CFD Lecture (4/8): Compressible Flow- Basics

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Compressible Flow: Basics
Treatment of Compressible Flow in CFD

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©ZeusNumerixPvtLtd:ConfidentialDocument
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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26-Feb-2009 Training on Mesh Generation

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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
𝜕𝜌
𝜕𝑡
+ ∇ ⋅ 𝜌𝐕 =0

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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 =
𝜌
𝜌𝑢
𝜌𝑣
𝜌𝑤
𝜌𝐸
𝐹 =
𝜌𝑢
𝜌𝑢2 + 𝑝
𝜌𝑢𝑣 + 𝑝
𝜌𝑢𝑤 + 𝑝
𝜌𝐸 + 𝑝 𝑢

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Difference
 Differential means the function is continuous
 Not able to capture physical discontinuities like shock wave
 Integral form there is called more fundamental
 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
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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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www.zeusnumerix.com
+91 72760 31511
Abhishek Jain
abhishek@zeusnumerix.com
Thank You !

CFD Lecture (4/8): Compressible Flow- Basics

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CFD Lecture (4/8): Compressible Flow- Basics