Embed Size (px)
Citation preview
Numerical experiments for turbulent flows
Jirı Trefilık1,2,a, Karel Kozel1,2,b, and Jaromır Prıhoda2,c
1 Institute of Thermomechanics AS CR, v. v. i.2 Faculty of Mechanical Engineering, CVUT Prague
Abstract. The aim of the work is to explore the possibilities of modelling transonic flows in the internal andexternal aerodynamics. Several configurations were analyzed and calculations were performed using both invis-cid and viscous models of flow. Viscous turbulent flows have been simulated using either zero equation algebraicBaldwin-Lomax model and two equation k−ω model in its basic version and improved TNT variant. The numer-ical solution was obtained using Lax-Wendroff scheme in the MacCormack form on structured non-ortogonalgrids. Artificial dissipation was added to improve the numerical stability. Achieved results are compared withexperimental data.
1 Mathematical models
1.1 Navier-Stokes equations
The two-dimensional laminar flow of a viscous compress-ible liquid is described by the system of Navier-Stokesequations
Wt + Fx + Gy = Rx + S y, (1)
where
W =
ρρuρve
, F =
ρu
ρu2 + pρuv
(e + p)u
, G =
ρvρuv
ρv2 + p(e + p)v
(2)
and
R =
0
τxxτxy
uτxx + vτxy + λTx
, S =
0
τxyτyy
uτxy + vτyy + λTy
(3)
with shear stresses given for the laminar flow by equations
τxx =23η(2ux − vy), τxy = η(uy + vx), τyy =
23η(−ux + 2vy).
(4)This system is enclosed by the equation of state
p = (κ − 1)[e −
12ρ(u2 + v2)
]. (5)
In the above given equations, ρ denotes density, u, vare components of velocity in the direction of axis x, y, pis pressure, e is total energy per a unit volume, T is tem-perature, η is dynamical viscosity and λ is thermal conduc-tivity coefficient. The parameter κ = 1.4 is the adiabaticexponent.
a e-mail: [email protected] e-mail: [email protected] e-mail: [email protected]
1.2 Reynolds averaged Navier-Stokes equations
For the modelling of a turbulent flow, the system of RANS(Reynolds Averaged Navier-Stokes) equations enclosed bya turbulence model is used. Two different turbulence mod-els with the turbulent viscosity were tested, one algebraic,Baldwin-Lomax and the two-equation k − ω model ac-cording to Wilcox. The system of averaged Navier-Stokesequations is formally the same as (1), but this time the flowparameters represent only mean values in the Favre sense,see [3]. The shear stresses are given for the turbulent flowsby equations
τxx =23
(η + ηt)(2ux − vy), (6)
τxy = (η + ηt)(uy + vx),
τyy =23
(η + ηt)(−ux + 2vy),
where ηt denotes the turbulent dynamic viscosity accord-ing to the Boussinesq hypothesis. The Reynolds number is
defined by Re = u∞Lη∞
and the Mach number by M =
√qa
where q =√
(u2 + v2) and a is the local speed of sound.All the computations were carried out using dimen-
sionless variables with reference variables given by inflowvalues. The reference length L is given by the width of thecomputational domain.
2 Turbulence models
2.1 Baldwin-Lomax model
Algebraic models are based on the model proposed forthe boundary-layer flows by Cebeci and Smith. Baldwin-Lomax model is its modification applicable for general tur-bulent shear flows. The boundary layer is divided into tworegions. In the inner (nearest to the wall) part, the turbulentviscosity is given by
ηt = ρF2Dκ
2y2|Ω|, (7)
EPJ Web of ConferencesDOI: 10.1051/C© Owned by the authors, published by EDP Sciences, 2013
,epjconf 201/
01090 (2013)4534501090
This is an Open Access article distributed under the terms of the Creative Commons Attribution License 2 0 , which . permits unrestricted use, distributiand reproduction in any medium, provided the original work is properly cited.
on,
Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/20134501090
EPJ Web of Conferences
whereΩ is the vorticity, which is in the 2D flow determinedby
Ω =∂u∂y−∂v
∂x(8)
and
FD = 1 − exp(−y+
A+
). (9)
y+ =u∗yν
denotes dimensionless distance from the wall, ν =
µρ
is kinematic viscosity, u∗ =
√(τwρ
)is so called friction
velocity and τw = µ(∂u∂y
)y=0
.The turbulent viscosity in the outer region is given by
ηto = αρCcpFwFk, (10)
where Ccp is a constant. Function Fw is determined by therelation
Fw = ymaxFmax (11)
for Fw being the maximum of the function
F = yFD|Ω| (12)
and ymax the distance from the wall in which F(ymax) =Fmax holds and
Fk =
1 + 5.5(CKL
y
ymax
)6−1
. (13)
The Baldwin-Lomax model (1978) contains following val-ues of the constants: κ = 0.4, A+ = 26, α = 0.0168,Ccp = 1.6, CKL = 0.3.
2.2 k − ω model
Two-equation models are based on transport equations fortwo characteristic scales of turbulent motion, mostly forthe turbulent energy k and dissipation rate ε, often used inthe form of specific dissipation rateω = ε/k. These charac-teristics are computed from transport equations. Turbulentviscosity is defined as
ηt = ρkω. (14)
The standard Wilcox k − ω model is formed by the equa-tions
∂
∂t(ρk) +
∂
∂x j(ρu jk) = Pk +
∂
∂x j
[(η + σ∗ηt)
∂k∂x j
]− β∗ρkω,
(15)
∂
∂t(ρω)+
∂
∂x j(ρu jω) = γ
ω
kPk +
∂
∂x j
[(η + σηt)
∂ω
∂x j
]−βρω2,
(16)where Pk = τi j∂ui/∂x j represents the production of
turbulent energy. Model coefficients are given by values:α = 5/9, β = 3/40, β∗ = 9/100, σ = 1/2 and σ∗ = 1/2,i, j ∈ 1, 2.
So called TNT modification of k-ω model:
∂(ρk)∂t
+∂(ρu jk)∂x j
= Pk +∂
∂x j
[(µ + σkµt)
∂k∂x j
]− β∗ρkω,
(17)
∂(ρω)∂t
+∂(ρu jω)∂x j
= αω
kPk+
∂
∂x j
[(µ + σωµt)
∂ω
∂x j
]−βρkω2+CD,
(18)where
CD = σdρ
ωmax
(∂k∂xi
∂ω
∂xi, 0
)(19)
and αω =ββ∗−
σωκ2
√β∗
, κ = 0.41, β = 340 , β∗ = 9
100 ,σω = 0.5, σk = 0.666.
3 Numerical methods
For the modelling of the flow cases Lax-Wendroff finitevolume method scheme was used on non-orthogonal struc-tured grids of quadrilateral and hexahedral cells Di j(k).– Predictor step:
Wn+1/2i, j = Wn
i, j−∆tµi, j
4∑k=1
[(Fn
k −1
ReRn
k
)∆yk −
(Gn
k −1
ReS n
k
)∆xk
].
(20)– Corrector step:
Wn+1i, j =
12
(Wni, j + Wn+1/2
i, j ) −∆t
2µi, j
4∑k=1
[(Fn+1/2
k −1
ReRn+1/2
k
)∆yk
−
(Gn+1/2
k −1
ReS n+1/2
k
)∆xk
]+ AD(Wn
i, j). (21)
The Mac Cormack scheme in the cell centered formwas applied to solving the system of RANS equations. Con-vective terms F, G are considered in predictor step in for-ward form and in the corrector step in upwind form of thefirst order of accuracy, dissipative terms in central form ofthe second order of accuracy. To indicate this we denotetheir numerical approximation as F, G.
The value µi j(k) represents the surface (volume) of thecell. Figure 14 shows evaluation of derivatives on edges ofcells: We imagine a virtual cell as shown in the picture. Weknow the values in the centers of the cells and we define theother two as a mean value of its surrounding cells. Fromthat we extrapolate to its edges and then we apply Green’sformula.
The scheme was extended to include Jameson’s artifi-cial dissipation because of the stability of the method
AD(Wni, j) = C1ψ1(Wn
i−1, j − 2Wni, j + Wn
i+1, j) (22)
+ C2ψ2(Wni, j−1 − 2Wn
i, j + Wni, j+1),
where
ψ1 =
∣∣∣∣pni−1, j − 2pn
i, j + pni+1, j
∣∣∣∣∣∣∣∣pni−1, j
∣∣∣∣ +∣∣∣∣pn
i, j
∣∣∣∣ +∣∣∣∣pn
i+1, j
∣∣∣∣ , (23)
ψ2 =
∣∣∣∣pni, j−1 − 2pn
i, j + pni, j+1
∣∣∣∣∣∣∣∣pni, j−1
∣∣∣∣ +∣∣∣∣pn
i, j
∣∣∣∣ +∣∣∣∣pn
i, j+1
∣∣∣∣ .
01090-p.2
EFM 2012
Inlet boundary conditions IBC A were used for invis-cid compressible flows; IBCBwere used for inviscid com-pressible flows. A nonzero angle of attack α1 was usedonly in the case of flows through DCA cascade.
On the outlet we prescribed only pressure p2 = p1 andthe other values were extrapolated from the flow field.
Further on there are three other types of boundary con-ditions: solid wall, symmetry axis and periodicity. Theseconditions are implemented by using virtual cells. Suchcells adjoin from outside on the boundary cells and weprescribe values of unknowns inside of them to obtain thedesired effect.
Solid wall (an inviscid flow): velocity components pre-scribed so that the sum of velocity vectors equals to zeroin its tangential component. Solid wall (a viscous flow):velocity components were prescribed so that the sum ofvelocity vectors equals zero. In both cases the rest of un-knowns is the same in both the virtual and the boundarycell.
Symmetry axis: this condition was realized by the sameway as the wall condition for an inviscid flow.
Periodicity condition: taking two corresponding seg-ments of boundary we prescribe into virtual cells of thefirst segment the values of unknowns contained in the bound-ary cells of the second and vice-versa.
Initial conditions were prescribed to comply with theinlet conditions.
5 Results
The following pictures ilustrate results obtained during thecalculations. First three of them present simulation of theflow through the DCA cascade with the inlet Mach num-ber M∞ = 0.832 and angle of attack α = 0. Result ob-tained by employing TNT scheme shows very good agree-ment with experiment. Then similar configuration with in-let Mach number M∞ = 0.863 and angle of attack α = 0follows. In this case for the modelling of turbulence theBaldwin-Lomax model was used. The last configuration ofthis type is for inlet Mach number M∞ = 0.982 with TNTmodel and again there is a good agreement. Second typedomain DEIW calculations follow. First there is an invis-cid flow in the same configuration as the reference resultcomplemented by the viscous flow computation on a sym-metrical domain. But the value of Reynolds number is 10times bigger, which might be the of cause why the separa-tion is more distinct.
Acknowledgment
The work was partly supported by by the institutional sup-port RVO 61388998 and by the grant projects GA AS CRIAA 2007 60 81, GA CR P101/10/1329, P 101/12/1271and SGS 10/243/OHK2/3T/12.
References
1. R. Dvorak, Transonic Flows, Academia, Prague(1986, in Czech).
2. R. Dvorak, K. Kozel, Mathematical modelling in aero-dynamics, CTU in Prague, Prague (1996, in Czech)
3. A. Favre, Equations des gaz turbulents compressibles,Jour. de Mecanique, 4, 361-390 (1965)
4. M. Feistauer, J. Felcman, I. Straskraba, Mathematicaland Computational Methods for Compressible Flow,Oxford University Press (2003)
5. C. Hirsch, Numerical Computation of Internal and Ex-ternal Flows, Volume II, - Computational Methods forInviscid and Viscous Flows, John Willey&Sons (1990)
6. J. Holman, J. Furst, Proceedings: Colloquium FluidDynamics 2008, 11-12, Institute of Thermomechan-ics, AS CR, v. v. i., Prague (2008)
7. J. Huml, J. Furst, K. Kozel, J. Prıhoda, Proceedings:Topical Problems of Fluid Dynamics 2009, 45-48, In-stitute of Thermodynamics, AS CR, v. v. i., Prague(2009, in Czech).
8. J. Huml, J. Holman, J. Furst, K. Kozel, Proceedings:Topical Problems of Fluid Dynamics 2010, 73-76, In-stitute of Thermomechanics, AS CR, v. v. i., Prague(2010)
9. P. Porızkova, Numerical Solution of CompressibleFlows Using Finite Volume Method, PhD DissertationCTU in Prague, Faculty of Mechanical Engineering,CTU in Prague, Prague (2009, in Czech).
10. J. Prıhoda, P. Louda, Mathematical modelling of tur-bulent flow, CTU in Prague, Prague 2007 (in Czech)
11. J. Simonek, K. Kozel, J. Trefilık, Proceedings: TopicalProblems of Fluid Mechanics 2007, 169-172, Instituteof Thermomechanics, AS CR, v. v. i., Prague (2007)
01090-p.3
EPJ Web of Conferences
Fig. 3. DCA cascade, inviscid flow, M∞ = 0.88, α = 0.6, Mach number isolines
Fig. 4. DCA cascade, viscous flow, M∞ = 0.88, α = 0.6, Re = 107, k − ω model, Mach number isolines
Fig. 5. DCA cascade, viscous flow, M∞ = 0.89, α = 0.5, Re = 107, k − ω model (TNT variant), Mach number isolines
Fig. 6. Experimental data, M∞ = 0.832, α = 0
Fig. 7. DCA cascade, inviscid flow, M∞ = 0.98, α = 3.0, Mach number isolines
Fig. 8. DCA cascade, viscous flow, M∞ = 0.98, α = 3.0, Re = 107, Baldwin-Lomax model, Mach number isolines
01090-p.4
EFM 2012
Fig. 9. Experimental data, M∞ = 0.863, α = 0
Fig. 10. DCA cascade, viscous flow, M∞ = 1.07, α = 2.3, Re = 107, k − ω model (TNT variant), Mach number isolines
Fig. 11. Experimental data, M∞ = 0.982, α = 0
Fig. 12. DEIW configuration, inviscid flow, M∞ = 0.775, α = 0, Mach number isolines
01090-p.5
EPJ Web of Conferences
Fig. 13. symmetric DEIW configuration, viscous flow, M∞ = 0.775, α = 0, Re = 106, k−ω model (TNT variant), Mach number isolines
Fig. 14. DEIW configuration reference result, viscous flow, M∞ = 0.775, α = 0, Re = 107, simple mixing length model, Mach numberisolines
01090-p.6
