Numerical experiments modelling turbulent flows

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 1.1 Navier-Stokes equations The two-dimensional laminar flow of a viscous compressible liquid is described by the system of Navier-Stokes equations Wt + Fx +Gy = Rx + S y, (1) where W = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ρ ρu ρv e ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ , F = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ρu ρu2 + p ρuv (e + p)u ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ , G = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ρv ρuv ρv2 + p (e + p)v ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (2)


Mathematical models 1.Navier-Stokes equations
The two-dimensional laminar flow of a viscous compressible liquid is described by the system of Navier-Stokes equations where with shear stresses given for the laminar flow by equations (4) This system is enclosed by the equation of state 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.

Reynolds averaged 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 where η t denotes the turbulent dynamic viscosity according to the Boussinesq hypothesis.The Reynolds number is where q = (u 2 + v 2 ) 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.

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 ) where P k = τ i j ∂u i /∂x j represents the production of turbulent energy, and α = 0.553, β = 3 40 , β * = 9 100 , σ ω = 0.5, σ k = 0.666.

S S T model
S S T model is another modification of k − ω model: The last term in the second equation expresses lateral diffusion.Turbulent viscosity is given by the formula and the functions F 1 a F 2 are defined as follows: where y is the distance from the wall and Model's constants are given by the formula where κ = 0.41, β * = 9/100, σ * = 0.85, a 1 = 0.31, (20) and -Inlet: - k = 0.
ω asymptotically approaches +∞, very high value prescribed.-Outlet: k and ω extrapolated from the flow.

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 D i j(k) .
-Predictor step: -Corrector step: 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 their numerical approximation as F, G.
The scheme was extended to include Jameson's artificial dissipation because of the stability of the method where .
EPJ Web of Conferences 02118-p.2The convergence to the steady state is followed by log L 2 residual defined by where M is a number of all cells in the computational domain.

Formulation of the problem
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.

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).

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.

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Boundary conditions for k and ω: