conical flow

1. ▪ Axisymmetric forebodies:
▪ Single Cone
▪ Multi-Cone
▪ Single Cone Forebody:
▪ Governing Equation to analytically model flow around Single-Cone Forebody:
ϒ−1
2
1 − 𝑣𝑟
2
− 𝑣𝑟
′2
2𝑣𝑟 + 𝑣′
𝑟𝑐𝑜𝑡𝜃 + 𝑣′′
𝑟 − 𝑣′
𝑟(𝑣𝑟 𝑣′
𝑟 + 𝑣′
𝑟 𝑣′′
𝑟) = 0
▪ Derived from continuity equation in spherical coordinates.
▪ It is called the Taylor-Maccoll ODE.
▪ Assumptions:
▪ Zero angle of attack
▪ Irrotational flow
▪ Right-circular cone – sharp point vortex (singular point)
2018 AIAA Space Planes and Hypersonics Conference 11/13/2022 1
INTRODUCTION – Background Theory
Singular Point

2. ▪ Multi-cone Forebody:
▪ Why do we need Multiple cones? – To achieve higher compression.
▪ No singularity after the first cone. Hence, cannot apply Taylor Maccoll ODE.
▪ MOTIVATION: Introduce a method to tackle the singularity assumption in Taylor-Maccoll so that the
flow around an axisymmetric forebody can be analytically modelled.
▪ This method will simplify the design process of an inlet and will give scope to Design Optimization!
2018 AIAA Space Planes and Hypersonics Conference 11/13/2022 2
INTRODUCTION – Motivation
H
θc2
θc1
θ
β

3. ▪ Consider a right-circular cone at zero angle of attack:
▪ Governing Equation for the flow is given by:
Where, 𝑣𝑟 = radial velocity; 𝑣𝜃= tangential velocity. We also know that, 𝑣θ = 𝑣𝑟
′
=
𝑑𝑣𝑟
𝑑𝜃
.
▪ Therefore, ODE with one unknown, 𝑣𝑟.
2018 AIAA Space Planes and Hypersonics Conference 11/13/2022 3
METHODOLOGY – Single Cone Forebody
ϒ−1
2
1 − 𝑣𝑟
2
− 𝑣𝑟
′2
2𝑣𝑟 + 𝑣′
𝑟𝑐𝑜𝑡𝜃 + 𝑣′′
𝑟 − 𝑣′
𝑟(𝑣𝑟 𝑣′
𝑟 + 𝑣′
𝑟 𝑣′′
𝑟) = 0

4. ▪ Given: Free-stream Mach number (𝑀∞).
▪ ALGORITHM:
1) Initialize shock angle (𝜃𝑠). Calculate Mach number (𝑀2) and deflection angle (𝛿) using 𝜃 − 𝛽 − 𝑀 relation.
2) Calculate 𝑣𝑟 and 𝑣θ using:
𝑉 =
1
1 +
2
(𝛾 − 1)𝑀2
2
𝑉2
= 𝑣𝑟
2
+ 𝑣𝜃
2
3) Value of 𝑣𝑟 is substituted in the governing equation and solved numerically for different values of 𝜃𝑠 while marching away from
the shock.
4) The algorithm is terminated when 𝑣θ = 0. (when it reaches the surface of the cone)
2018 AIAA Space Planes and Hypersonics Conference 11/13/2022 4
METHODOLOGY – Single Cone Forebody
ϒ−1
2
1 − 𝑣𝑟
2 − 𝑣𝑟
′2 2𝑣𝑟 + 𝑣′
𝑟𝑐𝑜𝑡𝜃 + 𝑣′′
𝑟 − 𝑣′
𝑟(𝑣𝑟 𝑣′
𝑟 + 𝑣′
𝑟 𝑣′′
𝑟) = 0

5. ▪ The algorithm is solved using MATLAB*
▪ ODE 15s is used to solve the Taylor Maccoll ODE.
▪ Output of the algorithm:
▪ Deflection angle or cone angle (𝜽𝒄)
▪ Resultant velocity at every intermediate shock (Vc).
▪ Mach number at every intermediate shock. (𝑴𝒊).
2018 AIAA Space Planes and Hypersonics Conference 11/13/2022 5
METHODOLOGY – Single Cone Forebody
* Lassaline, D. J. V., “Supersonic Right Circular Cone at Zero Angle of Attack,” Tech. rep., Ryerson University, AE, 2009
θs
θc
δ
vr
vθ
M∞
θi
M1
Mi
Mn

6. ▪ Basic Idea behind proposed method:
▪ We know that conical shocks take a curvilinear path.
▪ Any curved line/arc can be approximated as a collection of ‘n’ differential straight line elements.
▪ The second conical shock originating from the second cone is approximated as a collection of
‘n’ differential oblique shocks.
2018 AIAA Space Planes and Hypersonics Conference 11/13/2022 6
METHODOLOGY – Double Cone Forebody
Isolator
Cowl
H
θs1
θs2
θc2
θc1

7. ▪ ALGORITHM:
▪ Given: Second cone angle (𝜽𝒄𝟐) and inlet radius (H).
▪ Data from Taylor-Maccoll method: first shock angle (𝜽𝒔), intermediate Mach number (𝑴𝒊), intermediate shock
angles (𝜽𝒊)
▪ The properties in the intermediate shocks remain constant along the shock (isentropic). Mach number upstream of
the intermediate shocks are calculated as:
𝑀 = (𝑀𝑖+𝑀𝑖−1)/2
▪ Calculate shock angle (𝜷𝒊) downstream of every intermediate shock location using 𝜃𝑖 − 𝛽𝑖 − 𝑀.
𝜷 = 𝒇(𝑴, 𝜽𝒊)
Where, 𝜽𝒊 = 𝜽𝒄𝟐 − 𝜽𝒊 + 𝜽𝒄𝟏
2018 AIAA Space Planes and Hypersonics Conference 11/13/2022 7
METHODOLOGY – Double Cone Forebody

8. ▪ ALGORITHM Cont’d:
▪ Using simple trigonometry, the shock lengths are computed.
𝑙𝑖−1
sin(180−𝛽𝑖)
=
𝑠𝑖
sin(𝜃𝑖−𝜃𝑖−1)
=
𝑙𝑖
sin(𝛽𝑖−𝜃𝑖−1+𝜃𝑖)
(Sine Rule)
where, at i = 1
𝑆1 =
𝐻
𝑠𝑖𝑛𝜃𝑠
▪ 𝑖 marches from the first shock angle (𝜃𝑠) to the cone angle (𝜃𝑐1). The coordinates of the second shock
downstream of each intermediate shock is computed using the lengths from the above calculation.
▪ This gives the shock intersection point and the length of the fist cone.
▪ The Mach number is simply the average of ‘n’ downstream Mach numbers (𝑴𝟐𝒊).
▪ If the length of the first cone is given instead of the inlet radius, the same algorithm can be adopted, except
the calculation is carried out by marching in the increasing order of the intermediate shock angles.
2018 AIAA Space Planes and Hypersonics Conference 11/13/2022 8
METHODOLOGY – Double Cone Forebody
𝜃𝑖−1
θi
βi