CPCE Computational Fluid Dynamics Theory & Principles 2 — Questions and Answers
Question 1: Which numerical scheme is known for being unconditionally stable for diffusion problems regardless of time step size?
- Explicit Euler
- Crank-Nicolson
- Fully implicit (backward Euler) (Correct answer)
- Leapfrog
Correct answer: Fully implicit (backward Euler)
The fully implicit (backward Euler) scheme is unconditionally stable for diffusion equations, allowing large time steps without numerical instability.
Question 2: In CFD, the term 'upwinding' refers to:
- Solving in the direction opposite to flow
- Biasing the differencing stencil in the upstream flow direction (Correct answer)
- Adding artificial viscosity at outflow boundaries
- Extrapolating values from downstream nodes
Correct answer: Biasing the differencing stencil in the upstream flow direction
Upwinding biases the finite-difference or finite-volume stencil to use information from upstream, stabilizing convection-dominated flows.
Question 3: The Peclet number in CFD discretization is defined as the ratio of:
- Inertial forces to viscous forces
- Convective transport rate to diffusive transport rate on the mesh (Correct answer)
- Pressure gradient to velocity gradient
- Turbulent diffusivity to molecular diffusivity
Correct answer: Convective transport rate to diffusive transport rate on the mesh
The cell Peclet number (Pe = u·Δx/ν) compares convective to diffusive transport at the mesh cell level, guiding scheme selection.
Question 4: Which pressure-velocity coupling algorithm uses a staggered arrangement of pressure and velocity unknowns on separate grids?
- SIMPLE
- PISO
- MAC (Marker-and-Cell) (Correct answer)
- Fractional step
Correct answer: MAC (Marker-and-Cell)
The MAC method, one of the earliest incompressible solvers, stores pressure at cell centers and velocity at cell faces on a staggered grid.
Question 5: False diffusion in CFD is primarily caused by:
- Using too large a time step
- Flow not aligned with the grid lines in upwind schemes (Correct answer)
- Insufficient convergence of iterative solvers
- Applying periodic boundary conditions
Correct answer: Flow not aligned with the grid lines in upwind schemes
False (numerical) diffusion in upwind schemes is greatest when flow is oblique to the grid, adding artificial smearing of gradients.
Question 6: The Von Neumann stability analysis is applicable to:
- Nonlinear PDEs on unstructured grids
- Linear PDEs with constant coefficients on uniform Cartesian grids (Correct answer)
- Turbulence closure models
- Multiphase interface tracking methods
Correct answer: Linear PDEs with constant coefficients on uniform Cartesian grids
Von Neumann stability analysis uses Fourier mode decomposition and is strictly valid for linear PDEs with constant coefficients on uniform Cartesian meshes.
Question 7: In finite element methods for CFD, the Galerkin formulation requires that residuals be orthogonal to:
- The boundary conditions
- The basis (shape) functions (Correct answer)
- The pressure field
- The test mesh
Correct answer: The basis (shape) functions
In the standard Galerkin method, the weighted residual is set to zero with test functions chosen equal to the shape functions, enforcing orthogonality of the residual to the function space.
CPCE Computational Fluid Dynamics Theory & Principles 2
Which numerical scheme is known for being unconditionally stable for diffusion problems regardless of time step size?