The present study aims at relating lift and drag to flow structures around a delta wing of elliptic section. Aerodynamic forces are analysed in terms of fluid elements of non-zero vorticity and density gradient. The flow regime considered is Mα = 0.6 ∼ 1.8 and α = 5° ∼ 19°, where Mα denotes the free-stream Mach number and α the angle of attack. Let ρ denote the density, u velocity, and ω vorticity. It is found that there are two major source elements Re(x) and Ve(x) which contribute about 95% or even more to the aerodynamic forces for all the cases under consideration,
\[R_e({\bm x})=-\frac{1}{2} {\bm u}^2 \nabla\rho \cdot \nabla\phi\quad {\rm and}\quad V_e ({\bm x}) = -\rho{\bm u}\times {\bm \omega}\cdot \nabla\phi,\]
where θ is an acyclic potential, generated by the delta wing moving with unit velocity in the negative direction of the force (lift or drag). All the physical quantities are non-dimensionalized. Detailed force contributions are analysed in terms of the flow structures and the elements Re(x) and Ve(x). The source elements Re(x) and Ve(x) are concentrated in the following regions: the boundary layer in front of (below) the delta wing, the primary and secondary vortices over the delta wing, and a region of expansion around the leading edge. It is shown that Ve(x) due to vorticity prevails as the source of forces at relatively low Mach number, Mα < 0.7. Above about Mα = 0.75, Re(x) due to compressibility generally becomes the dominating contributor to the lift, while the overall contribution from Ve(x) decreases with increasing Mα, and even becomes negative at Mα = 1.2 for the lift, and at a higher Mα for the drag. The analysis is carried out with the aid of detailed numerical results by solving the Reynolds-averaged Navier–Stokes equations, which are in close agreement with experiments in comparisons of the surface pressure distributions.
Numerical Flow Simulation in a Centrifugal Pump at Design and Off-Design Conditions
