I . Verification

2. Definition of the numerical benchmark for thermal and viscous flows

A steady-state, two-dimensional natural convection of water in the differentially heated square cavity of a height L=38mm. Two vertical walls are isothermal, kept at temperatures TH = 10oC, TC= 0oC.  Top and bottom walls are assumed to be adiabatic. 
Natural convection in the cavity characterises variation of the temperature in the range of  ∆T=10oC. For simplicity we decided to define our numerical benchmark for a simplified case, assuming adiabatic top and bottom walls and constant fluid properties. Our main aim is to verify performance of the numerical models and to estimate the accuracy of the discrete approximate solutions in the presence of strong velocity and temperature gradients generated by the non-linear buoyancy term.
The basic equations describing the flow driven by natural convection consist of  conservation of mass, momentum and energy, and are given by:

eq_21_24                                             

Above equations describe the two-dimensional flow of an incompressible viscous fluid, where eq_tmp1 denote respectively the horizontal and the vertical velocity, the reference density of fluid, the pressure, the dynamic viscosity, the gravitational acceleration, the temperature, the thermal condactivity and the specific heat. Water physical properties, like dynamic viscosity, specific heat, thermal conductivity and reference density are assumed constant and their value at the reference temperature Tref = 0oC are assumed. The values applied to the numerical models are collected in Table 1. The anomalous thermal variation of the water density is implemented in buoyancy term only (eq. 2.3). The fourth order polynomial was used to describe variation of water density with temperature:
eq_25
(2.5)
Thermal boundary conditions for isothermal walls were taken Th = 10oC for the hot,and Tc = 0oC for the cold wall, respectively. For the adiabatic walls the zero heat flux thermal boundary condition is set.  The standard no-slip boundary conditions at all walls are adopted for the velocity components. Dimension of the cavity L was 38 mm. A steady state solution is searched for. Hence, the initial conditions play a secondary role and were not investigated. In most cases a uniform temperature of the fluid and zero velocity was assumed as an initial condition.

Table 1. Physical properties of  water used in the simulations.

Material properties of water at 0oC Value Unit
rodensity of water at reference temperature 999.8 kg/m3
midynamic viscosity 0.0017888 kg/ms
kapathermal conductivity 0.566 W/mK
cpspecific heat 4212.0 J/kgK
ggravitational acceleration 9.81 m/s2
betathermal expansion coefficient -6.733353E-05 1/K
The Rayleigh (Ra) and Prandtl (Pr) numbers describing investigated configuration are based on the fluid properties taken at the reference temperature and the cavity height. Their values are:
eq_tmp2
Dimensionless variables are used in most of the tested codes. Hence, results of the tests are given in non-dimensional form using for non-dimensional temperature teta, horizontal and vertical coordinates X,Y, and horizontal and vertical velocities U,W  the following scales:
eq_2_6 
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