EPJ Web of Conferences 67, 0 2118 (2014) DOI: 10.1051/epjconf / 2014 6702118 C Owned by the authors, published by EDP Sciences, 2014
Numerical experiments modelling turbulent flows Jiˇr´ı Trefil´ık1,a , Karel Kozel1,2,b , and Jarom´ır Pˇr´ıhoda1,c 1 2
Institute of Thermomechanics AS CR, v. v. i. Faculty of Mechanical Engineering, CVUT Prague Abstract. The work aims at investigation of the possibilities of modelling transonic flows mainly in external aerodynamics. New results are presented and compared with reference data and previously achieved results. For the turbulent flow simulations two modifications of the basic k − ω model are employed: SST and TNT. The numerical solution was achieved by using the MacCormack scheme on structured non-ortogonal grids. Artificial dissipation was added to improve the numerical stability.
1 Mathematical models
1.2 Reynolds averaged Navier-Stokes equations
1.1 Navier-Stokes equations
For the modelling of a turbulent flow, the system of RANS (Reynolds Averaged Navier-Stokes) equations enclosed by a turbulence model is used. Two different turbulence models with the turbulent viscosity were tested, one algebraic, Baldwin-Lomax and the two-equation k − ω model according to Wilcox. The system of averaged Navier-Stokes equations is formally the same as (1), but this time the flow parameters represent only mean values in the Favre sense, see [3]. The shear stresses are given for the turbulent flows by equations
The two-dimensional laminar flow of a viscous compressible liquid is described by the system of Navier-Stokes equations Wt + F x + Gy = R x + S y ,
(1)
⎛ ⎛ ⎛ ⎞ ⎞ ⎞ ⎜⎜⎜ ρu ⎟⎟⎟ ⎜⎜⎜ ρv ⎟⎟⎟ ⎜⎜⎜ ρ ⎟⎟⎟ ⎜⎜ ρu2 + p ⎟⎟⎟ ⎜ ⎜⎜ ρu ⎟⎟ ⎟⎟⎟ ⎟⎟⎟ ⎟⎟⎟ , G = ⎜⎜⎜⎜⎜ ρuv W = ⎜⎜⎜⎜⎜ ⎟⎟⎟⎟⎟ , F = ⎜⎜⎜⎜⎜ 2 ⎜ ⎟⎟⎠ ⎟ ρv ρv + p ρuv ⎜⎝ ⎜⎝ ⎜⎝ ⎟⎠ ⎟⎠ e (e + p)v (e + p)u
(2)
where
τ xy
and ⎛ ⎜⎜⎜ ⎜⎜ R = ⎜⎜⎜⎜⎜ ⎜⎝
0
τ xx τ xy uτ xx + vτ xy + λT x
⎛ ⎞ ⎜⎜⎜ ⎟⎟⎟ ⎜ ⎟⎟⎟ ⎟⎟⎟ , S = ⎜⎜⎜⎜⎜ ⎜⎜⎝ ⎟⎟⎠
0
τ xy τyy uτ xy + vτyy + λT y
⎞ ⎟⎟⎟ ⎟⎟⎟ ⎟⎟⎟ (3) ⎟⎟⎠
with shear stresses given for the laminar flow by equations 2 2 η(2u x − vy ), τ xy = η(uy + v x ), τyy = η(−u x + 2vy ). 3 3 (4) This system is enclosed by the equation of state 1 2 2 (5) p = (κ − 1) e − ρ(u + v ) . 2 τ xx =
In the above given equations, ρ denotes density, u, v are components of velocity in the direction of axis x, y, p is pressure, e is total energy per a unit volume, T is temperature, η is dynamical viscosity and λ is thermal conductivity coefficient. The parameter κ = 1.4 is the adiabatic exponent. a b c
e-mail:
[email protected] e-mail:
[email protected] e-mail:
[email protected]
2 (η + ηt )(2u x − vy ), 3 = (η + ηt )(uy + v x ), 2 = (η + ηt )(−u x + 2vy ), 3
τ xx =
τyy
(6)
where ηt denotes the turbulent dynamic viscosity according to the Boussinesq hypothesis. The Reynolds number
is
q defined by Re = uη∞∞L and the Mach number by M = a 2 2 where q = (u + v ) and a is the local speed of sound. All the computations were carried out using dimensionless variables with reference variables given by inflow values. The reference length L is given by the width of the computational domain.
2 Turbulence models 2.1 T NT model
So called TNT model is a modification of the classical k−ω model in which the turbulent viscosity is given as k (7) ηt = ρ . ω k denotes turbulent energy and ω is a specific dissipation rate. Values of k and ω are obtained as a solution of equations
This is an Open Access article distributed under the terms of the Creative Commons Attribution License 2.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/20146702118
EPJ Web of Conferences
∂(ρk) ∂(ρu j k) ∂ ∂k (μ + σk μt ) − β∗ ρkω (8) = Pk + + ∂t ∂x j ∂x j ∂x j ∂ω ω ∂ ∂(ρω) ∂(ρu j ω) (μ + σω μt ) −βρkω2 +C D = α Pk + + ∂t ∂x j k ∂x j ∂x j (9) where Pk = τi j ∂ui /∂x j represents the production of turbulent energy,
ρ ∂k ∂ω ,0 (10) max CD = 2ω ∂xi ∂xi and α = 0.553, β =
3 40 ,
β∗ =
9 100 ,
σω = 0.5, σk = 0.666.
2.2 S S T model
β1 = 3/40, σ1 = 1/2, σ∗1 = 1/2, γ1 = β1 /β∗ − σ1 κ2 / β∗ , (21) and β2 = 0.0828, σ2 = 0.85, σ∗2 = 1, γ2 = β2 /β∗ − σ2 κ2 / β∗ . (22) Boundary conditions for k and ω: – Inlet: ¯ 2. – k∞ = 32 (0.001U) k∞ – ω∞ = 90 ν – Wall: – k = 0. – ω asymptotically approaches +∞, very high value prescribed. – Outlet: k and ω extrapolated from the flow.
S S T model is another modification of k − ω model: ∂ ∂ ∂k ∂ (μ + σ∗ μt ) − β∗ ρkω (ρU˜ j k) = Pk + (ρk) + ∂t ∂x j ∂x j ∂x j (11) ∂ ω ∂ ∂ω ∂ ˜ (μ + σμt ) (12) (ρU j ω) = γ Pk + (ρω) + ∂t ∂x j k ∂x j ∂x j 1 ∂k ∂ω ¯ − F1 )σ2 , − βρω2 + 2ρ(1 ω xj xj The last term in the second equation expresses lateral diffusion. Turbulent viscosity is given by the formula
k a1 k (13) νt = min , ω F2 Ω
3 Numerical methods For the modelling of the flow cases Lax-Wendroff finite volume method scheme was used on non-orthogonal structured grids of quadrilateral and hexahedral cells Di j(k) . – Predictor step:
Wi,n+1/2 = Wi,n j − j
– Corrector step:
Wi,n+1 j =
and the functions F1 a F2 are defined as follows: F1 = tanh(Γ14 ), ⎞ ⎤ ⎡ ⎛ √ ⎢⎢⎢ ⎜⎜⎜ k 500ν ⎟⎟⎟ 4ρσ ¯ 2 k ⎥⎥⎥ ⎟⎠ ; ⎥⎦ , Γ1 = min ⎣⎢max ⎜⎝ ∗ ; β ωy ωy2 Dω y2 where y is the distance from the wall
1 ∂k ∂ω −20 , ¯ 2 ; 10 Dω = max 2ρσ ω xj xj F2 = tanh(Γ22 ),
(14)
(15)
(16) (17)
n n n AD(Wi,n j ) = C1 ψ1 (Wi−1, j − 2Wi, j + Wi+1, j )
+ (18)
where
Model’s constants are given by the formula φ = F1 φ1 + (1 − F1 )φ2 ,
(19)
where κ = 0.41, β∗ = 9/100, σ∗ = 0.85, a1 = 0.31,
4 1 n Δt ˜ n+1/2 1 n+1/2 ) − − F Δyk (Wi, j + Wi,n+1/2 R j k 2 2μi, j k=1 Re k
1 n+1/2 n Δx − G˜ n+1/2 − (24) S k + AD(Wi, j ). k Re k
The Mac Cormack scheme in the cell centered form was applied to solving the system of RANS equations. Convective terms F, G are considered in predictor step in forward form and in the corrector step in upwind form of the first order of accuracy, dissipative terms in central form of the second order of accuracy. To indicate this we denote ˜ G. ˜ their numerical approximation as F, The scheme was extended to include Jameson’s artificial dissipation because of the stability of the method
and ⎞ ⎛ √ ⎜⎜⎜ 2k 500ν ⎟⎟⎟ ⎟⎠ . ; Γ2 = max ⎜⎝ 0.09ωy ωy2
4 Δt ˜ n 1 n 1 n Fk − Rk Δyk − G˜ nk − S k Δxk . μi, j k=1 Re Re (23)
(20) 02118-p.2
C2 ψ2 (Wi,n j−1
−
2Wi,n j
+
(25)
Wi,n j+1 ),
pn − 2pn + pn i, j i−1, j i+1, j , ψ1 = n n p n i−1, j + pi, j + pi+1, j pn − 2pn + pn i, j i, j−1 i, j+1 . ψ2 = pn + pn + pn i, j i, j−1 i, j+1
(26)
EFM 2013
Acknowledgment The work was partly supported by by the institutional support RVO 61388998 and by the grant projects GA AS CR IAA 2007 60 81, GA CR P101/10/1329, P 101/12/1271 and SGS 10/243/OHK2/3T/12.
References Fig. 1. Computational domain
1. The convergence to the steady state is followed by log L2 residual defined by ⎞ ⎛ 1 ⎜⎜⎜ Wkn+1 − Wkn ⎟⎟⎟ n (27) ResW = ⎠⎟ ⎝⎜ M k Δt
3. 4.
where M is a number of all cells in the computational domain.
5. 6. 7.
4 Formulation of the problem
8.
The computational domain is demonstrated on the picture (1). The line AF designates the inlet boundary, DE is the outlet, BC is the wall and the rest is defined as symmetric boundary.
9.
2.
10.
5 Results The figure (2) represents the result of simulation obtained with the S S T model for the same parameters as those of the reference result (5). The agreement is very good and the symmetric simulation (4) indicates that small perturbations do not spread in the calculation domain which would destroy the symmetry. Figure (3) shows similar result obtained previously with the usage of the T NT modification of k − ω model. It can be noted that the S S T simulation leads to developement of a smaller wake and also the maximum value of velocity that exists in the flow field is significantly higher (around 1.32) compared to the one in the T NT model simulation ( around 1.26).
11. 12. 13. 14. 15. 16.
6 Conclusion A new method for the simulation of turbulent transonic flows has been developed and tested. The results obtained appear to be more realistic compared to those of the T NT method. At the same time the implementation of the S S T method is considerably more complex and therefore more error prone. The method is also approximately 25% slower in terms of the time needed to perform the calculation. Further simulations especially for the internal aerodynamics will be carried out to able to better assess its characteristics.
02118-p.3
R. Dvoˇra´ k, Transonic Flows, Academia, Prague (1986, in Czech). R. Dvoˇra´ k, K. Kozel, Mathematical modelling in aerodynamics, CTU in Prague, Prague (1996, in Czech) A. Favre, Jour. de Mecanique, 4, 361 (1965) M. Feistauer, J. Felcman, I. Straˇskraba, Mathematical and Computational Methods for Compressible Flow, Oxford University Press (2003) A. Hellsten, AIAA J., 36, 2528 (1998) A. Hellsten, AIAA J., 43, 1857 (2005) C. Hirsch, Numerical Computation of Internal and External Flows, Volume II, - Computational Methods for Inviscid and Viscous Flows, John Willey&Sons (1990) J. Holman, J. Furst, Proceedings: Colloquium Fluid Dynamics 2008, 11, Institute of Thermomechanics, AS CR, v. v. i., Prague (2008) J. Huml, J. Furst, K. Kozel, J. Pˇr´ıhoda, Proceedings: Topical Problems of Fluid Dynamics 2009, 45 (2009, in Czech). J. Huml, J. Holman, J. Furst, K. Kozel, Proceedings: Topical Problems of Fluid Dynamics 2010, 73 (2010) J.C. Kok, AIAA J., 38, 1292 (2000) F.R. Menter, AIAA J., 30, 1657 (1992) F.R. Menter, AIAA J., 32, 1598 (1994) P. Poˇr´ızkov´a, Numerical Solution of Compressible Flows Using Finite Volume Method, PhD Dissertation CTU in Prague (2009, in Czech). J. Pˇr´ıhoda, P. Louda, Mathematical modelling of turbulent flow, CTU in Prague (2007, in Czech) ˇ J. Simonek, K. Kozel, J. Trefil´ık, Proceedings: Topical Problems of Fluid Mechanics 2007, 169 (2007)
EPJ Web of Conferences
Fig. 2. Symmetric DEIW configuration, viscous flow, M∞ = 0.775, α = 0◦ , Re = 107 , k − ω model (SST variant), Mach number isolines
Fig. 3. Symmetric DEIW configuration, viscous flow, M∞ = 0.775, α = 0◦ , Re = 106 , k − ω model (TNT variant), Mach number isolines
Fig. 4. Symmetric DEIW configuration, viscous flow, M∞ = 0.775, α = 0◦ , Re = 107 , k − ω model (SST variant), Mach number isolines
Fig. 5. DEIW configuration reference result, viscous flow, M∞ = 0.775, α = 0◦ , Re = 107 , simple mixing length model, Mach number isolines
02118-p.4
