CPCE Computational Fluid Dynamics Theory & Principles 5 — Questions and Answers
Question 1: In Direct Numerical Simulation (DNS), the minimum mesh spacing required scales approximately as:
- Re^(1/2)
- Re^(3/4) (Correct answer)
- Re^(-1)
- Re^(2)
Correct answer: Re^(3/4)
DNS must resolve down to the Kolmogorov length scale η ~ Re^(-3/4)L, so mesh point counts scale as Re^(9/4) in 3D, making DNS prohibitively expensive at high Re.
Question 2: The Lax-Wendroff scheme achieves second-order accuracy in both space and time by:
- Using a midpoint rule for time integration with central differencing
- Replacing the time derivative with spatial derivatives via the PDE itself before discretizing (Correct answer)
- Applying a flux limiter to a first-order scheme
- Iterating the Euler explicit scheme twice per time step
Correct answer: Replacing the time derivative with spatial derivatives via the PDE itself before discretizing
Lax-Wendroff substitutes the second time derivative with spatial derivatives using the governing PDE, enabling a single-step second-order scheme.
Question 3: Which boundary condition type specifies the gradient (flux) of a variable normal to the boundary, rather than its value?
- Dirichlet condition
- Neumann condition (Correct answer)
- Robin (mixed) condition
- Periodic condition
Correct answer: Neumann condition
A Neumann boundary condition prescribes the normal derivative (flux) of the variable at the boundary, commonly used for symmetry planes or fully developed outflow.
Question 4: The concept of 'numerical diffusion' as opposed to 'numerical dispersion' refers to:
- Smearing of sharp gradients caused by truncation error in the dissipative part of the scheme (Correct answer)
- Phase errors causing different frequency components to travel at incorrect speeds
- Aliasing of high-frequency modes onto lower frequencies
- Non-conservation of mass in the discretized equations
Correct answer: Smearing of sharp gradients caused by truncation error in the dissipative part of the scheme
Numerical diffusion arises from even-order truncation error terms that act like additional viscosity, smearing gradients, while numerical dispersion comes from odd-order terms causing phase errors.
Question 5: In the Finite Volume Method, the divergence theorem is used to convert the volume integral of a divergence term into:
- A time derivative at cell centers
- A surface (face) integral of fluxes (Correct answer)
- A line integral along cell edges
- A source term in the cell
Correct answer: A surface (face) integral of fluxes
Gauss's divergence theorem converts ∫_V ∇·F dV into ∮_S F·n dS, transforming volume divergence integrals into face flux integrals in FVM.
Question 6: The 'realizability' requirement in turbulence modeling ensures that:
- The turbulent kinetic energy k remains non-negative and normal stresses are positive (Correct answer)
- The eddy viscosity is always larger than the molecular viscosity
- The turbulence model reproduces the exact DNS solution
- The pressure-strain term vanishes at solid walls
Correct answer: The turbulent kinetic energy k remains non-negative and normal stresses are positive
Realizability constraints guarantee physically admissible turbulence statistics, such as non-negative normal stresses and satisfaction of the Schwarz inequality for shear stresses.
Question 7: Which method is commonly used to handle moving boundaries or fluid-structure interaction in CFD by re-meshing the domain at each time step?
- Immersed Boundary Method (IBM)
- Arbitrary Lagrangian-Eulerian (ALE) method (Correct answer)
- Volume of Fluid (VOF) method
- Level Set Method
Correct answer: Arbitrary Lagrangian-Eulerian (ALE) method
The ALE method moves the mesh with the boundary to maintain mesh conformity while allowing interior nodes to move independently, combining Lagrangian and Eulerian descriptions.
CPCE Computational Fluid Dynamics Theory & Principles 5
In Direct Numerical Simulation (DNS), the minimum mesh spacing required scales approximately as: