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CHAPTER 4 Methods This chapter will start with a very brief overview of the governing equations; the continuity equation, the momentum equation and the energy equation. These equations form a model of incompressible and compressible flows of gases and liquids. These equations are non-linear, and there exist only a few laminar flow cases where these equations can be solved analytically. Instead, one can use numerical techniques to solve the governing equations, which will also be briefly discussed in this chapter. 4.1. Governing equations The governing equations that describe the conservation of mass (4.1), momen- tum (4.2) and energy (4.3) are the Navier-Stokes (NS) equations and the energy equation complemented with the equation of state (4.4): ∂ρ+ ∂ (ρ·uj)=0 (4.1) ∂t ∂xj ∂(ρ·ui)+ ∂ (ρ·ui·uj)=−∂p +∂τij +ρfi (4.2) ∂t ∂xj ∂xi ∂xj ∂(ρ·h)+ ∂ (ρ·u ·h)=∂p+u ∂p +τ ∂ui −∂qj +W +q (4.3) ∂t ∂x j ∂t j∂x ij∂x ∂x ext H jjjj p=ρ· T·R (4.4) where ρ is the density, p the pressure, T the temperature, xi the cartesian coordinates, t the time, U = ui, i=1,3 is the velocity, fi the specific force, τij is the viscous shear stress tensor, h is the specific enthalpy, R is the gas constant and qi is the heat flux. Wext is the work of external volume forces and qH is the heat source. The heat flux qi is modelled through Fourier’s law. The viscous shear stress tensor, τij is defined as: τij =μ·(∂ui +∂uj −2δij ·∂uk) (4.5) ∂xj ∂xi 3 ∂xk 25PDF Image | Numerical computations of the unsteady flow in a radial turbine
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