does Quasilinear mean?
linear in highest order derivatives but perhaps not in other terms.
What are the three type of of quasilinear PDE classifications?
Parabolic
Hyperbolic
Elliptic
How are the three type of PDEs defined?
Parabolic if B^2-4AC = 0
Hyperbolic if B^2-4AC > 0
Elliptic if B^2-4AC
Can you remember the single equation of a second order function of two variables? (to find a b c etc)
Look it up ;)
What does it mean when a problem is “well posed”?
It is consistent and has a unique solution. It has to have properly set boundary conditions.
What is a time independent model?
Where the properties do not significantly change with time.It is assumed that the distributed properties are functions of space but not time, and the time derivatives in the governing equations are replaced by zeros.
What is meant by a time dependant model?
evolution towards the equilibrium state is of interest, or the equilibrium state does not exist.
What type of PDEs are the Heat Equation, Wave Equation, LaPlace Equation?
heat = Parabolic wave = hyperbolic LaPlace = elliptic
What are the three fundamental physical principles?
- mass is conserved;
- F = ma (Newton’s second law);
- energy is conserved.
What is inviscid flow and what do you do with it?
Where the dissipative, transport phenomena of viscosity, mass diffusion and thermal conductivity are neglected.
Drop the viscous term in equations
What are the 2 extra equations to make the system 7 equations with 7 unknown variables?
Perfect Gas assumption: p=rhoRT
Thermodynamic relation: e=e(T,p)
for example for a calorically perfect gas e=Cv*T
What i a Newtonian fluid?
shear stress is proportional to the rates of change of the
fluid’s velocity vector or time rate of strain.
What is referred to the “complete Navier Stokes equations”?
In modern CFD this refers to a numerical solution of the complete system of equations
What are the three main steps of a numerical approach?
Modelling, Dicretisation, Soution
What are the two types of mesh?
Regular (structured) and Irregular (Unstructured)
What happens to the PDEs during discretisation?
They are also discretised on a domain or grid and transformed into linear algebraic equations.
They are valid at any position
What are the the three most common PDE discretisation methods?
- the finite difference method (FD)
- the finite element method (FE)
- the finite volume method (FV)
What happens with the FD method?
Cartesian grid with computational points at the nodes. It approximates the partial derivatives directly. The steps are the delta x. Uses Taylor expansion with the finite step size.
What happens in the FV method?
Computational points at centre of cells. Uses an integral form of the fluid equation.
What is first order forward difference and first order backward difference?
Forward when Delta X = +ve
f(xi+1)-f(x) / DeltaX = df/dx(xi)
Backward when Delta x = -ve
f(x)-f(xi-1) / DeltaX = df/dx(xi)
What is the second order central difference?
Calculated using a point in front and behind.:
f(xi+1)-f(xi-1) / 2*DeltaX = df/dx(xi)
What is the second order central second difference?
f(xi+1)-2*f(xi) +f(xi-1) / DeltaX ^2 = d2f/dx2(xi)
What is a Truncation Error?
It comes from the part of the taylor expansion series which was dropped for simplification
What is the order of truncation error or order of approximation of the finite difference scheme?
Its the lowest order term of the truncation error. It defines how fast the error of approximation decreases with grid step
What is meant by the term consistency?
It means that the truncation error vanishes as the step size goes to zero.
What should be considered when choosing grid spacing?
It should be chosen to minimise discretisation error
What is a Dirchlet boundary condition?
When some values of the functions are known at the edges.
What is a Neumann boundary condition
Values of gradient (flux) are known at the edge
How are the position and time variables in FD Method discretised?
They are independent and therefore are discretised separately
What is an Explicit Scheme?
When the value of the variable at depends only on values known from the previous time steps.
What is an Implicit Scheme?
When there are more than one value of the variable at its highest time step.
What are Modelling Errors?
The difference between the actual solution and the exact solution of the mathematical model.
What are the discretisation errors?
The difference between the exact solution of the system of algebraic equations generated by the numerical scheme and the exact analytical solution of the PDE problem.
What are round off errors?
The difference between the exact solution of algebraic equations and the solution by iterative model (i.e., solution given by the computer)
What is a stable numerical scheme?
When the error does not increase as we
progress from time step n to step n+1
In CFD how do we visualise the error decreasing with time steps?
A decreasing residual graph indicating
convergence of numerical solution
What causes the oscillations of a converging model at the start of the iterations?
Iteration errors cause oscillation at early stage of the solution before smoothing out
What is the amplification factor?
G = abs(error at new time step / error of previous time step)
G
What is a CFL condition?
Gives relation between time and spatial steps for which the Numerical scheme is stable
Name the 3 main classes of errors arising when a problem is solved
numerically, and describe them.
Modeling errors: difference between real solution and the exact solution of the mathematical mode.
Discretisation errors: difference between the exact solution of thesystem of algebraic equations generated by the numerical scheme and the exact analytical solution of the PDE problem
Round-off errors: difference between exact solution of algebraic equations and the solution given by the computer.
Underline the main steps followed when the problem is approached
through a Finite Volume method.
Subdivision of the flow domain into small non-overlapping volumes.
Application of the integral equation(s) to each one of the small control volumes.
Numerical evaluation of integrals and discretisation of the derivatives
In the modern CFD literature, the terminology “Navier–Stokes equations” refers to a set of three different PDEs. State their names.
- Continuity equation
- Momentum equation(s)
- Energy equation
