Naiver strokes equations :- These balance equations arise from applying Issac Newton’s second law to fluid motion , together with assumption that the stress in the fluid is the sum of a diffusing viscous and a pressure term- hence describing viscous flow.

1. Veltech Rangarajan Dr.Sagunthala R&D Institute of
Science and Technology
School of Mechanical and Construction
Department of Aeronautical Engineering
1151AE125 – Hypersonic Aerodynamics
(PRESENTATION)
TOPIC: VISCOUS HYPERSONIC FLOW THEORY
By
Dipranjan Gupta, 3rd year.
Dept. Aeronautical Engg.
1

2. 5/20/2020 2Online Class – Unmanned Systems
Naiver-strokes equation
Governing equations for viscous flow
Naiver strokes equations :- These balance equations arise from applying Issac
Newton’s second law to fluid motion , together with assumption that the stress in
the fluid is the sum of a diffusing viscous and a pressure term- hence describing
viscous flow.
These equations are given below
……………equation 1

3. 3
………………….equation 2
……………… equation 3
……………….equation 4
…………………equation 6

4. 4
……………….equation.7a
……………….equation.7b
.........................equation.7c
……………… equation.7d
……………… equation.7e
………………..equation.7f
• The preceding equations are written for an unsteady compressible, viscous, three-
dimensional flow in Cartesian coordinates in form of Naiver-stroke equation

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Boundary layers equations for hypersonic flow
• The energy equation under the assumption of very thin boundary layer for very high
Reynolds hypersonic flows.
• This energy equation is for total energy which means summation of kinetic energy and
internal energy .But it’s have neglected potential energy of the fluid particle.
• Derive the energy equation for kinetic energy alone.
The x-momentum – i and y- momentum- ii equations is given
………. Equation i
……….Equation ii
…………Equation iii

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………….equation iv

7. 6

8. 8

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Hypersonic Aerodynamics heating:-
• Aerodynamic heating is the heating of a solid body produced by its high-speed passage
through air (or by the passage of air past a static body), whereby its kinetic energy is
converted to heat by adiabatic heating, and (less significantly) by skin friction on the
surface of the object at a rate that depends on the viscosity and speed of the air.
• The main purposes for studying hypersonic viscous flow is for design of hypersonic
vehicles .For Hypersonic Vehicles the aspect of design is surfaces Heat transfer and skin
friction.Surfaces Heat transfer play role for design the conventional hypersonic Vehicles
• Skin friction is important for aerodynamics efficiency of slender vehicles. Its because skin
friction is important of aerodynamics heating at hypersonic speed.
• For demonstrations of aerodynamics heating at hypersonic heating is descried by Stanton
number in the case of flat plate in parallel flows flow . The Stanton number can be
defineds
qw = ẟ∞V∞(haw- hw)CH …….. eqn 1
• Assuming an approximate recovery factor of unity, haw =h0, where h0 is the total
enthalpy, defined as
h0 =h∞+(V^2∞)/ 2 ......... eqn 2

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• At hypersonic speeds,𝑣∞2/2 is much larger than h∞, and from Eq. (6.164) h0 is
essentially given by…..eqn 2
h0 ≈𝑣2/2……eqn 3
• Moreover, the surface temperature, although hot by normal standards, still must
remain less than the melting or decomposition temperature of the surface material.
Hence, the surface enthalpy hw is usually much less than h0 at hypersonic speeds.
h0 >>hw…………………eqn 4
• the approximate relation is
qw ≈ 1 /2ẟ∞𝑣3
∞𝐶H…..eqn 5
• The main purpose of eqn 5 is to demonstrate that aerodynamic heating increases with
the cube of the velocity and hence increases very rapidly in the hypersonic flight
regime

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• By comparison, aerodynamic drag is given by
D= 1 /2ẟ∞V∞SCD……….eqn 6
which increases as the square of the velocity. Hence, at hypersonic speeds,
aerodynamic heating increases much more rapidly with velocity than drag, and this is the
primary reason why aerodynamic heating is a dominant aspect of hypersonic vehicle design.

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Entropy layers effect on Aerodynamics Heating
Consider the the inviscid hypersonic flow over a blunt-nosed body. The surface streamline,
which has passed through the normal portion of the bow shock wave. Because the flow is
inviscid and adiabatic, the entropy is constant along this streamline, and equal to the entropy
behind a normal shock wave.
According to the usual boundary-layer method, this streamline with its normal shock entropy
would constitute the boundary condition at the outer edge of the boundary layer and the blunt
nose the thin boundary layer will be growing inside the entropy layer, and then the boundary
layer will eventually “swallow” the entropy layer far enough downstream. The conventional
boundary-layer assumption that the outer-edge boundary condition is given by the inviscid
surface streamline as shown in figure when dealing with blunt-nosed hypersonic bodies is not
appropriate.

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Figure :-Illustration of the surface streamline containing the normal shock entropy
• Within the framework of boundary-layer analysis, current practice is to estimate the
boundary-layer thickness d and then utilize the inviscid-flow properties located a
distance d from the wall as outer boundary conditions for the boundary layer. the
aerodynamic heat-transfer distribution along the space shuttle windward ray is shown at
the velocity and altitude corresponding to maximum heating along the entry trajectory.
The open circles are experimental data extrapolated from wind-tunnel data, the presence
of the entropy layer on a blunt-nosed hypersonic body has an important effect on
aerodynamic heating predictions using boundary-layer techniques. However, the simple
method just stated appears to be a reasonable approach to including the effect of the
entropy layer.

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Figure:-2 Comparison of predicted shuttle windward-ray heat-transfer
distributions; illustration of the entropy-layer effects
• The solid circles are from the calculations of which also account for the entropy layer.
Note two important aspects from Fig.2 the presence of the entropy layer increases the
predicted values of qw by at least 50%—a nontrivial amount .
• The taking into account of the entropy layer by using boundary-layer outer edge
properties associated with the inviscid flow a distance d from the wall gives good
agreement with the experimental data.

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Illustrate strong and weak viscous interactions

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For Knowledge only Not for exam:-
The strong interactions region is one where the following physicals effects occur.

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For the weak interactions is the one which following physical effect occur :-

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Explain with a schematic the shock wave boundary layer interactions occur:-
Figure of the schematic shock wave boundary layer interactions
Shock Wave boundary layer interactions categories with laminar boundary layer and
Turbulent boundary layer. Shock wave–boundary-layer interactions occur when a
shock wave and a boundary layer converge and, since both can be found in almost
every supersonic flow, these interactions are common place.

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BOUNDARY CONDITIONS
An important difference between inviscid and viscous flows can be seen explicitly
in the boundary conditions at the wall. The usual boundary condition for an
inviscid flow is no mass transfer through the wall which mathematically gets
expressed as the normal components of velocity to be zero at the wall.
This boundary condition is termed as “ free slip along the wall". Therefore both
components of velocity becomes zero for viscous wall boundary condition

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Hypersonic Boundary layer theory
Self-similar solutions
Although the title section involves the word “hypersonic” in reality it will be dealing with
compressible boundary layer theory , and the results will apply to both subsonic and
supersonic , as swell as hypersonic conditions. The concepts of self-similar boundary layers
is illustrated in figure,

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The transformed plane of (x , y) to ( ξ , ƞ ) in the transformation form where in the
velocity profile is independent of the transformed surface distance ξ , say , ξ1 and ξ2 .
Thus , in the transformed plane the velocity profile is given by u=u(ƞ), independent of ξ.
Boundary layers that exhibit this property are called self-similar boundary layers , and
solutions for these boundary layers are called self-similar solutions-the subject of this
section.
NON-SIMILAR HYPERSONIC BOUNDARY LARYERS
In general, even though these profiles are calculated in the transformed ξ-ƞ space, they
will be different profiles at each different value of ξ. Boundary layers that exhibits this
behaviour, which is the case in general, are called non-similar boundary layer

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Figure of Qualitative sketches of non-similar boundary-layer profiles
There are three methods for solving general non similar boundary layers
1) Local similarity
The method of local similarity is not a precisely exact solution for general non
similar boundary layers ,but it is an important bridges between the exact self-similar technique
discussed and the exact non similar solutions in the present Section.

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The concept of local similarity is outlined next

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2) Difference-Differential method
The difference differential method is inherently in
exact solution of the general boundary –layer equations. The general idea was originated
in1937 by Hartree and Womersley. Smith utilized the difference-differential method
extensively and with success ; a typical example of his work is represented.
Figure of schematic for finite difference solution of the boundary layer

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Heat transfer distribution over a flate –faced cylinder

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3) Finite difference method.
The ξ derivatives are replaced by finite differences .The next logical step is to replace both
ξ and ƞ derivatives by finite differences. Such finite difference solution are discussed here
, they represent the current state of the art in hypersonic boundary – laryer solutions.
In summery , a finite difference solution of a general non similar boundary laryer proceeds
as follows.
1) The solution must be started from a given solution at the leading edge , or at a
stagnation point. It stated earlier ,this can be obtained from appropriate self-similar
solutions.
2) The next downstream station, the finite difference produces reflected yields a solution
of the flow field variables across the boundary layer .

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3) Once the boundary-layer profiles of u and T are obtained the skin friction and heat
transfer at the wall

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Hypersonic Aerodynamics heating
The thermal design of hypersonic vehicles involves accurately and reliably predicting the
convective heating over the surface of the vehicle. Such results may be obtained by
numerically solving the Navier strokes equations or one of their subsets such as the
parabolized Navier-strokes (PNS) and viscous shock layer (VSL) equations for the flow
field surrounding the vehicle.
However , due to the excessive computer storage requirements and run times of there
detailed approaches , they are impractical for the preliminary design environment where a
range of geometries and flow parameters are to be studied.
On the other hand , engineering inviscid- viscous method have been demonstrated to
adequately predict the heating range over a wide range of geometries and aero thermal
environments.