Compressible flow basics

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Treatment of compressible flow in CFD
Abhishek Jain
abhishek@zeusnumerix.com
Compressible Flow: Basics

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2
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
3

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The Equations
 Equations in Conservation form (Differential)
4
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
6
U =
ρ
ρu
ρv
ρw
ρE
F =
ρu
ρ u2 + p
ρuv + p
ρuw + 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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Compressible flow basics

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