p sweeney - boundary layer on a slender body
TRANSCRIPT
Boundary Layer on a Slender Body
Paul Sweeney
September 2013
Imperial College London
Department of Mathematics
Supervised by Professor Anatoly Ruban
Submitted in part fulfilment of the requirements for the degree of
Master of Science in Applied Mathematics of Imperial College London
I herewith certify that all material in this dissertation which is not my
own work has been properly acknowledged
Abstract
This paper concerns high Reynolds number laminar flow past a slender
body of revolution, which consists of a needle with a cone attached to its
trailing edge. Assuming the radius of the needle is comparable with the
thickness of the boundary layer, self-similar solutions are found. From here,
the case in which the needle radius is much smaller than the boundary layer
thickness is then analysed. Attaching a cone to the trailing edge of the
needle, the second half of the paper focuses on studying the shock wave
produced by the cone, again seeking a self-similar solution. The asymptotic
behaviour of small values of the cone angle downstream of the shock is
then investigated. Finally the interaction of the boundary layer with the
shock wave is studied and a boundary-value problem is presented, providing
a numerical investigation to find the conditions in which boundary-layer
separation takes place.
2
Acknowledgments
I would like to thank my supervisor, Professor Anatoly Ruban, for his time,
expertise and patience, without which this project would not be complete.
I would like to thank the Department of Mathematics at Imperial College
London for providing an extraordinary learning experience throughout my
masters course.
I would like to thank my family for their constant encouragement and
guidance throughout my education.
Finally, I would like to thank Andy, Chris, Ed and Charly for their
unwavering support and humour throughout my time at Imperial College.
I can only imagine what antics are in store for this coming year.
3
Contents
1 Introduction 8
2 Hypersonic Boundary Layer on a Needle 11
2.1 Numerical Observations of the Solution . . . . . . . . . . . . 23
2.2 Asymptotic Behaviour of the Solution . . . . . . . . . . . . . 25
3 Shock Wave/Boundary Layer Interaction 28
3.1 Hypersonic Flow Past a Circular Cone . . . . . . . . . . . . . 30
3.1.1 Conical Inviscid Flow Equation . . . . . . . . . . . . . 30
3.1.2 Jump Condition Analysis . . . . . . . . . . . . . . . . 34
3.1.3 Strong and Weak Viscous Interactions . . . . . . . . . 36
3.2 Inspection Analysis of the Interaction Process . . . . . . . . . 38
3.3 Compression Ramp . . . . . . . . . . . . . . . . . . . . . . . . 44
4 Conclusion 48
Appendix 51
Bibliography 56
4
Nomenclature
Λ Inner variable for the inner asymptotic expansion
δ Boundary layer thickness
ε Growth rate of parabola
η Similarity variable
γ Ratio of specific heat
µ Dynamic viscosity
µF Dynamic viscosity at temperature fluctuations
φ Stream function
ρ Fluid density
ρ∞ Free-stream density
ξ Dorodnitsyn transformation variable
a∞ Speed of sound
cp Specific heat
f Self-similar function for U0
g Self-similar component of V0
h Enthalpy
L Characteristic length scale of needle
M∞ Free-stream Mach number
p Fluid pressure
5
p∞ Free-stream pressure
Pr Prandtl number - a ratio of momentum diffusivity to thermal diffu-
sivity
R Scaled radial distance
r Radial distance
R0 Function defining the surface of the needle
Re Reynolds number - a ratio of inertial to viscous forces
T Function defining the temperature at the surface of the needle
u Longitudinal velocity component
v Radial velocity component
V∞ Free-stream velocity
x Longitudinal coordinate
Superscripts
¯ Affine transformation for compression ramp problem
ˆ Dimensional component
˜ Component subject to the Mangler transformation
6
List of Figures
1.1 The North American Aviation X-43 . . . . . . . . . . . . . . 8
1.2 The NASA X-15 (Hyper-X) . . . . . . . . . . . . . . . . . . . 9
2.1 Cylindrical coordinate system . . . . . . . . . . . . . . . . . . 11
2.2 The flow past a slender body . . . . . . . . . . . . . . . . . . 12
2.3 Numerical solution for U0 and V0 with different values of ε . . 24
2.4 Comparison of the numerical, inner and outer asymptotic
solutions for φ′ . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1 Flow past a slender body . . . . . . . . . . . . . . . . . . . . 28
3.2 Boundary-layer separation . . . . . . . . . . . . . . . . . . . . 29
3.3 Compression ramp . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4 Flow pas a circular cone . . . . . . . . . . . . . . . . . . . . . 31
3.5 Cone flow solution . . . . . . . . . . . . . . . . . . . . . . . . 34
3.6 Viscous interaction regimes on a cone . . . . . . . . . . . . . 37
3.7 Progression of two streamlines within the boundary layer . . 41
3.8 Triple deck structure . . . . . . . . . . . . . . . . . . . . . . . 43
4.1 Compression ramp interaction region . . . . . . . . . . . . . . 49
7
1 Introduction
An important part of studying viscous flow is analysing the immediate vicin-
ity of a surface in which the effects of viscosity are significant, termed, the
boundary layer. Research into boundary layers began in the early part of
the 20th century by its pioneer, Ludwig Prandtl, who is often referred to as
the father of modern aerodynamics. At the Third International Mathemat-
ics Congress in Heidelberg, Germany, Prandtl postulated that the effect of
friction was caused by no-slip, a condition which states that fluid immedi-
ately adjacent to a surface sticks to the said surface. He theorised that the
no-slip effect is only present in the boundary layer. The presentation of his
theory at Heidelberg and later in his paper, Prandtl [11], paved the way for
boundary-layer theory.
In the 1950s − 1960s the physics of hypersonic flows became a real in-
terest to aerodynamicists. The space programmes that began in the mid
20th century required knowledge of the detailed processes that occur during
hypersonic flight. The space race was an important part of the advances
made in hypersonic craft, from the X-15 (see Figure 1.1), which to this day,
holds the record for the fastest speed ever achieved by a manned aircraft; to
NASA’s Space Shuttle program, which marked the first winged hypersonic
reentry of a manned spacecraft.
Figure 1.1: The North American Aviation X-15. Reproduced from [1].
8
Figure 1.2: The computational fluid dynamics image of the Hyper-X ve-hicle at a Mach 7 test condition. The figure illustrates surfaceheat transfer and flowfield contours at local Mach Number -a method for predicting vehicle performance and structural,pressure and thermal design loads. Reproduced from [2].
Although the definition of hypersonic flow is quite vague due to a lack of
discontinuity between itself and supersonic flows, it is generally considered
to be speeds of Mach 5 and above. Generally hypersonic flows have the fol-
lowing characteristics - thin shock layers, entropy layers, viscous interaction
phenomena (where the viscous flow greatly affects the inviscid flow, which
in turn affects the boundary layer) and high temperatures.
Hypersonic flow can approximately be separated into a number of regimes,
such as a radiation-dominated regime and ionised gas. In this paper, we shall
be studying a perfect gas. In this regime, the gas is regarded as an ideal
gas†.
A major complication of hypersonic flow are the shock waves produced
by a body and their interaction with the boundary layer. Shock-wave
boundary-layer interactions (SWBLI) play a big part in designing hyper-
sonic craft due to intense local heating. Understanding and controlling
SWBLI requires analytics or numerics. Initially studies of SWBLI were
approached experimentally but recently due to giant leaps in computing ca-
pabilities, numerical simulations have given great insight into the interactive
† An ideal gas obeys the law given by the Clapeyron equation
p = ρRT
where T is the absolute temperature measured in degrees Kelvin and R is the gasconstant.
9
processes (see Figure 1.2).
In this paper we are only concerned with laminar flow. However, there
have been many studies into turbulent flow, or more currently, laminar-
turbulent transition. This is the complicated process in which laminar flow
becomes turbulent and presently remains to be fully understood. Presently,
research areas are aiming to enhance current theoretical framework, while
developing strategies for transition control with the aim to develop aerody-
namic efficiency in vehicles.
10
2 Hypersonic Boundary Layer on
a Needle
In this section we study hypersonic laminar flow past a body of revolution
with a small radius, we shall refer to this body as a ”needle”. We shall
assume that the oncoming flow is parallel to the axis of the needle. Since
the boundary layer upstream of the trailing edge is not affected by the
edge itself, the boundary-layer behaves equally as that on a semi-infinite
needle. In which case, the problem does not have a characteristic length
scale, implying that self-similar solutions can be sought.
It is convenient to represent the problem in terms of cylindrical polar
coordinates (x, r, θ) (see Figure 2.1). Due to the symmetry of the needle
(see Figure 2.2), we go on to assume axisymmetric flow with respect to the
x-axis and flow is independent of θ, that is, the flow velocity is given by
(Vx, Vr, Vθ) = (u(x, r), v(x, r), 0).
Figure 2.1: Cylindrical polar coordinate system defining point M in theflow field. Reproduced and altered from Fig 1.29 of Rubanand Gajjar [16].
11
Figure 2.2: The flow past a needle.
The leading edge of the needle is located at the origin and the x-axis is
placed along its centre (see Figure 2.2). Assuming the flow is steady, the
dimensional Navier-Stokes equations governing this flow are
∂
∂x(ρru) +
∂
∂r(ρrv) = 0, (2.1a)
ρ
(u∂u
∂x+ v
∂u
∂r
)=− ∂p
∂x+
1
r
∂
∂r
[µr
(∂v
∂x+∂u
∂r
)]+
∂
∂x
2µ∂u
∂x− 2
3µ
[1
r
∂
∂r(rv) +
∂u
∂x
],
(2.1b)
ρ
(u∂v
∂x+ v
∂v
∂r
)=− ∂p
∂r+
∂
∂r
2µ∂v
∂r− 2
3µ
[1
r
∂
∂r(rv) +
∂u
∂x
]+ µ
∂
∂x
(∂v
∂x+∂u
∂r
)+ 2
µ
r
(∂v
∂r− v
r
),
(2.1c)
ρ
(u∂h
∂x+ v
∂h
∂r
)=ρ
(u∂p
∂x+ v
∂p
∂r
)
+1
Pr
1
r
∂
∂r
(µr∂h
∂r
)+ µ
(∂u
∂r
)2
,
(2.1d)
h =γ
γ − 1
p
ρ, (2.1e)
where (2.1a)-(2.1e) are the dimensional continuity, x-momentum, r-momentum,
energy and state equations respectively. We also note the quantities u, v,
ρ, p, h and µ are the velocity components in the x and y directions, gas
density, pressure, enthalpy and viscosity, respectively.
12
In order to express the Navier-Stokes equations (2.1), in non-dimensional
form, we use the following scalings
x = Lx, r = Lr, u = V∞u, v = V∞v
ρ =ρ∞M2∞ρ, p = p∞ +
ρ∞V2∞
M2∞
p, h = V 2∞h, µ = µFµ
(2.2)
where L is the length of the needle, µF is the viscosity in the region of
temperature fluctuations and V∞, ρ∞ and p∞ are the fluid velocity, density
and pressure in the free stream flow respectively. Note that in flows that
exceed the Mach number 0.3, the fluid undergoes large pressure changes
and in response, density varies significantly. Therefore, it is useful to scale
density with respect to the Mach number and as a consequence of balancing
the first convective term with the pressure gradient, we find the appropriate
scaling for pressure.
As a result of substituting the scalings (2.2), into the Navier-Stokes equa-
tions (2.1), we find
∂
∂x(ρru) +
∂
∂r(ρrv) = 0, (2.3a)
ρ
(u∂u
∂x+ v
∂u
∂r
)=− ∂p
∂x+M2∞
Re
1
r
∂
∂r
[µr
(∂v
∂x+∂u
∂r
)]+∂
∂x
2M2∞
Reµ∂u
∂x− 2
3
M4∞
Re
[1
r
∂
∂r(rv) +
∂u
∂x
],
(2.3b)
ρ
(u∂v
∂x+ v
∂v
∂r
)=− ∂p
∂r
+M2∞
Re
∂
∂r
2µ∂v
∂r− 2
3µ
[1
r
∂
∂r(rv) +
∂u
∂x
]+M2∞
Reµ∂
∂x
(∂v
∂x+∂u
∂r
)+ 2
M2∞
Re
µ
r
(∂v
∂r− v
r
),
(2.3c)
ρ
(u∂h
∂x+ v
∂h
∂r
)=ρ
(u∂p
∂x+ v
∂p
∂r
)+
M2∞
RePr
1
r
∂
∂r
(µr∂h
∂r
)+M2∞
Reµ
(∂u
∂r
)2
,
(2.3d)
h =1
γ − 1
1
ρ+
γ
γ − 1
p
ρ, (2.3e)
where Pr is the Prandtl number and the free-stream Mach number M∞, is
13
defined as
M∞ =V∞a∞
, where a∞ =
√γp∞ρ∞
and Re =ρ∞V∞L
µF.
To perform asymptotic analysis of the Navier-Stokes equations, (2.3), we
assume M∞ is finite and greater than one, we also assume
Re→∞.
Asymptotic analysis of the boundary layer is based on the limit procedure
x = O(1), R = δ−1r = O(1), Re→∞,
where δ is the thickness of the boundary layer to be determined. The
solution to the Navier-Stokes equations may be sought in the form of the
asymptotic expansions
u(x, r;Re) = U0(x,R) + . . . , v(x, r;Re) = σ(Re)V0(x,R) + . . .
ρ(x, r;Re) = ρ0(x,R) + . . . , p(x, r;Re) = χ(Re)P0(x,R) + . . .
h(x, r;Re) = h0(x,R) + . . . , µ(x, r;Re) = µ0(x,R) + . . .
(2.4)
Note coefficients σ(Re) and χ(Re) are unknown, and shall be found in the
following analysis. Also take note that the asymptotic expansion for u
should be an order one quantity as u increases from u = 0 at the surface of
the needle to u = 1 at the outer edge of the boundary layer.
Substituting the asymptotic expansions (2.4) into the continuity equation
(2.3a) yields∂
∂x(ρ0RU0) +
σ(Re)
δ(Re)
∂
∂R(ρ0RV0) = 0. (2.5)
If we assume δ(Re) σ(Re), the continuity equation would degenerate to
∂
∂x(ρ0RU0) = 0, U0
∣∣∣∣x=0
= 1, ⇒ U0 ≡ 1, (2.6)
which is comparable to dealing with the inviscid region, and so does not
satisfy the condition on the needle surface.
Alternatively, if σ(Re) δ(Re), the continuity equation would degener-
14
ate to∂
∂R(ρ0RV0) = 0. (2.7)
Since V0 = 0 on the plate surface, we can conclude V0 = 0 in the inner
region. Hence, according to the principle of least degeneration, we choose
σ(Re) = δ(Re), (2.8)
which reduces the continuity equation to
∂
∂x(ρ0RU0) +
∂
∂R(ρ0RV0) = 0. (2.9)
Now substituting (2.4) and (2.8) into the x-momentum equation (2.3b),
yields
ρ0
(U0∂U0
∂x+ V0
∂U0
∂R
)=− χ(Re)
∂P0
∂x
+M2∞
Reδ21
R
∂
∂R
[µ0R
(δ2∂V0∂x
+∂U0
∂R
)].
(2.10)
If δ2Re M2∞, the fluid would appear inviscid and U0 = 0 on the plate
surface. If on the other hand, M2∞ δ2Re, (2.10) degenerates to
∂
∂R
(µ0R
∂U0
∂R
)= 0. (2.11)
Note, we have disregarded the O(δ2) term as we have assumed δ2 1.
However, the problem does not have a solution when solved using the no-
slip condition on the plate surface and the matching condition with the
inviscid region. Thus, by the principle of least degeneration we choose
δ = M∞Re−1/2. (2.12)
To analyse χ(Re) we use (2.12) and based on Fitzhugh [4], we make a
further assumption that
δ(Re)→ 0 as M∞ →∞.
15
Now substituting (2.4) into the r-momentum equation (2.3c), yields
ρ0
(U0∂V0∂x
+ V0∂V0∂R
)=− χ(Re)
δ(Re)2∂P0
∂R
+ µ0M2∞
Re
∂
∂R
[2∂V0∂R
+1
R
∂
∂R(RV0) +
∂U0
∂x
]+ µ0
M2∞
Re
∂
∂x
(∂V0∂x
+1
δ(Re)2∂U0
∂R
)+ 2
M2∞
δ(Re)2Re
µ0R
(∂V0∂R− V0R
).
(2.13)
We note that for any χ(Re) 1, the pressure term can be disregarded. For
δ2 χ(Re), the pressure does not change across the boundary layer, that
is∂P0
∂R= 0. (2.14)
In which case, we can find the boundary condition for (2.14) by matching the
asymptotic expansion for pressure in the boundary layer with the inviscid
solution at the outer edge of the boundary layer and so
P0
∣∣∣∣R=∞
= 0. (2.15)
Solving (2.14) with (2.15), we can easily see that P0 = 0 inside the boundary
layer. We conclude by choosing a solution for χ(Re). Using our analysis to
find the solution for P0 and (2.12), by the principle of least degeneration,
we choose†
χ(Re) = M∞Re−1/2, (2.16)
which we note is equivalent to the boundary layer width (2.12).
We finish by summarising our analysis to form a closed set of equations
for the boundary layer form on the surface of a needle in high Reynolds
number flow. The solution of the Navier-Stokes equations, (2.3), may be
† The argument to choose the scaling for pressure is presented in Section 3.2 usingthe Ackeret formula, (3.53). Along with (3.47) and (3.52) we deduce pressure isO(M∞Re
−1/2).
16
sought in the form of the asymptotic expansions
u(x, r;Re) = U0(x,R) + . . . , v(x, r;Re) = M∞Re−1/2V0(x,R) + . . .
ρ(x, r;Re) = ρ0(x,R) + . . . , p(x, r;Re) = M∞Re−1/2P0(x,R) + . . .
h(x, r;Re) = h0(x,R) + . . . , µ(x, r;Re) = µ0(x,R) + . . .
(2.17)
Substitution of (2.17) into the Navier-Stokes equations (2.3) leads to the
boundary layer equations
∂
∂x(ρ0RU0) +
∂
∂R(ρ0RV0) = 0, (2.18a)
ρ0
(U0∂U0
∂x+ V0
∂U0
∂R
)=
1
R
∂
∂R
(µ0R
∂U0
∂R
), (2.18b)
ρ0
(U0∂V0∂x
+ V0∂V0∂R
)=µ0
∂2U0
∂x∂R+ 2
µ0R
(∂V0∂R− V0R
), (2.18c)
ρ0
(U0∂h0∂x
+ V0∂h0∂R
)=
1
Pr
1
R
∂
∂R
(µ0R
∂h0∂R
)+ µ0
(∂U0
∂R
)2
, (2.18d)
h0 =1
γ − 1
1
ρ0, (2.18e)
which are solved using the following set of boundary conditions. The free-
stream condition at the leading edge of the needle
U0 = 1, h0 =1
γ − 1at x = 0, R ∈ [0,∞) (2.19)
and at the outer edge of the boundary layer
U0 = 1, h0 =1
γ − 1at R =∞, x ∈ [0,∞) . (2.20)
We also have the no-slip condition on the needle surface
U0 = V0 = 0 at R = R0(x), x ∈ [0, 1] , (2.21)
where R0 is the radius of the needle as a function of x. We also add a
thermal condition and state the needle temperature is given as a function
of x,
h0 = T (x) at R = R0(x), R ∈ [0,∞) . (2.22)
17
To simplify the boundary value problem, (2.18)-(2.22), we can apply what
is known as the Mangler Transformation. The purpose of the Mangler
transformation is to alter an axisymmetric flow into a two-dimensional flow
problem. Its non-dimensional form is defined as,
x =
∫ x
0R2
0(x)dx, R = R0 [R−R0(x)] , U0 = U0
V0 =1
R0
(V0 +
U0R
R0
dR0
dx
), ρ0 = ρ0, µ0 = µ0, h0 = h0.
(2.23)
In order to apply the Mangler Transformation to (2.18a), (2.18b) and (2.18d),
we need to find expressions for the derivatives of the dependent variables.
They are found to be
∂
∂x= R2
0
∂
∂x+R
dR0
dx
∂
∂R, (2.24a)
∂
∂R= R0
∂
∂R. (2.24b)
Substitution of (2.23) and (2.24) into equations (2.18a), (2.18b) and (2.18d)
yields,
∂
∂x
(ρ0U0
)+∂
∂R
(ρ0V0
)= 0, (2.25a)
ρ0
(1 +
R
R20
)(U0∂U0
∂x+ V0
∂U0
∂R
)=∂
∂R
[(1 +
R
R20
)µ0∂U0
∂R
], (2.25b)
ρ0
(1 +
R
R20
)(U0∂h0∂x
+ V0∂h0
∂R
)=
1
Pr
∂
∂R
[(1 +
R
R20
)µ0∂h0
∂R
]
+
(1 +
R
R20
)µ0
(∂U0
∂R
)2
.
(2.25c)
To formulate the boundary conditions for (2.25) we simply apply the
Mangler Transformation to (2.19), (2.20) and (2.21). Hence, we see at the
leading edge of the boundary layer
U0 = 1 at x = 0, R ∈ [0,∞) (2.26)
18
as well as the outer edge of the boundary layer
U0 = 1 at R =∞, x ∈ [0,∞) (2.27)
and the no-slip condition on the needle surface
U0 = V0 = 0 at R = 0, (2.28)
essentially reducing the needle surface to a flat plate problem. We also have
h0 = hw at η = 0, (2.29)
where hw is the temperature on the body surface.
The solution to the boundary value problem can be sought in terms of
a self-similar solution. The form of the solution is found by seeking an
invariant affine transformation of (2.25) and boundary conditions (2.26)-
(2.29). Doing so, the form of the solution is found to be
U0(x, R) = f(η), V0(x, R) =1√xg(η), ρ0(x, R) = ρ(η),
µ0(x, R) = µ(η), h0 = h(η), R0(x) =√εx1/4,
(2.30)
where ε is a constant and η is the similarity variable, which is defined as
η =R√x. (2.31)
Hence∂η
∂x= −1
2
η
x,
∂η
∂R=
1√x, (2.32)
and so†
∂U0
∂x= −1
2
η
xf ′(η),
∂U0
∂R=
1√xf ′(η). (2.33)
Substitution of (2.33) into the x-momentum (2.25b) and the energy equa-
tion (2.25c) reduces them to
ρf ′(
1 +η
ε
)(−1
2ηf + g
)=[(
1 +η
ε
)µf ′]′, (2.34)
† Note the dashed notation represents the derivative with respect to η.
19
ρh′(
1 +η
ε
)(−1
2ηf + g
)=
1
Pr
[(1 +
η
ε
)µh′]′
+(
1 +η
ε
)µ(f ′)2 (2.35)
and the continuity equation, (2.25a), transforms into
−1
2η (ρf)′ + (ρg)′ = 0. (2.36)
We also need to formulate the boundary conditions for the self-similar
equations (2.34)-(2.36). This is done by substituting (2.30) into (2.26)-
(2.28). Note we can not apply (2.30) to (2.19) as we have a singularity at
x = 0. Thus, we see at the outer edge of the boundary layer
f = 1, at η =∞. (2.37)
Similarly, the no-slip condition on the needle’s surface transforms to
f = g = 0 at η = 0, x ∈[0,
∫ 1
0R2
0dx
]. (2.38)
The next steps seek to simplify the equations (2.34), (2.35) and (2.36).
Through simple manipulations, the continuity equation (2.36) can be rear-
ranged in the form
−1
2(ρηf)′ +
1
2ρf + (ρg)′ = 0. (2.39)
If we introduce a new function φ(η) such that
φ′ = ρf and φ(0) = 0, (2.40)
the continuity equation (2.39) becomes
−1
2(ρηf)′ +
1
2φ′ + (ρg)′ = 0. (2.41)
Integrating equation (2.41) with respect to η and using φ(0) = 0 to solve
for the integration constant, we find
−1
2ρηf + ρg = −1
2φ. (2.42)
20
We notice that (2.42) can be substituted into (2.34) and (2.35), the com-
bination produces the following equations
−1
2φf ′
(1 +
η
ε
)=[(
1 +η
ε
)µf ′]′, (2.43)
−1
2φh′
(1 +
η
ε
)=[(
1 +η
ε
)µh′]′
+(
1 +η
ε
)µ(f ′)2. (2.44)
For further simplification, it is convenient to introduce variable ξ in terms
of the Dorodnitsyn transformation
ξ =
∫ η
0ρ(η)dη, (2.45)
which transforms equations (2.40), (2.43) and(2.44) into the following set
of equations
dφ
dξ=f, (2.46a)
−1
2φdf
dξ
(1 +
η
ε
)=d
dξ
[(1 +
η
ε
)µρdf
dξ
], (2.46b)
−1
2φdh
dξ
(1 +
η
ε
)=d
dξ
[(1 +
η
ε
)µρdh
dξ
]+(
1 +η
ε
)µρ
(df
dξ
)2
.
(2.46c)
In order to close the set of equations, we need to specify the dependence
of the viscosity on temperature using work presented by Ruban and Gajjar
[14] on a compressible flat plate problem. For simplicity we assume a linear
dependence,
µ = CT =C
cph, (2.47)
for some constant C. We can apply (2.47) in order to express the gas in the
free stream by using the state equation (2.18e), to find
µ∞ =C
cph∞ =
C
cp
γ
γ − 1
p∞ρ∞
=C
cp
a2∞γ − 1
(2.48)
Now remembering h = V 2∞h and dividing (2.47) by (2.48), we form the
equation
µ = (γ − 1)h, (2.49)
21
which combined with the state equation (2.18e) in self-similar form, shows
that
µρ = 1. (2.50)
As a result, equation (2.46b) and (2.46c) can be written as†
−1
2φφ(
1 +η
ε
)=d
dξ
[(1 +
η
ε
)φ], (2.51a)
−1
2φh(
1 +η
ε
)=
1
Pr
d
dξ
[(1 +
η
ε
)h]
+(
1 +η
ε
)(φ)2, (2.51b)
where the boundary conditions are found using equation (2.46a). It follows
that,
φ(0) = φ(0) = 0 and φ(∞) = 1. (2.52)
The boundary-value problem has no known analytical solution and so
needs to be solved numerically. With the numerical solution for φ(η), the
velocity components in the boundary layer may simply be found by using
(2.46a) and by rearranging (2.41), resulting in
U0 = ρφ′(η), V0 =1
2√x
(ηφ′ − φ
ρ
)(2.53)
† Note the dot notation represents the derivative with respect to ξ .
22
2.1 Numerical Observations of the Solution
It can easily be seen from (2.51a), that if the radius of needle is much greater
than the boundary layer thickness i.e. δ R0, the problem is equivalent to
compressible flow past a flat plate. Hence, reducing (2.51a) to the Blasius
equation
On the other hand, if we assume the radius of the needle is much smaller
than the boundary layer thickness, R0 → 0, we can easily see the right hand
side of equation (2.51a) becomes the dominant term.
Under our Affine transformation we have assumed flow is past a paraboloid
of the form
R0 =√εx1/4, (2.54)
where ε is a constant and 0 < ε 1. At this stage, for simplicity, we shall
analyse the horizontal velocity component at a point where the density, ρ,
is constant i.e. an incompressible flow†. We can therefore rearrange (2.51a)
into
−1
2ρ (ε+ η)φφ′′ =
[(ε+ η)φ′′
]′(2.55)
In order to see accurate behaviours of the system (2.55), we perform
numerical simulations. To compute the solution φ′, we use the shooting
method combined with the 4th order Runge-Kutta stepping scheme and
the bisection method. The shooting method reduces the boundary-value
problem, (2.55) and (2.52), to a system of initial value problems, which is
then solved using the RK4 scheme - an iterative method for approximation
of the solutions. We note that since we do not have a boundary condition
for φ′′, the bisection method is used to approximate φ′′(0) to a desired level
of accuracy.
Figure (2.3)‡ shows profiles of the horizontal and vertical velocity com-
ponents, U0 and V0, at a fixed density value of ρ = 1 and different values
of ε. We observe three distinct stages in the transition from a large needle
radius to a small needle radius:
(a) ε 1: In this case the radius of the needle is large in comparison to the
width of the boundary layer. As expected, this solution is essentially
† A more accurate numerical solution can be found by solving the energy equation(2.51b), for h and using the state equation (2.18e), to solve for ρ.
‡ See Appendix for numerical coding.
23
that of Blasius equation for incompressible flow past a flat plate.
(b) ε = 1: Here the radius of the needle is comparable to that of the
boundary layer thickness. Comparing to Blasius solution, we see that
U0 increases at a higher rate and V0 has a lower velocity at the edge of
the boundary layer.
(c) ε 1: In this scenario the radius of the needle is much smaller than
the width of the boundary layer. In comparison to Figures (2.3a) and
(2.3b), U0 increases at a much higher rate. On the other hand, V0 has
a significantly lower velocity profile.
(a) ε = 100 (b) ε = 1
(c) ε = 0.001
Figure 2.3: Numerical solutions of equation (2.55) for different values ofε. The continuous curve indicates the solution of U0 = φ′ andthe dashed curve indicates the solution of V0 ∼ ηφ′ − φ.
24
2.2 Asymptotic Behaviour of the Solution
It is easily seen that (2.55) is a singular perturbation problem, which we shall
now proceed to solve using the method of matched asymptotic expansion.
Again, as in the case of our numerical solutions, for simplicity, we assume
density is constant.
Based on the boundary condition, φ′(∞) = 1, we introduce an asymptotic
expansion of the form
φ(η) = η + α(ε)φ1 + . . . as ε→ 0 , η = O(1), (2.56)
where α is a function of ε to be found. Substituting (2.56) into (2.55) yields
O(α) :(ηφ′′
)′+
1
2ρη2φ′′ = 0 , φ′1(∞) = 0, (2.57)
which can be solved for φ′1(η) using separation of variables and so
φ′1 = A1 Ei
(−1
4ρη2), (2.58)
where Ei is the exponential integral. Without loss of generality, the constant
factor can be absorbed into α(ε). We therefore find the outer expansion for
φ′ to be
φ′(η) = 1 + α(ε) Ei
(−1
4ρη2)
+ . . . (2.59)
We note that as η → 0, we find the inner limit of the outer expansion to be
φ′(η) = 1 + α(ε)
[ln
(1
4ρ
)+ 2 ln η − 1
4ρη2 + . . .
]. (2.60)
The ’boundary layer’ is near η = 0 as the outer solution breaks down
here. Introducing the inner variable
Λ =η
κ(ε), (2.61)
where the width of the layer, κ(ε), is to be found in the following analysis.
25
Using the inner variable it can easily be seen that
−ρ (ε+ η)φφ′′︸ ︷︷ ︸O
(ε · 1/κ2
)and O (1/κ)
= 2[(ε+ η)φ′′
]′︸ ︷︷ ︸O
(ε · 1/κ3
)and O
(1/κ2
) . (2.62)
Dominant balance leads us to the conclusion that κ = ε, thus
Λ =η
ε. (2.63)
In terms of Λ, the momentum equation, (2.55), reads
−1
2ρ (1 + Λ)φ(Λ)φ′′ =
[(1 + Λ)φ′′
]′. (2.64)
If we introduce the inner expansion
φ′ = β(ε)Φ′0 + . . . , (2.65)
where β is a function of ε yet to be found. Then
O(β) :[(1 + Λ) Φ′′0
]′= 0 , Φ′0(0) = 0, (2.66)
which has the solution
Φ′0 = B0 ln(1 + Λ). (2.67)
Without loss of generality, the factors B0 can be absorbed into β(ε) and so
substitution of (2.67) into (2.65) yields
φ′ = β(ε) ln(1 + Λ) + . . . (2.68)
We note that as Λ→∞, the outer limit of the inner expansion, (2.68), is
φ′ ∼ β(ε)
[ln Λ +
1
Λ+ . . .
], (2.69)
which can be written in terms of the outer variable η
φ′ ∼ β(ε)
[ln
1
ε+ ln η +
ε
η+ . . .
](2.70)
26
Matching the O(1) constant terms with the outer expansion (2.60) we find
1 + α ln
(1
4ρ
)= β ln
1
ε. (2.71)
Now matching with the logarithmic term in (2.60)
2α = β. (2.72)
Solving these simultaneous equations we find
α =1
ln(
4ε2ρ
) and β =2
ln(
4ε2ρ
) . (2.73)
Summarising our analysis, we find the outer solution
φ′(η) ∼ 1 +1
ln(
4ε2ρ
) Ei
(−1
4ρη2), (2.74)
and the inner solution
φ′ ∼ 2
ln(
4ε2ρ
) ln(1 + Λ). (2.75)
Figure 2.4: The solution of the problem of (2.55) & (2.52) with ε = 0.001and ρ = 1. The contintuous curve gives the exact numericalsolution. The dotted curve is the outer asymptotic expansionand the dashed curve is the inner asymptotic expansion.
27
3 Shock Wave/Boundary Layer
Interaction
In this section of the paper, we attached a cone to the trailing edge of the
needle (see Figure 3.1) with the aim to study the interaction of the boundary
layer with the shock wave. We place the origin at the needle/cone junction
and aligning the x-axis with the centre of the needle such that the leading
edge of the needle is placed at x = −L.
Classical boundary-layer theory is based on the idea that the effect of the
viscous layer on the outer inviscid region is negligible. However, it has been
shown numerically that the inviscid pressure distribution has a significant
affect on the behaviour of the boundary layer and so gives us an insight
into the boundary layer separation phenomenon. If the pressure gradient is
favourable, that is, the pressure decreases downstream, the boundary layer
stays attached to the wall. However with an adverse pressure gradient, that
is, the pressure increases downstream, the boundary layer tends to separate
Figure 3.1: The flow past a slender body, with a cone of angle Θ madewith the x-axis.
28
from the body surface.
On the surface of a flat plate aligned with the x-axis, the point of sepa-
ration may be defined as a point of zero skin friction i.e.
τw = µ∂u
∂y
∣∣∣∣y=0
= 0 (3.1)
We can easily see that if τw is positive upstream of the point itself, the fluid
particles within the boundary layer move downstream along the wall and
thereby remaining attached to the body surface. On the other hand, when
τw becomes negative an area of reversed flow (u < 0) appears near the wall
as seen in Figure 3.2.
Figure 3.2: Boundary-layer separation occurring at point S. Reproducedfrom Fig 9.4 of Ruban and Gajjar [17].
Assuming a small cone angle, Θ, in the case of hypersonic flow past thin
bodies, the boundary layer shrinks to a small vicinity. Since we have as-
sumed Vθ = 0 and that u and v are independent of θ, we shall consider a
plane aligned with the x and y axes in the upper-half place. The result then
resembles a compression ramp problem as seen in Figure 3.3.
The following work seeks to analyse the orders of terms in the Navier-
Stokes equations, combined with the shock jump conditions, to find a rela-
tion between pressure difference and the wedge angle, Θ. We then seek an
equation governing the flow in the inviscid region downstream of the shock
and seek a self-similar solution for this equation. Finally, we analyse the
boundary-layer equations in a small vicinity of the zero skin friction point.
The axisymmetric oblique shock being produced at the needle/cone junc-
29
Figure 3.3: The compression ramp problem.
tion, impinges upon the boundary layer at some point. Close to this point,
we aim to study the behaviour of perturbations induced in the flow through
the interaction between the shock wave and the boundary layer.
3.1 Hypersonic Flow Past a Circular Cone
A shock wave is a region of discontinuity across which causes a rapid rise in
pressure, temperature and density of the flow. In high speed flows, shock
waves are often oblique, that is, the flow velocity is not perpendicular to
the shock wave. The result is upstream and downstream flow directions are
changed across the shock.
3.1.1 Conical Inviscid Flow Equation
We shall now present the exact solution for a shock impinging on the leading
edge of an axisymmetric cone in the inviscid region. To do this we use the
Euler equations for compressible gas flows, which can be combined to form
a single equation for the velocity vector V
V · ∇(V 2
2
)= a2∇ ·V. (3.2)
Remembering we have assumed the flow is axisymmetric and Vθ = 0, equa-
tion (3.2) in cylindrical coordinates is given by
u
(u∂u
∂x+ v
∂v
∂x
)+ v
(u∂u
∂r+ v
∂v
∂r
)+
1
2
v
r
(u2 + v2
)= a2
[∂u
∂x+
1
r
∂
∂r(rv)
],
(3.3)
30
which can be rewritten in the form(1− u2
a2
)∂u
∂x+
(1− v2
a2
)∂v
∂r− uv
a2
(∂u
∂r+∂v
∂x
)+v
r
[1− 1
2a2(u2 + v2
)]= 0.
(3.4)
We now need a second equation relating the velocity components u and v.
Assuming the shock has a conical form, the strength of the shock is the same
for all streamlines crossing the shock, meaning that while entropy increases
at the shock, it then remains constant in the entire region between the shock
and body surface. Hence, using Crocco’s formula (assuming steady flow and
zero body forces), we have
[ω ×V] = 0, (3.5)
where ω is vorticity. Equation (3.5) leads us to conclude the flow is irrota-
tional, ω = 0, which is expressed by the equation
∂v
∂x− ∂u
∂r= 0. (3.6)
When solving (3.4) and (3.6), we also need to use the Bernoulli equation,
Figure 3.4: Flow past a circular cone. Reproduced and altered from Fig4.28 of Ruban and Gajjar [13].
31
which relates the local speed of sound to the flow velocity, we have
a2
γ − 1+V 2
2=
a2∞γ − 1
+V 2∞2
(3.7)
andV 2c
γ − 1+V 2c
2=
a2∞γ − 1
+V 2∞2, (3.8)
where Vc is the critical velocity defined as the velocity at a point in the flow
field in which the fluid speed is equivalent to the local value of the speed of
sound. Equations (3.7) and (3.8) can be combined to form
a2
γ − 1+V 2
2=
γ + 1
2(γ − 1)V 2c . (3.9)
The solution of equations (3.4), (3.6) and (3.9) can be sought in self-
similar form with the velocity components being written in terms of a sim-
ilarity variable ζ
u(x, r) = u(ζ) and v(x, r) = v(ζ), (3.10)
where the similarity variable is defined as
ζ =x
r. (3.11)
Rewriting (3.4) and (3.6) in terms of the similarity variable, we find(1− u2
a2
)du
dζ− ζ
(1− v2
a2
)dv
dζ+ 2ζ
uv
a2du
dζ
+ v
[1− 1
2a2(3u2 + v2
)]= 0,
(3.12a)
ζdu
dζ+dv
dζ= 0. (3.12b)
We can now seek to use u as the independent variable instead of ζ, with
the task to find the function v(u) which is related to v(ζ) by
v(ζ) = v[u(ζ)]. (3.13)
32
Using the chain rule, we can rewrite (3.12a) and (3.12b) in the form(1− u2
a2
)− ζ
(1− v2
a2
)dv
du+ 2ζ
uv
a2
+ vdζ
du
[1− 1
2a2(3u2 + v2
)]= 0,
(3.14a)
ζ =− dv
du. (3.14b)
It remains to use (3.14b) in order to rewrite (3.14a), doing so, we arrive at
the following equation for v(u)
vd2v
du2
[1− 1
2a2(3u2 + v2
)]= 1 +
(dv
du
)2
−(u
a+v
a
dv
du
)2
. (3.15)
It can be more conveniently written using the normalised velocity com-
ponents
λx =u
Vcand λr =
v
Vc. (3.16)
We have
λrd2λrdλ2x
[1− V 2
c
2a2(3λ2x + λ2r
)]= 1 +
(dλrdλx
)2
−(
λxa/Vc
+λra/Vc
dλrdλx
)2
,
(3.17)
where a/Vc can be found using (3.9)
a
Vc=
√γ + 1
2− γ − 1
2(λ2x + λ2r). (3.18)
Mathematically in order to calculate (3.17) it is convenient to consider
the problem in which the shock angle, α, is known and the cone angle, Θ, is
sought as part of the solution for the flow past the cone. We start by con-
structing a shock polar for a given velocity vector in the free-stream, where
the shock polar shows all possible positions of the tip of the normalised
velocity vector behind the shock (see Figure 3.5). Point W corresponds to
the cone surface. The straight line OS depicts the shock position, the per-
pendicular line DS intersects the shock polar at point B which represents
33
Figure 3.5: Cone flow solution. Reproduced and altered from Fig 4.30 ofRuban and Gajjar [13].
the values of λx and λr immediately behind the shock. At this point
dλrdλx
= −xr⇒ (r · dλ) = 0, (3.19)
where λ= (λx, λr) and r = (x, r).
Equation (3.17) can now be calculated from the shock towards the cone
surface. The impermeability condition implies that on the cone surface, λ
should be parallel to dλ, that is
(λ · dλ) = 0, (3.20)
at which point, numerical calculations should be terminated.
3.1.2 Jump Condition Analysis
It is known that the pressure, density and temperature relations across the
shock can be calculated by the jump conditions
p2p1
= 1 +2γ
γ + 1
(M2
1 sin2 α− 1), (3.21a)
ρ2ρ1
=(γ + 1)M2
1 sin2 α
(γ − 1)M21 sin2 α+ 2
, (3.21b)
T2T1
=p2p1
ρ1ρ2, (3.21c)
34
where the subscripts 1 and 2 represent the flow upstream and downstream
of the shock respectively and α is the shock angle. It follows that the Mach
number in the downstream region can be calculated using
M22 sin2(α−Θ) =
2 + (γ − 1)M21 sin2 α
γM21 sin2 α− (γ − 1)
, (3.22)
where Θ is the angle between the needle and cone at the needle/cone junc-
tion (see Figure 3.3).
We seek to find a relation between the wedge angle and the pressure
jump created by the shock. Using known† wedge/shock angle relations we
can write
M21 sin2 α− 1 =
γ + 1
2M2
1
sinα sin Θ
cos(α−Θ). (3.23)
By the definition of the Mach angle‡
sinαm =1
M1⇒ cosαm =
√M2
1 − 1
M1, (3.24)
as Θ is small we can say
cos(α−Θ) ∼ cosαm as α→ αm. (3.25)
Hence, we can rewrite (3.23) as
M21 sin2 α− 1 =
γ + 1
2
M31√
M21 − 1
Θ sinα, (3.26)
where we have used sin Θ ∼ Θ for Θ 1. Equation (3.26) is a quadratic
equation for sinα, and so
sinα
Θ=γ + 1
4+
[(γ + 1
4
)2
+4
M2∞Θ2
]1/2. (3.27)
† The θ − β −M equation defined as
tan θ = 2 cotβM2
1 sin2 β − 1
M21 (γ + cos 2β) + 2
,
where β is commonly used as the shock angle.‡ It is known that in the case of an oblique shock, for the shock to disturb fluid motion, a
necessary condition is M1 sinα ≥ 1. Thus the minimum angle to satisfy this conditionis the Mach angle αm = 1/M1. At this point, the oblique shock wave is know as aMach wave.
35
Note we have taken the positive root.
Substituting (3.26) into our jump condition for pressure, (3.21a), we find
p2p1
= 1 +γM3
1√M2
1 − 1Θ sinα. (3.28)
Now substituting (3.27) into (3.28) and approximating for large values of
M1, results in
p2p1∼ 1 +
γ(γ + 1)
4K2 + γK2
√(γ + 1
4
)2
+1
K2, (3.29)
where K = M1Θ is known as the hypersonic similarity parameter.
3.1.3 Strong and Weak Viscous Interactions
Figure 3.6 illustrates hypersonic viscous flow over a cone. We see there are
two regions, the strong interaction region in the immediate vicinity of the
leading edge and the weak interaction region further downstream. These
two regions are defined by the physical effects which occur.
Within the strong interaction region the growth rate of the boundary
layer thickness is large. The rapid growth strongly effects in the inviscid
flow region as its streamlines are deflected into oncoming flow. These large
changes feedback into the boundary layer, affecting its growth.
On the other hand, the weak interaction region experiences small growth
of the boundary layer, which has a weak affect on the inviscid flow region.
As a result, these changes have a negligible effect on the boundary layer.
For the purpose of this paper, we shall be considering the weak interaction
region.
It is known that K2 1 in the weak interaction region†. Thus, in the
weak interaction regionp2p1∼ 1 + γK, (3.30)
and sop2 − p1p1
=∆p
p1∼ K = M1Θ. (3.31)
Hence, we can conclude that in the weak interaction region of the boundary
† Known from the tangent cone approximation, that is, local pressure is equal to acone with the same slope. In contrast, K2 1 in the strong interaction region.
36
Figure 3.6: Viscous interaction regimes on a cone.
layer, ∆p is proportional to M∞Θ.‡
Returning to the x-momentum equation from the Navier-Stokes equa-
tions, (2.3b). An increase of pressure results in the deceleration of fluid
particles within the boundary layer, hence we need to compare the first
convective term with the pressure gradient
ρu∂u
∂x∼ ∂p
∂x. (3.32)
Based on the fact that upstream of the shock u and ρ are O(1) quantities
within the boundary layer and that we are studying weak interaction be-
tween with the shock; we also assume u and ρ remaining O(1) quantities in
the main part of the boundary layer downstream of the shock. We arrive
at the follow result
∆u ∼M∞Θ. (3.33)
Since the boundary layer width and the wedge angle is small, asymptotic
analysis of the boundary layer on the cone is based on the limit procedure
x = O(1), R = δ−1c r = O(1), Θ0 = Θ−1Θ = O(1), Re→∞,
where δc is the boundary layer width on the cone and Θ is a scaling param-
eter for the wedge angle which will be found in the following analysis.
The solution to the Navier-Stokes equations in the main part of the
boundary layer, downstream of the shock, may be sought in the form of
‡ The same method can be applied to proportionality relation between Θ and changesin density and temperature.
37
the asymptotic expansions
u(x, r;Re) = 1 +M∞ΘU0(x,R) . . . ,
v(x, r;Re) = σ(Re)V0(x,R) + . . . ,
p(x, r;Re) = M∞ΘP0(x,R) + . . . ,
ρ(x, r;Re) = ρ0(x,R) + . . . ,
h(x, r;Re) = h0(x,R) + . . . ,
µ(x, r;Re) = µ0(x,R) + . . . .
(3.34)
From the continuity equation (2.3a), substituting in the asymptotic ex-
pansions (3.34), by the principle of least degeneration and without loss of
generality we choose†
σ(Re) = M∞Θ2. (3.35)
To conclude, in the case of the boundary layer on the surface of the cone
downstream of the shock, without loss of generality, the solution for the
main part of the boundary layer may be sought in the form of the following
asymptotic expansions
u(x, r;Re) = 1 +M∞ΘU0(x,R) . . . ,
v(x, r;Re) = M∞Θ2V0(x,R) + . . . ,
p(x, r;Re) = M∞ΘP0(x,R) + . . . ,
ρ(x, r;Re) = ρ0(x,R) + . . . ,
h(x, r;Re) = h0(x,R) + . . . ,
µ(x, r;Re) = µ0(x,R) + . . . .
(3.36)
3.2 Inspection Analysis of the Interaction Process
The analysis of the interaction process is done by analysing the x-momentum
equation which we have previously seen. However, we choose to study the
effect of the pressure gradient as we know it plays a role in separation.
Hence, we once again use the x-momentum equation
ρ
(u∂u
∂x+ v
∂u
∂r
)= −dp
dx+M2∞
Re
1
r
∂
∂r
(µr∂u
∂r
). (3.37)
†Assuming we scale the radial component with respect to the cone angle, Θ.
38
We note again that we have assumed Vθ = 0 and the remaining velocity
components, u and v, are independent of θ. Hence, we can focus on a two-
dimensional plane as seen in our jump condition analysis, and treat this as
the interaction process for a compression ramp.
Since we are interested in a weak interaction with the shock, using our
jump condition analysis, we conclude the boundary layer continues up the
wedge and so allowing us to use the asymptotic expansions, (2.17). To
proceed, it must be stated that we know the solution of the wedge flow re-
mains smooth when the leading edge of the wedge is approached. Therefore,
we shall assume U0, ρ0 and µ0 may be represented in the form of Taylor
expansions
U0(x,R) = U00(R) + (−s)U01(R) + . . .
ρ0(x,R) = ρ00(R) + (−s)ρ01(R) + . . .
µ0(x,R) = µ00(R) + (−s)µ01(R) + . . .
as s = x− 1→ 0−. (3.38)
The leading order terms in (3.38) exhibit the following behaviour near the
surfaceU00(R) = λR+ . . .
ρ00(R) = ρw + . . .
µ00(R) = µw + . . .
as R→ 0, (3.39)
where λ, ρw and µw are positive constants representing the dimensionless
skin friction, density and viscosity on the wall surface.
Due to the shock emanating from the needle/cone junction, we assume a
small pressure rise ∆p 1 occurs at the outer edge of the boundary layer
over a short distance, ∆x 1. The change in pressure arises as a result of
deceleration of fluid particles within the boundary layer, hence we need to
compare the first convective term with the pressure gradient in (3.37)
ρu∂u
∂x∼ dp
dx. (3.40)
The perturbations in the flow are small so we can represent U0 and ρ0 by
their initial profiles
ρ00U00∂u
∂x∼ dp
dx. (3.41)
39
Approximating the derivatives by finite differences
ρ00U00∆u
∆x∼ ∆p
∆x⇒ ∆u ∼ ∆p
ρ00U00. (3.42)
Excluding near the surface of the body, everywhere within the boundary
layer ρ00 and U00 are order one quantities. Therefore we can write
∆u ∼ ∆p. (3.43)
If we now consider a space within the boundary layer bounded by two
neighbouring streamlines separated by an initial distance τ as seen in Figure
3.7. Using the conservation of mass we can write
ρ00U00τi = (ρ00 + ∆ρ) (U00 + ∆u) (τi + ∆τi) . (3.44)
Linearising, we deduce that
∆τiτi∼ ∆ρ
ρ00+
∆u
U00. (3.45)
Since we know U00 and ρ00 are order one quantities within main part of the
boundary layer, combined with (3.43), we find the thickness of the filament
increases by
∆τi ∼ τi∆p. (3.46)
Hence, we can deduce across the boundary layer the thickening effect is
given by
∆τ =∑i
∆τi ∼∑i
(τi∆p) ∼ ∆p∑i
τi ∼M∞Re−1/2∆p. (3.47)
Obviously our filament analysis is not valid near the bottom of the bound-
ary layer since U00 → 0 as R→ 0 and (3.42) predicts unbounded growth of
the perturbed velocity. In a thin layer close to the wall, ∆u ∼ U00 which
combined with (3.42) maybe written as
∆u ∼√
∆p. (3.48)
40
Comparing (3.48) with (3.39) leads us to conclude
R ∼√
∆p. (3.49)
Thus, the thickness of the thin sub-layer is given by
r = M∞Re−1/2R ∼M∞Re−1/2
√∆p. (3.50)
We now want to estimate the displacement effect of the sub-layer using
the mass conservation law. Using the same method as before except treating
the sub-layer as a single filament and that the variation of this layer is the
same order of its initial value (3.50)
∆τ ∼M∞Re−1/2√
∆p. (3.51)
Compared to (3.47) we see that for ∆p 1 the sublayer displacement
effect of the boundary layer is larger than that of the main boundary layer.
Hence, the slope angle, θ, of the streamlines at the outer edge of the bound-
ary layer should be based of the displacement of the sub-layer. We can
write
θ ∼ ∆τ
∆x∼ M∞Re
−1/2√∆p
∆x, (3.52)
Figure 3.7: Progression of two streamlines within the boundary layer.
41
which combined with the Ackeret formula
p =M2∞θ√
M2∞ − 1
, (3.53)
we deduce the pressure perturbations in the interaction region are give by
∆p ∼M∞θ ⇒ θ ∼ Re−1/2√
∆p
∆x. (3.54)
From our previous analysis of the boundary layer on the surface of the
needle, we know that the sub-layer should be viscous i.e.
ρu∂u
∂x∼ M2
∞Re
1
r
∂
∂r
(µr∂u
∂r
). (3.55)
Since ρ and µ are order one quanities within the boundary layer, writing
the above equation in finite differences, we have
U00∆u
∆x∼ M2
∞Re
∆u
(∆r)2. (3.56)
Since in the sub-layer ∆r ∼ r, this equation reduces to
U00
∆x∼ M2
∞Re
1
r2. (3.57)
Summarising our analysis, we have a system of simultaneous equations.
The velocity u at the bottom of the boundary layer approaching the inter-
action region
u ∼√
∆p. (3.58)
The thickness of the viscous sublayer
r ∼M∞Re−1/2√
∆p. (3.59)
The pressure influence within the interaction region
√∆p ∼ M∞Re
−1/2
∆x, (3.60)
42
Figure 3.8: Triple deck structure.
and in order to close the set of equations we use (3.57) and write it as
u
∆x∼ M2
∞Re
1
r2. (3.61)
Solving the sytem of algebraic equations (3.58) - (3.61) we find the following
which describe the characteristics of the flow within the interaction region
u ∼M1/4∞ Re−1/8 , r ∼M5/4
∞ Re−5/8,
∆p ∼M1/2∞ Re−1/4 , ∆x ∼M3/4
∞ Re−3/8.(3.62)
It is well documented that the interaction process in the vicinity of a
separation point has a three tier structure, or a triple deck (see Figure 3.8).
The interaction region is of O(M3/4∞ Re−3/8) and composed of three distinct
layers
1. Lower-deck: A viscous sub-layer close to the surface of the body of
O(M5/4∞ Re−5/8). In this region, flow is very sensitive to pressure varia-
tions which can significantly effect fluid particles, which in turn causes
deformation of streamlines - known as the displacement effect of the
boundary layer.
2. Middle-deck: This region is of O(M∞Re−1/2) and represents a con-
tinuation of the upstream boundary layer. Here, flow is less sensitive
to pressure variations as a result the displacement effect is minimal.
3. Upper-deck: A region of O(M3/4∞ Re−3/8) which occupies the invis-
43
cid flow region. It serves to covert deformations in streamlines into
perturbations of pressure, which are then transmitted through to the
lower-deck. The process amplifies fluid deceleration, which in turn
enhances boundary layer separation.
3.3 Compression Ramp
Using our analysis of the interactive process, we can now proceed to anal-
yse the compression ramp as seen in Figure 3.3 in order to establish the
conditions for separation.
We have defined Θ as angle the cone makes with the x-axis. Since the
pressure in region 1 is the same as region 2, using (3.31) and (3.62), we
choose the compression ramp angle such that
Θ = M−1/2∞ Re−1/4Θ0 where Θ0 = O(1). (3.63)
Now defining the axisymmetric body as
f(x) =
√ε(x+ 1)1/4 , x < 0√ε+ Θx , x ≥ 0
(3.64)
Assuming flow is over a thin parabolic needle with its leading edge located
at x = −1. Studying the problem for a thin needle i.e. ε 1, and since the
interaction region is very small, we can approximate f(x) by
f(x) ∼
0 , x < 0
Θx , x ≥ 0(3.65)
We shall now proceed to analyse the flow in the lower deck based on
research by Ruban and Gajjar [15], Korolev et al. [8] and Yapalparvi [19].
The solution in the inviscid region 3 is simply given by the interaction
law
P ∗ = − M2∞
λ√M2∞ − 1
dA
dx∗(3.66)
where A(x∗) is known as the displacement function and has its origins from
the asymptotic expansion of the stream function in the viscous sublayer as
we tend towards the outer edge of the region.
For the viscous sublayer, region 1, (3.62) suggests that asymptotic analysis
44
of the Navier-Stokes equations should be based on the limit procedure
x∗ = M−3/4∞ Re3/8x = O(1)
R∗ = M−5/4∞ Re5/8r = O(1)
as Re→∞. (3.67)
The solution to the Navier-Stokes equations in this region will be sought in
the form of the asymptotic expansions
u(x, r;Re) = M1/4∞ Re−1/8U∗(x∗, R∗) + . . . ,
v(x, r;Re) = M3/4∞ Re−3/8V ∗(x∗, R∗) + . . . ,
p(x, r;Re) = M1/2∞ Re−1/4P ∗(x∗, R∗) + . . . ,
ρ(x, r;Re) = ρ∗(x∗, R∗) + . . . ,
µ(x, r;Re) = µ∗(x∗, R∗) + . . . .
(3.68)
The estimations for u and p come from direct substitution of (3.62) whereas
v is found by using the continuity equation. Plus we know that ρ, µ and
h remain O(1) quanitites within the boundary layer. In fact Ruban and
Gajjar [15] found that ρ, µ and h are all constants within region 1. We
therefore choose
ρ∗ = ρw, µ∗ = µw, h∗ = hw (3.69)
We now proceed to substitute (3.68) and (3.69) into the Navier-Stokes
equations, (2.3). We find
∂
∂x∗(R∗U∗) +
∂
∂R∗(R∗V ∗) = 0, (3.70a)
ρw
(U∗
∂U∗
∂x∗+ V ∗
∂U∗
∂R∗
)=− dP ∗
dx∗+
1
R∗∂
∂R∗
(µwR
∗∂U∗
∂R∗
), (3.70b)
where we have used (3.39) to simplify.
Equations (3.70) are subject to the following boundary conditions. The
no-slip condition
U∗ = V ∗ = 0 at R∗ = f(x∗) (3.71)
We also have the matching condition with the main portion of the boundary
layer
U∗ → λR∗ as x∗ → −∞. (3.72)
In Prandtl’s formulation of the boundary layer, the pressure gradient is
45
known. In this case, we need to use the interaction law
P ∗ = − M2∞
λ√M2∞ − 1
dA
dx∗, (3.73)
where at the outer edge of the viscous sub-layer
U∗(x∗, R∗) = λR∗ +A(x∗) + . . . as R∗ →∞. (3.74)
Based on work done by Korolev et al. [8] and Yapalparvi [19], we can now
seek an affine transformation of the form
x∗ =µ−1/4w ρ
−1/2w
λ5/4β3/4X, R∗ =
µ1/4w ρ
−1/2w
λ3/4β1/4(R+ f(X)
),
U∗ =µ1/4w ρ
−1/2w
λ−1/4β1/4U , V ∗ =
µ3/4w ρ
−1/2w
λ−3/4β−1/4
(V + U
df
dX
),
A =µ1/4w ρ
−1/2w
λ−1/4β1/4A, P ∗ =
µ1/2w ρ
−1/2w
λ−1/2β1/2X
(3.75)
where β =√M2∞ − 1/M2
∞. Allowing us to represent the interaction prob-
lem, (3.70) - (3.74), in the following form
∂
∂X(RU) +
∂
∂R(RV ) = 0,
U∂U
∂X+ V
∂U
∂R=− dP
dX+
1
R
∂
∂R
(R∂U
∂R
),
P =− dA
dX,
U = V = 0 at R = 0,
U = R+ . . . as X → −∞,
U = R+ A(X) + . . . as R→∞.
(3.76)
Note the affine transformation (3.75) allows us to express the interaction
problem in terms of one controlling variable
Ω =θ0
µ1/2w λ1/2β1/2
. (3.77)
Also, (3.75) includes Prandtl’s transposition, which introduces a curvilinear
46
coordinate system with X measured along the body contour and R in the
normal direction. This leads to a simplification of the no-slip condition,
(3.71).
Subject to the affine transformation, the body contour, (3.65) is defined
as
f(X) = ΩH(X) where H(X) =
0 , x < 0
x , x ≥ 0(3.78)
The solution to the interaction problem (3.76), cannot be solved analyt-
ically and needs special numerical techniques. Using these techniques and
together with the definition for skin friction, (3.1), we can find the condition
for which separation takes place. Note, the numerics are beyond the scope
of this project.
47
4 Conclusion
This paper essentially has two parts. The first presented an in depth anal-
ysis of the boundary layer on a slender body of revolution created by a
hypersonic unperturbed free stream with zero angle of attack. The second
section discussed the addition of a slender cone attached to the trailing edge
of the needle. The key parameters in the first half of the paper were the
Reynolds number, the Mach number, the needle radius and the boundary
layer thickness - using order of magnitude analysis and under the assump-
tion that the radius of the needle is comparable with the thickness of the
boundary layer, a relationship between the four was found. On the other
hand, in the second half of the paper, the key parameter was the angle of
the cone and finding the change in pressure due to the shock in terms of
this angle.
Using asymptotic analysis, the Navier-stokes equations were reduced to a
set of equations governing the boundary layer on the surface of the needle.
Applying the Mangler transformation, the axisymmetric system was reduced
to a flat plate problem. Manipulating the resulting equations, we gain a
coupled set of third and second order ordinary differential equations. The
relationship between the boundary layer thickness and the needle radius was
then examined using numerical methods. The resulting equation was solved
and we were able to draw an analogy between a large needle radius, with
respect to the boundary layer width, and the classical Blasius boundary
layer equation. In comparison, with the case of a small needle radius, a
significantly lower vertical velocity and a sharper increase in the horizontal
velocity was found across the boundary layer.
The first part of the paper was concluded by seeking the asymptotic
behaviour of the small radius solution using the method of matched asymp-
totic expansions. In the asymptotic limit, the inner and outer expansions
matched up well with the numerical solution to the problem. However, as
we extended out through the boundary layer, the solution did degrade in
48
the overlapping region of the two expansions. A next step could be to find
higher order asymptotic approximations.
The second part of the paper studied the interaction of the boundary
layer with the shock wave. The Euler equations, are used to describe flow
in the inviscid flow region downstream of the shock. Under the assumption
that the shock has a conical form, the governing equations are reduced to
an ordinary differential equation. A self-similar equation was sought and
a single equation dependent on the normalised velocity components was
found. The resulting equation provides a further opportunity to study the
inviscid region with respect to small cone angles in the hodograph plane.
The jump conditions across a shock wave are known. Using order of mag-
nitude analysis, these conditions were manipulated in order to find the order
of the change in pressure with respect to the Mach number and the cone
angle. With these facts, the same was done for horizontal velocity varia-
tions with the outcome of finding the asymptotic expansions for studying
the main part of the boundary layer on the cone in the weak interaction
region.
The final part of the paper sought to study separation close to the needle/-
cone junction. Taking advantage of the symmetries of the problem, the in-
teraction region was modelled as a compression ramp problem and a triple
deck structure of this region was found. Focusing on the viscous sublayer,
a self-similar boundary value problem was derived. This boundary value
problem provides a future goal to be solve numerically, in order to find the
Figure 4.1: Schematic of a hypersonic shock-wave boundary-layer interac-tion on a compression ramp.
49
conditions in which separation of the boundary layer takes place.
Further to studying the separation problem numerically, the reattach-
ment shock created by the separation region reattaching to the cone surface
can be studied (see Figure 4.1). As a result, the separation and reattach-
ment shocks interact. Combined with the shock produced by the leading
edge of the needle, further study is needed in order to fully understand the
interaction problem.
50
Appendix
The numerics for solving (2.55) where calculated using Matlab. The coding
by Geubelle [5] was reproduced and altered based on our problem.
Coding:
clc
clear
% file: shooting.m
% This Matlab code combines the RK4 scheme and the bisection scheme
% to solve the self−similar boundary layer equation for steady, laminar,
% hypersonic, compressible flow. Defined by
%
% −0.5epsilon*ff''=[(epsilon+2eta)f'']' on 0<=eta<=10 and where epsilon<<1
%
% with the boundary conditions
% f(0)=0
% f'(0)=0
% f'(10)=1
% using the shooting method.
%
N=1000; % Number of intervals for RK4
tolerance=0.00001; % tolerance for bisection scheme
%
% First enter first guess a for f''(0) and compute associated error on
% f'(10)
a=input('Enter a lower bound of bisection interval on f"(0):');
errora=compute errorRK4(a,N);
% Next enter another guess b for f''(0) and compute associated error on
% f'(10)
b=input('Enter a upper bound of bisection interval on f"(0):');
errorb=compute errorRK4(b,N);
if errora*errorb >=0
sprintf( 'The interval for the guess for f"(0) does not contain a root')
else
51
sprintf('a=%17.9f error(a)=%17.9f b=%17.9f error(b)=%17.9f\n',a,errora,b,errorb)while b−a > tolerance % bisection loop
c=(a+b)/2; % get middle value c
errorc=compute errorRK4(c,N); % compute error on f'(10) corresponding to c
if errora*errorc > 0
a=c; % root is between c and b
errora=errorc;
else
b=c; % root is between a and c
errorb=errorc;
end
sprintf('a=%17.9f error(a)=%17.9f b=%17.9f error(b)=%17.9f\n',a,errora,b,errorb)end
sprintf(' the correct guess for f"(0) is %17.9f\n',(a+b)/2)end
%
% now that we have found the correct guess for f''(0), recompute the final
% solution in the boundary layer and plot the solution
%
% Defining the vector for eta
etavector=linspace(0,10,N+1); % eta coordinate between 0 and 10
% Defining the velocity vectors
U 0=zeros(N+1,1); % U 0 = f'
V 0=zeros(N+1,1);% V 0 ˜ eta*f'−fphi=zeros(N+1,1);
% Defining the solution vector
f=zeros(3,1);
f=[ 0; 0; (a+b)/2]; % initial values f(0) f'(0) and f''(0)
h=10/N;
% RK4 scheme loop
for i=1:N
eta=etavector(i+1);
k1=fprime(eta,f);
k2=fprime(eta+h/2,f+h/2*k1);
k3=fprime(eta+h/2,f+h/2*k2);
k4=fprime(eta+h,f+h*k3);
f=f+h/6*(k1+2*k2+2*k3+k4);
U 0(i+1)=f(2);
V 0(i+1)=eta*f(2)−f(1);phi(i+1)=f(1);
end
% Plotting the U 0 and V 0 velocity components against eta
52
figure(1)
plot(etavector,U 0,etavector,V 0,'linewidth',1.5)
xlabel '\phi'ylabel '\eta'title 'Blasius boundary layer solution'
legend('u velocity','v velocity')
grid on
figure(3)
plot(phi,etavector,'linewidth',1,'color','black')
ylabel '\eta'grid on
axis ([0 10 0 10])
figure(1)
plot(U 0,etavector,V 0,etavector,'−−','linewidth',1,'color','black')ylabel '\eta'grid on
axis ([0 1.5 0 10])
epsilon=0.001;
outer=1+1/(log(4/(epsilonˆ2))).*real(−expint(0.25.*etavector.ˆ2));inner=2/(log(4/(epsilonˆ2))).*log(1+etavector./epsilon);
figure(2)
plot(U 0,etavector,'linewidth',1,'color','black')
ylabel '\eta'grid on
axis ([0 1.2 0 3])
hold on
plot(inner,etavector,'−−','linewidth',1,'color','black')hold on
plot(outer,etavector,':','linewidth',1,'color','black')
matlab2tikz('solution.tikz','height','\figureheight','width','\figurewidth');
% file: fprime.m
% Computes the derivatives for the system of ODEs governing the boundary
% layer equation
% f1'=f2
% f2'=f3
53
% f(3)'=−((epsilon+eta)*f(1)*f(3)+f(3))/(epsilon+eta)
function df=fprime(eta, f)
epsilon=0.001;
df=zeros(3,1);
df(1)=f(2);
df(2)=f(3);
df(3)=−((epsilon+eta)*f(1)*f(3)+f(3))/(epsilon+eta);
% file: computer errorRK4.m
function erroryprime10=compute errorRK4(a,N)
% Computes the derivatives for the system of ODEs governing the boundary
% layer equation
% f1'=f2
% f2'=f3
% f3'=−f(3)*(2+0.5*epsilon*f(1))/(epsilon+2*eta)% over 0<=eta<=10 and epsilon<<1
% with boundary conditions,
% f1(0)=0
% f2(0)=0
% f3(0)=a
% using the 4th−order Runge Kutta scheme (with N steps).
% Then returns the "error" at eta=10 defined as
% erroryprime10=f2(eta=10)−1%
% set initial conditions and iteration interval
f=zeros(3,1); %solution vector with f, f' and f''
f=[0; 0; a]; % initial values of f, f' and f'' at eta=0
h=10/N;
eta=0;
% RK4 Scheme
for i=1:N
eta=eta+h;
k1=fprime(eta,f);
k2=fprime(eta+h/2,f+h/2*k1);
k3=fprime(eta+h/2,f+h/2*k2);
k4=fprime(eta+h,f+h*k3);
f=f+h/6*(k1+2*k2+2*k3+k4);
end
erroryprime10=f(2)−1; % error on f'(10) (should be = 1)
54
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