ch10 conical flow
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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.