TAU similarity rules –
correct usage of TAU in SI units
TAU Training
Bernhard Eisfeld
Folie 1 > TAU Training > Non-Dimensionalisation
Eisfeld> 25.-29.02.2008
Overview
•
Introduction
•
Theoretical background
•
Computation with TAU
•
Procedures for dimensional solutions
Folie 2 > TAU Training > Non-Dimensionalisation
Eisfeld> 25.-29.02.2008
Overview
•
Introduction
•
Theoretical background
•
Computation with TAU
•
Procedures for dimensional solutions
Folie 3 > TAU Training > Non-Dimensionalisation
Eisfeld> 25.-29.02.2008
Introduction
Difficulties in TAU application
TAU solves Navier-Stokes equations in non-dimensional form
• TAU allows for non-dimensional as well as dimensional output
Î Various parameters available for setting up a computation
•
Arising questions
•
•
•
What is the meaning of the parameters?
Which parameters must be set?
How to achieve consistent parameter settings?
Requirement
Understand the underlying principles of non-dimensionalisation
(similarity rules)
Folie 4 > TAU Training > Non-Dimensionalisation
Eisfeld> 25.-29.02.2008
Overview
•
Introduction
•
Theoretical Background
•
Computation with TAU
•
Procedures for dimensional solutions
Folie 5 > TAU Training > Non-Dimensionalisation
Eisfeld> 25.-29.02.2008
Theoretical background
Assumptions about the fluid (transonic flow)
• Newtonian fluid
• Viscosity follows Sutherland law
• Heat flux follows Fourier’s law
• Prandtl number is constant
• Perfect gas: ideal gas law + constant specific heats
Assumptions define
• Material laws
• Thermodynamic relations
TAU checks consistency of parameters with these assumptions!
Folie 6 > TAU Training > Non-Dimensionalisation
Eisfeld> 25.-29.02.2008
Theoretical background
Non-dimensionalisation of the governing equations
• Introduce (arbitrary) dimensional reference quantities φref
ˆ
• Define non-dimensional quantities φ =
φ
φref
→ φ = φˆ φref
• Introduce non-dimensional quantities into all equations
Î Non-dimensional equations with non-dim. reference quantity groups
Example: Continuity equation
⎛ Lref
⎜
⎜t U
⎝ ref ref
⎞ ∂ρˆ
∂
⎟
ˆUˆ k = 0
+
ρ
⎟ ∂tˆ ∂xˆ
k
⎠
( )
• Require formal identity of dimensional and non-dimensional equations
Î All non-dimensional groups are equal to 1
Example: Continuity equation
⎛ Lref
∂ρˆ
∂
ˆ
+
ρˆU k = 0 Î ⎜
⎜t U
∂tˆ ∂xˆk
⎝ ref ref
( )
⎞
⎟ =1
⎟
⎠
Î Not all reference quantities are independent (only 4)
Folie 7 > TAU Training > Non-Dimensionalisation
Eisfeld> 25.-29.02.2008
Theoretical background
Reference quantities
• (Arbitrary) choice in TAU:
- Reference pressure
pref
Note: Ideal gas law holds
- Reference temperature Tref
- Reference density
ρ ref
- Reference length
Lref
pref = ρ ref Rref Tref
(Rref is dependent variable)
Note: Values of reference quantities are arbitrary
• Sutherland law requires additional reference state (μ S , TS )
where
μS
T S+ S
= CS = const.
3/ 2
TS
Note: Sutherland reference state is independent (depends on fluid only)
Folie 8 > TAU Training > Non-Dimensionalisation
Eisfeld> 25.-29.02.2008
Theoretical background
Dimensional TAU reference state (1)
Assumptions:
• Reference quantities belong to the treated fluid
• Reference quantities belong to a true physical state
Î Reference gas constant = fluid gas constant
Rref = R
Î Reference state is defined by only two thermodynamic state variables
- Reference pressure
p * = pref
- Reference density
ρ * = ρ ref
Note: Reference state is still fictive !
Folie 9 > TAU Training > Non-Dimensionalisation
Eisfeld> 25.-29.02.2008
Theoretical background
Dimensional TAU reference state (2)
Dimensional reference state can be expressed in terms of
Fluid parameters
Specific gas constant
R
Isentropic exponent
γ
Prandtl number of the fluid
Pr = μ*Cp/λ*
Pressure of reference state
p*
Density of reference state
ρ*
Mach number of reference state
Ma* = U*/a*
Reynolds number of reference state
Re* = ρ*U*L*/μ*
Characteristic length
L*
Sutherland law parameters Sutherland constant
S
Thermodyn. state param‘s
Flow parameters
Geometric parameter
Constant
CS
Folie 10 > TAU Training > Non-Dimensionalisation
Eisfeld> 25.-29.02.2008
Theoretical background
Dimensional TAU reference state (3)
Sutherland law constraints choice of reference quantities
(Reference state fulfills Sutherland law)
μ*
T +S
*
(T )
* 3/ 2
ρ
=
*
p*
p*
γ * Ma L * + S
ρ
ρ R
= CS
3/ 2
*
Re*
⎛ p ⎞
⎜⎜ * ⎟⎟
⎝ρ R⎠
* *
In TAU in general Re* is considered a dependent quantity !
Folie 11 > TAU Training > Non-Dimensionalisation
Eisfeld> 25.-29.02.2008
Theoretical background
Non-dimensional TAU reference state
Apply non-dimensionalisation to the reference state
ρ
R
p
T
=1
=1
ρˆ * =
=1
=1
Rˆ =
pˆ * =
Tˆ * =
Rref
pref
ρ ref
Tref
Î Non-dimensional reference state can be expressed in terms of
Fluid parameters
Flow parameters
Geometric parameter
Isentropic exponent
γ
Prandtl number of the fluid
Pr = μ*Cp/λ*
Mach number of reference state
Ma* = U*/a*
Reynolds number of reference state
Re* = ρ*U*L*/μ*
Non-dim. char. length
(Reynolds length)
L*/Lref
Sutherland law parameters Non-dim. Sutherland constant
Ŝ = S/Tref
similarity parameters (equal values Î identical non-dim. solution)
Folie 12 > TAU Training > Non-Dimensionalisation
Eisfeld> 25.-29.02.2008
Theoretical background
Association of the reference state with the flow field
Reference state is still arbitrary!
Î Assumption (default): Reference state = far field state
• Pressure of reference state
p * = p∞
• Density of reference state
ρ * = ρ∞
• Mach number of reference state
Ma * = Ma∞ =
*
• Reynolds number of reference state Re = Re ∞ =
• Prandtl number of fluid
μ∞C p
Pr =
λ∞
U∞
a∞
ρ ∞U ∞ L*
μ∞
Caution: TAU references always refer to reference state.
TAU far field state may differ from reference state (user input) !
Folie 13 > TAU Training > Non-Dimensionalisation
Eisfeld> 25.-29.02.2008
Theoretical background
Non-dimensional coefficients
Definition of force coefficients
(with respect to far field state)
Definiton of pressure coefficient
(with respect to far field state)
C Force =
Cp =
Force
1
ρ ∞U ∞2 A
2
p − p∞
1
ρ ∞U ∞2
2
Deviating definitions in TAU
Definition of force coefficients
(with respect to reference state)
Definiton of pressure coefficient
(with respect to reference state)
TAU
C Force
=
C
TAU
p
=
Force
( )
1 * *2
ρ U A
2
p − p*
( )
1 * *
ρ U
2
2
Differences if
far field state
≠ reference state !
Folie 14 > TAU Training > Non-Dimensionalisation
Eisfeld> 25.-29.02.2008
Overview
•
Introduction
•
Theoretical background
•
Computation with TAU
•
Procedures for dimensional solutions
Folie 15 > TAU Training > Non-Dimensionalisation
Eisfeld> 25.-29.02.2008
Computation with TAU
TAU
parameters
Parameter
TAU key word string
Default value
γ
Gas constant gamma
1.4
Pr
Prandtl number
0.72
Ma*
Reference Mach number
--
U*
Reference velocity
--
Re*
Reynolds number
--
L*/Lref
Reynolds length
--
T*
Reference temperature
273.15 K
p*
Reference pressure
101325.0 Pa
ρ*
Reference density
1.29290880647192 kg/m2
R
Gas constant R
287 m2 / (s2 K)
S
Sutherland constant
110.4 K
TS
Sutherland reference temperature 273 K
μS
Sutherland reference viscosity
1.716e-5 Pa s
sgrid
Grid scale
-Folie 16 > TAU Training > Non-Dimensionalisation
Eisfeld> 25.-29.02.2008
Computation with TAU
Obtaining a non-dimensional solution
Parameter settings (1)
• Fluid parameters: TAU assumes air (default)
• Flow parameters: Specify
γ = 1.4, Pr =0.72
Ma* , Re*
• Parameters of the Sutherland law:
TAU assumes air (default)
(Values for other gases below)
S =110.4 K
TS = 273K
μ S = 1.716 ⋅10 −5 Pa s
⇒ CS = 1.458 ⋅10 −6 Pa s / K 1/ 2
Specify reference state temperature
T * =Tref
S
→ Sˆ =
Tref
Folie 17 > TAU Training > Non-Dimensionalisation
Eisfeld> 25.-29.02.2008
Computation with TAU
Obtaining a non-dimensional solution
Parameter settings (2)
• Geometric parameters:
Specify non-dimensional characteristic length (Reynolds length) L* / Lref
r
r
Note: Grid coordinates are non-dimensional, i.e. xˆ grid = x grid / Lref
*
Î Take L / Lref from grid
Folie 18 > TAU Training > Non-Dimensionalisation
Eisfeld> 25.-29.02.2008
Computation with TAU
Obtaining a non-dimensional solution
Solution output settings
• Information is not sufficient for dimensional output
Î Write output dimensionless (0/1) : 1
Note: Default is 0 (dimensional output) !
• Non-dimensional coefficients are valid
(if far field state = reference state!)
• Dimensional reference quantities in stdout are meaningless !
Folie 19 > TAU Training > Non-Dimensionalisation
Eisfeld> 25.-29.02.2008
Computation with TAU
Obtaining a dimensional solution
Parameter settings (1)
• Fluid parameters: TAU assumes air (default)
m2
γ = 1.4, Pr =0.72, R = 287 2
s K
• Thermodynamic parameters: In general specify only two of them
(p , ρ )
*
*
or
(p ,T )
*
*
or
(T
*
, ρ*
*
• Flow parameters: Specify Ma
)
or U *
*
Do not specify Re
(is defined by Sutherland law)
Folie 20 > TAU Training > Non-Dimensionalisation
Eisfeld> 25.-29.02.2008
Computation with TAU
Obtaining a dimensional solution
Parameter settings (2)
• Parameters of the Sutherland law
(according to White, “Viscous fluid flow”)
Gas
TS
in K
μs
in Pa s
S
CS
Temp.range
in K
in Pa s / K1/2
in K
for 2% error
Air (TAU)
273
1.716e-5
110.4
1.458e-6
170 – 1900
Air (White)
273
1.716e-5
111
1.460e-6
170 – 1900
CO2
273
1.370e-5
222
1.503e-6
190 – 1900
N2
273
1.663e-5
107
1.401e-6
100 – 1500
O2
273
1.919e-5
139
1.753e-6
190 – 2000
H2
273
8.411e-6
97
0.690e-6
220 - 1100
Default
Folie 21 > TAU Training > Non-Dimensionalisation
Eisfeld> 25.-29.02.2008
Computation with TAU
Obtaining a dimensional solution
Parameter settings (3)
• Geometric parameters:
Specify non-dimensional characteristic length (Reynolds length) L* / Lref
Specify the grid scale: TAU assumes Lref = 1m
L / 1m
s grid = 1m
Definition of grid scale
Lgrid / Lref
where
L1m / 1m
Lgrid / Lref
(“grid in meters”)
Some char. length in real world in m
Same char. length in grid units
Example: Grid for full scale aircraft, span:
Compute wind tunnel experiment, where
bmod el / 1m 2.50
=
= 0.05
Grid scale s grid =
bgrid / Lref
50
bgrid / Lref = 50
bmod el = 2.50m
Folie 22 > TAU Training > Non-Dimensionalisation
Eisfeld> 25.-29.02.2008
Computation with TAU
Obtaining a dimensional solution
Solution output settings
• Information is always sufficient for non-dimensional output
Î Write output dimensionless (0/1) : 1
• Correct dimensional output (forces in N) requires correct grid scale !
Î Write output dimensionless (0/1) : 0 (default setting)
• Non-dimensional coefficients are independent of the grid scale
Folie 23 > TAU Training > Non-Dimensionalisation
Eisfeld> 25.-29.02.2008
Overview
•
Introduction
•
Theoretical background
•
Computation with TAU
•
Procedures for dimensional solutions
Folie 24 > TAU Training > Non-Dimensionalisation
Eisfeld> 25.-29.02.2008
Procedures for dimensional solutions
Standard procedure for air
1.
2.
3.
4.
5.
6.
7.
Rely on TAU fluid properties for air
Specify 2 thermodynamic state quantities, i.e. (p*, ρ*) or (p*, T*) or (ρ*, T*)
Specify either Ma* or U*
Do not specify Re*
Specify Reynolds length L*/Lref
In case of dimensional solution specify grid scale correctly
Check stdout for
•
Unspecified thermodynamic quantity
•
Unspecified Mach number or velocity
•
Reynolds number
Possible adaptations:
•
Mach number/velocity:
Adapt sound speed via thermodynamic state quantities
•
Reynolds number:
•
Specify directly + only one thermodynamic state quantity
•
Adapt parameters of Sutherland law
Folie 25 > TAU Training > Non-Dimensionalisation
Eisfeld> 25.-29.02.2008
Procedures for dimensional solutions
Procedure for ICAO standard atmosphere
Specify all 3 thermodynamic state quantities, i.e. p*, ρ* and T*
Î R ≈ 287.053 m2/(s2K) ≠ 287.000 m2/(s2K) (default)
2. Specify either Ma* or U*
3. Do not specify Re*
4. Specify Reynolds length L*/Lref
5. In case of dimensional solution specify grid scale correctly
6. Check stdout for
•
Unspecified Mach number or velocity
•
Reynolds number
Possible adaptations:
•
Not possible.
Î Inconsistency with ICAO standard atmosphere
1.
Folie 26 > TAU Training > Non-Dimensionalisation
Eisfeld> 25.-29.02.2008
Procedures for dimensional solutions
Procedure for high pressure/low temperature wind tunnel (ETW)
Specify isentropic exponent of the gas (Default: γ = 1.4)
Specify 3 thermodynamic state quantities:
•
p*, ρ*, T* Î R
•
Gas constant R + 2 thermodynamic state quantities
3. Specify either Ma* or U*
4. Do not specify Re*
5. Specify Reynolds length L*/Lref
6. In case of dimensional solution specify grid scale correctly
7. Provide Sutherland reference state for the respective gas (TS, μS)
8. Check stdout for
•
Unspecified thermodynamic quantity
•
Unspecified Mach number or velocity
•
Reynolds number
Possible adaptations:
•
As in standard procedure for air
1.
2.
Folie 27 > TAU Training > Non-Dimensionalisation
Eisfeld> 25.-29.02.2008
Procedures for dimensional solutions
Procedure for unknown wind tunnel model size
Specify fluid properties in experiment (R, γ, Pr). Default is air.
Specify parameters of Sutherland law. Default is air.
Specify 2 thermodynamic state quantities, i.e. (p*, ρ*) or (p*, T*) or (ρ*, T*)
Specify either Ma* or U*
Do not specify Re*
Specify Reynolds length L*/Lref
Set grid scale sgrid = 1
Start TAU without providing the grid Î program stops
Î check stdout for Reynolds number ReTAU
9. Specify Reynolds number Re* according to experiment
10. Re-specify grid scale as sgrid = Re*/ReTAU
Î Grid coordinates interpreted as in meters
Grid scale determines model size
Possible adaptations:
•
Adjust sgrid for removing inconsistencies due to round-off errors
1.
2.
3.
4.
5.
6.
7.
8.
Folie 28 > TAU Training > Non-Dimensionalisation
Eisfeld> 25.-29.02.2008
Procedures for dimensional solutions
Documentation
Internal report (IB) in preparation:
“Non-dimensionalisation of the governing equations
and parameter specification for numerical simulations
with the DLR TAU code”
Folie 29 > TAU Training > Non-Dimensionalisation
Eisfeld> 25.-29.02.2008