Ch10 Conical Flow

1. Ch10 Conical Flow －Application in the aerodynamics of supersonic missiles, inlet diffusers with conical centerbodies for supersonic airplanes, … 10.1 Introduction －Axisymmetric supersonic flow over a sharp cone at zero angle attack －A cylindrical coordinate system －the exact nonlinear solution for a special degenerate case of 3-D flow (quasi-2D, ∵ u=f (r,z) only)

2. In this chapter, further specialize to “a sharp right-circular cone in a supersonic flow”

3. 10.2 Physical Aspects of Conical Flow (infinitely extended) ∵ the cone surface represents the stream surface (streamlines) ＆ no change after the conical shock and no meaningful length scale ∴flow properties are constant along the cone surface ∵the cone surface is simply a ray from the vertex ∴From geometrical reasoning, it only makes sense to assume the flow properties are constant along the rays.(experimental proven) Conical flow ≡ all flow properties are constant along rays from a given vertex.

4. 10.3 Quantitative Formulation (after Taylor and Maccoll) ( x, y ) → ( r, θ ) (axisymmetric flow) (constant along a ray from the vertex) Continuity equation for steady flow in terms of spherical coordinates

5. －(*) －continuity equation for axisymmetric conical flow ∵The shock wave is straight. ∴△s across the shock is the same. ↓ ▽s=0 throughout the conical flowfield. ＆ adiabatic + steady →△h0=0 Crocco’s eqn. ( a conbination of the momentum and energy equations )

6. －the conical flow field is irrotational. In spherical coordinates, ( Note：r is a coordinate. Not a flow parameter )

7. －(**) irrotational condition for axisymmetrical conical flow Euler’s equation in any direction For isentropic flow Define a new reference velocity Vmax －the maximum theoretical velocity obtainable from a fixed reservoir condition the flow has expanded to T=0°K

8. For a calorically perfect gas －(***) From Eqns. (*), (**), (***) 3 eqns. 3 variables (ρ， Vr，Vθ) only 1 independent variable θ (*) → (***) →

9. then or －Taylor-Maccoll equation an O.D.E. of no closed-form solution, solved numerically.

10. To expedite the numerical solution, －non-dimensional T-M equation only

11. inverse approach 10.4 Numerical Procedure a given shock wave is given → the cone surface is calculated. 1. Given θs＆ M∞ → M2＆ δ (flow deflection angle) right behind the shock is calculated with oblique shock relations. 2. 3. Solve non-dimensional T-M eqn. (marching away from the shock) at each △ θ increment, using any standard numerical solution technique, e.g. Runge-Kutta method. 4.

12. 5. T, P, ρ solved

13. 10.5 Physical Aspects of Supersonic Flow over cones similar to the θ-β-M relation for 2-D wedges 1. For a given cone angle θc & M∞ → 2 possible θs (strong ＆ weak solutions) 2. θc,max (θc > θc,max →shock detached) Note: 3-D relieving effect :the shock wave on a cone of given angle is weaker than that on a wedge of the same angle. →lower surface P, T, ρ＆ △s, (θmax)cone > (θmax)wedge for a given M∞

14. Note: For most cases, the complete flowfield between the shock and the cone (shock layer) is supersonic. However, if θc is large enough, but θc < θc,max, there are some cases where the flow becomes subsonic near the surface. →a supersonic flowfield is isentropically compressed to subsonic velocities.