JNTU
JNTU B.Tech III COMPUTATIONAL AERODYNAMICS Previous Question Paper (November/December - 2020 - R15)
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Main
Derive continuity equation in conservation and non conservation forms.
Answer
The continuity equation represents the principle of conservation of mass. For a fluid flow, it states that the rate of change of mass within a fixed control volume is equal to the net rate of mass flow across the control surface.
1. Conservation Form:
Consider a fixed control volume in space with volume V and surface area S. The total mass inside is the integral of density ρ over the volume. The rate of change of mass equals the negative flux of mass through the surface:
Applying the divergence theorem to the surface integral, we convert it to a volume integral:
Since this holds for any arbitrary volume, the integrand must be zero, yielding the differential conservation form:
2. Non-Conservation Form:
The non-conservation form is derived by expanding the divergence term using the vector identity ∇ ⋅ (ρV⃗) = ρ(∇ ⋅ V⃗) + V⃗ ⋅ ∇ρ:
Recognizing the substantial derivative (or material derivative) operator D/Dt = ∂/∂t + V⃗ ⋅ ∇, the equation simplifies to:
In this form, the equation describes the change in density of a fluid element as it moves along a streamline, rather than the change of properties at a fixed point in space.
Discuss about significance of conservation and non conservation forms of governing equations and its effect on CFD.
Answer
The conservation form of governing equations is derived by applying physical laws to a fixed control volume in space. These equations are written in terms of flux variables, such as mass, momentum, and energy flux, which are directly related to the physical conservation laws. In CFD, the conservation form is preferred for capturing discontinuities like shock waves because the integral form of the equations allows for the correct jump conditions (Rankine-Hugoniot relations) to be satisfied across the shock.
The non-conservation form is derived by applying physical laws to a fluid element moving with the flow. While mathematically equivalent to the conservation form for smooth, continuous flows, it does not naturally account for the conservation of mass, momentum, and energy across a discontinuity. Using the non-conservation form in regions with strong shocks often leads to incorrect shock speeds and strengths, as the numerical scheme may fail to conserve the integral properties of the flow.
In CFD applications, the choice significantly impacts numerical stability and accuracy. The conservation form is essential for shock-capturing methods to ensure that the numerical solution converges to the physically correct weak solution of the governing partial differential equations.
Write a short note on Shock fitting and Shock capturing methods.
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Find the behavior of first order wave equation, second order wave equation, one dimensional and two dimensional heat conduction equations.
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Elaborate different classes of PDE’s and their importance in understanding physical and CFD aspects of aerodynamic problems.
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Derive the stability condition for one dimensional heat conduction equation using von neumann stability analysis.
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Explain the structured grid generation for complex geometries with neat sketches.
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Write a short note on difference between FDM and FVM techniques with pros and cons.
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Write about the conditions on finite volume selections with neat sketch.
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Discuss why Relaxation technique is suitable elliptic form of PDE.
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Write a short note on aspects of numerical dissipation and dispersion.
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