JNTU B.Tech III COMPUTATIONAL AERODYNAMICS Previous Question Paper (August/September - 2021 - R18)

To derive the continuity equation in non-conservation form, consider a fixed fluid element of volume V with surface S. The rate of change of mass within the element is equal to the net mass flow out of the element. The mass of the element is given by the integral of density ρ over the volume V. By the Reynolds Transport Theorem, the rate of change of mass is:

Applying the Divergence Theorem to the surface integral, we convert the surface integral into a volume integral:

Substituting this back into the original equation:

Since this holds for any arbitrary volume, the integrand must be zero:

Expanding the divergence term using the vector identity ∇ · (ρV) = ρ(∇ · V) + (V · ∇)ρ, we obtain the non-conservation form:

This can also be expressed using the substantial derivative Dρ/Dt as:

The distinction between conservative and non-conservative forms lies in the mathematical formulation of the governing equations for fluid flow.

• Conservative Form: These equations are derived by applying physical laws to a fixed control volume in space. The dependent variables are conserved quantities (e.g., mass, momentum, total energy). These forms are mathematically expressed as the divergence of a flux, making them superior for capturing shock waves and discontinuities because they satisfy the integral form of the conservation laws.
• Non-Conservative Form: These equations are derived by applying physical laws to a fluid element moving with the flow. The dependent variables are primitive variables (e.g., pressure, density, velocity). These forms are generally easier to derive and implement for simple flows but fail to accurately capture the jump conditions across shocks without specific modifications.

In computational practice, the conservative form is preferred for high-speed compressible flows involving shocks, while the non-conservative form is often used in simpler aerodynamic applications or when high-order accuracy in smooth regions is the primary focus.