PSI - Issue 73

Vladimira Michalcova et al. / Procedia Structural Integrity 73 (2025) 106–111 Author name / Structural Integrity Procedia 00 (2025) 000–000

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Fig. 1. Scheme of the task.

3. Numerical modelling This task is solved as a 2D non-stationary problem. The turbulent statistical RANS model Standard k -  is used for the calculation, because it includes modifications for boundary layer solutions. The basic equations are the continuity equation (mass balance), the Navier-Stokes equation of motion (momentum balance) and the energy equation. The parameters that enter the energy equation as dependent on the current temperature are density   [kg.m - 3 ], specific heat capacity of the air c p [J. kg -1 K -1 ], thermal conductivity  [W. m -1 K -1 ] and turbulent viscosity  t [kg.m - 1 s - 1 ]. As a result of time averaging, the so-called Reynolds (Re) stresses appear in the momentum equation, for the solution of which additional equations are needed. RANS models use the Bussinesq theory to calculate the Re stresses, in which the so-called turbulent viscosity  t occurs. This is a function expressing complex functional dependencies of the state of the flowing fluid and the position of the point under consideration, i.e. a function describing a given turbulent flow. The model Standard k - ω models this quantity as dependent on the distance from the wall and thus takes into account the modelling in the boundary layer. At larger distances from the wall μ t is the same as μ t in k - ε models. Turbulent viscosity appears in the Navier-Stokes and energy equations and also in two additional transport equations for the turbulent kinetic energy transfer k (1) and for the specific rate of dissipation of turbulent kinetic energy  (2). Production members are included in both of these equations to improve the modelling of free shear flows. Transport equations of kinetic energy and dissipation for the presented problem: �( � � � �) + �(��� � �� ) � = � � � � � � � � � � � �+ � − � + � (1) �( � � � �) + �(��� � �� ) � = � � � � � � � � � � � �+ � − � + �� (2) where k is kinetic energy [m 2 .s -2 ],  is specific dissipation rate [s -1 ], G k is the generation of turbulence kinetic energy due to mean velocity gradients [kg.m -1 s -3 ], G  is generation of   [ kg.m -3 s -2 ],  k ,   are effective diffusivity of k [kg.m - 1 s -3 ], [kg.m -3 s -2 ], and Y k , Y  are dissipation of k and  due to turbulence [kg.m -1 s -3 ], [kg.m -3 s -2 ], G b , G  b  account for buoyancy terms [kg.m -1 s -3 ], [kg.m -3 s -2 ]. Turbulent viscosity μ t , is included here in the members  k ,   , Y k , Y  ,  Y k , Y  and G b , G  b  . To calculate the buoyancy forces, the Boussinsq model is used, which adds a source term to the momentum equation describing the change in current density according to the equation (3). In this relation   and  0 are actual and operating density [kg.m -3 ], g is gravity [m.s -2 ],  is thermal expansion coefficient [K -1 ], T and T 0 are actual and