JNTU
JNTU B.Tech IV COMPUTATIONAL FLUID DYNAMICS Previous Question Paper (February/March - 2022 - R18)
Answer any Five Questions. All Questions Carry Equal Marks.
Main
Explain in detail the similarities, differences, advantages and disadvantages between finite difference methods, finite volume methods and finite element methods used for solving fluid flow problems.
Answer
Computational Fluid Dynamics (CFD) relies on three primary discretization techniques to transform partial differential equations into algebraic systems. While all methods aim to approximate the governing Navier-Stokes equations, they differ fundamentally in their mathematical formulation and grid requirements.
Finite Difference Method (FDM): FDM approximates derivatives using Taylor series expansions at discrete grid points. It is conceptually the simplest method, highly efficient for structured, orthogonal grids, and easy to implement for simple geometries. However, it struggles significantly with complex, irregular boundaries and does not inherently guarantee conservation laws unless specifically formulated.
Finite Volume Method (FVM): FVM is the most widely used approach in commercial CFD codes. It discretizes the integral form of the conservation equations over control volumes. Its primary advantage is the inherent satisfaction of global and local conservation laws, even on coarse meshes. It is highly flexible for complex geometries using unstructured grids, though achieving higher-order accuracy can be mathematically complex compared to FDM.
Finite Element Method (FEM): FEM uses a variational or weighted residual formulation, dividing the domain into elements where the solution is approximated by polynomial shape functions. It provides a rigorous mathematical framework for error estimation and is excellent for handling complex geometries and coupled physics problems. However, it is computationally expensive, requires significant memory, and is often more difficult to implement for high-speed, convection-dominated flows compared to FVM.
| Method | Grid Flexibility | Conservation | Complexity |
|---|---|---|---|
| FDM | Low | Not inherent | Low |
| FVM | High | Inherent | Moderate |
| FEM | Very High | Requires specific formulation | High |
In summary, FDM is preferred for fundamental research and simple geometries, FVM is the industry standard for general-purpose fluid flow due to its robust conservation properties, and FEM is favored in structural-fluid interaction and specialized research where high mathematical rigor and complex domain handling are paramount.
Classify the following Partial Differential Equation for different real values of the parameter M.
Answer
The given second-order linear partial differential equation is of the form A∂2φ/∂x2 + B∂2φ/∂x∂y + C∂2φ/∂y2 + D = 0. Here, A = (1 - M2), B = 0, and C = 1. The classification depends on the discriminant B2 - 4AC:
- Elliptic: Occurs when B2 - 4AC < 0. This implies -4(1 - M2) < 0, which simplifies to M2 < 1, or |M| < 1 (subsonic flow).
- Parabolic: Occurs when B2 - 4AC = 0. This implies -4(1 - M2) = 0, which simplifies to M2 = 1, or |M| = 1 (sonic flow).
- Hyperbolic: Occurs when B2 - 4AC > 0. This implies -4(1 - M2) > 0, which simplifies to M2 > 1, or |M| > 1 (supersonic flow).
Compare the Gauss Seidel and Jordan Methods using the following set of equations: 5x + y = 10; and 2x + 3y = 4.
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Elliptic equations are typically used to characterize steady-state systems. Justify the statement with suitable examples with the prescribed boundary conditions.
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Explain discretization methods using Taylor’s series expansion for 1st order forward, backward and 2nd order central difference schemes.
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Consider a steady conduction in 1-dimensional bar with known temperature at x = 0 and x = L are 0 K and 200 K respectively. The length of the bar is 8 m; conductivity is constant at 1.5 W/K-m. Find the temperature at the center of 3 equal parts.
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Explain the explicit and implicit approach with the help of one dimensional heat conduction equation. Perform von-Neumann stability analysis.
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Prove that the Wave equation is a hyperbolic.
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What is CFL Condition? Discuss its significance.
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Enumerate the various steps involved in Alternating-direction implicit method.
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Reduce the general form of Navier-Stokes and continuity equations for the case of 2D Viscous Incompressible flow problems.
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Discuss geometry, boundary conditions and solution methods for the Lid Driven Cavity Problem.
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What is meant by staggered grid? Explain the importance of staggered grids for flow computations.
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Describe the first order forward time and centered space method of solving one dimensional transient diffusion equation.
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