Continuum Mechanics Analysis of Supersonic Flow

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Continuum Mechanics Analysis of Supersonic Flow

• Submitted by
• Rajendra B Dubagunta
• Siva Prasad Rao Batchu
• Visweswara Mudiam
• Sandeep Kancharla Narayana
• Kalpana Thota

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OVERVIEW

• Introduction
• Supersonic Flow
• Continuum Mechanics Applied to Supersonic Flow
• Vorticity Surfaces and Bernoulli Function
• P,? relationship for inviscid flow
• Conclusions
• Example

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INTRODUCTION

• What is a wave?
• a wave can be described as a disturbance that
  travels through a medium, transporting energy
  from one location to another location.
• Types of wave propagation based on speed
• Subsonic
• Transonic
• Supersonic

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INTRODUCTION contd.

• Subsonic
• Mach number less than unity.
• Generally ranges form 0ltMlt1
• Transonic
• For this type of flow the Mach number M is
  approximately unity
• Transonic flow.
• Compressibility effects like the flow choking
  becomes very important.

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INTRODUCTION contd.

• Supersonic
• Mach number for this kind of flow is greater than
  unity.
• Mach number for this flow ranges from 1ltMlt3.

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SUPERSONIC FLOW

• Mach number greater than unity
• Reynolds number is greater than unity
• For supersonic flow we can regard the fluid as
  ideal.
• This behavior is exemplified by the supersonic
  flow associated with shock waves.
• A disturbance starting for any point in a
  supersonic flow is propagated only downstream
  within a cone whose opening angles decreases.
  Where as in subsonic flow it propagates in
  upstream.

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CONTINUUM MECHANICS APPLIED TO SUPERSONIC FLOW

• Reynolds number in supersonic flows is much
  larger than unity.
• When the Reynolds number is large viscous forces
  are small compared to inertial forces, and can be
  ignored, except in highly localized regions of
  the flow. For supersonic flow we can regard the
  fluid as ideal. This behavior is exemplified by
  the supersonic flow associated with shock waves.
  The thickness of the shock front is , and
  elsewhere the fluid can be treated as inviscid.

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VORTICITY STREAM SURFACES FOR INVISCID FLOW

• VORTICITY STREAM SURFACES FOR INVISCID FLOW

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P,? RELATIONSHIP FOR INVISCID FLOW

• P,? RELATIONSHIP FOR INVISCID FLOW

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Example

• A column of fluid is contained with in a
  vertical circular cylinder as shown below. The
  cylinder and the fluid rotate as a rigid body
  about the axis of the cylinder with a constant
  angular velocity. What are the vorticity-stream
  surfaces? Show that the Bernoulli function varies
  across the vorticity-stream surfaces. Also prove
  that the isobaric stream surfaces are paraboloids
  of revolution. Assume the fluid has a constant
  density.

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SOLUTION

• SOLUTION

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CONCLUSIONS

• In this project we studied different kinds of
  flows like the Sub-Sonic, Sonic, Supersonic flows
  and the Supersonic flow is discussed with
  Continuum Mechanics behavior of the flow. In this
  project we discussed the Vorticity-Stream
  functions and the P, relationship of the flow
  using Continuum Mechanics.

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HOME WORK

• Find the speed of propagation of a plane of small
  disturbances in the direction normal to the plane
  in an infinite, compressible, in viscid fluid
  initially at rest with a unique p, -relationship
  in the absence of body forces.

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• THANK YOU

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• QUESTIONS