BME062 – COMPUTATIONAL FLUID DYNAMICS
Section-Wise Solved Answers (2024–25)
SECTION A
Attempt all questions in brief (2 × 7 = 14 marks)
(a) Can CFD be used as a research tool? Justify with examples.
Yes, Computational Fluid Dynamics (CFD) is a powerful research tool. It allows researchers to analyze complex fluid flow, heat transfer, and mass transfer problems that are difficult or expensive to study experimentally. For example, CFD is widely used in aerospace engineering to study airflow over aircraft wings, in biomedical research to analyze blood flow in arteries, and in automotive engineering to improve vehicle aerodynamics. CFD enables visualization of flow fields and parametric studies with high accuracy.
(b) Define conservation of mass and momentum in fluid flow.
Conservation of mass states that mass can neither be created nor destroyed; hence, the mass entering a control volume must equal the mass leaving it. In fluid mechanics, this principle is represented by the continuity equation.
Conservation of momentum states that the rate of change of momentum of a fluid particle equals the sum of external forces acting on it. This principle forms the basis of the Navier–Stokes equations.
(c) Difference between explicit and implicit approach.
In the explicit approach, the solution at the next time step is calculated directly from known values at the current time step. It is simple to implement but conditionally stable.
In the implicit approach, the solution involves unknown future values and requires solving simultaneous equations. Although computationally intensive, it is unconditionally stable and suitable for larger time steps.
(d) Define grid transformation. Why is it required?
Grid transformation is the process of mapping a physical domain with complex geometry into a simpler computational domain. It is required to simplify governing equations, improve numerical accuracy, and enable easier discretization of irregular geometries in CFD simulations.
(e) Key difference between finite element and finite volume methods.
The finite element method (FEM) is based on variational principles and uses shape functions, making it suitable for complex geometries and structural problems.
The finite volume method (FVM) is based on conservation laws and ensures strict conservation of fluxes across control volumes, making it widely used in fluid flow problems.
(f) State Newton’s second law and its application in fluid mechanics.
Newton’s second law states that the rate of change of momentum of a body is proportional to the applied force and occurs in the direction of that force.
In fluid mechanics, this law is applied to fluid elements to derive momentum equations, leading to the Navier–Stokes equations.
(g) Define compressed grid.
A compressed grid is a non-uniform grid where grid points are clustered in regions with high gradients, such as boundary layers. It improves accuracy without increasing the total number of grid points significantly.
SECTION B
Attempt any three (7 × 3 = 21 marks)
(a) Derivation of Navier–Stokes equations for incompressible flow.
For an incompressible fluid, density remains constant. Applying Newton’s second law to a fluid element and considering viscous stresses, pressure forces, and body forces leads to the Navier–Stokes equations. These equations describe the balance between inertial forces, pressure forces, viscous forces, and external forces in fluid motion.
(b) Derivation of 1-D unsteady heat conduction equation using explicit and implicit approach.
The 1-D unsteady heat conduction equation is derived from energy conservation. Using finite difference approximation, the explicit scheme computes future temperature directly, while the implicit scheme involves solving algebraic equations. The explicit scheme is conditionally stable, whereas the implicit scheme is unconditionally stable.
(c) Grid clustering and comparison of structured and unstructured grids.
Grid clustering is the technique of concentrating grid points in regions with steep gradients.
Structured grids have regular connectivity and are easy to generate but less flexible for complex geometries. Unstructured grids use irregular connectivity and are more suitable for complex shapes but require more computational effort.
(d) Typical errors in FEM and methods to minimize them.
Errors in FEM include discretization error, numerical integration error, and round-off error. These can be minimized by refining the mesh, using higher-order elements, and improving numerical integration techniques.
(e) Challenges in applying FVM to complex geometries.
Complex geometries require unstructured meshes, which increase computational cost and complexity. However, FVM remains popular because it strictly enforces conservation laws. Structured meshes are simpler, while unstructured meshes offer flexibility.
SECTION C
Q3. Attempt any one (7 marks)
(a) Boundary layer equations and initial/boundary conditions.
Boundary layer equations simplify the Navier–Stokes equations near solid surfaces where viscous effects dominate. They help analyze velocity and temperature gradients near walls.
Initial conditions define the state of the flow at the start of simulation, while boundary conditions specify flow behavior at domain boundaries such as inlet, outlet, and walls.
(b) Conservation of energy and classification of second-order PDEs.
The conservation of energy principle states that energy change equals heat added minus work done. In fluid dynamics, it governs heat transfer and temperature distribution.
Second-order PDEs are classified as elliptic, parabolic, and hyperbolic. For example, Laplace’s equation is elliptic, diffusion equation is parabolic, and wave equation is hyperbolic.
