Compressible flow basics
-
Upload
george-mathew-thekkekara -
Category
Engineering
-
view
135 -
download
0
Transcript of Compressible flow basics
© Z
eus N
um
erix
Defense | Nuclear Power | Aerospace | Infrastructure | Industry
Treatment of compressible flow in CFD
Abhishek [email protected]
Compressible Flow: Basics
© Z
eus N
um
erix
2
Overview
Conservation Laws
Conservation form of equations
Governing equations: Hyperbolic
The Wave theory: CFL Condition
Schemes and their types
Eigen values
Boundary conditions
© Z
eus N
um
erix
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
© Z
eus N
um
erix
The Equations
Equations in Conservation form (Differential)
4
Similarly for y and z direction
© Z
eus N
um
erix
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
5
© Z
eus N
um
erix
Integral Form of Equations
Conservation form
F, G, H are similar
S depends on the type of flow solved
6
U =
ρρuρvρwρE
F =
ρuρ u2 + pρuv + pρuw + p(ρE + p)u
© Z
eus N
um
erix
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
7
© Z
eus N
um
erix
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
8
© Z
eus N
um
erix
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
9
© Z
eus N
um
erix
Equations
Hyperbolic
Disturbance propagates from domain of dependence (Brown) to range of influence (Gray)
Characteristics APC and BPD
10
P
C
D
B
A
© Z
eus N
um
erix
Mach Cone
Supersonic flow means signal does not travel in all directions
Signal does not reach upstream
11
Zone of Silence
Mach Cone
Motion
Zone of Silence
© Z
eus N
um
erix
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
12
© Z
eus N
um
erix
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)
13
© Z
eus N
um
erix
The Wave Theory
Flow is traveling of waves
Slope of the wave matters
14
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
© Z
eus N
um
erix
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)
15
Numerical domain of dependence
True domain of dependence
True wave direction
i i+1
© Z
eus N
um
erix
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
16
© Z
eus N
um
erix
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
17
© Z
eus N
um
erix
Thank You!
3 November 2014 18