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Defense | Nuclear Power | Aerospace | Infrastructure | Industry

Treatment of compressible flow in CFD

Abhishek Jainabhishek@zeusnumerix.com

Compressible Flow: Basics

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Overview

Conservation Laws

Conservation form of equations

Governing equations: Hyperbolic

The Wave theory: CFL Condition

Schemes and their types

Eigen values

Boundary conditions

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Conservation Laws

Mass is Conserved

Net mass flowing out of the system = Net mass decreased in the system

Momentum is Conserved

Rate of change of momentum = Momentum transfer through the surfaces – Forces (surface and body)

Surface forces – Shear stress, pressure, surface tension

Body forces – Gravity, centrifugal or electromagnetic etc

Energy is Conserved

Rate of change of energy = Neat heat flux + work done by body and surface forces

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The Equations

Equations in Conservation form (Differential)

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Similarly for y and z direction

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Conservation and Non-Conservation Forms

Conservation Form

Easy to code as all equations look similar

More physical as “can be simply stated in English”

Primary variable are calculated from the flux variables

Captures the shock; (shock produced by solution)

Non-Conservation Form

Equation given above are expanded

Has a shock fitting approach; solution is a forethought i.e. shock location must be approximately known

Captures the shock better

There have been instances where shock fitting and capturing methods have been used with either forms

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Integral Form of Equations

Conservation form

F, G, H are similar

S depends on the type of flow solved

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U =

ρρuρvρwρE

F =

ρuρ u2 + pρuv + pρuw + p(ρE + p)u

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Difference

Integral form means that if we were to add up the properties in the whole domain there will be an equilibrium

Does not assume if quantities are a part of continuous function or discontinuous

Differential means the function is continuous

Not able to capture physical discontinuities like shock wave

Integral form there is called more fundamental

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Completing the Loop

The number of equations is FIVE

Assuming calorically perfect gas E=CvT

Number of unknowns

ρ, u, v, w, p, T

Hence the thermal equation of state is used for closure

p= ρRT

R is the Gas Constant

Please remember that the above equation was not used in incompressible flow

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Equation Types

Eigen values are calculated using the coefficients of the equations

In case the Eigen values of the equations are

Real and distinct – equation is Hyperbolic

Real and Zero – equation is Parabolic

Imaginary – equation is elliptic

Characteristic lines are curves where slope of dependent variable is indeterminant

Slope of the characteristic line can be real, zero or imaginary making the equations Hyperbolic, parabolic or elliptic

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Equations

Hyperbolic

Disturbance propagates from domain of dependence (Brown) to range of influence (Gray)

Characteristics APC and BPD

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P

C

D

B

A

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Mach Cone

Supersonic flow means signal does not travel in all directions

Signal does not reach upstream

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Zone of Silence

Mach Cone

Motion

Zone of Silence

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Equations

Parabolic – Effect travels through one direction only

Elliptic – Effect travels in all directions

For complex equations like Navier Stokes the behavior may be mixed

Examples

Supersonic inviscid flow – hyperbolic

Subsonic inviscid flow – elliptic

Boundary layer flow – parabolic

Scheme that works for one set fails for another

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The Time Marching

Supersonic blunt body problem

Flow inside blue circle is subsonic

At other places supersonic

Problem is elliptic in circle & hyperbolic outside

First technique to make problem hyperbolic

Introduce time derivative in steady problem

March in time to ‘reach’ at steady state

Most widely used method (Finite Volume)

Marching explicit or implicit (next lecture)

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The Wave Theory

Flow is traveling of waves

Slope of the wave matters

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X

Time

i i+1 i+2 i+3 i+4 i+5 i+6

n+4

n+3

n+2

n+1

n

Area of physical domain covered

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CFL Condition

Stencil are the points used for simulation of flow

In case below it is i(n), i+1(n) and i(n+1)

For stable simulation Numerical domain must be greater than physical domain

Δt = CFL * Δx/Wave speed (CFL acts as a factor of safety)

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Numerical domain of dependence

True domain of dependence

True wave direction

i i+1

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Schemes

Problems in CFD can be finally simplified to

dU/dt +dF/dx = 0

Method of solving for [F] is called a scheme

Flux vector splitting schemes

Equations contains waves that travel in forward and backward direction

Flux vector split in such a fashion that waves are split in forward moving and backward moving

Solved independently to get solution

Van Leer scheme – M = M++M-

Steger Warming Method – Λ = Λ++Λ-

van Leer better at sonic points as M is second derivative

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Schemes

Flux vector splitting schemes are diffusive and do not capture boundary layer properly

Easy to code with faster turn around time

Roe Averaged – solves a local Reimann problem and give better result at boundary layer but produces expansion shock

Entropy Fix Roe – Forced condition put on such that Entropy never decreases

AUSM – Combines the goodness of flux vector splitting schemes and Roe type schemes

Has many modifications for time decrease or better performance

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Thank You!

3 November 2014 18