Ch10 Conical Flow

10.1 Introduction

－ Application in the aerodynamics of supersonic missiles, inlet diffusers with conical centerbodies for supersonic airplanes, …

－ Axisymmetric supersonic flow over a sharp cone at zero

angle attack

－ A cylindrical coordinate system

0

, ,r z

－ the exact nonlinear solution for a special degenerate case of

3-D flow (quasi-2D, u=f (r,z) only)∵

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

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.

10.3 Quantitative Formulation (after Taylor and Maccoll)

( x, y ) → ( r, θ )

0

(axisymmetric flow)

0r

(constant along a ray from the vertex)

Continuity equation for steady flow

( ) 0V ��������������

in terms of spherical coordinates

22

1

1 1 sin 0

sin sin

rV r Vr r

VV

r r

��������������

22

1 1 12 cos sin 0

sin sin

2 1cot

rr

r

VV Vr V r V

r r r r

V VVV

r r r

2 cot 0r

VV V V

－ (*) － 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 → h△ 0=0Crocco’s eqn. ( a conbination of the momentum and energy equations )

0T s h V V ����������������������������

0V ��������������

－ the conical flow field is irrotational.In spherical coordinates,

2

sin

1

sin

sin

r

r

e re r e

Vr r

V rV r V

������������������������������������������

��������������

2

1sin

sin

sin sin }

r

rr

e rV rVr

Vre rV V r e rV

r r

��������������

����������������������������

｛

sinsin r

V VV r

r

rV VV r

r

0V

( Note ： r is a coordinate. Not a flow parameter )

rVV

－ (**)

irrotational condition for axisymmetrical conical flow

Euler’s equation in any direction

2 2 2r

dp VdV

V V V

r rdp V dV V dV

For isentropic flow

2

s

dp pa

d

2

1r r

dV dV V dV

a

Define a new reference velocity Vmax － the maximum theoretical velocity obtainable from a fixed reservoir condition the flow has expanded to T=0°K

22max

0 .2 2

VVh const h

For a calorically perfect gas22 2

max

1 2 2

Va V

2 2 2 2 2 2max max

1 1

2 2 ra V V V V V

2 2 2max

2

1r r

r

V dV V dVd

V V V

－ (***)

From Eqns. (*), (**), (***) 3 eqns. 3 variables (ρ ， Vr ， Vθ)only 1 independent variable θ

(*) → 2 cot 0r

dV V dV V

d d

(***) → 2 2 2max

2

1

rr

r

dVdVV Vd d d

d V V V

then

2 2 2max

22 cot 0

1

rr

rr

dVdVV VdV V d dV V

d V V V

or

2 2 2max

12 cot 0

2r

r r r

dV dVdVV V V V V V V V

d d d

2

2r rdVdV d V

Vd d d

2 2 22 2

max 2 2

12 cot 0

2r r r r r r r

r r r

dV dV d V dV dV dV d VV V V V

d d d d d d d

－ Taylor-Maccoll equation

an O.D.E. of

no closed-form solution, solved numerically.

, rr

dVV f V

d

To expedite the numerical solution, '

max

VV

V

2

2' ' 2 ' ' ' ' 2 '' ' '

2 2

11 2 cot 0

2r r r r r r r

r r r

dV dV d V dV dV dV d VV V V

d d d d d d d

－ non-dimensional T-M equation

'V f M only

22max

2 2

VVh

22 2max

1 2 2

Va V

22

max1 1 1

1 2 2

Va

V V

22

max2 11

1

V

M V

1

2'

2max

21

1

VV

V M

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.

10.4 Numerical Procedure inverse approach

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.

' ' 'right behind the shock max

1

2

2

&

2 1

1

& geometry (δ angle)

r

VV V V

V

M

4. 'Until 0 , i.e., cone surface.cV

5.

2 2' ' ' ' '

1

2

2

& solved M is solved.

2 1

1

r rV V V V V

M

20

120

1

120

1 1

2

1 1

2

1 1

2

TM

T

PM

P

M

T, P, ρ solved

10.5 Physical Aspects of Supersonic Flow over conessimilar 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∞

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.